1

Derived Categories in Algebraic GeometryA first glance

Paolo Stellari

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

3

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

3

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

3

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

3

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

2

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

3

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

3

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

4

The geometric setting

Let X be a smooth projectivevariety defined over analgebraically closed field K

Example

Consider for example the zerolocus of

xd0 + + xd

n

in the projective space PnK for

an integer d ge 1

(Vague) Question

How do we categorize X

Find a categoryassociated to X which encodesimportant bits

of its geometry

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

5

The geometric setting

In a more precise formconsider the (abelian)category Coh(X ) of coherentsheaves on X

They have locally a finitepresentation

The abelian structure onCoh(X ) provides therelevant notion of shortexact sequence

0 rarr E1 rarr E2 rarr E3 rarr 0

Example

The local descriptionmentioned before can bethought as follows

Let R be a noetherian ring AnR-module M is coherent ifthere exists an exact sequence

Roplusk1 rarr Roplusk2 rarr M rarr 0

for some non-negative integersk1 k2

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

6

Coherent sheaves are too much

We then have the following (simplified version of a) classicalresult

Theorem (Gabriel)

Let X1 and X2 be smooth projective schemes over K ThenX1 sim= X2 if and only if there is an equivalence of abeliancategories Coh(X1) sim= Coh(X2)

It has been generalized invarious contexts by AntieauCanonaco-S Perego

Coh(X ) encodes too muchof the geometry of X

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

7

Looking for alternatives

Aim

We would like to associate to X a category with a nicestructure which weakly encodes the geometry of X (not justup to isomorphism)

For example the categories associated to X1 and to X2 must beclose if

X1 and X2 are nicely related by birational transformations

One of them is a moduli space on the other one (ie itparametrizes objects defined on the second one)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

8

Looking for alternatives

Let us enlarge the category and look for the bounded derivedcategory of coherent sheaves

Db(X ) = Db(Coh(X ))

on X

Objects bounded complexes of coherent sheaves

rarr 0 rarr Eminusi rarr Eminusi+1 rarr rarr Ekminus1 rarr Ek rarr 0 rarr

Morphisms slightly more complicated then morphisms ofcomplexes (but we do not care here)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

9

Basic operations

Given a complex E isin Db(X ) we can shift it to the left(E E [1]) or to the right (E E [minus1])

We can take direct sums E1 oplus E2 and direct summands ofE isin Db(X )

Db(X ) is not abelian but it is triangulated short exactsequences are replaced by (non functorial) distinguishedtriangles

E1 rarr E2 rarr E3 rarr E1[1]

We say that E2 is an extension of E1 and E3

Important construction

For A sube Db(X ) we take the category 〈A〉 generated by A (iethe smallest full triang subcat of Db(X ) containing A andclosed under shifts extensions direct sums and summands)

10

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

11

Summary

Coh(X )is too much

A new hope Db(X )

How do wedescribeobjects of

Db(X )

How do weunderstand

the structureof Db(X )

Stability

Applications

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

12

How do we describe objects of Db(X )

Let us go back to a special instance of the first example

Example (n = 2 d ge 1 K = C)

Consider the planar curve C which is the zero locus of

xd0 + xd

1 + xd2

in the projective space P2

Take E isin Db(C) of the form

rarr E iminus1 d iminus1minusrarr E i d i

minusrarr E i+1 rarr

and define

Hi(E) =ker (d i)

Im(d iminus1)isin Coh(C)

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

13

How do we describe objects of Db(X )

In our example we then have an isomorphism in Db(C)

E sim=983120

i

Hi(E)[minusi]

On the other hand each F isin Coh(C) splits asF sim= Ftf oplus Ftor where Ftf is locally free and Ftor is supportedon points

In conclusion each E isin Db(C) is a direct sum of shiftedlocally free or torsion sheaves

Warning 1

For higher dimensional varieties this is false

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

14

How do we understand the structure of Db(X )

We say that Db(X ) has a semiorthogonal decompisition

Db(X ) = 〈A1 Ak 〉

if

Db(X ) is generated by extensions shifts direct sums andsummands by the objects in A1 Ak

There are no Homs from right to left between the ksubcategories

A1

1003534983577983577A2

1003534983574983574

1003536983640983640

middot middot middot1003534983576983576

1003536983640983640

Ak

1003536983638983638

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

15

How do we understand the structure of Db(X )

Consider again the case of the planar curve C

xd0 + xd

1 + xd2 = 0

d = 1 2Genus g = 0

We have

Db(C) = 〈OC OC(1)〉

where

〈OC(i)〉 sim= Db(pt)

for i = 0 1

d = 3Genus g = 1

Db(C) is indec

But interestingautoequivalecegroup

d ge 4Genus g ge 2

Db(C) is indec

But uninterestingautoequivalecegroup

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

16

How do we understand the structure of Db(X )

Conclusion

In general when X is a Fano variety (in our standing examplewe want d minus n minus 1 lt 0) we look for interesting decompositionshoping that

The components are simpler and of lsquosmaller dimensionrsquo

Get a dimension reduction a component encodes much ofthe geometry of X

Warning 2

Semiorthogonal decompositions are in general non-canonical

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

17

Summary

Warning 1Difficult to describe

objects

Warning 2Difficult to describe

the structure of Db(X )

We needmore structure

on Db(X )

Stability

18

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

19

Back to sheaves

Take n = 2 in our example (but this can make it work for anysmooth projective variety)

The idea is that we want to filter any coherent sheaf in acanonical way

More precisely

We have an abelian category Coh(C)

We define a function

microslope(minus) =deg(minus)

rk (minus)

(or +infin when the denominator is 0) defined on Coh(C)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such that

The quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

20

Back to sheaves

Definition

A sheaf E isin Coh(C) is (semi)stable if for all non-trivialsubsheaves F rarr E such that rk(F ) lt rk(E) we have

microslope(F ) lt (le)microslope(E)

HarderndashNarasimhan filtration

Any sheaf E has a filtration

0 = E0 rarr E1 rarr rarr Enminus1 rarr En = E

such thatThe quotient Ei+1Ei is semistable for all i microslope(E1E0) gt gt microslope(EnEnminus1)

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

21

From sheaves to complexes

Question

Can we have something similar for objects in Db(X )

Mathematical PhysicsHomological Mirror Symmetry

(Kontsevich)

Birational geometryHomological MMP

(Bondal Orlov Kawamata)

Bridgelandstability

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

22

Stability conditions

Let us start discussing the general setting we do not evenneed to work with the specific triangulated category Db(X )

Let T be a triangulated category

Let Γ be a free abelian group offinite rank with a surjective mapv K (T) rarr Γ

Example

T = Db(C) for C asmooth projective curve

Γ = N(C) = H0 oplus H2

with

v = (rk deg)

A Bridgeland stability condition on T is a pair σ = (AZ )

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

23

Stability conditions

A is the heart of abounded t-structure on T

Z Γ rarr C is a grouphomomorphism

Example

A = Coh(C)

Z (v(minus)) = minusdeg +radicminus1rk

such that for any 0 ∕= E isin A

1 Z (v(E)) isin Rgt0e(01]πradicminus1

2 E has a Harder-Narasimhan filtration with respect tomicroσ(minus) = minusRe(Z )(minus)

Im(Z )(minus) (or +infin)

3 Support property (Kontsevich-Soibelman) wall andchamber structure with locally finitely many walls

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

24

Stability conditions

Warning 3

The example is somehow misleading it only works indimension 1

The following is a remarkable result

Theorem (Bridgeland)

If non-empty the space Stab(T) parametrizing stabilityconditions on T is a complex manifold of dimension rk(Γ)

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

25

Stability conditions

Warning 4

It is a very difficult problem to construct stability conditions

It is solved mainly in these cases

Curves (Bridgeland Macrı)

Surfaces over C (Bridgeland Arcara Bertram) surface inpositive characteristic (only partially solved)

Fano threefolds (Bernardara-Macrı-Schmidt-Zhao Li)

Threefolds with trivial canonical bundle abelian 3-folds(Maciocia-Piyaratne Bayer-Macrı-S) Calabi-Yau(Bayer-Macrı-S Li)

Fourfolds in general is a big mistery

26

Outline

1 What are they

2 How to handle them

3 Stability

4 Applications

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

27

Main idea

We want to combine all the ideas we discussed so far

Semiorthogonal decompositions

Stability conditions

Geometry ofcubic 4-folds

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

28

The setting

Let X be a cubic fourfold (ie a smooth hypersurface ofdegree 3 in P5) Let H be a hyperplane section

Most of the time defined over C but for some results definedover a field K = K with char(K) ∕= 2

Example (d = 3 and n = 5)

Consider for example the zero locus of

x30 + + x3

5 = 0

in the projective space P5

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈

Ku(X )

OX OX (H)OX (2H)

〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

= 983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

29

Homological algebra

Let us now look at the bounded derived category of coherentsheaves on X

Db(X )

=

〈 Ku(X ) OX OX (H)OX (2H) 〉

Ku(X )

=983069E isin Db(X )

Hom (OX (iH)E [p]) = 0i = 0 1 2 forallp isin Z

983070

Kuznetsov component of X

Exceptional objects

〈OX (iH)〉 sim= Db(pt)

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

30

The idea

X (and Db(X )) arecomplicated objects

Semiorthogonal decompositionreduce to a rsquosurfacersquo Ku(X )

Stabilityconditions

New appli-cations to

hyperkahlergeometry

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

31

Noncommutative K3s

Remark

For every cubic 4-folds X Ku(X ) behaves like Db(S) for S aK3 surface (ie a simply connected smooth projective varietyof dimension 2 with trivial canonical bundle)

But almost never Ku(X ) sim= Db(S) for S a K3 surface

This is related to the following

Conjecture (Kuznetsov)

A cubic 4-fold X is rational (ie birational to P4 rsquo=rsquo a big opensubset of X is isomorphic to an open subset of P4) if and only ifKu(X ) sim= Db(S) for some K3 surface S

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

32

The results existence of stability conditions

We have seen that if S is a K3 surface then Db(S) carriesstability conditions

Question (Addington-Thomas Huybrechts)

If X is a cubic 4-fold does Ku(X ) carry stability conditions

Theorem 1 (Bayer-Lahoz-Macrı-S BLMS+Nuer-Perry)

For any cubic fourfold X we have Stab(Ku(X )) ∕= empty Moreoverwe can explicitly describe a connected componentStabdagger(Ku(X )) of Stab(Ku(X ))

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

33

The results moduli spaces

Whenever we have a good notion of stability we would like toparametrize all objects which are (semi)stable with respect to it

We fix some topological invariants of the objects in Ku(X ) thatwe want to parametrize These invariants are encoded by avector v in cohomology that we usually call Mukai vector

Fix a stability condition σ isin Stabdagger(Ku(X )) which is nice withrespect to v (always possible)

Denote byMσ(Ku(X ) v)

the space parametrizing σ-stable objects in Ku(X ) with Mukaivector v Call it moduli space

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

34

The results moduli spaces

Question

What is the geometry of Mσ(Ku(X ) v)

This is a non-trivial question as moduli spaces of Bridgelandstable objects are in general lsquostrangersquo objects

Theorem 2 (BLMNPS)

Mσ(Ku(X ) v) is non-empty if and only if v2 + 2 ge 0 Moreoverin this case it is a smooth projective hyperkahler manifold ofdimension v2 + 2 deformation equivalent to a Hilbert schemeof points on a K3 surface

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

35

The results moduli spaces

Definition

A hyperkahler manifold is a simply connected compact kahlermanifold X such that H0(X Ω2

X ) is generated by an everywherenon-degenerate holomorphic 2-form

There are very few examples (up to deformation)

1 K3 surfaces

2 Hilbert schemes of points on K3 surface (denoted byHilbn(K3)

3 Generalized Kummer varieties (from abelian surfaces)

4 Two sporadic examples by OrsquoGrady

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

36

Hyperkahler geometry and cubic 4-folds

The fact that any cubic 4-fold X has a very interestinghyperkahler geometry is classical This is related to rationalcurves in X

Beauville-Donagi the variety parametrizing lines in X is aHK 4-fold

Lehn-Lehn-Sorger-van Strated if X does not contain aplane the moduli space of (generalized) twisted cubics inX is after a fibration and the contraction of a divisor a HK8-fold

Question

Can we recover these classical HK manifolds in our newframework

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

37

Hyperkahler geometry and cubic 4-folds

Theorem(s) (Li-Pertusi-Zhao Lahoz-Lehn-Macrı-S)

The variety of lines and the one of twisted cubics areisomorphic to moduli spaces of stable objects in the Kuznetsovcomponent

New (BLMNPS)

In these two cases we can explicitely describe the birationalmodels via variation of stability

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family

38

Hyperkahler geometry and cubic 4-folds

But the most striking application is the following

Corollary (BLMNPS)

For any pair (a b) of coprime integers there is a unirationallocally complete 20-dimensional family over an open subset ofthe moduli space of cubic fourfolds of polarized smoothprojective HKs of dimension 2n + 2 where n = a2 minus ab + b2

This follows from a new theory of Bridgeland stability in families+ Theorem 2 in families + deformations of cubics in their20-dim family