derived torelli theorem and orientation · derived torelli theorem and orientation paolo stellari...
TRANSCRIPT
![Page 1: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/1.jpg)
Derived Torelli Theorem and Orientation
Paolo Stellari
Dipartimento di Matematica “F. Enriques”Universita degli Studi di Milano
Joint work with D. Huybrechts and E. Macrı (math.AG/0608430 + work in progress)
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 2: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/2.jpg)
Outline
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 3: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/3.jpg)
Outline
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 4: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/4.jpg)
Outline
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 5: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/5.jpg)
Outline
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 6: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/6.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 7: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/7.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Aim of the talk
Let X be a smooth projective K3 surface.
ProblemDescribe the group of autoequivalences of the tringulatedcategory
Db(X ) := DbCoh(OX -Mod),
of bounded complexes of OX -modules with coherentcohomologies.
More generally: Consider the derived category of twistedsheaves!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 8: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/8.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Aim of the talk
Let X be a smooth projective K3 surface.
ProblemDescribe the group of autoequivalences of the tringulatedcategory
Db(X ) := DbCoh(OX -Mod),
of bounded complexes of OX -modules with coherentcohomologies.
More generally: Consider the derived category of twistedsheaves!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 9: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/9.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Aim of the talk
Let X be a smooth projective K3 surface.
ProblemDescribe the group of autoequivalences of the tringulatedcategory
Db(X ) := DbCoh(OX -Mod),
of bounded complexes of OX -modules with coherentcohomologies.
More generally: Consider the derived category of twistedsheaves!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 10: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/10.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Aim of the talk
Let X be a smooth projective K3 surface.
ProblemDescribe the group of autoequivalences of the tringulatedcategory
Db(X ) := DbCoh(OX -Mod),
of bounded complexes of OX -modules with coherentcohomologies.
More generally:
Consider the derived category of twistedsheaves!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 11: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/11.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Aim of the talk
Let X be a smooth projective K3 surface.
ProblemDescribe the group of autoequivalences of the tringulatedcategory
Db(X ) := DbCoh(OX -Mod),
of bounded complexes of OX -modules with coherentcohomologies.
More generally: Consider the derived category of twistedsheaves!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 12: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/12.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 13: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/13.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
Theorem (Torelli Theorem)Let X and Y be K3 surfaces. Suppose that there exists aHodge isometry
g : H2(X , Z) → H2(Y , Z)
which maps the class of an ample line bundle on X into theample cone of Y . Then there exists a unique isomorphismf : X ∼−→ Y such that f∗ = g.
Lattice theory + Hodge structures + ample cone
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 14: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/14.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
Theorem (Torelli Theorem)Let X and Y be K3 surfaces.
Suppose that there exists aHodge isometry
g : H2(X , Z) → H2(Y , Z)
which maps the class of an ample line bundle on X into theample cone of Y . Then there exists a unique isomorphismf : X ∼−→ Y such that f∗ = g.
Lattice theory + Hodge structures + ample cone
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 15: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/15.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
Theorem (Torelli Theorem)Let X and Y be K3 surfaces. Suppose that there exists aHodge isometry
g : H2(X , Z) → H2(Y , Z)
which maps the class of an ample line bundle on X into theample cone of Y .
Then there exists a unique isomorphismf : X ∼−→ Y such that f∗ = g.
Lattice theory + Hodge structures + ample cone
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 16: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/16.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
Theorem (Torelli Theorem)Let X and Y be K3 surfaces. Suppose that there exists aHodge isometry
g : H2(X , Z) → H2(Y , Z)
which maps the class of an ample line bundle on X into theample cone of Y . Then there exists a unique isomorphismf : X ∼−→ Y such that f∗ = g.
Lattice theory + Hodge structures + ample cone
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 17: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/17.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
Theorem (Torelli Theorem)Let X and Y be K3 surfaces. Suppose that there exists aHodge isometry
g : H2(X , Z) → H2(Y , Z)
which maps the class of an ample line bundle on X into theample cone of Y . Then there exists a unique isomorphismf : X ∼−→ Y such that f∗ = g.
Lattice theory
+ Hodge structures + ample cone
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 18: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/18.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
Theorem (Torelli Theorem)Let X and Y be K3 surfaces. Suppose that there exists aHodge isometry
g : H2(X , Z) → H2(Y , Z)
which maps the class of an ample line bundle on X into theample cone of Y . Then there exists a unique isomorphismf : X ∼−→ Y such that f∗ = g.
Lattice theory + Hodge structures
+ ample cone
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 19: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/19.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
Theorem (Torelli Theorem)Let X and Y be K3 surfaces. Suppose that there exists aHodge isometry
g : H2(X , Z) → H2(Y , Z)
which maps the class of an ample line bundle on X into theample cone of Y . Then there exists a unique isomorphismf : X ∼−→ Y such that f∗ = g.
Lattice theory + Hodge structures + ample cone
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 20: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/20.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
All K3 surfaces are diffeomorphic. Fix X and let Λ := H2(X , Z).
Problem: diffeomorphismsDescribe the group of diffeomorphisms of X .
Theorem (Borcea, Donaldson)Consider the natural map
ρ : Diff(X ) −→ O(H2(X , Z)).
Then im (ρ) = O+(H2(X , Z)), where O+(H2(X , Z)) is the groupof orientation preserving Hodge isometries.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 21: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/21.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
All K3 surfaces are diffeomorphic. Fix X and let Λ := H2(X , Z).
Problem: diffeomorphismsDescribe the group of diffeomorphisms of X .
Theorem (Borcea, Donaldson)Consider the natural map
ρ : Diff(X ) −→ O(H2(X , Z)).
Then im (ρ) = O+(H2(X , Z)), where O+(H2(X , Z)) is the groupof orientation preserving Hodge isometries.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 22: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/22.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
All K3 surfaces are diffeomorphic. Fix X and let Λ := H2(X , Z).
Problem: diffeomorphismsDescribe the group of diffeomorphisms of X .
Theorem (Borcea, Donaldson)Consider the natural map
ρ : Diff(X ) −→ O(H2(X , Z)).
Then im (ρ) = O+(H2(X , Z)), where O+(H2(X , Z)) is the groupof orientation preserving Hodge isometries.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 23: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/23.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Motivations: geometry
All K3 surfaces are diffeomorphic. Fix X and let Λ := H2(X , Z).
Problem: diffeomorphismsDescribe the group of diffeomorphisms of X .
Theorem (Borcea, Donaldson)Consider the natural map
ρ : Diff(X ) −→ O(H2(X , Z)).
Then im (ρ) = O+(H2(X , Z)), where O+(H2(X , Z)) is the groupof orientation preserving Hodge isometries.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 24: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/24.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 25: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/25.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);3 there exists a Hodge isometry g : T (X ) → T (Y );4 Y is isomorphic to a smooth compact 2-dimensional fine
moduli space of stable sheaves on X .
Lattice theory + Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 26: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/26.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);3 there exists a Hodge isometry g : T (X ) → T (Y );4 Y is isomorphic to a smooth compact 2-dimensional fine
moduli space of stable sheaves on X .
Lattice theory + Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 27: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/27.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );
2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);3 there exists a Hodge isometry g : T (X ) → T (Y );4 Y is isomorphic to a smooth compact 2-dimensional fine
moduli space of stable sheaves on X .
Lattice theory + Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 28: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/28.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);
3 there exists a Hodge isometry g : T (X ) → T (Y );4 Y is isomorphic to a smooth compact 2-dimensional fine
moduli space of stable sheaves on X .
Lattice theory + Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 29: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/29.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);3 there exists a Hodge isometry g : T (X ) → T (Y );
4 Y is isomorphic to a smooth compact 2-dimensional finemoduli space of stable sheaves on X .
Lattice theory + Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 30: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/30.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);3 there exists a Hodge isometry g : T (X ) → T (Y );4 Y is isomorphic to a smooth compact 2-dimensional fine
moduli space of stable sheaves on X .
Lattice theory + Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 31: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/31.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);3 there exists a Hodge isometry g : T (X ) → T (Y );4 Y is isomorphic to a smooth compact 2-dimensional fine
moduli space of stable sheaves on X .
Lattice theory
+ Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 32: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/32.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The settingThe geometric caseThe derived case
The statement
Derived Torelli Theorem (Orlov, Mukai)Let X and Y be K3 surfaces. Then the following conditions areequivalent:
1 Db(X ) ∼= Db(Y );2 there exists a Hodge isometry f : H(X , Z) → H(Y , Z);3 there exists a Hodge isometry g : T (X ) → T (Y );4 Y is isomorphic to a smooth compact 2-dimensional fine
moduli space of stable sheaves on X .
Lattice theory + Hodge structures
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 33: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/33.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 34: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/34.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Brauer groups
DefinitionThe Brauer group of X is
Br (X ) := H2(X ,O∗X )tor
in the analytic topology.
If X is a K3 surface, the Universal Coefficient Theorem, yields anice description of the Brauer group:
Br (X ) ∼= Hom(T (X ), Q/Z)
where T (X ) := NS(X )⊥.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 35: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/35.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Brauer groups
DefinitionThe Brauer group of X is
Br (X ) := H2(X ,O∗X )tor
in the analytic topology.
If X is a K3 surface, the Universal Coefficient Theorem, yields anice description of the Brauer group:
Br (X ) ∼= Hom(T (X ), Q/Z)
where T (X ) := NS(X )⊥.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 36: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/36.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Brauer groups
DefinitionThe Brauer group of X is
Br (X ) := H2(X ,O∗X )tor
in the analytic topology.
If X is a K3 surface, the Universal Coefficient Theorem, yields anice description of the Brauer group:
Br (X ) ∼= Hom(T (X ), Q/Z)
where T (X ) := NS(X )⊥.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 37: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/37.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Brauer groups
DefinitionThe Brauer group of X is
Br (X ) := H2(X ,O∗X )tor
in the analytic topology.
If X is a K3 surface, the Universal Coefficient Theorem, yields anice description of the Brauer group:
Br (X ) ∼= Hom(T (X ), Q/Z)
where T (X ) := NS(X )⊥.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 38: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/38.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
DefinitionA pair (X , α) where X is a smooth projective variety andα ∈ Br (X ) is a twisted variety.
Represent α ∈ Br(X ) as a Cech 2-cocycle
{αijk ∈ Γ(Ui ∩ Uj ∩ Uk ,O∗X )}
on an analytic open cover X =⋃
i∈I Ui .
RemarkThe following definitions depend on the choice of the cocyclerepresenting α ∈ Br (X ).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 39: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/39.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
DefinitionA pair (X , α) where X is a smooth projective variety andα ∈ Br (X ) is a twisted variety.
Represent α ∈ Br(X ) as a Cech 2-cocycle
{αijk ∈ Γ(Ui ∩ Uj ∩ Uk ,O∗X )}
on an analytic open cover X =⋃
i∈I Ui .
RemarkThe following definitions depend on the choice of the cocyclerepresenting α ∈ Br (X ).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 40: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/40.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
DefinitionA pair (X , α) where X is a smooth projective variety andα ∈ Br (X ) is a twisted variety.
Represent α ∈ Br(X ) as a Cech 2-cocycle
{αijk ∈ Γ(Ui ∩ Uj ∩ Uk ,O∗X )}
on an analytic open cover X =⋃
i∈I Ui .
RemarkThe following definitions depend on the choice of the cocyclerepresenting α ∈ Br (X ).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 41: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/41.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
DefinitionA pair (X , α) where X is a smooth projective variety andα ∈ Br (X ) is a twisted variety.
Represent α ∈ Br(X ) as a Cech 2-cocycle
{αijk ∈ Γ(Ui ∩ Uj ∩ Uk ,O∗X )}
on an analytic open cover X =⋃
i∈I Ui .
RemarkThe following definitions depend on the choice of the cocyclerepresenting α ∈ Br (X ).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 42: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/42.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
An α-twisted coherent sheaf E is a collection of pairs
({Ei}i∈I , {ϕij}i,j∈I)
where
Ei is a coherent sheaf on the open subset Ui ;ϕij : Ej |Ui∩Uj → Ei |Ui∩Uj is an isomorphism
such that1 ϕii = id,2 ϕji = ϕ−1
ij and3 ϕij ◦ ϕjk ◦ ϕki = αijk · id.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 43: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/43.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
An α-twisted coherent sheaf E is a collection of pairs
({Ei}i∈I , {ϕij}i,j∈I)
whereEi is a coherent sheaf on the open subset Ui ;
ϕij : Ej |Ui∩Uj → Ei |Ui∩Uj is an isomorphismsuch that
1 ϕii = id,2 ϕji = ϕ−1
ij and3 ϕij ◦ ϕjk ◦ ϕki = αijk · id.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 44: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/44.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
An α-twisted coherent sheaf E is a collection of pairs
({Ei}i∈I , {ϕij}i,j∈I)
whereEi is a coherent sheaf on the open subset Ui ;ϕij : Ej |Ui∩Uj → Ei |Ui∩Uj is an isomorphism
such that1 ϕii = id,2 ϕji = ϕ−1
ij and3 ϕij ◦ ϕjk ◦ ϕki = αijk · id.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 45: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/45.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
An α-twisted coherent sheaf E is a collection of pairs
({Ei}i∈I , {ϕij}i,j∈I)
whereEi is a coherent sheaf on the open subset Ui ;ϕij : Ej |Ui∩Uj → Ei |Ui∩Uj is an isomorphism
such that
1 ϕii = id,2 ϕji = ϕ−1
ij and3 ϕij ◦ ϕjk ◦ ϕki = αijk · id.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 46: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/46.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
An α-twisted coherent sheaf E is a collection of pairs
({Ei}i∈I , {ϕij}i,j∈I)
whereEi is a coherent sheaf on the open subset Ui ;ϕij : Ej |Ui∩Uj → Ei |Ui∩Uj is an isomorphism
such that1 ϕii = id,
2 ϕji = ϕ−1ij and
3 ϕij ◦ ϕjk ◦ ϕki = αijk · id.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 47: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/47.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
An α-twisted coherent sheaf E is a collection of pairs
({Ei}i∈I , {ϕij}i,j∈I)
whereEi is a coherent sheaf on the open subset Ui ;ϕij : Ej |Ui∩Uj → Ei |Ui∩Uj is an isomorphism
such that1 ϕii = id,2 ϕji = ϕ−1
ij and
3 ϕij ◦ ϕjk ◦ ϕki = αijk · id.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 48: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/48.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
An α-twisted coherent sheaf E is a collection of pairs
({Ei}i∈I , {ϕij}i,j∈I)
whereEi is a coherent sheaf on the open subset Ui ;ϕij : Ej |Ui∩Uj → Ei |Ui∩Uj is an isomorphism
such that1 ϕii = id,2 ϕji = ϕ−1
ij and3 ϕij ◦ ϕjk ◦ ϕki = αijk · id.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 49: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/49.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
In this way we get the abelian category Coh(X , α).
Pass to the category of bounded complexes.
Localize: require that any quasi-isomorphism is invertible.
We get the bounded derived category Db(X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 50: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/50.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
In this way we get the abelian category Coh(X , α).
Pass to the category of bounded complexes.
Localize: require that any quasi-isomorphism is invertible.
We get the bounded derived category Db(X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 51: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/51.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
In this way we get the abelian category Coh(X , α).
Pass to the category of bounded complexes.
Localize: require that any quasi-isomorphism is invertible.
We get the bounded derived category Db(X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 52: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/52.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Twisted sheaves
In this way we get the abelian category Coh(X , α).
Pass to the category of bounded complexes.
Localize: require that any quasi-isomorphism is invertible.
We get the bounded derived category Db(X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 53: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/53.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Motivations
M is a 2-dimensional, irreducible, smooth and projective coarsemoduli space of stable sheaves on X .
Caldararu: the obstructionA special element α ∈ Br (M) is the obstruction to the existenceof a universal family on M.
Proposition (Mukai, Caldararu)Let X be a K3 surface and let M be a coarse moduli space ofstable sheaves on X as above. Then Db(X ) ∼= Db(M, α−1) (viathe twisted universal/quasi-universal family.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 54: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/54.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Motivations
M is a 2-dimensional, irreducible, smooth and projective coarsemoduli space of stable sheaves on X .
Caldararu: the obstructionA special element α ∈ Br (M) is the obstruction to the existenceof a universal family on M.
Proposition (Mukai, Caldararu)Let X be a K3 surface and let M be a coarse moduli space ofstable sheaves on X as above. Then Db(X ) ∼= Db(M, α−1) (viathe twisted universal/quasi-universal family.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 55: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/55.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Motivations
M is a 2-dimensional, irreducible, smooth and projective coarsemoduli space of stable sheaves on X .
Caldararu: the obstructionA special element α ∈ Br (M) is the obstruction to the existenceof a universal family on M.
Proposition (Mukai, Caldararu)Let X be a K3 surface and let M be a coarse moduli space ofstable sheaves on X as above.
Then Db(X ) ∼= Db(M, α−1) (viathe twisted universal/quasi-universal family.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 56: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/56.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Motivations
M is a 2-dimensional, irreducible, smooth and projective coarsemoduli space of stable sheaves on X .
Caldararu: the obstructionA special element α ∈ Br (M) is the obstruction to the existenceof a universal family on M.
Proposition (Mukai, Caldararu)Let X be a K3 surface and let M be a coarse moduli space ofstable sheaves on X as above. Then Db(X ) ∼= Db(M, α−1) (viathe twisted universal/quasi-universal family.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 57: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/57.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The twisted derived case
Twisted Derived Torelli Theorem (Huybrechts-S.)
Let X and X ′ be two projective K3 surfaces endowed withB-fields B ∈ H2(X , Q) and B′ ∈ H2(X ′, Q).
1 If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence, then thereexists a naturally defined Hodge isometryΦB,B′∗ : H(X , B, Z) ∼= H(X ′, B′, Z).
2 Suppose there exists a Hodge isometryg : H(X , B, Z) ∼= H(X ′, B′, Z) that preserves the naturalorientation of the four positive directions. Then there existsan equivalence Φ : Db(X , αB) ∼= Db(X ′, αB′) such thatΦB,B′∗ = g.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 58: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/58.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The twisted derived case
Twisted Derived Torelli Theorem (Huybrechts-S.)
Let X and X ′ be two projective K3 surfaces endowed withB-fields B ∈ H2(X , Q) and B′ ∈ H2(X ′, Q).
1 If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence, then thereexists a naturally defined Hodge isometryΦB,B′∗ : H(X , B, Z) ∼= H(X ′, B′, Z).
2 Suppose there exists a Hodge isometryg : H(X , B, Z) ∼= H(X ′, B′, Z) that preserves the naturalorientation of the four positive directions. Then there existsan equivalence Φ : Db(X , αB) ∼= Db(X ′, αB′) such thatΦB,B′∗ = g.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 59: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/59.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The twisted derived case
Twisted Derived Torelli Theorem (Huybrechts-S.)
Let X and X ′ be two projective K3 surfaces endowed withB-fields B ∈ H2(X , Q) and B′ ∈ H2(X ′, Q).
1 If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence, then thereexists a naturally defined Hodge isometryΦB,B′∗ : H(X , B, Z) ∼= H(X ′, B′, Z).
2 Suppose there exists a Hodge isometryg : H(X , B, Z) ∼= H(X ′, B′, Z) that preserves the naturalorientation of the four positive directions. Then there existsan equivalence Φ : Db(X , αB) ∼= Db(X ′, αB′) such thatΦB,B′∗ = g.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 60: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/60.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The twisted derived case
Twisted Derived Torelli Theorem (Huybrechts-S.)
Let X and X ′ be two projective K3 surfaces endowed withB-fields B ∈ H2(X , Q) and B′ ∈ H2(X ′, Q).
1 If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence, then thereexists a naturally defined Hodge isometryΦB,B′∗ : H(X , B, Z) ∼= H(X ′, B′, Z).
2 Suppose there exists a Hodge isometryg : H(X , B, Z) ∼= H(X ′, B′, Z) that preserves the naturalorientation of the four positive directions. Then there existsan equivalence Φ : Db(X , αB) ∼= Db(X ′, αB′) such thatΦB,B′∗ = g.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 61: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/61.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 62: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/62.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Lattice structure
Using the cup product, we get the Mukai pairing on H∗(X , Z):
〈α, β〉 := −α1 · β3 + α2 · β2 − α3 · β1,
for every α = (α1, α2, α3) and β = (β1, β2, β3) in H∗(X , Z).
H∗(X , Z) endowed with the Mukai pairing is called Mukai latticeand we write H(X , Z) for it.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 63: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/63.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Lattice structure
Using the cup product, we get the Mukai pairing on H∗(X , Z):
〈α, β〉 := −α1 · β3 + α2 · β2 − α3 · β1,
for every α = (α1, α2, α3) and β = (β1, β2, β3) in H∗(X , Z).
H∗(X , Z) endowed with the Mukai pairing is called Mukai latticeand we write H(X , Z) for it.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 64: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/64.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Lattice structure
Using the cup product, we get the Mukai pairing on H∗(X , Z):
〈α, β〉 := −α1 · β3 + α2 · β2 − α3 · β1,
for every α = (α1, α2, α3) and β = (β1, β2, β3) in H∗(X , Z).
H∗(X , Z) endowed with the Mukai pairing is called Mukai latticeand we write H(X , Z) for it.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 65: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/65.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Lattice structure
Using the cup product, we get the Mukai pairing on H∗(X , Z):
〈α, β〉 := −α1 · β3 + α2 · β2 − α3 · β1,
for every α = (α1, α2, α3) and β = (β1, β2, β3) in H∗(X , Z).
H∗(X , Z) endowed with the Mukai pairing is called Mukai latticeand we write H(X , Z) for it.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 66: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/66.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 67: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/67.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 68: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/68.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 69: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/69.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 70: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/70.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 71: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/71.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 72: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/72.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 73: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/73.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Any α ∈ Br (X ) is determined by some B ∈ H2(X , Q) andvice-versa.
Actually α is determined by B ∈ T (X )∨ ⊗Q/Z!
More precisely, given α ∈ Br (X ), there exists B ∈ H2(X , Q)such that
α = exp(B).
Let H2,0(X ) = 〈σ〉 and let B be a B-field on X .
ϕ = exp(B) · σ = σ + B ∧ σ ∈ H2(X , C)⊕ H4(X , C)
is a generalized Calabi-Yau structure.
Hitchin, Huybrechts: complete classification of generalized CYstructures on K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 74: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/74.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Definition
Let X be a K3 surface with a B-field B ∈ H2(X , Q). We denoteby H(X , B, Z) the weight-two Hodge structure on H∗(X , Z) with
H2,0(X , B) := exp(B)(
H2,0(X ))
and H1,1(X , B) its orthogonal complement with respect to theMukai pairing.
This gives a generalization of the notion of Picard lattice andtranscendental lattice.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 75: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/75.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Definition
Let X be a K3 surface with a B-field B ∈ H2(X , Q).
We denoteby H(X , B, Z) the weight-two Hodge structure on H∗(X , Z) with
H2,0(X , B) := exp(B)(
H2,0(X ))
and H1,1(X , B) its orthogonal complement with respect to theMukai pairing.
This gives a generalization of the notion of Picard lattice andtranscendental lattice.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 76: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/76.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Definition
Let X be a K3 surface with a B-field B ∈ H2(X , Q). We denoteby H(X , B, Z) the weight-two Hodge structure on H∗(X , Z) with
H2,0(X , B) := exp(B)(
H2,0(X ))
and H1,1(X , B) its orthogonal complement with respect to theMukai pairing.
This gives a generalization of the notion of Picard lattice andtranscendental lattice.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 77: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/77.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Hodge structure
Definition
Let X be a K3 surface with a B-field B ∈ H2(X , Q). We denoteby H(X , B, Z) the weight-two Hodge structure on H∗(X , Z) with
H2,0(X , B) := exp(B)(
H2,0(X ))
and H1,1(X , B) its orthogonal complement with respect to theMukai pairing.
This gives a generalization of the notion of Picard lattice andtranscendental lattice.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 78: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/78.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Orientation
Let σX be a generator of H2,0(X ) and let ω be a Kahler class.Then
〈Re(σX ), Im(σX ), 1− ω2/2, ω〉
is a positive four-space in H(X , R).
Remark1 It comes, by the choice of the basis, with a natural
orientation.2 It is easy to see that this orientation is independent of the
choice of σX and ω.3 The isometry j := idH0⊕H4 ⊕(− id)H2 is not orientation
preserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 79: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/79.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Orientation
Let σX be a generator of H2,0(X ) and let ω be a Kahler class.
Then〈Re(σX ), Im(σX ), 1− ω2/2, ω〉
is a positive four-space in H(X , R).
Remark1 It comes, by the choice of the basis, with a natural
orientation.2 It is easy to see that this orientation is independent of the
choice of σX and ω.3 The isometry j := idH0⊕H4 ⊕(− id)H2 is not orientation
preserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 80: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/80.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Orientation
Let σX be a generator of H2,0(X ) and let ω be a Kahler class.Then
〈Re(σX ), Im(σX ), 1− ω2/2, ω〉
is a positive four-space in H(X , R).
Remark1 It comes, by the choice of the basis, with a natural
orientation.2 It is easy to see that this orientation is independent of the
choice of σX and ω.3 The isometry j := idH0⊕H4 ⊕(− id)H2 is not orientation
preserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 81: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/81.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Orientation
Let σX be a generator of H2,0(X ) and let ω be a Kahler class.Then
〈Re(σX ), Im(σX ), 1− ω2/2, ω〉
is a positive four-space in H(X , R).
Remark
1 It comes, by the choice of the basis, with a naturalorientation.
2 It is easy to see that this orientation is independent of thechoice of σX and ω.
3 The isometry j := idH0⊕H4 ⊕(− id)H2 is not orientationpreserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 82: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/82.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Orientation
Let σX be a generator of H2,0(X ) and let ω be a Kahler class.Then
〈Re(σX ), Im(σX ), 1− ω2/2, ω〉
is a positive four-space in H(X , R).
Remark1 It comes, by the choice of the basis, with a natural
orientation.
2 It is easy to see that this orientation is independent of thechoice of σX and ω.
3 The isometry j := idH0⊕H4 ⊕(− id)H2 is not orientationpreserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 83: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/83.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Orientation
Let σX be a generator of H2,0(X ) and let ω be a Kahler class.Then
〈Re(σX ), Im(σX ), 1− ω2/2, ω〉
is a positive four-space in H(X , R).
Remark1 It comes, by the choice of the basis, with a natural
orientation.2 It is easy to see that this orientation is independent of the
choice of σX and ω.
3 The isometry j := idH0⊕H4 ⊕(− id)H2 is not orientationpreserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 84: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/84.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Orientation
Let σX be a generator of H2,0(X ) and let ω be a Kahler class.Then
〈Re(σX ), Im(σX ), 1− ω2/2, ω〉
is a positive four-space in H(X , R).
Remark1 It comes, by the choice of the basis, with a natural
orientation.2 It is easy to see that this orientation is independent of the
choice of σX and ω.3 The isometry j := idH0⊕H4 ⊕(− id)H2 is not orientation
preserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 85: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/85.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 86: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/86.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier-Mukai functors
Definition
F : Db(X , α) → Db(Y , β) is of Fourier-Mukai type if there existsE ∈ Db(X × Y , α−1 � β) and an isomorphism of functors
F ∼= Rp∗(EL⊗ q∗(−)),
where p : X × Y → Y and q : X × Y → X are the naturalprojections.
The complex E is called the kernel of F and a Fourier-Mukaifunctor with kernel E is denoted by ΦE .
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 87: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/87.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier-Mukai functors
Definition
F : Db(X , α) → Db(Y , β) is of Fourier-Mukai type if there existsE ∈ Db(X × Y , α−1 � β) and an isomorphism of functors
F ∼= Rp∗(EL⊗ q∗(−)),
where p : X × Y → Y and q : X × Y → X are the naturalprojections.
The complex E is called the kernel of F and a Fourier-Mukaifunctor with kernel E is denoted by ΦE .
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 88: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/88.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier-Mukai functors
Definition
F : Db(X , α) → Db(Y , β) is of Fourier-Mukai type if there existsE ∈ Db(X × Y , α−1 � β) and an isomorphism of functors
F ∼= Rp∗(EL⊗ q∗(−)),
where p : X × Y → Y and q : X × Y → X are the naturalprojections.
The complex E is called the kernel of F and a Fourier-Mukaifunctor with kernel E is denoted by ΦE .
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 89: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/89.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem (Orlov)
Any exact functor F : Db(X ) → Db(Y ) which1 is fully faithful2 admits a left adjoint
is a Fourier-Mukai functor.
Remark (Bondal-Van den Bergh)Item (2) is automatic!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 90: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/90.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem (Orlov)
Any exact functor F : Db(X ) → Db(Y ) which
1 is fully faithful2 admits a left adjoint
is a Fourier-Mukai functor.
Remark (Bondal-Van den Bergh)Item (2) is automatic!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 91: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/91.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem (Orlov)
Any exact functor F : Db(X ) → Db(Y ) which1 is fully faithful
2 admits a left adjointis a Fourier-Mukai functor.
Remark (Bondal-Van den Bergh)Item (2) is automatic!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 92: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/92.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem (Orlov)
Any exact functor F : Db(X ) → Db(Y ) which1 is fully faithful2 admits a left adjoint
is a Fourier-Mukai functor.
Remark (Bondal-Van den Bergh)Item (2) is automatic!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 93: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/93.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem (Orlov)
Any exact functor F : Db(X ) → Db(Y ) which1 is fully faithful2 admits a left adjoint
is a Fourier-Mukai functor.
Remark (Bondal-Van den Bergh)Item (2) is automatic!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 94: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/94.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem (Orlov)
Any exact functor F : Db(X ) → Db(Y ) which1 is fully faithful2 admits a left adjoint
is a Fourier-Mukai functor.
Remark (Bondal-Van den Bergh)Item (2) is automatic!
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 95: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/95.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem. (Canonanco-S.)Let (X , α) and (Y , β) be twisted varieties. Let
F : Db(X , α) → Db(Y , β)
be an exact functor such that, for any F ,G ∈ Coh(X , α),
Hom Db(Y ,β)(F (F), F (G)[j]) = 0 if j < 0.
Then there exist E ∈ Db(X × Y , α−1 � β) and an isomorphismof functors F ∼= ΦE . Moreover, E is uniquely determined up toisomorphism.
Any equivalence is of Fourier–Mukai type.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 96: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/96.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem. (Canonanco-S.)Let (X , α) and (Y , β) be twisted varieties. Let
F : Db(X , α) → Db(Y , β)
be an exact functor
such that, for any F ,G ∈ Coh(X , α),
Hom Db(Y ,β)(F (F), F (G)[j]) = 0 if j < 0.
Then there exist E ∈ Db(X × Y , α−1 � β) and an isomorphismof functors F ∼= ΦE . Moreover, E is uniquely determined up toisomorphism.
Any equivalence is of Fourier–Mukai type.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 97: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/97.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem. (Canonanco-S.)Let (X , α) and (Y , β) be twisted varieties. Let
F : Db(X , α) → Db(Y , β)
be an exact functor such that, for any F ,G ∈ Coh(X , α),
Hom Db(Y ,β)(F (F), F (G)[j]) = 0 if j < 0.
Then there exist E ∈ Db(X × Y , α−1 � β) and an isomorphismof functors F ∼= ΦE . Moreover, E is uniquely determined up toisomorphism.
Any equivalence is of Fourier–Mukai type.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 98: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/98.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem. (Canonanco-S.)Let (X , α) and (Y , β) be twisted varieties. Let
F : Db(X , α) → Db(Y , β)
be an exact functor such that, for any F ,G ∈ Coh(X , α),
Hom Db(Y ,β)(F (F), F (G)[j]) = 0 if j < 0.
Then there exist E ∈ Db(X × Y , α−1 � β)
and an isomorphismof functors F ∼= ΦE . Moreover, E is uniquely determined up toisomorphism.
Any equivalence is of Fourier–Mukai type.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 99: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/99.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem. (Canonanco-S.)Let (X , α) and (Y , β) be twisted varieties. Let
F : Db(X , α) → Db(Y , β)
be an exact functor such that, for any F ,G ∈ Coh(X , α),
Hom Db(Y ,β)(F (F), F (G)[j]) = 0 if j < 0.
Then there exist E ∈ Db(X × Y , α−1 � β) and an isomorphismof functors F ∼= ΦE .
Moreover, E is uniquely determined up toisomorphism.
Any equivalence is of Fourier–Mukai type.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 100: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/100.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem. (Canonanco-S.)Let (X , α) and (Y , β) be twisted varieties. Let
F : Db(X , α) → Db(Y , β)
be an exact functor such that, for any F ,G ∈ Coh(X , α),
Hom Db(Y ,β)(F (F), F (G)[j]) = 0 if j < 0.
Then there exist E ∈ Db(X × Y , α−1 � β) and an isomorphismof functors F ∼= ΦE . Moreover, E is uniquely determined up toisomorphism.
Any equivalence is of Fourier–Mukai type.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 101: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/101.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
Fourier–Mukai equivalences
Theorem. (Canonanco-S.)Let (X , α) and (Y , β) be twisted varieties. Let
F : Db(X , α) → Db(Y , β)
be an exact functor such that, for any F ,G ∈ Coh(X , α),
Hom Db(Y ,β)(F (F), F (G)[j]) = 0 if j < 0.
Then there exist E ∈ Db(X × Y , α−1 � β) and an isomorphismof functors F ∼= ΦE . Moreover, E is uniquely determined up toisomorphism.
Any equivalence is of Fourier–Mukai type.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 102: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/102.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Chern character
Take E ∈ Db(X , α) and its image E = [E ] in the Grothendieckgroup
K (X , α) := K (Db(X , α)).
One defines a twisted Chern character
ch B(E) ∈ H∗(X , Z)
depending on the choice of a B-field representing the elementα ∈ Br (X ).
RemarkThe notion of c1(E) is well-defined.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 103: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/103.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Chern character
Take E ∈ Db(X , α) and its image E = [E ] in the Grothendieckgroup
K (X , α) := K (Db(X , α)).
One defines a twisted Chern character
ch B(E) ∈ H∗(X , Z)
depending on the choice of a B-field representing the elementα ∈ Br (X ).
RemarkThe notion of c1(E) is well-defined.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 104: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/104.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Chern character
Take E ∈ Db(X , α) and its image E = [E ] in the Grothendieckgroup
K (X , α) := K (Db(X , α)).
One defines a twisted Chern character
ch B(E) ∈ H∗(X , Z)
depending on the choice of a B-field representing the elementα ∈ Br (X ).
RemarkThe notion of c1(E) is well-defined.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 105: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/105.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
The Chern character
Take E ∈ Db(X , α) and its image E = [E ] in the Grothendieckgroup
K (X , α) := K (Db(X , α)).
One defines a twisted Chern character
ch B(E) ∈ H∗(X , Z)
depending on the choice of a B-field representing the elementα ∈ Br (X ).
RemarkThe notion of c1(E) is well-defined.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 106: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/106.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
A commutative diagram
Using the Chern character one gets the commutative diagram:
Db(X , α)
[−]
��
Φ// Db(Y , β)
[−]
��K (X , α)
ch B(−)·√
td(X)
��
// K (Y , β)
ch B′(−)·
√td(Y )
��
H(X , B, Z)ΦB,B′
∗ // H(Y , B′, Z).
Paolo Stellari Derived Torelli Theorem and Orientation
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MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
Twisted sheavesThe structuresIdeas form the proof
A commutative diagram
Using the Chern character one gets the commutative diagram:
Db(X , α)
[−]
��
Φ// Db(Y , β)
[−]
��K (X , α)
ch B(−)·√
td(X)
��
// K (Y , β)
ch B′(−)·
√td(Y )
��
H(X , B, Z)ΦB,B′
∗ // H(Y , B′, Z).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 108: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/108.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 109: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/109.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Orienatation
The orientation preserving requirement is missing in item (i) ofthe Twisted Derived Torelli Theorem.
Proposition (Huybrechts-S.)Any known twisted or untwisted equivalence is orientationpreserving.
Conjecture (Szendroi, Huybrechts-S.)
Let X and X ′ be two algebraic K3 surfaces with B-fields B andB′. If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence thenΦB,B′∗ : H(X , B, Z) → H(X ′, B′, Z) preserves the natural
orientation of the four positive directions.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 110: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/110.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Orienatation
The orientation preserving requirement is missing in item (i) ofthe Twisted Derived Torelli Theorem.
Proposition (Huybrechts-S.)Any known twisted or untwisted equivalence is orientationpreserving.
Conjecture (Szendroi, Huybrechts-S.)
Let X and X ′ be two algebraic K3 surfaces with B-fields B andB′. If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence thenΦB,B′∗ : H(X , B, Z) → H(X ′, B′, Z) preserves the natural
orientation of the four positive directions.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 111: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/111.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Orienatation
The orientation preserving requirement is missing in item (i) ofthe Twisted Derived Torelli Theorem.
Proposition (Huybrechts-S.)Any known twisted or untwisted equivalence is orientationpreserving.
Conjecture (Szendroi, Huybrechts-S.)
Let X and X ′ be two algebraic K3 surfaces with B-fields B andB′. If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence thenΦB,B′∗ : H(X , B, Z) → H(X ′, B′, Z) preserves the natural
orientation of the four positive directions.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 112: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/112.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Orienatation
The orientation preserving requirement is missing in item (i) ofthe Twisted Derived Torelli Theorem.
Proposition (Huybrechts-S.)Any known twisted or untwisted equivalence is orientationpreserving.
Conjecture (Szendroi, Huybrechts-S.)
Let X and X ′ be two algebraic K3 surfaces with B-fields B andB′.
If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence thenΦB,B′∗ : H(X , B, Z) → H(X ′, B′, Z) preserves the natural
orientation of the four positive directions.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 113: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/113.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Orienatation
The orientation preserving requirement is missing in item (i) ofthe Twisted Derived Torelli Theorem.
Proposition (Huybrechts-S.)Any known twisted or untwisted equivalence is orientationpreserving.
Conjecture (Szendroi, Huybrechts-S.)
Let X and X ′ be two algebraic K3 surfaces with B-fields B andB′. If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence
thenΦB,B′∗ : H(X , B, Z) → H(X ′, B′, Z) preserves the natural
orientation of the four positive directions.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 114: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/114.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Orienatation
The orientation preserving requirement is missing in item (i) ofthe Twisted Derived Torelli Theorem.
Proposition (Huybrechts-S.)Any known twisted or untwisted equivalence is orientationpreserving.
Conjecture (Szendroi, Huybrechts-S.)
Let X and X ′ be two algebraic K3 surfaces with B-fields B andB′. If Φ : Db(X , αB) ∼= Db(X ′, αB′) is an equivalence thenΦB,B′∗ : H(X , B, Z) → H(X ′, B′, Z) preserves the natural
orientation of the four positive directions.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 115: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/115.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 116: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/116.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Evidence and problem
Proposition (Huybrechts-S.)
Let X and X ′ be K3 surfaces with large Picard number and letB ∈ H2(X , Q) and B′ ∈ H2(X ′, Q). Then the following areequivalent:
1 There exists an exact equivalence
Φ : Db(X , αB)∼−→ Db(X ′, αB′).
2 There exists an orientation preserving Hodge isometry
g : H(X , B, Z) ∼= H(X ′, B′, Z).
Problem: Is this always true?In the untwisted case, this is always true.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 117: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/117.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Evidence and problem
Proposition (Huybrechts-S.)
Let X and X ′ be K3 surfaces with large Picard number and letB ∈ H2(X , Q) and B′ ∈ H2(X ′, Q).
Then the following areequivalent:
1 There exists an exact equivalence
Φ : Db(X , αB)∼−→ Db(X ′, αB′).
2 There exists an orientation preserving Hodge isometry
g : H(X , B, Z) ∼= H(X ′, B′, Z).
Problem: Is this always true?In the untwisted case, this is always true.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 118: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/118.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Evidence and problem
Proposition (Huybrechts-S.)
Let X and X ′ be K3 surfaces with large Picard number and letB ∈ H2(X , Q) and B′ ∈ H2(X ′, Q). Then the following areequivalent:
1 There exists an exact equivalence
Φ : Db(X , αB)∼−→ Db(X ′, αB′).
2 There exists an orientation preserving Hodge isometry
g : H(X , B, Z) ∼= H(X ′, B′, Z).
Problem: Is this always true?In the untwisted case, this is always true.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 119: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/119.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Evidence and problem
Proposition (Huybrechts-S.)
Let X and X ′ be K3 surfaces with large Picard number and letB ∈ H2(X , Q) and B′ ∈ H2(X ′, Q). Then the following areequivalent:
1 There exists an exact equivalence
Φ : Db(X , αB)∼−→ Db(X ′, αB′).
2 There exists an orientation preserving Hodge isometry
g : H(X , B, Z) ∼= H(X ′, B′, Z).
Problem: Is this always true?In the untwisted case, this is always true.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 120: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/120.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Evidence and problem
Proposition (Huybrechts-S.)
Let X and X ′ be K3 surfaces with large Picard number and letB ∈ H2(X , Q) and B′ ∈ H2(X ′, Q). Then the following areequivalent:
1 There exists an exact equivalence
Φ : Db(X , αB)∼−→ Db(X ′, αB′).
2 There exists an orientation preserving Hodge isometry
g : H(X , B, Z) ∼= H(X ′, B′, Z).
Problem: Is this always true?In the untwisted case, this is always true.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 121: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/121.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Evidence and problem
Proposition (Huybrechts-S.)
Let X and X ′ be K3 surfaces with large Picard number and letB ∈ H2(X , Q) and B′ ∈ H2(X ′, Q). Then the following areequivalent:
1 There exists an exact equivalence
Φ : Db(X , αB)∼−→ Db(X ′, αB′).
2 There exists an orientation preserving Hodge isometry
g : H(X , B, Z) ∼= H(X ′, B′, Z).
Problem: Is this always true?
In the untwisted case, this is always true.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 122: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/122.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementEvidence
Evidence and problem
Proposition (Huybrechts-S.)
Let X and X ′ be K3 surfaces with large Picard number and letB ∈ H2(X , Q) and B′ ∈ H2(X ′, Q). Then the following areequivalent:
1 There exists an exact equivalence
Φ : Db(X , αB)∼−→ Db(X ′, αB′).
2 There exists an orientation preserving Hodge isometry
g : H(X , B, Z) ∼= H(X ′, B′, Z).
Problem: Is this always true?In the untwisted case, this is always true.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 123: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/123.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 124: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/124.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The ‘final’ Twisted Derived Torelli Theorem
Theorem (Huybrechts-Macrı-S.)
Let X be a K3 surface and let B ∈ H2(X , Q) be such that(X , αB) is generic. Then there exists a short exact sequence
1 −→ Z[2] −→ Aut (Db(X , αB))ϕ−→ O+ −→ 1,
where O+ is the group of the Hodge isometries of H(X , B, Z)preserving the orientation.
A twisted K3 surface (X , α) is generic meaning that it is genericin the moduli space of twisted K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 125: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/125.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The ‘final’ Twisted Derived Torelli Theorem
Theorem (Huybrechts-Macrı-S.)
Let X be a K3 surface and let B ∈ H2(X , Q) be such that(X , αB) is generic.
Then there exists a short exact sequence
1 −→ Z[2] −→ Aut (Db(X , αB))ϕ−→ O+ −→ 1,
where O+ is the group of the Hodge isometries of H(X , B, Z)preserving the orientation.
A twisted K3 surface (X , α) is generic meaning that it is genericin the moduli space of twisted K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 126: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/126.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The ‘final’ Twisted Derived Torelli Theorem
Theorem (Huybrechts-Macrı-S.)
Let X be a K3 surface and let B ∈ H2(X , Q) be such that(X , αB) is generic. Then there exists a short exact sequence
1 −→ Z[2] −→ Aut (Db(X , αB))ϕ−→ O+ −→ 1,
where O+ is the group of the Hodge isometries of H(X , B, Z)preserving the orientation.
A twisted K3 surface (X , α) is generic meaning that it is genericin the moduli space of twisted K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 127: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/127.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The ‘final’ Twisted Derived Torelli Theorem
Theorem (Huybrechts-Macrı-S.)
Let X be a K3 surface and let B ∈ H2(X , Q) be such that(X , αB) is generic. Then there exists a short exact sequence
1 −→ Z[2] −→ Aut (Db(X , αB))ϕ−→ O+ −→ 1,
where O+ is the group of the Hodge isometries of H(X , B, Z)preserving the orientation.
A twisted K3 surface (X , α) is generic meaning that it is genericin the moduli space of twisted K3 surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 128: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/128.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
PropertiesA generic K3 surface (X , α) is, in the rest of this talk,characterized by the fact that:
1 The underlying K3 surface X is generic and projective (i.e.Picard number 1).
2 The twist α is generic (i.e. restrictions on the order of α).
WarningWe have no idea about how to use the generic case to provethe conjecture for any twisted K3 surface (X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 129: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/129.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
PropertiesA generic K3 surface (X , α) is, in the rest of this talk,characterized by the fact that:
1 The underlying K3 surface X is generic and projective (i.e.Picard number 1).
2 The twist α is generic (i.e. restrictions on the order of α).
WarningWe have no idea about how to use the generic case to provethe conjecture for any twisted K3 surface (X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 130: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/130.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
PropertiesA generic K3 surface (X , α) is, in the rest of this talk,characterized by the fact that:
1 The underlying K3 surface X is generic and projective (i.e.Picard number 1).
2 The twist α is generic (i.e. restrictions on the order of α).
WarningWe have no idea about how to use the generic case to provethe conjecture for any twisted K3 surface (X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 131: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/131.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
PropertiesA generic K3 surface (X , α) is, in the rest of this talk,characterized by the fact that:
1 The underlying K3 surface X is generic and projective (i.e.Picard number 1).
2 The twist α is generic (i.e. restrictions on the order of α).
WarningWe have no idea about how to use the generic case to provethe conjecture for any twisted K3 surface (X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 132: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/132.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
PropertiesA generic K3 surface (X , α) is, in the rest of this talk,characterized by the fact that:
1 The underlying K3 surface X is generic and projective (i.e.Picard number 1).
2 The twist α is generic (i.e. restrictions on the order of α).
WarningWe have no idea about how to use the generic case to provethe conjecture for any twisted K3 surface (X , α).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 133: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/133.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
The equivalent formulation would be:
Take an equivalence Φ : Db(X , α)∼−→ Db(Y , β), of generic
twisted K3 surfaces (X , α) and (Y , β). Then the natural map Φ∗induced on cohomology is an orientation preserving Hodgeisometry.
RemarkWe proved Bridgeland’s Conjecture for generic twisted K3surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 134: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/134.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
The equivalent formulation would be:
Take an equivalence Φ : Db(X , α)∼−→ Db(Y , β), of generic
twisted K3 surfaces (X , α) and (Y , β). Then the natural map Φ∗induced on cohomology is an orientation preserving Hodgeisometry.
RemarkWe proved Bridgeland’s Conjecture for generic twisted K3surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 135: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/135.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
The equivalent formulation would be:
Take an equivalence Φ : Db(X , α)∼−→ Db(Y , β), of generic
twisted K3 surfaces (X , α) and (Y , β). Then the natural map Φ∗induced on cohomology is an orientation preserving Hodgeisometry.
RemarkWe proved Bridgeland’s Conjecture for generic twisted K3surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 136: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/136.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Remarks
The equivalent formulation would be:
Take an equivalence Φ : Db(X , α)∼−→ Db(Y , β), of generic
twisted K3 surfaces (X , α) and (Y , β). Then the natural map Φ∗induced on cohomology is an orientation preserving Hodgeisometry.
RemarkWe proved Bridgeland’s Conjecture for generic twisted K3surfaces.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 137: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/137.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Outline1 Motivations
The settingThe geometric caseThe derived case
2 Twisted Derived Torelli TheoremTwisted sheavesThe structuresIdeas form the proof
3 The conjectureThe statementEvidence
4 The twisted generic caseThe statementIdeas from the proof
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 138: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/138.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The genericity hypothesis
Definition
An object E ∈ Db(X , α) is spherical if
Hom (E , E [i]) ∼={
C if i ∈ {0, 2}0 otherwise.
PropositionThere exists a dense subset of twisted K3 surfaces (X , α) suchthat Db(X , α) does not contain spherical objects.
In the untwisted case, Db(X ) always contains spherical objects.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 139: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/139.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The genericity hypothesis
Definition
An object E ∈ Db(X , α) is spherical if
Hom (E , E [i]) ∼={
C if i ∈ {0, 2}0 otherwise.
PropositionThere exists a dense subset of twisted K3 surfaces (X , α) suchthat Db(X , α) does not contain spherical objects.
In the untwisted case, Db(X ) always contains spherical objects.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 140: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/140.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The genericity hypothesis
Definition
An object E ∈ Db(X , α) is spherical if
Hom (E , E [i]) ∼={
C if i ∈ {0, 2}0 otherwise.
PropositionThere exists a dense subset of twisted K3 surfaces (X , α) suchthat Db(X , α) does not contain spherical objects.
In the untwisted case, Db(X ) always contains spherical objects.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 141: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/141.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
The genericity hypothesis
Definition
An object E ∈ Db(X , α) is spherical if
Hom (E , E [i]) ∼={
C if i ∈ {0, 2}0 otherwise.
PropositionThere exists a dense subset of twisted K3 surfaces (X , α) suchthat Db(X , α) does not contain spherical objects.
In the untwisted case, Db(X ) always contains spherical objects.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 142: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/142.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
For simplicity, we restrict ourselves to the case of stabilityconditions on derived categories Db(X , α)! Any triangulatedcategory would fit.
A stability condition on Db(X , α) is a pair σ = (Z ,P) where
Z : N (X , α)⊗ C → C is a linear map (the central charge;here N (X , α) is the sublattice of H(X , Z) orthogonal to thegeneralized complex structure)
P(φ) ⊂ Db(X , α) are full additive subcategories for eachφ ∈ R
satisfying the following conditions:
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 143: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/143.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
For simplicity, we restrict ourselves to the case of stabilityconditions on derived categories Db(X , α)! Any triangulatedcategory would fit.
A stability condition on Db(X , α) is a pair σ = (Z ,P) where
Z : N (X , α)⊗ C → C is a linear map (the central charge;here N (X , α) is the sublattice of H(X , Z) orthogonal to thegeneralized complex structure)
P(φ) ⊂ Db(X , α) are full additive subcategories for eachφ ∈ R
satisfying the following conditions:
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 144: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/144.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
For simplicity, we restrict ourselves to the case of stabilityconditions on derived categories Db(X , α)! Any triangulatedcategory would fit.
A stability condition on Db(X , α) is a pair σ = (Z ,P) where
Z : N (X , α)⊗ C → C is a linear map
(the central charge;here N (X , α) is the sublattice of H(X , Z) orthogonal to thegeneralized complex structure)
P(φ) ⊂ Db(X , α) are full additive subcategories for eachφ ∈ R
satisfying the following conditions:
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 145: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/145.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
For simplicity, we restrict ourselves to the case of stabilityconditions on derived categories Db(X , α)! Any triangulatedcategory would fit.
A stability condition on Db(X , α) is a pair σ = (Z ,P) where
Z : N (X , α)⊗ C → C is a linear map (the central charge;here N (X , α) is the sublattice of H(X , Z) orthogonal to thegeneralized complex structure)
P(φ) ⊂ Db(X , α) are full additive subcategories for eachφ ∈ R
satisfying the following conditions:
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 146: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/146.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
For simplicity, we restrict ourselves to the case of stabilityconditions on derived categories Db(X , α)! Any triangulatedcategory would fit.
A stability condition on Db(X , α) is a pair σ = (Z ,P) where
Z : N (X , α)⊗ C → C is a linear map (the central charge;here N (X , α) is the sublattice of H(X , Z) orthogonal to thegeneralized complex structure)
P(φ) ⊂ Db(X , α) are full additive subcategories for eachφ ∈ R
satisfying the following conditions:
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 147: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/147.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
(a) If 0 6= E ∈ P(φ), then Z (E) = m(E) exp(iπφ) for somem(E) ∈ R>0.
(b) P(φ + 1) = P(φ)[1] for all φ.
(c) Hom (E1, E2) = 0 for all Ei ∈ P(φi) with φ1 > φ2.
(d) Any 0 6= E ∈ Db(X , α) admits a Harder–Narasimhanfiltration given by a collection of distinguished triangles
Ei−1 → Ei → Ai
with E0 = 0 and En = E such that Ai ∈ P(φi) withφ1 > . . . > φn.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 148: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/148.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
(a) If 0 6= E ∈ P(φ), then Z (E) = m(E) exp(iπφ) for somem(E) ∈ R>0.
(b) P(φ + 1) = P(φ)[1] for all φ.
(c) Hom (E1, E2) = 0 for all Ei ∈ P(φi) with φ1 > φ2.
(d) Any 0 6= E ∈ Db(X , α) admits a Harder–Narasimhanfiltration given by a collection of distinguished triangles
Ei−1 → Ei → Ai
with E0 = 0 and En = E such that Ai ∈ P(φi) withφ1 > . . . > φn.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 149: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/149.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
(a) If 0 6= E ∈ P(φ), then Z (E) = m(E) exp(iπφ) for somem(E) ∈ R>0.
(b) P(φ + 1) = P(φ)[1] for all φ.
(c) Hom (E1, E2) = 0 for all Ei ∈ P(φi) with φ1 > φ2.
(d) Any 0 6= E ∈ Db(X , α) admits a Harder–Narasimhanfiltration given by a collection of distinguished triangles
Ei−1 → Ei → Ai
with E0 = 0 and En = E such that Ai ∈ P(φi) withφ1 > . . . > φn.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 150: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/150.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
(a) If 0 6= E ∈ P(φ), then Z (E) = m(E) exp(iπφ) for somem(E) ∈ R>0.
(b) P(φ + 1) = P(φ)[1] for all φ.
(c) Hom (E1, E2) = 0 for all Ei ∈ P(φi) with φ1 > φ2.
(d) Any 0 6= E ∈ Db(X , α) admits a Harder–Narasimhanfiltration given by a collection of distinguished triangles
Ei−1 → Ei → Ai
with E0 = 0 and En = E such that Ai ∈ P(φi) withφ1 > . . . > φn.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 151: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/151.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
(a) If 0 6= E ∈ P(φ), then Z (E) = m(E) exp(iπφ) for somem(E) ∈ R>0.
(b) P(φ + 1) = P(φ)[1] for all φ.
(c) Hom (E1, E2) = 0 for all Ei ∈ P(φi) with φ1 > φ2.
(d) Any 0 6= E ∈ Db(X , α) admits a Harder–Narasimhanfiltration given by a collection of distinguished triangles
Ei−1 → Ei → Ai
with E0 = 0 and En = E such that Ai ∈ P(φi) withφ1 > . . . > φn.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 152: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/152.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
(a) If 0 6= E ∈ P(φ), then Z (E) = m(E) exp(iπφ) for somem(E) ∈ R>0.
(b) P(φ + 1) = P(φ)[1] for all φ.
(c) Hom (E1, E2) = 0 for all Ei ∈ P(φi) with φ1 > φ2.
(d) Any 0 6= E ∈ Db(X , α) admits a Harder–Narasimhanfiltration given by a collection of distinguished triangles
Ei−1 → Ei → Ai
with E0 = 0 and En = E such that Ai ∈ P(φi) withφ1 > . . . > φn.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 153: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/153.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
(a) If 0 6= E ∈ P(φ), then Z (E) = m(E) exp(iπφ) for somem(E) ∈ R>0.
(b) P(φ + 1) = P(φ)[1] for all φ.
(c) Hom (E1, E2) = 0 for all Ei ∈ P(φi) with φ1 > φ2.
(d) Any 0 6= E ∈ Db(X , α) admits a Harder–Narasimhanfiltration given by a collection of distinguished triangles
Ei−1 → Ei → Ai
with E0 = 0 and En = E such that Ai ∈ P(φi) withφ1 > . . . > φn.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 154: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/154.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
To exhibit a stability condition on Db(X , α), it is enough to givea bounded t-structure on Db(X , α) with heart A;a group homomorphism Z : K (A) → C such that Z (E) ∈ H,for all 0 6= E ∈ A, and with the Harder–Narasimhanproperty (H := {z ∈ C : z = |z|exp(iπφ), 0 < φ ≤ 1}).
All stability conditions are assumed to be locally-finite. Henceevery object in P(φ) has a finite Jordan–Holder filtration.Stab (Db(X , α)) is the set of locally finite stability conditions.
The group Aut (Db(X , α)) acts on Stab (Db(X , α)).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 155: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/155.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
To exhibit a stability condition on Db(X , α), it is enough to give
a bounded t-structure on Db(X , α) with heart A;a group homomorphism Z : K (A) → C such that Z (E) ∈ H,for all 0 6= E ∈ A, and with the Harder–Narasimhanproperty (H := {z ∈ C : z = |z|exp(iπφ), 0 < φ ≤ 1}).
All stability conditions are assumed to be locally-finite. Henceevery object in P(φ) has a finite Jordan–Holder filtration.Stab (Db(X , α)) is the set of locally finite stability conditions.
The group Aut (Db(X , α)) acts on Stab (Db(X , α)).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 156: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/156.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
To exhibit a stability condition on Db(X , α), it is enough to givea bounded t-structure on Db(X , α) with heart A;
a group homomorphism Z : K (A) → C such that Z (E) ∈ H,for all 0 6= E ∈ A, and with the Harder–Narasimhanproperty (H := {z ∈ C : z = |z|exp(iπφ), 0 < φ ≤ 1}).
All stability conditions are assumed to be locally-finite. Henceevery object in P(φ) has a finite Jordan–Holder filtration.Stab (Db(X , α)) is the set of locally finite stability conditions.
The group Aut (Db(X , α)) acts on Stab (Db(X , α)).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 157: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/157.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
To exhibit a stability condition on Db(X , α), it is enough to givea bounded t-structure on Db(X , α) with heart A;a group homomorphism Z : K (A) → C such that Z (E) ∈ H,for all 0 6= E ∈ A, and with the Harder–Narasimhanproperty (H := {z ∈ C : z = |z|exp(iπφ), 0 < φ ≤ 1}).
All stability conditions are assumed to be locally-finite. Henceevery object in P(φ) has a finite Jordan–Holder filtration.Stab (Db(X , α)) is the set of locally finite stability conditions.
The group Aut (Db(X , α)) acts on Stab (Db(X , α)).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 158: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/158.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
To exhibit a stability condition on Db(X , α), it is enough to givea bounded t-structure on Db(X , α) with heart A;a group homomorphism Z : K (A) → C such that Z (E) ∈ H,for all 0 6= E ∈ A, and with the Harder–Narasimhanproperty (H := {z ∈ C : z = |z|exp(iπφ), 0 < φ ≤ 1}).
All stability conditions are assumed to be locally-finite. Henceevery object in P(φ) has a finite Jordan–Holder filtration.Stab (Db(X , α)) is the set of locally finite stability conditions.
The group Aut (Db(X , α)) acts on Stab (Db(X , α)).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 159: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/159.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
To exhibit a stability condition on Db(X , α), it is enough to givea bounded t-structure on Db(X , α) with heart A;a group homomorphism Z : K (A) → C such that Z (E) ∈ H,for all 0 6= E ∈ A, and with the Harder–Narasimhanproperty (H := {z ∈ C : z = |z|exp(iπφ), 0 < φ ≤ 1}).
All stability conditions are assumed to be locally-finite. Henceevery object in P(φ) has a finite Jordan–Holder filtration.Stab (Db(X , α)) is the set of locally finite stability conditions.
The group Aut (Db(X , α)) acts on Stab (Db(X , α)).
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 160: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/160.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
Bridgeland’s main results are the following:
1 For each connected component Σ ⊆ Stab (Db(X , α)) thereis a linear subspace V (Σ) ⊆ (N (X , α)⊗ C)∨ with awell-defined linear topology such that the natural map
Z : Σ −→ V (Σ), (Z ,P) 7−→ Z
is a local homeomorphism.
2 The manifold Stab (Db(X , α)) is finite dimensional.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 161: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/161.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
Bridgeland’s main results are the following:
1 For each connected component Σ ⊆ Stab (Db(X , α)) thereis a linear subspace V (Σ) ⊆ (N (X , α)⊗ C)∨ with awell-defined linear topology such that the natural map
Z : Σ −→ V (Σ), (Z ,P) 7−→ Z
is a local homeomorphism.
2 The manifold Stab (Db(X , α)) is finite dimensional.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 162: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/162.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
Bridgeland’s main results are the following:
1 For each connected component Σ ⊆ Stab (Db(X , α)) thereis a linear subspace V (Σ) ⊆ (N (X , α)⊗ C)∨ with awell-defined linear topology such that the natural map
Z : Σ −→ V (Σ), (Z ,P) 7−→ Z
is a local homeomorphism.
2 The manifold Stab (Db(X , α)) is finite dimensional.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 163: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/163.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: Bridgeland
Bridgeland’s main results are the following:
1 For each connected component Σ ⊆ Stab (Db(X , α)) thereis a linear subspace V (Σ) ⊆ (N (X , α)⊗ C)∨ with awell-defined linear topology such that the natural map
Z : Σ −→ V (Σ), (Z ,P) 7−→ Z
is a local homeomorphism.
2 The manifold Stab (Db(X , α)) is finite dimensional.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 164: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/164.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Denote byP(X , α) ⊂ N (X , α)⊗ C
the open subset of vectors ϕ such that real part and imaginarypart of ϕ generate a positive plane in N (X , α)⊗ R.
As in the untwisted case, P(X , α) has two connectedcomponents.
We shall denote by P+(X , α) the one that contains
ϕ = exp(B + iω)
with B rational B-field and ω rational Kahler class.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 165: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/165.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Denote byP(X , α) ⊂ N (X , α)⊗ C
the open subset of vectors ϕ such that real part and imaginarypart of ϕ generate a positive plane in N (X , α)⊗ R.
As in the untwisted case, P(X , α) has two connectedcomponents.
We shall denote by P+(X , α) the one that contains
ϕ = exp(B + iω)
with B rational B-field and ω rational Kahler class.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 166: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/166.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Denote byP(X , α) ⊂ N (X , α)⊗ C
the open subset of vectors ϕ such that real part and imaginarypart of ϕ generate a positive plane in N (X , α)⊗ R.
As in the untwisted case, P(X , α) has two connectedcomponents.
We shall denote by P+(X , α) the one that contains
ϕ = exp(B + iω)
with B rational B-field and ω rational Kahler class.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 167: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/167.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Denote byP(X , α) ⊂ N (X , α)⊗ C
the open subset of vectors ϕ such that real part and imaginarypart of ϕ generate a positive plane in N (X , α)⊗ R.
As in the untwisted case, P(X , α) has two connectedcomponents.
We shall denote by P+(X , α) the one that contains
ϕ = exp(B + iω)
with B rational B-field and ω rational Kahler class.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 168: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/168.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Fix ϕ ∈ P+(X , α). We define:
1 The function Zϕ(E) := 〈vα(E), ϕ〉.
2 The full additive subcategory T (ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) such that every non-trivialtorsion free quotient E � E ′ satisfies im(Zϕ(E ′)) > 0.
3 The full additive subcategory F(ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) which are torsion free andare such that every non-zero subsheaf E ′ satisfiesim(Zϕ(E ′)) ≤ 0.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 169: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/169.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Fix ϕ ∈ P+(X , α). We define:
1 The function Zϕ(E) := 〈vα(E), ϕ〉.
2 The full additive subcategory T (ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) such that every non-trivialtorsion free quotient E � E ′ satisfies im(Zϕ(E ′)) > 0.
3 The full additive subcategory F(ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) which are torsion free andare such that every non-zero subsheaf E ′ satisfiesim(Zϕ(E ′)) ≤ 0.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 170: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/170.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Fix ϕ ∈ P+(X , α). We define:
1 The function Zϕ(E) := 〈vα(E), ϕ〉.
2 The full additive subcategory T (ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) such that every non-trivialtorsion free quotient E � E ′ satisfies im(Zϕ(E ′)) > 0.
3 The full additive subcategory F(ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) which are torsion free andare such that every non-zero subsheaf E ′ satisfiesim(Zϕ(E ′)) ≤ 0.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 171: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/171.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Fix ϕ ∈ P+(X , α). We define:
1 The function Zϕ(E) := 〈vα(E), ϕ〉.
2 The full additive subcategory T (ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) such that every non-trivialtorsion free quotient E � E ′ satisfies im(Zϕ(E ′)) > 0.
3 The full additive subcategory F(ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) which are torsion free andare such that every non-zero subsheaf E ′ satisfiesim(Zϕ(E ′)) ≤ 0.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 172: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/172.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Fix ϕ ∈ P+(X , α). We define:
1 The function Zϕ(E) := 〈vα(E), ϕ〉.
2 The full additive subcategory T (ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) such that every non-trivialtorsion free quotient E � E ′ satisfies im(Zϕ(E ′)) > 0.
3 The full additive subcategory F(ϕ) ⊂ Coh(X , α) of thenon-trivial objects E ∈ Db(X , α) which are torsion free andare such that every non-zero subsheaf E ′ satisfiesim(Zϕ(E ′)) ≤ 0.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 173: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/173.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Define the abelian category
A(ϕ) :=
E ∈ Db(X , α) :• Hi(E) = 0 for i 6∈ {−1, 0},• H−1(E) ∈ F(ϕ),• H0(E) ∈ T (ϕ)
.
For good choices of B and ω, the pair (Zϕ,A(ϕ)) defines astability condition.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 174: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/174.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Define the abelian category
A(ϕ) :=
E ∈ Db(X , α) :• Hi(E) = 0 for i 6∈ {−1, 0},• H−1(E) ∈ F(ϕ),• H0(E) ∈ T (ϕ)
.
For good choices of B and ω, the pair (Zϕ,A(ϕ)) defines astability condition.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 175: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/175.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: examples
Define the abelian category
A(ϕ) :=
E ∈ Db(X , α) :• Hi(E) = 0 for i 6∈ {−1, 0},• H−1(E) ∈ F(ϕ),• H0(E) ∈ T (ϕ)
.
For good choices of B and ω, the pair (Zϕ,A(ϕ)) defines astability condition.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 176: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/176.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: the connected component
Following Bridgeland one defines a connected component
Stab †(Db(X , α)) ⊆ Stab (Db(X , α))
containing the stability conditions previously described.
Theorem (Huybrechts-Macrı-S.)
If (X , α) is a generic twisted K3 surface, then Stab †(Db(X , α)) isthe unique connected component of maximal dimension inStab (Db(X , α)). Such a component is also simply-connected.
Paolo Stellari Derived Torelli Theorem and Orientation
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MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: the connected component
Following Bridgeland one defines a connected component
Stab †(Db(X , α)) ⊆ Stab (Db(X , α))
containing the stability conditions previously described.
Theorem (Huybrechts-Macrı-S.)
If (X , α) is a generic twisted K3 surface, then Stab †(Db(X , α)) isthe unique connected component of maximal dimension inStab (Db(X , α)). Such a component is also simply-connected.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 178: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/178.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Stability conditions: the connected component
Following Bridgeland one defines a connected component
Stab †(Db(X , α)) ⊆ Stab (Db(X , α))
containing the stability conditions previously described.
Theorem (Huybrechts-Macrı-S.)
If (X , α) is a generic twisted K3 surface, then Stab †(Db(X , α)) isthe unique connected component of maximal dimension inStab (Db(X , α)). Such a component is also simply-connected.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 179: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/179.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Concluding the proof
1 There exists a continuous map
π : Stab †(Db(X , α)) → P+(X , α)
which is a covering map.
2 Since Stab †(Db(X , α)) is the unique connected componentof maximal dimension, the action of Aut (Db(X , α)) onStab (Db(X , α)) preserves it.
3 This action corresponds to the action of decktransformations, and P+(X , α) corresponds to the positivedirections. One gets that any Φ ∈ Aut (Db(X , α)) isorientation preserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 180: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/180.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Concluding the proof
1 There exists a continuous map
π : Stab †(Db(X , α)) → P+(X , α)
which is a covering map.
2 Since Stab †(Db(X , α)) is the unique connected componentof maximal dimension, the action of Aut (Db(X , α)) onStab (Db(X , α)) preserves it.
3 This action corresponds to the action of decktransformations, and P+(X , α) corresponds to the positivedirections. One gets that any Φ ∈ Aut (Db(X , α)) isorientation preserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 181: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/181.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Concluding the proof
1 There exists a continuous map
π : Stab †(Db(X , α)) → P+(X , α)
which is a covering map.
2 Since Stab †(Db(X , α)) is the unique connected componentof maximal dimension, the action of Aut (Db(X , α)) onStab (Db(X , α)) preserves it.
3 This action corresponds to the action of decktransformations, and P+(X , α) corresponds to the positivedirections. One gets that any Φ ∈ Aut (Db(X , α)) isorientation preserving.
Paolo Stellari Derived Torelli Theorem and Orientation
![Page 182: Derived Torelli Theorem and Orientation · Derived Torelli Theorem and Orientation Paolo Stellari Dipartimento di Matematica “F. Enriques” Universita degli Studi di Milano` Joint](https://reader030.vdocuments.mx/reader030/viewer/2022041021/5ed03cb17d4cb6261160d12f/html5/thumbnails/182.jpg)
MotivationsTwisted Derived Torelli Theorem
The conjectureThe twisted generic case
The statementIdeas from the proof
Concluding the proof
1 There exists a continuous map
π : Stab †(Db(X , α)) → P+(X , α)
which is a covering map.
2 Since Stab †(Db(X , α)) is the unique connected componentof maximal dimension, the action of Aut (Db(X , α)) onStab (Db(X , α)) preserves it.
3 This action corresponds to the action of decktransformations, and P+(X , α) corresponds to the positivedirections. One gets that any Φ ∈ Aut (Db(X , α)) isorientation preserving.
Paolo Stellari Derived Torelli Theorem and Orientation