birational geometry and moduli spaces of varieties of

Birational geometry and moduli

spaces of varieties of general type

James McKernan

UCSD

Birational geometry and moduli spaces of varieties of general type – p. 1

Review

We would like to construct the moduli space of

varieties of general type— by analogy with Mg.


Review



Even if we are interested in irreducible normalvarieties with no boundary we are obliged toconsider non-normal, reducible varieties.


Review




We have to consider semi log canonical pairs whichmeans we have to deal with log canonical pairs. Thedouble locus appears with coefficient one.


Review





We will see that the precise definition of the modulifunctor is unclear for pairs.


Review





We will see that the precise definition of the modulifunctor is unclear for pairs.

We only know how to construct the moduli space ofsemi log canonical models and not slc pairs but wewill still prove result for pairs.


Log canonical pairs (X,∆)

A log resolution π : Y −→ X is a projective

birational map s.t. (Y,Γ = ∆ + E) is log smooth

and E =∑

Ei supports an ample divisor over X .





and E =∑


We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.





and E =∑


We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.

ai = a(Ei, X,∆) is the log discrepancy of Ei.





and E =∑


We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.


a = infi,Y ai is the log discrepancy of (X,∆).(X,∆) is log canonical if a ≥ 0.





and E =∑


We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.


a = infi,Y ai is the log discrepancy of (X,∆).(X,∆) is log canonical if a ≥ 0.

(X,∆) is klt if ⌊∆⌋ = 0 and a > 0.


Klt vs lc

Varieties come in three types: KX is -ve, 0, +ve.


Klt vs lc


Singularities come in three types: klt, lc, not lc.


Klt vs lc



Let S be the cone over a curve C, genus g, degree d.


Klt vs lc




Minimal resolution π : T −→ S exceptional divisor

E ≃ C, E2 = −d. Write KT + E = π∗KS + aE.


Klt vs lc






2g−2 = degKE = (KT +E) ·E = π∗KS ·E+aE2.


Klt vs lc







2g − 2 = −ad =⇒ a =2− 2g

d.


Klt vs lc







2g − 2 = −ad =⇒ a =2− 2g

d.

S is klt, lc, not lc if and only if KC is -ve, 0, +ve.


Klt vs lc







2g − 2 = −ad =⇒ a =2− 2g

d.

S is klt, lc, not lc if and only if KC is -ve, 0, +ve.

KS is -ve, 0, +ve if and only if a < 2, a = 2, a > 2.


Deformation woes

Two families of surfaces in P5, Veronese P2 ⊂ P5

and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.


Deformation woes


and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.

Both surfaces degenerate to the cone S0 over a

rational normal curve of degree four, K2S0

= 9.


Deformation woes


and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.



= 9.

Both families are flat and the central fibre iskawamata log terminal so semi log canonical.


Deformation woes


and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.



= 9.


Take ramified covers to get families of general type.


Deformation woes


and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.



= 9.



To get around this we have to fix the degree in themoduli functor.


Deformation woes


and P1 × P1, type (2, 1): K2S1

= 9 and K2S2

= 8.



= 9.



To get around this we have to fix the degree in themoduli functor.

(Hacking, Abramovich-Hassett) We also requirereflexive powers of ωX to commute with basechange (over a non-reduced base).


Moduli functor

Fix the degree d. The moduli functor of slc models

assigns to every scheme S the set Mslcd (S):


Moduli functor



flat projective morphisms X −→ S; the fibres are slc, with

ample canonical class of degree d; ωX is flat over S and

all reflexive powers of ωX commute with base change.


Moduli functor






Viehweg: only one power commutes with base change.


Moduli functor







If we allow pairs, for example in the case of the cone overa rational normal curve, we even get embedded points.


Moduli functor







If we allow pairs, for example in the case of the cone overa rational normal curve, we even get embedded points.

There are many ways around this; does not occur if

coefficients > 1/2 (Kollár); float the coefficients(Hacking); branch varieties (Alexeev-Knutson).


Adjunction and the different

If X is smooth and S is a prime divisor then by

adjunction (KX + S)|S = KS.





If X is singular then (KX +∆)|S = KS +Θ, wheremultS ∆ = 1 and Θ ≥ 0 is called the different.






(X,∆) is divisorially log terminal if there is a logresolution such that ai > 0 all i. plt if x∆y = S.







(X,∆) (lc dlt plt) implies (S,Θ) (slc dlt klt).








If ∆ = S then Θ has coefficients (r − 1)/r.









If coefficients of ∆ belong to I then coefficients of

Θ belong to D(I) = { (r − 1 + s)/r | s ∈ I+ }.









If coefficients of ∆ belong to I then coefficients of

Θ belong to D(I) = { (r − 1 + s)/r | s ∈ I+ }.

Hence if I satisfies DCC then D(I) satisfies DCC.


From slc to lc

One can view a stable curve as a set of smoothcurves together with a set of pairs of points.


From slc to lc


Given a semi log canonical pair (X,∆) by defn one

can associate a log canonical pair (Xν, D +∆ν).Associate to D an involution τ : D −→ D, whichrepresents the gluing data. Fixes different on D.


From slc to lc




Theorem: (Kollár) Natural bijection between sets:


From slc to lc





(i) slc pairs (X,∆) such that KX +∆ is ample.


From slc to lc






(ii) lc pairs (Y,Γ +G) plus an involution τ : Gν −→ Gν

fixing the different such that KY + Γ +G is ample.


From slc to lc






(ii) lc pairs (Y,Γ +G) plus an involution τ : Gν −→ Gν

fixing the different such that KY + Γ +G is ample.

Correspondence no longer holds if we drop theampleness condition.


Moduli space of pairs

Focus on one aspect of the construction of themoduli space of semi log canonical varieties.




Most steps go through as in the construction of thesmooth components.





New issue is boundedness of the moduli functor.






For example, if you fix the degree d how to boundthe number of components of an slc pair?







In the smooth case, simply take the closure in theChow variety.








A stable curve of genus g has at most 2g − 2components. Each component contributes ≥ 1.








A stable curve of genus g has at most 2g − 2components. Each component contributes ≥ 1.

Canonical models of varieties are singular, KnX ∈ Q.


DCC for the volume

Theorem: (Hacon,-,Xu) Fix n, I ⊂ [0, 1] satisfyingthe DCC. Let D be the set of log smooth pairs

(X,∆), where X is a projective variety ofdimension n and the coefficients of ∆ belong to I .Then the set

{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.


DCC for the volume



{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.

Alexeev n = 2.


DCC for the volume



{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.

Alexeev n = 2.

If (X,∆) is semi log canonical then let (Xi,∆i) bethe components of the normalisation.


DCC for the volume



{ vol(X,KX +∆) | (X,∆) ∈ D }

satisfies the DCC.

Alexeev n = 2.

If (X,∆) is semi log canonical then let (Xi,∆i) bethe components of the normalisation.

d = (KX +∆)n =∑

(KXi+∆i)

n =∑

di.


Log canonical woes

Why is the log canonical case harder than thekawamata log terminal case?


Log canonical woes


Run into a brick wall called abundance:


Log canonical woes



Conjecture: (Abundance) If (X,∆) is kawamata logterminal and KX +∆ is nef then KX +∆ issemiample (some multiple is base point free).


Log canonical woes




Key case: X not uniruled implies κ(X,KX) 6= −∞.


Log canonical woes





Let Y be the cone over X . Finite generation of the

canonical ring R(Y,KY ) implies κ(X,KX) 6= −∞.


Log canonical woes





Let Y be the cone over X . Finite generation of the

canonical ring R(Y,KY ) implies κ(X,KX) 6= −∞.

Cascini-Corti-Lazic: New approach to Mori theory.First prove finite generation and use this to derivestandard results of the MMP.


Two key results

If π : X −→ U is a morphism then (X,∆) is log smooth

over U if (X,∆) is log smooth and all strata dominate U .


Two key results



Theorem: (Berndtsson-Paun) If π : X −→ D is amorphism, (X,∆) is log smooth over D and x∆y = 0then h0(Xt,m(KXt

+∆t)) is independent of t for allm ≥ 0.


Two key results





A good minimal model of a log canonical pair (X,∆) isa minimal model f : X 99K Y such that KY + Γ issemiample, where Γ = f∗∆.


Two key results





A good minimal model of a log canonical pair (X,∆) isa minimal model f : X 99K Y such that KY + Γ issemiample, where Γ = f∗∆.

Theorem: (Birkar; Hacon-Xu) If π : X −→ U is amorphism, (X,∆) is dlt, every non klt centre dominates

U and (X,∆) has a good model over an open subset of

U then (X,∆) has a good model over U .Birational geometry and moduli spaces of varieties of general type – p. 12

Boundedness of moduli functor

Theorem: Fix an integer n, a positive rational

number d and a set I ⊂ [0, 1] which satisfies DCC.

Then the set Fslc(n, d, I) of all log pairs (X,∆) s.t.






X is projective of dimension n,







(X,∆) is semi log canonical,








the coefficients of ∆ belong to I ,









KX +∆ is an ample Q-divisor, and










(KX +∆)n = d,is bounded. In particular there is a finite set I0 such

that Fslc(n, d, I) = Fslc(n, d, I0).










(KX +∆)n = d,is bounded. In particular there is a finite set I0 such

that Fslc(n, d, I) = Fslc(n, d, I0).

Alexeev: n = 2.


Generic abundance

A key result to prove boundedness is:


Generic abundance


Theorem: (Hacon,-,Xu) If π : X −→ U is morphism

and (X,∆) is log smooth over U then (X,∆) has agood minimal model over U if and only if one fibrehas a good minimal model over U .


Generic abundance




In particular to prove abundance it suffices to prove

abundance for some deformation of (X,∆).


Generic abundance




In particular to prove abundance it suffices to prove

abundance for some deformation of (X,∆).

We now give some of the ideas behind some resultswhich are a warm up to a proof of boundedness.


IOU argument

Theorem: (HM, T, T) Fix n. There is a positiveinteger m0 such that φmKX

is birational for allm ≥ m0, where X is of general type, dim. n.


IOU argument



Tsuji’s key observation. It is enough to prove:


IOU argument




Theorem: If vol(X, rKX) > 1 then φmrKYis

birational for all m ≥ m0.


IOU argument






Indeed, if vol(X,KX) > 1 there is nothing to prove.


IOU argument







Otherwise pick r such that 1 < vol(X, rKX) ≤ 2n.


IOU argument








vol(X,mrKX) < (2m)n and so X belongs to abirationally bounded family, using the Chow variety.


IOU argument








vol(X,mrKX) < (2m)n and so X belongs to abirationally bounded family, using the Chow variety.

We may find ǫ > 0 such that vol(X,KX) > ǫ and

vol(X, rKX) > 1, r = ⌈1/ǫ⌉. m0r works.Birational geometry and moduli spaces of varieties of general type – p. 15

Isolated non klt centres







A non klt centre of the pair (X,∆) is the image of adivisor of log discrepancy at most zero.






Lemma 1: Given x ∈ X general, find D ∈ |krKX |such that x is an isolated non kawamata log terminal

centre of (X,B = D/kr).








Lemma 2: Given x, y ∈ X general, find B ∼Q rKX

such that x is an isolated lc centre of (X,B) and ybelongs to a non klt centre V .








Lemma 2: Given x, y ∈ X general, find B ∼Q rKX

such that x is an isolated lc centre of (X,B) and ybelongs to a non klt centre V .

It is enough to prove Lemma 2; lift section (1, 0)∈ Γ(X,Ox ⊕OV ), using the theory of multiplierideal sheaves. Birational geometry and moduli spaces of varieties of general type – p. 16

Lifting sections

Lemma 1: Given x ∈ X general, find B ∼Q rKX

such such that x is an isolated non kawamata log

terminal centre of (X,D).


Lifting sections




By assumption vol(X,nrKX) > (rn)n.


Lifting sections





By asymptotic Riemann-Roch find D ∈ |krKX |such that multX D > kn.


Lifting sections






(X,B) is not kawamata log terminal at x; let x ∈ Vbe a non kawamata log terminal centre.


Lifting sections







As x ∈ V is general, V is of general type.


Lifting sections








Kawamata subadjunction, (KX +B)|V = KV +Θ.


Lifting sections








Kawamata subadjunction, (KX +B)|V = KV +Θ.

Lift sections by Serre vanishing and proceed byinduction on the dimension.


Birational automorphisms

Theorem: (Hacon,-,Xu) Fix n. There exists c = cnsuch that |Bir(X)| < c · vol(X,KX).




If X is the canonical model then Aut(X) = Bir(X).





Take a G = Aut(X)-equivariant resolution.






Let π : X −→ Y = X/G be the natural morphism.







KX = π∗(KY + Γ), Γ has coefficients (r − 1)/r.








Run IOU argument in orbifold case: volumebounded from below.









If vol(Y,KY + Γ) ≥ δ then c = 1

δ.










δ.

n = 1, (P1, 12p+ 2

3q + 6

7r), δ = 1/42, c = 42.










δ.

n = 1, (P1, 12p+ 2

3q + 6

7r), δ = 1/42, c = 42.

n = 2 Alexeev, Xiao.Birational geometry and moduli spaces of varieties of general type – p. 18

New directions

What general statements apply to all moduli spaces?


New directions


Let Msm be the open locus of normal varieties.


New directions



Theorem: Every subvariety V of Msm has loggeneral type.


New directions




Vast generalisation of Shafarevich’s conjecture that

every family of curves over P1, A1, C∗ and everyelliptic curve, is isotrivial.


New directions






Combined work, including Campana, Kebekus,Kovács, Paun, Popa, Schnell, Taji, Viehweg, Zuo.


New directions






Combined work, including Campana, Kebekus,Kovács, Paun, Popa, Schnell, Taji, Viehweg, Zuo.

Idle speculation: To construct the moduli space ofsemi log canonical pairs we will need anotherinsight, akin to the idea of dropping GIT.


Kähler geometry

Mori theory is only supposed to apply to thecategory of projective varieties.


Kähler geometry


Theorem: (Höring-Peternell) We can run theKM -MMP for terminal Kähler threefolds.


Kähler geometry



Theorem: (Campana-Höring-Peternell) Abundanceholds for terminal Kähler threefolds.


Kähler geometry



Theorem: (Campana-Höring-Peternell) Abundanceholds for terminal Kähler threefolds.

Therefore every smooth Kähler manifold has a goodminimal model and we get very strong classificationresults for Kähler manifolds.


Characteristic p

Missing Kodaira vanishing and we only haveresolution of singularities for 3-folds.


Characteristic p


Keel proved a strong base point free theorem, usingFrobenius instead of Kawamata-Viehweg vanishing(inspired by results of Smith, which in turn use thetheory of tight closure, Hochster-Huneke).


Characteristic p



Hacon and Xu proved existence of threefold flipsp > 5 , using Grothendieck duality and trace ofFrobenius (Schwede) and termination for KX-flips.


Characteristic p




Birkar established MMP for threefolds, p > 5.


Characteristic p




Birkar established MMP for threefolds, p > 5.

Patakfalvi proved a positivity result for thepushforward of the dualising sheaf.


Foliations

A foliation is a subsheaf of F ⊂ TX closed underLie bracket.


Foliations


Associate a canonical divisor KF = −c1(F).


Foliations



Bogomolov-McQuillan: Cone theorem for rank onefoliations.


Foliations




Brunella-McQuillan: Classification of rank onefoliations on surfaces. Abundance fails (for varietieswith universal cover D× D).


Foliations





Araujo-Druel: results for Fano foliations.


Foliations






Pereira-Touzet: results for K-trivial foliations.


Foliations






Pereira-Touzet: results for K-trivial foliations.

Spicer: partial results for a cone theorem for ranktwo foliations on threefolds. theorem


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