Moduli and Positivity of Push-Forward Sheaves

Weak positivity The multiplication map Moduli and compactiﬁcation

Eckart Viehweg

http://www.uni-due.de/˜mat903/koeln06.pdf

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f : X → Y a surjective morphism between complex projective −1 irreducible varieties. f0 : X0 = f (Y0) → Y0 smooth part. For simplicity: Assume that the general ﬁbre F is irreducible.

ν The general theme: f∗ωX /Y should be “(weakly) positive”.

Problems:

I The sheaf ωX /Y might not exist. ν I Even if it exist, f∗ωX /Y is not necessarily locally free.

I What is the right deﬁnition of “(weakly) positive”?

So imagine a world where all morphisms are ﬂat and all push forward sheaves locally free, i.e. assume: Y a non-singular curve.

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Deﬁnition. Y a non-singular curve, F a locally free sheaf.

A. F is nef ⇐⇒ O (1) is a nef line bundle on (F). df P(F) P ⇐⇒ ∀ morphisms τ : Y 0 → Y and ∀ invertible quotients N of τ ∗F one has deg(N ) ≥ 0.

B. F is ample (= positive) ⇐⇒ O (1) is an ample line df P(F) bundle on P(F). α ⇐⇒ For some α > 0 the sheaf S (F) ⊗ OY (−pt) is nef.

Theorem. (Fujita, 77) X , Y non-singular, dim(Y ) = 1 =⇒ f∗ωX /Y is nef.

Method: Curvature of Hodge bundles.

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Corollary. “The positivity package” (X , Y non-singular, dim(Y ) = 1) ν I. f∗ωX /Y is nef ∀ ν ≥ 1. −1 I May assume: f semistable, i.e. X non-singular and f (y) reduced NCD for all y ∈ Y . Assume f −1(y) NCD. Then

0 normalization 0 X −−−−−−−−→ X ×Y Y −−−−→ X    0   f y y yf Y 0 −−−−→= Y 0 −−−−→σ Y 0 ν ∗ ν and f∗ωX 0/Y 0 ⊂ σ f∗ωX /Y for ν ≥ 1. η II. (f semistable) If det(f∗ωX /Y ) is ample for some η ≥ 1 then ν + f∗ωX /Y is ample ∀ ν ≥ 2 (or zero).

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I Better: Replace “semistable” by “mild”, i.e.: ﬂat and total space normal with rational Gorenstein singularities, and the same holds for all pullbacks under dominant morphisms σ : Y 0 → Y with Y 0 normal with rational Gorenstein singularities. i. “Easy proof of I. for ν = 1”: 1 Koll´ar’s vanishing =⇒ H (Y , f∗ωX ⊗ A) = 0 for A invertible, ample.

=⇒ f∗ωX ⊗ OY (2 · pt) = f∗ωX /Y ⊗ ωY (2 · pt) generated by global sections. f mild =⇒ r-fold ﬁbre product f r : X r → Y is mild. r Nr I Base change =⇒ f∗ ωX r /Y = f∗ωX /Y . Nr =⇒ ωY (2 · pt) ⊗ f∗ωX /Y globally generated =⇒ I.

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ii. Multiplier ideals. D ≥ 0 divisor on X , and N ∈ N. D D (88-90) ωX {− N } = τ∗ωX 0 (−[ N ]) where τ : X 0 → X is a log-resolution. D D Today’s notation: J (− N ) ⊗ ωX = τ∗ωX 0 (−x N y).

I. for ν = 1 =⇒ N Corollary. L invertible on X , E ⊂ f∗(L ⊗ OX (−D)) with E nef and

∗ ∗ N N f E −−→ f f∗(L ⊗ OX (−D)) −−→L ⊗ OX (−D)

D surjective. Then f∗(ωX /Y ⊗ L ⊗ J (− N )) is nef. +

6 / 32 Weak positivity The multiplication map Moduli and compactiﬁcation iii. Proof of I for ν > 1. Choose ν ρ = Min{µ ≥ 0; f∗ωX /Y ⊗ OY ((µν − 1) · pt) nef }. ∗ ν−1 ν (ν−1)ν D such that f S (f∗ωX /Y ) → ωX /Y ⊗ OX (−D) ν−1 ∗ is surjective, N = ν and L = ωX /Y ⊗ f OY (ρ(ν − 1) · pt).

ν−1 ν ∗ E = S (f∗ωX /Y ⊗ OY (ρν · pt)) is ample, and f E generates ν (ν−1)ν ∗ L (−D) = ωX /Y ⊗ f OY (ρν(ν − 1) · pt) ⊗ OX (−D). So D ν f∗(ωX /Y ⊗ L ⊗ J (− N )) = f∗ωX /Y ⊗ OY (ρ(ν − 1) · pt) is nef. =⇒ ρ(ν − 1) > (ρ − 1)ν − 1 =⇒ ρ < ν + 1.

ν 2 Hence f∗ωX /Y ⊗ OY (ν · pt) nef and every invertible quotient ν 2 of f∗ωX /Y has degree ≥ −ν . ν Same on all coverings of Y (+ base change) =⇒ f∗ωX /Y nef.

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η ν iv. Proof of II. (det(f∗ωX /Y ) ample =⇒ f∗ωX /Y ample.) Def. (88-90) D ≥ 0 Cartier divisor, L invertible on X . D e(D) = Min{N ≥ 0; J (− N ) = OX } e(L) = Sup{e(D); D ≥ 0; OX (D) = L}

Observations: a. e(L) < ∞. −1 b. e(L|F ) is semi-continuous in families, F = f (y). D c. N ≥ e(L|F ) =⇒ J (− N ) = OX in a neighborhood of F . r O ∗ d. e( pri L) on X × · · · × X (r-times) is independent of r, hence = e(L).

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η Assume OY (α · pt) = det(f∗ωX /Y ). r r Consider f : X = X ×Y · · · ×Y X −−→ Y η (r = rk(f∗ωX /Y )-times). r η O η r η det(f∗ωX /Y ) −−→ f∗ωX /Y = f∗ ωX r /Y gives a divisor 0 η 0 r Γ + α · F ∼ ωX r /Y for some ﬁbre F of f . r β For all β f∗ ωX r /Y is nef, hence repeat the calculations of the ﬁrst part without twist. r ν D+Γ+α·F 0 =⇒ f∗ (ωX r /Y ⊗ J (− η+β )) nef, where D is a relative base locus. D+Γ+α·F 0 D β 1 =⇒ J (− η+β )|F = J (− η+β )|F . Replacing Y by a covering: α ≥ η + β =⇒ D+Γ+α·F 0 0 J (− η+β ) ⊂ OX (−F ). r ν r ν Hence f∗ ωX r /Y ⊗ OY (−pt) is nef =⇒ f∗ ωX r /Y ample.

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X , Y non-singular, dim(Y ) = 1, f semistable: “The positivity package” ν I. f∗ωX /Y is nef ∀ ν ≥ 1. η ν II. If det(f∗ωX /Y ) is ample for some η ≥ 1 then f∗ωX /Y is ample ∀ ν ≥ 2 (or zero).

η III. Conjecture. f not birationally isotrivial =⇒ det(f∗ωX /Y ) is ample for some η ≥ 1.

III. known for: F of general type (Koll´ar, V.) and if F has a minimal model (Kawamata). (F general ﬁbre)

From now on: Assume that ωF is semi-ample for all ﬁbres F of f0 : X0 → Y0.

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Problems for dim(Y ) > 1:

I No semi-stable reduction, not even a ﬂat CM-reduction. ν I Even if f is ﬂat and CM, f∗ωX /Y is not necessarily locally free.

I What is the right deﬁnition of “(weakly) positive”? We used several times, that F is positive if its pullback is positive.

Deﬁnition: Z a quasiprojective variety, U a dense open subscheme, H an ample invertible and F a locally free sheaf. a. F is nef ⇐⇒ ∀ curves τ : C → Z and for all invertible quotients L of τ ∗F one has deg(L) ≥ 0. b. F is ample with respect to U ⇐⇒ for some ν > 0 there is a M morphism H −−→ Sν(F) surjective over U. c. F is weakly positive over U ⇐⇒ ∀ β > 0 the sheaf Sβ(F) ⊗ H is ample w.r.t. U.

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Comments:

I nef only makes sense for Z proper.

I nef =⇒ weakly positive.

I Assume Z is proper. One wants: (∗) holds for F ⇐⇒ (∗) holds for τ ∗F.

Not true for: (∗) = “ample w.r.t. U”. Not true for: (∗) = “weakly positive over U”. Obviously true for: (∗) = “nef”, and τ proper, surjective. True for: (∗) = “nef and ample w.r.t. U”, and τ proper, surjective and ﬁnite over U.

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Return to f : X → Y , with smooth part f0 : X0 → Y0. (Y0 might be singular) Theorem. (1989) I . f ων is weakly positive over Y . 0 0∗ X0/Y0 0 η0 ν II0. det(f0∗ω ) is ample =⇒ f0∗ω is ample for ν ≥ 2. X0/Y0 X0/Y0

Proof is quite hard, in particular for Y0 singular. In between: More conceptional proof, and extension to compactiﬁcations.

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A replacement for semistable reduction: Assume: f0 : X0 → Y0 a smooth projective morphism of quasi-projective reduced schemes. ωF is semiample for all ﬁbres F of f0.

Theorem (2006). Given a finite set I of positive integers there exists a projective compactification Y of Y0, a finite (ν) covering φ : W → Y , and for ν ∈ I a locally free sheaf FW on W with: −1 i. For W0 = φ (Y0) and φ0 = φ|W0 one has φ∗f ων = F (ν)| . 0 0∗ X0/Y0 W W0 (ν) ii. FW is “natural” and compatible with products, i.e.

14 / 32 Weak positivity The multiplication map Moduli and compactiﬁcation ii. i.e.: Let ξ : Y 0 → W be a morphism from a non-singular 0 0 −1 0 variety Y with Y0 = ξ (W0) dense in Y . Assume either that Y 0 is a curve, or that Y 0 → W is dominant. For some r ≥ 1 let X (r) be a non-singular model of the r-fold product family

0 0 X0 = (X0 ×Y0 · · · ×Y0 X0) ×Y0 Y0

which allows a projective morphism f (r) : X (r) → Y 0. Then r (r) ν O ∗ (ν) f∗ ωX (r)/Y 0 = ξ FW .

Addendum. One may assume that OY0 → φ0∗OW0 splits. Harder to get: Similar results for multiplier ideals.

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The proof is based on:

I The Weak Semistable Reduction Theorem of Abramovich-Karu.

I The Extension Theorem of Gabber.

Corollary: The Positivity Package. (ν) I. The sheaf FW is nef. (η0) II. Assume that det(FW ) is ample with respect to W0, for some ∗ η0 η0 η0 ∈ I , and that the evaluation map f f0∗ω → ω is 0 X0/Y0 X0/Y0 surjective. (ν) Then FW is ample with respect to W0 (or zero), for ν ≥ 2 with ν ∈ I .

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Some details from the proof of the Theorem: Start with any compactiﬁcation f : X → Y of f0. By the Weak Semistable Reduction of Abramovich-Karu one ﬁnds:

ϕ0 X ←−−−− Z 0    0 f y g y ϕ Y ←−−−− Y 0 0 0 I ϕ, ϕ are alterations (= generically finite, proper), with Y smooth. 0 0 0 0 I g : Z → Y is mild (= flat, Gorenstein, and Z rational singularities + compatibility of those properties with certain pullbacks). 0 I But: For Z one might have blown up the general fibre.

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Complete the diagram

ϕ0 0 ρ X ←−−−− Z 0 ←−−−−δ Z −−−−→δ X 0 −−−−→ X alt       0 g 0  f y g ymild y f y yf ϕ ϕ Y ←−−−− Y 0 ←−−−−= Y 0 −−−−→= Y 0 −−−−→ Y , alt

0 0 0 I f : X → Y is a desingularization of the pullback family. 0 I δ and δ are modiﬁcations, with Z smooth.

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0 ν 0 ν I “Mild” =⇒ g∗ωZ 0/Y 0 = f∗ωX 0/Y 0 . 0 0 ν I Modify Y such that g∗ωZ 0/Y 0 is locally free. 0 0 I Show that for some open dense set Yg ⊂ Y and for all curves 0 0 τ : C → Y meeting Yg one has compatibility with base change. 0 0 I Repeat the construction over Y \ Yg and “glue”. (ν) I One gets a nice candidate for FY 0 over some large alteration Y 0 of Y , and this is the right one on all curves.

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Gabber’s Extension Theorem =⇒ (ν) The sheaf FY 0 is the pullback of a locally free sheaf on a compactiﬁcation of the Stein factorization.

For the “Positivity Package” one needs in addition some ﬂattening construction for multiplier ideals and one has to repeat the whole construction for certain cyclic coverings.

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(ν) III. missing: When is FW ample with respect to −1 W0 = φ (Y0)? (F) One needs: Fibres of f not all isomorphic along a positive dimensional subvariety of Y0. Keep assumption: Smooth ﬁbres F of f : X → Y are minimal models. Theorem. (V., Kawamata, Koll´ar) (ν) Under those assumptions det(FW ) is ample with respect to W0.

Methods. F curve: use existence of projective moduli.

I F general type + minimal model: use local Torelly for coverings.

I F minimal model: use local Torelly for coverings (Kawamata).

I F general type: Use the multiplication map (Koll´ar,V.).

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Here: F general type + minimal model. Use the multiplication map and the Hilbert-scheme. A variant of this proof shows the existence of quasi-projective moduli schemes.

µ ν νµ The multiplication map m : S (f0∗ω ) −−→ f0∗ω X0/Y0 X0/Y0 extends to µ (ν) (νµ) m : S (FW ) −−→FW Properties. For µ ν 1 one has: a. m is surjective over W0. b. For φ(w) = y ∈ Y0 the kernel of µ (ν) (νµ) −1 S (FW ) ⊗ C(w) −−→FW ⊗ C(w) determines F = f (y).

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(ν) a) + b) + FW nef + (F) (νµ) =⇒ det(FW ) ample with respect to W0.

(ν) (νµ) Main idea: Write F = FW , r = rk(F), Q = FW , ρ = rk(Q). ∨⊕r π Consider the projective bundle P = P(F ) −−→ W . ∗ ⊕r On P one has OP(−1) → π F or the universal bases ⊕r ∗ OP(−1) → π F. ∗ So on U = P \ Γ, for Γ = zero locus of OP(−r) → det(π F), ⊕r ∼ ∗ one has OU (−1) = π F|U . Claim 1. For some a, b ≥ 0 the sheaf det(F)a ⊗ det(Q)b is ample with respect to W0. ⇐= Claim 2. For some eﬀective divisor ∆ supported in P \ U ∗ a b the sheaf π (det(F) ⊗ det(Q) ) ⊗ OP(∆) is ample with −1 respect to U0 = U ∩ π (W0).

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µ ⊕r ∗ Consider m : S (OU (−1) ) −−→ π Q|U . Pl¨ucker ρ µ ⊕r This gives U0 −−→ Grass −−−−−→ P(∧ S (C )). ρ µ ⊕r (F) =⇒ U0 −−→ P(∧ S (C )) is quasi ﬁnite. 0 ρ µ ⊕r Blowing up, this extends to P → P(∧ S (C )) and the pullback of an ample invertible sheaf is of the form 0∗ −µρ r 0 π (det(F) ⊗ det(Q) ) ⊗ OP (E) 0 for some divisor E, supported in P \ U. ∗ r ∗ r−1 Claim 3. OP(r) ⊗ π det(F) = OP(Γ) ⊗ π det(F) is nef. Then replace E by E + γ · Γ. ∗ For Claim 3.: OP(Γ) = OP(r) ⊗ π det(F) and Lr r−1 ∗ Lr ∗ ∨ ∗ ∧ π F = π (F ⊗ det(F)) −−→OP(1) ⊗ π det(F).

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Comments. “Claim 2 =⇒ Claim 1” is a baby case of a stability criterion in GIT for the PGl bundle U → W .

Mh = moduli functor of canonically polarized manifolds with Hilbert polynomial h. H the corresponding Hilbert scheme of ν canonically embedded manifolds in Mh(C), with universal family N (g : X → H, X ⊂ P × H). ν ν Write λν = det(g∗ωX /H ) and recall that h(ν) = rk(g∗ωX /H ). Pl¨ucker coordinates for H give ample sheaf (ν, µ 0) −µ·h(ν·µ) h(ν·µ) Lν,µ = λν ⊗ λµν .

“Positivity Package I0” =⇒ λµν ample.

“Positivity Package II0” =⇒ λη ample for η ≥ 2 with h(η) 6= 0.

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Mumford’s GIT: G = PGl(n, C) acts on H. Consider a G-linearized ample invertible sheaf L on H. Assume for all x ∈ H the “Hilbert-Mumford Criterion holds”

=⇒ the quotient π : H → Mh = H/PGl(n, C) exists. Mh is a coarse moduli scheme for Mh. ∗ There exists an ample Q-divisor D on Mh with L = OH (π D). Problem: How to verify the Hilbert-Mumford Criterion? And for which invertible G-linearized sheaf?

Candidates: Lν,µ, the sheaf from the Pl¨ucker embedding.

Or: λν.

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For the sheaf: Lν,µ λν Ampleness: By deﬁnition Hard part, needs Pos. P. I0 &II0 H.-M. Criterion: Hard part Easy, using Pos. P. I0 Advantage: ”Better” ample sheaves

(ν) The proof of the ampleness of FW translates to a H.-M. Criterion for λν on H, using:

A. The ampleness of λν.

B. For one special family f0 : X0 → Y0 ∈ Mh(Y0), with H ⊂ Y0 open and dense, the sheaf F (ν) is weakly positive over Y . Y0 0

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Theorem (1990) There exists a coarse quasiprojective moduli scheme Mh for Mh. A similar argument shows: Theorem (1990) There exists a coarse quasiprojective moduli scheme Mh for the moduli functor of polarized minimal models with Hilbert polynomial h.

I Up to now we only used the corollaries I0 and II0. Using I. and II. one gets more: “nice” compactiﬁcations M¯ h for:

∗ The moduli scheme Mh of canonically polarized manifolds.

∗∗ The moduli scheme Mh of polarized manifolds F with ν ωF = OF . (In fact: only for the reduced moduli schemes.)

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Hence: For the sheaf: Lν,µ λν Advantage: ”Better” Some “nice” ample sheaves compactiﬁcation

One gets “nice” (?) compactiﬁcations M¯ h Recall: Given ν ≥ 2 in (∗) or (∗∗), there exists an ample invertible sheaf λ0 on Mh with: (>) Let Ψ : Y0 → Mh be a morphism induced by a family f : X → Y . Then Ψ∗λ = det(f ων ). 0 0 0 0 0∗ X0/Y0

Cheating again: Only λ0 ∈ Pic(Mh) ⊗ Q.

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Theorem (2006) Let Mh be the reduced moduli scheme (∗) of canonically polarized manifolds and ν > 1, or ν (∗∗) of polarized manifolds F with ωF = OF . Then there exists a compactiﬁcation M¯ h of Mh and an extension of λ0 to an invertible sheaf λ on M¯ h with:

λ is nef and ample with respect to Mh. ¯ −1 (>) If Y is a curve, Ψ : Y → Mh a morphism with Y0 = Ψ (Mh) dense, and if Y0 → Mh is induced by the smooth part of a ∗ ν semistable family f : X → Y , then Ψ λ = det(f∗ωX /Y ).

∗ “Corollary”: In (>) deg(Ψ λ) is a natural geometric height function.

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For a height function one would like to have upper bounds: Theorem (V.-Zuo, 2005): For a semistable family f : X → Y of n-folds over a curve Y , with discriminant S one has ν deg(f∗ωX /Y ) n · ν (0 ≤) ν ≤ · (2g(Y ) − 2 + #S). rank(f∗ωX /Y ) 2

Corollary (V.-Zuo, 2002/04): Given (Y , S) there are only ﬁnitely many families of canonically polarized manifolds of of polarized manifolds with given Hilbert polynomial, up to deformations (of the morphism Y0 = Y \ S to the moduli stack Mh). As indicated by the dates: First proof was with a weaker inequality, and without using the sheaf λ on M¯ h.

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Comments, Hopes and Questions: a. Existence of good extensions of direct image sheaves is the unpleasant part of the construction of Mh. b. A modiﬁed version for algebraic spaces should allow to use the Weil-Petersson metric to get ample sheaves on Mh. c. It can be used to understand the relation between diﬀerent stability concepts (K-, Hilbert-Mumford-, and Chow-stability) d. The existence of a height function has more implications:

I Uniform boundedness (For curves: Caporaso).

I Construction of moduli schemes (or stacks) of morphisms from curves to the moduli stacks Mh with (∗) or (∗∗) (For compact moduli problems: Abramovich-Vistoli). e. For Y a curve, what are the slopes of the Harder-Narasimhan ν ﬁltration for f∗ωX /Y ?.

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