Weak positivity The multiplication map Moduli and compactification Moduli and Positivity of Push-Forward Sheaves Eckart Viehweg http://www.uni-due.de/˜mat903/koeln06.pdf 1 / 32 Weak positivity The multiplication map Moduli and compactification 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. 2 / 32 Weak positivity The multiplication map Moduli and compactification 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. 3 / 32 Weak positivity The multiplication map Moduli and compactification 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). 4 / 32 Weak positivity The multiplication map Moduli and compactification 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. 5 / 32 Weak positivity The multiplication map Moduli and compactification 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. 7 / 32 Weak positivity The multiplication map Moduli and compactification 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). 8 / 32 Weak positivity The multiplication map Moduli and compactification 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. 9 / 32 Weak positivity The multiplication map Moduli and compactification 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. 10 / 32 Weak positivity The multiplication map Moduli and compactification 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. 11 / 32 Weak positivity The multiplication map Moduli and compactification 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. 12 / 32 Weak positivity The multiplication map Moduli and compactification 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. 13 / 32 Weak positivity The multiplication map Moduli and compactification 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.