Classical varishing theorems New framework of vanishing theorems Comments on the proof Vanishing theorems for complex projective varieties Osamu Fujino Osaka University July 26, 2016 Comments on the proof Classical vanishing theorems New framework of vanishing theorems Comments on the proof Classical vanishing theorems New framework of vanishi Kodaira The Kodaira vanishing theorem is one of the most famous and most important vanishing theorems for complex projective varieties. Kodaira The Kodaira vanishing theorem is one of the most famous and most important vanishing theorems for complex projective varieties. Theorem 1.1 (Kodaira) ● X: smooth projective variety ● D: ample Cartier divisor Then H 9 (X, O x (K x + D)) = 0 for every q > 0.
Osamu Fujino
Varishing theorems for complex projective varieties
Kodaira The Kodaira vanishing theorem is one of the most famous and most important vanishing theorems for complex projective varieties. Theorem 1.1 (Kodaira) ● X: smooth projective variety ● D: ample Cartier divisor Then H 9 (X, O x (K x + D)) = 0 for every q > 0. Note: ● Kχ: canonical divisor of X ●  ✓dim XΩ x = Ox(Kx)
Osamu Fujino
Vanishing theorems for complex projective varieties
Classical vanishing theorems Now framework of vanishing theorems Kawamata–Viehweg As is well-known, the Kawamata–Viehweg vanishing theorem is a powerful generalization of the Kodaira vanishing theorem.
As is well-known, the Kawamata–Viehweg vanishing theorem is a powerful generalization of the Kodaira vanishing theorem. Theorem 1.2 (Kawamata–Viehweg) ● X: smooth projective variety ● D : nef and big Q-divisor ● Supp{D}: SNC divisor Then H 9 (X, O x (K  x  + [D])) = 0 for every q > 0.
Osamu Fujino
Varishing theorems for complex projective varieties

As is well-known, the Kawamata–Viehweg vanishing theorem is a powerful generalization of the Kodaira vanishing theorem. Theorem 1.2 (Kawamata–Viehweg) ● X: smooth projective variety ● D : nef and big Q-divisor ● Supp{D}: SNC divisor Then H 9 (X, O x (K  x  + [D])) = 0 for every q > 0. Note: ● {D}: the fractional part of D, [D]: the round-up of D ●  D : nef and big  ⇐⇒  Ddim × > 0 and  D · C ≥  0 for any curve C
Osamu Fujino
Varishing theorems for complex projective varieties
Classical vanishing theorems New framework of vanishing theorems Comments on the proof Nadel The (algebraic version of) Nadel vanishing theorem is also a useful generalization of Kodaira. Nadel The (algebraic version of) Nadel vanishing theorem is also a useful generalization of Kodaira. Theorem 1.3 (Nadel) ● X: smooth projective variety ● L: Cartier divisor ● D : effective Q-divisor ● L – D: nef and big Then H 9 (X, Ox (Kx + L) & J(X, D)) = 0 for every q > 0, where J (X, D): multiplier ideal sheaf of (X, D).
Osamu Fujino
Varishing theorems for complex projective varieties
Classical vanishing theorems Comments on the proof Kollár Kollár’s results contain Kodaira as a special case. Kollár Kollár’s results contain Kodaira as a special case. Theorem 1.4 (Kollár) ● X: smooth projective variety ● Y: projective variety ● f : X → Y: surjective morphism
Varishing theorems for complex projective varieties
Kollár Kollár’s results contain Kodaira as a special case. Theorem 1.4 (Kollár) ● X: smooth projective variety ● Y: projective variety ● f : X → Y: surjective morphism Then we have: (i) Rq f*Ox(Kx): torsion-free for every q (ii)  HP(Y,R9 f*Ox(Kx) ® Oy(H)) = 0 for every p > 0 and q ≥ 0, where H : ample Cartier divisor on Y .
Osamu Fujino
Vanishing theorems for complex projective varieties
Classical vanishing theorems New framework of vanish Comments on the proof We need more general vanishing theorems for the minimal model program (MMP, for short) for higher-dimensional algebraic varieties.
● M: smooth variety /C ● X: SNC divisor on M ● B: R-divisor on M such that Supp B: SNC divisor ● B and X have no common components, Supp(B + X): SNC divisor ● D = B|x
Osamu Fujino

● M: smooth variety /C ● X: SNC divisor on M ● B: R-divisor on M such that Supp B: SNC divisor ● B and X have no common components, Supp(B + X): SNC divisor o  D = B|x Definition 2.1 (GESNC pair) (X, D ): globally embedded simple normal crossing (GESNC) pair
Osamu Fujino
Varishing theorems for complex projective varieties

● M: smooth variety /C ● X: SNC divisor on M ● B: R-divisor on M such that Supp B: SNC divisor ● B and X have no common components, Supp(B + X): SNC divisor o D = B|x Definition 2.1 (GESNC pair) (X, D ): globally embedded simple normal crossing (GESNC) pair Definition 2.2 (SNC pair) (Y, Δ): simple normal crossing (SNC) pair (Y, Δ): Zariski locally isomorphic to a GESNC pair
Osamu Fujino
Vanishing theorems for complex projective varieties
Classical vanishing theorems Comments on the proof Stratum of SNC pair ● (X, D): SNC pair ● D ∈ [0, 1] ● ν : X ′ → X: normalization ● K v + O = v* (K + D)
● (X, D): SNC pair ● D ∈ [0, 1] ● ν : X ′ → X: normalization ● K x + O = v*(K + D) Definition 2.3 (Stratum) ● W: closed subvariety of X W: stratum of (X, D) def ←→  W = v(C), where C is a log canonical center of (X ′′,Θ), or W is an irreducible component of X
Osamu Fujino
Varishing theorems for complex projective varieties

Theorem 2.4 (Relative Hodge theoretic injectivity theorem) ● (X, Δ): SNC pair, Δ ∈ [0, 1], π : X → S : proper o  L: Cartier divisor on X ● D: effective Weil divisor on X ● Supp D ⊂ Supp Δ ●  L ~R,π Kχ + Δ
Varishing theorems for complex projective varieties

Theorem 2.4 (Relative Hodge theoretic injectivity theorem) ● (X, Δ): SNC pair, Δ ∈ [0, 1], π : X → S : proper o  L: Cartier divisor on X ● D: effective Weil divisor on X ● Supp D ⊂ Supp Δ ● L ~R,π Kχ + Δ Then R 9 π , O χ ( L ) → R 9 π , O χ ( L + D ) is injective for every q.
Vanishing theorems for complex projective varieties

Theorem 2.5 (Injectivity for SNC pair) ● (X, Δ): SNC pair, Δ ∈ [0, 1], π : X → S : proper, as before ● L: Cartier divisor on X ● D : effective Cartier, permissible with respect to (X, Δ)
Osamu Fujino
Varishing theorems for complex projective varieties

Theorem 2.5 (Injectivity for SNC pair) ● (X, Δ): SNC pair, Δ ∈ [0, 1], π : X → S : proper, as before ● L: Cartier divisor on X ● D : effective Cartier, permissible with respect to (X, Δ) We further assume: (i)  L ~ R π  K x + Δ + H (ii)  H :  π -semi-ample R -divisor (iii) tH ~ π D + D ′ , t ∈ R ≠ 0, D': effective R-Cartier R-divisor, permissible with respect to (X, Δ)
Varishing theorems for complex projective varieties

Theorem 2.5 (Injectivity for SNC pair) ● (X, Δ): SNC pair, Δ ∈ [0, 1], π : X → S : proper, as before ● L: Cartier divisor on X ● D : effective Cartier, permissible with respect to (X, Δ) We further assume: (i)  L ~R,π Kχ + Δ + H (ii)  H :  π -semi-ample R -divisor (iii) tH ~ π D + D ′ , t ∈ R o , D': effective R-Cartier R-divisor, permissible with respect to (X, Δ) Then Rqπ*Oχ(L) → Rqπ*Oχ(L + D) is injective for every q.
Varishing theorems for complex projective varieties
Torsion-freeness and Vanishing for SNC pair Theorem 2.6 (Torsion-freeness and Vanishing thereom) ● (Y, Δ): SNC pair, Δ ∈ [0, 1], f : Y → X: proper ● L: Cartier divisor on Y such that L – ( K γ + Δ ): f-semi-ample
Varishing theorems for complex projective varieties

Theorem 2.6 (Torsion-freeness and Vanishing thereom) ● (Y,Δ): SNC pair, Δ ∈ [0,1], f : Y → X: proper ● L: Cartier divisor on Y such that L – ( K γ + Δ): f-semi-ample Then we have: (i) Every associated prime of R 9 f ⋆ O r ( L ) is the generic point of the f-image of some stratum of (Y, Δ). о π : X → V : projective (ii) ●  L − (K Y + Δ) ~  f * H, H: π-ample R-divisor on X ⇒⇒ RP π xR9 f .Oy (L) = 0 for every p > 0 and q ≥ 0.
Osamu Fujino
Varishing theorems for complex projective varieties
New framework of vanishing theorems Main statement Our result for SNC pairs contains Kodaira, Kawamata–Viehweg, Nadel, Kollár, and many other powerful and useful vanishing results as very special cases. MHS for cohomology with compact support Almost all the classical vanishing theorems (Kawamata–Viehweg, Kollár, etc.) can be proved by the E 1 -degeneration of
Osamu Fujino
MHS for cohomology with compact support Almost all the classical vanishing theorems (Kawamata–Viehweg, Kollár, etc.) can be proved by the E 1 -degeneration of
My idea is to use the E 1 -degeneration of
where X: smooth projective variety, D: SNC divisor.
MHS for cohomology with compact support
Almost all the classical vanishing theorems (Kawamata–Viehweg, Kollár, etc.) can be proved by the E 1 -degeneration of
My idea is to use the E 1 -degeneration of
where X: smooth projective variety, D: SNC divisor. In my framework, Ox(Kx + D) = Hom(Ω (log D) ® Ox(−D), ωx). We do not see O x ( K x + D ) as /\dim x Ω Vanishing theorems for complex projective varieties Comments on the proof Final remarks Precisely speaking: ● (X, D): SNC pair, or finite cyclic cover of SNC pair We have to consider MHS on H ξ ( X \ [ D ], C). Classical vanishing theorems New framework of vanishing theorems Comments on the proof Thank you Thank you very much! Osamu Fujino