Hodge theoretic injectivity theorems The main theorem of this section is

5.4. HODGE THEORETIC INJECTIVITY THEOREMS 169

Let us recall the following well-known results on Du Bois singular- ities.

Theorem 5.3.7. Let X be a normal algebraic variety with only quotient singularities. Then X is Du Bois. Note that X has only rational singularities.

Theorem 5.3.7 follows from, for example, [Du, 5.2. Th´eor`eme], [Kv1], and so on.

Lemma 5.3.8. Let X be a variety with closed subvarieties X1 and X₂ such that X = X₁ ∪X₂. Assume that X₁, X₂, and X₁ ∩X₂ are Du Bois. Note that, in particular, we assume that X₁∩X₂ is reduced.

Then X is Du Bois.

For the proof of Lemma 5.3.8, see, for example, [Schw, Lemma 3.4].

Although it is dispensable, we will use the notion of Du Bois com- plexes for the proof of the Hodge theoretic injectivity theorem: Theo- rem 5.4.1.

5.4. Hodge theoretic injectivity theorems

170 5. INJECTIVITY AND VANISHING THEOREMS

Theorem 5.4.2 is a very useful generalization of Lemma3.1.1. It is sufficient for many applications of the minimal model program. How- ever, we need Theorem 5.4.1 for the theory of quasi-log schemes dis- cussed in Chapter 6. For various applications of Theorem 5.4.2, see [F36, Section 5].

First, let us prove Theorem 5.4.2.

Proof of Theorem 5.4.2. Without loss of generality, we may assume that X is connected. We set S = b∆c and B = {∆}. By perturbingB, we may assume thatB is aQ-divisor (cf. Lemma5.2.13).

We setM=OX(L−K_X −S). Let N be the smallest positive integer such thatN L∼N(K_X+S+B). In particular,N B is an integral Weil divisor. We take the N-fold cyclic cover

π⁰ :Y⁰ = Spec_X

N−1

⊕

i=0

M⁻ⁱ →X

associated to the sectionN B ∈ |M^N|. More precisely, lets ∈H⁰(X,M^N) be a section whose zero divisor isN B. Then the dual ofs:OX → M^N defines an OX-algebra structure on ⊕_N−1

i=0 M⁻ⁱ. Let Y → Y⁰ be the normalization and let π :Y →X be the composition morphism. It is well known that

Y = Spec_X

N⊕−1 i=0

M⁻ⁱ(biBc).

For the details, see [EsVi3, 3.5. Cyclic covers]. Note that Y has only quotient singularities by construction. We set T = π^∗S. Let T =∑

i∈IT_i be the irreducible decomposition. Then every irreducible component of Ti1 ∩ · · · ∩Ti_k has only quotient singularities for every {i1,· · · , ik} ⊂ I. Hence it is easy to see that both Y and T have only Du Bois singularities by Theorem5.3.7 and Lemma5.3.8 (see also [I]).

Therefore, the pair (Y, T) is a Du Bois pair by Proposition5.3.6. This means that OY(−T) → Ω⁰_Y,T is a quasi-isomorphism (see also [FFS, 3.4]). We note that T is Cartier. Hence OY(−T) is the defining ideal sheaf ofT onY. TheE₁-degeneration of

E₁^p,q =H^q(Y,Ω^p_Y,T)⇒H^p+q(Y, j_!CY−T) implies that the homomorphism

H^q(Y, j_!CY−T)→H^q(Y,OY(−T)) induced by the natural inclusion

j_!CY−T ⊂ OY(−T)

5.4. HODGE THEORETIC INJECTIVITY THEOREMS 171

is surjective for every q(see Remark5.3.4). By taking a suitable direct summand

C ⊂ M⁻¹(−S) of

π_∗(j_!CY−T)⊂π_∗OY(−T), we obtain a surjection

H^q(X,C)→H^q(X,M⁻¹(−S))

induced by the natural inclusion C ⊂ M⁻¹(−S) for every q. We can check the following simple property by examining the monodromy ac- tion of the Galois group Z/NZof π :Y →X onC around SuppB.

Lemma 5.4.3 (cf. [KoMo, Corollary 2.54]). Let U ⊂ X be a con- nected open set such that U ∩Supp ∆6=∅. Then H⁰(U,C|U) = 0.

Proof of Lemma 5.4.3. If U ∩SuppB 6= ∅, then H⁰(U,C|U) = 0 since the monodromy action on C around SuppB is nontrivial. If U ∩SuppS 6= ∅, then H⁰(U,C|U) = 0 since C is a direct summand of

π_∗(j_!CY−T) and T =π^∗S.

This property is utilized by the following fact. The proof of Lemma 5.4.4 is obvious.

Lemma 5.4.4 (cf. [KoMo, Lemma 2.55]). Let F be a sheaf of Abelian groups on a topological spaceV and letF₁ andF₂ be subsheaves of F. Let Z be a closed subset of V. Assume that

(1) F₂|V−Z =F|V−Z, and

(2) ifU is connected, open and U∩Z 6=∅, then H⁰(U, F₁|U) = 0.

Then F₁ is a subsheaf of F₂. As a corollary, we obtain:

Corollary5.4.5 (cf. [KoMo, Corollary 2.56]). LetM ⊂ M⁻¹(−S) be a subsheaf such thatM|X−Supp ∆ =M⁻¹(−S)|X−Supp ∆. Then the in- jection

C → M⁻¹(−S) factors as

C →M → M⁻¹(−S).

Therefore,

H^q(X, M)→H^q(X,M⁻¹(−S)) is surjective for every q.

172 5. INJECTIVITY AND VANISHING THEOREMS

Proof of Corollary 5.4.5. The first part is clear from Lemma 5.4.3 and Lemma 5.4.4. This implies that we have maps

H^q(X,C)→H^q(X, M)→H^q(X,M⁻¹(−S)).

As we saw above, the composition is surjective. Hence so is the map

on the right.

Therefore,H^q(X,M⁻¹(−S−D))→H^q(X,M⁻¹(−S)) is surjective for every q. By Serre duality, we obtain that

H^q(X,OX(KX)⊗ M(S))→H^q(X,OX(KX)⊗ M(S+D)) is injective for every q. This means that

H^q(X,OX(L))→H^q(X,OX(L+D))

is injective for every q.

Next, let us prove Theorem5.4.1, the main theorem of this section.

The proof of Theorem5.4.2 given above works for Theorem5.4.1 with some minor modifications.

Proof of Theorem 5.4.1. Without loss of generality, we may assume that X is connected. We can take an effective Cartier divisor D⁰onXsuch thatD⁰−Dis effective and SuppD⁰ ⊂Supp ∆. Therefore, by replacing D with D⁰, we may assume that D is a Cartier divisor.

We set S = b∆c and B = {∆}. By Lemma 5.2.13, we may assume that B is a Q-divisor. We set M= OX(L−K_X −S). Let N be the smallest positive integer such that N L∼N(K_X +S+B). We define anOX-algebra structure of⊕N−1

i=0 M⁻ⁱ(biBc) by s∈H⁰(X,M^N) with (s= 0) =N B. We set

π:Y = Spec_X

N−1

⊕

i=0

M⁻ⁱ(biBc)→X and T = π^∗S. Let Y = ∑

j∈JY_j be the irreducible decomposition.

Then every irreducible component ofY_j1∩· · ·∩Y_jlhas only quotient sin- gularities for every {j₁,· · · , j_l} ⊂ J by construction. Let T =∑

i∈IT_i be the irreducible decomposition. Then every irreducible component of Ti1∩ · · · ∩Ti_k has only quotient singularities for every {i1,· · ·, ik} ⊂I by construction. Hence it is easy to see that both Y and T are Du Bois by Theorem 5.3.7 and Lemma 5.3.8 (see also [I]). Therefore, the pair (Y, T) is a Du Bois pair by Proposition 5.3.6. This means that OY(−T) → Ω⁰_Y,T is a quasi-isomorphism (see also [FFS, 3.4]). We

5.4. HODGE THEORETIC INJECTIVITY THEOREMS 173

note that T is Cartier. Hence OY(−T) is the defining ideal sheaf ofT onY. The E₁-degeneration of

E₁^p,q =H^q(Y,Ω^p_Y,T)⇒H^p+q(Y, j_!CY−T) implies that the homomorphism

H^q(Y, j!CY−T)→H^q(Y,OY(−T)) induced by the natural inclusion

j!CY−T ⊂ OY(−T)

is surjective for every q(see Remark5.3.4). By taking a suitable direct summand

C ⊂ M⁻¹(−S) of

π_∗(j_!CY−T)⊂π_∗OY(−T), we obtain a surjection

H^q(X,C)→H^q(X,M⁻¹(−S))

induced by the natural inclusion C ⊂ M⁻¹(−S) for every q. It is easy to see that Lemma 5.4.3 holds for this new setting. Hence Corollary 5.4.5 also holds without any modifications. Therefore,

H^q(X,M⁻¹(−S−D))→H^q(X,M⁻¹(−S)) is surjective for everyq. By Serre duality, we obtain that

H^q(X,OX(L))→H^q(X,OX(L+D))

is injective for every q.

We close this section with an easy application of Theorem 5.4.1.

Corollary 5.4.6 (Kodaira vanishing theorem for simple normal crossing varieties). LetXbe a projective simple normal crossing variety and letLbe an ample line bundle onX. ThenH^q(X,OX(K_X)⊗L) = 0 for every q >0.

Proof. We take a general member ∆ ∈ |L^l| for some positive large number l. Then we can find a Cartier divisor M on X such that M ∼Q K_X+¹_l∆ and that OX(K_X)⊗ L ' OX(M). Then, by Theorem 5.4.1,

H^q(X,OX(M))→H^q(X,OX(M +m∆))

is injective for every q and any positive integer m. Since ∆ is ample, Serre’s vanishing theorem implies that H^q(X,OX(M)) = 0 for every

q >0.

174 5. INJECTIVITY AND VANISHING THEOREMS