Simple normal crossing pairs

160 5. INJECTIVITY AND VANISHING THEOREMS

simple normal crossing divisor on X. The decomposition H_cⁱ(X−∆,C) = ⊕

p+q=i

H^q(X,Ω^p_X(log ∆)⊗ OX(−∆)).

is suitable for our purposes. The dual statement H²ⁿ⁻ⁱ(X−∆,C) = ⊕

p+q=i

H^n−q(X,Ωⁿ_X^−p(log ∆)),

which is well known and is commonly used is not useful for our pur- poses. Note that the paper [FF] supports our approach in this chapter.

Anyway, [Am1, 3. Vanishing theorems] seems to be quite short.

In this chapter, we establish the injectivity, vanishing, and torsion- free theorems sufficient for the theory of quasi-log schemes discussed in Chapter 6. This chapter covers all the results in [Am1, Section 3] and contains several nontrivial generalizations. In [Am1, Section 3], Ambro closely followed Esnault–Viehweg’s arguments in [EsVi2]

(see also [F17, Chapter 2]). On the other hand, our approach in this chapter is more similar to Koll´ar’s (see, for example, [Ko6, Chapter 9]

and [KoMo, Section 2.4]).

5.2. SIMPLE NORMAL CROSSING PAIRS 161

such that fi ∈Γ(X,K^∗X) and ri ∈Z (resp. ri ∈Q, or ri ∈R) for every i. We note that (f_i) is aprincipal Cartier divisorassociated tof_i, that is, the image off_i by

Γ(X,K^∗X)→Γ(X,KX^∗/OX^∗ ).

Letf :X →Y be a morphism. If there is anR-Cartier divisorB onY such that D₁ ∼_R D₂+f^∗B, then D₁ is said to be relatively R-linearly equivalent to D₂. It is denoted by D₁ ∼_R,f D₂ orD₁ ∼_R,Y D₂.

5.2.4 (Supports). Let D be a Cartier divisor on X. Thesupport of D, denoted by SuppD, is the subset of X consisting of points x such that a local equation forDis not inO^∗X,x. The support ofDis a closed subset of X.

5.2.5 (Weil divisors,Q-divisors, andR-divisors). LetXbe an equidi- mensional variety. We note that X is not necessarily regular in codi- mension one. A (Weil) divisor D onX is a finite formal sum

∑n i=1

d_iD_i

whereD_i is an irreducible reduced closed subscheme ofX of pure codi- mension one and d_i is an integer for every i such that D_i 6= D_j for i6=j.

If d_i ∈ Q (resp. d_i ∈ R) for every i, then D is called a Q-divisor (resp.R-divisor). We define theround-updDe=∑r

i=1ddieDi(resp. the round-down bDc = ∑_r

i=1bdicDi), where for every real number x, dxe (resp.bxc) is the integer defined byx≤ dxe< x+1 (resp.x−1<bxc ≤ x). The fractional part {D} of D denotes D− bDc. We define D^<1 =

∑

di<1d_iD_i, and so on. We call D a boundary (resp. subboundary) R- divisor if 0≤d_i ≤1 (resp. d_i ≤1) for everyi.

Let us define simple normal crossing pairs.

Definition 5.2.6 (Simple normal crossing pairs). We say that the pair (X, D) issimple normal crossingat a pointa ∈XifXhas a Zariski open neighborhood U of a that can be embedded in a smooth variety Y, where Y has regular system of parameters (x₁,· · · , x_p, y₁,· · · , y_r) ata = 0 in which U is defined by a monomial equation

x1· · ·xp = 0 and

D=

∑r i=1

α_i(y_i = 0)|U, α_i ∈R.

162 5. INJECTIVITY AND VANISHING THEOREMS

We say that (X, D) is a simple normal crossing pairif it is simple nor- mal crossing at every point of X. If (X,0) is a simple normal crossing pair, then X is called a simple normal crossing variety. IfX is a sim- ple normal crossing variety, then X has only Gorenstein singularities.

Thus, it has an invertible dualizing sheafω_X. Therefore, we can define the canonical divisorK_X such thatω_X ' OX(K_X) (cf. [Li, Section 7.1 Corollary 1.19]). It is a Cartier divisor on X and is well-defined up to linear equivalence.

We say that a simple normal crossing pair is embedded if there exists a closed embeddingι:X ,→M, where M is a smooth variety of dimension dimX+ 1. We call M the ambient space of (X,∆).

The author learned the following interesting example from Kento Fujita (cf. [Ko13, Remark 1.9]).

Example 5.2.7. Let X1 = P² and let C1 be a line on X1. Let X2 =P² and let C2 be a smooth conic onX2. We fix an isomorphism τ : C₁ → C₂. By gluing X₁ and X₂ along τ : C₁ → C₂, we obtain a simple normal crossing surface X such that−K_X is ample (cf. [Fk1]).

We can check that X can not be embedded into any smooth varieties as a simple normal crossing divisor.

We note that a simple normal crossing pair is called asemi-snc pair in [Ko13, Definition 1.9].

Definition 5.2.8 (Strata and permissibility). Let X be a simple normal crossing variety and let X = ∪

i∈IX_i be the irreducible de- composition of X. A stratum of X is an irreducible component of X_i1 ∩ · · · ∩X_ik for some {i₁,· · · , i_k} ⊂I. A Cartier divisor D onX is permissibleifDcontains no strata ofX in its support. A finiteQ-linear (resp. R-linear) combination of permissible Cartier divisors is called a permissible Q-divisor(resp. R-divisor) on X.

5.2.9. Let X be a simple normal crossing variety. Let PerDiv(X) be the abelian group generated by permissible Cartier divisors on X and let Weil(X) be the abelian group generated by Weil divisors onX.

Then we can define natural injective homomorphisms of abelian groups

ψ : PerDiv(X)⊗ZK→Weil(X)⊗ZK

5.2. SIMPLE NORMAL CROSSING PAIRS 163

for K= Z, Q, and R. Let ν : Xe →X be the normalization. Then we have the following commutative diagram.

Div(X)e ⊗ZK ^∼_e

ψ

//Weil(eX)⊗ZK

ν_∗

PerDiv(X)⊗_ZK _ψ ^//

ν^∗

OO

Weil(X)⊗_ZK

Note that Div(X) is the abelian group generated by Cartier divisorse onXe and that ψeis an isomorphism since Xe is smooth.

Byψ, every permissible Cartier divisor (resp.Q-divisor orR-divisor) can be considered as a Weil divisor (resp. Q-divisor or R-divisor). For Q-divisors and R-divisors, see 5.2.5. Therefore, various operations, for example, bDc, D^<1, and so on, make sense for a permissible R-divisor D onX.

We note the following easy example.

Example 5.2.10. Let X be a simple normal crossing variety in C³ = SpecC[x, y, z] defined by xy = 0. We set D₁ = (x+z = 0)∩X and D₂ = (x−z = 0)∩X. Then D = ¹₂D₁ + ¹₂D₂ is a permissible Q-divisor on X. In this case, bDc = (x = z = 0) on X. Therefore, bDc is not a Cartier divisor onX.

Definition 5.2.11 (Simple normal crossing divisors). Let X be a simple normal crossing variety and letD be a Cartier divisor on X. If (X, D) is a simple normal crossing pair and D is reduced, then D is called asimple normal crossing divisor onX.

Remark 5.2.12. Let X be a simple normal crossing variety and let D be a K-divisor on X where K = Q or R. If SuppD is a simple normal crossing divisor on X and D is K-Cartier, then bDc and dDe (resp. {D}, D^<1, and so on) are Cartier (resp. K-Cartier) divisors on X (cf. [BVP, Section 8]).

The following lemma is easy but important.

Lemma5.2.13. LetX be a simple normal crossing variety and letB be a permissible R-divisor on X such thatbBc= 0. Let A be a Cartier divisor on X. Assume that A ∼R B. Then there exists a permissible Q-divisorC onX such that A∼Q C, bCc= 0, and SuppC = SuppB.

Proof. We can write B = A+∑_k

i=1r_i(f_i), where f_i ∈ Γ(X,K^∗X) and r_i ∈R for every i. Here, KX is the sheaf of total quotient rings of

164 5. INJECTIVITY AND VANISHING THEOREMS

OX (see 5.2.1). Let P ∈X be a scheme theoretic point corresponding to some stratum of X. We consider the following affine map

K^k →H⁰(X_P,KX^∗P/O^∗XP)⊗ZK given by (a₁,· · ·, a_k) 7→ A+∑_k

i=1a_i(f_i), where X_P = SpecOX,P and K=Qor R. Then we can check that

P ={(a₁,· · · , a_k)∈R^k|A+∑

i

a_i(f_i) is permissible} ⊂R^k is an affine subspace of R^k defined over Q. Therefore, we see that

S ={(a1,· · · , ak)∈ P | Supp(A+∑

i

ai(fi))⊂SuppB} ⊂ P is an affine subspace of R^k defined over Q. Since (r₁,· · · , r_k) ∈ S, we know that S 6= ∅. We take a point (s₁,· · · , s_k) ∈ S ∩Q^k which is general in S and sufficiently close to (r₁,· · · , r_k) and set

C=A+

∑k i=1

si(fi).

By construction,Cis a permissibleQ-divisor such thatC ∼_Q A,bCc=

0, and SuppC = SuppB.

We need the following important definition in Section5.6.

Definition5.2.14 (Strata and permissibility for pairs). Let (X, D) be a simple normal crossing pair. Letν :X^ν →Xbe the normalization.

We define Θ by the formula

K_X^ν + Θ =ν^∗(K_X +D).

Then a stratum of (X, D) is an irreducible component of X or the ν- image of a log canonical center of (X^ν,Θ). WhenD= 0, this definition is compatible with Definition 5.2.8. A Cartier divisor B on X is per- missible with respect to (X, D) if B contains no strata of (X, D) in its support. A finite Q-linear (resp. R-linear) combination of permissible Cartier divisors with respect to (X, D) is called apermissibleQ-divisor (resp. R-divisor) with respect to (X, D).

5.2.15 (Partial resolution of singularities for reducible varieties). In this chapter, we will repeatedly use the following results on the partial resolution of singularities for reducible varieties.

Theorem 5.2.16 is a special case of [BM, Theorem 1.5].