Viehweg vanishing theorem

56 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Theorem 3.3.2. Let π : X → S be a proper surjective morphism from a smooth variety X, let L be an invertible sheaf on X, and let D be an effective Cartier divisor on X such that SuppD is normal crossing. Assume that L^N(−D) is π-nef for some positive integer N and that κ(X_η,(L⁽¹⁾)_η) = m, where X_η is the generic fiber of π, (L⁽¹⁾)_η =L⁽¹⁾|Xη, and

L⁽¹⁾ =L(−bD Nc).

Then we have

Rⁱπ_∗(L⁽¹⁾⊗ω_X) = 0 for i >dimX−dimS−m.

We note that SuppDis not necessarilysimple normal crossing. We only assume that SuppD is normal crossing.

Remark 3.3.3. In Theorem 3.3.2, we assume that S is a point for simplicity. We note that κ(X,L⁽¹⁾) = m does not necessarily imply κ(X,L⁽ⁱ⁾) = m for 2≤i≤N −1, where

L⁽ⁱ⁾ =L^⊗i(−biD N c).

Therefore, Viehweg’s original arguments in [V1] depending on Bogo- molov’s vanishing theorem do not seem to work in our setting.

Let us reformulate Theorem 3.2.1 for the proof of Theorem 3.3.2.

Theorem 3.3.4 (Kawamata–Viehweg vanishing theorem). Let f : Y →X be a proper surjective morphism from a smooth variety Y and let M be a Cartier divisor on Y. Let ∆ be an effective Q-divisor on Y such that Supp ∆ is normal crossing and b∆c = 0. Assume that M −(K_Y + ∆) is f-nef and f-big. Then

Rⁱf_∗OY(M) = 0 for every i >0.

Proof. We put D = M −(K_Y + ∆). Then D is an f-nef and f-big Q-divisor on Y such that {D}=d∆e −∆ and dDe=M −K_Y. By Theorem 3.2.1, we obtainRⁱf_∗OY(K_X +dDe) = 0 for everyi >0.

Therefore,Rⁱf_∗OY(M) = 0 for every i >0.

Remark 3.3.5. It is obvious that Theorem3.3.4is a special case of Theorem 3.3.2. By applying Theorem 3.3.2, the assumption in Theo- rem3.3.4can be weaken as follows:M−(KX+∆) isf-nef andM−KX

is f-big. We note that M −K_X is f-big if M −(K_X + ∆) is f-big.

In this section, we give a quick proof of Theorem 3.3.2 only by using

3.3. VIEHWEG VANISHING THEOREM 57

Theorem 3.3.4 and Hironaka’s resolution. Therefore, Theorem 3.3.2 is essentially the same as Theorem 3.3.4.

Let us start the proof of Theorem 3.3.2.

Proof of Theorem 3.3.2. Without loss of generality, we may assume that S is affine. Let f : Y → X be a proper birational mor- phism from a smooth quasi-projective varietyY such that Suppf^∗D∪ Exc(f) is a simple normal crossing divisor. We write

K_Y =f^∗(K_X + (1−ε){D

N}) +E_ε.

Then F = dE_εe is an effective exceptional Cartier divisor on Y and independent of ε for 0 < ε 1. Therefore, the coefficients of F −E_ε are continuous for 0 < ε 1. Let L be a Cartier divisor on X such that L ' OX(L). We may assume that κ(X_η,(L− b^D_Nc)_η) = m ≥ 0.

Let Φ :X 99KZ be the relative Iitaka fibration overS with respect to l(L− b^D_Nc), wherel is a sufficiently large and divisible positive integer.

We may further assume that f^∗(L− bD

Nc)∼Q ϕ^∗A+E,

whereE is an effectiveQ-divisor such that SuppE∪Suppf^∗D∪Exc(f) is simple normal crossing, ϕ = Φ◦f :Y →Z is a morphism, and A is a ψ-ampleQ-divisor on Z with ψ :Z →S.

Y

f

ϕ

@

@@

@@

@@

X ^{_ Φ_ _//}

π@@@@@@

@@ Z

ψ

S

Let ∑

i

E_i = SuppE∪Suppf^∗D∪Exc(f) be the irreducible decomposition. We can write E_ε = ∑

ia^ε_iE_i and E =∑

ib_iE_i. We note that a^ε_i is continuous for 0< ε1. We put

∆_ε =F −E_ε+εE.

By definition, we can see that every coefficient of ∆_ε is in [0,2) for 0 < ε 1. Thus, b∆_εc is reduced. If a^ε_i < 0, then a^ε_i ≥ −1 + _N¹ for 0< ε1. Therefore, ifda^ε_ie −a^ε_i+εb_i ≥1 for 0< ε1, thena^ε_i >0.

58 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Thus,F⁰ =F − b∆εc is effective andf-exceptional for 0< ε1. On the other hand, (Y,{∆_ε}) is obviously klt for 0< ε1. We note that

f^∗(K_X +L− bD

Nc) +F⁰−(K_Y +{∆_ε})

=f^∗(K_X +L− bD

Nc) +F −f^∗(K_X + (1−ε){D

N})−E_ε

−(F −E_ε+εE)

∼_Q (1−ε)f^∗(L− D

N) +εϕ^∗A

for a rational number ε with 0< ε1. We put M =f^∗(K_X +L− bD

Nc) +F⁰.

Let H be a p-ample general smooth Cartier divisor on Y, where p = ψ◦ϕ=π◦f :Y →S. Since

(M+H)−(KY +{∆ε})∼Q (1−ε)f^∗(L− D

N) +εϕ^∗A+H, is p-ample, we obtain

Rⁱp_∗OY(M +H) = 0

for every i >0 by Theorem 3.3.4. By the long exact sequence

· · · →Rⁱp_∗OY(M)→Rⁱp_∗OY(M +H)→Rⁱp_∗OH(M+H)→ · · · obtained from

0→ OY(M)→ OY(M+H)→ OH(M +H)→0, we obtain

Rⁱp_∗OH(M +H)'Rⁱ⁺¹p_∗OY(M) for every i >0. We note that

M −(K_Y +{∆_ε})∼_Q (1−ε)f^∗(L− D

N) +εϕ^∗A and

(M +H)|H −(K_H +{∆_ε}|H)∼Q (1−ε)f^∗(L− D

N)|H +εϕ^∗A|H. We also note that (H,{∆ε}|H) is klt and

κ(Hη,(ϕ^∗A)|Hη)≥min{m,dimHη}.

By repeating the above argument, that is, taking a general smooth hyperplane cut, and by Theorem 3.3.4, we obtain

Rⁱp_∗OY(M) = Rⁱp_∗OY(f^∗(K_X +L− bD

Nc) +F⁰) = 0

3.3. VIEHWEG VANISHING THEOREM 59

for every i >dimY −dimS−m= dimX−dimS−m (see also [V1, Remark 0.2]). On the other hand,

Rⁱf_∗OY(M) = Rⁱf_∗OY(f^∗(K_X +L− bD

Nc) +F⁰) = 0 for every i >0 by Theorem 3.3.4. We note that

f_∗OY(f^∗(K_X +L− bD

Nc) +F⁰)' OX(K_X +L− bD Nc)

by the projection formula because F⁰ is effective and f-exceptional.

Therefore, we obtain

Rⁱπ_∗OX(K_X +L− bD

Nc) =Rⁱp_∗OY(M) = 0

for every i >dimX−dimS−m.

We give an obvious corollary of Theorem 3.3.2.

Corollary 3.3.6. Let X be an n-dimensional smooth complete variety and letLbe an invertible sheaf onX. Assume thatD∈ |L^N|for some positive integer N and that SuppD is a simple normal crossing divisor on X. Then we have

Hⁱ(X,L⁽¹⁾⊗ω_X) = 0 for i > n−κ(X,{^D_N}).

We think that Theorem 3.3.7, which is similar to Theorem 3.3.2 and easily follows from the usual Kawamata–Viehweg vanishing the- orem: Theorem 3.2.1 (see also Theorem 3.3.4), is easier to use than Theorem 3.3.2. So we contain it for the reader’s convenience.

Theorem 3.3.7 (Kawamata–Viehweg vanishing theorem). Let f : Y → X be a projective morphism from a smooth variety Y onto a variety X. Let ∆ be an effective Q-divisor on Y such that Supp ∆ is a normal crossing divisor and that b∆c= 0. Let M be a Cartier divisor onY such thatM−(K_Y+∆) isf-nef andν(X_η,(M−(K_Y+∆))|Xη) = m, where X_η is the generic fiber of f. Then Rⁱf_∗OY(M) = 0 for every i >dimY −dimX−m.

Proof. We use induction on dimY−dimX. If dimY−dimX = 0, thenM−(K_Y + ∆) isf-big. Therefore, Theorem3.3.7is a special case of Theorem3.3.4when dimY −dimX = 0. Whenm= dimY −dimX, Theorem3.3.7 follows from Theorem3.3.4. Thus, we may assume that m <dimY −dimX. Without loss of generality, we may assume that X is affine by shrinking X. Let A be an f-very ample Cartier divisor

60 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

onY. We take a general member H of |A|. We consider the following short exact sequence

0→ OY(M)→ OY(M+H)→ OH(M +H)→0.

Since M +H−(K_Y + ∆) is f-ample, we obtain Rⁱf_∗OY(M+H) = 0 for every i >0 by Theorem 3.3.4. This implies that

Rⁱf_∗OH(M +H)'Rⁱ⁺¹f_∗OY(M)

holds for every i≥1. Since (M +H)|H −(K_H + ∆|H) is f-nef and ν(H_η,((M+H)|H −(K_H + ∆|H))|Hη)≥m,

where H_η is the generic fiber of H→f(H), we obtain Rⁱf_∗OH(M +H) = 0

fori >dimH−dimX−m = dimY −dimX−m−1 by induction on dimY −dimX. Therefore, we have

Rⁱf_∗OY(M) = 0

for i >dimY −dimX−m.