Classical vanishing theorems and some applications
3.2. Kawamata–Viehweg vanishing theorem
48 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS
In general, we can write D ∼ L1 −L2 where L1 and L2 are both very ample Cartier divisors. By using the above argument twice, we obtain f : Y → X such that f∗Li ∼mL0i for some Cartier divisors L0i for i= 1,2. Thus we obtain the desired morphism f :Y →X.
3.2. KAWAMATA–VIEHWEG VANISHING THEOREM 49
forq >0 by Theorem3.1.7. SinceHis ample onS andmis sufficiently large,
Rqπ∗OX(KX +dDe+mπ∗H) 'Rqπ∗OX(KX +dDe)⊗ OS(mH) is generated by global sections. Therefore, we obtain
Riπ∗OX(KX +dDe) = 0 for every i >0.
Step 2. In this step, we treat the general case by using the result obtained in Step 1.
Now we prove the theorem under the conditions (i) and (ii). We may assume that S is affine since the statement is local. By Kodaira’s lemma (see Lemma 2.1.18) and Hironaka’s resolution theorem, we can construct a projective birational morphism f : Y → X from another smooth varietyY which is projective overSand divisorsFα’s onY such that Suppf∗D∪(∪
αFα)∪Exc(f) is a simple normal crossing divisor onY and thatf∗D−∑
δαFα isπ◦f-ample for someδα ∈Qwith 0< δα 1 (see also [KMM, Corollary 0-3-6]). Then by applying the result proved in Step1 to f, we obtain
0 =Rif∗OY(KY +df∗D−∑
δαFαe) = Rif∗OY(KY +df∗De) for every i >0. We can also see that
f∗OY(KY +df∗De)' OX(KX +dDe)
by Lemma3.2.2below. So, we have, by the special case treated in Step 1,
0 =Ri(π◦f)∗OY(KY +df∗D−∑
δαFαe)
=Riπ∗(f∗OY(KY +df∗De))
=Riπ∗OX(KX +dDe)
for every i >0.
Lemma 3.2.2. LetX be a smooth variety and letDbe anR-divisor on X such that Supp{D} is a simple normal crossing divisor on X.
Let f :Y →X be a proper birational morphism from a smooth variety Y such that Suppf∗{D} ∪Exc(f) is a simple normal crossing divisor on Y. Then we have
f∗OY(KY +df∗De)' OX(KX +dDe).
50 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS
Proof. We put ∆ = dDe − bDc. Then ∆ is a reduced simple normal crossing divisor on X. We can write
KY +f∗−1∆ =f∗(KX + ∆) + ∑
Ei:f-exceptional
a(Ei, X,∆)Ei. We have a(Ei, X,∆)∈Z and a(Ei, X,∆)≥ −1 for every i. Then
KY +f∗−1∆ +f∗bDc=f∗(KX +dDe) + ∑
Ei:f-exceptional
a(Ei, X,∆)Ei. We can easily check that
multEi(
df∗De −(f∗−1∆ +f∗bDc))
≥1
for everyf-exceptional divisorEi witha(Ei, X,∆) =−1. Thus we can write
KY +df∗De=f∗(KX +dDe) +F,
where F is an effective f-exceptional Cartier divisor onY. Therefore, we havef∗OY(KY +df∗De)' OX(KX +dDe).
We used the following lemma in the proof of Theorem3.2.1. We give a detailed proof for the reader’s convenience (see also Lemma 5.5.2).
Lemma 3.2.3. Let π : X → S be a projective surjective morphism from a smooth variety X to an affine variety S. Let D be a Q-divisor onX such thatDisπ-ample andSupp{D}is a simple normal crossing divisor on X. Then there exist a completion π :X →S of π :X →S where X and S are both projective with π|X = π and a π-ample Q- divisor D on X with D|X =D such that Supp{D} is a simple normal crossing divisor on X.
Proof. Let m be a sufficiently large and divisible positive integer such that the natural surjection
π∗π∗OX(mD)→ OX(mD)
induces an embedding ofX intoPS(π∗OX(mD)) over S. Letπ0 :X0 → S be an arbitrary completion of π : X → S such that X0 and S are both projective and X0 is smooth. We can construct suchπ0 :X0 →S by Hironaka’s resolution theorem. Let D0 be the closure of D on X0. We consider the natural map
π0∗π∗0OX0(mD0)→ OX0(mD0).
The image of the above map can be written as J ⊗ OX0(mD0)⊂ OX0(mD0),
where J is an ideal sheaf on X0 such that SuppOX0/J ⊂X0\X. Let X00 be the normalization of the blow-up ofX0 by J and f :X00 →X0
3.2. KAWAMATA–VIEHWEG VANISHING THEOREM 51
the natural map. We note that f is an isomorphism overX ⊂X0. We can write f−1J · OX00 = OX00(−E) for some effective Cartier divisor E on X00. By replacing X0 with X00 and mD0 with mf∗D0 −E, we may assume that mD0 isπ-very ample over S and is π-generated over S. Therefore, we can consider the morphism ϕ : X0 → X00 over S associated to the surjection
π0∗π0∗OX0(mD0)→ OX0(mD0)→0.
We note thatϕis an isomorphsim overS by construction. By replacing X0 withX00again, we may assume thatD0 isπ0-ample. By using Hiron- aka’s resolution theorem, we may further assume thatX0 is smooth. By Szab´o’s resolution lemma (see Lemma 2.3.19), we can make Supp{D0} a simple normal crossing divisor. Thus, we obtain desired completions
π :X →S and D.
Remark 3.2.4. In Lemma3.2.3, we used Szab´o’s resolution lemma (see Lemma 2.3.19), which was obtained after [KMM] was written.
See 3.2.5 below.
3.2.5 (Kawamata–Viehweg vanishing theorem without using Szab´o’s resolution lemma). Here, we explain how to prove the Kawamata–
Viehweg vanishing theorem (see Theorem 3.2.1) without using Szab´o’s resolution lemma (see Lemma 2.3.19). The following proof is due to Noboru Nakayama.
Proof of Theorem 3.2.1 without Szab´o’s lemma. It is suf- ficient to prove Step 1 in the proof of Theorem 3.2.1. Let π : X → S be a projective surjective morphism from a smooth variety X to an affine varietyS and letD be a Q-divisor onX such that Dis π-ample and that Supp{D} is a simple normal crossing divisor on X. Then, by taking completions, we have projective varieties X and S with a projective morphism π:X →S such that
• X is a Zariski open dense subset of X,
• S is a Zariski open dense subset of S, and
• π|X is the composition of π and the open immersionS →S.
X
π
//X
π
S //S Note thatπ−1(S) =X since π is proper.
Claim. In the above setting, there exist a birational morphism µ: Y → X from another smooth projective variety Y and a Q-divisor C on Y such that
52 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS
(i) C is relatively nef and relatively big over S,
(ii) Supp{C} ∪Exc(µ) is a simple normal crossing divisor on Y, and
(iii) C|Y =µ∗D, where Y =µ−1(X) and µ= µ|Y :Y →X is the induced birational morphism.
Here, we have an isomorphism
OX(KX +dDe)'µ∗OY(KY +dCe).
(♣)
Proof of Claim. Note that, by Lemma 3.2.2, we can check the isomorphism (♣) by (ii) and (iii). By taking a resolution ofX, we may assume thatXis smooth. Then the closureDofDinXis a Q-Cartier Q-divisor. Let us consider the natural homomorphism
ϕm :π∗π∗OX(mD)→ OX(mD)
for a sufficiently large positive fixed integermsuch thatmDis Cartier.
Then ϕm is surjective on X since D is π-ample. By taking some fur- ther blow-ups, we may assume that the image of ϕm is expressed as OX(mD−E) for an effective Cartier divisorE onX with E∩X =∅. Thus, we have a projective varietyP overSand a morphismf :X →P over S such that
f∗H ∼mD−E
for a Cartier divisorH onP which is relatively ample overS. SinceD is π-ample and E∩X =∅, the induced morphism
f|X :X =X×SS →P :=P ×SS
is finite. In particular, f is a generically finite morphism. We set D0 :=D− 1
mE.
Then D0 is π-nef and π-big, and D0|X = D. We can take a birational morphism µ :Y →X from another smooth projective variety Y such that the union of theµ-exceptional locus and Suppµ∗({D0}) is a simple normal crossing divisor on Y. We set C := µ∗D0. Then we have a
desired µ:Y →X with C.
We can easily see that we can prove Theorem 3.2.1 without using Lemma 3.2.3 when S is projective (see the proof of Theorem 3.2.1).
Note that we do not have to shrink S and assume that S is affine in the proof of Theorem 3.2.1 if S is projective. From now on, we will freely use Theorem 3.2.1 when the target space is projective.
By applying Theorem3.2.1, we obtain Riµ∗OY(KY +dCe) = 0
3.2. KAWAMATA–VIEHWEG VANISHING THEOREM 53
for everyi >0 sinceX is projective. By applying Theorem3.2.1again, we obtain
Ri(π◦µ)∗OY(KY +dCe) = 0
for every i > 0 since S is projective. Hence, by the Leray spectral sequence, we have
Riπ∗(µ∗OY(KY +dCe)) = 0
for everyi >0. By considering the restriction toS and by the isomor- phism (♣), we have
Riπ∗OX(KX +dDe) = 0
for every i > 0. It is the desired Kawamata–Viehweg vanishing theo-
rem.
Remark 3.2.6. In [Nak1, Theorem 3.7], Nakayama proved a gen- eralization of the Kawamata–Viehweg vanishing theorem (see Theorem 3.2.1) in the analytic category. Of course, the proof of [Nak1, Theo- rem 3.7] does not need Szab´o’s resolution lemma. For several related results in the analytic category, we recommend the reader to see [F31].
As a very special case of Theorem3.2.1, we have:
Theorem3.2.7 (Grauert–Riemenschneider vanishing theorem).Let f :X →Y be a generically finite morphism from a smooth variety X.
Then Rif∗OX(KX) = 0 for every i >0.
Proof. Note thatKX−KX isf-nef andf-big sincefis generically finite. Therefore, we obtain Theorem3.2.7as a special case of Theorem
3.2.1.
For a related result, see Lemma 3.8.7, Remark 3.8.8, and Theorem 3.8.9 below.
Viehweg’s formulation of the Kawamata–Viehweg vanishing theo- rem is slightly different from Theorem 3.2.1.
Theorem 3.2.8 (Viehweg). Let X be a smooth variety and let π : X →S be a proper surjective morphism onto a variety S. Assume that a Q-divisor D on X satisfies the following conditions:
(i0) D is π-nef and dDe isπ-big, and
(ii) {D} has support with only normal crossings.
Then Riπ∗OX(KX +dDe) = 0 for every i >0.
We note that the condition (i0) in Theorem 3.2.8 is slightly weaker than (i) in Theorem 3.2.1. We discuss a generalization of Theorem 3.2.8 in Section 3.3.
54 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS
Let us generalize Theorem 3.2.1 for R-divisors. We will repeatedly use it in the subsequent chapters.
Theorem3.2.9 (Kawamata–Viehweg vanishing theorem forR-divisors). Let X be a smooth variety and let π : X → S be a proper surjective morphism onto a varietyS. Assume that anR-divisorD onX satisfies the following conditions:
(i) D is π-nef and π-big, and
(ii) {D} has support with only normal crossings.
Then Riπ∗OX(KX +dDe) = 0 for every i >0.
Proof. When D isπ-ample, we perturb the coefficients of D and may assume thatDis aQ-divisor. Then, by Theorem 3.2.1, we obtain Riπ∗OX(KX +dDe) = 0 for every i > 0. By using this special case, Step 2 in the proof of Theorem 3.2.1 works without any changes. So,
we obtain this theorem.
As a corollary of Theorem 3.2.9, we obtain the vanishing theorem of Reid–Fukuda type. It will play important roles in the subsequent chapters. Before we state it, we prepare the following definition.
Definition 3.2.10 (Nef and log big divisors). Let f : V → W be a proper surjective morphism from a smooth variety V to a varietyW and letB be a boundaryR-divisor onV such that SuppB is a simple normal crossing divisor. We put T = bBc. Let T = ∑m
i=1Ti be the irreducible decomposition. Let G be an R-divisor on V. We say that G is f-nef and f-log big with respect to (V, B) if and only if G is f- nef, f-big, and G|C is f|C-big for every C, where C is an irreducible component of Ti1 ∩ · · · ∩Tik for some {i1,· · · , ik} ⊂ {1,· · · , m}.
Of course, Definition 3.2.10 is compatible with Definition 5.7.2 be- low.
Theorem 3.2.11 (Vanishing theorem of Reid–Fukuda type). Let V be a smooth variety and let B be a boundary R-divisor on V such that SuppB is a simple normal crossing divisor. Let f :V → W be a proper morphism onto a varietyW. Assume thatDis a Cartier divisor on V such that D−(KV +B) is f-nef and f-log big with respect to (V, B). Then Rif∗OV(D) = 0 for every i >0.
Proof. We use induction on the number of irreducible components ofbBcand on the dimension ofV. IfbBc= 0, then Theorem3.2.11fol- lows from the Kawamata–Viehweg vanishing theorem: Theorem 3.2.9.
Therefore, we may assume that there is an irreducible divisorS ⊂ bBc.