Injectivity and vanishing theorems
5.7. Vanishing theorems of Reid–Fukuda type
182 5. INJECTIVITY AND VANISHING THEOREMS
becauseH1(Y, f∗OX(−L)) =H2(Y, f∗OX(−L)) = 0 by the Kawamata–
Viehweg vanishing theorem (see Theorem 3.2.1). On the other hand, we have
Rqf∗OY 'Hq(E,OE)
for everyq >0 sinceRqf∗OY(−E) = 0 for everyq >0 by the Grauert–
Riemenschneider vanishing theorem (see Theorem 3.2.7). Thus, we obtain H2(X,OX(−L)) ' C2. In particular, H2(X,OX(−L)) 6= 0.
We note that X is not Cohen–Macaulay. In the above example, if we assume that E is a K3-surface, then Hq(X,OX(−L)) = 0 for q < 3 and X is Cohen–Macaulay. For the details, see Section 7.2, especially, Lemma 7.2.7.
5.7. VANISHING THEOREMS OF REID–FUKUDA TYPE 183
(i) Every associated prime of Rqf∗OY(L) is the generic point of the f-image of some stratum of (Y,∆).
(ii) We have
Rpπ∗Rqf∗OY(L) = 0 for every p >0.
Proof. Note that L−(KY + ∆) isf-semi-ample. Therefore, (i) is a special case of Theorem 5.6.3 (i).
From now on, we will prove (ii). We note that we may assume that V is affine without loss of generality.
Step 1. We assume that every stratum of (Y,∆) dominates some irreducible component ofX. By taking the Stein factorization, we may assume that f has connected fibers. Then we may further assume that X is irreducible and every stratum of (Y,∆) dominates X. By Chow’s lemma, there exists a projective birational morphism µ : X0 → X such that π0 : X0 → V is projective. By taking a proper birational morphism ϕ : Y0 → Y that is an isomorphism over the generic point of any stratum of (Y,∆), we have the following commutative diagram.
Y0 ϕ //
g
Y
f
X0
πBB0BBBBB B µ // X
π
V
Then, by Theorem5.2.17(see also [BVP, Theorem 1.4]), we can write KY0+ ∆0 =ϕ∗(KY + ∆) +E,
where
(1) (Y0,∆0) is an embedded simple normal crossing pair such that
∆0 is a boundary R-divisor.
(2) E is an effectiveϕ-exceptional Cartier divisor.
(3) Every stratum of (Y0,∆0) dominates X0.
We note that every stratum of (Y,∆) dominates X. Therefore, ϕ∗L+E ∼R KY0 + ∆0+ϕ∗f∗H.
We note that
ϕ∗OY0(ϕ∗L+E)' OY(L) and
Riϕ∗OY0(ϕ∗L+E) = 0
184 5. INJECTIVITY AND VANISHING THEOREMS
for everyi >0 by Theorem5.6.3 (i). Thus, by replacingY and Lwith Y0 and ϕ∗L+E, we may assume that ϕ :Y0 →Y is the identity, that is, we have
Y
g
Y
f
X0 µ //
πBB0BBBBB
B X
π
V.
We put F = Rqg∗OY(L). Since µ∗H is nef and big over V and π0 : X0 →V is projective, we can writeµ∗H =E+A, whereAis aπ0-ample R-divisor on X0 and E is an effective R-Cartier R-divisor by Kodaira (see Lemma 2.1.18). By the same arguments as above, we take some blow-ups and may further assume that (Y,∆ +g∗E) is an embedded simple normal crossing pair. If k is a sufficiently large positive integer, then
b{∆}+ 1
kg∗Ec= 0, µ∗H = 1
kE+ 1
kA+k−1 k µ∗H,
and 1
kA+k−1 k µ∗H
isπ0-ample. Thus,F isµ∗-acyclic and (π◦µ)∗ =π∗0-acyclic by Theorem 5.6.3 (ii). We note that
L−(
KY + ∆ + 1 kg∗E
)∼R g∗ (1
kA+ k−1 k µ∗H
) .
So, we have Rqf∗OY(L) ' µ∗F and Rqf∗OY(L) is π∗-acyclic. Thus, we finish the proof when every stratum of (Y,∆) dominates some irre- ducible component of X.
Step 2. We treat the general case by induction on dimf(Y). By taking some embedded log transformations (see Lemma 5.7.4 below), we can decomposeY =Y0∪Y00 as follows:Y0 is the union of all strata of (Y,∆) that are not mapped to irreducible components of X and Y00 =Y −Y0. We put
KY00+ ∆Y00 = (KY + ∆)|Y00−Y0|Y00.
Thenf : (Y00,∆Y00)→X andL00 =L|Y00−Y0|Y00satisfy the assumption in Step1. We consider the following short exact sequence
0→ OY00(L00)→ OY(L)→ OY0(L)→0.
5.7. VANISHING THEOREMS OF REID–FUKUDA TYPE 185
By taking Rqf∗, we have short exact sequence
0→Rqf∗OY00(L00)→Rqf∗OY(L)→Rqf∗OY0(L)→0 for every q. This is because the connecting homomorphisms
Rqf∗OY0(L)→Rq+1f∗OY00(L00)
are zero maps for everyq by (i). Since (ii) holds for the first and third members by Step1and by induction on the dimension, respectively, it also holds for Rqf∗OY(L).
So, we finish the proof.
We have already used Lemma 5.7.4 in the proof of Theorem 5.7.3.
Lemma 5.7.4 is easy to check. So we omit the proof.
Lemma 5.7.4 (cf. [Am1, p.218 embedded log transformation]). Let (X,∆) be an embedded simple normal crossing pair and let M be an ambient space of (X,∆). Let C be a smooth stratum of (X,∆). Let σ :N →M be the blow-up along C. LetY denote the reduced structure of the total transform of X in N. We put
KY + ∆Y =f∗(KX + ∆),
where f =σ|Y. Then we have the following properties.
(i) (Y,∆Y) is an embedded simple normal crossing pair with an ambient space N.
(ii) f∗OY ' OX and Rif∗OY = 0 for every i >0.
(iii) The strata of (X,∆) are exactly the images of the strata of (Y,∆Y).
(iv) σ−1(C)is a maximal (with respect to the inclusion)stratum of (Y,∆Y).
(v) If ∆ is a boundary R-divisor on X, then ∆Y is a boundary R-divisor on Y.
Remark 5.7.5. We need the notion of embedded simple normal crossing pairs to prove Theorem 5.7.3 even when Y is smooth. It is a key point of the proof of Theorem 5.7.3. Note that we do not need the assumption that Y is embedded in Step 1in the proof of Theorem 5.7.3.
As a corollary of Theorem 5.7.3, we can prove the following van- ishing theorem. It is the culmination of the works of several au- thors: Kawamata, Viehweg, Nadel, Reid, Fukuda, Ambro, Fujino, and others. To the author’s best knowledge, we can not find it in the literature except [F17]. Note that Theorem 5.7.6 is a complete gener- alization of [KMM, Theorem 1-2-5].
186 5. INJECTIVITY AND VANISHING THEOREMS
Theorem 5.7.6 (see [F17, Theorem 2.48]). Let (X,∆) be a log canonical pair such that ∆ is a boundary R-divisor and let L be a Q- Cartier Weil divisor on X. Assume that L−(KX + ∆) is nef and log big over V with respect to (X,∆), where π : X → V is a proper morphism. Then Rqπ∗OX(L) = 0 for every q >0.
Proof. Let f :Y →X be a log resolution of (X,∆) such that KY =f∗(KX + ∆) +∑
i
aiEi with ai ≥ −1 for every i. We may assume that ∑
iEi ∪Suppf∗L is a simple normal crossing divisor on Y. We put
E =∑
i
aiEi and
F = ∑
aj=−1
(1−bj)Ej,
where bj = multEj{f∗L}. We note that A=L−(KX + ∆) is nef and log big over V with respect (X,∆) by assumption. So, we have
f∗A=f∗L−f∗(KX + ∆)
=df∗L+E+Fe −(KY +F +{−(f∗L+E+F)}).
We can easily check that
f∗OY(df∗L+E+Fe)' OX(L)
and thatF +{−(f∗L+E+F)}has a simple normal crossing support and is a boundaryR-divisor onY. By the above definition ofF,Ais nef and log big overV with respect tof : (Y, F+{−(f∗L+E+F)})→X.
Therefore, by Theorem 5.7.3 (ii), we obtain that OX(L) is π∗-acyclic.
Thus, we have Rqπ∗OX(L) = 0 for every q >0.
As a special case, we have the Kawamata–Viehweg vanishing theo- rem for klt pairs.
Corollary5.7.7 (Kawamata–Viehweg vanishing theorem, see [KMM, Remark 1-2-6]). Let (X,∆) be a klt pair and let Lbe a Q-Cartier Weil divisor on X. Assume thatL−(KX+ ∆) is nef and big overV, where π : X → V is a proper morphism. Then Rqπ∗OX(L) = 0 for every q >0.
We add one example.
5.7. VANISHING THEOREMS OF REID–FUKUDA TYPE 187
Example 5.7.8. Let Y be a projective surface which has the fol- lowing properties: (i) there exists a projective birational morphism f : X → Y from a smooth projective surface X, and (ii) the ex- ceptional locus E of f is an elliptic curve with KX +E =f∗KY. For example, Y is a cone over a smooth plane cubic curve andf :X →Y is the blow-up at the vertex of Y. We note that (X, E) is a plt pair.
LetH be an ample Cartier divisor onY. We consider a Cartier divisor L=f∗H+KX +E onX. ThenL−(KX+E) is nef and big, but not log big with respect to (X, E). By the short exact sequence
0→ OX(f∗H+KX)→ OX(f∗H+KX +E)→ OE(KE)→0, we obtain
R1f∗OX(f∗H+KX +E)'H1(E,OE(KE))'C(P), where P =f(E). By the Leray spectral sequence, we have
0→H1(Y, f∗OX(KX +E)⊗ OY(H))→H1(X,OX(L))
→H0(Y,C(P))→H2(Y, f∗OX(KX +E)⊗ OY(H))
→ · · · .
IfH is sufficiently ample, thenH1(X,OX(L))'H0(Y,C(P))'C(P).
In particular, H1(X,OX(L))6= 0.
Remark5.7.9. In Example5.7.8, there exists an effectiveQ-divisor B onX such thatL−k1B is ample for everyk >0 by Kodaira’s lemma (see Lemma2.1.18). Since L·E = 0, we haveB·E <0. In particular,
(X, E+ 1 kB)
is not log canonical for any k > 0. This is the main reason why H1(X,OX(L)) 6= 0. If (X, E + 1kB) were log canonical, then the am- pleness of L−(KX +E + 1kB) would imply H1(X,OX(L)) = 0 by Theorem 5.6.4.
If Y is quasi-projective in Theorem 5.7.3, we do not need the as- sumption that the pair (Y,∆) is embedded.
Theorem 5.7.10 ([FF, Theorem 6.3]). Let f : (Y,∆) → X be a proper morphism from a quasi-projective simple normal crossing pair (Y,∆) to a scheme X such that ∆is a boundary R-divisor. Let L be a Cartier divisor on Y and letπ :X →V be a proper morphism between schemes. Assume that
f∗H ∼RL−(KY + ∆),
188 5. INJECTIVITY AND VANISHING THEOREMS
where H is nef and log big over V with respect to f : (Y,∆) → X.
Let q be an arbitrary non-negative integer. Then we have the following properties.
(i) Every associated prime of Rqf∗OY(L) is the generic point of the f-image of some stratum of (Y,∆).
(ii) We have
Rpπ∗Rqf∗OY(L) = 0 for every p >0.
We can easily reduce Theorem 5.7.10 to Theorem 5.7.3. For the proof of Theorem 5.7.3, see the proof of [FF, Theorem 6.3]. We used Theorem 5.7.10 for the proof of the main theorem of [FF].