Classical vanishing theorems and some applications
3.1. Kodaira vanishing theorem
In this section, we give a proof of Kodaira’s vanishing theorem for projective varieties based on the theory of mixed Hodge structures.
Let us start with the following easy lemma (see [Am2] and [F36]).
We will prove generalizations of Lemma 3.1.1in Chapter 5 (see Theo- rem 5.4.1 and Theorem 5.4.2).
Lemma 3.1.1. Let X be a smooth projective variety and let ∆ be a reduced simple normal crossing divisor on X. Let D be an effective Cartier divisor on X such that SuppD⊂Supp ∆. Then the map
Hi(X,OX(−D−∆))→Hi(X,OX(−∆)),
which is induced by the natural inclusion OX(−D)⊂ OX, is surjective for every i. Equivalently, by Serre duality,
Hi(X,OX(KX + ∆))→Hi(X,OX(KX + ∆ +D)),
which is induced by the natural inclusionOX ⊂ OX(D), is injective for every i.
Proof. In this proof, we use the classical topology and Serre’s GAGA. We consider the following Hodge to de Rham type spectral sequence:
E1p,q=Hq(X,ΩpX(log ∆)⊗ OX(−∆)) ⇒Hcp+q(X\∆,C).
It is well known that it degenerates atE1 by the theory of mixed Hodge structures (see Remark 3.1.4 and Remark 3.1.5 below). This implies that the natural inclusion
ι!CX\∆ ⊂ OX(−∆), where ι:X\∆→X, induces the surjections
Hci(X\∆,C) = Hi(X, ι!CX\∆)−→αi Hi(X,OX(−∆)) for all i. Note that
ι!CX\∆ ⊂ OX(−D−∆)⊂ OX(−∆).
3.1. KODAIRA VANISHING THEOREM 43
Therefore,αi factors as
αi :Hi(X, ι!CX\∆)→Hi(X,OX(−D−∆))→Hi(X,OX(−∆)) for every i. This implies that
Hi(X,OX(−D−∆)) →Hi(X,OX(−∆))
is surjective for everyi.
As an obvious application, we have:
Corollary 3.1.2. Let X be a smooth projective variety and let ∆ be a reduced simple normal crossing divisor onX. Assume that there is an ample Cartier divisor D on X such that SuppD⊂Supp ∆. Then
Hi(X,OX(−∆)) = 0 for every i <dimX, equivalently,
Hi(X,OX(KX + ∆)) = 0 for every i >0.
Proof. By Serre duality and Serre’s vanishing theorem, we have Hi(X,OX(−aD−∆)) = 0
for a sufficiently large and positive integer a and for every i < dimX.
By Lemma 3.1.1, we obtain that Hi(X,OX(−∆)) = 0 for every i <
dimX. By Serre duality, we see that Hi(X,OX(KX + ∆)) = 0 for
everyi >0.
For an application of Corollary 3.1.2, we will prove Ambro’s van- ishing theorem: Theorem 3.11.1
By using a standard covering trick, we can recover Kodaira’s van- ishing theorem for projective varieties from Lemma3.1.1. We will treat the Kodaira vanishing theorem for compact complex manifold in The- orem 3.7.4.
Theorem 3.1.3 (Kodaira vanishing theorem). Let X be a smooth projective variety and let H be an ample Cartier divisor on X. Then
Hi(X,OX(KX +H)) = 0 for every i >0, equivalently, by Serre duality,
Hi(X,OX(−H)) = 0 for every i <dimX.
44 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS
Proof. We take a smooth divisor B ∈ |mH| for some positive integer m. Let f :V → X be the m-fold cyclic cover ramifying along B. Then
f∗OV =
m−1
⊕
k=0
OX(−kH).
Therefore, it is sufficient to prove thatHi(V,OV(−f∗H)) = 0 for every i <dimV = dimX. This is because OX(−H) is a direct summand of f∗OV(−f∗H). Note that OV(f∗H) has a section, which is the reduced preimage of B, by construction. By iterating this process, we obtain a tower of cyclic covers:
Vn→ · · · →V0 →X.
By suitable choice of the ramification divisors, we may assume that the pull-back of H on Vn has no base points. Therefore, we reduced Theorem 3.1.3 to the case when the linear system |H| has no base points. Let ∆∈ |H|be a reduced smooth divisor on X. Then
Hi(X,OX(−l∆))→Hi(X,OX(−∆))
is surjective for every i and every l ≥ 1 by Lemma 3.1.1. Therefore, Hi(X,OX(−∆)) = 0 for i < dimX by Serre duality and Serre’s van-
ishing theorem.
For the reader’s convenience, we give remarks on theE1-degeneration of the Hodge to de Rham type spectral sequence in the proof of Lemma 3.1.1.
Remark 3.1.4. For the proof of Theorem 3.1.3, it is sufficient to assume that ∆ is smooth in Lemma3.1.1. When ∆ is smooth, we can easily construct the mixed Hodge complex of sheaves on X giving a natural mixed Hodge structure on Hc•(X\∆,Z). From now on, we use the notation and the framework in [PS, §3.3 and §3.4]. Let Hdg•(X) (resp.Hdg•(∆)) be a Hodge complex of sheaves onX (resp. ∆) giving a natural pure Hodge structure on H•(X,Z) (resp. H•(∆,Z)). Then the mixed cone
Hdg•(X,∆) := Cone(Hdg•(X)→i∗Hdg•(∆))[−1],
wherei: ∆ →X is the natural inclusion, gives a natural mixed Hodge structure on Hc•(X \∆,Z). For the details, see [PS, Example 3.24].
We note that
0→ΩpX(log ∆)⊗ OX(−∆)→ΩpX →Ωp∆→0 is exact for every p. Therefore, we can easily see that
E1p,q =Hq(X,ΩpX(log ∆)⊗ OX(−∆))⇒Hcp+q(X\∆,C)
3.1. KODAIRA VANISHING THEOREM 45
degenerates at E1 by the theory of mixed Hodge structures.
Remark 3.1.5. We putd= dimX. In the proof of Lemma 3.1.1, Hq(X,ΩpX(log ∆)⊗ OX(−∆))
is dual to
Hd−q(X,ΩdX−p(log ∆)) by Serre duality. By Poincar´e duality,
Hcp+q(X\∆,C) is dual to
H2d−(p+q)(X\∆,C).
By Deligne (see [Del]), it is well known that
E1p,q =Hq(X,ΩpX(log ∆))⇒Hp+q(X\∆,C) degenerates at E1. This implies that
dimHk(X\∆,C) = ∑
p+q=k
Hq(X,ΩpX(log ∆)) for every k. Therefore, by the above observation, we have
dimHck(X\∆,C) = ∑
p+q=k
Hq(X,ΩpX(log ∆)⊗ OX(−∆)) for every k. Thus the Hodge to de Rham type spectral sequence
E1p,q=Hq(X\∆,ΩpX(log ∆)⊗ OX(−∆))⇒Hcp+q(X\∆,C) degenerates at E1.
Anyway, we will completely generalize Lemma 3.1.1 in Chapter5.
Remark 3.1.6. It is well known that the Kodaira vanishing the- orem for projective varieties follows from the theory of pure Hodge structures. For the details, see, for example, [KoMo, 2.4 The Kodaira vanishing theorem].
By using a covering trick, we have a slight but very important gen- eralization of Kodaira’s vanishing theorem. Theorem 3.1.7 is usually called Kawamata–Viehweg vanishing theorem.
Theorem 3.1.7 (Kawamata–Viehweg vanishing theorem). Let X be a smooth projective variety and let D be an ample Q-divisor on X such that Supp{D} is a simple normal crossing divisor on X. Then
Hi(X,OX(KX +dDe)) = 0 for every i >0.
46 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS
Remark 3.1.8. In this book, there are various formulations of the Kawamata–Viehweg vanishing theorem in order to make it useful for many applications. See, for example, Theorem 3.1.7, Theorem 3.2.1, Theorem 3.2.8, Theorem 3.2.9, Theorem 3.3.1, Theorem 3.3.2, Theo- rem 3.3.4, Theorem 3.3.7, Theorem 4.1.1, Corollary 5.7.7, and so on.
Before we start the proof of Theorem 3.1.7, let us recall an easy lemma.
Lemma 3.1.9. Let f : Y → X be a finite morphism between n- dimensional normal irreducible varieties. Then the natural inclusion OX →f∗OY is a split injection.
Proof. It is easy to see that n1TraceY /X splits the natural inclusion OX →f∗OY, where TraceY /X is the trace map.
Proof of Theorem 3.1.7. We put L = dDe and ∆ = L−D.
We put ∆ =∑
jaj∆j such that ∆j is a reduced and smooth (possibly disconnected) divisor onX for everyj and thataj is a positive rational number for every j. It is sufficient to prove Hi(X,OX(−L)) = 0 for every i <dimX by Serre duality. We use induction on the number of divisors ∆j. If ∆ = 0, then Theorem 3.1.7 is nothing but Kodaira’s vanishing theorem: Theorem 3.1.3. We put a1 = b/m such that b is an integer and m is a positive integer. We can construct a finite surjective morphism p1 : X1 → X such that X1 is smooth and that p∗1∆1 ∼mBfor some Cartier divisorB onX1 (see Lemma3.1.10). We may further assume that everyp∗1∆j is smooth and∑
jp∗1∆j is a simple normal crossing divisor onX1(see Lemma3.1.10). It is easy to see that Hi(X,OX(−L)) is a direct summand ofHi(X1, p∗1OX(−L)) by Lemma 3.1.9. By construction,p∗1∆1is a member of|mB|. Letp2 :X2 →X1be the corresponding cyclic cover. Then X2 is smooth, p∗2p∗1∆j is smooth for every j, and ∑
jp∗2p∗1∆j is a simple normal crossing divisor on X2. Since
p2∗OX2 =
m⊕−1 k=0
OX1(−kB), we obtain
Hi(X2, p∗2(p∗1OX(−L)⊗ OX1(bB)))
=
m−1
⊕
k=0
Hi(X1, p∗1OX(−L)⊗ OX1((b−k)B)).
3.1. KODAIRA VANISHING THEOREM 47
The k =b case shows that Hi(X1, p∗1OX(−L)) is a direct summand of Hi(X2, p∗2(p∗1OX(−L)⊗ OX1(bB))). Note that
p∗2(p∗1L−bB)∼p∗2p∗1D+∑
j>1
aip∗2p∗1∆j.
Therefore, by induction on the number of divisors ∆j, we obtain Hi(X2, p∗2(p∗1OX(−L)⊗ OX1(bB))) = 0
for every i < dimX2 = dimX. Thus we obtain the desired vanishing
theorem.
The following covering trick is due to Bloch–Gieseker (see [BlGi]) and is well known (see, for example, [KoMo, Proposition 2.67]).
Lemma 3.1.10. Let X be a projective variety, let D be a Cartier divisor on X, and let m be a positive integer. Then there is a normal varietyY, a finite surjective morphismf :Y →X, and a Cartier divisor D0 on Y such that f∗D ∼mD0.
Furthermore, ifX is smooth and ∑
jFj is a simple normal crossing divisor on X, then we can choose Y to be smooth such that f∗Fj is smooth for every j and∑
jf∗Fj is a simple normal crossing divisor on Y.
Proof. Let π:Pn →Pn be the morphism given by (x0 :x1 :· · · , xn)7→(xm0 :xm1 :· · ·:xmn).
Then π∗OPn(1)' OPn(m).
Let L be a very ample Cartier divisor on X. Then there is a mor- phism h : X → Pn such that OX(L) ' h∗OPn(1). Let Y be the normalization of the fiber product X×PnPn sitting in the diagram:
Y hY //
f
Pn
π
X h
//Pn.
If Dis very ample, then we put L=D. In this case, f∗OX(D)'h∗Y(π∗OPn(1))'h∗YOPn(m).
If X is smooth, then we consider π0 : Pn → Pn which is the com- position of π with a general automorphism of the target spacePn. By Kleiman’s Bertini type theorem (see, for example, [Har4, Chapter III Theorem 10.8]), we can makeY smooth and ∑
jf∗Fj a simple normal crossing divisor on Y.
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.