Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

TL;DR

This paper extends vanishing theorems for nef vector bundles on projective manifolds, establishing new conditions under which certain cohomology groups vanish, especially for indecomposable nef rank 2 bundles.

Contribution

It proves a general vanishing result for nef vector bundles depending on specific cohomology, and shows vanishing for indecomposable nef rank 2 bundles on certain manifolds.

Findings

Vanishing of $H^{n,n}(X, E)$ for specific nef vector bundles.

Vanishing holds for indecomposable nef rank 2 bundles on certain projective manifolds.

Vanishing also applies to nef and big rank 2 bundles on varieties with trivial canonical bundle.

Abstract

In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2 vector bundles $E$ on some specific type of $n$ dimensional projective manifold $X.$ The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.

Equations

$$H^{p,q}(X,L)=0\text{mbox}forp+q-n>0.$$

$$H^{p,q}(X,E)=0\text{mbox}forp+q-n>r-1.$$

$$H^{n,q}(X,E)=0\text{mbox}forq>r-1?$$

$$H^{n,q}(X,S^k(E)\otimes \det (E)\otimes L)=0,\text{mbox}forq>0.$$

$$s_n(E)=p_{\ast }(c_1({\mathcal{O}}_{\mathbb{P}(E)}(1){)}^{n+r-1}>0,$$

$$\sigma :E\to I_Z\to 0.$$

$$\tau :E{|}_C\to {\mathcal{O}}_C(-D)\to 0.$$

$$0\to F\to E\to Q\to 0$$

$$\sigma :{\mathcal{O}}_X\to E^{\vee }$$

$$0\to {\mathcal{O}}_X\to E^{\vee }\to F^{\vee }\to 0,$$

$$0\to F\to E\to {\mathcal{O}}_X\to 0.$$

$$0\to H^0(X,{\mathcal{O}}_X)\to H^0(X,E^{\vee })\to H^0(X,F^{\vee })\to \cdots .$$

$$k(E)-1\le k(F)\le r-1,$$

$$0\to {\mathcal{O}}_X\to E^{\vee }\to \det (E{)}^{\vee }\to 0.$$

$$0\to F\to E\to {\mathcal{O}}_X\to 0.$$

$$0\to \det (E)\to E\to {\mathcal{O}}_X\to 0.$$

$$H^2(X,{\Omega }_X^2\otimes L)\simeq H^{2,2}({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}\oplus {\mathcal{O}}_{{\mathbb{P}}^2}(1))$$

Full text

1. Introduction

Let $X$ be a smooth projective complex manifold of dimension $n.$ For any coherent sheaf $E$ on $X,$ we denote $H^{p,q}(X,E)$ the cohomology group $H^{q}(X,E\otimes\Omega^{p}_{X}),$ where $\Omega^{p}_{X}$ is the sheaf of holomorphic differential forms of degree $p$ on $X.$

Akizuki-Kodaira-Nakano famous vanishing theorem says:

If $L$ is an ample line bundle on a projective manifold $X$ of dimension $n,$ then

[TABLE]

The particular case $p=n$ is the Kodaira vanishing theorem. The Kodaira vanishing theorem was extended to nef and big line bundle on a smooth surface by Ramanujam [7] and for higher dimension by Kawamata [3] and Viehweg [9].

Ramanujam has given in [7] an example showing that in general, one does not expect Akizuki-Kodaira-Nakano type vanishing result for nef and big line bundle.

Le Potier [6] generalized the Akizuki-Kodaira-Nakano type vanishing theorem to the case of ample vector bundle as follows:

If $E$ is an ample vector bundle of rank $r$ on a projective manifold $X$ of dimension $n,$ then

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The vanishing results of Ramanujam-Kawamata-Viehweg and Le Potier naturally led to ask the following question:

Let $E$ be a nef and big vector bundle of rank $r$ on a projective manifold $X$ of dimension $n.$ Is

[TABLE]

The example given by Ramanujam in [7] shows that one can not expect in general ”Akizuki-Kodaira-Nakano“ type of vanishing for nef and big line bundle. The same example can also be used to show that the question (2) has a negative answer (see Example (4.2)).

Regarding the question (2) for a nef and big rank two vector bundle $E$ on a smooth surface $X$ the only group which one hope to vanish is the group $H^{2,2}(X,E).$ In trying to investigate this problem we obtained the following:

Theorem 1.1**.**

Let $E$ be a nef vector bundle of rank $r$ on a projective manifold $X$ of dimension $n.$ Set $k(E):={\rm dim}H^{n,n}(X,E).$ Then $k(E)\leq r$ and $E$ admits a trivial bundle of rank $k(E)$ as quotient. In particular, $k(E)=r$ if and only if $E$ is isomorphic to trivial vector bundle of rank $r.$

Corollary 1.2**.**

Let $E$ be an indecomposable nef vector bundle of rank $r$ on a projective manifold $X$ of dimension $n.$ Assume that $c_{r}(E)\neq 0.$ Then $H^{n,n}(X,E)=0.$

For the case of rank 2 vector bundles we have the following:

Theorem 1.3**.**

Let $E$ be an indecomposable nef vector bundle of rank $2$ on a projective manifold $X$ of dimension $n.$ If $H^{1}(X,\det(E))=0,$ then $H^{n,n}(X,E)=0.$

As a consequence we obtain:

Corollary 1.4**.**

*Let $X$ be a Grassmannian of dimension $n\geq 2$ or a complete intersection of dimension $n\geq 3$ in a Grassmannian.

If $E$ is an indecomposable nef vector bundle of rank $2$ on $X,$ then $H^{n,n}(X,E)=0.$*

Corollary 1.5**.**

Let $X$ be a projective manifold of dimension $n\geq 2$ with $K_{X}={\mathcal{O}}_{X}.$ If $E$ is an indecomposable nef and big vector bundle of rank $2$ on $X,$ then $H^{n,n}(X,E)=0.$

We recall a vanishing theorem of Schneider [8] related to nef and big vector bundle, in a slightly different version from the original one, but follows from the proof given there.

Theorem 1.6**.**

Let $E$ (resp. $L$) be a vector bundle (resp. line bundle) on a projective manifold $X$ of dimension $n.$ If $E\otimes L$ is nef and big then

[TABLE]

2. Notations and Definitions

Throughout we work over the field of complex numbers.

For a vector bundle $E$ on a projective manifold $X,$ we will denote by $E^{\vee}$ the dual of $E,$ $c_{i}(E)\in H^{2i}(X,{\mathbb{Z}})$ is the $i$-th chern class of $E,$ ${\mathbb{P}}(E)$ is the projective bundle whose fiber over a point $x\in X$ is the projective space of $1-$dimensional quotients of the vector space $E_{x},$ and ${\mathcal{O}}_{{\mathbb{P}}(E)}(1)$ the universal quotient line bundle on ${\mathbb{P}}(E).$

Definition 2.1**.**

Let $X$ be a projective manifold of dimension $n.$ A line bundle $L$ on $X$ is called nef, if for every irreducible curve $C$ in $X$ degree of $L|_{C}$ is non negative. A nef line bundle $L$ is called big if $c_{1}(L)^{n}>0.$

A vector bundle $E$ on $X$ is said to be nef if the line bundle ${\mathcal{O}}_{{\mathbb{P}}(E)}(1)$ on ${\mathbb{P}}(E)$ is nef.

A nef vector bundle $E$ is said to be big if ${\mathcal{O}}_{{\mathbb{P}}(E)}(1)$ on ${\mathbb{P}}(E)$ is big or equivalently

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where $s_{n}(E)$ is the $n$-th Segre class of $E$ and $p:{\mathbb{P}}(E)\to X$ be the natural projection.

3. Proof of the results

First we recall some results which we need.

Proposition 3.1**.**

[5*, Proposition 6.1.18 (i)]** A vector bundle $E$ on $X$ is nef if and only if the following condition is satisfied:

Given any morphism $f:C\to X$ finite onto its image from an irreducible smooth curve $C$ to $X,$ and given any quotient line bundle $L$ of $f^{*}(E),$ then one has ${\rm deg}L\geq 0.$*

Lemma 3.2**.**

[1, Proposition 1.16]** Let $E$ be a nef vector bundle on a projective manifold $X$ of dimension $n.$ If $\sigma$ is a non-zero section of $E^{\vee}$ then $\sigma$ is nowhere vanishing on $X.$

Proof: The proof given in [1] uses analytic methods. Here we give an algebraic proof. First we prove the lemma when $X$ is a curve. In this case if $\sigma$ vanishes at some points, we get a positive degree line sub bundle of $E^{\vee}.$ By dualizing we see that $E$ has a line bundle quotient of negative degree. This is a contradiction to the Proposition(3.1). Thus $\sigma$ is nowhere vanishing on $X.$

For the general case, assume $\sigma$ vanishes at some points and dimension of $X$ is greater than one. Let $Z$ be the subscheme of $X$ defined by the vanishing of $\sigma$ and $I_{Z}$ denotes its sheaf of ideals. The section $\sigma$ induces surjection

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Let $C$ be a smooth curve in $X$ with the property $D=C\cap Z$ is a non-empty proper closed subscheme of $C.$ Then by restricting the surjective map $\sigma$ to $C$ and going modulo torsion we get a surjection:

[TABLE]

Since $C$ is a smooth curve ${\mathcal{O}}_{C}(-D)$ is a line bundle of negative degree, which is a contradiction to the fact that $E$ is nef. Hence we must have $Z=\emptyset.$ $\hfill{\Box}$

Lemma 3.3**.**

[10, see, Proposition 4.8.]** If $E$ is a nef and big vector bundle on a Kähler manifold $X,$ then the line bundle $\det(E)$ on $X$ is big.

The Dominance theorem [theorem 3.3] in [4] ensures that $\det(E)$ is nef.

We also need to recall the proposition [Prop. 1.15 (iii)] in [1]. We will state it in a different version, which follows immediatly from the proof given there.

Lemma 3.4**.**

Let

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*be an exact sequence of holomorphic vector bundles and $rank(E)=r,rank(F)=f.$

If $\wedge^{r-f+1}E\otimes\det Q^{-1}$ is nef (resp. ample), then $F$ is nef (resp. ample).*

Proof of Theorem(1.1):

The proof is by induction on the $rank(E)=r.$ If $r=1$ and $k(E)=0$ then there is nothing to prove.

If $k(E)>0,$ then by Lemma(3.2) there is a non zero homomorphism

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which is nowhere vanishing. This implies that $E$ is a trivial bundle of rank one. Since $k({\mathcal{O}}_{X})=1,$ the Theorem follows in this case.

Let $r>1.$ We assume our Theorem holds for all nef vector bundles of rank less than or equal $r-1.$ Again, if $k(E)=0$ there is nothing to prove. So we assume $k(E)>0.$ Then applying Lemma(3.2) we get an exact sequence

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where $F^{\vee}$ is a dual of vector bundle $F$ of rank $r-1.$ Dualizing (5) we get an exact sequence

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By Lemma(3.4) $F$ is a nef vector bundle. Now since $rank(F)=r-1,$ we have by induction assumption $k(F)\leq r-1$ and $F$ admits a trivial quotient of rank $k(F).$ This implies by duality $F^{\vee}$ admits trivial subbundle of rank $k(F).$ Now from the long cohomology exact sequence

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assosiated to the exact sequence (5), we deduce

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and the image of global sections of $E^{\vee}$ generate a trivial subbundle $V$ of rank $k(E)-1$ in $F^{\vee}.$ Taking the inverse image of this $V$ we see that $E^{\vee}$ admits a subbundle $S^{\vee}$ of rank $k(E).$ Note that $S^{\vee}$ is an extension of ${{\mathcal{O}}_{X}}^{k(E)-1}$ by ${\mathcal{O}}_{X}.$ The dual $S$ of $S^{\vee}$ is nef, since it is an extension of trivial bundle of rank $k(E)-1$ by a trivial bundle of rank $1.$ If $k(E)<r$ then it follows by induction $S$ is trivial. This proves the result.

If $k(E)=r$ then again by induction $F={{\mathcal{O}}_{X}}^{r-1}$ and all the sections of $F^{\vee}$ lifts to sections of $E^{\vee},$ hence $E^{\vee}$ and $E$ are isomorphic to ${{\mathcal{O}}_{X}}^{r}.$ $\hfill{\Box}$

Proof of Theorem(1.3):

Assume $H^{n,n}(X,E)\neq 0,$ then we get by Serre duality $H^{0,0}(X,E^{\vee})\neq 0.$ Let $\sigma$ be a non-zero section of $E^{\vee}.$ Since $E$ is nef by Lemma(3.2) the section $\sigma$ is nowhere vanishing, and gives an exact sequence

[TABLE]

This extension gives a class in the cohomology group $H^{1}(X,{\rm det}(E)).$ But by our assumption this group is zero and hence the extension splits. Thus $E^{\vee}$ splits and hence $E$ splits too, this is a contadiction. $\hfill{\Box}$

Proof of Corollary (1.2):

If $H^{n,n}(X,E)\neq 0,$ then by Theorem(1.1) we get an exact sequence

[TABLE]

This implies $c_{r}(E)=c_{r}(F)=0,$ this is a contradiction. $\hfill{\Box}$

Proof of Corollary (1.4):

If $X$ is a Grassmannian of dimension $\geq 2$ or a complete intersection of dimension $\geq 3$ in a Grassmannian, then for any line bundle $L$ $H^{1}(X,L)=0.$ Hence if $E$ is an indecomposable vector bundle of rank two on $X,$ then the hypothesis of Theorem(1.1) holds for $E.$ $\hfill{\Box}$

Proof of Corollary (1.5)

Assume $H^{n,n}(X,E)\neq 0.$ Since $E$ is nef and big, ${\rm det}(E)$ is nef and big by the Lemma(3.3). Hence we have an exact sequence:

[TABLE]

But $K_{X}$ is trivial implies $H^{1}(X,\det(E))=0$ by Kawamata-Ramanujam-Viehweg vanishing theorem. Hence that the exact sequence (10) splits and hence $E$ is decompsable, which is a contradiction. $\hfill{\Box}$

Remark 3.5**.**

Corollary 1.5 applies for example to complex algebraic torus, $K3$ surfaces and Calabi-Yau manifolds.

$\hfill{\Box}$

4. Counter examples of Ramanujam

Example 4.1**.**

The following example is due to Ramanujam [7].

Denote ${\mathbb{P}}^{3}$ blown up at a point by $X$ and $\pi:X\to{\mathbb{P}}^{3}$ be the natural morphism and $L=\pi^{*}({\mathcal{O}}_{{\mathbb{P}}^{3}}(1)).$ Clearly the line bundle $L$ is nef and big and hence $H^{1}(X,\Omega^{1}_{X}\otimes L^{-1})\neq 0.$

Example 4.2**.**

Note that the variety $X$ in the Example(4.1) can be identified with ${\mathbb{P}}(E)$ in such a way that $L\simeq{\mathcal{O}}_{{\mathbb{P}}(E)}(1),$ where $E={\mathcal{O}}_{{\mathbb{P}}^{2}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{2}}(1).$ Clearly the bundle $E$ on ${\mathbb{P}}^{2}$ is nef and big and $H^{2,2}({\mathbb{P}}^{2},E)\neq 0.$ This shows that one can not expect Le Potier type vanishing result for nef and big vector bundle even for $p=n.$

More general example: if $Y$ is a projective manifold of dimention $n$ and $H$ is an ample line bundle on $Y,$ then the vector bundle $E={\mathcal{O}}_{Y}\oplus H$ is nef and big vector bundle but $H^{n,n}(Y,E)\neq 0.$

Remark 4.3**.**

The non vanishing of $H^{1,1}(X,L^{-1})$ of Example(4.1) can be deduced from the non vanishing of the group $H^{2,2}({\mathbb{P}}^{2},E)$ in Example(4.2). Indeed:

[TABLE]

by Le Potier isomorphism [6, Lemma 8].

Acknowledgements: Last named author would like to thank University of Lille1 and University of Artois at Lens and would also like to thank IRSES-Moduli program for their hospitality and the support.