## Vanishing theorems for ample vector bundles.(English)Zbl 0906.14011

The main result of the article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety. Namely, let $$E$$ be a holomorphic vector bundle of rank $$e$$, and $$L$$ a line bundle on a smooth projective complex variety $$X$$ of dimension $$n$$. Suppose that $$E$$ is ample and $$L$$ nef, or that $$E$$ is nef and $$L$$ ample. Then, for any sequences of integers $$k_1, \dots, k_l$$ and $$j_1, \dots, j_m$$, $H^{p,q} \Bigl(X,S^{k_1} E\otimes \dots S^{k_\ell} E\otimes \wedge^{j_1} E\otimes \cdots\otimes \wedge^{j_m} E\otimes (\text{det} E)^{\ell+n-p} \otimes L\Bigr) =0$ as soon as $$p+q>n+\sum^m_{s=1}(\ell-j_s)$$.
In particular this extends a classical theorem of Griffiths and Le Potier to the Dolbeault cohomology. As an application the author proves conjectures of Debarre and Kim for branched coverings of Grassmannians and extends a well-known Barth-Lefschetz type theorem for branched covers of projective spaces, due to Lazarsfeld.

### MSC:

 14F17 Vanishing theorems in algebraic geometry 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

### Keywords:

vanishing theorem; ample vector bundle
Full Text: