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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)
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