The main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety. In particular, we extend classical theorems of Griffiths and Le Potier to the whole Dolbeault cohomology, prove a variant of an uncorrect conjecture of Sommese, and answer a question of Demailly. As an application, we prove conjectures of Debarre and Kim for branched coverings of grassmannians, and extend a well-known Barth-Lefschetz type theorem for branched covers of projective spaces, due to Lazarsfeld. We also obtain new restriction theorems for certain degeneracy loci.
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