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 classica
l theorems of Griffiths and Le Potier to the whole Dolbeault cohomolog
y, prove a variant of an uncorrect conjecture of Sommese, and answer a
question of Demailly. As an application, we prove conjectures of Deba
rre 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.