Vanishing theorems for tensor powers of an ample vector bundle

Abstract

LetE be a holomorphic vector bundle of rankr on a compact complex manifoldX of dimensionn. It is shown that the cohomology groupsHp,q(X, E⊗k⊗(detE)l) vanish ifE is ample andp+q≧n+1,l≧n−p+r−1. The proof rests on the well-known fact that every tensor powerE⊗k, splits into irreducible representations of Gl(E). By Bott's theory, each component is canonically isomorphic to the direct image onX of a homogeneous line bundle over a flag manifold ofE. The proof is then reduced to the Kodaira-Akizuki-Nakano vanishing theorem for line bundles by means of the Leray spectral sequence, using backward induction onp. We also obtain a generalization of Le Potier's isomorphism theorem and a counterexample to a vanishing conjecture of Sommese.