Some remarks on vanishing theorems for holomorphic vector bundles (Q795974)

The main result of this paper is the following precise vanishing theorem: Let X be a projective manifold of dimension n and let E and F be vector bundles of rank r, resp. of rank 1 on X. Suppose that \({\mathcal O}_{{\mathbb{P}}(E)}(1)\) and \(A=(\det E)^{-1}\otimes K_ X^{-1}\otimes F\) are numerically semi-positive line bundles. Assume moreover that either \(c_ 1(A)^ n>0\) or that \(\tilde c_ n(E)>0.\) Then \(H^ q(X,S^{\mu}(E)\otimes F)=0\) for \(q>0,\quad \mu \geq 0.\) Here \(\tilde c{}_ n(E)\) is the component in \(H^{2n}(X,{\mathbb{Z}})\) of \(c(E^*)^{- 1}\). - This generalizes results of Griffiths. Similar improvements can be given for the vanishing theorems of Le Potier and Faltings.