Positivity and vanishing theorems for ample vector bundles

Abstract

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if \(E\) is an ample vector bundle over a compact K\"ahler manifold \(X\), \(S^kE\ts \det E\) is both Nakano-positive and dual-Nakano-positive for any \(k\geq 0\). Moreover, \(H^{n,q}(X,S^kE\ts \det E)=H^{q,n}(X,S^kE\ts \det E)=0\) for any \(q\geq 1\). In particular, if \((E,h)\) is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle \((S^kE\ts \det E, S^kh\ts \det h)\) is both Nakano-positive and dual-Nakano-positive for any \(k\geq 0\).