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 E is an ample vector bundle over a compact Kähler manifold X X , S k E ⊗ det E S^kE\otimes \det E is both Nakano-positive and dual-Nakano-positive for any k ≥ 0 k\geq 0 . Moreover, H n , q ( X , S k E ⊗ det E ) = H q , n ( X , S k E ⊗ det E ) = 0 H^{n,q}(X,S^kE\otimes \det E)=H^{q,n}(X,S^kE\otimes \det E)=0 for any q ≥ 1 q\geq 1 . In particular, if ( E , h ) (E,h) is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle ( S k E ⊗ det E , S k h ⊗ det h ) (S^kE\otimes \det E, S^kh\otimes \det h) is both Nakano-positive and dual-Nakano-positive for any k ≥ 0 k\geq 0 .