L2 Estimates and Vanishing Theorems for Holomorphic Vector Bundles Equipped with Singular Hermitian Metrics

Abstract

We investigate singular hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ estimates and vanishing theorems usually require an assumption that the vector bundles are Nakano positive. However, there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theorems by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.