Hodge and Vanishing Theorems on Complex Finsler Vector Bundles

Abstract

In this paper, we study the Hodge and vanishing theorems on strongly pseudoconvex complex Finsler vector bundles. First, we give definitions of various Laplace operators on strongly pseudoconvex complex Finsler vector bundles and obtain the Hodge theorems about them. Second, we deduce some useful formulas on strongly pseudoconvex complex Finsler vector bundles, such as the Bochner–Kodaira–Nakano identity and $$\partial {\bar{\partial }}$$ Bochner–Kodaira technique which is obtained by Siu on Hermitian vector bundles. Finally, we prove some vanishing theorems on strongly pseudoconvex complex Finsler vector bundles. In particular, on strongly pseudoconvex complex Berwald vector bundles, we get the Hodge isomorphism theorems and vanishing theorems for cohomology groups.