A vanishing theorem on Kaehler Finsler manifolds

Abstract

Let M be a connected compact complex manifold endowed with a strongly pseudoconvex complex Finsler metric F. In this paper, we first define the complex horizontal Laplacian h and complex vertical Laplacian rectangle(v) on the holomorphic tangent bundle T(1.0)M of M, and then we obtain a precise relationship among rectangle(h), rectangle(v), and the Hodge-Laplace operator Delta on (T(1.0)M, ), where ) is the induced Hermitian metric on T(1.0)M by F. As an application, we prove a vanishing theorem of holornorphic p-forms on M under the condition that F is a Kaehler Finsler metric on M. (C) 2009 Elsevier B.V. All rights reserved