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 □ v on the holomorphic tangent bundle T 1 , 0 M of M, and then we obtain a precise relationship among □ h , □ v and the Hodge–Laplace operator △ 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 holomorphic p-forms on M under the condition that F is a Kaehler Finsler metric on M.