Laplacians on the holomorphic tangent bundle of a Kaehler manifold

Abstract

Let M be a connected complex manifold endowed with a Hermitian metric g. In this paper, the complex horizontal and vertical Laplacians associated with the induced Hermitian metric 〈·, ·〉 on the holomorphic tangent bundle T1,0M of M are defined, and their explicit expressions are obtained. Using the complex horizontal and vertical Laplacians associated with the Hermitian metric 〈·, ·〉 on T1,0M, we obtain a vanishing theorem of holomorphic horizontal p forms which are compactly supported in T1,0M under the condition that g is a Kaehler metric on M.