Laplacians for the holomorphic tangent bundles with g-nature metrics on complex Finsler manifolds

Let [Formula: see text] be a strongly pseudoconvex compact complex Finsler manifold. We first introduce a class of [Formula: see text]-nature metric [Formula: see text] for the slit holomorphic tangent bundle [Formula: see text] on [Formula: see text]. Then, we define the complex horizontal Laplacian [Formula: see text], and complex vertical Laplacian [Formula: see text], and obtain a precise relationship among [Formula: see text], [Formula: see text] and the Hodge–Laplace operator [Formula: see text] on [Formula: see text]. As an application, we discuss the holomorphic Killing vector fields associated to [Formula: see text].