The Chern-Finsler connection and Finsler-Kähler manifolds

Abstract

In this paper, we shall discuss the theory of connection in complex Finsler geometry, i.e., the Chern-Finsler connection $\nabla$ and its applications. In particular, we shall investigate (1) the ampleness of holomorphic vector bundles over a compact complex manifold which is based on the study due to [Ko1], (2) some special class of complex Finsler metrics and its characterization in terms of torsion and curvature of $\nabla$, and in the last section, (3) the characterization of Finsler-Kähler manifolds in terms of the Cartan connection $D$ which is naturally induced on the real tangent bundle from $\nabla$.