Positivities and vanishing theorems on complex Finsler manifolds.

Abstract In this paper, we study the relations between curvatures and geometries of a compact complex Finsler manifold. We first introduce various definitions of curvatures and then vanishing theorems are established under some positivity assumptions of curvatures. In particular, we get some criteria of a compact complex Finsler manifold to be of negative Kodaira dimension. [ABSTRACT FROM AUTHOR]