On K\"ahler manifolds with non-negative mixed curvature

Abstract

In this work, we investigate compact K\"ahler manifolds with non-negative or quasi-positive mixed curvature coming from a linear combination of the Ricci and holomorphic sectional curvature, which covers various notions of curvature considered in the literature. Specifically, we prove a splitting theorem, analogous to the Cheeger-Gromoll splitting theorem, for complete K\"ahler manifolds with non-negative mixed curvature containing a line, and then establish a structure theorem for compact K\"ahler manifolds with non-negative mixed curvature. We also show that the Hodge numbers of compact K\"ahler manifolds with quasi-positive mixed curvature must vanish. Both results are based on the conformal perturbation method.