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

TL;DR

This paper studies compact K"ahler manifolds with non-negative or quasi-positive mixed curvature, proving a splitting theorem and structure results, and showing vanishing of certain Hodge numbers, using conformal perturbation methods.

Contribution

It introduces new splitting and structure theorems for K"ahler manifolds with mixed curvature, extending classical results and analyzing Hodge number vanishing.

Findings

Proved a Cheeger-Gromoll type splitting theorem for K"ahler manifolds with non-negative mixed curvature.

Established a structure theorem for compact K"ahler manifolds with non-negative mixed curvature.

Showed that Hodge numbers vanish for compact K"ahler manifolds with quasi-positive 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.