On mixed curvature for Hermitian manifolds

TL;DR

This paper introduces the concept of mixed curvature in Hermitian manifolds, proving conditions under which such manifolds are Kähler and classifying certain cases with constant mixed curvature, revealing links to complex geometry and curvature properties.

Contribution

It extends previous results by analyzing mixed curvature, providing classification results for Hermitian and locally conformal Kähler manifolds, and establishing conditions for Kodaira dimension based on curvature positivity.

Findings

Hermitian surface with constant mixed curvature is Kähler unless specific conditions hold.

Partial classification of locally conformal Kähler manifolds with constant mixed curvature.

Semi-positive mixed curvature implies Kodaira dimension -infinity under certain conditions.

Abstract

In this paper, we consider {\em mixed curvature} $\mathcal{C}_{\alpha,\beta}$ for Hermitian manifolds, which is a convex combination of the first Chern Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam \cite{CLT}. We prove that if a compact Hermitian surface with constant mixed curvature $c$, then the Hermitian metric must be K\"{a}hler unless $c=0$ and $2\alpha+\beta=0$, which extends a previous result by Apostolov-Davidov-Mu\v{s}karov. For the higher-dimensional case, we also partially classify compact locally conformal K\"{a}hler manifolds with constant mixed curvature. Lastly, we prove that if $\beta\geq0, \alpha(n+1)+2\beta>0$, then a compact Hermitian manifold with semi-positive but not identically zero mixed curvature has Kodaira dimension $-\infty$.