On mixed curvature for Hermitian manifolds

Abstract

In this paper, we consider {\em mixed curvature} $\mathcal{C}_{α,β}$ 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 Kähler unless $c=0$ and $2α+β=0$, which extends a previous result by Apostolov-Davidov-Muškarov. For the higher-dimensional case, we also partially classify compact locally conformal Kähler manifolds with constant mixed curvature. Lastly, we prove that if $β\geq0, α(n+1)+2β>0$, then a compact Hermitian manifold with semi-positive but not identically zero mixed curvature has Kodaira dimension $-\infty$.