On mixed curvature for Hermitian manifolds

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 Kähler unless $c=0$ and $2\alpha +\beta =0$, 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 $\beta \ge 0,\alpha (n+1)+2\beta >0$, then a compact Hermitian manifold with semi-positive but not identically zero mixed curvature has Kodaira dimension $-\mathrm{\infty }$.