Constant $k$th-mixed curvature

In this paper, we consider general $k$th-mixed curvature ${\mathcal{C}}_{\alpha ,\beta }^{(k)}$ ($\beta \mathrm{\ne }0$) for Hermitian manifolds, which is a convex combination of the $k$th Chern Ricci curvature and holomorphic sectional curvature. We prove that any compact Hermitian surface with constant $k$th-mixed curvature is self-dual. Furthermore, we show that if a compact Hermitian surface has constant 2th-mixed curvature $c$, then the Hermitian metric must be Kähler. For the higher-dimensional case, when the parameters $\alpha$ and $\beta$ satisfy certain conditions, we can also obtain partial results.