The Projectivity of Compact Kähler Manifolds with Mixed Curvature Condition

In a recent paper, Li–Ni–Zhu study the nefness and ampleness of the canonical line bundle of a compact Kähler manifold with \(\textrm{Ric}_k\leqslant 0\) ${\text{Ric}}_k⩽0$ and provide a direct alternate proof to a recent result of Chu–Lee–Tam. In this paper, we generalize the method of Li–Ni–Zhu to a more general setting which concerning the connection between the mixed curvature condition and the positivity of the canonical bundle. The key point is to do some a priori estimates to the solution of a Mong-Ampère type equation.