$(\epsilon ,\delta )$--Quasi-Negative Curvature and Positivity of the Canonical Bundle

A recent theorem of Diverio–Trapani and Wu–Yau asserts that a compact Kähler manifold with a Kähler metric of quasi-negative holomorphic sectional curvature is projective and canonically polarized. This confirms a long-standing conjecture of Yau. We consider the notion of $(\epsilon ,\delta )$–quasi-negativity, generalizing quasi-negativity, and obtain gap-type theorems for ${\int }_X{c}_1(K_X{)}^n>0$ in terms of the real bisectional curvature and weighted orthogonal Ricci curvature. These theorems are also a generalization of that results in \cite{ZhangZheng} by Zhang-Zheng and in \cite{ChuLeeTam} by Chu-Lee-Tam.