\((\varepsilon ,\delta )\)–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 \((\varepsilon ,\delta )\) $(\epsilon ,\delta )$ –quasi-negativity, generalizing quasi-negativity, and obtain gap-type theorems for \(\int _X c_1(K_X)^n>0\) ${\int }_X{c}_1{(K_X)}^n>0$ in terms of the real bisectional curvature and weighted orthogonal Ricci curvature. These theorems are also a generalization of that results by Zhang-Zheng (arXiv:2010.01314v4 ) and Chu-Lee-Tam (Trans Am Math Soc 375(11):7925–7944, 2022).