Abstract

A recent theorem of Wu-Yau, Tosatti-Yang, and Diverio-Trapani states that a compact Kaehler manifold admitting a Kaehler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of Yau. In this talk, we shall introduce a natural notion of almost quasi-negative holomorphic sectional curvature and extend this theorem to compact Kaehler manifolds of almost quasi-negative holomorphic sectional curvature.