摘要: Negativity of the holomorphic sectional curvature (HSC) plays fundamental roles in several topics in complex geometry. For example, a conjecture of Yau in 1970's predicts that the negativity of HSC should be intimately related to the positivity of the canonical line bundle. In this talk we shall first recall recent progresses on Yau's conjecture. We then, with natural motivations, introduce an "almost" nonpositivity notion for HSC on compact Kaehler manifolds and its applications, including positivity of the canonical line bundle, Chern number inequalities and several new rigidity theorems for holomorphic maps.