A recent breakthrough of Wu and Yau asserts that a compact projective Kahler
manifold with negative holomorphic sectional curvature must have ample
canonical line bundle. In the talk, we will talk about some of the recent
advances along this direction. In particular, we will discuss the case
where the manifold is a noncompact Kahler manifold. We will also discuss
the case when the Kahlerity is a priori unknown. Part of these are joint
work with S. Huang, L.-F. Tam, F. Tong.