# Hermitian manifolds with non-positive curvature

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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.