4-11【Shiyu Zhang】Compact Kaehler manifolds with quasi-positive holomorphic sectional curvature

We show that a compact Kaehler manifold with quasi-positive holomorphic sectional curvature is projetive and rationally connected. This result extends Yau's conjecture on positive holomorphic sectional curvature and answers a problem raised by Yang and Matsumura.  Firstly, I will recap past progress in understanding the structure of compact Kaehler manifold with nonnegative holomorphic sectional curvature. Subsequently, I will briefly explain our method. This is a joint work with Xi Zhang.