On projective manifolds with semi-positive holomorphic sectional curvature

We establish structure theorems for a smooth projective variety X with semi-positive holomorphic sectional curvature. We first prove that X is rationally connected if X has no truly flat tangent vectors at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive). This result solves Yau's conjecture on positive holomorphic sectional curvature in a strong form. Moreover, we prove that X admits a locally trivial morphism: X Y such that the fiber F is rationally connected and the image Y has a finite \'etale cover A Y by an abelian variety A. We also show that the universal cover of X is biholomorphic and isometric to the product Cᵐ F of the complex Euclidean space Cᵐ with the flat metric and the rationally connected fiber F with the induced K\"ahler metric. Our structure theorem is a natural generalization of the structure theorem established by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature.