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. first prove that is rationally connected if has no truly flat tangent vectors at some point (which satisfied when the curvature quasi-positive). This result solves Yau's conjecture on positive in strong form. Moreover, we admits locally trivial morphism $\phi:X\to Y$ such fiber $F$ and image $Y$ finite \'etale cover $A\to by an abelian $A$. also show universal of biholomorphic isometric to product $\Bbb{C}^m\times F$ complex Euclidean space $\Bbb{C}^m$ metric induced K\"ahler metric. Our theorem natural generalization established Howard-Smyth-Wu Mok bisectional