On projective manifolds with semi-positive holomorphic sectional curvature

Abstract

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. Specifically, we prove that $X$ admits a locally trivial morphism $\varphi :X\to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite étale cover $A\to Y$ by an abelian variety $A$, by combining the author's previous work with the theory of holomorphic foliations. Moreover, we show that the universal cover of $X$ is biholomorphic and isometric to the product ${\mathbb{C}}^m\times F$ of the universal cover ${\mathbb{C}}^m$ of $Y$ with a flat metric and the rationally connected fiber $F$ with a Kähler metric whose holomorphic sectional curvature is semi-positive.

arXiv comment20pages