On projective manifolds with semi-positive holomorphic sectional curvature
Shin‐ichi Matsumura
Abstract
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 $\phi:X\to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite \'etale cover $A\to Y$ by an abelian variety $A$. We also show that the universal cover of $X$ is biholomorphic and isometric to the product $\Bbb{C}^m\times F$ of the complex Euclidean space $\Bbb{C}^m$ 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.