On the structure of compact K\"{a}hler manifolds with nonnegative holomorphic sectional curvature

Abstract

In this paper, we establish a "pseudo-effective" version of the holonomy principle for compact K\"{a}hler manifolds with nonnegative holomorphic sectional curvature. As applications, we prove that if a compact complex manifold $M$ admits a K\"{a}hler metric $\omega$ with nonnegative holomorphic sectional curvature and $(M,\omega)$ has no nonzero truly flat tangent vector at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive), then $M$ must be projective and rationally connected. This answers a problem raised by Matsumura and Yang and extends Yau's conjecture. We also prove that a compact simply connected K\"{a}hler manifold with nonnegative holomorphic sectional curvature is projective and rationally connected. Additionally, we classify non-projective K\"{a}hler 3-dimensional manifolds with nonnegative holomorphic sectional curvature. Furthermore, we show that a compact K\"{a}hler manifold admits a Hermitian metric with positive real bisectional curvature is a projective and rationally connected manifold.