Uniform weak RC-positivity and rational connectedness

TL;DR

This paper proves that uniform weak RC-positivity of the tangent bundle on a compact Kähler manifold implies projectiveness and rational connectedness, extending previous results with weaker assumptions.

Contribution

It generalizes earlier work by showing that uniform weak RC-positivity suffices for rational connectedness, and establishes Hermitian metrics with positive mean curvature for such bundles.

Findings

Uniform weak RC-positivity implies projectiveness and rational connectedness.

Holomorphic vector bundles with this property admit Hermitian metrics with positive mean curvature.

A quasi-positive version of the result is also established.

Abstract

In this paper, we show that if the holomorphic tangent bundle $TX$ of a compact K\"ahler manifold $X$ is uniformly weakly RC-positive, then $X$ is projective and rationally connected. This result is previously established by Xiaokui Yang under the stronger assumption that $TX$ is uniformly RC-positive. The result we obtain is, in fact, more general. If a holomorphic vector bundle $E$ is uniformly weakly RC-positive, then $E$ admits a Hermitian metric whose mean curvature is positive. A quasi-positive version is also proved in this paper.