Mean curvature positivity and rational connectedness

TL;DR

This paper establishes a link between mean curvature positivity and rational connectedness in compact Kähler manifolds, providing a new differential geometric criterion for rational connectedness based on holomorphic tangent bundle properties.

Contribution

It introduces a novel criterion connecting mean curvature positivity with rational connectedness, using Uhlenbeck-Yau's continuity method on holomorphic vector bundles.

Findings

A compact Kähler manifold is projective and rationally connected if and only if its holomorphic tangent bundle is mean curvature positive.

The paper proves the equivalence between mean curvature positivity and HN-positivity on holomorphic vector bundles.

Provides a differential geometric criterion for rational connectedness in complex geometry.

Abstract

In this paper, we use Uhlenbeck-Yau's continuity method to establish the correspondence between the mean curvature positivity and the HN-positivity on holomorphic vector bundles over compact Hermitian manifolds. As its application, we get a differential geometric criterion for rational connectedness, i.e. we prove that a compact K\"ahler manifold is projective and rationally connected if and only if its holomorphic tangent bundle is mean curvature positive.