RC-positive metrics on rationally connected manifolds

TL;DR

This paper establishes a link between RC-positivity of metrics on rationally connected manifolds and their projectivity, providing new criteria for rational connectedness based on curvature conditions.

Contribution

It proves that RC-positivity of a Hermitian metric implies projectivity and rational connectedness, and vice versa, connecting curvature conditions with algebraic properties.

Findings

RC-positive metrics imply projectivity and rational connectedness

Rationally connected projective manifolds admit RC-positive metrics

Certain curvature conditions ensure rational connectedness

Abstract

In this paper, we prove that if a compact K\"ahler manifold $X$ has a smooth Hermitian metric $\omega$ such that $(T_X,\omega)$ is uniformly RC-positive, then $X$ is projective and rationally connected. Conversely, we show that, if a projective manifold $X$ is rationally connected, then the tautological line bundle $\mathscr{O}_{T_X^*}(-1)$ is uniformly RC-positive (which is equivalent to the existence of some RC-positive complex Finlser metric on $X$). As an application, we prove that if $(X,\omega)$ is a compact K\"ahler manifold with certain quasi-positive holomorphic sectional curvature, then $X$ is projective and rationally connected.