Balanced metrics on uniruled manifolds

TL;DR

This paper establishes a characterization of uniruled manifolds via the existence of balanced metrics with positive scalar Chern curvature, extending results to certain complex manifolds of dimension three.

Contribution

It provides a new criterion for uniruledness based on balanced metrics and scalar curvature, including for class  manifolds in dimension three.

Findings

Uniruled Moishezon manifolds support balanced metrics with positive scalar Chern curvature.

The characterization extends to class  manifolds of dimension three.

Balanced metrics are linked to the uniruled property in complex geometry.

Abstract

We show that an $n-$dimensional Moishezon manifold is uniruled if and only if it supports a balanced metric $\omega^{n-1}$ of positive total scalar Chern curvature. A similar statement also holds true for class $\mathcal C$ manifolds of dimension three.