Scalar curvature and uniruledness on projective manifolds

TL;DR

This paper establishes a link between the sign of total scalar curvature on projective manifolds and their uniruledness, showing positive scalar curvature implies uniruledness, while zero scalar curvature implies either uniruledness or torsion canonical bundle.

Contribution

It proves new results connecting scalar curvature signs with uniruledness and canonical bundle properties on projective manifolds, advancing understanding in complex geometry.

Findings

Positive total scalar curvature implies the manifold is uniruled.

Zero total scalar curvature implies the manifold is either uniruled or has torsion canonical bundle.

All Kähler metrics on a non-uniruled manifold have scalar curvature of the same sign, zero or negative.

Abstract

It is a basic tenet in complex geometry that {\it negative} curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while {\it positive} curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifold $M$ with a K\"ahler metric with positive total scalar curvature is uniruled, which is equivalent to every point of $M$ being contained in a rational curve. We also prove that if $M$ possesses a K\"ahler metric of total scalar curvature equal to zero, then either $M$ is uniruled or its canonical line bundle is torsion. The proof of the latter theorem is partially based on the observation that if $M$ is not uniruled, then the total scalar curvatures of all K\"ahler metrics on $M$ must have the same sign, which is either zero or negative.