On the proportionality of Chern and Riemannian scalar curvatures

TL;DR

This paper investigates the relationship between Chern and Riemannian scalar curvatures on Hermitian manifolds, exploring conditions under which they are proportional, extending known Kähler characterizations to non-Kähler settings.

Contribution

It introduces new conditions for proportionality of scalar curvatures in non-Kähler Hermitian manifolds, broadening understanding beyond the Kähler case.

Findings

Proves existence of non-Kähler metrics with proportional scalar curvatures.

Characterizes when scalar curvature proportionality implies Kählerity.

Identifies obstructions to proportionality in non-Kähler geometries.

Abstract

On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler setting, one can seek to understand extensions of these characterizations to general Hermitian manifolds. This idea has been the subject of much study from the cohomological side, however, the focus here is to address such a question from the perspective of curvature relationships. In particular, on compact manifolds the K\"ahler condition is characterized by the relationship that the Chern scalar curvature is equal to half the Riemannian scalar curvature. What we study here is the existence, or lack thereof, of non-Kahler Hermitian metrics for which a more general proportionality relationship between these scalar curvatures holds.