An equivalence of scalar curvatures on Hermitian manifolds

TL;DR

This paper investigates whether the scalar curvature equivalence known for Kähler metrics extends to non-Kähler Hermitian metrics, finding that certain noncompact manifolds admit such metrics and providing a classification of these cases.

Contribution

The paper demonstrates the existence of noncompact Hermitian manifolds where scalar curvature equivalence holds and classifies these metrics, extending understanding beyond Kähler geometry.

Findings

Certain noncompact Hermitian manifolds admit metrics with scalar curvature equivalence.

A classification theorem for these metrics is established.

The equivalence does not generally hold in compact cases, only in specific noncompact settings.

Abstract

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu-Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kahler. However, we prove that a certain class of noncompact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.