Hermitian metrics with vanishing second Chern Ricci curvature

TL;DR

This paper investigates the properties of Hermitian metrics with zero second Chern Ricci curvature on compact complex manifolds, revealing rigidity phenomena and characterizing when such metrics exist or do not.

Contribution

It introduces a rigidity phenomenon for second Chern Ricci curvature and characterizes second Chern Ricci-flat Hermitian metrics on various manifolds.

Findings

Characterization of second Chern Ricci-flat Hermitian metrics on certain manifolds

Non-existence results for such metrics in specific cases

Identification of rigidity phenomena related to second Chern Ricci curvature

Abstract

We describe a rigidity phenomenon exhibited by the second Chern Ricci curvature of a Hermitian metric on a compact complex manifold. This yields a characterisation of second Chern Ricci-flat Hermitian metrics on several types of manifolds as well as a range of non-existence results for such metrics.