Special Hermitian metrics

TL;DR

This paper investigates the stability and deformation properties of a special class of Hermitian metrics characterized by a specific differential condition on their fundamental two-form, motivated by their existence on certain compact complex manifolds.

Contribution

It introduces and analyzes the stability and deformation behavior of Hermitian metrics satisfying the condition \\partial ar \\partial \\omega^k=0 for all relevant k, expanding understanding of their geometric properties.

Findings

Analysis of stability at blow-up points

Deformation behavior of these Hermitian metrics

Conditions under which such metrics exist on compact manifolds

Abstract

We study the stability at blow-up and deformations of a class of Hermitian metrics whose fundamental two-form $\omega$ satisfies the condition $\partial \bar \partial \omega^k=0$, for any $k$ between 1 and $n-1$ (where $n$ is the complex dimension of the manifold). We are motivated by the existence of compact complex manifold supporting such metrics.