A Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic bundles

TL;DR

This paper extends the Kobayashi-Hitchin correspondence to $I_ ext{±}$-holomorphic bundles on bi-Hermitian manifolds, introducing $ ext{α}$-stability and $ ext{α}$-Hermitian-Einstein metrics, with examples on generalized Kähler manifolds.

Contribution

It establishes a Kobayashi-Hitchin correspondence for $I_ ext{±}$-holomorphic bundles, incorporating $ ext{α}$-stability that varies with the parameter $ ext{α}$.

Findings

$ ext{α}$-stability depends on the parameter $ ext{α}$

Established a correspondence between $ ext{α}$-Hermitian-Einstein metrics and $ ext{α}$-stability

Included examples of generalized holomorphic bundles on generalized Kähler manifolds

Abstract

In this paper, we introduce the notions of $\alpha$-Hermitian-Einstein metric and $\alpha$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized K\"ahler manifolds. We also show that $\alpha$-stability of a vector bundle, in this sense, can depend on the parameter $\alpha$.