# The Hitchin--Kobayashi Correspondence for Quiver Bundles over Generalized K\"ahler Manifolds

## TL;DR

This paper proves a correspondence between stability and Hermitian--Einstein metrics for quiver bundles over generalized K"ahler manifolds, extending classical results to a broader geometric setting.

## Contribution

It establishes the Hitchin--Kobayashi correspondence for $I_ ext{pm}$-holomorphic quiver bundles on generalized K"ahler manifolds with Gauduchon metrics, generalizing known results.

## Key findings

- Proves the equivalence of polystability and existence of Hermitian--Einstein metrics for these bundles.
- Extends the Hitchin--Kobayashi correspondence to a new class of complex manifolds.
- Provides a framework for analyzing quiver bundles in generalized K"ahler geometry.

## Abstract

In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian--Einstein metric.