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

In this paper, we establish the Hitchin–Kobayashi correspondence for the $I_{\pm }$-holomorphic quiver bundle $\mathcal{E}=(E,\varphi )$ 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.