Canonical metrics on holomorphic quiver bundles over compact generalized Kähler manifolds

The purpose of this paper is twofold. By introducing the \((\alpha ,\sigma ,\tau )\) $(\alpha ,\sigma ,\tau )$ -Hermite–Yang–Mills flow, we first solve the Dirichlet problem for \((\alpha ,\sigma ,\tau )\) $(\alpha ,\sigma ,\tau )$ -Hermite–Yang–Mills equations on holomorphic quiver bundles over compact generalized Kähler manifolds. Second, by using Uhlenbeck–Yau’s continuity method, we prove that the \((\alpha ,\sigma ,\tau )\) $(\alpha ,\sigma ,\tau )$ -semi-stability implies the existence of approximate \((\alpha ,\sigma ,\tau )\) $(\alpha ,\sigma ,\tau )$ -Hermite–Yang–Mills structure on holomorphic quiver bundles over compact generalized Kähler manifolds.