Canonical metrics on holomorphic bundles over compact bi-Hermitian manifolds

The purpose of this paper is twofold. We first solve the Dirichlet problem for α -Hermitian–Einstein equations on I ± -holomorphic bundles over bi-Hermitian manifolds. Second, by using Uhlenbeck–Yau’s continuity method, we prove that the existence of approximate α -Hermitian–Einstein structure and the α -semi-stability on I ± -holomorphic bundles over compact bi-Hermitian manifolds are equivalent.