The Kobayashi–Hitchin Correspondence of Generalized Holomorphic Vector Bundles Over Generalized Kähler Manifolds of Symplectic Type

Abstract

Abstract In this paper, we establish the Kobayashi–Hitchin correspondence, that is, the equivalence of the existence of an Einstein–Hermitian metric and $\psi $-polystability of a generalized holomorphic vector bundle over a compact generalized Kähler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles, and we obtain $\psi $-stable Poisson modules over complex surfaces that are not stable in the ordinary sense.