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In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitianmetric of a generalized holomorphic vector bundle over a generalized Kahlermanifold of symplectic type was introduced from the moment map framework. Inthis paper we establish a Kobayashi-Hitchin correspondence, that is, theequivalence of the existence of an Einstein-Hermitian metric and$\psi$-polystability of a generalized holomorphic vector bundle over a compactgeneralized Kahler manifold of symplectic type. Poisson modules provideintriguing generalized holomorphic vector bundles and we obtain $\psi$-stablePoisson modules over complex surfaces which are not stable in the ordinarysense.

Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type