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

## TL;DR

This paper proves a correspondence between Einstein-Hermitian metrics and stability conditions for generalized holomorphic vector bundles on symplectic type generalized Kahler manifolds, extending classical results to a broader geometric setting.

## Contribution

It establishes a Kobayashi-Hitchin correspondence for generalized holomorphic vector bundles, linking geometric structures with stability notions in generalized Kahler geometry.

## Key findings

- Proves the equivalence between Einstein-Hermitian metrics and $\psi$-polystability.
- Constructs $\psi$-stable Poisson modules on complex surfaces.
- Extends classical stability results to generalized Kahler manifolds.

## Abstract

In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a 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 Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $\psi$-stable Poisson modules over complex surfaces which are not stable in the ordinary sense.