Stable Bundles on Irregular Vaisman Manifolds

A locally conformally K\"ahler (LCK) manifold is a complex manifold whose universal cover is K\"ahler with monodromy group acting on the universal cover by holomorphic homotheties. A Vaisman manifold $M$ is a compact non-K\"ahler LCK manifold admitting an action of a holomorphic conformal flow lifting to an action on a K\"ahler cover by nontrivial homotheties. When the orbits of the action on $M$ are compact, it is known that every stable holomorphic vector bundle over $M$, $\dim(M) \geq 3$, is equivariant and filtrable. In the present paper we generalize this result to irregular Vaisman manifolds.