Stable Bundles on Irregular Vaisman Manifolds

TL;DR

This paper extends the understanding of stable holomorphic vector bundles on Vaisman manifolds, showing that even irregular cases maintain equivariance and filtrability properties, broadening the class of manifolds with known bundle stability characteristics.

Contribution

It generalizes the stability and equivariance results of holomorphic vector bundles from regular to irregular Vaisman manifolds.

Findings

Stable holomorphic vector bundles are equivariant on irregular Vaisman manifolds.

Bundles are filtrable even in the irregular case.

Results extend known properties from regular to irregular Vaisman manifolds.

Abstract

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.