Families of stable bundles on the fibres of the hyperk\"ahler twistor projection

TL;DR

This paper investigates the stability properties of holomorphic vector bundles on the twistor space of hyperk"ahler manifolds, establishing conditions under which fiberwise stability implies global stability, with implications for the structure of such bundles.

Contribution

It extends Teleman's argument to twistor spaces and provides a partial converse to Kaledin and Verbitsky's results, linking fiberwise stability to global stability in specific cases.

Findings

Zariski openness of stability applies to these bundles.

Irreducible bundles of rank 2 or 3 are generically fiberwise stable.

Presence of a simple fiber implies global stability under certain conditions.

Abstract

Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperk\"ahler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres $\pi^{-1}(I)$ of the twistor projection $\pi : \mathrm{Tw}(M) \to \mathbb{CP}^1$, and study the relationship between stability of $E$ and its fibrewise stability. We verify that the argument of Teleman establishing the Zariski openness of stability and semi-stability in families of bundles applies in the case of the family $\left\{E_I\right\}$. We prove a partial converse to a result of Kaledin and Verbitsky, showing that an irreducible bundle $E$ on $\mathrm{Tw}(M)$ is generically fibrewise stable if the rank of $E$ is 2 or 3, or at least one element of the family $\left\{E_I\right\}$ is a simple bundle, in the sense that $\mathrm{Hom}(E_I, E_I) = \mathbb{C}$.