# An example of a stable but fibrewise nonstable bundle on the twistor space of a hyperk\"ahler manifold

## TL;DR

This paper constructs an explicit example of a stable bundle on the twistor space of a hyperk"ahler manifold that restricts to nonstable bundles on all fibres, revealing nuanced stability behavior in this geometric context.

## Contribution

It provides the first explicit example of a stable bundle on the twistor space with nonstable restrictions to all fibres, and explores the relationship between bundle stability properties.

## Key findings

- Constructed an explicit stable bundle with nonstable fibre restrictions.
- Described the relationship between subsheaf-free bundles and stable restrictions.
- Announced a forthcoming result on bundle stability criteria.

## Abstract

We construct an explicit example of a stable bundle on the twistor space $\mathrm{Tw}(M)$ of a hyperk\"ahler manifold $M$ whose restrictions to all the fibres of the natural twistor projection $\pi : \mathrm{Tw}(M) \to \mathbb{CP}^1$ are nonstable. We also describe the relationship between bundles on $\mathrm{Tw}(M)$ that do not have subsheaves of strictly lower rank and bundles that stably restrict to the fibres of $\pi$, and announce a result whose proof will appear in a forthcoming paper.