An example of a stable but fibrewise nonstable bundle on the twistor space of a hyperkähler manifold

Abstract

We construct an explicit example of a stable bundle on the twistor space $\mathrm{Tw}(M)$ of a hyperkähler 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.

6 pages; to appear in Math. Notes