On vector bundles over hyperk\"ahler twistor spaces

We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperk"ahler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection \pi :Tw(M)\rightarrow CP^1. It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we prove that its restriction to the generic fibre of \pi is stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on Tw(M) whose restriction to every fibre of \pi is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperk"ahler twistor spaces; this method is employed in constructing these examples.