On vector bundles over hyperk\"ahler twistor spaces

TL;DR

This paper investigates the properties of holomorphic vector bundles over hyperk"ahler twistor spaces, providing stability criteria, triviality conditions, and new construction methods for irreducible bundles with specific stability behaviors.

Contribution

It offers a new characterization of semistability, proves triviality of bundles with holomorphic connections, and introduces a novel method for constructing irreducible vector bundles on hyperk"ahler twistor spaces.

Findings

Bundles with holomorphic connections are trivial.

Generic fiber restrictions of prime rank bundles are stable.

Constructed examples of non-stable restrictions for composite rank bundles.

Abstract

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.