On vector bundles over hyperk\"ahler twistor spaces

Indranil Biswas; Artour Tomberg

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arXiv · 1911.00833

On vector bundles over hyperk\"ahler twistor spaces

Indranil Biswas·Artour Tomberg

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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.

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Indranil Biswas, Artour Tomberg. 2019-11-03. On vector bundles over hyperk\"ahler twistor spaces. https://arxiv.org/abs/1911.00833

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