Abstract
We study the holomorphic vector bundles E over the twistor space \({{\,\mathrm{Tw}\,}}(M)\) of a compact simply connected hyperkähler 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 \(\pi \,:\, {{\,\mathrm{Tw}\,}}(M)\,\longrightarrow \, {\mathbb {CP}}^1\). 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 \({{\,\mathrm{Tw}\,}}(M)\) 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 \({{\,\mathrm{Tw}\,}}(M)\) whose restriction to every fibre of \(\pi \) is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples.
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Acknowledgements
The authors would like to thank Ajneet Dhillon, Jacques Hurtubise and Misha Verbitsky for the many valuable discussions and suggestions. The study of the second author has been funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project ’5-100’. The first author is supported by a J. C. Bose Fellowship.
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Biswas, I., Tomberg, A. On vector bundles over hyperkähler twistor spaces. Math. Z. 300, 3143–3170 (2022). https://doi.org/10.1007/s00209-021-02893-6
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DOI: https://doi.org/10.1007/s00209-021-02893-6