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Hyper-holomorphic connections on vector bundles on hyper-Kähler manifolds

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We study infinitesimal deformations of autodual and hyper-holomorphic connections on complex vector bundles on hyper-Kähler manifolds of arbitrary dimension. In particular, we describe the DG Lie algebra controlling this deformation problem. Moreover, we prove associative formality for derived endomorphisms of a holomorphic vector bundle admitting a projectively hyper-holomorphic connection.

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Acknowledgements

This work grew up from our attempt to understand Verbitsky’s work on hyper-holomorphic connections, in particular his quadraticity result. We would like to thank him for his seminal papers on the subject, to which the present paper owes a lot. We are grateful to Simone Diverio, Marco Manetti and Kieran O’Grady for useful conversations on the subject of this paper. We also warmly thank the referee for the keen proofreading and useful comments. Both authors acknowledge support from Ateneo 2017 Varietà speciali, spazi di moduli e teoria delle deformazioni, that made possible the conference “hyper-Kähler varieties in Rome” organised in September 2021. The second author is also supported by the PRIN grant with CUP E84I19000500006.

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Authors and Affiliations

  1. Dipartimento di Matematica, Alma Mater studiorum Università di Bologna, Piazza di porta San Donato 5, 40126, Bologna, Italy

    Francesco Meazzini

  2. Dipartimento di Matematica, Università degli studi di Roma Tor Vergata, Viale della ricerca scientifica 1, 00133, Roma, Italy

    Claudio Onorati

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  1. Francesco Meazzini
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  2. Claudio Onorati
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Meazzini, F., Onorati, C. Hyper-holomorphic connections on vector bundles on hyper-Kähler manifolds. Math. Z. 303, 17 (2023). https://doi.org/10.1007/s00209-022-03176-4

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