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Rigid hyperholomorphic sheaves remain rigid along twistor deformations of the underlying hyparkähler manifold

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Abstract

Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over \(M\times M\) there exists a rank \(2n-2\) reflexive hyperholomorphic sheaf \(E_M\), whose fiber over a non-diagonal point \((F_1,F_2)\) is \(\mathrm{Ext}^1_S(F_1,F_2)\). The sheaf \(E_M\) can be deformed along some twistor path to a sheaf \(E_X\) over the Cartesian square \(X\times X\) of every Kähler manifold X deformation equivalent to M. We prove that \(E_X\) is infinitesimally rigid, and the isomorphism class of the Azumaya algebra is independent of the twistor path chosen. This verifies conjectures in Markman and Mehrotra (A global Torelli theorem for rigid hyperholomorphic sheaves, 2013. arXiv:1310.5782v1; Integral transforms and deformations of K3 surfaces, 2015. arXiv:1507.03108v1) and renders the results of these two papers unconditional.

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Notes

  1. The assumptions of [22, Theorem 1.4] are preserved under twistor deformations of E, by [24, Proposition 3.2] and [24, Lemma 7.6]. The proofs of [24, Proposition 3.2] and [24, Lemma 7.6] are unconditional.

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Acknowledgements

The work of E. Markman was partially supported by a grant from the Simons Foundation (# 427110) and his work during March 2017 by the Max Planck Institute in Bonn. M. Verbitsky is partially supported by the Russian Academic Excellence Project ‘5-100’. S. Mehrotra acknowledges support from CONICYT by way of the grant FONDECYT Regular 1150404. This grant also partially funded the visit of E. Markman and M. Verbitsky to Pontificia Universidad Católica de Chile in March 2016. The authors thank the referee for his help in improving the exposition.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Eyal Markman

  2. Chennai Math. Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam, 603103, India

    Sukhendu Mehrotra

  3. Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil

    Misha Verbitsky

  4. Department of Mathematics, Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 9 Usacheva Street, Moscow, Russia, 117312

    Misha Verbitsky

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  1. Eyal Markman
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  2. Sukhendu Mehrotra
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  3. Misha Verbitsky
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Correspondence to Sukhendu Mehrotra.

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Markman, E., Mehrotra, S. & Verbitsky, M. Rigid hyperholomorphic sheaves remain rigid along twistor deformations of the underlying hyparkähler manifold. European Journal of Mathematics 5, 964–1012 (2019). https://doi.org/10.1007/s40879-019-00323-w

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  • DOI: https://doi.org/10.1007/s40879-019-00323-w

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