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European Journal of Mathematics

, Volume 5, Issue 3, pp 964–1012 | Cite as

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

  • Eyal Markman
  • Sukhendu MehrotraEmail author
  • Misha Verbitsky
Research Article
  • 10 Downloads

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 Open image in new window 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.

Keywords

Holomorphic symplectic manifolds Hyperholomorphic sheaves Twistor families Monodromy 

Mathematics Subject Classification

14J28 14J60 14D05 14D99 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Eyal Markman
    • 1
  • Sukhendu Mehrotra
    • 2
    Email author
  • Misha Verbitsky
    • 3
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Chennai Math. InstituteKelambakkamIndia
  3. 3.Instituto Nacional de Matemática Pura e AplicadaRio de JaneiroBrazil
  4. 4.Department of Mathematics, Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia

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