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


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.


Holomorphic symplectic manifolds Hyperholomorphic sheaves Twistor families Monodromy 

Mathematics Subject Classification

14J28 14J60 14D05 14D99 



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.


  1. 1.
    Amerik, E., Verbitsky, M.: Rational curves on hyperkähler manifolds. Int. Math. Res. Not. IMRN 2015(23), 13009–13045 (2015)zbMATHGoogle Scholar
  2. 2.
    Bando, S., Siu, Y.-T.: Stable sheaves and Einstein–Hermitian metrics. In: Mabuchi, T., et al. (eds.) Geometry and Analysis on Complex Manifolds, pp. 39–50. World Scientific, River Edge (1994)Google Scholar
  3. 3.
    Bayer, A., Hassett, B., Tschinkel, Yu.: Mori cones of holomorphic symplectic varieties of K3 type. Ann. Sci. Éc. Norm. Supér. (4) 48(4), 941–950 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differential Geom. 18(4), 755–782 (1984)zbMATHGoogle Scholar
  5. 5.
    Beauville, A.: Some remarks on Kähler manifolds with \(c_1=0\). In: Ueno, K. (ed.) Classification of Algebraic and Analytic Manifolds. Progress in Mathematics, vol. 39, pp. 1–26. Birkhäuser, Boston (1983)Google Scholar
  6. 6.
    Bragg, D., Lieblich, M.: Twistor spaces for supersingular K3 surfaces (2018). arXiv:1804.07282
  7. 7.
    Buskin, N.: Every rational Hodge isometry between two K3 surfaces is algebraic. J. Reine Angew. Math.
  8. 8.
    Charles, F., Markman, E.: The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of \(K3\) surfaces. Compositio Math. 149(3), 481–494 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chevalley, C.C.: The Algebraic Theory of Spinors. Columbia University Press, New York (1954)zbMATHGoogle Scholar
  10. 10.
    Hitchin, N.J., Karlhede, A., Lindström, U., Roček, M.: Hyper-Kähler metrics and supersymmetry. Comm. Math. Phys. 108(4), 535–589 (1987)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Huybrechts, D.: Compact Hyperkähler Manifolds: Basic results. Invent. Math. 135(1), 63–113 (1999). Erratum: Invent. Math. 152(1), 209–212 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd edn. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  13. 13.
    Huybrechts, D., Schröer, S.: The Brauer group of analytic \(K3\) surfaces. Int. Math. Res. Not. IMRN 2003(50), 2687–2698 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kobayashi, R.: Moduli of Einstein metrics on a \(K3\) surface and degenerations of type I. In: Ochiai, T. (ed.) Kähler Metric and Moduli Spaces. Advanced Studies in Pure Mathematics, 18-II, pp. 257–311. Academic Press, Boston (1990)Google Scholar
  15. 15.
    Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures, I. Ann. Math. 67(2), 328–401 (1958)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures III. Stability theorems for complex analytic structures. Ann. Math. 71(1), 43–76 (1960)MathSciNetzbMATHGoogle Scholar
  17. 17.
    MacLane, S.: Categories for the Working Mathematician. 2nd edn. Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)Google Scholar
  18. 18.
    Markman, E.: On the monodromy of moduli spaces of sheaves on \(K3\) surfaces. J. Algebraic Geom. 17(1), 29–99 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Markman, E.: Integral constraints on the monodromy group of the hyperKähler resolution of a symmetric product of a \(K3\) surface. Internat. J. Math. 21(2), 169–223 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Markman, E.: The Beauville–Bogomolov class as a characteristic class (2011). arXiv:1105.3223v3
  21. 21.
    Markman, E.: A survey of Torelli and monodromy results for holomorphic-symplectic varieties. In: Ebeling, W., et al. (eds.) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol. 8, pp. 257–323. Springer, Heidelberg (2011). arXiv:1101.4606
  22. 22.
    Markman, E.: Naturality of the hyperholomorphic sheaf over the cartesian square of a manifold of \(K3^{[n]}\)-type (2016). arXiv:1608.05798.v1
  23. 23.
    Markman, E.: On the existence of universal families of marked hyperkähler varieties (2017). arXiv:1701.08690
  24. 24.
    Markman, E., Mehrotra, S.: A global Torelli theorem for rigid hyperholomorphic sheaves (2013). arXiv:1310.5782v1
  25. 25.
    Markman, E., Mehrotra, S.: Integral transforms and deformations of \(K3\) surfaces (2015). arXiv:1507.03108v1
  26. 26.
    Mongardi, G.: A note on the Kähler and Mori cones of hyperkähler manifolds. Asian J. Math. 19(4), 583–591 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mukai, S.: On the moduli space of bundles on \(K3\) surfaces. I. Vector Bundles on Algebraic Varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 11, pp. 341–413. Oxford University Press, New York (1987)Google Scholar
  28. 28.
    Mukai, S.: Fourier functor and its application to the moduli of bundles on an abelian variety. Algebraic Geometry. Advanced Studies in Pure Mathematics, vol. 10, pp. 515–550. North-Holland, Amsterdam (1987)Google Scholar
  29. 29.
    O’Grady, K.G.: The weight-two Hodge structure of moduli spaces of sheaves on a \(K3\) surface. J. Algebraic Geom. 6(4), 599–644 (1997)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Orlov, D.O.: Derived categories of coherent sheaves and equivalences between them. Russian Math. Surveys 58(3), 511–591 (2003)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Siu, Y.: Extension of locally free analytic sheaves. Math. Ann. 179, 285–294 (1969)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Schumacher, G., Toma, M.: Moduli of Kähler manifolds equipped with Hermite–Einstein vector bundles. Rev. Roumaine Math. Pures Appl. 38(7–8), 703–719 (1993)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Verbitsky, M.: Hyperholomorphic sheaves and new examples of hyperkaehler manifolds. In: Verbitsky, M., Kaledin, D. (eds.) Hyperkähler Manifolds. Mathematical Physics (Somerville), vol. 12. International Press, Somerville (1999)Google Scholar
  34. 34.
    Verbitsky, M.: Coherent sheaves on general \(K3\) surfaces and tori. Pure Appl. Math. Q. 4(3), Part 2, 651–714 (2008)Google Scholar
  35. 35.
    Verbitsky, M.: Hyperholomorphic bundles over a hyper-Kähler manifold. J. Algebraic Geom. 5(4), 633–669 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Verbitsky, M.: Ergodic complex structures on hyperkähler manifolds. Acta Math. 215(1), 161–182 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Verbitsky, M.: Mapping class group and a global Torelli theorem for hyperkähler manifolds. Duke Math. J. 162(15), 2929–2986 (2013)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Verbitsky, M.: Ergodic complex structures on hyperkähler manifolds: an erratum (2017). arXiv:1708.05802
  39. 39.
    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampére equation I. Comm. Pure Appl. Math. 31(3), 339–411 (1978)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001)MathSciNetzbMATHGoogle Scholar

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

Personalised recommendations