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Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type


We classify the cohomology classes of Lagrangian 4-planes ℙ4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = −2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that (C, C) = −γ, for (·,·) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associated to C. In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n points on a K3 surface, the class C = of a line in a smooth Lagrangian n-plane ℙn must satisfy (,) = −(n + 3)/2. We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X, and showing there is a unique monodromy orbit of Lagrangian 4-planes.

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  1. Bayer A., Hassett B., Tschinkel Yu., Mori cones of holomorphic symplectic varieties of K3 type, preprint available at

  2. Bayer A., Macrì E., Projective and birational geometry of Bridgeland moduli spaces, preprint avaliable at

  3. Bayer A., Macrì E., MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, preprint available at

  4. Beauville A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom., 1983, 18(4), 755–782

    MATH  MathSciNet  Google Scholar 

  5. Bosma W., Cannon J., Playoust C., The Magma algebra system. I. The user language, In: Computational Algebra and Number Theory, London, August 23–27, 1993, J. Symbolic Comput., 1997, 24(3–4), 235–265

    MATH  MathSciNet  Google Scholar 

  6. Cohen H., A Course in Computational Algebraic Number Theory, Grad. Texts in Math., 138, Springer, Berlin, 1993

    Google Scholar 

  7. Ellingsrud G., Göttsche L., Lehn M., On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom., 2001, 10(1), 81–100

    MATH  MathSciNet  Google Scholar 

  8. Ellingsrud G., Strømme S.A., On the homology of the Hilbert scheme of points in the plane, Invent. Math., 1987, 87(2), 343–352

    Article  MATH  MathSciNet  Google Scholar 

  9. Fujiki A., On the de Rham cohomology group of a compact Kähler symplectic manifold, In: Algebraic Geometry, Sendai, June 24–29, 1985, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, 105–165

    Google Scholar 

  10. Fulton W., Harris J., Representation Theory, Grad. Texts in Math., 129, Springer, New York, 1991

    Google Scholar 

  11. Grigorov G., Jorza A., Patrikis S., Stein W.A., Tarniţă C., Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves, Math. Comp., 2009, 78(268), 2397–2425

    Article  MATH  MathSciNet  Google Scholar 

  12. Harvey D., Hassett B., Tschinkel Yu., Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces, preprint available at

  13. Hassett B., Tschinkel Yu., Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Anal., 2009, 19(4), 1065–1080

    Article  MATH  MathSciNet  Google Scholar 

  14. Hassett B., Tschinkel Yu., Intersection numbers of extremal rays on holomorphic symplectic varieties, Asian J. Math., 2010, 14(3), 303–322

    Article  MATH  MathSciNet  Google Scholar 

  15. Hassett B., Tschinkel Yu., Hodge theory and Lagrangian planes on generalized Kummer fourfolds, preprint availabe at

  16. Lehn M., Sorger C., The cup product of Hilbert schemes for K3 surfaces, Invent. Math., 2003, 152(2), 305–329

    Article  MATH  MathSciNet  Google Scholar 

  17. Looijenga E., Peters C., Torelli theorems for Kähler K3 surfaces, Compositio Math., 1980/81, 42(2), 145–186

    MATH  MathSciNet  Google Scholar 

  18. Markman E., On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Algebr. Geom., 2008, 17(1), 29–99

    Article  MATH  MathSciNet  Google Scholar 

  19. Markman E., The Beauville-Bogomolov class as a characteristic class, preprint availabe at

  20. Markman E., Private communication

  21. Mongardi G., A note on the Kähler and Mori cones of manifolds of K3[n] type, preprint available at

  22. Ran Z., Hodge theory and deformations of maps, Compositio Math., 1995, 97(3), 309–328

    MATH  MathSciNet  Google Scholar 

  23. Stein W.A. et al., Sage Mathematics Software, Version 5.2, The Sage Development Team, 2013, available at

    Google Scholar 

  24. Voisin C., Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, In: Complex Projective Geometry, Trieste, June 19–24, Bergen, July 3–6, 1989, London Math. Soc. Lecture Note Ser., 179, Cambridge University Press, Cambridge, 1992, 294–303

    Google Scholar 

  25. The PARI Group, Bordeaux, PARI/GP, Version 2.5.4, 2012, available at

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Correspondence to Benjamin Bakker.

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Bakker, B., Jorza, A. Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type. centr.eur.j.math. 12, 952–975 (2014).

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  • 14C25
  • 14G05
  • 14J282


  • Holomorphic symplectic variety
  • Cone of curves