Abstract
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 = ℓ of a line in a smooth Lagrangian n-plane ℙn must satisfy (ℓ,ℓ) = −(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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References
Bayer A., Hassett B., Tschinkel Yu., Mori cones of holomorphic symplectic varieties of K3 type, preprint available at http://arxiv.org/abs/1307.2291
Bayer A., Macrì E., Projective and birational geometry of Bridgeland moduli spaces, preprint avaliable at http://arxiv.org/abs/1203.4613
Bayer A., Macrì E., MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, preprint available at http://arxiv.org/abs/1301.6968
Beauville A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom., 1983, 18(4), 755–782
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
Cohen H., A Course in Computational Algebraic Number Theory, Grad. Texts in Math., 138, Springer, Berlin, 1993
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
Ellingsrud G., Strømme S.A., On the homology of the Hilbert scheme of points in the plane, Invent. Math., 1987, 87(2), 343–352
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
Fulton W., Harris J., Representation Theory, Grad. Texts in Math., 129, Springer, New York, 1991
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
Harvey D., Hassett B., Tschinkel Yu., Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces, preprint available at http://arxiv.org/abs/1011.1285
Hassett B., Tschinkel Yu., Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Anal., 2009, 19(4), 1065–1080
Hassett B., Tschinkel Yu., Intersection numbers of extremal rays on holomorphic symplectic varieties, Asian J. Math., 2010, 14(3), 303–322
Hassett B., Tschinkel Yu., Hodge theory and Lagrangian planes on generalized Kummer fourfolds, preprint availabe at http://arxiv.org/abs/1004.0046
Lehn M., Sorger C., The cup product of Hilbert schemes for K3 surfaces, Invent. Math., 2003, 152(2), 305–329
Looijenga E., Peters C., Torelli theorems for Kähler K3 surfaces, Compositio Math., 1980/81, 42(2), 145–186
Markman E., On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Algebr. Geom., 2008, 17(1), 29–99
Markman E., The Beauville-Bogomolov class as a characteristic class, preprint availabe at http://arxiv.org/abs/1105.3223
Markman E., Private communication
Mongardi G., A note on the Kähler and Mori cones of manifolds of K3[n] type, preprint available at http://arxiv.org/abs/1307.0393
Ran Z., Hodge theory and deformations of maps, Compositio Math., 1995, 97(3), 309–328
Stein W.A. et al., Sage Mathematics Software, Version 5.2, The Sage Development Team, 2013, available at http://www.sagemath.org
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
The PARI Group, Bordeaux, PARI/GP, Version 2.5.4, 2012, available at http://pari.math.u-bordeaux.fr
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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). https://doi.org/10.2478/s11533-013-0389-3
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DOI: https://doi.org/10.2478/s11533-013-0389-3