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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0389-3,
author = {Benjamin Bakker and Andrei Jorza},
title = {Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {952-975},
zbl = {1307.14014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0389-3}
}
Benjamin Bakker; Andrei Jorza. Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type. Open Mathematics, Tome 12 (2014) pp. 952-975. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0389-3/
[1] Bayer A., Hassett B., Tschinkel Yu., Mori cones of holomorphic symplectic varieties of K3 type, preprint available at http://arxiv.org/abs/1307.2291 | Zbl 06502670
[2] Bayer A., Macrì E., Projective and birational geometry of Bridgeland moduli spaces, preprint avaliable at http://arxiv.org/abs/1203.4613 | Zbl 1314.14020
[3] 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 | Zbl 1308.14011
[4] Beauville A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom., 1983, 18(4), 755–782 | Zbl 0537.53056
[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 | Zbl 0898.68039
[6] Cohen H., A Course in Computational Algebraic Number Theory, Grad. Texts in Math., 138, Springer, Berlin, 1993
[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 | Zbl 0976.14002
[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 http://dx.doi.org/10.1007/BF01389419 | Zbl 0625.14002
[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
[10] Fulton W., Harris J., Representation Theory, Grad. Texts in Math., 129, Springer, New York, 1991 | Zbl 0744.22001
[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 http://dx.doi.org/10.1090/S0025-5718-09-02253-4 | Zbl 1209.11059
[12] 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
[13] Hassett B., Tschinkel Yu., Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Anal., 2009, 19(4), 1065–1080 http://dx.doi.org/10.1007/s00039-009-0022-6 | Zbl 1183.14058
[14] Hassett B., Tschinkel Yu., Intersection numbers of extremal rays on holomorphic symplectic varieties, Asian J. Math., 2010, 14(3), 303–322 http://dx.doi.org/10.4310/AJM.2010.v14.n3.a2 | Zbl 1216.14012
[15] Hassett B., Tschinkel Yu., Hodge theory and Lagrangian planes on generalized Kummer fourfolds, preprint availabe at http://arxiv.org/abs/1004.0046 | Zbl 1296.14008
[16] Lehn M., Sorger C., The cup product of Hilbert schemes for K3 surfaces, Invent. Math., 2003, 152(2), 305–329 http://dx.doi.org/10.1007/s00222-002-0270-7 | Zbl 1035.14001
[17] Looijenga E., Peters C., Torelli theorems for Kähler K3 surfaces, Compositio Math., 1980/81, 42(2), 145–186 | Zbl 0477.14006
[18] Markman E., On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Algebr. Geom., 2008, 17(1), 29–99 http://dx.doi.org/10.1090/S1056-3911-07-00457-2 | Zbl 1185.14015
[19] Markman E., The Beauville-Bogomolov class as a characteristic class, preprint availabe at http://arxiv.org/abs/1105.3223
[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 http://arxiv.org/abs/1307.0393
[22] Ran Z., Hodge theory and deformations of maps, Compositio Math., 1995, 97(3), 309–328 | Zbl 0845.14007
[23] Stein W.A. et al., Sage Mathematics Software, Version 5.2, The Sage Development Team, 2013, available at http://www.sagemath.org
[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
[25] The PARI Group, Bordeaux, PARI/GP, Version 2.5.4, 2012, available at http://pari.math.u-bordeaux.fr