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2012 | 10 | 4 | 1472-1485
Tytuł artykułu

Symplectic involutions on deformations of K3[2]

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let X be a hyperkähler manifold deformation equivalent to the Hilbert square of a K3 surface and let φ be an involution preserving the symplectic form. We prove that the fixed locus of φ consists of 28 isolated points and one K3 surface, and moreover that the anti-invariant lattice of the induced involution on H 2(X, ℤ) is isomorphic to E 8(−2). Finally we show that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
4
Strony
1472-1485
Opis fizyczny
Daty
wydano
2012-08-01
online
2012-05-31
Twórcy
Bibliografia
  • [1] Amerik E., On an automorphism of Hilb[2] of certain K3 surfaces, Proc. Edinb. Math. Soc., 2011, 54(1), 1–7 http://dx.doi.org/10.1017/S0013091509001138
  • [2] Beauville A., Some remarks on Kähler manifolds with c 1 = 0, In: Classification of Algebraic and Analytic Manifolds, Katata, July 7–13, 1982, Progr. Math., 39, Birkäuser, Boston, 1983, 1–26
  • [3] Beauville A., Antisymplectic involutions of holomorphic symplectic manifolds, J. Topol., 2011, 4(2), 300–304 http://dx.doi.org/10.1112/jtopol/jtr002
  • [4] Boissière S., Automorphismes naturels de l’espace de Douady de points sur une surface, preprint available at http://arxiv.org/abs/0905.4367
  • [5] Boissière S., Nieper-Wißkirchen M., Sarti A., Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties, J. Math. Pures Appl., 2011, 95(5), 553–563
  • [6] Boissière S., Sarti A., A note on automorphisms and birational transformations of holomorphic symplectic manifolds, Proc. Amer. Math. Soc. (in press), DOI: 10.1090/S0002-9939-2012-11277-8
  • [7] Camere C., Symplectic involutions of holomorphic symplectic fourfolds, preprint available at http://arxiv.org/abs/1010.2607
  • [8] Gritsenko V.A., Hulek K., Sankaran G., Moduli of K3 surfaces and irreducible symplectic manifolds, In: Handbook of Moduli, International Press (in press)
  • [9] Huybrechts D., Compact hyperkähler manifolds, In: Calabi-Yau Manifolds and Related Geometries, Nordfjordeid, June, 2001, Universitext, Springer, Berlin, 2003, 161–225 http://dx.doi.org/10.1007/978-3-642-19004-9_3
  • [10] Huybrechts D., A global Torelli theorem for hyperkähler manifolds (after Verbitsky), preprint available at http://arxiv.org/abs/1106.5573
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  • [12] Markman E., A survey of Torelli and monodromy results for holomorphic-symplectic varieties, In: Complex and Differential Geometry, Hannover, September 14–18, 2009, Springer Proc. Math., 8, Springer, Heidelberg, 2011, 257–322
  • [13] Markman E., Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections, preprint available at http://arxiv.org/abs/0912.4981
  • [14] Morrison D.R., On K3 surfaces with large Picard number, Invent. Math., 1984, 75(1), 105–121 http://dx.doi.org/10.1007/BF01403093
  • [15] Mukai S., Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math., 1988, 94(1), 183–221 http://dx.doi.org/10.1007/BF01394352
  • [16] Namikawa Y., Deformation theory of singular symplectic n-folds, Math. Ann., 2011, 319(3), 597–623 http://dx.doi.org/10.1007/PL00004451
  • [17] Nikulin V.V., Finite groups of automorphisms of Kählerian surfaces of type K3, Uspehi Mat. Nauk, 1976, 31(2), 223–224 (in Russian)
  • [18] Nikulin V.V., Integral symmetric bilinear forms and some of their applications, Izv. Math., 1980, 14(1), 103–167 http://dx.doi.org/10.1070/IM1980v014n01ABEH001060
  • [19] O’Grady K.G., Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics, Duke Math. J., 2006, 134(1), 99–137 http://dx.doi.org/10.1215/S0012-7094-06-13413-0
  • [20] O’Grady K.G., Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal., 2005, 15(6), 1223–1274 http://dx.doi.org/10.1007/s00039-005-0538-3
  • [21] Oguiso K., Automorphism of hyperkähler manifolds in the view of topological entropy, In: Algebraic Geometry, Seoul, July 5–9, 2004, Contemp. Math., 422, 2007, American Mathematical Society, Providence, 173–185
  • [22] Verbitsky M., A global Torelli theorem for hyperkähler manifolds, preprint available at http://arxiv.org/abs/0908.4121
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-012-0073-z
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