Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T16:21:54.134Z Has data issue: false hasContentIssue false
  • English
  • Français

Moduli spaces of irreducible symplectic manifolds

Part of: Surfaces and higher-dimensional varieties Compact analytic spaces Forms and linear algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 January 2010

V. Gritsenko ,
K. Hulek  and
G. K. Sankaran
V. Gritsenko
Affiliation:
Université Lille 1, Laboratoire Paul Painlevé, F-59655 Villeneuve d’Ascq, Cedex, France (email: valery.gritsenko@math.univ-lille1.fr)
K. Hulek
Affiliation:
Institut für Algebraische Geometrie, Leibniz Universität Hannover, D-30060 Hannover, Germany (email: hulek@math.uni-hannover.de)
G. K. Sankaran
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK (email: gks@maths.bath.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2] manifolds with polarisation of degree 2d and split type is of general type if d≥12.

Keywords

irreducible symplectic manifoldmoduliorthogonal groupsSiegel formula for quadratic forms.

MSC classification

Secondary: 14J15: Moduli, classification 11E25: Sums of squares and representations by other particular quadratic forms 11F55: Other groups and their modular and automorphic forms (several variables) 14J35: $4$-folds 32J27: Compact Kähler manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 404 - 434
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), 755–782.CrossRefGoogle Scholar
[2]Bogomolov, F., Hamiltonian Kählerian manifolds, (Russian) Dokl. Akad. Nauk SSSR 243 (1978), 1101–1104.Google Scholar
[3]Borcherds, R. E., Automorphic forms on Os+2,2(ℝ) and infinite products, Invent. Math. 120 (1995), 161–213.CrossRefGoogle Scholar
[4]Borcherds, R. E., Katzarkov, L., Pantev, T. and Shepherd-Barron, N. I., Families of K3 surfaces, J. Algebraic Geom. 7 (1998), 183–193.Google Scholar
[5]Borel, A., Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differential Geom. 6 (1972), 543–560.CrossRefGoogle Scholar
[6]Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4, 5 et 6 (Hermann, Paris, 1968).Google Scholar
[7]Cohen, H., Sums involving the values at negative integers of L functions of quadratic characters, Math. Ann. 217 (1975), 271–285.CrossRefGoogle Scholar
[8]Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften, vol. 290 (Springer, New York, 1988).CrossRefGoogle Scholar
[9]Debarre, O., Un contre-exemple au théorème de Torelli pour les variétés symplectiques irréductibles, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 681–684.Google Scholar
[10]Debarre, O. and Voisin, C., Hyper-Kähler fourfolds and Grassmann geometry, arXiv:0904.3974 [math.AG], 27 pp.Google Scholar
[11]Eichler, M., Quadratische Formen und orthogonale Gruppen, Grundlehren der mathematischen Wissenschaften, vol. 63 (Springer, Berlin, 1974).CrossRefGoogle Scholar
[12]Freitag, E., Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften, vol. 254 (Springer, Berlin, 1983).CrossRefGoogle Scholar
[13]Fujiki, A., On the de Rham cohomology group of a compact Kähler symplectic manifold, in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 105–165.CrossRefGoogle Scholar
[14]Gritsenko, V. and Hulek, K., Minimal Siegel modular threefolds, Math. Proc. Camb. Philos. Soc. 123 (1998), 461–485.CrossRefGoogle Scholar
[15]Gritsenko, V., Hulek, K. and Sankaran, G. K., The Kodaira dimension of the moduli of K3 surfaces, Invent. Math. 169 (2007), 519–567.CrossRefGoogle Scholar
[16]Gritsenko, V., Hulek, K. and Sankaran, G. K., The Hirzebruch–Mumford volume for the orthogonal group and applications, Doc. Math. 12 (2007), 215–241.CrossRefGoogle Scholar
[17]Gritsenko, V., Hulek, K. and Sankaran, G. K., Hirzebruch–Mumford proportionality and locally symmetric varieties of orthogonal type, Doc. Math. 13 (2008), 1–19.CrossRefGoogle Scholar
[18]Gritsenko, V., Hulek, K. and Sankaran, G. K., Abelianisation of orthogonal groups and the fundamental group of modular varieties, J. Algebra 322 (2009), 463–478.CrossRefGoogle Scholar
[19]Gritsenko, V. and Schulze-Pillot, R., Eisenstein series on four-dimensional hyperbolic space, Acta Arith. 67 (1994), 241–268.CrossRefGoogle Scholar
[20]Hassett, B., Special cubic fourfolds, Compositio Math. 120 (2000), 1–23.CrossRefGoogle Scholar
[21]Hassett, B. and Tschinkel, Y., Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Anal. 19 (2009), 1065–1080.CrossRefGoogle Scholar
[22]Huybrechts, D., Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), 63–113.CrossRefGoogle Scholar
[23]Huybrechts, D., Erratum: Compact hyper-Kähler manifolds: basic results, Invent. Math. 152 (2003), 209–212.CrossRefGoogle Scholar
[24]Huybrechts, D., Compact hyperkähler manifolds, in Calabi–Yau manifolds and related geometries (Nordfjordeid, 2001) (Universitext, Springer, Berlin, 2003), 161–225.CrossRefGoogle Scholar
[25]Huybrechts, D., Finiteness results for compact hyperkähler manifolds, J. Reine Angew. Math. 558 (2003), 15–22.Google Scholar
[26]Huybrechts, D., Moduli spaces of hyperkähler manifolds and mirror symmetry, in Intersection theory and moduli, ICTP Lecture Notes, vol. XIX (Abdus Salam Int. Cent. Theor. Phys., Trieste, 2004), 185–247.Google Scholar
[27]Iwaniec, H., Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
[28]Kawamata, Y., Unobstructed deformations. A remark on a paper of Z. Ran, J. Algebraic Geom. 1 (1992), 183–190.Google Scholar
[29]Kneser, M., Erzeugung ganzzahliger orthogonaler Gruppen durch Spiegelungen, Math. Ann. 255 (1981), 453–462.CrossRefGoogle Scholar
[30]Kollár, J. and Matsusaka, T., Riemann–Roch type inequalities, Amer. J. Math. 105 (1983), 229–252.CrossRefGoogle Scholar
[31]Kondo, S., On the Kodaira dimension of the moduli space of K3 surfaces. II, Compositio Math. 116 (1999), 111–117.CrossRefGoogle Scholar
[32]Malyshev, A. V., On the representation of integers by positive quadratic forms, Trudy Mat. Inst. Steklov 65 (1962), (in Russian).Google Scholar
[33]Markman, E., On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Algebraic Geom. 17 (2008), 29–99.CrossRefGoogle Scholar
[34]Markman, E., Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface, Internat. J. Math. (to appear), math.AG/0601304.Google Scholar
[35]Namikawa, Y., Counter-example to global Torelli problem for irreducible symplectic manifolds, Math. Ann. 324 (2002), 841–845.CrossRefGoogle Scholar
[36]Nikulin, V. V., Integral symmetric bilinear forms and some of their applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177; English translation in Math. USSR, Izvestiya 14 (1980), 103–167.Google Scholar
[37]O’Grady, K., Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512 (1999), 49–117.CrossRefGoogle Scholar
[38]O’Grady, K., A new six-dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003), 435–505.CrossRefGoogle Scholar
[39]O’Grady, K., Irreducible symplectic 4-folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134 (2006), 99–137.Google Scholar
[40]O’Grady, K., Irreducible symplectic 4-folds numerically equivalent to (K3)[2], Commun. Contemp. Math. 10 (2008), 553–608.CrossRefGoogle Scholar
[41]Ran, Z., Deformations of manifolds with torsion or negative canonical bundle, J. Algebraic Geom. 1 (1992), 279–291.Google Scholar
[42]Shimura, G., The representation of integers as sums of squares, Amer. J. Math. 124 (2002), 1059–1081.CrossRefGoogle Scholar
[43]Siegel, C. L., Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), 527–606.CrossRefGoogle Scholar
[44]Szendrői, B., Some finiteness results for Calabi–Yau threefolds, J. Lond. Math. Soc. (2) 60 (1999), 689–699.CrossRefGoogle Scholar
[45]Tian, G., Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, in Mathematical aspects of string theory, Advanced Series in Mathematical Physics, vol. 1, ed. Yau, S.-T. (World Scientific, Singapore, 1987), 629–646.CrossRefGoogle Scholar
[46]Todorov, A. N., The Weil–Petersson geometry of the moduli space of SU(n≥3) (Calabi–Yau) manifolds I, Commun. Math. Phys. 126 (1989), 325–346.CrossRefGoogle Scholar
[47]Viehweg, E., Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 30 (Springer, Berlin, 1995).CrossRefGoogle Scholar
[48]Voisin, C., Théorème de Torelli pour les cubiques de ℙ5, Invent. Math. 86 (1986), 577–601.CrossRefGoogle Scholar
[49]Voisin, C., Géométrie des espaces de modules de courbes et de surfaces K3 [d’après Gritsenko–Hulek–Sankaran, Farkas–Popa, Mukai, Verra …]. Séminaire BOURBAKI 59ème année, 2006–2007, n 981.Google Scholar
[50]Yang, T., An explicit formula for local densities of quadratic forms, J. Number Theory 72 (1998), 309–356.CrossRefGoogle Scholar
[51]Zagier, D., Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, in Modular functions of one variable, VI (Bonn, 1976), Lecture Notes in Mathematics, vol. 627 (Springer, Berlin, 1977), 105–169.CrossRefGoogle Scholar