Uniruledness of orthogonal modular varieties

Gritsenko, V.; Hulek, K.

doi:10.1090/S1056-3911-2014-00632-9

Authors: V. Gritsenko and K. Hulek
Journal: J. Algebraic Geom. 23 (2014), 711-725
DOI: https://doi.org/10.1090/S1056-3911-2014-00632-9
Published electronically: February 21, 2014
MathSciNet review: 3263666
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Abstract | References | Additional Information

Abstract: A strongly reflective modular form with respect to an orthogonal group of signature determines a Lorentzian Kac-Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than , then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised $\textup {K3}$ surfaces. Finally, we prove that the moduli space of Kummer surfaces associated to -polarised abelian surfaces is uniruled.

[BPT] F. Bogomolov, T. Petrov, and Y. Tschinkel, Rationality of moduli of elliptic fibrations with fixed monodromy, Geom. Funct. Anal. 12 (2002), no. 6, 1105–1160. MR 1952925, https://doi.org/10.1007/s00039-002-1105-9
[B] Richard E. Borcherds, Automorphic forms on 𝑂_{𝑠+2,2}(𝑅) and infinite products, Invent. Math. 120 (1995), no. 1, 161–213. MR 1323986, https://doi.org/10.1007/BF01241126
[BKPS] Richard E. Borcherds, Ludmil Katzarkov, Tony Pantev, and N. I. Shepherd-Barron, Families of 𝐾3 surfaces, J. Algebraic Geom. 7 (1998), no. 1, 183–193. MR 1620702
[BDPP] S. Boucksom, J.-P. Demailly, M. Paun, and Th. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22 (2013), 201-248., https://doi.org/10.1090/S1056-3911-2012-00574-8.
[CS] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369
[Do] I. V. Dolgachev, Mirror symmetry for lattice polarized 𝐾3 surfaces, J. Math. Sci. 81 (1996), no. 3, 2599–2630. Algebraic geometry, 4. MR 1420220, https://doi.org/10.1007/BF02362332
[EZ] Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735
[GG] B. Grandpîerre and V. Gritsenko, The baby functions of the Borcherds form $\Phi _{12}$ . In preparation..
[G1] V. Gritsenko, Modular forms and moduli spaces of abelian and 𝐾3 surfaces, Algebra i Analiz 6 (1994), no. 6, 65–102 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 6 (1995), no. 6, 1179–1208. MR 1322120
[G2] Valeri Gritsenko, Irrationality of the moduli spaces of polarized abelian surfaces, Internat. Math. Res. Notices 6 (1994), 235 ff., approx. 9 pp.}, issn=1073-7928, review=\MR{1277050}, doi=10.1155/S1073792894000267,.
[G3] V. Gritsenko, Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms, Algebra i Analiz 11 (1999), no. 5, 100–125; English transl., St. Petersburg Math. J. 11 (2000), no. 5, 781–804. MR 1734348
[G4] V. Gritsenko, Reflective modular forms in algebraic geometry, 28. arXiv:1005.3753
[GH1] Valeri Gritsenko and Klaus Hulek, Minimal Siegel modular threefolds, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 3, 461–485. MR 1607981, https://doi.org/10.1017/S0305004197002259
[GH2] V. Gritsenko and K. Hulek, Commutator coverings of Siegel threefolds, Duke Math. J. 94 (1998), no. 3, 509–542. MR 1639531, https://doi.org/10.1215/S0012-7094-98-09421-2
[GHS1] V. A. Gritsenko, K. Hulek, and G. K. Sankaran, The Kodaira dimension of the moduli of 𝐾3 surfaces, Invent. Math. 169 (2007), no. 3, 519–567. MR 2336040, https://doi.org/10.1007/s00222-007-0054-1
[GHS2] V. Gritsenko, K. Hulek, and G. K. Sankaran, Moduli spaces of irreducible symplectic manifolds, Compos. Math. 146 (2010), no. 2, 404–434. MR 2601632, https://doi.org/10.1112/S0010437X0900445X
[GHS3] V. Gritsenko, K. Hulek, and G. K. Sankaran, Moduli spaces of $\mathrm {K3}$ surfaces and holomorphic symplectic varieties: Handbook of Moduli (ed. G. Farkas and I. Morrison), Advanced Lect. in Math., IP, vol. 1, Somerville, MA, 2012.
[GHS4] V. Gritsenko, K. Hulek, and G. K. Sankaran, Abelianisation of orthogonal groups and the fundamental group of modular varieties, J. Algebra 322 (2009), no. 2, 463–478. MR 2529099, https://doi.org/10.1016/j.jalgebra.2009.01.037
[GN1] Valeri A. Gritsenko and Viacheslav V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras. II, Internat. J. Math. 9 (1998), no. 2, 201–275. MR 1616929, https://doi.org/10.1142/S0129167X98000117
[GN2] V. A. Gritsenko and V. V. Nikulin, On the classification of Lorentzian Kac-Moody algebras, Uspekhi Mat. Nauk 57 (2002), no. 5(347), 79–138 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 5, 921–979. MR 1992083, https://doi.org/10.1070/RM2002v057n05ABEH000553
[GN3] Valeri A. Gritsenko and Viacheslav V. Nikulin, The arithmetic mirror symmetry and Calabi-Yau manifolds, Comm. Math. Phys. 210 (2000), no. 1, 1–11. MR 1748167, https://doi.org/10.1007/s002200050769
[Ma1] Shouhei Ma, Rationality of the moduli spaces of -elementary $\mathrm {K3}$ surfaces, 57. arXiv:1110.5110
[Ma2] Shouhei Ma, The unirationality of the moduli spaces of 2-elementary 𝐾3 surfaces, Proc. Lond. Math. Soc. (3) 105 (2012), no. 4, 757–786. With an appendix by Ken-Ichi Yoshikawa. MR 2989803, https://doi.org/10.1112/plms/pds008
[Mi] Yoichi Miyaoka, Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 245–268. MR 927960
[MM] Yoichi Miyaoka and Shigefumi Mori, A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), no. 1, 65–69. MR 847952, https://doi.org/10.2307/1971387
[N] V. V. Nikulin, Finite groups of automorphisms of Kählerian 𝐾3 surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137 (Russian). MR 544937

Additional Information

V. Gritsenko
Affiliation: Université Lille 1, Laboratoire Paul Painlevé, F-59655 Villeneuve d’Ascq, Cedex, France; and Institut Universitaire de 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

DOI: https://doi.org/10.1090/S1056-3911-2014-00632-9
Received by editor(s): February 16, 2012
Received by editor(s) in revised form: August 24, 1012
Published electronically: February 21, 2014
Article copyright: © Copyright 2014 University Press, Inc.