Hostname: page-component-6bf8c574d5-9f2xs Total loading time: 0 Render date: 2025-03-09T13:48:32.485Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of automorphisms of hyperkähler manifolds

Part of: Deformations of analytic structures Forms and linear algebraic groups Global differential geometry Cycles and subschemes

Published online by Cambridge University Press:  31 May 2017

Ekaterina Amerik  and
Misha Verbitsky
Ekaterina Amerik
Affiliation:
Université Paris-11, Laboratoire de Mathématiques, Campus d’Orsay, Bâtiment 425, 91405 Orsay, France Laboratory of Algebraic Geometry, National Research University HSE, Department of Mathematics, 6 Usacheva Str., Moscow, Russia email Ekaterina.Amerik@gmail.com
Misha Verbitsky
Affiliation:
Laboratory of Algebraic Geometry, National Research University HSE, Department of Mathematics, 6 Usacheva Str., Moscow, Russia Université Libre de Bruxelles, CP 218, Bd du Triomphe, 1050 Brussels, Belgium email verbit@mccme.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_{2}(M)\geqslant 5$, we construct a deformation $M^{\prime }$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$-classes is hyperbolic. If $b_{2}(M)\geqslant 14$, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

Keywords

hyperkähler manifoldKähler conequadratic forms

MSC classification

Primary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 32G13: Analytic moduli problems
Secondary: 14C22: Picard groups 11E12: Quadratic forms over global rings and fields
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 8 , August 2017 , pp. 1610 - 1621
Copyright
© The Authors 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amerik, E. and Verbitsky, M., Morrison–Kawamata cone conjecture for hyperkähler manifolds, Ann. Sci. Éc. Norm. Supér. (4), to appear, arXiv:1408.3892.Google Scholar
Amerik, E. and Verbitsky, M., Rational curves on hyperkähler manifolds , Int. Math. Res. Not. IMRN 2015 (2015), 13009–13045.Google Scholar
Amerik, E. and Verbitsky, M., Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkähler geometry, Preprint (2016), arXiv:1604.03927.Google Scholar
Bayer, A., Hassett, B. and Tschinkel, Yu., Mori cones of holomorphic symplectic varieties of K3 type , Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), 941–950.CrossRefGoogle Scholar
Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle , J. Differential Geom. 18 (1983), 755–782.Google Scholar
Bogomolov, F. A., On the decomposition of Kähler manifolds with trivial canonical class , Math. USSR-Sb. 22 (1974), 580–583.CrossRefGoogle Scholar
Bogomolov, F. A., Hamiltonian Kähler manifolds , Sov. Math. Dokl. 19 (1978), 1462–1465.Google Scholar
Borel, A. and Harish-Chandra, Arithmetic subgroups of algebraic groups , Ann. of Math. (2) 75 (1962), 485–535.CrossRefGoogle Scholar
Cantat, S., Dynamique des automorphismes des surfaces K3 , Acta Math. 187 (2001), 1–57.CrossRefGoogle Scholar
Cantat, S. and Dupont, C., Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy, Preprint (2014), arXiv:1410.1202.Google Scholar
Dickson, L. E., Introduction to the theory of numbers (Dover, New York, 1954).Google Scholar
Fujiki, A., On the de Rham cohomology group of a compact Kähler symplectic manifold , Adv. Stud. Pure Math. 10 (1987), 105–165.CrossRefGoogle Scholar
Hassett, B. and Tschinkel, Yu., Extremal rays and automorphisms of holomorphic symplectic varieties, Preprint (2015), arXiv:1506.08153.Google Scholar
Huybrechts, D., Finiteness results for hyperkähler manifolds , J. Reine Angew. Math. 558 (2003), 15–22.Google Scholar
Kapovich, M., Kleinian groups in higher dimensions , in Geometry and dynamics of groups and spaces. In memory of Alexander Reznikov, Progress in Mathematics, vol. 265, eds Kapranov, M. et al. (Birkhauser, Basel, 2007), 485–562, available athttp://www.math.ucdavis.edu/%7Ekapovich/EPR/klein.pdf.Google Scholar
Markman, E., A survey of Torelli and monodromy results for holomorphic–symplectic varieties , in Proc. conf. on complex and differential geometry, Springer Proceedings in Mathematics, vol. 8 (Springer, Berlin, Heidelberg, 2011), 257–322.CrossRefGoogle Scholar
McMullen, C. T., Dynamics on K3 surfaces: Salem numbers and Siegel disks , J. Reine Angew. Math. 545 (2002), 201–233.Google Scholar
Meyer, A., Mathematische Mittheilungen , Vierteljahrschr. Naturforsch. Ges. Zürich 29 (1884), 209–222.Google Scholar
Nikulin, V., Integral symmetric bilinear forms and some of their applications , Math. USSR Izv. 14 (1980), 103–167.CrossRefGoogle Scholar
Oguiso, K., Some aspects of explicit birational geometry inspired by complex dynamics , in Proc. int. congress of mathematicians, Seoul, Vol. II (Kyung Moon SA, Seoul, 2014), 695–721.Google Scholar
Participants of Mathoverflow, 2-dimensional sublattices with all vectors having very big square (in absolute value) (1 September 2015), http://mathoverflow.net/questions/215636/2-dimensional-sublattices-with-all-vectors-having-very-big-square-in-absolute-v/.Google Scholar
Serre, J.-P., A course in arithmetic, Graduate Texts in Mathematics, vol. 7 (Springer, New York, 1973).CrossRefGoogle Scholar
Verbitsky, M., Quaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds , Compos. Math. 143 (2007), 1576–1592.CrossRefGoogle Scholar
Verbitsky, M., A global Torelli theorem for hyperkähler manifolds , Duke Math. J. 162 (2013), 2929–2986.CrossRefGoogle Scholar
Verbitsky, M., Ergodic complex structures on hyperkähler manifolds , Acta Math. 215 (2015), 161–182.CrossRefGoogle Scholar