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Construction of automorphisms of hyperkähler manifolds

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

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 [email protected]
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 [email protected]
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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 

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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