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Construction of automorphisms of hyperkähler manifolds

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

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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