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Finiteness of Klein actions and real structures on compact hyperkähler manifolds

Abstract

One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperkähler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperkähler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperkähler manifold is finitely presented.

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Fig. 1

Notes

  1. 1.

    This means that the G-homomorphism from \(A'\) to A is injective and identifies \(A'\) with a normal subgroup of A such that the quotient group is isomorphic to \(A''\) via the G-homomorphism from A to \(A''\).

  2. 2.

    Recall that a sequence of morphisms of pointed sets is called exact, if the image of a morphism is equal to the fiber of the next morphism over the base point.

  3. 3.

    A codimension-one face would be called a facet.

  4. 4.

    A hyperbolic lattice is a free abelian group of finite rank endowed with a non-degenerate bilinear form of signature \((1, {{\,\mathrm{rank}\,}}-1)\).

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Acknowledgements

We are grateful to Ekaterina Amerik, Samuel Boissière, Kenneth Brown, Grégoire Menet, Giovanni Mongardi and Jean-Yves Welschinger for helpful discussions. The work started during the second Japanese-European Symposium on symplectic varieties and moduli spaces at Levico Terme in September 2017. We would like to thank the organizers and other participants of the conference.

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Correspondence to Lie Fu.

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Andrea Cattaneo is supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR) and is member of GNSAGA of INdAM. Lie Fu is supported by ECOVA (ANR-15-CE40-0002), HodgeFun (ANR-16-CE40-0011), LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon and Projet Inter-Laboratoire 2017, 2018 by Fédération de Recherche en Mathématiques Rhône-Alpes/Auvergne CNRS 3490.

Communicated by Jean-Yves Welschinger.

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Cattaneo, A., Fu, L. Finiteness of Klein actions and real structures on compact hyperkähler manifolds. Math. Ann. 375, 1783–1822 (2019). https://doi.org/10.1007/s00208-019-01876-7

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Mathematics Subject Classification

  • 14P99
  • 14J50
  • 53G26