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Mathematische Annalen

, Volume 375, Issue 3–4, pp 1783–1822 | Cite as

Finiteness of Klein actions and real structures on compact hyperkähler manifolds

  • Andrea Cattaneo
  • Lie FuEmail author
Article

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.

Mathematics Subject Classification

14P99 14J50 53G26 

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut Camille Jordan UMR 5208Université Claude Bernard Lyon 1Villeurbanne CedexFrance

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