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Jordan Groups and Automorphism Groups of Algebraic Varieties

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Automorphisms in Birational and Affine Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 79))

Abstract

The first section of this paper is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. The appendix contains formulations of some open problems and the relevant comments.

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Notes

  1. 1.

    It is proved in the recent preprint I. Mundet i Riera, Finite group actions on spheres, Euclidean spaces, and compact manifolds with χ ≠ 0 (March 2014) [arXiv:1403.0383] that if M is a sphere, an Euclidean space \(\mathbb{R}^{n}\), or a compact manifold (possibly with boundary) with nonzero Euler characteristic, then Diff(M) is Jordan.

  2. 2.

    Recently I found that in some papers toral varieties are called very affine varieties.

  3. 3.

    The answer is obtained in the recent preprint I. Mundet i Riera, Finite group actions on spheres, Euclidean spaces, and compact manifolds with χ ≠ 0 (March 2014) [arXiv:1403.0383] (see also the footnote at the end of Sect. 1.2.4): \(\mathrm{Diff}(\mathbb{R}^{n})\) is Jordan for every n. Given Theorem 3(1)(i), this yields the following

    Theorem. Aut(A n) is Jordan for every n.

  4. 4.

    See the footnote at the end of Sect. 2.2.3

  5. 5.

    It is proved in the recent preprint T.I. Bandman, Y.G. Zarhin, Jordan groups and algebraic surfaces (April 2014) [arXiv:1404.1581] that if X is birationally isomophic P 1 × E, where E is an elliptic curve, then Aut(X) is Jordan. The proof is based on the other recent preprint Y.G. Zarhin, Jordan groups and elliptic ruled surfaces (January 2014) [arXiv:1401.7596], where this is proved for projective X. Given Theorem 11, we then obtain

    Theorem. If X is a variety of dimension ⩽2, then Aut(X) is Jordan.

  6. 6.

    See the footnote at the end of Sect. 2.2.3

  7. 7.

    The proof in [53] should be corrected as follows. Assume that there is a faithful action of G of a smooth projective curve Y and a dominant G-equivariant morphism \(\varphi: X \rightarrow Y\) of degree n > 1. By the construction, X and Y have the same genus g > 1, and the Hurwitz formula yields that the number of branch points of \(\varphi\) (counted with positive multiplicities) is the integer \((n - 1)(2 - 2g)\). But the latter is negative—a contradiction.

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Acknowledgements

Supported by grants , , and the programme Contemporary Problems of Theoretical Mathematics of the Branch of Mathematics in the Russian Academy of Sciences.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow, 119991, Russia

    Vladimir L. Popov

  2. National Research University Higher School of Economics, Myasnitskaya 20, Moscow, 101000, Russia

    Vladimir L. Popov

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  1. Vladimir L. Popov
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Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom

    Ivan Cheltsov

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Ciro Ciliberto

  3. Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany

    Hubert Flenner

  4. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    James McKernan

  5. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Yuri G. Prokhorov

  6. Institute Fourier, UMR 5582, CNRS-UJF, University of Grenoble I, Saint Martin D’Hères, France

    Mikhail Zaidenberg

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Popov, V.L. (2014). Jordan Groups and Automorphism Groups of Algebraic Varieties. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_11

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