Finite groups acting on elliptic surfaces

  • Constantin ShramovEmail author
Research Article


We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration.


Elliptic surface Hopf surface Kodaira surface Automorphism group 

Mathematics Subject Classification




The author is grateful to Sergey Gorchinskiy, Stefan Nemirovski, Yuri Prokhorov, and Egor Yasinsky for useful discussions.


  1. 1.
    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces. 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 4. Springer, Berlin (2004)CrossRefGoogle Scholar
  2. 2.
    Bennett, C., Miranda, R.: The automorphism groups of the hyperelliptic surfaces. Rocky Mountain J. Math. 20(1), 31–37 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Birkar, C.: Singularities of linear systems and boundedness of Fano varieties (2016). arXiv:1609.05543
  4. 4.
    Borcea, C.: Moduli for Kodaira surfaces. Compositio Math. 52(3), 373–380 (1984)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Claudon, B., Höring, A., Lin, H.-Y.: The fundamental group of compact Kähler threefolds (2016). arXiv:1612.04224
  6. 6.
    Fujimoto, Y., Nakayama, N.: Compact complex surfaces admitting non-trivial surjective endomorphisms. Tohoku Math. J. (2) 57(3), 395–426 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kato, M.: Topology of Hopf surfaces. J. Math. Soc. Japan 27, 222–238 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kodaira, K.: On the structure of compact complex analytic surfaces I. Amer. J. Math. 86, 751–798 (1964)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kodaira, K.: On the structure of compact complex analytic surfaces II. Amer. J. Math. 88, 682–721 (1966)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Matumoto, T., Nakagawa, N.: Explicit description of Hopf surfaces and their automorphism groups. Osaka J. Math. 37(2), 417–424 (2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Oguiso, K.: Bimeromorphic automorphism groups of non-projective hyperkähler manifolds—a note inspired by C. T. McMullen. J. Differential Geom. 78(1), 163–191 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Prokhorov, Yu., Shramov, C.: Jordan property for groups of birational selfmaps. Compositio Math. 150(12), 2054–2072 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Prokhorov, Yu., Shramov, C.: Automorphism groups of compact complex surfaces (2017). arXiv:1708.03566
  14. 14.
    Prokhorov, Yu., Shramov, C.: Automorphism groups of Inoue and Kodaira surfaces (2018). arXiv:1812.02400
  15. 15.
    Prokhorov, Yu., Shramov, C.: Finite groups of birational selfmaps of threefolds. Math. Res. Lett. 25(3), 957–972 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shramov, C., Vologodsky, V.: Automorphisms of pointless surfaces (2018). arXiv:1807.06477

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia

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