Email this article Bounded automorphism groups of compact complex surfaces
- Authors: Prokhorov Y.G.1, Shramov C.A.1
-
Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 211, No 9 (2020)
- Pages: 105-118
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133348
- DOI: https://doi.org/10.4213/sm9335
- ID: 133348
Cite item
Open Access
Access granted
Subscription Access
Abstract
We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kähler manifold of nonnegative Kodaira dimension, always has bounded finite subgroups. Bibliography: 23 titles.
Keywords
About the authors
Yuri Gennadievich Prokhorov
Steklov Mathematical Institute of Russian Academy of Sciences
Email: prokhoro@mi-ras.ru
Doctor of physico-mathematical sciences, Professor
Constantin Aleksandrovich Shramov
Steklov Mathematical Institute of Russian Academy of Sciences
Email: costya.shramov@gmail.com
Doctor of physico-mathematical sciences, no status
References
- D. N. Akhiezer, Lie group actions in complex analysis, Aspects Math., E27, Friedr. Vieweg & Sohn, Braunschweig, 1995, viii+201 pp.
- C. Bennett, R. Miranda, “The automorphism groups of the hyperelliptic surfaces”, Rocky Mountain J. Math., 20:1 (1990), 31–37
- C. Birkar, Singularities of linear systems and boundedness of Fano varieties, 2016
- W. P. Barth, K. Hulek, C. A. M. Peters, A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), 4, 2nd ed., Springer-Verlag, Berlin, 2004, xii+436 pp.
- A. Fujiki, “On automorphism groups of compact Kähler manifolds”, Invent. Math., 44:3 (1978), 225–258
- D. Huybrechts, Lectures on K3 surfaces, Cambridge Stud. Adv. Math., 158, Cambridge Univ. Press, Cambridge, 2016, xi+485 pp.
- Jin Hong Kim, “Jordan property and automorphism groups of normal compact Kähler varieties”, Commun. Contemp. Math., 20:3 (2018), 1750024, 9 pp.
- K. Kodaira, “On compact analytic surfaces. II”, Ann. of Math. (2), 77:3 (1963), 563–626
- J. Kollar, “The structure of algebraic threefolds: an introduction to Mori's program”, Bull. Amer. Math. Soc. (N.S.), 17:2 (1987), 211–273
- R. Miranda, The basic theory of elliptic surfaces, Notes of lectures, Dottorato Ric. Mat., ETS Editrice, Pisa, 1989, vi+108 pp.
- Sh. Mukai, Yu. Namikawa, “Automorphisms of Enriques surfaces which act trivially on the cohomology groups”, Invent. Math., 77:3 (1984), 383–397
- Yu. Prokhorov, C. Shramov, “Jordan property for groups of birational selfmaps”, Compos. Math., 150:12 (2014), 2054–2072
- Yu. Prokhorov, C. Shramov, “Automorphism groups of compact complex surfaces”, Int. Math. Res. Not. IMRN, Publ. online: 2019, rnz124, 22 pp.
- Yu. Prokhorov, C. Shramov, “Finite groups of birational selfmaps of threefolds”, Math. Res. Lett., 25:3 (2018), 957–972
- J. Sawon, Isotrivial elliptic $K3$ surfaces and Lagrangian fibrations, 2014
- J.-P. Serre, “Bounds for the orders of the finite subgroups of $G(k)$”, Group representation theory, EPFL Press, Lausanne, 2007, 405–450
- C. Shramov, “Finite groups acting on elliptic surfaces”, Eur. J. Math., Publ. online: 2019, 12 pp.
- C. Shramov, V. Vologodsky, Automorphisms of pointless surfaces, 2018
- K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Notes written in collaboration with P. Cherenack, Lecture Notes in Math., 439, Springer-Verlag, Berlin–New York, 1975, xix+278 pp.
- В. В. Никулин, “Классификация решеток Пикара К3-поверхностей”, Изв. РАН. Сер. матем., 82:4 (2018), 115–177
- M. Maruyama, “On automorphism groups of ruled surfaces”, J. Math. Kyoto Univ., 11 (1971), 89–112
- В. В. Пржиялковский, И. А. Чельцов, К. А. Шрамов, “Трехмерные многообразия Фано с бесконечными группами автоморфизмов”, Изв. РАН. Сер. матем., 83:4 (2019), 226–280
- А. В. Пухликов, “Автоморфизмы некоторых аффинных дополнений в проективном пространстве”, Матем. сб., 209:2 (2018), 138–152
Supplementary files
