Jordan property for groups of bimeromorphic automorphisms of compact Kähler threefolds

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

Let $X$ be a nonuniruled compact Kähler space of dimension $3$. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.Bibliography: 29 titles.

About the authors

Alexey Sergeevich Golota

Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE); Steklov Mathematical Institute of Russian Academy of Sciences

without scientific degree

References

  1. J. Deserti, The Cremona group and its subgroups, Math. Surveys Monogr., 252, Amer. Math. Soc., Providence, RI, 2021, xii+187 pp.
  2. C. Jordan, “Memoire sur les equations differentielles lineaires à integrale algebrique”, J. Reine Angew. Math., 1878:84 (1878), 89–215
  3. V. L. Popov, “On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties”, Affine algebraic geometry: the Russell festschrift (McGill Univ., Montreal, QC, 2009), CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011, 289–311
  4. Sheng Meng, De-Qi Zhang, “Jordan property for non-linear algebraic groups and projective varieties”, Amer. J. Math., 140:4 (2018), 1133–1145
  5. Jin Hong Kim, “Jordan property and automorphism groups of normal compact Kahler varieties”, Commun. Contemp. Math., 20:3 (2018), 1750024, 9 pp.
  6. Sheng Meng, F. Perroni, De-Qi Zhang, “Jordan property for automorphism groups of compact spaces in Fujiki's class $mathscr{C}$”, J. Topol., 15:2 (2022), 806–814
  7. V. L. Popov, “The Jordan property for Lie groups and automorphism groups of complex spaces”, Math. Notes, 103:5 (2018), 811–819
  8. J.-P. Serre, “Bounds for the orders of the finite subgroups of $G(k)$”, Group representation theory, EPFL Press, Lausanne, 2007, 405–450
  9. Yu. G. Zarhin, “Theta groups and products of abelian and rational varieties”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 299–304
  10. Yu. Prokhorov, C. Shramov, “Jordan property for groups of birational selfmaps”, Compos. Math., 150:12 (2014), 2054–2072
  11. Yu. Prokhorov, C. Shramov, “Jordan property for Cremona groups”, Amer. J. Math., 138:2 (2016), 403–418
  12. Yu. Prokhorov, C. Shramov, “Automorphism groups of compact complex surfaces”, Int. Math. Res. Not. IMRN, 2021:14 (2021), 10490–10520
  13. Yu. Prokhorov, C. Shramov, “Finite groups of birational selfmaps of threefolds”, Math. Res. Lett., 25:3 (2018), 957–972
  14. Ю. Г. Прохоров, К. А. Шрамов, “Конечные группы бимероморфных автоморфизмов унилинейчатых трехмерных кэлеровых многообразий”, Изв. РАН. Сер. матем., 84:5 (2020), 169–196
  15. Ю. Г. Прохоров, К. А. Шрамов, “Конечные группы бимероморфных автоморфизмов неунилинейчатых трехмерных кэлеровых многообразий”, Матем. сб., 213:12 (2022), 86–108
  16. A. Fujiki, “A theorem on bimeromorphic maps of Kähler manifolds and its applications”, Publ. Res. Inst. Math. Sci., 17:2 (1981), 735–754
  17. A. Höring, T. Peternell, “Minimal models for Kähler threefolds”, Invent. Math., 203:1 (2016), 217–264
  18. Ч. Кэртис, И. Райнер, Теория представлений конечных групп и ассоциативных алгебр, Наука, М., 1969, 668 с.
  19. R. Elkik, “Rationalite des singularites canoniques”, Invent. Math., 64:1 (1981), 1–6
  20. J. Kollar, Sh. Mori, Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens, A. Corti, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998, viii+254 pp.
  21. S. Boucksom, “Divisorial Zariski decompositions on compact complex manifolds”, Ann. Sci. Ecole Norm. Sup. (4), 37:1 (2004), 45–76
  22. S. Boucksom, J.-P. Demailly, M. Păun, T. Peternell, “The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension”, J. Algebraic Geom., 22:2 (2013), 201–248
  23. M. Brunella, “A positivity property for foliations on compact Kähler manifolds”, Internat. J. Math., 17:1 (2006), 35–43
  24. H. Grauert, R. Remmert, Coherent analytic sheaves, Grundlehren Math. Wiss., 265, Springer-Verlag, Berlin, 1984, xviii+249 pp.
  25. M. Hanamura, “On the birational automorphism groups of algebraic varieties”, Compos. Math., 63:1 (1987), 123–142
  26. J. Kollar, “Flops”, Nagoya Math. J., 113 (1989), 15–36
  27. J. R. King, “The currents defined by analytic varieties”, Acta Math., 127:3-4 (1971), 185–220
  28. Jia Jia, Sheng Meng, Moishezon manifolds with no nef and big classes
  29. Junyan Cao, A. Höring, “Rational curves on compact Kähler manifolds”, J. Differential Geom., 114:1 (2020), 1–39

Supplementary files

Supplementary Files
Action
1. JATS XML

Copyright (c) 2023 Golota A.S.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).