Finite groups of~bimeromorphic selfmaps of~uniruled K\"ahler threefolds
Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 978-1001.

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We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.
Keywords: Kähler manifold, rationally connected fibration.
Mots-clés : Jordan group, bimeromorphic map
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Yu. G. Prokhorov; K. A. Shramov. Finite groups of~bimeromorphic selfmaps of~uniruled K\"ahler threefolds. Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 978-1001. http://geodesic.mathdoc.fr/item/IM2_2020_84_5_a6/

[1] 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 | DOI | MR | Zbl

[2] Ch. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Pure Appl. Math., XI, Interscience Publishers, a division of John Wiley Sons, New York–London, 1962, xiv+685 pp. | MR | MR | Zbl

[3] J.-P. Serre, “A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field”, Mosc. Math. J., 9:1 (2009), 183–198 | DOI | MR | Zbl

[4] J.-P. Serre, “Le groupe de Cremona et ses sous-groupes finis”, Séminaire Bourbaki, Exposés 997–1011, v. 2008/2009, Astérisque, 332, Soc. Math. France, Paris, 2010, Exp. No. 1000, vii, 75–100 | MR | Zbl

[5] Yu. G. Zarhin, “Theta groups and products of Abelian and rational varieties”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 299–304 | DOI | MR | Zbl

[6] Yu. Prokhorov, C. Shramov, “Jordan property for Cremona groups”, Amer. J. Math., 138:2 (2016), 403–418 | DOI | MR | Zbl

[7] Yu. Prokhorov, C. Shramov, “Jordan property for groups of birational selfmaps”, Compos. Math., 150:12 (2014), 2054–2072 | DOI | MR | Zbl

[8] C. Birkar, Singularities of linear systems and boundedness of Fano varieties, 2016, arXiv: 1609.05543

[9] E. Yasinsky, “The Jordan constant for Cremona group of rank $2$”, Bull. Korean Math. Soc., 54:5 (2017), 1859–1871 | DOI | MR | Zbl

[10] Yu. Prokhorov, C. Shramov, “Jordan constant for Cremona group of rank 3”, Mosc. Math. J., 17:3 (2017), 457–509 | DOI | MR | Zbl

[11] Yu. Prokhorov, C. Shramov, “$p$-subgroups in the space Cremona group”, Math. Nachr., 291:8-9 (2018), 1374–1389 | DOI | MR | Zbl

[12] V. L. Popov, “Three plots about the Cremona groups”, Izv. Math., 83:4 (2019), 830–859 | DOI | DOI | MR | Zbl

[13] V. L. Popov, “Finite subgroups of diffeomorphism groups”, Proc. Steklov Inst. Math., 289 (2015), 221–226 | DOI | DOI | MR | Zbl

[14] B. Csikós, L. Pyber, E. Szabó, Diffeomorphism groups of compact 4-manifolds are not always Jordan, 2014, arXiv: 1411.7524

[15] I. Mundet i Riera, “Finite group actions on 4-manifolds with nonzero Euler characteristic”, Math. Z., 282:1-2 (2016), 25–42 | DOI | MR | Zbl

[16] I. Mundet i Riera, “Finite group actions on homology spheres and manifolds with nonzero Euler characteristic”, J. Topol., 12:3 (2019), 744–758 | DOI | Zbl

[17] I. Mundet i Riera, Finite group actions on manifolds without odd cohomology, 2013, arXiv: 1310.6565

[18] I. Mundet i Riera, “Finite groups acting symplectically on $T^2\times S^2$”, Trans. Amer. Math. Soc., 369:6 (2017), 4457–4483 | DOI | MR | Zbl

[19] I. Mundet i Riera, “Finite subgroups of Ham and Symp”, Math. Ann., 370:1-2 (2018), 331–380 | DOI | MR | Zbl

[20] Shengkui Ye, “Symmetries of flat manifolds, Jordan property and the general Zimmer program”, J. Lond. Math. Soc. (2), 100:3 (2019), 1065–1080 | DOI | MR | Zbl

[21] Yu. Prokhorov, C. Shramov, “Finite groups of birational selfmaps of threefolds”, Math. Res. Lett., 25:3 (2018), 957–972 | DOI | MR | Zbl

[22] Yu. Prokhorov, C. Shramov, “Automorphism groups of compact complex surfaces”, Int. Math. Res. Not. IMRN, 2019, rnz124 | DOI

[23] Yu. Prokhorov, C. Shramov, “Automorphism groups of Inoue and Kodaira surfaces”, Asian J. Math., 24:2 (2020), 355–368 | DOI

[24] Yu. G. Prokhorov, K. A. Shramov, “Automorphism groups of Moishezon threefolds”, Math. Notes, 106:4 (2019), 651–655 | DOI | DOI | MR | Zbl

[25] Jin Hong Kim, “Jordan property and automorphism groups of normal compact Kähler varieties”, Commun. Contemp. Math., 20:3 (2018), 1750024, 9 pp. | DOI | MR | Zbl

[26] Yu. G. Zarhin, “Complex tori, theta groups and their Jordan properties”, Proc. Steklov Inst. Math., 307 (2019), 22–50 ; arXiv: 1902.06184 | DOI | DOI | MR | Zbl

[27] 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. | MR | Zbl

[28] Several complex variables VII. Sheaf-theoretical methods in complex analysis, Encyclopaedia Math. Sci., 74, eds. H. Grauert, Th. Peternell, R. Remmert, Springer-Verlag, Berlin, 1994, vi+369 pp. | DOI | MR | MR | Zbl

[29] C. Voisin, Hodge theory and complex algebraic geometry, Transl. from the French, v. I, Cambridge Stud. Adv. Math., 76, Cambridge Univ. Press, Cambridge, 2007, x+322 pp. | DOI | MR | Zbl

[30] K. Kodaira, “On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)”, Ann. of Math. (2), 60:1 (1954), 28–48 | DOI | MR | Zbl

[31] B. G. Moĭshezon, “On $n$-dimensional compact complex varieties with $n$-algebraically independent meromorphic functions. I”, Amer. Math. Soc. Transl. Ser. 2, 63, Amer. Math. Soc., Providence, RI, 1967, 51–177 | DOI | MR | Zbl

[32] C. Shramov, “Fiberwise bimeromorphic maps of conic bundles”, Internat. J. Math., 30:11 (2019), 1950059, 12 pp. | DOI | MR | Zbl

[33] W. P. Barth, K. Hulek, C. A. M. Peters, A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), 4, 2nd enlarged ed., Springer-Verlag, Berlin, 2004, xii+436 pp. | DOI | MR | Zbl

[34] Yu. G. Prokhorov, C. A. Shramov, “Bounded automorphism groups of compact complex surfaces”, Sb. Math., 211:9 (2020) | DOI | DOI | MR

[35] A. Fujiki, “Deformation of uniruled manifolds”, Publ. Res. Inst. Math. Sci., 17:2 (1981), 687–702 | DOI | MR | Zbl

[36] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996, viii+320 pp. | DOI | MR | Zbl

[37] J.-P. Demailly, Th. Peternell, “A Kawamata–Viehweg vanishing theorem on compact Kähler manifolds”, J. Differential Geom., 63:2 (2003), 231–277 | DOI | MR | Zbl

[38] A. Höring, Th. Peternell, “Minimal models for Kähler threefolds”, Invent. Math., 203:1 (2016), 217–264 | DOI | MR | Zbl

[39] E. Bierstone, P. D. Milman, “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant”, Invent. Math., 128:2 (1997), 207–302 | DOI | MR | Zbl

[40] H. Hironaka, “Flattening theorem in complex-analytic geometry”, Amer. J. Math., 97:2 (1975), 503–547 | DOI | MR | Zbl

[41] F. Campana, “On twistor spaces of the class $\mathscr{C}$”, J. Differential Geom., 33:2 (1991), 541–549 | DOI | MR | Zbl

[42] F. Campana, “Connexité rationnelle des variétés de Fano”, Ann. Sci. École Norm. Sup. (4), 25:5 (1992), 539–545 | DOI | MR | Zbl

[43] J. Varouchas, “Kähler spaces and proper open morphisms”, Math. Ann., 283:1 (1989), 13–52 | DOI | MR | Zbl

[44] T. Graber, J. Harris, J. Starr, “Families of rationally connected varieties”, J. Amer. Math. Soc., 16:1 (2003), 57–67 | DOI | MR | Zbl

[45] F. Campana, J. Winkelmann, “Rational connectedness and order of non-degenerate meromorphic maps from $\mathbb C^n$”, Eur. J. Math., 2:1 (2016), 87–95 | DOI | MR | Zbl

[46] A. Fujiki, “On the structure of compact complex manifolds in $\mathscr C$”, Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, 231–302 | DOI | MR | Zbl

[47] F. Campana, Th. Peternell, “Complex threefolds with non-trivial holomorphic $2$-forms”, J. Algebraic Geom., 9:2 (2000), 223–264 | MR | Zbl

[48] F. Campana, A. Höring, Th. Peternell, “Abundance for Kähler threefolds”, Ann. Sci. Éc. Norm. Supér. (4), 49:4 (2016), 971–1025 | DOI | MR | Zbl

[49] Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surveys, 73:3 (2018), 375–456 | DOI | DOI | MR | Zbl

[50] V. G. Sarkisov, “On conic bundle structures”, Math. USSR-Izv., 20:2 (1983), 355–390 | DOI | MR | Zbl

[51] A. A. Avilov, “Existence of standard models of conic fibrations over non-algebraically-closed fields”, Sb. Math., 205:12 (2014), 1683–1695 | DOI | DOI | MR | Zbl

[52] Sh. Mori, Yu. G. Prokhorov, “Threefold extremal curve germs with one non-Gorenstein point”, Izv. Math., 83:3 (2019), 565–612 | DOI | DOI | MR | Zbl

[53] Hsueh-Yung Lin, Algebraic approximations of compact Kähler threefolds, 2017, arXiv: 1710.01083

[54] V. A. Iskovskikh, “On the rationality problem for conic bundles”, Math. USSR-Sb., 72:1 (1992), 105–111 | DOI | MR | Zbl

[55] A. Fujiki, “On the Douady space of a compact complex space in the category $\mathscr C$”, Nagoya Math. J., 85 (1982), 189–211 | DOI | MR | Zbl

[56] T. Bandman, Yu. G. Zarhin, Bimeromorphic automorphisms groups of certain conic bundles, 2019, arXiv: 1910.05849