Uniqueness of birational structures on Inoue surfaces
Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 873-917.

We prove that the natural (Aff 2 (C),C 2 )-structure on an Inoue surface is the unique (Bir( 2 ), 2 (C))-structure, generalizing a result of Bruno Klingler which asserts that the natural (Aff 2 (C),C 2 )-structure is the unique (PGL 3 (C), 2 (C))-structure.

Nous prouvons que la (Aff 2 (C),C 2 )-structure naturelle sur une surface d’Inoue est l’unique (Bir( 2 ), 2 (C))-structure. Ceci généralise un résultat de Bruno Klingler qui affirme que la (Aff 2 (C),C 2 )-structure est l’unique (PGL 3 (C), 2 (C))-structure.

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DOI: 10.5802/aif.3537
Classification: 57M50, 14E07
Keywords: Inoue surfaces, birational structures, Cremona group
Mot clés : surfaces d’Inoue, structures birationnelles, groupe de Cremona

Zhao, Shengyuan 1

1 Institute for Mathematical Sciences, Stony Brook University, Stony Brook NY 11794-3660 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Zhao, Shengyuan. Uniqueness of birational structures on Inoue surfaces. Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 873-917. doi : 10.5802/aif.3537. https://aif.centre-mersenne.org/articles/10.5802/aif.3537/

[1] Ahlfors, Lars V. Zur Theorie der Überlagerungsflächen, Acta Math., Volume 65 (1935) no. 1, pp. 157-194 | DOI | MR | Zbl

[2] Blanc, Jérémy; Déserti, Julie Degree growth of birational maps of the plane, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 14 (2015) no. 2, pp. 507-533 | MR | Zbl

[3] Blanc, Jérémy; Furter, Jean-Philippe Topologies and structures of the Cremona groups, Ann. Math., Volume 178 (2013) no. 3, pp. 1173-1198 | DOI | MR | Zbl

[4] Blanc, Jérémy; Yasinsky, Egor Quotients of groups of birational transformations of cubic del Pezzo fibrations (2020) (preprint, https://arxiv.org/abs/1907.04696)

[5] Bogomolov, Fedor; Böhning, Christian On uniformly rational varieties, Topology, geometry, integrable systems and mathematical physics. Novikov’s seminar: 2012–2014. Selected papers of the seminar, Moscow, Russia, 2012–2014. Dedicated to S. P. Novikov on the occasion of his 75th birthday, American Mathematical Society, 2014, pp. 33-48 | Zbl

[6] Brunella, Marco Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. Éc. Norm. Supér., Volume 30 (1997) no. 5, pp. 569-594 | DOI | Numdam | MR | Zbl

[7] Brunella, Marco Courbes entières et feuilletages holomorphes, Enseign. Math., Volume 45 (1999) no. 1-2, pp. 195-216 | MR | Zbl

[8] Brunella, Marco Birational geometry of foliations, Monografías de Matemática, IMPA - Instituto de Matemática Pura e Aplicada, 2000, 138 pages | MR | Zbl

[9] Cantat, Serge Dynamique des automorphismes des surfaces K3, Acta Math., Volume 187 (2001) no. 1, pp. 1-57 | DOI | MR | Zbl

[10] Cantat, Serge Sur les groupes de transformations birationnelles des surfaces, Ann. Math., Volume 174 (2011) no. 1, pp. 299-340 | DOI | MR | Zbl

[11] Cantat, Serge Dynamics of automorphisms of compact complex surfaces, Frontiers in complex dynamics (Princeton Mathematical Series), Volume 51, Princeton University Press, 2014, pp. 463-514 | DOI | MR | Zbl

[12] Cantat, Serge The Cremona group, Algebraic geometry: Salt Lake City 2015 (de Fernex, Tommaso et al., eds.) (Proceedings of Symposia in Pure Mathematics), Volume 97, American Mathematical Society, 2018, pp. 101-142 | MR | Zbl

[13] Cantat, Serge; Favre, Charles Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math., Volume 561 (2003), pp. 199-235 | DOI | MR | Zbl

[14] Delzant, Thomas; Py, Pierre Kähler groups, real hyperbolic spaces and the Cremona group. with an appendix by Serge Cantat, Compos. Math., Volume 148 (2012) no. 1, pp. 153-184 | DOI | MR | Zbl

[15] Demailly, Jean-Pierre Variétés hyperboliques et équations différentielles algébriques, Gaz. Math., Soc. Math. Fr. (1997) no. 73, pp. 3-23 | MR | Zbl

[16] Deroin, Bertrand Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive, Ann. Sci. Éc. Norm. Supér., Volume 38 (2005) no. 1, pp. 57-75 | DOI | Numdam | MR | Zbl

[17] Déserti, Julie On solvable subgroups of the Cremona group, Ill. J. Math., Volume 59 (2015) no. 2, pp. 345-358 | MR | Zbl

[18] Diller, Jeffrey; Favre, Charles Dynamics of bimeromorphic maps of surfaces, Am. J. Math., Volume 123 (2001) no. 6, pp. 1135-1169 | DOI | MR | Zbl

[19] Dinh, Tien-Cuong; Sibony, Nessim Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Am. Math. Soc., Volume 18 (2005) no. 2, pp. 291-312 | DOI | MR | Zbl

[20] Dinh, Tien-Cuong; Sibony, Nessim Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebr. Geom., Volume 19 (2010) no. 3, pp. 473-529 | DOI | MR | Zbl

[21] Dloussky, Georges Special birational structures on non-Kählerian complex surfaces, J. Math. Pures Appl., Volume 106 (2016) no. 1, pp. 76-122 | DOI | MR | Zbl

[22] Ehresmann, Charles Sur les espaces localement homogènes, Enseign. Math., Volume 35 (1936), pp. 317-333 | Zbl

[23] Fornæss, John Erik; Sibony, Nessim Riemann surface laminations with singularities, J. Geom. Anal., Volume 18 (2008) no. 2, pp. 400-442 | DOI | MR | Zbl

[24] Ghys, Étienne Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) (Cerveau, Dominique et al., eds.) (Panoramas et Synthèses), Volume 8, Société Mathématique de France, 1999, pp. ix, xi, 49-95 | MR | Zbl

[25] Gromov, Mikhail Oka’s principle for holomorphic sections of elliptic bundles, J. Am. Math. Soc., Volume 2 (1989) no. 4, pp. 851-897 | MR | Zbl

[26] Inoue, Masahisa On surfaces of Class VII 0 , Invent. Math., Volume 24 (1974), pp. 269-310 | DOI | MR | Zbl

[27] Klingler, Bruno Structures affines et projectives sur les surfaces complexes, Ann. Inst. Fourier, Volume 48 (1998) no. 2, pp. 441-477 | DOI | Numdam | MR | Zbl

[28] Kwon, Alice; Sullivan, Dennis Geometry on all prime Three Manifolds (2020) (preprint, https://arxiv.org/abs/1906.10820)

[29] McQuillan, Michael Diophantine approximations and foliations, Publ. Math., Inst. Hautes Étud. Sci. (1998) no. 87, pp. 121-174 | DOI | Numdam | MR | Zbl

[30] Moncet, Arnaud Géométrie et dynamique sur les surfaces algébriques réelles, Ph. D. Thesis, Université de Rennes 1, Rennes, France (2012)

[31] Plante, Joseph F. Foliations with measure preserving holonomy, Ann. Math., Volume 102 (1975) no. 2, pp. 327-361 | DOI | MR | Zbl

[32] Sullivan, Dennis Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., Volume 36 (1976), pp. 225-255 | DOI | MR | Zbl

[33] Teleman, Andrei D. Projectively flat surfaces and Bogomolov’s theorem on class VII 0 surfaces, Int. J. Math., Volume 5 (1994) no. 2, pp. 253-264 | DOI | MR | Zbl

[34] Ueno, Kenji Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, 439, Springer, 1975 | DOI | Zbl

[35] Urech, Christian Subgroups of elliptic elements of the Cremona group, J. Reine Angew. Math., Volume 770 (2021), pp. 27-57 | DOI | MR | Zbl

[36] Zhao, Shengyuan Birational Kleinian groups (2021) (to appear, https://arxiv.org/abs/2103.09350)

[37] Zhao, Shengyuan Centralizers of elements of infinite order in plane Cremona groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 23 (2022) no. 2, pp. 915-957 | DOI | MR | Zbl

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