Levi-flat hypersurfaces and their complement in complex surfaces
Annales de l'Institut Fourier, Volume 67 (2017) no. 6, p. 2423-2462
In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of the complement of the hypersurface are modifications of Stein domains. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse invariant measure. This leads us to conjecture that Levi-flat hypersurfaces in complex algebraic surfaces that are diffeomorphic to a hyperbolic torus bundle over the circle are fibrations by algebraic curves.
Dans ce travail nous étudions les hypersurfaces Levi-plates analytiques dans les surfaces algébriques complexes. Dans un premier temps nous montrons que si leur feuilletage admet une dynamique chaotique (c’est à dire, s’il n’admet pas de mesure transverse invariante) alors les composantes connexes de l’extérieur de l’hypersurface sont des modifications de domaines de Stein. Ceci permet d’étendre le feuilletage CR en un feuilletage algébrique singulier sur la surface complexe ambiante. Nous appliquons ce résultat pour montrer, par l’absurde, qu’une hypersurface Levi-plate analytique qui admet une structure affine transverse dans une surface algébrique complexe possède une mesure transverse invariante. Ceci nous amène à conjecturer que les hypersurfaces Levi-plates dans les surfaces algébriques complexes qui sont difféomorphes à un fibré hyperbolique en tores sur le cercle sont des fibrations par courbes algébriques.
Received : 2016-10-24
Accepted : 2017-04-28
Published online : 2017-12-14
DOI : https://doi.org/10.5802/aif.3139
Classification:  32V40,  32T15,  32D15,  37F75,  37C40
Keywords: complex analysis and complex geometry, theory of foliations, Levi-flat hypersurfaces, invariant measure, Stein manifold, holomorphic convexity, analytic extension.
@article{AIF_2017__67_6_2423_0,
     author = {Canales Gonz\'alez, Carolina},
     title = {Levi-flat hypersurfaces and their complement in complex surfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     pages = {2423-2462},
     doi = {10.5802/aif.3139},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_6_2423_0}
}
Levi-flat hypersurfaces and their complement in complex surfaces. Annales de l'Institut Fourier, Volume 67 (2017) no. 6, pp. 2423-2462. doi : 10.5802/aif.3139. https://aif.centre-mersenne.org/item/AIF_2017__67_6_2423_0/

[1] ArnolʼD, Vladimir Igorevich Chapitres supplémentaires de la théorie des équations différentielles ordinaires, “Mir”, Moscow (1984), 329 pages (Translated from the Russian by Djilali Embarek, Reprint of the 1980 edition) | MR 898218 (88e:58027) | Zbl 0956.34506

[2] Beardon, Alan F. The geometry of discrete groups, Springer, Graduate Texts in Mathematics, Tome 91 (1995), xii+337 pages (Corrected reprint of the 1983 original) | MR 1393195 (97d:22011) | Zbl 0528.30001

[3] Brunella, Marco On the dynamics of codimension one holomorphic foliations with ample normal bundle, Indiana Univ. Math. J., Tome 57 (2008) no. 7, pp. 3101-3113 | Article | MR 2492227 (2010g:32051) | Zbl 1170.37023

[4] Camacho, César; Scárdua, Bruno Holomorphic foliations with Liouvillian first integrals, Ergodic Theory Dyn. Syst., Tome 21 (2001) no. 3, pp. 717-756 (Erratum in ibid. 23 (2003), no. 3, p. 985-987) | Article | MR 1836428 (2002k:37080) | Zbl 1051.37022

[5] Candel, Alberto; Conlon, Lawrence Foliations. II, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 60 (2003), xiv+545 pages | MR 1994394 (2004e:57034) | Zbl 1035.57001

[6] Cartan, Elie Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, Ann. Mat. Pura Appl., Tome 11 (1933) no. 1, pp. 17-90 | Article | MR 1553196 | Zbl 0005.37304

[7] Colţoiu, Mihnea; Mihalache, Nicolae Strongly plurisubharmonic exhaustion functions on 1-convex spaces, Math. Ann., Tome 270 (1985) no. 1, pp. 63-68 | Article | MR 769607 (86a:32037) | Zbl 0533.32009

[8] Cousin, Gaël; Pereira, Jorge Vitório Transversely affine foliations on projective manifolds, Math. Res. Lett., Tome 21 (2014) no. 5, pp. 985-1014 | Article | MR 3294560 | Zbl 1306.32025

[9] Deroin, Bertrand; Dupont, Christophe Topology and dynamics of laminations in surfaces of general type, J. Am. Math. Soc., Tome 29 (2016) no. 2, pp. 495-535 | Article | MR 3454381 | Zbl 1334.32014

[10] Deroin, Bertrand; Kleptsyn, Victor Random conformal dynamical systems, Geom. Funct. Anal., Tome 17 (2007) no. 4, pp. 1043-1105 | Article | MR 2373011 (2010j:37012) | Zbl 1143.37008

[11] Deroin, Bertrand; Kleptsyn, Victor; Navas, Andrés On the question of ergodicity for minimal group actions on the circle, Mosc. Math. J., Tome 9 (2009) no. 2, p. 263-303, back matter | MR 2568439 (2010m:37041) | Zbl 1193.37034

[12] Deroin, Bertrand; Kleptsyn, Victor; Navas, Andrés Towards the solution of some fundamental questions concerning group actions on the circle and codimension-one foliations (2013) (https://arxiv.org/abs/1312.4133v1 )

[13] Diederich, Klas; Ohsawa, Takeo Harmonic mappings and disc bundles over compact Kähler manifolds, Publ. Res. Inst. Math. Sci., Tome 21 (1985) no. 4, pp. 819-833 | Article | MR 817167 (87g:32017) | Zbl 060.32023

[14] Diederich, Klas; Ohsawa, Takeo On the displacement rigidity of Levi flat hypersurfaces—the case of boundaries of disc bundles over compact Riemann surfaces, Publ. Res. Inst. Math. Sci., Tome 43 (2007) no. 1, pp. 171-180 http://projecteuclid.org/euclid.prims/1199403813 | Article | MR 2319541 (2008m:32023) | Zbl 1141.53040

[15] Garnett, Lucy Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., Tome 51 (1983) no. 3, pp. 285-311 | Article | MR 703080 (84j:58099) | Zbl 0524.58026

[16] Ghys, Etienne Flots transversalement affines et tissus feuilletés, Mém. Soc. Math. France (1991) no. 46, pp. 123-150 (Analyse globale et physique mathématique (Lyon, 1989)) | Article | MR 1125840 (92i:57026) | Zbl 0761.57016

[17] Ghys, Etienne; Sergiescu, Vlad Stabilité et conjugaison différentiable pour certains feuilletages, Topology, Tome 19 (1980) no. 2, pp. 179-197 | Article | MR 572582 (81k:57022) | Zbl 0478.57017

[18] Godbillon, Claude Feuilletages : Études géométriques, Birkhäuser, Basel, Progress in Mathematics, Tome 98 (1991), xiv+474 pages | MR 1120547 (93i:57038) | Zbl 0724.58002

[19] Grauert, Hans On Levi’s problem and the imbedding of real-analytic manifolds, Ann. Math., Tome 68 (1958), pp. 460-472 | Article | MR 0098847 (20 #5299) | Zbl 0108.07804

[20] Grauert, Hans Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Tome 146 (1962), pp. 331-368 | Article | MR 0137127 (25 #583) | Zbl 0173.33004

[21] Henkin, Gennadi; Leiterer, Jürgen Theory of functions on complex manifolds, Birkhäuser, Monographs in Mathematics, Tome 79 (1984), 226 pages | MR 774049 (86a:32002) | Zbl 0573.32001

[22] Ivashkovich, Sergej Bochner-Hartogs type extension theorem for roots and logarithms of holomorphic line bundles, Tr. Mat. Inst. Steklova, Tome 279 (2012), pp. 269-287 | MR 3086770 | Zbl 1316.32006

[23] Ivashkovich, Sergej Extension properties of complex analytic objects, Max Planck Institüt für Mathematik (2013)

[24] Levi, Eugenio Elia Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse, Ann. Mat. Pura Appl., Tome 17 (1910) no. 1, pp. 61-87 http://link.springer.com/article/10.1007%2FBF02419336 | Article | Zbl 41.0487.01

[25] Lins Neto, Alcides A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier, Tome 49 (1999) no. 4, pp. 1369-1385 | Article | MR 1703092 (2000h:32047) | Zbl 0963.32022

[26] Merker, Joël; Porten, Egmont The Hartogs extension theorem on (n-1)-complete complex spaces, J. Reine Angew. Math., Tome 637 (2009), pp. 23-39 | Article | MR 2599079 (2011a:32013) | Zbl 1185.32001

[27] Narasimhan, Raghavan The Levi problem for complex spaces. II, Math. Ann., Tome 146 (1962), pp. 195-216 | Article | MR 0182747 (32 #229) | Zbl 0131.30801

[28] Nemirovskiĭ, S. Y. Stein domains with Levi-plane boundaries on compact complex surfaces, Mat. Zametki, Tome 66 (1999) no. 4, pp. 632-635 | Article | MR 1747093 (2001d:32025)

[29] Peternell, Thomas Martin Pseudoconvexity, the Levi problem and vanishing theorems, Several complex variables, VII, Springer, Berlin (Encyclopaedia Math. Sci.) Tome 74 (1994), pp. 221-257 | MR 1326622 | Zbl 0811.32011

[30] Scárdua, Bruno Transversely affine and transversely projective holomorphic foliations, Ann. Sci. Éc. Norm. Supér., Tome 30 (1997) no. 2, pp. 169-204 | Article | MR 1432053 (97k:32049) | Zbl 0889.32031

[31] Schoen, Richard; Yau, Shing-Tung On univalent harmonic maps between surfaces, Invent. Math., Tome 44 (1978) no. 3, pp. 265-278 | Article | MR 0478219 (57 #17706) | Zbl 0388.58005

[32] Siu, Yum-Tong; Trautmann, Günther Gap-sheaves and extension of coherent analytic subsheaves, Springer, Berlin, Lecture Notes in Mathematics, Tome 172 (1971), v+172 pages | MR 0287033 (44 #4240) | Zbl 0208.10403

[33] Takeuchi, A. Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif, J. Math. Soc. Japan, Tome 16 (1964), pp. 159-181 | Article | MR 0173789 (30 #3997) | Zbl 0141.08804

[34] Troyanov, Marc Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc., Tome 324 (1991) no. 2, pp. 793-821 | Article | MR 1005085 (91h:53059) | Zbl 0724.53023