no. 6
Levi-flat hypersurfaces and their complement in complex surfaces
[Hypersurfaces Levi-plates et leur complément dans les surfaces complexes]
Canales González, Carolina1
1 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay (France)
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2423-2462.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 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.
Mot clés : analyse et géométrie complexes, théorie des feuilletages, hypersurfaces Levi-plates, mesure transverse invariante, variété Stein, convexité holomorphe, prolongement analytique.

Canales González, Carolina 1

1 Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay 91405 Orsay (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Canales González, Carolina. Levi-flat hypersurfaces and their complement in complex surfaces. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2423-2462. doi : 10.5802/aif.3139. https://aif.centre-mersenne.org/articles/10.5802/aif.3139/

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