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 Ufimsk. Mat. Zh., 2017, Volume 9, Issue 3, Pages 172–185 (Mi ufa397)

Levi-flat world: a survey of local theory

A. Sukhov

Université de Lille (Sciences et Technologies), U.F.R. de Mathématiques, 59655 Villeneuve d’Ascq, Cedex, France

Abstract: This expository paper concerns local properties of Levi-flat real analytic manifolds with singularities. Levi-flat manifolds arise naturally in Complex Geometry and Foliation Theory. In many cases (global) compact Levi-flat manifolds without singularities do not exist. These global obstructions make natural the study of Levi-flat objects with singularities because they always exist. The present expository paper deals with some recent results on local geometry of Levi-flat singularities. One of the main questions concerns an extension of the Levi foliation as a holomorphic foliation to a full neighborhood of singularity. It turns out that in general such extension does not exist. Nevertheless, the Levi foliation always extends as a holomorphic web (a foliation with branching) near a non-dicritical singularity. We also present an efficient criterion characterizing these singularities.

Keywords: CR structure, Levi-flat manifold.

 Funding Agency Grant Number Labex The author is partially supported by Labex CEMPI.

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English version:
Ufa Mathematical Journal, 2017, 9:3, 172–185 (PDF, 481 kB); https://doi.org/10.13108/2017-9-3-172

Bibliographic databases:

UDC: 517.55
MSC: 37F75,34M,32S,32D
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Citation: A. Sukhov, “Levi-flat world: a survey of local theory”, Ufimsk. Mat. Zh., 9:3 (2017), 172–185; Ufa Math. J., 9:3 (2017), 172–185

Citation in format AMSBIB
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