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Singular Levi-flat hypersurfaces and codimension one foliations. (English) Zbl 1214.32012
A Levi-flat hypersurface \(M\) in a complex manifold \(X\) is a smooth real hypersurface such that the field of its tangent complex hyperplanes forms an integrable distribution, i.e., a real hypersurface foliated by complex hypersurfaces. The latter foliation is called the Levi foliation. A theorem of E. Cartan says that, if \(M\) is real analytic and smooth, then the Levi foliation always extends to a holomorphic foliation by hypersurfaces on a complex neighborhood of \(M\).
The paper under review extends Cartan’s theorem to singular real analytic Levi-flat hypersurfaces \(M\subset X\). Namely, Theorem 1.3 of the paper says that there always exist another complex manifold \(Y\) of the same dimension as \(X\) and holomorphically projected onto \(X\), and a real analytic Levi-flat hypersurface \(N\subset Y\) that is extendable to a (singular) holomorphic foliation by hypersurfaces on \(Y\) such that (1) the projection sends an open subset \(N_0\subset N\) isomorphically onto the regular part of the initial singular Levi-flat hypersurface \(M\), and (2) the restriction to \(\overline N_0\) of the projection is a proper map onto the closure of the regular part of \(M\).
This is the first step towards desingularization of Levi flat hypersurfaces. It is a remarkable result that will have important applications in holomorphic foliations, complex analysis and geometry.
The proof is based on the following nice idea: to lift the Levi foliation to the projectivized cotangent bundle and prove the result for the lifted Levi-flat surface (which now has codimension greater than 1 in the ambient complex manifold). The key argument is given by Theorem 2.5, which deals with a (singular) Levi-flat real analytic subset \(N\) of arbitrary codimension in a complex manifold. Let \(N_{\text{reg}}\) denote the regular part of \(N\). Theorem 2.5 says that every point of the closure \(\overline{N_{\text{reg}}}\) has a neighborhood where \(\overline{N_{\text{reg}}}\) is locally a Levi-flat hypersurface in some complex submanifold (whose real dimension is thus equal to \(\dim N +1\)).

32V25 Extension of functions and other analytic objects from CR manifolds
32S65 Singularities of holomorphic vector fields and foliations
32C05 Real-analytic manifolds, real-analytic spaces
Full Text: arXiv