# zbMATH — the first resource for mathematics

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 $$\dim N +1$$).

##### MSC:
 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: