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Local Levi-flat hypersurfaces invariants by a codimension one holomorphic foliation. (English) Zbl 1225.32038
From the foliation theory point of view, a Levi-flat hypersurface $$M$$ in $$\mathbb{C}^n$$ is a real analytic hypersurface such that the field of its tangent complex hyperplanes forms an integrable distribution on its smooth part $$M^*$$, i.e., a real hypersurface foliated by complex hypersurfaces on its smooth part. (It is called the Levi foliation of $$M$$.)
The authors study the local singular situation.
Given $$\mathcal{F}$$, a germ of a codimension one singular holomorphic foliation at $$(\mathbb{C}^n, 0)$$, and $$M$$, a germ of a real Levi-flat hypersurface, the authors say that $$\mathcal{F}$$ and $$M$$ are tangent if the leaves of the Levi foliation of $$M$$ are also leaves of $$\mathcal{F}$$.
The first result is:
Let $$\mathcal{F}$$ be a germ at $$0 \in \mathbb{C}^n$$, $$n \geq 2$$, of a codimension one singular holomorphic foliation tangent to a germ at $$0$$ of a real codimension one and irreducible analytic variety $$M$$. Then $$\mathcal{F}$$ has a non-constant meromorphic first integral.
Moreover, in dimension $$n=2$$, the following dichotomy holds:
If $$\mathcal{F}$$ is dicritical, then it has a non-constant meromorphic first integral.
If $$\mathcal{F}$$ is non-dicritical, then it has a non-constant holomorphic first integral.
The second result concerns the existence of a holomorphic foliation tangent to the germ of a singular Levi-flat hypersurface. Some natural concepts are required.
Let $$M = F^{-1}(0)$$ be a germ at $$(\mathbb{C}^n, 0)$$ of a real analytic Levi-flat hypersurface. The complexification of $$M$$ is $$M^{}_\mathbb{C} = F^{-1}_{\mathbb{C}} (0)$$, considering the natural complexification of $$F$$ on $$(\mathbb{C}^{2n}, 0)$$. The algebraic dimension of $$\text{sing\,} (M)$$ is the complex dimension of the singular set of $$M_{\mathbb{C}}$$. Denote by $$\eta^{}_\mathbb{C}$$ the complexification of the Levi form of $$F$$ on $$(\mathbb{C}^{2n}, 0)$$.
The second result is:
Let $$M$$ be a germ of an irreducible real analytic Levi-flat hypersurface at $$(\mathbb{C}^n, 0)$$, $$n \geq 2$$. Assume that the algebraic dimension of $$\text{sing\,}(M)$$ is $$\leq 2n-4$$. Then there exists a unique germ at $$0 \in \mathbb{C}^n$$ of a codimension one singular holomorphic foliation $$\mathcal{F}_M$$ which is tangent to $$M$$, if one of the following conditions is fullfiled:
(a) $$n \geq 3$$ and $$\text{cod}_{M^*_\mathbb{C}} \big(\text{sing\,} (\eta_\mathbb{C} |_{M^*_\mathbb{C}})\big) \geq 3.$$
(b) $$n \geq 2$$, $$\text{cod}_{M^*_\mathbb{C}} \big(\text{sing\,} (\eta_\mathbb{C} |_{M^*_\mathbb{C}})\big) \geq 2$$, and the complexification of the Levi-foliation given by $$\eta_\mathbb{C}=0$$ has a non-constant holomorphic first integral.
Moreover, in both cases the foliation $$\mathcal{F}_M$$ has a non-constant holomorphic first integral $$f$$ such that $$M= \{ \text{Im\,}(f) = 0 \}$$.
In a certain sense the second main result asserts that if the singularities of $$M$$ are sufficiently small, then $$M$$ is given by the zeros of the imaginary part of a holomorphic function.
Relations with previous results by D. Burns and X. Gong [ibid. 121, No. 1, 23–53 (1999; Zbl 0931.32009)] and M. Brunella [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 4, 661–672 (2007; Zbl 1214.32012)] are discussed. Many explicit examples are provided, showing the richness of the subject.

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
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