Local Levi-flat hypersurfaces invariants by a codimension one holomorphic foliation.

*(English)*Zbl 1225.32038From 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.

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.

Reviewer: Jesus Muciño-Raymundo (Morelia)

##### 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 |