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On singular real analytic Levi-flat foliations
(International Press, Inc., 2020-12-01)
A singular real analytic foliation $\mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $\mathcal{L}$ which is tangent to $\mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(\mathbb{C}^{n},0)$ under the hypothesis that $\mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $\mathcal{L}$, from which the classification of $\mathcal{F}$ derives: either it has a meromorphic first integral or is defined by a closed rational $1-$form. Our local results also allow us to classify real algebraic Levi-flat foliations on the complex projective space $\mathbb{P}^{n} = \mathbb{P}^{n}_{\mathbb{C}}$.
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Pull-back of singular Levi-flat hypersurfaces
(Mittag-Leffler Institute, 2021-01-01)
We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat hypersurface in a complex projective surface under a rational map or to be the pull-back of a real algebraic curve under a meromorphic function. In particular, we give an application to the case of a singular real analytic Levi-flat hypersurface. Our results improve previous ones due to Lebl and Bretas-Fernndez-Prez-Mol.
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Chow's theorem for real analytic Levi-flat hypersurfaces
(Elsevier Masson s.r.l., 2022-10-01)
In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space Pn, n≥2. More specifically, we prove that a real analytic Levi-flat hypersurface M⊂Pn, with singular set of real dimension at most 2n−4 and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in Pn. As a consequence, M is a semialgebraic set. We also prove that a Levi foliation on Pn — a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one — satisfying similar conditions — singular set of real dimension at most 2n−4 and all leaves algebraic — is defined by the level sets of a rational function.
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On the Milnor Number of Non-Isolated Singularities of Holomorphic Foliations and Its Topological Invariance
(John Wiley and Sons Ltd, 2023-03-01)
We define the Milnor number of a one-dimensional holomorphic foliation (Formula presented.) as the intersection number of two holomorphic sections with respect to a compact connected component (Formula presented.) of its singular set. Under certain conditions, we prove that the Milnor number of (Formula presented.) on a three-dimensional manifold with respect to (Formula presented.) is invariant by (Formula presented.) topological equivalences.
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On Briançon–Skoda Theorem for Foliations
(Elsevier GmbH, 2023-12-01)
We generalize Mattei's result relative to the Briançon–Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.
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On Milnor and Tjurina Numbers of Foliations
(Springer Science+Business Media, 2025-06-01)
We study the relationship between the Milnor and Tjurina numbers of a singular foliation F, in the complex plane, with respect to a balanced divisor of separatrices B for F. For that, we associate with F a new number called the χ-number and we prove that it is a C1 invariant for holomorphic foliations. We compute the polar excess number of F with respect to a balanced divisor of separatrices B for F, via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of B and the χ-number of F. On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of F. Finally, we state numerical relationships between some classic indices, as Baum–Bott, Camacho–Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.
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