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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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Foliations on the projective plane with finite group of symmetries
(Mattioli 1885, 2020-03-09)
Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine the maximal value that $|\mbox{aut}(\mathcal{F})|$ can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorphism group.
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Characteristic directions of two-dimensional biholomorphisms
(Cambridge University Press, 2020-05-01)
We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$ , and that at least one of these parabolic manifolds is or contains a parabolic curve.
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Differentiable invariants of holomorphic foliations
(Springer Science and Business Media Deutschland GmbH, 2022-12-01)
In this paper we study differentiable equivalences of germs of singular holomorphic foliations in dimension two. We prove that the Camacho–Sad indices are invariant by such equivalences. We also prove that the Baum–Bott index is a differentiable invariant for some classes of foliations. As a corollary we show that generic degree two holomorphic foliations of P2 are differentiably rigid.
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Foliations on P2 with only one singular point
(Springer Science and Business Media B.V., 2022-10-01)
In this paper we study holomorphic foliations on P2 with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples of such foliations in degree three.
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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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Multiply Hölder functions
(Pontificia Universidad Católica del Perú, 2024-12-27)
In this article we show several properties about multiply Hölder functions. We study the Hölder class of a composition of multiply Hölder functions and prove that a map and its inverse belong — under certain hypotheses — to the same Hölder class. We also prove some extension properties of multiply Hölder functions; for example, we show that a multiply Hölder functions always extends, in the same Hölder class, to “exceptional” sets that are codimension one manifolds.
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Transversely Product Singularities of Foliations in Projective Spaces
(Academie des sciences, 2023-01-01)
We prove that a transversely product component of the singular set of a holomorphic foliation on ℙ n is necessarily a Kupka component.
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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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A flower theorem in dimension two
(Cambridge University Press, 2025-10-01)
We prove a two-dimensional analog of the Leau–Fatou flower theorem for non-degenerate reduced biholomorphisms tangent to the identity.
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