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Introduction
Publications
Publications (30)
In this paper we study germs of holomorphic foliations, at the origin of the complex plane, tangent to Pfaffian hypersurfaces - integral hypersurfaces of real analytic 1-forms - satisfying the Rolle-Khovanskii condition. This hypothesis leads us to conclude that such a foliation is defined by a closed meromorphic 1-form, also allowing the classific...
We prove a two-dimensional analog of Leau-Fatou flower theorem for non-degenerate reduced tangent to the identity biholomorphisms.
In this paper we study holomorphic foliations on P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}^2$$\end{document} with only one singular point. If the...
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 lev...
In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space $\mathbb{P}^{n}$, $n \geq 2$. More specifically, we prove that a real analytic Levi-flat hypersurface $M \subset \mathbb{P}^{n}$, with singular set of real dimension at most $2n-4$ and whose Levi leaves are contained in a...
We define the Milnor number -- as the intersection number of two holomorphic sections -- of a one-dimensional holomorphic foliation $\mathscr{F}$ with respect to a compact connected component $C$ of its singular set. Under certain conditions, we prove that the Milnor number of $\mathscr{F}$ on a three-dimensional manifold with respect to $C$ is inv...
Let F denote a singular holomorphic foliation on P2 having a finite automorphism group Aut(F). We determine the maximal value that | Aut(F)| can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorphism group.
In this paper we study holomorphic foliations on $\mathbb{P}^2$ 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.
Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\aut(\F)$. Fixed the degree of $\F$, we determine the maximal value that $|\aut(\F)|$ can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorph...
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 prove that for each characteristic direction [v] of a tangent to the identity diffeomorphism of order k+1 in C^2 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.
We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $\mathbb{C}^2$ 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 c...
Myopic economic agents are well studied in economics. They are impatient. A myopic topology is a topology such that every continuous prefer- ence relation is myopic. If the space is l∞, the Mackey topology T M(l∞; l∞¹), is the largest locally convex such topology. However there is a growing interest in patient consumers. In this paper we analyze th...
We study a special kind of local invariant sets of singular holomorphic foliations called nodal separators. We define notions of equisingularity and topological equivalence for nodal separators as intrinsic objects and, in analogy with the celebrated theorem of Zariski for analytic curves, we prove the equivalence of these notions. We give some app...
We prove that a $C^{\infty}$ equivalence between germs holomorphic foliations at $({\mathbb C}^2,0)$ establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.
We prove that a $C^{\infty}$ equivalence between germs holomorphic foliations at $({\mathbb C}^2,0)$ establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.
We study compact invariant sets for holomorphic foliations on Stein manifold. As application, we show some dynamical properties concerning minimal sets (with singularities) of foliations and real analytic Levi-flat hypersurfaces in projective spaces.
We define the polar curves and the polar family associated to a projective web and obtain some results about the geometry of the generic element of this family. We also deal with the particular case of foliations and prove the constancy of the topological embedded type of the generic polar.
Given topologically equivalent germs of holomorphic foliations F and F, under some hypothesis, we construct topological equivalences extending to some regions of the divisor after resolution of singularities. As an application we study the topological invariance of the projective holonomy representation.
We define the polar curves and the polar family associated to a projective web and obtain some results about the geometry of the generic element of this family. We also deal with the particular case of foliations and prove the constancy of the topological embedded type of the generic polar.
Let $\mathcal{F}$ be a codimension one holomorphic foliation in
$\mathbb{P}^n$, $n\geq 2$, leaving invariant a real-analytic Levi-flat
hypersurface $M$ with regular part $M^{*}$. Then every leaf of $\mathcal{F}$
outside $\overline{M^{*}}$ accumulates in $\overline{M^{*}}$.
Consider a complex one dimensional foliation on a complex surface near a
singularity $p$. If $\mathcal{I}$ is a closed invariant set containing the
singularity $p$, then $\mathcal{I}$ contains either a separatrix at $p$ or an
invariant real three dimensional manifold singular at $p$.
In this paper we study bilipschitz equivalences of germs of holomorphic
foliations in $(\mathbb{C}^2,0)$. We prove that the algebraic multiplicity of a
singularity is invariant by such equivalences. Moreover, for a large class of
singularities, we show that the projective holonomy representation is also a
bilipschitz invariant.
In this paper we give a description of the sets of accumulation of secants for orbits of real analytic vector fields in dimension three with the origin as only ω-limit point. It is an infinitesimal version of the Poincaré-Bendixson problem in dimension three. These sets have structure of cyclic graph when the singularities are isolated under one bl...
We describe the sets of accumulation of secants for orbits of real analytic
vector fields in dimension three having the origin as only {\omega}-limit
point. It is a kind of infinitesimal Poincar\'e-Bendixson problem in dimension
three. These sets have structure of cyclic graph when the singularities are
isolated under one blow-up. In the case of hy...
We prove that the algebraicmultiplicity of a holomorphic vector
field at an isolated singularity is invariant by topological equivalences which are differentiable at the singular point.
We explicitly define a family of seminorms on the space of all bounded real sequence l ∞ . This family gives rise to a Hausdorff lo-cally convex topology which is not equivalent to the usual ones: the weak topology σ(l ∞ , l 1), the norm topology τ ∞ , the Mackey topology m(l ∞ , l 1) and the strict topology β. We show that this new topology, denot...