arXiv subjects

Javier Ribón

Publications and source records attributed to Javier Ribón.

At least 19 recordsLinked to original sources

arXiv2026

Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs

Consider a compact leaf $\mathcal{L}$ of a holomorphic foliation $\mathcal{F}$ such that all the leaves of the restriction of $\mathcal{F}$ to some neighborhood $U$ of $\mathcal{L}$ are closed in $U$. We show that there exists an invariant germ of analytic set $V$ in a neighborhood of $\mathcal{L}$, of dimension higher than $\dim (\mathcal{F})$, consisting of compact leaves, such that the volume of the leaves in $V$ is uniformly bounded by above. We also provide a globalization of this result if the ambient manifold is projective. In order to study closed leaves foliations, and via the holonomy representation, we introduce a new concept, of independent interest, for subgroups $G$ of germs of holomorphic diffeomorphisms, the so called {\it torsion locus}. We show that it is non-trivial if $G$ has finite orbits.

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arXiv2025

Explicit Computation of The Generic Component of the Analytic Moduli of a Plane Branch

Let ${\mathcal C}$ be a fixed equisingularity class of irreducible germs of complex analytic plane curves. We compute a basis of the ${\mathbb C}[[x]]$-module of Kähler differentials for generic $Γ\in {\mathcal C}$, algorithmically, and study its behaviour under blow-up. As a first application, we give an algorithm providing the generic semimodule in an equisingularity class in terms of its multiplicity and its Puiseux characteristic exponents. As another application, we give an alternative proof for a formula of Genzmer, that provides the dimension of the moduli of analytic classes in the equisingularity class of $Γ$.

math.AG↗

arXiv2025

Completely Integrable Foliations: Singular Locus, Invariant Curves and Topological Counterparts

We study codimension $q \geq 2$ holomorphic foliations defined in a neighborhood of a point $P$ of a complex manifold that are completely integrable, i.e. with $q$ independent meromorphic first integrals. We show that either $P$ is a regular point, a non-isolated singularity or there are infinitely many invariant analytic varieties through $P$ of the same dimension as the foliation, the so called separatrices. Moreover, we see that this phenomenon is of topological nature. Indeed, we introduce topological counterparts of completely integrable local holomorphic foliations and tools, specially the concept of total holonomy group, to build holomorphic first integrals if they have isolated separatrices. As a result, we provide a topological characterization of completely integrable non-degenerated elementary isolated singularities of vector fields with an isolated separatrix.

math.CV↗

arXiv2024

Constructive solution of Zariski's Moduli Problem for Plane Branches

In this paper we give an explicit solution to Zariski's moduli problem for plane branches. We compute (in an algorithmic way) the set of Kähler differentials of an irreducible germ of holomorphic plane curve. We show that there is a basis of this set whose main elements correspond to dicritical foliations. Indeed, we discuss several concepts of generation for the semimodule of values of Kähler differentials of the curve and provide basis of Kähler differentials, for every of these concepts, whose geometric properties are described. Moreover, we give an algorithmic construction of the bases.

math.AG↗

arXiv2023

Complexity of Puiseux solutions of differential and $q$-difference equations of order and degree one

We relate the complexity of both differential and $q$-difference equations of order one and degree one and their solutions. Our point of view is to show that if the solutions are complicated, the initial equation is complicated too. In this spirit, we bound from below an invariant of the differential or $q$-difference equation, the height of its Newton polygon, in terms of the characteristic factors of a solution. The differential and the $q$-difference cases are treated in a unified way.

math.CV↗

arXiv2023

The Poincaré problem for reducible curves

We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are isolated, i.e. they are never contained in non-trivial analytic families of equisingular invariant curves. In this case we show that the multiplicity of an invariant curve is at most twice the multiplicity of the foliation. Finally, we apply the local methods to foliations in the complex projective plane.

math.CV↗

arXiv2023

Nondegenerate germs of holomorphic foliations with prescribed holonomy

We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of reduced singularities. The most intricate case in the 2 dimensional setting corresponds to (Siegel) saddle singularities. This work treats the analogous problem for saddles in higher dimension. We show that any germ of holomorphic biholomorphism, in any dimension, can be obtained as the holonomy map associated to an integral curve of a saddle singularity. A natural question is whether we can prescribe the linear part of the saddle germ of vector field provided the holonomy map. The answer to this question is known to be positive in dimension 2. We see that this is not the case in higher dimension. In spite of this, we provide a positive result under a natural condition for the holonomy map.

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arXiv2022

Finite orbits for nilpotent actions on the torus

A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.

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arXiv2022

The solvable length of groups of local diffeomorphisms

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we show that $2n+1$ is the sharpest upper bound for the derived length of solvable subgroups of the group $\mathrm{Diff}({\mathbb C}^{n},0)$ of local complex analytic diffeomorphisms for $n=2,3,4,5$.

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arXiv2022

Stable manifolds of biholomorphisms in $\mathbb{C}^n$ asymptotic to formal curves

Given a germ of biholomorphism $F\in\mathrm{Diff}(\mathbb{C}^n,0)$ with a formal invariant curve $Γ$ such that the multiplier of the restricted formal diffeomorphism $F|_Γ$ is a root of unity or satisfies $|(F|_Γ)'(0)|<1$, we prove that either $Γ$ is contained in the set of periodic points of $F$ or there exists a finite family of stable manifolds of $F$ where all the orbits are asymptotic to $Γ$ and whose union eventually contains every orbit asymptotic to $Γ$. This result generalizes to the case where $Γ$ is a formal periodic curve.

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arXiv2021

Fixed points of nilpotent actions on surfaces of negative Euler characteristic

We prove that a locally nilpotent group $G$ of $C^{1}$ diffeomorphisms of a compact surface $S$ of non-vanishing Euler characteristic has a finite orbit ${\mathcal O}$ whose cardinal is bounded by above by a function of the characteristic of Euler of $S$. We focus on the case of negative Euler characteristic $χ(S)$. Then we can choose ${\mathcal O}$ so that it consists of global contractible fixed points of thesubgroup $G_0$ of $G$ consisting of isotopic to the identity elements. In particular $G$ has a global contractible fixed point if it consists of isotopic to the identity elements.

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arXiv2021

A fixed point curve theorem for finite orbits local diffeomorphisms

We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a local biholomorphism $F$, in dimension 2 with finite orbits, there exists an analytic curve passing through the origin and contained in the fixed point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of $F$ at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of local biholomorphisms, with finite orbits, such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they can not be embedded in one parameter groups.

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arXiv2020

The local Poincaré problem for irreducible branches

Let ${\mathcal F}$ be a germ of holomorphic foliation defined in a neighborhood of the origin of ${\mathbb C}^{2}$ that has a germ of irreducible holomorphic invariant curve $γ$. We provide a lower bound for the vanishing multiplicity of ${\mathcal F}$ at the origin in terms of the equisingularity class of $γ$. Moreover, we show that such a lower bound is sharp. Finally, we characterize the types of dicritical singularities for which the multiplicity of $\mathcal{F}$ can be bounded in terms of that of $γ$ and provide an explicit bound in this case.

math.AG↗

arXiv2019

Boundness of intersection numbers for actions by two-dimensional biholomorphisms

We say that a group $G$ of local (maybe formal) biholomorphisms satisfies the uniform intersection property if the intersection multiplicity $(ϕ(V), W)$ takes only finitely many values as a function of $G$ for any choice of analytic sets $V$ and $W$. In dimension $2$ we show that $G$ satisfies the uniform intersection property if and only if it is finitely determined, i.e. there exists a natural number $k$ such that different elements of $G$ have different Taylor expansions of degree $k$ at the origin. We also prove that $G$ is finitely determined if and only if the action of $G$ on the space of germs of analytic curves have discrete orbits.

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arXiv2018

Analytic Moduli of Plane Branches and Holomorphic Flows

We study the behaviour (in the infinitesimal neighbourhood of the singularity) of a singular plane branch under the action of holomorphic flows. The techniques we develop provide a new elementary, geometric and dynamical solution to Zariski's moduli problem for singular branches in $({\mathbb C}^{2},0)$. Furthermore, we study whether elements of the same class of analytic conjugacy are conjugated by a holomorphic flow; in particular we show that there exists an analytic class that is not complete: meaning that there are two elements of the class that are not analytically conjugated by a local diffeomorphism embedded in a one-parameter flow.

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arXiv2017

Stable manifolds of two-dimensional biholomorphisms asymptotic to formal curves

Let $F\in\mathrm{Diff}(\mathbb{C}^2,0)$ be a germ of a holomorphic diffeomorphism and let $Γ$ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F|_Γ$ is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism $F|_Γ$ should satisfy, if $Γ$ were convergent, in order to have orbits converging to the origin). Then we prove that $F$ has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to $Γ$. Our results generalize to the case where $Γ$ is a formal periodic curve of $F$.

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arXiv2017

Global fixed points for nilpotent actions on the torus

An isotopic to the identity map of the $2$-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the $2$-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the $2$-torus has a global fixed point.

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arXiv2016

Finite dimensional groups of local diffeomorphisms

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic group up to add extra formal diffeomorphisms. We show that this is the case for virtually polycyclic subgroups and in particular finitely generated virtually nilpotent groups of local biholomorphisms. We provide several methods to identify this property and build examples. As a consequence we generalize results of Arnold, Seigal-Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups.

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