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Rudy Rosas

Publications and source records attributed to Rudy Rosas.

At least 19 recordsLinked to original sources
arXiv2026

Neighborhoods of curves with a prescribed number of foliations

Given a connected projective curve $C \subset \mathbb{P}^n$, $n \geq 2$, and an integer $0 \leq \ell \leq n$, we construct an n-dimensional (non compact) complex manifold, obtained as a neighborhood of an embedded copy of $C$, which carries exactly $\ell$ codimension one holomorphic foliations; moreover, every codimension one distribution on it is one of these foliations. We also determine the field of meromorphic functions of these manifolds: it can be prescribed to be $\mathbb{C}$ or a purely transcendental extension of transcendence degree one, and no larger field is possible as soon as the number of foliations is finite. This extends to arbitrary dimension, and refines, previous constructions of neighborhoods of curves in surfaces without foliations or without non-constant meromorphic functions.

math.AG↗
arXiv2025

Indices of holomorphic foliations and the bifurcation conjecture

In this paper, we revisit local invariants (Gómez-Mont-Seade-Verjovsky, variation, Camacho-Sad and Baum-Bott indices) associated with singular holomorphic foliations on $(\mathbb{C}^2 , 0)$ and we provide semi-global formulas for them in terms of the reduction of singularities of the foliation. A key technical ingredient is the Cholesky-type factorization of the intersection matrix of the exceptional divisor, which allows for an explicit control of multiplicities and indices along the resolution process. Using this factorization, we express the Milnor number and other indices as quadratic forms in intersection vectors associated to balanced divisors introduced by Y. Genzmer. As a main application, we address a conjecture posed by A. Szawlowski concerning pencils of plane holomorphic germs. We prove that the excess of Milnor numbers along the pencil is precisely captured by the invariants derived from our formulas, thereby confirming the conjecture in full generality. This also yields a new expression for the dimension of the parameter space of universal unfoldings of meromorphic functions in the sense of T. Suwa.

math.AG↗
arXiv2024

Holomorphic foliations tangent to Rolle-pfaffian hypersurfaces

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 classification of the simple models in its reduction of singularities.

math.CV↗
arXiv2024

Distributions and Legendrian foliations in dimension 3

We study the field of rational first integrals of distributions. We show that for a distribution on 3 dimensional manifolds there exists a tangent vector field with the same field of first integrals. We also show a similar result for integrable distributions in any dimension.

math.AG↗
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.

math.DS↗
arXiv2022

A flower theorem in dimension two

We prove a two-dimensional analog of Leau-Fatou flower theorem for non-degenerate reduced tangent to the identity biholomorphisms.

math.DS↗
arXiv2022

Distributions, first integrals and Legendrian foliations

We study germs of holomorphic distributions with "separated variables'. In codimension one, a well know example of this kind of distribution is given by the canonical contact structure on $\mathbb{P}^{2m+1}$ . Another example is the Darboux distribution, which gives the normal local form of any contact structure. Given a germ $D$ of holomorphic distribution with separated variables in $(\mathbb{C}^n,0)$, we show that there exists , for some $κ\in \mathbb{Z}_{\geq 0}$ related to the Taylor coefficients of $D$, a holomorphic submersion $H_{D}: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^κ,0)$ such that $D$ is completely non-integrable on each level of $H_{D}$. Furthermore, we show that there exists a holomorphic vector field $Z$ tangent to $D$, such that each level of $H_{D}$ contains a leaf of $Z$ that is somewhere dense in the level. In particular, the field of meromorphic first integrals of $Z$ and that of $D$ are the same.

math.CV↗
arXiv2021

On the Milnor number of non-isolated singularities of holomorphic foliations and its topological invariance

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 invariant by $C^1$ topological equivalences.

math.CV↗
arXiv2021

Chow's theorem for real analytic Levi-flat hypersurfaces

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 algebraic hypersurfaces, is tangent to the levels of a rational function in $\mathbb{P}^{n}$. As a consequence, $M$ is a semialgebraic set. We also prove that a Levi foliation on $\mathbb{P}^{n}$ - 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.

math.CV↗
arXiv2021

Foliations on $\mathbb{P}^2$ with only one singular point

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.

math.DS↗
arXiv2020

Foliations on the projective plane with finite group of symmetries

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.

math.AG↗
arXiv2018

On singular real analytic Levi-flat foliations

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 $\mathbb{P}^{n} = \mathbb{P}^{n}_{\mathbb{C}}$.

math.DS↗
arXiv2018

Characteristic directions of two-dimensional biholomorphisms

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 curve.

math.DS↗
arXiv2017

Nodal separators of holomorphic foliations

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 applications in the study of topological equivalences of holomorphic foliations. In particular, we show that the nodal singularities and its eigenvalues in the resolution of a generalized curve are topological invariants.

math.DS↗
arXiv2016

Differentiable equisingularity of holomorphic foliations

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.

math.DS↗
arXiv2016

Bilipschitz invariants for germs of holomorphic foliations

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.

math.DS↗