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Rogério Mol

Publications and source records attributed to Rogério Mol.

15 recordsLinked to original sources
arXiv2026

Logarithmic models and meromorphic functions in dimension two

In this article we describe the construction of logarithmic models in both real and complex cases. A logarithmic model is a germ of closed meromorphic 1-form with simple poles - and the analytic foliation defined by it - produced upon some specified geometric data: the structure of dicritical (non-invariant) components in the exceptional divisor of its reduction of singularities, a prescribed finite set of separatrices - invariant analytic branches at the origin - and Camacho-Sad indices with respect to these separatrices. As an application, we use logarithmic models in order to construct real and complex germs of meromorphic functions with a given indeterminacy structure and prescribed sets of zeroes and poles. Also, in the real case, in the specific case where all trajectories accumulating at the origin are contained in analytic curves, logarithmic models are used in order to build germs of analytic vector fields with a given Bendixson's sectorial decomposition of a neighborhood of $0 \in \R^{2}$ into hyperbolic, parabolic and elliptic sectors. As a consequence, we can produce real meromorphic functions with prescribed sectorial decompositions.

math.CV↗
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↗
arXiv2023

On the algebraicity of germs of meromorphic functions

In this article we prove that every germ of analytic meromorphic function at $(\mathbb{C}^{2},0)$ is equivalent, under the right composition by a germ of biholomorphism, to a germ of algebraic meromorphic function. An analogous result is also true for real analytic meromorphic functions.

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↗
arXiv2019

Separatrices for real analytic vector fields in the plane

Let $X$ be a germ of real analytic vector field at $({\mathbb R}^{2},0)$ with an algebracally isolated singularity. We say that $X$ is a topological generalized curve if there are no topological saddle-nodes in its reduction of singularities. In this case, we prove that if either the order $ν_{0}(X)$ or the Milnor number $μ_{0}(X)$ is even, then $X$ has a formal separatrix, that is, a formal invariant curve at $0 \in {\mathbb R}^{2}$. This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.

math.DS↗
arXiv2018

Holomorphic vector fields tangent to foliations in dimension three

This article studies germs of holomorphic vector fields at the origin of C3 that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the vector field - for instance, that the final models belong to a family of vector fields whose linear parts have eigenvalues satisfying a condition of non-resonance - in order to conclude that the foliation is of complex hyperbolic type, that is, without saddle-nodes in its reduction of singularities. In the second part, we prove that a vector field that is tangent to three independent foliations is tangent to a whole pencil of foliations - hence, to infinitely many foliations - and, as a consequence, it leaves invariant a germ of analytic surface. This final part is based on a local version of a well-known characterization of pencils of foliations of codimension one in projective spaces.

math.DS↗
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

Real Analytic Vector Fields with First Integral and Separatrices

We prove that a germ of analytic vector field at $(\mathbb{R}^3,0)$ that possesses a non-constant analytic first integral has a real formal separatrix. We provide an example which shows that such a vector field does not necessarily have a real analytic separatrix.

math.DS↗
arXiv2017

Second type foliations of codimension one

In this article, for holomorphic foliations of codimension one at $(\mathbb{C}^{3},0)$, we define the family of second type foliations. This is formed by foliations having, in the reduction process by blow-up maps, only well oriented singularities, meaning that the reduction divisor does not contain weak separatrices of saddle-node singularities. We prove that the reduction of singularities of a non-dicritical foliation of second type coincides with the desingularization of its set of separatrices.

math.DS↗
arXiv2017

Residue-type indices and holomorphic foliations

We investigate residue-type indices for germs of holomorphic foliations in the plane and characterize second type foliations - those not containing tangent saddle-nodes in the reduction of singularities - by an expression involving the Baum-Bott, variation and polar excess indices. These local results are applied in the study of logarithmic foliations on compact complex surfaces.

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

The Poincaré problem in the dicritical case

We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the $GSV$-index. We apply it to the Poincaré problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, ${\rm deg}(S) \leq {\rm deg}(\mathcal{F}) + 2$.

math.DS↗
arXiv2015

Local polar invariants for plane singular foliations

In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the GSV-index in terms of polar curves. We also interpret the proofs concerning the Poincaré problem with polar invariants.

math.DS↗