Related papers: Analytic varieties invariant by foliations and Pfa…
We solve the Poincar\'e problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give an algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We…
We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable…
We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or…
In this paper, we study the analytic classification of a class of nilpotent singularities of holomorphic foliations in $(\mathbb{C}^2,0)$, those exhibiting a Poincar\'e-Dulac type singularity in their reduction process. This analytic…
We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance…
In this paper, we address one of the most basic and fundamental problems in the theory of foliations and ODEs, the topological invariance of the algebraic multiplicity of a holomorphic foliation. For instance, we prove an adapted version of…
We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criteria for the…
We give a bounding of degree of quasi-smooth hypersurfaces which are invariant by a one dimensional holomorphic foliation of a given degree on a weighted projective space.
Let M be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field. Let f be an analytic diffeomorphism from M onto itself and p be a fixed point of f. We discuss invariant manifolds around p, like…
Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincar\'e-Hopf when the ambient…
In this paper we give a geometric description of the foliation of a generic real analytic family unfolding a real analytic vector field with a weak focus at the origin, and show that two such families are orbitally analytically equivalent…
We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological…
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…
The leaves in singular holomorphic foliation theory are examples of quasi-analytic layers. In the first part of our publication we are concerned with a theory of these subjects. A quasi-analytic decomposition of a complex manifold is a…
This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac-Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the…
In this work we consider holomorphic foliations of degree two on the projective plane $\mathbb{P}^2$ having an invariant line. In a suitable choice of affine coordinates these foliations are induced by a quadratic vector field over the…
We consider one dimensional holomorphic foliations with isolated singularities that leave invariant a local complete intersection. We establish explicit formulas for the total GSV index of such foliations and obtain bounds for this index.…
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th…
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some…
Geometrical properties of holonomic and non holonomic varieties defined by the Pfaff equations connected with a first order systems of differential equations are studied. The Riemann extensions of affine connected spaces for investigation…