Related papers: Analytic varieties invariant by foliations and Pfa…
In this expository article, we study and discuss invariants of vector fields and holomorphic foliations that intertwine the theories of complex analytic singular varieties and singular holomorphic foliations on complex manifolds: two…
In this work, we study inequalities and enumerative formulas for flags of Pfaff systems on $\mathbb{P}^n_{\mathbb{C}}$. More specifically, we find the number of independent Pfaff systems that leave invariant a one-dimensional holomorphic…
We present results expressing conditions for the existence of meromorphic first integrals for Pfaff equations of arbitrary codimension, integrable or not, on complex manifolds. These results are in the same vein as previous ones by J-P.…
This work concerns the problem of relating characteristic numbers of one-dimensional holomorphic foliations of P^n to those of algebraic varieties invariant by them. More precisely: if M is a connected complex manifold, a one-dimensional…
Let F be a holomorphic foliation of general type on CP(2) which admits a rational first integral. We provide bounds for the degree of the first integral of F just in function of the degree, the birational invariants of F and the geometric…
Integrals of the Pfaffian form over the nonsingular part of a projective variety compute information closely related to the Mather-Chern class of the variety and to other invariants such as the local Euler obstruction along strata of its…
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…
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…
We show in the framework of Pfaff systems theory, the functional dependences of the general analytic solutions of a suitable system of involutive differential equations describing the differences between the analytic solutions of the…
In this paper we consider the question of bounding the degree of an divisor $D$ invariant by a $\F$ holomorphic foliation, without rational first integral, on smooth algebraic variety $X$ in terms of degree of $\F$ and some invariants of…
We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in \mathbb{D}_{\epsilon}}$ be topological trivial (in $\mathbb{C}^2$) analytic deformation of a…
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…
In this work we study some problems related with algebraic hypersurfaces invariant by foliations on weighted projective spaces $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ generalizing some results known for $\p$, as for example: the…
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…
We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf…
We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and…
A singular foliation $\mathcal{F}$ on a complete Riemannian manifold $M$ is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of $M$ into the orbits of a Lie group action by…
We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincar\'{e}-Dulac normal forms for autonomous systems of ODEs with diagonal matrix of the linear part. Using tools from…
One of the various versions of the classical Lyapunov-Poincar\'e center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R.…
In this paper, we are concerned with studying the existence of invariant complex manifolds of two-dimensional holomorphic systems. From the geometric singular perturbation theory we know that if a slow-fast system has associated a normally…