We present a constructive proof of the fact that the set of algebraic Pfaff equations without algebraic solutions over the complex projective plane is dense in the set of all algebraic Pfaff equations of a given degree.

Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...

We give an intersection theoretic proof of M. Soares’ bounds for the Poincaré-Hopf index of an isolated singularity of a foliation of ${\mathrm{ℂ\mathbb{P}}}^n$.

In this paper we prove that holomorphic codimension one singular foliations on $ℂ{\mathbb{P}}^n,n\ge 3$ have no non trivial minimal sets. We prove also that for $n\ge 3$, there is no real analytic Levi flat hypersurface in $ℂ{\mathbb{P}}^n$.

Here we show that a Kupka component $K$ of a codimension 1 singular foliation $F$ of $\mathbf{C}{\mathbf{P}}^n,n\ge 6$ with $\mathrm{deg}\left(K\right)$ not a square is a complete intersection. The result implies the existence of a meromorphic first integral of $F$.

Here we show that a Kupka component $K$ of a codimension 1 singular foliation $F$ of $ℂ{\mathbb{P}}^n,n\ge 6$ is a complete intersection. The result implies the existence of a meromorphic first integral of $F$. The result was previously known if $\mathrm{deg}\left(K\right)$ was assumed to be not a square.

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some...

The aim of this note is to give a clearer and more direct proof of the main result of another paper of the author. Moreover, we give some complementary results related to R-complete algebraic foliations with R a rational function of type ℂ*.

Let $(V,0)$ be a germ of a complete intersection variety in ${ℂ}^{n+k}$, $n\>0$, having an isolated singularity at $0$ and $X$ be the germ of a holomorphic vector field having an isolated zero at $0$ and tangent to $V$. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of $X$ is also isolated in the ambient space ${ℂ}^{n+k}$ we give a formula for the homological index in terms of local linear algebra.