Related papers: Characterization of second type plane foliations u…
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…
We characterize nondicrital generalized curve foliations with fixed reduced separatrix. Moreover, we give suficient conditions when a plane analytic curve is its reduced separatrix. For that, we introduce a distinguished expression for a…
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…
In this paper, we study nilpotent holomorphic foliations in complex dimension $n+1$, at the origin, defined by germs of integrable 1-forms whose linear part is given by \(zdz\). These foliations generalize the classical nilpotent foliations…
We conclude the classification of isoparametric (or equivalently, polar) foliations of complex and quaternionic projective spaces. This is done by investigating the projections of certain inhomogeneous isoparametric foliations of the…
A holomorphic foliation on $\mathbb P^2_{\mathbb C}$, or a real analytic foliation on $\mathbb{P}^{2}_{\mathbb{R}},$ is said to be convex if its leaves other than straight lines have no inflection points. The classification of the convex…
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…
We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a…
We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria…
We proof a foliated version of the Poincare-Hopf theorem and other results which clarify the geometric and ergodic meaning of the Euler characteristic of a measured foliation.
We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor…
Consider all moduli points corresponding with polarized abelian varieties in characteristic p such that the associated quasi-polarized p-divisible group is geometrically isomorphic with a given one. This defines a subset C of the moduli…
We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface,…
The polar curves of foliations $\mathcal F$ having a curve $C$ of separatrices generalize the classical polar curves associated to hamiltonian foliations of $C$. As in the classical theory, the equisingularity type ${\wp}({\mathcal F})$ of…
In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two…
These are lecture notes of a course given in Pisa, SNS, in february 2002. They provide a classification of holomorphic foliations of nongeneral type on compact Kaehler surfaces.
We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf…
In this article we study the analytic classification of certain types of quasi-homogeneous cuspidal holomorphic foliations in $(\CC^3,{\bf 0})$ via the essential holonomy defined over one of the components of the exceptional divisor that…
We give a survey of the approaches to classifying foliations, starting with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of foliations are…
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…