Related papers: Topology of singular holomorphic foliations along …

This work deals with the topological classification of singular foliation germs on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the…

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…

Let $X$ be a complex surface and $Y$ be an elliptic curve embedded in $X$. Assume that there exists a non-singular holomorphic foliation $\mathcal{F}$ with $Y$ as a compact leaf, defined on a neighborhood of $Y$ in $X$. We investigate the…

We study those real $\mathcal{C}^\infty$ foliations in complex surfaces whose leaves are holomorphic curves. The main motivation is to try and understand these foliations in neighborhoods of curves: can we expect the space of foliations in…

We consider a non-dicritic germ of foliations defined in some ball, with finite number of separatrices and satisfying some additional but generic hypothesis. We prove that there exists an open neighborhood U of the total separatice set S…

A holomorphic foliation is defined as an integrable coherent subsheaf of the tangent sheaf. The structure of the leaves around a singularity is read off from the structure of the stalks. This was done by Baum when the dimension of the…

We consider holomorphic foliations by curves on compact complex manifolds, for which we investigate the existence of projective structures along the leaves varying holomorphically (foliated projective structures), that satisfy particular…

This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and…

Let $\mathcal{F}$ be the germ at $\mathbf{0} \in \mathbb{C}^n$ of a holomorphic foliation of dimension $d$, $1 \leq d < n$, with an isolated singularity at $\mathbf{0}$. We study its geometry and topology using ideas that originate in the…

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0 \in \mathbb{C}$ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of…

We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy…

We consider an embedded general complex torus $C_n$ into a complex manifold $M_{n+d}$ with a unitary flat normal bundle $N_C$. We show the existence of (non-singular) holomorphic foliation in a neighborhood of $C$ in $M$ having $C$ as leaf…

We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no…

We proof here the existence of a topological thick and thin decomposition of any closed definable thick isolated singularity germ in the spirit of the recently discovered metric thick and thin decomposition of complex normal surface…

The object of this survey is to give an overview on the topology of singularities of holomorphic foliation germs on $(\mathbb C^2,0)$.

Let $M$ be a two dimensional complex manifold, $p \in M $ and \Fl a germ of holomorphic foliation of \M at $p$. Let $S\subset M$ be a germ of an irreducible, possibly singular, curve at $p$ in $M$ which is a separatrix for \Fl. We prove…

In this work we use our previous results on the topological classification of generic singular foliation germs on $(\mathbb C^{2},0)$ to construct complete families: after fixing the semi-local topological invariants we prove the existence…

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…

The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations…

We prove that every topological conjugation between two germs of singular holomorphic curves in the complex plane is homotopic to another conjugation which extends homeomorphically to the exceptional divisors of their minimal…