Emmanuel Paul

• Université Toulouse III - Paul Sabatier

The leaves of the Painlevé foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections. Therefore they are the irreducible components of the fibers of the Riemann-Hilbert correspondence that sends each connection on its monodromy data, and this correspondence induces a conjugation between the d...

The leaves of the Painlev{\'e} foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections. Therefore they are the fibers of the Riemann-Hilbert correspondence that sends each connection on its monodromy data, and this correspondence induces a conjugation between the dynamics of the foliation a...

In the present paper, we will first present briefly a general research program about the study of the "natural dynamics" on character varieties and wild character varieties. Afterwards, we will illustrate this program in the context of the Painlev\'e differential equations $P_{\rm VI}$ and $P_{\rm V}$.

We consider the topological class of a germ of 2-variables quasi-homogeneous complex analytic function. Each element f in this class induces a germ of foliation (f = constants) and a germ of curve (f = 0). We first describe the moduli space of the foliations in this class and give analytic normal forms. The classification of curves induces a distri...

We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in each class. Then we study on th...

We give geometric and algorithmic criterions in order to have of a proper Galois closure for a codimension one germ of quasi-homogeneous foliation. We recall this notion recently introduced by B. Malgrange, and describe the Galois envelope of a group of germs of analytic diffeomorphisms. The geometric criterions are obtained from transverse analyti...

From recent developments in non linear dieren tial Galois theory, we have now three equivalent denitions for the Galois reducibility of a codimension one foliation dened by a germ of holomorphic one-form !: - the rst one is related to Godbillon-Vey sequences: there exists a nite sequence of lenght at most three of meromorphic one forms !0, !1, and...

We classify up to a formal change of variable the vector fields of two complex variables together with the foliations that they de.ne which are a perturbation of a quasi-homogeneous Hamiltonian vector field X0by terms of higher degree of quasi-homogeneity. We do not require anything about the degree 0of the initial vector field X0, but we assume th...

On classifie \`a changement de variable formel pr\`es les champs de vecteurs --ainsi que les feuilletages qu'ils d\'efinissent-- qui sont des perturbations d'un champ hamiltonien quasi-homog\`ene $X_0$ par des termes de degr\'e de quasi-homog\'en\'eit\'e sup'erieur. Le degr\'e $\delta_0$ du champ $X_0$ est quelconque, mais on demande que le champ p...

On considère un feuilletage holomorphe sur une variété complexe de dimension deux, qui laisse invariant un diviseur compact D. On introduit une notion d'holonomie globale le long du diviseur D qui généralise la notion usuelle d'holonomie d'une feuille compacte. OnétablitOnétablit une correspondance entre la résolubilité de cette holonomie et l'inté...

Let D be a normal crossing divisor in a complex analytic manifold of dimension n, and let Ω be a closed logarithmic one-form, with poles on D. Under appropriate hypothesis, we prove the connectedness of the fibers for a primitive of Ω in "good" neighborhoods of D. We deduce the connectedness of the fibers of Liouvillian functions of type f=f1λ1 ⋯ f...

On donne une description du feuilletage dfini sur une varit complexe par une forme logarithmique ferme, ples sur un diviseur croisements normaux. On obtient un rsultat de trivialit topologique pour une famille de telles formes.