Characteristic Numbers and invariant subvarieties for Projective Webs

We define the characteristic numbers of a holomorphic k-distribution of any dimension on $mathbb P^n$ and obtain relations between these numbers and the characteristic numbers of an invariant subvariety. As an application we bound the degree of a smooth invariant hypersurface

Projective structures and neighborhoods of rational curves

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries

Foliations and webs inducing Galois coverings

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 assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations

Positive Neighborhoods of Rational Curves

International audienceWe study neighborhoods of rational curves in surfaces with self-intersection number 1 that can be linearised

Sur le nombre de fibrations transverses à une courbe rationnelle dans une surface

International audienceWe investigate the existence, and lack of uniqueness, of a holomorphic fibration by discs transverse to a rational curve in a complex surface.Nous étudions l'existence et le défaut d'unicité de fibrations holomorphes en disques transverse à une courbe rationnelle dans une surface complexe

Neighborhoods of rational curves without functions

International audienceWe prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along the curve. In particular, at arbitray small neighborhood of the curve, any meromorphic function is constant. This implies that the Picard group is not countably generated

Projective structures and neighborhoods of rational curves

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries

Automorphisms of projective structures

We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of flat projective structures having positive dimensional Lie algebra of projective vector fields

Submanifolds with Ample Normal Bundle

We construct germs of complex manifolds of dimension m along projective submanifolds of dimension n with ample normal bundle and without non-constant meromorphic functions whenever m ≥ 2n. We also show that our methods do not allow the construction of similar examples when m < 2n by establishing an algebraicity criterion for foliations on projective spaces which generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence