Frank Loray

Réduction formelle des singularités cuspidales de champs de vecteurs analytiques

We are interested in singularities of analytic vector fields in (real or complex) dimension 2, up to changes of coordinates and reparametrization of time. Two integers p , q ∈ ℕ * being fixed, we give a list of models for small perturbations of the Hamiltonian vector field X = py p −1 (∂/∂x) + qx q −1 (∂/∂y) associated with the function H(x, y) = y p −x q . Each of these perturbations reduces by formal changes of coordinates to one and only one of our models, by an explicit algorithm. These perturbations are no more Hamiltonian. We derive by this way an uncountable dimensional space of distinct analytic classes of such perturbations. The Introduction ends with questions about problems of convergence.

Singular foliations with trivial canonical class

This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds.

TRANSVERSELY PROJECTIVE FOLIATIONS ON SURFACES: EXISTENCE OF MINIMAL FORM AND PRESCRIPTION OF MONODROMY

We introduce a notion of minimal form for transversely projective structures of singular foliations on complex manifolds. Our first main result says that this minimal form exists and is unique when ambient space is two-dimensional. From this result, one obtains a natural way to produce invariants for transversely projective foliations on surfaces. Our second main result says that on projective surfaces one can construct singular transversely projective foliations with prescribed monodromy.

Algebraic Reduction Theorem for complex codimension one singular foliations

Let

Classification de certains feuilletages associés à un cusp

M

Flat parabolic vector bundles on elliptic curves

be a compact complex manifold equipped with

Counting fixed points of a finitely generated subgroup of Aff[C]

n=\dim(M)

Hitchin Hamiltonians in Genus 2

meromorphic vector fields that are linearly independent at a generic point. The main theorem is the following. If

Pseudo-groupe d'une singularité de feuilletage holomorphe en dimension deux

M

A preparation theorem for codimension-one foliations

is not bimeromorphic to an algebraic manifold, then any codimension one complex foliation