Paris, 17-19 October 2016


A foliation of a manifold of dimension 2, can be described either as the integral curves of a given vector field, or as the tangent curves of a given 1-form.
The singularities of a foliation are given by the zeroes of such vector field or 1-form. Up to analytically extending the foliation, the set of singularities has codimension at least 2 (hence points for foliations of surfaces).
A classical theorem by Seidenberg states that, given an isolated singularity of a foliation of a surface, there exists a sequence of blow-ups so that the lifted foliation has only "simple" singularities.
In this case, "simple" means that the linear part of the vector field or 1-form describing the lifted foliation has at least a non-zero eigenvalue.
This process is called "resolution" (or "reduction") of the singularities of a foliation.
Together with the study of reduced singularities, it allows to describe very precisely the geometry of the leaves of the foliation.

For manifolds of dimension 3, foliations described by vector fields and 1-forms give different objects (foliations by curves and foliations by surfaces).
These cases are much more involved than the situation in dimension 2. They share some difficulties with the resolution of singularities of varieties in positive characteristic, and of the study of minimal models.

The aim of this meeting is to explain the recent results of resolutions of singularities of foliations by surfaces and
by curves, as well as some applications of these results.


Mini-courses:

Felipe Cano (Valladolid): Local uniformization of codimension one singular foliations (following M. Fernandez Duque).
Yohann Genzmer (Toulouse): 
Germs of curves in the complex plane and the problem of Zariski: a foliated approach.
Daniel Panazzolo (Haute-Alsace): 
Resolution of singularities of foliations by curves.


Talks:

Enrica Floris (Basel):
Invariance of plurigenera for foliations on surfaces.
Jean-Philippe Rolin (Dijon):
Non-oscillating integral curves and valuations.


Organizers:


Hussein Mourtada, Matteo Ruggiero, Bernard Teissier.