Movable singularities of ODEs: a topological approach
Résumé
Near a regular point of a complex, scalar and first-order ODE, a local system of solutions defines a trivial fibration $\text{disk}\times\text{disk}$. The holomorphic foliation associated to the ODE is obtained by patching up all these local systems, giving a partition of the ambient space into connected Riemann surfaces called leaves (maximal solutions) and singularities of the ODE. There is no reason why this object should continue to be a locally trivial fibration near a singular point, because nothing guarantees that neighboring leaves all have the same topology. Of course in the simplest example, where the foliation is defined by the level sets of a holomorphic submersion $H~:~\left(\mathbb{C}^{2},0\right)\to\left(\mathbb{C},0\right)$, the theorem of J.~\noun{Milnor} ensures that $H$ is a holomorphic fibration with total space a well-chosen complement $U$ of the singular fiber $H^{-1}\left(0\right)$. In that case the topology of the other leaves is constant, and the natural morphism $\pi_{1}\left(H^{-1}\left(\text{cst}\right)\right)\to\pi_{1}\left(U\right)$ is injective: the ``holes'' in the leaves can only be caused by a set of finitely many ``special'' leaves (separatrices). Any foliation satisfying this property is deemed \emph{incompressible}. D.~\noun{Mar\'in} and J.F.~\noun{Mattei} have generalized Milnor theorem to most singular planar holomorphic foliations: under generic assumptions a germ of a foliation is incompressible. We will explain the principle of their proof and how to weaken their assumptions down to an almost sharp characterization of incompressible foliations, then use this study to exhibit examples of compressible foliations. While we understand how to guarantee incompressibility, it is not clear how the extra topology can be accounted for in case of compressibility. We will observe on some examples that it is produced by so-called \emph{movable singularities} (poles of the solutions which do not come from singularities of the ODE). This explanation is satisfying only in a global context, for what does it mean for a solution to ``tend to $\infty$'' when the ODE is only defined in a polydisk? In this talk we propose to use compressibility failures to represent persistent movable singularities in a local context, and we will present some (hopefully) convincing arguments in favor of such a definition.
Domaines
| Origine | Fichiers produits par l'(les) auteur(s) |
|---|---|
| Licence |