The Galois groupoid of a foliation. (Le groupoïde Galois d’un feuilletage.) (French) Zbl 1033.32020

Malgrange, Bernard

Ghys, Étienne (ed.) et al., Essays on geometry and related topics. Mémoires dédiés à André Haefliger. Vol. 2. Genève: L’Enseignement Mathématique (ISBN 2-940264-04-X/pbk). Monogr. Enseign. Math. 38, 465-501 (2001).

The article is an attempt to define for nonlinear differential equations an analogue of the differential Galois group (or Picard-Vessiot group) for linear differential equations. The author proposes a definition in the context of foliations with singularities on a complex analytic variety, by considering the Zariski closure (in an appropriate sense) of the holonomy of the foliation. In the linear case this corresponds to the usual definition. In a previous work, H. Umemura [Nagoya Math. J. 144, 59–135 (1996; Zbl 0878.12002)] had given another definition which however, in the linear case, does not completely recover the differential Galois group but only the corresponding formal group. It is unclear what is the relation between Umemura’s definition and that presented in this paper.
For the entire collection see [Zbl 0988.00115].

Reviewer: Alessandra Iozzi (Zürich)

MSC:

32S65	Singularities of holomorphic vector fields and foliations
32C38	Sheaves of differential operators and their modules, \(D\)-modules
58H05	Pseudogroups and differentiable groupoids
34M45	Ordinary differential equations on complex manifolds
12H05	Differential algebra
37F75	Dynamical aspects of holomorphic foliations and vector fields

Keywords:

differential Galois group; Picard-Vessiot group; singular foliation; holonomy

Citations:

Zbl 0878.12002