La classification des connexions irrégulières à une variable. (French) Zbl 0531.58015

Malgrange, Bernard

Mathématique et physique, Sémin. Ec. Norm. Supér., Paris 1979-1982, Prog. Math. 37, 381-399 (1983).

[For the entire collection see Zbl 0516.00021.]
This is the development of an unpublished work by Deligne, on the classification of connections with irregular singularities on a smooth curve. The idea, which goes back to Birkhoff, consists in associating to such a connection a ”Stokes structure” at each singular point, given by the growth of solutions in sectors and the so-called Stokes phenomenon. A crucial point is a theorem on holomorphic invertible matrices due to Y. Sibuya. Examples are given, which are in fact special cases of the work of Jurkat on this subject. Finally, a problem of moduli is considered, for analytic connections near a singular point having a prescribed formalization. The following results are sketched: 1) The problem of moduli has a solution (in the strong sense). 2) The problem can be stated in algebraic terms; but, in that case, even the problem of coarse moduli has a negative answer.

MSC:

58C25	Differentiable maps on manifolds
58K99	Theory of singularities and catastrophe theory
32Sxx	Complex singularities
53C05	Connections (general theory)

Keywords:

holomorphic map germs; Stokes structure; classification of connections with irregular singularities on a smooth curve

Citations:

Zbl 0516.00021