Stokes Cocycle and Differential Galois Groups

M. Loday-Richaud

Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 2, Tome 2 (2003), pp. 103-115

Résumé

The classification of germs of ordinary linear differential systems with meromorphic coefficients at 0 under convergent gauge transformations and fixed normal form is essentially given by the non-Abelian 1-cohomology set of Malgrange–Sibuya. (Germs themselves are actually classified by a quotient of this set.) It is known that there exists a natural isomorphism $h$ between a unipotent Lie group (called the Stokes group) and the 1-cohomology set of Malgrange–Sibuya; the inverse map which consists of choosing, in each cohomology class, a special cocycle called a Stokes cocycle is proved to be natural and constructive. We survey here the definition of the Stokes cocycle and give a combinatorial proof for the bijectivity of $h$. We state some consequences of this result, such as Ramis, density theorem in linear differential Galois theory; we note that such a proof based on the Stokes cocycle theorem and the Tannakian theory does not require any theory of (multi-)summation.