Differential equations with irregular singular points and Stokes phenomenon in dimension 2. (Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2.) (French) Zbl 0947.32005

Astérisque. 263. Paris: Société Mathématique de France. vii, 190 p. (2000).
This is a book on an asymptotic property of singularity of holonomic differential system of several complex variables, especially with irregular singularity. The asymptotic behavior at singular points of ordinary linear differential equations has been sufficiently well understood while the multivariate version of it – the asymptotic theory of holonomic system with irregular singularity has yet been known very little. The author tries to fill the gap by introducing new notions and showing some consequences in this area. It is the “good formal structure (bonne structure formale)” of a meromorphic connection on an analytic variety \(X\) with poles along a divisor \(Z\subset X\). The author considers the case that \(X\) is a complex analytic surface and \(Z\) is a closed curve in \(X\), and gives a conjecture that there exists a sequence of complex blowing-up of a meromorphic connection such that the good formal structure is transmitted by the pull-back morphisms. There are some consequences of this conjecture and the proof of this conjecture is given for the bundles of rank \(\leq 5\). The existence of a lifting of a good formal structure at the level of asymptotic expansions in bisectors is also shown. Applications are given to complex conjugation of holonomic \({\mathcal D}\)-modules.
Reviewer: M.Muro (Yanagido)


32C38 Sheaves of differential operators and their modules, \(D\)-modules
35A20 Analyticity in context of PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs