Local moduli for meromorphic differential equations. (English) Zbl 0683.34003

Babbitt, D. G.; Varadarajan, V. S.

Centre National de la Recherche Scientifique. Astérisque, 169-170. Paris: Société Mathématique de France. 217 p. FF 160.00; $ 27.00 (1989).

This monograph is devoted to the local structure of linear meromorphic differential systems of the form \(du/dz=A(z)u\) in a neighbourhood of an irregular singular point, in the framework of vector bundles and meromorphic connections.
In part I, “Meromorphic connections and their Stokes phenomena”, the fundamental objects of study are the germs of pairs (V,\(\nabla)\) where V is a holomorphic vector bundle on a disk in the complex plane and \(\nabla\) is a meromorphic connection. The formal aspects of the theory are represented by a functor from the category of germs of pairs to the category of formal differential modules over the corresponding formalizations \({\mathcal F}=C[[z]][z^{-1}];\) the structure theory of formal differential modules is presented in categorical terms, essentially in the form given by P. Deligne.
In Part II a detailed study of the Stokes sheaf and its cohomology is presented and a fundamental result stating that the functor of the first cohomology is representable by an affine space of dimension equal to the irregularity of the endomorphism bundle is proved. Part III is dedicated to the study of loca moduli space for marked and, separately, for unmarked pairs, proving that the analytic complex space corresponding to the above mentioned affine space is a local moduli space for the local isoformal deformations of a pair. A brief historical survey of the main themes and results in the monograph is presented in the Appendix.

Reviewer: S.Mirica

MSC:

34M99	Ordinary differential equations in the complex domain
14H15	Families, moduli of curves (analytic)

Keywords:

irregular singular point; formal differential modules; local moduli space; local isoformal deformations; historical survey