Vanishing cycles of irregular \(\mathfrak D\)-modules. (English) Zbl 0940.32005

Laurent, Yves

Compos. Math. 116, No. 3, 241-310 (1999).

The vanishing cycles of regular holonomic \({\mathcal D}\)-modules were originally introduced by Kashiwara and Malgrange, and later extended to non-holonomic \({\mathcal D}\)-modules, complexes of \({\mathcal D}\)-modules and microdifferential equations. The definition of vanishing cycles of nonregular \({\mathcal D}\)-modules and its application to the study of the solutions of the \({\mathcal D}\)-modules are the purpose of this paper.
This paper consists of five sections. After some preparations on basic notions and notations in §1, the author proves that microfunctions can be restricted to a hypersurface in some special cases (§2). The key result of this paper is Theorem 3.1.1 in §3, which asserts that an operator with the principal symbol \(t^m\) is equivalent to the operator \(t^m\). In §4, the vanishing cycle is defined and their properties are studied. The microlocal Cauchy theorem is also proved and some results about non-holonomic \({\mathcal D}\)-modules are deduced. In the last section (§5), the results on holonomic modules, their solutions and the index theorems are discussed.

Reviewer: M.Muro (Yanagido)

MSC:

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

Keywords:

irregularity; \({\mathcal D}\)-modules; vanishing cycles