On the Irregularity of the Holonomic Modules

Abstract

A key result of the last decade is the equivalence of categories between the triangulated category of holonomic regular complexes and the triangulated category of constructible complexes on a smooth complex algebraic or analytic variety [K], [Me 1]. The proof of this result depended heavily on Hironaka's theorem on the resolution of the singularities. This chapter discusses the Deligne's Principle, and reviews the semi-continuity theorem. The semi-continuity theorem for the irregularity is the analogue of the semi-continuity theorem of the Swan conductor due to Deligne. It describes the behavior of the irregularity in a family of differential equations. Various theorems are proven. The chapter discusses the irregular vanishing cycles, and presents the comparison theorem. Grothendieck's motivation [G] for the comparison theorem between the De Rham cohomologies is that the De Rham cohomology is a good or a Weil cohomology for algebraic varieties over a field of characteristic zero. This motivation is completely independent of Riemann's existence theorem.