Microlocal study of sheaves. (English) Zbl 0589.32019

Kashiwara, Masaki; Schapira, Pierre

Astérisque, 128. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. 235 p. FF 135.00; $ 16.00 (1985).

This book deals with a microlocalization of a sheaf \({\mathcal F}\) on a real manifold M. Originally microlocal analysis means an analysis of a hyperfunction f(x) on a real analytic manifold by considering it as a microfunction on the cotangent bundle \(T^*M\) and studying it as a section on \(T^*M\). The sheaves \({\mathcal B}_ M\) of hyperfunctions on M and \({\mathcal C}_ M\) of microfunctions on \(T^*M\) are constructed through cohomological operations of \({\mathcal O}_ X:\) the sheaf of holomorphic functions on the complexification X of M. For a section f(x) of a hyperfunction, the support of f(x) as a microfunction on \(T^*M\) is called the singular support or singular spectrum and plays a fundamental role in microlocal analysis.
The authors define the same notion as singular support for any sheaf \({\mathcal F}\) on a real manifold M and call it the micro-support. It is a closed conic involutive subset of \(T^*X\), describing the set of codirections where \({\mathcal F}\) and its cohomology do not propagate. Functorial properties of the micro-support are studied, and the derived category of sheaves is localized in \(T^*M\), which gives a meaning to the action of contact transformations on sheaves. In particular the shift of simple sheaves along smooth Lagrangian manifolds is calculated by means of the Maslov index. Applications are given to real or complex analytic constructible sheaves, regular holonomic modules, and micro- differential systems.

Reviewer: M.Muro

MSC:

32C05	Real-analytic manifolds, real-analytic spaces
32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
58J40	Pseudodifferential and Fourier integral operators on manifolds
32A45	Hyperfunctions
46F15	Hyperfunctions, analytic functionals
18F99	Categories in geometry and topology
55R40	Homology of classifying spaces and characteristic classes in algebraic topology
35N99	Overdetermined problems for partial differential equations and systems of partial differential equations
32L99	Holomorphic fiber spaces

Keywords:

microlocalization of sheaves on real analytic manifolds; hyperfunction; singular support; micro-support