\(D\)-modules and microlocal calculus. Translated from the Japanese by Mutsumi Saito. (English) Zbl 1017.32012

Translations of Mathematical Monographs. 217. Providence, RI: American Mathematical Society (AMS). xvi, 254 p. (2003).
The systematic study of \(D\)-modules began in 1960 by Mikio Sato when he gave a talk at a colloquium of the Department of mathematics of the University of Tokyo and was mainly developed from 1970 to 1980. One of the main contributor to this area is Masaki Kashiwara, who is the author of this book. The theory of \(D\)-modules has naturally adjusted itself to the scene of contemporary mathematics and holds hidden power of application in both pure and applied mathematics.
This book is a compact introduction to the basic theory of \(D\)-modules and its application to micro-local calculus. The author begins this book with an exposition about fundamental properties of \(D\)-modules and introduces neatly a clear picture of the notion of characteristic varieties and holonomic systems. Then he gives an explanation about micro-differential operators and micro-local analysis of holonomic systems. The final section is devoted to the explicit computation of \(b\)-functions by using micro-local calculus. In the appendix he gives a description about derived category and symplectic geometry for the theoretical foundation.
This book seems to be written toward the micro-localization of \(D\)-modules, (regular) holonomic systems, hyperfunction solutions to them and their application. The author also stresses that the micro-local calculus is not only beautiful but useful in practical calculus like Cauchy’s theorem in complex function theory. His aim is fully attained in showing that the micro-local method has power to further develop practical calculus.


32C38 Sheaves of differential operators and their modules, \(D\)-modules
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
58J15 Relations of PDEs on manifolds with hyperfunctions
32A27 Residues for several complex variables