
Author(s)  Björk, JanEric 

Title  Analitic DModules and Applications 
Publisher  Kluwer Academic Publishers 
Year of publication  1993 
Reviewed by  VALERII V.TROFIMOV 
Let X be complex analitic manifold, DX be the sheaf of differential operators with holomorphic coefficients on X. The book under review contains the foundations of the DXmodules. Much of the topics included appear in book form for the first time. RiemannHilbert correspondence, BernsteinSata polynomials, microdifferential analysis are presented in detail. In this book, emphasis given to the study of the algebraic and sheaftheoretic properties of the complexes of the DXmodules and recent its applications. The author uses the modern tools such as homological algebra, complex analysis, and symplectic geometry.
The rough contents of the book are as follows, 1. The sheaf DX and its modules, 2. Operations on Dmodules, 3. Holonomic Dmodules, 4. Deligne modules, 5. Regular holonomic Dmodules, 6. bFunctions, 7. Distributions and regular holonomic systems, 8. Microdifferential operators.
The book is not selfcontained. With a view to help readers, there are seven extensive appendices: I. Derived Categories, 2. Sheaf theory, 3. Filtered rings, 4. Homological algebra, 5. Complex analysis, 6. Analitic geometry, 7. Symplectic analysis. There are an extensive bibliography, list of notations and index will be helpfull for readers.
Analitic Dmoduls and Applications is unique comprehensive monograph concerning with the fundamental properties of DXmodules. Graduate students and active researchers alike with interest in differential systems, infinite dimensional representations of Lie groups, calculus of residues in complex manifolds can make a good investment purchasing Björk's book.