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Acta Applicandae

Book review

Author(s) Björk, Jan-Eric
Title Analitic D-Modules and Applications
Publisher Kluwer Academic Publishers
Year of publication 1993

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 DX-modules. Much of the topics included appear in book form for the first time. Riemann-Hilbert correspondence, Bernstein-Sata polynomials, microdifferential analysis are presented in detail. In this book, emphasis given to the study of the algebraic and sheaf-theoretic properties of the complexes of the DX-modules 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 D-modules, 3. Holonomic D-modules, 4. Deligne modules, 5. Regular holonomic D-modules, 6. b-Functions, 7. Distributions and regular holonomic systems, 8. Microdifferential operators.

The book is not self-contained. 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 D-moduls and Applications is unique comprehensive monograph concerning with the fundamental properties of DX-modules. 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.