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Algebraic $$\mathcal D$$-modules. (English) Zbl 0642.32001
Perspectives in Mathematics, Vol. 2, Boston etc.: Academic Press, Inc., Harcourt Brace Jovanovich, Publishers. xii, 355 p.; \$ 29.95; £25.00 (1987).
Let $$X$$ be a complex manifold. We denote by $$\mathcal D_X$$ the sheaf (of non-commutative rings) of holomorphic linear partial differential operators of finite order. The study of $$\mathcal D_X$$-modules, as well as the study of modules over other sheaves of differential operators (the so called “algebra analysis”) developed tremendously in the last decades. A central result of the theory, due to Kashiwara and Mebkhout, is the Riemann-Hilbert (short R-H) correspondence (the twenty-first problem of Hilbert in higher dimensions). In the beginning of the eighties the $$\mathcal D$$-modules (more specifically, the R-H correspondence) played a fundamental role in the infinite dimensional representation theory of semisimple real Lie groups, namely in the solution of the Kazhdan-Lusztig conjecture. In fact, for the applications to the representation theory, only the algebraic frame was needed: when $$X$$ is algebraic and the differential operators are assumed to be algebraic.
The book, which is an outgrowth of a seminar organized by A. Borel in 1984 in Bern, is a self-contained presentation of the theory of $$\mathcal D$$-modules in the algebraic frame. It is particularly valuable for the mathematicians working in algebraic geometry and representation theory.
The content of the eight chapters is briefly the following. Chapter I by P.-P. Grivel deals with derived categories. On 107 pages one gives a detailed introduction into the subject, proving all the facts which are needed further in the book.
Chapter II by B. Kaup contains the proof of the coherence of $${\mathcal D}$$ and study of the first properties of coherent $${\mathcal D}$$-modules (filtrations).
Chapter III by A. Haefliger is a detailed presentation of the classical theory of systems of meromorphic differential equations with regular singular points at the origin of the punctured disk (Fuchs theory), as well as the modern formulation of it (in language of connections).
Chapter IV by B. Malgrange deals with the solution given by Deligne to the Riemann-Hilbert problem in higher dimensions in the context of integral connections on holomorphic bundles. Firstly one develops the theory in the holomorphic case (replacing the punctured disk by the complement of an effective divisor on a complex manifold), then one proves the R-H correspondence on algebraic manifolds (not necessarily projective).
Chapter V by F. Ehless, dealing with the algebras of algebraic differential operators over affine spaces (Weyl algebras), contains the properties of modules over such algebras: Krull dimension, good filtrations, homological algebra, holonomic modules. Here the main research references are the papers of J. Bernstein and J. E. Ross.
Chapter VI, VII and VIII by A. Borel, based on the prerequisites of chapters I–V, gives the fundaments of the theory of algebraic $$\mathcal D$$-modules. One defines and studies the basic notions: coherent $$\mathcal D$$-modules, the characteristic variety, holonomic $$\mathcal D$$-modules, the dualizing $$\mathcal D$$-module, regular holonomic modules, direct and inverse images, as well as extensions of these to complexes of $$\mathcal D$$-modules. The culmination of the theory in the book is the proof of (algebraic) R-H correspondence between (bounded) complexes of $$\mathcal D$$-modules having regular holonomic cohomology and (bounded) complexes of sheaves over $$\mathbb C$$ having constructible cohomology.
Reviewer: C.Bǎnicǎ

##### MSC:
 32-06 Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 35Q15 Riemann-Hilbert problems in context of PDEs