This book is at the same time an introduction, addressed to sophisticated mathematicians, to a purely algebro-geometric theory of \(D\)-modules, and a summary of the main results, most of which due to the author, in that field. The necessary and sufficient prerequisite requested of the reader, aside from basic experience with algebraic and analytic sheaves, is a good knowledge of homological algebra, and familiarity with the formalism and basic properties of derived categories. Relying on this background, the proofs are complete, with two exceptions: Kashiwara’s constructibility theorem and the theorem of faithful flatness of the ring of differential operators of infinite order over the ring of differential operators (due to Sato-Kawai-Kashiwara). The original micro-differential proofs of those results, did not in fact fit into the spirit of this book: proofs based purely on the algebro-geometric formalism of \(D\)-modules were later provided by the author, but only the first could be included, as a final note (with L. Narváez-Macarro) in the present book.
According to Grothendieck’s philosophy, \(D\)-modules are regarded in this book as general coefficients for the cohomology of smooth analytic or algebraic varieties over the complex numbers. (A Monsky-Washnitzer type theory of \(D\)-modules is, however, also in the author’s mind, as well as a generalization to singular varieties.) On the model of the theory of quasi-coherent coefficients over a scheme, developed in R. Hartshorne’s book “Residues and duality” [Berlin etc.: Springer-Verlag (1966; Zbl 0212.26101) and of the theory of discrete coefficients contained in \(SGA 4 and 5\) (SGA \(=\) Sémin. Géom. Algébr.), the author establishes in the first chapter of this book a complete formalism for the present type of coefficients including “les six opérations de Grothendieck” for a smooth morphism \(f: X\to Y,\) and Grothendieck’s local and global duality. In particular, the global duality results here obtained include as special cases the most general known formulations of Serre and Poincaré dualities. While the author makes a point of treating the algebraic and analytic situations in parallel, some asymmetries still remain: only the hypothesis of existence of a good global filtration allows one to prove the coherence of the direct image via a proper morphism of a coherent \(D\)-module in the analytic case (see section \(I\quad 5.4).\) In the algebraic case that hypothesis is automatically verified.
The main result of the second chapter is the interpretation of discrete constructible coefficients in terms of regular holonomic \(D\)-module coefficients. This is “Mebkhout’s equivalence” between the derived category of complexes of \(D\)-modules with bounded regular-holonomic cohomology and the derived category of complexes of sheaves of vector spaces with bounded constructible cohomology. This equivalence, obtained via the (contravariant) solution functor, and the dual equivalence via the (covariant) de Rham functor, represent the widest known generalization of the so-called Riemann-Hilbert correspondence; they are compatible with Grothendieck’s operations and are interchanged by duality (in any of the two categories). The previous statements hold in both the analytic and the algebraic case; their proofs depend strongly on Hironaka’s resolution of singularities, since they rely on Deligne’s work [cf. P. Deligne, “Équations différentielles à points singuliers réguliers”, Lect. Notes Math. 163. Berlin etc.: Springer Verlag (1970; Zbl 0244.14004)] on connections with regular singularities. All of the known comparison theorems between algebraic and complex-analytic cohomology follow here from more general statements of equivalence between several seemingly independent definitions of regularity (the author proves that the structural sheaf is a regular \(D\)-module!), together with their behaviour under direct images. Here, relevant tools are the local algebraic cohomology of analytic \(D\)-modules, and the functor of their solutions in formal functions along a closed analytic subspace. The author has recently been able to free all of the previous results of dependence upon Hironaka’s resolution of singularities (see the author’s article in [Publ. Math., Inst. Hautes Étud. Sci. 69, 47–89 (1989; Zbl 0709.14015)], opening the way to an extension of the theory to schemes in positive characteristic: it is clear that, if a new systematic treatment of the theory of D-modules were to be written, this new approach by the author should be followed.
These main results are complemented by a variety of topics: the problem of Cauchy-Kowalewski in the \(D\)-module setting, the study of \(D\)-modules over a 1-dimensional complex disk, the characterization of perverse sheaves and finally the theory of vanishing cycles (in the sense \(of SGA 7)\). Considerable attention is devoted to this last topic in chapter three, where the author and C. Sabbah, show how to recover the properties of the \(V\)-filtration of Fuchs-Malgrange-Kashiwara, from the general results on \(D\)-modules obtained in the first two chapters.
We found this book extremely interesting, rich and pleasant, and recommend its reading to any mathematician possessing the necessary background.