##
**Analytic \(D\)-modules and applications.**
*(English)*
Zbl 0805.32001

Mathematics and its Applications (Dordrecht). 247. Dordrecht: Kluwer Academic Publishers. xiii, 581 p. (1993).

This book contains the analytic theory of \({\mathcal D}_ X\)-modules, i.e. sheaves of modules over the sheaf of rings \({\mathcal D}_ X\) of linear holomorphic differential operators on a given complex manifold \(X\).

In Chapter I the author introduces the ring \({\mathcal D}_ n\) of linear differential operators with coefficients in \({\mathcal O}_ n\), the local ring of convergent power series in \(n\) variables and the sheaf \({\mathcal D}_ X\) of holomorphic differential operators on the complex analytic manifold \(X\). He studies the category \(\text{Coh}({\mathcal D}_ X)\) of (left) coherent \({\mathcal D}_ X\)-modules. In particular, for each \({\mathcal M}\in \text{Coh}({\mathcal D}_ X)\) its characteristic variety \(SS({\mathcal M})\) is constructed. The involutivity of \(SS({\mathcal M})\) (as an analytic set of \(T^* X\)) is proved in Appendix III [following O. Gabber, Am. J. Math. 103, 445-468 (1981; Zbl 0492.16002)]. Applying some results of Appendix IV it is proved that \({\mathcal D}_ n\) is an Auslander-regular ring and that its global homological dimension is equal to \(n\). The chapter ends with some results about twisted rings of differential operators.

Chapter II is devoted to the study of the fundamental operations on \({\mathcal D}\)-modules and more generally on the derived category \(D^ b ({\mathcal D})\) of bounded complexes of \({\mathcal D}\)-modules. A systematic study of the direct and inverse image of complexes of \({\mathcal D}\)-modules is given, developing, in particular, images of \({\mathcal D}\)-modules under closed imbeddings and under projections. It is proved the coherence preservation in the proper case. The chapter contains the study of the category of spezializable \({\mathcal D}_ X\)-modules along a submanifold \(Y\) in \(X\) and the construction of the Malgrange-Kashiwara filtration on specializable \({\mathcal D}\)-modules. The chapter ends with the Mebkhout’s theorem on relative duality for coherent \({\mathcal D}\)-modules.

Chapter III is centered on the study of holonomic \({\mathcal D}\)-modules and more generally of objects in the derived category \(D^ b_{\text{hol}} ({\mathcal D})\) of bounded complexes of \({\mathcal D}\)-modules with holonomic cohomology. The first fundamental result is the existence of the Bernstein-Sato polynomial associated to sections of holonomic \({\mathcal D}\)- modules with respect to a global section of \({\mathcal O}\), and its application to the preservation of holonomicity by the fundamental operations on \({\mathcal D}\)-modules. The second main result is the constructibility of the solution complex \(\text{Sol} ({\mathcal M})\) for \({\mathcal M}\) in \(D^ b_{\text{hol}} ({\mathcal D})\) (Kashiwara’s constructibility theorem) which uses some results about prolongation of solutions under non-characteristic deformations and the Mittag-Leffler theorem. Some results on \({\mathcal D}^ \infty\) and Mebkhout’s local duality are used to characterize holonomic \({\mathcal D}\)-modules like the objects in \(D^ b_{\text{hol}} ({\mathcal D})\) whose solution complex is a perverse complex. The chapter ends with a duality formula for the \(V\)- filtration of Malgrange-Kashiwara for holonomic \({\mathcal D}\)-modules.

Chapter IV deals with Deligne modules associated to local systems in the complement of a divisor. It is proved that every Deligne module is a meromorphic connection (with poles along the divisor). First it is proved for the normal crossing case and then, using Hironaka’s resolution of singularities, in the general one. The chapter contains also a study of the \(L^ 2\)-lattice introduced by Barlet and Kashiwara, a Hartog’s theorem for Deligne modules and it ends with the interplay between Deligne modules and Nilsson class functions.

Chapter V develops the central topic of the book: regular holonomic \({\mathcal D}\)-modules. In the one-dimensional case, the index theorem of Malgrange is proved. The study of general regular holonomic \({\mathcal D}\)- modules is reduced to the case of cyclic modules of type \({\mathcal D}/{\mathcal D}P\) and following B. Malgrange [“Equations différentielles à coefficients polynomiaux”, Birkhäuser (1991; Zbl 0764.32001)] a classification of regular holonomic modules is given. In any dimensions it is proved that regularity is preserved by the fundamental operations on \({\mathcal D}\)-modules and, using the resolution of singularities of Hironaka, the author gives a proof of the Riemann-Hilbert correspondence of Kashiwara and Mebkhout. Two sections are devoted to the study of intersection complexes. The Mebkhout’s irregularity complex of a holonomic \({\mathcal D}\)-module along a hypersurface is defined and following Z. Mebkhout [Prog. Math. 88, 83-132 (1990; Zbl 0731.14007)] it is proved that this complex is perverse. The chapter ends with some results about algebraic \({\mathcal D}\)-modules.

Chapter VI treats the theory of \(b\)-functions for regular holonomic \({\mathcal D}\)-modules. Some results on coherence are proved and used to study the characteristic varieties of regular holonomic \({\mathcal D}\)- modules under localisation along hypersurfaces, including results of Kashiwara-Kawai and Lê-Mebkhout. It is proved that the roots of the Bernstein-Sato polynomial associated to a germ of holomorphic function (with isolated singularities) are determined, up to integer shifts, by the eigenvalues of the monodromy of the corresponding Milnor fibration. This theorem is due to Malgrange. The nearby cycle and the vanishing cycle of a perverse complex \(\text{DR}({\mathcal M})\) are exhibited as the de Rham complex of regular holonomic \({\mathcal D}\)-modules, constructed from the \(V\)-filtration of Malgrange-Kashiwara on \({\mathcal M}\). The chapter ends with some results (due to D. Barlet) on the effective contribution of zeros of \(b\)-functions to poles of the meromorphic extension of the corresponding distribution valued function.

Chapter VII deals with analytic \({\mathcal D}\)-modules theory on real manifolds. It includes results about extendible distributions and develops the theory of regular holonomic distributions. It is shown that every regular holonomic \({\mathcal D}\)-modules on a complex manifold is locally isomorphic to a cyclic module generated by a distribution on the underlying real manifold. Some problems related to meromorphic continuation of distributions are treated. Some sections are devoted to a complete study of the Kashiwara’s conjugation functor and the construction, using the Kashiwara’s temperature Hom-functor, of an inverse to the de Rham functor in the Riemann-Hilbert correspondence.

The last chapter (Chapter VIII) deals with \({\mathcal E}_ X\)-modules where \({\mathcal E}_ X\) is the sheaf of micro-differential operators on the cotangent bundle \(T^* X\). This sheaf and its filtrations are introduced in the first section. The basic results (some of them without proofs) about coherent \({\mathcal E}\)-modules and \({\mathcal E}\)-modules with regular singularities are then represented. A special section is devoted to holonomic \({\mathcal E}\)-modules. The main result is the structure theorem of the holonomic \({\mathcal E}\)-module with support in generic position. The one-dimensional case was proved first by B. Malgrange [Adv. Math., Suppl. Stud. 7B, 513-530 (1981; Zbl 0468.46026)]. The general case has been treated by the author and by B. Abdel Gadir [“Analyse microlocale des systèmes différentiels holonomes”, Ph. D. Thesis, Univ. Joseph Fourier, Grenoble, June (1992)]. Regular holonomic \({\mathcal E}\)-modules and \(b\)-functions are also studied. The sheaf of micro-local operators \({\mathcal E}^ \mathbb{R}\) is introduced and used to prove the local index formula (due to Kashiwara) for holonomic \({\mathcal D}\)-modules. The chapter ends with the construction of the micro-local conjugation functor of Andronikof.

The book includes seven appendixes (more than 160 pages) treating the following subjects: 1) the properties of derived categories and the construction of derived functors, 2) sheaf theory, 3) filtered noetherian rings and sheaves of filtered rings, 4) homological algebra (including Auslander’s condition, the bidualizing complex in the non-commutative case and various results on pure modules due to O. Gabber), 5) complex analysis (with some vanishing theorems for local cohomology and the presentation of the local Milnor filtration), 6) analytic geometry and 7) simplectic analysis.

The book ends with a long list of references (incomplete in some cases). Unfortunately, some cross-references in the text are wrong and the citations to the references are, in many cases, not complete, which makes difficult further possible lectures for the reader.

In the reviewer’s opinion the monograph is reasonably self-contained and offers to researchers and graduate students the complete proof of the fundamental results of the theory which, in general, can be only found in articles from journals.

In Chapter I the author introduces the ring \({\mathcal D}_ n\) of linear differential operators with coefficients in \({\mathcal O}_ n\), the local ring of convergent power series in \(n\) variables and the sheaf \({\mathcal D}_ X\) of holomorphic differential operators on the complex analytic manifold \(X\). He studies the category \(\text{Coh}({\mathcal D}_ X)\) of (left) coherent \({\mathcal D}_ X\)-modules. In particular, for each \({\mathcal M}\in \text{Coh}({\mathcal D}_ X)\) its characteristic variety \(SS({\mathcal M})\) is constructed. The involutivity of \(SS({\mathcal M})\) (as an analytic set of \(T^* X\)) is proved in Appendix III [following O. Gabber, Am. J. Math. 103, 445-468 (1981; Zbl 0492.16002)]. Applying some results of Appendix IV it is proved that \({\mathcal D}_ n\) is an Auslander-regular ring and that its global homological dimension is equal to \(n\). The chapter ends with some results about twisted rings of differential operators.

Chapter II is devoted to the study of the fundamental operations on \({\mathcal D}\)-modules and more generally on the derived category \(D^ b ({\mathcal D})\) of bounded complexes of \({\mathcal D}\)-modules. A systematic study of the direct and inverse image of complexes of \({\mathcal D}\)-modules is given, developing, in particular, images of \({\mathcal D}\)-modules under closed imbeddings and under projections. It is proved the coherence preservation in the proper case. The chapter contains the study of the category of spezializable \({\mathcal D}_ X\)-modules along a submanifold \(Y\) in \(X\) and the construction of the Malgrange-Kashiwara filtration on specializable \({\mathcal D}\)-modules. The chapter ends with the Mebkhout’s theorem on relative duality for coherent \({\mathcal D}\)-modules.

Chapter III is centered on the study of holonomic \({\mathcal D}\)-modules and more generally of objects in the derived category \(D^ b_{\text{hol}} ({\mathcal D})\) of bounded complexes of \({\mathcal D}\)-modules with holonomic cohomology. The first fundamental result is the existence of the Bernstein-Sato polynomial associated to sections of holonomic \({\mathcal D}\)- modules with respect to a global section of \({\mathcal O}\), and its application to the preservation of holonomicity by the fundamental operations on \({\mathcal D}\)-modules. The second main result is the constructibility of the solution complex \(\text{Sol} ({\mathcal M})\) for \({\mathcal M}\) in \(D^ b_{\text{hol}} ({\mathcal D})\) (Kashiwara’s constructibility theorem) which uses some results about prolongation of solutions under non-characteristic deformations and the Mittag-Leffler theorem. Some results on \({\mathcal D}^ \infty\) and Mebkhout’s local duality are used to characterize holonomic \({\mathcal D}\)-modules like the objects in \(D^ b_{\text{hol}} ({\mathcal D})\) whose solution complex is a perverse complex. The chapter ends with a duality formula for the \(V\)- filtration of Malgrange-Kashiwara for holonomic \({\mathcal D}\)-modules.

Chapter IV deals with Deligne modules associated to local systems in the complement of a divisor. It is proved that every Deligne module is a meromorphic connection (with poles along the divisor). First it is proved for the normal crossing case and then, using Hironaka’s resolution of singularities, in the general one. The chapter contains also a study of the \(L^ 2\)-lattice introduced by Barlet and Kashiwara, a Hartog’s theorem for Deligne modules and it ends with the interplay between Deligne modules and Nilsson class functions.

Chapter V develops the central topic of the book: regular holonomic \({\mathcal D}\)-modules. In the one-dimensional case, the index theorem of Malgrange is proved. The study of general regular holonomic \({\mathcal D}\)- modules is reduced to the case of cyclic modules of type \({\mathcal D}/{\mathcal D}P\) and following B. Malgrange [“Equations différentielles à coefficients polynomiaux”, Birkhäuser (1991; Zbl 0764.32001)] a classification of regular holonomic modules is given. In any dimensions it is proved that regularity is preserved by the fundamental operations on \({\mathcal D}\)-modules and, using the resolution of singularities of Hironaka, the author gives a proof of the Riemann-Hilbert correspondence of Kashiwara and Mebkhout. Two sections are devoted to the study of intersection complexes. The Mebkhout’s irregularity complex of a holonomic \({\mathcal D}\)-module along a hypersurface is defined and following Z. Mebkhout [Prog. Math. 88, 83-132 (1990; Zbl 0731.14007)] it is proved that this complex is perverse. The chapter ends with some results about algebraic \({\mathcal D}\)-modules.

Chapter VI treats the theory of \(b\)-functions for regular holonomic \({\mathcal D}\)-modules. Some results on coherence are proved and used to study the characteristic varieties of regular holonomic \({\mathcal D}\)- modules under localisation along hypersurfaces, including results of Kashiwara-Kawai and Lê-Mebkhout. It is proved that the roots of the Bernstein-Sato polynomial associated to a germ of holomorphic function (with isolated singularities) are determined, up to integer shifts, by the eigenvalues of the monodromy of the corresponding Milnor fibration. This theorem is due to Malgrange. The nearby cycle and the vanishing cycle of a perverse complex \(\text{DR}({\mathcal M})\) are exhibited as the de Rham complex of regular holonomic \({\mathcal D}\)-modules, constructed from the \(V\)-filtration of Malgrange-Kashiwara on \({\mathcal M}\). The chapter ends with some results (due to D. Barlet) on the effective contribution of zeros of \(b\)-functions to poles of the meromorphic extension of the corresponding distribution valued function.

Chapter VII deals with analytic \({\mathcal D}\)-modules theory on real manifolds. It includes results about extendible distributions and develops the theory of regular holonomic distributions. It is shown that every regular holonomic \({\mathcal D}\)-modules on a complex manifold is locally isomorphic to a cyclic module generated by a distribution on the underlying real manifold. Some problems related to meromorphic continuation of distributions are treated. Some sections are devoted to a complete study of the Kashiwara’s conjugation functor and the construction, using the Kashiwara’s temperature Hom-functor, of an inverse to the de Rham functor in the Riemann-Hilbert correspondence.

The last chapter (Chapter VIII) deals with \({\mathcal E}_ X\)-modules where \({\mathcal E}_ X\) is the sheaf of micro-differential operators on the cotangent bundle \(T^* X\). This sheaf and its filtrations are introduced in the first section. The basic results (some of them without proofs) about coherent \({\mathcal E}\)-modules and \({\mathcal E}\)-modules with regular singularities are then represented. A special section is devoted to holonomic \({\mathcal E}\)-modules. The main result is the structure theorem of the holonomic \({\mathcal E}\)-module with support in generic position. The one-dimensional case was proved first by B. Malgrange [Adv. Math., Suppl. Stud. 7B, 513-530 (1981; Zbl 0468.46026)]. The general case has been treated by the author and by B. Abdel Gadir [“Analyse microlocale des systèmes différentiels holonomes”, Ph. D. Thesis, Univ. Joseph Fourier, Grenoble, June (1992)]. Regular holonomic \({\mathcal E}\)-modules and \(b\)-functions are also studied. The sheaf of micro-local operators \({\mathcal E}^ \mathbb{R}\) is introduced and used to prove the local index formula (due to Kashiwara) for holonomic \({\mathcal D}\)-modules. The chapter ends with the construction of the micro-local conjugation functor of Andronikof.

The book includes seven appendixes (more than 160 pages) treating the following subjects: 1) the properties of derived categories and the construction of derived functors, 2) sheaf theory, 3) filtered noetherian rings and sheaves of filtered rings, 4) homological algebra (including Auslander’s condition, the bidualizing complex in the non-commutative case and various results on pure modules due to O. Gabber), 5) complex analysis (with some vanishing theorems for local cohomology and the presentation of the local Milnor filtration), 6) analytic geometry and 7) simplectic analysis.

The book ends with a long list of references (incomplete in some cases). Unfortunately, some cross-references in the text are wrong and the citations to the references are, in many cases, not complete, which makes difficult further possible lectures for the reader.

In the reviewer’s opinion the monograph is reasonably self-contained and offers to researchers and graduate students the complete proof of the fundamental results of the theory which, in general, can be only found in articles from journals.

Reviewer: F.Castro-Jiménez (Sevilla)

### MSC:

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

58J15 | Relations of PDEs on manifolds with hyperfunctions |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

16S32 | Rings of differential operators (associative algebraic aspects) |

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |