##
**Differential equations with polynomial coefficients.
(Équations différentielles à coefficients polynomiaux.)**
*(French)*
Zbl 0764.32001

Progress in Mathematics (Boston, Mass.). 96. Boston, MA etc.: Birkhäuser. 232 p. (1991).

The modern algebraic theory of differential systems, or “theory of \(\mathcal D\)-modules”, brings us new relations between two mathematical areas traditionally far apart: the theory of the systems of linear partial differential equations and the algebraic geometry. The first benefited from the geometric point of view and the algebraic treatment of singularities. The second has seen it endowed with new structures, which explain some depth relations between discrete and continuous cohomological theories of varieties (Riemann-Hilbert correspondence). In the book under review, the author shows how to incorporate fully some new elements of classical analysis and mathematical physics to the theory of \({\mathcal D}\)-modules, especially the Laplace and stationary phase methods. The use of “sophisticated” tools, with respect to the traditional methods in classical analysis, as sheaf theory and derived categories, allows us to understand many exception phenomena of the classical theory. The book is mainly devoted to cover two topics: Geometric description of one-dimensional holonomic \(\mathcal D\)-modules (regular and irregular), and the effect of the Fourier transform on this description. We will describe the contents of the book.

Chapter I gives a summary, without proofs, of the theory of \(\mathcal D\)- modules: holonomic \(\mathcal D\)-modules, direct and inverse images by closed (resp. open) embeddings, De Rham complex, meromorphic connections and algebraization of germs of holonomic \(\mathcal D\)-modules of one variable.

In chapter II the author studies the regularity condition for the meromorphic connections and the holonomic \(\mathcal D\)-modules of 1 variable. He gives the algebraic description of germs of regular holonomic \(\mathcal D\)-modules of 1 variable, in terms of diagrams of vector spaces, and the role of “vanishing cycles” and “microsolutions” (see L. Boutet de Monvel in [Mathématique et physique, Sémin. Ec. Norm. Supér., Paris 1979-1982 Prog. Math. 37, 313–321 (1983; Zbl 0578.35080)] and J. Briançon and Ph. Maisonobe [Enseign. Math., II. Ser. 30, 7–38 (1984; Zbl 0542.14008)]). The chapter ends with the convergent-formal equivalence in the regular case.

Chapters III and IV deal with the one dimensional irregular case. First, the author studies the classification of formal meromorphic connections, by using the Newton polygon [W. Wasov “Asymptotic expansions for ordinary differential equations. New York etc.: Interscience Publishers (1965; Zbl 0133.35301) and A. Levelt, Ark. Mat. 13, 1–27 (1975; Zbl 0305.34008)], and of formal holonomic \(\mathcal D\)-modules of one variable. Second, the author studies the classification of convergent meromorphic connections, based on the notion of “Stokes structure” of Deligne-Malgrange [see D. Bertrand, Séminaire Bourbaki, Vol. 1978/79, Exp. No. 538, Lect. Notes Math. 770, 228–243 (1980; Zbl 0445.12012) and the author in [Mathématique et physique, Sémin. Éc. Norm. Supér., Paris 1979–1982), Prog. Math. 37, 381–391 (1983; Zbl 0531.58015)], and the irregular Riemann-Hilbert correspondence in dimension 1. In this material, one can see how the classical theory of asymptotic expansions can be enriched by sheaf theory and cohomology, to obtain strong results under a very conceptual form. The chapter ends with the author’s index and comparison theorems [Enseign. Math., II. Sér. 20, 147–176 (1974; Zbl 0299.34011)].

Chapters V, VI, VII are concerned with formal, geometric and local Fourier transforms [see J.-L. Brylinski, the author and J.-L. Verdier, C. R. Acad. Sci., Paris, Sér. I 297, 55–58 (1983; Zbl 0553.14005); ibid. 303, 193–198 (1986; Zbl 0601.32010) and the author, Adv. Math., Suppl. Stud. 78, Math. 513–530 (1981; Zbl 0468.46026)]. The main topic treated is how the geometric data describing the one dimensional holonomic \(\mathcal D\)-modules, i.e. De Rham complexes, vanishing cycles and Stokes structures, are changed by the classical formal Fourier transform: \(x\mapsto \partial_{\xi}, \partial_x \mapsto -\xi\).

Chapters VIII, IX, X, XI deal with the phase stationary method, the Laplace integral, the study of growth conditions at infinity, and the general inversion formula.

Chapter XII analyzes the topics treated before in a concrete example: the equations of exponential type.

The book ends with two appendices. The first one sums up the study of asymptotic expansions with sheaf theory. The second one recalls the algebraic Fourier transform of Deligne-Katz-Laumon of \(\mathcal D\)-modules.

Chapter I gives a summary, without proofs, of the theory of \(\mathcal D\)- modules: holonomic \(\mathcal D\)-modules, direct and inverse images by closed (resp. open) embeddings, De Rham complex, meromorphic connections and algebraization of germs of holonomic \(\mathcal D\)-modules of one variable.

In chapter II the author studies the regularity condition for the meromorphic connections and the holonomic \(\mathcal D\)-modules of 1 variable. He gives the algebraic description of germs of regular holonomic \(\mathcal D\)-modules of 1 variable, in terms of diagrams of vector spaces, and the role of “vanishing cycles” and “microsolutions” (see L. Boutet de Monvel in [Mathématique et physique, Sémin. Ec. Norm. Supér., Paris 1979-1982 Prog. Math. 37, 313–321 (1983; Zbl 0578.35080)] and J. Briançon and Ph. Maisonobe [Enseign. Math., II. Ser. 30, 7–38 (1984; Zbl 0542.14008)]). The chapter ends with the convergent-formal equivalence in the regular case.

Chapters III and IV deal with the one dimensional irregular case. First, the author studies the classification of formal meromorphic connections, by using the Newton polygon [W. Wasov “Asymptotic expansions for ordinary differential equations. New York etc.: Interscience Publishers (1965; Zbl 0133.35301) and A. Levelt, Ark. Mat. 13, 1–27 (1975; Zbl 0305.34008)], and of formal holonomic \(\mathcal D\)-modules of one variable. Second, the author studies the classification of convergent meromorphic connections, based on the notion of “Stokes structure” of Deligne-Malgrange [see D. Bertrand, Séminaire Bourbaki, Vol. 1978/79, Exp. No. 538, Lect. Notes Math. 770, 228–243 (1980; Zbl 0445.12012) and the author in [Mathématique et physique, Sémin. Éc. Norm. Supér., Paris 1979–1982), Prog. Math. 37, 381–391 (1983; Zbl 0531.58015)], and the irregular Riemann-Hilbert correspondence in dimension 1. In this material, one can see how the classical theory of asymptotic expansions can be enriched by sheaf theory and cohomology, to obtain strong results under a very conceptual form. The chapter ends with the author’s index and comparison theorems [Enseign. Math., II. Sér. 20, 147–176 (1974; Zbl 0299.34011)].

Chapters V, VI, VII are concerned with formal, geometric and local Fourier transforms [see J.-L. Brylinski, the author and J.-L. Verdier, C. R. Acad. Sci., Paris, Sér. I 297, 55–58 (1983; Zbl 0553.14005); ibid. 303, 193–198 (1986; Zbl 0601.32010) and the author, Adv. Math., Suppl. Stud. 78, Math. 513–530 (1981; Zbl 0468.46026)]. The main topic treated is how the geometric data describing the one dimensional holonomic \(\mathcal D\)-modules, i.e. De Rham complexes, vanishing cycles and Stokes structures, are changed by the classical formal Fourier transform: \(x\mapsto \partial_{\xi}, \partial_x \mapsto -\xi\).

Chapters VIII, IX, X, XI deal with the phase stationary method, the Laplace integral, the study of growth conditions at infinity, and the general inversion formula.

Chapter XII analyzes the topics treated before in a concrete example: the equations of exponential type.

The book ends with two appendices. The first one sums up the study of asymptotic expansions with sheaf theory. The second one recalls the algebraic Fourier transform of Deligne-Katz-Laumon of \(\mathcal D\)-modules.

Reviewer: Luis Narváez-Macarro (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 |

34M99 | Ordinary differential equations in the complex domain |

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 |