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The monodromy group. (English) Zbl 1103.32015

The aim of the book under review is to present a very powerful approach based on modern methods of monodromy theory to the study of many problems from singularity theory, algebraic geometry and theory of differential equations.
In the preface the author outlines the historical background of development of the monodromy theory from B. Riemann till present time and briefly describes the contents of his book. He also underlines that essential parts of the book are based on a two-year course at Warsaw University given by the author more than ten years ago.
In the first four chapters the author gives an introduction to the notion of monodromy in applications to multi-valued holomorphic functions and their Riemann surfaces, to the Morse theory in the real domain and the theory of normal forms for functions, he also describes some basic facts from the algebraic topology of manifolds and fibre bundles as well as from the topology and monodromy of analytic and algebraic functions including the Milnor theorem, Picard-Lefschetz formula, root systems, Coxeter groups, resolution and normalization of singularities, etc.
The next chapter 5 is devoted to the study of integrals along vanishing cycles, including basic results from the theory of Gauss-Manin connection, Picard-Fuchs systems, oscillating integrals and their relations with singularity theory. Then a method of abelian integrals is explained in detail; this chapter contains a few good examples and description of results Khovanski, Gabrielov, Petrov, and others.
It should be remarked that most of the above materials can also be found in the book [V. I. Arnol’d, et al., Singularities of differentiable maps. Volumes I, II. Monographs in Mathematics, Vol. 82, 83. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001, 1988; Zbl 0659.58002)] .
In chapter 7 general ideas of the theory of Hodge structures and period mappings are described. Starting from classical results on the Hodge structure on algebraic manifolds, the author goes into the theory of mixed Hodge structures on incomplete manifolds and on the cohomological Milnor bundle, he also discusses the notions of limit Hodge structures in the sense of Schmid and Steenbrink, relations with the monodromy theory, etc. In conclusion, some basic constructions due to D. Mumford and Ph. Griffiths from the theory of period mappings in algebraic geometry are considered [see, for example, P. A. Griffiths, Bull. Am. Math. Soc. 76, 228–296 (1970; Zbl 0214.19802)].
The subject of the next chapter 8 are the non-autonomous linear differential systems \(\dot z = A(t)z\), \( z\in \mathbb{C}^m,\) and the linear higher order differential equations \(x^{(n)} + a_1(t)x^{(n-1)} + \cdots + a_n(t)x = 0, \, x\in \mathbb{C},\) where \(t\in {\mathbb C}\) and all the entries of the matrix \(A(t)\) as well as the coefficients \(a_i(t)\) are meromorphic functions. The author discusses the notion of regular and irregular singularities, some aspects of the global theory of linear differential equations, the Riemann-Hilbert problem with a detailed analysis of Bolibruch’s counter-example, the notion of isomonodromic deformations and some applications and relations with quantum field theory.
The chapters 9 and 10 are mainly concentrated on the local and global theory of holomorphic foliations in \(\mathbb{C P}^2,\) respectively. They contain the theory of resolution of vector fields, the theory of resurgent functions in the sense of Ecalle, the theory of Martinet-Ramis modules, theorems of Bryuno and Yoccoz, some results concerning the nonlinear Riemann-Hilbert problem [P. M. Elizarov, et al., Nonlinear Stokes phenomena. Providence, RI: American Mathematical Society. Adv. Sov. Math. 14, 57–105 (1993; Zbl 1010.32501)], Ziglin theory, and a lot of other very interesting material with a number of useful examples.
In chapter 11 the author presents the basic notions and tools of differential Galois theory including the theory of Picard-Vessiot extensions with applications to the problem of integration of polynomial vector fields (Singer’s theorem). He then discusses the monodromy theory of algebraic functions including the topological proof of Abel-Ruffini theorem, Khovanski’s generalization of the monodromy group for large class of functions, the monodromy properties of Singer’s first integrals.
The concluding chapter 12 is mainly based on the famous books of F. Klein [see ‘Vorlesungen über die hypergeometrische Funktion’. Berlin: Springer (1933; Zbl 0007.12202 and JFM 59.0375.11), and so on.] It contains classical results from the theory of hypergeometric functions and explicit calculations of the monodromy and solutions of hypergeometric equations in quadratures. In conclusion two kinds of generalizations of hypergeometric functions are described. The first one is based on the Picard-Deligne-Mostow approach while the second – on the approach of I. Gelfand, A. Varchenko, and others.
The book under review is written in a very clear and concise style, almost all key topics are followed by carefully chosen examples, non-formal remarks and comments. It contains a large number of pictures which may be considered as real visualizations of the discussed ideas. The bibliography includes 359 selected references. This makes the exposition not only accessible to beginning graduate students but highly interesting and useful for advanced research workers in algebraic and differential topology, algebraic geometry, complex analysis, differential equations and related fields of pure and applied mathematics and mathematical physics.

MSC:

32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
58K10 Monodromy on manifolds
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
32G20 Period matrices, variation of Hodge structure; degenerations
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
32S30 Deformations of complex singularities; vanishing cycles
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
32S55 Milnor fibration; relations with knot theory
58K05 Critical points of functions and mappings on manifolds
58K45 Singularities of vector fields, topological aspects
58K65 Topological invariants on manifolds
12H05 Differential algebra
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