The present book is a sequel to a previous one by the same authors [Singularities of differentiable mappings. Classification of critical points, caustics and wave fronts (1982; Zbl 0513.58001)].
In the authors’ words: ”If the previous book contained the basics of the zoology of the singularities of differentiable mappings, i.e. it was devoted to the description of where and what kind of singularities can be met, this book contains elements of the anatomy and physiology of the singularities of differentiable mappings. This means that in it are considered the problems of construction of singularities and of their functioning. Another distinctive feature of the present book is the emphasis on those problems for which it is important to get out into the complex field, while the first part is devoted to topics for the most of which it is not essential over which field (real or complex) they are considered. Problems like, for example, the distribution of singularities, the relation between singularities and Lie algebras, the asymptotics of various parameter depending integrals, become clearer in the complex domain.”
The text is divided into three chapters. The first chapter deals with the topological construction of isolated critical points of holomorphic functions. It starts with an introduction to Picard-Lefschetz theory (vanishing cycles, the monodromy and variation operators, Picard- Lefschetz operators) and proceeds with the description of the basic topological characteristics of these critical points and of their interrelationship and methods of computation (nonsingular level sets, the monodromy groups, the Seifert form, the Picard-Lefschetz theorem, intersection matrices, stabilization, bifurcation diagrams, resolution, distinguished bases, polar curves, intersection forms of unimodal and bimodal singularities, intersection matrices of singularities of functions of two variables, intersection forms of boundary singularities and the topology of complete intersections).
Chapter II is devoted to the study of oscillatory integrals and of their asymptotics by the method of stationary phase. It includes a discussion of the main results and a detailed description of methods for computing asymptotics, as well as a discussion of the relations between asymptotics and the various characteristics of critical points of phases of integrals (resolution of singularities, Newton polygons). Many examples and tables of orders of asymptotics for critical points of phases (in particular, for simple, unimodal and bimodal ones) are added.
Chapter III contains a treatment of the integral calculus on the level manifolds of critical points of holomorphic mappings. The main object of study is the integral of a holomorphic form, defined in a neighborhood of a critical point, on a cycle lying in a level manifold of a function and vanishing into the critical point. First, the simplest properties (holomorphic dependence on parameters, series decompositions) are presented; then, the complex oscillatory integrals are studied as a particular case of the integrals in the saddlepoint method. Further, the relationship with differential equations is investigated (the determinant theorem, the Gauss-Manin connection). The exposition proceeds with the study of the asymptotic decomposition coefficients for continuously parameter dependent integrals of holomorphic forms on cycles, of the Hodge and weight filtrations, and of the relations between the mixed Hodge structure of a critical point and other characteristics; it concludes with constructions and examples concerning the period mapping and the intersection form.
The authors also included a large reference list, updated to the last moment.