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Über dieses Buch

The present. volume is the second volume of the book "Singularities of Differentiable Maps" by V.1. Arnold, A. N. Varchenko and S. M. Gusein-Zade. The first volume, subtitled "Classification of critical points, caustics and wave fronts", was published by Moscow, "Nauka", in 1982. It will be referred to in this text simply as "Volume 1". Whilst the first volume contained the zoology of differentiable maps, that is it was devoted to a description of what, where and how singularities could be encountered, this volume contains the elements of the anatomy and physiology of singularities of differentiable functions. This means that the questions considered in it are about the structure of singularities and how they function. Another distinctive feature of the present volume is that we take a hard look at questions for which it is important to work in the complex domain, where the first volume was devoted to themes for which, on the whole, it was not important which field (real or complex) we were considering. Such topics as, for example, decomposition of singularities, the connection between singularities and Lie algebras and the asymptotic behaviour of different integrals depending on parameters become clearer in the complex domain. The book consists of three parts. In the first part we consider the topological structure of isolated critical points of holomorphic functions. We describe the fundamental topological characteristics of such critical points: vanishing cycles, distinguished bases, intersection matrices, monodromy groups, the variation operator and their interconnections and method of calculation.

Inhaltsverzeichnis

Frontmatter

The topological structure of isolated critical points of functions

Introduction

Abstract
In the topological investigation of isolated critical points of complex-analytic functions the problem arises of describing the topology of its level sets. The topology of the level sets or infra-level sets of smooth real-valued functions on manifolds may be investigated with the help of Morse theory (see [255]). The idea there is to study the change of structure of infra-level sets and level sets of functions upon passing critical values. In the complex case passing through a critical value does not give rise to an interesting structure, since all the non-singular level sets near one critical point are not only homeomorphic but even diffeomorphic. The complex analogue of Morse theory, describing the topology of level sets of complex analytic functions, is the theory of Picard-Lefschetz (which historically precedes Morse theory). In Picard-Lefschetz theory the fundamental principle is not passing through a critical point but going round it in the complex plane.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 1. Elements of the theory of Picard-Lefschetz

Abstract
In this chapter we shall define concepts of Picard-Lefschetz theory such as vanishing cycles, the monodromy and variation operators, the Picard-Lefschetz operators, etc. As we have already said, they are used to investigate the topology of critical points of holomorphic functions.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 2. The topology of the non-singular level set and the variation operator of a singularity

Abstract
Let ƒ:(ℂn, 0)→(ℂ, 0) be a singularity, that is the germ of a holomorphic function, with an isolated critical point at the origin. It follows from implicit function theorem that in a neighbourhood of the origin in the space ℂn the level set ƒ-1 (ε) for ε≠0 is a non-singular analytic manifold and the level set ƒ-1(0) is a non-singular manifold away from the origin. At the point 0∈ℂn the level set has a singular point.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 3. The bifurcation sets and the monodromy group of a singularity

Abstract
The characteristics of a singularity which were discussed in the second chapter (the multiplicity of a singularity, its intersection matrix, its monodromy group…) are closely linked to such objects as the level and function bifurcation sets of the singularity, its resolution and its polar curve. Several of these links will be discussed in this chapter.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 4. The intersection matrices of singularities of functions of two variables

Abstract
The method of calculating the intersection matrix of a singularity of a function of two variables described in this chapter is due to S. M. Gusein-Zade ([147], [148]) and N. A’Campo ([7], [8]). It applies to all singularities of two variables. Using it allows us substantially to simplify many calculations connected with the quadratic form of a singularity (for example, the calculation of its signature).
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chaptrer 5. The intersection forms of boundary singularities and the topology of complete intersections

Abstract
In this chapter we shall give a short exposition of some generalizations concerned, principally, with the concepts of the intersection form and vanishing cycles for singularities of functions on manifolds with boundary, for complete intersections,…
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Oscillatory integrals

Frontmatter

Chapter 6. Discussion of results

Abstract
The problems of optics, acoustics and quantum mechanics, the theory of partial differential equations, probability theory and number theory lead to the need to study oscillatory integrals with large values of the parameter.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 7. Elementary integrals and the resolution of singularities of the phase

Abstract
In this chapter we shall study the asymptotics of an oscillatory integral, the phase of which is a monomial. We shall indicate the connection between the asymptotics of an oscillatory integral and the poles of the meromorphic function
$$F(\lambda)=\smallint{f^\lambda}(x)\Phi(x)dx,$$
, Where ƒ is the phase, and ϕ is the amplitude of the oscillatory integral. We shall introduce the discrete characteristics of the resolution of the singularity of a critical point of the phase: the weight of the resolution and the multiplicity set. We shall describe the connection between these characteristics and the basic characteristics of the asymptotic behaviour of the oscillatory integral: the oscillation index, its multiplicity and the index set.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 8. Asymptotics and Newton polyhedra

Abstract
We shall consider the class of critical points of the phase, the Taylor series of which have fixed Newton polyhedron. If the Newton polyhedron is remote, then almost all the critical points of the class have the same oscillation index. This common oscillation index is equal to the remoteness of the Newton polyhedron. A critical point of the class has the typical oscillation index if the principal part of its Taylor series is ℝ-nondegenerate. (Remember that the condition of ℝ-nondegeneracy is an explicitly written-out algebraic condition on a finite set of Taylor coefficients, see § 6.2). This assertion was formulated as Theorem 6.4. Its proof occupies the whole of the present chapter. The proof uses the resolution of the singularity of the critical point of the phase. In the previous chapter we defined a numerical characteristic of the resolution of a singularity, namely the weight, and we proved that if the weight is greater than —1 then the oscillation index of the critical point is equal to the weight (Theorem 7.5). In this section we construct a manifold and a map of it into ℝn, which resolves the singularity of almost all the critical points of the class we are considering. We shall show that the weight of the constructed resolution of the singularity is equal to the remoteness of the Newton polyhedron. In this way we shall prove Theorem 6.4.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 9. The singular index, examples

Abstract
In this chapter we shall prove the additivity of the oscillation index, and describe explicitly the calculation of the singular index in the tables in § 6.1.10. In the second part of the chapter we give an example of the deformation of a critical point. This example illustrates several phenomena. First, the absence of semicontinuity of the oscillation index. Second, the existence of critical points which are complex equivalent but which have distinct singular indices. Third, the existence of a critical point in which the singular index is not equal to the remoteness. Finally, the existence of a critical point in which the principal part of the Taylor series is ℝ-nondegenerate but the remoteness of the Newton polyhedron is greater than the oscillation index.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Integrals of holomorphic forms over vanishing cycles

Frontmatter

Chapter 10. The simplest properties of the integrals

Abstract
In this chapter we shall prove the holomorphic dependence of the integral on the parameters; we shall explain the connection between branches of integrals and the monodromy group in homology; we shall prove that the integral can be expanded in a series in a neighbourhood of the given value of the parameter.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 11. Complex oscillatory integrals

Abstract
In the study of the asymptotic behaviour of functions one frequently has to study the asymptotic behaviour of integrals of the form
$$\int\limits_\Gamma {{e^{\tau f(x)}}\Phi (x)d{x_1} \wedge \ldots \wedge d{x_n}} $$
for large values of the parameter τ. Here ƒ, ϕ are holomorphic functions on ℂ n , Γ is a real n-dimensional chain, lying in ℂn, and τ is a large real parameter. The function ƒ is called the phase, the function ϕ is called the amplitude. Such integrals are called integrals of the saddle-point method. For examples of problems in which the need to study these arises see, for example, the book of M. B. Fedoryuk, “The saddle-point method” [110].
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 12. Integrals and differential equations

Abstract
In this chapter we shall prove that many-valued functions, given as integrals of a holomorphic differential form over classes of continuous families of homologies, vanishing at the critical point of a holomorphic function, are all solutions of an ordinary homogeneous linear differential equation, the order of which is not greater than the multiplicity of the critical point. The analysis of this phenomenon leads to the concept of the Gauss-Manin connection in the fibration of vanishing cohomologies associated with the Milnor fibration of the critical point.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 13. The coefficients of series expansions of integrals, the weight and Hodge filtrations and the spectrum of a critical point

Abstract
Let us consider the integral of a holomorphic differential form over a homology class of a continuous family of integral homologies of fibres of the Milnor fibration of a critical point. The function given by the integral can be expanded in a series of powers of the parameter and powers of the logarithm of the parameter of the family (Chapter 10). Each coefficient of the series depends linearly both on the form and on the continuous family of integral homologies. If the form is fixed but the continuous family varies, then each coefficient of the series is a linear function of the continuous families. Linear combinations, over ℂ of continuous families of integral homology classes form the space of covariantly constant (with respect to the Gauss-Manin connection) sections of the homological Milnor fibration. Therefore (if the form is fixed) each coefficient of the series is a linear function on the space of covariantly constant sections of the homological Milnor fibration, that is it is a covariantly constant section of the cohomological Milnor fibration.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 14. The mixed Hodge structure of an isolated critical point of a holomorphic function

Abstract
A mixed Hodge structure in a vector space is two filtrations of the space, satisfying the axioms indicated below. In the space of cohomologies, vanishing at the critical point of the holomorphic function, there is a natural mixed Hodge structure. The role of the above-mentioned filtrations is played by the weight and Hodge filtrations, introduced in Chapter 13. The weight filtration is constructed from the Jordan structure of the monodromy operator and reflects the behaviour of integrals over vanishing cycles under analytic continuation of the integrals round critical values of the parameter. The Hodge filtration is constructed by starting from a comparison of the rates of convergence to zero of integrals over vanishing cycles as the parameter of the integrals converges to the critical value. As is well-known, there are in geometry two theories which study a function in a neighbourhood of a critical point: Morse theory and Picard-Lefschetz theory. Morse theory studies the reconstruction of a level hypersurface of the function as the level tends to the critical value. Picard-Lefschetz theory studies the transformation of a level hypersurface of a function as the level goes round the critical value in the complex plane. In this sense the theory of mixed Hodge structures of critical points is a synthesis of Morse theory and Picard-Lefschetz theory. The mixed Hodge structure in the vanishing cohomology plays an outstanding role in the local theory of singularities.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Chapter 15. The period map and the intersection form

Abstract
Let us be given a smooth fibration and a differential form on the space of fibration which is closed on the fibres. In such a situation there arises the period map of the form — a many-valued map from the base of the fibration to the cohomology of the fibre. A point of the base is mapped to the cohomology class of the form in the fibre over the point translated to the cohomology of a distinguished fibre. The fact that it is many-valued arises from the fact that there is not a unique choice of path for the translation.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Backmatter

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