**Mathematical Surveys and Monographs**

Volume: 97;
2002;
324 pp;
Hardcover

MSC: Primary 14; 31; 32; 35;
Secondary 33

**Print ISBN: 978-0-8218-2948-6
Product Code: SURV/97**

List Price: $102.00

AMS Member Price: $81.60

MAA Member Price: $91.80

**Electronic ISBN: 978-1-4704-1324-8
Product Code: SURV/97.E**

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# Applied Picard–Lefschetz Theory

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*V. A. Vassiliev*

Many important functions of mathematical physics are defined as integrals
depending on parameters. The Picard–Lefschetz theory studies how analytic
and qualitative properties of such integrals (regularity, algebraicity,
ramification, singular points, etc.) depend on the monodromy of corresponding
integration cycles. In this book, V. A. Vassiliev presents several versions of
the Picard–Lefschetz theory, including the classical local monodromy
theory of singularities and complete intersections, Pham's generalized
Picard–Lefschetz formulas, stratified Picard–Lefschetz theory, and
also twisted versions of all these theories with applications to integrals of
multivalued forms.

The author also shows how these versions of the Picard–Lefschetz
theory are used in studying a variety of problems arising in many areas of
mathematics and mathematical physics.

In particular, he discusses the following classes of functions:

- volume functions arising in the Archimedes–Newton problem of integrable bodies;
- Newton–Coulomb potentials;
- fundamental solutions of hyperbolic partial differential equations;
- multidimensional hypergeometric functions generalizing the classical Gauss hypergeometric integral.

The book is geared toward a broad audience of graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.

#### Readership

Graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.

#### Reviews & Endorsements

This is a book rich in ideas …

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Applied Picard-Lefschetz Theory

- CONTENTS v6 free
- PREFACE ix10 free
- INTRODUCTION 114 free
- CHAPTER I. LOCAL MONODROMY THEORY OF ISOLATED SINGULARITIES OF FUNCTIONS AND COMPLETE INTERSECTIONS 2942
- 1. Gauß–Manin connection in homological bundles. Monodromy and variation operators 2942
- 2. Picard–Lefschetz formula 3245
- 3. Monodromy theory of isolated function singularities 3750
- 4. Dynkin diagrams of real singularities of functions of two variables (after S. M. Gusein-Zade and N. A'Campo) 5164
- 5. Classification of singularities of smooth functions 5669
- 6. Lyashko–Looijenga covering 6275
- 7. Complements of discriminants of real simple singularities (after E. Looijenga) 6578
- 8. Pham singularities 6780
- 9. Singularities and local monodromy of complete intersections 7184

- CHAPTER II. STRATIFIED PICARD–LEFSCHETZ THEORY AND MONODROMY OF HYPERPLANE SECTIONS 7588
- CHAPTER III. NEWTON'S THEOREM ON THE NON-INTEGRABILITY OF OVALS 111124
- 1. Introduction 111124
- 2. Reduction to monodromy theory 117130
- 3. The class "cap" 119132
- 4. Ramification of integration chains at non-singular points 121134
- 5. Examples 124137
- 6. Obstructions to integrability arising from cuspidal edges. Proof of Theorem 1.8 126139
- 7. Ramification close to asymptotic hyperplanes. Proof of Theorem 1.9 133146
- 8. Open problems 136149

- CHAPTER IV. LACUNAS AND LOCAL PETROVSKII CONDITIONFOR HYPERBOLIC DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS 137150
- 1. Introduction 137150
- 2. Hyperbolic polynomials 140153
- 3. Hyperbolic operators and hyperbolic polynomials. Sharpness, diffusion, and lacunas 142155
- 4. Generating functions and generating families of wave fronts. Classification of singular points of wave fronts 146159
- 5. Local lacunas close to non-singular points of fronts and close tosingular points of types A[sub(2)] and A[sub(3)] (after Davydova, Borovikov and G°arding) 149162
- 6. Petrovskil and Leray cycles. Herglotz–Petrovskh–Leray formula. Petrovskil condition for global lacunas 151164
- 7. Local Petrovskil condition and local Petrovskil cycle. Local Petrovskil condition implies sharpness 155168
- 8. Sharpness implies the local Petrovskil condition close to the finite type points of wave fronts 159172
- 9. Local Petrovskil condition can be stronger than sharpness 162175
- 10. Normal forms of non-sharpness at the singularities of wave fronts (after A.N. Varchenko) 162175
- 11. Problems 164177

- CHAPTER V. CALCULATION OF LOCAL PETROVSKII CYCLES AND ENUMERATION OF LOCAL LACUNAS CLOSE TO REAL SINGULARITIES 165178
- 1. Main theorems 165178
- 2. Local lacunas close to table singularities 174187
- 3. Calculation of the even local Petrovskil class 182195
- 4. Calculation of the odd local Petrovskil class 187200
- 5. Stabilization of local Petrovskil classes 191204
- 6. Local lacunas close to simple singularities 192205
- 7. Geometric characterization of local lacunas at simple singularities 207220
- 8. A program enumerating topologically distinct morsifications of real function singularities 209222

- CHAPTER VI. HOMOLOGY OF LOCAL SYSTEMS, TWISTED MONODROMY THEORY, AND REGULARIZATION OF IMPROPER INTEGRATION CYCLES 215228
- 1. Local systems and their homology groups 215228
- 2. Twisted vanishing homology of functions and complete intersections 218231
- 3. Regularization of non-compact cycles 224237
- 4. The "double loop" cycle 226239
- 5. Monodromy of twisted vanishing homology for Pham singularities 234247
- 6. Stratified Picard–Lefschetz theory with twisted coefficients 240253

- CHAPTER VII. ANALYTIC PROPERTIES OF SURFACE POTENTIALS 251264
- 1. Introduction 251264
- 2. Theorems of Newton and Ivory 254267
- 3. Hyperbolic potentials are regular in the hyperbolicity domain (after V.I. Arnold and A.B. Givental) 256269
- 4. Reduction to monodromy theory 260273
- 5. Ramification of potentials and monodromy of complete intersections 265278
- 6. Examples: curves, quadrics, and Ivory's second theorem 272285
- 7. Description of the small monodromy group 274287
- 8. Proof of Theorem 1.4 283296
- 9. Proof of Theorem 1.3 284297

- CHAPTER VIII. MULTIDIMENSIONAL HYPERGEOMETRIC FUNCTIONS, THEIR RAMIFICATION, SINGULARITIES, AND RESONANCES 287300
- BIBLIOGRAPHY 313326
- INDEX 321334