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Applied Picard–Lefschetz Theory http://dx.doi.org/10.1090/surv/097 Applied Picard-Lefschetz Theory Mathematical Surveys and Monographs Volume 97 Applied Picard-Lefschetz Theory V. A. Vassiliev AVAEM^ American Mathematical Society Editorial Board Peter S. Landweber Tudor Stefan Ratiu Michael P. Loss, Chair J. T. Stafford 2000 Mathematics Subject Classification. Primary 14D05, 14B05, 31B10, 32S40, 35B60; Secondary 33C70, 35L67. Library of Congress Cataloging-in-Publication Data Vasil'ev, V. A., 1956- Applied Picard-Lefschetz theory / V. A. Vassiliev. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 97) Includes bibliographical references and index. ISBN 0-8218-2948-3 (alk. paper) 1. Picard-Lefschetz theory. 2. Singularities (Mathematics) 3. Integral representations. I. Title. II. Mathematical surveys and monographs ; no. 97. QA564.V37 2002 516.3'5—dc21 2002066541 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 Contents PREFACE ix INTRODUCTION 1 1. Monodromy and its localization 1 2. Newton's problem on the integrability of ovals 6 3. Surface potentials 13 4. Petrovskh theory of lacunas for hyperbolic operators 18 5. Hypergeometric integrals 22 Chapter I. LOCAL MONODROMY THEORY OF ISOLATED SINGULARITIES OF FUNCTIONS AND COMPLETE INTERSECTIONS 29 1. GauB-Manin connection in homological bundles. Monodromy and variation operators 29 2. Picard-Lefschetz formula 32 3. Monodromy theory of isolated function singularities 37 4. Dynkin diagrams of real singularities of functions of two variables (after S.M. Gusein-Zade and N. A'Campo) 51 5. Classification of singularities of smooth functions 56 6. Lyashko-Looijenga covering 62 7. Complements of discriminants of real simple singularities (after E. Looijenga) 65 8. Pham singularities 67 9. Singularities and local monodromy of complete intersections 71 Chapter II. STRATIFIED PICARD-LEFSCHETZ THEORY AND MONODROMY OF HYPERPLANE SECTIONS 75 1. Stratifications of semianalytic and subanalytic sets 76 2. Monodromy of hyperplane sections 79 3. Simplest facts on intersection homology theory 89 4. Stratified Picard-Lefschetz theory 91 Chapter III. NEWTON'S THEOREM ON THE NON-INTEGRABILITY OF OVALS 111 1. Introduction 111 2. Reduction to monodromy theory 117 3. The class "cap" 119 4. Ramification of integration chains at non-singular points 121 5. Examples 124 vi CONTENTS 6. Obstructions to integrability arising from cuspidal edges. Proof of Theorem 1.8 126 7. Ramification close to asymptotic hyperplanes. Proof of Theorem 1.9 133 8. Open problems 136 Chapter IV. LACUNAS AND LOCAL PETROVSKII CONDITION FOR HYPERBOLIC DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS 137 1. Introduction 137 2. Hyperbolic polynomials 140 3. Hyperbolic operators and hyperbolic polynomials. Sharpness, diffusion, and lacunas 142 4. Generating functions and generating families of wave fronts. Classification of singular points of wave fronts 146 5. Local lacunas close to non-singular points of fronts and close to singular points of types A2 and A3 (after Davydova, Borovikov and Garding) 149 6. Petrovskil and Leray cycles. Herglotz-Petrovskh-Leray formula. Petrovskil condition for global lacunas 151 7. Local Petrovskil condition and local Petrovskil cycle. Local Petrovskil condition implies sharpness 155 8. Sharpness implies the local Petrovskil condition close to the finite type points of wave fronts . 159 9. Local Petrovskil condition can be stronger than sharpness 162 10. Normal forms of non-sharpness at the singularities of wave fronts (after A.N. Varchenko) 162 11. Problems 164 Chapter V. CALCULATION OF LOCAL PETROVSKII CYCLES AND ENUMERATION OF LOCAL LACUNAS CLOSE TO REAL SINGULARITIES 165 1. Main theorems 165 2. Local lacunas close to table singularities 174 3. Calculation of the even local Petrovskh class 182 4. Calculation of the odd local Petrovskil class 187 5. Stabilization of local Petrovskil classes 191 6. Local lacunas close to simple singularities 192 7. Geometric characterization of local lacunas at simple singularities 207 8. A program enumerating topologically distinct morsifications of real function singularities 209 Chapter VI. HOMOLOGY OF LOCAL SYSTEMS, TWISTED MONODROMY THEORY, AND REGULARIZATION OF IMPROPER INTEGRATION CYCLES 215 1. Local systems and their homology groups 215 2. Twisted vanishing homology of functions and complete intersections 218 3. Regularization of non-compact cycles 224 4. The "double loop" cycle 226 5. Monodromy of twisted vanishing homology for Pham singularities 234 CONTENTS vii 6. Stratified Picard-Lefschetz theory with twisted coefficients 240 Chapter VII. ANALYTIC PROPERTIES OF SURFACE POTENTIALS 251 1. Introduction 251 2. Theorems of Newton and Ivory 254 3. Hyperbolic potentials are regular in the hyperbolicity domain (after V.I. Arnold and A.B. Givental) 256 4. Reduction to monodromy theory 260 5. Ramification of potentials and monodromy of complete intersections 265 6. Examples: curves, quadrics, and Ivory's second theorem 272 7. Description of the small monodromy group 274 8. Proof of Theorem 1.4 283 9. Proof of Theorem 1.3 284 Chapter VIII. MULTIDIMENSIONAL HYPERGEOMETRIC FUNCTIONS, THEIR RAMIFICATION, SINGULARITIES, AND RESONANCES 287 1. Introduction 287 2. Proof of the meromorphy theorem 291 3. The hypergeometric function and its one-dimensional generalizations 295 4. Homology of complements of plane arrangements. Basic strata 297 5. The number of independent hypergeometric integrals on basic strata 305 Bibliography 313 Index 321 PREFACE Many important functions of mathematical physics have integral representa­ tions, i.e. are defined by integrals depending on parameters. Such functions include, in particular, the fundamental solutions of a majority of classical partial differential equations, Newton-Coulomb potentials, integral Fourier transforms, initial data of inverse tomography problems, hypergeometric functions, Feynman integrals, etc. The general construction of these integral representations is as follows. Suppose we have an analytic fiber bundle E —> T, an exterior differential form uo on E, whose restrictions on the fibers are closed, and a family of integration cycles in these fibers, parametrized by the corresponding points of the base T and depending con­ tinuously on these points. Then the integral UJ along these cycles is a function on the base. Analytic and qualitative properties of such functions depend on the monodromy of these cycles, i.e. on the natural action of the fundamental group of the base in the homology groups of the fibers: this action defines the ramification of the analytic continuation of our integral function. The study of this action (which is a purely topological problem) allows us to answer questions on the analytic behavior of the integral function, in particular whether this function is single-valued or at least algebraic, what are the singular points of this function, and (partially) what is its asymptotics close to these points. Ramification of integral functions arising in different problems is described by different (but having some common features) versions of the Picard-Lefschetz theory. Our book contains a list of such versions, including the classical local mon­ odromy theory of singularities of functions and complete intersections, F. Pham's generalized Picard-Lefschetz formulas, stratified version of the theory (studying in particular the monodromy of homology groups of hyperplane sections of singular varieties), and also twisted versions of all these theories (related to integrals of multivalued forms). Using them, we study four famous classes of functions: — volume functions arising in the Archimedes-Newton problem on integrable bodies; — Newton-Coulomb potentials, — fundamental solutions of hyperbolic partial differential equations (studied, in particular, in the lacuna theory of Hadamard-Petrovskh-Atiyah-Bott- Garding), and — multidimensional (Gelfand-Aomoto) hypergeometric functions generalizing the Gaufi hypergeometric integral. Some of main results described in the book are as follows. 1. Newton's theorem on the algebraic nonintegrability of plane ovals (stating that the function on the space of lines in R2, associating with any line ix x PREFACE the area cut by it from a convex domain with smooth boundary, cannot be algebraic) is extended to non-convex domains in M2 and to convex domains in any even-dimensional space. In the odd-dimensional case we find numerous geometric obstructions to the algebraicity
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