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 of such a function, showing that this algebraicity (taking place for a ball, accordingly to Archimedes) is a very exceptional situation. 2. In the theory of hyperbolic partial differential equations, for almost all opera­ tors with constant coefficients we prove the Atiyah-Bott-Garding conjecture on the equivalence of the local regularity ("sharpness") of the fundamental solution to the local topological Petrovskh condition; on the other hand we present a very degenerate operator, for which the conjecture fails. Also we find all domains of regularity close to all simple singularities of the wave front (in particular, to all singularities arising on generic fronts in < 7-dimensional spaces) and to many non-simple singularities. A combinatorial algorithm for finding such domains is described (its computer realization is available via http://www.pdmi.ras.ru/~arnsem/papers). 3. In potential theory, analyzing the results of Newton, Ivory and Arnold, we reduce the study of analytic properties of the Newtonian potential of an algebraic hypersurface to the monodromy theory of complete intersections, and find all cases when such potentials of generic hyperbolic hypersurfaces are algebraic. 4. In the Gelfand-Aomoto theory of general hypergeometric functions we cal­ culate the exact number of linearly independent solutions of generalized hypergeometric equations for the most important types of such functions, and prove that all these solutions have integral representations. (This part is written following our joint work with I.M. Gelfand and A.V. Zelevinskh.) 5. The "stratified" Picard-Lefschetz formulas are written explicitly; they great­ ly reduce the calculation of the ramification of homology groups of hyper- plane sections of singular algebraic varieties and their complements. The background of these results includes: — standard Picard-Lefschetz theory (describing the ramification of cycles and integrals in smooth varieties), — classification of singular points of smooth functions (essentially coinciding with the classification of singularities of wave fronts), — stratification theory and intersection homology theory (in the terms of which the ramification of cycles on singular varieties is described), — topological study of complete intersections (which is the natural tool for the study of the ramification of potentials), — theory of plane arrangements, — homology groups of local systems (to which the integration cycles of multi­ valued differential forms belong, in particular of hypergeometric forms and of kernels of the Laplace operator), and — generalizations of the Picard-Lefschetz theory for these homology groups. The necessary information about all these subjects is collected in Chapters I, II, VI, and VIII.

Acknowledgments. I thank V.I. Arnold, who attracted my attention to the three problems of integral geometry studied in Chapters III—V and VII. PREFACE xi

I thank W. Ebeling, A.M. Gabrielov, I.M. Gelfand, V.A. Ginzburg, V.V. Gor- yunov, S.M. Gusein-Zade, V.P. Palamodov, A.V. Zelevinskii, and B.Z. Shapiro for useful conversations. In preparing this manuscript I was supported in part by grants from RFBR (project 01-01-00660), NWO-RFBR (047-008-005) and INTAS (00-0259). Bibliography

[1] N. A'Campo, he groupe de monodromie du deploiement des singularites isolees de courbes planes. I, Math. Ann. 213:1 (1975), 1-32. [2] N. A'Campo, Le groupe de monodromie du deploiement des singularites isolees de courbes planes. II, In: Actes du Congres Internationale des Mathematiciens (Vancouver, 1974), 1975, vol.1, 395-404. [3] N. A'Campo, Tresses, monodromie et le groupe symplectique. Comm. Math. Helv. 54 (1979), 318-327. [4] K. Aomoto, Les equations aux differences lineaires et les integrales des fonctions multi- formes, J. Fac. Sci. Univ. of Tokyo 22 (1975), 271-297. [5] K. Aomoto, On the structure of integrals of power products of linear functions, Sci. papers, Coll. Gen. Ed., Univ. Tokyo., 26 (1977), 49-61. [6] V. I. Arnold, Cohomology ring of the colored braid group. Matem. Zametki 5:2 (1969), 227-231; Engl, transl., Math. Notes 5 (1969), 138-140. [7] V. I. Arnold, On the disposition of ovals of real plane algebraic curves, involutions of four- dimensional smooth manifolds and arithmetic of integer quadratic forms, Funkts. Anal, i Prilozh., 5:3 (1971), 1-9; Engl, translation in Functional Anal. Appl., 5 (1971). [8] V. I. Arnold, Normal forms of functions close to degenerate critical points, the Weyl groups Ak,Dk,Ek, and Lagrange singularities. Funkts. Anal, i Prilozh., 6:4 (1972), 3-25; Engl. translation in Functional Anal. Appl., 6 (1972), 254-272. [9] V. I. Arnold, Remarks on the stationary phase method and Coxeter numbers, Uspekhi Mat. Nauk, 28:5, 1973, 17-44; Engl, transl. in Russian Math. Surveys, 28:5 (1973), 19-48. [10] V. I. Arnold, Normal forms of functions in neighborhoods of degenerate critical points, Uspekhi Mat. Nauk 29:2 (1974), 11-49, Engl, transl. in Russian Math. Surveys 29:2 (1974), 10-50. [11] V. I. Arnold, Critical points of smooth functions and their normal forms, Uspekhi Mat. Nauk 30:5 (1975), 3-65, Engl, transl. in Russian Math. Surveys 30:5 (1975), 1-75. [12] V. I. Arnold, Additional chapters of the theory of ordinary differential equations, Nauka, Moscow, 1978, 304 p. Engl, transl.: see [20]. [13] V. I. Arnold, On the Newtonian potential of hyperbolic layers, Selecta Math. Soviet., 4:2, 1985, 103-106. [14] V. I. Arnold, Singularities of systems of rays, Uspekhi Mat. Nauk 38 (1983), no.2, 77-147, Engl, transl. in Russian Math. Surveys 38 (1983), no.2, 87-176. [15] V. I. Arnold, Magnetic field analogues of the theorems of Newton and Ivory, Uspekhi Mat. Nauk 1983, 38:5, 145-146. [16] V. I. Arnold, Kepler's second law and the topology of Abelian integrals, Kvant, 1987, 12, 17-21. Engl, transl. in Kvant Selecta: Algebra and Analysis II, Math. World, vol. 15, Amer. Math. Soc, Providence, RI, 1999, 131-140. [17] V. I. Arnold, The 300-th anniversary of mathematical natural philosophy and celestial me­ chanics, Priroda, 1987, No. 8, 5-15. (Russian) [18] V. I. Arnold, Surfaces defined by hyperbolic equations, Matem. Zametki 44:1 (1988), 3-18; Engl, transl.: Math. Notes 44 (1988), 489-497. [19] V. I. Arnold, Ramified covering QP2 —> S4, hyperbolicity and projective topology, Sibirsk. Mat. Zh., 29:5 (1988), 36-47; Engl, transl.: Siberian Math. J., 29:5 (1988), 717-726. [20] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer, New York, 1988. [21] V. I. Arnold, Mathematical methods in classical mechanics, "Nauka", Moscow, 1989; English transl. of 1st edition: Springer-Verlag, Berlin and New York, 1978.

313 314 BIBLIOGRAPHY

[22] V. I. Arnold, Huygens and Barrow, and Newton and Hooke — the first steps of mathematical analysis and catastrophe theory, from the evolvents to quasicrystals, "Nauka", Moscow, 1989; English translation: Birkhauser, Basel, 1990. [23] V. I. Arnold, Singularities of caustics and wave fronts. Dorderecht-Boston-London: Kluwer Academic Publ., 1990. [24] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of differentiable maps, vol.1: Classification of critical points, caustics, and wave fronts. "Nauka", Moscow, 1982; Engl, transl.: Birkhauser, Basel, 1985. [25] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of differentiable maps, vol.2: Monodromy and asymptotics of integrals, "Nauka", Moscow, 1984; Engl, transl.: Birkhauser, Basel, 1988. [26] V. I. Arnold, V. A. Vassiliev, V.V. Goryunov, and O.V. Lyashko, Singularities, 1. Dynamical systems, 6, Itogi Nauki i Tekhniki: Sovr. Probl. Mat.: Fungamentalnye Napravleniya, vol. 6, VINITI, Moscow, 1988; English transl.: Encyclopaedia Math. Sci., vol. 6, Springer-Verlag, Berlin and New York, 1993+1997. [27] V. I. Arnold, V. A. Vassiliev, V.V. Goryunov, and O.V. Lyashko, Singularities, 2. Dynamical systems, 8, Itogi Nauki i Tekhniki: Sovr. Probl. Mat.: Fungamentalnye Napravleniya, vol. 39, VINITI, Moscow, 1989; English transl.: Encyclopaedia Math. Sci., vol. 39, Springer- Verlag, Berlin and New York, 1993. [28] V. I. Arnold and V. A. Vassiliev, Newton's Principia read 300 years later, Notices Amer. Math. Soc, 36:9 (1989), 1148-1154 and 37:2 (1990), 144. [29] M. F. Atiyah, R. Bott, and L. Garding, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math., 124 (1970), 109-189. [30] M. F. Atiyah, R. Bott, and L. Garding, Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Math., 131 (1973), 145-206. [31] M. F. Atiyah, Resolution of singularities and division of distributions, Comm. Pure and Appl. Math., 23:2 (1970), 145-150. [32] Yu. Yu. Berest and A. P. Veselov, The Hadamard problem and Coxeter groups: new exam­ ples of Huygens' equations, Funkts. Anal, i Prilozh., 28:1 (1994), 3-15. Engl, translation in Functional Anal. Appl., 28:1 (1994). [33] A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Asterisque 100, Soc. Mat. de France, Paris 1982, 5-171. [34] I. N. Bernstein and S. I. Gelfand, Meromorphy of the function Px, Funkts. Anal, i Prilozh., 3:4 (1969), 84-85, Engl, translation in Functional Anal. Appl., 3:4 (1969), 68-69. [35] A. Bjorner, On the homology of geometric lattices, Algebra Univ. 14:1 (1982), 107-128. [36] V. A. Borovikov, Fundamental solutions of linear partial equations with constant coeffi­ cients, Trudy Moskov. Mat. Obshch. 8 (1959), 199-257. English transl: Amer. Math. Soc. Translations, Ser.2, vol. 25, 11-76, 1963. [37] V. A. Borovikov, Some sufficient conditions for the absence of lacunas, Mat. Sbornik, 55 (97) (1961), 237-254. (Russian) [38] N. Bourbaki, Groupes et algebres de Lie, Chapitres 4, 5, 6, Hermann, Paris, 1968. [39] E. Brieskorn, Die Monodromie der isolierten Singularitaten von Hyperflachen, Manuscr. Math., 2(1970), 103-161. [40] E. Brieskorn, Sur les groupes de tresses (d'apres V. I. ArnoVd). In: Seminaire Bourbaki, 24, No. 401 (1971/72). Berlin: Springer, 1973. 21-44. (Lect. Notes Math, 317). [41] J. Cheeger, M. Goresky, and R. MacPherson, L2-cohomology and intersection homology of singular algebraic varieties, in Seminar on Differential Geometry, Ann. of Math. Studies, vol. 102, Princeton Univ. Press, Princeton, 1982, pp. 303-340. [42] D. Cheniot, Sur les sections transversales d'un ensemble stratifie, C. R. Ac. Sci. Paris, ser.A, 275 (1972), 915-916. [43] Yu. S. Chislenko, Decompositions of simple singularities of real functions, Funkts. Anal. i Prilozh., 22:4 (1988), 52-67; Engl, translation in Functional Anal. Appl., 22:4 (1988), 297-310. [44] A. M. Davydova, A sufficient condition for the absence of a lacuna for a partial differential equation of hyperbolic type, Ph.D.Thesis, Moscow State Univ., 1945, 43 pp. [45] P. Deligne, Equations differentielles a points singuliers reguliers. Lect. Notes Math. 163, Springer, Berlin, 1970. BIBLIOGRAPHY 315

[46] P. Deligne and G. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IHES 63 (1986), 5-89. [47] P. Deligne, Les immeubles des groupes de tresses generalises, Invent. Math., 17 (1972), 273-302. [48] A. Dimca and C. G. Gibson, Contact germs from the plane to the plane, in: [149], Part 1, 277-282. [49] W. Ebeling, The monodromy groups of isolated singularities of complete intersections. Lect. Notes Math., vol.1293, Springer, Berlin, 1987. [50] W. Ebeling, Appendix to [175]. [51] H. Esnault, V. Shechtman, and E. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math., 109:3 (1992), 557-561; Erratum: Ibid. 112 (1993), 447. [52] D. B. Fuchs, A. T. Fomenko, and V. L. Gutenmakher, Homotopy topology, "Nauka", Moscow, 1970. English transl.: Academiai Kiado, Budapest, 1986. [53] D. Fotiadi, M. Froissart, J. Lascoux, F. Pham, Applications of an isotopy theorem, Topology, 4 (1965), 159-191. [54] D. B. Fuchs, Cohomology of the braid group mod 2, Funkts. Anal, i Prilozh., 4:2 (1970), 62-73; Engl, translation in Functional Anal. Appl., 4:2 (1970), 143-151. [55] A. M. Gabrielov, On projections of semianalytic sets, Funkts. Anal, i Prilozh., 2:4 (1968), 18-30; Engl, translation in Functional Anal. Appl., 2 (1968), 282-291. [56] A. M. Gabrielov, Intersection matrices for some singularities, Funkts. Anal, i Prilozh., 7:3 (1973), 18-32; Engl, translation in Functional Anal. Appl., 7 (1973), 182-193. [57] A. M. Gabrielov, Bifurcations, Dynkin diagrams and modality of isolated singularities, Funkts. Anal, i Prilozh., 8:2 (1974), 7-12; Engl, translation in Functional Anal. Appl., 8 (1974), 94-98. [58] A. M. Gabrielov, Polar curves and intersection matrices of singularities, Invent. Math., 54:1 (1979), 15-22. [59] A. M. Gabrielov, Proof of a theorem of I. G. Petrovskii, in: [134], pp. 456-465 (Russian), 496-508 (English). [60] L. Garding, Local hyperbolicity, Israel J. Math., 13 (1972), 65-81. [61] L. Garding, Sharp fronts of paired oscillatory integrals, Publ. Res. Inst, for Math. Sci. Kyoto Univ., 12 (1977), 53-68. [62] I. M. Gelfand, A general theory of hypergeometric functions, Doklady AN SSSR, 288:1 (1986), 14-18, English transl.: Sov. Math. Doklady, 33, 573-577. [63] I. M. Gelfand and G. E. Shilov, Generalized Functions, vol. 1: Properties and Operations, Moscow, Fizmatlit, 1959. English translation: Academic Press, New York, 1964. [64] I. M. Gelfand and S. I. Gelfand, Generalized hypergeometric equations, Doklady AN SSSR, 288:2, (1986), 279-283. English transl.: Sov. Math. Doklady 33 (1986), 643-646. [65] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinskii, Discriminants, resultants, and multidimensional determinants, Birkhauser, Boston, 1994. [66] I. M. Gelfand and G. L. Rybnikov, Algebraic and topological invariants of oriented matroids, Doklady AN SSSR, 307:4 (1989), 791-795, English transl.: Sov. Math. Doklady, 40 (1990), 148-152. [67] I. M. Gelfand and A. V. Zelevinskii, Algebraic and combinatorial aspects of the general theory of hypergeometric functions, Funkts. Anal, i Prilozh., 1986, 20:3, 17-34; Engl, translation in Functional Anal. Appl., 20:3 (1986), 183-197. [68] M. Giusti, Classification des singularites isolees ^intersections completes simples, C.R. Ac. Sci. Paris, Ser. A, 284 (1977), 167-170. [69] M. Giusti, Classification des singularites isolees simples d'intersections completes, in [149], Part 1, 457-494. [70] A. B. Givental, P olynomiality of electro statical potentials, Uspekhi Mat. Nauk, 39:5 (1984), 253-254 (in Russian). [71] A. B. Givental, Twisted Picard-Lefschetz formulae. Funkts. Anal, i Prilozh., 22:1 (1988), 12-22; Engl, translation in Functional Anal. Appl., 22:1 (1988), 10-18. [72] M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135- 162. [73] M. Goresky and R. MacPherson, Stratified Morse theory, Springer-Verlag, Berlin and New York, 1986. 316 BIBLIOGRAPHY

[74] V. V. Goryunov, Monodromy of the image of a map C2 —» C3, Funkts. Anal, i Prilozh., 25:3 (1991), 12-18; Engl, translation in Functional Anal. Appl, 25:3 (1991), 174-180. H. Grauert and R. Remmert, Komplexe Raume, Math. Ann., 136:2 (1958), 245-318. G. M. Greuel, Dualitdt in der lokalen Kohomologie isolierter Singular it at en, Math. Ann., 250 (1980), 157-173. G. M. Greuel and H. A. Hamm, Invarianten quasihomogener vollstandiger Durchschnitte, Invent. Math., 49:1 (1978), 67-86. Ph. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley &; Sons, New York, 1978. D. A. Gudkov, Topology of real projective algebraic varieties, Uspekhi Mat. Nauk, 29:4 (1974), 3-79. English transl. in Russian Math. Surveys 29:4 (1974), 1-79. V. Guillemin and Sh. Sternberg, Geometric Asymptotics, AMS, Providence RI, 1977. S. M. Gusein-Zade, Intersection matrices for some singularities of functions of two vari­ ables, Funkts. Anal, i Prilozh., 8:1 (1974), 11-15; Engl, translation in Functional Anal. Appl., 8 (1974), 10-13. S. M. Gusein-Zade, Dynkin diagrams of singularities of functions of two variables, Funkts. Anal, i Prilozh., 8:4 (1974), 23-30; Engl, translation in Functional Anal. Appl., 8 (1974). S. M. Gusein-Zade, Monodromy groups of isolated singularities of hyper surf aces, Uspekhi Mat. Nauk 32:2 (1977), 23-65, Engl, transl. in Russian Math. Surveys 32:2 (1977), 23-69. S. M. Gusein-Zade, Index of a singular point of a gradient vector field, Funkts. Anal, i Prilozh., 18:4 (1984), 7-12; Engl, translation in Functional Anal. Appl., 18 (1984), 6-10. J. Hadamard, he probleme de Cauchy et les equations aux derivees partielles lineaires hy- perboliques, Hermann, Paris, 1932. H. Hamm, Lokale topologische Eigenschaften komplexer Raume, Math. Ann., 191:3 (1971), 235-252. H. Hamm, Lefschetz theorems for singular varieties, in [149], Part II, 575-586. H. Hamm and Le Dung Trang, Un theoreme de Zariski du type de Lefschetz, Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 317-355. R. Hardt, Topological properties of subanalytic sets, Trans. Amer. Math. Soc, 211 (1975), 57-70. R. Hartshorne, Algebraic geometry, Springer, Berlin and New York, 1977. A. Hattori, Topology of Cn minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. of Tokyo 22 (1975), 205-219. A. Hefez and F. Lazzeri, The intersection matrix of Brieskorn singularities, Invent. Math., 25:2 (1974), 143-157. Higher transcendental functions. (H. Bateman project) Vol. 1. (A. Erdelyi et al., editors), McGraw-Hill, New York-Toronto-London, 1953. H. Hironaka, Resolution of an algebraic variety over a field of characteristic zero, Ann. Math., 79:1(1964), 109-203 and 79:2 (1964), 205-326. H. Hironaka, Subanalytic sets, in: Number Theory, Algebraic Geometry and Commutative Algebra, volume in honor of Y. Akizuki, Kinokunya Tokyo 1973, 453-493. L. Hormander, Fourier integral operators. I, Acta Math. 127 (1971), 71-183. L. Hormander, The Analysis of Linear Partial Differential Operators I, II, Springer, Berlin, 1983. R. C. Hwa and V. L. Toeplitz, Homology and Feynman integrals, Behjamin, Amsterdam, 1966. J. Ivory, On the attraction of homogeneous ellipsoids, Philos. Trans. Roy. Soc. London, 99 (1809), 345-372. W. A. M. Janssen, Skew-symmetric vanishing lattices and their monodromy groups, Math. Ann., 266:1 (1983), 115-133. W. A. M. Janssen, Skew-symmetric vanishing lattices and their monodromy groups. II, Math. Ann., 272:1 (1985), 17-22. P. Jaworski, Distribution of critical values of miniver sal deformations of parabolic singular­ ities, Invent. Math., 86:1 (1986), 19-33. V. M. Kharlamov, Topology of real algebraic manifolds, in: [134], 465-493. S. L. Kleiman, The enumerative theory of singularities, in: Real and Complex Singularities. Nordic Summer School, Oslo, 1976. Sijthoff and Noordhoff, Groningen, 1977, 297-396. BIBLIOGRAPHY 317

[105] A. G. Kushnirenko, On the multiplicity of the solution of a system of holomorphic equations, Optimal Control, no.2, Izdat. Moskov. Univ., Moscow, 1977, 62-65 (in Russian). [106] K. Lamotke, Die Homologie isolierter Singularitaten. Math. Z., 143 (1975), 27-44. [107] Le Dung Trang, Calcul du nombre de cycles evanouissants d'une hypersurface complexe, Ann. Inst. Fourier, Grenoble, 23:4, 1973, 261-270. [108] Le Dung Trang, Calculation of the Milnor number of an isolated singularity of a complete intersection, Funkts. Anal, i Pril., 8:2, 1974, 45-49; Engl, transl. in Functional Anal. Appl., 8 (1974), 127-131. [109] Le Dung Trang, Vanishing cycles on complex analytic sets, Various Problems in Algebraic Analysis (Proc. Sympos., Res. Inst. Math. Sci., Kyoto Univ., 1975), Suri kaiseki kenkyusho Kokyuroku, 266 (1976), 299-318. [110] J. Leray, Un prolongement de la transformation de Laplace qui transforme la solution uni- taire d'un operateur hyperbolique en sa solution elementaire (Probleme de Cauchy, IV), Bull. Soc. Math. France, 90 (1962), 39-156. [Ill] S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. di Pisa (3) 18 (1964), 449-474. [112] S. Lojasiewicz, Ensembles semi-analytiques, preprint, Inst. Hautes Etudes Sci., 1972. [113] E. J. N. Looijenga, The complement of the bifurcation variety of a simple singularity, Invent. Math., 23:2 (1974), 105-116. [114] E. J. N. Looijenga, On the semi-universal deformations of a simple elliptic singularity, Topology 16:3 (1977), 257-262, and 17:1 (1978), 23-40. [115] E. J. N. Looijenga, The discriminant of a real simple singularity, Compositio Math., 37:1 (1978), 51-62. [116] E. J. N. Looijenga, Isolated singular points of complete intersections, Cambridge Univ. Press, Cambridge, 1984, 200 pp. [117] O. V. Lyashko, Decompositions of simple singularities of functions, Funk. Anal, i Prilozh., 10:2 (1976), 49-56; Engl, translation in Functional Anal. Appl., 10 (1976), 122-128. [118] B. Malgrange, Integrates asymptotiques et monodromie, Ann. Sci. Ec. Norm. Super. IV Ser. 7 (1974), 405-430. [119] J. Mather, Stability of C°°-mappings. I—VI. Ann. Math. 87:1 (1968), 89-104, and 89:2 (1969), 254-291; Publ. Math. IHES 35 (1969), 127-156, and 37 (1970), 223-248; Adv. Math., 4 (1970), 301-335; Lect. Notes Math. 192 (1971), 207-253. [120] J. Mather, Notes on topological stability. Mimeographed notes, Harvard University 1970. [121] J. Milnor, Morse theory, Princeton, N. J.: Princeton Univ. Press. 1963. 155 pp. (Annals of Math. Studies, 51) [122] J. Milnor, Lectures on the h-cobordism theorem, Princeton, N. J.: Princeton Univ. Press, 1965. 116 pp. [123] J. Milnor, Singular points of complex hyper surf aces, Princeton Univ. Press and Univ. of Tokyo Press, Princeton, NJ, and Tokyo: 1968. (Ann. of Math. Studies, 61) [124] J. Milnor, Introduction to algebraic K-theory, Princeton Univ. Press and Univ. of Tokyo Press, Princeton, NJ, and Tokyo, 1971, 184 p. (Ann. of Math. Studies, 72) [125] J. Milnor and J. Stasheff, Characteristic classes. Princeton Univ. Press and Univ. of Tokyo Press, Princeton, NJ, and Tokyo, 1974. [126] G. D. Mostow, Generalized Picard lattices arising from half-integer conditions, Publ. Math. IHES 63 (1986), 91-106. [127] N. E. Mnev, On manifolds of combinatorial types of projective configurations and convex polyhedra, Doklady Akad. Nauk SSSR, 283:6 (1985), 1312-1314. English transl: Soviet Math. Doklady 32 (1985), 335-337. [128] D. Mumford, Algebraic Geometry. I. Complex Projective Varieties. Springer; Berlin- Heidelberg-NewYork, 1976. [129] I. Newton, Philosophiae Naturalis Principia Mathematica, London, 1687. [130] W. Nuij, A note on hyperbolic polynomials, Math. Scand. 23 (1968), 69-72. [131] V. P. Palamodov, On the multiplicity of a holomorphic map, Funkts. Anal, i Prilozh., 1:3 (1967), 54-65; Engl, translation in Functional Anal. Appl., 1 (1967), 218-226. [132] V. P. Palamodov, Generalized functions and harmonic analysis, in: Itogi Nauki i Tekhn. VINITI. Sovremennye Problemy Matematiky, Fundamentalnye Napravleniya, vol. 72; VINITI, 1991, 5-134. English transl.: Commutative Harmonic Analysis, vol. 3, Encyclopae­ dia Math. Sci., vol. 72, Springer-Verlag, Berlin and New York, 1995, 1-127. 318 BIBLIOGRAPHY

[133] I. G. Petrovsky, On the diffusion of waves and the lacunas for hyperbolic equations, Matem. Sbornik, 17(59) (1945), 289-370. See also [134]. [134] I. G. PetrovskiT, Selected works. Systems of partial differential equations. Algebraic geom­ etry, Moscow, "Nauka", 1986, 504 pp.; English transl., Gordon and Breach, Amsterdam, 1996, 557 pp. [135] F. Pham, Formules de Picard-Lefschetz generalisees et ramification des integrales, Bull. Soc. Math. France, 93 (1965), 333-367. [136] F. Pham, Introduction a Vetude topologique des singularites de Landau, Gauthier-Villars, Paris, 1967. [137] F. Pham, Singularites des processus de diffusion multiple, Ann. de l'lnstitut Henri Poincare Sect. A., Physique Theorique, 6 (1967), 89-204. [138] A. Pickl, Die Homologie der Enhdngung eines vollstdndigen Durchschnitts mit isolierter Singularitat, Preprint, Math. Gottingensis Schriftenr., Sonderforschungbereich Geom. Anal. 15, 1985. [139] R. Remmert, Projectionen analytischer Mengen, Math. Ann., 130 (1956), 410-441. [140] R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Raume, Math. Ann., 133 (1957), 338-370. [141] V. A. Rokhlin, Complex topological characteristics of real algebraic curves, Uspekhi Mat. Nauk 33:5 (1978), 77-89, Engl, transl. in Russian Math. Surveys 33:5 (1978), 85-98. [142] G. L. Rybnikov, On the fundamental group of the complex hyperplane arrangement, math.AG/9805056. [143] M. Salvetti, Topology of the complement of real hyperplanes in CN, Inventiones Math. 88 (1987), 603-618. [144] V. Schechtman and A. Varchenko, Quantum groups and homology of local systems, Algebraic Geometry and Analytic Geometry (Tokyo 1990), ICM-90 Satellite Conf. Proa, Springer, Tokyo, 1991, 182-197. [145] V. Schechtman and A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106:1 (1991), 139-194. [146] J.-P. Serre, Homologie singuliere des espanes fibres. Ann. Math., 54 (1951), 425-505. [147] S. Smale, Generalized Poincare conjecture in dimensions greater than four, Ann. of Math. 74 (1961), 391-406. [148] S. Smale, On the structure of manifolds, Amer. J. of Math. 84 (1962), 387-399. [149] Singularities. (P. Orlik, ed.) Proceedings of Symposia in Pure Mathematics, Vol. 40, Parts I, II. Amer. Math. Soc, Providence RI, 1983. [150] L. Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat., 8 (1969), 145-162. [151] B. Teissier, The hunting of invariants in the geometry of discriminants, in: Real and Com­ plex Singularities. Nordic Summer School, Oslo, 1976. Sijthoff and Noordhoff, Groningen, 1977, 565-677. [152] B. Teissier, Varietes polaires. I. Invariants polaires des singularites d'hyper surf aces, Invent. Math. 40:3 (1977), 267-292. [153] R. Thorn, Ensembles et morphismes stratifies, Bull. Amer. Math. Soc. 75:2 (1969), 240-284. [154] D. Trotman, Counterexamples in stratification theory: two discordant horns, in: Proc. Nordic Summer School, Oslo 1976. Sijthoff and Noordhoff, Groningen 1977, pp. 679-686. [155] D. Trotman, Stability of transversality to a stratification implies Whitney (a)-regularity, Inv. Math. 50 (1979), 273-277. [156] D. Trotman, Comparing regularity conditions of stratifications, in [149], Part II, pp. 575- 586. [157] G. N. Tyurina, Locally semiuniver sal flat deformations of isolated singularities of complex spaces, Izv. Akad. Nauk SSSR, Ser. Mat. 33 (1969), 1026-1058; Engl, transl. in Math. USSR Izv. 3 (1970), 967-999. [158] A. D. Vainshtein and B. Z. Shapiro, Multidimensional analogues of the Newton and Ivory theorems, Funkts. Anal, i Prilozh., 19:1 (1985), 20-24; Engl, translation in Functional Anal. Appl., 19:1 (1985), 17-20. [159] A. D. Vainshtein and B. Z. Shapiro, Singularities of the boundary of a domain of hyper­ bolicity, Itogi Nauki i Tekhn., Ser. Sovrem. Probl. Mat., Nov. Dostizh. 33, 55-78; Engl. translation: J. Sov. Math., 52 (1990), 3326-3337. BIBLIOGRAPHY 319

[160] A. N. Varchenko, Theorems on the topological equisingularity of families of algebraic vari­ eties and families of polynomial maps, Izv. Akad. Nauk SSSR, Ser. Mat., 36 (1972), 957- 1019; Engl, transl. in Mat. USSR Izv. 6 (1972), 949-1008. [161] A. N. Varchenko, Normal forms of nonsmoothness of solutions of hyperbolic equations, Izv. Akad. Nauk SSSR, Ser. Mat., 51:3 (1987), 652-665; Engl, transl. in Mat. USSR Izv. 30:3 (1988), 615-628. [162] A. N. Varchenko, Combinatorics and topology of dispositions of affine hyperplanes in a real space, Funkts. Anal, i ego Prilozh., 21:1 (1987), 11-22. English transl.: Funct. Anal, and its Appl., 21:1 (1987), 11-22. [163] A. N. Varchenko, Multidimensional hypergeometric functions in conformal field theory, alge­ braic K-theory, algebraic geometry, Proc. Intern. Congress of Mathematicians, Kyoto 1990. Tokyo: Math. Soc. of Japan, 1991. V.l, pp. 281-300. [164] V. A. Vassiliev, Asymptotics of exponential integrals in the complex domain, Funkts. Anal. i ego Pril. 13:4 (1979), 1-12. English transl. in Funct. Anal and its Appl., 13:4 (1979). [165] V. A. Vassiliev, Sharpness and the local Petrovskii condition for strictly hyperbolic equations with constant coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 242-283; Engl, transl. in Math. USSR Izv. 28 (1987), 233-273. [166] V. A. Vassiliev, Lagrange and Legendre characteristic classes, Second edition, Adv. Studies in Contemp. Math., vol.3, Gordon and Breach, New York, 1993. [167] V. A. Vassiliev, Lacunas of hyperbolic partial differential operators and singularity theory, in: Theory of Operators in Functional Spaces, Saratov Univ. Press, Kuybyshev Branch, Kuybyshev, 1989, pp. 30-43; Engl, transl: Singularity Theory and some Problems of Func­ tional Analysis. S. G. Gindikin, ed. Amer. Math. Soc. Translations, Ser.2, vol. 153, 1992, pp. 25-37. [168] V. A. Vassiliev, Geometry of local lacunas of hyperbolic operators with constant coefficients, Mat. Sbornik, 183:1 (1992), 114-129. English transl.: Russian Acad. Sci. Sbornik Math. 75 (1993), 111-123. [169] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, 2d revised edition. Transl. of Math. Monographs, vol. 98, Amer. Math. Soc, 1994, 274 pp. [170] V. A. Vassiliev, Complexes of connected graphs, The I. M. Gel'fand's mathematical seminars 1990-1992, L. Corvin, I. Gel'fand, and J. Lepovsky, Eds.; Birkhauser, Basel, 1993, 223-235. [171] V. A. Vassiliev, Ramified Integrals, Singularities and Lacunas, Dorderecht: Kluwer Aca­ demic Publishers, 1994, 304 pp. [172] V. A. Vassiliev, Stratified Picard-Lefschetz theory, Selecta Math., New Series. 1:3 (1995), 597-621. [173] V. A. Vassiliev, Topology of complements of discriminants, Phasis, Moscow, 1997, 552 pp. (in Russian) [174] V. A. Vassiliev, Stratified Picard-Lefschetz theory with twisted coefficients, Amer. Math. Soc. Transl. (2) 180 (1997), 241-255. [175] V. A. Vassiliev, Monodroray of complete intersections and surface potentials, in: Singular­ ities. The Brieskorn Anniversary Volume. Progress in Mathematics, Vol. 162. Birkhauser Verlag, Basel-Boston-Berlin, 1998, 205-237. [176] V. A. Vassiliev, Introduction to Topology, AMS, Providence RI, 2001. [177] V. A. Vassiliev, Topology of plane arrangements and their complements, Russian Math. Surveys, 56:2 (2001), 167-203. [178] V. A. Vassiliev, I. M. Gelfand, and A. V. Zelevinskii, Behavior of general hypergeometric finctions in the complex domain, Doklady Akad. Nauk SSSR, 290:2 (1986), 277-281; English transl.: Soviet Math. Doklady, 34 (1987), 268-272. [179] V. A. Vassiliev, I. M. Gelfand, and A. V. Zelevinskii, General hypergeometric functions on complex Grassmannians, Funkts. Anal, i ego Pril., 21:1 (1987), 23-38; English transl: Funct. Anal, and its Appl., 21:1 (1987), 19-31. [180] V. A. Vassiliev and V. V. Serganova, On the number of real and complex moduli of singu­ larities of smooth functions and realizations of matroids, Mat. Zametki 49:1 (1991), 19-27; Engl, transl. in Math. Notes, 49:1 (1991), 15-20. [181] B. Wainryb, On the monodromy group of plane curve singularities, Math. Ann., 246 (1980), 141-154. [182] C. T. C. Wall, Regular stratifications, Lecture Notes in Math., vol. 468, Springer-Verlag, Berlin, 1975, 332-344. 320 BIBLIOGRAPHY

[183] C. T. C. Wall, Classification of unimodular isolated singularities of complete intersections, in: [149], Part 2, 625-640. [184] E. T. Whittaker, G. N. Watson, A course of modern analysis, Cambridge Univ. Press., 1927. [185] F. L. Zak, Structure of Gaufi maps, Funkts. Anal, i Prilozh., 21:1 (1987), 39-50; Engl. translation: Functional Anal, and its Appl., 21:1 (1987), 39-50. [186] V. M. Zakalyukin, On Lagrange and Legendre singularities, Funkts. Anal, i Prilozh., 10:1 (1976), 26-36; Engl, translation: Functional Anal, and its Appl., 10 (1976), 23-31. [187] O. Zariski, On the Poincare group of a projective hypersurface, Ann. Math. 38 (1937), 131-141. [188] G. M. Ziegler, Combinatorial models for subspace arrangements: Habilitationsschrift. Berlin: TU, 1992. [189] G. M. Ziegler and R. T. Zivaljevic, Homotopy type of arrangements via diagrams of spaces, Math. Ann., 295 (1993), 527-548. Index

-«, 76 ar/(B(F)), 291 I, 95, 101 D-equivalence, 56 ir, 95 D0, 56 oo-operator, 102 Do-equivalence, 56 -S 101 stable, 57 ^, 101 EP, 143 (-), 32 i?(p,g)? 5i d-B(r+l)^ 249 241 3, 100 H, 241 l miJU(M,L), 216 A(F), 63 Hyp(iM), 141 £(F), 46 7C*(M), 90 SR(F), 65 IH*(M), 90 /3(*,2/), 157 /i, 101 Xz,7> 241 (/*)', 101 241 Xz,75 .J(x), 269 /.(/), 38, 71 ^=const stratum, 60 r-degenerate manifolds, 127 309 -7[j] LJF, 255 Pc, 75 P (a), 82 ^x,29 c J(x), 268 PR, H3 /C(X), 117 5-non-degenerate polynomial, 266 /C-equivalence, 72 S_(x), 260 £(cr), 83 S(x), 260 M(x), 275 Vk, HI M2, 56 W(P), 145 W(P), 146 £, 35 WF, 261 Ker J(x), 269 VKF, 261 Re A(P), 140 XMJ 62 Var, 30 /(P'9), 50

Var(0), 241 /(«), 44 red, 84 /i(a;), 157 red', 84 i7, 241 reg(V), 80 i7, 241 tang(V), 80 j7, 241 tang(a), 79 j7, 241 p.A, 215 A(P), 140 *(/), 39 A*, 116 uj, 299 B -< A, 76 ^(C1,^), 63 adapted coordinates, 98, 150 C°°-lacuna, 145 adjoint strata, 76 C°°-sharpness, 145 admissible chain, 89 C~\ 240 algebraic body, 111

321 322 INDEX algebraic integrability, 111 elliptic polynomial, 258 Arnold class, 261 elliptic singularities, 60 localized, 270 equivalent deformations, 47 projective, 261 reduced, 266, 271 first type cycle, 279 Arnold orientation, 257 first type point, 279 formal lacuna, 166 base of deformation, 45 even, odd, 166 basic stratum, 299 fundamental solution, 142 bifurcation diagram of functions, 63 principal, 143 big monodromy group, 268 GauB-Manin connection, 29 bistable equivalence, 172 generating family, 122 body, 111 projective, 148 algebraic, 111 generating function, 122 non-singular, 111 projective, 146 cap, 120 generic cuspidal edge, 116 Cauchy problem, 142 generic hyperplane, 82 caustic, 63 gradient ideal, 47 chain subanalytic, 89 graph of separatrices, 51 admissible, 89 harm, 96 characterizing function, 193 Herglotz-PetrovskiT-Leray integrals, 151 charge, standard, 255 holomorphic lacuna, 145 Clemens structure, 232 holomorphic sharpness, 145 codimension of a complete intersection, 71 homological bundle, 29 cohomological bundle, 29 homological covering, 291 cohomology group of a local system, 216 homology group of a local system, 216 complete intersection, 71 homology group with coefficients in the local completely infinite reflection group, 284 system, 216 completely real singularity, 51 hyperbolic operator, 142 complex link, 82 hyperbolic polynomial, 140, 256 conditional integral, 291 hyperbolic potential, 258 corank of a singularities, 57 hyperbolic singularities, 60 cuspidal edge, generic, 126 hyperbolicity domain, 141, 257 hyperplane arrangement, 298 Davydova-Borovikov condition, 149, 150 basic, 299 deformation, 45 essential, 298 equivalent, 47 hypersurface, 35 induced, 47 projectively dual, 116 miniversal, 48 of a complete intersection, 72 imaginary cone, 299 versal, 47 induced deformations, 47 degenerate point, 115 infertile segment of the queue, 213 diffusion of waves, 144, 145 integration cycle, 291 direct image of a local system, 215 tame, 291 discriminant, 46 intersection matrix, 44 of a complete intersection, 73 isolated singularity, 37 real, 65 of a complete intersection, 71 distinguished basis, 42 distinguished system of paths, 40 Le number, 83 perfect, 278 lacuna, 144 dual local system, 218 C°°-, 145 dual projective space, 116 holomorphic, 145 dual set, 79 local, 137, 145 Dynkin diagram, 45 strong, 145 Laplace operator, standard solution of, 255 element representable by a cap, 120 Lefschetz thimble, 50 element representable by a vanishing cycle, Leray class, 153 120 Leray coboundary operator, 35 INDEX 323

Leray cycle, 153 projective generating family, 148 local algebra, 48 projective generating function, 146 local lacuna, 137, 145 proper cone, 142 projectivized, 149 proper modality, 61 local Petrovskii class, 157 purse, 206 local Petrovskii condition, 156 pyramid, 195 local system, 215 dual, 218 quasihomogeneous function, 59 localized homology group, 157 locally finite chain, 216 real discriminant, 65 real function, 51 Maxwell set, 63, 196 regular adjacency of strata, 76, 77 Milnor fibration, 33 resolution of singularities, 225 Milnor number, 38 resonance, 292 of a complete intersection, 71 Riemannian covering, 217 miniversal deformation, 48 mirage, 306 Sabirization, 51 modality, 58 second type cycle, 279 monodromy group, 29 second type point, 279 monodromy representation, 29 semialgebraic subset, 78 Morse function, 38 semianalytic subset, 78 Morse lemma, 38 separatrices, 51 Morse singularity, 38 lower and upper, 51 Modification, 39 separatrix diagram, 51 strict, 39 sharpness, C°°-, 145 virtual, 209 sharpness, holomorphic, 145 simple loop, 40 natural orientation, 255 simple singularities, 57 negative index, 173 simple tangency, 80, 266 Newton-Coulomb potential, 255 simplicial resolution, 302 non-parabolic point of a stratum, 80 singularity, 37 non-singular body, 111 completely real, 51 obstructing germ of a curve, 127 elliptic, 60 obstructing stratum, 127 hyperbolic, 60 open stratum, 298 isolated, 37 order complex, 301 of a complete intersection, 71 orientation sheaf, 215 Morse, 38 stratified, 92 parabolic point, 115 parabolic, 60 parabolic singularities, 60 simple, 57 parts of a partition, 79 unimodal, 60 perversity, 91 small monodromy group, 268 Petrovskii class, even, 165 space of momenta, 137, 144 Petrovskii class, local, 157 stabilization, 44 Petrovskii class, odd, 165 stable front, 148 Petrovskii condition, 154 stable singularity As, 148 local, 156 standard charge, 255 Petrovskii cycle, 153 with density, 256 Petrovskii local cycle, even, 165 stratified Morse function, 92 Petrovskii minus-condition, 193 stratified Morse singularity, 92 Petrovskii plus-condition, 193 stratified submersion, 78 Picard-Lefschetz formula, 33 stratum, 76 point of finite type, 156 strictly hyperbolic polynomial, 140, 256 potential, 252 strictly Morse function, 38 potential function, 255 strong lacuna, 145 primary stratification, 76 subanalytic chain, 89 principal fundamental solution, 143 subanalytic subset, 78 projective duality, 115 swallowtail, 46 324

Tarski-Seidenberg lemma, 78 Thorn isomorphism, 217 Thom isotopy lemma, 78 transversality, 77 to a stratified set, 77, 80 trivial extension, 224 trivial point, 117 tube operator, 35 two-index polynomial, 200 Tyurina number, 73 unimodal singularities, 60 vanishing cell, 35 vanishing cycle, 42 vanishing sphere, 33, 41 versal deformation, 47 vertex of an arrangement, 299 Vieta map, 62 virtual Morsification, 209 wave front, 137, 145 stable, 148 Whitney conditions, 76, 77 Whitney stratification, 77 Whitney umbrella, 88 zone, 252