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Monodromy of Complete Intersections and Surface Potentials

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Singularities

Part of the book series: Progress in Mathematics ((PM,volume 162))

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Abstract

Following Newton, Ivory and Arnold, we study the Newtonian potentials of algebraic hypersurfaces in R n. The ramification of (analytic continuations of) these potential depends on a monodromy group, which can be considered as a proper subgroup of the local monodromy group of a complete intersection (acting on a twisted vanishing homology group if n is odd). Studying this monodromy group we prove, in particular, that the attraction force of a hyperbolic layer of degree d in R n coincides with appropriate algebraic vector-functions everywhere outside the attracting surface if n = 2 or d = 2, and is non-algebraic in all domains other than the hyperbolicity domain if the surface is generic and (d ≥ 3)&(n ≥ 3)&(n + d ≥ 8).

Recently W. Ebeling removed the last restriction d + n ≥ 8, see his Appendix to this article.

Research supported by the Russian Fund of Basic Investigations (project 95-01-00846a) and INTAS grant (Project # 4373)

To Egbert Brieskorn with admiration

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References

  1. V.I. Arnold: On the Newtonian potential of hyperbolic layers. Trudy Tbilisskogo Universiteta, Ser. Math./Mekh./Astron., V. 232–233, # 13–14, 23–29, (1982). English transi, in: Selecta Math. Soviet., 4:2, 103–106, (1985).

    Google Scholar 

  2. V.I. Arnold: Magnetic field analogues of the theorems of Newton and Ivory. Uspekhi Mat. Nauk 38:5, 145–146, (1983).

    Google Scholar 

  3. V.I. Arnold: Kepler’s second law and the topology of Abelian integrals. Kvant, No. 12,17–21 (Russian); A topological proof of transcendence of Abelian integrals in Newton’s Principia, Quant, No. 12, 1–15, (1987); The 300-th anniversary of mathematical natural philosophy and celestial mechanics, Priroda, No. 8, 5–15, (1987) (in Russian).

    Google Scholar 

  4. M.F. Atiyah; R. Bott; L. Gârding: Lacunas for hyperbolic differential operators with constant coefficients. Acta Math., 124, 109–189, (1970) and 131, 145–206, (1973).

    Article  MathSciNet  MATH  Google Scholar 

  5. V.I. Arnold; V.A. Vassiliev: Newton’s Principia read 300 years later. Notices Amer. Math. Soc., 36:9, 1148–1154, (1989).

    MathSciNet  Google Scholar 

  6. V.I. Arnold; A.N. Varchenko; S.M. Gusein-Zade: Singularities of differentiate maps. V 1,2; “Nauka”, Moscow, (1982, 1984); Engl, transl: Birkhäuser, Basel, (1985, 1988).

    Google Scholar 

  7. V.I. Arnold; V.A. Vassiliev; V.V. Goryunov; O. V. Lyashko: Singularities, 1 and 2. Dynamical systems, 6;39, VINITI, Moscow, (1988; 1989); English transi.: Encyclopaedia Math. Sci., V. 6; 39, Springer-Verlag, Berlin, New York, (1993).

    Google Scholar 

  8. W. Ebeling: The monodromy groups of isolated singularities of com-plete intersections. Lect. Notes Math., vol.1293, Springer, Berlin a.o., (1987).

    Google Scholar 

  9. A.B. Givental: Polynomiality of the electrostatic potentials. Uspekhi Mat. Nauk, 39:5, 253–254 (in Russian), (1984).

    Google Scholar 

  10. A.B. Givental: Twisted Picard-Lefschetz formulae. Funkts. Anal, i Prilozh., 22:1, 12–22, (1988); Engl, translation in Functional Anal. Appl., 22:1, 10–18 (1988).

    MathSciNet  Google Scholar 

  11. A.M. Gabrielov, Bifurcations, Dynkin diagrams and modality of isolated singularities, Funkts. Anal, i Prilozh., 8:2, 7–12, (1974); Engl, transi, in Funct. Anal. Appl., 8, 94–98, (1974).

    MathSciNet  Google Scholar 

  12. G.-M. Greuel; H.A. Hamm: Invarianten quasihomogener vollständiger Durchschnitte. Invent. Math., 49:1, 67–86, (1978).

    Article  MathSciNet  MATH  Google Scholar 

  13. Ph. Griffiths; J. Harris: Principles of algebraic geometry. John Wiley & Sons, New York a.o., (1978).

    MATH  Google Scholar 

  14. M. Goresky; R. MacPherson: Stratified Morse theory. Springer-Verlag, Berlin and New York, (1986).

    Google Scholar 

  15. H. Grauert; R. Remmert: Komplexe Räume. Math. Ann., 136:2, 245–318, (1958).

    Article  MathSciNet  MATH  Google Scholar 

  16. S.M. Gusein-Zade: Monodromy groups of isolated singularities of hy-persurfaces. Uspekhi Mat. Nauk 32, no.2, 23–65, (1977); Engl, transl, in Russian Math. Surveys 32, No.2, 23–69, (1977).

    MathSciNet  Google Scholar 

  17. H. Hamm: Locale topologische Eigenschaften komplexer Räume. Math. Ann., 191, 235–252, (1971).

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Ivory: On the attraction of homogeneous ellipsoids. Philos. Trans., 99, 345–372, (1809).

    Google Scholar 

  19. J. Milnor: Singular points of complex hypersurf aces. Princeton Univ. Press, Princeton, NJ, and Univ. of Tokyo Press, Tokyo, (1968).

    Google Scholar 

  20. J. Milnor; P. Orlik: Isolated singularities, defined by weighted isolated polynomials. Topology, 9:2, 385–393, (1970).

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Nuij: A note on hyperbolic polynomials. Math. Scand. 23, 69–72, (1968).

    MathSciNet  MATH  Google Scholar 

  22. I. Newton: Philosophiae Naturalis Principia Mathematiea. London, (1687).

    Google Scholar 

  23. I.G. Petrovsky: On the diffusion of waves and the lacunas for hyperbolic equations. Matem. Sbornik, 17(59), 289–370, (1945).

    Google Scholar 

  24. F. Pham: Formules de Picard-Lefschetz génèralisées et ramification des intégrales. Bull. Soc. Math. France, 93, 333–367, (1965).

    MathSciNet  MATH  Google Scholar 

  25. F. Pham: Introduction à l’étude topologique des singularités de Landau. Gauthier-Villars, Paris, (1967).

    MATH  Google Scholar 

  26. A.D. Vainshtein; B.Z. Shapiro: Multidimensional analogues of the Newton and Ivory theorems. Funkts. Anal, i Prilozh., 19:1, 20–24, (1985); Engl, translation in Functional Anal. Appl., 19:1, 17–20, (1985).

    MathSciNet  Google Scholar 

  27. V.A. Vassiliev: Sharpness and the local Petrovskii condition for hyperbolic equations with constant coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 50, 242–283, (1986); Engl, transl. in Mat. USSR Izv. 28, 233–273, (1987).

    MathSciNet  Google Scholar 

  28. V.A. Vassiliev: Ramified Integrals, Singularities and Lacunas. Kluwer, (1994).

    Google Scholar 

  29. O. Zariski: On the Poincaré group of a projective hypersurface. Ann. Math. 38, 131–141, (1937).

    Article  MathSciNet  Google Scholar 

References

  1. P. Deligne: La conjecture de Weil. II, Publ. Math. IHES 52, 137–252, (1980).

    MathSciNet  MATH  Google Scholar 

  2. W. Ebeling; S.M. Gusein-Zade: Suspensions of fat points and their intersection forms. These proceedings.

    Google Scholar 

  3. W. Ebeling; Ch. Okonek: Donaldson invariants, monodromy groups, and singularities. Intern. J. Math. 1, 233–250, (1990).

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Friedman; J.W. Morgan: Smooth Four-Manifolds and Complex Surfaces. Springer, Berlin etc., (1994).

    MATH  Google Scholar 

  5. V.A. Vassiliev: Monodromy of complete intersections and surface potentials. These proceedings.

    Google Scholar 

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Authors and Affiliations

  1. Steklov Math. Institute, Vavilova Str. 42, Moscow, Russia

    Victor A. Vassiliev

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  1. Victor A. Vassiliev
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Editor information

Editors and Affiliations

  1. Department of Geometry and Topology, Steklov Mathematical Institute, 8, Gubkina Stree, 117966, Moscow GSP-1, Russia

    V. I. Arnold

  2. CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, Postfach 3049, F-75775, Paris Cedex 16e, France

    V. I. Arnold

  3. Fachbereich Mathematik, Universität Kaiserslautern, D-67653, Kaiserslautern, Germany

    G.-M. Greuel

  4. Subfaculteit Wiskunde, Katholieke Universiteit Nijmegen Toernooiveld, NL-6525 ED, Nijmegen, The Netherlands

    J. H. M. Steenbrink

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Vassiliev, V.A. (1998). Monodromy of Complete Intersections and Surface Potentials. In: Arnold, V.I., Greuel, GM., Steenbrink, J.H.M. (eds) Singularities. Progress in Mathematics, vol 162. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8770-0_11

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  • DOI: https://doi.org/10.1007/978-3-0348-8770-0_11

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9767-9

  • Online ISBN: 978-3-0348-8770-0

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