Duality for Regular Systems of Weights: A Précis

  • Kyoji Saito
Chapter
Part of the Progress in Mathematics book series (PM, volume 160)

Abstract

The theory of a regular system of weights was originally developed in order to understand the flat structure in the period maps for primitive forms [S9]. In the present work, the theory is applied to understand the self-duality of ADE and the strange duality of Arnold. Beyond the original purpose, the theory yields further a class of dual weight systems, which seem to have close connection with the Conway group and to be interesting to be studied yet. On the other hand, the duality of weight systems has an interpretation in terms of the Dedekind eta function. But its meaning is not yet clear.

Keywords

Stein Lution Reso Mirror Symmetry 

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References

  1. [A1]
    Arnold, Vladimir I., Critical Points of Smooth Functions, Proc. Internat. Congress Math., Vancouver, I, (1974), p.19.Google Scholar
  2. [A2]
    Arnold, Vladimir I., Critical Points of Smooth Functions and their Normal Forms, Usp. Mat. Nauk. 30;5 (1975), 3–65. (Eng. translation in Russ. Math. Surv. 30; 5 (1975), 1-75.)Google Scholar
  3. [A3]
    Arnold, Vladimir I., Critical Points of Smooth Functions and their Normal Forms, Usp. Mat. Nauk. 30;5 (1975), 3–65. (Eng. translation in Russ. Math. Surv. 30; 5 (1975), 1-75.)Google Scholar
  4. [Bo]
    Bourbaki, Nicolas, Groupes et algèbres de Lie, chap. 4, 5 et 6, Hermann, Paris, 1968.Google Scholar
  5. [B1]
    Brieskorn, Egbert, Die Habilitationsschrift, 1965, Bonn.Google Scholar
  6. [B2]
    Brieskorn, Egbert, Beispiele zur Differential topology von Singularitäten, Invent. Math. 2 (1966), 1–14.MathSciNetMATHCrossRefGoogle Scholar
  7. [B3]
    Brieskorn, Egbert, Die Milnorgitter der exzeptionellen und unimodularen Singularitäten, Bonner Math. Schriften, No. 150, Bonn 1983.Google Scholar
  8. [C-N]
    Conway, John H. and Norton, Simon P., Monstrous Moonshine, Bull. London Math. 11 (1979), 308–339.MathSciNetMATHCrossRefGoogle Scholar
  9. [D1]
    Dolgachev, Igor V., Quotient-Conical singularities on complex surfaces, Funk. Anal. i Prilozen, Translated in Funct. Anal. Appl. 8 (1974), 160–161.CrossRefGoogle Scholar
  10. [D2]
    Dolgachev, Igor V., Automorphic forms and quasihomogeneous singularities, Funk. Anal. i Prilozen 9;2 (1975), 67–68. Translated in Funct. Anal. Appl. 9 (1975), 149-151.Google Scholar
  11. [D3]
    Dolgachev, Igor V., On the link space of Gorenstein quasihomogeneous surface singularities, Math. Ann. 232 (1978), 85–98.MathSciNetCrossRefGoogle Scholar
  12. [E1]
    Ebeling, Wolfgang, Quadratische Formen und Monodrmie gruppen von Singularitäten, Math. Ann. 255 (1981), 463–498.MathSciNetMATHCrossRefGoogle Scholar
  13. [E2]
    Ebeling, Wolfgang, Milnor lattices and geometric bases of some special singularities, L’Enseignement Math. 29 (1983), 263–280.MathSciNetMATHGoogle Scholar
  14. [E3]
    Ebeling, Wolfgang, An arithmetic characterization of the symmetric monodromy groups of singularities, Invent. Math. 77 (1984), 85–99.MathSciNetMATHCrossRefGoogle Scholar
  15. [E-W]
    Ebeling, Wolfgang and Terry Wall, Kodaira Singularities and an Extension of Arnold’s Strange Duality, Compositio Math. 56;1 (1985), 3–77.MathSciNetMATHGoogle Scholar
  16. [G]
    Gabrielov, Andrei M., Dynkin diagrams of unimodal singularities, Funck. Anal. i Prilozh. 8;3 (1974), 1–6, (Engl. translation in Funct. Anal. Appl. 8 (1974), 192-196.)MathSciNetGoogle Scholar
  17. [H]
    Hall, Marshall Jr., Combinatorial Theory, Blaisdell Publishing Company, 1967.Google Scholar
  18. [H-L]
    Harada, Koichiro and Mong Lung Lang, Some Elliptic curves arising from the Leech lattice, J. Alg. 125 (1989), 298–310.MATHCrossRefGoogle Scholar
  19. [H-M]
    Honda, Taira and Miyawaki, Isao, Zeta functions of elliptic curves of 2-power conductor, J. Math. Soc. Japan 26 (1974), 362–373.MathSciNetMATHCrossRefGoogle Scholar
  20. [K]
    Kondo, Takeshi, The automorphism group of the Leech lattice and elliptic modular functions, J. Math. Soc. Japan 37 (1985), 337–362.MathSciNetMATHCrossRefGoogle Scholar
  21. [Ko]
    Koike, Masao, Moonshines of PSL2(Fq) and the automorphism group of Leech lattice, Japan. J. Math. 122 (1986), 283–323.MathSciNetMATHGoogle Scholar
  22. [Le]
    Lê Dũng Tráng, Calcul des cycles evanouissants des hypersurfaces complexes, Ann. Inst. Fourier 23 (1973), 261–270.MATHCrossRefGoogle Scholar
  23. [Lo]
    Looijenga, Eduard, Rational surfaces with an anti-canonical cycle, Ann. of Math. 114 (1981), 267–322.MathSciNetMATHCrossRefGoogle Scholar
  24. [Ma]
    Mason, Geoffrey, Frame shapes and rational characters of finite groups, J. Alg. 89 (1984), 237–246.MATHCrossRefGoogle Scholar
  25. [M]
    Milnor, John, Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton Univ. Press, 1968.Google Scholar
  26. [M-O]
    Milnor, John and Orlik, Peter, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385–393.MathSciNetMATHCrossRefGoogle Scholar
  27. [N]
    Nakamura, Iku, Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities I, Math. Ann. 252 (1980), 221–235.MathSciNetMATHCrossRefGoogle Scholar
  28. [Oc]
    Ochiai, Hiroyuki, A personal letter to I. Satake (1992 Feb.).Google Scholar
  29. [Od]
    Oda, Tadao, Introduction to Algebraic Singularities, (with an Appendix by Taichi Ambai) Preprint, Lecture at Tata. Inst.Google Scholar
  30. [O-W]
    Orlik, Peter and Wagreich, Philip, Isolated singularities with C*-actions, Ann. Math. 93 (1971), 205–228.MathSciNetMATHCrossRefGoogle Scholar
  31. [O]
    Oka, Mutsuo, Principal zeta-function of nondegenerate complete intersection singularity, J. Fac. Sci., Univ. Tokyo, Sect. IA Math. 31;1 (1990), 11–32.MathSciNetGoogle Scholar
  32. [P]
    Peterson, H., Über die arithmetischen Eigenschaften eines Systems multiplikativer Modulefunkionen von Primzahlstufe, Acta Math. 95 (1956), 57–110.MathSciNetCrossRefGoogle Scholar
  33. [P1]
    Pinkham, Henry, Singularités exceptionelles, la dualité étrange d’Arnol’d et les surfaces K 3, C. R. Acad. Sci. Paris 284(A) (1977), 615–618.MathSciNetMATHGoogle Scholar
  34. [P2]
    Pinkham, Henry, Normal surface singularities with C*-action, Math. Ann. 227 (1977), 183–193.MathSciNetCrossRefGoogle Scholar
  35. [P3]
    Pinkham, Henry, Deformation of Normal Surface Singularities with C*-action, Math. Ann. 232 (1978), 65–84.MathSciNetMATHCrossRefGoogle Scholar
  36. [R]
    Eds. Andrews, George E., Askey, Richard, A., Berndt, Bruce, C., Ramanathan, K. G. and Rankin, Robert, A., Ramanujan Revisited, Proceedings of the Centenary Conference, Univ. of Illinois at Urbana-Champaign, June 1–5, 1987, Academic Press.Google Scholar
  37. [S1]
    Saito, Kyoji, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142.MathSciNetMATHCrossRefGoogle Scholar
  38. [S2]
    Saito, Kyoji, Einfach-elliptische Singularitäten, Invent. Math. 23 (1974), 289–325.MathSciNetMATHCrossRefGoogle Scholar
  39. [S3]
    Saito, Kyoji, Period Mapping Associated to a Primitive form, Publ. RIMS, Kyoto Univ. 19 (1983), 1231–1264.MATHCrossRefGoogle Scholar
  40. [S4]
    Saito, Kyoji, Extended Affine Root Systems, I, Publ. RIMS, Kyoto Univ. 21 (1985), 75–179; II, ibid 26 (1990), 15-78; III, ibid 33 (1997), 301-329; Part IV in preparation; V, preprint.MATHCrossRefGoogle Scholar
  41. [S5]
    Saito, Kyoji, A New Relation among Cartan Matrix and Coxeter Matrix, J. Alg. 105 (1987), 149–158.MATHCrossRefGoogle Scholar
  42. [S6]
    Saito, Kyoji, Regular system of weights and their associated singularities, Complex Analytic Singularities, Adv. Stud. Pure Math. 8 (1987), 472–596.Google Scholar
  43. [S7]
    Saito, Kyoji, Algebraic surfaces associated to regular systems of weights, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata 2 (1988), 517–614.Google Scholar
  44. [S8]
    Saito, Kyoji, On the existence of exponents prime to the Coxeter number, J. Alg. 114 (1988), 333–356.MATHCrossRefGoogle Scholar
  45. [S9]
    Saito, Kyoji, On a Linear Structure of a Quotient Variety by a Finite Reflexion Group, Publ. RIMS, Kyoto Univ. 29 (1993), 535–579.MATHCrossRefGoogle Scholar
  46. [S10]
    Saito, Kyoji, On the Characteristic Polynomial for a Regular System of Weights, Preprint RIMS-875, (1992). On a Duality of Characteristic Polynomials for Regular Systems of Weights, Preprint RIMS-993, (1994).Google Scholar
  47. [S11]
    Saito, Kyoji, Around the theory of the generalized weight system: relations with singularity theory, the generalized Weyl group and its invariant theory, etc., Amer. Math. Soc. Transl. (2), 1997.Google Scholar
  48. [Sh]
    Shioda, Tetsuji, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. of Math. 108 (1986), 415–432.MathSciNetMATHCrossRefGoogle Scholar
  49. [T]
    Takahashi, Atsushi, Master Thesis at RIMS 1998.Google Scholar
  50. [Wa1]
    Wagreich, Philip, The structure of quasihomogeneous singularities, Proc. Symp. Pure Math. AMS 40 (1973), 593–624.MathSciNetGoogle Scholar
  51. [Wa2]
    Wagreich, Philip, Algebra of automorphic form with few generators, Trans. AMS 262 (1980), 367–389.MathSciNetMATHCrossRefGoogle Scholar
  52. [Wah]
    Wahl, Jonathan, A characterization of Quasi-homogeneous Gorenstein surface singularities, Compositio Math. 55 (1985) 269–288.MathSciNetMATHGoogle Scholar
  53. [Va]
    Varchenko, Alexander N., Zeta-function of Monodromy and Newton’s Diagram, Invent. Math. 37 (1976), 253–262.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Kyoji Saito
    • 1
  1. 1.RIMSKyoto UniversityJapan

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