Advertisement

On the Link Space of a ℚ-Gorenstein Quasi-Homogeneous Surface Singularity

  • Anna Pratoussevitch
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In this paper we prove the following theorem: Let M be the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein surface singularity. Then M is diffeomorphic to a coset space \( \tilde \Gamma _1 \backslash \tilde G/\tilde \Gamma _2 \), where \( \tilde G \) is the 3-dimensional Lie group \( \widetilde{PSL}(2,\mathbb{R}) \), while \( \tilde \Gamma _1 \) and \( \tilde \Gamma _2 \) are discrete subgroups of \( \tilde G \), the subgroup \( \tilde \Gamma _1 \) is co-compact and \( \tilde \Gamma _2 \) is cyclic. Conversely, if M is diffeomorphic to a coset space as above, then M is diffeomorphic to the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein singularity. We also prove the following characterisation of quasi-homogeneous ℚ-Gorenstein surface singularities: A quasi-homogeneous surface singularity is ℚ-Gorenstein of index r if and only if for the corresponding automorphy factor (U, Γ, L) some tensor power of the complex line bundle L is Γ-equivariantly isomorphic to the rth tensor power of the tangent bundle of the Riemannian surface U.

Keywords

ℚ-Gorenstein singularity quasi-homogeneous singularity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BPR03]
    Egbert Brieskorn, Anna Pratoussevitch, and Frank Rothenhäusler, The Combinatorial Geometry of Singularities and Arnold’s Series E, Z, Q, Moscow Mathematical Journal 3 (2003), no. 2, 273–333, the special issue dedicated to Vladimir I. Arnold on the occasion of his 65th birthday.MathSciNetGoogle Scholar
  2. [Dol75]
    Igor V. Dolgachev, Automorphic forms and quasihomogeneous singularities, Funct. Anal. Appl. 9 (1975), 149–151.CrossRefGoogle Scholar
  3. [Dol77]
    _____, Automorphic forms and weighted homogeneous equations, provisional version, unpublished typed manuscript, 1977.Google Scholar
  4. [Dol83]
    _____, On the Link Space of a Gorenstein Quasihomogeneous Surface Singularity, Math. Ann. 265 (1983), 529–540.CrossRefMathSciNetGoogle Scholar
  5. [Ish87]
    Shihoko Ishii, Isolated Q-Gorenstein singularities of dimension three, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 165–198.Google Scholar
  6. [Ish00]
    Shihoko Ishii, The quotients of log-canonical singularities by finite groups, Singularities — Sapporo 1998, Adv. Stud. Pure Math., vol. 29, Kinokuniya, Tokyo, 2000, pp. 135–161.Google Scholar
  7. [KR85]
    Ravi S. Kulkarni and Frank Raymond, 3-dimensional Lorentz space forms and Seifert fiber spaces, J. Diff. Geometry 21 (1985), 231–268.MathSciNetGoogle Scholar
  8. [LV80]
    Gérard Lion and Michèle Vergne, The Weil representation, Maslov index and theta series, Progress in Mathematics, vol. 6, Birkhäuser Boston, Mass., 1980.Google Scholar
  9. [Mil75]
    John Milnor, On the 3-dimensional Brieskorn manifolds M (p, q, r), Knots, groups and 3-manifolds (L. P. Neuwirth, ed.), Annals of Math. Studies, vol. 84, Princeton University Press, Princeton, 1975, pp. 175–225.Google Scholar
  10. [Neu77]
    Walter D. Neumann, Brieskorn complete intersections and automorphic forms, Invent. Math. 42 (1977), 285–293.CrossRefMathSciNetGoogle Scholar
  11. [Pin77]
    Henry Pinkham, Normal surface singularities with ℂ*-action, Math. Ann. 227 (1977), 183–193.CrossRefMathSciNetGoogle Scholar
  12. [Pra]
    Anna Pratoussevitch, Fundamental Domains in Lorentzian Geometry, to appear in Geom. Dedicata, math.DG/0308279.Google Scholar
  13. [Rei87]
    Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (S. Bloch, ed.), Proceedings of Symposia in Pure Mathematics, vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414.Google Scholar
  14. [Wat81]
    Kei-ichi Watanabe, Some remarks concerning Demazure’s construction of normal graded rings, Nagoya Math. J. 83 (1981), 203–211.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Anna Pratoussevitch
    • 1
  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

Personalised recommendations