μ-Constant Monodromy Groups and Torelli Results for Marked Singularities, for the Unimodal and Some Bimodal Singularities

  • Falko Gauss
  • Claus HertlingEmail author


This paper is a sequel to Hertling (Ann Inst Fourier (Grenoble) 61(7):2643–2680, 2011). There a notion of marking of isolated hypersurface singularities was defined, and a moduli space M μ mar for marked singularities in one μ-homotopy class of isolated hypersurface singularities was established. One can consider it as a global μ-constant stratum or as a Teichmüller space for singularities. It comes together with a μ-constant monodromy group \(G^{mar} \subset G_{\mathbb{Z}}\). Here \(G_{\mathbb{Z}}\) is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that M μ mar is connected. This is equivalent to \(G^{mar} = G_{\mathbb{Z}}\). Also Torelli-type conjectures were formulated. All conjectures were proved for the simple singularities and 22 of the exceptional unimodal and bimodal singularities. In this paper, the conjectures are proved for the remaining unimodal singularities and the remaining exceptional bimodal singularities.


μ-Constant monodromy group Hyperbolic singularities Marked singularity Moduli space Simple elliptic singularities Torelli-type problem 



This work was supported by the DFG grant He2287/4-1 (SISYPH).


  1. 1.
    Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Boston, Basel, Stuttgart (1985)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. II. Birkhäuser, Boston (1988)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ebeling, W.: Milnor lattices and geometric bases of some special singularities. In: Noeuds, Tresses et Singularités. Monographie de l’Enseignement Mathématique, vol. 31, pp. 129–146. Genève, Enseignement Mathématique (1983)Google Scholar
  4. 4.
    Ebeling, W.: Functions of Several Complex Variables and Their Singularities. Graduate Studies in Mathematics, vol. 83. American Mathematical Society, Providence, RI (2007)Google Scholar
  5. 5.
    Gabrielov, A.M.: Dynkin diagrams of unimodal singularities. Funct. Anal. Appl. 8, 192–196 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hertling, C.: Analytische Invarianten bei den unimodularen und bimodularen Hyperflächensingularitäten. Doctoral thesis. Bonner Mathematische Schriften 250, Bonn (1992)Google Scholar
  7. 7.
    Hertling, C.: Ein Torellisatz für die unimodalen und bimodularen Hyperflächensingularitäten. Math. Ann. 302, 359–394 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hertling, C.: Brieskorn lattices and Torelli type theorems for cubics in \(\mathbb{P}^{3}\) and for Brieskorn-Pham singularities with coprime exponents. In: Singularities, the Brieskorn Anniversary Volume. Progress in Mathematics, vol. 162, pp. 167–194. Birkhäuser Verlag, Basel-Boston-Berlin (1998)Google Scholar
  9. 9.
    Hertling, C.: Classifying spaces and moduli spaces for polarized mixed Hodge structures and for Brieskorn lattices. Compos. Math. 116, 1–37 (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hertling, C.: Generic Torelli for semiquasihomogeneous singularities. In: Libgober, A., Tibar, M. (eds.) Trends in Singularities, pp. 115–140. Birkhäuser Verlag, Basel (2002)CrossRefGoogle Scholar
  11. 11.
    Hertling, C.: Frobenius Manifolds and Moduli Spaces for Singularities. Cambridge Tracts in Mathematics, vol. 151. Cambridge University Press, Cambridge (2002)Google Scholar
  12. 12.
    Hertling, C.: μ-Constant monodromy groups and marked singularities. Ann. Inst. Fourier (Grenoble) 61 (7), 2643–2680 (2011)Google Scholar
  13. 13.
    Kaenders, R.: The Seifert form of a plane curve singularity determines its intersection multiplicities. Ind. Math. N.S. 7 (2), 185–197 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kluitmann, P.: Ausgezeichnete Basen erweiterter affiner Wurzelgitter. Doctoral thesis. Bonner Mathematische Schriften 185, Bonn (1987)Google Scholar
  15. 15.
    Lê, D.T., Ramanujam, C.P.: The invariance of Milnor’s number implies the invariance of the topological type. Am. J. Math. 98, 67–78 (1973)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Michel, F., Weber, C.: Sur le rôle de la monodromie entière dans la topologie des singularités. Ann. Inst. Fourier (Grenoble) 36, 183–218 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Milanov, T., Shen, Y.: The modular group for the total ancestor potential of Fermat simple elliptic singularities. Commun. Number Theor. Phys. 8, 329–368 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton (1968)Google Scholar
  19. 19.
    Orlik, P.: On the homology of weighted homogeneous polynomials. In: Lecture Notes in Mathematics, vol. 298. Springer, Berlin (1972)Google Scholar
  20. 20.
    Saito, K.: Einfach-elliptische Singularitäten. Invent. Math. 23, 289–325 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Saito, M.: Period mapping via Brieskorn modules. Bull. Soc. Math. France 119, 141–171 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Lehrstuhl für Mathematik VIUniversität MannheimMannheimGermany

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