Some Open Problems in Complex Singularities

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 222)


We discuss some open problems and questions related with five different topics in complex singularities. These are: (i) Topological and holomorphic ranks of an isolated singularity germ and the Zariski-Lipman conjecture; (ii) Graph manifolds and links of surface singularities. (iii) Milnor’s fibration for complex singularities and the topology of analytic foliations near an isolated singularity. (iv) Rochlin’s signature theorem and Gorenstein surface singularities. (v) The index of a vector field on a singular variety. These are all topics on which I have been interested for a long time.


Zariski-Lipman conjecture Graph manifolds Gorenstein singularities Milnor fibrations Foliations Rochlin signature theorem Indices of vector fields 


  1. 1.
    Arnold, V.I.: Remarks on singularities of finite codimension in complex dynamical systems. Funct. Anal. Appl. 3, 1–5 (1969)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atiyah, M.F.: Riemann Surfaces and Spin Structures. Ann. Scient. Ec. Norm. Sup. 4, 47–62 (1971)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Biswas, I., Gurjar, R.V., Kolte, S.U.: On the Zariski-Lipman conjecture for normal algebraic surfaces. J. Lond. Math. Soc., II. Ser. 90(1), 270–286 (2014)Google Scholar
  4. 4.
    Bonatti, Ch., Gómez-Mont, X.: The index of a holomorphic vector field on a singular variety. Astérisque 222, 9–35 (1994)MATHGoogle Scholar
  5. 5.
    Brasselet, J.P., Seade, J., Suwa, T.: Vector Fields on Singular Varieties. Lecture notes in mathematics # 1987. Springer, Berlin (2009)Google Scholar
  6. 6.
    Camacho, C., Kuiper, N., Palis, J.: The topology of holomorphic flows with singularity. Inst. Hautes Et. Sci. Publ. Math. 48, 5–38 (1978)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chung, F., Xu, Y.-J., Yau, S.T.: Classification of weighted dual graphs with only complete intersection singularities structures. Trans. AMS 261, 3535–3596 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cisneros-Molina, J.L.: Join theorem for polar weighted homogeneous singularities. In Singularities II. Geometric and topological aspects. Proceedings Internatinal Conference School and workshop on the geometry and topology of singularities. A. M. S. Contemporary Mathematics 475, 43–59 (2008)Google Scholar
  9. 9.
    Durfee, A.H.: The signature of smoothings of complex surface singularities. Math. Ann. 232, 85–98 (1978)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Esnault, H., Seade, J., Viehweg, E.: Characteristic divisors on complex manifolds. J. Reine Angew. Math. 424, 17–30 (1992)MathSciNetMATHGoogle Scholar
  11. 11.
    Freedman, M., Kirby, R.: A geometric proof of Rochlins theorem. In: Algebraic and Geometric Topology. Proceedings of Symposia Pure Mathematics, vol. XXXII, pp. 85–97. A.M.S (1978)Google Scholar
  12. 12.
    Gómez-Mont, X.: An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity. J. Algebraic Geom. 7, 731–752 (1998)MathSciNetMATHGoogle Scholar
  13. 13.
    Gómez-Mont, X., Seade, J., Verjovsky, A.: The index of a holomorphic flow with an isolated singularity. Math. Ann. 291, 737–751 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gómez-Mont, X., Seade, J., Verjovsky, A.: Topology of a holomorphic vector field around an isolated singularity. Funct. Anal. Appl. 27, 97–103 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grauert, H.: Über Modifikationen und exzeptionnelle analytische Mengen. Math. Ann. 146, 331–368 (1962)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Greuel, G.-M., Steenbrink, J.: On the Topology of Smoothable Singularities. Proceedings of sympsia pure mathematics Part 1, pp. 535–545. A.M.S. (1983)Google Scholar
  17. 17.
    Hochster, M.: The Zariski-Lipman conjecture in the graded case. J. Algebra 47(2), 411–424 (1977)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ishida, H.: Torus invariant transverse Kähler foliations. Trans. A. M. S. 369, 5137–5155 (2017)CrossRefMATHGoogle Scholar
  19. 19.
    Källström, R.: The Zariski-Lipman conjecture for complete intersections. J. Algebra 337, 169–180 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Larrión, F., Seade, J.: Complex surface singularities from the combinatorial point of view. Topol. Appl. 66, 251–265 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Laufer, H.B.: On \(\mu \) for surface singularities. In: Several Complex Variables Proceedings of SymposIa Pure Mathematics Part 1, vol. XXX, pp. 45–49. AMS (1977)Google Scholar
  22. 22.
    Lê, D.T.: Computation of the milnor number of an isolated singularity of a complete intersection. Funct. Anal. Appl. 8, 127–131 (1974)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Libgober, A.: Theta characteristics on singular curves, spin structures and Rokhlin theorem. Ann. Sci. Éc. Norm. Supér. 4(21)(4), 623–635 (1988)Google Scholar
  24. 24.
    Limón, B., Seade, J.: Morse theory and the topology of holomorphic foliations near an isolated singulariry. J. Topol. 4(3), 667–686 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lins Neto, A.: Holomorphic Rank of Hypersurfaces with an Isolated Singularity. Bol. Soc. Bras. Mat. 29(1), 145–161 (1998)Google Scholar
  26. 26.
    Lipman, J.: Free derivation modules on algebraic varieties. Am. J. Math. 87, 874–898 (1965)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    López de Medrano, S.: Singularities of real homogeneous quadratic mappings Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matemáticas, pp. 1–18 (2012)Google Scholar
  28. 28.
    Meersseman, L.: A new geometric construction of compact complex manifolds in any dimension. Math. Ann. 317, 79–115 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of mathematics studies. Princeton University Press, New York (1968)MATHGoogle Scholar
  30. 30.
    Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. IHES 9, 229–246 (1961)CrossRefMATHGoogle Scholar
  31. 31.
    Oka, M.: Non-degenerate mixed functions. Kodai Math. J. 33, 1–62 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Popescu-Pampu, P., Seade, J.: A finitness theorem for dual graphs of surface singularities. Int. J. Maths. 20(8), 1057–1068 (2009)CrossRefMATHGoogle Scholar
  33. 33.
    Popescu-Pampu, P.: Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones. Duke Math. J. 159(3), 539–559 (2011)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Ruas, M.A.S., Seade, J., Verjovsky, A.: On real singularities with a Milnor fibration. In: Libgober, A., Tibǎr, M. (eds.) Trends in Singularities, pp. 191–213 (2002)Google Scholar
  35. 35.
    Scheja, G., Storch, U.: Differentielle Eigenschaften der Lokalisierungen analytischer Algebren. Math. Ann. 197, 137–170 (1972)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Seade, J.: A cobordism invariant for surface singularities. Proceedings Symposia Pure Mathematics part 2, vol. 40, pp. 479–484 (1983)Google Scholar
  37. 37.
    Seade, J.: Vector fields on smoothings of complex singularities. In Ramírez de Arellano, E., Sundararaman, D. (eds.) Topics in Several Complex Variables, Research Notes in Mathematics, vol. 112, pp. 152–157. Pitman Advanced Publishing Program (1985)Google Scholar
  38. 38.
    Seade, J.: The index of a vector field on a complex surface with singularities. In: Verjovsky, A. (ed.) The Lefschetz Centennial Conference Contemporary Mathematics Part III, American Mathematical Society, vol. 58, pp. 225–232 (1987)Google Scholar
  39. 39.
    Seade, J.: On the Topology of Isolated Singularities in Analytic Spaces. Progress in mathematics. Birkhauser, Basel (2006)Google Scholar
  40. 40.
    Seade, J.: Remarks on Laufer’s formula for the Milnor number, Rochlin’s signature theorem and the analytic Euler characteristic of compact complex manifolds. Meth. Appl. Anal. 24, 105–123. Special issue in honor of Henry Laufer’s 70th Birthday (2017)Google Scholar
  41. 41.
    Seade, J.: A note on 3-manifolds and complex surface singularities. Preprint 2018, to be publishedGoogle Scholar
  42. 42.
    Steenbrink, J.: Mixed hodge structures associated with isolated singularities. Proceedings of Symposia Pure Mathematics, In: Singularities Part 2 (Arcata, Calif., 1981), vol. 40, pp. 513–536 (1983)Google Scholar
  43. 43.
    Thom, R.: Généralization de la théorie de Morse et variétés feuilletées. Ann. Inst. Fourier (Grenoble) 14, 173–190 (1964)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico CityMexico

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