Skip to main content
SpringerLink
Log in

Some Meeting Points of Singularity Theory and Low Dimensional Topology

  • Chapter
Deformations of Surface Singularities

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 23))

Abstract

We review some basic facts which connect the deformation theory of normal surface singularities with the topology of their links. The presentation contains some explicit descriptions for certain families of singularities (cyclic quotients, sandwiched singularities).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akhmedov, A., Etnyre, J. B., Mark, T. E. and Smith, I., A note on Stein fillings of contact manifolds, Math. Res. Lett., 15, no. 6 (2008), 1127–1132.

    Article  MATH  MathSciNet  Google Scholar 

  2. Arndt, J., Verselle Deformationen zyklischer Quotientensingularitäten, Diss. Hamburg, 1988.

    Google Scholar 

  3. Artin, M., Some numerical criteria for contractibility of curves on algebraic surfaces, Amer. J. of Math., 84 (1962), 485–496.

    Article  MATH  MathSciNet  Google Scholar 

  4. Artin, M., On isolated rational singularities of surfaces, Amer. J. of Math., 88 (1966), 129–136.

    Article  MATH  MathSciNet  Google Scholar 

  5. Artin, M., Algebraic construction of Brieskorn’s resolutions, J. of Algebra, 29 (1974), 330–348.

    Article  MATH  MathSciNet  Google Scholar 

  6. Balke, L., Smoothings of cyclic quotient singularities from a topological point of view, arXiv:math/9911070.

    Google Scholar 

  7. Barth, W., Peters, C. and Van de Ven, A., Compact Complex Surfaces, Springer-Verlag, 1984.

    Google Scholar 

  8. Behnke, K. and Knörrer, H., On infinitesimal deformations of rational surface singularities, Comp. Math., 61 (1987), 103–127.

    MATH  Google Scholar 

  9. Behnke, K. and Riemenschneider, O., Quotient surface singularities and their deformations, in: Singularity theory, D. T. Lê, K. Saito & B. Teissier eds., World Scientific, 1995, 1–54.

    Google Scholar 

  10. Bhupal, M. and Ono, K., Symplectic fillings of links of quotient surface singularities, arXiv:0808.3794.

    Google Scholar 

  11. Bogomolov, F. A. and de Oliveira, B., Stein Small Deformations of Strictly Pseudoconvex Surfaces, Contemporary Mathematics, 207 (1997), 25–41.

    Article  Google Scholar 

  12. Braun, G. and Némethi, A., Surgery formula for Seiberg-Witten invariants of negative definite plumbed 3-manifolds, J. reine angew. Math., 638 (2010), 189–208.

    MATH  MathSciNet  Google Scholar 

  13. Briançon J. and Speder, J., Les conditions de Whitney impliquent “μ* constant”, Ann. Inst. Fourier (Grenoble), 26 (1976), 153–164.

    Article  MATH  MathSciNet  Google Scholar 

  14. Brieskorn, E., Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann., 178 (1968), 255–270.

    Article  MATH  MathSciNet  Google Scholar 

  15. Brieskorn, E., Singular elements in semi-simple algebraic groups, Proc. Int. Con. Math. Nice, 2 (1971), 279–284.

    MathSciNet  Google Scholar 

  16. Caubel, C., and Popescu-Pampu, P., On the contact boundaries of normal surface singularities, C. R. Acad. Sci. Paris, Ser. I 339 (2004), 43–48.

    Article  MATH  MathSciNet  Google Scholar 

  17. Caubel, C., Némethi, A. and Popescu-Pampu, P., Milnor open books and Milnor fillable contact 3-manifolds, Topology, 45 (2006), 673–689.

    Article  MATH  MathSciNet  Google Scholar 

  18. Christophersen, J. A., On the components and discriminant of the versal base space of cyclic quotient singularities; in: Singularity theory and its applications, Warwick 1989, Part I, D. Mond, J. Montaldi eds., LNM 1462, Springer, 1991.

    Google Scholar 

  19. Christophersen, J. A. and Gustavsen, T. S., On infinitesimal deformations and obstructions for rational surface singularities, J. Algebraic Geometry, 10 (1) (2001), 179–198.

    MATH  MathSciNet  Google Scholar 

  20. Colin, V., Giroux, E., Honda, K., Finitude homotopique et isotopique des structures de contact tendues, Publ. Math. Inst. Hautes Études Sci., 109 (2009), 245–293.

    Article  MATH  MathSciNet  Google Scholar 

  21. Demazure, M., Pinkham, H., Teissier, B. (editors), Séminaire sur les Singularités des Surfaces, Lecture Notes of Math., 777, Springer-Verlag, 1980.

    Google Scholar 

  22. Durfee, A., The Signature of Smoothings of Complex Surface Singularities, Math. Ann., 232 (1978), 85–98.

    Article  MATH  MathSciNet  Google Scholar 

  23. Durfee, A., Fifteen characterizations of rational double points and simple critical points, L’enseignement Math., 25 (1979), 131–163.

    MathSciNet  Google Scholar 

  24. Eisenbud, D. and Neumann, W., Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Studies, 110, Princeton University Press, 1985.

    Google Scholar 

  25. Eliashberg, Y., Filling by holomorphic discs and its applications, Geometry of lowdimensional manifolds, 2 (Durham, 1989), 45–67, London Math. Soc. Lecture Note Ser., 151, Cambridge Univ. Press, 1990.

    Google Scholar 

  26. Elkik, R., Singularités rationelles et Déformations, Inv. Math., 47 (1978), 139–147.

    Article  MATH  MathSciNet  Google Scholar 

  27. Etnyre, J. and Ozbagci, B., Invariants of contact structures from open books, Trans. AMS, 360 (6) (2008), 3133–3151.

    Article  MATH  MathSciNet  Google Scholar 

  28. Giroux, E., Structures de contact en dimension trois et bifurcations des foilletages de surfaces, Invent. Math., 141 (2000), 615–689.

    Article  MATH  MathSciNet  Google Scholar 

  29. Giroux, E., Structures de contact sur les variétés fibrées en cercles au-dessus d’une surface, Comment. Math. Helv., 76 (2001), 218–262.

    Article  MATH  MathSciNet  Google Scholar 

  30. Giroux, E., Géometrie de contact: de la dimension trois vers les dimensions supérieures, Proc. ICM, Beijing 2002, Vol. II., 405–414.

    MathSciNet  Google Scholar 

  31. Grauert, H., Über Modifikationen und exceptionelle analytische Mengen, Math. Annalen, 146 (1962), 331–368.

    Article  MATH  MathSciNet  Google Scholar 

  32. Grauert, H., Über die Deformationen Isolierten Singularitäten Analytischer Mengen, Inv. Math., 15 (1972), 171–198.

    Article  MATH  MathSciNet  Google Scholar 

  33. Greuel, G.-M. and Steenbrink, J., On the topology of smoothable singularities, Proc. of Symp. in Pure Math., 40, Part 1 (1983), 535–545.

    Article  MathSciNet  Google Scholar 

  34. Hartshorne, R., Algebraic Geometry, Graduate Texts in Math., 52, Springer-Verlag 1977.

    Google Scholar 

  35. Honda, K., On the classification of tight contact structures I., Geom. Topol., 4 (2000), 309–368.

    Article  MATH  MathSciNet  Google Scholar 

  36. Honda, K., On the classification of tight contact structures II., J. Differential Geom. 55 (2000), 83–143.

    MATH  MathSciNet  Google Scholar 

  37. de Jong, T. and van Straten, D., On the base space of a semi-universal deformation of rational quadruple points, Annals of Math., 134 (2) (1991), 653–678.

    Article  MATH  Google Scholar 

  38. de Jong, T. and van Straten, D., On the deformation theory of rational surface singularities with reduced fundamental cycle, J. Alg. Geom., 3 (1994), 117–172.

    MATH  Google Scholar 

  39. de Jong, T. and van Straten, D., Deformation theory of sandwiched singularities, Duke Math. J., 95 (3) (1998), 451–522.

    Article  MATH  MathSciNet  Google Scholar 

  40. Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. Math., 91 (1988), 299–338.

    Article  MATH  MathSciNet  Google Scholar 

  41. Kollár, J., Flips, flops, minimal models, etc., Surveys in Diff. Geom., 1 (1991), 113–199.

    Article  Google Scholar 

  42. Laufer, H. B., Normal two-dimensional singularities, Annals of Math. Studies, 71, Princeton University Press, 1971.

    Google Scholar 

  43. Laufer, H. B., On rational singularities, Amer. J. of Math., 94 (1972), 597–608.

    Article  MATH  MathSciNet  Google Scholar 

  44. Laufer, H. B., Taut two-dimensional singularities, Math. Ann., 205 (1973), 131–164.

    Article  MATH  MathSciNet  Google Scholar 

  45. Laufer, H. B., On minimally elliptic singularities, Amer. J. of Math., 99 (1977), 1257–1295.

    Article  MATH  MathSciNet  Google Scholar 

  46. Laufer, H. B., On μ for surface singularities, Proceedings of Symposia in Pure Math., 30 (1977), 45–49.

    Article  MathSciNet  Google Scholar 

  47. Laufer, H. B., Weak simultaneous resolution for deformations of Gorenstein surface singularities, Proc. of Symp. in Pure Math., 40, Part 2 (1983), 1–29.

    Article  MathSciNet  Google Scholar 

  48. Laufer, H. B., Strong Simultaneous Resolution for Surface Singularities, Adv. Studies in Pure Math., 8 (1986), 207–214. Complex Analytic Singularities.

    MathSciNet  Google Scholar 

  49. Laufer, H. B., The multiplicity of isolated two-dimensional hypersurface singularities, Transactions of the AMS, 302, Number 2 (1987), 489–496.

    Article  MATH  MathSciNet  Google Scholar 

  50. Lê Dũng Tráng, Topologie des singularités des hypersurfaces complexes, Astérisque, 7–8 (1973), 171–182.

    Google Scholar 

  51. Lipman, J., Double point resolutions of deformations of rational singularities, Compositio Math., 38 (1979), 37–42.

    MATH  MathSciNet  Google Scholar 

  52. Lisca, P., On lens spaces and their symplectic fillings, Math. Res. Letters, 1, vol. 11 (2004), 13–22.

    Article  MathSciNet  Google Scholar 

  53. Lisca, P., On symplectic fillings of lens spaces, Trans. Amer. Math. Soc., 360 (2008), 765–799.

    Article  MATH  MathSciNet  Google Scholar 

  54. Lisca, P. and Stipsicz, A. I., On the existence of tight contact structures on Seifert fibered 3-manifolds, Duke Math. J., 148m no. 2, (2009), 175–209.

    Article  MATH  MathSciNet  Google Scholar 

  55. Looijenga, E., The smoothing components of a triangle singularity. I, Proc. of Symp. in Pure Math., 40, Part 2, (1983), 173–183.

    Article  MathSciNet  Google Scholar 

  56. Looijenga, E. J. N., Isolated Singular Points on Complete Intersections, London Math. Soc. Lecture Note Series, 77, Cambridge University Press 1984.

    Google Scholar 

  57. Looijenga, E., Riemann-Roch and smoothing of singularities, Topology, 25 (3) (1986), 293–302.

    Article  MATH  MathSciNet  Google Scholar 

  58. Looijenga, E. and Wahl, J., Quadratic functions and smoothing surface singularities, Topology, 25 (1986), 261–291.

    Article  MATH  MathSciNet  Google Scholar 

  59. McDuff, D., The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc., 3, no. 3, (1990), 679–712.

    MATH  MathSciNet  Google Scholar 

  60. Milnor, J., Singular points of complex hypersurfaces, Annals of Math. Studies, 61, Princeton University Press, 1968.

    Google Scholar 

  61. Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, IHES Publ. Math., 9 (1961), 5–22.

    Article  MATH  MathSciNet  Google Scholar 

  62. Némethi, A., Five lectures on normal surface singularities, lectures delivered at the Summer School in Low dimensional topology, Budapest, Hungary, 1998; Bolyai Society Math. Studies, 8 (1999), 269–351.

    Google Scholar 

  63. Némethi, A., “Weakly” Elliptic Gorenstein Singularities of Surfaces, Inventiones Math., 137 (1999), 145–167.

    Article  MATH  Google Scholar 

  64. Némethi, A., The resolution of some surface singularities, I., (cyclic coverings); Proceedings of the AMS Conference, San Antonio, 1999; Contemporary Mathematics, 266, 89–128.

    Article  Google Scholar 

  65. Némethi, A., , August 2002; Contemporary Mathematics, 354 (2004), 161–208.

    Google Scholar 

  66. Némethi, A., The cohomology of line bundles of splice-quotient singularities, arXiv:0810.4129.

    Google Scholar 

  67. Némethi, A. and Nicolaescu, L. I., Seiberg-Witten invariants and surface singularities, Geometry and Topology, Volume 6 (2002), 269–328.

    Article  MATH  MathSciNet  Google Scholar 

  68. Némethi, A. and Nicolaescu, L. I., Seiberg-Witten invariants and surface singularities II (singularities with good C*-action), Journal of London Math. Soc. (2), 69 (2004), 593–607.

    Article  MATH  Google Scholar 

  69. Némethi, A. and Nicolaescu, L. I., Seiberg-Witten invariants and surface singularities III (splicings and cyclic covers), Selecta Mathematica, New series, Vol. 11, Nr. 3–4 (2005), 399–451.

    MATH  MathSciNet  Google Scholar 

  70. Némethi, A. and Okuma, T., On the Casson invarint conjecture of Neumann-Wahl, J. of Algebraic Geometry, 18 (2009), 135–149.

    Article  MATH  Google Scholar 

  71. Némethi, A. and Okuma, T., The Seiberg-Witten invariant conjecture for splice-quotients, J. of London Math. Soc., 28 (2008), 143–154.

    Article  Google Scholar 

  72. Némethi, A. and Popescu-Pampu, P., On the Milnor fibers of cyclic quotient singularities, Proc. London Math. Soc., 101(2) (2010), 497–553.

    Article  Google Scholar 

  73. Némethi, A. and Popescu-Pampu, P., On the Milnor fibers of sandwiched singularities, Int. Math. Res. Not., 6 (2010), 1041–1061.

    Google Scholar 

  74. Némethi, A. and Tosun, M., Invariants of open books of links of surface singularities, Studia Sc. Math. Hungarica, 48(1) (2011), 135–144.

    MATH  Google Scholar 

  75. Neumann, W. D., A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Transactions of the AMS, 268, Number 2, (1981), 299–344.

    Article  MATH  Google Scholar 

  76. Neumann, W. D. and Pichon, A., Complex analytic realization of links, Intelligence of low dimensional topology 2006, 231–238, Ser. Knots Everything, 40, World Sci. Publ., Hackensack, NJ, 2007.

    Google Scholar 

  77. Neumann, W. and Wahl, J., Complex surface singularities with integral homology sphere links, Geometry and Topology, 9 (2005), 757–811.

    Article  MATH  MathSciNet  Google Scholar 

  78. Neumann, W. and Wahl, J., Complete intersection singularities of splice type as universal abelian covers, Geometry and Topology, 9 (2005), 699–755.

    Article  MATH  MathSciNet  Google Scholar 

  79. Okuma, T., The geometric genus of splice-quotient singularities, Transaction AMS, 360 (2008), 6643–6659.

    Article  MATH  MathSciNet  Google Scholar 

  80. Ohta, H. and Ono, K., Symplectic fillings of the link of simple elliptic singularities, J. reine angew. Math., 565 (2003), 183–205.

    MATH  MathSciNet  Google Scholar 

  81. Ohta, H. and Ono, K., Simple singularities and symplectic fillings, J. Differential Geom., 69 (2005), 1–42.

    MATH  MathSciNet  Google Scholar 

  82. Ohta, H. and Ono, K., Examples of isolated surface singularities whose links have infinitely many symplectic fillings, J. Fixed Point Theory Appl., 3 (2008), 51–56.

    Article  MATH  MathSciNet  Google Scholar 

  83. Orlik, P. and Wagreich, Ph., Isolated singularities of algebraic surfaces with ℂ* action, Ann. of Math. (2), 93 (1971), 205–228.

    Article  MATH  MathSciNet  Google Scholar 

  84. Orlik, P. and Wagreich, P., Algebraic surfaces with k*-action, Acta Math., 138 (1977), 43–81.

    Article  MATH  MathSciNet  Google Scholar 

  85. Ozbagci, B. and Stipsicz, A., Contact 3-manifolds with infinitely many Stein fillings, Proc. AMS, 132 (2004), 1549–1558.

    Article  MATH  MathSciNet  Google Scholar 

  86. Pinkham, H., Deformations of algebraic varieties with G m action, Astérisque, 20 (1974), 1–131.

    MATH  MathSciNet  Google Scholar 

  87. Pinkham, H., Normal surface singularities with ℂ* action, Math. Ann., 117 (1977), 183–193.

    Article  MathSciNet  Google Scholar 

  88. Pinkham, H., Smoothing of the D pqr singularities, p + q + r = 22, Proc. of Symp. in Pure Math., 40, Part 2, (1983), 373–377.

    Article  MathSciNet  Google Scholar 

  89. Popescu-Pampu, P., The geometry of continued fractions and the topology of surface singularities, in Singularities in Geometry and Topology 2004, Advanced Studies in Pure Mathematics, 46 (2007), 119–195.

    Google Scholar 

  90. Popescu-Pampu, P., Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones, Duke Math. Journal, 159, No. 3, (2011), 539–559.

    Article  MATH  MathSciNet  Google Scholar 

  91. Reid, M., Chapters on Algebraic Surfaces, in: Complex Algebraic Geometry, IAS/Park City Mathematical Series, Volume 3 (J. Kollár editor), 3–159, 1997.

    Google Scholar 

  92. Riemenschneider, O., Bemerkungen zur Deformationstheorie Nichtrationaler Singularitäten, Manus. Math., 14 (1974), 91–99.

    Article  MATH  MathSciNet  Google Scholar 

  93. Riemenschneider, O., Deformationen von Quotintensingularitäten (nach zyklischen Gruppen), Math. Ann., 209 (1974), 211–248.

    Article  MATH  MathSciNet  Google Scholar 

  94. Schaps, M., Deformations of Cohen-Macauley Schemes of codimension 2 and Non-Singular Deformations of Space Curves, Am. J. Math., 99 (1977), 669–685.

    Article  MATH  MathSciNet  Google Scholar 

  95. Schlessinger, M., Functors of Artin Rings, Trans. AMS, 130 (1968), 208–222.

    Article  MATH  MathSciNet  Google Scholar 

  96. Seade, J. A., A cobordism invariant for surface singularities, Proc. of Symp. in Pure Math., 40(2) (1983), 479–484.

    Article  MathSciNet  Google Scholar 

  97. Smith, I., Torus fibrations on symplectic four-manifolds, Turkish J. Math., 25, no. 1, (2001), 69–95.

    MATH  MathSciNet  Google Scholar 

  98. Spivakovsky, M., Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Annals of Math., 131 (1990), 411–491.

    Article  MATH  MathSciNet  Google Scholar 

  99. Steenbrink, J. H. M., Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., 40, Part 2 (1983), 513–536.

    Article  MathSciNet  Google Scholar 

  100. Stevens, J., Elliptic Surface Singularities and Smoothings of Curves, Math. Ann., 267 (1984), 239–247.

    Article  MATH  MathSciNet  Google Scholar 

  101. Stevens, J., On the versal deformation of cyclic quotient singularities, LNM, 1462 (1991), 302–319. (Singularity theory and its applications, Warwick 1989)

    MathSciNet  Google Scholar 

  102. Stevens, J., Partial resolutions of rational quadruple points, Int. J. of Math., 2 (2) (1991), 205–221.

    Article  MATH  MathSciNet  Google Scholar 

  103. Stevens, J., Deformations of singularities, Springer LNM 1811, 2003.

    Google Scholar 

  104. Teissier, B., Cycles évanescents, sections planes et conditions de Whitney, Asterisque, 7–8 (1973), 285–362.

    MathSciNet  Google Scholar 

  105. Teissier, B., Déformation à type topologique constant II, Séminaire Douady-Verdier 1972.

    Google Scholar 

  106. Teissier, B., Résolution simultanée I, II, LNM, 777 (1980), 71–146.

    MathSciNet  Google Scholar 

  107. Tjurina, G.-N., Locally Flat Deformations of Isolated Singularities of Complex Spaces, Math. USSR Izvestia, 3 (1969), 967–999.

    Article  Google Scholar 

  108. Tomari, M., A p g-formula and elliptic singularities, Publ. R. I. M. S. Kyoto University, 21 (1985), 297–354.

    Article  MATH  MathSciNet  Google Scholar 

  109. Ustilovsky, I., Infinitely many contact structures on S4m+1, I.M.R.N., 14 (1999), 781–792.

    MathSciNet  Google Scholar 

  110. Vaquié, M., Résolution simultanée de surfaces normales, Ann. Inst. Fourier, 35 (1985), 1–38.

    Article  MATH  Google Scholar 

  111. Wagreich, Ph., Elliptic singularities of surfaces, Amer. J. of Math., 92 (1970), 419–454.

    Article  MATH  MathSciNet  Google Scholar 

  112. Wahl, M. J., Equisingular deformations of normal surface singularities, I, Ann. of Math., 104 (1976), 325–356.

    Article  MATH  MathSciNet  Google Scholar 

  113. Wahl, M. J., Simultaneous resolution of rational singularities, Compositio Math., 38 (1) (1979), 43–54.

    MATH  MathSciNet  Google Scholar 

  114. Wahl, M. J., Elliptic Deformations of Minimally Elliptic Singularities, Math. Ann., 253 (1980), 241–262.

    Article  MATH  MathSciNet  Google Scholar 

  115. Wahl, J., Smoothings of normal surface singularities, Topology, 20 (1981), 219–246.

    Article  MATH  MathSciNet  Google Scholar 

  116. Yau, S. S.-T., On maximally elliptic singularities, Transactions of the AMS, 257, Number 2 (1980), 269–329.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rényi Institute of Mathematics, Budapest, Hungary

    Andras Némethi

Authors
  1. Andras Némethi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    András Némethi  & ágnes Szilárd  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2013 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Némethi, A. (2013). Some Meeting Points of Singularity Theory and Low Dimensional Topology. In: Némethi, A., Szilárd, á. (eds) Deformations of Surface Singularities. Bolyai Society Mathematical Studies, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39131-6_4

Download citation

Publish with us

Policies and ethics

Search

Navigation