Deformations of Surface Singularities pp 109-162 | Cite as
Some Meeting Points of Singularity Theory and Low Dimensional Topology
Chapter
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).
Keywords
Minimal Resolution Open Book Milnor Number Plane Curve Singularity Rational Homology Sphere
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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.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [4]Artin, M., On isolated rational singularities of surfaces, Amer. J. of Math., 88 (1966), 129–136.CrossRefMATHMathSciNetGoogle Scholar
- [5]Artin, M., Algebraic construction of Brieskorn’s resolutions, J. of Algebra, 29 (1974), 330–348.CrossRefMATHMathSciNetGoogle 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.MATHGoogle 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.CrossRefGoogle 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.MATHMathSciNetGoogle Scholar
- [13]Briançon J. and Speder, J., Les conditions de Whitney impliquent “μ* constant”, Ann. Inst. Fourier (Grenoble), 26 (1976), 153–164.CrossRefMATHMathSciNetGoogle Scholar
- [14]Brieskorn, E., Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann., 178 (1968), 255–270.CrossRefMATHMathSciNetGoogle Scholar
- [15]Brieskorn, E., Singular elements in semi-simple algebraic groups, Proc. Int. Con. Math. Nice, 2 (1971), 279–284.MathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [17]Caubel, C., Némethi, A. and Popescu-Pampu, P., Milnor open books and Milnor fillable contact 3-manifolds, Topology, 45 (2006), 673–689.CrossRefMATHMathSciNetGoogle 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.MATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [23]Durfee, A., Fifteen characterizations of rational double points and simple critical points, L’enseignement Math., 25 (1979), 131–163.MathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [27]Etnyre, J. and Ozbagci, B., Invariants of contact structures from open books, Trans. AMS, 360 (6) (2008), 3133–3151.CrossRefMATHMathSciNetGoogle Scholar
- [28]Giroux, E., Structures de contact en dimension trois et bifurcations des foilletages de surfaces, Invent. Math., 141 (2000), 615–689.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [30]Giroux, E., Géometrie de contact: de la dimension trois vers les dimensions supérieures, Proc. ICM, Beijing 2002, Vol. II., 405–414.MathSciNetGoogle Scholar
- [31]Grauert, H., Über Modifikationen und exceptionelle analytische Mengen, Math. Annalen, 146 (1962), 331–368.CrossRefMATHMathSciNetGoogle Scholar
- [32]Grauert, H., Über die Deformationen Isolierten Singularitäten Analytischer Mengen, Inv. Math., 15 (1972), 171–198.CrossRefMATHMathSciNetGoogle 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.CrossRefMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [36]Honda, K., On the classification of tight contact structures II., J. Differential Geom. 55 (2000), 83–143.MATHMathSciNetGoogle 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.CrossRefMATHGoogle 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.MATHGoogle Scholar
- [39]de Jong, T. and van Straten, D., Deformation theory of sandwiched singularities, Duke Math. J., 95 (3) (1998), 451–522.CrossRefMATHMathSciNetGoogle Scholar
- [40]Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. Math., 91 (1988), 299–338.CrossRefMATHMathSciNetGoogle Scholar
- [41]Kollár, J., Flips, flops, minimal models, etc., Surveys in Diff. Geom., 1 (1991), 113–199.CrossRefGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [44]Laufer, H. B., Taut two-dimensional singularities, Math. Ann., 205 (1973), 131–164.CrossRefMATHMathSciNetGoogle Scholar
- [45]Laufer, H. B., On minimally elliptic singularities, Amer. J. of Math., 99 (1977), 1257–1295.CrossRefMATHMathSciNetGoogle Scholar
- [46]Laufer, H. B., On μ for surface singularities, Proceedings of Symposia in Pure Math., 30 (1977), 45–49.CrossRefMathSciNetGoogle 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.CrossRefMathSciNetGoogle Scholar
- [48]Laufer, H. B., Strong Simultaneous Resolution for Surface Singularities, Adv. Studies in Pure Math., 8 (1986), 207–214. Complex Analytic Singularities.MathSciNetGoogle Scholar
- [49]Laufer, H. B., The multiplicity of isolated two-dimensional hypersurface singularities, Transactions of the AMS, 302, Number 2 (1987), 489–496.CrossRefMATHMathSciNetGoogle 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.MATHMathSciNetGoogle Scholar
- [52]Lisca, P., On lens spaces and their symplectic fillings, Math. Res. Letters, 1, vol. 11 (2004), 13–22.CrossRefMathSciNetGoogle Scholar
- [53]Lisca, P., On symplectic fillings of lens spaces, Trans. Amer. Math. Soc., 360 (2008), 765–799.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [55]Looijenga, E., The smoothing components of a triangle singularity. I, Proc. of Symp. in Pure Math., 40, Part 2, (1983), 173–183.CrossRefMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [58]Looijenga, E. and Wahl, J., Quadratic functions and smoothing surface singularities, Topology, 25 (1986), 261–291.CrossRefMATHMathSciNetGoogle Scholar
- [59]McDuff, D., The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc., 3, no. 3, (1990), 679–712.MATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHGoogle 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.CrossRefGoogle 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.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHGoogle 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.MATHMathSciNetGoogle Scholar
- [70]Némethi, A. and Okuma, T., On the Casson invarint conjecture of Neumann-Wahl, J. of Algebraic Geometry, 18 (2009), 135–149.CrossRefMATHGoogle Scholar
- [71]Némethi, A. and Okuma, T., The Seiberg-Witten invariant conjecture for splice-quotients, J. of London Math. Soc., 28 (2008), 143–154.CrossRefGoogle 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.CrossRefGoogle 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.MATHGoogle 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.CrossRefMATHGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [78]Neumann, W. and Wahl, J., Complete intersection singularities of splice type as universal abelian covers, Geometry and Topology, 9 (2005), 699–755.CrossRefMATHMathSciNetGoogle Scholar
- [79]Okuma, T., The geometric genus of splice-quotient singularities, Transaction AMS, 360 (2008), 6643–6659.CrossRefMATHMathSciNetGoogle Scholar
- [80]Ohta, H. and Ono, K., Symplectic fillings of the link of simple elliptic singularities, J. reine angew. Math., 565 (2003), 183–205.MATHMathSciNetGoogle Scholar
- [81]Ohta, H. and Ono, K., Simple singularities and symplectic fillings, J. Differential Geom., 69 (2005), 1–42.MATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [83]Orlik, P. and Wagreich, Ph., Isolated singularities of algebraic surfaces with ℂ* action, Ann. of Math. (2), 93 (1971), 205–228.CrossRefMATHMathSciNetGoogle Scholar
- [84]Orlik, P. and Wagreich, P., Algebraic surfaces with k*-action, Acta Math., 138 (1977), 43–81.CrossRefMATHMathSciNetGoogle Scholar
- [85]Ozbagci, B. and Stipsicz, A., Contact 3-manifolds with infinitely many Stein fillings, Proc. AMS, 132 (2004), 1549–1558.CrossRefMATHMathSciNetGoogle Scholar
- [86]Pinkham, H., Deformations of algebraic varieties with G m action, Astérisque, 20 (1974), 1–131.MATHMathSciNetGoogle Scholar
- [87]Pinkham, H., Normal surface singularities with ℂ* action, Math. Ann., 117 (1977), 183–193.CrossRefMathSciNetGoogle 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.CrossRefMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [93]Riemenschneider, O., Deformationen von Quotintensingularitäten (nach zyklischen Gruppen), Math. Ann., 209 (1974), 211–248.CrossRefMATHMathSciNetGoogle 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.CrossRefMATHMathSciNetGoogle Scholar
- [95]Schlessinger, M., Functors of Artin Rings, Trans. AMS, 130 (1968), 208–222.CrossRefMATHMathSciNetGoogle Scholar
- [96]Seade, J. A., A cobordism invariant for surface singularities, Proc. of Symp. in Pure Math., 40(2) (1983), 479–484.CrossRefMathSciNetGoogle Scholar
- [97]Smith, I., Torus fibrations on symplectic four-manifolds, Turkish J. Math., 25, no. 1, (2001), 69–95.MATHMathSciNetGoogle Scholar
- [98]Spivakovsky, M., Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Annals of Math., 131 (1990), 411–491.CrossRefMATHMathSciNetGoogle Scholar
- [99]Steenbrink, J. H. M., Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., 40, Part 2 (1983), 513–536.CrossRefMathSciNetGoogle Scholar
- [100]Stevens, J., Elliptic Surface Singularities and Smoothings of Curves, Math. Ann., 267 (1984), 239–247.CrossRefMATHMathSciNetGoogle Scholar
- [101]Stevens, J., On the versal deformation of cyclic quotient singularities, LNM, 1462 (1991), 302–319. (Singularity theory and its applications, Warwick 1989)MathSciNetGoogle Scholar
- [102]Stevens, J., Partial resolutions of rational quadruple points, Int. J. of Math., 2 (2) (1991), 205–221.CrossRefMATHMathSciNetGoogle 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.MathSciNetGoogle 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.MathSciNetGoogle Scholar
- [107]Tjurina, G.-N., Locally Flat Deformations of Isolated Singularities of Complex Spaces, Math. USSR Izvestia, 3 (1969), 967–999.CrossRefGoogle Scholar
- [108]Tomari, M., A p g-formula and elliptic singularities, Publ. R. I. M. S. Kyoto University, 21 (1985), 297–354.CrossRefMATHMathSciNetGoogle Scholar
- [109]Ustilovsky, I., Infinitely many contact structures on S4m+1, I.M.R.N., 14 (1999), 781–792.MathSciNetGoogle Scholar
- [110]Vaquié, M., Résolution simultanée de surfaces normales, Ann. Inst. Fourier, 35 (1985), 1–38.CrossRefMATHGoogle Scholar
- [111]Wagreich, Ph., Elliptic singularities of surfaces, Amer. J. of Math., 92 (1970), 419–454.CrossRefMATHMathSciNetGoogle Scholar
- [112]Wahl, M. J., Equisingular deformations of normal surface singularities, I, Ann. of Math., 104 (1976), 325–356.CrossRefMATHMathSciNetGoogle Scholar
- [113]Wahl, M. J., Simultaneous resolution of rational singularities, Compositio Math., 38 (1) (1979), 43–54.MATHMathSciNetGoogle Scholar
- [114]Wahl, M. J., Elliptic Deformations of Minimally Elliptic Singularities, Math. Ann., 253 (1980), 241–262.CrossRefMATHMathSciNetGoogle Scholar
- [115]Wahl, J., Smoothings of normal surface singularities, Topology, 20 (1981), 219–246.CrossRefMATHMathSciNetGoogle Scholar
- [116]Yau, S. S.-T., On maximally elliptic singularities, Transactions of the AMS, 257, Number 2 (1980), 269–329.CrossRefMATHGoogle Scholar
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