Centre de diffusion de revues académiques mathématiques


Winter Braids Lecture Notes

Table des matières de ce volume | Article précédent | Article suivant
Patrick Popescu-Pampu
Complex singularities and contact topology
Winter Braids Lecture Notes, 3 : Winter Braids VI (Lille, 2016) (2016), Exp. No. 3, 74 p., doi: 10.5802/wbln.14
Article PDF
Mots clés: Contact structure, complex singularity, cyclic quotient singularity, Gorenstein singularity, graph manifold, Hirzebruch-Jung singularity, Milnor fiber, minimally elliptic singularity, modification, normal singularity, plumbing, plurisubharmonic function, quotient singularity, Stein filling, rational surface singularity, resolution of singularities, smoothing, versal deformation.

Résumé - Abstract

This text is a greatly expanded version of the mini-course I gave during the school Winter Braids VI organized in Lille between 22–25 February 2016. It is an introduction to the study of interactions between singularity theory of complex analytic varieties and contact topology. I concentrate on the relation between the smoothings of singularities and the Stein fillings of their contact boundaries. I tried to explain basic intuitions and facts in both fields, for the sake of the readers who are not accustomed with one of them.


[1] Akhmedov, A., Ozbagci, B. Singularity links with exotic Stein fillings. J. Singul. 8 (2014), 39-49.
[2] Arnold, V. I., Gusein-Zade, S. M., Varchenko, A. N. Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Monographs in Mathematics 83. Birkhäuser, Boston, MA., 1988.
[3] Aroca, J.-M., Hironaka, H., Vicente, J. L. The theory of the maximal contact. Memorias de Matemática del Instituto “Jorge Juan” 29. Consejo Superior de Investigaciones Científicas, Madrid, 1975.
[4] Aroca, J.-M., Hironaka, H., Vicente, J. L. Desingularization theorems. Memorias de Matemática del Instituto “Jorge Juan” 30. Consejo Superior de Investigaciones Científicas, Madrid, 1977.
[5] Artin, M. Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J. Math. 84 (1962), 485-496.
[6] Artin, M. On isolated rational singularities of surfaces. Amer. J. Math. 88 (1966), 129-136.
[7] Artin, M. Algebraic construction of Brieskorn’s resolutions. J. Algebra 29 (1974), 330-348.
[8] Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A. Compact complex surfaces. Second enlarged edition, Springer-Verlag, Berlin, 2004.
[9] Baykur, R. I., Van Horn-Morris, J. Families of contact 3-manifolds with arbitrarily large Stein fillings. With an appendix by Samuel Lisi and Chris Wendl. J. Differential Geom. 101 (2015), no. 3, 423-465.
[10] Bădescu, L. Algebraic surfaces. Universitext. Springer-Verlag, New York, 2001.
[11] Behnke, K., Riemenschneider, O. Quotient surface singularities and their deformations. In Singularity theory, 1-54. D. T. Lê, K. Saito & B. Teissier eds. World Scientific, Hackensack, N.J., 1995.
[12] Bennequin, D. Entrelacements et équations de Pfaff. Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), 87-161, Astérisque, 107-108, Soc. Math. France, Paris, 1983.
[13] Bennequin, D. Topologie symplectique, convexité holomorphe et structures de contact [d’après Y. Eliashberg, D. Mc Duff et al.] Séminaire Bourbaki no. 725, Astérisque 189-190, Soc. Math. France, 1990.
[14] Bhupal, M., Ono, K. Symplectic fillings of links of quotient surface singularities. Nagoya Math. J. 207 (2012), 1-45.
[15] Bhupal, M., Ozbagci, B. Milnor open books of links of some rational surface singularities. Pacific J. Math. 254 (2011), no. 1, 47-65.
[16] Bhupal, M., Ozbagci, B. Canonical contact structures on some singularity links. Bull. Lond. Math. Soc. 46 (2014), no. 3, 576-586.
[17] Bhupal, M., Stipsicz, A. Smoothings of singularities and symplectic topology. In Deformations of surface singularities, 57-97, Bolyai Soc. Math. Stud. 23, János Bolyai Math. Soc., Budapest, 2013.
[18] Bogomolov, F.A., de Oliveira, B. Stein Small Deformations of Strictly Pseudoconvex Surfaces. Contemporary Mathematics 207 (1997), 25-41.
[19] Borman, M. S., Eliashberg, Y., Murphy, E. Existence and classification of overtwisted contact structures in all dimensions. Acta Math. 215 (2015), no. 2, 281-361.
[20] Brieskorn, E. Beispiele zur Differentialtopologie von Singularitäten. Invent. Math. 2 (1966), 1-14.
[21] Brieskorn, E. Rationale Singularitäten Komplexer Flächen. Invent. Math. 4 (1968), 336-358.
[22] Brieskorn, E. Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta Math. 2 (1970), 103-161.
[23] Brieskorn, E. Singularities in the work of Friedrich Hirzebruch. In Surveys in differential geometry 7, Int. Press, Somerville, MA., 2000, 17-60.
[24] Bruns, W., Herzog, J. Cohen-Macaulay rings. Cambridge Univ. Press, Cambridge, 1993.
[25] Buchweitz, R.-O., Greuel, G.-M. The Milnor number and deformations of complex curve singularities. Invent. Math. 58 (1980), no. 3, 241-281.
[26] Budur, N. Singularity invariants related to Milnor fibers: survey. In Zeta functions in algebra and geometry, 161-187, Contemp. Math. 566, American Mathematical Society, Providence, R.I., 2012.
[27] Cartan, H. Quotient d’un espace analytique par un groupe d’automorphismes. In A symposium in honor of S. Lefschetz, Algebraic geometry and topology, 90-102. Princeton University Press, Princeton, N.J., 1957.
[28] Cassens, H., Slodowy, P. On Kleinian singularities and quivers. In Singularities. The Brieskorn anniversary volume. V. I. Arnold, G.-M. Greuel and J. H. M. Steenbrink eds. 263-288. Progress in Mathematics 162, Birkhäuser, Basel, 1998.
[29] Caubel, C. Contact structures and non-isolated singularities. In Singularity theory. Dedicated to Jean-Paul Brasselet on his 60-th birthday. D. Chéniot et al. eds., 475-485. World Scientific, Hackensach, N.J., 2007.
[30] Caubel, C., Némethi, A., Popescu-Pampu, P. Milnor open books and Milnor fillable contact 3-manifolds. Topology 45 (2006), 673-689.
[31] Chevalley, C. Invariants of finite groups generated by reflections. Amer. J. Math. 77 (1955), 778-782.
[32] Christophersen, J. A. On the components and discriminant of the versal base space of cyclic quotient singularities. In Singularity theory and its applications, Part I (Coventry, 1988/1989), 81-92, Lecture Notes in Mathematics 1462 Springer, Berlin, 1991.
[33] Cieliebak, K., Eliashberg, Y. From Stein to Weinstein and back: symplectic geometry and affine complex manifolds. American Mathematical Society, Providence, RI, 2012.
[34] Cutkosky, S.D. Valuations in algebra and geometry. In Singularities in algebraic and analytic geometry (San Antonio, TX, 1999), 45-63, Contemp. Math. 266, American Mathematical Society, Providence, R.I., 2000.
[35] Cutkosky, S.D. Resolution of singularities. Graduate Studies in Mathematics 63. American Mathematical Society, Providence, R.I., 2004.
[36] Danilov, V. I. Polyhedra of schemes and algebraic varieties. Math. USSR-Sb. 26 (1975), no. 1, 137-149.
[37] Ding, F., Geiges, H. Symplectic fillability of tight contact structures on torus bundles. Algebr. Geom. Topol. 1 (2001), 153-172.
[38] Durfee, A.H. The signature of smoothings of complex surface singularities. Math. Ann. 232 (1978), 85-98.
[39] Durfee, A.H. Fifteen characterizations of rational double points and simple critical points. Enseignem. Math. 25 (1979), 131-163.
[40] Durfee, A.H. Neighborhoods of algebraic sets. Trans. Amer. Math. Soc. 276 (1983), no. 2, 517-530.
[41] Durfee, A.H. Singularities. In History of Topology. I.M. James ed., North Holland, Amsterdam, 1999, 417-434.
[42] Durfee, A.H., Hain, R.M. Mixed Hodge structures on the homotopy of links. Math. Ann. 280 (1988), no. 1, 69-83.
[43] Dutertre, N. Semi-algebraic neighborhoods of closed semi-algebraic sets. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 1, 429-458.
[44] Eisenbud, D. Commutative algebra with a view toward algebraic geometry. Springer-Verlag, New York, 1995.
[45] Eliashberg, Y. Classification of overtwisted contact structures on $3$-manifolds. Invent. Math. 98 (1989), no. 3, 623-637.
[46] Eliashberg, Y. Filling by holomorphic discs and its applications. Geometry of low-dimensional manifolds, 2 (Durham, 1989), 45-67, London Math. Soc. Lecture Note Ser., 151, Cambridge Univ. Press, Cambridge, 1990.
[47] Eliashberg, Y. Unique holomorphically fillable contact structure on the 3-torus. Internat. Math. Res. Notices 2 (1996), 77-82.
[48] Eliashberg, Y., Gromov, M. Convex symplectic manifolds. In Several complex variables and complex geometry. Part 2, Proc. Sympos. Pure Math. 52, 135-162. Amer. Math. Soc., Providence, R.I., 1991.
[49] Etnyre, J. B. Lectures on open book decompositions and contact structures. In Floer homology, gauge theory, and low-dimensional topology, 103-141, Clay Math. Proc. 5, American Mathematical Society, Providence, R.I., 2006.
[50] Etnyre, J. B., Honda, K. On the nonexistence of tight contact structures. Annals of Maths. 153 (2001), 749-766.
[51] Etnyre, J. B., Honda, K. Tight contact structures with no symplectic fillings. Invent. Math. 148 (2002), 609-626.
[52] Faber, E., Hauser, H. Today’s menu: geometry and resolution of singular algebraic surfaces. Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 3, 373-417.
[53] Fernández de Bobadilla, J., Menegon Neto, A. The boundary of the Milnor fibre of complex and real analytic non-isolated singularities. Geom. Dedicata 173 (2014), 143-162.
[54] Fintushel, R., Stern, R. J. Rational blowdowns of smooth 4-manifolds. J. Differential Geom. 46 (1997), no. 2, 181-235.
[55] Fischer, G. Complex analytic geometry. in Lecture Notes in Mathematics 538, Springer, Berlin-New York, 1976.
[56] Fowler, J. Rational homology disk smoothing components of weighted homogeneous surface singularities. PhD Thesis, University of North Carolina at Chapel Hill, 2013. ISBN: 978-1303-63900-5.
[57] Geiges, H. An introduction to contact topology. Cambridge University Press, Cambridge, 2008.
[58] Ghiggini, P. Strongly fillable contact 3-manifolds without Stein fillings. Geom. Topol. 9 (2005), 1677-1687.
[59] Giroux, E. Géométrie de contact: de la dimension trois vers les dimensions supérieures. Proceedings of the ICM 2002 (Beijing), Higher Ed. Press, vol.II, 405-414.
[60] Giroux, E., Goodman, N. On the stable equivalence of open books in three-manifolds. Geom. Topol. 10 (2006), 97-114.
[61] Grauert, H. On Levi’s problem and the imbedding of real-analytic manifolds. Ann. of Math. (2) 68 (1958), 460-472.
[62] Grauert, H. Über Modifikationen und exzeptionnelle analytische Mengen. Math. Ann. 146 (1962), 331-368.
[63] Grauert, H. Über die Deformationen isolierter Singularitäten analytische Mengen. Invent. Math. 15 (1972), 171-198.
[64] Grauert, H., Remmert, R. Theory of Stein spaces. Springer-Verlag, Berlin-New York, 1979.
[65] Gray, J.W. Some global properties of contact structures. Ann. of Math. (2) 69 (1959), 421-450.
[66] Greuel, G.-M., Looijenga, E.N. The dimension of smoothing components. Duke Math. J. 52 (1985), no. 1, 263-272.
[67] Greuel, G.-M., Steenbrink, J. On the topology of smoothable singularities. In Singularities, Part 1 (Arcata, Calif., 1981) Proc. of Symp. in Pure Maths. 40, 535-545. American Mathematical Society, Providence, R.I., 1983.
[68] Greuel, G.-M., Lossen, C., Shustin, E. Introduction to singularities and deformations. Springer, Berlin, 2007.
[69] Griffiths, Ph., Harris, J. Principles of algebraic geometry. John Wiley & Sons, New York, 1978.
[70] Gromov, M. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307-347.
[71] Hamm, H. Lokale topologische Eigenschaften komplexer Räume. Math. Ann. 191 (1971), 235-252.
[72] Hamm, M. Die verselle Deformation zyklischer quotienten Singularitäten: Gleichungen und torische Struktur. Diss. Hamburg 2008.
[73] Hartshorne, R. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977.
[74] Hauser, H. La construction de la déformation semi-universelle d’un germe de variété analytique complexe. Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 1, 1-56.
[75] Hauser, H. Seven short stories on blowups and resolutions. In Proceedings of Gökova Geometry-Topology Conference 2005, 1-48, 2006.
[76] Hazewinkel, M., Hesselink, W., Siersma, D., Veldkamp, F.D. The ubiquity of Coxeter-Dynkin diagrams. (An introduction to the A-D-E problem). Nieuw Archief voor Wiskunde (3), XXV (1977), 257-307.
[77] Hind, R. Stein fillings of lens spaces. Commun. Contemp. Math. 5 (2003), no. 6, 967-982.
[78] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 (1964) 205-326.
[79] Hironaka, H. Introduction to the theory of infinitely near singular points. Memorias de Matemática del Instituto “Jorge Juan” 28. Consejo Superior de Investigaciones Científicas, Madrid, 1974.
[80] Hirzebruch, F. Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen. Math. Ann. 126 (1953), 1-22.
[81] Hirzebruch, F. The topology of normal singularities of an algebraic surface. Séminaire Bourbaki, Tome 8 (1962-1964), Exposé no. 250, 129-137.
[82] Hirzebruch, F. Singularities and exotic spheres. Séminaire Bourbaki, Tome 10 (1966-1968), Exposé no. 314, 13-32.
[83] Hirzebruch, F. Hilbert modular surfaces. Enseignem. Math. 19 (1973), 183-281.
[84] Hirzebruch, F., Mayer, K. H. $O(n)$-Mannigfaltigkeiten, exotische Sphären und Singularitäten. Lecture Notes in Mathematics 57, Springer-Verlag, Berlin-New York, 1968.
[85] Hörmander, L. Notions of convexity. Progress in Maths. 127, Birkhäuser, Boston, 1994.
[86] Ishii, S. Introduction to singularities. Springer, Tokyo, 2014.
[87] Jaco, W.H., Shalen, P.B. Seifert Fibered Spaces in Three-manifolds. Memoirs of the American Mathematical Society 21 (1979), no. 220, viii+ 192 p.
[88] Johannson, K. Homotopy Equivalences of 3-Manifolds with Boundaries. Lecture Notes in Mathematics 761, Springer, Berlin, 1979.
[89] de Jong, T., van Straten, D. Deformation theory of sandwiched singularities. Duke Math. Journal 95, no. 3 (1998), 451-522.
[90] de Jong, T., Pfister, G. Local analytic geometry. Friedr. Vieweg $\&$ Sohn, Braunschweig, 2000.
[91] Jung, H.W.E. Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x,y in der Umgebung einer Stelle x=a, y=b. J. Reine Angew. Math. 133 (1908), 289-314.
[92] Klein, F. Lectures on the icosahedron and the solution of equations of the fifth degree. Dover Publ., INC., 2003. First German edition: 1884.
[93] Kollár, J. Flips, flops, minimal models, etc. In Surveys in differential geometry 1 (1991), 113-199.
[94] Kollár, J. Lectures on resolution of singularities. Annals of Mathematics Studies 166. Princeton University Press, Princeton, N.J., 2007.
[95] Kollár, J. Links of complex analytic singularities. In Surveys in differential geometry 18, Int. Press, Somerville, MA., 2013, 157-193.
[96] Kollár, J., Shepherd-Barron, N. I. Threefolds and deformations of surface singularities. Invent. Math. 91 (1988), 299-338.
[97] Kontsevich, M., Soibelman, Y. Affine structures and non-Archimedean analytic spaces. In The unity of mathematics, 321-385, Progr. Math. 244, Birkhäuser Boston, MA., 2006.
[98] Kwon, M., van Koert, O. Brieskorn manifolds in contact topology. Bull. Lond. Math. Soc. 48 (2016), no. 2, 173-241.
[99] Laufer, H.B. Normal two-dimensional Singularities. Princeton Univ. Press, Princeton, N.J., 1971.
[100] Laufer, H.B. On rational singularities. Amer. J. Math. 94 (1972), 597-608.
[101] Laufer, H.B. Taut two-dimensional singularities. Math. Ann. 205, 131-164 (1973).
[102] Laufer, H.B. On minimally elliptic singularities. Amer. J. Math. 99 (6), 1257-1295 (1977).
[103] Laufer, H.B. On $\mu $ for surface singularities. In Several complex Variables I, 45-49. Proc. Sympos. Pure Math. 30. R.O. Wells ed. American Mathematical Society, Providence, R.I., 1977.
[104] Lekili, Y., Ozbagci, B. Milnor fillable contact structures are universally tight. Math. Res. Lett. 17 (2010), no. 6, 1055-1063.
[105] Lê D.T. Topologie des singularités des hypersurfaces complexes. In Singularités à Cargèse (Rencontre Singularités Géom. Analyt., Inst. Études Sci., Cargèse, 1972), 171-182. Astérisque 7-8, Soc. Math. France, Paris, 1973.
[106] Lê, D.T., Seade, J., Verjovsky, A. Quadrics, orthogonal actions and involutions in complex projective spaces. Enseign. Math. (2) 49 (2003), no. 1-2, 173-203.
[107] Lê, D.T., Teissier, B., Limites d’espaces tangents en géométrie analytique. Comment. Math. Helv. 63 (1988), 540-578.
[108] Lê D. T., Tosun, M. Combinatorics of rational singularities. Comment. Math. Helv. 79 (2004), 582-604.
[109] Lipman, J. Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. no. 36 (1969), 195-279.
[110] Lipman, J. Introduction to resolution of singularities. In Algebraic geometry. Proc. Sympos. Pure Math. 29, 187- 230. American Mathematical Society, Providence, R.I., 1975.
[111] Lipman, J. Double point resolutions of deformations of rational singularities. Compositio Math. 38 (1979), no. 1, 37-42.
[112] Lisca, P. On lens spaces and their symplectic fillings. Math. Res. Letters 1, vol. 11 (2004), 13-22.
[113] Lisca, P. On symplectic fillings of lens spaces. Trans. Amer. Math. Soc. 360 (2008), 765-799.
[114] Looijenga, E.N. Isolated singular points on complete intersections. London Math. Soc. Lecture Notes Series 77, Cambridge Univ. Press, Cambridge, 1984.
[115] Looijenga, E.N. Riemann-Roch and smoothings of singularities. Topology 25 no. 3 (1986), 293-302.
[116] Looijenga, E.N., Wahl, J. Quadratic functions and smoothing surface singularities. Topology 25 (1986), no. 3, 261-291.
[117] Lutz, R. Sur quelques propriétés des formes différentielles en dimension trois. Thèse, Strasbourg, 1971.
[118] Martinet, J. Formes de contact sur les variétés de dimension $3$. Proceedings of Liverpool Singularities Symposium, II (1969/1970), 142-163. Lecture Notes in Math., 209, Springer, Berlin, 1971.
[119] Massot, P., Niederkrüger, K., Wendl, C. Weak and strong fillability of higher dimensional contact manifolds. Invent. Math. 192 (2013), no. 2, 287-373.
[120] Matsuki, K. Introduction to the Mori program. Springer-Verlag, New York, 2002.
[121] McDuff, D. The structure of rational and ruled symplectic 4-manifolds. J. Amer. Math. Soc. 3 (1990), 679-712. Erratum. J. Amer. Math. Soc. 5 (1992), 987-988.
[122] McLean, M. Reeb orbits and the minimal discrepancy of an isolated singularity. Invent. Math. 204 (2016) no. 2, 505-594.
[123] McLean, M. Floer Cohomology, multiplicity and the log canonical threshold. arXiv:1608.07541.
[124] Mendris, R., Némethi, A. The link of $\lbrace f (x, y) + z^n = 0\rbrace $ and Zariski’s conjecture. Compositio Math. 141 (2005), 502-524.
[125] Michel, F., Pichon, A. On the boundary of the Milnor fibre of nonisolated singularities. Int. Math. Res. Not. 2003 no. 43, 2305-2311. Erratum. Int. Math. Res. Not. 2004, no. 6, 309-310.
[126] Michel, F., Pichon, A. Carrousel in family and non-isolated hypersurface singularities in $\mathbb{C}^3$. J. Reine Angew. Math. 720 (2016), 1-32.
[127] Milnor, J. On manifolds homeomorphic to the $7$-sphere. Ann. of Math. (2) 64 (1956), 399-405.
[128] Milnor, J. Morse theory. Annals of Mathematics Studies, no. 51, Princeton University Press, Princeton, N.J. 1963.
[129] Milnor, J. Singular points of complex hypersurfaces. Princeton Univ. Press, Princeton, N.J., 1968.
[130] Mumford, D. The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Études Sci. Publ. Math. no. 9 (1961), 5-22.
[131] Mumford, D. The red book of varieties and schemes. Second expanded edition. Lecture Notes in Mathematics 1358, Springer, 1999.
[132] Némethi, A. Five lectures on normal surface singularities. In Low Dimensional Topology. Bolyai Soc. Math. Stud. 8, 269-351, János Bolyai Math. Soc., Budapest, 1999.
[133] Némethi, A. Some topological invariants of isolated hypersurface singularities. In Low dimensional topology, Bolyai Soc. Math. Stud. 8, 353-413, János Bolyai Math. Soc., Budapest, 1999.
[134] Némethi, A. Invariants of normal surface singularities. Contemporary Mathematics 354, 161-208 AMS, 2004.
[135] Némethi, A. Some meeting points of singularity theory and low dimensional topology. In Deformations of surface singularities, 109-162, Bolyai Soc. Math. Stud. 23, János Bolyai Math. Soc., Budapest, 2013.
[136] Némethi, A. Links of rational singularities, L-spaces and LO fundamental groups. arXiv:1510.07128.
[137] Némethi, A., Popescu-Pampu, P. On the Milnor fibers of cyclic quotient singularities. Proc. London Math. Society (3) 101 (2010), 554-588.
[138] Némethi, A., Popescu-Pampu, P. On the Milnor fibers of sandwiched singularities. Int. Maths. Research Notices 2010, no. 6, 1041-1061.
[139] Némethi, A., Szilárd, Á. Milnor fiber boundary of a non-isolated surface singularity. Lecture Notes in Mathematics 2037. Springer, Heidelberg, 2012.
[140] Némethi, A., Szilard, Á. eds. Deformations of surface singularities, Bolyai Soc. Math. Stud. 23, János Bolyai Math. Soc., Budapest, 2013.
[141] Némethi, A., Tosun, M. Invariants of open books of links of surface singularities. Studia Sci. Math. Hungar. 48 (2011), no. 1, 135-144.
[142] Neumann, W. A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Amer. Math. Soc. 268, 2 (1981), 299-344.
[143] Neumann, W., Swarup, G. Canonical decompositions of 3-manifolds. Geom. Topol. 1 (1997), 21-40.
[144] Ohta, H., Ono, K. Symplectic fillings of the link of simple elliptic singularities. J. Reine Angew. Math. 565 (2003), 183-205.
[145] Ohta, H., Ono, K. Simple singularities and symplectic fillings. J. Differential Geom. 69 (2005), no. 1, 1-42.
[146] Ohta, H., Ono, K. Examples of isolated surface singularities whose links have infinitely many symplectic fillings. J. Fixed Point Theory Appl. 3 (2008), no. 1, 51-56.
[147] Okuma, T. Universal abelian covers of certain surface singularities. Math. Ann. 334 (2006), no. 4, 753-773.
[148] Orlik, P. The multiplicity of a holomorphic map at an isolated critical point. In Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), 405-474. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.
[149] Ozbagci, B. On the topology of fillings of contact $3$-manifolds. Geom. and Topology Monographs 19 (2015), 73-123.
[150] Ozbagci, B., Stipsicz, A.I. Surgery on contact 3-manifolds and Stein surfaces. Springer, 2004.
[151] Park, H., Park, J., Shin, D., Urzúa, G. Milnor fibers and symplectic fillings of quotient surface singularities. arXiv:1507.06756.
[152] Park, H., Shin, D., Stipsicz, A. I. Normal complex surface singularities with rational homology disk smoothings. arXiv:1311.1929.
[153] Park, H., Stipsicz, A. I. Smoothings of singularities and symplectic surgery. J. Symplectic Geom. 12 (2014), no. 3, 585-597.
[154] Peternell, Th. Pseudoconvexity, the Levi problem and vanishing theorems. Chapter V of Several complex variables VII. Encyclopaedia of Math. Sciences 74, H. Grauert, Th. Peternell, R. Remmert eds., Springer-Verlag, 1994, 221-257.
[155] Peternell, Th. Modifications. Chapter VII of Several complex variables VII. Encyclopaedia of Math. Sciences 74, H. Grauert, Th. Peternell, R. Remmert eds., Springer-Verlag, 1994, 285-317.
[156] Peternell, Th., Remmert, R. Differential calculus, holomorphic maps and linear structures on complex spaces. Chapter II of Several complex variables VII. Encyclopaedia of Math. Sciences 74, H. Grauert, Th. Peternell, R. Remmert eds., Springer-Verlag, 1994, 97-144.
[157] Pham, F. Formules de Picard-Lefschetz généralisées et ramification des intégrales. Bull. Soc. Math. France 93 (1965), 333-367.
[158] Pinkham, H.C. Deformations of algebraic varieties with $\mathbb{G}_m$-action. Astérisque 20, S.M.F. 1974.
[159] Plamenevskaya, O., Van Horn-Morris, J. Planar open books, monodromy factorizations and symplectic fillings. Geom. Topol. 14 (2010), no. 4, 2077-2101.
[160] Popescu-Pampu, P. The geometry of continued fractions and the topology of surface singularities. Advanced Stud. in Pure Maths 46 (2007), 119-195.
[161] Popescu-Pampu, P. On the cohomology rings of holomorphically fillable manifolds. In Singularities II. Geometric and Topological Aspects. J. P. Brasselet et al. eds., Contemporary Mathematics 475, AMS, 2008, 169-188.
[162] Popescu-Pampu, P. Topologie de contact et singularités complexes. Mémoire d’habilitation, Univ. Paris 7, 2008. Available at http://math.univ-lille1.fr/ popescu/Habilit-PPP.pdf
[163] Popescu-Pampu, P. Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones. Duke Math. Journal 159, no. 3 (2011), 539-559.
[164] Popescu-Pampu, P. Introduction to Jung’s method of resolution of singularities. In Topology of Algebraic Varieties and Singularities. J. I. Cogolludo-Agustin and E. Hironaka eds. Contemporary Mathematics 538, AMS, 2011, 401-432.
[165] Popescu-Pampu, P. On the smoothings of non-normal isolated surface singularities. Journal of Singularities 12 (2015), 164-179.
[166] Popescu-Pampu, P. What is the genus? History of Mathematics subseries. Lecture Notes in Mathematics 2162, Springer, 2016.
[167] Prill, D. Local classification of quotients of complex manifolds by discontinuous groups. Duke Math. Journal 34 (1967) 375-386.
[168] Reid, M. Chapters on Algebraic Surfaces. In Complex Algebraic Geometry, 3-159. J. Kollár editor, American Mathematical Society, Providence, R.I., 1997.
[169] Riemenschneider, O. Deformationen von Quotientensingularitäten (nach Zyklischen Gruppen). Math. Ann. 209, 211-248 (1974).
[170] Riemenschneider, O. A note on the toric duality between the cyclic quotient surface singularities $A_{n,q}$ and $A_{n,n-q}$. In Singularities in geometry and topology, 161-179, IRMA Lect. Math. Theor. Phys. 20, Eur. Math. Soc., Zürich, 2012.
[171] Saito, K. Quasihomogene isolierte Singularit�ten von Hyperflächen. Invent. Math. 14 (1971), 123-142.
[172] Saito, K. Einfach-elliptische Singularitäten. Invent. Math. 23 (1974), 289-325.
[173] Samuel, P. La notion de multiplicité en algèbre et en géométrie algébrique. J. Math. Pures Appl. (9) 30 (1951). 159-205.
[174] Schlessinger, M. Functors of Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208-222.
[175] Schlessinger, M. Rigidity of quotient singularities. Invent. Math. 14 (1971), 17-26.
[176] Seade, J. On the topology of isolated singularities in analytic spaces. Progress in Mathematics 241. Birkhäuser Verlag, Basel, 2006.
[177] Slodowy, P. Groups and special singularities. In Singularity theory, 731-799, D.T. Lê, K. Saito, B. Teissier eds., World Scientific, 1995.
[178] Steenbrink, J. Mixed Hodge structures associated with isolated singularities. Proc. of Symposia in Pure Maths. 40 (1983), Part 2, 513-536.
[179] Stepanov, D. A. A remark on the dual complex of a resolution of singularities. Russian Math. Surveys 61 (2006), no. 1, 181-183.
[180] Stevens, J. On the versal deformation of cyclic quotient singularities. In Singularity theory and its applications, Part I (Coventry, 1988/1989), 302-319. Lecture Notes in Mathematics 1462, Springer, Berlin, 1991.
[181] Stevens, J. Partial resolutions of quotient singularities. Manuscripta Math. 79 (1993), no. 1, 7-11.
[182] Stevens, J. Deformations of singularities. Lecture Notes in Mathematics 1811, Springer, 2003.
[183] Stevens, J. On the classification of rational surface singularities. Journal of Singularities 7 (2013), 108-133.
[184] Stevens, J. The versal deformation of cyclic quotient singularities. In Deformations of surface singularities, 163-201, Bolyai Soc. Math. Stud. 23, János Bolyai Math. Soc., Budapest, 2013.
[185] Stipsicz, A. I., Szabó, Z., Wahl, J. Rational blowdowns and smoothings of surface singularities. J. Topol. 1 (2008), no. 2, 477-517.
[186] Sullivan, D. On the intersection ring of compact three manifolds. Topology 14 (1975), 275-277.
[187] Teissier, B. The hunting of invariants in the geometry of discriminants. In Real and complex singularities. (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), 565-678. Sijthoff and Noordhoff, 1977.
[188] Teissier, B. A bouquet of bouquets for a birthday. In Topological methods in modern mathematics (Stony Brook, NY, 1991), 93-122, Publish or Perish, Houston, TX, 1993.
[189] Thom, R. Les structures différentiables des boules et des sphères. Colloque Géom. Diff. Globale (Bruxelles, 1958), 27-35. Centre Belge Rech. Math., Louvain, 1959.
[190] Tyurina, G.N. Absolute isolatedness of rational singularities and triple rational points. Functional Analysis and its Applications 2 (1968), 324-332.
[191] Tyurina, G.N. Locally semiuniversal flat deformations of isolated singularities of complex spaces. Math. USSR-Izv., 33 (1969), no. 5, 967-999.
[192] Ustilovsky, I. Infinitely many contact structures on $\mathbb{S}^{4m+1}$. Internat. Math. Res. Notices 1999, no. 14, 781-791.
[193] Vakil, R. Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164 (2006), no. 3, 569-590.
[194] Du Val, P. On isolated singularities of surfaces which do not affect the conditions of adjunction. Part I, Proc. Cambridge Phil. Soc. 30 (1933-34), 483-491.
[195] Du Val, P. On absolute and non-absolute singularities of algebraic surfaces. Revue de la Faculté des Sciences de l’Univ. d’Istanbul (A) 91 (1944), 159-215.
[196] Varchenko, A.N. Contact structures and isolated singularities. Mosc. Univ. Math. Bull. 35 (1980), no.2, 18-22.
[197] Wagreich, P. Elliptic singularities of surfaces. Amer. J. of Math. 92 (2) (1970), 419-454.
[198] Wahl, J. Smoothings of normal surface singularities. Topology 20 (1981), no. 3, 219-246.
[199] Wahl, J. Milnor and Tjurina numbers for smoothings of surface singularities. Algebr. Geom. 2 (2015), no. 3, 315-331.
[200] Waldhausen, F. Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, II. Invent. Math. 3 (1967), 308-333 and 4 (1967), 87-117.
[201] Wall, C.T.C. Quadratic forms and normal surface singularities. In Quadratic forms and their applications. (Dublin 1999), 293-311. Contemp. Math. 272, American Mathematical Society, Providence, R.I., 2000.
[202] Wall, C.T.C. Singular points of plane curves. Cambridge Univ. Press, Cambridge, 2004.
[203] Whitney, H. The general type of singularity of a set of $2n- 1$ smooth functions of $n$ variables. Duke Math. J. 10, no. 1 (1943), 161-172.
[204] Whitney, H. The singularities of a smooth $n$-manifold in $(2n - 1)$-space. Ann. Math. 45, no. 2 (1944), 247-293.
[205] Winkelnkemper, H. E. Manifolds as open books. Bull. Amer. Math. Soc. 79 (1973), no. 1, 45-51.
[206] Zariski, O. The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. 76 no. 3 (1962), 560-615.
Copyright Cellule MathDoc 2018 | Crédit | Plan du site