Complutense University Library

Links and analytic invariants of superisolated singularities

Luengo Velasco, Ignacio and Melle Hernández, Alejandro and Némethi , A. (2005) Links and analytic invariants of superisolated singularities. Journal of algebraic geometry, 14 (3). pp. 543-565. ISSN 1056-3911

[img] PDF
Restricted to Repository staff only until 31 December 2020.

321kB

Official URL: http://www.ams.org/journals/jag/2005-14-03/S1056-3911-05-00397-8/

View download statistics for this eprint

==>>> Export to other formats

Abstract

Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show that the ``Seiberg-Witten invariant conjecture''(of Nicolaescu and the third author), the ``Universal abelian cover conjecture'' (of Neumann and Wahl) and the ``Geometric genus conjecture'' fail (at least at that generality in which they were formulated). Moreover, we also show that for Gorenstein singularities (even with integral homology sphere links) besides the geometric genus, the embedded dimension and the multiplicity (in particular, the Hilbert-Samuel function) also fail to be topological; and in general, the Artin cycle does not coincide with the maximal (ideal) cycle.


Item Type:Article
Additional Information:

Received March 29, 2004 and, in revised form, June 19, 2004. The first two authors are partially supported by BFM2001-1488-C02-01. The third author is partially supported by NSF grant DMS-0304759.

Uncontrolled Keywords:Superisolated singularities; Normal surface singularity; Seiberg-Witten invariants
Subjects:Sciences > Mathematics > Set theory
ID Code:17089
References:

Enrique Artal-Bartolo, Forme de Jordan de la monodromie des singularités superisolées de surfaces, Mem. Amer. Math. Soc. 109 (1994), no. 525, x+84 (French, with English summary).

E. Artal Bartolo, P. Cassou-Noguès, I. Luengo, and A. Melle Hernández, Monodromy conjecture for some surface singularities, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 4, 605–640 (English, with English and French summaries).

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

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

Olivier Collin and Nikolai Saveliev, A geometric proof of the Fintushel-Stern formula, Adv. Math. 147 (1999), no. 2, 304–314.

Olivier Collin, Equivariant Casson invariant for knots and the Neumann-Wahl formula, Osaka J. Math. 37 (2000), no. 1, 57–71.

Torsten Fenske, Rational 1- and 2-cuspidal plane curves, Beiträge Algebra Geom. 40 (1999), no. 2, 309–329.

Fernández de Bobadilla, J.; Luengo-Valesco, I.; Melle-Hernández, A. and Némethi, A.: On rational cuspidal projective plane curves, manuscript in preparation.

Ronald Fintushel and Ronald J. Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. (3) 61 (1990), no. 1, 109–137.

G. Fujita, A splicing formula for Casson-Walker’s invariant, Math. Ann. 296 (1993), no. 2, 327–338.

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

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

Henry B. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), no. 6, 1257–1295.

Henry B. Laufer, On μ for surface singularities, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), Amer. Math. Soc., Providence, R. I., 1977, pp. 45–49.

Christine Lescop, Global surgery formula for the Casson-Walker invariant, Annals of Mathematics Studies, vol. 140, Princeton University Press, Princeton, NJ, 1996.

Ignacio Luengo, The μ-constant stratum is not smooth, Invent. Math. 90 (1987), no. 1, 139–152.

A. Melle-Hernández, Milnor numbers for surface singularities, Israel J. Math. 115 (2000), 29–50.

Makoto Namba, Geometry of projective algebraic curves, Monographs and Textbooks in Pure and Applied Mathematics, vol. 88, Marcel Dekker Inc., New York, 1984.

A. Némethi, Five lectures on normal surface singularities, Low dimensional topology (Eger, 1996/Budapest, 1998) Bolyai Soc. Math. Stud., vol. 8, János Bolyai Math. Soc., Budapest, 1999, pp. 269–351. With the assistance of Ágnes Szilárd and Sándor Kovács.

Némethi, A.: Dedekind sums and the signature of f(x, y)+zN, Selecta Mathematica, New series 4 (1998), 361–376.

Némethi, A.: Dedekind sums and the signature of f(x, y) + zN, II., Selecta Mathematica, New series 5 (1999), 161–179.

András Némethi, “Weakly” elliptic Gorenstein singularities of surfaces, Invent. Math. 137 (1999), no. 1, 145–167.

Némethi, A.: On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds, arXiv:math.AG/0310083.

Némethi, A.: Line bundles associated with normal surface singularities, arXiv:math.AG/0310084.

András Némethi, Invariants of normal surface singularities, Real and complex singularities, Contemp. Math., vol. 354, Amer. Math. Soc., Providence, RI, 2004, pp. 161–208.

András Némethi and Liviu I. Nicolaescu, Seiberg-Witten invariants and surface singularities, Geom. Topol. 6 (2002), 269–328 (electronic).

András Némethi and Liviu I. Nicolaescu, Seiberg-Witten invariants and surface singularities. II. Singularities with good ℂ*-action, J. London Math. Soc. (2) 69 (2004), no. 3, 593–607.

Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities III (splicings and cyclic covers), arXiv:math.AG/0207018.

Walter D. Neumann, Abelian covers of quasihomogeneous surface singularities, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 233–243.

Walter Neumann and Jonathan Wahl, Casson invariant of links of singularities, Comment. Math. Helv. 65 (1990), no. 1, 58–78.

Walter D. Neumann and Jonathan Wahl, Universal abelian covers of surface singularities, Trends in singularities, Trends Math., Birkhäuser, Basel, 2002, pp. 181–190.

Walter D. Neumann and Jonathan Wahl, Universal abelian covers of quotient-cusps, Math. Ann. 326 (2003), no. 1, 75–93.

Neumann, W. and Wahl, J.: Complex surface singularities with integral homology sphere links, arXiv:math.AG/0301165.

Okuma, T.: Numerical Gorenstein elliptic singularities, preprint.

Pinkham, H.: Normal surface singularities with C∗-action, Math. Ann. 227 (1977),183–193

G.-M. Greuel, G. Pfister, and H. Schönemann. SINGULAR 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de.

Tomari, M.: A pg–formula and elliptic singularities, Publ. R.I.M.S. Kyoto University 21 (1985), 297–354.

Turaev, V.G.: Torsion invariants of Spinc-structures on 3-manifolds, Math. Res. Letters 4 (1997), 679–695.

Jonathan M. Wahl, Equisingular deformations of normal surface singularities. I, Ann. of Math. (2) 104 (1976), no. 2, 325–356.

Stephen Shing Toung Yau, On almost minimally elliptic singularities, Bull. Amer. Math. Soc. 83 (1977), no. 3, 362–364.

Stephen Shing Toung Yau, On strongly elliptic singularities, Amer. J. Math. 101 (1979), no. 4, 855–884.

Stephen Shing Toung Yau, On maximally elliptic singularities, Trans. Amer. Math. Soc. 257 (1980), no. 2, 269–329.

Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491.

Deposited On:13 Nov 2012 09:47
Last Modified:07 Feb 2014 09:41

Repository Staff Only: item control page