Complete intersection singularities of splice type as universal abelian covers

Neumann, Walter D¹ ; Wahl, Jonathan²

¹ Department of Mathematics, Barnard College, Columbia University, New York, New York 10027, USA
² Department of Mathematics, The University of North Carolina, Chapel Hill, North Carolina 27599-3250, USA

Geometry & topology, Tome 9 (2005) no. 2, pp. 699-755.

Résumé

It has long been known that every quasi-homogeneous normal complex surface singularity with $\mathbb{Q}$–homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called “splice type singularities,” which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with $\mathbb{Q}$–homology sphere links, called “splice-quotient singularities.” According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with $\mathbb{Q}$–homology sphere links. As quotients of complete intersections, they are necessarily $\mathbb{Q}$–Gorenstein, and many $\mathbb{Q}$–Gorenstein singularities with $\mathbb{Q}$–homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with $\mathbb{Q}$–homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture.

Détail

DOI : 10.2140/gt.2005.9.699

Keywords: surface singularity, Gorenstein singularity, rational homology sphere, complete intersection singularity, abelian cover

Bibliographie

[1] D Eisenbud, W Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies 110, Princeton University Press (1985)

[2] R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer (1977)

[3] H B Laufer, On rational singularities, Amer. J. Math. 94 (1972) 597

[4] H B Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977) 1257

[5] I Luengo-Velasco, A Melle-Hernández, A Némethi, Links and analytic invariants of superisolated singularities, J. Algebraic Geom. 14 (2005) 543

[6] W D Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981) 299

[7] W D Neumann, Abelian covers of quasihomogeneous surface singularities, from: "Singularities, Part 2 (Arcata, Calif., 1981)", Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 233

[8] W D Neumann, G A Swarup, Canonical decompositions of 3–manifolds, Geom. Topol. 1 (1997) 21

[9] W D Neumann, J Wahl, Universal abelian covers of surface singularities, from: "Trends in singularities", Trends Math., Birkhäuser (2002) 181

[10] W D Neumann, J Wahl, Universal abelian covers of quotient-cusps, Math. Ann. 326 (2003) 75

[11] W D Neumann, J Wahl, Complex surface singularities with integral homology sphere links, Geom. Topol. 9 (2005) 757

[12] T Okuma, Universal abelian covers of rational surface singularities, J. London Math. Soc. $(2)$ 70 (2004) 307

[13] T Okuma, Universal abelian covers of certain surface singularities, Math. Ann. 334 (2006) 753

[14] M Reid, Chapters on algebraic surfaces, from: "Complex algebraic geometry (Park City, UT, 1993)", IAS/Park City Math. Ser. 3, Amer. Math. Soc. (1997) 3