Geometry & Topology 9 (2005) 699–755

DOI: 10.2140/gt.2005.9.699

Abstract

It has long been known that every quasi-homogeneous normal complex surface singularity with 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 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 Q–homology sphere links. As quotients of complete intersections, they are necessarily Q–Gorenstein, and many Q–Gorenstein singularities with Q–homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with Q–homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture.

Keywords

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

Mathematical Subject Classification

Primary: 14B05, 32S50

Secondary: 57M25, 57N10

References

Publication

Received: 31 October 2004
Revised: 18 April 2005
Accepted: 6 March 2005
Published: 28 April 2005
Proposed: Robion Kirby
Seconded: Ronald Fintushel, Ronald Stern

Authors