Geometry & Topology, Vol. 9 (2005)
Paper no. 17, pages 699--755.
Complete intersection singularities of splice type as universal abelian covers
Walter D Neumann, Jonathan Wahl
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
AMS subject classification.
Primary: 32S50, 14B05.
Secondary: 57M25, 57N10.
DOI: 10.2140/gt.2005.9.699
E-print: arXiv:math.AG/0407287
Submitted to GT on 31 October 2004.
(Revised 18 April 2005.)
Paper accepted 6 March 2005.
Paper published 28 April 2005.
Notes on file formats
Walter D Neumann, Jonathan Wahl
Department of Mathematics, Barnard College, Columbia University
New York,
NY 10027, USA
and
Department of Mathematics, The University of
North Carolina
Chapel Hill, NC 27599-3250, USA
Email: neumann@math.columbia.edu, jmwahl@email.unc.edu
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