Universal abelian covers of certain surface singularities

Abstract

Every normal complex surface singularity with $\mathbb{Q}$-homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations''. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If $(X,o)$ is a rational or minimally elliptic singularity, then its universal abelian cover $(Y,o)$ is an equisingular deformation of an isolated complete intersection singularity $(Y_0,o)$ defined by a Neumann-Wahl system. Furthermore, if $G$ denotes the Galois group of the covering $Y\to X$, then $G$ also acts on $Y_0$ and $X$ is an equisingular deformation of the quotient $Y_0/G$.

18 pages

Cited by in corpus (5)

• Complete intersection singularities of splice type as universal abelian covers
• On the Casson Invariant Conjecture of Neumann--Wahl
• Ultrametric spaces of branches on arborescent singularities
• The multiplicity of abelian covers of splice quotient singularities
• Local tropicalizations of splice type surface singularities