Complete intersection singularities of splice type as universal abelian covers

Abstract

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.