Complete intersection singularities of splice type as universal abelian covers

By Walter D. Neumann, J. Wahl. 2004

86 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1be9174dc660d4a749cbf9aa2078a7b36cc8c89d

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Citations in the corpus

On the Casson Invariant Conjecture of Neumann–Wahl (2006, unknown): “…type singularity whose link is Σ (via Neumann-Wahl system/construction), and the geometric genus of any such analytic structure is independent of the choice of the Neumann-Wahl system (by 4.3 of [21] and 10.1 of [19]), hence it depends only on the combinatorics of Γ — it will be denoted by p g (Γ).”
Lattice cohomology of normal surface singularities (2007, background): “[21]), they include all the rational, minimal elliptic singularities, and all singularities which admit a good C∗-action.”
Complex surface singularities with integral homology sphere links (2003, background): “The converse direction, that equations of splice type lead to abelian covers of Q–Gorenstein singularities with expected topological type, is the main content of [30] (in particular, as already mentioned, the equations of splice type of the current paper give singularities with the expected homology”
Rational blowdowns and smoothings of surface singularities (2006, methodology): “k that a given group action is free on Y −{o}. Describing the resolution graph of X is achieved by lifting the action of G to the weighted blow–up of Y , then dividing and resolving the quotient, cf. [24]. One needs to compute the Milnor number µY for Y , to show that it satisﬁes 1 + µY = |G|. But, ”