Asian Journal of Mathematics

Volume 19 (2015)

Number 4

Topological classification of simplest Gorenstein non-complete intersection singularities of dimension 2

Pages: 651 – 792

DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n4.a4

Authors

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Mingyi Zhang (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Huaiqing Zuo (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Abstract

Let $p$ be normal singularity of the 2-dimensional Stein space $V$. Let $\pi : M \to V$ be a minimal good resolution of $V$, such that the irreducible components $A_i$ of $A = \pi^{-1}(p)$ are nonsingular and have only normal crossings. Associated to $A$ is weighted dual graph $\Gamma$ which, along with the genera of the $A_i$, fully describes the topology and differentiable structure of $A$ and the topological and differentiable nature of the embedding of $A$ in $M$. It is well known that the simplest Gorenstein non-complete intersection singularities of dimension two are exactly those minimal elliptic singularities with fundamental cycle self intersection number $-5$. In this paper we classify all weighted dual graphs of these singularities. In particular, we prove that there is no integral homology link structure in the class of simplest Gorenstein non-complete intersection singularities of dimension two.

Keywords

normal singularities, topological classification, weighted dual graph

2010 Mathematics Subject Classification

Primary 32S25. Secondary 14B05, 58K65.

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Published 4 November 2015