On the Link Space of a ℚ-Gorenstein Quasi-Homogeneous Surface Singularity

Abstract

In this paper we prove the following theorem: Let M be the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein surface singularity. Then M is diffeomorphic to a coset space $\tilde \Gamma _1 \backslash \tilde G/\tilde \Gamma _2$, where $\tilde G$ is the 3-dimensional Lie group $\widetilde{PSL}(2,\mathbb{R})$, while $\tilde \Gamma _1$ and $\tilde \Gamma _2$ are discrete subgroups of $\tilde G$, the subgroup $\tilde \Gamma _1$ is co-compact and $\tilde \Gamma _2$ is cyclic. Conversely, if M is diffeomorphic to a coset space as above, then M is diffeomorphic to the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein singularity. We also prove the following characterisation of quasi-homogeneous ℚ-Gorenstein surface singularities: A quasi-homogeneous surface singularity is ℚ-Gorenstein of index r if and only if for the corresponding automorphy factor (U, Γ, L) some tensor power of the complex line bundle L is Γ-equivariantly isomorphic to the rth tensor power of the tangent bundle of the Riemannian surface U.

Research partially supported by SFB 611 of the DFG.