A geometric definition of Gabrielov numbers

Abstract

Gabrielov numbers describe certain Coxeter–Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold’s strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $G$ of $\mathrm{SL}(3,{\mathbb {C}})$ using the Gabrielov numbers of the cusp singularity and data of the group $G$. Here we consider a crepant resolution $Y \rightarrow {\mathbb {C}}^3/G$ and the preimage $Z$ of the image of the Milnor fibre of the cusp singularity under the natural projection ${\mathbb {C}}^3 \rightarrow {\mathbb {C}}^3/G$. Using the McKay correspondence, we compute the homology of the pair $(Y,Z)$. We construct a basis of the relative homology group $H_3(Y,Z;{\mathbb {Q}})$ with a Coxeter–Dynkin diagram where one can read off the Gabrielov numbers.