A derived approach to geometric McKay correspondence in dimension three

TL;DR

This paper extends the geometric McKay correspondence to three dimensions, establishing a connection between group representations and geometric structures in the context of isolated singularities.

Contribution

It provides a detailed three-dimensional generalization of the McKay correspondence for abelian groups with isolated singularities, linking derived categories to geometric subschemes.

Findings

Derived category equivalence induces a correspondence between representations and subschemes.

The correspondence relates to Reid's recipe in the context of G-Hilb (C^3).

The work extends known two-dimensional results to three dimensions.

Abstract

We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail when G is abelian and C^3/G has a single isolated singularity. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G-Hilb (C^3). This correspondence appears to be related to Reid's recipe.