McKay correspondence

TL;DR

This paper discusses the McKay correspondence, exploring its relation to mirror symmetry, providing numerical examples, and introducing Nakamura's results on G-clusters, with a focus on ongoing work and fun calculations.

Contribution

It offers a detailed exploration of the McKay correspondence, including numerical examples and connections to mirror symmetry, along with an introduction to Nakamura's work on G-clusters.

Findings

Numerical examples of McKay correspondences

Relation of McKay correspondence to mirror symmetry

Introduction to Nakamura's results on G-clusters

Abstract

This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how the McKay correspondence for finite subgroups of SL(n,C) relates to mirror symmetry. The main aim is to give numerical examples of how the 2 McKay correspondences (1) representations of G <--> cohomology of resolution (2) conjugacy classes of G <--> homology must work, and to restate my 1992 Conjecture as a tautology, like cohomology or K-theory of projective space. Another aim is to give an introduction to Nakamura's results on the Hilbert scheme of G-clusters, following his preprints and his many helpful explanations. This is partly based on joint work with Y. Ito, and has benefited from encouragement and invaluable suggestions of S. Mukai.