La correspondance de McKay

TL;DR

This paper investigates the geometric and topological properties of quotient varieties formed by finite automorphism groups acting on Calabi-Yau manifolds, using diverse methods from algebraic geometry and string theory.

Contribution

It introduces multiple approaches to analyze the homology of resolutions of quotient varieties, linking algebraic cycles to group representations and conjugacy classes.

Findings

Constructed bases of homology corresponding to group representations

Developed methods from string theory and algebraic geometry for quotient analysis

Enhanced understanding of resolutions with trivial canonical class

Abstract

Let M be a quasiprojective algebraic manifold with K_M=0 and G a finite automorphism group of M acting trivially on the canonical class K_M; for example, a subgroup G of SL(n,C) acting on C^n in the obvious way. We aim to study the quotient variety X=M/G and its resolutions Y -> X (especially under the assumption that Y has K_Y=0) in terms of G-equivariant geometry of M. At present we know 4 or 5 quite different methods of doing this, taken from string theory, algebraic geometry, motives, moduli, derived categories, etc. For G in SL(n,C) with n=2 or 3, we obtain several methods of cobbling together a basis of the homology of Y consisting of algebraic cycles in one-to-one correspondence with the conjugacy classes or the irreducible representations of G.