Euler characteristic of crepant resolutions of specific modular quotient singularities

TL;DR

This paper extends the McKay correspondence to positive characteristic, proving it for certain groups and proposing a conjecture for the general modular case, with applications to Euler characteristics of crepant resolutions.

Contribution

It generalizes the McKay correspondence to positive characteristic for specific group structures and introduces a conjectural framework for the modular case.

Findings

Proved the generalized McKay correspondence for groups with a semidirect product structure.

Applied wild McKay correspondence over finite fields as mass formulas.

Provided additional examples with complex group structures.

Abstract

In this paper, we consider a generalization of the McKay correspondence in positive characteristic regarding the Euler characteristic of crepant resolutions of quotient singularities given by finite subgroups of the special linear group. As the main result, we prove that this generalization holds for groups with a specific semidirect product structure, using the wild McKay correspondence over finite fields as mass formulas. Furthermore, two additional examples with more complicated structures are also given. Based on our main result, we propose a conjectural form of the generalized McKay correspondence in the modular case.