Euler characteristic of crepant resolutions of specific modular quotient singularities
Linghu Fan
TL;DR
This work extends the McKay correspondence to positive characteristic by relating the Euler characteristic of crepant resolutions to representation-theoretic data for certain modular quotient singularities. It develops and employs the wild McKay mass-formula framework over finite fields, introducing $f_G$ and $F_G$ as generating series tied to $G$-étale algebras and Galois actions, and shows that for groups with a non-modular abelian normal subgroup of index $p$, the Euler characteristic equals the number of conjugacy classes, which also equals the number of indecomposable $kG$-modules. The main result is established separately in the abelian and non-abelian cases, with corollaries and two explicit examples; a conjectural modular generalization is proposed. The paper’s computations in characteristic $2$ for $A^4/C_2^2$ and $A^4/A_4$ provide concrete validations, matching known stringy motives and demonstrating the interplay between arithmetic, representation theory, and birational geometry in the modular setting.
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.