The Wild McKay Correspondence for Cyclic Groups of Prime Power Order

TL;DR

This paper provides a formula for the v-function in the wild McKay correspondence for cyclic groups of prime power order, and applies it to analyze singularities and stringy motives of quotient varieties.

Contribution

It introduces a valuation-based formula for the v-function and offers criteria for singularity types and convergence of stringy motives in this context.

Findings

Derived a valuation-based formula for the v-function.

Established criteria for singularity classifications of quotient varieties.

Determined conditions for convergence of stringy motives.

Abstract

The $\boldsymbol{v}$-function is a key ingredient in the wild McKay correspondence. In this paper, we give a formula to compute it in terms of valuations of Witt vectors, when the given group is a cyclic group of prime power order. We apply it to study singularities of a quotient variety by a cyclic group of prime square order. We give a criterion whether the stringy motive of the quotient variety converges or not. Furthermore, if the given representation is indecomposable, then we also give a simple criterion for the quotient variety being terminal, canonical, log canonical, and not log canonical.