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The Wild McKay Correspondence for Cyclic Groups of Prime Power Order

Mahito Tanno, Takehiko Yasuda

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.

The Wild McKay Correspondence for Cyclic Groups of Prime Power Order

Abstract

The -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.

Paper Structure

This paper contains 24 sections, 42 theorems, 139 equations, 1 table.

Table of Contents

  1. Introduction
  2. G-covers of the formal punctured disk
  3. The Artin--Schreier--Witt theory
  4. Stratification and parameterization
  5. Explicit description of G-actions on G-covers
  6. v-functions
  7. The indecomposable case
  8. Ramification jumps
  9. Discrepancies of singularities
  10. Motivic integration
  11. Stringy motives
  12. The wild McKay correspondence
  13. Discrepancies of singularities
  14. The case G=Z/p2Z
  15. Some invariants
  16. ...and 9 more sections

Key Result

Theorem 1.1

Assume that $G$ acts on an affine space $\mathbb{A}_{k}^{d}$ linearly and effectively and that $G$ has no pseudo-reflection. Then we have Here $\mathnormal{M}_{\text{st}}(X)$ denotes the stringy motive of the quotient variety $X$, $\operatorname{\mathnormal{G}-Cov}(D)$ denotes the moduli space of $G$-covers of $D \coloneq \mathop{\mathrm{Spec}}\nolimits k[[t]]$, and $\bm{v}$ is the $\bm{v}$-funct

Theorems & Definitions (78)

  • Theorem 1.1: Yasuda2019:Motivic
  • Theorem 1.2: \ref{['thm:v-function']}
  • Theorem 1.3: \ref{['thm:cond-converge']}
  • Corollary 1.4: \ref{['prop:cond-klt']}
  • Theorem 1.5: \ref{['thm:estimate-discrep']}
  • Proposition 2.1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 68 more