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The $p$-cyclic McKay correspondence via motivic integration

Takehiko Yasuda

Abstract

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ like in the tame case. Also, we link a crepant resolution with a count of Artin-Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

The $p$-cyclic McKay correspondence via motivic integration

Abstract

We study the McKay correspondence for representations of the cyclic group of order in characteristic . The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic like in the tame case. Also, we link a crepant resolution with a count of Artin-Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

Paper Structure

This paper contains 42 sections, 63 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Convention
  3. $G$-covers of the formal disk
  4. $G$-covers of the formal punctured disk
  5. The stratification by the ramification jump
  6. Moduli spaces of $G$-covers of $D^{*}$
  7. The stratified moduli space of $G$-covers of the formal disk
  8. Details of the $G$-actions on $\mathcal{O}_{E^{*}}$ and $\mathcal{O}_{E}$
  9. Twisted arcs and jets
  10. Ordinary arcs and jets of a scheme
  11. A $G$-representation
  12. $G$-arcs and jets.
  13. Twisted arcs and jets
  14. Push-forward maps for twisted arcs and jets
  15. Motivic integration
  16. ...and 27 more sections

Key Result

Theorem 1.1

If $D_{V}\ge p,$ then

Theorems & Definitions (162)

  • Theorem 1.1: Proposition \ref{['prop:explict M_st']} and Corollary \ref{['cor:McKay-non-pair']}
  • Lemma 2.1
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 152 more