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Wilder McKay correspondences

Takehiko Yasuda

Abstract

A conjectural generalization of the McKay correspondence in terms of stringy invariants to arbitrary characteristic, including the wild case, was recently formulated by the author in the case where the given finite group linearly acts on an affine space. In cases of very special groups and representations, the conjecture has been verified and related stringy invariants have been explicitly computed. In this paper, we try to generalize the conjecture and computations to more complicated situations such as non-linear actions on possibly singular spaces and non-permutation representations of non-abelian groups.

Wilder McKay correspondences

Abstract

A conjectural generalization of the McKay correspondence in terms of stringy invariants to arbitrary characteristic, including the wild case, was recently formulated by the author in the case where the given finite group linearly acts on an affine space. In cases of very special groups and representations, the conjecture has been verified and related stringy invariants have been explicitly computed. In this paper, we try to generalize the conjecture and computations to more complicated situations such as non-linear actions on possibly singular spaces and non-permutation representations of non-abelian groups.

Paper Structure

This paper contains 24 sections, 17 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Convention and notation
  4. Motivic integration and stringy motifs
  5. Centered log varieties
  6. Crepant morphisms
  7. Motivic integration
  8. Stringy invariants
  9. Group actions
  10. $G$-arcs
  11. The untwisting technique revisited
  12. The change of variables formula
  13. The McKay correspondence for linear actions
  14. The McKay correspondence for non-linear actions
  15. Computing boundaries of untwisting varieties
  16. ...and 9 more sections

Key Result

Proposition 1.4

Let $C=\bigsqcup_{j=1}^{l}C_{j}$ be the decomposition of $C$ into the connected components. Then with $[C_{j}:D]$ the degree of $C_{j}\to D$.

Theorems & Definitions (73)

  • Remark 1.1
  • Conjecture 1.2: The wild McKay correspondence conjecture Yasuda:2013fk
  • Conjecture 1.3: Conjecture \ref{['conj: non-linear McKay-1']}
  • Proposition 1.4: See Corollary \ref{['cor:weight non-permutation']} for a slightly more general result
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Conjecture 2.5
  • Proposition 2.6
  • ...and 63 more