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Open problems in the wild McKay correspondence and related fields

Takehiko Yasuda

Abstract

The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and arithmetic on extensions of local fields. The principal purpose of this article is to collect open problems on the wild McKay correspondence, as well as those in related fields that the author believes are interesting or important. It also serves as a survey on the present state of these fields.

Open problems in the wild McKay correspondence and related fields

Abstract

The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and arithmetic on extensions of local fields. The principal purpose of this article is to collect open problems on the wild McKay correspondence, as well as those in related fields that the author believes are interesting or important. It also serves as a survey on the present state of these fields.

Paper Structure

This paper contains 17 sections, 33 equations.

Table of Contents

  1. Introduction
  2. Brief guide to the wild McKay correspondence and stringy motives
  3. Wild actions and wild quotients
  4. Stringy motives
  5. The wild McKay correspondence
  6. Log-terminal singularities
  7. Crepant resolutions
  8. Rationality
  9. Duality
  10. The $v$-function/Fröhlich's module resolvent
  11. The moduli space $\Delta_{G}$
  12. Non-linear actions
  13. Generalization in various directions
  14. Relation to other theories on wild ramification
  15. The derived wild McKay correspondence
  16. ...and 2 more sections

Theorems & Definitions (3)

  • Remark 1.1
  • Remark 3.5
  • Remark 5.2