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Toward motivic integration over wild Deligne-Mumford stacks

Takehiko Yasuda

Abstract

We discuss how the motivic integration will be generalized to wild Deligne-Mumford stacks, that is, stabilizers may have order divisible by the characteristic of the base or residue field. We pose several conjectures on this topic. We also present some possible applications concerning stringy invariants, resolution of singularities, and weighted counts of extensions of local fields.

Toward motivic integration over wild Deligne-Mumford stacks

Abstract

We discuss how the motivic integration will be generalized to wild Deligne-Mumford stacks, that is, stabilizers may have order divisible by the characteristic of the base or residue field. We pose several conjectures on this topic. We also present some possible applications concerning stringy invariants, resolution of singularities, and weighted counts of extensions of local fields.

Paper Structure

This paper contains 30 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Conventions
  4. The ring of values
  5. Motivic integration over varieties
  6. Varieties over $k$
  7. Varieties over $D$
  8. Motivic integration over Deligne-Mumford stacks
  9. Spaces of $G$-covers of a formal disk
  10. $G$-arcs and jets
  11. Twisted arcs and jets
  12. Formulating the change of variables formula for DM stacks
  13. Maps of twisted arc spaces
  14. Order functions associated to submodules
  15. The change of variables formula
  16. ...and 15 more sections

Key Result

Lemma 4.5

Let $E\in G\text{-}\mathrm{Cov}(D)$ and $G'\subset G$ the stabilizer of a connected component $E'$ of $E$, that is, $G':=\{g\in G\mid g(E')=E'\}.$ Let $M_{0}:=M\times_{D} D_0$. (Recall $D_0 =\mathrm{Spec}\, k$.) Then Here $N_{G}(G')$ is the normalizer of $G'$ in $G.$

Theorems & Definitions (61)

  • Conjecture 1.1: Conjecture \ref{['conj--main']}
  • Conjecture 4.1
  • Definition 4.2
  • Definition 4.3
  • Conjecture 4.4
  • Lemma 4.5
  • proof
  • Definition 4.6
  • Conjecture 4.7
  • Lemma 4.8
  • ...and 51 more