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Motivic integration over wild Deligne-Mumford stacks

Takehiko Yasuda

Abstract

We develop the motivic integration theory over formal Deligne-Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne-Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava's mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms.

Motivic integration over wild Deligne-Mumford stacks

Abstract

We develop the motivic integration theory over formal Deligne-Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne-Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava's mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms.

Paper Structure

This paper contains 37 sections, 114 theorems, 331 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Twisted arcs
  4. The change of variables formula
  5. Stringy motives
  6. The wild McKay correspondence
  7. Untwisting by Hom stacks
  8. Related topics
  9. Terminology, notation and convention
  10. Acknowledgments
  11. Ind-DM stacks and formal DM stacks
  12. Galoisian group schemes
  13. Frobenius morphisms and ind-perfection
  14. Formal torsors
  15. Moduli stack $\Delta$
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\mathcal{Y},\mathcal{X}$ be DM stacks of finite type over $k$ and suppose that they have the same pure dimension and are generically smooth over $k$. Let $f\colon\mathcal{Y}\to\mathcal{X}$ be a proper birational morphism. Then, for a measurable function $h$ on $\mathcal{J}_{\infty}\mathcal{X}$,

Theorems & Definitions (297)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm:Mst-crepant-nonformal']}
  • Theorem 1.3: Corollaries \ref{['cor:wild-McKay-linear']} and \ref{['cor:wild-McKay-linear-1']}
  • Corollary 1.4: Corollary \ref{['cor:discrep']}
  • Corollary 1.5: Corollary \ref{['cor:motivic-Bhargava']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 287 more