Motivic integration for singular Artin stacks

TL;DR

This paper extends the motivic integration change of variables formula to singular Artin stacks, enabling new applications to warped stacks and broadening the scope of motivic integration techniques.

Contribution

It generalizes the change of variables formula for motivic integrals from smooth to singular Artin stacks, including applications to warped stacks.

Findings

Extended the change of variables formula to singular stacks

Applied the formula to warped stacks of Artin stacks

Provided a canonical expression for motivic integrals over arcs

Abstract

Let $\mathcal{X} \to Y$ be a birational modification of a variety by an Artin stack. In previous work, under the assumption that $\mathcal{X}$ is smooth, we proved a change of variables formula relating motivic integrals over arcs of $Y$ to motivic integrals over arcs of $\mathcal{X}$. In this paper, we extend that result to the case where $\mathcal{X}$ is singular. We may therefore apply this generalized formula to the so-called warping stack $\mathscr{W}(\mathcal{X})$ of $\mathcal{X}$, which may be singular even when $\mathcal{X}$ is smooth. We thus obtain a change of variables formula \emph{canonically} expressing any given motivic integral over arcs of $Y$ as a motivic integral over \emph{warped arcs} of $\mathcal{X}$.