A motivic change of variables formula for Artin stacks

TL;DR

This paper establishes a motivic change of variables formula for Artin stacks, linking integrals over arcs of a variety and a smooth Artin stack, with implications for stringy Hodge numbers and crepantness.

Contribution

It introduces a new motivic change of variables formula for Artin stacks and proposes a novel notion of crepantness extending the classical case.

Findings

Derived a change of variables formula for motivic integrals on Artin stacks.

Connected the formula to stringy Hodge numbers and crepantness.

Generalized crepantness to Artin stacks beyond schemes.

Abstract

Let $\mathcal{X} \to Y$ be a birational map from a smooth Artin stack to a (possibly singular) variety. We prove a change of variables formula that relates motivic integrals over arcs of $Y$ to motivic integrals over arcs of $\mathcal{X}$. With a view toward the study of stringy Hodge numbers, this change of variables formula leads to a new notion of crepantness for the map $\mathcal{X} \to Y$ that coincides with the usual notion in the special case that $\mathcal{X}$ is a scheme.