Stringy invariants and toric Artin stacks

TL;DR

This paper introduces a conjectural framework for computing stringy invariants of singular varieties using motivic integration over arcs of smooth Artin stacks, and verifies it for a class of toric stacks called fantastacks.

Contribution

It proposes conjectures relating motivic measures of Artin stacks to stringy invariants and proves these conjectures for fantastacks, a specific class of toric Artin stacks.

Findings

Conjectured formulas for motivic measures and stringy Hodge numbers.

Verification of conjectures for fantastacks.

Connections between non-separated resolutions and motivic integration.

Abstract

We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin stacks, and we verify this framework in the case of fantastacks, which are certain toric Artin stacks that provide (non-separated) resolutions of singularities for toric varieties. Specifically, let $\mathcal{X}$ be a smooth Artin stack admitting a good moduli space $\pi: \mathcal{X} \to X$, and assume that $X$ is a variety with log-terminal singularities, $\pi$ induces an isomorphism over a nonempty open subset of $X$, and the exceptional locus of $\pi$ has codimension at least 2. We conjecture a formula for the motivic measure for $\mathcal{X}$ in terms of the Gorenstein measure for $X$ and a function measuring the degree to which $\pi$ is non-separated. We also conjecture that if the stabilizers of $\mathcal{X}$ are special groups in the sense of Serre, then almost all arcs of $X$ lift to arcs of $\mathcal{X}$, and we explain how in this case, our conjectures imply a formula for the stringy Hodge numbers of $X$ in terms of a certain motivic integral over the arcs of $\mathcal{X}$. We prove these conjectures in the case where $\mathcal{X}$ is a fantastack.