Stringy invariants and toric Artin stacks

• Published in: Forum of mathematics. Sigma Vol. 10
• Main Authors: Satriano, Matthew, Usatine, Jeremy
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.01.2022
• Subjects: Algebraic and Complex Geometry; Integrals; Isomorphism; Singularities; Stacks; 14D23; 14E18; 14M25
• ISSN: 2050-5094; 2050-5094
• DOI: 10.1017/fms.2022.3

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 (nonseparated) 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 change-of-variables formula relating the motivic measure for $\mathcal {X}$ to the Gorenstein measure for X and functions measuring the degree to which $\pi $ is nonseparated. We also conjecture that if the stabilisers 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 (assuming a finiteness hypothesis satisfied by fantastacks) 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.