Stringy invariants and toric Artin stacks

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 X be a smooth Artin stack admitting a good moduli space π : X → X, and assume that X is a variety with log-terminal singularities, π induces an isomorphism over a nonempty open subset of X, and the exceptional locus of π has codimension at least 2. We conjecture a formula for the motivic measure for X in terms of the Gorenstein measure for X and a function measuring the degree to which π is non-separated. We also conjecture that if the stabilizers of X are special groups in the sense of Serre, then almost all arcs of X lift to arcs of 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 X . We prove these conjectures in the case where X is a fantastack. Contents 15 5. Fibers of the maps of jets 19 6. Gorenstein measure and toric varieties 26 7. Motivic measure and canonical stacks 29 8. Stringy invariants and toric Artin stacks: proof of Theorem 1.5 33 9. Fantastacks with special stabilizers: proof of Theorem 1.8 35 References 37necessarily satisfy the "strict valuative criterion", i.e., there may exist arcs of X that do not lift to arcs of X . Thus in general, we cannot use this conjecture to compute the total Gorenstein measure µ Gor X (L (X)), which specializes to the stringy Hodge numbers of X. This issue already occurs in the case where X is a Deligne-Mumford stack. For this reason, Yasuda uses a notion of "twisted arcs" of X instead of usual arcs of X , and this is why the inertia of X and orbifold Hodge numbers appear in Yasuda's setting. We take a different approach, emphasizing a setting in which the next conjecture predicts that almost all arcs of X lift to (finitely many) arcs of X .Conjecture 1.2. Let X be a finite type Artin stack over k admitting a good moduli space π : X → X. Assume X is an irreducible k-scheme and that π induces an isomorphism over a nonempty open subset of X. If the stabilizers of X are all special groups, then sep −1 π (0) ⊂ L (X) is measurable and µ X (sep −1 π (0)) = 0, where we note that µ X is the usual (non-Gorenstein) motivic measure on L (X).Remark 1.3. All special groups are connected, so if X is a Deligne-Mumford stack whose stabilizers are special groups, then the stabilizers of X are all trivial. Thus Conjecture 1.2 highlights a setting that is "orthogonal" to the setting considered by Yasuda.We now apply our conjectures to computing stringy Hodge numbers. In subsection 3.4, we introduce a function sep X : |L (X )| → Q ≥0 , which is essentially given by 1/(sep π •L (π)) except we are pedantic about dividing by 0. We think of its integral L (X ) sep X dµ X as a kind of motivic class of L (X ) corrected by sep X to acco...

A motivic change of variables formula for Artin stacks

“…However, the good moduli space map X → Y is a natural candidate for a nice resolution of singularities by a smooth Artin stack. In [SU20], the authors introduced a conjectural (proven in the toric case) framework that uses motivic integration for the stack X to study the stringy Hodge numbers of the singular variety Y . This framework suggested that when X → Y is birational and has exceptional locus with codimension at least 2, there should be a motivic change of variables formula making X behave like a crepant resolution of Y (see [SU20, Conjecture 1.1 and Theorem 1.7] for precise details).…”

“…We expect a better understanding of this condition will have applications to studying stringy Hodge numbers. As an example of such an application, we end this introduction by posing the following natural question, which explicitly ties Theorem 1.3 and Definition 1.6 back to the framework proposed in [SU20].…”