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Stringy invariants and toric Artin stacks

Published online by Cambridge University Press:  07 February 2022

Matthew Satriano
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada; E-mail: msatriano@uwaterloo.ca
Jeremy Usatine
Affiliation:
Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI, 02912, USA; E-mail: jeremy_usatine@brown.edu

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 (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.

Type
Algebraic and Complex Geometry
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

Let X be a variety with log-terminal singularities. Motivated by mirror symmetry for singular Calabi–Yau varieties, Batyrev introduced stringy Hodge numbers for X in [Reference BatyrevBat98], which are defined in terms of a resolution of singularities. In particular, if X admits a crepant resolution $Y \to X$ by a smooth projective variety Y, then the stringy Hodge numbers of X are equal to the usual Hodge numbers of Y. In [Reference Denef and LoeserDL02], Denef and Loeser defined the Gorenstein measure $\mu ^{\mathrm {Gor}}_X$ on the arc scheme $\mathscr {L}(X)$ of X and used it to prove a McKay correspondence that refines the McKay correspondence conjectured by Reid in [Reference ReidRei92] and proved by Batyrev in [Reference BatyrevBat99]. The measure $\mu ^{\mathrm {Gor}}_X$ takes values in a modified Grothendieck ring of varieties $\widehat {\mathscr {M}}_k\left [\mathbb {L}^{1/m}\right ]$ and is a refinement of the stringy Hodge numbers of X. If X admits a crepant resolution $Y \to X$ , then $\mu ^{\mathrm {Gor}}_X$ is essentially equivalent to the usual motivic measure $\mu _Y$ on $\mathscr {L}(Y)$ as introduced by Kontsevich in [Reference KontsevichKon95].

A major open question asks whether or not the stringy Hodge numbers of projective varieties are nonnegative, as conjectured by Batyrev in [Reference BatyrevBat98, Conjecture 3.10]. A stronger conjecture predicts that stringy Hodge numbers of projective varieties are equal to the dimensions of some kind of cohomology groups. In [Reference YasudaYas04], these conjectures were proved in the case where X has quotient singularites. Yasuda showed that in that case, if $\mathcal {X}$ is the canonical smooth Deligne–Mumford stack over X, then the stringy Hodge numbers of X are equal to the orbifold Hodge numbers of $\mathcal {X}$ in the sense of Chen and Ruan [Reference Chen and RuanCR04]. To prove this result, Yasuda introduced a notion of motivic integration (further developed in [Reference YasudaYas06Reference YasudaYas19]) for Deligne–Mumford stacks and proved a formula expressing $\mu ^{\mathrm {Gor}}_X$ in terms of certain motivic integrals over arcs of $\mathcal {X}$ . When X is projective, those integrals over arcs of $\mathcal {X}$ compute the orbifold Hodge numbers of $\mathcal {X}$ .

In this paper, we initiate a similar program for varieties with singularities that are worse than quotient singularities. Such varieties never arise as the coarse space of a smooth Deligne–Mumford stack, so one is instead forced to consider Artin stacks. A major technicality is that such stacks are not separated. This leads us to define new functions $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}$ , discussed later, which measure the degree to which an Artin stack is not separated. These functions play a key role in our theory.

The class of varieties we consider are those X occurring as the good moduli space (in the sense of [Reference AlperAlp13]) of a smooth Artin stack $\mathcal {X}$ ; varieties of this form arise naturally in the context of geometric invariant theory. We require that the map $\pi \colon \mathcal {X} \to X$ induce an isomorphism over a nonempty open subset of X and that the exceptional locus of $\pi $ have codimension at least $2$ . In other words, we want $\mathcal {X}$ to be a ‘small’ resolution of X. We conjecture a relationship between $\mu ^{\mathrm {Gor}}_X$ and a motivic measure $\mu _{\mathcal {X}}$ on the arc stack $\mathscr {L}(\mathcal {X})$ of $\mathcal {X}$ . This relationship involves integrating $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}: \mathscr {L}(X) \to \mathbb {N}$ against $\mu ^{\mathrm {Gor}}_X$ . This function $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}$ counts the number of arcs of $\mathcal {X}$ (in some auxiliary measurable subset $\mathcal {C}$ ), up to isomorphism, above each arc of X, and can therefore be thought of as an invariant which measures the nonseparatedness of $\pi $ . We emphasise that this conjectural relationship is not ‘built into’ our definition of $\mu _{\mathcal {X}}$ . In fact, our notion of $\mu _{\mathcal {X}}$ is straightforward: it is more or less Kontsevich’s original motivic measure, except that various notions for schemes are replaced with the obvious analogues for Artin stacks. When the stabilisers of $\mathcal {X}$ are special groups in the sense of SerreFootnote 1 and $\mathscr {L}(\pi ): \mathscr {L}(\mathcal {X}) \to \mathscr {L}(X)$ has finite fibres outside a set of measure $0$ , our conjectures imply a formula expressing the stringy Hodge numbers of X in terms of a certain motivic integral over $\mathscr {L}(\mathcal {X})$ .

We prove that our conjectures hold when X is a toric variety and $\mathcal {X}$ is a fantastack – that is, a type of smooth toric Artin stack in the sense of [Reference Geraschenko and SatrianoGS15aReference Geraschenko and SatrianoGS15b]. Fantastacks are a broad class of toric stacks that allow one to simultaneously have any specified toric variety X as a good moduli space while also obtaining stabilisers with arbitrarily large dimension. An important special case of fantastacks (and their products with algebraic tori) is the so-called canonical stack $\mathcal {X}$ over a toric variety X. When X has quotient singularities, $\mathcal {X}$ is the canonical smooth Deligne–Mumford stack over X; when X has worse singularities, the good moduli space of $\mathcal {X}$ is still X, but $\mathcal {X}$ is an Artin stack that is not Deligne–Mumford.

1.1 Conventions

Throughout this paper, k will be an algebraically closed field with characteristic $0$ . All Artin stacks will be assumed to have affine (geometric) stabilisers, and all toric varieties will be assumed to be normal. For any stack $\mathcal {X}$ over k, we will let $\lvert \mathcal {X}\rvert $ denote the topological space associated to $\mathcal {X}$ , and for any k-algebra R, we will let $\overline {\mathcal {X}}(R)$ denote the set of isomorphism classes of the category $\mathcal {X}(R)$ .

1.2 Conjectures

Our first conjecture predicts a relationship between $\mu ^{\mathrm {Gor}}_X$ and $\mu _{\mathcal {X}}$ . As already mentioned, our formula involves integrals weighted by functions $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}$ that measure the degree to which $\pi $ is not separated. We refer the reader to section 3 for precise definitions of the arc stack $\mathscr {L}(\mathcal {X})$ and its motivic measure $\mu _{\mathcal {X}}$ , and to subsection 3.4 for the definition of $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}$ and its integral $\int _C \operatorname {\mathrm {sep}}_{\pi , \mathcal {C}} \mathrm {d}\mu ^{\mathrm {Gor}}_X$ .

Conjecture 1.1. Let $\mathcal {X}$ be a smooth irreducible Artin stack over k admitting a good moduli space $\pi : \mathcal {X} \to X$ , where X is a separated k-scheme and has log-terminal singularities. Assume that $\pi $ induces an isomorphism over a nonempty open subset of X, and that the exceptional locus of $\pi $ has codimension at least $2$ .

If $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ is a measurable subset such that $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}: \mathscr {L}(X) \to \mathbb {N} \cup \{\infty \}$ is finite outside a set of measure $0$ , then $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}: \mathscr {L}(X) \to \mathbb {N} \cup \{\infty \}$ has measurable fibres, and for any measurable subset $C \subset \mathscr {L}(X)$ , the set $\mathcal {C} \cap \mathscr {L}(\pi )^{-1}(C) \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ is measurable and satisfies

$$ \begin{align*} \mu_{\mathcal{X}}\left(\mathcal{C} \cap \mathscr{L}(\pi)^{-1}(C)\right) = \int_C \operatorname{\mathrm{sep}}_{\pi, \mathcal{C}} \mathrm{d}\mu^{\mathrm{Gor}}_X \in \widehat{\mathscr{M}}_k\left[\mathbb{L}^{1/m}\right], \end{align*} $$

where $m \in \mathbb {Z}_{>0}$ is such that $mK_X$ is Cartier.

This conjecture predicts that for the purpose of computing $\mu ^{\mathrm {Gor}}_X$ , the stack $\mathcal {X}$ behaves like a crepant resolution of X, except we need to correct by $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}$ to account for the fact that $\mathcal {X}$ is not separated over X. Set

$$ \begin{align*} \operatorname{\mathrm{sep}}_\pi = \operatorname{\mathrm{sep}}_{\pi, \lvert\mathscr{L}(\mathcal{X})\rvert}. \end{align*} $$

Notice that Conjecture 1.1 implies, in particular, that the motivic measure $\mu _{\mathcal {X}}$ ‘does not see’ how $\mu ^{\mathrm {Gor}}_X$ behaves on the set $\operatorname {\mathrm {sep}}_\pi ^{-1}(0) \subset \mathscr {L}(X)$ . This set can have nonzero measure because $\pi : \mathcal {X} \to X$ does not necessarily satisfy the ‘strict valuative criterion’ – that is, there may exist arcs of X (even outside a set of measure $0$ ) that do not lift to arcs of $\mathcal {X}$ . Thus in general we cannot use this conjecture to compute the total Gorenstein measure $\mu ^{\mathrm {Gor}}_X(\mathscr {L}(X))$ , which specialises to the stringy Hodge numbers of X. This issue already occurs in the case where $\mathcal {X}$ is a Deligne–Mumford stack. For this reason, Yasuda uses a notion of ‘twisted arcs’ of $\mathcal {X}$ instead of usual arcs of $\mathcal {X}$ , and this is why the inertia of $\mathcal {X}$ and orbifold Hodge numbers appear in Yasuda’s setting. We take a different approach, emphasising a setting in which the next conjecture predicts that almost all arcs of X lift to arcs of $\mathcal {X}$ .

Conjecture 1.2. Let $\mathcal {X}$ be a finite-type Artin stack over k admitting a good moduli space $\pi : \mathcal {X} \to X$ . Assume that X is an irreducible k-scheme and that $\pi $ induces an isomorphism over a nonempty open subset of X. If the stabilisers of $\mathcal {X}$ are all special groups, then $\operatorname {\mathrm {sep}}_\pi ^{-1}(0) \subset \mathscr {L}(X)$ is measurable and

$$ \begin{align*} \mu_X\left( \operatorname{\mathrm{sep}}_\pi^{-1}(0) \right) = 0, \end{align*} $$

where we note that $\mu _X$ is the usual (non-Gorenstein) motivic measure on $\mathscr {L}(X)$ .

Remark 1.3. All special groups are connected, so if $\mathcal {X}$ is a Deligne–Mumford stack whose stabilisers are special groups, then its stabilisers are all trivial. Thus Conjecture 1.2 highlights a setting that is ‘orthogonal’ to the setting considered by Yasuda.

Our next question is motivated by the fact that if $\operatorname {\mathrm {sep}}_\pi $ is finite outside a set of measure $0$ , we may then consider the special case of Conjecture 1.1 where $\mathcal {C} = \lvert \mathscr {L}(\mathcal {X})\rvert $ .

Question 1.4. Let $\mathcal {X}$ be a finite-type Artin stack over k admitting a good moduli space $\pi : \mathcal {X} \to X$ . Assume that X is an irreducible k-scheme and that $\pi $ induces an isomorphism over a nonempty open subset of X. When is

$$ \begin{align*} \mu_X\left(\operatorname{\mathrm{sep}}_\pi^{-1}(\infty)\right) = 0 \end{align*} $$

satisfied?

We now give an application of this framework to computing stringy Hodge numbers. In subsection 3.4, we introduce the function $\operatorname {\mathrm {sep}}_{\mathcal {X}} = 1/(\operatorname {\mathrm {sep}}_\pi \circ \mathscr {L}(\pi )): \lvert \mathscr {L}(\mathcal {X})\rvert \to \mathbb {Q}_{\geq 0} \cup \{\infty \}$ . We think of its integral $\int _{\mathscr {L}(\mathcal {X})} \operatorname {\mathrm {sep}}_{\mathcal {X}} \mathrm {d}\mu _{\mathcal {X}}$ as a kind of motivic class of $\mathscr {L}(\mathcal {X})$ corrected by $\operatorname {\mathrm {sep}}_{\mathcal {X}}$ to account for the fact that $\mathcal {X}$ is not separated. We refer the reader to subsection 3.4 for the precise definitions of $\int _{\mathscr {L}(\mathcal {X})} \operatorname {\mathrm {sep}}_{\mathcal {X}} \mathrm {d}\mu _{\mathcal {X}}$ and the ring $\widehat {\mathscr {M}_k \otimes _{\mathbb {Z}} \mathbb {Q}}$ . The next proposition is then immediate:

Proposition 1.5. With hypotheses as in Conjecture 1.1, if the stablisers of $\mathcal {X}$ are special groups and $\mu _X\left (\operatorname {\mathrm {sep}}_\pi ^{-1}(\infty )\right ) = 0$ , then Conjecture 1.1 and Conjecture 1.2 imply that the fibres of $\operatorname {\mathrm {sep}}_{\mathcal {X}}: \lvert \mathscr {L}(\mathcal {X})\rvert \to \mathbb {Q}_{\geq 0}$ are measurable and

$$ \begin{align*} \mu^{\mathrm{Gor}}_X(\mathscr{L}(X)) = \int_{\mathscr{L}(\mathcal{X})} \operatorname{\mathrm{sep}}_{\mathcal{X}} \mathrm{d}\mu_{\mathcal{X}} \in \widehat{\mathscr{M}_k \otimes_{\mathbb{Z}} \mathbb{Q}}. \end{align*} $$

Since the stringy Hodge–Deligne invariant of X is a specialisation of the image of $\mu ^{\mathrm {Gor}}_X(\mathscr {L}(X))$ in $\left (\widehat {\mathscr {M}_k \otimes _{\mathbb {Z}} \mathbb {Q}}\right )\left [\mathbb {L}^{1/m}\right ] \supset \widehat {\mathscr {M}_k \otimes _{\mathbb {Z}} \mathbb {Q}}$ , Proposition 1.5 provides a conjectural formula for the stringy Hodge numbers of X (when the stringy Hodge numbers exist – that is, when the stringy Hodge–Deligne invariant is a polynomial).

We envision a few potential applications of this framework. Noting that the good moduli space map $\pi : \mathcal {X} \to X$ is intrinsic to the stack $\mathcal {X}$ and therefore so is the integral $\int _{\mathscr {L}(\mathcal {X})} \operatorname {\mathrm {sep}}_{\mathcal {X}} \mathrm {d}\mu _{\mathcal {X}}$ , we hope that a cohomological interpretation of $\int _{\mathscr {L}(\mathcal {X})} \operatorname {\mathrm {sep}}_{\mathcal {X}} \mathrm {d}\mu _{\mathcal {X}}$ will lead to progress on Batyrev’s conjecture on the nonnegativity of stringy Hodge numbers. We also hope that by considering Proposition 1.5 as a kind of McKay correspondence, our conjectures will lead to new representation-theoretic statements for positive-dimensional algebraic groups.

Remark 1.6. The hypothesis $\mu _X\left (\operatorname {\mathrm {sep}}_\pi ^{-1}(\infty )\right ) = 0$ in Proposition 1.5 allows us to make a canonical choice for $\mathcal {C}$ in Conjecture 1.1, specifically the choice $\mathcal {C} = \lvert \mathscr {L}(\mathcal {X})\rvert $ . We hope that even when this hypothesis does not hold, one can still (after an appropriate generalisation of the notion of an arc) make a canonical choice for $\mathcal {C}$ . This is a subject of our ongoing research.

1.3 Main results

Our first main result is that Conjecture 1.1 holds, and $\mu _X\left (\operatorname {\mathrm {sep}}_\pi ^{-1}(\infty )\right ) = 0$ , when $\mathcal {X}$ is a fantastack and $\mathcal {C} = \lvert \mathscr {L}(\mathcal {X})\rvert $ . In particular, our framework applies to the Gorenstein measure of any toric variety X with log-terminal singularities.

Theorem 1.7. Conjecture 1.1 holds and $\mu _X\left (\operatorname {\mathrm {sep}}_\pi ^{-1}(\infty )\right ) = 0$ when $\mathcal {X}$ is a fantastack and $\mathcal {C} = \lvert \mathscr {L}(\mathcal {X})\rvert $ .

Remark 1.8. In fact, our techniques prove a more general result: the conclusions of Conjecture 1.1 hold when $\mathcal {C} = \lvert \mathscr {L}(\mathcal {X})\rvert $ and $\mathcal {X}$ is a fantastack satisfying a certain combinatorial condition analogous to $\mathcal {X} \to X$ being ‘crepant’ (see Remark 2.20 for more details). It is important to note here that unlike the case of Deligne–Mumford stacks, defining $K_{\mathcal {X}}$ for Artin stacks is a subtle issue, so there is no a priori obvious definition one can take for $\mathcal {X}\to X$ to be crepant.

Remark 1.9. We note that the stacks $\mathcal {X}$ in Theorem 1.7 all have commutative stabilisers. In order to provide evidence that Conjecture 1.1 should not be limited to the setting of commutative stabilisers, we also verify that it holds in examples that involve $\mathrm {SL}_2$ as a stabiliser (see section 10). These examples also demonstrate the flexibility in choosing the auxiliary set $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ .

Remark 1.10. Theorem 1.7 can be thought of as a motivic change-of-variables formula. We note that Balwe introduced versions of motivic integration for Artin n-stacks [Reference BalweBal08Reference BalweBal15] and proved a change-of-variables formula [Reference BalweBal08, Theorem 7.2.5]. However, Theorem 1.7 cannot be obtained from Balwe’s result, as the map $\pi : \mathcal {X} \to X$ does not satisfy Balwe’s hypotheses: specifically, $\pi $ is not ‘ $0$ -truncated’.

The three main steps of proving Theorem 1.7 are as follows. First, we give a combinatorial description of the fibres of the map $\mathscr {L}(\pi ): \mathscr {L}(\mathcal {X}) \to \mathscr {L}(X)$ . Second, we show that for sufficiently large n, the map of jets $\mathscr {L}_n(\pi ): \mathscr {L}_n(\mathcal {X}) \to \mathscr {L}_n(X)$ has constant fibres (after taking the fibres’ reduced structure) over certain combinatorially defined pieces of $\mathscr {L}_n(X)$ . These two steps allow us to reduce Theorem 1.7 to the final step: verifying the case where the measurable sets C are certain combinatorially defined subsets of $\mathscr {L}(X)$ . A key ingredient in this final step is Theorem 3.9 and its corollary, Corollary 3.16, which show how to compute the motivic measure of the stack quotient of a variety by the action of a special group.

Our second main result is that Conjecture 1.2 holds for fantastacks.

Theorem 1.11. Conjecture 1.2 holds when $\mathcal {X}$ is a fantastack.

An essential ingredient in proving Theorem 1.11 is Theorem 9.1, which may be of independent interest, as it provides a combinatorial criterion to check whether or not the stabilisers of a fantastack are special groups.

2 Preliminaries

In this section, we introduce notation and recall some facts about motivic integration for schemes and the Gorenstein measure, the Grothendieck ring of stacks and constructible subsets and toric Artin stacks.

2.1 Motivic integration for schemes

If X is a k-scheme, for each $n \in \mathbb {N}$ we will let $\mathscr {L}_n(X)$ denote the nth jet scheme of X; for each $n \geq m$ we will let $\theta ^n_m: \mathscr {L}_n(X) \to \mathscr {L}_m(X)$ denote the truncation morphism; we will let $\mathscr {L}(X) = \varprojlim _n\mathscr {L}_n(X)$ denote the arc scheme of X; and for each $n \in \mathbb {N}$ we will let $\theta _n: \mathscr {L}(X) \to \mathscr {L}_n(X)$ denote the canonical morphism, which is also referred to as a truncation morphism. For any k-algebra R and k-scheme X, the map is bijective by [Reference BhattBha16, Theorem 1.1], and we will often implicitly make this identification.

We will let $K_0(\mathbf {Var}_k)$ denote the Grothendieck ring of finite type k-schemes; for each finite-type k-scheme X we will let $\mathrm {e}(X) \in K_0(\mathbf {Var}_k)$ denote its class; we will let $\mathbb {L} = \mathrm {e}\left (\mathbb {A}_k^1\right ) \in K_0(\mathbf {Var}_k)$ denote the class of the affine line; and for each constructible subset C of a finite-type k-scheme we will let $\mathrm {e}(C) \in K_0(\mathbf {Var}_k)$ denote its class.

We will let $\mathscr {M}_k$ denote the ring obtained by inverting $\mathbb {L}$ in $K_0(\mathbf {Var}_k)$ . For each $\Theta \in \mathscr {M}_k$ , let $\dim (\Theta ) \in \mathbb {Z} \cup \{-\infty \}$ denote the infimum over all $d \in \mathbb {Z}$ such that $\Theta $ is in the subgroup of $\mathscr {M}_k$ generated by elements of the form $\mathrm {e}(X)\mathbb {L}^{-n}$ with $\dim (X) - n \leq d$ , and let $\lVert \Theta \rVert = \exp (\dim (\Theta ))$ . We will let $\widehat {\mathscr {M}}_k$ denote the separated completion of $\mathscr {M}_k$ with respect to the non-Archimedean seminorm $\lVert \cdot \rVert $ , and we will also let $\lVert \cdot \rVert $ denote the non-Archimedean norm on $\widehat {\mathscr {M}}_k$ . For any $m \in \mathbb {Z}_{>0}$ , we will let $\widehat {\mathscr {M}}_k\left [\mathbb {L}^{1/m}\right ] = \widehat {\mathscr {M}}_k[t]/(t^m-\mathbb {L})$ , we will let $\mathbb {L}^{1/m}$ denote the image of t in $\widehat {\mathscr {M}}_k\left [\mathbb {L}^{1/m}\right ]$ and we will endow $\widehat {\mathscr {M}}_k\left [\mathbb {L}^{1/m}\right ]$ with the topology induced by the equality

$$ \begin{align*} \widehat{\mathscr{M}}_k\left[\mathbb{L}^{1/m}\right] = \bigoplus_{\ell = 0}^{m-1} \widehat{\mathscr{M}}_k \cdot \left(\mathbb{L}^{1/m}\right)^\ell, \end{align*} $$

where each summand $\widehat {\mathscr {M}}_k \cdot \left (\mathbb {L}^{1/m}\right )^\ell $ has the topology induced by the bijection

$$ \begin{align*} \widehat{\mathscr{M}}_k \to \widehat{\mathscr{M}}_k \cdot \left(\mathbb{L}^{1/m}\right)^\ell: \Theta \mapsto \Theta \cdot \left(\mathbb{L}^{1/m}\right)^\ell. \end{align*} $$

We note that in the foregoing and throughout this paper, if $\Theta $ is an element of $K_0(\mathbf {Var}_k)$ , $\mathscr {M}_k$ or $\widehat {\mathscr {M}}_k$ , we slightly abuse notation by also using $\Theta $ to refer to its image under any of the ring maps $K_0(\mathbf {Var}_k) \to \mathscr {M}_k \to \widehat {\mathscr {M}}_k \to \widehat {\mathscr {M}}_k\left [\mathbb {L}^{1/m}\right ]$ .

If X is an equidimensional finite-type k-scheme and $C \subset \mathscr {L}(X)$ is a cylinder – that is, $C =(\theta _n)^{-1}(C_n)$ for some $n \in \mathbb {N}$ and some constructible subset $C_n \subset \mathscr {L}_n(X)$ – we will let $\mu _X(C) \in \widehat {\mathscr {M}}_k$ denote the motivic measure of C, so by definition

$$ \begin{align*} \mu_X(C) = \lim_{n \to \infty} \mathrm{e}(\theta_n(C))\mathbb{L}^{-(n+1)\dim X} \in \widehat{\mathscr{M}}_k, \end{align*} $$

where we note that each $\theta _n(C)$ is constructible (for example, by [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 5, Corollary 1.5.7(b)]) and this limit exists (for example, by [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Theorem 2.5.1]). The motivic measure $\mu _X$ can be extended to the class of so-called measurable subsets of $\mathscr {L}(X)$ , whose definition we now recall.

Definition 2.1. Let X be an equidimensional finite-type scheme over k, set $C \subset \mathscr {L}(X)$ and $\varepsilon \in \mathbb {R}_{>0}$ , let I be a set, let $C^{(0)} \subset \mathscr {L}(X)$ be a cylinder and let $\left \{C^{(i)}\right \}_{i \in I}$ be a collection of cylinders in $\mathscr {L}(X)$ .

The data $\left (C^{(0)}, \left (C^{i}\right )_{i \in I}\right )$ is called a cylindrical $\varepsilon $ -approximation of C if

$$ \begin{align*} \left(C \cup C^{(0)}\right) \setminus \left(C \cap C^{(0)}\right) \subset \bigcup_{i \in I} C^{(i)} \end{align*} $$

and, for all $i \in I$ ,

$$ \begin{align*} \left\lVert \mu_X\left(C^{(i)}\right) \right\rVert < \varepsilon. \end{align*} $$

Definition 2.2. Let X be an equidimensional finite-type scheme over k, and set $C \subset \mathscr {L}(X)$ . The set C is called measurable if for any $\varepsilon \in \mathbb {R}_{>0}$ there exists a cylindrical $\varepsilon $ -approximation of C. The motivic measure of a measurable subset $C \subset \mathscr {L}(X)$ is defined to be the unique element $\mu _X(C) \in \widehat {\mathscr {M}}_k$ such that for any $\varepsilon \in \mathbb {R}_{>0}$ and any cylindrical $\varepsilon $ -approximation $\left (C^{(0)}, \left (C^{(i)}\right )_{i \in I}\right )$ of C, we have

$$ \begin{align*} \left\lVert \mu_X(C) - \mu_X\left(C^{(0)}\right) \right\rVert < \varepsilon. \end{align*} $$

Such an element $\mu _X(C)$ exists by [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Theorem 3.3.2].

For the remainder of this subsection, let X be an integral finite-type separated k-scheme with log-terminal singularities. We will set notation relevant for the Gorenstein measure associated to X. We will let $K_X$ denote the canonical divisor on X. If $m \in \mathbb {Z}_{>0}$ is such that $mK_X$ is Cartier, we will let $\omega _{X,m} = \iota _*\left ( \left (\Omega _{X_{\mathrm {sm}}}^{\dim X}\right )^{\otimes m} \right )$ , where $\iota : X_{\mathrm {sm}} \hookrightarrow X$ is the inclusion of the smooth locus of X, and we will let $\mathscr {J}_{X,m}$ denote the unique ideal sheaf on X such that the image of $\left (\Omega _X^{\dim X}\right )^{\otimes m} \to \omega _{X,m}$ is equal to $\mathscr {J}_{X,m} \omega _{X,m}$ . If $C \subset \mathscr {L}(X)$ is measurable, we will let $\mu ^{\mathrm {Gor}}_X(C)$ denote the Gorenstein measure of C, so by definition,

$$ \begin{align*} \mu^{\mathrm{Gor}}_X(C) &= \int_C \left(\mathbb{L}^{1/m}\right)^{\operatorname{\mathrm{ord}}_{\mathscr{J}_{X,m}}} \mathrm{d} \mu_X \\ &= \sum_{n = 0}^\infty \left(\mathbb{L}^{1/m}\right)^n \mu_X\left(\operatorname{\mathrm{ord}}_{\mathscr{J}_{X,m}}^{-1}(n) \cap C\right) \in \widehat{\mathscr{M}}_k\left[\mathbb{L}^{1/m}\right], \end{align*} $$

where $m \in \mathbb {Z}_{>0}$ is such that $mK_X$ is Cartier and $\operatorname {\mathrm {ord}}_{\mathscr {J}_{X,m}}: \mathscr {L}(X) \to \mathbb {N} \cup \{\infty \}$ is the order function of the ideal sheaf $\mathscr {J}_{X,m}$ . The following proposition is easy to check using the definition of $\mu ^{\mathrm {Gor}}_X$ and standard properties of $\mu _X$ given in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 3.4.3]:

Proposition 2.3. Let $\left \{C^{(i)}\right \}_{i \in \mathbb {N}}$ be a sequence of pairwise disjoint measurable subsets of $\mathscr {L}(X)$ such that $C = \bigcup _{i = 0}^\infty C^{(i)}$ is measurable. Then

$$ \begin{align*} \lim_{i \to \infty} \mu^{\mathrm{Gor}}_X\left(C^{(i)}\right) = 0, \end{align*} $$

and

$$ \begin{align*} \mu^{\mathrm{Gor}}_X(C) = \sum_{i =0}^\infty \mu^{\mathrm{Gor}}_X\left(C^{(i)}\right). \end{align*} $$

2.2 The Grothendieck ring of stacks and constructible subsets

We will let $K_0(\mathbf {Stack}_k)$ denote the Grothendieck ring of stacks in the sense of [Reference EkedahlEke09], and for each finite-type Artin stack $\mathcal {X}$ over k, we will let $\mathrm {e}(\mathcal {X}) \in K_0(\mathbf {Stack}_k)$ denote the class of $\mathcal {X}$ . If $K_0(\mathbf {Var}_k)\left [\mathbb {L}^{-1}, \left \{(\mathbb {L}^n - 1)^{-1}\right \}_{n \in \mathbb {Z}_{>0}}\right ]$ is the ring obtained from $K_0(\mathbf {Var}_k)$ by inverting $\mathbb {L}$ and $(\mathbb {L}^n-1)$ for all $n \in \mathbb {Z}_{>0}$ , then the obvious ring map $K_0(\mathbf {Var}_k) \to K_0(\mathbf {Stack}_k)$ induces an isomorphism

$$ \begin{align*} K_0(\mathbf{Var}_k)\left[\mathbb{L}^{-1}, \left\{(\mathbb{L}^n - 1)^{-1}\right\}_{n \in \mathbb{Z}_{>0}}\right] \cong K_0(\mathbf{Stack}_k), \end{align*} $$

by [Reference EkedahlEke09, Theorem 1.2]. Therefore there exists a unique ring map

$$ \begin{align*} K_0(\mathbf{Stack}_k) \to \widehat{\mathscr{M}}_k, \end{align*} $$

whose composition with $K_0(\mathbf {Var}_k) \to K_0(\mathbf {Stack}_k)$ is the usual map $K_0(\mathbf {Var}_k) \to \widehat {\mathscr {M}}_k$ . If $\Theta \in K_0 (\mathbf {Stack}_k)$ , we will slightly abuse notation by also using $\Theta $ to refer to its image under $K_0(\mathbf {Stack}_k) \to \widehat {\mathscr {M}}_k$ . By [Reference EkedahlEke09, Propositions 1.1(iii) and 1.4(i)], if G is a special group over k, then $\mathrm {e}(G) \in K_0(\mathbf {Stack}_k)$ is a unit, and for any finite-type k-scheme X with G-action, the class of the stack quotient is

$$ \begin{align*} \mathrm{e}([X/G]) = \mathrm{e}(X)\mathrm{e}(G)^{-1} \in K_0(\mathbf{Stack}_k). \end{align*} $$

Remark 2.4. Let G be an algebraic group over k. For each $n \in \mathbb {N}$ , we give $\mathscr {L}_n(G)$ the group structure induced by applying the functor $\mathscr {L}_n$ to the group law $G \times _k G \to G$ . It is easy to verify that for each $n \in \mathbb {N}$ , we have a short exact sequence

$$ \begin{align*} 1 \to \mathfrak{g} \to \mathscr{L}_{n+1}(G) \xrightarrow{\theta^{n+1}_n} \mathscr{L}_n(G) \to 1, \end{align*} $$

where $\mathfrak {g}$ is the Lie algebra of G. Thus by induction on n, the fact that $\mathbb {G}_a$ is special, the fact that extensions of special groups are special and the fact that $\mathscr {L}_0(G) \cong G$ , we see that if G is a special group, then each jet scheme $\mathscr {L}_n(G)$ is a special group.

To state the next result, we recall that if $\mathcal {X}$ is a finite-type Artin stack over k, then the topological space $\lvert \mathcal {X}\rvert $ is Noetherian, so its constructible subsets are precisely those subsets that can be written as a finite union of locally closed subsets.

Proposition 2.5. Let $\mathcal {X}$ be a finite-type Artin stack over k and let $\mathcal {C} \subset \lvert \mathcal {X}\rvert $ be a constructible subset. Then there exists a unique $\mathrm {e}(\mathcal {C}) \in K_0(\mathbf {Stack}_k)$ that satisfies the following property. If $\{\mathcal {X}_i\}_{i \in I}$ is a finite collection of locally closed substacks $\mathcal {X}_i$ of $\mathcal {X}$ such that $\mathcal {C}$ is equal to the disjoint union of the $\lvert \mathcal {X}_i\rvert $ , then

$$ \begin{align*} \mathrm{e}(\mathcal{C}) = \sum_{i \in I}\mathrm{e}(\mathcal{X}_i) \in K_0(\mathbf{Stack}_k). \end{align*} $$

Proof. The proposition holds by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 2, Corollary 1.3.5].

If $\mathcal {X}$ is a finite-type Artin stack and $\mathcal {C} \subset \lvert \mathcal {X}\rvert $ is a constructible subset, we will let $\mathrm {e}(\mathcal {C})$ denote the class of $\mathcal {C}$ – that is, $\mathrm {e}(\mathcal {C})$ is as in the statement of Proposition 2.5.

We end this subsection with a useful tool to compute the class of a stack.

Definition 2.6. Let S be a scheme, let Z be scheme over S, let $\mathcal {Y}$ and $\mathcal {F}$ be Artin stacks over S and let $\xi : \mathcal {Y} \to Z$ be a morphism over S. We say $\xi $ is a piecewise trivial fibration with fibre $\mathcal {F}$ if there exists a finite cover $\{Z_i\}_{i \in I}$ of Z consisting of pairwise disjoint locally closed subschemes $Z_i \subset Z$ such that for all $i \in I$ ,

$$ \begin{align*} (\mathcal{Y} \times_Z Z_i)_{\mathrm{red}} \cong (\mathcal{F} \times_S Z_i)_{\mathrm{red}} \end{align*} $$

as stacks over $(Z_i)_{\mathrm {red}}$ .

Remark 2.7. Let Z be a finite-type scheme over k, let $\mathcal {Y}$ and $\mathcal {F}$ be finite-type Artin stacks over k and let $\xi : \mathcal {Y} \to Z$ be a piecewise trivial fibration with fibre $\mathcal {F}$ . Then by Proposition 2.5,

$$ \begin{align*} \mathrm{e}(\mathcal{Y}) = \mathrm{e}(\mathcal{F})\mathrm{e}(Z) \in K_0(\mathbf{Stack}_k). \end{align*} $$

The next proposition is well known in the case where $\mathcal {Y}$ is a scheme.

Proposition 2.8. Let S be a Noetherian scheme, let Z be a finite-type scheme over S, let $\mathcal {Y}$ and $\mathcal {F}$ be finite-type Artin stacks over S and let $\xi : \mathcal {Y} \to Z$ be a morphism over S. Then $\xi $ is a piecewise trivial fibration with fibre $\mathcal {F}$ if and only if for all $z \in Z$ , there exists an isomorphism

$$ \begin{align*} (\mathcal{Y} \times_Z \operatorname{\mathrm{Spec}}(k(z)))_{\mathrm{red}} \cong (\mathcal{F} \times_S \operatorname{\mathrm{Spec}}(k(z)))_{\mathrm{red}} \end{align*} $$

of stacks over $k(z)$ , where $k(z)$ denotes the residue field of z.

Proof. If $\xi $ is a piecewise trivial fibration with fibre $\mathcal {F}$ , then for every $z\in Z$ , there is a locally closed subset $Z'\subseteq Z$ containing z for which $(\mathcal {Y}\times _Z Z')_{\mathrm {red}}\cong (\mathcal {F}\times _S Z')_{\mathrm {red}}$ as $Z^{\prime }_{\mathrm {red}}$ -stacks. Then

$$ \begin{align*} \begin{split} (\mathcal{Y}\times_Z \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}} &=\left((\mathcal{Y}\times_Z Z')_{\mathrm{red}}\times_{Z^{\prime}_{\mathrm{red}}} \operatorname{\mathrm{Spec}} k(z)\right)_{\mathrm{red}}\\ &\cong\left((\mathcal{F}\times_S Z')_{\mathrm{red}}\times_{Z^{\prime}_{\mathrm{red}}} \operatorname{\mathrm{Spec}} k(z)\right)_{\mathrm{red}}=(\mathcal{F}\times_S \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}}. \end{split} \end{align*} $$

We now show that the converse holds. Since

$$ \begin{align*} (\mathcal{Y}_{\mathrm{red}}\times_Z \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}}=(\mathcal{Y}\times_Z \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}} \end{align*} $$

for every $z\in Z$ , we can assume $\mathcal {Y}$ is reduced. By Noetherian induction on Z, we need only find a nonempty open subset $U\subseteq Z$ for which $(\mathcal {Y}\times _Z U)_{\mathrm {red}}\cong (\mathcal {F}\times _S U)_{\mathrm {red}}$ . Let $z\in Z$ be the generic point of an irreducible component of Z; replacing Z by an open affine neighbourhood of z, we may further assume Z is affine. Since $\mathcal {O}_{Z,z}$ is a field, $\mathcal {Y}\times _Z \operatorname {\mathrm {Spec}} k(z)$ is reduced and we hence have a surjective closed immersion

$$ \begin{align*} \iota\colon\mathcal{Y}\times_Z \operatorname{\mathrm{Spec}} k(z)\cong(\mathcal{F}\times_S \operatorname{\mathrm{Spec}} k(z))_{\mathrm{red}}\to\mathcal{F}\times_S \operatorname{\mathrm{Spec}} k(z). \end{align*} $$

Now, $\operatorname {\mathrm {Spec}}\mathcal {O}_{Z,z}=\lim _\lambda U_\lambda $ is the inverse limit of open affine neighbourhoods $U_\lambda \subseteq Z$ of z. Since Z is affine, each map $U_\lambda \to Z$ is affine. Note also that $\mathcal {Y}$ is Noetherian, hence quasicompact and quasiseparated, and that $\mathcal {F}\times _S \operatorname {\mathrm {Spec}} k(z)\to \operatorname {\mathrm {Spec}} k(z)$ is locally of finite presentation. [Reference RydhRyd15, Proposition B.2] then shows there is some index $\lambda $ and a morphism $\iota _\lambda \colon \mathcal {Y}\times _Z U_\lambda \to \mathcal {F}\times _S U_\lambda $ whose base change to $\operatorname {\mathrm {Spec}}\mathcal {O}_{Z,z}$ is $\iota $ . Furthermore, since $\mathcal {F}\times _S \operatorname {\mathrm {Spec}} k(z)\to \operatorname {\mathrm {Spec}} k(z)$ and $\xi $ are both of finite presentation, [Reference RydhRyd15, Proposition B.3] shows that after replacing $\lambda $ by a larger index if necessary, we can assume $\iota _\lambda $ is a surjective closed immersion, and hence defines an isomorphism $(\mathcal {Y}\times _Z U_\lambda )_{\mathrm {red}}\cong (\mathcal {F}\times _S U_\lambda )_{\mathrm {red}}$ .

2.3 Toric Artin stacks

In this subsection, we briefly review the theory of toric stacks introduced in [Reference Geraschenko and SatrianoGS15a], as well as establish some notation. Since the focus in our paper is on the toric variety X, and the toric stack $\mathcal {X}$ is viewed as a stacky resolution of X, we introduce some notational changes to emphasise this focus.

Definition 2.9. A stacky fan is a pair $\left (\widetilde {\Sigma },\nu \right )$ , where $\widetilde {\Sigma }$ is a fan on a lattice $\widetilde {N}$ and $\nu \colon \widetilde {N}\to N$ is a homomorphism to a lattice N so that the cokernel $\operatorname {\mathrm {cok}}\nu $ is finite.

A stacky fan $\left (\widetilde {\Sigma },\nu \right )$ gives rise to a toric stack as follows. Let $X_{\widetilde \Sigma }$ be the toric variety associated to $\widetilde {\Sigma }$ . Since $\operatorname {\mathrm {cok}}\nu $ is finite, $\nu ^*$ is injective, so we obtain a surjective homomorphism of tori

$$ \begin{align*} \widetilde{T}:=\operatorname{\mathrm{Spec}} k\left[\widetilde{N}^*\right]\longrightarrow\operatorname{\mathrm{Spec}} k[N^*]=:T. \end{align*} $$

Let $G_\nu $ denote the kernel of this map. Since $\widetilde {T}$ is the torus of $X_{\widetilde \Sigma }$ , we obtain a $G_\nu $ -action on $X_{\widetilde \Sigma }$ via the inclusion $G_\nu \subset \widetilde {T}$ .

Definition 2.10. With notation as in the previous paragraph, if $\left (\widetilde {\Sigma },\nu \right )$ is a stacky fan, the associated toric stack is defined to be

$$ \begin{align*} \mathcal{X}_{\widetilde{\Sigma},\nu}:=\left[X_{\widetilde{\Sigma}}/G_\nu\right]. \end{align*} $$

When $\widetilde {\Sigma }$ is the fan generated by the faces of a single cone $\widetilde {\sigma }$ , we denote $\mathcal {X}_{\widetilde {\Sigma },\nu }$ by $\mathcal {X}_{\widetilde {\sigma },\nu }$ .

Example 2.11. If $\Sigma $ is a fan on a lattice N and we let $\nu $ be the identity map, then $\mathcal {X}_{\Sigma ,\nu }=X_\Sigma $ . Thus, every toric variety is an example of a toric stack.

In this paper, we concentrate in particular on fantastacks introduced in [Reference Geraschenko and SatrianoGS15a, Section 4]. These play a particularly important role for us because they allow us to start with a toric variety $X_\Sigma $ and produce a smooth stack $\mathcal {X}$ with arbitrary degree of stackyness while maintaining the property that X is the good moduli space of $\mathcal {X}$ . In the following, we let $e_1,\dotsc ,e_r$ be the standard basis for $\mathbb {Z}^r$ .

Definition 2.12. Let $\Sigma $ be a fan on a lattice N, and let $\nu \colon \mathbb {Z}^r\to N$ be a homomorphism with finite cokernel so that every ray of $\Sigma $ contains some $v_i:=\nu (e_i)$ and every $v_i$ lies in the support of $\Sigma $ . For a cone $\sigma \in \Sigma $ , let $\widetilde \sigma =\mathrm {cone}(\{e_i\mid v_i\in \sigma \})$ . We define the fan $\widetilde \Sigma $ on $\mathbb {Z}^r$ as the fan generated by all the $\widetilde \sigma $ . We define

$$ \begin{align*} \mathcal{F}_{\Sigma,\nu} := \mathcal{X}_{\widetilde\Sigma,\nu}. \end{align*} $$

Any toric stack isomorphic to some $\mathcal {F}_{\Sigma ,\nu }$ is called a fantastack. When $\Sigma $ is the fan generated by the faces of a cone $\sigma $ , we denote $\mathcal {F}_{\Sigma ,\nu }$ by $\mathcal {F}_{\sigma ,\nu }$ .

Remark 2.13. By [Reference Geraschenko and SatrianoGS15a, Example 6.24] (compare [Reference SatrianoSat13, Theorem 5.5]), the natural map

$$ \begin{align*} \mathcal{F}_{\Sigma,\nu}\longrightarrow X_\Sigma \end{align*} $$

is a good moduli space morphism. Furthermore, fantastacks have moduli interpretations in terms of line bundles and sections, analogous to the moduli interpretation for $\mathbb {P}^n$ [Reference Geraschenko and SatrianoGS15a, Section 7].

The next two results will be useful later on.

Proposition 2.14. Let $\sigma $ be a pointed full-dimensional cone and suppose that the good moduli space map $\pi \colon \mathcal {F}_{\sigma ,\nu }\to X_\sigma $ is an isomorphism over the torus T of $X_\sigma $ . Then for any $f \in F:=\widetilde {\sigma }^\vee \cap \widetilde {N}^*$ , there exists some $f' \in F$ such that

$$ \begin{align*} f + f' \in P:=\sigma^\vee\cap N^*. \end{align*} $$

In particular, if $\psi \colon F\to \mathbb {N}\cup \{\infty \}$ is a morphism of monoids and $\psi (P)\subset \mathbb {N}$ , then $\psi (F)\subset \mathbb {N}$ .

Proof. Let $v_i=\beta (e_i)$ for $1\leq i\leq r$ . Since $\pi $ is an isomorphism over T, each $v_i\neq 0$ . As $\sigma $ is pointed, there exists some $p \in P$ such that $\langle v_i, p \rangle> 0$ for all i. Viewing p as an element of F via the inclusion $P\subset F$ , we have $\langle e_i, p \rangle> 0$ .

Let $f_1, \dotsc , f_r$ be the basis of $\widetilde {M}$ dual to $e_1, \dotsc , e_r$ . Since the $f_i$ are generators of F, it suffices to prove the proposition for each $f_i$ . Note that

$$ \begin{align*} \langle e_1, p \rangle f_1 + \dotsb + \langle e_r, p \rangle f_r = p \in P. \end{align*} $$

Since $\langle e_i, p \rangle> 0$ , we see that

$$ \begin{align*} f^{\prime}_i:=(\langle e_i,p\rangle-1)f_i+\sum_{j\neq i}\left\langle e_j,p\right\rangle f_j\in F \end{align*} $$

and that $f_i+f^{\prime }_i\in P$ .

Proposition 2.15. Keep the notation and hypotheses of Proposition 2.14 and let $\beta \colon \widetilde \sigma \cap \widetilde {N}\to \sigma \cap N$ be the induced map. If $w \in \sigma \cap N$ , then $\beta ^{-1}(w)$ is a finite set.

Proof. Let $f_1, \dotsc , f_r$ be the minimal generators of the monoid F. By Proposition 2.14, there exist $f^{\prime }_1, \dotsc , f^{\prime }_r$ such that $f_i + f^{\prime }_i \in P$ for all $i \in \{1, \dotsc , r\}$ . For any $\widetilde {w} \in \beta ^{-1}(w)$ ,

$$ \begin{align*} \left\langle \widetilde{w}, f_i \right\rangle \leq \left\langle \widetilde{w}, f_i + f^{\prime}_i \right\rangle = \left\langle w, f_i+f^{\prime}_i\right\rangle, \end{align*} $$

so there are only finitely many possible values for each $\left \langle \widetilde {w}, f_i \right \rangle $ . Thus $\beta ^{-1}(w)$ is a finite set.

We end this section by discussing canonical stacks as defined in [Reference Geraschenko and SatrianoGS15a, Section 5].

Definition 2.16. If $\Sigma $ is a fan on a lattice N, let $v_1,\dotsc ,v_r\in N$ be the first lattice points on the rays of $\Sigma $ , let $\nu \colon \mathbb {Z}^r\to N$ be the map $\nu (e_i):=v_i$ and let $\widetilde \Sigma $ be as in Definition 2.12. If $N'$ is a direct complement of the support of $\Sigma $ and $\nu '\colon \mathbb {Z}^r\oplus N'\to N$ is given by $\nu '(v,n')=\nu (v)+n'$ , then $\mathcal {X}_{\widetilde \Sigma ,\nu '}$ is the canonical stack of $X_\Sigma $ .

Remark 2.17. With notation as in Definition 2.16, if the support of $\Sigma $ is N, the canonical stack of $X_\Sigma $ is the fantastack $\mathcal {F}_{\widetilde {\Sigma },\nu }$ .

The next proposition, which is straightforward from the definition, says that canonical stacks are compatible with open immersions. This will be useful for us, as this proposition will allow us to reduce most of our work to the case of affine toric varieties defined by a d-dimensional cone in $N_{\mathbb {R}}$ .

Proposition 2.18. Let $\Sigma $ be a fan consisting of pointed rational cones in $N_{\mathbb {R}}$ , let $\sigma $ be a cone in $\Sigma $ , let $X(\Sigma )$ and $X(\sigma )$ be the T-toric varieties associated to $\Sigma $ and $\sigma $ , respectively, and let $\iota : X(\sigma ) \hookrightarrow X(\Sigma )$ be the open inclusion. If $\mathcal {X}(\Sigma )$ and $\mathcal {X}(\sigma )$ are the canonical stacks over $X(\Sigma )$ and $X(\sigma )$ , respectively, and $\pi (\Sigma ): \mathcal {X}(\Sigma ) \to X(\Sigma )$ and $\pi (\sigma ): \mathcal {X}(\sigma ) \to X(\sigma )$ are the canonical maps, then there exists a map $\mathcal {X}(\sigma ) \to \mathcal {X}(\Sigma )$ such that

is a fibre product diagram.

For the remainder of this subsection, let $\sigma $ be a d-dimensional pointed rational cone in $N_{\mathbb {R}}$ , let X be the affine T-toric variety associated to $\sigma $ , let $\mathcal {X}$ be the canonical stack over X and let $\pi : \mathcal {X} \to X$ be the canonical map. At points later in this paper, we will refer to the following list of notations when we want to set it, and we also set it for the remainder of this subsection:

Notation 2.19.

  • Let $M = N^*$ .

  • Let $\widetilde {N}$ be the free abelian group with generators indexed by the rays of $\sigma $ .

  • Let $\widetilde {M} = \widetilde {N}^*$ .

  • Let $\langle \cdot , \cdot \rangle $ denote both pairings $N \otimes _{\mathbb {Z}} M \to \mathbb {Z}$ and $\widetilde {N} \otimes _{\mathbb {Z}} \widetilde {M} \to \mathbb {Z}$ .

  • Let $\widetilde {T} = \operatorname {\mathrm {Spec}}\left (k\left [\widetilde {M}\right ]\right )$ be the algebraic torus with cocharacter lattice $\widetilde {N}$ .

  • Let $\widetilde {\sigma }$ be the positive orthant of $\widetilde {N}_{\mathbb {R}}$ – that is, $\widetilde {\sigma }$ is the positive span of those generators of $\widetilde {N}$ that are indexed by the rays of $\sigma $ .

  • Let $\widetilde {X}$ be the affine $\widetilde {T}$ -toric variety associated to $\widetilde {\sigma }$ .

  • Let $\beta : \widetilde {\sigma } \cap \widetilde {N} \to \sigma \cap N$ be the monoid map taking the generator of $\widetilde {N}$ indexed by a ray of $\sigma $ to the first lattice point of that ray.

  • Let $\widetilde {\pi }: \widetilde {X} \to X$ be the toric map associated to $\beta ^{\mathrm {gp}}: \widetilde {N} \to N$ .

  • Let $P = \sigma ^\vee \cap M$ . Note that $X = \operatorname {\mathrm {Spec}}(k[P])$ .

  • Let $F = \widetilde {\sigma }^\vee \cap \widetilde {M}$ . Note that $\widetilde {X} = \operatorname {\mathrm {Spec}}(k[F])$ .

  • Identify P with its image under the injection $P \hookrightarrow F$ given by dualising $\beta $ . Note that $P \hookrightarrow F$ is injective because $\sigma $ is full-dimensional.

  • Let $A = F^{\mathrm {gp}}/P^{\mathrm {gp}} = \widetilde {M}/ M$ .

  • Let $G = \operatorname {\mathrm {Spec}}(k[A])$ be the kernel of the algebraic group homomorphism $\widetilde {T} \to T$ obtained by restricting $\widetilde {\pi }$ , and let G act on $\widetilde {X}$ by restricting the toric action of $\widetilde {T}$ on $\widetilde {X}$ .

By definition, the canonical stack $\mathcal {X}$ is equal to the stack quotient $\left [\widetilde {X}/G\right ]$ and the morphism $\widetilde {\pi }: \widetilde {X} \to X$ is the composition $\widetilde {X} \to \left [\widetilde {X} / G\right ] = \mathcal {X} \xrightarrow {\pi } X$ .

We note that because our focus is on singular varieties instead of on stacks, we simplify our exposition by focusing on canonical stacks over toric varieties instead of all fantastacks. The expositional advantage is that canonical stacks depend only on the toric variety and not on additional data, as is the case for other fantastacks. We end this section with two remarks which explain why we have not lost any generality by making this expositional simplification and discuss a generalisation of Theorem 1.7.

Remark 2.20. For Theorem 1.7, it is sufficient to consider canonical stacks, as these are precisely the fantastacks satisfying the hypotheses of Conjecture 1.1. Nonetheless, we note that with only superficial modifications to our techniques, one can actually prove a more general statement than Theorem 1.7, which we explain here.

With notation as in Definition 2.16, let $\mathcal {X}=\mathcal {F}_{\Sigma ,\nu }$ be a fantastack. Assume that $X=X_\Sigma $ is $\mathbb {Q}$ -Gorenstein, so for each maximal cone $\sigma \in \Sigma $ , there exist $q_\sigma \in N^*$ and $m_\sigma \in \mathbb {Z}_{>0}$ such that the set

$$ \begin{align*} \mathcal{H}_\sigma:=\{v\in \sigma\cap N\mid\langle q_\sigma,v\rangle=m_\sigma\} \end{align*} $$

contains the first lattice point of every ray of $\sigma $ . We say the good moduli space map $\pi \colon \mathcal {X}\to X$ is combinatorially crepant if $\nu (e_i)\in \bigcup _\sigma \mathcal {H}_\sigma $ for every $i\in \{1,\dotsc ,r\}$ .

For example, the canonical stack is combinatorially crepant over X. Since Lemma 7.9 holds for all fantastacks that are combinatorially crepant over their good moduli space, the conclusions of Conjecture 1.1 hold for any fantastack that is combinatorially crepant over its good moduli space.

Remark 2.21. If $\mathcal {F}_{\Sigma , \nu }$ is a fantastack over X, then $\mathcal {F}_{\Sigma , \nu } \to X$ is an isomorphism over a nonempty open subset of X if and only if $\nu $ does not send any standard basis vector to $0$ . Since Proposition 2.14 holds for every fantastack satisfying the hypotheses of Conjecture 1.2, our proofs show that Theorem 1.11 holds for any fantastack as well.

3 Motivic integration for stacks

For the remainder of this paper, by a quotient stack over k we will mean an Artin stack over k that is isomorphic to the stack quotient of a k-scheme by the action of a linear algebraic group over k.

Remark 3.1. Let G be a linear algebraic group over k acting on a k-scheme $\widetilde {X}$ , and let $G \hookrightarrow G'$ be an inclusion of G as a closed subgroup of a linear algebraic group $G'$ over k. Then we have an isomorphism

$$ \begin{align*} \left[\widetilde{X} / G\right] \cong \left[\left(\widetilde{X} \times^G G'\right)/G'\right], \end{align*} $$

where $\widetilde {X} \times ^G G'$ is the k-scheme with $G'$ -action obtained from $\widetilde {X}$ by pushout along $G \hookrightarrow G'$ . Thus any quotient stack is isomorphic to a stack quotient of a scheme by a general linear group, which in particular is a special group.

In this section we define a notion of motivic integration for quotient stacks. On the one hand, our definition is straightforward: it is more or less identical to motivic integration for schemes, but in various places we need to replace notions for schemes with the obvious analogues for Artin stacks; in particular, our motivic integration for quotient stacks does not depend on a choice of presentation for the stack as a quotient. On the other hand, our notion allows explicit computations in terms of motivic integration for schemes, as long as one first writes the stack as a stack quotient of a scheme by a special group.

Definition 3.2. Let $\mathcal {X}$ be an Artin stack over k, and set $n \in \mathbb {N}$ . The nth jet stack of $\mathcal {X}$ , denoted $\mathscr {L}_n(\mathcal {X})$ , is the Weil restriction of $\mathcal {X} \otimes _k k[t]/\left (t^{n+1}\right )$ with respect to the morphism $\operatorname {\mathrm {Spec}}\left (k[t]/\left (t^{n+1}\right )\right ) \to \operatorname {\mathrm {Spec}}(k)$ .

Remark 3.3. Each jet stack $\mathscr {L}_n(\mathcal {X})$ is an Artin stack by [Reference RydhRyd11, Theorem 3.7(iii)].

Remark 3.4. Each jet stack $\mathscr {L}_n(\mathcal {X})$ has affine (geometric) stabilisers by the following argument. Let $y: \operatorname {\mathrm {Spec}}(k') \to \mathscr {L}_n(\mathcal {X})$ be a geometric point corresponding to $\psi _n: \operatorname {\mathrm {Spec}}\left (k'[t]/\left (t^{n+1}\right )\right ) \to \mathcal {X}$ . Because $\mathcal {X}$ has affine (geometric) stabilisers, the reduction of the stabiliser of $\psi _n$ is affine, so the stabiliser of $\psi _n$ is affine. Thus the stabiliser of y, which is the Weil restriction of the stabiliser of $\psi _n$ , is affine.

The morphisms $k[t]/\left (t^{n+1}\right ) \to k[t]/\left (t^{m+1}\right )$ , when $n \geq m$ , induce truncation morphisms $\theta ^n_m: \mathscr {L}_n(\mathcal {X}) \to \mathscr {L}_m(\mathcal {X})$ for any Artin stack $\mathcal {X}$ over k. Like in the case of schemes, we use these truncation morphisms to define arcs of $\mathcal {X}$ and a stack parametrising them.

Definition 3.5. Let $\mathcal {X}$ be an Artin stack over k. The arc stack of $\mathcal {X}$ is the inverse limit $\mathscr {L}(\mathcal {X}) = \varprojlim _n \mathscr {L}_n(\mathcal {X})$ , where the inverse limit is taken with respect to the truncation morphisms $\theta ^n_m: \mathscr {L}_n(\mathcal {X}) \to \mathscr {L}_m(\mathcal {X})$ .

Remark 3.6. The name arc stack is justified by the fact that $\mathscr {L}(\mathcal {X})$ is indeed a stack (see, for example, [Reference TalpoTal14, Proposition 2.1.9]). Since $\mathscr {L}(\mathcal {X})$ is a stack as opposed to an Artin stack, we use the symbol $\lvert \mathscr {L}(\mathcal {X})\rvert $ to denote equivalence classes of points but do not define a topology on this set.

Remark 3.7. Let $\mathcal {X}$ be an Artin stack over k, and let $k'$ be a field extension of k. The truncation morphism induces a functor for each $n \in \mathbb {N}$ , and these functors induce a functor . Since $\mathcal {X}$ is an Artin stack, the functor is an equivalence of categories, for example by Artin’s criterion for algebraicity. Throughout this paper, we will often implicitly make this identification.

We will let each $\theta _n: \mathscr {L}(\mathcal {X}) \to \mathscr {L}_n(\mathcal {X})$ denote the canonical morphism, and we will also call these truncation morphisms.

We will eventually define a notion of measurable subsets of $\lvert \mathscr {L}(\mathcal {X})\rvert $ and a motivic measure $\mu _{\mathcal {X}}$ that assigns an element of $\widehat {\mathscr {M}}_k$ to each of these measurable subsets. We begin with an important special case of measurable subsets. Note that when $\mathcal {X}$ is finite type over k, so is each $\mathscr {L}_n(\mathcal {X})$ , by [Reference RydhRyd11, Proposition 3.8(xv)].

Definition 3.8. Let $\mathcal {X}$ be a finite-type Artin stack over k, and set $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ . We call the subset $\mathcal {C}$ a cylinder if there exist some $n \in \mathbb {N}$ and a constructible subset $\mathcal {C}_n \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ such that $\mathcal {C} = (\theta _n)^{-1}(\mathcal {C}_n)$ .

The next theorem, which we will prove later in this section, allows us to define a motivic integration for quotient stacks that is closely related to motivic integration for schemes.

Theorem 3.9. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a cylinder. Then the set $\theta _n(\mathcal {C}) \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ is constructible for each $n \in \mathbb {N}$ , and the sequence

$$ \begin{align*} \left\{\mathrm{e}(\theta_n(\mathcal{C})) \mathbb{L}^{-(n+1)\dim\mathcal{X}}\right\}_{n \in \mathbb{N}} \subset \widehat{\mathscr{M}}_k \end{align*} $$

converges.

Furthermore, suppose that G is a special group over k and $\widetilde {X}$ is a k-scheme with G-action such that there exists an isomorphism $\left [\widetilde {X} / G\right ] \xrightarrow {\sim } \mathcal {X}$ , let $\rho : \widetilde {X} \to \mathcal {X}$ be the composition of the quotient map $\widetilde {X} \to \left [\widetilde {X} / G\right ]$ with the isomorphism $\left [\widetilde {X} / G\right ] \xrightarrow {\sim } \mathcal {X}$ and let $\widetilde {C} = \mathscr {L}(\rho )^{-1}(\mathcal {C})$ . Then $\widetilde {C} \subset \mathscr {L}(\widetilde {X})$ is a cylinder, and

$$ \begin{align*} \lim_{n \to \infty} \mathrm{e}(\theta_n(\mathcal{C}))\mathbb{L}^{-(n+1)\dim\mathcal{X}} = \mu_{\widetilde{X}} \left(\widetilde{C}\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \in \widehat{\mathscr{M}}_k. \end{align*} $$

Remark 3.10. Let G and $\widetilde {X}$ be as in the statement of Theorem 3.9. Since $\left [\widetilde {X}/G\right ]$ is equidimensional and G is geometrically irreducible, $\widetilde {X}$ is equidimensional as well, and hence $\mu _{\widetilde {X}}$ is well defined.

Before we prove Theorem 3.9, we will discuss some useful consequences. First, we can define the motivic measure $\mu _{\mathcal {X}}$ on cylinders.

Definition 3.11. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a cylinder. The motivic measure of $\mathcal {C}$ is

$$ \begin{align*} \mu_{\mathcal{X}}(\mathcal{C}) = \lim_{n \to \infty} \mathrm{e}(\theta_n(\mathcal{C}))\mathbb{L}^{-(n+1)\dim\mathcal{X}} \in \widehat{\mathscr{M}}_k. \end{align*} $$

Remark 3.12. Let $\mathcal {X}$ be an equidimensional smooth Artin (not necessarily quotient) stack over k and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a cylinder. One can verify that $\theta _n(\mathcal {C}) \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ is constructible for each $n \in \mathbb {N}$ and that $\left \{\mathrm {e}(\theta _n(\mathcal {C})) \mathbb {L}^{-(n+1)\dim \mathcal {X}}\right \}_{n \in \mathbb {N}}$ stabilises for sufficiently large n, so Definition 3.11 also makes sense here. Although this is not used for the main results of this paper, our main conjectures are stated in the generality, so we provide the argument for completeness in subsection 3.5.

We now define measurable subsets analogously to the case of schemes.

Definition 3.13. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, set $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ and $\varepsilon \in \mathbb {R}_{>0}$ , let I be a set, let $\mathcal {C}^{(0)} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a cylinder and let $\left \{\mathcal {C}^{(i)}\right \}_{i \in I}$ be a collection of cylinders in $\lvert \mathscr {L}(\mathcal {X})\rvert $ .

We say that $\left (\mathcal {C}^{(0)}, \left (\mathcal {C}^{(i)}\right )_{i \in I}\right )$ is a cylindrical $\varepsilon $ -approximation of $\mathcal {C}$ if

$$ \begin{align*} \left(\mathcal{C} \cup \mathcal{C}^{(0)}\right) \setminus \left(\mathcal{C} \cap \mathcal{C}^{(0)}\right) \subset \bigcup_{i \in I} \mathcal{C}^{(i)} \end{align*} $$

and, for all $i \in I$ ,

$$ \begin{align*} \left\lVert\mu_{\mathcal{X}}\left(\mathcal{C}^{(i)}\right)\right\rVert < \varepsilon. \end{align*} $$

Definition 3.14. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and set $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ . We say that $\mathcal {C}$ is measurable if for any $\varepsilon \in \mathbb {R}_{>0}$ , there exists a cylindrical $\varepsilon $ -approximation of $\mathcal {C}$ .

Remark 3.15. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a cylinder. Then for any $\varepsilon \in \mathbb {R}_{>0}$ , we have that $(\mathcal {C}, \emptyset )$ is a cylindrical $\varepsilon $ -approximation of $\mathcal {C}$ . In particular, $\mathcal {C}$ is measurable.

We now see that Theorem 3.9 allows us to extend $\mu _{\mathcal {X}}$ to measurable subsets.

Corollary 3.16. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a measurable subset. Then there exists a unique $\mu _{\mathcal {X}}(\mathcal {C}) \in \widehat {\mathscr {M}}_k$ such that for any $\varepsilon \in \mathbb {R}_{>0}$ and any cylindrical $\varepsilon $ -approximation $\left (\mathcal {C}^{(0)}, \left (\mathcal {C}^{(i)}\right )_{i \in I}\right )$ of $\mathcal {C}$ ,

$$ \begin{align*} \left\lVert\mu_{\mathcal{X}}(\mathcal{C}) - \mu_{\mathcal{X}}\left(\mathcal{C}^{(0)}\right)\right\rVert < \varepsilon. \end{align*} $$

Furthermore, suppose that $G, \widetilde {X}, \rho $ are as in the statement of Theorem 3.9 and $\widetilde {C}=\mathscr {L}(\rho )^{-1}(\mathcal {C})$ . Then $\widetilde {C} \subset \mathscr {L}\left (\widetilde {X}\right )$ is measurable, and

$$ \begin{align*} \mu_{\mathcal{X}}(\mathcal{C}) = \mu_{\widetilde{X}} \left(\widetilde{C}\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \in \widehat{\mathscr{M}}_k. \end{align*} $$

Proof. Let $G, \widetilde {X}, \rho , \widetilde {C}$ be as in the second part of the corollary. For any $\varepsilon \in \mathbb {R}_{>0}$ and any cylindrical $\varepsilon $ -approximation $\left (\mathcal {C}^{(0)}, \left (\mathcal {C}^{(i)}\right )_{i \in I}\right )$ of $\mathcal {C}$ , Theorem 3.9 implies that $\left (\mathscr {L}(\rho )^{-1}\left (\mathcal {C}^{(0)}\right ), \left (\mathscr {L}(\rho )^{-1}\left (\mathcal {C}^{(i)}\right )\right )_{i \in I}\right )$ is a cylindrical $\varepsilon \left \lVert \mathrm {e}(G) \mathbb {L}^{-\dim G}\right \rVert $ -approximation of $\widetilde {C}$ . Thus $\widetilde {C}$ is measurable, and for any cylindrical $\varepsilon $ -approximation $\left (\mathcal {C}^{(0)}, \left (\mathcal {C}^{(i)}\right )_{i \in I}\right )$ of $\mathcal {C}$ ,

$$ \begin{align*} \left\lVert\mu_{\widetilde{X}} \left(\widetilde{C}\right)\right.& \left.\mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} - \mu_{\mathcal{X}}\left(\mathcal{C}^{(0)}\right)\right\rVert\\ &= \left\lVert\mu_{\widetilde{X}} \left(\widetilde{C}\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} - \mu_{\widetilde{X}} \left(\mathscr{L}(\rho)^{-1}\left(\mathcal{C}^{(0)}\right)\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G}\right\rVert\\ &\leq \left\lVert \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \right\rVert \left\lVert \mu_{\widetilde{X}}\left(\widetilde{C}\right) - \mu_{\widetilde{X}}\left(\mathscr{L}(\rho)^{-1}\left(\mathcal{C}^{(0)}\right)\right) \right\rVert\\ &< \varepsilon \left\lVert \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \right\rVert \left\lVert\mathrm{e}(G) \mathbb{L}^{-\dim G}\right\rVert, \end{align*} $$

where the first equality follows from Theorem 3.9. Once $\mu _{\mathcal {X}}(\mathcal {C})$ is shown to exist, this chain of inequalities proves $\mu _{\mathcal {X}}(\mathcal {C}) = \mu _{\widetilde {X}} \left (\widetilde {C}\right ) \mathrm {e}(G)^{-1} \mathbb {L}^{\dim G}$ . To show the existence of $\mu _{\mathcal {X}}(\mathcal {C})$ , it suffices by Remark 3.1 to assume G is a general linear group, so this chain of inequalities and Lemma 3.17 finish the proof.

Lemma 3.17. Let G be a general linear group over k. Then

$$ \begin{align*} \left\lVert \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \right\rVert \left\lVert\mathrm{e}(G) \mathbb{L}^{-\dim G}\right\rVert = 1. \end{align*} $$

Proof. Using Euler–Poincaré polynomials, it is straightforward to check (see, for example, the proof of [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 2, Lemma 4.1.3]) that if $n_0 \in \mathbb {Z}$ and $\{c_n\}_{n \geq n_0}$ is a sequence of integers with $c_{n_0} \neq 0$ , then

$$ \begin{align*} \left\lVert \sum_{n \geq n_0} c_n \mathbb{L}^{-n} \right\rVert = \exp(-n_0). \end{align*} $$

The lemma then follows from the fact that $\mathrm {e}(G)$ is a polynomial in $\mathbb {L}$ (see, for example, the proof of [Reference JoyceJoy07, Lemma 4.6]).

Definition 3.18. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a measurable subset. The motivic measure of $\mathcal {C}$ is defined to be $\mu _{\mathcal {X}}(\mathcal {C}) \in \widehat {\mathscr {M}}_k$ , as in the statement of Corollary 3.16.

Remark 3.19. Remark 3.15 implies that Definition 3.18 generalises Definition 3.11.

In the next two subsections, we will prove Theorem 3.9.

3.1 Jet schemes of quotient stacks

In this subsection, we describe the jet schemes of a stack quotient as stack quotients themselves. This is the first step in providing the relationship between motivic integration for quotient stacks and motivic integration for schemes. This description, Corollary 3.22, is a special case of the next proposition, which describes the Weil restriction of a stack quotient.

If $S'$ and S are schemes and $S' \to S$ is a finite flat morphism of finite presentation, we will let $\mathscr {R}_{S'/S}$ denote the functor taking each stack over $S'$ to its Weil restriction with respect to $S' \to S$ , and we note that if $\mathcal {X}$ is an Artin stack over $S'$ , then $\mathscr {R}_{S'/S}(\mathcal {X})$ is an Artin stack over S [Reference RydhRyd11, Theorem 3.7(iii)].

Proposition 3.20. Let $S'$ and S be schemes and $S' \to S$ be a finite flat morphism of finite presentation. If $\widetilde {X}'$ is an $S'$ -scheme with an action by a linear algebraic group $G'$ over $S'$ , then there exists an isomorphism

$$ \begin{align*} \mathscr{R}_{S'/S}\left(\left[\widetilde{X}'/G'\right]\right)\xrightarrow{\sim} \left[\mathscr{R}_{S'/S}\left(\widetilde{X}'\right)/\mathscr{R}_{S'/S}(G')\right] \end{align*} $$

such that

commutes.

Remark 3.21. In the statement of Proposition 3.20, the action of $\mathscr {R}_{S'/S}(G')$ on $\mathscr {R}_{S'/S}\left (\widetilde {X}'\right )$ is obtained by applying $\mathscr {R}_{S'/S}$ to the map $G' \times _{S'} \widetilde {X}' \to \widetilde {X}'$ defining the action of $G'$ on $\widetilde {X}'$ .

Proof. We let $\mathcal {X}'=\left [\widetilde {X}'/G'\right ]$ , $\rho ': \widetilde {X}' \to \mathcal {X}'$ be the quotient map, $\mathcal {X}=\mathscr {R}_{S'/S}(\mathcal {X}')$ , $\widetilde {X}=\mathscr {R}_{S'/S}\left (\widetilde {X}'\right )$ and $G=\mathscr {R}_{S'/S}(G')$ . Since $\rho '\colon \widetilde {X}'\to \mathcal {X}'$ is a smooth cover, $\mathscr {R}_{S'/S}(\rho ')\colon \widetilde {X}\to \mathcal {X}$ is as well, by [Reference RydhRyd11, Proposition 3.5(v)]. Since $\rho '$ is a $G'$ -torsor, the natural map $G'\times _{S'} \widetilde {X}'\to \widetilde {X}'\times _{\mathcal {X}'} \widetilde {X}'$ induced by the $G'$ -action $G' \times _{S'} \widetilde {X}' \to \widetilde {X}'$ is an isomorphism, and applying Weil restriction, we see that the map $G\times _S \widetilde {X}\to \widetilde {X}\times _{\mathcal {X}} \widetilde {X}$ induced by the G-action $G \times _S \widetilde {X} \to \widetilde {X}$ is an isomorphism as well. Thus, $\mathscr {R}_{S'/S}(\rho ')\colon \widetilde {X}\to \mathcal {X}$ is a G-torsor, thereby inducing an isomorphism $\mathcal {X}\xrightarrow {\sim }\left [\widetilde {X}/G\right ]$ which makes the diagram in the statement of the proposition commute.

By the definition of jet stacks, the following is a special case of Proposition 3.20:

Corollary 3.22. Let G be a linear algebraic group over k acting on a k-scheme $\widetilde {X}$ , and set $n \in \mathbb {N}$ . There exists an isomorphism

$$ \begin{align*} \mathscr{L}_n\left(\left[\widetilde{X}/G\right]\right) \xrightarrow{\sim} \left[\mathscr{L}_n\left(\widetilde{X}\right)/\mathscr{L}_n(G)\right], \end{align*} $$

such that

commutes.

Remark 3.23. In the statement of Corollary 3.22, the action of $\mathscr {L}_n(G)$ on $\mathscr {L}_n\left (\widetilde {X}\right )$ is obtained by applying $\mathscr {L}_n$ to the map $G \times _{k} \widetilde {X} \to \widetilde {X}$ defining the G-action on $\widetilde {X}$ .

3.2 Truncation morphisms and quotient stacks

Lemma 3.24. Let $\mathcal {X}$ be an Artin stack over k, let $\widetilde {X}$ be a scheme over k and let $\rho : \widetilde {X} \to \mathcal {X}$ be a smooth covering. Set $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ and $\widetilde {C} = \mathscr {L}(\rho )^{-1}(\mathcal {C}) \subset \mathscr {L}\left (\widetilde {X}\right )$ . Then for all $n \in \mathbb {N}$ ,

$$ \begin{align*} \mathscr{L}_n(\rho)^{-1}(\theta_n(\mathcal{C})) = \theta_n\left(\widetilde{C}\right). \end{align*} $$

Proof. Set $n \in \mathbb {N}$ . Clearly, $\theta _n\left (\widetilde {C}\right ) \subset \mathscr {L}_n(\rho )^{-1}(\theta _n(\mathcal {C}))$ .

To prove the opposite inclusion, let $k'$ be a field extension of k, and let $\widetilde {\psi }_n \in \mathscr {L}_n\left (\widetilde {X}\right )(k')$ and $\psi \in \mathscr {L}(\mathcal {X})(k')$ be such that the class of $\psi $ in $\lvert \mathscr {L}(\mathcal {X})\rvert $ is contained in $\mathcal {C}$ and $\mathscr {L}_n(\rho )\left (\widetilde {\psi }_n\right ) \cong \theta _n(\psi )$ . We must show $\widetilde {\psi }_n\in \theta _n\left (\widetilde {C}\right )$ . Since $\rho $ is smooth, by the infinitesimal lifting criterion, we have a dotted arrow filling in the following diagram:

Then $\widetilde {\psi }\in \widetilde {C}$ , so $\widetilde {\psi }_n\in \theta _n\left (\widetilde {C}\right )$ .

We may now prove the next proposition, which by Remark 3.1 and Remark 3.10 implies Theorem 3.9.

Proposition 3.25. Let G be a special group over k, let $\widetilde {X}$ be an equidimensional finite-type scheme over k with G-action, let $\mathcal {X} = \left [\widetilde {X}/G\right ]$ , let $\rho : \widetilde {X} \to \mathcal {X}$ be the quotient map, let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a cylinder and let $\widetilde {C} = \mathscr {L}(\rho )^{-1}(\mathcal {C})$ . Then $\widetilde {C} \subset \mathscr {L}(\widetilde {X})$ is a cylinder, the set $\theta _n(\mathcal {C}) \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ is constructible for each $n \in \mathbb {N}$ , and the sequence

$$ \begin{align*} \left\{\mathrm{e}(\theta_n(\mathcal{C})) \mathbb{L}^{-(n+1)\dim\mathcal{X}}\right\}_{n \in \mathbb{N}} \subset \widehat{\mathscr{M}}_k \end{align*} $$

converges to

$$ \begin{align*} \mu_{\widetilde{X}} \left(\widetilde{C}\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \in \widehat{\mathscr{M}}_k. \end{align*} $$

Remark 3.26. In the statement of Proposition 3.25, because G is irreducible, the irreducible components of $\widetilde {X}$ are G-invariant, so $\mathcal {X}$ is equidimensional.

Proof. We first show that $\widetilde {C} \subset \mathscr {L}\left (\widetilde {X}\right )$ is a cylinder. Because $\mathcal {C}$ is a cylinder, there exist some $n \in \mathbb {N}$ and some constructible subset $\mathcal {C}_n \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ such that $\mathcal {C} = (\theta _n)^{-1}(\mathcal {C}_n)$ . Then $\widetilde {C} = (\theta _n)^{-1}\left (\mathscr {L}_n(\rho )^{-1}(\mathcal {C}_n)\right )$ is a cylinder.

Now we will show that for all $n \in \mathbb {N}$ , the set $\theta _n(\mathcal {C})$ is a constructible subset of $\mathscr {L}_n(\mathcal {X})$ . Each $\theta _n\left (\widetilde {C}\right )$ is a constructible subset of $\mathscr {L}_n\left (\widetilde {X}\right )$ . Therefore each $\theta _n(\mathcal {C}) \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ is constructible, by Chevalley’s theorem for Artin stacks [Reference Hall and RydhHR17, Theorem 5.2], Corollary 3.22 and Lemma 3.24.

Then since G is a special group, $\mathscr {L}_n(G)$ is as well, by Remark 2.4. Then Corollary 3.22 and Lemma 3.24 imply that for each $n \in \mathbb {N}$ ,

$$ \begin{align*} \mathrm{e}(\theta_n(\mathcal{C})) = \mathrm{e}\left(\theta_n\left(\widetilde{C}\right)\right)\mathrm{e}(\mathscr{L}_n(G))^{-1} = \mathrm{e}\left(\theta_n\left(\widetilde{C}\right)\right) \mathrm{e}(G)^{-1} \mathbb{L}^{-n\dim G}, \end{align*} $$

where the second equality holds because G is smooth. Therefore,

$$ \begin{align*} \mu_{\widetilde{X}}(\widetilde{C})\mathrm{e}(G)^{-1}\mathbb{L}^{\dim G} &= \lim_{n \to \infty} \mathrm{e}\left(\theta_n\left(\widetilde{C}\right)\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G -(n+1)\dim \widetilde{X}}\\ &= \lim_{n \to \infty} \mathrm{e}(\theta_n(\mathcal{C})) \mathbb{L}^{-(n+1)\dim\mathcal{X}}.\\[-36pt] \end{align*} $$

3.3 Properties of motivic integration for quotient stacks

We now state some basic properties of motivic integration for quotient stacks. We will use these properties later in this paper.

Proposition 3.27. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, let $\left \{\mathcal {C}^{(i)}\right \}_{i \in \mathbb {N}}$ be a sequence of pairwise disjoint measurable subsets of $\lvert \mathscr {L}(\mathcal {X})\rvert $ and let $\mathcal {C} = \bigcup _{i= 0}^\infty \mathcal {C}^{(i)}$ . If $\lim _{i \to \infty } \mu _{\mathcal {X}}\left (\mathcal {C}^{(i)}\right ) = 0$ , then $\mathcal {C}$ is measurable and

$$ \begin{align*} \mu_{\mathcal{X}}(\mathcal{C}) = \sum_{i = 0}^\infty \mu_{\mathcal{X}}\left(\mathcal{C}^{(i)}\right). \end{align*} $$

Proof. The set $\mathcal {C}$ is measurable by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 3.4.2]. The remainder of the proposition follows from Corollary 3.16 and the analogous statement for schemes [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 3.4.3] applied to the scheme $\widetilde {X}$ in the statement of Corollary 3.16.

Proposition 3.28. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and set $\mathcal {C} \subset \mathcal {D} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ . If $\mathcal {D}$ is measurable and $\mu _{\mathcal {X}}(\mathcal {D}) = 0$ , then $\mathcal {C}$ is measurable and $\mu _{\mathcal {X}}(\mathcal {C}) = 0$ .

Proof. The proposition holds by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Corollary 3.5.5(a)].

Proposition 3.29. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and let $\mathcal {C}, \mathcal {D}$ be measurable subsets of $\lvert \mathscr {L}(\mathcal {X})\rvert $ . If $\mathcal {C} \subset \mathcal {D}$ , then

$$ \begin{align*} \lVert \mu_{\mathcal{X}}(\mathcal{C}) \rVert \leq \lVert \mu_{\mathcal{X}}(\mathcal{D}) \rVert. \end{align*} $$

Proof. By Remark 3.1, there exist $G, \widetilde {X}, \rho $ as in the statement of Theorem 3.9 such that G is a general linear group. Let $\widetilde {C} = \mathscr {L}(\rho )^{-1}(\mathcal {C})$ and $\widetilde {D} = \mathscr {L}(\rho )^{-1}(\mathcal {D})$ . Then

$$ \begin{align*} \lVert \mu_{\mathcal{X}}(\mathcal{C}) \rVert &= \left\lVert \mu_{\widetilde{X}}\left(\widetilde{C}\right)\mathrm{e}(G)^{-1}\mathbb{L}^{\dim G} \right\rVert\\ &\leq \left\lVert \mathrm{e}(G)^{-1}\mathbb{L}^{\dim G}\right\rVert \left\lVert \mu_{\widetilde{X}}\left(\widetilde{C}\right)\right\rVert\\ &\leq \left\lVert \mathrm{e}(G)^{-1}\mathbb{L}^{\dim G}\right\rVert \left\lVert \mu_{\widetilde{X}}\left(\widetilde{D}\right)\right\rVert\\ &= \left\lVert \mathrm{e}(G)^{-1}\mathbb{L}^{\dim G}\right\rVert \left\lVert \mu_{\mathcal{X}}(\mathcal{D}) \mathrm{e}(G)\mathbb{L}^{-\dim G}\right\rVert\\ &\leq \left\lVert \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G} \right\rVert \left\lVert\mathrm{e}(G) \mathbb{L}^{-\dim G}\right\rVert \lVert \mu_{\mathcal{X}}(\mathcal{D}) \rVert\\ &= \lVert \mu_{\mathcal{X}}(\mathcal{D}) \rVert, \end{align*} $$

where the first and fourth lines follow from Corollary 3.16, the third line follows from the analogous statement [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Corollary 3.3.5] for schemes applied to $\widetilde {X}$ and the last line follows from Lemma 3.17.

Proposition 3.30. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, let $\mathcal {Y}$ be a closed substack of $\mathcal {X}$ with $\dim \mathcal {Y} < \dim \mathcal {X}$ and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be the image of $\lvert \mathscr {L}(\mathcal {Y})\rvert $ in $\lvert \mathscr {L}(\mathcal {X})\rvert $ . Then $\mathcal {C}$ is measurable and $\mu _{\mathcal {X}}(\mathcal {C}) = 0$ .

Proof. For each $n \in \mathbb {N}$ , let $\mathcal {C}_n \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ be the image of $\lvert \mathscr {L}_n(\mathcal {Y})\rvert $ in $\lvert \mathscr {L}_n(\mathcal {X})\rvert $ , and let $\mathcal {C}^{(n)} = (\theta _n)^{-1}(\mathcal {C}_n)$ . By [Reference RydhRyd11, Proposition 3.5(vi)], each $\mathcal {C}_n$ is a closed subset of $\mathscr {L}_n(\mathcal {X})$ , so each $\mathcal {C}^{(n)}$ is a cylinder in $\mathscr {L}(\mathcal {X})$ .

By Remark 3.1, there exist $G, \widetilde {X}, \rho $ as in the statement of Theorem 3.9. Let $\widetilde {Y} = \widetilde {X} \times _{\mathcal {X}} \mathcal {Y}$ . Then $\mathscr {L}_n(\rho )^{-1}(\mathcal {C}_n)$ is the underlying set of $\mathscr {L}_n\left (\widetilde {Y}\right )$ . Thus by Theorem 3.9,

$$ \begin{align*} \mu_{\mathcal{X}}\left(\mathcal{C}^{(n)}\right) &= \mu_{\widetilde{X}} \left(\mathscr{L}(\rho)^{-1}\left(\mathcal{C}^{(n)}\right)\right) \mathrm{e}(G)^{-1} \mathbb{L}^{\dim G}\\ &= \mu_{\widetilde{X}}\left((\theta_n)^{-1}\left(\mathscr{L}_n\left(\widetilde{Y}\right)\right)\right)\mathrm{e}(G)^{-1} \mathbb{L}^{\dim G}. \end{align*} $$

By [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 2.3.1],

$$ \begin{align*} \lim_{n \to \infty} \mu_{\widetilde{X}}\left((\theta_n)^{-1}\left(\mathscr{L}_n\left(\widetilde{Y}\right)\right)\right) = 0, \end{align*} $$

so

$$ \begin{align*} \lim_{n \to \infty} \mu_{\mathcal{X}}\left(\mathcal{C}^{(n)}\right) = 0. \end{align*} $$

Therefore for any $\varepsilon \in \mathbb {R}_{>0}$ , we get that $\left (\emptyset , \left (\mathcal {C}^{(n)}\right )\right )$ is a cylindrical $\varepsilon $ -approximation of $\mathcal {C}$ for sufficiently large n, and we are done, by the definition of $\mu _{\mathcal {X}}$ .

Proposition 3.31. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, and let $\mathcal {C}$ and $\mathcal {D}$ be measurable subsets of $\lvert \mathscr {L}(\mathcal {X})\rvert $ . Then the intersection $\mathcal {C} \cap \mathcal {D}$ , the union $\mathcal {C} \cup \mathcal {D}$ and the complement $\mathcal {C} \setminus \mathcal {D}$ are all measurable subsets of $\lvert \mathscr {L}(\mathcal {X})\rvert $ .

Proof. The proposition holds by the exact same proof used for the analogous statement for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 6, Proposition 3.2.8].

Proposition 3.32. Let $\mathcal {X}$ be an equidimensional finite-type quotient stack over k, let $\iota : \mathcal {U} \hookrightarrow \mathcal {X}$ be the inclusion of an open substack and set $\mathcal {C} \subset (\theta _0)^{-1}(\lvert \mathcal {U}\rvert ) \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ . Then $\mathcal {C}$ is a measurable subset of $\lvert \mathscr {L}(\mathcal {X})\rvert $ if and only if $\mathscr {L}(\iota )^{-1}(\mathcal {C})$ is a measurable subset of $\lvert \mathscr {L}(\mathcal {U})\rvert $ , and in that case,

$$ \begin{align*} \mu_{\mathcal{X}}(\mathcal{C}) = \mu_{\mathcal{U}}\left(\mathscr{L}(\iota)^{-1}(\mathcal{C})\right). \end{align*} $$

Proof. As in the case of schemes, this is an easy consequence of the definitions and the fact that for all $n \in \mathbb {N}$ , the morphism $\mathscr {L}_n(\iota ): \mathscr {L}_n(\mathcal {U}) \to \mathscr {L}_n(\mathcal {X})$ is an open immersion by [Reference RydhRyd11, Proposition 3.5(vii)].

3.4 Nonseparatedness functions

We now introduce notation for the nonseparatedness functions $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}$ , $\operatorname {\mathrm {sep}}_\pi $ and $\operatorname {\mathrm {sep}}_{\mathcal {X}}$ that were used in the statements of the main conjectures and theorems of this paper. Throughout this subsection, let $\mathcal {X}$ be an Artin stack over k, set $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ , let X be a scheme over k and let $\pi : \mathcal {X} \to X$ be a map. For any field extension $k'$ of k, we will let $\overline {\mathcal {C}}(k')$ denote the subset of $\overline {\mathscr {L}(\mathcal {X})}(k')$ consisting of arcs whose classes in the set $\lvert \mathscr {L}(\mathcal {X})\rvert $ are contained in $\mathcal {C}$ .

If $k'$ is a field extension of k and $\varphi \in \mathscr {L}(X)(k')$ , we set

$$ \begin{align*} \operatorname{\mathrm{sep}}_{\pi, \mathcal{C}}(\varphi) = \#\left(\overline{\mathcal{C}}(k') \cap \overline{\left(\mathscr{L}(\pi)^{-1}(\varphi)\right)}(k')\right) \in \mathbb{N} \cup \{\infty\}, \end{align*} $$

which induces a map $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}: \mathscr {L}(X) \to \mathbb {N} \cup \{\infty \}$ by considering each $\varphi \in \mathscr {L}(X)$ as a point valued in its residue field. If, furthermore, we assume that X is integral, finite type and separated and has log-terminal singularities, that $\operatorname {\mathrm {sep}}_{\pi , \mathcal {C}}: \mathscr {L}(X) \to \mathbb {N} \cup \{\infty \}$ has measurable fibres and that $C \subset \mathscr {L}(X)$ is a measurable subset, then we can consider the motivic integral

$$ \begin{align*} \int_C \operatorname{\mathrm{sep}}_{\pi, \mathcal{C}} \mathrm{d}\mu^{\mathrm{Gor}}_X = \sum_{n \in \mathbb{N}} n \mu^{\mathrm{Gor}}_X\left(\operatorname{\mathrm{sep}}_{\pi, \mathcal{C}}^{-1}(n) \cap C\right) \in \widehat{\mathscr{M}}_k\left[\mathbb{L}^{1/m}\right], \end{align*} $$

where $m \in \mathbb {Z}_{>0}$ is such that $mK_X$ is Cartier. Note that with these assumptions, the series defining $\int _C \operatorname {\mathrm {sep}}_{\pi , \mathcal {C}} \mathrm {d}\mu ^{\mathrm {Gor}}_X$ converges because

$$ \begin{align*} \lim_{n \to \infty} n \mu^{\mathrm{Gor}}_X\left(\operatorname{\mathrm{sep}}_{\pi, \mathcal{C}}^{-1}(n) \cap C\right) = 0, \end{align*} $$

which follows from

$$ \begin{align*} \lim_{n \to \infty} \mu^{\mathrm{Gor}}_X\left(\operatorname{\mathrm{sep}}_{\pi, \mathcal{C}}^{-1}(n) \cap C\right) = 0, \end{align*} $$

which itself, for example, is a consequence of Proposition 2.3.

Set

$$ \begin{align*} \operatorname{\mathrm{sep}}_\pi = \operatorname{\mathrm{sep}}_{\pi, \lvert\mathscr{L}(\mathcal{X})\rvert} \end{align*} $$

and

$$ \begin{align*} \operatorname{\mathrm{sep}}_{\mathcal{X}} = 1/(\operatorname{\mathrm{sep}}_\pi \circ \mathscr{L}(\pi)): \lvert\mathscr{L}(\mathcal{X})\rvert \to \mathbb{Q}_{\geq 0} \cup \{\infty\}. \end{align*} $$

If, furthermore, we assume that $\mathcal {X}$ is an equidimensional and finite-type quotient stack over k and that $\operatorname {\mathrm {sep}}_{\mathcal {X}}: \lvert \mathscr {L}(\mathcal {X})\rvert \to \mathbb {Q}_{\geq 0} \cup \{\infty \}$ has measurable fibres, we can consider the motivic integral

$$ \begin{align*} \int_{\mathscr{L}(\mathcal{X})} \operatorname{\mathrm{sep}}_{\mathcal{X}} \mathrm{d}\mu_{\mathcal{X}} = \sum_{n \in \mathbb{Z}_{\geq 1}} (1/n) \mu_{\mathcal{X}}\left( \operatorname{\mathrm{sep}}_{\mathcal{X}}^{-1}(1/n) \right) \in \widehat{\mathscr{M}_k \otimes_{\mathbb{Z}} \mathbb{Q}}, \end{align*} $$

where the ring $\widehat {\mathscr {M}_k \otimes _{\mathbb {Z}} \mathbb {Q}}$ is defined like $\widehat {\mathscr {M}}_k$ in subsection 2.1 by replacing any mention of $K_0(\mathbf {Var}_k)$ with $K_0(\mathbf {Var}_k) \otimes _{\mathbb {Z}} \mathbb {Q}$ and any mention of ‘subgroup’ with ‘ $\mathbb {Q}$ -subspace’. With these assumptions, the series defining $\int _{\mathscr {L}(\mathcal {X})} \operatorname {\mathrm {sep}}_{\mathcal {X}} \mathrm {d}\mu _{\mathcal {X}}$ converges because

$$ \begin{align*} \lim_{n \to \infty} (1/n) \mu_{\mathcal{X}}\left( \operatorname{\mathrm{sep}}_{\mathcal{X}}^{-1}(1/n) \right) = 0, \end{align*} $$

which by the definition of the norm on $\widehat {\mathscr {M}_k \otimes _{\mathbb {Z}} \mathbb {Q}}$ follows from

$$ \begin{align*} \lim_{n \to \infty} \mu_{\mathcal{X}}\left( \operatorname{\mathrm{sep}}_{\mathcal{X}}^{-1}(1/n) \right) = 0, \end{align*} $$

which itself follows from Corollary 3.16 and properties of motivic measures for schemes.

3.5 Motivic integration for smooth stacks via the cotangent complex

In this subsection, we prove that the motivic measure $\mu _{\mathcal {X}}$ is also well defined when $\mathcal {X}$ is an equidimensional smooth Artin (not necessarily quotient) stack over k. We only explicitly verify this for cylinders, but by a standard argument (identical to the one for schemes in [Reference Chambert-Loir, Nicaise and SebagCLNS18]), it leads to well-defined notions (that coincide with our definitions in the case where $\mathcal {X}$ is a quotient stack) of measurable subsets of $\lvert \mathscr {L}(\mathcal {X})\rvert $ and their motivic measures. The main result of this subsection is the following theorem, which immediately implies that Definition 3.11 makes sense in this setting:

Theorem 3.33. Let $\mathcal {X}$ be an equidimensional smooth Artin stack over k, and let $\mathcal {C} \subset \lvert \mathscr {L}(\mathcal {X})\rvert $ be a cylinder. Then the set $\theta _n(\mathcal {C}) \subset \lvert \mathscr {L}_n(\mathcal {X})\rvert $ is constructible for each $n \in \mathbb {N}$ , and the sequence

$$ \begin{align*} \left\{\mathrm{e}(\theta_n(\mathcal{C})) \mathbb{L}^{-(n+1)\dim\mathcal{X}}\right\}_{n \in \mathbb{N}} \subset K_0(\mathbf{Stack}_k) \end{align*} $$

stabilises for sufficiently large n.

We first prove two lemmas, after which we prove this theorem.

Lemma 3.34. Let $\mathcal {X}$ be an equidimensional smooth Artin stack over k, and set $n \in \mathbb {N}$ . There exist some $\ell \in \mathbb {Z}_{>0}$ , a partition $\lvert \mathscr {L}_n(\mathcal {X})\rvert = \bigsqcup _{i=1}^\ell \mathcal {C}_i$ of $\lvert \mathscr {L}_n(\mathcal {X})\rvert $ into constructible subsets $\mathcal {C}_i$ and some $r_1, \dotsc , r_\ell , j_1, \dotsc , j_\ell \in \mathbb {N}$ such that

  • for any $i \in \{1, \dotsc , \ell \}$ , we have $r_i - j_i = \dim \mathcal {X}$ , and

  • for any $i \in \{1, \dotsc , \ell \}$ , any field extension $k'$ of k and any $\psi _n \in \mathscr {L}_n(\mathcal {X})(k')$ whose class in $\lvert \mathscr {L}_n(\mathcal {X})\rvert $ is contained in $\mathcal {C}_i$ , we have

    $$ \begin{align*} \left(\theta^{n+1}_n\right)^{-1}(\psi_n) \cong \mathbb{A}^{r_i}_{k'} \times_{k'} B\mathbb{G}_{a}^{j_i}. \end{align*} $$

Proof. Fix $\xi _n\colon \operatorname {\mathrm {Spec}} k'\to \mathscr {L}_n(\mathcal {X})$ and let $\mathcal {Y}_{\xi _n}$ denote the fibre of the truncation map $\mathscr {L}_{n+1}(\mathcal {X})\to \mathscr {L}_n(\mathcal {X})$ over $\xi _n$ . For any $\alpha \colon \operatorname {\mathrm {Spec}} A\to \operatorname {\mathrm {Spec}} k'$ , the A-valued points $\mathcal {Y}_{\xi _n}(A)$ are the category of lifts of $\xi _n\otimes _{k'}A$ to $\mathscr {L}_{n+1}(\mathcal {X})$ . For all $m\geq 0$ , let $\mathcal {X}_m=\mathcal {X}\otimes _k k[t]/\left (t^{m+1}\right )$ and let $\alpha _m\colon \operatorname {\mathrm {Spec}} A[t]/\left (t^{m+1}\right )\to \operatorname {\mathrm {Spec}} k'[t]/\left (t^{m+1}\right )$ be the map induced by $\alpha $ ; for $m\leq n$ , let $\varphi _m\colon \operatorname {\mathrm {Spec}} k'[t]/\left (t^{m+1}\right )\to \mathcal {X}_m$ denote the map induced by $\xi _n$ . We then obtain a cartesian diagram

where the curved arrow is the structure map. Let $\mathcal {J}_n$ denote the ideal sheaf of $\operatorname {\mathrm {Spec}} k'[t]/\left (t^{n+1}\right )\to \operatorname {\mathrm {Spec}} k'[t]/\left (t^{n+2}\right )$ considered as a $k'$ -module. By [Reference OlssonOls06, Theorem 1.5] and the fact that $\mathcal {X}_{n+1}$ and $A[t]/\left (t^{n+2}\right )$ are flat over $k[t]/(t^{n+2})$ , the obstruction to the existence of a dotted arrow in this diagram lives in

$$ \begin{align*} \operatorname{\mathrm{Ext}}^1\left(L(\alpha_0\varphi_0)^*L_{\mathcal{X}/k},\alpha_0^*\mathcal{J}_n\right)=\operatorname{\mathrm{Ext}}^1\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)\otimes_{k'}\mathcal{J}_n\otimes_{k'}A. \end{align*} $$

We will show that this group vanishes, and so by [Reference OlssonOls06, Theorem 1.5], the objects (resp., automorphisms) of $\mathcal {Y}_{\xi _n}(A)$ are parametrised by $\operatorname {\mathrm {Ext}}^n\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )\otimes _{k'}\mathcal {J}_n\otimes _{k'}A$ where $n=0$ (resp., $n=-1$ ). In particular, if V (resp., G) denotes the affine space (resp., algebraic vector group) over $k'$ associated to the vector space $\operatorname {\mathrm {Ext}}^n\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )\otimes _{k'}\mathcal {J}_n$ with $n=0$ (resp., $n=-1$ ), then we have

$$ \begin{align*} \mathcal{Y}_{\xi_n}\cong V\times_{k'} G\cong \mathbb{A}^{r\left(\xi_n\right)}\times B\mathbb{G}_a^{j\left(\xi_n\right)}, \end{align*} $$

where $r(\xi _n)=\dim \operatorname {\mathrm {Ext}}^0\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )$ and $j(\xi _n)=\dim \operatorname {\mathrm {Ext}}^1\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )$ . Note that this implies

$$ \begin{align*} e\left(\mathcal{Y}_{\xi_n}\right)=\mathbb{L}^{r\left(\xi_n\right)-j\left(\xi_n\right)}. \end{align*} $$

Therefore, to finish the proof of the theorem, it suffices to show that

(1) $$ \begin{align} \operatorname{\mathrm{Ext}}^1\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)=0,\qquad r(\xi_n)-j(\xi_n)=\dim\mathcal{X}, \end{align} $$

and that the locus of $\varphi _0\in \lvert \mathcal {X}\rvert $ where $r(\xi _n)$ is constant is given by a constructible set. Since these remaining statements depend only on the dimension of the $\operatorname {\mathrm {Ext}}$ -groups over $k'$ , it suffices to replace $k'$ with an extension field, and hence we can assume $k'$ is algebraically closed.

Let $\rho \colon \widetilde {X}\to \mathcal {X}$ be a smooth cover. Since $k'$ is algebraically closed, we may fix a lift $\phi _0\colon \operatorname {\mathrm {Spec}} k'\to \widetilde {X}$ of $\varphi _0$ . Since $\widetilde {X}$ and $\rho $ are smooth, we have an exact triangle

$$ \begin{align*} p^*L_{\mathcal{X}/k}\to\Omega^1_{\widetilde{X}/k}\to\Omega^1_{\widetilde{X}/\mathcal{X}}, \end{align*} $$

from which we obtain an exact triangle

$$ \begin{align*} \Gamma\left(\phi_0^*T_{\widetilde{X}/\mathcal{X}}\right)\to\Gamma\left(\phi_0^*T_{\widetilde{X}/k}\right)\to \mathrm{RHom}\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right). \end{align*} $$

In particular, $\operatorname {\mathrm {Ext}}^n\left (L\varphi _0^*L_{\mathcal {X}/k},\mathcal {O}_{k'}\right )=0$ for $n\neq 0,-1$ , and there is an exact sequence

(2) $$ \begin{align} 0\to\operatorname{\mathrm{Ext}}^{-1}\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)\to\Gamma\left(\phi_0^*T_{\widetilde{X}/\mathcal{X}}\right)\to\Gamma\left(\phi_0^*T_{\widetilde{X}/k}\right)\to\operatorname{\mathrm{Ext}}^0\left(L\varphi_0^*L_{\mathcal{X}/k},\mathcal{O}_{k'}\right)\to0. \end{align} $$

Thus,

$$ \begin{align*} r(\xi_n)-j(\xi_n)=\dim\Gamma\left(\phi_0^*T_{\widetilde{X}/k}\right)-\dim\Gamma\left(\phi_0^*T_{\widetilde{X}/\mathcal{X}}\right)= \dim\mathcal{X}, \end{align*} $$

thereby establishing equation (1). Finally, note from formula (2) that the cokernel of $\Gamma \left (\phi _0^*T_{\widetilde {X}/\mathcal {X}}\right )\to \Gamma \left (\phi _0^*T_{\widetilde {X}/k}\right )$ depends only on $\varphi _0$ and not on the choice of lift $\phi _0$ . So the locus of $\varphi _0\in \lvert \mathcal {X}\rvert $ where $r(\xi _n)$ is constant is the image under $\rho $ of the locus of $\phi _0\in \left \lvert \widetilde {X}\right \rvert $ where the dimension is constant. By Chevalley’s theorem for Artin stacks [Reference Hall and RydhHR17, Theorem 5.2], it is therefore enough to show that the locus of such $\phi _0$ is constructible. This follows by applying [Sta21, Lemma 0BDI] to the $2$ -term complex $T_{\widetilde {X}/\mathcal {X}}\to T_{\widetilde {X}/k}$ and using the fact that $\left \lvert \widetilde {X}\right \rvert $ is Noetherian, so that all locally constructible sets are constructible.

Lemma 3.35. Let $\mathcal {Y}$ , $\mathcal {Z}$ and $\mathcal {F}$ be finite-type Artin stacks over k, let $\mathcal {Y} \to \mathcal {Z}$ be a k-morphism and assume that for any field extensions $k'$ of k and any k-morphism $\operatorname {\mathrm {Spec}}(k') \to \mathcal {Z}$ , there exists a $k'$ -isomorphism

$$ \begin{align*} \left(\mathcal{Y} \times_{\mathcal{Z}} \operatorname{\mathrm{Spec}}(k')\right)_{\mathrm{red}} \cong \left(\mathcal{F} \times_{\operatorname{\mathrm{Spec}}(k)} \operatorname{\mathrm{Spec}}(k')\right)_{\mathrm{red}}. \end{align*} $$

Then

$$ \begin{align*} \mathrm{e}(\mathcal{Y}) = \mathrm{e}(\mathcal{F})\mathrm{e}(\mathcal{Z}) \in K_0(\mathbf{Stack}_k). \end{align*} $$

Proof. Because $\mathcal {Z}$ can be stratified by quotient stacks [Reference KreschKre99, Proposition 3.5.9], we may assume that $\mathcal {Z} = [Z / G]$ for some finite-type scheme Z over k with an action by a general linear group G over k. Let $\mathcal {Y}' = \mathcal {Y} \times _{\mathcal {Z}} Z$ . Because $Z \to \mathcal {Z}$ and $\mathcal {Y}' \to \mathcal {Y}$ are G-torsors and G is a special group, [Reference EkedahlEke09, Proposition 1.1(ii)] gives

$$ \begin{align*} \mathrm{e}(Z) &= \mathrm{e}(G)\mathrm{e}(\mathcal{Z}),\\ \mathrm{e}(\mathcal{Y}') &= \mathrm{e}(G)\mathrm{e}(\mathcal{Y}). \end{align*} $$

By the hypotheses on $\mathcal {Y} \to \mathcal {Z}$ , Proposition 2.8 implies that $\mathcal {Y}' \to Z$ is a piecewise trivial fibration with fibre $\mathcal {F}$ , so in particular,

$$ \begin{align*} \mathrm{e}(\mathcal{Y}') = \mathrm{e}(\mathcal{F})\mathrm{e}(Z). \end{align*} $$

Thus,

$$ \begin{align*} \mathrm{e}(G)\mathrm{e}(\mathcal{Y}) = \mathrm{e}(G)\mathrm{e}(\mathcal{F})\mathrm{e}(\mathcal{Z}). \end{align*} $$

Because G is a special group, $\mathrm {e}(G)^{-1} \in K_0(\mathbf {Stack}_k)$ , so we are done.

We may now prove Theorem 3.33.

Proof of Theorem 3.33

By definition, there exist some $n_0 \in \mathbb {N}$ and some constructible subset $\mathcal {C}_{n_0} \subset \left \lvert \mathscr {L}_{n_0}(\mathcal {X})\right \rvert $ such that $\mathcal {C} = \left (\theta _{n_0}\right )^{-1}\left (\mathcal {C}_{n_0}\right )$ . Because $\mathcal {X}$ is smooth, infinitesimal lifting implies that the truncation maps $\theta _n: \lvert \mathscr {L}(\mathcal {X})\rvert \to \lvert \mathscr {L}_n(\mathcal {X})\rvert $ are all surjective, so

$$ \begin{align*} \theta_n(\mathcal{C}) = \begin{cases} \left(\theta^n_{n_0}\right)^{-1}\left(\mathcal{C}_{n_0}\right), & n \geq n_0, \\ \theta^{n_0}_n\left(\mathcal{C}_{n_0}\right), & n < n_0. \end{cases} \end{align*} $$

Thus all $\theta _n(\mathcal {C})$ are constructible – immediately for $n \geq n_0$ , and by Chevalley’s theorem for Artin stacks [Reference Hall and RydhHR17, Theorem 5.2] for $n < n_0$ .

The remainder of the theorem then follows from the fact that Lemma 3.34 and Lemma 3.35 imply that for any $n \geq n_0$ ,

$$ \begin{align*} \mathrm{e}(\theta_n(\mathcal{C})) = \left(\theta^n_{n_0}\right)^{-1}\left(\mathcal{C}_{n_0}\right) = \mathrm{e}\left(\mathcal{C}_{n_0}\right)\mathbb{L}^{\left(n-n_0\right)\dim\mathcal{X}}.\\[-39pt] \end{align*} $$

4 Fibres of the map of arcs

Our goal in this section is to give a combinatorial characterisation of the fibres of $\mathscr {L}(\pi )\colon \mathscr {L}(\mathcal {X})\to \mathscr {L}(X)$ , where $\mathcal {X}$ is a fantastack and $\pi \colon \mathcal {X}\to X$ is its good moduli space map (see Theorem 4.9). We accomplish this goal by first defining the tropicalisation of arcs both for toric varieties and for toric stacks.

4.1 Tropicalising arcs of toric stacks

Given a toric variety $X=\operatorname {\mathrm {Spec}}(k[P])$ , a k-algebra R and an arc $\varphi \in \mathscr {L}\left (\widetilde {X}\right )(R)$ , we denote by $\varphi ^*(p)$ the image of p under , where the latter map is the pullback corresponding to $\varphi $ .

Definition 4.1. If $\sigma $ is a pointed rational cone on a finite-rank lattice N and $k'$ is a field extension of k, we define the tropicalisation map

$$ \begin{align*} \operatorname{\mathrm{trop}}: \mathscr{L}(X_\sigma)(k') \to \operatorname{\mathrm{Hom}}(\sigma^\vee\cap N^*, \mathbb{N} \cup \{\infty\}) \end{align*} $$

by $\operatorname {\mathrm {trop}}(\varphi )(p):=\operatorname {\mathrm {ord}}_t\varphi ^*(p)$ , where $\operatorname {\mathrm {ord}}_t$ denotes the order of vanishing at t.

More generally, if $\left (\sigma ,\nu \colon \widetilde {N}\to N\right )$ is a stacky fan with $\sigma $ a pointed cone and $\mathcal {X}:=\mathcal {X}_{\sigma ,\nu }:=[X_\sigma /G_\nu ]$ is the corresponding toric stack, then we define the tropicalisation map on isomorphism classes of arcs

$$ \begin{align*} \operatorname{\mathrm{trop}}: \overline{\mathscr{L}(\mathcal{X})}(k') \to \operatorname{\mathrm{Hom}}\left(\sigma^\vee\cap \widetilde{N}^*, \mathbb{N} \cup \{\infty\}\right) \end{align*} $$

as follows. If $\psi \in \mathscr {L}(\mathcal {X})(k')$ , then fix a finite field extension $k''$ of $k'$ and a lift $\widetilde {\psi }\in \mathscr {L}(X_\sigma )(k'')$ of $\psi $ . We define $\operatorname {\mathrm {trop}}(\psi ):=\operatorname {\mathrm {trop}}\left (\widetilde {\psi }\right )$ . We show in Lemma 4.4 that this is well defined.

Remark 4.2. Note that we have a natural inclusion

$$ \begin{align*} \sigma\cap N=\operatorname{\mathrm{Hom}}(\sigma^\vee\cap N^*, \mathbb{N})\subset\operatorname{\mathrm{Hom}}(\sigma^\vee\cap N^*, \mathbb{N} \cup \{\infty\}). \end{align*} $$

Lemma 4.3. Let $\Omega $ be any field of characteristic $0$ and let $G=\mathbb {G}_m^r\times \prod _{i=1}^N\mu _{n_i}$ . Then every G-torsor over $\operatorname {\mathrm {Spec}}(\Omega [[t]])$ is isomorphic to the pullback of a G-torsor over $\operatorname {\mathrm {Spec}}(\Omega )$ .

Proof. Let $q\colon \operatorname {\mathrm {Spec}}(\Omega [[t]])\to \operatorname {\mathrm {Spec}}(\Omega )$ denote the structure map. Since G is an étale group scheme, G-torsors on any $\Omega $ -scheme Y are classified up to isomorphism by $H^1_{et}(Y,G)=H^1_{et}(Y,\mathbb {G}_m)^{\oplus r}\oplus \bigoplus _{i=1}^N H^1_{et}\left (Y,\mu _{n_i}\right )$ . In particular, it suffices to show that the pullback map $q^*\colon H^1_{et}(\operatorname {\mathrm {Spec}}(\Omega ),G)\to H^1_{et}(\operatorname {\mathrm {Spec}}(\Omega [[t]]),G)$ is an isomorphism when G is either $\mathbb {G}_m$ or $\mu _n$ .

We first handle the case $G=\mathbb {G}_m$ . Since $H^1_{et}(Y,\mathbb {G}_m)=\mathrm {Pic}(Y)$ and since both $\mathrm {Pic}(\operatorname {\mathrm {Spec}}(\Omega ))$ and $\mathrm {Pic}(\operatorname {\mathrm {Spec}}(\Omega [[t]]))$ are trivial, we see that $q^*$ is an isomorphism.

We next handle the case $G=\mu _n$ . From the Kummer sequence

$$ \begin{align*} 1\to\mu_n\to\mathbb{G}_m\xrightarrow{\times n}\mathbb{G}_m\to1, \end{align*} $$

we see that if Y is any $\Omega $ -scheme with trivial Picard group, we have

$$ \begin{align*} H^1_{et}(Y,\mu_n)=\mathcal{O}_Y(Y)^*/(\mathcal{O}_Y(Y)^*)^n \end{align*} $$

(see, for example, [Reference MilneMil80, p. 125]). So it remains to show that $k^*/(k^*)^n\to k[[t]]^*/(k[[t]]^*)^n$ is an isomorphism. Since every element $f(t)\in k[[t]]^*$ can be written uniquely as $ag(t)$ with $a\in k^*$ and $g(t)\in k[[t]]^*$ with $g(t)-1\in tk[[t]]$ , it is enough to prove that such $g(t)$ are in $(k[[t]]^*)^n$ . This follows immediately from Hensel’s lemma: the polynomial $P(x)=x^n-g(t)\in k[[t]][x]$ has a root, since $P(1)=0$ mod t and $P'(1)\neq 0$ mod t.

Lemma 4.4. With notation as in Definition 4.1, such a lift $\widetilde {\psi }$ exists, and $\operatorname {\mathrm {trop}}(\psi )$ is independent of both $k''$ and $\widetilde {\psi }$ .

Proof. For ease of notation, write $F:=\sigma ^\vee \cap \widetilde {N}^*$ and $G:=G_\nu =\operatorname {\mathrm {Spec}}(k[A])$ , where A is a finitely generated abelian group. Note that the G-action on $X_\sigma $ corresponds to a monoid map $\eta \colon F\to A$ . The arc $\psi $ corresponds to a G-torsor and G-equivariant map $Q\to X_\sigma $ . By Lemma 4.3, Q is isomorphic to the pullback of a G-torsor over $\operatorname {\mathrm {Spec}}(k')$ , which can be itself be trivialised after a finite field extension $k''$ . Thus, after base change to , we obtain a trivialisation of Q and hence a lift $\widetilde {\psi }$ .

Next, it is clear that if $\widetilde {\psi }\in \mathscr {L}(X_\sigma )(k'')$ is a lift of