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Motivic integration over wild Deligne-Mumford stacks

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Pith's one-line read A motivic integration theory for wild Deligne-Mumford stacks proves the wild McKay correspondence for arbitrary finite groups.

desk verdict Real advance: motivic integration over wild DM stacks plus the wild McKay correspondence for arbitrary finite groups, but the central formula's right-hand side is defined in a companion paper [TY19], so the theorem is conditional on that well-definedness result. read the letter →

arxiv 1908.02932 v2 pith:5O6LQOV3 submitted 2019-08-08 math.AG math.NT

Takehiko Yasuda This is my paper · ORCID
classification math.AGmath.NT MSC 14E1814D2314E1614B0511G2511S15
keywords motivicintegrationDeligne-MumfordstackswildMcKaycorrespondencestringymotivesformallogpairsGrothendieckringofvarietiesramification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops motivic integration over Deligne-Mumford stacks whose stabilizers may have order divisible by the characteristic, a case previously out of reach, and uses it to prove the wild McKay correspondence for linear actions of arbitrary finite groups. The central formula says that for a finite group G acting linearly on affine space A^d_k without pseudo-reflections, the stringy motive of the quotient A^d_k/G is the integral over the moduli space Δ_G of G-torsors over the punctured formal disk of $L^{{d-v}}$, where v is a constructible function tied to ramification. Before this paper, the wild case was known only for cyclic groups of prime order. If correct, the result gives a motivic analogue of Bhargava's mass formula and a way to compute discrepancies of quotient singularities from arithmetic data.

What carries the argument

The load-bearing construction is the untwisting stack. For a formal DM stack X over the power series ring, one fixes the moduli stack Θ of Galoisian group schemes and the universal integral model E_Θ of a G-torsor over the punctured formal disk, whose quotient is the universal twisted formal disk. The untwisting stack is Utg_Γ(X)=Hom^rep(E_Γ,X_Γ), the Hom stack of representable morphisms, modified by passing to the rig-pure part along a flattening stratification Γ→Θ. The stack of twisted arcs is then the arc stack J_∞ Utg_Γ(X)^pur. The paper's change-of-variables formula for proper birational morphisms compares motivic integrals on J_∞Y and J_∞X, with the new shift function s_X measuring the difference between the Jacobian orders of the coarse-modulus projection and the untwisting morphism; in the linear case s_X(γ)=-v(E), where v is the length of a module of equivariant maps, essentially the Artin conductor.

What would settle it

Take G=Z/$p^{2}$ acting diagonally on $A^{2}$_k over a perfect field of characteristic p, compute the right side ∫$L^{{2-v}}$ by stratifying Δ_G according to ramification data, compute the left side from a log resolution or from the change-of-variables formula, and compare the two classes in the completed Grothendieck ring; any mismatch would refute the wild McKay correspondence in this case.

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Extended reading notes

Core claim

At the paper's core is the claim that twisted arcs on a wild stack can be untwisted: every representable morphism from a twisted formal disk to a stack is controlled by an ordinary arc on an 'untwisting stack', so that motivic integration on wild stacks reduces to integration on schemes. The paper constructs the untwisting stack as a Hom stack of representable morphisms from the universal integral model of a G-cover, stratifies it to make it flat, and proves a change-of-variables formula carrying two correction terms: the usual Jacobian order and a new shift function s_X. For the quotient stack [A^d_k/G], the shift function equals -v on the moduli space of G-torsors, yielding the wild McKay correspondence M_st(A^d_k/G)=∫_{Δ_G}$L^{{d-v}}$. The same machinery yields invariance of stringy motives of stacky log pairs under crepant morphisms and, by specializing to the symmetric group, the motivic version of Bhargava's mass formula.

Load-bearing premise

The formula's right-hand side presupposes that the integral over Δ_G is well defined in the completed Grothendieck ring; the paper invokes a companion result for that well-definedness rather than proving it here.

Editorial extensions

If this is right

  • For every finite group G with a linear action on A^d_k having no pseudo-reflection, the quotient's stringy motive equals the motivic integral over Δ_G of L^{d-v}, generalizing the known cyclic-prime-order case.
  • The motivic version of Bhargava's mass formula holds: ∫_{Δ_{S_n}}L^{-a}=Σ_{j=0}^{n-1}P(n,n-j)L^{-j}, with a the Artin conductor.
  • Crepant birational morphisms of stacky log pairs preserve stringy motives, including in characteristic dividing the stabilizer orders and for singular DM stacks.
  • Discrepancies of quotient singularities are computable from the integral: discrep(centers⊂X_sing;X)=d-1-max{dim X_sing, dim∫_{Δ_G\{o}}L^{d-v}}, and convergence of the integral is equivalent to log terminality when a log resolution exists.
  • Specializing to characteristic zero recovers the classical McKay sum over conjugacy classes Σ L^{d-age(g)}, and specializing to tame stacks recovers the earlier tame motivic McKay correspondence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the correspondence holds, stringy invariants of wild quotients become arithmetic objects: each stratum of G-torsors contributes a class determined by ramification data, which a cohomological realization would turn into congruences between counts of local field extensions and stringy Hodge numbers.
  • The invariance under crepant morphisms suggests the stringy motive defines a K-equivalence invariant for singular DM stacks in arbitrary characteristic, a property not established before for wild stabilizers.
  • A concrete testable extension is the case G=Z/p^2 in dimension two: writing the integral over Δ_G explicitly from Artin-Schreier data and comparing it with a resolution-theoretic computation would verify the whole mechanism in the first genuinely new wild case.
  • The untwisting technique appears to extend beyond linear actions to arbitrary finite group actions on formal schemes, since the local structure theorem reduces to linear actions only after passing to tangent representations; checking non-linear wild actions would widen the correspondence's scope.
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Referee Report

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Summary. The paper develops a theory of motivic integration for formal Deligne-Mumford stacks over a power series ring in arbitrary characteristic. It introduces untwisting stacks, constructs a space of twisted arcs, proves a change-of-variables formula for twisted arcs, and defines stringy motives for stacky log pairs and their invariance under crepant pseudo-modifications. The main application, Theorem 1.3 (Corollaries 14.4 and 16.3), establishes the wild motivic McKay correspondence for linear actions of arbitrary finite groups, generalizing the previously known cyclic-prime-order case, and yields a motivic version of Bhargava's mass formula as a special case.

Significance. If correct, this is a substantial contribution: it provides the first general wild motivic McKay correspondence for arbitrary finite groups and a motivic Bhargava mass formula, while also unifying and extending earlier tame and formal-scheme theories. The paper is detailed and proof-based, with explicit statements, an appendix on general stack-theoretic results, and careful treatment of the completed Grothendieck ring and its realizations. It is transparent about its reliance on the companion works [TY17] and [TY19]; in particular, the well-definedness of the integral over Δ_G in Theorem 1.3 is cited to [TY19] rather than proved here. I found no internal inconsistency in how that dependency is used, and the reliance is standard mathematical practice when the cited results are available.

minor comments (4)
  1. [Section 1.5 and the definition before Corollary 14.4] The right-hand side of Theorem 1.3 is an integral over Δ_G whose well-definedness as an element of the completed Grothendieck ring is cited to [TY19]. Since this is a necessary precondition for the statement, please state explicitly which theorem of [TY19] is being invoked and include its statement, or at least its theorem number, so that the reader can verify the hypotheses without consulting the companion paper.
  2. [Lemma 14.3] The proof of Lemma 14.3 argues that after removing a constructible subset one obtains an open dense substack on which the boundary is vertical, and then concludes that v is constant on each connected component of Γ_G. The passage from the open dense substack to the full local constructibility statement should mention a noetherian induction or an explicit refinement of the stratification, because the argument as written only establishes the open dense substep.
  3. [Section 16] When passing from formal DM stacks over Df to DM stacks over k, the notation J∞X and Mst(X,A) is defined by base change to X×Df; it would help the reader if the text explicitly noted that the motivic measure and the shift function are the ones constructed in Section 11 after this base change, since the notation is reused for the non-formal setting.
  4. [Corollary 16.4] The proof of Corollary 16.4 is only a sketch and uses the quantity dim ∫_{Δ_G\{o}} L^{d-v} defined by a supremum over a countably infinite stratification. The authors state that this is independent of the stratification; a brief justification would be useful, especially because the integral may diverge and the supremum may be infinite.

Circularity Check

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No significant circularity: the wild McKay formula is derived from the untwisting change-of-variables theorem, and the integral over Δ_G is a computed identification, not an assumed input.

full rationale

The central derivation of Theorem 1.3 (Corollaries 14.4 and 16.3) is self-contained rather than circular. The stringy motive Mst(A^d/G) is expanded through the untwisting formalism (Theorem 13.3, Corollary 13.4) into a sum over connected components of the stratified parameter stack Γ, with weights L^{-v_{G,i}}. The function v is not defined as the shift function; it is defined independently as a normalized length 1/♯G · length(Hom(M,O_E)/(O_E·Ξ_{E,ι})) attached to a G-cover E, and Lemma 14.2 then proves s_X = -v. Corollary 14.4 identifies the computed sum with ∫_{Δ_G} L^{d-v} by exhibiting a geometrically bijective map from the untwisting strata to Δ_G and using the explicit definition of the integral as ∑_s {v^{-1}(s)} L^{d-s}; this is a derived equality, not a definitional tautology. The only external dependence of the statement is the well-definedness of the right-hand integral, which is delegated to the companion paper [TY19] both in §1.5 ('The well-definedness of this integral was proved in [TY19]') and before Corollary 14.4 ('In [TY19], we study this kind of integrals more systematically and especially proves the well-definedness of the above integral'). That is a precondition on the object in the formula, not a recycled version of the equality being proved, and as a stated theorem of prior work it functions as independent support rather than as a circular assumption. The proof of Lemma 14.3 on local constructibility of v is also supplied in the paper, even while a prior proof in [TY19] is acknowledged. The application to Bhargava's mass formula (Corollary 16.5) imports the separately established identification of Artin conductor with v from [WY15, Th. 4.8], again a citation to an earlier theorem, not a renaming of the present conclusion. Overall, no step in the derivation forces the conclusion by definition, by fitted data, or by a self-citation chain; the paper's main theorem is a genuine theorem resting on the developed motivic integration machinery.

Assumptions & free parameters 1 free parameters · 7 assumptions · 4 invented entities

The central claims rest on a large body of standard algebraic geometry plus several results from the author's prior works, especially [TY17, TY19] for the moduli of torsors and well-definedness of the main integral. The new entities are rigorous constructions central to the proof, not empirically fitted parameters.

free parameters (1)
  • r
    Positive integer used to adjoin the formal root L^{1/r} in the completed Grothendieck ring of varieties (Section 9). This is a technical choice, not fitted to data; all relevant functions take values in (1/r)Z.
assumptions (7)
  • standard math Keel-Mori theorem for existence of coarse moduli spaces of DM stacks
    Used in Lemma 2.12 and elsewhere to define coarse moduli spaces of formal DM stacks.
  • standard math Olsson's representability and finiteness of Hom stacks (Theorem 6.2)
    Used to construct untwisting stacks as Hom stacks in Section 6.
  • domain assumption Results of [TY17] on moduli of torsors over punctured formal disks, including uniformizability after ind-perfection (Prop. 5.19)
    Provides the moduli stack Theta of Galoisian torsors and the universal integral model E_Theta used throughout Sections 5-7.
  • domain assumption Well-definedness of the integral over Delta_G of L^{d-v} proved in [TY19]
    Used to define the right-hand side of the wild McKay correspondence (Theorem 1.3, Corollary 14.4).
  • domain assumption Results of [WY15] relating v to Artin and Swan conductors, and [Yas16] on tuning modules
    Used in Section 14 to identify the shift function with -v(E).
  • standard math Mittag-Leffler and approximation theorem for linear equations [Ron06, Th. 3.1]
    Used in Lemma 2.15 for invariant subrings.
  • standard math Invariance of small etale site under thickenings [Gro67, Sta20]
    Used in Proposition 2.18 and Lemma A.8.
invented entities (4)
  • Untwisting stack UtgGamma(X)^pur
    purpose: Reduces twisted arcs on X to untwisted arcs on a rigid-pure Hom stack, enabling the motivic measure and change of variables formula.
    Core new construction of the paper; no external empirical handle, but used to prove the wild McKay correspondence.
  • Shift function s_X on twisted arcs
    purpose: Corrects the change of variables formula for twisted arcs and appears in stringy motives; generalizes age and fermion shift.
    Defined via Jacobian orders of rig-etale morphisms in Definition 11.6.
  • Universal twisted formal disk E_Theta
    purpose: Parametrizes all twisted formal disks uniformly, enabling definition of the twisted arc stack J_infinity X.
    Constructed from the moduli of torsors Theta in Section 5.
  • Stringy motive Mst(X,A) for stacky log pairs independent evidence
    purpose: Generalizes Batyrev's stringy E-function and Denef-Loeser's motivic integrals; central invariant studied in the paper.
    Specializes to known invariants in the tame and characteristic-zero cases, providing consistency checks.

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Pith. "Pith review of Motivic integration over wild Deligne-Mumford stacks." pith.science (2026). https://pith.science/paper/5O6LQOV3

@misc{pith2026190802932,
  author       = {Pith},
  title        = {Pith review of: Motivic integration over wild Deligne-Mumford stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5O6LQOV3}},
  note         = {Machine review of arXiv:1908.02932}
}
read the original abstract

We develop the motivic integration theory over formal Deligne-Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne-Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava's mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms.

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