[Paper Review] Toward motivic integration over wild Deligne-Mumford stacks

Takehiko Yasuda|arXiv (Cornell University)|Feb 13, 2013

Algebraic Geometry and Number Theory30 references10 citations

TL;DR

This paper proposes a generalization of motivic integration to wild Deligne-Mumford stacks—where stabilizers may have order divisible by the characteristic—by formulating a conjectural change of variables formula involving twisted arcs and weight functions. The key contribution is a conjectural link between stringy invariants of quotient singularities and motivic counts of extensions of local fields, offering a motivic McKay correspondence in the wild setting.

ABSTRACT

We discuss how the motivic integration will be generalized to wild Deligne-Mumford stacks, that is, stabilizers may have order divisible by the characteristic of the base or residue field. We pose several conjectures on this topic. We also present some possible applications concerning stringy invariants, resolution of singularities, and weighted counts of extensions of local fields.

Motivation & Objective

• To extend motivic integration to wild Deligne-Mumford stacks, where group actions have orders divisible by the characteristic.
• To formulate a change of variables formula for twisted arcs in the context of wild quotient stacks.
• To connect stringy invariants of quotient singularities with motivic counts of extensions of local fields.
• To explore applications to the non-existence of crepant or functorial resolutions of singularities.
• To provide a motivic analogue of Serre’s and Bhargava’s mass formulas for local field extensions.

Proposed method

• Define twisted arcs of a Deligne-Mumford stack as representable morphisms from quotient stacks associated to Galois covers of the formal disk.
• Introduce canonical weight functions $ w_{\mathcal{X}} $ and $ w_{\mathcal{Y}} $ on the spaces of twisted arcs $ \mathcal{J}_{\infty}\mathcal{X} $ and $ \mathcal{J}_{\infty}\mathcal{Y} $.
• Propose a conjectural change of variables formula involving the Jacobian order $ \mathrm{ord}\,\mathrm{Jac}_{f} $, motivic measure $ \mu_{\mathcal{X}} $, and the Lefschetz motive $ \mathbb{L} $.
• Use the technique of untwisting to reduce twisted arc spaces to non-twisted ones, providing justification for the conjecture in the linear case.
• Analyze the case where $ \mathcal{Y} = [\mathbb{A}_D^d / G] $ and $ \mathcal{X} = \mathbb{A}_D^d / G $, deriving explicit conjectural expressions for weight functions.
• Relate the motivic mass of extensions of a local field to stringy invariants of the quotient singularity $ \mathbb{A}_D^d / G $.

Experimental results

Research questions

• RQ1 How can motivic integration be generalized to wild Deligne-Mumford stacks where stabilizers have order divisible by the characteristic?
• RQ2 What is the correct form of the change of variables formula in the wild setting, and how do weight functions and Jacobian orders behave?
• RQ3 Can stringy invariants of quotient singularities be expressed as motivic counts of local field extensions?
• RQ4 What constraints do stringy invariants impose on the existence of crepant or functorial resolutions?
• RQ5 Under what conditions does the motivic mass formula for extensions of local fields coincide with the stringy invariant of the associated singularity?

Key findings

• The conjectural change of variables formula relates motivic integrals on twisted arc spaces of wild DM stacks via Jacobian order and weight functions.
• For a cyclic group $ G $ of prime order $ p $ acting on a representation $ V $ with $ D_V = p $, the stringy invariant at $ T=0 $ evaluates to 1, indicating potential topological obstructions to resolution.
• The paper derives a condition for the existence of a crepant resolution: $ D_V = p $ is necessary, as otherwise the Euler characteristic realization fails to be an integer.
• The stringy invariant $ P_{\mathrm{st}}(X) $ must be a polynomial in $ T $, not in $ T^{1/r} $, if a crepant resolution exists.
• The motivic mass of extensions of a local field with residue field size $ q $ is conjecturally equal to the stringy invariant of the quotient singularity $ \mathbb{A}^d / G $, generalizing Serre’s and Bhargava’s mass formulas.
• The dual complex of the exceptional divisor in a resolution must have Euler characteristic equal to the stringy invariant evaluated at $ T=0 $, providing a topological obstruction to resolution.