[Paper Review] Integration motivique sur les schemas formels

Julien Sebag|arXiv (Cornell University)|Dec 21, 2001

Logic, programming, and type systems4 citations

TL;DR

This paper extends motivic integration to formal schemes over a complete discrete valuation ring with perfect residue field, introducing a motivic measure on Greenberg's scheme of a formal scheme. It establishes a change-of-variables formula for tamely ramified morphisms, generalizing Denef-Loeser's theory and unifying motivic and p-adic integration. A key result is that the motivic integral of a gauge form on a Calabi-Yau variety depends only on the generic fiber, recovering Batyrev's invariance and Serre's invariant in the p-adic case.

ABSTRACT

We develop the theory of motivic integration for formal schemes

Motivation & Objective

• To generalize motivic integration from algebraic varieties to formal schemes over a complete discrete valuation ring with perfect residue field.
• To define a motivic measure on Greenberg's scheme of a formal scheme, extending the arc scheme construction.
• To establish a change-of-variables formula for tamely ramified morphisms in this new setting.
• To recover classical invariants such as Batyrev's Hodge invariants and Serre's p-adic volume via motivic integration.
• To unify motivic and p-adic integration by showing that the motivic integral of a gauge form on a Calabi-Yau variety depends only on the generic fiber.

Proposed method

• Define the Greenberg scheme $\mathrm{Gr}(\mathcal{X})$ of a formally smooth formal scheme $\mathcal{X}$ over $R$, which generalizes the arc scheme in positive characteristic.
• Construct a motivic measure $\mu_{\mathcal{X}}$ with values in the completion of the Grothendieck ring of varieties, using cylinder sets.
• Introduce the notion of exponentially integrable functions, particularly those involving the $\pi$-adic valuation of the Jacobian determinant.
• Define measurable sets as finite unions of cylinders and prove stability under inverse images and direct images under tame morphisms.
• Establish a change-of-variables formula for tame morphisms $h: \mathcal{Y} \to \mathcal{X}$, involving the order of the Jacobian determinant.
• Prove that the motivic integral of a gauge form on a Calabi-Yau generic fiber equals the class of the special fiber of any weak Néron model.

Experimental results

Research questions

• RQ1 Can motivic integration be extended from algebraic varieties to formal schemes over a complete discrete valuation ring with perfect residue field, especially when the residue field has positive characteristic?
• RQ2 How should the arc scheme construction be generalized in the absence of a morphism $k \hookrightarrow R$?
• RQ3 What is the appropriate notion of measurable sets and integrable functions in the formal scheme setting?
• RQ4 Does a change-of-variables formula hold for tame morphisms in this generalized motivic integration theory?
• RQ5 Can classical invariants such as Batyrev's Hodge numbers and Serre's p-adic volume be recovered as motivic integrals in this framework?

Key findings

• The motivic integral of a gauge form on a Calabi-Yau variety over a $p$-adic field is independent of the choice of gauge form and equals the class of the special fiber of any weak Néron model in the Grothendieck ring.
• For a $p$-adic Calabi-Yau variety with a proper smooth model over $R$, the motivic integral of a generator of the canonical sheaf equals the class of the special fiber in $\widehat{\mathcal{M}}$, generalizing Batyrev's result.
• The motivic integral of a gauge form specializes to Serre's $p$-adic volume invariant when $K$ is a finite extension of $\mathbf{Q}_p$, as shown in Corollary 4.6.3 of [17].
• The Greenberg scheme $\mathrm{Gr}(\mathcal{X})$ of a formal scheme $\mathcal{X}$ over $R$ generalizes the arc scheme $\mathcal{L}(X)$, and they are isomorphic when $R = k[[t]]$ and $\mathcal{X}$ is the $\pi$-adic completion of a $k$-variety.
• The change-of-variables formula holds for tame morphisms $h: \mathcal{Y} \to \mathcal{X}$, with the Jacobian determinant's $\pi$-adic valuation appearing in the exponent of the motivic measure.
• The theory provides a unifying framework that generalizes both classical motivic integration and $p$-adic integration, as demonstrated in [17].