Equivariant motivic integration on special formal schemes

TL;DR

This paper develops an equivariant motivic integration framework for special formal schemes, extending previous algebraic results and introducing a motivic Milnor fiber concept for formal power series.

Contribution

It constructs an equivariant motivic integration for special formal schemes and establishes a change of variables formula, connecting to prior work on algebraic varieties.

Findings

Established an equivariant Néron smoothening for formal schemes.

Proved the change of variable formula in the motivic integration context.

Introduced the motivic Milnor fiber for formal power series.

Abstract

We construct, based on Nicaise's article in Math. Ann. in 2009, an equivariant geometric motivic integration for special formal schemes, such that when applying to algebraizable formal schemes, we can revisit our previous work in 2020 on equivariant motivic integration for algebraic varieties. We prove the change of variable formula for the integral by pointing out the existence of an equivariant N\'eron smoothening for a flat generically smooth special formal scheme. We also define the motivic Milnor fiber of a formal power series and predict that it is the right quantity to define the motivic Milnor fiber of a germ of complex analytic functions.