Motivic integration on special rigid varieties and the motivic integral identity conjecture

TL;DR

This paper proves the motivic integral identity conjecture for formal functions by developing a new equivariant motivic integration framework on special rigid varieties, establishing motivic volumes and their properties.

Contribution

It introduces a novel equivariant motivic integration theory for special rigid varieties and proves the motivic integral identity conjecture.

Findings

Established a motivic volume for special smooth rigid varieties.

Proved the motivic integral identity conjecture for formal functions.

Extended motivic volume to a Grothendieck ring homomorphism.

Abstract

We prove in this paper the original version of Kontsevich and Soibelman's motivic integral identity conjecture for formal functions by developing a novel framework for equivariant motivic integration on special rigid varieties. This theory is built upon our recent research on equivariant motivic integration within the realm of special formal schemes. The central element of our approach lies in demonstrating that two formal models of a given smooth rigid variety can be dominated by a third formal model. Notably, a similar assertion for quasi-compact rigid varieties was obtained by Bosch, L\"utkebohmert, and Raynaud in 1993. Consequently, we establish a concept of motivic volume for a special smooth rigid variety, ensuring independence from the selection of its models. We demonstrate that this motivic volume can be extended to a homomorphism from a certain Grothendieck ring of special smooth rigid varieties to the classical Grothendieck ring of varieties. Moreover, our developed motivic volume exhibits a Fubini-type property, which recovers Nicaise and Payne's motivic Fubini theorem for the tropicalization map.