Motivic integration on Berkovich spaces

TL;DR

This paper develops a geometric motivic measure and integration theory on Berkovich spaces, extending previous frameworks to trivially valued fields and connecting with existing nontrivial valuation theories.

Contribution

It introduces a new motivic integration framework on Berkovich spaces over trivially valued fields, aligning with Kontsevich's approach and relating to Hrushovski-Kazhdan's work.

Findings

Defined a motivic measure on Berkovich analytifications

Established a functorial motivic integration theory

Connected trivial and nontrivial valuation cases

Abstract

We define a motivic measure on the Berkovich analytification of an algebraic variety defined over a trivially valued field, and introduce motivic integration in this setting. The construction is geometric with a similar spirit as Kontsevich's original definition, and leads to the formulation of a functorial theory which mirrors, in this aspect, the approach of Cluckers and Loeser via constructibe motivic functions. A version of the integral over nontrivially valued fields and its relation to Hrushovski and Kazhdan's integration are also discussed.