Grothendieck rings of definable subassignments and equivariant motivic measures

TL;DR

This paper explores the structure of definable subassignments in algebraic geometry, establishing connections between different motivic measures and providing a framework for comparing them within arc spaces.

Contribution

It introduces a new categorical framework linking definable subassignments to arc spaces, enabling comparison of motivic measures by Cluckers-Loeser and Denef-Loeser.

Findings

Categories of definable subassignments are equivalent to semi-algebraic and constructible subsets of arc spaces.

The paper establishes a method to compare different motivic measures within these classes.

Results facilitate a unified understanding of motivic integration in algebraic geometry.

Abstract

The paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These materials allow us to compare the motivic measure of Cluckers-Loeser and the one of Denef-Loeser in certain classes of definable subassignments.