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5 - The Grothendieck ring of varieties

Published online by Cambridge University Press:  07 October 2011

By
Johannes Nicaise  and
Julien Sebag
Edited by
Raf Cluckers ,
Johannes Nicaise  and
Julien Sebag
Johannes Nicaise
Affiliation:
Katholieke Universiteit Leuven
Julien Sebag
Affiliation:
Université Rennes
Raf Cluckers
Affiliation:
Université de Lille
Johannes Nicaise
Affiliation:
Katholieke Universiteit Leuven, Belgium
Julien Sebag
Affiliation:
Université de Rennes I, France
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Summary

Introduction

Since its creation in the middle of the nineties, the theory of motivic integration has been developed in different directions, following a geometric and/or model-theoretic approach. The theory has profound applications in several areas of mathematics, such as algebraic geometry, singularity theory, number theory and representation theory.

A common feature of the different versions of motivic integration is that the integrals take their values in an appropriate Grothendieck ring, often the Grothendieck ring of varieties. Many applications of motivic integration involve equalities of certain motivic integrals, and hence equalities in the Grothendieck ring of varieties; see, for example, the Batyrev-Kontsevich Theorem [6], which motivated the introduction of motivic integration. Therefore, it is natural to ask for the geometric meaning behind such equalities in the Grothendieck ring.

Unfortunately, the Grothendieck ring of varieties is quite hard to grasp, and little is known about it; many basic and fundamental questions remain unanswered. The central question is arguably the one raised by Larsen and Lunts (see Section 6.2), for which only partial results have been obtained so far.

The present paper is a survey on the Grothendieck ring of varieties. We recall its definition (Section 3), and its main realization maps (Section 4). These realizations constitute the motivic nature of the Grothendieck ring. Besides, we motivate the study of the Grothendieck ring by listing the principal known results, and formulating some challenging open problems (Section 5), which are connected to fundamental questions in algebraic geometry.

Type
Chapter
Information
Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry , pp. 145 - 188
Publisher: Cambridge University Press
Print publication year: 2011

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