Since its creation by Kontsevich in 1995, motivic integration has developed quickly in several directions and has found applications in various domains, such as singularity theory and the Langlands program. In its development, it incorporated tools from model theory and nonarchimedean geometry. The aim of the present book is to give an introduction to different theories of motivic integration and related topics.

Motivic integration is a theory of integration for various classes of geometric objects. One term in this “trade-name” seems unusual: motivic. What is a motive and in what sense is this theory of integration motivic? These questions lie at the heart of the theory. We will try to answer them in Section 4. First, we briefly introduce the two other protagonists of the present book: model theory and non-archimedean geometry. We will explain below how they interact with the theory of motivic integration.

Model theory

A central notion in model theory is language. A language ℒ is a collection of symbols, divided into three types: function symbols, relation symbols, and constant symbols. For every function symbol and relation symbol, one specifies the number of arguments. A formula in the language ℒ is built from these symbols, variables, Boolean combinations and quantifiers.

In his seminar notes [5] on Rigid Analytic Spaces, Tate developed a powerful analogue of classical complex analysis over complete non-Archimedean fields. The toplogy of such a field K is quite weak, as it is totally disconnected. For example, any disk in K may be viewed as a disjoint union of smaller disks. Therefore a local definition of analyticity cannot yield useful global results of the type we know them from complex analysis. For example, this concerns global identity theorems and also the fact that analytic functions on classical projective spaces are constant.

Following the method from complex analysis and looking at functions admitting local power series expansions, we end up with so-called locally analytic spaces, or wobbly analytic spaces in the terminology of [5]. Since these are lacking connectivity in the non-Archimedean case, Tate equipped them with an additional structure, a so-called h-structure, which provides a certain substitute for connectivity. All this is guided by the idea that polydisks in affine n-spaces should be viewed as being “connected” and that their analytic functions should admit globally convergent power series expansions.

Having such an idea in mind, there is another theory, which provides useful guidance, namely, the theory of Grothendieck's schemes (of locally finite type over K).

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.

Introduction

This paper gives a partial survey of the joint project [7] with K. Fujiwara (Nagoya Univ.), dealing with the part consisting of topological-ring theoretical aspects in rigid geometry, which has not been presented in our previous survey [6].

In classical algebraic geometry, finite type algebras over a field play a cornerstone role as the so-called ‘coordinate rings’, that is, the rings of regular functions on affine varieties. Scheme theory replaces affine varieties by affine schemes, and thus deals with arbitrary rings as basic building blocks. Still in scheme theory, however, fields and finite type algebras over a field keep their privileged position; fields are ‘point objects’, and finite type algebras over a field are ‘fiber objects’ over a point for locally of finite type morphisms between schemes.

In rigid geometry, on the other hand, we usually start with the so-called affinoids, that is, certain ‘affine-like’ objects, which come from topologically of finite type algebras over a complete non-archimedean valued field. This situation can be seen as an analogue of classical algebraic geometry, and thus one wants to ask for a scheme-theory-like generalization of rigid geometry. There are already several attempts to this goal; one of such attempts is via the relative rigid spaces by Bosch and Lükebohmert [1]. The most important question in these attempts is: what kind of topological rings should one start with?

These notes will give some very basic definitions and results from model theory. They contain many examples, and in particular discuss extensively the various languages used to study valued fields. They are intended as giving the necessary background to read the papers by Cluckers-Loeser, Delon, Halupczok and Kowalski in this volume. We also mention a few recent results or directions of research in the model theory of valued fields, but omit completely those themes which will be discussed elsewhere in this volume. So for instance, we do not even mention motivic integration.

People interested in learning more model theory should consult standard model theory books. For instance: D. Marker, Model Theory: an Introduction, Graduate Texts in Mathematics 217, Springer-Verlag New York, 2002; C.C. Chang, H.J. Keisler, Model Theory, North-Holland Publishing Company, Amsterdam 1973; W. Hodges, A shorter model theory, Cambridge University Press, 1997.

Languages, structures, satisfaction

Languages and structures

Languages. A language is a collection ℒ, finite or infinite, of symbols. These symbols are of three kinds:

1. – function symbols,

2. – relation symbols,

3. – constant symbols.

To each function symbol f is associated a number n(f) ∈ ℕ>0, and to each relation symbol R a number n(R) ∈ ℕ>0. The numbers n(f) and n(R) are called the arities of the function f, resp., the relation R.