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Über dieses Buch

This monograph focuses on the geometric theory of motivic integration, which takes its values in the Grothendieck ring of varieties. This theory is rooted in a groundbreaking idea of Kontsevich and was further developed by Denef & Loeser and Sebag. It is presented in the context of formal schemes over a discrete valuation ring, without any restriction on the residue characteristic. The text first discusses the main features of the Grothendieck ring of varieties, arc schemes, and Greenberg schemes. It then moves on to motivic integration and its applications to birational geometry and non-Archimedean geometry. Also included in the work is a prologue on p-adic analytic manifolds, which served as a model for motivic integration.
With its extensive discussion of preliminaries and applications, this book is an ideal resource for graduate students of algebraic geometry and researchers of motivic integration. It will also serve as a motivation for more recent and sophisticated theories that have been developed since.



Chapter 1. Prologue: p-Adic Integration

Motivic integration and some of its applications take they very inspiration from results of p-adic integration, that is, integration on analytic manifolds over non-Archimedean locally compact fields.
Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag

Chapter 2. The Grothendieck Ring of Varieties

In this chapter, we define the Grothendieck ring of varieties over an arbitrary base scheme. This is a ring of virtual varieties up to cut-and-paste relations; it takes a central place in the theory of motivic integration, because (after a suitable localization and/or completion) it serves as the ring where motivic integrals take their values. After the basic definitions in section 1, we define the notion of motivic measures, which are ring morphisms from the Grothendieck ring to other rings with a more explicit structure. Motivic measures are fundamental both for the understanding of Grothendieck ring itself and for extracting geometric information from its elements. Among the motivic measures, we develop in sections 3 and 5 the cohomological and motivic realizations. In sections 5 and 6, we study the main structure theorems for the Grothendieck ring over a field of characteristic zero: the theorems of Bittner and Larsen-Lunts. Bittner’s theorem gives a presentation of the Grothendieck ring in terms of smooth projective varieties and blow-up relations, which is quite useful to construct motivic measures. The theorem of Larsen and Lunts relates equalities in the Grothendieck ring to the notion of stable birational equivalence. In section 4 we discuss a process of dimensional completion for the Grothendieck ring of varieties.
Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag

Chapter 3. Arc Schemes

This chapter is devoted to the study of the arc schemes associated with schemes X defined over arbitrary base schemes S. Informally speaking, an arc on a scheme X is a formal germ of a curve on X, and the arc scheme \(\mathcal{L}_{\infty }(X/S)\) parameterizes the arcs in the fibers of XS. The arc scheme was originally defined by Nash (1995) to obtain information about the structure of algebraic singularities and their resolutions. It also takes the spotlight in the theory of motivic integration, as the measure space over which functions are integrated. In section 2, we construct the spaces of jets, which are approximate arcs up to finite order. The construction consists of a process of restriction of scalars à la Weil, presented in section 1. We then explain in section 3 why arc schemes exist and how to recover them as limits of jet schemes. We study their topology in section 4 and their differential properties in section 3. Finally, in section 5 we explain a local structure theorem for arc schemes due to Grinberg and Kazhdan (2000) and Drinfeld (2002).
Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag

Chapter 4. Greenberg Schemes

Let R be a complete discrete valuation ring, let \(\mathfrak{m}\) be its maximal ideal, and let k be its residue field. When R = k[​[t]​] and X is a k-scheme, we defined in chapter 3 the schemes of jets \(\mathcal{L}_{n}(X/k)\) and the scheme of arcs \(\mathcal{L}_{\infty }(X/k)\) on X whose k-points are in canonical bijection with \(X(R/\mathfrak{m}^{n+1})\) and X(R), respectively.
Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag

Chapter 5. Structure Theorems for Greenberg Schemes

Throughout this chapter, we denote by R a complete discrete valuation ring with maximal ideal \(\mathfrak{m}\) and residue field k. For every integer n⩾0, we set \(R_{n} = R/\mathfrak{m}^{n+1}\).
Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag

Chapter 6. Motivic Integration

In this chapter we develop the theory of motivic integration on formal schemes \(\mathfrak{X}\) over a complete discrete valuation ring R, introduced by Sebag (2004a) and generalizing the constructions of Kontsevich (1995), Denef and Loeser (1999), and Looijenga (2002).
Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag

Chapter 7. Applications

This final chapter is devoted to a selection of notable applications of motivic integration.
Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag


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