Motivic Integration
1st Edition
Published on 15. September 2018
Book
Hardback
XX, 526 pages
978-1-4939-7885-4 (ISBN)
Description
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.
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.
Reviews / Votes
"The introduction contains a very succinct presentation of the contents of each chapter, which efficiently gives a first rough idea of how everything is presented and organised. . , the author of this review believes that a more detailed account of some of the chapters, somehow designed as a mathematical tour guide, could be useful to others, keeping it as much readable as possible and focusing on the main concepts introduced in this very rich book." (Loïs Faisant, zbMATH 1545.14021, 2025)
More details
Series
Edition
1st ed. 2018
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Illustrations
47 s/w Abbildungen
XX, 526 p. 47 illus.
Dimensions
Height: 241 mm
Width: 160 mm
Thickness: 35 mm
Weight
980 gr
ISBN-13
978-1-4939-7885-4 (9781493978854)
DOI
10.1007/978-1-4939-7887-8
Schweitzer Classification
Other editions
Additional editions
Antoine Chambert-Loir | Johannes Nicaise | Julien Sebag
Motivic Integration
Book
12/2019
Birkhauser Boston Inc
€149.79
Shipment within 15-20 days
Antoine Chambert-Loir | Johannes Nicaise | Julien Sebag
Motivic Integration
E-Book
09/2018
1st Edition
Birkhäuser
€139.09
Available for download
Persons
Antoine Chambert-Loir (1971- ) is Professor of Mathematics at Université de Paris. He previously taught at the École normale supérieure, University Pierre-et-Marie-Curie, École polytechnique, University of Rennes and Université Paris-Sud. Besides coauthoring a 3-volume book of exercises in analysis, his books include A Field Guide to Algebra and, as co-author, the prize-winning Motivic Integration.
His current research is in arithmetic geometry.
Content
Introduction.- Prologue:
p
-adic Integration.- Analytic Manifolds.- The Theorem of Batyrev-Kontsevich.- Igusa's Local Zeta Function.- The Grothendieck Ring of Varieties.- Additive Invariants on Algebraic Varieties.- Motivic Measures.- Cohomolical Realizations.- Localization, Completion, and Modification.- The Theorem of Bittner.- The Theorem of Larsen-Lunts and Its Applications.- Arc Schemes.- Weil Restriction.- Jet Schemes.- The Arc Scheme of a Variety.- Topological Properties of Arc Schemes.- The Theorem of Grinberg-Kazhdan-Drinfeld.- Greenberg Schemes.- Complete Discrete Valuation Rings.- The Ring Schemes
R
n.- Greenberg Schemes.- Topological Properties of Greenberg Schemes.- Structure Theoremes for Greenberg Schemes.- Greenberg Approximation on Formal Schemes.- The Structure of the Truncation Morphisms.- Greenberg Schemes and Morphisms of Formal Schemes.- Motivic Integration.- Motivic Integration in the Smooth Case.- The Volume of a Constructibel Subset.- Measurable Subsets of Greenberg Schemes.- Motivic Integrals.- Semi-algebraic Subsets of Greenberg Schemes.- Applications.- Kapranov's Motivic Zeta Function.- Valuations and the Space of Arcs.- Motivic Volume and Birational Invariants.- Denef-Loeser's Zeta Function and the Monodromy Conjecture.- Motivic Invariants of Non-Archimedean Analytic Spaces.- Motivic Zeta Functions of Formal Shemes and Analytic Spaces.- Motivic Serre Invariants of Algebraic Varieties.- Appendix.- Constructibility in Algebraic Geometry.- Birational Geometry.- Formal and Non-Archimedean Geometry.- Index.- Bibliography.