Motivic Integration

Motivic Integration

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 scheme...

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Bibliographic Details
Main Authors: Chambert-Loir, Antoine (Author, http://id.loc.gov/vocabulary/relators/aut), Nicaise, Johannes (http://id.loc.gov/vocabulary/relators/aut), Sebag, Julien (http://id.loc.gov/vocabulary/relators/aut)
Corporate Author: SpringerLink (Online service)
Format: Electronic eBook
Language:English
Published: New York, NY : Springer New York : Imprint: Birkhäuser, 2018.
Edition:1st ed. 2018.
Series:Progress in Mathematics, 325
Subjects:
Algebraic geometry.
K-theory.
Algebraic Geometry.
K-Theory.
Online Access:Full Text via HEAL-Link
Table of Contents:
  • 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 Rn
  • 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.

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