Motivic Integration

“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)

• 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.