Book contents
- Frontmatter
- Contents
- List of Contributors
- Preface
- 1 Birational Geometry of Algebraic Varieties and Shokurov’s Work
- 2 ACC for Log Canonical Thresholds for Complex Analytic Spaces
- 3 Conjectures on the Kodaira Dimension
- 4 Characterizing Terminal Fano Threefolds with the Smallest Anti-Canonical Volume, II
- 5 Uniform Rational Polytopes for Iitaka Dimensions
- 6 MMP for Algebraically Integrable Foliations
- 7 On Toric Fano Fibrations
- 8 Q-Fano Threefolds of Fano Index 13
- 9 Reflective Hyperbolic 2-Elementary Lattices, K3 Surfaces and Hyperkahler Manifolds
- 10 The Relative Du Bois Complex – on a Question of S. Zucker
- 11 Factorization Presentations
- 12 Spectrum Bounds in Geometry
- 13 On the DCC of Iitaka Volumes
- 14 Shokurov’s Index Conjecture for Quotient Singularities
- 15 A Note on the Sarkisov Program
- 16 Cluster Varieties and Toric Specializations of Fano Varieties
- 17 Birational Rigidity and Alpha Invariants of Fano Varieties
- 18 On F-Pure Inversion of Adjunction
- 19 On Termination of Flips and Fundamental Groups
- 20 Motivic Integration on Berkovich Spaces
20 - Motivic Integration on Berkovich Spaces
Published online by Cambridge University Press: 06 December 2024
Book contents
- Frontmatter
- Contents
- List of Contributors
- Preface
- 1 Birational Geometry of Algebraic Varieties and Shokurov’s Work
- 2 ACC for Log Canonical Thresholds for Complex Analytic Spaces
- 3 Conjectures on the Kodaira Dimension
- 4 Characterizing Terminal Fano Threefolds with the Smallest Anti-Canonical Volume, II
- 5 Uniform Rational Polytopes for Iitaka Dimensions
- 6 MMP for Algebraically Integrable Foliations
- 7 On Toric Fano Fibrations
- 8 Q-Fano Threefolds of Fano Index 13
- 9 Reflective Hyperbolic 2-Elementary Lattices, K3 Surfaces and Hyperkahler Manifolds
- 10 The Relative Du Bois Complex – on a Question of S. Zucker
- 11 Factorization Presentations
- 12 Spectrum Bounds in Geometry
- 13 On the DCC of Iitaka Volumes
- 14 Shokurov’s Index Conjecture for Quotient Singularities
- 15 A Note on the Sarkisov Program
- 16 Cluster Varieties and Toric Specializations of Fano Varieties
- 17 Birational Rigidity and Alpha Invariants of Fano Varieties
- 18 On F-Pure Inversion of Adjunction
- 19 On Termination of Flips and Fundamental Groups
- 20 Motivic Integration on Berkovich Spaces
Summary
We define a motivic measure on the Berkovich analytification of an algebraic variety defined over a trivially valued field, and introduce motivic integration in this setting. The construction is geometric with a similar spirit to Kontsevich’s original definition and leads to the formulation of a functorial theory that mirrors, in this aspect, the approach of Cluckers and Loeser via constructible motivic functions. A version of the integral over nontrivially valued fields and its relation to Hrushovski and Kazhdan’s integration are also discussed.
- Type
- Chapter
- Information
- Higher Dimensional Algebraic GeometryA Volume in Honor of V. V. Shokurov, pp. 382 - 440Publisher: Cambridge University PressPrint publication year: 2025