Berkovich Spaces and Applications

  • Antoine Ducros
  • Charles Favre
  • Johannes Nicaise

Part of the Lecture Notes in Mathematics book series (LNM, volume 2119)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Berkovich Analytic Spaces

    1. Front Matter
      Pages 1-1
    2. Michael Temkin
      Pages 3-66
  3. Applications to Geometry

    1. Front Matter
      Pages 133-133
    2. Bertrand Rémy, Amaury Thuillier, Annette Werner
      Pages 141-202
  4. Valuation Spaces and Dynamics

    1. Front Matter
      Pages 203-203
    2. Mattias Jonsson
      Pages 205-366
  5. Back Matter
    Pages 415-416

About this book


We present an introduction to Berkovich’s theory of non-archimedean analytic spaces that emphasizes its applications in various fields. The first part contains surveys of a foundational nature, including an introduction to Berkovich analytic spaces by M. Temkin, and to étale cohomology by A. Ducros, as well as a short note by C. Favre on the topology of some Berkovich spaces. The second part focuses on applications to geometry. A second text by A. Ducros contains a new proof of the fact that the higher direct images of a coherent sheaf under a proper map are coherent, and B. Rémy, A. Thuillier and A. Werner provide an overview of their work on the compactification of Bruhat-Tits buildings using Berkovich analytic geometry.

The third and final part explores the relationship between non-archimedean geometry and dynamics. A contribution by M. Jonsson contains a thorough discussion of non-archimedean dynamical systems in dimension 1 and 2. Finally a survey by J.-P. Otal gives an account of Morgan-Shalen's theory of compactification of character varieties.

This book will provide the reader with enough material on the basic concepts and constructions related to Berkovich spaces to move on to more advanced research articles on the subject. We also hope that the applications presented here will inspire the reader to discover new settings where these beautiful and intricate objects might arise.


Berkovich analytic spaces Bruhat-Tits buildings Etale cohomology Morgan-Shalen's compactification Non-Archimedean dynamics

Editors and affiliations

  • Antoine Ducros
    • 1
  • Charles Favre
    • 2
  • Johannes Nicaise
    • 3
  1. 1.Pierre and Marie Curie UniversityParisFrance
  2. 2.Ecole PolytechniquePalaiseau CedexFrance
  3. 3.University of LeuvenHeverleeBelgium

Bibliographic information