Les espaces de Berkovich sont mod\'er\'es, d'apr\`es E. Hrushovski et F. Loeser

TL;DR

This paper discusses the tameness properties of algebraic Berkovich spaces, showing they have a homotopy type of a compact polyhedron, based on model-theoretic methods by Hrushovski and Loeser.

Contribution

It applies model-theoretic tools, especially stably dominated types, to establish topological tameness of Berkovich spaces, a novel approach in non-Archimedean geometry.

Findings

Berkovich spaces have the homotopy type of a compact polyhedron.

Model-theoretic methods reveal tameness properties of these spaces.

The approach bridges model theory and non-Archimedean geometry.

Abstract

This is the (revised) printed version of the talk no 1056 (june 2012) of the Bourbaki seminar, which will be published in an Ast\'erisque volume. This is a report on a paper by Hrushovski and Loeser (/arxiv:1009.0252). In this paper they establish, using in a crucial way model-theoretic tools and especially the notion of a stably dominated type, various tameness properties of the topology of algebraic Berkovich spaces (e.g. they prove that such a space has the homotopy type of a compact polyhedron).