Non-archimedean tame topology and stably dominated types

TL;DR

This paper introduces topological methods into the model theory of valued fields, constructing an analogue of Berkovich analytification that reveals new topological properties of algebraic varieties over non-archimedean fields.

Contribution

It defines a new topological space for algebraic varieties over non-archimedean fields, proving properties like contractibility and finite homotopy types, without smoothness assumptions.

Findings

Berkovich space retracts to a finite simplicial complex

Berkovich space is locally contractible

Homotopy types vary finitely in algebraic families

Abstract

Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the Berkovich analytification $V^{an}$ of $V$, and deduce several new results on Berkovich spaces from it. In particular we show that $V^{an}$ retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on $V$. When $V$ varies in an algebraic family, we show that the homotopy type of $V^{an}$ takes only a finite number of values. The space $\hat {V}$ is obtained by defining a topology on the pro-definable set of stably dominated types on $V$. The key result is the construction of a pro-definable strong retraction of $\hat {V}$ to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.