Berkovich spaces embed in Euclidean spaces

TL;DR

This paper proves that Berkovich analytifications of quasi-projective schemes over certain nonarchimedean fields can be embedded into Euclidean spaces, and characterizes their topology in special cases.

Contribution

It establishes Euclidean embeddings for Berkovich spaces over fields with countable dense subsets, extending understanding of their topological structure.

Findings

Berkovich analytifications embed in R^{2d+1} for d-dimensional schemes.

For curves over dense value groups, the homeomorphism type is described using local dendrites.

The results apply to fields with countable dense subsets, broadening the class of spaces with known Euclidean embeddings.

Abstract

Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V^an of any d-dimensional quasi-projective scheme V over K embeds in R^{2d+1}. If, moreover, the value group of K is dense in R_{>0} and V is a curve, then we describe the homeomorphism type of V^an by using the theory of local dendrites.