Berkovich spaces embed in Euclidean spaces

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.

💡 Research Summary

The paper investigates the topological embedding of Berkovich analytifications of algebraic varieties into Euclidean spaces. Let K be a complete non‑archimedean field equipped with a non‑archimedean absolute value, and assume that K possesses a countable dense subset. Under these hypotheses the authors prove that for any d‑dimensional quasi‑projective K‑scheme V, its Berkovich analytification V^{an} admits a topological embedding into the real Euclidean space ℝ^{2d+1}. The result rests on two main ingredients. First, the countable dense subset guarantees that V^{an} is a second‑countable, metrizable space, which is essential for applying classical embedding theorems. Second, the authors show that V^{an} can be expressed as an inverse limit of finite simplicial complexes of dimension at most d. This is achieved by using formal models and the associated skeleta in the sense of Berkovich and Temkin. Each finite complex can be embedded into ℝ^{2d+1} by the Menger–Nöbeling theorem, which states that any compact metric space of topological dimension ≤ d embeds in ℝ^{2d+1}. By carefully choosing compatible embeddings at each stage of the inverse system, they obtain a continuous injective map f : V^{an}→ℝ^{2d+1} that is a homeomorphism onto its image.

A second, more refined theorem concerns the case of curves (d = 1) when the value group |K^×| is dense in ℝ_{>0}. In this situation the authors prove that V^{an} is a local dendrite, i.e., a compact, connected, locally contractible metric space in which every point has a neighbourhood that is a dendrite (a tree‑like continuum without simple closed curves). They give a complete description of the homeomorphism type of V^{an} by invoking the theory of local dendrites: each point of V^{an} is either an endpoint or a branch point of finite order, and the global structure is obtained by gluing together a finite or countable collection of arcs and trees according to the combinatorics of the reduction graph of a semistable model of the curve. This classification recovers the familiar picture of Berkovich curves as a “skeleton” (a metric graph) together with a family of open balls attached at the vertices, and it shows that, under the density hypothesis on the value group, the metric on the skeleton can be realized with arbitrary positive lengths, yielding a full continuum of possible dendritic shapes.

The paper also discusses several examples and applications. For projective space ℙ^n_K, the analytification ℙ^n_K^{an} embeds in ℝ^{2n+1}, providing a concrete visualisation of the non‑archimedean projective line and higher‑dimensional projective spaces. For Tate curves and Mumford curves, the authors illustrate how the dendritic structure appears explicitly, with the skeleton being a circle (for Tate curves) or a finite graph (for Mumford curves) and the attached “ends” corresponding to type‑1 points. They note that when the value group is not dense (e.g., K = ℚ_p), the same construction yields an embedding but possibly into a higher‑dimensional Euclidean space, and they leave the optimal dimension problem for future work.

In summary, the authors establish a bridge between non‑archimedean analytic geometry and classical Euclidean topology: any quasi‑projective Berkovich space over a countable‑dense non‑archimedean field can be faithfully represented inside a Euclidean space of dimension 2d+1, and in the one‑dimensional case with dense value group the space is completely classified as a local dendrite. This advances our geometric intuition for Berkovich spaces and opens the way for further investigations into embeddings, dimension bounds, and the interplay with tropical and metric graph theories.