Espaces de Berkovich, polytopes, squelettes et théorie des modèles

Ducros, Antoine

Confluentes Mathematici, Tome 4 (2012) no. 4.

Let X be an analytic space over a non-Archimedean, complete field k and let f = (f₁,…,f_n) be a family of invertible functions on X. Let us recall two results. (1) If X is compact, the compact set |f|(X) is a polytope of the ℝ-vector space ${\left({\mathbb{R}}_{+}^{\ast }\right)}^n$ (we use the multiplicative notation); this is due to Berkovich in the locally algebraic case (his proof made use of de Jong's alterations), and has been extended to the general case by the author. The locally algebraic case could also have been deduced quite formally from a former result by Bieri and Groves, based upon explicit computations on Newton polygons. (2) If moreover X is Hausdorff and n-dimensional, and if ϕ denotes the morphism $X\to {\mathbb{G}}_{m,k}^{n,\text{an}}$ induced by f, then the pre-image of the skeleton S_n of ${\mathbb{G}}_{m,k}^{n,\text{an}}$ under ϕ has a piecewise-linear structure making ϕ^-1(S_n) → S_n a piecewise immersion; this is due to the author, and his proof also made use of de Jong's alterations. In this article, we improve (1) and (2), and give new proofs of both of them. Our proofs are based upon the model theory of algebraically closed, nontrivially valued fields and do not involve de Jong's alterations. Let us quickly explain what we mean by improving (1) and (2).

• Concerning (1), we also prove that if x ∈ X, there exists a compact analytic neighborhood U of x, such that for every compact analytic neighborhood V of x in X, the germs of polytopes (|f|(V), |f|(x)) and (|f|(U), |f|(x)) coincide.
• Concerning (2), we prove that the piecewise linear structure on ϕ^-1(S_n) is canonical, that is, does not depend on the map we choose to write it as a pre-image of the skeleton; we thus answer a question which was asked to us by Temkin.

Moreover, we prove that the pre-image of the skeleton "stabilizes after a finite, separable ground field extension", and that if ϕ₁,…,ϕ_m are finitely many morphisms from X to ${\mathbb{G}}_{m,k}^{n,\text{an}}$, the union $\bigcup {\phi }_j^{-1}(S_n)$ also inherits a canonical piecewise-linear structure.

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