Espaces de Berkovich, polytopes, squelettes et th\'eorie des mod\`eles

Abstract

Let $X$ be an analytic space over a non-Archimedean, complete field $k$ and let $(f_1,..., f_n)$ be a family of invertible functions on $X$. Let $\phi$ the morphism $X\to G_m^n$ induced by the $f_i$'s, and let $t$ be the map $X\to (R^*_+)^n$ induced by the norms of the $f_i$'s. Let us recall two results. 1) The compact set $t(X)$ is a polytope of the $R$-vector space $(R^*_+)^n$ (we use the multiplicative notation) ; this is due to Berkovich in the locally algebraic case, and has been extended to the general case by the author. 2) If moreover $X$ is Hausdorff and $n$-dimensional, then the pre-image under $\phi$ of the skeleton $S_n$ of $G_m^n$ has a piecewise-linear structure making $\phi^{-1}(S_n)\to S_n$ a piecewise immersion ; this is due to the author. 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, non-trivially valued fields. Let us quickly explain what we mean by improving 1) and 2). - Concerning 1), we also prove that if $x\in X$, 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 $(t(U),t(x))$ and $(t(V),t(x))$ coincide. - Concerning 2), we prove that the piecewise linear structure on $\phi^{-1}(S_n)$ is canonical, that is, doesn't 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 $\phi_1,..., \phi_m$ are finitely many morphisms from $X\to G_m^n$, the union $\bigcup \phi_j(S_n)$ also inherits a canonical piecewise-linear structure.