Espaces de Berkovich, polytopes, squelettes et théorie des modèles

Details Espaces de Berkovich, polytopes, squelettes et théorie des modèles Espaces de Berkovich, polytopes, squelettes et théorie des modèles

2012-03-29

arxiv.org/abs/1203.6498v2

Mathematics

Algebraic Geometry

In the first version I forgot to mention a bound on the dimension in th. 3.3, assertion 1); I've added it

Scientific paper

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. 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 a new proof of both of them, based upon model-theoretic tools instead of de Jong's alterations, which were used in the former proofs. 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.

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