MODNET
Research Training Network in Model Theory
 Publications > Preprint server > Preprint Number 434 Preprint Number 434 434. Antoine Ducros Espaces de Berkovich, polytopes, squelettes et théorie des modèles E-mail: (email address protected by JavaScript. Please enable JavaScript to contact) Submission date: 30 March 2012. 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. 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. Mathematics Subject Classification: 14 G 14G22, 12J10 Keywords and phrases: Full text arXiv 1203.6498: pdf, ps.

 Last updated: March 31 2012 20:58 Please send your corrections to: The e-mail address is protected, enable Javascript to see it