On the structure of nonarchimedean analytic curves

Abstract

Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space X^an. We define semistable vertex sets of X^an and their associated skeleta, which are essentially finite metric graphs embedded in X^an. We prove a folklore theorem which states that semistable vertex sets of X are in natural bijective correspondence with semistable models of X, thus showing that our notion of skeleton coincides with the standard definition of Berkovich. We use the skeletal theory to define a canonical metric on H(X^an) := X^an - X(K), and we give a proof of Thuillier's nonarchimedean Poincaré-Lelong formula in this language using results of Bosch and Lütkebohmert.

23 pages. This an expanded version of section 5 of arXiv:1104.0320 which appears in the conference proceedings "Tropical and Non-Archimedean Geometry"