Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton

Abstract

We associate a weight function to pairs (X,!) consisting of a smooth and proper variety X over a complete discretely valued field and a pluricanonical form ! on X. This weight function is a real-valued function on the non-archimedean analytification of X. It is piecewise affine on the skeleton of any regular model with strict normal crossings of X, and strictly ascending as one moves away from the skeleton. We apply these properties to the study of the Kontsevich-Soibelman skeleton of (X, !), and we prove that this skeleton is connected when X has geometric genus one and ! is a canonical form on X. This result can be viewed as an analog of the Shokurov-Kollar connectedness theorem in birational geometry.