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

We associate a weight function to pairs consisting of a smooth and proper variety X over a complete discretely valued field and a differential form on X of maximal degree. 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 such a pair, and we prove that this skeleton is connected when X has geometric genus one. This result can be viewed as an analog of the Shokurov-Kollar connectedness theorem in birational geometry.