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We associate a weight function to pairs consisting of a smooth and propervariety X over a complete discretely valued field and a differential form on Xof maximal degree. This weight function is a real-valued function on thenon-archimedean analytification of X. It is piecewise affine on the skeleton ofany regular model with strict normal crossings of X, and strictly ascending asone moves away from the skeleton. We apply these properties to the study of theKontsevich-Soibelman skeleton of such a pair, and we prove that this skeletonis connected when X has geometric genus one. This result can be viewed as ananalog of the Shokurov-Kollar connectedness theorem in birational geometry.

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