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

TL;DR

This paper introduces a weight function on non-archimedean analytic spaces associated with algebraic varieties and differential forms, revealing properties of the Kontsevich-Soibelman skeleton and proving its connectedness for genus one varieties.

Contribution

It defines a new weight function on non-archimedean spaces and demonstrates its properties, including the connectedness of the Kontsevich-Soibelman skeleton for genus one varieties.

Findings

The weight function is piecewise affine on the skeleton.

The weight function is strictly ascending away from the skeleton.

The Kontsevich-Soibelman skeleton is connected for genus one varieties.

Abstract

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.