Tropical structures on Berkovich skeleta and wild ramification (Art Waeterschoot, KU Leuven)

Résumé: Smooth Berkovich analytic spaces often deformation retract onto polyhedral subsets called skeleta. Over a discretely valued ground field these skeleta are commonly constructed as dual intersection complexes of toroidal formal models. I will explain how to equip such skeleta with 'tropical' structure (roughly, the datum of a metrized balanced polyhedral complex), extending earlier work of Gubler-Werner-Rabinoff beyond the semistable setting. The benefit is that one can now develop potential theory directly on the skeleton even when wild ramification is present. The main result is a Riemann-Hurwitz formula for finite-to-one covers of skeleta, showing that the change of curvature is measured by the Laplacian of a wild ramification function known as the 'different', complementing earlier work of Temkin and collaborators. We illustrate the formula with examples coming from arithmetic curves. Based on ArXiv 2503.13875 and 2504.16543.