Lifting harmonic morphisms of tropical curves, metrized complexes, and Berkovich skeleta
Omid Amini, Matthew H. Baker, Erwan Brugallé, Joseph Rabinoff · arXiv (Cornell University) · 2013
Let K be a complete and algebraically closed field with value group Λ and residue field k, and let ϕ: X ′ → X be a finite morphism of smooth, proper, irreducible, stable marked algebraic curves over K. We show that ϕ gives rise in a canonical way to a finite and effective harmonic morphism of Λ-metric graphs, and more generally to a finite harmonic morphism of Λ-metrized complexes of k-curves. These canonical “abstract tropicalizations ” are constructed using Berkovich’s notion of the skeleton of an analytic curve. Our arguments give analytic proofs of stronger “skeletonized ” versions of some foundational results of Liu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metric graphs/tropical curves and metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of Λ-metrized complexes of k-curves lifts to a finite morphism of K-curves. If in addition the ramification points are marked, we obtain a complete classification of all such lifts along with their automorphisms. This generalizes and provides new analytic proofs of earlier results of Saïdi and Wewers. We prove a similar result concerning the existence of liftings for morphisms of tropical curves, except the genus of the source curve can no longer be fixed. From this point of view, morphisms of metrized complexes are better behaved than morphisms of tropical curves. The caveat on the genus in the lifting