Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

TL;DR

This paper studies how finite morphisms of algebraic curves over non-Archimedean fields are controlled by combinatorial skeleta, proving lifting theorems for harmonic morphisms and providing new analytic insights into curve reduction and tropical geometry.

Contribution

It introduces a framework linking finite morphisms of algebraic curves to harmonic morphisms of metrized complexes, with new lifting theorems and analytic proofs of classical results.

Findings

Finite harmonic morphisms of metrized complexes lift to morphisms of algebraic curves.

Established analytic proofs for semistable reduction theorems.

Provided a classification of lifts with automorphisms for tamely ramified morphisms.

Abstract

Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger "skeletonized" versions of some foundational results ofLiu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of \Lambda-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 Sa\"idi and Wewers. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of N\'eron models, providing a negative answer to a question of Ribet motivated by number theory. This article is the first in a series of two. The second article contains several applications of our lifting results to questions about lifting morphisms of tropical curves.