Open Access

Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

  • Omid Amini1,
  • Matthew Baker2Email author,
  • Erwan Brugallé3 and
  • Joseph Rabinoff2
Research in the Mathematical Sciences20152:7

DOI: 10.1186/s40687-014-0019-0

Received: 9 December 2014

Accepted: 30 December 2014

Published: 10 June 2015


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 of Liu-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 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. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Néron 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.