Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

  • Omid Amini
  • Matthew Baker
  • Erwan Brugallé
  • Joseph Rabinoff
Open Access


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.


Open Ball Harmonic Morphism Tropical Curf Finite Morphism Admissible Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank Amaury Thuillier and Antoine Ducros for their help with some of the descent arguments in (5.2). We are grateful to Andrew Obus for a number of useful comments based on a careful reading of the first arXiv version of this manuscript. We are also very grateful to the referee for quickly and thoroughly checking all of the technical arguments in the paper, pointing out several errors and suggesting many improvements. MB was partially supported by NSF grant DMS-1201473. EB was partially supported by the ANR-09-BLAN-0039-01.


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Copyright information

© Amini et al.; licensee Springer. 2015

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors and Affiliations

  • Omid Amini
    • 1
  • Matthew Baker
    • 2
  • Erwan Brugallé
    • 3
  • Joseph Rabinoff
    • 2
  1. 1.CNRS - Département de mathématiques et applicationsRue d’UlmParis
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlanta GA 30332-0160USA
  3. 3.Université Pierre et Marie CurieParisFrance

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