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 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.
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Acknowledgments
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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Amini, O., Baker, M., Brugallé, E. et al. Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Mathematical Sciences 2, 7 (2015). https://doi.org/10.1186/s40687-014-0019-0
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DOI: https://doi.org/10.1186/s40687-014-0019-0