Math @ Duke

Publications [#345770] of Joseph D Rabinoff
Papers Published
 Amini, O; Baker, M; Brugallé, E; Rabinoff, J, Lifting harmonic morphisms i: Metrized complexes and berkovich skeleta,
Research in Mathematical Sciences, vol. 2 no. 1
(December, 2015) [doi]
(last updated on 2021/08/04)
Abstract: Let K be an algebraically closed, complete nonArchimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic Kcurves 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 Kcurves 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 LiuLorenzini, 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 Ωmetrized complexes of kcurves lifts to a finite morphism of Kcurves. 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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