Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

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 -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.

Degeneration of Linear Series from the Tropical Point of View and Applications

“…A more direct proof of Corollary 7.2, which in fact proves a stronger statement replacing Γ with an arbitrary metrized complex of curves, and the field K with any complete and algebraically closed non-Archimedean field whose value group contains all edge lengths in some model for Γ, can be found in Theorem 3.24 of [ABBR14a]. The proof uses formal and rigid geometry.…”

Lifting matroid divisors on tropical curves

“…4 We recall from [1] that a lift of Ŵ M to a metrized complex means associating a P 1 k for each vertex v of the graph, which we denote C v , and a point on C v for each edge incident to v. A lift of the divisor D M is a choice of a point on C e for each element e of M.…”

“…4,5,6, and 7 to give a matroid whose realization space projects to SpecR ⊂ SpecS n . The realization of this matroid is determined by the values of the y i , together with a number of parameters, such as the height of the horizontal lines, which are allowed to be generic, and thus the realization space is an open subset of A N × SpecR.…”

“…A sufficient condition for lifting a rank 1 divisor is for it to be the underlying graph of a metrized complex which has a tame harmonic morphism to a tree (see [4,Sec. 2] for precise definitions).…”

A metric graph satisfying w41=1$w_4^1 = 1$ that cannot be lifted to a curve satisfying dim⁡ (W41)=1$\dim \;(W_4^1 ) = 1$

“…E) such that induces a finite harmonic morphism Q ! of degree 2 (Section 4 of [16] uses no assumption on the characteristic of k). Since Q is a metric graph (augmentation map identically zero) this is also the case for hence is a metric graph.…”