We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the analytification of a curve to the tropicalizations of its toric embeddings stabilize to isometries on finite subgraphs. Other applications include generalizations of Speyer's well-spacedness condition and the KatzMarkwig-Markwig results on tropical j-invariants.

ABSTRACT. Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space X an . We define semistable vertex sets of X an and their associated skeleta, which are essentially finite metric graphs embedded in X an . We prove a folklore theorem which states that semistable vertex sets of X are in natural bijective correspondence with semistable models of X, thus showing that our notion of skeleton coincides with the standard definition of Berkovich [Ber90]. We use the skeletal theory to define a canonical metric on H(X an ) ≔ X an X(K), and we give a proof of Thuillier's nonarchimedean Poincaré-Lelong formula in this language using results of Bosch and Lütkebohmert.

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.

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ABSTRACT. Let K be a complete, algebraically closed non-archimedean field with ring of integers K • and let X be a K-variety. We associate to the data of a strictly semistable K • -model X of X plus a suitable horizontal divisor H a skeleton S(X , H) in the analytification of X. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker-Payne-Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on S(X , H). For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a wellknown result by Sturmfels-Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.