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.

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f 1 , . . . , fn are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f 1 , . . . , fn. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind [EKL06] and Gubler [Gub07b]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers. 1 1.4. From the perspective of a tropical geometer, the theory of rigid spaces is useful because the analytic topology on R n is much better approximated by the rigid-analytic topology on the torus G n m . For example, the unit box [0, 1] n is an analytic neighborhood in the Euclidean space R n , yet trop −1 ([0, 1] n ) ⊂ |G n m | is the n-fold product of the annulus {ξ ∈ K × : val(ξ) ∈ [0, 1]}, which is a very nicely behaved rigid-analytic object (it is a smooth affinoid space), but is not the set of points underlying a subscheme. Similarly, (R ∪ {∞}) n can naturally be regarded as the tropicalization of the affine space A n (see §5), under which identification the unit ball B n is the inverse image under trop of (R ≥0 ∪ {∞}) n (a neighborhood of the point (∞, . . . , ∞)). 1.5.The following example is an application of rigid-analytic methods to a tropically-local problem. Let U {0} = trop −1 ({0}) ⊂ |G n m |. This is an affinoid space, which implies (see §4) that it is the maximal spectrum of the algebrais the associated subscheme then |Y | ∩ U {0} is identified with the set of maximal ideals of K U {0} containing the ideal aK U {0} , so to show that 0 ∈ Trop(Y ) is equivalent to showing that a does not generate the unit ideal in K U {0} . This ends up being equivalent to the well-known criterion that the initial ideal of a at 0 contain no monomials. The characterization of the tropicalization (or rather the Bieri-Groves set) of a scheme by initial ideals was proved by Einsiedler-Kapranov-Lind [EKL06] using these methods; we give a treatment below (7.9) which also applies to tropicalizations of analytic spaces. (The first complete proof of this theorem was given by Draisma [Dra08, Theorem 4.2] and also uses affinoid algebras, albeit in a different way; see 7.13.)1.6. A family of translations of a tropical variety parameterized by an interval corresponds to a family of subvarieties of a torus parametrized by a rigid-analytic annulus. We study such families in order to prove the theorem indicated in (1.1,ii); to illustrate the main idea we will sketch a special case., and suppose that Trop(f 1 ) ∩ ...

ABSTRACT. We prove that if X, X ′ are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X)∩ Trop(X ′ ) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(∆) and its associated extended tropicalization N R (∆); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the closure of C in N R (∆). The proof involves a result on continuity of intersection numbers in the context of non-Archimedean analytic spaces.

ABSTRACT. In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (nonaugmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve.This article is the second in a series of two.Throughout this paper, unless explicitly stated otherwise, K denotes a complete algebraically closed nonArchimedean field with nontrivial valuation val : K → R ∪ {∞}. Its valuation ring is denoted R, its maximal ideal is mR, and the residue field is k = R/mR. We denote the value group of K by Λ = val(K × ) ⊂ R.