Let K be a complete and algebraically closed field with value group Lambda and residue field k,
and let phi : X' -> X be a finite morphism of smooth, proper, irreducible, stable marked algebraic curves over K.
We show that phi gives rise in a canonical way to a finite and effective harmonic morphism of Lambda-metric graphs,
and more generally to a finite harmonic morphism of Lambda-metrized complexes of k-curves.
These canonical ``abstract tropicalizations'' are constructed using
Berkovich's notion of the skeleton of an analytic curve. Our arguments
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 metric graphs/tropical curves
and metrized complexes to morphisms of curves over K. We prove that
every tamely ramified finite harmonic morphism of Lambda-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.
We prove a similar result concerning the existence of liftings for
morphisms of tropical curves,
except
the genus of the source curve
can no longer be fixed.
From this point of view, morphisms of metrized complexes are better behaved than morphisms of tropical curves.
The caveat on the genus in the lifting result for tropical
curves is necessary: we show by example that the gonality of a
tropical curve C can be strictly smaller than the gonality of any smooth proper curve X of the same genus lifting C.
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 as an application classify all (augmented) metric graphs arising as
the tropicalization of a hyperelliptic curve.
We also 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.