Which morphisms between tropical curves come from algebraic geometry?
Which morphisms between tropical curves come from algebraic geometry?
A finite morphism between stable marked curves over a non-archimedean field gives rise in a natural way to a morphism of metric graphs (also known as "abstract tropical curves"). 听We study the question of which morphisms between abstract tropical curves arise from this construction. 听This leads us naturally to the notion of听metrized complexes of curves (which are tropical curves endowed with some extra structure) and of harmonic morphisms between metrized complexes. 听We show using Berkovich's theory of analytic spaces that every tamely ramified finite harmonic morphism of metrized complexes of curves arises from a听morphism of algebraic curves. 听This generalizes and provides new analytic proofs of earlier results of Saidi and Wewers. 听We also present counterexamples to some possible strengthenings of this result; for example, the gonality of a tropical curve can be strictly smaller than the gonality of any smooth proper lift. 听As an application of the above considerations, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Neron models, providing a negative answer to a question of Ribet.