Abstract
We associate to an analytic subvariety of a torus a tropical variety. In the first part, we generalize the results from tropical algebraic geometry to this non-archimedean analytic situation. The periodic case is applied to a totally degenerate abelian variety A. We obtain a dimensionality bound for the range of an algebraic morphism from a smooth variety Y to A in terms of the singularities of a strictly semistable model of Y. The main result is that Chambert-Loir's canonical measures on a closed subvariety of A induce piecewise Haar measures on the associated tropical variety and we give an explicit description of these measures in terms of tropical geometry. In a subsequent paper, this is the key step in the proof of Bogomolov's conjecture for totally degenerate abelian varieties over function fields.
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