Tropical varieties for non-archimedean analytic spaces

Details Tropical varieties for non-archimedean analytic spaces Tropical varieties for non-archimedean analytic spaces

2006-09-14

arxiv.org/abs/math/0609383v2

Invent. Math. 169, No.2, 321-376 (2007)

Mathematics

Number Theory

46 pages; submitted. Improved formulations, minor errors corrected, Example 9.8 added

Scientific paper

10.1007/s00222-007-0048-z

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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