Tropical varieties for non-archimedean analytic spaces

摘要

Generalizing the construction from tropical algebraic geometry, we associate to every ( irreducible d-dimensional) closed analytic subvariety of G(m)(n) m a tropical variety in R-n with respect to a complete non-archimedean place. By methods of analytic and formal geometry, we prove that the tropical variety is a totally concave locally finite union of d-dimensional polytopes. For an algebraic morphism f : X'. A to a totally degenerate abelian variety A, we give a bound for the dimension of f( X') in terms of the singularities of a strictly semistable model of X'. A closed d-dimensional subvariety X of A induces a periodic tropical variety. A generalization of Mumford's construction yields models of X and A which can be handled with the theory of toric varieties. For a canonically metrized line bundle (L) over bar on A, the measures c(1)((L) over bar vertical bar x)(lambda d) are piecewise Haar measures on X. Using methods of convex geometry, we give an explicit description of these measures in terms of tropical geometry. In a subsequent paper, this is a key step in the proof of Bogomolov's conjecture for totally degenerate abelian varieties over function fields.