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Tropical varieties for non-archimedean analytic spaces

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Abstract

Generalizing the construction from tropical algebraic geometry, we associate to every (irreducible d-dimensional) closed analytic subvariety of \(\mathbb{G}_{m}^{n}\) a tropical variety in ℝn 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 L̄ on A, the measures c 1(L̄| X )∧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.

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  1. Fachbereich Mathematik, Universität Dortmund, Vogelpothsweg 87, Dortmund, D-44221, Germany

    Walter Gubler

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  1. Walter Gubler
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Gubler, W. Tropical varieties for non-archimedean analytic spaces. Invent. math. 169, 321–376 (2007). https://doi.org/10.1007/s00222-007-0048-z

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