Tropical varieties for non-archimedean analytic spaces
arXiv:math/0609383 · doi:10.1007/s00222-007-0048-z
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.
46 pages; submitted. Improved formulations, minor errors corrected, Example 9.8 added
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Cited by in corpus (21)
- Stable pair, tropical, and log canonical compact moduli of del Pezzo surfaces
- The Bogomolov conjecture for totally degenerate abelian varieties
- Functorial tropicalization of logarithmic schemes: The case of constant coefficients
- Non-archimedean canonical measures on abelian varieties
- A tropical approach to non-archimedean Arakelov theory
- Faithful realizability of tropical curves
- Special divisors on marked chains of cycles
- Lifting divisors on a generic chain of loops
- Strict supports of canonical measures and applications to the geometric Bogomolov conjecture
- Skeletons of Prym varieties and Brill--Noether theory
- Differentiability of relative volumes over an arbitrary non-Archimedean field
- Tropicalization of the moduli space of stable maps
- Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration (with an appendix by Walter Gubler: The minimal dimension of a canonical measure)
- Prym-Brill-Noether Loci of special curves
- Heights of hypersurfaces in toric varieties
- Faltings height and Néron-Tate height of a theta divisor
- Rank of divisors on hyperelliptic curves and graphs under specialization
- The arithmetic Hodge Index Theorem and rigidity of dynamical systems over function fields
- Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
- Higher dimensional essential minima and equidistribution of cycles
- Monge-Ampère measures for toric metrics on abelian varieties