Tropical varieties for non-archimedean analytic spaces

By Walter Gubler. 2006

122 citations recorded by the source corpus.

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 c1(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.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1870e97eb2743d72a034e2f53912457d6b152bef

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Citations in the corpus

The Bogomolov conjecture for totally degenerate abelian varieties (2006, background): “REM 13 5.4 Let X be an irreducible closed subvariety of A of dimension d. The main theorem of tropical analytic geometry says that val(Xan v ) is a rational polytopal set in Rn/Λ of pure dimension d ([Gu3], Theorem 6.9). For P ∈ X(K), the Gal(K/K)-orbit is denoted by O(P). Let (Pm)m∈I be a net in X(”
Skeletons and Fans of Logarithmic Structures (2015, background): “bspace of (A1)an consisting of the points j:j a;r for a2k and r2[0;1) as well as the Gauss point j:j 1;1. 0 A1 (A1)i 7.2. The case of toric varieties. Let k be a non-Archimedean eld. As observed in [Gub07], [EKL06], [Gub13], and [BPR], there is an intricate relationship between non-Archimedean anal”
On the structure of non-archimedean analytic curves (2013, background): “This is a polytopal domain in Gan m [20,23]; it is therefore an affinoid space whose ring of analytic functions is”
Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “Following [16] as well as [26] and [27], one of the many ways of defining Trop(Y ) is by taking it to be the projection of the non-Archimedean analytic space Y an (see Section 5 below) associated to Y into NR via a natural continuous tropicalization map trop : T an −→ NR.”
The arithmetic Hodge index theorem for adelic line bundles (2013, methodology): “Then there is a theory of integrable metrized line bundles and their intersection numbers developed in [8,17,19,40].”
A guide to tropicalizations (2011, methodology): “Proposition 6.9. If v is trivial, then the sums in the denition of A are nite and hence A = K[M] which immediately yields the claims. Remark 6.18 The special case of polytopal domains was studied in [Gub3]. Using Hochster’s theorem for toric varieties, Rabino has shown that A is CohenMacauley for an”
Analytification is the limit of all tropicalizations (2008, background): “Arguments close to the spirit of this paper also appear in Gubler’s elegant study of tropicalization of nonarchimedean analytic spaces [15].”
Equidistribution over function fields (2008, methodology): “follows from Theorem 3.5 and the compatibility of Chambert-Loir’s measures with base change ([ Gu3 ], Remark 3.10) that the height remains invariant under base change to K ′.”
A non-Archimedean approach to K-stability (2018, background): “It is continuous under monotone limits and extends the operator introduced by Chambert-Loir [Cha06] and Gubler [Gub07].”