Tropical Skeletons

Gubler, Walter; Rabinoff, Joseph; Werner, Annette

Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 1905-1961.

Nous étudions les relations entre la géométrie tropicale et la géométrie analytique pour les sous-schémas fermés des variétés toriques. Soit $K$ un corps non-archimédien et complet et soit $X$ un sous-schéma fermé d’une variété torique sur $K$. Nous définissons le squelette tropical de X comme le sous-ensemble de l’espace de Berkovich associé $X^{\mathrm{an}}$ qui est composé de tous les points du bord de Shilov dans les fibres du morphisme de tropicalisation de Kajiwara–Payne. Nous développons des critères polyèdraux pour que des points limite appartiennent au squelette tropical, et pour que cet espace soit fermé. Nous appliquons ce critère pour les points limite à la question de la continuité de la section canonique du morphisme de tropicalisation sur le lieu de multiplicité un. On sait que cette section est continue sur chaque orbite du tore ; nous donnons des critères de continuité au croisement des orbites. Quand $X$ est schön et défini sur un corps discrètement valué, nous montrons que la squelette tropical coïncide avec le squelette d’une paire strictement semistable, et qu’il est naturellement isomorphe au complexe paramétrisant de Helm–Katz.

In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the tropical skeleton of $X$ as the subset of the associated Berkovich space $X^{\mathrm{an}}$ which collects all Shilov boundary points in the fibers of the Kajiwara–Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When $X$ is schön and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm–Katz.

Classification : 14G22, 14T05
Keywords: Tropical geometry, Kajiwara–Payne tropicalization, Berkovich spaces, skeletons
Mot clés : géométrie tropicale, tropicalisation de Kajiwara–Payne, espaces de Berkovich, squelettes

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