Publications Mathématiques de Besançon

Algèbre et Théorie des Nombres
Local Heights of Toric Varieties over Non-archimedean Fields
[Hauteurs locales des variétés toriques sur un corps ultramétrique complet]
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2017), pp. 5-77.

Nous généralisons des résultats concernant les hauteurs locales prouvés précédemment pour une valuation discrète au cas d’une valeur absolue ultramétrique quelconque. Nous traitons tout d’abord le case de la formule de récurrence de Chambert-Loir et Thuillier. Ensuite nous généralisons la formule de Burgos–Philippon–Sombra pour la hauteur locale torique d’une variété torique normale propre. Nous appliquons la formule correspondante de Moriwaki pour les hauteurs globales sur un corps de type fini au cas d’une fibration qui est génériquement torique. Nous illustrons ce dernier résultat par un exemple naturel où des valuations non discrètes jouent un rôle important.

We generalize results about local heights previously proved in the case of discrete absolute values to arbitrary non-archimedean absolute values. First, this is done for the induction formula of Chambert-Loir and Thuillier. Then we prove the formula of Burgos–Philippon–Sombra for the toric local height of a proper normal toric variety in this more general setting. We apply the corresponding formula for Moriwaki’s global heights over a finitely generated field to a fibration which is generically toric. We illustrate the last result in a natural example where non-discrete non-archimedean absolute values really matter.

Publié le :
DOI : 10.5802/pmb.15
Classification : 14M25, 14G40, 14G22
Mots clés : Toric geometry, local heights, berkovich spaces, Chambert-Loir measure, heights of varieties over finitely generated fields
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Walter Gubler; Julius Hertel. Local Heights of Toric Varieties over Non-archimedean Fields. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2017), pp. 5-77. doi : 10.5802/pmb.15. https://pmb.centre-mersenne.org/articles/10.5802/pmb.15/

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