Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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Walter Gubler; Julius Hertel
Local Heights of Toric Varieties over Non-archimedean Fields
(Hauteurs locales des variétés toriques sur un corps ultramétrique complet)
Publications mathématiques de Besançon (2017), p. 5-77, doi: 10.5802/pmb.15
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Class. Math.: 14M25, 14G40, 14G22
Keywords: Toric geometry, local heights, berkovich spaces, Chambert-Loir measure, heights of varieties over finitely generated fields


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.


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