Successive minima of toric height functions
[Minimums successifs des fonctions hauteurs toriques]
Burgos Gil, José Ignacio ; Philippon, Patrice ; Sombra, Martín
Annales de l'Institut Fourier, Tome 65 (2015), p. 2145-2197 / Harvested from Numdam
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Étant donné un ℝ-diviseur torique métrisé d’une variété torique sur un corps global, nous démontrons une formule pour le minimum essentiel de la fonction hauteur associée. Sous des hypothèses de positivité convenables, nous donnons également des formules pour tous les minimums successifs. Nous appliquons ces résultats à l’étude, dans le cadre torique, des relations entre les minimums successifs et d’autres invariants arithmétiques comme la hauteur et le volume arithmétique. Nous appliquons aussi nos formules au calcul des minimums successifs de plusieurs familles d’exemples, incluant les espaces projectifs pondérés, les fibrés toriques et les translatés de sous-tores.

Given a toric metrized ℝ-divisor on a toric variety over a global field, we give a formula for the essential minimum of the associated height function. Under suitable positivity conditions, we also give formulae for all the successive minima. We apply these results to the study, in the toric setting, of the relation between the successive minima and other arithmetic invariants like the height and the arithmetic volume. We also apply our formulae to compute the successive minima for several families of examples, including weighted projective spaces, toric bundles and translates of subtori.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2985
Classification:  14G40,  14M25,  52A41
Mots clés: Hauteur, minimum essentiel, minimums successifs, variété torique, ℝ-diviseur métrisé torique, fonction concave, dualité de Legendre-Fenchel 
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     title = {Successive minima of toric height functions},
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     volume = {65},
     year = {2015},
     pages = {2145-2197},
     doi = {10.5802/aif.2985},
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Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín. Successive minima of toric height functions. Annales de l'Institut Fourier, Tome 65 (2015) pp. 2145-2197. doi : 10.5802/aif.2985. http://gdmltest.u-ga.fr/item/AIF_2015__65_5_2145_0/

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