Successive minima of toric height functions
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, p. 2145-2197
Given a toric metrized -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.
Étant donné un -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.
Received : 2014-03-17
Accepted : 2014-09-30
Published online : 2015-11-09
DOI : https://doi.org/10.5802/aif.2985
Classification:  14G40,  14M25,  52A41
Keywords: Height, essential minimum, successive minima, toric variety, toric metrized -divisor, concave function, Legendre-Fenchel duality
@article{AIF_2015__65_5_2145_0,
     author = {Burgos Gil, Jos\'e Ignacio and Philippon, Patrice and Sombra, Mart\'\i n},
     title = {Successive minima of toric height functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     pages = {2145-2197},
     doi = {10.5802/aif.2985},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2015__65_5_2145_0}
}
Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín. Successive minima of toric height functions. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2145-2197. doi : 10.5802/aif.2985. https://aif.centre-mersenne.org/item/AIF_2015__65_5_2145_0/

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