no. 5
Successive minima of toric height functions
[Minimums successifs des fonctions hauteurs toriques]
Burgos Gil, José Ignacio1 ; Philippon, Patrice2 ; Sombra, Martín3
1 Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3) Calle Nicolás Cabrera 15 Campus UAB, Cantoblanco 28049 Madrid (Spain)
2 Institut de Mathématiques de Jussieu U.M.R. 7586 du CNRS Équipe de Théorie des Nombres. Case 247, 4 place Jussieu 75252 Paris cedex 05 (France)
3 ICREA & Universitat de Barcelona Departament d’Àlgebra i Geometria. Gran Via 585 08007 Barcelona (Spain)
Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2145-2197.

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.

DOI : 10.5802/aif.2985
Classification : 14G40, 14M25, 52A41
Keywords: Height, essential minimum, successive minima, toric variety, toric metrized $\mathbb{R}$-divisor, concave function, Legendre-Fenchel duality
Mot clés : Hauteur, minimum essentiel, minimums successifs, variété torique, $\mathbb{R}$-diviseur métrisé torique, fonction concave, dualité de Legendre-Fenchel

Burgos Gil, José Ignacio 1 ; Philippon, Patrice 2 ; Sombra, Martín 3

1 Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3) Calle Nicolás Cabrera 15 Campus UAB, Cantoblanco 28049 Madrid (Spain)
2 Institut de Mathématiques de Jussieu U.M.R. 7586 du CNRS Équipe de Théorie des Nombres. Case 247, 4 place Jussieu 75252 Paris cedex 05 (France)
3 ICREA & Universitat de Barcelona Departament d’Àlgebra i Geometria. Gran Via 585 08007 Barcelona (Spain) 
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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) no. 5, pp. 2145-2197. doi : 10.5802/aif.2985. https://aif.centre-mersenne.org/articles/10.5802/aif.2985/

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