[Minimums successifs des fonctions hauteurs toriques]
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.
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.
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
@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},
pages = {2145--2197},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {65},
number = {5},
year = {2015},
doi = {10.5802/aif.2985},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2985/}
}
TY - JOUR AU - Burgos Gil, José Ignacio AU - Philippon, Patrice AU - Sombra, Martín TI - Successive minima of toric height functions JO - Annales de l'Institut Fourier PY - 2015 SP - 2145 EP - 2197 VL - 65 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2985/ DO - 10.5802/aif.2985 LA - en ID - AIF_2015__65_5_2145_0 ER -
%0 Journal Article %A Burgos Gil, José Ignacio %A Philippon, Patrice %A Sombra, Martín %T Successive minima of toric height functions %J Annales de l'Institut Fourier %D 2015 %P 2145-2197 %V 65 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2985/ %R 10.5802/aif.2985 %G en %F 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, Tome 65 (2015) no. 5, pp. 2145-2197. doi : 10.5802/aif.2985. http://www.numdam.org/articles/10.5802/aif.2985/
[1] Small points on subvarieties of a torus, Duke Math. J., Volume 150 (2009) no. 3, pp. 407-442 | Zbl
[2] Factoring bivariate sparse (lacunary) polynomials, J. Complexity, Volume 23 (2007) no. 2, pp. 193-216 | DOI | Zbl
[3] Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 3, pp. 625-688 | DOI | Numdam | MR | Zbl
[4] Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math., Volume 181 (2010) no. 2, pp. 337-394 | DOI | Zbl
[5] Limit distribution of small points on algebraic tori, Duke Math. J., Volume 89 (1997) no. 3, pp. 465-476 | DOI | Zbl
[6] Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, Cambridge, 2006, pp. xvi+652 | DOI | Zbl
[7] Heights of projective varieties and positive Green forms, J. Amer. Math. Soc., Volume 7 (1994) no. 4, pp. 903-1027 | DOI | Zbl
[8] Convex optimization, Cambridge University Press, Cambridge, 2004, pp. xiv+716 | DOI | Zbl
[9] Fake weighted projective spaces, Warsaw Univ. (Poland) (2002) (Masters thesis http://arxiv.org/abs/0805.1211v1)
[10] Arithmetic positivity on toric varieties (to appear in J. Alg. Geom., http://arxiv.org/abs/1210.7692)
[11] Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque (2014) no. 360, pp. vi+222
[12] Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | DOI | Zbl
[13] Mesures de Mahler et équidistribution logarithmique, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 3, pp. 977-1014 | Numdam | Zbl
[14] Differentiability of the arithmetic volume function, J. Lond. Math. Soc. (2), Volume 84 (2011) no. 2, pp. 365-384 | DOI | Zbl
[15] Minorations des hauteurs normalisées des sous-variétés des tores, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 28 (1999) no. 3, pp. 489-543 | Numdam | Zbl
[16] Density results for Hilbert subsets, Indian J. Pure Appl. Math., Volume 30 (1999) no. 1, pp. 109-127 | Zbl
[17] Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., Volume 335 (2006) no. 2, pp. 311-361 | DOI | Zbl
[18] Local and canonical heights of subvarieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 2 (2003) no. 4, pp. 711-760 | Numdam | Zbl
[19] Diophantine geometry, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000, pp. xiv+558 (An introduction) | DOI | Zbl
[20] Algebra, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002, pp. xvi+914 | DOI | Zbl
[21] Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford University Press, Oxford, 2002, pp. xvi+576 (Translated from the French by Reinie Erné, Oxford Science Publications) | Zbl
[22] Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322, Springer-Verlag, Berlin, 1999, pp. xviii+571 (Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder) | DOI | Zbl
[23] Quelques aspects diophantiens des variétés toriques projectives, Diophantine approximation (Dev. Math.), Volume 16, SpringerWienNewYork, Vienna, 2008, pp. 295-338 | DOI | Zbl
[24] Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970, pp. xviii+451 | Zbl
[25] Minimums successifs des variétés toriques projectives, J. Reine Angew. Math., Volume 586 (2005), pp. 207-233 | DOI | Zbl
[26] Équirépartition des petits points, Invent. Math., Volume 127 (1997) no. 2, pp. 337-347 | DOI | Zbl
[27] Basic number theory, Springer-Verlag, New York-Berlin, 1974, pp. xviii+325 (Die Grundlehren der Mathematischen Wissenschaften, Band 144) | Zbl
[28] Big line bundles over arithmetic varieties, Invent. Math., Volume 173 (2008) no. 3, pp. 603-649 | DOI | Zbl
[29] Positive line bundles on arithmetic varieties, J. Amer. Math. Soc., Volume 8 (1995) no. 1, pp. 187-221 | DOI | Zbl
[30] Small points and adelic metrics, J. Algebraic Geom., Volume 4 (1995) no. 2, pp. 281-300 | Zbl
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