Successive minima of toric height functions
[Minimums successifs des fonctions hauteurs toriques]
Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2145-2197.

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.

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. http://www.numdam.org/articles/10.5802/aif.2985/

[1] Amoroso, Francesco; Viada, Evelina Small points on subvarieties of a torus, Duke Math. J., Volume 150 (2009) no. 3, pp. 407-442 | Zbl

[2] Avendaño, Martin; Krick, Teresa; Sombra, Martin Factoring bivariate sparse (lacunary) polynomials, J. Complexity, Volume 23 (2007) no. 2, pp. 193-216 | DOI | Zbl

[3] Baker, Matthew H.; Rumely, Robert 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] Berman, Robert; Boucksom, Sébastien Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math., Volume 181 (2010) no. 2, pp. 337-394 | DOI | Zbl

[5] Bilu, Yuri Limit distribution of small points on algebraic tori, Duke Math. J., Volume 89 (1997) no. 3, pp. 465-476 | DOI | Zbl

[6] Bombieri, Enrico; Gubler, Walter Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, Cambridge, 2006, pp. xvi+652 | DOI | Zbl

[7] Bost, J.-B.; Gillet, H.; Soulé, C. Heights of projective varieties and positive Green forms, J. Amer. Math. Soc., Volume 7 (1994) no. 4, pp. 903-1027 | DOI | Zbl

[8] Boyd, Stephen; Vandenberghe, Lieven Convex optimization, Cambridge University Press, Cambridge, 2004, pp. xiv+716 | DOI | Zbl

[9] Buczynska, Weronika Fake weighted projective spaces, Warsaw Univ. (Poland) (2002) (Masters thesis http://arxiv.org/abs/0805.1211v1)

[10] Burgos Gil, J. I.; Moriwaki, A.; Philippon, P.; Sombra, M. Arithmetic positivity on toric varieties (to appear in J. Alg. Geom., http://arxiv.org/abs/1210.7692)

[11] Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque (2014) no. 360, pp. vi+222

[12] Chambert-Loir, Antoine Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | DOI | Zbl

[13] Chambert-Loir, Antoine; Thuillier, Amaury Mesures de Mahler et équidistribution logarithmique, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 3, pp. 977-1014 | Numdam | Zbl

[14] Chen, Huayi Differentiability of the arithmetic volume function, J. Lond. Math. Soc. (2), Volume 84 (2011) no. 2, pp. 365-384 | DOI | Zbl

[15] David, Sinnou; Philippon, Patrice 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] Dèbes, Pierre Density results for Hilbert subsets, Indian J. Pure Appl. Math., Volume 30 (1999) no. 1, pp. 109-127 | Zbl

[17] Favre, Charles; Rivera-Letelier, Juan Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., Volume 335 (2006) no. 2, pp. 311-361 | DOI | Zbl

[18] Gubler, Walter 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] Hindry, Marc; Silverman, Joseph H. Diophantine geometry, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000, pp. xiv+558 (An introduction) | DOI | Zbl

[20] Lang, Serge Algebra, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002, pp. xvi+914 | DOI | Zbl

[21] Liu, Qing 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] Neukirch, Jürgen 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] Philippon, Patrice; Sombra, Martin Quelques aspects diophantiens des variétés toriques projectives, Diophantine approximation (Dev. Math.), Volume 16, SpringerWienNewYork, Vienna, 2008, pp. 295-338 | DOI | Zbl

[24] Rockafellar, R. Tyrrell Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970, pp. xviii+451 | Zbl

[25] Sombra, Martin Minimums successifs des variétés toriques projectives, J. Reine Angew. Math., Volume 586 (2005), pp. 207-233 | DOI | Zbl

[26] Szpiro, L.; Ullmo, E.; Zhang, S. Équirépartition des petits points, Invent. Math., Volume 127 (1997) no. 2, pp. 337-347 | DOI | Zbl

[27] Weil, André Basic number theory, Springer-Verlag, New York-Berlin, 1974, pp. xviii+325 (Die Grundlehren der Mathematischen Wissenschaften, Band 144) | Zbl

[28] Yuan, Xinyi Big line bundles over arithmetic varieties, Invent. Math., Volume 173 (2008) no. 3, pp. 603-649 | DOI | Zbl

[29] Zhang, Shouwu Positive line bundles on arithmetic varieties, J. Amer. Math. Soc., Volume 8 (1995) no. 1, pp. 187-221 | DOI | Zbl

[30] Zhang, Shouwu Small points and adelic metrics, J. Algebraic Geom., Volume 4 (1995) no. 2, pp. 281-300 | Zbl

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