Inégalité d’indice de Hodge en géométrie et arithmétique : une approche probabiliste
Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 231-262.

En utilisant l’approche probabiliste en géométrie arithmétique, nous donnons une nouvelle démonstration de l’inégalité d’indice de Hodge pour les -diviseurs adéliques, et nous proposons une nouvelle voie pour sa généralisation au cas de dimension supérieure.

By using the probabilistic approach in arithmetic geometry, one gives a new proof of the Hodge index inequality for adelic -divisors, and proposes a new way of generalizing it to higher dimensional case.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.33
Classification : 14G40, 11G30
Mot clés : Inégalité d’indice de Hodge, géométrie d’Arakelov, diviseur adélique, corps d’Okounkov, système linéaire gradué, $\mathbb{R}$-filtration
Keywords: Hodge index inequality, Arakelov geometry, adelic divisor, Okounkov body, graded linear series, $\mathbb{R}$-filtration
Huayi Chen 1

1 Université Grenoble Alpes, Institut Fourier F-38000 Grenoble, France
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{JEP_2016__3__231_0,
     author = {Huayi Chen},
     title = {In\'egalit\'e d{\textquoteright}indice de {Hodge} en g\'eom\'etrie et arithm\'etique~: une approche probabiliste},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {231--262},
     publisher = {ole polytechnique},
     volume = {3},
     year = {2016},
     doi = {10.5802/jep.33},
     zbl = {06670707},
     mrnumber = {3522823},
     language = {fr},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.33/}
}
TY  - JOUR
AU  - Huayi Chen
TI  - Inégalité d’indice de Hodge en géométrie et arithmétique : une approche probabiliste
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2016
SP  - 231
EP  - 262
VL  - 3
PB  - ole polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.33/
DO  - 10.5802/jep.33
LA  - fr
ID  - JEP_2016__3__231_0
ER  - 
%0 Journal Article
%A Huayi Chen
%T Inégalité d’indice de Hodge en géométrie et arithmétique : une approche probabiliste
%J Journal de l’École polytechnique — Mathématiques
%D 2016
%P 231-262
%V 3
%I ole polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.33/
%R 10.5802/jep.33
%G fr
%F JEP_2016__3__231_0
Huayi Chen. Inégalité d’indice de Hodge en géométrie et arithmétique : une approche probabiliste. Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 231-262. doi : 10.5802/jep.33. https://jep.centre-mersenne.org/articles/10.5802/jep.33/

[1] P. Barbe & M. Ledoux - Probabilité, Collection Enseignement Sup Mathématiques, vol. 33, EDP Sciences, Les Ulis, 2007 | Zbl

[2] V. G. Berkovich - Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys and Monographs, vol. 33, American Mathematical Society, Providence, R.I., 1990 | MR | Zbl

[3] S. Bobkov & M. Madiman - “Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures”, J. Functional Analysis 262 (2012) no. 7, p. 3309-3339 | DOI | MR | Zbl

[4] J.-B. Bost - “Potential theory and Lefschetz theorems for arithmetic surfaces”, Ann. Sci. École Norm. Sup. (4) 32 (1999) no. 2, p. 241-312 | DOI | Numdam | MR | Zbl

[5] S. Boucksom & H. Chen - “Okounkov bodies of filtered linear series”, Compositio Math. 147 (2011) no. 4, p. 1205-1229 | DOI | MR | Zbl

[6] J. I. Burgos Gil, P. Philippon & M. Sombra - Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, vol. 360, Société Mathématique de France, Paris, 2014 | Zbl

[7] H. Chen - “Arithmetic Fujita approximation”, Ann. Sci. École Norm. Sup. (4) 43 (2010) no. 4, p. 555-578 | DOI | Numdam | MR | Zbl

[8] H. Chen - Géométrie d’Arakelov : théorèmes de limite et comptage des points rationnels, Université Paris Diderot, 2011, Mémoire d’habilitation à diriger des recherches

[9] H. Chen - “Majorations explicites des fonctions de Hilbert-Samuel géométrique et arithmétique”, Math. Z. 279 (2015) no. 1-2, p. 99-137 | DOI | Zbl

[10] T. M. Cover & Z. Zhang - “On the maximum entropy of the sum of two dependent random variables”, IEEE Trans. Information Theory 40 (1994) no. 4, p. 1244-1246 | DOI | MR | Zbl

[11] S. D. Cutkosky - “Asymptotic multiplicities of graded families of ideals and linear series”, Advances in Math. 264 (2014), p. 55-113 | DOI | MR | Zbl

[12] C. Dellacherie & P.-A. Meyer - Probabilités et potentiel, Hermann, Paris, 1975, Chap. I à IV

[13] A. Ducros - “Espaces analytiques p-adiques au sens de Berkovich”, in Séminaire Bourbaki. Vol. 2005/06, Astérisque, vol. 311, Société Mathématique de France, Paris, 2007, p. 137-176, Exp. No. 958 | Numdam | Zbl

[14] D. Eisenbud - The geometry of syzygies, Graduate Texts in Math., vol. 229, Springer-Verlag, New York, 2005 | MR

[15] G. Faltings - “Calculus on arithmetic surfaces”, Ann. of Math. (2) 119 (1984) no. 2, p. 387-424 | DOI | MR | Zbl

[16] T. Fujita - “Approximating Zariski decomposition of big line bundles”, Kodai Math. J. 17 (1994) no. 1, p. 1-3 | DOI | MR | Zbl

[17] J. Gallier - Geometric methods and applications, Texts in Appl. Math., vol. 38, Springer, New York, 2011 | MR

[18] É. Gaudron - “Pentes de fibrés vectoriels adéliques sur un corps global”, Rend. Sem. Mat. Univ. Padova 119 (2008), p. 21-95 | DOI | Zbl

[19] H. Gillet & C. Soulé - “On the number of lattice points in convex symmetric bodies and their duals”, Israel J. Math. 74 (1991) no. 2-3, p. 347-357 | DOI | MR | Zbl

[20] P. Hriljac - “Heights and Arakelov’s intersection theory”, Amer. J. Math. 107 (1985) no. 1, p. 23-38 | DOI | MR | Zbl

[21] H. Ikoma - “Boundedness of the successive minima on arithmetic varieties”, J. Algebraic Geom. 22 (2013) no. 2, p. 249-302 | MR | Zbl

[22] K. Kaveh & A. G. Khovanskii - “Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory”, Ann. of Math. (2) 176 (2012) no. 2, p. 925-978 | DOI | MR | Zbl

[23] K. Künnemann - “Some remarks on the arithmetic Hodge index conjecture”, Compositio Math. 99 (1995) no. 2, p. 109-128 | Numdam | MR | Zbl

[24] R. Lazarsfeld - Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), vol. 48, Springer-Verlag, Berlin, 2004 | MR

[25] R. Lazarsfeld & M. Mustaţă - “Convex bodies associated to linear series”, Ann. Sci. École Norm. Sup. (4) 42 (2009) no. 5, p. 783-835 | DOI | Numdam | MR | Zbl

[26] T. Luo - “Riemann-Roch type inequalities for nef and big divisors”, Amer. J. Math. 111 (1989) no. 3, p. 457-487 | DOI | MR | Zbl

[27] A. Moriwaki - “Hodge index theorem for arithmetic cycles of codimension one”, Math. Res. Lett. 3 (1996) no. 2, p. 173-183 | DOI | MR | Zbl

[28] A. Moriwaki - Adelic divisors on arithmetic varieties, Mem. Amer. Math. Soc., vol. 242, no. 1144, American Mathematical Society, Providence, R.I., 2016 | Zbl

[29] A. Okounkov - “Brunn-Minkowski inequality for multiplicities”, Invent. Math. 125 (1996) no. 3, p. 405-411 | MR | Zbl

[30] C. A. Rogers & G. C. Shephard - “Convex bodies associated with a given convex body”, J. London Math. Soc. (2) 33 (1958), p. 270-281 | DOI | MR | Zbl

[31] D. Roy & J. L. Thunder - “An absolute Siegel’s lemma”, J. reine angew. Math. 476 (1996), p. 1-26 | MR | Zbl

[32] C. E. Shannon - “A mathematical theory of communication”, AT&T Bell Labs. Tech. J. 27 (1948), p. 379-423, 623–656 | MR | Zbl

[33] A. J. Stam - “Some inequalities satisfied by the quantities of information of Fisher and Shannon”, Inform. and Control 2 (1959), p. 101-112 | DOI | MR | Zbl

[34] S. Takagi - “Fujita’s approximation theorem in positive characteristics”, J. Math. Kyoto Univ. 47 (2007) no. 1, p. 179-202 | DOI | MR | Zbl

[35] J. L. Thunder - “An adelic Minkowski-Hlawka theorem and an application to Siegel’s lemma”, J. reine angew. Math. 475 (1996), p. 167-185 | MR | Zbl

[36] J. Yan - Lectures on measure theory, Lect. series Chinese Acad. Sci., Science Press, Beijing, 2004

[37] X. Yuan - “On volumes of arithmetic line bundles”, Compositio Math. 145 (2009) no. 6, p. 1447-1464 | DOI | MR | Zbl

[38] X. Yuan - “Algebraic dynamics, canonical heights and Arakelov geometry”, in Fifth International Congress of Chinese Mathematicians, AMS/IP Stud. Adv. Math., vol. 51, 2e partie, American Mathematical Society, Providence, R.I., 2012, p. 893-929 | Zbl

[39] X. Yuan & S.-W. Zhang - “The arithmetic Hodge index theorem for adelic line bundles I : number fields”, Math. Ann. (2016), online, arXiv :1304.3538

[40] S.-W. Zhang - “Positive line bundles on arithmetic varieties”, J. Amer. Math. Soc. 8 (1995) no. 1, p. 187-221 | DOI | MR | Zbl

Cité par Sources :