Inégalité d’indice de Hodge en géométrie et arithmétique : une approche probabiliste
[Hodge index inequality in geometry and arithmetic: a probabilistic approach]
Chen, Huayi1
1 Université Grenoble Alpes, Institut Fourier F-38000 Grenoble, France
Journal de l’École polytechnique — Mathématiques, Volume 3 (2016), pp. 231-262.

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

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

Received:
Accepted:
Published online:
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
Chen, Huayi 1

1 Université Grenoble Alpes, Institut Fourier F-38000 Grenoble, France 
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Chen, Huayi. Inégalité d’indice de Hodge en géométrie et arithmétique : une approche probabiliste. Journal de l’École polytechnique — Mathématiques, Volume 3 (2016), pp. 231-262. doi : 10.5802/jep.33. http://archive.numdam.org/articles/10.5802/jep.33/

[1] Barbe, Ph.; Ledoux, M. Probabilité, Collection Enseignement Sup Mathématiques, 33, EDP Sciences, Les Ulis, 2007, v+241 pages | Zbl

[2] Berkovich, V. G. Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys and Monographs, 33, American Mathematical Society, Providence, R.I., 1990, x+169 pages | MR | Zbl

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

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

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

[6] Burgos Gil, J. I.; Philippon, P.; Sombra, M. Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, 360, Société Mathématique de France, Paris, 2014, vi+222 pages | Zbl

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

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

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

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

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

[12] Dellacherie, C.; Meyer, P.-A. Probabilités et potentiel, Hermann, Paris, 1975, x+291 pages (Chap. I à IV)

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

[14] Eisenbud, D. The geometry of syzygies, Graduate Texts in Math., 229, Springer-Verlag, New York, 2005, xvi+243 pages | MR

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

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

[17] Gallier, J. Geometric methods and applications, Texts in Appl. Math., 38, Springer, New York, 2011, xxviii+680 pages | MR

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

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

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

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

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

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

[24] Lazarsfeld, R. Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), 48, Springer-Verlag, Berlin, 2004, xviii+387 pages | MR

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

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

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

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

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

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

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

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

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

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

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

[36] Yan, J. Lectures on measure theory, Lect. series Chinese Acad. Sci., Science Press, Beijing, 2004, ix+289 pages

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

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

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

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

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