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Big line bundles over arithmetic varieties. (English) Zbl 1146.14016
The paper under review provides the closing chapter to the story of arithmetic equidistribution begun in 1997 by L. Szpiro, E. Ullmo, and S. Zhang [Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)]. Its author has written a lucid introduction in which he — with remarkable clarity — states the goals of his work, his main technical results on arithmetically big line bundles, the consequences for dynamical and arithmetic equidistribution, the context into which these results fit in the literature, and all of the requisite definitions. The interested researcher would be best served by perusing the extended introduction to the paper.
For the reader who would like a more hasty description of the results in the article, I offer the following summary (which assumes some familiarity with the objects of Arakelov theory).
Theorem. (Generic Equidistribution of Small Points) Suppose $$X$$ is a projective variety of dimension $$n-1$$ over a number field $$K$$, and $$\overline{{\mathcal L}}$$ is an adelic metrized line bundle over $$X$$ such that $$\mathcal{L}$$ is ample and the metric is semipositive. Let $$\{x_m\}$$ be an infinite sequence of algebraic points in $$X(\overline{K})$$ that is generic and small. Then for any place $$v$$ of $$K$$, the Galois orbits of the sequence $$\{x_m\}$$ are equidistributed in the analytic space $$X_{\mathbb{C}_v}^{\text{an}}$$ with respect to the probability measure $$d\mu_v = c_1\left(\overline{\mathcal{L}}\right)_v^{n-1} / \text{deg}_{\mathcal{L}}(X)$$.
The seminal work of Szpiro/Ullmo/Zhang mentioned above proves this result in the case where $$v$$ is an archimedean place of $$K$$, and the curvature form $$c_1\left(\overline{\mathcal{L}}\right)_v^{n-1}$$ is strictly positive. A. Chambert-Loir [J. Reine Angew. Math. 595, 215–235 (2006; Zbl 1112.14022)] constructed the nonarchimedean analogue of this curvature form (supported on a $$v$$-adic analytic space as defined by Berkovich). He then proved the above result for $$v$$ nonarchimedean under the added hypothesis that $$X$$ is a curve, or that the metric on $$\overline{\mathcal{L}}$$ is defined by an ample model. The contribution of the paper under review is to remove these positivity hypotheses by introducing the notion of an arithmetically big line bundle. Some large tensor power of such a line bundle has many small sections, and this is exactly the key to extending the proof of Szpiro/Ullmo/Zhang. The author’s main result is an Arakelov-theoretic generalization of Siu’s theorem. Essentially, it gives a numerical criterion for the existence of small sections of the difference of two ample line bundles. Here is a simplified version of it:
Theorem. (Arithmetic Siu’s Theorem) Let $$\overline{\mathcal{L}}$$ and $$\overline{\mathcal{M}}$$ be two Hermitian line bundles over an arithmetic variety $$X$$ of dimension $$n$$. Assume that $$\overline{\mathcal{L}}$$ and $$\overline{\mathcal{M}}$$ are ample. Then \begin{aligned} h^0\left( \left(\overline{\mathcal{L}} \otimes \overline{\mathcal{M}}^{\vee}\right)^{\otimes N}\right) &:= \log \#\left\{s \in \Gamma\left(X, \left(\mathcal{L} \otimes \mathcal{M}^{\vee}\right)^{\otimes N} \right): \|s \|_{\text{sup}} < 1 \right\} \\ &\geq \frac{\widehat{c}_1\left(\overline{\mathcal{L}}\right)^n - n \cdot \widehat{c}_1\left(\overline{\mathcal{L}}\right)^{n-1} \widehat{c}_1\left(\overline{\mathcal{M}}\right)}{n!} N^n + o(N^n). \end{aligned} In particular, $$\overline{\mathcal{L}} \otimes \overline{\mathcal{M}}^{\vee}$$ is arithmetically big if $$\widehat{c}_1\left(\overline{\mathcal{L}}\right)^n - n \cdot \widehat{c}_1\left(\overline{\mathcal{L}}\right)^{n-1} \widehat{c}_1\left(\overline{\mathcal{M}}\right) > 0$$.
The paper is divided into three sections: Introduction, Arithmetic bigness, and Equidistribution theory. The second section forms the technical heart of the paper; it is quite difficult, although carefully written. The third section deals with applications of the bigness result, including how to deduce equidistribution from the arithmetic version of Siu’s Theorem. It can be read independently from the second section.

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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##### References:
  Abbes, A., Bouche, T.: Théorème de Hilbert–Samuel ”arithmétique”. Ann. Inst. Fourier 45, 375–401 (1995) · Zbl 0818.14011  Arakelov, S.J.: Intersection theory of divisors on an arithmetic surface. Math. USSR Izv. 8, 1167–1180 (1974) · Zbl 0355.14002  Autissier, P.: Points entiers sur les surfaces arithmétiques. J. Reine Angew. Math. 531, 201–235 (2001) · Zbl 1007.11041  Autissier, P.: Équidistribution des sous-variétés de petite hauteur. J. Théor. Nombres Bordx. 18(1), 1–12 (2006) · Zbl 1228.11087  Berkovich, V.G.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Math. Surv. Monogr., vol. 33. Am. Math. Soc., Providence, RI (1990) · Zbl 0715.14013  Bombieri, E., Gubler, W.: Heights in Diophantine Geometry. New Math. Monogr., vol. 4. Cambridge University Press, Cambridge (2006) · Zbl 1115.11034  Baker, M., Ih, S.: Equidistribution of small subvarieties of an abelian variety. New York J. Math. 10, 279–285 (2004) · Zbl 1131.11035  Bilu, Y.: Limit distribution of small points on algebraic tori. Duke Math. J. 89(3), 465–476 (1997) · Zbl 0918.11035  Bosch, S., Lütkebohmert, W.: Formal and rigid geometry. I. Rigid spaces. Math. Ann. 295(2), 291–317 (1993) · Zbl 0808.14017  Bouche, T.: Convergence de la métrique de Fubini–Study d’un fibré linéaire positif. Ann. Inst. Fourier 40(1), 117–130 (1990) · Zbl 0685.32015  Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1–2), 1–40 (1982) · Zbl 0547.32012  Bismut, J.-M., Vasserot, E.: The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle. Commun. Math. Phys. 125, 355–367 (1989) · Zbl 0687.32023  Chambert-Loir, A.: Points de petite hauteur sur les variétés semi-abéliennes. Ann. Sci. Éc. Norm. Supér., IV. Sér. 33(6), 789–821 (2000) · Zbl 1018.11034  Chambert-Loir, A.: Mesures et équidistribution sur les espaces de Berkovich. J. Reine Angew. Math. 595, 215–235 (2006) · Zbl 1112.14022  Chambert-Loir, A.: Arakelov Geometry, Variational Principles and Equidistribution of Small Points. Available at http://perso.univ-rennes1.fr/antoine.Chambert-Loir/publications/papers/cmi2.pdf · Zbl 0944.14010  Demailly, J.-P.: Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. Fr., Nouv. Sér. 19, 1–125 (1985)  Dieudonné, J., Grothendieck, A.: Éléments de géométrie algébrique. Publ. Math., Inst. Hautes Étud. Sci. 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967)  Faltings, G.: Calculus on arithmetic surfaces. Ann. Math. (2) 119, 387–424 (1984) · Zbl 0559.14005  Gillet, H., Soulé, C.: Arithmetic intersection theory. Publ. Math., Inst. Hautes Étud. Sci. 72, 93–174 (1990) · Zbl 0741.14012  Gillet, H., Soulé, C.: An arithmetic Riemann–Roch theorem. Invent. Math. 110, 473–543 (1992) · Zbl 0777.14008  Gillet, H., Soulé, C.: On the number of lattice points in convex symmetric bodies and their duals. Isr. J. Math. 74(2–3), 347–357 (1991) · Zbl 0752.52008  Gubler, W.: Local heights of subvarieties over non-archimedean fields. J. Reine Angew. Math. 498, 61–113 (1998) · Zbl 0906.14013  Hartshorne, R.: Algebraic Geometry. Springer, New York (1977) · Zbl 0367.14001  Lazarsfeld, R.: Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series. Ergeb. Math. Grenzgeb., 3. Folge, vol. 48. Springer, Berlin (2004) · Zbl 1093.14501  Lipman, J.: Desingularization of two-dimensional schemes. Ann. Math. (2) 107(1), 151–207 (1978) · Zbl 0369.14005  Maillot, V.: Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables. Mém. Soc. Math. Fr., Nouv. Sér., vol. 80, (2000)  Moriwaki, A.: Arithmetic Bogomolov–Gieseker’s inequality. Am. J. Math. 117, 1325–1347 (1995) · Zbl 0854.14013  Moriwaki, A.: Arithmetic height functions over finitely generated fields. Invent. Math. 140(1), 101–142 (2000) · Zbl 1007.11042  Moriwaki, A.: Continuity of volumes on arithmetic varieties. arXiv:math.AG/0612269 · Zbl 1167.14014  Raynaud, M.: Géométrie analytique rigide d’après Tate, Kiehl, $$\cdots$$ . Bull. Soc. Math. Fr. 39–40, 319–327 (1974) · Zbl 0299.14003  Serre, J.-P.: Local fields. Translated from the French by Marvin Jay Greenberg. Grad. Texts Math., vol. 67. Springer, New York, Berlin (1979) · Zbl 0423.12016  Siu, Y.-T.: An effective Mastusaka big theorem. Ann. Inst. Fourier 43(5), 1387–1405 (1993) · Zbl 0803.32017  Szpiro, L., Ullmo, E., Zhang, S.: Équidistribution des petits points. Invent. Math. 127, 337–348 (1997) · Zbl 0991.11035  Tian, G.: On a set of polarized Kahler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990) · Zbl 0706.53036  Ullmo, E.: Positivité et discrétion des points algébriques des courbes. Ann. Math. (2) 147, 167–179 (1998) · Zbl 0934.14013  Zhang, S.: Positive line bundles on arithmetic varieties. J. Am. Math. Soc. 8, 187–221 (1995) · Zbl 0861.14018  Zhang, S.: Small points and adelic metrics. J. Algebr. Geom. 4, 281–300 (1995) · Zbl 0861.14019  Zhang, S.: Equidistribution of small points on abelian varieties. Ann. Math. 147(1), 159–165 (1998) · Zbl 0991.11034  Zhang, S.: Small points and Arakelov theory. In: Proceedings of ICM (Berlin 1998), vol. II. Doc. Math., pp. 217–225. Universität Bielefeld, Bielefeld (1998) · Zbl 0912.14008  Zhang, S.: Distributions in algebraic dynamics. In: A Tribute to Professor S.-S. Chern. Surv. Differ. Geom., vol. X, pp. 381–430. Int. Press, Boston, MA (2006) · Zbl 1207.37057
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