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.


14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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