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Equidistribution of small subvarieties of an Abelian variety. (English) Zbl 1131.11035
Let $$A$$ be an Abelian variety over a number field $$K$$. Using the choice of an embedding of $$\overline K$$ into $$\mathbb{C}$$, the set $$A(\overline K)$$ may then be viewed as a subset of $$A(\mathbb{C})$$, and any symmetric ample line bundle $$L$$ on $$A$$ can be equipped with a so-called cubical metric, i.e., a smooth Hermitean metric whose curvature form is invariant under translations in $$A$$ [cf. L. Moret-Bailly, Exp. IV, Astérisque 127, 29–87 (1985; Zbl 0588.14028)]. The resulting metrized line bundle gives rise to a canonical height function $$\widehat h_L$$ on the set of closed subvarieties of $$A$$, which actually does not depend on the choice of a cubical metric on $$L$$ [cf. S. Zhang, J. Algebr. Geom. 4, 281–300 (1995; Zbl 0861.14019)] and coincides with the classical Néron-Tate canonical height with respect to $$L$$ on closed points $$x\in A(\overline K)$$. Finally, a small strict sequence of closed subvarieties of $$A$$ is a sequence $$(X_n)_{n\in\mathbb{N}}$$ of closed subvatieties such that $$\lim_{n\to\infty}\,\widehat h(X_n)= 0$$ and such that no proper closed torsion subvariety of $$A$$ contains an infinite subsequence of $$(X_n)_{n\in\mathbb{N}}$$.
With respect to this general set-up, the authors of the paper under review provide a detailed proof of the following equidistribution result:
Theorem 1.1. Let $$A$$ be an Abelian variety over a number field $$K$$, let $$L$$ be a symmetric ample line bundle on $$A$$, and let $$(X_n)_{n\in\mathbb{N}}$$ be a small strict sequence of closed subvarieties of $$A$$. Then for each complex embedding of $$\overline K$$ into $$\mathbb{C}$$ and for every real-valued function $$f$$ on $$A(\mathbb{C})$$ the sequence of integrals $$\int_{A(\mathbb{C})} f\mu_n$$ converges to the integral $$\int_{A(\mathbb{C})}f\mu$$ as $$n\to\infty$$, where setting $$d_n= \dim X_n$$ and $$g= \dim A$$ the occurring measures are $\mu_n= (c_1(L_{X_n})^{-d_n}\cdot c_1(L)^{d_n}\cdot \delta_{X_n}\quad \text{and}\quad \mu= c_1(\overline L)^g\cdot c_1(L)^{-g}.$ This amounts to saying that the sequence $$(\mu_n)_{n\in\mathbb{N}}$$ of measures weakly converges to the Haar measure $$\mu$$ of total mass 1 on $$A(\mathbb{C})$$.
This result generalizes the corresponding equidistribution theorem for small strict sequences of points in an Abelian variety $$A$$ over $$K$$, which was proven by L. Szpiro, E. Ullmo and S. Zhang [Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)].
In fact, the authors’ proof of their higher-dimensional equidistribution theorem is partially based on the foregoing Szpiro-Ullmo-Zhang approach. Namely, they first show how to approximate the height of each subvariety $$X_n$$ by Néron-Tate heights of points on $$X_n$$ thereby using S.-W. Zhang’s theorem [Ann. Math. (2) 147, No. 1, 159–165 (1998; Zbl 0991.11034)], and apply then the method of proof of Szpiro-Ullmo-Zhang to a particular subsequence of such points.

MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 11G35 Varieties over global fields 14K15 Arithmetic ground fields for abelian varieties 14K12 Subvarieties of abelian varieties 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
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