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Small points and adelic metrics. (English) Zbl 0861.14019
In this paper, we prove the Bogomolov conjecture for a subvariety \(Y\) of an abelian variety \(A\), provided \(Y-Y\) generates A, and the map \(NS(A)_\mathbb{R} \to NS(Y)_\mathbb{R}\) is not injective.
To prove the main result as above, we first extend Gillet-Soulé’s intersection theory [H. Gillet and C. Soulé, Publ. Math., Inst. Hautes Étud. Sci. 72, 93-174 (1990; Zbl 0741.14012)] to so-called integrable metrized line bundles, then for a dynamic system, construct so-called admissible metrized line bundles, and finally prove the positivity of the normalized height of \(y\) in \(A\).

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14K05 Algebraic theory of abelian varieties