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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 $$A$$.

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