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Arithmetic height functions over finitely generated fields. (English) Zbl 1007.11042
Author’s summary: Let $$K$$ be a finitely generated field over $$\mathbb{Q}$$, $$X$$ a projective variety over $$K$$, and $$L$$ an ample line bundle on $$X$$. In this paper, we propose a new height function $h_L^{\text{arith}}: X(\overline{\mathbb{Q}})\to \mathbb{R}$ of arithmetic type. The first main result of this paper is the following:
Theorem 1 (Northcott’s theorem). For any numbers $$M$$ and any positive integers $$e$$, the set $\{P\in X(\overline{K})\mid h_L^{\text{arith}} (P)\leq M,\;[K(P):K]\leq e\}$ is finite.
If $$A$$ is an Abelian variety over $$K$$ and $$L$$ is a symmetric ample line bundle on $$A$$, then the canonical height function $\widehat{h}_L^{\text{arith}}: A(\overline{\mathbb{Q}})\to \mathbb{R}$ is given as a quadratic form. The second main result is the following solution of Bogomolov’s conjecture, which is a generalization of results due to E. Ullmo [Ann. Math. (2) 147, 167-179 (1998; Zbl 0934.14013)] and S. Zhang, Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11034)].
Theorem 2 (Bogomolov’s conjecture). Let $$X$$ be a subvariety of $$A_{\overline{K}}$$. If the set $\{P\in X(\overline{K})\mid \widehat{h}_L^{\text{arith}} (P)\leq \varepsilon\}$ is Zariski dense in $$X$$ for any positive numbers $$\varepsilon$$, then $$X$$ is a translation of an Abelian subvariety of $$A_{\overline{K}}$$ by a torsion point.

##### MSC:
 11G50 Heights 11G10 Abelian varieties of dimension $$> 1$$ 11G35 Varieties over global fields 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14K15 Arithmetic ground fields for abelian varieties
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