Let $X$ be a projective smooth curve of genus $g\geq 2$ and let $\overline{L}$ be an ample line bundle on $X$ with a canonical metric inducing the Néron–Tate height. When $F$ is a number field, Bogomolov conjectured in [14] that $e(\overline{L})>0$ ; this conjecture has been shown by Ullmo [53]. Its generalization to a subvariety $X$ of an Abelian variety $A$, $L$ being an ample line bundle on $A$ with a canonical metric, asserts that $e(X,\overline{L})>0$ when $X$ is not the translate of an abelian subvariety by a torsion point ; it has been shown by Zhang [60].

Since $h_{\overline{L}}(P)=0$ for any algebraic point $P\in A(\overline{F})$ which is a torsion point, these theorems imply in turn a theorem of Raynaud [46, 47] (formerly, a conjecture of Manin and Mumford) that the torsion points lying in a subvariety $X$ of an abelian variety are not Zariski dense in $X$, unless $X$ is itself the translate of an abelian subvariety by a torsion point.

The analogues of Bogomolov’s and Zhang’s conjecture in the geometric case is still open in general ; see [37, 21] and the references therein for partial results.