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.

The proofs by Ullmo and Zhang of Bogomolov’s conjecture make a fundamental use of an equidistribution principle which had been discovered together with Szpiro [50]. Let us first introduce a terminology : say a sequence (or a net) of algebraic points in a variety $X$ over a number field is generic if any strict subvariety of $X$ contains at most finitely terms of the sequence.

Let $\overline{L}$ be a line bundle on $X$ with a semi-positive adelic metric. The idea of the equidistribution principle is to consider a generic sequence $(x_{j})$ such that $h_{\overline{L}}(x_{j})\rightarrow e(\overline{L})$, i.e., realizing the equality in Zhang’s inequality, and to use this inequality further, as a variational principle. Let $v$ be a place of $F$ ; for any $n$, let $\delta(x_{j})_{v}$ be the probability measure on $X_{v}$ which gives any conjugate of $x_{j}$ the same mass, $1/[F(x_{j}):F]$. The equidistribution theorem states that for a generic sequence $(x_{j})$, the sequence of measures $(\delta(x_{j})_{v})$ on $X_{v}$ converges vaguely towards the measure $c_{1}(\overline{L})^{n}_{v}/(c_{1}(L)^{n}|X)$.

In these papers, the equidistribution property was only investigated at an archimedean place, but the introduction of the measures on Berkovich spaces was motivated by potential equidistribution theorems on those. In [18], I was able to prove general results on curves only. Indeed, unless $X$ is a curve, I needed an ampleness assumption on the metrized line bundle $\overline{L}$ in order to apply Zhang’s inequality to slight variations of it. This requirement has been removed by a paper of Yuan [55] who could understand arithmetic volumes beyond the ample case. Yuan’s proof is an arithmetic analogue of an inequality of Siu [49] which Faber [28] and Gubler [38] used to prove the geometric case of the equidistribution theorem.

In [19], we considered more general variations of the metrized line bundles. The discussion in that article was restricted to the arithmetic case but the arguments extend to the geometric case.

The particular case stated above is equivalent to loc.cit., Lemma 6.1, as one can see by by multiplying the metric on $\overline{L}$ by an adequate constant at some place of $F$. Taking for $M$ the trivial line bundle $\mathscr{O}_{X}$, with an admissible metric, one recovers the equidistribution theorems of Yuan, Faber and Gubler.