Small points and Arakelov theory (Q1126721)

The following conjecture was posed by \textit{F. A. Bogomolov} [Math. USSR, Izv. 17, 55-72 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 782-804 (1980; Zbl 0466.14015)]: Let \(A\) be an abelian variety over \(\overline{\mathbb Q}\), let \(X\) be a closed irreducible subvariety of \(A\), and let \(\widehat h\) be a Néron-Tate canonical height on \(A\) with respect to an ample symmetric divisor. Suppose that \(X\) is not a translate of an abelian subvariety of \(A\) by a torsion point. Then there is an \(\varepsilon>0\) such that the set \(\{x\in X(\overline{\mathbb Q}):\widehat h(x)<\varepsilon\}\) is not Zariski-dense in \(X\). This paper discusses the proof of this conjecture by \textit{S. Zhang} [Ann. Math., II. Ser. 147, 159-165 (1998)]. The proof relies heavily on work on equidistribution of points of small height due to \textit{E. Ullmo} [Ann. Math., II. Ser. 147, 167-179 (1998)]. The ideas in this proof also showed the positivity of the dualizing sheaf of an arithmetic surface: \(\bigl(\omega_{X/B}^2\bigr)>0\). The paper concludes by formulating a conjectural generalization of the Bogomolov conjecture for abelian schemes over curves.