The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998.[1]
Generalization
In 1998, Zhang[2] proved the following generalization:
Let A be an
abelian variety defined over K, and let be the Néron-Tate height on A associated to an ample symmetric divisor. A
subvariety is called a torsion subvariety if it is the translate of an abelian subvariety of A by a torsion point. If X is not a torsion subvariety, then there is an such that the set
Chambert-Loir, Antoine (2013). "Diophantine geometry and analytic spaces". In Amini, Omid; Baker, Matthew; Faber, Xander (eds.). Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011. Contemporary Mathematics. Vol. 605. Providence, RI:
American Mathematical Society. pp. 161–179.
ISBN978-1-4704-1021-6.
Zbl1281.14002.