Heights on a subvariety of an abelian variety

Extending Ullmo–Zhangʹs result on the Bogomolov conjecture, we give conditions that a closed subvariety of an abelian variety A defined over a number field is isomorphic to an abelian variety in terms of the value distribution of a Neron–Tate height function on the subvariety. As a corollary of the result, we prove the Bogomolov conjecture which claims that if an irreducible curve X in A is not isomorphic to an elliptic curve, then for the pseudodistance defined by the Neron–Tate height, the distribution of algebraic points on X is uniformly discrete. These results can be extended in the case where base fields are finitely generated over Q via Moriwakiʹs height theory.