Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

On an abelian scheme over a smooth curve over Q̄ a symmetric relatively ample line bundle defines a fiberwise Néron--Tate height. If the base curve is inside a projective space, we also have a height on its Q̄-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the function field of a curve defined over Q̄. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic $0$.