Heights in families of abelian varieties and the geometric Bogomolov conjecture

On an abelian scheme over a smooth curve over $\bar{\mathbb{Q}}$ 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 $\bar{\mathbb{Q}}$-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 $\bar{\mathbb{Q}}$. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic 0.