In Collaboration with Ziyang Gao (CNRS, Princeton) we aim at proving new height inequalities on families of abelian varieties. Our bounds will apply to higher dimension and go beyond the case of curves on which the focus has been. These should lead to new cases of the Bogomolov Conjecture for function fields, a problem that has remained open since breakthrough work of Ullmo and Zhang in the 1990s. A preprint is now available on the arxiv (1801.05762). Partially supported by SNF grant "Diophantine Problems, o-Minimality, and Heights".