Around the Property (B) for fields of algebraic numbers

The Bogomolov Property arose after Amoroso and Dvornicich gave a sharp lower bound for the height on the abelian closure of the rationals. It is related to a relative (and stronger) version of the Lehmer problem, which has applications towards ambitious conjectures in arithmetic geometry. Interesting examples of fields with this Property have been found recently. I will describe these results and try to say a word about the proofs.