  We extend to the case of semi-abelian varieties the statements of various
variants of the conjecture alla Bogomolov about the non-density of small points
of small height in abelian varieties. Inspired by recent work of Ullmo, Zhang
and Bilu, we then prove these conjectures when the semi-abelian variety is
almost split. The proof uses Arakelov geometry and equirepartition arguments.
