  The Equidistribution Conjecture is proved for general semiabelian varieties
over number fields. Previously, this conjecture was only known in the special
case of almost split semiabelian varieties through work of Chambert-Loir. The
general case has remained intractable so far because the height of a
semiabelian variety is negative unless it is almost split. In fact, this places
the conjecture outside the scope of Yuan's equidistribution theorem on
algebraic dynamical systems. To overcome this, an asymptotic adaption of the
equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here.
It also allows a new proof of the Bogomolov Conjecture and hence a
self-contained proof of the Strong Equidistribution Conjecture in the same
general setting.
