  We extend to the general codimension a lower bound for the essential minimum
on abelian varieties found in a previous work, under a conjecture about
ordinary primes in abelian varieties. This lower bound is the best expected,
``up to an epsilon'', in the degree of the subvariety. Following a strategy of
Amoroso, we change the transcendance phasis in order to simplify the zero lemma
and its combinatorics. The last argument is a descent procedure on varieties,
which is far more intricate in the abelian setting since there is not, in
general, a lifting of the Frobenius in characteristic zero.
