  Let T be a positive plurisubharmonic current of bidimension (p,p) and let
$\delta>0$. Assume that the Lelong number of T satisfies $\nu(T,a)\geq \delta$
on a dense subset of supp(T) (rectifiable currents satisfy this condition).
Then $T=\phi[X]$, where X is a complex subvariety of pure dimension p and
$\phi$ is a weakly plurisubharmonic function on X. We have an analogous result
for plurisuperharmonic currents. We also introduce and study a notion of
polynomial p-hull.
