Polynomial hulls and positive currents

Abstract

Let T be a positive plurisubharmonic current of bidimension (p,p) and let $δ>0$. Assume that the Lelong number of T satisfies $ν(T,a)\geq δ$ on a dense subset of supp(T) (rectifiable currents satisfy this condition). Then $T=ϕ[X]$, where X is a complex subvariety of pure dimension p and $ϕ$ 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.