Geometric properties of upper level sets of Lelong numbers on projective spaces

Let $T$ be a positive closed current of unit mass on the complex projective space $\mathbb P^n$. For certain values $\alpha<1$, we prove geometric properties of the set of points in $\mathbb P^n$ where the Lelong number of $T$ exceeds $\alpha$. We also consider the case of positive closed currents of bidimension (1,1) on multiprojective spaces.