  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.
