On Lelong Numbers of Positive Closed Currents on ${\mathbb{P}}^n$

Let $T$ be a positive closed current of bidimension $(p,p)$ with unit mass on the complex projective space ${\mathbb{P}}^n$. For certain values of $\alpha$ and $\beta =\beta (p,\alpha )$ we show that if $T$ has enough points where the Lelong number is at least $\alpha$, then the upper level set $E_{\beta }^{+}(T)$ of points where $T$ has Lelong number strictly larger than $\beta$ has certain geometric properties.