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Let $T$ be a positive closed current of unit mass on the complex projectivespace $\mathbb P^n$. For certain values $\alpha<1$, we prove geometricproperties 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 ofbidimension (1,1) on multiprojective spaces.

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