  Let $T$ be a positive closed current of bidegree $(1,1)$ on a multiprojective
space $X={\mathbb P}^{n_1}\times\ldots\times{\mathbb P}^{n_k}$. For certain
values of $\alpha$, which depend on the cohomology class of $T$, we show that
the set of points of $X$ where the Lelong numbers of $T$ exceed $\alpha$ have
certain geometric properties. We also describe the currents $T$ that have the
largest possible Lelong number in a given cohomology class, and the set of
points where this number is assumed.
