Lelong Numbers of Bidegree (1,1) Currents on Multiprojective Spaces

Abstract

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.