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

## TL;DR

This paper investigates the geometric properties of points with high Lelong numbers of positive closed (1,1) currents on multiprojective spaces, characterizing extremal currents and their singularity sets.

## Contribution

It provides a detailed analysis of Lelong number thresholds and characterizes currents with maximal Lelong numbers in specific cohomology classes.

## Key findings

- Sets where Lelong numbers exceed certain thresholds have specific geometric structures.
- Currents with maximal Lelong numbers are explicitly described.
- The paper identifies the points where the maximal Lelong number is attained.

## 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.