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

TL;DR

This paper investigates the geometric properties of the upper level sets of Lelong numbers for positive closed currents on complex projective space, establishing conditions under which these sets exhibit specific geometric structures.

Contribution

It introduces new conditions relating Lelong number thresholds to the geometry of their upper level sets for positive closed currents on projective space.

Findings

Upper level sets can have specific geometric properties under certain Lelong number conditions

Thresholds for Lelong numbers influence the structure of the sets

Results extend understanding of the distribution of singularities in positive currents

Abstract

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.