A Property of Upper Level Sets of Lelong Numbers of Currents on $\mathbb{P}^2$

Abstract

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on the complex projective space $\mathbb P^2$. For $\alpha > 2/5$ and $\beta = (2-2\alpha)/3$ we show that if $T$ has four point with Lelong number greater than $\alpha$, the upper level set $E_{\beta}^+ (T)$ of points of $T$ with Lelong number strictly larger than $\beta$ is contained within a conic with the exception of at most one point.