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

TL;DR

This paper investigates the geometric structure of upper level sets of Lelong numbers for positive closed currents on the complex projective plane, revealing they are mostly contained within a conic under certain conditions.

Contribution

It establishes a new property linking high Lelong number points to conic containment, advancing understanding of current singularities in complex geometry.

Findings

Upper level set $E_{eta}^+(T)$ is contained within a conic with at most one exception.

The result applies for currents with four points of Lelong number greater than $rac{2}{5}$.

Provides geometric constraints on the distribution of singularities of currents.

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.