Upper level sets of Lelong numbers on $\mathbb P^2$ and cubic curves

TL;DR

This paper investigates the structure of positive closed currents on the complex projective plane, showing that for certain Lelong number thresholds, the upper level sets are mostly contained within a cubic curve, with at most two exceptions.

Contribution

It establishes a bound on the number of points outside a cubic curve where the Lelong numbers exceed a given threshold, revealing geometric constraints on such currents.

Findings

Upper level sets are mostly contained in a cubic curve for lpha  .

At most two points outside the cubic curve have high Lelong numbers.

Provides geometric bounds on the distribution of Lelong numbers on .

Abstract

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_{\alpha}(T)$ be the upper level sets of Lelong numbers $\nu(T,x)$ of $T$. For any $\alpha\geq \frac{1}{3}$, we show that $|V_{\alpha}(T)\setminus C|\leq 2$ for some cubic curve $C$.