Geometric properties of upper level sets of Lelong numbers on projective spaces

TL;DR

This paper investigates the geometric structure of upper level sets of Lelong numbers for positive closed currents on complex projective and multiprojective spaces, revealing new properties for certain thresholds.

Contribution

It establishes geometric properties of Lelong number level sets for positive closed currents on projective spaces, extending to multiprojective spaces for bidimension (1,1) currents.

Findings

Characterization of upper level sets for Lelong numbers exceeding certain thresholds

Extension of geometric properties to multiprojective spaces

Insights into the structure of positive closed currents

Abstract

Let $T$ be a positive closed current of unit mass on the complex projective space $\mathbb P^n$. For certain values $\alpha<1$, we prove geometric properties of the set of points in $\mathbb P^n$ where the Lelong number of $T$ exceeds $\alpha$. We also consider the case of positive closed currents of bidimension (1,1) on multiprojective spaces.