# The Lelong number, the Monge-Amp\`ere mass and the Schwarz symmetrization of plurisubharmonic functions

## TL;DR

This paper investigates how Schwarz symmetrization affects key invariants of $S^1$-invariant plurisubharmonic functions, revealing precise relationships between the Lelong number, integrability index, and Monge-Ampère mass.

## Contribution

It establishes exact formulas linking the Lelong number and integrability index under symmetrization, and shows Monge-Ampère mass decreases for toric functions with a single pole.

## Key findings

- $n$ times the integrability index equals the Lelong number of the symmetrized function
- Monge-Ampère mass decreases under symmetrization for certain toric functions
- Provides new insights into invariants of plurisubharmonic functions under symmetrization

## Abstract

The aim of this paper is to study the Lelong number, the integrability index and the Monge-Amp\`ere mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge-Amp\`ere mass is always decreasing under the symmetrization