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

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