The Lelong number, the Monge–Ampère mass, and the Schwarz symmetrization of plurisubharmonic functions

Abstract

The aim of this paper is to study the Lelong number, the integrability index and the Monge–Ampère 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–Ampère mass is always decreasing under the symmetrization.