Symmetrization of plurisubharmonic functions on the Fano manifolds

Abstract

Given a compact complex manifold $Y$ with a negative line bundle $L\to Y$, we study the Schwarz-type symmetrization on the total space of $L$. We prove that this symmetrization does not increase the Monge-Ampère energy for the fibrewise $S^1$-invariant plurisubharmonic functions in the "unit ball" under some assumptions. As an application we generalize the sharp Moser-Trudinger inequality on the unit ball.