Symmetrization of plurisubharmonic functions on the Fano manifolds

TL;DR

This paper investigates a Schwarz-type symmetrization for plurisubharmonic functions on line bundles over Fano manifolds, showing it does not increase Monge-Ampère energy and applying this to generalize a Moser-Trudinger inequality.

Contribution

It introduces a symmetrization technique on line bundles over Fano manifolds and proves its energy non-increasing property, extending classical inequalities.

Findings

Symmetrization does not increase Monge-Ampère energy for certain functions.

Generalization of the sharp Moser-Trudinger inequality.

Application to plurisubharmonic functions on line bundles over Fano manifolds.

Abstract

Given a compact complex manifold $Y$ with a negative line bundle $L\rightarrow Y$, we study the Schwarz-type symmetrization on the total space of $L$. We prove that this symmetrization does not increase the Monge-Amp\`{e}re 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.