Symmetrization of plurisubharmonic functions and Fano manifolds

Abstract

Let L be a negative line bundle on a compact complex manifold Y, we define a Schwarz-type symmetrization on the total space X of L. We prove that this symmetrization does not increase the Monge–Ampere energy for fibrewise $$S^{1}$$ -invariant plurisubharmonic functions defined on the ”unit ball” in X. As a result, we generalize the sharp Moser–Trudinger inequality to this ”unit ball”.