Projective Logarithmic Potentials

TL;DR

This paper investigates the regularity and properties of projective logarithmic potentials of probability measures on complex projective space, revealing conditions for absolute continuity and regularity based on measure characteristics.

Contribution

It extends the understanding of the Green operator's regularizing effects and characterizes the regularity of potentials in relation to measure properties on complex projective space.

Findings

The Green operator $G$ has strong regularizing properties.

The complex Monge-Ampère measure of the potential is absolutely continuous iff the measure has no atoms.

Regularity of the potential depends on the measure's dimension.

Abstract

We study the projective logarithmic potential $G_\mu$ of a Probability measure $\mu$ on the complex projective space ${P}^{n}$ equiped with the Fubini-Study metric $\omega$. We prove that the Green operator $G $ has strong regularizing properties. It was shown by the second author that the range of the operator $G$ is contained in the (local) domain of definition of the complex Monge-Amp\`ere operator on $P^n$. This result extends earlier results by Carlehed. We will show that the complex Monge-Amp\`ere measure of the logarithmic potential of $\mu$ is absolutely continuous with respect to the Lebesgue measure on $P^n$ if and only if the measure $\mu$ has no atoms. Moreover when the measure $\mu$ has a "positive dimension", we give more precise results on regularity properties of the potential $G_\mu$ in terms of the dimension of $\mu$.