Logarithmic potentials on $\mathbb{P}^n$

TL;DR

This paper investigates the properties of logarithmic potentials of probability measures on complex projective space, establishing their relation to the complex Monge-Ampère operator and showing absolute continuity under certain conditions.

Contribution

It demonstrates that the logarithmic potential operator maps measures into the domain of the Monge-Ampère operator and proves absolute continuity of the Monge-Ampère measure for atomless measures.

Findings

Range of the potential operator is within the Monge-Ampère domain.

Logarithmic potentials of atomless measures have Monge-Ampère measures absolutely continuous w.r.t. Fubini-Study volume.

Provides new insights into the regularity of potentials on complex projective space.

Abstract

We study the projective logarithmic potential $\mathbb{G}_{\mu}$ of a Probability measure $\mu$ on the complex projective space $\mathbb{P}^{n}$. We prove that the Range of the operator $\mu\longrightarrow \mathbb{G}_{\mu}$ is contained in the (local) domain of definition of the complex Monge-Amp\`ere operator acting on the class of quasi-plurisubharmonic functions on $\mathbb{P}^n$ with respect to the Fubini-Study metric. Moreover, when the measure $\mu $ has no atom, we show that the complex Monge-Amp\`ere measure of its Logarithmic potential is an absolutely continuous measure with respect to the Fubini-Study volume form on $\mathbb{P}^{n}$