# Some applications of Projective Logarithmic Potentials

## TL;DR

This paper explores applications of projective logarithmic potentials, comparing capacities, defining transfinite diameter, and establishing analogs of classical theorems in complex geometry.

## Contribution

It introduces notions of projective logarithmic energy and capacity, compares them with Monge-Ampère capacity, and proves an Evans-type theorem in this context.

## Key findings

- Projective logarithmic capacity is quantitatively compared with Monge-Ampère capacity.
- The transfinite diameter of a set coincides with its logarithmic capacity.
- An Evans-type theorem for sets of zero projective logarithmic capacity is established.

## Abstract

We continue the study in \cite{As18, AAZ18} by giving a multitude of applications of projective logarithmic potentials. First we introduce the notions of projective logarithmic energy and capacity associated to projective kernel that was introduced and studied in \cite{As18, AAZ18}. We compare quantitatively the projective logarithmic capacity with the complex Monge-Amp\`ere capacity on $\mathbb P^n$ and we deduce that the set of zero logarithmic capacity is of Monge-Amp\`ere capacity zero. Further, we define transfinite diameter of a compact set and we show that it coincides with logarithmic capacity. Finally we deduce that there is an analogous of classical Evans's theorem that for any compact set $K$ of zero projective logarithmic capacity shows the existence of Probability measure whose potential admits $K$ as polar set.