Some applications of Projective Logarithmic Potentials

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.