Arithmetic capacities on \(\mathbb{P}^ N\) (Q1323466)

We prove the existence of Chinburg's ``sectional capacity'' for adelic sets in \(\mathbb{P}^ N\), by decomposing it as a product of local sectional capacities. On \(\mathbb{C}^ N\), we show that the local sectional capacity coincides with the \(N\)th power of Leja's generalized transfinite diameter. We prove that for an arbitrary set, the local sectional capacity is the limit of the local sectional capacities of rational domains containing the original set. Finally, we give several examples of local sectional capacities.