A local version of the arithmetic Hodge index theorem over quasiprojective varieties

We define a local intersection number for metrised line bundles over quasiprojective varieties with compact support and show the local arithmetic Hodge index theorem for this intersection number. As a consequence we obtain a uniqueness result for the Monge-Ampère equation over quasiprojective varieties within a certain class of solutions both in the archimedean and non-archimedean setting.