The local heights of Definition 2.39 agree with the local heights introduced by Gubler, see [Gub03, Proposition 3.5] for the Archimedean case and [Gub03, Remark 9.4] for the non-Archimedean case. In the Archimedean case, the local height in [Gub03] is defined in terms of a refined star product of Green currents based on [Bur94]. The hypothesis needed in Gubler’s definition of local heights are weaker than the ones we use. We have chosen the current definition because it is more elementary and suffices for our purposes.