The space of finite-energy metrics over a degeneration of complex manifolds

Abstract

Given a degeneration of projective complex manifolds X→𝔻 * with meromorphic singularities, and a relatively ample line bundle L on X, we study spaces of plurisubharmonic metrics on L, with particular focus on (relative) finite-energy conditions. We endow the space ℰ ^ 1 (L) of relatively maximal, relative finite-energy metrics with a d 1 -type distance given by the Lelong number at zero of the collection of fiberwise Darvas d 1 -distances. We show that this metric structure is complete and geodesic. Seeing X and L as schemes X K , L K over the discretely-valued field K=ℂ((t)) of complex Laurent series, we show that the space ℰ 1 (L K an ) of non-Archimedean finite-energy metrics over L K an embeds isometrically and geodesically into ℰ ^ 1 (L), and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.