THE SPACE OF FINITE-ENERGY METRICS OVER A DEGENERATION OF COMPLEX MANIFOLDS

Journal article, 2023

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 ϵ 1(L) of relatively maximal, relative finite-energy metrics with a d1-type distance given by the Lelong number at zero of the collection of fiberwise Darvas d1-distances. We show that this metric structure is complete and geodesic. Seeing X and L as schemes XK, LK over the discretely-valued field K = C((t)) of complex Laurent series, we show that the space ϵ1(Lan K ) of non-Archimedean finite-energy metrics over Lan K embeds isometrically and geodesically into ϵ 1(L), and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.