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

2021/07/10 by Reboulet, Rémi · 3 citations
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.04841

Abstract

Given a degeneration of compact projective complex manifolds X over the punctured disc, 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 \cE¹(L) of relatively maximal, relative finite-energy metrics with a d₁-type distance given by the Lelong number at zero of the collection of fibrewise Darvas d₁-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 \cE¹(L_\K\an) of non-Archimedean finite-energy metrics over L_\K\an embeds isometrically and geodesically into \cE¹(L), and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case. We investigate consequences regarding convexity of non-Archimedean functionals.

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