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

TL;DR

This paper studies finite-energy metrics over degenerating complex manifolds, establishing a complete geodesic metric space structure and linking it with non-Archimedean geometry, extending previous trivially-valued results.

Contribution

It introduces a new metric structure on finite-energy plurisubharmonic metrics over degenerations and embeds non-Archimedean metrics into this framework, generalizing prior work.

Findings

The metric space of finite-energy metrics is complete and geodesic.

Non-Archimedean metrics embed isometrically into the complex setting.

The work extends previous trivially-valued cases to more general degenerations.

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 $\hat \cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise 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 $\K=\mathbb{C}((t))$ of complex Laurent series, we show that the space $\cE^1(L_\K\an)$ of non-Archimedean finite-energy metrics over $L_\K\an$ embeds isometrically and geodesically into $\hat \cE^1(L)$, 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.