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

DOI10.5802/JEP.229arXiv2107.04841MaRDI QIDQ6372471

Rémi Reboulet

Publication date: 10 July 2021

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

Mathematics Subject Classification ID

Kähler manifolds (32Q15) Plurisubharmonic functions and generalizations (32U05) Birational geometry (14E99) Non-Archimedean analysis (32P05)

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