Infinite-dimensional flats in the space of positive metrics on an ample line bundle

2023/11/22 by Rémi Reboulet, Reboulet, Rémi, Nyström, David Witt · 2 citations
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2311.13451

Abstract

We show that any continuous positive metric on an ample line bundle L lies at the apex of many infinite-dimensional Mabuchi-flat cones. More precisely, given any bounded graded filtration F of the section ring of L, the set of bounded decreasing convex functions on the support of the Duistermaat--Heckman measure of F embeds L^p-isometrically into the space of bounded positive metrics on L with respect to Darvas' d_p distance for all p∈[1,∞), and in particular with respect to the Mabuchi metric (p=2).

Cited by

• Filtrations and asymptotic geometry of non-Archimedean norms on section rings
• Lines in the space of Kähler metrics

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