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

TL;DR

The paper proves that any continuous positive metric on an ample line bundle $L$ is the apex of many infinite-dimensional Mabuchi-flat cones. By quantising positive metrics through spaces of norms on $H^0(X,kL)$ and employing bounded graded filtrations with associated Duistermaat--Heckman measures, the authors construct, for each filtration, an embedding of the cone of bounded convex decreasing functions on the DH-support into the space of bounded positive metrics, preserving the $d_p$-distance for all $p\in[1,\infty)$. This yields infinite-dimensional flats in the metric space $(\mathcal{E}^p(X,L), d_p)$, generalising known geodesic rays from test configurations. The work bridges non-Archimedean norm geometry with complex-analytic metric geometry, suggesting an asymptotic building structure (apartments given by filtrations) in the space of Kähler metrics and connecting to Okounkov bodies and toric degenerations. It provides a robust framework to study flat subspaces and opens questions about aligning metrics via filtrations to realize a richer apartment-like geometry in the space of positive metrics.

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\in[1,\infty), and in particular with respect to the Mabuchi metric (p=2).

Theorems & Definitions (20)

• Proposition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4: fin1
• Definition 1.5
• Example 1.6: bb
• Proposition 1.7
• proof
• Example 2.1
• Theorem 2.2