Infinite-Dimensional Flats in the Space of Positive Metrics on an Ample Line Bundle

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 $\mathcal{F}$ of the section ring of $L$, the set of bounded decreasing convex functions on the support of the Duistermaat–Heckman measure of $\mathcal{F}$ 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 ($p=2$).