$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds

Abstract

Given $(X,ω)$ compact Kähler manifold and $ψ\in\mathcal{M}^{+}\subset PSH(X,ω)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that $\int_{X}(ω+dd^{c}ψ)^{n}>0$, we prove that the $ψ-$relative finite energy class $\mathcal{E}^{1}(X,ω,ψ)$ becomes a complete metric space if endowed with a distance $d$ which generalizes the well-known $d_{1}$ distance on the space of Kähler potentials. Moreover, for $\mathcal{A}\subset \mathcal{M}^{+}$ total ordered, we equip the set $X_{\mathcal{A}}:=\bigsqcup_{ψ\in\overline{\mathcal{A}}}\mathcal{E}^{1}(X,ω,ψ)$ with a natural distance $d_{\mathcal{A}}$ which coincides with the distance $d$ on $\mathcal{E}^{1}(X,ω,ψ)$ for any $ψ\in\overline{\mathcal{A}}$. We show that $\big(X_{\mathcal{A}},d_{\mathcal{A}}\big)$ is a complete metric space. As a consequence, assuming $ψ_{k}\searrow ψ$ and $ψ_{k},ψ\in \mathcal{M}^{+}$, we also prove that $\big(\mathcal{E}^{1}(X,ω,ψ_{k}),d\big)$ converges in a Gromov-Hausdorff sense to $\big(\mathcal{E}^{1}(X,ω,ψ),d\big)$ and that there exists a direct system $\Big\langle\big(\mathcal{E}^{1}(X,ω,ψ_{k}),d\big),P_{k,j}\Big\rangle$ in the category of metric spaces whose direct limit is dense into $\big(\mathcal{E}^{1}(X,ω,ψ),d\big)$.