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

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