$$L^{1}$$ Metric Geometry of Potentials with Prescribed Singularities on Compact Kähler Manifolds

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