Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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A. Trusiani

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

created by trusiani on 11 Dec 2019



Inserted: 11 dec 2019
Last Updated: 11 dec 2019

Year: 2019

ArXiv: 1909.03897 PDF


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 of a distance $d$ which generalizes the well-known $d_{1}$ distance on the space of K\"ahler potentials. Later, for $\mathcal{A}\subset \mathcal{M}^{+}$ total ordered, we equip the set $X_{\mathcal{A}}:=\bigsqcup_{\psi\in\overline{\mathcal{A}}}\mathcal{E}^{1}(X,\omega,\psi)$ of 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 it is possible to define a direct system $\Big(\mathcal{E}^{1}(X,\omega,\psi_{k}),P_{k,j}\Big)$ in the category of metric spaces whose direct limit is dense into $\big(\mathcal{E}^{1}(X,\omega,\psi),d\big)$.

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