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

Abstract

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. a fixed potential determing some singularities such that ${\int }_X(\omega +d{d}^c\psi {)}^n>0$, 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 Kähler potentials. Moreover, for $\mathcal{A}\subset {\mathcal{M}}^{+}$ total ordered, we equip the set $X_{\mathcal{A}}:={⨆}_{\psi \in \overline{\mathcal{A}}}{\mathcal{E}}^1(X,\omega ,\psi )$ 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 $(X_{\mathcal{A}},d_{\mathcal{A}})$ is a complete metric space. As a consequence, assuming ${\psi }_k\searrow \psi$ and ${\psi }_k,\psi \in {\mathcal{M}}^{+}$, we also prove that $({\mathcal{E}}^1(X,\omega ,{\psi }_k),d)$ converges in a Gromov-Hausdorff sense to $({\mathcal{E}}^1(X,\omega ,\psi ),d)$ and that there exists a direct system $⟨({\mathcal{E}}^1(X,\omega ,{\psi }_k),d),P_{k,j}⟩$ in the category of metric spaces whose direct limit is dense into $({\mathcal{E}}^1(X,\omega ,\psi ),d)$.

Old Lemmas 4.2-4.3 has been modified and moved to section 2 (to Lemma 2.6 and Proposition 2.7). Other minor changes. Final version published in The Journal of Geometric Analysis

References in corpus (6)

• Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity
• On the constant scalar curvature Kähler metrics, existence results
• On the constant scalar curvature Kähler metrics, general automorphism group
• Integration by parts formula for non-pluripolar product
• Envelopes with Prescribed Singularities
• Mabuchi geometry of big cohomology classes with prescribed singularities

Cited by in corpus (5)

• Kähler-Einstein metrics with prescribed singularities on Fano manifolds
• Mabuchi geometry of big cohomology classes with prescribed singularities
• A complete metric topology on relative low energy spaces
• Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting
• Geodesics in the space of $m$-subharmonic functions with bounded energy