The strong topology of ω-plurisubharmonic functions

Abstract

On $(X,\omega)$ compact K\"ahler manifold, given a model type envelope $\psi\in PSH(X,\omega)$ (i.e. a singularity type) we prove that the Monge-Amp\`ere operator is an homeomorphism between the set of $\psi$-relative finite energy potentials and the set of $\psi$-relative energy measures endowed with their strong topologies given as the coarsest refinements of the weak topologies such that the relative energies become continuous. Moreover, given a totally ordered family $\mathcal{A}$ of model type envelopes with positive total mass representing different singularities types, the sets $X_{\mathcal{A}}, Y_{\mathcal{A}}$ given respectively as the union of all $\psi$-relative finite energy potentials and of all $\psi$-relative finite energy measures varying $\psi\in\overline{\mathcal{A}}$ have two natural strong topologies which extends the strong topologies on each component of the unions. We show that the Monge-Amp\`ere operator produces an homeomorphism between $X_{\mathcal{A}}$ and $Y_{\mathcal{A}}$. As an application we also prove the strong stability of a sequence of solutions of prescribed complex Monge-Amp\`ere equations when the measures have uniformly $L^{p}$-bounded densities for $p>1$ and the prescribed singularities are totally ordered.