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The strong topology of $\omega$-plurisubharmonic functions

Antonio Trusiani

Vol. 16 (2023), No. 2, 367–405

On a compact Kähler manifold (X,ω), given a model-type envelope ψ PSH (X,ω) (i.e., a singularity type) we prove that the Monge–Ampère operator is a homeomorphism between the set of ψ-relative finite energy potentials and the set of ψ-relative finite 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 𝒜 of model-type envelopes with positive total mass representing different singularity types, the sets X𝒜 and Y 𝒜, given as the union of all ψ-relative finite energy potentials and of all ψ-relative finite energy measures with varying ψ 𝒜 ¯, respectively, have two natural strong topologies which extend the strong topologies on each component of the unions. We show that the Monge–Ampère operator produces a homeomorphism between X𝒜 and Y 𝒜.

As an application we also prove the strong stability of a sequence of solutions of complex Monge–Ampère equations when the measures have uniformly Lp-bounded densities for p > 1 and the prescribed singularities are totally ordered.

complex Monge-Ampère equations, compact Kähler manifolds, quasi-psh functions
Mathematical Subject Classification
Primary: 32W20
Secondary: 32Q15, 32U05
Received: 14 May 2020
Revised: 4 March 2021
Accepted: 10 June 2021
Published: 3 May 2023
Antonio Trusiani
University of Rome Tor Vergata

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