Quasi-monotone convergence of plurisubharmonic functions

TL;DR

This paper introduces a natural quasi-monotone topology for plurisubharmonic functions and demonstrates its equivalence to other stronger topologies, enhancing understanding of the complex Monge-Ampère operator's convergence properties.

Contribution

The paper defines a new quasi-monotone topology for plurisubharmonic functions and proves its equivalence to existing stronger topologies, clarifying convergence behavior.

Findings

The quasi-monotone topology is essentially equivalent to other strong topologies.

This topology provides a natural framework for studying convergence of plurisubharmonic functions.

The results improve understanding of the Monge-Ampère operator's continuity properties.

Abstract

The complex Monge-Amp\`ere operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded quasi-plurisubharmonic functions by various authors. As this operator is not continuous for the $L^{1}$-topology, several stronger topologies have been introduced over the last decades to remedy this, while maintaining efficient compactness criteria. The purpose of this note is to show that these stronger topologies are essentially equivalent to the natural quasi-monotone topology that we introduce and study here.