Domains of definition of Monge-Amp\`ere operators on compact K\"ahler manifolds

TL;DR

This paper defines and analyzes the largest domain of $ ext{PSH}$ functions on compact K"ahler manifolds where the Monge-Amp e operator is well defined, extending previous local definitions and exploring energy finiteness.

Contribution

It introduces the maximal domain for Monge-Amp e operators on compact K"ahler manifolds and studies properties of twisted operators and energy conditions.

Findings

The domain $DMA(X,\omega)$ is larger than the local domain but still proper.

Twisted Monge-Amp e operators are well defined for all $ ext{PSH}$ functions.

Functions with slightly attenuated singularities have finite weighted Monge-Amp e energy.

Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold. We introduce and study the largest set $DMA(X,\omega)$ of $\omega$-plurisubharmonic (psh) functions on which the complex Monge-Amp\`ere operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all $\om$-psh functions. We prove that certain twisted Monge-Amp\`ere operators are well defined for all $\omega$-psh functions. As a consequence, any $\om$-psh function with slightly attenuated singularities has finite weighted Monge-Amp\`ere energy.