Domains of definition of Monge-Ampère operators on compact Kähler manifolds

Abstract

Let (X, ω) be a compact Kahler manifold. We introduce and study the largest set DMA(X, ω) of ω-plurisubharmonic (psh) functions on which the complex Monge-Ampere 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, ω) of all ω-psh functions. We prove that certain twisted Monge-Ampere operators are well defined for all ω-psh functions. As a consequence, any ω-psh function with slightly attenuated singularities has finite weighted Monge-Ampere energy.