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

Coman, Dan; Guedj, Vincent; Zeriahi, Ahmed

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

Authors

Coman, Dan

Guedj, Vincent

Zeriahi, Ahmed

Description

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