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

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

2007-05-31

arxiv.org/abs/0705.4529v1

Mathematics

Complex Variables

Scientific paper

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.

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