Monge-Amp\`ere operators on complex varieties in $\C^n

The main goal of this article is to find, following the approach given in [Ce1] and [Ce2], the largest possible sub-class of plurisubharmornic functions on a complex variety on which the complex Monge-Amp`ere operator can be reasonably defined. Moreover, we are also interested in giving sufficient condition for solvability of the Monge-Amp`ere operator Monge-Amp`ere operators on complex varieties.