Definition 13.9.32AF01ACCF985

Definition 13.9.3.2AF01ACCF985 A closed positive $(1,1)$-current $T\in \alpha $ has full Monge–Ampère mass if $\int _ X\langle T^ n\rangle =\operatorname {vol}(\alpha )$. We write $\mathcal E(X,\theta )$ for the set of $u\in \mathrm{PSH}(X,\theta )$ such that $\theta +\mathrm{dd}^{\mathrm{c}}u$ has full Monge–Ampère mass. For $\phi \in \mathrm{PSH}(X,\theta )$, the class $\mathcal E(X,\theta ,\phi )$ consists of the potentials $u\in \mathrm{PSH}(X,\theta )$ with $u\leq \phi +O(1)$ and

these potentials have full mass relative to $\phi $. In a big class, $\mathcal E(X,\theta )=\mathcal E(X,\theta ,V_\theta )$, because every $u\in \mathrm{PSH}(X,\theta )$ satisfies $u\leq V_\theta +\sup _ Xu$ and $\operatorname {vol}(\{ \theta \} )$ is the mass of $V_\theta $.