Definition 10.3.79499229CD425

Definition 10.3.7.9499229CD425 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Recall from section 6.18 that $\varphi $ is more singular than $\psi $, written $\varphi \preceq \psi $, if $\varphi \leq \psi +C$ on $X$ for some constant $C$; we then also say that $\psi $ is less singular than $\varphi $. A potential $\psi \in \mathrm{PSH}(X,\theta )$ has minimal singularities if every $\varphi \in \mathrm{PSH}(X,\theta )$ is more singular than $\psi $. If $\mathrm{PSH}(X,\theta )\ne \varnothing $, put

where ${}^*$ denotes the upper semicontinuous regularization.