Definition 16.21.199168E437529

Definition 16.21.1.99168E437529 For $u\in \mathrm{PSH}(X,\theta )$, the singularity envelope of $u$ is

The functions $v$ in this supremum are the competitors for $P_\theta [u]$. A model potential is a potential $\phi $ with $P_\theta [\phi ]=\phi $ and positive mass $\int _ X\langle \theta _\phi ^ n\rangle >0$. Since $u-\sup _ Xu$ is a competitor and all competitors are nonpositive, the singularity envelope is always defined, belongs to $\mathrm{PSH}(X,\theta )$, and satisfies $u-\sup _ Xu\leq P_\theta [u]\leq V_\theta $ and $\sup _ XP_\theta [u]=0$.