Definition 16.22.13D3EBF9A66D06

Definition 16.22.13.D3EBF9A66D06 Let $u,\phi \in \mathrm{PSH}(X,\theta )$ with $u\preceq \phi $. The singularity envelope of $u$ relative to $\phi $ is

Choose $C_0$ with $u-C_0\leq \phi $. For $C\geq -C_0$, the potential $u-C_0$ is a common minorant of $u+C$ and $\phi $, so the rooftop envelope $P_\theta (u+C,\phi )$ lies between $u-C_0$ and $u+C$ and has the singularity type of $u$. These rooftop envelopes increase with $C$ and are bounded above by $\phi $, so the regularized limit is a potential between $u-C_0$ and $\phi $.