The same bounded-shift preorder on quasi-plurisubharmonic functions on a compact manifold. The Hermitian source distinguishes ordinary type from multiplier-ideal type.
Definition 16.7.1.3E4D57445698Assume that $X$ is compact. Given $\varphi ,\psi \in \mathrm{QPSH}(X)$, we say that $\varphi $ is more singular than $\psi $ and write $\varphi \preceq \psi $1 if there is $C\in \mathbb {R}$ such that $\varphi \leq \psi +C$. We also say $\psi $ is less singular than $\varphi $ and write $\psi \succeq \varphi $.
In case $\varphi \preceq \psi $ and $\psi \preceq \varphi $, we say $\varphi $ and $\psi $ have the same singularity type. We write $\varphi \sim \psi $ in this case.
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Definition 16.7.1.3E4D57445698 Let $u,v$ be quasi-plurisubharmonic functions on $X$. We write $u\preceq v$ if there is a constant $C\in \mathbb R$ such that $u\leq v+C$ on $X$, and $u\sim v$ if both $u\preceq v$ and $v\preceq u$; in the latter case $u$ and $v$ have the same singularity type. When this has to be distinguished from the multiplier-ideal type introduced below, we speak of the ordinary singularity type, or simply the ordinary type.
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