PomeProposition 5.33.1.EFC5ED339FC3 The relation $\preceq $ is a preorder, and $\sim $ is an equivalence relation. If $u$ is quasi-plurisubharmonic and $f$ is a smooth function on $X$, then $u+f\sim u$. In particular, all smooth functions on $X$ have the same singularity type.
Proposition 5.33.1EFC5ED339FC3
Proof. Reflexivity follows with $C=0$. If $u\leq v+C$ and $v\leq w+D$, then $u\leq w+C+D$, which proves transitivity. As for every preorder, comparison in both directions defines an equivalence relation. Since $X$ is compact, a smooth function $f$ satisfies $|f|\leq A$ for some constant $A$, and then $u-A\leq u+f\leq u+A$. Taking $u=0$ shows that every smooth function has the singularity type of the constants. $\square$
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