PomeCorollary 6.9.3.A3BCC047C1A7 Let $x\in X$, and let $u$ be a function on a neighborhood of $x$ that near $x$ is the sum $u=v+h$ of a plurisubharmonic function $v$ and a bounded measurable function $h$. Put $\nu (u,x)=\nu (v,x)$; by proposition 6.7.1 this does not depend on the decomposition. If $\nu (u,x)=0$, then $\mathrm e^{-tu}$ is integrable on a neighborhood of $x$ for every $t>0$. This applies in particular to a function $u$ that is quasi-plurisubharmonic near $x$, that is, the sum of a plurisubharmonic function and a smooth real function near $x$. For a weight $\varphi $ on $X$ with $\nu (\varphi ,x)=0$ it says that $\mathcal I(t\varphi )_ x=\mathcal O_{X,x}$ for every $t>0$.
Corollary 6.9.3A3BCC047C1A7
Proof. If $u=v+h=v'+h'$ are two such decompositions near $x$, then $v$ and $v'$ take the value $-\infty $ at the same points, and $v\leq v'+C$ and $v'\leq v+C$ near $x$, where $C$ bounds $|h|+|h'|$; so $\nu (v,x)=\nu (v',x)$ by proposition 6.7.1. If $X$ has dimension zero at $x$, then $v(x)$ is finite and there is nothing to prove. Otherwise let $U$ be a connected neighborhood of $x$ on which $v$ is plurisubharmonic and $|h|\leq M$ for a constant $M$. Applying lemma 6.9.1 to the weight $v$ on $U$ and $m=t$, we obtain $\operatorname {ord}_ x\mathcal I(tv)\leq t\nu (v,x)=0$. Hence $\mathcal I(tv)_ x$ contains a germ that does not vanish at $x$; this germ is a unit, so $\mathcal I(tv)_ x=\mathcal O_{X,x}$, that is, $\mathrm e^{-tv}$ is integrable on a neighborhood of $x$. Since $\mathrm e^{-tu}\leq \mathrm e^{tM}\mathrm e^{-tv}$ on $U$, the same holds for $\mathrm e^{-tu}$. For a weight $\varphi $, the integrability of $\mathrm e^{-t\varphi }$ near $x$ means that $1\in \mathcal I(t\varphi )_ x$. $\square$
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