Lemma 5.38.123CF548646A75

Proof. If $s$ is bounded, upper bounds for $U_ m$ and $V_ m$ give the assertion with $\delta _ m=1/m$ for $m>1$. Otherwise principalize the ideal of $s$ by a log resolution $\pi :\widetilde X\to X$ with smooth $\widetilde X$. Let $E_ i$ be the components of the resulting divisor, with positive coefficients $c_ i$ in $s\circ \pi $; in local coordinates $z_ i$ defining these components, the singular part of $s\circ \pi $ is $\sum _ i c_ i\log |z_ i|^2$. Set $A_ i=1+\operatorname {ord}_{E_ i}(\operatorname {Jac}\pi )$. The hypothesis and lemma 5.38.9 give

Choose a positive integer $N\geq \max _ i2A_ i/c_ i$ and put $\delta _ m=N/m$, which is less than one for large $m$. Then the right-hand side above is at least $(1-\delta _ m)c_ i$. After a smooth local correction, divisorial removal from the positive Hessian of $(U_ m+V_ m)\circ \pi $ shows that

has a quasi-plurisubharmonic representative, which is locally bounded above. A finite cover of $\widetilde X$ by charts and descent along the proper map $\pi $, as in the proof of lemma 5.38.11, therefore give $U_ m+V_ m\leq (1-\delta _ m)s+O(1)$. The bounded constant may depend on $m$. $\square$