PomeProposition 11.8.2.26C9605156BE Suppose an effective Cartier divisor agrees, with multiplicity one, with $\{ \zeta =0\} $ throughout the chosen coordinate chart. Then $C_\delta $ is uniformly bounded for all sufficiently small $\delta $.
Proposition 11.8.226C9605156BE
Proof. Let $s$ be the canonical section of its divisor line bundle. In the chart write $s=\zeta e$ with $e$ a nonvanishing holomorphic frame. Choose a smooth Hermitian metric $h$ with $|e|_ h=1$ on a neighborhood of $\operatorname {supp}\chi \times \{ |\zeta |\leq \delta _0\} $ for a fixed small $\delta _0>0$. The smooth global functions $\psi _\delta =\log (|s|_ h^2+\delta ^2)$ satisfy $\mathrm{dd}^{\mathrm{c}}\psi _\delta \geq -C\omega $ with one $C\geq 1$. Off the divisor their Hessians are a bounded scalar multiple of the negative curvature of $h$, plus a semipositive gradient term; continuity gives the same lower bound at the divisor. Fix $0<\epsilon \leq (2C)^{-1}$, so $\omega +\epsilon \mathrm{dd}^{\mathrm{c}}\psi _\delta \geq \omega /2$. On the flat-frame region,
\[ \mathrm{dd}^{\mathrm{c}}\psi _\delta =\frac{\delta ^2}{(|\zeta |^2+\delta ^2)^2}\sigma _\zeta \geq \frac1{4\delta ^2}\sigma _\zeta \quad (|\zeta |\leq \delta ). \]
If $P=\| p\| _\infty $, then $B_\delta \leq P\delta ^{-2}\sigma _\zeta $ on its support. Therefore
\[ \omega +\epsilon \mathrm{dd}^{\mathrm{c}}\psi _\delta \geq \frac{\epsilon }{4P}B_\delta \]
globally for $\delta \leq \delta _0$. Every $\Theta \in \mathcal T_\omega $ gives, by smooth Stokes,
\[ 1=\int _ X(\omega +\epsilon \mathrm{dd}^{\mathrm{c}}\psi _\delta )\wedge \Theta \geq \frac{\epsilon }{4P}\int _ XB_\delta \wedge \Theta . \]
Taking the supremum proves $C_\delta \leq 4P/\epsilon $. $\square$
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