Lemma 5.18.152E0432C779A0

Proof. We shall prove a more precise local result. Take $x\in X$, take an open neighborhood $U\subseteq X$ of $x$, on which

where $c\in \mathbb {Q}_{>0}$ and $g_1,\ldots ,g_ N$ are holomorphic functions on $U$. Then we claim that for any $\lambda >0$, we have

where for any $a\in \mathbb {R}$, $a_+$ means $a\lor 0$. Note that (5.18.16) is a straightforward consequence of (5.18.17). The global result follows easily from the local result by the compactness of $X$.

Fix a smooth volume form $\mathrm{d}V$ on $X$. Take $f\in \mathcal{I}(\lambda \varphi )$. By strong openness theorem 5.16.21, we can find $\epsilon >0$ so that $f\in \mathcal{I}((\lambda +\epsilon /c)\varphi )$. Therefore, we can find an open neighborhood $W\subseteq U$ of $x$ so that

But it follows from the Briançon–Skoda division theorem ([Sko72], [SB74], [Theorem 11.17, Dem12]) that

where $\alpha =\left(\lfloor c \lambda \rfloor -n+1\right)_+$. Here $\left(g_{1,x},\ldots ,g_{N,x} \right)$ denotes the ideal in $\mathcal{O}_{X,x}$ generated by the germs of $g_1,\ldots ,g_ N$ at $x$. Note that $\alpha \geq (c\lambda -n)_+$.

It follows that on a neighborhood of $x$, we have

Hence (5.18.17) follows. $\square$