Proposition 6.10.6675539E4A76C

Proof. The second assertion is immediate, since $|F|^2\mathrm e^{-\varphi }\leq C^2\sum _ j|g_ j|^2\mathrm e^{-\varphi }$ near $x$. For the first, choose generators $g_1,\ldots ,g_ r$ of $\mathcal I(\varphi )_ x$ (theorem 6.8.9), representatives of the $g_ j$, of $F$ and of the $a_ j$ on a common neighborhood, and put $|g|=(\sum _ j|g_ j|^2)^{1/2}$. Each $a_ j$ is a finite sum of products of $j$ generators with holomorphic coefficients, which are bounded on a smaller neighborhood; hence $|a_ j|\leq C_ j|g|^ j$ there. At a point where $|g|>0$, put $\lambda =|F|/|g|$. Dividing the equation by $|g|^ d$ gives $\lambda ^ d\leq \sum _ jC_ j\lambda ^{d-j}$, which forces $\lambda \leq C$ with $C=\max \{ 1,\sum _ jC_ j\} $. At a point where $|g|=0$, all the $a_ j$ vanish, so $F^ d=0$ and $F=0$. Thus $|F|\leq C|g|$ near $x$, and the second assertion applies. $\square$