Lemma 6.11.1642017869F45

Proof. We first construct a nonconstant holomorphic map $\delta $ from a disc around $0\in \mathbb C$ to $\mathbb C^ n$ with $\delta (0)=0$ such that $F\circ \delta $ has no zeros other than $\tau =0$ and each $(g_ j/F)\circ \delta $ extends holomorphically across $\tau =0$ with value $0$.

If $r=0$, choose a vector $v$ such that $\zeta \mapsto F(\zeta v)$ does not vanish identically near $0$; for instance, $v$ may be any vector at which the first nonzero homogeneous term of $F$ at $0$ does not vanish. The zeros of this one-variable function are isolated, so on a small disc it vanishes at most at $0$, and $\delta (\tau )=\tau v$ has the required properties.

Let $r\geq 1$. The bound $|F|\leq i|g|$ fails on the ball $B(0,1/i)$ for every large integer $i$, so there are points $z_ i$ with $|z_ i|<1/i$ and $|F(z_ i)|>i|g(z_ i)|$; in particular $F(z_ i)\neq 0$ and $|g(z_ i)|/|F(z_ i)|<1/i$. We may take $z_ i\neq 0$: if $z_ i=0$, the strict inequality $|F|>i|g|$ at $0$ persists at points $z\neq 0$ near $0$, because $F$ and $|g|$ are continuous.

Let $U$ be a connected neighborhood of $0$ on which all functions are defined, and let $\sigma \colon \widetilde U\to U$ be the blowup of the ideal $(F,g_1,\ldots ,g_ r)$, which is not identically zero on $U$ (definition 9.6.1). Thus $\widetilde U\subset U\times \mathbb P^ r$ is the closure of the graph of $z\mapsto [F(z):g_1(z):\cdots :g_ r(z)]$ over the complement of the common zeros, and it is a complex space by proposition 9.6.2. The graph points $\tilde z_ i=(z_ i,[F(z_ i):g(z_ i)])$ converge to $b=(0,[1:0:\cdots :0])$, because $g(z_ i)/F(z_ i)\to 0$. On the part of $\widetilde U$ lying over the chart $\{ \xi _0\neq 0\} $ of $\mathbb P^ r$, the functions $\xi _ j/\xi _0$ are holomorphic and vanish at $b$, and the identities $g_ j\circ \sigma =(\xi _ j/\xi _0)(F\circ \sigma )$ hold on the graph, hence, both sides being continuous, on its closure, which is this part of $\widetilde U$.

The germ of $\widetilde U$ at $b$ is the union of finitely many irreducible branches (proposition 9.3.1); passing to a subsequence, we may assume that all $\tilde z_ i$ lie on a representative $W$ of one branch. The analytic subgerm of $(W,b)$ formed by $\sigma ^{-1}(0)\cup \{ F\circ \sigma =0\} $ is different from $(W,b)$, because $W$ contains the points $\tilde z_ i$, arbitrarily close to $b$, at which $z_ i\neq 0$ and $F(z_ i)\neq 0$; for the same reason $(W,b)$ is not the point $b$ and has positive dimension. By proposition 9.5.2 there is a nonconstant holomorphic map $c$ from a disc into $W$ with $c(0)=b$ such that $c(\tau )\notin \sigma ^{-1}(0)$ and $F(\sigma (c(\tau )))\neq 0$ for $\tau \neq 0$; shrinking the disc, we may assume that $c$ takes values over the chart $\{ \xi _0\neq 0\} $. Put $\delta =\sigma \circ c$. Then $\delta (0)=0$, and for $\tau \neq 0$ we have $\delta (\tau )\neq 0$ and $F(\delta (\tau ))\neq 0$; so $\delta $ is not constant, and $F\circ \delta $ has no zeros other than $\tau =0$. For $\tau \neq 0$ the identities above give $(g_ j/F)(\delta (\tau ))=(\xi _ j/\xi _0)(c(\tau ))$, and the right side is holomorphic on the whole disc with value $(\xi _ j/\xi _0)(b)=0$ at $\tau =0$.

Finally we choose the coordinates and the parameter. Let $q$ be the smallest vanishing order at $0$ of the coordinate functions of $\delta $; it is finite because $\delta $ is not constant, and $q\geq 1$ because $\delta (0)=0$. Then $\delta (\tau )=\tau ^ qe+O(\tau ^{q+1})$ with a vector $e\neq 0$. Choose a linear form $\ell $ with $\ell (e)\neq 0$ and linear coordinates in which $\ell $ is the first coordinate; then $\delta _1(\tau )=\tau ^ qu(\tau )$ with $u$ holomorphic and $u(0)=\ell (e)\neq 0$, while the other coordinates satisfy $|\delta _ j(\tau )|\leq C|\tau |^ q$ near $0$. On a disc around $0$ on which $u$ has no zeros, $u$ has a holomorphic logarithm $h$, and $\kappa =\mathrm e^{h/q}$ is a holomorphic $q$th root of $u$. The function $s(\tau )=\tau \kappa (\tau )$ satisfies $s'(0)=\kappa (0)\neq 0$, so by the holomorphic inverse function theorem in one variable (proposition 5.2.4(1) with $n=1$) it maps a neighborhood of $0$ biholomorphically onto a disc around $0$. Its inverse $s^{-1}$ is holomorphic and vanishes at $0$, so that $|s^{-1}(t)|\leq C'|t|$ after shrinking the disc. Put $\gamma =\delta \circ s^{-1}$ on this disc. Then $\gamma _1(t)=s^{-1}(t)^ qu(s^{-1}(t))=s(s^{-1}(t))^ q=t^ q$, and $|\gamma _ j(t)|\leq CC'^ q|t|^ q$ for $j\geq 2$. Since $s^{-1}$ maps the punctured disc onto a punctured neighborhood of $0$, $F\circ \gamma $ has no zeros other than $t=0$, and each $(g_ j/F)\circ \gamma =((g_ j/F)\circ \delta )\circ s^{-1}$ extends holomorphically across $t=0$ with value $0$. $\square$