Lemma 9.4.558E87F89D557

Proof. Step 1: the construction for an irreducible germ. Suppose first that $p=1$, so that $(X,x)$ is irreducible. Choose generators $f_1,\ldots ,f_ m$ of the $\mathcal O_{X,x}$-module $\widetilde{\mathcal O}_{X,x}$ and a denominator $h$ as in proposition 9.4.3, so that $hf_ j=g_ j$ with $g_ j$ holomorphic. Choose a neighborhood $W\subset \Omega $ of $x$ on which $h$ and the $g_ j$ are holomorphic, the $f_ j$ are weakly holomorphic, and each $f_ j$ satisfies a monic equation $P_ j(z,f_ j)=0$ whose coefficients are holomorphic on $W$. On $(X\cap W)\setminus \{ h=0\} $ the functions $g_ j/h$ are holomorphic and agree with $f_ j$ on $X_{\mathrm{reg}}$. Let $Y$ be the closure in $W\times \mathbb C^ m$ of the graph of $(g_1/h,\ldots ,g_ m/h)$ over $(X\cap W)\setminus \{ h=0\} $. This graph is the part outside $\{ h=0\} $ of the analytic set

By lemma 9.3.7, applied in $W\times \mathbb C^ m$ to $Z$ and the analytic set $\{ h=0\} $, the closure $Y$ of $Z\setminus \{ h=0\} $ is analytic; near each point it is the union of the branches of $Z$ on which $h$ does not vanish identically.

The monic equations $P_ j(z,g_ j/h)=0$ hold on $X_{\mathrm{reg}}\setminus \{ h=0\} $, which is dense in $X\setminus \{ h=0\} $, and their left-hand sides are continuous there; so they hold on the graph, hence on its closure $Y$: $P_ j(z,w_ j)=0$. By (9.2.1) they bound $w$ over compact subsets of $X\cap W$ and leave finitely many possible values of $w$ over each point. Hence the projection $\nu \colon Y\to X\cap W$ is proper with finite fibers.

Over $X_{\mathrm{reg}}$, the set $Y$ is the graph of $f=(f_1,\ldots ,f_ m)$: a point of $Y$ over $z_0\in X_{\mathrm{reg}}$ is a limit of graph points over points $z_ k\to z_0$, and since the $f_ j$ are continuous at $z_0$, it equals $(z_0,f(z_0))$. Conversely, $(z_0,f(z_0))$ lies in $Y$, because $z_0$ is a limit of points of $X_{\mathrm{reg}}\setminus \{ h=0\} $: the germ of $h$ at $z_0$ is not identically zero on the manifold germ $(X,z_0)$, so $\{ h=0\} $ is nowhere dense in $X_{\mathrm{reg}}$ near $z_0$, by lemma 9.2.8. Hence $\nu $ is an isomorphism over $X_{\mathrm{reg}}\cap W$, with inverse $z\mapsto (z,f(z))$. This graph is dense in $Y$, because $X_{\mathrm{reg}}\setminus \{ h=0\} $ is dense in $X\setminus \{ h=0\} $ and $(g_1/h,\ldots ,g_ m/h)$ is continuous there; so $Y$ is the closure of the graph of $f$ over $X_{\mathrm{reg}}\cap W$, the set $\nu ^{-1}(X_{\mathrm{sing}})$ is nowhere dense, and $\nu $ is bimeromorphic. Every point of $Y$ over $x$ is a limit of points $(z_ k,f(z_ k))$ with $z_ k\in X_{\mathrm{reg}}$ and $z_ k\to x$; by the last assertion of proposition 9.4.3, the fiber over $x$ is the single point $y=(x,\lim _{z\to x}f(z))$.

Step 2: normality of $Y$ at $y$. Let $a$ be a weakly holomorphic function on $Y$ near $y$. Since $\nu ^{-1}(X_{\mathrm{reg}})$ is a manifold, it is contained in $Y_{\mathrm{reg}}$, and $b=a\circ (\nu |_{\nu ^{-1}(X_{\mathrm{reg}})})^{-1}$ is holomorphic on $X_{\mathrm{reg}}$ near $x$. It is bounded near $x$: by properness and since the fiber over $x$ is $\{ y\} $, the inverse image of a small neighborhood of $x$ lies in a neighborhood of $y$ on which $a$ is bounded. Thus $b\in \widetilde{\mathcal O}_{X,x}$, and $b=\sum _ ja_ jf_ j$ with $a_ j\in \mathcal O_{X,x}$. Lift the $a_ j$ to holomorphic functions $\widetilde a_ j$ near $x$ in $\Omega $. The function $\sum _ j\widetilde a_ j(z)w_ j$ is holomorphic near $y$, and it equals $a$ on $\nu ^{-1}(X_{\mathrm{reg}})$ near $y$, which is dense in $Y_{\mathrm{reg}}$; since both functions are continuous on $Y_{\mathrm{reg}}$, they agree on $Y_{\mathrm{reg}}$ near $y$. Hence $\widetilde{\mathcal O}_{Y,y}=\mathcal O_{Y,y}$, and $y$ is a normal point. By lemma 9.4.4, all points of $Y$ near $y$ are normal, and by properness the inverse image of a small neighborhood $U$ of $x$ lies in this set of normal points. Replacing $W$ by $U$ and $Y$ by $\nu ^{-1}(U)$, we obtain the assertions of the lemma for $p=1$. They persist when $W$ is replaced by a smaller open neighborhood of $x$, because the closure of the graph over the smaller set is the part of $Y$ over it, and normality, finiteness, bimeromorphy and being an isomorphism over the regular locus are preserved under restriction over open subsets of $X$.

Step 3: several branches. In general, apply Steps 1 and 2 to each irreducible germ $(X_ i,x)$, with $X_ i$ regarded as an analytic subset of a ball around $x$ in $\Omega $, and let $W$ be the intersection of the resulting neighborhoods; by the persistence just noted, the assertions about the individual $Y_ i$ hold over $W$. The disjoint union of the $Y_ i$ is normal, and its map to $X\cap W$ is finite, with $p$ points over $x$. It is an isomorphism over $X_{\mathrm{reg}}\cap W$, because, by corollary 9.3.13(1), $X_{\mathrm{reg}}\cap W$ is the disjoint union of the open sets $X_ i\cap X_{\mathrm{reg}}=(X_ i)_{\mathrm{reg}}\setminus \bigcup _{j\neq i}X_ j$, over which $\nu _ i$ is an isomorphism. It is bimeromorphic, with the nowhere dense analytic subset $X_{\mathrm{sing}}\cap W$ of $X\cap W$. Indeed, by corollary 9.3.13(1), $X_{\mathrm{sing}}\cap X_ i=(X_ i)_{\mathrm{sing}}\cup (X_ i\cap \bigcup _{j\neq i}X_ j)$, which is a nowhere dense analytic subset of $X_ i\cap W$ by corollary 9.3.6; so its inverse image under the bimeromorphic map $\nu _ i$ is nowhere dense in $Y_ i$, by lemma 9.4.1. These properties again persist for smaller neighborhoods of $x$. $\square$