PomeCorollary 9.4.7.8D2ABD4CAC6A Let $X$ be an analytic subset of a domain $\Omega \subset \mathbb C^ N$, with coordinates $z$, let $\nu \colon \widetilde X\to X$ be its normalization, and let $x\in X$. Let $W$ and $\nu _ i\colon Y_ i\to X_ i\cap W$, $1\leq i\leq p$, be as in lemma 9.4.5, for sufficiently small representatives $X_1,\ldots ,X_ p$ of the branches of $(X,x)$. Then each $\nu _ i$ is a normalization of $X_ i\cap W$ whose fiber over $x$ is a single point, and there is an isomorphism of $\nu ^{-1}(W)$ onto the disjoint union of $Y_1,\ldots ,Y_ p$ under which $\nu $ corresponds to the maps $\nu _ i$. In particular, every holomorphic function on $\widetilde X$ near a point of $\nu ^{-1}(x)$ corresponds, near that point, to the restriction to some $Y_ i$ of a holomorphic function of $(z,w)$.
Corollary 9.4.78D2ABD4CAC6A
Proof. By lemma 9.4.5, each $Y_ i$ is normal, and $\nu _ i$ is finite, bimeromorphic and an isomorphism over $(X_ i)_{\mathrm{reg}}\cap W$, with a single point over $x$; so $\nu _ i$ is a normalization of $X_ i\cap W$. Likewise, the disjoint union of the $Y_ i$ is a normalization of $X\cap W$. So is $\nu ^{-1}(W)\to X\cap W$, because being finite, normal, bimeromorphic and an isomorphism over the regular locus are local properties over $X$. By the uniqueness in theorem 9.4.6, there is an isomorphism over $X\cap W$ between them. The last assertion holds because the holomorphic functions on the analytic set $Y_ i$ are, locally, restrictions of holomorphic functions of the ambient coordinates $(z,w)$. $\square$
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