Theorem 9.4.6 (Normalization)12A47DB401E2

Proof. Step 1: local normalizations. Let $x\in X$, and choose a local model of a neighborhood of $x$. By lemma 9.4.5, applied to sufficiently small representatives of the branches of $(X,x)$, there is a neighborhood $W$ of $x$ such that $X\cap W$ has a normalization, namely the disjoint union of the normal spaces $Y_ i$ with the maps $\nu _ i$, whose fiber over $x$ has one point for each branch of $(X,x)$.

Step 2: uniqueness. Let $\nu _1\colon Y_1\to X$ and $\nu _2\colon Y_2\to X$ be finite bimeromorphic maps from normal complex spaces that are isomorphisms over $X_{\mathrm{reg}}$. Over $X_{\mathrm{reg}}$ there is a unique isomorphism $\phi _0=\nu _2^{-1}\circ \nu _1$ commuting with the maps to $X$; we show that it extends holomorphically. Let $p\in Y_1$ and $x=\nu _1(p)$. Choose disjoint open neighborhoods $W_ l$ of the finitely many points $q_ l$ of $\nu _2^{-1}(x)$, each relatively compact in the domain $W'_ l$ of a local model of $Y_2$, so that the coordinate functions of these models are bounded on $W_ l$. By properness of $\nu _2$, there is a neighborhood $U$ of $x$ with $\nu _2^{-1}(U)\subset \bigcup _ lW_ l$. The analytic subset $E=\nu _1^{-1}(X_{\mathrm{sing}})$ of $Y_1$ is nowhere dense, because $\nu _1$ is bimeromorphic and $X_{\mathrm{sing}}$ is nowhere dense in $X$ (lemma 9.4.1).

Since $Y_1$ is normal, the germ $(Y_1,p)$ is irreducible, and a sufficiently small parametrization neighborhood $V$ of $p$ with $\nu _1(V)\subset U$ has the property that $V\setminus E$ is connected: its connected covering locus remains connected after removing the analytic subset $E$, which is nowhere dense in it, by lemma 9.2.9, and the result is dense in $V\setminus E$. Hence $\phi _0(V\setminus E)$ lies in a single $W_ l$. The coordinate functions of $W'_ l$ composed with $\phi _0$ are bounded holomorphic functions on $V\setminus E$, and they extend holomorphically to $V$ by lemma 9.4.4. The extended map takes values in $\overline{W_ l}\subset W'_ l$, and it satisfies the equations of the local model, which hold on the dense set $V\setminus E$; so it is a holomorphic map $V\to Y_2$ extending $\phi _0$. These local extensions are unique, by density of $V\setminus E$, and therefore glue to a holomorphic map $\phi \colon Y_1\to Y_2$ with $\nu _2\circ \phi =\nu _1$. Extending $\phi _0^{-1}$ in the same way, we obtain the inverse map, since the compositions are the identity on dense open sets. Any isomorphism over $X$ agrees with $\phi _0$ over $X_{\mathrm{reg}}$, so it equals $\phi $. This proves uniqueness.

Step 3: gluing. By the uniqueness just proved, applied over the overlaps of their domains, the local normalizations of Step 1 are canonically isomorphic there, and these isomorphisms are compatible on triple overlaps, because an isomorphism over $X$ between two normalizations is unique; so they glue to a space $\widetilde X$ with a map $\nu \colon \widetilde X\to X$. The space $\widetilde X$ is Hausdorff, since points with different images are separated by inverse images of disjoint neighborhoods and points with the same image lie in one local normalization. Since $X$ has a countable basis, countably many of the open sets over which local normalizations were constructed cover $X$, and each of these local normalizations, a finite disjoint union of analytic subsets of open subsets of the spaces $\mathbb C^{N+m_ i}$, has a countable basis; hence so has $\widetilde X$. Being locally compact and Hausdorff with a countable basis, $\widetilde X$ is paracompact. Being finite, normal and bimeromorphic are local properties over $X$, so $\nu $ is a normalization. The description of the fibers follows from Step 1.

Step 4: the direct image. Let $U\subset X$ be open. If $g$ is holomorphic on $\nu ^{-1}(U)$, then $g\circ \nu ^{-1}$ is holomorphic on $U\cap X_{\mathrm{reg}}$ and locally bounded near $U\cap X_{\mathrm{sing}}$, because $\nu $ is proper; so it is weakly holomorphic. Conversely, if $f$ is weakly holomorphic on $U$, then $f\circ \nu $ is holomorphic on $\nu ^{-1}(U\cap X_{\mathrm{reg}})$ and locally bounded near the nowhere dense analytic subset $\nu ^{-1}(U\cap X_{\mathrm{sing}})$; by lemma 9.4.4, applied on the normal space $\nu ^{-1}(U)$, it extends holomorphically to $\nu ^{-1}(U)$. These two maps are inverse to each other, which proves $\nu _*\mathcal O_{\widetilde X}=\widetilde{\mathcal O}_ X$. $\square$