9.4 Reduced spaces and normalizationC16A2F5D9879

Proof. Let $E$ be as in the definition, and let $G\subset F^{-1}(B)$ be open in $Y$. Then $G\setminus F^{-1}(E)$ is an open subset of $Y\setminus F^{-1}(E)$ contained in $F^{-1}(B\setminus E)$, which corresponds under the isomorphism $Y\setminus F^{-1}(E)\to X\setminus E$ to the nowhere dense subset $B\setminus E$ of $X\setminus E$; so $G\setminus F^{-1}(E)$ is empty, and $G$, an open subset of the nowhere dense set $F^{-1}(E)$, is empty as well. In particular $F$ is also an isomorphism over $X\setminus (E\cup B)$, and $F^{-1}(E\cup B)=F^{-1}(E)\cup F^{-1}(B)$ is nowhere dense, as the union of two closed nowhere dense sets; so $E$ may be replaced by $E\cup B$. If $U=X\setminus B$, then $F^{-1}(U)$ is the complement of the closed nowhere dense set $F^{-1}(B)$, hence dense. Finally, a nonempty open set $G\subset Y$ meets the dense open set $Y\setminus F^{-1}(E)$, and $F$ maps the nonempty open set $G\setminus F^{-1}(E)$ isomorphically onto an open subset of $X\setminus E$. $\square$

Proof. By corollary 9.3.13(1), $X_{\mathrm{reg}}\cap U$ is the disjoint union of the sets $X_ i\cap X_{\mathrm{reg}}=(X_ i)_{\mathrm{reg}}\setminus \bigcup _{j\neq i}X_ j$, each of which is open in $X_{\mathrm{reg}}$. The analytic subset $(X_ i)_{\mathrm{reg}}\cap \bigcup _{j\neq i}X_ j$ of the manifold $(X_ i)_{\mathrm{reg}}$ is nowhere dense in it, by corollary 9.3.6, applied to $(X_ i,x)$ and its subgerm $(X_ i\cap \bigcup _{j\neq i}X_ j,x)$, which differs from $(X_ i,x)$ because the branches are irredundant. Hence a bounded holomorphic function on $X_ i\cap X_{\mathrm{reg}}\cap V$ extends uniquely across this analytic subset to a holomorphic function on $(X_ i)_{\mathrm{reg}}\cap V$, by lemma 9.2.9 applied on connected coordinate balls of $(X_ i)_{\mathrm{reg}}$, and the extension is bounded by the same bound. Applied near each point of $X_ i\cap V$, this shows that the restriction of $f$ has a unique extension $f_ i$, which is weakly holomorphic on $X_ i\cap V$. Conversely, if $f_ i$ is weakly holomorphic on $X_ i\cap V$ for each $i$, the function equal to $f_ i$ on $X_ i\cap X_{\mathrm{reg}}\cap V$ is holomorphic and locally bounded on $X_{\mathrm{reg}}\cap V$, hence weakly holomorphic on $X\cap V$; by the uniqueness of the extensions, the two constructions are inverse to each other. Passing to germs at $x$ gives the last assertion. $\square$

Proof. Step 1: an irreducible germ. Let $(X,0)$ be irreducible, with a local parametrization as in theorem 9.3.4, its degree $q$, its primitive linear form $u$, its Weierstrass polynomial $W$ and $\delta =\operatorname {Disc}(W)$. We show that $\delta _ y\widetilde{\mathcal O}_{X,y}\subset \mathcal O_{X,y}$ for every $y$ in the parametrization polydisc. Let $f$ be weakly holomorphic near $y$ and bounded on $V\cap X_{\mathrm{reg}}$ for a neighborhood $V$ of $y$. Let $y=y^{(1)},y^{(2)},\ldots ,y^{(s)}$ be the points of $X$ over $y'$, choose disjoint neighborhoods $V_1\subset V,V_2,\ldots ,V_ s$ of them, and, using properness, a connected neighborhood $N'$ of $y'$ such that all points of $X$ over $N'$ lie in $\bigcup _ lV_ l$. Define $F$ on $X_{\mathrm{reg}}\cap \pi ^{-1}(N')$ by $F=f$ on $V_1$ and $F=0$ on the other $V_ l$.

For $z'\in N'\setminus \{ \delta =0\} $, let $z^{(1)},\ldots ,z^{(q)}$ be the points of $X$ over $z'$; the values $u(z^{(i)})$ are the distinct roots of $W(z',\cdot )$, by theorem 9.3.4, so there is a unique polynomial $\sum _{j<q}b_ j(z')T^ j$ taking the value $F(z^{(i)})$ at $T=u(z^{(i)})$ for every $i$. By Cramer's rule $\delta b_ j=D\cdot D_ j$, where $D$ is the Vandermonde determinant of the $u(z^{(i)})$, so that $D^2=\delta $ by (9.3.2), and $D_ j$ is obtained by replacing its $j$th column with the values $F(z^{(i)})$. The products $D\cdot D_ j$ do not depend on the order of the points; they are holomorphic, because the points depend holomorphically on $z'$ locally, and bounded, because $u$ and $F$ are bounded. By lemma 9.2.9 they extend holomorphically to $N'$. The holomorphic function $H(z)=\sum _ j(\delta b_ j)(z')u(z'')^ j$ on $N'\times \Delta ''$ equals $\delta F$ on the covering locus over $N'$, hence $\delta f$ on $X_{\mathrm{reg}}\cap V_1\cap \pi ^{-1}(N')$, by density of the covering locus and continuity. Thus $\delta f$ extends to the holomorphic germ of $H|_ X$ at $y$.

Step 2: several branches. Fix $x\in X$, and let $X_1,\ldots ,X_ p$ be sufficiently small representatives of the branches of $(X,x)$, and let $\delta _ j$ be a denominator for $X_ j$ as in Step 1, lifted to a holomorphic function of the ambient coordinates. Adding to $\delta _ j$ a linear combination $\sum _ la_ le_{jl}$ of local generators $e_{jl}$ of the ideal of $X_ j$ does not change it on $X_ j$. For each $k\neq j$ some $e_{jl}$ does not vanish identically on $X_ k$, so the coefficients $a$ for which $\delta _ j+\sum _ la_ le_{jl}$ vanishes identically on $X_ k$ form a proper affine subspace or the empty set. Choosing $a$ outside these finitely many subspaces, we obtain $\widetilde\delta _ j$, equal to $\delta _ j$ on $X_ j$ and not identically zero on any $X_ k$. Then $\eta =\prod _ j\widetilde\delta _ j$ is not identically zero on any $X_ k$, and it is a denominator for each $X_ j$ separately: if $f$ is weakly holomorphic on $X$ near a point $y$, the restriction of $f$ to $X_ j\cap X_{\mathrm{reg}}$ extends to a weakly holomorphic function $f_ j$ on $X_ j$ near $y$, by lemma 9.4.2, so by Step 1 $\eta f_ j=g_ j|_{X_ j}$ for some holomorphic $g_ j$ near $y$.

For each $j$ choose a product $v_ j$ of local equations of the branches $X_ k$, $k\neq j$, that does not vanish identically on $X_ j$, and put $h=\eta \sum _ jv_ j$. On $X_ j$ we have $h=\eta v_ j$, which is not identically zero, and $hf=\sum _ jv_ jg_ j$ holds on every $X_ j\cap X_{\mathrm{reg}}$, hence on $X_{\mathrm{reg}}$ near $y$. Thus $h_ y\widetilde{\mathcal O}_{X,y}\subset \mathcal O_{X,y}$ for all $y$ near $x$. Finally, $\{ h=0\} \cap X_ j$ has smaller dimension than $X_ j$ at $x$, hence at all points near $x$, by upper semicontinuity; every branch of $(X,y)$ is a branch of some $(X_ j,y)$ and has dimension $\dim X_ j$, by purity. So, after shrinking $U$, the germ of $h$ at every $y\in U$ is not identically zero on any branch of $(X,y)$.

Step 3: integrality, finiteness and limits. The map $f\mapsto (hf)/h$ embeds $\widetilde{\mathcal O}_{X,x}$ into the total ring of fractions of $\mathcal O_{X,x}$; it does not depend on the choice of $h$, and it is injective because $X_{\mathrm{reg}}\setminus \{ h=0\} $ is dense in $X_{\mathrm{reg}}$. Multiplication by $h$ identifies $\widetilde{\mathcal O}_{X,x}$ with an ideal of the Noetherian ring $\mathcal O_{X,x}$, so it is a finitely generated $\mathcal O_{X,x}$-module.

Every weakly holomorphic germ is integral. On an irreducible germ, in the notation of Step 1, the elementary symmetric functions of the values of $f$ at the $q$ points of $X$ over $z'\in \Delta '\setminus \{ \delta =0\} $ are bounded holomorphic functions, which extend to $\Delta '$ by lemma 9.2.9. They are the coefficients of a monic polynomial $P(z',T)$ with $P(z',f)=0$ on the covering locus, hence on $X_{\mathrm{reg}}$, and these coefficients belong to $\mathcal O_ d\subset \mathcal O_{X,x}$. For several branches, the product of the polynomials of the branches, with coefficients lifted to holomorphic functions, is a monic equation for $f$. Conversely, let $a/b$ be an element of the total ring of fractions, with $b$ a nonzerodivisor, that is integral over $\mathcal O_{X,x}$. On $X_{\mathrm{reg}}\setminus \{ b=0\} $ near $x$ the function $a/b$ satisfies a monic equation with bounded coefficients, so it is bounded by (9.2.1). The set $\{ b=0\} $ meets each connected open subset of the manifold $X_{\mathrm{reg}}$ in a nowhere dense analytic subset, so by lemma 9.2.9 $a/b$ extends to a holomorphic function on $X_{\mathrm{reg}}$ near $x$, which is weakly holomorphic and is mapped to $a/b$.

Let $(X,x)$ be irreducible and $f$ weakly holomorphic near $x$, satisfying the monic equation $P(z',f)=0$ just constructed. As $Q$ runs through the parametrization polydiscs around $x$, the sets $X_{\mathrm{reg}}\cap Q$ are the traces on $X_{\mathrm{reg}}$ of a basis of neighborhoods of $x$, and each of them is connected, because it lies between the connected covering locus and its closure. The cluster set of $f$ at $x$ is the intersection of the decreasing compact connected nonempty sets $\overline{f(X_{\mathrm{reg}}\cap Q)}$, hence it is nonempty, compact and connected. If $w$ is a cluster value, then $w=\lim f(z_ k)$ for points $z_ k\to x$ of $X_{\mathrm{reg}}$, and $P(x',w)=\lim P(z_ k',f(z_ k))=0$; so the cluster set is contained in the finite set of roots of $P(x',\cdot )$. Being connected, it is a single point, and $f$ has a finite limit at $x$. $\square$

Proof. We show that the set of nonnormal points is the support of a coherent sheaf. We first record two facts about coherent sheaves. If $\mathcal F$ and $\mathcal G$ are coherent $\mathcal O_ X$-modules, the sheaf $\mathcal Hom(\mathcal F,\mathcal G)$ of module homomorphisms is coherent, and its stalk at $x$ is $\operatorname {Hom}_{\mathcal O_{X,x}}(\mathcal F_ x,\mathcal G_ x)$. Indeed, $\mathcal F$ has local presentations $\mathcal O_ X^ a\to \mathcal O_ X^ b\to \mathcal F\to 0$, and $\mathcal Hom(\mathcal F,\mathcal G)$ is the kernel of the induced morphism $\mathcal G^ b\to \mathcal G^ a$; we apply lemma 9.2.6.

Next, the support of a coherent module $\mathcal F$ is analytic. In a local presentation as above, given by a $b\times a$ matrix $\Phi $ of holomorphic functions, $\mathcal F_ x=0$ exactly when the matrix $\Phi (x)$ has rank $b$. If $\Phi (x)$ has rank $b$, then $\mathcal F_ x=\mathfrak m_ x\mathcal F_ x$, where $\mathfrak m_ x$ is the maximal ideal, and the following determinant argument gives $\mathcal F_ x=0$: writing each of finitely many generators of $\mathcal F_ x$ as a combination of them with coefficients in $\mathfrak m_ x$, with coefficient matrix $C$, the adjugate identity shows that $\det (I-C)$ annihilates the generators, and this determinant is a unit. Conversely, if $\mathcal F_ x=0$, then $\Phi $ maps $\mathcal O_{X,x}^ a$ onto $\mathcal O_{X,x}^ b$, and reducing modulo $\mathfrak m_ x$ shows that $\Phi (x)$ has rank $b$. Hence the support of $\mathcal F$ is the common zero set of the minors of size $b$ of $\Phi $.

Let $h$ be a denominator as in proposition 9.4.3 on an open set $U$, let $\mathcal I$ be the coherent ideal sheaf of $\{ h=0\} $ in $U$, and put $\mathcal F=\mathcal Hom(\mathcal I,\mathcal I)$. Multiplication embeds $\mathcal O_ U$ in $\mathcal F$, because $\mathcal I$ contains the nonzerodivisor $h$. Every $\varphi \in \mathcal F_ y$ is multiplication by an element of the total ring of fractions: if $a=\varphi (h)$, then $h\varphi (g)=\varphi (hg)=ag$ for $g\in \mathcal I_ y$, so $\varphi $ is multiplication by $a/h$. Let $g_1,\ldots ,g_ N$ generate $\mathcal I_ y$, and write $(a/h)g_ i=\varphi (g_ i)=\sum _ jc_{ij}g_ j$ with $c_{ij}\in \mathcal O_{X,y}$. Multiplying the matrix $(a/h)I-(c_{ij})$ by its adjugate, we see that its determinant, a monic polynomial in $a/h$ with coefficients in $\mathcal O_{X,y}$, annihilates every $g_ i$, hence $\mathcal I_ y$ and in particular $h$; since $h$ is invertible in the total ring of fractions, this polynomial vanishes, and $a/h$ is integral over $\mathcal O_{X,y}$. Hence, by proposition 9.4.3,

We claim that the support of the coherent sheaf $\mathcal F/\mathcal O_ U$ is the set of nonnormal points of $U$; this proves the first assertion. At a normal point $\widetilde{\mathcal O}_{X,y}=\mathcal O_{X,y}$, so $\mathcal F_ y=\mathcal O_{X,y}$. Let $y$ be a nonnormal point, so that $\widetilde{\mathcal O}_{X,y}\neq \mathcal O_{X,y}$. By the analytic Nullstellensatz (corollary 9.3.8), applied in a local model, every element of $\mathcal I_ y$ has a power in $h\mathcal O_{X,y}$. Let $g_1,\ldots ,g_ N$ generate $\mathcal I_ y$, and choose $e$ with $g_ i^ e\in h\mathcal O_{X,y}$ for all $i$. Every monomial of degree $\ell =N(e-1)+1$ in the $g_ i$ contains some $g_ i$ at least $e$ times, so $\mathcal I_ y^\ell \subset h\mathcal O_{X,y}$, and therefore $\mathcal I_ y^\ell \widetilde{\mathcal O}_{X,y}\subset h\widetilde{\mathcal O}_{X,y}\subset \mathcal O_{X,y}$.

Let $k$ be the least integer with $\mathcal I_ y^ k\widetilde{\mathcal O}_{X,y}\subset \mathcal O_{X,y}$; then $k\geq 1$, and we may choose $w\in \mathcal I_ y^{k-1}\widetilde{\mathcal O}_{X,y}$ with $w\notin \mathcal O_{X,y}$. Then $w\mathcal I_ y\subset \mathcal O_{X,y}$, and every product $wg$, $g\in \mathcal I_ y$, vanishes on $\{ h=0\} $: at regular points because $g$ vanishes there and $w$ is holomorphic, and at singular points because $w$ is locally bounded, $g$ tends to zero, and $wg$ is continuous. Thus $w\mathcal I_ y\subset \mathcal I_ y$, so multiplication by $w$ belongs to $\mathcal F_ y$; it is not multiplication by an element of $\mathcal O_{X,y}$, since $w\notin \mathcal O_{X,y}$ and $h$ is a nonzerodivisor. This proves the claim.

For the last assertion, let $f$ be a holomorphic function on $X\setminus E$ that is locally bounded near $E$. Since $E$ is nowhere dense, an extension is unique. The set $E\cap X_{\mathrm{reg}}$ meets every connected open subset of the manifold $X_{\mathrm{reg}}$ in a nowhere dense analytic subset, so $f$ extends holomorphically to $X_{\mathrm{reg}}$ by lemma 9.2.9. The extension is locally bounded near $X_{\mathrm{sing}}$, because on a neighborhood of each point it is bounded by a bound of $f$ on the dense subset $X_{\mathrm{reg}}\setminus E$; hence it is weakly holomorphic, and therefore holomorphic, because $X$ is normal. It agrees with $f$ on $X\setminus E$, since both functions are continuous there and agree on the dense subset $X_{\mathrm{reg}}\setminus E$. $\square$

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 $x$. $\square$

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 $\nu _*\mathcal O_{\widetilde X}=\widetilde{\mathcal O}_ X$. $\square$

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 $(z,w)$. $\square$

Proof. On a unibranch space the normalization has one-point fibers, so it is a continuous bijection; being proper, it is closed, hence a homeomorphism. For the second assertion, let $U$ be the intersection of $X$ with a parametrization polydisc of $(X,x)$. Its covering locus is a connected manifold, dense in $U$; it remains connected after deleting $A$, by lemma 9.2.9, because $A$ is nowhere dense, and the result is dense in $U\setminus A$. Hence $U\setminus A$ is connected. $\square$