Proposition 9.4.336422A732754

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 $x$. $\square$