Lemma 9.4.4D73CC9F08C05

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 $X_{\mathrm{reg}}\setminus E$. $\square$