Lemma 9.3.19113991711A43

Proof. By corollary 9.3.17, $D_{\mathrm{reg}}$ is a connected manifold, dense in $D$, and all branches of $D$ at all points have the dimension $n-1$ of $D_{\mathrm{reg}}$. Let $x\in D$ and let $D_1,\ldots ,D_ k$ be representatives of the branches of $(D,x)$. We construct a generator $f_ i$ of the ideal of each $D_ i$ near $x$, and show that $f=f_1\cdots f_ k$ has the required properties.

Step 1: an equation of one branch. Fix $i$, and take a parametrization of $(D_ i,x)$ as in theorem 9.3.4, in coordinates centered at $x$. Its base dimension is $d=n-1$, by proposition 9.3.9, so $z''=z_ n$ is a single coordinate. For $z'\in \Delta '\setminus \{ \delta =0\} $, the $q$ points of $D_ i$ over $z'$ are distinct, and therefore so are their coordinates $z_ n$; put $P(z',T)=\prod _{z\in \pi ^{-1}(z')}(T-z_ n)$. The coefficients of $P$ are, up to sign, the elementary symmetric functions of these coordinates; they are holomorphic, because the sheets are graphs of holomorphic maps, and bounded, because $\Delta ''$ is bounded, so they extend holomorphically to $\Delta '$ by lemma 9.2.9. As $z'\to 0$, the points of $D_ i$ over $z'$ tend to $0$, because $\pi $ is proper and $\pi ^{-1}(0)=\{ 0\} $; hence the extended coefficients vanish at $0$, and $P$ is a Weierstrass polynomial of degree $q$ in $z_ n$. The function $P(z',z_ n)$ vanishes on the covering locus, hence on $D_ i\cap (\Delta '\times \Delta '')$, by density and continuity.

The germ of $P(z',z_ n)$ generates the prime ideal $I_{D_ i,x}$. Indeed, let $g\in I_{D_ i,x}$, and let $R$ be the remainder of the division of $g$ by $P$ (proposition 9.2.2), a polynomial in $z_ n$ of degree less than $q$ with coefficients holomorphic near $0$ in $\mathbb C^{n-1}$; it vanishes on $D_ i$ near $0$, since $g$ and $P(z',z_ n)$ do. For $z'$ near $0$ with $\delta (z')\neq 0$, the $q$ points of $D_ i$ over $z'$ are close to $0$, so $R(z',\cdot )$ vanishes at their $q$ distinct coordinates $z_ n$ and is zero. The coefficients of $R$ thus vanish on a dense subset of a neighborhood of $0$, and $R=0$.

Let $f_ i$ be a holomorphic function near $x$ with the germ of $P(z',z_ n)$. The sheaf $\mathcal I_{D_ i}$ has finitely many generators near $x$ by theorem 9.3.10, and each of them is a multiple of $f_ i$ at $x$, hence on a neighborhood; so $f_ i$ generates $\mathcal I_{D_ i,y}$ for $y$ near $x$. In particular, at every point near $x$ outside $D_ i$ the ideal of $D_ i$ is the unit ideal, so $f_ i$ does not vanish there, and $f_ i^{-1}(0)=D_ i$ near $x$.

Step 2: the product $f=f_1\cdots f_ k$. Let $y\in D_{\mathrm{reg}}$ be close to $x$. By corollary 9.3.13(1), $y$ lies on only one $D_ i$, and $D=D_ i=\{ \tau =0\} $ near $y$ for a local coordinate $\tau $; then $f_ i$ is a unit times $\tau $, since both generate $\mathcal I_{D_ i,y}$ (lemma 9.3.12), and $f_ j(y)\neq 0$ for $j\neq i$, by Step 1. Thus $f$ vanishes to order one along $D_{\mathrm{reg}}$ near $x$, and $f^{-1}(0)=D$ near $x$. Choose a neighborhood $V$ of $x$ on which all this holds and on which, in addition, $D_ i\setminus \bigcup _{j\neq i}D_ j$ is dense in $D_ i$ for every $i$; this is possible by corollary 9.3.6, since the branches are irredundant. Let $y\in V$ and $g\in \mathcal I_{D,y}$; we show by induction on $l$ that $g\in (f_1\cdots f_ l)_ y$. For $l=0$ there is nothing to prove. If $g=f_1\cdots f_ lh$ with $l<k$, then $h$ vanishes near $y$ on $D_{l+1}\setminus (D_1\cup \cdots \cup D_ l)$, where the $f_ i$, $i\leq l$, have no zeros; this set is dense in $D_{l+1}$, so $h$ vanishes on $D_{l+1}$ near $y$, and $h\in (f_{l+1})_ y$. Hence $f$ generates $\mathcal I_{D,y}$.

Finally, in the notation of section 5.5, the regular part of the zero set of $f$ is $D_{\mathrm{reg}}\cap V$, with multiplicity one. By theorem 5.5.6, $[D]=\mathrm{dd}^{\mathrm{c}}\log |f|^2$ on $V$; hence $[D]$ is closed and positive, with measure coefficients of locally finite mass. $\square$