Lemma 9.5.5776A532CD79E

Proof. For $d=0$ the germ is the point $x$, and there is nothing to prove. Let $d\geq 1$, embed a representative $X$ in a local model $X\subset \Omega \subset \mathbb C^ N$ such that $X$ is pure $d$-dimensional and $F$ is the restriction of a holomorphic map on $\Omega $, also denoted $F$, and choose a closed ball $\overline{B(x,\rho )}\subset \Omega $, with $X\cap B(x,\rho )\subset V$, whose intersection with $X\cap F^{-1}(0)$ is $\{ x\} $. The compact set $X\cap \partial B(x,\rho )$ does not meet $F^{-1}(0)$, so $|F|$ has a positive minimum $\eta $ on it (if it is empty, put $\eta =\infty $). Let $B_\varepsilon $ be the ball of radius $\varepsilon <\eta $ about $0\in \mathbb C^ d$, and let $X'=X\cap B(x,\rho )\cap F^{-1}(B_\varepsilon )$. Then $F\colon X'\to B_\varepsilon $ is proper: the inverse image of a compact set $K\subset B_\varepsilon $ is $X\cap \overline{B(x,\rho )}\cap F^{-1}(K)$, which is compact and contains no point of the boundary sphere. Its fibers are compact analytic subsets of the open ball $B(x,\rho )$, hence finite by corollary 9.5.3.

It remains to prove surjectivity. On each connected component of the manifold $X'_{\mathrm{reg}}$, which has dimension $d$, the differential of $F$ has rank $d$ at some point. Otherwise its maximal rank $r$ on some component would be smaller; near a point where the rank is $r$, the rank would be constantly $r$, and by lemma 9.3.14, applied in a chart of $X'_{\mathrm{reg}}$, the fibers near this point would be submanifolds of positive dimension $d-r$, contradicting their finiteness.

Let $C$ be the union of $X'_{\mathrm{sing}}$ and of the set of regular points where the rank of $F$ is less than $d$. In the local model, $C$ is the union of $X'_{\mathrm{sing}}$ and of the common zero set in $X'$ of the minors of size $N$ of the matrix formed by the differentials of local generators of the ideal of $X$ and the rows of $\mathrm dF$; so $C$ is analytic. It contains no branch of $X'$ at any point. Indeed, each branch contains a nonempty open subset of $X'_{\mathrm{reg}}$, consisting of regular points of the branch outside the other branches (corollary 9.3.6), and on each component of $X'_{\mathrm{reg}}$ the points of rank less than $d$ form a proper analytic subset, which is nowhere dense by lemma 9.2.8. Hence $C$ has dimension at most $d-1$ by proposition 9.3.9.

Its image $E=F(C)$ is closed in $B_\varepsilon $ by properness, and it is $\mathcal H^{2d-1}$-null. Indeed, by corollary 9.3.13(2), $C$ is a finite union of complex submanifolds of open subsets of $\Omega $ of dimensions at most $d-1$, on each of which $F$ has rank at most $d-1$, so lemma 9.3.15(1) applies.

Over $B_\varepsilon \setminus E$, the map $F$ is a local biholomorphism at every point of its source, by proposition 5.2.4(1) applied in charts of $X'_{\mathrm{reg}}$, and it is proper; so its image is open and closed in $B_\varepsilon \setminus E$. This image is nonempty: at a regular point of rank $d$, the image of a neighborhood is open, and it is not contained in $E$, which is Lebesgue null because it is $\mathcal H^{2d-1}$-null. Since $B_\varepsilon \setminus E$ is connected by lemma 9.5.4, the image contains $B_\varepsilon \setminus E$, which is dense in $B_\varepsilon $; being closed in $B_\varepsilon $, it equals $B_\varepsilon $. $\square$