Theorem 9.7.2 (Remmert)0FC675348681

Proof. We prove the two theorems together by induction on dimension. Let $R_ m$ denote the assertion of theorem 9.7.1 for sets $Z$ of dimension at most $m$, and $P_ m$ the assertion of theorem 9.7.2 for maps whose source has dimension at most $m$. Both assertions are local on $X$. A proper map has compact inverse image over a compact neighborhood of a point of the target, and near this compact set its source has finite dimension; so, locally on the target, every proper map falls under some $P_ m$. The assertion $P_0$ holds because a source of dimension zero is discrete, so the image of a proper map from it is closed and locally finite, hence analytic. The assertion $R_0$ holds trivially: a set $Z$ of dimension zero has no branches of dimension greater than $p\geq 0$, so $Z=\varnothing $. We show that $P_{m-1}$ implies $R_ m$, and that $R_ m$ and $P_{m-1}$ imply $P_ m$.

Step 1: $P_{m-1}$ implies $R_ m$; reductions. Since the assertion is local, we may assume that $X$ is an analytic subset of a domain $\Omega \subset \mathbb C^ N$; then $Z$ is an analytic subset of $\Omega \setminus A$, and its closure in $X$ is its closure in $\Omega $. We first reduce to the case where $A$ is a submanifold of dimension $p$, by induction on $p$. Let $A''$ be the union of $A_{\mathrm{sing}}$ and of the irreducible components of $A$ of dimension less than $p$; it is analytic of dimension at most $p-1$ (empty if $p=0$), and $A\setminus A''$ is a closed submanifold of $\Omega \setminus A''$ of dimension $p$. Suppose that the closure $Z_1$ of $Z$ in $\Omega \setminus A''$ is analytic. No branch $B$ of a germ of $Z_1$ is contained in $A$: since $Z$ is dense in $Z_1$, the branches not contained in $A$ already have union $Z_1$ near the base point, and the decomposition into branches is irredundant. Hence $B\setminus A$ is dense in $B$, and at its regular points outside the other branches $B$ coincides with $Z$, whose branches have dimension greater than $p$. Hence all branches of $Z_1$ have dimension greater than $p>p-1$, and by the induction hypothesis on $p$ the closure of $Z_1$ in $\Omega $, which is the closure of $Z$, is analytic.

Thus we may assume that $A$ is a $p$-dimensional submanifold. Fix a point of $A$ in the closure of $Z$; after a holomorphic change of coordinates it is the origin, and near it $A$ is a linear subspace $L$ of dimension $p$. Finally, the irreducible components of $Z$ are pure-dimensional by corollary 9.3.17(1), and their dimensions exceed $p$, because each component contains regular points of $Z$ near which $Z$ coincides with it. The union of the components of a given dimension $s$ is analytic in $\Omega \setminus A$, by corollary 9.3.16, and only the finitely many values $p<s\leq m$ occur; so we may assume that $Z$ is pure of dimension $s$, with $p<s\leq m$. We shall present $Z$ near the origin as a branched covering over an open subset of $\mathbb C^ s$ and write down equations for its closure.

Choice of a projection. We choose linear forms $\xi _1,\ldots ,\xi _ s$ on $\mathbb C^ N$ with the following properties: (i) for each $j<s$, the form $\xi _{j+1}$ does not vanish identically on any irreducible component of $Z\cap \{ \xi _1=\cdots =\xi _ j=0\} $; (ii) the restrictions of $\xi _1,\ldots ,\xi _ p$ to $L$ are linearly independent; (iii) at a point $z_\alpha $ chosen in each connected component $Z^\circ _\alpha $ of the manifold $Z_{\mathrm{reg}}$, the restrictions of $\xi _1,\ldots ,\xi _ s$ to the tangent space of $Z$ are linearly independent.

We choose the forms one at a time. Suppose that $\xi _1,\ldots ,\xi _ j$, $j<s$, have been chosen and that $Z_ j=Z\cap \{ \xi _1=\cdots =\xi _ j=0\} $ is pure of dimension $s-j$, which holds for $j=0$. Since $\Omega $ has a countable basis, $Z_ j$ has countably many irreducible components and $Z_{\mathrm{reg}}$ has countably many connected components $Z^\circ _\alpha $, by corollary 9.3.16. The three properties exclude, for $\xi _{j+1}$, countably many proper linear subspaces of the dual space of $\mathbb C^ N$: by (i), the forms vanishing on a given component of $Z_ j$, which has positive dimension $s-j$; by (ii), if $j<p$, the forms whose restriction to $L$ is a combination of those of $\xi _1,\ldots ,\xi _ j$; and by (iii), the forms whose restriction to the $s$-dimensional tangent space of $Z$ at $z_\alpha $ is a combination of those of $\xi _1,\ldots ,\xi _ j$. A countable union of proper linear subspaces has Lebesgue measure zero, so $\xi _{j+1}$ can be chosen. Then $Z_{j+1}$ is empty or pure of dimension $s-j-1$, by proposition 9.5.6; here we use that a holomorphic function vanishing identically on a local branch of $Z_ j$ vanishes on the irreducible component of $Z_ j$ containing this branch. Indeed, the branch contains regular points of $Z_ j$ outside the other branches, by corollary 9.3.6, and hence a nonempty open subset of the regular locus of this component, so corollary 9.3.16 applies.

Replacing $\xi _ j$, $j>p$, by $\xi _ j-\sum _{i\leq p}a_{ji}\xi _ i$ with suitable constants, we may assume that $\xi _{p+1},\ldots ,\xi _ s$ vanish on $L$; this triangular change does not affect the zero sets $\{ \xi _1=\cdots =\xi _ j=0\} $ or the independence at the points $z_\alpha $. Complete $\xi _1,\ldots ,\xi _ s$ by linear forms vanishing on $L$ to linear coordinates $z=(z',z'')$ with $z'=(\xi _1,\ldots ,\xi _ s)\in \mathbb C^ s$. Then $L=\mathbb C^ p\times \{ 0\} $, and we let $\pi (z)=z'$.

A proper projection. By (i), the set $Z\cap \pi ^{-1}(0)$ is discrete in $\Omega \setminus A$, and by (ii), $A\cap \pi ^{-1}(0)=\{ 0\} $ near $0$. Choose a ball $B''$ centered at $0$ in the $z''$-space whose boundary sphere does not meet $Z\cap \pi ^{-1}(0)$; this excludes countably many radii. The set $Z\cup A$ is closed near $0$ and does not meet the compact set $\{ 0\} \times \partial B''$, so there is a ball $B'\subset \mathbb C^ s$ centered at $0$ with $(Z\cup A)\cap (\overline{B'}\times \partial B'')=\varnothing $. Let $L'=B'\cap (\mathbb C^ p\times \{ 0\} )$, so that $A\cap (B'\times B'')$ lies over $L'$. The projection of $Z\cap (B'\times B'')$ over $B'\setminus L'$ is proper, because over a compact set $K\subset B'\setminus L'$ it is the compact set $(Z\cup A)\cap (K\times \overline{B''})$, and its fibers are compact analytic subsets of $B''$, hence finite by corollary 9.5.3.

The exceptional sets in the base. Let $S_1$ be the set of singular points of $Z$ over $B'\setminus L'$. It is analytic of dimension at most $s-1\leq m-1$, and its projection is proper, so $S_1'=\pi (S_1)$ is analytic in $B'\setminus L'$ by $P_{m-1}$. Over $B'\setminus (L'\cup S_1')$ the set $Z$ is a manifold of dimension $s$; let $S_2$ be the set of its points where $\pi |_ Z$ has rank less than $s$. It is analytic, given locally by a Jacobian determinant. Each connected component of this manifold is an open subset of some $Z^\circ _\alpha $, and on the connected manifold $Z^\circ _\alpha $ the set where the rank of $\pi $ is less than $s$ is a proper analytic subset, since the rank is $s$ at $z_\alpha $ by (iii); so it is nowhere dense, and $S_2$ has dimension at most $s-1$. By $P_{m-1}$, $S_2'=\pi (S_2)$ is analytic in $B'\setminus (L'\cup S_1')$.

By lemma 9.3.15(3), the sets $S_1'$ and $S_2'$ have dimension at most $s-1$, so they are nowhere dense in the open subsets of $\mathbb C^ s$ in which they are analytic. Applying lemma 9.2.9 first to $L'$, then to $S_1'$ and then to $S_2'$, we see that

is connected; applying it in the opposite order, we see that every bounded holomorphic function on $G$ extends holomorphically to $B'$.

The set $Z\cap \pi ^{-1}(G)$ is dense in $Z\cap (B'\times B'')$. Indeed, on each irreducible component $Z_\alpha =\overline{Z^\circ _\alpha }$, the functions $\xi _{p+1},\ldots ,\xi _ s$ do not all vanish identically, because $\pi |_ Z$ is open near $z_\alpha $ and $s>p$; so $Z_\alpha \cap \pi ^{-1}(L')$ is a proper analytic subset of $Z_\alpha $, and it is nowhere dense by corollary 9.3.17(2). Moreover, the points at which $\pi |_ Z$ has rank $s$ are dense in $Z$, by the previous paragraph, and near such points $\pi $ is open; hence $\pi ^{-1}(S_1')$ and $\pi ^{-1}(S_2')$, inverse images of nowhere dense sets, are nowhere dense in $Z$ over $B'\setminus L'$ and over $B'\setminus (L'\cup S_1')$, respectively.

Over $G$, the projection $Z\cap \pi ^{-1}(G)\to G$ is proper and, by proposition 5.2.4(1) applied in charts of the $s$-dimensional manifold $Z$ over $G$, a local biholomorphism, so its image is open and closed in the connected set $G$; it is nonempty, because $Z\cap (B'\times B'')$ is nonempty and $Z\cap \pi ^{-1}(G)$ is dense in it. Hence it is a covering of $G$ with a fixed number $q\geq 1$ of sheets. Indeed, let $z'\in G$ and let $y_1,\ldots ,y_ l$ be the points of the finite fiber over $z'$; choose disjoint neighborhoods $V_ i$ of the $y_ i$ mapped biholomorphically onto neighborhoods of $z'$, and, by properness, a connected neighborhood $N$ of $z'$ in $G$, contained in all these images, such that every point of $Z$ over $N$ lies in $\bigcup _ iV_ i$. Then the part of $Z$ over $N$ is the disjoint union of $l$ graphs over $N$, so the number of points in the fibers is locally constant, hence constant on $G$.

Equations for the closure. For a linear form $\lambda $ on the $z''$-space and $z'\in G$ put

The coefficient of $T^{q-k}$ is a homogeneous polynomial of degree $k$ in the coefficients of $\lambda $, whose coefficients are symmetric polynomials in the coordinates of the $q$ points of the fiber; these are bounded holomorphic functions on $G$, and they extend holomorphically to $B'$. Thus $P_\lambda (z',\lambda (z''))=\sum _\beta g_\beta (z)\lambda ^\beta $, where $\lambda ^\beta $ runs over the monomials in the coefficients of $\lambda $ and the $g_\beta $ are holomorphic on $B'\times B''$. Let $W=\{ g_\beta =0\text{ for all }\beta \} $, the set of points where $P_\lambda (z',\lambda (z''))=0$ for every $\lambda $; it is analytic. Over $G$, the set $W$ is $Z$: a point outside a finite fiber can be separated from all its points by a linear form. Every fiber of $W$ over $B'$ has at most $q$ points, because a linear form taking distinct values at $q+1$ points of one fiber would give $q+1$ distinct roots of a polynomial of degree $q$.

The closure. Choose a smaller ball $B''_1\subset B''$ whose boundary does not meet the finite set $W\cap \pi ^{-1}(0)$, and then a ball $B'_1\subset B'$ centered at $0$ with $W\cap (\overline{B'_1}\times \partial B''_1)=\varnothing $. Then $W_1=W\cap (B'_1\times B''_1)$ is proper with finite fibers over $B'_1$. Every branch of $W_1$ has dimension at most $s$: at a regular point of a branch of dimension $e$, the projection restricted to the branch has finite fibers, so by lemma 9.3.14, applied near a point where its rank is maximal, its maximal rank is $e$, and $e\leq s$.

Let $W_ s$ be the union of the irreducible components of $W_1$ of dimension $s$, and $W_{<s}$ the union of the others. We claim that the closure of $Z$ in $B'_1\times B''_1$ is $W_ s$, which is analytic; this proves $R_ m$. Each component of $W_ s$ is pure of dimension $s$, by corollary 9.3.17(1), and the image under $\pi $ of every neighborhood of each of its points in it contains a neighborhood of the image point, by lemma 9.5.5; so its points over the dense set $G$ are dense in it, and they belong to $Z$. Conversely, let $y\in Z\cap (B'_1\times B''_1)$ lie over $G$. The germ $(W,y)=(Z,y)$ is a manifold germ of dimension $s$, hence irreducible by lemma 9.3.12, so it equals $(W_ s,y)$ or $(W_{<s},y)$, and the latter has smaller dimension. Hence $y\in W_ s$, and since $Z\cap \pi ^{-1}(G)$ is dense in $Z$ and $W_ s$ is closed, the closure of $Z$ in $B'_1\times B''_1$ is contained in $W_ s$.

Step 2: $R_ m$ and $P_{m-1}$ imply $P_ m$. Let $F\colon Y\to X$ be proper, with $\dim Y\leq m$. Let $\rho \colon M\to Y$ be a resolution given by theorem 9.6.6. It is proper, and it is surjective, because its image is closed and contains the dense set $Y_{\mathrm{reg}}$; so replacing $F$ by $F\circ \rho $ does not change the image. Moreover $\dim M\leq m$: the set $\rho ^{-1}(Y_{\mathrm{sing}})$ is nowhere dense, because $\rho $ is bimeromorphic (lemma 9.4.1), so every connected component of $M$ contains a nonempty open set mapped isomorphically onto an open subset of $Y_{\mathrm{reg}}$. Fix a point of $X$, a relatively compact open neighborhood $U'$ of it and an open neighborhood $U''\Subset X$ of $\overline{U'}$ embedded in a domain of some $\mathbb C^ N$. The compact set $(F\circ \rho )^{-1}(\overline{U'})$ meets only finitely many connected components $M_1,\ldots ,M_ k$ of the manifold $(F\circ \rho )^{-1}(U'')$, and the restriction $F_ j\colon M_ j\to U''$ of $F\circ \rho $ is proper. It suffices to show that each $F_ j(M_ j)$ is analytic in $U''$.

Let $r$ be the maximal rank of the differential of $F_ j$ on the connected manifold $M_ j$. If $r=0$, the components of $F_ j$ have zero differential and are constant, so the image is a point. Let $r\geq 1$. The critical set $C=\{ \operatorname {rank}\mathrm dF_ j<r\} $ is an analytic subset of $M_ j$, defined by minors, and it is not all of $M_ j$; so it has dimension at most $m-1$, and $F_ j(C)$ is analytic by $P_{m-1}$. It has dimension at most $r-1$. Indeed, by corollary 9.3.13(2), $C$ is a finite union of complex submanifolds of open subsets of $M_ j$, and the restriction of $F_ j$ to each of them has rank at most $r-1$; so $F_ j(C)$ is $\mathcal H^{2r}$-null by lemma 9.3.15(1), and $\dim F_ j(C)\leq r-1$ by lemma 9.3.15(2).

If $F_ j(C)=F_ j(M_ j)$, we are done. Otherwise $F_ j^{-1}(F_ j(C))$ is an analytic subset of the connected manifold $M_ j$ different from $M_ j$, and on its complement $M'_ j$ the rank of $F_ j$ is constantly $r$. The restriction $M'_ j\to U''\setminus F_ j(C)$ is proper, and a proper map of constant rank has analytic image. Indeed, at a point $w$ of the image, cover the compact fiber by finitely many open sets $V_ l$ such that, by lemma 9.3.14 applied in charts of $M'_ j$, each $F_ j(V_ l)$ is a closed submanifold of dimension $r$ of a neighborhood of $w$. By properness, the inverse image of a small neighborhood $O$ of $w$ lies in $\bigcup _ lV_ l$, so the image meets $O$ in a finite union of closed submanifolds of dimension $r$. Thus $F_ j(M_ j)\setminus F_ j(C)=F_ j(M'_ j)$ is an analytic subset of $U''\setminus F_ j(C)$, pure of dimension $r\leq m$. Since $\dim F_ j(C)<r$, the assertion $R_ m$ shows that its closure in $U''$ is analytic. This closure is $F_ j(M_ j)$, because $M'_ j$ is dense in $M_ j$ and $F_ j(M_ j)$ is closed by properness. This proves $P_ m$ and completes the induction. $\square$