9.7 Proper images and relative principalization05BF24D5DFA6

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. $\square$

Proof. The composite is proper. Let $E_ X\subset X$ and $E_ Y\subset Y$ be nowhere dense analytic subsets as in the definition of bimeromorphic maps for $f$ and $g$. The image $f(E_ Y)$ is analytic by theorem 9.7.2, and it is nowhere dense: an open set $G\subset f(E_ Y)$ satisfies $G\setminus E_ X\subset f(E_ Y\setminus f^{-1}(E_ X))$, which corresponds under the isomorphism $f\colon Y\setminus f^{-1}(E_ X)\to X\setminus E_ X$ to the nowhere dense set $E_ Y\setminus f^{-1}(E_ X)$; so $G\subset E_ X$, and $G=\varnothing $. Hence $B=E_ X\cup f(E_ Y)$ is a nowhere dense analytic subset of $X$. The map $f$ is an isomorphism over $X\setminus B$, and $f^{-1}(X\setminus B)$ does not meet $E_ Y$, so $f\circ g$ is an isomorphism over $X\setminus B$. Finally, $f^{-1}(B)$ is a nowhere dense analytic subset of $Y$, and $(f\circ g)^{-1}(B)=g^{-1}(f^{-1}(B))$ is nowhere dense in $Z$, by two applications of lemma 9.4.1. $\square$

Proof. Let $\mathfrak b=\mathfrak a\mathcal O_ W$. We first principalize $\mathfrak b$ by blowups of $W$, perform the same blowups on $M$, and then principalize the remaining ideal away from the strict transform of $W$. If $\dim W=0$, then $W$ is a point, $\mathfrak b$ is the unit ideal, the first sequence below is empty, and the argument applies with $H=0$.

Step 1: principalizing the restriction. Apply theorem 9.6.7 to the ideal $\mathfrak b$ on the compact manifold $W$, with empty divisor. We obtain a finite sequence of blowups $W_ i\to W_{i-1}$ with smooth centers $C_ i\subset W_{i-1}$, $1\leq i\leq t$, and ideals $\mathfrak b_ i$ with $\mathfrak b_0=\mathfrak b$, $\mathfrak b_ i=\mathcal O_{W_ i}(E^ W_ i)\mathfrak b_{i-1}\mathcal O_{W_ i}$, where $E^ W_ i$ is the inverse image of $C_ i$, and $\mathfrak b_ t=\mathcal O_{W_ t}$. By induction on $i$, each $W_{i-1}$ is connected, $\mathfrak b_{i-1}$ is nonzero, and $C_ i$ has positive codimension in $W_{i-1}$ at each of its points; for $i=1$, the first two assertions are hypotheses of the theorem. Assume them for some $i\leq t$. A nonzero coherent ideal on a connected manifold has nonzero stalks at all points, since the points at which its stalk vanishes form an open set, and so, by the identity theorem, do the points at which it does not vanish; hence its zero set is nowhere dense. So $C_ i\subset V(\mathfrak b_{i-1})$ is nowhere dense in $W_{i-1}$, that is, of positive codimension at each of its points. Then $W_ i\setminus E^ W_ i\cong W_{i-1}\setminus C_ i$ is connected by lemma 9.2.9 and dense in $W_ i$, so $W_ i$ is connected; and $\mathfrak b_ i$ is nonzero, because it agrees with $\mathfrak b_{i-1}$ on this set.

We perform the same blowups on the ambient manifolds: let $M_0=M$ and let $\sigma _ i\colon M_ i\to M_{i-1}$ be the blowup along $C_ i$, where $W_{i-1}\subset M_{i-1}$ is the strict transform of $W$. The center $C_ i$ is closed in $M_{i-1}$, because $W_{i-1}$ is closed, and it has codimension at least two in $M_{i-1}$. By proposition 9.6.3, the strict transform of $W_{i-1}$ in $M_ i$ is $\operatorname {Bl}_{C_ i}W_{i-1}=W_ i$, and $E_ i\cap W_ i=E^ W_ i$, where $E_ i=\sigma _ i^{-1}(C_ i)$. More precisely, in the charts of that proposition whose pivot is one of the coordinates $z_1,\ldots ,z_ k$ along $W_{i-1}$, which are the only ones meeting $W_ i$, a local equation $e_ i$ of $E_ i$ restricts to a local equation of $E^ W_ i$. This sequence is adapted to $W$.

Step 2: the transformed ideals. Put $H_0=0$ and $\mathfrak a_0=\mathfrak a$, and define inductively

We show by induction that $\mathfrak a_ i$ is a holomorphic ideal and that $\mathfrak a_ i\mathcal O_{W_ i}=\mathfrak b_ i$. Every local section $h$ of $\mathfrak a_{i-1}$ vanishes on $C_ i$, because its restriction to $W_{i-1}$ belongs to $\mathfrak b_{i-1}$ and $C_ i\subset V(\mathfrak b_{i-1})$. Hence $h\in \mathcal I_{C_ i}$, and $h\circ \sigma _ i$ belongs to $\mathcal I_{C_ i}\mathcal O_{M_ i}=\mathcal O_{M_ i}(-E_ i)$; that is, it is divisible by the local equation $e_ i$ of $E_ i$, and $\mathfrak a_ i$ is holomorphic. Restricting $(h\circ \sigma _ i)/e_ i$ to $W_ i$ divides $h|_{W_{i-1}}\circ \sigma _ i|_{W_ i}$ by the local equation $e_ i|_{W_ i}$ of $E^ W_ i$, which gives $\mathfrak a_ i\mathcal O_{W_ i}=\mathfrak b_ i$; this holds also when $C_ i$ is a hypersurface in $W_{i-1}$. Iterating (9.6.5) and the definition of $\mathfrak a_ i$, we obtain

Since $\mathfrak b_ t=\mathcal O_{W_ t}$, the ideal $\mathfrak a_ t$ is the unit ideal at each point of $W_ t$: a local section whose restriction to $W_ t$ does not vanish at a point is a unit of the local ring of $M_ t$ there. Hence $V(\mathfrak a_ t)$ is disjoint from $W_ t$.

Step 3: principalizing away from $W_ t$. The manifold $M_ t$ is compact, and it is connected by the argument given for the $W_ i$ in Step 1; the ideal $\mathfrak a_ t$ is not zero, being the unit ideal near $W_ t$. Apply theorem 9.6.7 to $\mathfrak a_ t$ on $M_ t$, with empty divisor. Each center lies in the zero set of the current transformed ideal; since the blowups are isomorphisms near the successive copies of $W_ t$, which do not meet the centers, these transformed ideals remain the unit ideal near these copies, and all centers are disjoint from them. Let $g\colon M'\to M_ t$ be the composite, and let $H'$ be the effective divisor with $\mathfrak a_ t\mathcal O_{M'}=\mathcal O_{M'}(-H')$ obtained by iterating (9.6.8); put $H=g^*H_ t$ and $f=\sigma _1\circ \cdots \circ \sigma _ t\circ g$. The composition $f$ is adapted to $W$, and the strict transform $W'=g^{-1}(W_ t)$ is mapped isomorphically onto $W_ t$. By proposition 9.6.3, $\mathcal I_{W_ t}\mathcal O_{M'}=\mathcal I_{W'}$, and pulling back the two identities of Step 2 gives (9.7.6) and (9.7.7). The divisor $H'$ is supported on the exceptional divisors of $g$, which are disjoint from $W'$.

Step 4: exceptional divisors and restriction. Every component of the support of $H$ maps into the union of the images in $M$ of the centers $C_ i$; each such image lies in $W$, is analytic by theorem 9.7.2, and has dimension at most $\dim C_ i<\dim W\leq \dim M-1$ by lemma 9.3.15(3), so it has codimension at least two in $M$. Hence $H$ is exceptional for $f$, which is proper and bimeromorphic by proposition 9.6.2, corollary 9.7.3.

No component of $H$ contains $W'$: by Step 1, the restriction to $W_ i$ of a local equation of $E_ i$ is a local equation of the hypersurface $E^ W_ i$ of $W_ i$, so by induction the restriction to $W_ t$ of a local equation of $H_ t$ is not identically zero, and $g$ is an isomorphism near $W'$. Hence $H|_{W'}$ is defined, and restricting (9.7.7) to $W'$, where $H'$ vanishes, gives (9.7.8). Finally, $f|_{W'}$ is the composition of the blowups $W_ i\to W_{i-1}$ followed by the isomorphism $W'\to W_ t$. It is an isomorphism outside the union of the images in $W$ of the centers $C_ i$ that have codimension at least two in $W_{i-1}$; this union is a finite union of analytic subsets of dimension smaller than $\dim W$, hence a proper analytic subset of $W$. $\square$