I am studying the proof of Remmert's Theorem on the book Griffiths & Harris - Principles of Algebraic Geometry, Chapter 3 Section 2 page 395.
Theorem (Remmert's Proper Mapping Theorem) Let $U$ and $N$ be complex manifolds, $M \subset U$ an analytic subvariety and $f \colon U \longrightarrow N$ a holomorphic mapping whose restriction to $M$ is proper. Then the image $f(M)$ is an analytic subvariety.
The proof is by induction on $n = dim \, M$. After some preliminary reduction the essential point is to prove the theorem when
- $M$ is a irreducible variety of dimension $n$.
- $N = \triangle^{n+1} = \{ (z_1, \ldots, z_n) \in \mathbb{C}^{n+1} \, | \, |z_i| < 1 \, \, \, \forall i \, \, \, \} \subset \mathbb{C}^{n+1} $.
- $\exists \, p_0 \in M^{*} $ a smooth point such that the Jacobian matrix of $f \colon M \longrightarrow \mathbb{C}^{n+1}$ has maximum rank $n$.
So we want prove that $f(M)$ is an $n$-dimensional analytic subvariety of the polycylinder.
The idea is this. Let $W \subset M$ be the union of the singular set of M and the subvariety where the Jacobian of $f$ has rang $< n$. By induction assumption $f(W)$ is an analytic subvariety of codimension $\ge 2$ in $\triangle^{n+1}$. The image of a sufficiently small neighborhood of a point $p \in M - W$ is a piece of smooth analytic hypersurface in $\triangle^{n+1}$.
The book now says
$\overline{f(M - W)} = f(M - W) \cup f(W)$
and the problem is therefore to show that the two pieces $f(M- W)$ and $f(W)$ fit togheter nicely. So the proof go on with techniques on currents...
I do not understand last sentence. Actually I think the proof is concluded since we have proved that $f(M) = f(M - W) \cup f(W)$ and so $f(M)$ is the union of two analytic subvavierty and so is an analytic subvariety itself! What am I doing wrong? What does it mean with fit togheter nicely?
Thanks for attention