Proper mapping theorem
My professor mentioned a proper mapping theorem after the name of Remmert which says:
Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex analytic subvariety of $ X$, then $f(V)$ is a subvariety of $Y$.
I know this is a deep result, and the proof is not easy, but is there any simple reason which can convince me $f(V)$ is a subvariety, at least intuitively?
Besides, what is the analog result in algebraic geometry? In Hartshorne, he can only say the image is constructible set.
EDIT: One of intuitive explanation which mix the language of complex and algebraic geometry may be following: suppose X,Y are projective analytic varieties over $\mathbb{C}$, than they are algebraic varieties. Because proper morphism are closed, and also because the image is a constructible set, it has to be a variety.
The problems of above explanation are: (1) The question is obviously local on Y, but I have to assume X,Y are projective analytic varieties in order to translate back to algebraic varities(GAGA).(2)I still use the result that the image is constructible which is not obvious intuitively，and GAGA to connect analytic variety to algebraic variety. All in all, it is not a good idea to use algebraic geometry to explain complex geometry.
Solution 1:
Remmert’s proper mapping theorem fits into a series of subsequent generalizations:
The finite mapping theorem: It assumes the map $f: X \longrightarrow Y$ to be finite.
Remmert’s proper mapping theorem as stated in your question.
Grauert’s theorem on the coherence of the direct image sheaves: If $X$ and $Y$ are complex spaces, $f: X \longrightarrow Y$ a proper holomorphic map and $\mathscr F$ a coherent sheaf on $X$, then all direct image sheaves $R^if_* (\mathscr F), i \in \mathbb N,$ are coherent.
Grauert's theorem implies Remmert's theorem, because any analytic set is the support of its structure sheaf, which is coherent. In my opinion, Grauert's theorem and its different proofs belong to the deepest results of complex analysis.
The finite mapping theorem has both a topological aspect and an algebraic aspect because it considers a proper mapping with zero-dimensional fibres. The proof goes by induction on the dimension of $X$. Thanks to the properness of $f$ the induction step reduces to a local situation at points $x = 0 \in X$ and $f(x) = 0\in Y$: Consider
$$pr: \mathbb {C}^n \longrightarrow \mathbb {C}^{n-1},$$
the canonical projection onto the first $n-1$ coordinates and consider a proper ideal
$$\mathscr a \subsetneq \mathbb C {\{z_1,...,z_n}\},$$
regular in $z_n,$ which defines the germ of an analytic set $V(\mathscr a)$ at $x$. Intuitively, the map $$pr|V(\mathscr a) \longrightarrow pr(V(\mathscr a))$$
is a branched covering: By methods from local analytic algebra one proves
$$pr(V(\mathscr a)) = V(\mathscr a \cap \mathbb C \{z_1,.,,,z_{n-1}\}).$$
The analogue and guide of Grauert's theorem is the proper mapping theorem of Grothendieck (see Hartshorne III, Remark 8.8.1). The proof of Grothendieck's theorem is much easier than the proof of Grauert's theorem.