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.

• There's a notion of "proper morphism" native to algebraic geometry -- a morphism is said to be "proper" if it's separated and every base extension of it is a closed map. In particular, it itself is a closed map, so of course the image of any subvariety of $X$ is a subvariety of $Y$. Of course, this doesn't really answer your question, since it's not clear why this is the correct generalization of the analytic "proper." But you can check that a map of projective varieties over $\mathbb{C}$ is analytically proper iff it is algebraically proper. Dec 16, 2012 at 1:18
• As for the intuitive reason, take $V$ to be a manifold and $f$ to have a Jacobian matrix of constant rank; then for every $x\in V$ there exists a neighborhood $W_x$ such that $f(W_x)$ is a manifold. By compactness, for every $y\in V$ there exist only finitely many $x\in f^{-1}(y)$, so we can find $W\subset\bigcap f(W_x)$ a neighborhood of $y$ which is a manifold. To deal first with the non-constant rank case and then with the singular case, you just need extension theorems (namely, Remmert-Stein), which don't have anything to do with properness of maps, and induction. Dec 16, 2012 at 1:33

Remmert’s proper mapping theorem fits into a series of subsequent generalizations:

1. The finite mapping theorem: It assumes the map $f: X \longrightarrow Y$ to be finite.

2. Remmert’s proper mapping theorem as stated in your question.

3. 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.

• Great answer: I wonder why it was not upvoted. Anyway, +1.I share your opinion on Grauert's result: it might have won him a Fields medal if it hadn't been so badly written. But Grauert's proof was perfectly correct, as I was assured by Knorr, Forster, Malgrange and others who had deciphered it. Sep 16, 2014 at 7:57
• Sorry for the necromancy. Can you suggest some reference for the proof of Grauert's theorem on the coherence of the direct image sheafs over complex manifolds? Sep 17, 2021 at 20:12
• You are welcome: 1) Original proof, but difficult to read: Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen. Publ. Inst. Hautes Études Sci. No 5, 233-292 (1960). 2) Forster, O.; Knorr, K.: Ein Beweis des Grauertschen Bildgarbensatzes nach Ideen von B. Malgrange. Manuscripta Math. 5, 19-44 (1971). 3) Kiehl, R.; Verdier, J.L.: Ein einfacher Beweis des Kohärenzsatzes von Grauert. Mathematische Annalen 1971 (195), 24-50. Note that Grauert's theorem also holds for complex spaces, i.e. with singularities. Sep 19, 2021 at 18:59