Remmert's proper mapping theorem

Remarks

Properness is necessary. The projection $f \colon \C^2 \to \C$, $(z_1, z_2) \mapsto z_1$, maps the analytic set $A = \set{z_1 z_2 = 1}$ onto $\C \setminus \set{0}$, which is not closed and so not analytic; here $f$ restricted to $A$ is not proper.

Example: the projection $\mathbb{P}^m \times Y \to Y$ is proper because $\mathbb{P}^m$ is compact, so the image of any analytic subset of $\mathbb{P}^m \times Y$ in $Y$ is analytic. This is the analytic form of the main theorem of elimination theory.

One proof derives it from the coherence of direct images. Give $A$ its reduced structure. Then $f_* \mathcal{O}_A$ is a coherent sheaf on $Y$ by Grauert's direct image theorem, and its support is $f(A)$. The support of a coherent sheaf is analytic.

Going deeper

For a student

The theorem is a basic tool for building analytic sets. For example, the graph of a holomorphic map is analytic, and its image under a proper projection is analytic. Remmert proved it before the direct image theorem was known, and it was an early milestone of the global theory of complex spaces.