Chow's theorem

Remarks

A short proof uses the Remmert–Stein theorem. Let $p \colon \C^{n+1} \setminus \set{0} \to \mathbb{P}^n(\C)$ be the projection, and let $C = p^{-1}(A)$, the cone over $A$; it is analytic in $\C^{n+1} \setminus \set{0}$, and each irreducible component has dimension at least $1$. By the Remmert–Stein theorem, $C \cup \set{0}$ is an analytic subset of $\C^{n+1}$. Near $0$ it is the zero set of holomorphic functions $f_1, \dots, f_r$. Since $C$ is invariant under $z \mapsto \lambda z$, the homogeneous parts of the Taylor series of the $f_j$ also vanish on $C$. By Hilbert's basis theorem, finitely many of these homogeneous polynomials define $C \cup \set{0}$. Example: the image of $\mathbb{P}^1(\C)$ under $[s : t] \mapsto [s^3 : s^2 t : s t^2 : t^3]$ is a closed analytic subset of $\mathbb{P}^3(\C)$, and indeed it is the twisted cubic, defined by $z_0 z_2 = z_1^2$, $z_1 z_3 = z_2^2$ and $z_0 z_3 = z_1 z_2$. Compactness of projective space is essential: in $\C$, the analytic subset $\Z$ is not the zero set of polynomials, since a nonzero polynomial has finitely many zeros.

Going deeper

For a student

Chow's theorem is the first instance of the principle that compact analytic objects in projective space are algebraic, which Serre's GAGA theorems extend to coherent sheaves and their cohomology. Combined with the Kodaira embedding theorem, it shows that every compact Kähler manifold with an integral Kähler class is a projective algebraic variety. It also implies that holomorphic maps between projective varieties are algebraic, by applying the theorem to their graphs.