Chow’s theorem

For the purposes of this entry, let us define asany complex analytic variety of ${\mathbb{P}}^n,$ the $n$ dimensionalcomplex projective space.Let $\sigma :{ℂ}^{n+1}\setminus \{0\}\to {\mathbb{P}}^n$ be the natural projection. That is,the map that takes $(z_1,\mathrm{\dots },z_{n+1})$ to $[z_1:\mathrm{\dots }:z_{n+1}]$ in homogeneous coordinates.We define algebraic projective variety of ${\mathbb{P}}^n$ as a set $\sigma (V)$where $V\subset {ℂ}^{n+1}$ is the common zero set ofa finite family of homogeneous holomorphic polynomials. It is not hard to show that $\sigma (V)$is a in the above sense. Usually an algebraicprojective variety is just called a projective variety partly because of the following theorem.

Theorem (Chow).

Every complex analytic projective variety is algebraic.

We follow the proof by Cartan, Remmert and Stein. Note that the application of the Remmert-Stein theoremis the key point in this proof.

Proof.

Suppose that we have a complex analytic variety $X\in {\mathbb{P}}^n$. It is not hard to show thatthat ${\sigma }^{-1}(X)$ is a complex analytic subvariety of ${ℂ}^{n+1}\setminus \{0\}.$ Bythe theorem of Remmert-Stein the set $V={\sigma }^{-1}(X)\cup \{0\}$ is a subvariety of ${ℂ}^{n+1}.$Furthermore $V$ is a complex cone, that is if $z=(z_1,\mathrm{\dots },z_{n+1})\in V,$ then $tz\in V$ for all$t\in ℂ.$

Final step is to show that if a complex analytic subvariety $V\subset {ℂ}^{n+1}$ is a complex cone,then it is given by the vanishing of finitely many homogeneous polynomials.Take a finite set of defining functions of $V$ near the origin. I.e. take $f_1,\mathrm{\dots },f_k$defined in some open ball $B=B(0,ϵ),$ such that in$B\cap V=\{z\in B\mid f_1(z)=\mathrm{\cdots }=f_k(z)=0\}.$ We can suppose that $ϵ$is small enough that the power series for $f_j$ converges in $B$ for all $j.$Expand $f_j$ in a power series near the origin and group togetherhomogeneous terms as $f_j={\sum }_{m=0}^{\mathrm{\infty }}{f}_{jm}$, where $f_{jm}$ is a homogeneous polynomial ofdegree $m.$ For $t\in ℂ$ we write

$$f_j(tz)=\sum _{m=0}^{\mathrm{\infty }}{f}_{jm}(tz)=\sum _{m=0}^{\mathrm{\infty }}{t}^m{f}_{jm}(z)$$

For a fixed $z\in V$ we know that $f_j(tz)=0$ for all hence we have a power seriesin one variable that is identically zero, and so all coefficients are zero.Thus $f_{jm}$ vanishes on $V\cap B$ and hence on $V.$ It follows that$V$ is defined by a family of homogeneous polynomials. Since the ring of polynomials is Noetherian we need onlyfinitely many, and we are done.∎