Chow’s theorem

For the purposes of this entry, let us define as any complex analytic variety of ${\mathbb{P}}^n,$ the $n$ dimensional complex 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 of a finite family of homogeneous holomorphic polynomials. It is not hard to show that $\sigma (V)$ is a in the above sense. Usually an algebraic projective 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 theorem is 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 that that ${\sigma }^{-1}(X)$ is a complex analytic subvariety of ${ℂ}^{n+1}\setminus \{0\}.$ By the 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 together homogeneous terms as $f_j={\sum }_{m=0}^{\mathrm{\infty }}{f}_{jm}$, where $f_{jm}$ is a homogeneous polynomial of degree $m.$ 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 series in 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 only finitely many, and we are done. ∎

Title	Chow’s theorem
Canonical name	ChowsTheorem
Date of creation	2013-03-22 17:46:32
Last modified on	2013-03-22 17:46:32
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	9
Author	jirka (4157)
Entry type	Theorem
Classification	msc 14A10
Classification	msc 51N15
Classification	msc 32C25
Synonym	every complex analytic projective variety is algebraic
Related topic	RemmertSteinExtensionTheorem
Related topic	ProjectiveVariety
Related topic	RemmertSteinTheorem
Related topic	MeromorphicFunctionOnProjectiveSpaceMustBeRational
Defines	complex analytic projective variety