# 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 $\sigma\colon{\mathbb{C}}^{n+1}\setminus\{0\}\to{\mathbb{P}}^{n}$ be the natural projection. That is, the map that takes $(z_{1},\ldots,z_{n+1})$ to $[z_{1}:\ldots:z_{n+1}]$ in homogeneous coordinates. We define algebraic projective variety of ${\mathbb{P}}^{n}$ as a set $\sigma(V)$ where $V\subset{\mathbb{C}}^{n+1}$ 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).

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 ${\mathbb{C}}^{n+1}\setminus\{0\}.$ By the theorem of Remmert-Stein the set $V=\sigma^{-1}(X)\cup\{0\}$ is a subvariety of ${\mathbb{C}}^{n+1}.$ Furthermore $V$ is a complex cone, that is if $z=(z_{1},\ldots,z_{n+1})\in V,$ then $tz\in V$ for all $t\in{\mathbb{C}}.$

Final step is to show that if a complex analytic subvariety $V\subset{\mathbb{C}}^{n+1}$ 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 $f_{1},\ldots,f_{k}$ defined in some open ball $B=B(0,\epsilon),$ such that in $B\cap V=\{z\in B\mid f_{1}(z)=\cdots=f_{k}(z)=0\}.$ We can suppose that $\epsilon$ 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 $f_{j}=\sum_{m=0}^{\infty}f_{jm}$, where $f_{jm}$ is a homogeneous polynomial of degree $m.$ For $t\in{\mathbb{C}}$ we write

 $f_{j}(tz)=\sum_{m=0}^{\infty}f_{jm}(tz)=\sum_{m=0}^{\infty}t^{m}f_{jm}(z)$

For a fixed $z\in V$ we know that $f_{j}(tz)=0$ for all $\lvert t\rvert<1,$ 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. ∎

## References

• 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
 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