# 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 $\sigma :{\u2102}^{n+1}\setminus \{0\}\to {\mathbb{P}}^{n}$ 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 $V\subset {\u2102}^{n+1}$ 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 ${\u2102}^{n+1}\setminus \{0\}.$ By
the theorem of Remmert-Stein the set $V={\sigma}^{-1}(X)\cup \{0\}$ is a subvariety^{} of ${\u2102}^{n+1}.$
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
$t\in \u2102.$

Final step is to show that if a complex analytic subvariety $V\subset {\u2102}^{n+1}$ 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 $B=B(0,\u03f5),$ such that in
$B\cap V=\{z\in B\mid {f}_{1}(z)=\mathrm{\cdots}={f}_{k}(z)=0\}.$ We can suppose that $\u03f5$
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 $t\in \u2102$ 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. ∎

## 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 |