Chow’s theorem

For the purposes of this entry, let us define as any complex analytic variety of n, the n dimensional complex projective space. Let σ:n+1{0}n be the natural projection. That is, the map that takes (z1,,zn+1) to [z1::zn+1] in homogeneous coordinatesMathworldPlanetmath. We define algebraicPlanetmathPlanetmath projective variety of n as a set σ(V) where Vn+1 is the common zero setPlanetmathPlanetmath of a finite family of homogeneous holomorphic polynomialsMathworldPlanetmathPlanetmath. It is not hard to show that σ(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.


Suppose that we have a complex analytic variety Xn. It is not hard to show that that σ-1(X) is a complex analytic subvariety of n+1{0}. By the theorem of Remmert-Stein the set V=σ-1(X){0} is a subvarietyMathworldPlanetmath of n+1. Furthermore V is a complex cone, that is if z=(z1,,zn+1)V, then tzV for all t.

Final step is to show that if a complex analytic subvariety Vn+1 is a complex cone, then it is given by the vanishing of finitely many homogeneous polynomialsMathworldPlanetmath. Take a finite set of defining functionsMathworldPlanetmath of V near the origin. I.e. take f1,,fk defined in some open ball B=B(0,ϵ), such that in BV={zBf1(z)==fk(z)=0}. We can suppose that ϵ is small enough that the power series for fj converges in B for all j. Expand fj in a power series near the origin and group together homogeneous terms as fj=m=0fjm, where fjm is a homogeneous polynomial of degree m. For t we write


For a fixed zV we know that fj(tz)=0 for all |t|<1, hence we have a power series in one variable that is identically zero, and so all coefficients are zero. Thus fjm vanishes on VB 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. ∎


  • 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