For the purposes of this entry, let us define as any complex analytic variety of the dimensional complex projective space. Let be the natural projection. That is, the map that takes to in homogeneous coordinates. We define algebraic projective variety of as a set where is the common zero set of a finite family of homogeneous holomorphic polynomials. It is not hard to show that is a in the above sense. Usually an algebraic projective variety is just called a projective variety partly because of the following theorem.
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.
Suppose that we have a complex analytic variety . It is not hard to show that that is a complex analytic subvariety of By the theorem of Remmert-Stein the set is a subvariety of Furthermore is a complex cone, that is if then for all
Final step is to show that if a complex analytic subvariety is a complex cone, then it is given by the vanishing of finitely many homogeneous polynomials. Take a finite set of defining functions of near the origin. I.e. take defined in some open ball such that in We can suppose that is small enough that the power series for converges in for all Expand in a power series near the origin and group together homogeneous terms as , where is a homogeneous polynomial of degree For we write
For a fixed we know that for all hence we have a power series in one variable that is identically zero, and so all coefficients are zero. Thus vanishes on and hence on It follows that 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.
|Date of creation||2013-03-22 17:46:32|
|Last modified on||2013-03-22 17:46:32|
|Last modified by||jirka (4157)|
|Synonym||every complex analytic projective variety is algebraic|
|Defines||complex analytic projective variety|