Chow’s theorem

Chow’s theorem

For the purposes of this entry, let us define as
any

complex analytic variety

of

ℙ

n

,

the

n

dimensional
complex projective space.
Let

σ

:

ℂ

n

+

1

∖

{

0

}

→

ℙ

n

be the

natural projection

. That is,
the map that takes

(

z

1

,

…

,

z

n

+

1

)

to

[

z

1

:

…

:

z

n

+

1

]

in

homogeneous coordinates

.
We define

algebraic

projective variety

of

ℙ

n

as a set

σ

⁢

(

V

)

where

V

⊂

ℂ

n

+

1

is the common

zero set

of
a finite family of

homogeneous

holomorphic

polynomials

. 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).

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

∈

ℙ

n

. It is not hard to show that
that

σ

-

1

⁢

(

X

)

is a complex analytic subvariety of

ℂ

n

+

1

∖

{

0

}

.

By
the theorem of Remmert-Stein the set

V

=

σ

-

1

⁢

(

X

)

∪

{

0

}

is a

subvariety

of

ℂ

n

+

1

.

Furthermore

V

is a complex cone, that is if

z

=

(

z

1

,

…

,

z

n

+

1

)

∈

V

,

then

t

⁢

z

∈

V

for all

t

∈

ℂ

.

Final step is to show that if a complex analytic subvariety

V

⊂

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

,

…

,

f

k

defined in some open ball

B

=

B

⁢

(

0

,

ϵ

)

,

such that in

B

∩

V

=

{

z

∈

B

∣

f

1

⁢

(

z

)

=

⋯

=

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

=

∑

m

=

0

∞

f

j

⁢

m

, where

f

j

⁢

m

is a homogeneous polynomial of
degree

m

.

For

t

∈

ℂ

we write

f

j

⁢

(

t

⁢

z

)

=

∑

m

=

0

∞

f

j

⁢

m

⁢

(

t

⁢

z

)

=

∑

m

=

0

∞

t

m

⁢

f

j

⁢

m

⁢

(

z

)

For a fixed

z

∈

V

we know that

f

j

⁢

(

t

⁢

z

)

=

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

f

j

⁢

m

vanishes on

V

∩

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

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