analytic set

analytic set

Let

G

⊂

ℂ

N

be an open set.

Definition

.

A set

V

⊂

G

is said to be

locally analytic

if for every point

p

∈

V

there exists a neighbourhood

U

of

p

in

G

and

holomorphic functions

f

1

,

⋯

,

f

m

defined in

U

such that

U

∩

V

=

{

z

:

f

k

⁢

(

z

)

=

0

⁢

for all

⁢

1

≤

k

≤

m

}

.

This basically says that around each point of

V

,

the set

V

is

analytic

.
A stronger

definition

is required.

Definition

.

A set

V

⊂

G

is said to be an

analytic variety

in

G

(or

analytic set

in

G

)
if for every point

p

∈

G

there exists a neighbourhood

U

of

p

in

G

and holomorphic functions

f

1

,

⋯

,

f

m

defined in

U

such that

U

∩

V

=

{

z

:

f

k

⁢

(

z

)

=

0

⁢

for all

⁢

1

≤

k

≤

m

}

.

Note the change, now

V

is analytic around each point of

G

.

Since the

zero sets

of holomorphic functions are closed, this for example implies that

V

is relatively closed in

G

,

while a local

variety

need not be closed.
Sometimes an analytic variety is called an

analytic set

.

At most points an analytic variety

V

will in fact be a

complex
analytic manifold

. So

Definition

.

A point

p

∈

V

is called a

regular point

if there is a neighbourhood

U

of

p

such that

U

∩

V

is a complex analytic manifold. Any other
point is called a

singular point

.

The set of regular points of

V

is denoted by

V

-

or sometimes

V

*

.

For any regular point

p

∈

V

we can define the

dimension

as

dim

p

⁡

(

V

)

=

dim

ℂ

⁡

(

U

∩

V

)

where

U

is as above and thus

U

∩

V

is a manifold with a well defined
dimension. Here we of course take the complex dimension of these manifolds.

Definition

.

Let

V

be an analytic variety,
we define the dimension of

V

by

dim

⁡

(

V

)

=

sup

⁡

{

dim

p

⁡

(

V

)

:

p

⁢

a regular point of

⁢

V

}

.

Definition

.

The regular point

p

∈

V

such that

dim

p

⁡

(

V

)

=

dim

⁡

(

V

)

is called a

top
point

of

V

.

Similarly as for manifolds we can also talk about

subvarieties

. In this case we modify definition a little bit.

Definition

.

A set

W

⊂

V

where

V

⊂

G

is a local variety is said to be
a

subvariety

of

V

if for every point

p

∈

V

there exists a neighbourhood

U

of

p

in

G

and holomorphic functions

f

1

,

⋯

,

f

m

defined in

U

such that

U

∩

W

=

{

z

:

f

k

⁢

(

z

)

=

0

⁢

for all

⁢

1

≤

k

≤

m

}

.

That is, a subset

W

is a subvariety if it is definined by the vanishing of

analytic functions

near all points of

V

.

References

1

E. M. Chirka.
.
Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.

2

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

Title

analytic set

Canonical

name

AnalyticSet

Date of creation

2013-03-22 14:59:28

Last modified on

2013-03-22 14:59:28

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

10

Author

jirka (4157)

Entry type

Definition

Classification

msc 32C25

Classification

msc 32A60

Synonym

analytic variety

Synonym

complex analytic variety

Related topic

IrreducibleComponent2

Defines

regular point

Defines

simple point

Defines

top simple point

Defines

singular point

Defines

locally analytic

Defines

dimension of a variety

Defines

subvariety of a complex analytic variety

Defines

complex analytic subvariety

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