aarjaneiro/planetmath huggingface dataset:
Bergman kernel

Bergman kernel

Let

G

⊂

ℂ

n

be a domain (

http://planetmath.org/Domain2

). And let

A

2

⁢

(

G

)

be the

Bergman space

. For a fixed

z

∈

G

, the

functional

f

↦

f

⁢

(

z

)

is a

bounded

linear functional

. By the

Riesz representation theorem

(as

A

2

⁢

(

G

)

is a

Hilbert space

) there exists an element of

A

2

⁢

(

G

)

that represents it, and
let’s call that element

k

z

∈

A

2

⁢

(

G

)

. That is we have that

f

⁢

(

z

)

=

⟨

f

,

k

z

⟩

. So we can define the

Bergman kernel

.

Definition

.

The

function

K

⁢

(

z

,

w

)

:=

k

z

⁢

(

w

)

¯

is called the

Bergman kernel

.

By definition of the

inner product

in

A

2

⁢

(

G

)

we then have that
for

f

∈

A

2

⁢

(

G

)

f

⁢

(

z

)

=

∫

G

f

⁢

(

w

)

⁢

K

⁢

(

z

,

w

)

⁢

𝑑

V

⁢

(

w

)

,

where

d

⁢

V

is the volume measure.

As the

A

2

⁢

(

G

)

space is a

subspace

of

L

2

⁢

(

G

,

d

⁢

V

)

which is a separable Hilbert space then

A

2

⁢

(

G

)

also has a countable

orthonormal basis

, say

{

φ

j

}

j

=

1

∞

.

Theorem

.

We can compute the Bergman kernel as

K

⁢

(

z

,

w

)

=

∑

j

=

1

∞

φ

j

⁢

(

z

)

⁢

φ

j

⁢

(

w

)

¯

,

where the sum converges uniformly on

compact subsets

of

G

×

G

.

Note that integration against the Bergman kernel is just the

orthogonal
projection

from

L

2

⁢

(

G

,

d

⁢

V

)

to

A

2

⁢

(

G

)

. So not only is this kernel reproducing for

holomorphic functions

, but it will produce a holomorphic function when we just feed in any

L

2

⁢

(

G

,

d

⁢

V

)

function.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Bergman kernel

Canonical name

BergmanKernel

Date of creation

2013-03-22 15:04:45

Last modified on

2013-03-22 15:04:45

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A25

Related topic

BergmanSpace

Related topic

BergmanMetric

Generated on Fri Feb 9 20:32:12 2018 by

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Bergman space

Bergman space

Let

G

⊂

ℂ

n

be a domain and let

d

⁢

V

denote the Euclidean volume measure on

G

.

Definition

.

Let

A

2

⁢

(

G

)

:=

{

f

⁢

holomorpic in

⁢

G

|

∫

G

|

f

⁢

(

z

)

|

2

⁢

𝑑

V

⁢

(

z

)

<

∞

}

.

A

2

⁢

(

G

)

is called the

Bergman space

on

G

. The norm on this space
is defined as

∥

f

∥

:=

∫

G

|

f

⁢

(

z

)

|

2

⁢

𝑑

V

⁢

(

z

)

.

Further we define an

inner product

on

A

2

⁢

(

G

)

as

⟨

f

,

g

⟩

:=

∫

G

f

⁢

(

z

)

⁢

g

⁢

(

z

)

¯

⁢

𝑑

V

⁢

(

z

)

.

The inner product as defined above really is an inner product and further
it can be shown that

A

2

⁢

(

G

)

is

complete

since

convergence

in the above norm implies normal convergence (

uniform convergence

on

compact subsets

). The space

A

2

⁢

(

G

)

is therefore a

Hilbert space

.
Sometimes this space is also denoted by

L

a

2

⁢

(

G

)

.

References

1

D’Angelo, John P.
,
CRC Press, 1993.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Bergman space

Canonical name

BergmanSpace

Date of creation

2013-03-22 15:04:43

Last modified on

2013-03-22 15:04:43

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

10

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A36

Related topic

BergmanKernel

Generated on Fri Feb 9 20:32:14 2018 by

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intersection of complex analytic varieties is a complex analytic variety

intersection of complex analytic varieties is a complex analytic variety

A useful result allowing us to define the “smallest” analytic variety is
the following.

Theorem

.

Let

G

⊂

C

N

be an open set, then an arbitrary

intersection

of complex analytic varieties in

G

is a complex analytic variety in

G

.

References

1

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

Title

intersection of complex analytic varieties is a complex analytic variety

Canonical name

IntersectionOfComplexAnalyticVarietiesIsAComplexAnalyticVariety

Date of creation

2013-03-22 14:59:31

Last modified on

2013-03-22 14:59:31

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32A60

Generated on Fri Feb 9 20:32:16 2018 by

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proper mapping theorem

proper mapping theorem

Theorem

.

Let

X

and

Y

be

complex manifolds

,

f

:

X

→

Y

be a proper (

http://planetmath.org/ProperMap

) holomorphic map,
and

V

⊂

X

be a

complex analytic subvariety

of

X

.

Then

f

⁢

(

V

)

is a subvariety of

Y

.

References

1

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

Title

proper mapping theorem

Canonical name

ProperMappingTheorem

Date of creation

2013-03-22 17:41:01

Last modified on

2013-03-22 17:41:01

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32A60

Classification

msc 32A25

Generated on Fri Feb 9 20:32:18 2018 by

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Remmert-Stein theorem

Remmert-Stein theorem

For a

complex analytic subvariety

V

and

p

∈

V

a

regular point

, let

dim

p

⁡

V

denote the complex dimension of

V

near the point

p

.

Theorem

(Remmert-Stein).

Let

U

⊂

C

n

be a domain (

http://planetmath.org/Domain2

) and let

S

be a complex analytic subvariety of

U

of

dimension

m

<

n

.

Let

V

be a complex analytic subvariety of

U

\

S

such that

dim

p

⁡

V

>

m

for all
regular points

p

∈

V

.

Then the

closure

of

V

in

U

is an analytic variety in

U

.

The condition that

dim

p

⁡

V

>

m

for all

regular

p

is the same as saying that all the

irreducible
components

of

V

are of dimension

strictly

greater than

m

.

To show that the

restriction

on the dimension
of

S

is “sharp,”
consider the following example where the dimension of

V

equals the dimension of

S

.
Let

(

z

,

w

)

∈

ℂ

2

be our coordinates and let

V

be defined by

w

=

e

1

/

z

in

ℂ

2

∖

S

,

where

S

is defined by

z

=

0

.

The closure of

V

in

ℂ

2

cannot possibly be

analytic

. To see this look for example at

W

=

V

¯

∩

{

w

=

1

}

.

If

V

¯

is analytic then

W

ought to be a

zero dimensional

complex analytic set and thus a set of

isolated points

, but it has a

limit point

(

0

,

1

)

by Picard’s

theorem

.

Finally note that there are various

generalizations

of this theorem where the set

S

need not be a

variety

,
as long as it is of small enough dimension. Alternatively, if

V

is of finite volume, we can weaken the
restrictions on

S

even further.

References

1

Klaus Fritzsche, Hans Grauert.
,
Springer-Verlag, New York, New York, 2002.

2

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

Title

Remmert-Stein theorem

Canonical

name

RemmertSteinTheorem

Date of creation

2013-03-22 15:04:55

Last modified on

2013-03-22 15:04:55

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32A60

Classification

msc 32C25

Synonym

Remmert-Stein extension theorem

Related topic

ChowsTheorem

Generated on Fri Feb 9 20:32:20 2018 by

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Weierstrass preparation theorem

Weierstrass preparation theorem

The following

theorem

is known as the

Weierstrass preparation theorem

, though sometimes that name is reserved for the corollary and this theorem is then known as the Weierstrass division theorem.

In the following we use the standard notation for

coordinates

in

ℂ

n

that

z

=

(

z

1

,

…

,

z

n

)

=

(

z

′

,

z

n

)

. That is

z

′

is the first

n

-

1

coordinates.

Theorem

.

Let

f

:

C

n

→

C

be a

function

analytic

in a neighbourhood

U

of the origin such that

f

⁢

(

0

,

z

n

)

z

n

m

extends to be analytic at the origin and is not zero
at the origin for some positive integer

m

(in other words, as a function of

z

n

, the function has a zero of order

m

at the origin). Then there exists
a

polydisc

D

⊂

U

such that every function

g

holomorphic

and

bounded

in

D

can be written as

g

=

q

⁢

f

+

r

,

where

q

is an

analytic function

and

r

is a polynomial in the

z

n

variable
of degree less then

m

with the coefficients being

holomorphic functions

in

z

′

. Further there exists a

constant

C

independent

of

g

such that

sup

z

∈

D

⁡

|

q

⁢

(

z

)

|

≤

C

⁢

sup

z

∈

D

⁡

|

g

⁢

(

z

)

|

.

The representation

g

=

q

⁢

f

+

r

is unique.
Finally the coefficients of the

power series

expansions of

q

and

r

are
finite

linear combinations

of the coefficients of the power series of

g

.

Note that

r

is not necessarily a

Weierstrass polynomial

.

Corollary

.

Let

f

be as above, then there is a unique representation of

f

as

f

=

h

⁢

W

, where

h

is analytic in a neighbourhood of the origin and

h

⁢

(

0

)

≠

0

and

W

being a Weierstrass polynomial.

It should be noted that the condition that

f

⁢

(

0

,

z

n

)

z

n

m

extends to be analytic, which is

equivalent

to
saying that

f

⁢

(

0

,

z

n

)

≢

0

, is not an essential

restriction

.
In fact

f

⁢

(

0

,

z

n

)

≡

0

, then there exists a linear

change of coordinates

, arbitrarily close to the

identity

,
such that the condition of the theorem is satisfied in the new set of coordinates.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Weierstrass preparation theorem

Canonical name

WeierstrassPreparationTheorem

Date of creation

2013-03-22 15:04:28

Last modified on

2013-03-22 15:04:28

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32B05

Synonym

Weierstrass division theorem

Related topic

WeierstrassPolynomial

Generated on Fri Feb 9 20:32:22 2018 by

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Puiseux parametrization

Puiseux parametrization

Theorem

.

Suppose that

V

⊂

U

⊂

C

2

is an

irreducible

complex analytic

subset of (

complex

) dimension 1 where

U

is a domain. Suppose that

0

∈

V

. Then there exists an analytic (

holomorphic

) map

f

:

D

→

V

, where

D

is the

unit disc

,
such that

f

⁢

(

0

)

=

0

and

f

⁢

(

D

)

=

N

where

N

⊂

V

is a neighbourhood of

0

in

V

,

f

is one to one, and further

f

|

D

\

{

0

}

is a biholomorphism onto

N

\

{

0

}

.
In fact there exist suitable

local coordinates

(

z

,

w

)

in

C

2

such that

f

is then given by

ξ

↦

(

z

,

w

)

where

z

=

ξ

k

,

w

=

∑

n

=

m

∞

a

n

⁢

ξ

n

where

m

>

k

.

This is sometimes written as

w

=

∑

n

=

m

∞

a

n

⁢

z

n

/

k

and hence the name

Puiseux series parametrization

. If you do however write it like this, it must be properly interpreted, as the

Puiseux series

is in general not single valued.

A similar result for arbitrary complex analytic sets with

singularities

of codimension 1 in higher dimensional spaces under further conditions on the

singular set

was obtained by Stutz, see Chirka

[

1

]

page 98.

References

1

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

2

Alexandru Dimca.
.
Vieweg, Braunschweig, Germany, 1987.

Title

Puiseux parametrization

Canonical name

PuiseuxParametrization

Date of creation

2013-03-22 15:20:32

Last modified on

2013-03-22 15:20:32

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32B10

Synonym

Puiseux series parametrization

Synonym

Puiseux normalization

Synonym

Puiseux series normalization

Synonym

Puiseux parameterization

Synonym

Puiseux series parameterization

Related topic

PuiseuxSeries

Generated on Fri Feb 9 20:32:24 2018 by

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Puiseux series

Puiseux series

A

formal series

of the form

∑

n

=

m

∞

a

n

⁢

z

n

/

k

where

m

and

k

are integers such that

k

≥

1

is
is called a

Puiseux series

or a

fractional power series

. Note that if

k

>

1

, then

z

n

/

k

could be multivalued. One example of the use of such a

power series

is the

Puiseux parametrization

of one-dimensional

complex analytic varieties

.

References

1

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

2

Alexandru Dimca.
.
Vieweg, Braunschweig, Germany, 1987.

Title

Puiseux series

Canonical name

PuiseuxSeries

Date of creation

2013-03-22 15:20:28

Last modified on

2013-03-22 15:20:28

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32B10

Synonym

fractional power series

Related topic

PuiseuxParametrization

Related topic

GeneralPower

Generated on Fri Feb 9 20:32:25 2018 by

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subanalytic set

subanalytic set

Let

U

⊂

ℝ

n

.
Suppose

𝒜

⁢

(

U

)

is any ring of real valued

functions

on

U

.
Define

𝒮

⁢

(

𝒜

⁢

(

U

)

)

to be the smallest
set of subsets of

U

, which contain the sets

{

x

∈

U

⁢

∣

f

⁢

(

x

)

>

⁢

0

}

for all

f

∈

𝒜

⁢

(

U

)

,
and is

closed under

finite union, finite

intersection

and

complement

.

Definition

.

A set

V

⊂

ℝ

n

is

semianalytic

if and only if for each

x

∈

ℝ

n

, there exists a neighbourhood

U

of

x

, such that

V

∩

U

∈

𝒮

⁢

(

𝒪

⁢

(

U

)

)

, where

𝒪

⁢

(

U

)

denotes the real-analytic real valued functions.

Unlike for

semialgebraic sets

, there is no

Tarski-Seidenberg theorem

for semianalytic sets, and

projections

of semianalytic sets are in general not semianalytic.

Definition

.

We say

V

⊂

ℝ

n

is a

subanalytic

set if for
each

x

∈

ℝ

n

, there exists a relatively compact semianalytic set

X

⊂

ℝ

n

+

m

and a neighbourhood

U

of

x

, such that

V

∩

U

is the projection of

X

onto the first

n

coordinates

.

In particular all semianalytic sets are subanalytic.
On an open dense set
subanalytic sets are submanifolds and hence we
can define

dimension

. Hence at a point

p

, where a set

A

is a submanifold,
the dimension

dim

p

⁡

A

is the dimension of the submanifold. The

dimension

of the subanalytic set is the maximum

dim

p

⁡

A

for all

p

where

A

is a submanifold.
Semianalytic sets are

contained

in a real-analytic

subvariety

of the same dimension. However, subanalytic sets are not in general
contained in any subvariety of the same dimension. We do have however the
following.

Theorem

.

A subanalytic set

A

can be written as a

locally finite

union of
submanifolds.

The set of subanalytic sets is still not completely closed under projections however. Note that
a real-analytic subvariety that is not relatively compact can have a
projection which is not a locally finite union of submanifolds, and hence
is not subanalytic.

Definition

.

Let

U

⊂

ℝ

n

. A

mapping

f

:

U

→

ℝ

m

is said to be subanalytic (resp. semianalytic)
if the graph of

f

(i.e. the set

{

(

x

,

y

)

∈

U

×

ℝ

m

∣

x

,

y

=

f

⁢

(

x

)

}

) is subanalytic (resp. semianalytic)

References

1

Edward Bierstone and Pierre D. Milman,

Semianalytic and subanalytic
sets

, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42.

http://www.ams.org/mathscinet-getitem?mr=89k:32011

MR 89k:32011

Title

subanalytic set

Canonical name

SubanalyticSet

Date of creation

2013-03-22 16:46:16

Last modified on

2013-03-22 16:46:16

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 32B20

Classification

msc 14P15

Related topic

TarskiSeidenbergTheorem

Related topic

SemialgebraicSet

Defines

subanalytic

Defines

semianalytic set

Defines

semianalytic

Defines

semianalytic function

Defines

subanalytic function

Defines

semianalytic mapping

Defines

subanalytic mapping

Defines

dimension of a subanalytic set

Generated on Fri Feb 9 20:32:27 2018 by

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analytic space

analytic space

A

Hausdorff topological space

X

is said to be an

analytic space

if:

1.

There exists a

countable

number of

open sets

V

j

covering

X

.

2.

For each

V

j

there exists a

homeomorphism

φ

j

:

Y

j

→

V

j

,

where

Y

j

is a local

complex analytic subvariety

in some

ℂ

n

.

3.

If

V

j

and

V

k

overlap, then

φ

j

-

1

∘

φ

k

is a biholomorphism.

Usually one attaches to

X

a set of

coordinate systems

𝒢

, which is a set (now

uncountable

)
of triples

(

V

ι

,

φ

ι

,

Y

ι

)

as above, such that whenever

V

is an open set,

Y

a local complex analytic subvariety, and a homeomorphism

φ

:

Y

→

V

, such that

φ

ι

-

1

∘

φ

is a biholomorphism for some

(

V

ι

,

φ

ι

,

Y

ι

)

∈

𝒢

then

(

V

,

φ

,

Y

)

∈

𝒢

.

Basically

𝒢

is the set of all possible coordinate systems
for

X

.

We can also define the

singular set

of an analytic space. A point

p

is

if there exists (at least one) a coordinate system

(

V

ι

,

φ

ι

,

Y

ι

)

∈

𝒢

with

p

∈

V

ι

and

Y

ι

a

complex manifold

. All other points are the

singular points

.

Any local complex analytic subvariety is an analytic space, so this is a natural

generalization

of the

concept

of a

subvariety

.

References

1

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

Title

analytic space

Canonical name

AnalyticSpace

Date of creation

2013-03-22 17:41:43

Last modified on

2013-03-22 17:41:43

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 32C15

Synonym

complex analytic space

Related topic

LocallyCompactGroupoids

Generated on Fri Feb 9 20:32:29 2018 by

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weakly holomorphic

weakly holomorphic

Let

V

be a local

complex analytic variety

.
A

function

f

:

U

⊂

V

→

ℂ

(where

U

is open in

V

)
is said to be

weakly holomorphic

through

U

if there exists a nowhere dense complex analytic subvariety

W

⊂

V

and

W

contains the

singular points

of

V

and

V

∖

W

⊂

U

,
and such that

f

is

holomorphic

on

V

∖

W

and

f

is

locally bounded

on

V

.

It is not hard to show that we can then just take

W

to be the set of singular
points of

V

and have

U

=

V

∖

W

as we can extend

f

to all the
nonsingular points of

V

.

Usually we denote by

𝒪

w

⁢

(

V

)

the ring of weakly holomorphic functions
through

V

. Since any neighbourhood of a point

p

in

V

is a local

analytic

subvariety

,
we can define germs of weakly holomorphic functions at

p

in the obvious way. We usually
denote by

𝒪

p

w

⁢

(

V

)

the ring of germs at

p

of weakly holomorphic
functions.

References

1

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

Title

weakly holomorphic

Canonical name

WeaklyHolomorphic

Date of creation

2013-03-22 17:41:46

Last modified on

2013-03-22 17:41:46

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 32C15

Classification

msc 32C20

Synonym

w-holomoprhic

Related topic

NormalComplexAnalyticVariety

Generated on Fri Feb 9 20:32:40 2018 by

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irreducible component

irreducible component

Let

G

⊂

ℂ

N

be an

open set

.

Definition

.

A

locally analytic

set (or an analytic variety)

V

⊂

G

is said to be

irreducible

if whenever we have two locally analytic sets

V

1

and

V

2

such that

V

=

V

1

∪

V

2

, then either

V

=

V

1

or

V

=

V

2

. Otherwise

V

is
said to be . A maximal irreducible

subvariety

of

V

is said to be an

irreducible component

of

V

. Sometimes irreducible components are
called

ircomps

.

Note that if

V

is an analytic variety in

G

, then a subvariety

W

is an irreducible component of

V

if and only if

W

*

(the set of

regular points

of

W

) is a

connected

complex analytic manifold

. This means that the irreducible components of

V

are the

closures

of the

connected components

of

V

*

.

References

1

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

Title

irreducible component

Canonical name

IrreducibleComponent1

Date of creation

2013-03-22 15:04:58

Last modified on

2013-03-22 15:04:58

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32C25

Classification

msc 32A60

Synonym

ircomp

Related topic

AnalyticSet

Defines

irreducible analytic variety

Defines

irreducible locally analytic set

Defines

irreducible analytic variety

Defines

reducible locally analytic set

Defines

reducible analytic variety

Generated on Fri Feb 9 20:32:44 2018 by

LaTeXML
analytic sheaf

analytic sheaf

Let

M

be a

complex manifold

. Let

𝒪

be the sheaf of germs of

analytic functions

(which is a sheaf of rings) and let

ℱ

be a sheaf of

𝒪

-modules.

ℱ

is then called an

analytic sheaf

.

Example

.

Suppose

X

is an analytic vector bundle over

M

. Then the sheaf of germs of analytic sections of

X

is an analytic sheaf.

Example

.

The sheaf of germs of

meromorphic functions

on

M

is an analytic sheaf.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

analytic sheaf

Canonical name

AnalyticSheaf

Date of creation

2013-03-22 17:39:02

Last modified on

2013-03-22 17:39:02

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32C35

Generated on Fri Feb 9 20:32:45 2018 by

LaTeXML
coherent analytic sheaf

coherent analytic sheaf

Let

M

be a

complex manifold

and

ℱ

be an

analytic sheaf

.
For

z

∈

M

, denote by

ℱ

z

the stalk of

ℱ

at

z

.
By

𝒪

denote the sheaf of germs of

analytic functions

. For a

section

f

and a point

z

∈

M

denote by

f

z

the germ of

f

at

z

.

ℱ

is said to be

locally

finitely generated

if for every

z

∈

M

,
there is a neighbourhood

U

of

z

, a finite number of sections

f

1

,

…

,

f

k

∈

Γ

⁢

(

U

,

ℱ

)

such that for each

w

∈

U

,

ℱ

w

is a finitely generated module (as an

𝒪

w

-module).

Let

U

be a neighbourhood in

M

and
Suppose that

f

1

,

…

,

f

k

are sections in

Γ

⁢

(

U

,

ℱ

)

.
Let

ℛ

⁢

(

f

1

,

…

,

f

k

)

be the subsheaf of

𝒪

k

over

U

consisting of the germs

{

(

g

1

,

…

,

g

k

)

∈

𝒪

z

k

∣

∑

j

=

1

k

g

j

⁢

(

f

j

)

z

=

0

}

.

ℛ

⁢

(

f

1

,

…

,

f

k

)

is called the

sheaf of relations

.

Definition

.

ℱ

is called a

coherent analytic sheaf

if

ℱ

is locally finitely generated and if for every open subset

U

⊂

M

,
and

f

1

,

…

,

f

k

∈

Γ

⁢

(

U

,

ℱ

)

,
the
sheaf

ℛ

⁢

(

f

1

,

…

,

f

k

)

is locally finitely generated.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

coherent analytic sheaf

Canonical name

CoherentAnalyticSheaf

Date of creation

2013-03-22 17:39:05

Last modified on

2013-03-22 17:39:05

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 32C35

Defines

locally finitely generated sheaf

Defines

sheaf of relations

Generated on Fri Feb 9 20:32:47 2018 by

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