several complex variables

several complex variables

Several complex variables is a study of

holomorphic

(

http://planetmath.org/HolomorphicFunctionsOfSeveralVariables

)

functions

on the space

ℂ

n

or on abstract

complex manifolds

. It is also the study of objects and spaces under

biholomorphic mappings

. While the
theory of one complex variable (

http://planetmath.org/FundamentalTheoremsInComplexAnalysis

) is really a subset of the study of several complex variables, the study of several complex variables is substantially different. As an example, in one variable theory, the

Riemann mapping theorem

tells us that all

simply connected

domains (with the exception of the entire plane) are biholomorphically equivalent to each other, in just

ℂ

2

the open

unit ball

and the

open polydisc

are not
biholomorphic.

A large part of the study of several complex variables is the study of domains of

holomorphic functions

, namely, the

domains of holomorphy

. Topics related to this study include

•

Bergman space

,

•

pseudoconvex

and

strongly pseudoconvex

domains,

•

Reinhardt domains

,

•

∂

¯

-operator (

http://planetmath.org/BarpartialOperator

) and the solution to the inhomogeneous equation

∂

¯

⁢

u

=

f

,

•

Stein manifolds

.

Important theorems in the theory of several complex variables include,

•

Cauchy integral formula in several variables

,

•

Oka coherence theorem

,

•

Cartan theorem

A and theorem B (

http://planetmath.org/CartanTheoremB

),

•

Hartogs’ theorem on separate analyticity,

•

Hartogs’ phenomenon,

•

Remmert-Stein theorem

,

•

Weierstrass preparation theorem

.

Several complex variables also includes the study of the

zero sets

of

complex analytic functions

and these are called

complex analytic varieties

. Study of such objects in one complex dimension is invariably boring as zero sets of complex analytic functions of one variable are just isolated points. However,
the zero set of a holomorphic function of

n

complex variables is at most points an

n

-

1

dimensional complex manifold.

The study of

real manifolds

in complex spaces (such as the boundary of a domain) and their behaviour under biholomorphic mappings is called CR

geometry

. See the entries on

CR submanifold

and

CR function

for more information.

The topics in several complex variables have many

connections

to other parts of modern mathematics. There are many connections to both real and

complex

algebraic geometry

on one hand, and partial differential equations on the other.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

2

Albert Boggess.
,
CRC, 1991.

3

D’Angelo, John P.
,
CRC Press, 1993.

4

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

5

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

6

Hassler Whitney.
.
Addison-Wesley, 1972.

Title

several complex variables

Canonical name

SeveralComplexVariables

Date of creation

2013-03-22 15:11:41

Last modified on

2013-03-22 15:11:41

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Topic

Classification

msc 32-00

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