<s>
The	O
theory	O
of	O
functions	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
is	O
the	O
branch	O
of	O
mathematics	O
dealing	O
with	O
complex-valued	O
functions	O
.	O
</s>
<s>
The	O
name	O
of	O
the	O
field	O
dealing	O
with	O
the	O
properties	O
of	O
function	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
is	O
called	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
(	O
and	O
analytic	O
space	O
)	O
,	O
that	O
has	O
become	O
a	O
common	O
name	O
for	O
that	O
whole	O
field	O
of	O
study	O
and	O
Mathematics	O
Subject	O
Classification	O
has	O
,	O
as	O
a	O
top-level	O
heading	O
.	O
</s>
<s>
A	O
function	O
is	O
-tuples	O
of	O
complex	O
numbers	O
,	O
classically	O
studied	O
on	O
the	O
complex	B-Algorithm
coordinate	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Equivalently	O
,	O
they	O
are	O
locally	O
uniform	B-Algorithm
limits	I-Algorithm
of	O
polynomials	O
;	O
or	O
locally	O
square-integrable	O
solutions	O
to	O
the	O
-dimensional	O
Cauchy	O
–	O
Riemann	O
equations	O
.	O
</s>
<s>
For	O
one	O
complex	O
variable	O
,	O
every	O
domain( )	O
,	O
is	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
of	O
some	O
function	O
,	O
in	O
other	O
words	O
every	O
domain	O
has	O
a	O
function	O
for	O
which	O
it	O
is	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
For	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
this	O
is	O
not	O
the	O
case	O
;	O
there	O
exist	O
domains	O
(	O
)	O
that	O
are	O
not	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
of	O
any	O
function	O
,	O
and	O
so	O
is	O
not	O
always	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
,	O
so	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
is	O
one	O
of	O
the	O
themes	O
in	O
this	O
field	O
.	O
</s>
<s>
Also	O
,	O
the	O
interesting	O
phenomena	O
that	O
occur	O
in	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
are	O
fundamentally	O
important	O
to	O
the	O
study	O
of	O
compact	O
complex	O
manifolds	B-Architecture
and	O
complex	O
projective	O
varieties	O
(	O
)	O
and	O
has	O
a	O
different	O
flavour	O
to	O
complex	B-Algorithm
analytic	I-Algorithm
geometry	I-Algorithm
in	O
or	O
on	O
Stein	O
manifolds	B-Architecture
,	O
these	O
are	O
much	O
similar	O
to	O
study	O
of	O
algebraic	O
varieties	O
that	O
is	O
study	O
of	O
the	O
algebraic	O
geometry	O
than	O
complex	B-Algorithm
analytic	I-Algorithm
geometry	I-Algorithm
.	O
</s>
<s>
Naturally	O
the	O
analogues	O
of	O
contour	O
integrals	O
will	O
be	O
harder	O
to	O
handle	O
;	O
when	O
an	O
integral	O
surrounding	O
a	O
point	O
should	O
be	O
over	O
a	O
three-dimensional	O
manifold	B-Architecture
(	O
since	O
we	O
are	O
in	O
four	O
real	O
dimensions	O
)	O
,	O
while	O
iterating	O
contour	O
(	O
line	O
)	O
integrals	O
over	O
two	O
separate	O
complex	O
variables	O
should	O
come	O
to	O
a	O
double	O
integral	O
over	O
a	O
two-dimensional	O
surface	O
.	O
</s>
<s>
A	O
number	O
of	O
issues	O
were	O
clarified	O
,	O
in	O
particular	O
that	O
of	O
analytic	B-Algorithm
continuation	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
the	O
D	O
of	O
that	O
kind	O
are	O
rather	O
special	O
in	O
nature	O
(	O
especially	O
in	O
complex	B-Algorithm
coordinate	I-Algorithm
spaces	I-Algorithm
and	O
Stein	O
manifolds	B-Architecture
,	O
satisfying	O
a	O
condition	O
called	O
pseudoconvexity	O
)	O
.	O
</s>
<s>
The	O
natural	O
domains	O
of	O
definition	O
of	O
functions	O
,	O
continued	O
to	O
the	O
limit	O
,	O
are	O
called	O
Stein	O
manifolds	B-Architecture
and	O
their	O
nature	O
was	O
to	O
make	O
sheaf	O
cohomology	O
groups	O
vanish	O
,	O
also	O
,	O
the	O
property	O
that	O
the	O
sheaf	O
cohomology	O
group	O
disappears	O
is	O
also	O
found	O
in	O
other	O
high-dimensional	O
complex	O
manifolds	B-Architecture
,	O
indicating	O
that	O
the	O
Hodge	O
manifold	B-Architecture
is	O
projective	O
.	O
</s>
<s>
The	O
deformation	O
theory	O
of	O
complex	O
structures	O
and	O
complex	O
manifolds	B-Architecture
was	O
described	O
in	O
general	O
terms	O
by	O
Kunihiko	O
Kodaira	O
and	O
D	O
.	O
C	O
.	O
Spencer	O
.	O
</s>
<s>
C	O
.	O
L	O
.	O
Siegel	O
was	O
heard	O
to	O
complain	O
that	O
the	O
new	O
theory	O
of	O
functions	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
had	O
few	O
functions	O
in	O
it	O
,	O
meaning	O
that	O
the	O
special	O
function	O
side	O
of	O
the	O
theory	O
was	O
subordinated	O
to	O
sheaves	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
other	O
fields	O
,	O
such	O
as	O
Banach	O
algebra	O
theory	O
,	O
that	O
draw	O
on	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
The	O
complex	B-Algorithm
coordinate	I-Algorithm
space	I-Algorithm
is	O
the	O
Cartesian	O
product	O
of	O
copies	O
of	O
,	O
and	O
when	O
is	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
,	O
can	O
be	O
regarded	O
as	O
a	O
Stein	O
manifold	B-Architecture
,	O
and	O
more	O
generalized	O
Stein	O
space	O
.	O
</s>
<s>
is	O
also	O
considered	O
to	O
be	O
a	O
complex	O
projective	O
variety	O
,	O
a	O
Kähler	O
manifold	B-Architecture
,	O
etc	O
.	O
</s>
<s>
In	O
coordinate-free	O
language	O
,	O
any	O
vector	O
space	O
over	O
complex	O
numbers	O
may	O
be	O
thought	O
of	O
as	O
a	O
real	O
vector	O
space	O
of	O
twice	O
as	O
many	O
dimensions	O
,	O
where	O
a	O
complex	O
structure	O
is	O
specified	O
by	O
a	O
linear	B-Architecture
operator	I-Architecture
(	O
such	O
that	O
)	O
which	O
defines	O
multiplication	O
by	O
the	O
imaginary	O
unit	O
.	O
</s>
<s>
Likewise	O
,	O
if	O
one	O
expresses	O
any	O
finite-dimensional	O
complex	O
linear	B-Architecture
operator	I-Architecture
as	O
a	O
real	O
matrix	B-Algorithm
(	O
which	O
will	O
be	O
composed	O
from	O
2	O
×	O
2	O
blocks	O
of	O
the	O
aforementioned	O
form	O
)	O
,	O
then	O
its	O
determinant	O
equals	O
to	O
the	O
square	O
of	O
absolute	O
value	O
of	O
the	O
corresponding	O
complex	O
determinant	O
.	O
</s>
<s>
f	O
is	O
continuous	O
on	O
DUsing	O
Hartogs	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
on	I-Algorithm
separate	I-Algorithm
holomorphicity	I-Algorithm
,	O
If	O
condition	O
(	O
B	O
)	O
is	O
met	O
,	O
it	O
will	O
be	O
derived	O
to	O
be	O
continuous	O
.	O
</s>
<s>
Also	O
,	O
take	O
the	O
closed	O
polydisc	B-Algorithm
so	O
that	O
it	O
becomes	O
.	O
</s>
<s>
From	O
(	O
2	O
)	O
,	O
if	O
f	O
is	O
holomorphic	O
,	O
on	O
polydisc	B-Algorithm
and	O
,	O
the	O
following	O
evaluation	O
equation	O
is	O
obtained	O
.	O
</s>
<s>
If	O
function	O
f	O
is	O
holomorphic	O
,	O
on	O
polydisc	B-Algorithm
,	O
from	O
the	O
Cauchy	O
's	O
integral	O
formula	O
,	O
we	O
can	O
see	O
that	O
it	O
can	O
be	O
uniquely	O
expanded	O
to	O
the	O
next	O
power	O
series	O
.	O
</s>
<s>
We	O
have	O
already	O
explained	O
that	O
holomorphic	O
functions	O
on	O
polydisc	B-Algorithm
are	O
analytic	O
.	O
</s>
<s>
Also	O
,	O
from	O
the	O
theorem	O
derived	O
by	O
Weierstrass	O
,	O
we	O
can	O
see	O
that	O
the	O
analytic	O
function	O
on	O
polydisc	B-Algorithm
(	O
convergent	O
power	O
series	O
)	O
is	O
holomorphic	O
.	O
</s>
<s>
If	O
a	O
sequence	O
of	O
functions	O
which	O
converges	B-Algorithm
uniformly	I-Algorithm
on	O
compacta	O
inside	O
a	O
domain	O
D	O
,	O
the	O
limit	O
function	O
f	O
of	O
also	O
uniformly	O
on	O
compacta	O
inside	O
a	O
domain	O
D	O
.	O
Also	O
,	O
respective	O
partial	O
derivative	O
of	O
also	O
compactly	O
converges	O
on	O
domain	O
D	O
to	O
the	O
corresponding	O
derivative	O
of	O
f	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
define	O
a	O
combination	O
of	O
positive	O
real	O
numbers	O
such	O
that	O
the	O
power	O
series	O
converges	B-Algorithm
uniformly	I-Algorithm
at	O
and	O
does	O
not	O
converge	O
uniformly	O
at	O
.	O
</s>
<s>
The	O
integral	O
in	O
the	O
second	O
term	O
,	O
of	O
the	O
right-hand	O
side	O
is	O
performed	O
so	O
as	O
to	O
see	O
the	O
zero	O
on	O
the	O
left	O
in	O
every	O
plane	O
,	O
also	O
this	O
integrated	O
series	O
is	O
uniformly	B-Algorithm
convergent	I-Algorithm
in	O
the	O
annulus	O
,	O
where	O
and	O
,	O
and	O
so	O
it	O
is	O
possible	O
to	O
integrate	O
term	O
.	O
</s>
<s>
The	O
Cauchy	O
integral	O
formula	O
holds	O
only	O
for	O
polydiscs	B-Algorithm
,	O
and	O
in	O
the	O
domain	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
polydiscs	B-Algorithm
are	O
only	O
one	O
of	O
many	O
domain	O
,	O
so	O
we	O
introduce	O
the	O
Bochner	B-Algorithm
–	I-Algorithm
Martinelli	I-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
When	O
the	O
function	O
f	O
,	O
g	O
is	O
analytic	O
in	O
the	O
domain	O
D	O
,	O
even	O
for	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
the	O
identity	O
theorem	O
holds	O
on	O
the	O
domain	O
D	O
,	O
because	O
it	O
has	O
a	O
power	O
series	O
expansion	O
the	O
neighbourhood	O
of	O
point	O
of	O
analytic	O
.	O
</s>
<s>
At	O
this	O
time	O
,	O
is	O
called	O
a	O
U	O
,	O
V	O
biholomorphism	B-Algorithm
also	O
,	O
we	O
say	O
that	O
U	O
and	O
V	O
are	O
biholomorphically	B-Algorithm
equivalent	I-Algorithm
or	O
that	O
they	O
are	O
biholomorphic	B-Algorithm
.	O
</s>
<s>
When	O
,	O
open	O
balls	O
and	O
open	O
polydiscs	B-Algorithm
are	O
not	O
biholomorphically	B-Algorithm
equivalent	I-Algorithm
,	O
that	O
is	O
,	O
there	O
is	O
no	O
biholomorphic	B-Algorithm
mapping	I-Algorithm
between	O
the	O
two	O
.	O
</s>
<s>
This	O
was	O
proven	O
by	O
Poincaré	O
in	O
1907	O
by	O
showing	O
that	O
their	O
automorphism	B-Algorithm
groups	I-Algorithm
have	O
different	O
dimensions	O
as	O
Lie	O
groups	O
.	O
</s>
<s>
However	O
,	O
even	O
in	O
the	O
case	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
there	O
are	O
some	O
results	O
similar	O
to	O
the	O
results	O
of	O
the	O
theory	O
of	O
uniformization	O
in	O
one	O
complex	O
variable	O
.	O
</s>
<s>
If	O
then	O
f	O
is	O
said	O
to	O
be	O
connected	O
to	O
V	O
,	O
and	O
g	O
is	O
said	O
to	O
be	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
f	O
.	O
From	O
the	O
identity	O
theorem	O
,	O
if	O
g	O
exists	O
,	O
for	O
each	O
way	O
of	O
choosing	O
W	O
it	O
is	O
unique	O
.	O
</s>
<s>
When	O
n	O
>	O
2	O
,	O
the	O
following	O
phenomenon	O
occurs	O
depending	O
on	O
the	O
shape	O
of	O
the	O
boundary	O
:	O
there	O
exists	O
domain	O
U	O
,	O
V	O
,	O
such	O
that	O
all	O
holomorphic	O
functions	O
over	O
the	O
domain	O
U	O
,	O
have	O
an	O
analytic	B-Algorithm
continuation	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
researching	O
when	O
domain	O
boundaries	O
become	O
natural	O
boundaries	O
has	O
become	O
one	O
of	O
the	O
main	O
research	O
themes	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
In	O
polydisks	B-Algorithm
,	O
the	O
Cauchy	O
's	O
integral	O
formula	O
holds	O
and	O
the	O
power	O
series	O
expansion	O
of	O
holomorphic	O
functions	O
is	O
defined	O
,	O
but	O
the	O
unique	O
radius	O
of	O
convergence	O
is	O
not	O
defined	O
for	O
each	O
variable	O
.	O
</s>
<s>
Also	O
,	O
since	O
the	O
Riemann	O
mapping	O
theorem	O
does	O
not	O
hold	O
,	O
polydisks	B-Algorithm
and	O
open	O
unit	O
balls	O
are	O
not	O
biholomorphic	B-Algorithm
mapping	I-Algorithm
,	O
and	O
also	O
,	O
polydisks	B-Algorithm
was	O
possible	O
to	O
separation	O
of	O
variables	O
,	O
but	O
it	O
does	O
n't	O
always	O
hold	O
for	O
any	O
domain	O
.	O
</s>
<s>
Early	O
Knowledge	O
into	O
the	O
properties	O
of	O
field	O
of	O
study	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
such	O
as	O
Logarithmically-convex	O
,	O
Hartogs	B-Algorithm
's	I-Algorithm
extension	I-Algorithm
theorem	I-Algorithm
,	O
etc	O
.	O
</s>
<s>
On	O
the	O
polydisk	B-Algorithm
consisting	O
of	O
two	O
disks	O
when	O
.	O
</s>
<s>
Hartogs	B-Algorithm
's	I-Algorithm
extension	I-Algorithm
theorem	I-Algorithm
(	O
1906	O
)	O
;	O
Let	O
f	O
be	O
a	O
holomorphic	O
function	O
on	O
a	O
set	O
,	O
where	O
is	O
a	O
bounded	B-Algorithm
(	O
surrounded	O
by	O
a	O
rectifiable	O
closed	O
Jordan	O
curve	O
)	O
domain	O
on	O
(	O
)	O
and	O
K	O
is	O
a	O
compact	O
subset	O
of	O
G	O
.	O
If	O
the	O
complement	O
is	O
connected	O
,	O
then	O
every	O
holomorphic	O
function	O
f	O
regardless	O
of	O
how	O
it	O
is	O
chosen	O
can	O
be	O
each	O
extended	O
to	O
a	O
unique	O
holomorphic	O
function	O
on	O
G	O
.	O
</s>
<s>
It	O
is	O
also	O
called	O
Osgood	B-Algorithm
–	I-Algorithm
Brown	I-Algorithm
theorem	I-Algorithm
is	O
that	O
for	O
holomorphic	O
functions	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
the	O
singularity	O
is	O
a	O
accumulation	O
point	O
,	O
not	O
an	O
isolated	O
point	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
various	O
properties	O
that	O
hold	O
for	O
holomorphic	O
functions	O
of	O
one-variable	O
complex	O
variables	O
do	O
not	O
hold	O
for	O
holomorphic	O
functions	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
From	O
Hartogs	B-Algorithm
's	I-Algorithm
extension	I-Algorithm
theorem	I-Algorithm
the	O
domain	O
of	O
convergence	O
extends	O
from	O
to	O
.	O
</s>
<s>
Thullen	O
's	O
classical	O
result	O
says	O
that	O
a	O
2-dimensional	O
bounded	B-Algorithm
Reinhard	O
domain	O
containing	O
the	O
origin	O
is	O
biholomorphic	B-Algorithm
to	O
one	O
of	O
the	O
following	O
domains	O
provided	O
that	O
the	O
orbit	O
of	O
the	O
origin	O
by	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
has	O
positive	O
dimension	O
:	O
</s>
<s>
(	O
polydisc	B-Algorithm
)	O
;	O
</s>
<s>
Two	O
n-dimensional	O
bounded	B-Algorithm
Reinhardt	O
domains	O
and	O
are	O
mutually	O
biholomorphic	B-Algorithm
if	O
and	O
only	O
if	O
there	O
exists	O
a	O
transformation	O
given	O
by	O
,	O
being	O
a	O
permutation	O
of	O
the	O
indices	O
)	O
,	O
such	O
that	O
.	O
</s>
<s>
When	O
moving	O
from	O
the	O
theory	O
of	O
one	O
complex	O
variable	O
to	O
the	O
theory	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
depending	O
on	O
the	O
range	O
of	O
the	O
domain	O
,	O
it	O
may	O
not	O
be	O
possible	O
to	O
define	O
a	O
holomorphic	O
function	O
such	O
that	O
the	O
boundary	O
of	O
the	O
domain	O
becomes	O
a	O
natural	O
boundary	O
.	O
</s>
<s>
Considering	O
the	O
domain	O
where	O
the	O
boundaries	O
of	O
the	O
domain	O
are	O
natural	O
boundaries	O
(	O
In	O
the	O
complex	B-Algorithm
coordinate	I-Algorithm
space	I-Algorithm
call	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
)	O
,	O
the	O
first	O
result	O
of	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
was	O
the	O
holomorphic	O
convexity	O
of	O
H	O
.	O
Cartan	O
and	O
Thullen	O
.	O
</s>
<s>
Levi	O
's	O
problem	O
shows	O
that	O
the	O
pseudoconvex	O
domain	O
was	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
In	O
sheaf	O
cohomology	O
,	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
has	O
come	O
to	O
be	O
interpreted	O
as	O
the	O
theory	O
of	O
Stein	O
manifolds	B-Architecture
.	O
</s>
<s>
The	O
notion	O
of	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
is	O
also	O
considered	O
in	O
other	O
complex	O
manifolds	B-Architecture
,	O
furthermore	O
also	O
the	O
complex	B-Algorithm
analytic	I-Algorithm
space	I-Algorithm
which	O
is	O
its	O
generalization	O
.	O
</s>
<s>
When	O
a	O
function	O
f	O
is	O
holomorpic	O
on	O
the	O
domain	O
and	O
cannot	O
directly	O
connect	O
to	O
the	O
domain	O
outside	O
D	O
,	O
including	O
the	O
point	O
of	O
the	O
domain	O
boundary	O
,	O
the	O
domain	O
D	O
is	O
called	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
of	O
f	O
and	O
the	O
boundary	O
is	O
called	O
the	O
natural	O
boundary	O
of	O
f	O
.	O
In	O
other	O
words	O
,	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
D	O
is	O
the	O
supremum	O
of	O
the	O
domain	O
where	O
the	O
holomorphic	O
function	O
f	O
is	O
holomorphic	O
,	O
and	O
the	O
domain	O
D	O
,	O
which	O
is	O
holomorphic	O
,	O
cannot	O
be	O
extended	O
any	O
more	O
.	O
</s>
<s>
For	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
i.e.	O
</s>
<s>
Hartogs	B-Algorithm
 '	I-Algorithm
extension	I-Algorithm
theorem	I-Algorithm
gives	O
an	O
example	O
of	O
a	O
domain	O
where	O
boundaries	O
are	O
not	O
natural	O
boundaries	O
.	O
</s>
<s>
Formally	O
,	O
a	O
domain	O
D	O
in	O
the	O
n-dimensional	O
complex	B-Algorithm
coordinate	I-Algorithm
space	I-Algorithm
is	O
called	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
if	O
there	O
do	O
not	O
exist	O
non-empty	O
domain	O
and	O
,	O
and	O
such	O
that	O
for	O
every	O
holomorphic	O
function	O
f	O
on	O
D	O
there	O
exists	O
a	O
holomorphic	O
function	O
g	O
on	O
V	O
with	O
on	O
U	O
.	O
</s>
<s>
For	O
the	O
case	O
,	O
the	O
every	O
domain	O
(	O
)	O
was	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
;	O
we	O
can	O
define	O
a	O
holomorphic	O
function	O
with	O
zeros	O
accumulating	O
everywhere	O
on	O
the	O
boundary	O
of	O
the	O
domain	O
,	O
which	O
must	O
then	O
be	O
a	O
natural	O
boundary	O
for	O
a	O
domain	O
of	O
definition	O
of	O
its	O
reciprocal	O
.	O
</s>
<s>
If	O
are	O
domains	O
of	O
holomorphy	O
,	O
then	O
their	O
intersection	O
is	O
also	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
If	O
is	O
an	O
increasing	O
sequence	O
of	O
domains	O
of	O
holomorphy	O
,	O
then	O
their	O
union	O
is	O
also	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
(	O
see	O
Behnke	O
–	O
Stein	O
theorem	O
)	O
.	O
</s>
<s>
If	O
and	O
are	O
domains	O
of	O
holomorphy	O
,	O
then	O
is	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
The	O
first	O
Cousin	O
problem	O
is	O
always	O
solvable	O
in	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
,	O
also	O
Cartan	O
showed	O
that	O
the	O
converse	O
of	O
this	O
result	O
was	O
incorrect	O
for	O
.	O
</s>
<s>
Let	O
be	O
a	O
domain	O
,	O
or	O
alternatively	O
for	O
a	O
more	O
general	O
definition	O
,	O
let	O
be	O
an	O
dimensional	O
complex	O
analytic	O
manifold	B-Architecture
.	O
</s>
<s>
Also	O
,	O
at	O
this	O
time	O
,	O
D	O
is	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
every	O
convex	O
domain	O
is	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
If	O
such	O
a	O
relations	O
holds	O
in	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
it	O
looks	O
like	O
a	O
more	O
manageable	O
condition	O
than	O
a	O
holomorphically	O
convex	O
.	O
</s>
<s>
In	O
full	O
generality	O
,	O
the	O
notion	O
can	O
be	O
defined	O
on	O
an	O
arbitrary	O
complex	O
manifold	B-Architecture
or	O
even	O
a	O
Complex	B-Algorithm
analytic	I-Algorithm
space	I-Algorithm
as	O
follows	O
.	O
</s>
<s>
There	O
fore	O
,	O
if	O
is	O
of	O
class	O
,	O
then	O
is	O
plurisubharmonic	B-Algorithm
if	O
and	O
only	O
if	O
the	O
hermitian	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
semidefinite	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
-function	O
u	O
is	O
plurisubharmonic	B-Algorithm
if	O
and	O
only	O
if	O
is	O
a	O
positive	O
(	O
1	O
,	O
1	O
)	O
-form	O
.	O
</s>
<s>
When	O
the	O
hermitian	B-Algorithm
matrix	I-Algorithm
of	O
u	O
is	O
positive-definite	O
and	O
class	O
,	O
we	O
call	O
u	O
a	O
strict	O
plurisubharmonic	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
One	O
says	O
that	O
X	O
is	O
pseudoconvex	O
if	O
there	O
exists	O
a	O
continuous	O
plurisubharmonic	B-Algorithm
function	I-Algorithm
on	O
X	O
such	O
that	O
the	O
set	O
is	O
a	O
relatively	O
compact	O
subset	O
of	O
X	O
for	O
all	O
real	O
numbers	O
x	O
.	O
i.e.	O
</s>
<s>
there	O
exists	O
a	O
smooth	O
plurisubharmonic	B-Algorithm
exhaustion	O
function	O
.	O
</s>
<s>
Often	O
,	O
the	O
definition	O
of	O
pseudoconvex	O
is	O
used	O
here	O
and	O
is	O
written	O
as	O
;	O
Let	O
X	O
be	O
a	O
complex	O
n-dimensional	O
manifold	B-Architecture
.	O
</s>
<s>
Then	O
is	O
said	O
to	O
be	O
weeak	O
pseudoconvex	O
there	O
exists	O
a	O
smooth	O
plurisubharmonic	B-Algorithm
exhaustion	O
function	O
.	O
</s>
<s>
Let	O
X	O
be	O
a	O
complex	O
n-dimensional	O
manifold	B-Architecture
.	O
</s>
<s>
Strongly	O
pseudoconvex	O
if	O
there	O
exists	O
a	O
smooth	O
strictly	O
plurisubharmonic	B-Algorithm
exhaustion	O
function	O
,	O
i.e.	O
,	O
is	O
positive	O
definite	O
at	O
every	O
point	O
.	O
</s>
<s>
The	O
strong	O
Levi	O
pseudoconvex	O
domain	O
is	O
simply	O
called	O
strong	O
pseudoconvex	O
and	O
is	O
often	O
called	O
strictly	O
pseudoconvex	O
to	O
make	O
it	O
clear	O
that	O
it	O
has	O
a	O
strictly	O
plurisubharmonic	B-Algorithm
exhaustion	O
function	O
in	O
relation	O
to	O
the	O
fact	O
that	O
it	O
may	O
not	O
have	O
a	O
strictly	O
plurisubharmonic	B-Algorithm
exhaustion	O
function	O
.	O
</s>
<s>
For	O
arbitrary	O
complex	O
manifold	B-Architecture
,	O
Levi	O
(	O
–	O
Krzoska	O
)	O
pseudoconvexity	O
does	O
not	O
always	O
have	O
an	O
plurisubharmonic	B-Algorithm
exhaustion	O
function	O
,	O
i.e.	O
</s>
<s>
If	O
for	O
every	O
boundary	O
point	O
of	O
D	O
,	O
there	O
exists	O
an	O
analytic	B-Algorithm
variety	I-Algorithm
passing	O
which	O
lies	O
entirely	O
outside	O
D	O
in	O
some	O
neighborhood	O
around	O
,	O
except	O
the	O
point	O
itself	O
.	O
</s>
<s>
Oka	O
's	O
proof	O
of	O
Levi	O
's	O
problem	O
was	O
that	O
when	O
the	O
unramified	O
Riemann	O
domain	O
over	O
was	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
(	O
holomorphically	O
convex	O
)	O
,	O
it	O
was	O
proved	O
that	O
it	O
was	O
necessary	O
and	O
sufficient	O
that	O
each	O
boundary	O
point	O
of	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
is	O
an	O
Oka	O
pseudoconvex	O
.	O
</s>
<s>
i.e.	O
,	O
let	O
be	O
a	O
holomorphic	O
map	O
,	O
if	O
every	O
point	O
has	O
a	O
neighborhood	O
U	O
such	O
that	O
admits	O
a	O
-plurisubharmonic	O
exhaustion	O
function	O
(	O
weakly	O
1-complete	O
)	O
,	O
in	O
this	O
situation	O
,	O
we	O
call	O
that	O
X	O
is	O
locally	O
pseudoconvex	O
(	O
or	O
locally	O
Stein	O
)	O
over	O
Y	O
.	O
</s>
<s>
In	O
the	O
locally	O
pseudoconvex	O
domain	O
is	O
itself	O
a	O
pseudoconvex	O
domain	O
and	O
it	O
is	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
D	O
is	O
a	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
D	O
is	O
the	O
union	O
of	O
an	O
increasing	O
sequence	O
of	O
analytic	B-Algorithm
polyhedrons	I-Algorithm
in	O
D	O
.	O
</s>
<s>
The	O
notion	O
of	O
coherent	O
(	O
coherent	O
sheaf	O
cohomology	O
)	O
helped	O
solve	O
the	O
problems	O
in	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
However	O
,	O
these	O
theorems	O
do	O
not	O
hold	O
because	O
the	O
singularities	O
of	O
analytic	O
function	O
in	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
is	O
not	O
isolated	O
points	O
,	O
this	O
problem	O
is	O
called	O
the	O
Cousin	O
problem	O
and	O
is	O
formulated	O
in	O
sheaf	O
cohomology	O
terms	O
.	O
</s>
<s>
They	O
are	O
now	O
posed	O
,	O
and	O
solved	O
,	O
for	O
arbitrary	O
complex	O
manifold	B-Architecture
M	O
,	O
in	O
terms	O
of	O
conditions	O
on	O
M	O
.	O
M	O
,	O
which	O
satisfies	O
these	O
conditions	O
,	O
is	O
one	O
way	O
to	O
define	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
study	O
of	O
the	O
cousin	O
's	O
problem	O
made	O
us	O
realize	O
that	O
in	O
the	O
study	O
of	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
it	O
is	O
possible	O
to	O
study	O
of	O
global	O
properties	O
from	O
the	O
patching	O
of	O
local	O
data	O
,	O
that	O
is	O
it	O
has	O
developed	O
the	O
theory	O
of	O
sheaf	O
cohomology	O
.	O
</s>
<s>
In	O
particular	O
,	O
by	O
Cartan	O
's	O
theorem	O
B	O
,	O
the	O
Cousin	O
problem	O
is	O
always	O
solvable	O
if	O
M	O
is	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
When	O
M	O
is	O
a	O
Stein	O
manifold	B-Architecture
,	O
the	O
middle	O
arrow	O
is	O
an	O
isomorphism	O
because	O
for	O
so	O
that	O
a	O
necessary	O
and	O
sufficient	O
condition	O
in	O
that	O
case	O
for	O
the	O
second	O
Cousin	O
problem	O
to	O
be	O
always	O
solvable	O
is	O
that	O
(	O
This	O
condition	O
called	O
Oka	O
principle	O
.	O
)	O
</s>
<s>
Since	O
a	O
non-compact	O
(	O
open	O
)	O
Riemann	O
surface	O
always	O
has	O
a	O
non-constant	O
single-valued	O
holomorphic	O
function	O
,	O
and	O
satisfies	O
the	O
second	O
axiom	O
of	O
countability	O
,	O
the	O
open	O
Riemann	O
surface	O
is	O
in	O
fact	O
a	O
1-dimensional	O
complex	O
manifold	B-Architecture
possessing	O
a	O
holomorphic	O
mapping	O
into	O
the	O
complex	O
plane	O
.	O
</s>
<s>
The	O
Whitney	O
embedding	O
theorem	O
tells	O
us	O
that	O
every	O
smooth	O
n-dimensional	O
manifold	B-Architecture
can	O
be	O
embedded	O
as	O
a	O
smooth	O
submanifold	O
of	O
,	O
whereas	O
it	O
is	O
"	O
rare	O
"	O
for	O
a	O
complex	O
manifold	B-Architecture
to	O
have	O
a	O
holomorphic	O
embedding	O
into	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
an	O
arbitrary	O
compact	O
connected	O
complex	O
manifold	B-Architecture
X	O
,	O
every	O
holomorphic	O
function	O
on	O
it	O
is	O
constant	O
by	O
Liouville	O
's	O
theorem	O
,	O
and	O
so	O
it	O
cannot	O
have	O
any	O
embedding	O
into	O
complex	O
n-space	O
.	O
</s>
<s>
That	O
is	O
,	O
for	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
arbitrary	O
complex	O
manifolds	B-Architecture
do	O
not	O
always	O
have	O
holomorphic	O
functions	O
that	O
are	O
not	O
constants	O
.	O
</s>
<s>
So	O
,	O
consider	O
the	O
conditions	O
under	O
which	O
a	O
complex	O
manifold	B-Architecture
has	O
a	O
holomorphic	O
function	O
that	O
is	O
not	O
a	O
constant	O
.	O
</s>
<s>
Complex	O
manifolds	B-Architecture
that	O
can	O
be	O
holomorphic	O
embedded	O
into	O
are	O
called	O
Stein	O
manifolds	B-Architecture
.	O
</s>
<s>
Also	O
Stein	O
manifolds	B-Architecture
satisfy	O
the	O
second	O
axiom	O
of	O
countability	O
.	O
</s>
<s>
A	O
Stein	O
manifold	B-Architecture
is	O
a	O
complex	O
submanifold	O
of	O
the	O
vector	O
space	O
of	O
n	O
complex	O
dimensions	O
.	O
</s>
<s>
A	O
Stein	O
space	O
is	O
similar	O
to	O
a	O
Stein	O
manifold	B-Architecture
but	O
is	O
allowed	O
to	O
have	O
singularities	O
.	O
</s>
<s>
If	O
the	O
univalent	O
domain	O
on	O
is	O
connection	O
to	O
a	O
manifold	B-Architecture
,	O
can	O
be	O
regarded	O
as	O
a	O
complex	O
manifold	B-Architecture
and	O
satisfies	O
the	O
separation	O
condition	O
described	O
later	O
,	O
the	O
condition	O
for	O
becoming	O
a	O
Stein	O
manifold	B-Architecture
is	O
to	O
satisfy	O
the	O
holomorphic	O
convexity	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
Stein	O
manifold	B-Architecture
is	O
the	O
properties	O
of	O
the	O
domain	O
of	O
definition	O
of	O
the	O
(	O
maximal	O
)	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
an	O
analytic	O
function	O
.	O
</s>
<s>
Suppose	O
X	O
is	O
a	O
paracompact	O
complex	O
manifolds	B-Architecture
of	O
complex	O
dimension	O
and	O
let	O
denote	O
the	O
ring	O
of	O
holomorphic	O
functions	O
on	O
X	O
.	O
</s>
<s>
We	O
call	O
X	O
a	O
Stein	O
manifold	B-Architecture
if	O
the	O
following	O
conditions	O
hold	O
:	O
</s>
<s>
X	O
is	O
holomorphically	B-Algorithm
separable	I-Algorithm
,	O
From	O
this	O
condition	O
,	O
we	O
can	O
see	O
that	O
the	O
Stein	O
manifold	B-Architecture
is	O
not	O
compact	O
.	O
</s>
<s>
The	O
open	O
neighborhood	O
of	O
every	O
point	O
on	O
the	O
manifold	B-Architecture
has	O
a	O
holomorphic	O
chart	O
to	O
the	O
.	O
</s>
<s>
A	O
deep	O
theorem	O
of	O
Behnke	O
and	O
Stein	O
(	O
1948	O
)	O
asserts	O
that	O
X	O
is	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
Cartan	O
extended	O
Levi	O
's	O
problem	O
to	O
Stein	O
manifolds	B-Architecture
.	O
</s>
<s>
If	O
the	O
relative	O
compact	O
open	O
subset	O
of	O
the	O
Stein	O
manifold	B-Architecture
X	O
is	O
a	O
Locally	O
pseudoconvex	O
,	O
then	O
D	O
is	O
a	O
Stein	O
manifold	B-Architecture
,	O
and	O
conversely	O
,	O
if	O
D	O
is	O
a	O
Locally	O
pseudoconvex	O
,	O
then	O
X	O
is	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
Then	O
X	O
is	O
a	O
Stein	O
manifold	B-Architecture
if	O
and	O
only	O
if	O
D	O
is	O
locally	O
the	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
Also	O
,	O
Grauert	O
proved	O
for	O
arbitrary	O
complex	O
manifolds	B-Architecture
M	O
.	O
</s>
<s>
If	O
the	O
relative	O
compact	O
subset	O
of	O
a	O
arbitrary	O
complex	O
manifold	B-Architecture
M	O
is	O
a	O
strongly	O
pseudoconvex	O
on	O
M	O
,	O
then	O
M	O
is	O
a	O
holomorphically	O
convex	O
(	O
i.e.	O
</s>
<s>
Stein	O
manifold	B-Architecture
)	O
.	O
</s>
<s>
Also	O
,	O
D	O
is	O
itself	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
And	O
Narasimhan	O
extended	O
Levi	O
's	O
problem	O
to	O
Complex	B-Algorithm
analytic	I-Algorithm
space	I-Algorithm
,	O
a	O
generalized	O
in	O
the	O
singular	O
case	O
of	O
complex	O
manifolds	B-Architecture
.	O
</s>
<s>
A	O
Complex	B-Algorithm
analytic	I-Algorithm
space	I-Algorithm
which	O
admits	O
a	O
continuous	O
strictly	O
plurisubharmonic	B-Algorithm
exhaustion	O
function	O
(	O
i.e.strongly	O
pseudoconvex	O
)	O
is	O
Stein	O
space	O
.	O
</s>
<s>
This	O
means	O
that	O
Behnke	O
–	O
Stein	O
theorem	O
,	O
which	O
holds	O
for	O
Stein	O
manifolds	B-Architecture
,	O
has	O
not	O
found	O
a	O
conditions	O
to	O
be	O
established	O
in	O
Stein	O
space	O
.	O
</s>
<s>
Let	O
X	O
is	O
complex	O
manifold	B-Architecture
,	O
X	O
is	O
K-complete	O
if	O
,	O
to	O
each	O
point	O
,	O
there	O
exist	O
finitely	O
many	O
holomorphic	O
map	O
of	O
X	O
into	O
,	O
,	O
such	O
that	O
is	O
an	O
isolated	O
point	O
of	O
the	O
set	O
.	O
</s>
<s>
This	O
concept	O
also	O
applies	O
to	O
complex	B-Algorithm
analytic	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
standard	O
complex	O
space	O
is	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
Every	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
in	O
is	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
quite	O
easily	O
that	O
every	O
closed	O
complex	O
submanifold	O
of	O
a	O
Stein	O
manifold	B-Architecture
is	O
a	O
Stein	O
manifold	B-Architecture
,	O
too	O
.	O
</s>
<s>
The	O
embedding	O
theorem	O
for	O
Stein	O
manifolds	B-Architecture
states	O
the	O
following	O
:	O
Every	O
Stein	O
manifold	B-Architecture
X	O
of	O
complex	O
dimension	O
n	O
can	O
be	O
embedded	O
into	O
by	O
a	O
biholomorphic	B-Algorithm
proper	O
map	O
.	O
</s>
<s>
These	O
facts	O
imply	O
that	O
a	O
Stein	O
manifold	B-Architecture
is	O
a	O
closed	O
complex	O
submanifold	O
of	O
complex	O
space	O
,	O
whose	O
complex	O
structure	O
is	O
that	O
of	O
the	O
ambient	O
space	O
(	O
because	O
the	O
embedding	O
is	O
biholomorphic	B-Algorithm
)	O
.	O
</s>
<s>
Every	O
Stein	O
manifold	B-Architecture
of	O
(	O
complex	O
)	O
dimension	O
n	O
has	O
the	O
homotopy	O
type	O
of	O
an	O
n-dimensional	O
CW-Complex	O
.	O
</s>
<s>
In	O
one	O
complex	O
dimension	O
the	O
Stein	O
condition	O
can	O
be	O
simplified	O
:	O
a	O
connected	O
Riemann	O
surface	O
is	O
a	O
Stein	O
manifold	B-Architecture
if	O
and	O
only	O
if	O
it	O
is	O
not	O
compact	O
.	O
</s>
<s>
Every	O
Stein	O
manifold	B-Architecture
X	O
is	O
holomorphically	O
spreadable	O
,	O
i.e.	O
</s>
<s>
The	O
first	O
Cousin	O
problem	O
can	O
always	O
be	O
solved	O
on	O
a	O
Stein	O
manifold	B-Architecture
.	O
</s>
<s>
Being	O
a	O
Stein	O
manifold	B-Architecture
is	O
equivalent	O
to	O
being	O
a	O
(	O
complex	O
)	O
strongly	O
pseudoconvex	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
latter	O
means	O
that	O
it	O
has	O
a	O
strongly	O
pseudoconvex	O
(	O
or	O
plurisubharmonic	B-Algorithm
)	O
exhaustive	O
function	O
,	O
i.e.	O
</s>
<s>
The	O
function	O
invites	O
a	O
generalization	O
of	O
Stein	O
manifold	B-Architecture
to	O
the	O
idea	O
of	O
a	O
corresponding	O
class	O
of	O
compact	O
complex	O
manifolds	B-Architecture
with	O
boundary	O
called	O
Stein	O
domain	O
.	O
</s>
<s>
Some	O
authors	O
call	O
such	O
manifolds	B-Architecture
therefore	O
strictly	O
pseudoconvex	O
manifolds	B-Architecture
.	O
</s>
<s>
Numerous	O
further	O
characterizations	O
of	O
such	O
manifolds	B-Architecture
exist	O
,	O
in	O
particular	O
capturing	O
the	O
property	O
of	O
their	O
having	O
"	O
many	O
"	O
holomorphic	O
functions	O
taking	O
values	O
in	O
the	O
complex	O
numbers	O
.	O
</s>
<s>
In	O
the	O
GAGA	O
set	O
of	O
analogies	O
,	O
Stein	O
manifolds	B-Architecture
correspond	O
to	O
affine	O
varieties	O
.	O
</s>
<s>
Stein	O
manifolds	B-Architecture
are	O
in	O
some	O
sense	O
dual	O
to	O
the	O
elliptic	O
manifolds	B-Architecture
in	O
complex	O
analysis	O
which	O
admit	O
"	O
many	O
"	O
holomorphic	O
functions	O
from	O
the	O
complex	O
numbers	O
into	O
themselves	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
a	O
Stein	O
manifold	B-Architecture
is	O
elliptic	O
if	O
and	O
only	O
if	O
it	O
is	O
fibrant	O
in	O
the	O
sense	O
of	O
so-called	O
"	O
holomorphic	O
homotopy	O
theory	O
"	O
.	O
</s>
<s>
A	O
compact	O
one-dimensional	O
complex	O
manifold	B-Architecture
was	O
a	O
Riemann	O
sphere	O
.	O
</s>
<s>
However	O
,	O
for	O
the	O
high-dimensional	O
compact	O
complex	O
manifolds	B-Architecture
,	O
the	O
existence	O
of	O
meromorphic	O
functions	O
and	O
classification	O
of	O
meromorphic	O
function	O
cannot	O
be	O
easily	O
verified	O
because	O
in	O
several	B-Architecture
complex	I-Architecture
variable	I-Architecture
cannot	O
have	O
isolated	O
singularities	O
.	O
</s>
<s>
In	O
fact	O
,	O
Hopf	O
found	O
a	O
class	O
of	O
compact	O
complex	O
manifolds	B-Architecture
without	O
nonconstant	O
meromorphic	O
functions	O
..	O
However	O
,	O
there	O
is	O
a	O
Siegel	O
result	O
that	O
gives	O
the	O
necessary	O
conditions	O
for	O
compact	O
complex	O
manifolds	B-Architecture
to	O
be	O
algebraic	O
.	O
</s>
<s>
The	O
generalization	O
of	O
the	O
Riemann-Roch	O
theorem	O
to	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
was	O
first	O
extended	O
by	O
Kodaira	O
to	O
compact	O
analytic	O
surfaces	O
,	O
and	O
then	O
to	O
three-dimensional	O
,	O
and	O
then	O
n-dimensional	O
Kähler	O
varieties	O
.	O
</s>
<s>
Cartan-Serre	O
proved	O
the	O
following	O
property	O
:	O
the	O
cohomology	O
group	O
is	O
finite-dimensional	O
for	O
a	O
coherent	O
sheaf	O
on	O
a	O
compact	O
complex	O
manifold	B-Architecture
M	O
.	O
Hirzebruch	O
generalized	O
the	O
theorem	O
for	O
compact	O
complex	O
manifolds	B-Architecture
in	O
1994	O
(	O
The	O
Hirzebruch	O
–	O
Riemann	O
–	O
Roch	O
theorem	O
)	O
and	O
Grothendieck	O
more	O
generalized	O
it	O
(	O
The	O
Grothendieck	O
–	O
Hirzebruch	O
–	O
Riemann	O
–	O
Roch	O
theorem	O
)	O
.	O
</s>
<s>
In	O
the	O
high-dimensional	O
(	O
compact	O
)	O
complex	O
manifolds	B-Architecture
,	O
the	O
phenomenon	O
that	O
the	O
sheaf	O
cohomology	O
group	O
vanishing	O
occurs	O
,	O
then	O
the	O
existence	O
condition	O
of	O
meromorphic	O
function	O
can	O
be	O
given	O
by	O
calculating	O
the	O
numerical	O
value	O
of	O
the	O
topological	O
invariant	O
,	O
by	O
using	O
generalized	O
the	O
Riemann-Roch	O
theorem	O
,	O
and	O
it	O
is	O
the	O
Kodaira	O
vanishing	O
theorem	O
and	O
its	O
generalization	O
Nakano	O
vanishing	O
theorem	O
etc	O
.	O
</s>
<s>
i.e.	O
,	O
gives	O
the	O
conditions	O
when	O
a	O
compact	O
complex	O
manifold	B-Architecture
is	O
projective	O
.	O
</s>
<s>
For	O
example	O
,	O
Kodaira	O
embedding	O
theorem	O
says	O
that	O
a	O
compact	O
Kähler	O
manifold	B-Architecture
M	O
,	O
with	O
a	O
Hodge	O
metric	O
,	O
there	O
is	O
a	O
complex-analytic	O
embedding	O
of	O
M	O
into	O
complex	O
projective	O
space	O
of	O
enough	O
high-dimension	O
N	O
.	O
Chow	O
's	O
theorem	O
shows	O
that	O
the	O
complex	O
analytic	O
subspace	O
(	O
subvariety	O
)	O
of	O
a	O
closed	O
complex	O
projective	O
space	O
to	O
be	O
an	O
algebraic	O
that	O
is	O
,	O
so	O
it	O
is	O
the	O
common	O
zero	O
of	O
some	O
homogeneous	O
polynomials	O
,	O
such	O
a	O
relationship	O
is	O
one	O
example	O
of	O
what	O
is	O
called	O
Serre	O
's	O
GAGA	O
principle	O
.	O
</s>
<s>
Then	O
combined	O
with	O
Kodaira	O
's	O
result	O
,	O
a	O
compact	O
Kähler	O
manifold	B-Architecture
M	O
embeds	O
as	O
an	O
algebraic	O
variety	O
.	O
</s>
<s>
This	O
gives	O
an	O
example	O
of	O
a	O
complex	O
manifold	B-Architecture
with	O
enough	O
meromorphic	O
functions	O
.	O
</s>
<s>
Broadly	O
,	O
the	O
GAGA	O
principle	O
says	O
that	O
the	O
geometry	O
of	O
projective	O
complex	B-Algorithm
analytic	I-Algorithm
spaces	I-Algorithm
(	O
or	O
manifolds	B-Architecture
)	O
is	O
equivalent	O
to	O
the	O
geometry	O
of	O
projective	O
complex	O
varieties	O
.	O
</s>
<s>
Also	O
,	O
the	O
deformation	O
theory	O
of	O
compact	O
complex	O
manifolds	B-Architecture
has	O
developed	O
as	O
Kodaira	O
–	O
Spencer	O
theory	O
.	O
</s>
<s>
However	O
,	O
despite	O
being	O
a	O
compact	O
complex	O
manifold	B-Architecture
,	O
there	O
are	O
counterexample	O
of	O
that	O
cannot	O
be	O
embedded	O
in	O
projective	O
space	O
and	O
are	O
not	O
algebraic	O
.	O
</s>
