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O, n~z Since
u~'
principal
R'
R'
u~(~') 2
which
~'
be holomorphic
function~ in fi
S1 .
(The
small enough.) Now, the germs ~R'
there exist
qi
f, ~i' ~i
does
not
=
qi(b) ~ O 5(Sl),
can be found if
U
is
so that there
V on
and, for
x
i. near
=
f a. ga
V rU
of
such that S1
Then, b,
f
a
and holomorphic
induces the germ
and such that
a
on
be a point at which for each
such that
u
~i fi = ~i f b~SlnV
of holomorphic germs
~i~'
~ifa ,
on
vanish
(2.1) Let
Sl,
u = (fi) a"
there is a neighbourhood
functions
Let
U, vanishing on
(fi) a e ~ r
~i' q i ' % '
qi(fi) a
fa'
R').
such that ai
Hence,
u R' = ~,n,
we have
vanishing on
Hence,
into
is the only prime ideal in
on
~
R'
exists because
Then the principal ideal
5(S I)
exist
~'
is a Dedekind ring, and
which generate the ideal sheaf U
and
~' = u R', so that R' is a fa ideal domain. Let u - g , fa, ga~Oa ' ga$~"
fl .... 'fk
,
R'.
can be factored uniquely,
u~',
(since 2
R,
is
the only non-zero prime ideal in
~ '. Hence, by definition,
~' # (O), R').
R'
This is a local ring with maximal ideal
noetherian and integrally closed, and hence~so is Since
Let
V.
S1
since fi
is regular and fl ..... fk
generate
is a multiple of
f,
ii7
we
see
hood
that
the
b.
We
of
a regular
Thus,
3.
Let
b~X,
let
X
on
be
have
since
~.
is n o w h e r e at
x~X,
hs
any
regular
Proof.
Let
generate
U
the on
the
sheaf
X.
vanishing if
z~A
Hence
hi, . . . ,h m point
Theorem
3.
that
Proof. Theorem the
Let 2,
of
Let X
X
A x, of
set
A
of
~r
C
let
point
of
i ~ j
~ n
vanish
be
of
g
is
g = h(f)x , where
is a r e g u l a r
f,
point
of
h i ..... h m
generate
functions
IV,
Remark
after
Theorem
A,
if
= f,
we
h
o A
=
n
-
in 6,
have
p
+
i.
--Z
and
1 ,<j ~ n on
Xz
m
it
follows
that
z
is
a
'
a complex
space.
a
normal
at
a;
then
U
of
a
dim S b
X.
functions
of h o l o m o r p h i c
at
X,
,
f
X.
be
n
zeros
if
of h o l o m o r p h i c
By C h a p t e r
is n o r m a l X
set
>~n - p = d i m Xz,
X
b
to p r o v e
and
that
then
A
of g e r m s
{ m,
only
set
that
is a c o n t r a -
point,
and
hypothesis,
a neighbourhood
singular
the
3 below)
an o p e n
= n - dim ( i
regular
such
in
~(X)
J 1 ~ i ~m, since
that
(---~.11
rank
in
a neighbour-
holomorphic f u n c t i o n s on a n in C , v a n i s h i n g on X, which
By our
a regular
0h. (--~) 3 o
rank
set
in
be
sheaf
A.
have
at e a c h
on
point
b
~(A)
on
is
of
ideal
vanishing ideal
g = 0
Lemma this
2, w e
p
dense
hl,...,h m
neighbourhood
from
analytic
Suppose
holomorphic then
Theorem
an
~(S I)
bs
dimension
on
X
generates
(as f o l l o w s
X;
X
holomorphic f
of
f
to p r o v e
Lemma
of
claim
point
diction.
be
function
we
is o p e n
The
in
set
- 2
for
is,
that,
have
~ dim _~
aeX
X.
there
such
of
b~U.
by if
S
is
is
118
By T h e o r e m
i, the sheaf
is a s u b s h e a f ~a
= ~a .
~U %
Since
equals = %
at
of
~U'
~U
and since
b
near
subsheaf a; h e n c e
4.
an a n a l y t i c
subset of
for a n y
Let
aeX.
be e x t e n d e d Proof.
U;
further
is n o r m a l
~U
at
is open, b,
%
a,
the c o h e r e n t
for such
This
definition
X
be a n o r m a l X
sheaf
we have X
is n o r m a l
is an i m m e d i a t e
12; T h e o r e m
3. The n o r m a l i z a t i o n
theorem
Proof. point
space.
Because of
X
over
%' f. l hi _ gi'
and
set in
let
cn.
V
X - Y
points
is small
h(x)
= x,hl(x)
an a n a l y t i c
closure
Y
Y'
V • Cm.
To p r o v e
U
Oka)
can
III,
and the
that
seen
are h o l o m o r p h i c
gi
is n o w h e r e
of
V;
the
X
be
(Y,~). that a n y
U
X
admits
is an a n a l y t i c
(Lemma 1) that
h i,...,hm ~a
Let
Let
to p r o v e
such that
suppose
We have
enough.
3 above
a normalization
be a n e i g h b o u r h o o d
fi" gi
of e a c h
of
on
of C h a p t e r
of
2, it s u f f i c e s
let
V
clearly
theorem
has
H e n c e we m a y
where
set of r e g u l a r if
X
~a-m~
set of zeros
Y
spaces.
a neighbourhood
a normalization. set in an open
Then
of L e m m a
has
is a finite
function
consequence
(The n o r m a l i z a t i o n
any complex
and
dim --a Y ~ dim --a X - 2
2 and P r o p o s i t i o n
complex
4.
space
X.
of n o r m a l
Theorem
complex
such that
Then any h o l o m o r p h i c
to
Proposition
map
on
b.
Proposition
V'
X
the set of p o i n t s w h e r e
the c o h e r e n t
for
is c o h e r e n t
generate of
a
on
V,
dense.
Let
~a
~a
in w h i c h and the V'
be the
h0 are h o l o m o r p h i c on i m h : V' ~ V x C be the
. . h (x)). The image h(V') is #'#m set Y' ( V' • Cm . We c l a i m that the in
this,
V • Cm we m a y
is an a n a l y t i c suppose
that
V
subset
of
is i r r e d u c i b l e .
119
Let
x Cm
Yi r V'
~x,
be
the
z I .... ,zi_ I, h i ( x ) ,
z I,. ~.,9 i .... ,Zm
are
analytic
z i + I .... ,Zm)
arbitrary,
Y! in V x Cm . C l e a r l y l any a ~ Y i. Further, the
set
dim
Hence,
by Chapter
Y' l
in
V
Z. ia
x Cm
V • Cm
dim
=
IV,
in
V x C
closure
of
Y'
Y
Proposition
a;
the
hence
is a c o m p a c t is a p r o p e r analytic also
V
hi~ if
map.
V
set of
V,
same
and
Z. ) Y.. l l
4,
the closure Yi m Z = i~=I Y'~" T h e n
of
• C m)
the
= Y'.
is a n a l y t i c
Z
Hence
(by C h a p t e r
of
Cm
{x}
enough, Hence,
if
equation
and that
with
where
K
the projection ~ : Y ~ V -I ~ (x) is a c o m p a c t
x~V,
• C m,
Y r V x K,
so is f i n i t e . every
hi
coefficients
(One m a y
satisfies
a
holomorphic
on
e n o u g h . ) F u r t h e r , if S is t h e s i n g u l a r -I A = ~ (S), then V' = V - S and
Since
h
is h o l o m o r p h i c
isomorphism.
Further,
on
V',
to c o m p l e t e
the p r o o f
only
to p r o v e
that
is s u f f i c i e n t l y
of
Since
every
fibre
hoods
saturated
if
a,
then
of
~
with
has
respect
W ~
-I
(W)
is a n o r m a l
~,
: Y' ~ V' Y - A
of T h e o r e m
a fundamental to
~
by definition,
Y. Thus,
this
is n o r m a l
at a n y p o i n t
Yo
with
is
4, we
small complex
system follows
space.
of n e i g h b o u r at o n e
3 and Y
IV,
in a n e i g h b o u r h o o d
in
Theorem
for
property
a~Z)
are b o u n d e d
is s m a l l
of
and
neighbourhood
(3.1)
they
of
is s m a l l
is an a n a l y t i c
have
a,
the c l o s u r e - i a defined by
for
Zn(V'
this by observing
Y - A = Y'.
dense
- I
x C TM
Further,
polynomial
if
V
V
subset
subset
prove
monic
in
the
Let
and
and
4') .
Since of
has
a
xeV',
= d i m m V • Cm
Proposition
n
Y.l
Z i cV • C
is a n a l y t i c .
is a n a l y t i c
where
and
d i m ~Y.
Zi = {(x'z) I gi (x) z'l = f'l(x) } (that
set
~ ( y o ) = a.
from
120
Proof of
(3.1).
Let
yo(~
be anal y t i c
h(x)
exists
X~VvnV' We cla i m that
z
the function
(~)
I =
~ z
g = i
an e l e m e n t
Ik z l (v)
=
'"
on of
.. ' z(V)) m
'
it is clear
that
if
V~,
N / v.
= 0
on
he nc e a' we have 0 =
(a, z (v))
of
a
projects
into
an ele m e n t of
~a"
this m e a n s
that
(y) = x
near
This proves
(3.1)
Y
near
v~, by P r o p o s i t i o n
a, = 0
on
uihi,
X
so that it T h e o r e m
be any c o m p l e x
sheaf of qerms of w e a k l y h o l o m o r p h i c
Proof. seen that
Let ~X
sheaf of
(Y, ~
D
on the set of -I ~o~ is
a;
if we set
VN,
N ~ v,
ui~a.
above or directly).
~
d ef in es
But c l e a r l y
is normal
at
Yo"
4.
and
fu nc ti on s
~X on
the X.
Then
OX-mOdules.
either
Chapter
Y
space,
be a n o r m a l i z a t i o n
= ~* (~Y)
~x-coherent.
2,
and is a n e i g h b o u r h o o d
near
Yo'
and w i t h
Let
is a c o h e r e n t
If
small n e i g h b o u r h o o d
V'nV v
~ =
u i ~ a.
g
ui(~(y) ). zi, if y = (x,z I ..... Zm), i=l (u) Yo = (a, z ) on Y. Hence ~ is
on
4'
Hence m
Then
~ =
holomorphic
is
uihi,
V . Let n o w e be h o l o m o r p h i c u points of D n Y and bounded; then
h o l o m o r p h i c and b o u n d e d on -I = ~o~ on V ' n V v near
consider
v / ~.
on
regular
OX
V v,
Further,
Vv
v
(x,z (v))~Y.
In fact,
g =
that any s u f f i c i e n t l y
of
Hence
ui(a) z(U) i '
z (~) ~ z (u)
V
Proposition 2,
v.
for each
(v)
u i(a) z i(N) '
w h i c h proves
Theorem
U --v,a V be the components, and
of
z (u) = lim
z(~)
(a)) ~ a
V into i r r e d u c i b l e --a By sets r e p r e s e n t i n g V * --D,a
decomposition
defines
-I
of
X.
It is e a s i l y
from the proof of
IV, T h e o r e m
7 implies
(3.1) that
given ~X
i2i
Theorem of
5.
x~X
Let
at which
subset of Proof. %/~x
X
~ O,
Chapter
X
is not normal
the set
i.e.
IV,
Proposition
5. This
at any
dense
than
dues
for which
Since
~/O
is a
set by
analytic
Let
seen.
u
Since
C l e a r l y we have an inclusion ~
~
By Chapter
on
U,
U
on ~
5
U).
of any
S
is a
be the sheaf
which vanish
We claim
III,
is a universal
= O};
is coherent,
~ c~. (on
U
U
which
and let
functions
3 = Hom ~ (~,~).
5.
S = {x~Ulu(x)
set in
of
for the construction
is a n e i g h b o u r h o o d
function
x~U.
injection
x(X
to Grauert-Remmert,
sufficient
5, there
of germs of holomorphic
a natural
analytic
is an analytic
proof of Theorem
to Theorem
denominator
N
as we have
and a holomorphic
and let
(closed)
N
6.
is more
The Grauert-Remmert
nowhere
is a
is the set of
a proof,
of the normalization,
aaX
N
sheaf of ~ - m o d u l e s ,
Corollary
Then the set
N = {x~X I (~/~)x ~ O}.
We end by giving Theorem
space.
X. Clearly,
coherent
be a g o m p l e x
on
so is ~
that there
In fact,
~S, . is
let
%
fa~x be a non-zero divisor in ~s Hom~) (Jx' J• and let x is ~ - l i n e a r , we have a(g) = wg and w = ~ Since c~ f x lies in the complete ring of quotients for g~J . Clearly w x is a finite ~ - m o d u l e (containing since x x N x is integral over ~ x ' so that, b y non-zero divisors) , w Proposition
i,
We assert, of
X,
w s x. that
Thus, if
N
we have
satz
If there
'
~. points
we have
xeNnU;
Let
~c~=
is the set of non-normal
N~U = { ~ U I Ox
Proof
inclusions
t en then is
~x ~ % "
k >~I
%
~ F~}-
= Now,
such that
a fort or by the Hilbert ~k ~ x
r Ux~x c % x
%
= ;x
Nullstellen(since
u
122
is a universal denominator at that
~-I
~x ~ %
Then, we~x as yeU - S
and and
bounded near that w ~ ~x"
x.
of
~x
%
~ %"
since
x).
~x ~ % "
Further, we may suppose Let
w~ x "r now if ~e~x, y ~ x, sLnce ~(x) = O since Hence , if @~x'
Thus the map ~ f ~ w f
into tself,
and, since
W~x,
Y
-i % ,
w ~ x-
then (w~)(y) ~ O and w is w~e~x, it follows
is an ~ x -linear map a~%,
so that
Thus
= {x u r (/e)x SLnce
w~
o}.
is coherent, this set is analytic by Chapter IV,
Proposition 6. Theorem 5 is proved.
123
CHAPTER
VII.
- HOLOMORPHIC
In t h i s R.
Remmert
analytic of
and
The
K.
set, a n d
Remmert
space
chapter,
28,
under
last
shall
Stein then
30
32
use
prove
on
first
the
asserts
holomorphic
sections
OF COMPLEX
contain
a theorem
a proof
that map
some
SPACES
singularities
it to g i v e
which
a proper
two
we
MAPPINGS
the
of
of
of
an
the
image
theorem
of
a complex
in an a n a l y t i c
applications
of
set. these
theorems.
I.
The
theorem
All
complex
countable
Theorem
~ p
the
is
dim
Z
and
Y
that
for
aEA.
dim
Y
any A
the set
Y.
the
analytic
subset
of dimension
local
analytic
Z
be
is
sufficient
is
analytic
Y,IX'
the
set
to b e
Again,
if
with
~
Let of
A =
~ - Y
is t h e u n i o n m).
of
It
dim
an
analytic X,
X'
of
of
the
suffices
a,
Am , that
where
irreducible
Y,
cl~ure
that of
= m
the
of
that
analytic Y C
is
is
n
of
an for
components
that
theorem
suppose
an
Am
dim b Am
that
induction
we may
subspace
to p r o v e
the
(by o u r
by
X.
with
that
can,
the
points
suppose
affine
~ n~m~p such
we
in
suppose
Z c D
to p r o v e
Y = @.
that
singular
that
while
set
we may
subsets
in
Suppose
that
moreover,
of
an a n a l y t i c
Y ~ p - I,
means
on
a~Y,
an
Y
X - Y.
is n o n - s i n g u l a r ,
a neighbourhood
i < p.
(Am
Suppose,
all
of
is
~c C n.
X'
intersection
dimension
b~A m
of
i__nn X is
It
is n o n - s i n g u l a r .
A
and
~ -I
theorem
for
Since
automorphism
of
Let
in
hypothesis).
subset
such
proved
A
assumed
a comple x space
an a n a l y t i c
= ~ - Z.
of
be
p ~O
an o p e n
< dim
are
inteqer
Since
X'
A,~X'
A
closure
is a l r e a d y
considered
~.
X,
is an
Proof. X
spaces
X
of
dim a A Then
at
Remmert-Stein
i.
subset there
of
every of
A
closure
A
m
124
of
A
is a n a l y t i c .
m
(l.i) with
If
~r C
an a f f i n e
A r~ - Y beA, a
subspace
for a n y
such that
AnU
To p r o v e
II,
is o p e n ,
and
of
is an a n a l y t i c
then,
that
n
Thus we have
C n,
there
m
{x~
(ii)
For
a
point
Proof.
Let
constant
on a n y
=
... = i
(countable)
Let
{x~}
linear
of the 11
be
form,
positive
be
set
for a n y U
as
irreducible
a countable independent
follows.
We may
clearly dim
repeat
dim
~< m a x
dense of
suppose forms
this
C
n A
of
X
and
is n o t
component = o}
~ max
= 12(x)
countable
many
=
values
for w h i c h
12
a
constant
on a n y
of a n y of t h e
and
sets
for w h i c h
(0, d i m Y - 2) .
= ll(Xu) , 1 2 ( x ) = 12(xv) } ,< m -
change
e(z)
= Im(Xv) }"
is n o t
and
= O}
m
~ m - 2.
We have
of c o o r d i n a t e s ,
IZm+ll
only
to
times. we may
suppose that n in C , we have
z are the coordinate functions 'n ..... Im = Zm. Now, B = AuY is c l o s e d
function
is an
(0, d i m Y - 1)
z I .... z_l
xu
which
of
subset
{x~AIll(x)
construction
By a l i n e a r
is c o u n t a b l e .
{xu} cA,
II, w h i c h
{xs
{ x ~ A l l l(x)
set
f o r m on
irreducible
= l l ( x u) },
= O}n(AuY)
component
d i m Y n l ~ i(O) n12 I(0) Then,
(x)
m
of
{ x ~ A l l i ( x ) = li(xu),...,im(X)
a linear
dimensional
{x~AIli(x)
the
> 1
set.
dense
dim Ynlll(O)
=
of
and
is a n e i g h b o u r h o o d
we proceed
I ll(x)
isolated
11
l,
dim b A = m
is an a n a l y t i c
(I.i),
intersection
of d i m e n s i o n
set w i t h
aeY,
the
following.
There exist linearly independent linear n C w i t h the f o l l o w i n g p r o p e r t i e s : on
(i)
if
Y r
the
a = O.
. . . ,i
and
to p r o v e
2
on the
+
... +
set
_~zI
IZnl .
2 o
can .
in
Q.
take
Since only
= Zm = O } n B ,
2
i25
there
is
6
> 0
(arbitrarily
small)
such
that
B n K = @, where
K
is the
set
Since
K
is c o m p a c t
{z I = and
... = Zm = O,
B
closed,
I Z m + l l 2 + . . . + I Z n ~2 = 6}.
there
is
2
{lZll
,< ~ . . . .
Let
U
be
the
and
U"
= {z"r
BU = B n U
set
= ~(Y)
An(U
Corollary of
is the
- Z)
= closure
b
b~A. such
means,
in
Then, x
x A,
and
A
We g i v e o f the
The
theorem
complex exist
U'
- Z)
is an
of
theorem
Let
Hence,
by
the c l o s u r e
A,
in
U.
there
are
Hence
so t h a t
Toprove
points -1 x x(x).
of
3 in C h a p t e r
is c o m p l e t e l y
a
is p r o p e r ;
III
in
x, h e n c e
~(y) ~Y'
U),
X
U.
point
3 to T h e o r e m
< ~},
is p r o p e r .
Z ) Y.
is d e n s e
isolated
of
with
~ i < m.
is a n a l y t i c
immediately
above
of
by construction,
with
the
intersection
in
that
if
subsets B U,
is a n a l y t i c
- Z)
Then,
- Y'
such
< ~ .... , I Z m
to
since
> O
= 6}nB = @.
restricted
10 in C h a p t e r
in a n e i g h b o u r h o o d on
< 6}.
of d i m e s i o n
An(U
by Corollary
is o p e n near
An(U
that
{z'~cmllzl I
for o t h e r
= B n ( U - Z)
U
that
of
=
2
+...+IZnl
x : B n ( U - Z) ~ U'
- Z)
in
We c l a i m
let
Cm
3 to P r o p o s i t i o n
An(U
U'
notation
~ : U ~ U',
l i n e a r s u b s p a c e of -1 Z = ~ (Y'). Then moreover,
x U",
similar
the p r o j e c t i o n Y'
U'
,< c, IZm+ll
I IZm+ll 2 + . . . + i Z n l 2
(with
Further
,IZm~
e
there An(U
A =
this, x~A
near
But t h i s
III,
that
xl A
are
points
y
- Z)
is d e n s e
proved.
an a p p l i c a t i o n ,
due
to
H. C a r t a n ,
theorem.
of C h o w .
projective
finitely
Pl ..... P k ~ C Z o
many
Any
space
analytic
~N
such
A
is an a l g e b r a i c
homoqeneous
..... ZN
subset
polynomials
that
of t h e set,
i.e.
there
126
A =
Proof. whose in
p1(z
Let
X
images
C N+I -
we h a v e Hence,
,
belong
in
X
of
C
N+I
- {0}
is an a n a l y t i c
for a n y
z~X
X
1, the c l o s u r e
f '
D
A;
= o}.
and
A ~ O,
at a n y p o i n t
X = Y
of
set
X
is
in
C
A~C, > O. N+I
set.
fl ....
cyl inder
to
the d i m e n s i o n
by Theorem
Let
)
and since,
Az~X,
is an a n a l y t i c
=
be the set of p o i n t s
in ~ N 0
)
be h o l o m o r p h i c
functions
in a p o l y -
m
about
0
s u c h that
YnD = {z~D I fl(z)
=
=
f
m
(z)
=
o}.
Let CO
f'l(Z) =
Pvi(z) v=o
be the e x p a n s i o n nomials
in
degree
v.
D;
We a s s e r t
of P
vi
that
Z = {zEcN+I IPvi(z) Clearly
that
a
is
here
= 0
Let
for all
kz o ~YnD ;
hence
CO
polynomial
v ) O,
polyof
Y
i = I ..... m}. and
Z
are
b y a c o m p l e x number,
that
Zo~YnD.
of homogeneous
a homogeneous
and since b o t h
To p r o v e
ZnD ) YnD.
series
where
by multiplication
Z r Y.
Z ) Y,
Then,
for
it s u f f i c e s A~C,
lAi
left
this
implies
to p r o v e ~ 1,
that
we h a v e
CO
Pvi(AZo)
=
V=O
Clearly
in
Y = Z,
ZnD ( YnD,
invariant
f.
this
Since
AVp
(zo)
= 0
for
Pvi(Zo)
= 0
for e a c h
D =0
implies
that
Cz o .... ,ZN
many homogeneous
vi
is N o e t h e r i a n ,
polynomials
i = 1 .... m Ikl " '
v, i = l t . . . # m .
there e x i s t
Pl ..... P k ~ C Z o ' ' ' ' ' Z N
,< 1.
finitely such that
127
C n+i I P i (z) = O for all
Y = This
u, i} = {zecN+llpl(z)
= . . . = pk(z)
= O}.
is just the theorem. nk
Corollary.
Any analytic
algebraic.
This
follows
and the r e m a r k
that ~ n
N =
- i
(n+l) (m+l)
subset of
at o n c e xpm
~Ivnl• ... •
from the t h e o r e m
becomes
by means
is of C h o w
a submanifold
of
~N,
of the Segre m a p p i n g
(xO ..... x n) x (Yo .... 'Ym ) ~ (XoY O ..... X o Y m , X l Y O ..... x l Y m ..... X n Y O ..... X n Y m) For a q u i t e d i f f e r e n t deeper
s t u d y of a n a l y t i c
see Serre
24,
holomorphic Let
X
= k
theorem
25
be
map
a holomorphic
jacobian matrix
X
If
of of
X
= dimxX
f 9 X - + Cn
points
of
into a c o m p l e x
1.
is R e m m e r t ' s
of a n a l y t i c
sets u n d e r
X, f
ex(f) =
max xEX'
complex
x~X.
Let
space Y.
space,
proper
and
map, p
is a c o m p l e x m a n i f o l d .
be a
We set
- d i m x f-lf(x).
let
X'
complex
be a d e n s e
the m a x i m u m
at any point
Qx(f).
i.e.
f : X ~ Y
is a pure d i m e n s i o n a l
By the s e m i c o n t i n u i t y max x~X
section
a pure d i m e n s i o n a l
Proposition
have
varieties,
mappings.
Qx(f)
Proof.
of this
on the images
is i n d e p e n d e n t
holomorphic
regular
of a l g e b r a i c
and a m u c h
mappinqs
The principal
dim x x
properties
theorem,
35.
2. H o l o m o r p h i c
theorem
p r o o f of this
of
X'
subset
Then
p = max xs IV, we
we m a y s u p p o s e
The c o r o l l a r y
of
of the r a n k of the
t h e o r e m of C h a p t e r Hence
space
follows
then
that from
Qx(f) .
128
the r a n k t h e o r e m
(Chapter
I) .
We call the i n t e g e r If
m a x Qx(f) = Q(f) the rank of f. x~X is not pure d i m e n s i o n a l , Q(f) = m a x Q ( f l X ), where
X
X =~Jx~
is the d e c o m p o s i t i o n
ponents.
(When
X
of
X
into i r r e d u c i b l e
is p u r e d i m e n s i o n a l ,
com-
the two d e f i n i t i o n s
agree.)
Proposition
2.
the c o m p l e x
space ont____oan a n a l y t i c
in
Cn
Proof.
Let
Then
f : X ~ Y
Q(f)
such that the j a c o b i a n
Now,
let
Hence, Yo
isomorphism open
set
Qx(f)
m
matrix
We Cm,
Thus,
m a p of g =
X
is c o n s t a n t .
is a s u b s p a c e
gl
factor }, of
X
U
of
and Yo
for any
and
irreducible
Then,
c # cv
if
c~
for any
of d i m e n s i o n
~ Q(f). h
an a n a l y t i c Y
onto
that
if
< d i m X,
an
g : X ~ W m
We C ,
then
{c }, v = 1,... components
{projection u,
Xo~X
xEf-l(u),
set
let
= dim Y.
point
on
to prove
set
has r a n k
dim Y
o n t o an o p e n
on those
C n)
xo
Y,
Then,
(gl,...,gm) ,
on w h i c h
first
into
at
of
we h a v e
of
on the
point
m = dim Y.
be the v a l u e s gl
f
m a p of
in an open
is a r e g u l a r
of
of a n e i g h b o u r h o o d
Let
Y
b y the r a n k theorem,
= Wx(h o f).
~ ~(g).
I, there
be a r e g u l a r
is a h o l o m o r p h i c
set
(as a m a p p i n g
B9 P r o p o s i t i o n
p = Q(f).
be a h o l o m o r p h i c
then
of
X
of
W -I Xl = gl (c)
and
,. ) : X 1 ~ Cm - I maps X 1 o n t o an o p e n s u b s e t g' = (g~ "''gm of Cm- . It f o l l o w s (by induction) that t h e r e is a r e g u l a r point
xl~X I
m - l ~ dim(Xl) where
gl
of
XI
for w h i c h
- d i m x l g , - I g , (x)~ d i m x l X - i is the r e s t r i c t i o n
component
Z
m
and the p r o p o s i t i o n
~ Q(g),
of
X
with
of
dimxl
g
- d i m x l g - l g ( x i)
~ Qxi(gi~l,
to an i r r e d u c i b l e
Z = dimxl X(xI~Z);
is proved.
hence
129
Theorem
2.
Let
X
_a p r o p e r
holomorphic
Proof.
If
is a g a i n
proper.
given in
point
C N)
finite
of
family
proper,
of
S
suppose
be
induction,
the
A = f-lf(S)
is
an
is a n a l y t i c ,
if
A
= X,
Then
dim
that dim
A ~ X. f(A)
such
= dim
that
Since as
the
X
our
the
fibre
of
of
flA
f
: A ~
of
f
by
an
Proposition
in
a set
in
CN
(U)
has
(X',
Y')
family
{Xa}
of
It
Hence,
the
suffices
subset
Since
the
is a g a i n
and set
let in
n
p = Q(f). ~.
Since
Hence
f(A)
is p r o v e d .
be
such
Cn
Y.
induction,
analytic.
X.
a
C
in
pairs
By P r o p o s i t i o n
: X ~
~ -1
finite.
X,
theorem
Let
Y
irreducible.
in
< n.
and
is
set
of
set
finite.
analytic
set
= Qa(flA)
Q(flA)
the
: X ~
an o p e n
Y,
all
a closed
is an
A
for
is
in
suppose,
locally
set
analytic
of
fibre
that
T = Q(flA).
Q(fIA)
definition
to
in
We
f(X~)
map
set
an o p e n
locally
each
= T
X.
is
singular
f(S)
is
f
: f-1 (U) ~ U
f
a neighbourhood
compact
X
and
analytic
then
,
Further,
is
that
is an
Y
proved
of
spaces
analytic Y
< n.
a proper
we m a y
Let
X'
{ f ( X a) }
to p r o v e
restriction
that
already
dim
sets
therefore
in
n = dim
components
of
set
a closed
Let is
complex f(X)
is r e l a t i v e l y
with
irreducible
By
U
be
(imbedding
suppose
theorem
spaces
Then
as
dimension.
the
map.
Hence
yo~Y
if
Y
is an o p e n
we may
Moreover,
that
U
and
= f(S)
Suppose
then
2, we h a v e
a regular
point
a
by
exists
of
A
the
i. through
a
is the
a neighbourhood
of
same
A,
we
have Qa(fIA)
= d i m a A - d i m a f-lf(a)
Moreover, because an
since
of T h e o r e m
analytic
Since
f
X - A
suffices
subset is
to p r o v e
is p r o p e r , i, of
it
we
suffices
~ - T
irreducible, that
< d i m a X - d i m a f-lf(a)
f(X-A)
have
f(X)
to p r o v e
is
of
- A) .
that
f(X
p
at e v e r y
of d i m e n s i o n because
= f(X
=Qa(f) ~Q(f)=p.
Proposition
analytic
in
Thus,
- A)
2,
~ - T.
is
point. it
130
Let of
the
B r X - A
jacobian
analytic f
~-
- T.
If
Hence f(X
dimension and
2.
Theorem
3.
Proof.
the
is c o m p a c t
for
sets
is
then
closed,
is
x o.
Let
whose
X - Ku Hence,
{x v}
and
result
I,
of
an
in
is p r o v e d . we
I)
have
= Q(flX-A-C)
to p r o v e
that
Q - T - f(C)
rank
rank
is
an
analytic
case
f
Suppose
is
X,
of
Yo"
is
that
is
of
be
f
Kv
: X-~
Y
Y.
then
that
is
it
f - l f ( x o)
false,
and of
a fundamental is n o t dense
is n o t with
for w h i c h Theorem
f
irreducible.
is n o w h e r e it
proves
a sequence
{Vv}
is
is an
constant,
this
Since
xvcX-
Hence
in
to p r o v e
(since
set
set
X
and
~
This
is n o t
sufficient
this
spaces,
when
if
p,
theorem.
complex
is a c l o s e d
a contradiction.
is
set
above,
be
each
there
Since
as
constant
f - l f ( x o)
for
< n.
analytic
{Kv} , K v c ~ v + I
union
B
rank
constant, in
X
compact,
by
f ( x v) ~V v -
f({xu})
3 is p r o v e d
{yo}.
is n o t when
X
irreducible. The
and
it
Clearly
Y
that
irreducible,
assumption).
is
assert
each
the
the
: X - A - C -~ ~ - T - f(C)
the
of n e i g h b o u r h o o d s
and meets
But
f(X)
fact,
is an
subset
the
and
at w h i c h
(by P r o p o s i t i o n
has
first
= f(Xo)"
X
< p.
of T h e o r e m
f
of
X
we
In
and
= Q(f)
j acobian
Then
is p r o p e r .
compact
is
(X - A),
since
Let
case,
Yo
of p o i n t s
of d i m e n s i o n
because
Consider
In t h i s
system
- A)
But
a close d map.
let
< dim
consequence
Theorem
f
is an a n a l y t i c
p.
immediate
of
X - A
suffices,
- A - C)
proper
C
< Q(flX
it
set
is p r o p e r , f(B) -i C = f f(B) = X - A, dim
f(C)
of
the
T
Otherwise, dim
matrix
subset
: X ~
be
the
general following
case
of
lemma.
Theorem
3 follows
at o n c e
from
this
= p.
131
Lemma
I.
and
Let
X,
f : X ~ Y
yo~Y,
there
subset
K
Y
be
locally
a continuous
compact
closed
is a n e i g h b o u r h o o d
of
X
such
map.
U
of
Hausdorff
spaces
Then,
any
Yo
for
and
a compact
that
f(K) nU = f(X) nU.
Proof.
If t h i s
is false,
of n e i g h b o u r h o o d s subsets such
of
X
that
in
X
whose
image
Let
any complex
space
point such
x has
a~X that
Proof.
components (for all
x
be
and
point
at
assert
X , and --a near a on
f
of
Theorem
set
3.
complex
a holomorphic
system
- Kv,
a contradiction.
it,
independent
xv~X
is a c l o s e d
a pure dimensional
is a n a l y t i c
is a r e g u l a r
closed,
system
of c o m p a c t
is a p o i n t
{xv}
f : X ~ Y is
We
there
and with
a fundamental
a~X.
of
X
a sequence
then
is n o t
lemma,
is a f u n d a m e n t a l
{Kv}
But
Y
f-lf(x)
f(U) Let
and
{yo}.
in
3.
k = dim
{Uv}
K u c K u + i ~ X,
the
Proposition
x
with
proves
that
Yoo
f ( x v ) ~ U u-
This
Y
of
and
of
x~X.
space,
map
such
Then
any
of n e i g h b o u r h o o d s
U
f(a) .
that
if
X
are
the
irreducible
f - l f (x) = k t h e na dim = fl~ , --v, v v,a x v v Xu). In fact, dim f~Ifv(x) = k if X
on
X
.
Hence,
by
Proposition
i,
V
f~Ifv(x)v
dim
) k
for
all
x ~ X v.
In the
other
hand,
since
X
f-lf
(x)
= X
V
it s u f f i c e s Let S
n ( f-lf(x) ) V
U
be
to p r o v e
subset
there
S
exists;
set in
is a p l a n e
contains
a
as
Cn
,
H
U and
t
X --a
~ k.
Hence
is i r r e d u c i b l e .
if
X
by the
point,
of
a
and
that
dim a S + k = dima
of d i m e n s i o n
isolated
when
neighbourhood
such
in fact, then
3
small)
of
= {a}
Snf-lf(a)
an o p e n
Proposition
a (sufficiently
an a n a l y t i c
(Such an
d i m x f -vl f v (x)
we have '
X.
is an a n a l y t i c
local
set
representation
n - k
and we may
such take
that
in theorem
Hnf-lf(a)
S = HnX.)
r
132
Let
flS = g.
Hence,
Then
by Chapter
a
is an
IV,
Proposition
analytic
set
suitably
chosen.
and
S W' = f - l f (S) nW;
let
of
W,
on
g(S).
and
dim thus,
in an o p e n
f(S W) Hence,
S~ - k for
4.
of d i m e n s i o n dimension
m,
Proof.
Suppose
a~X, U
Y
dim
such
d i m f ( a ) f(U)
that
= p,
Suppose
in
to T h e o r e m Let
h
finite
=
by
such
the
of
p = I,
for
f
b
the
unions
of
when
the result
and
subset of
f(a) so t h a t
S = dima
f(U)
X;
= f(S)
complex
complex
all
of
is o p e n
if
x~X. for all
arbitrarily at
space
space
f : X ~ Y
= m - p
are
f(a) .
locally
is o p e n .
and
x~X.
small
open
By P r o p o s i ~ o n
irreducible,
of C h a p t e r
then fibres
Y = ~
g
the
fibres
of
f.
are
where
g
in
; if
b~Y, : V ~
there. ~c C p
Corollary
3
finite. of
Hence,
at o n c e
induction
X i = { x ~ f - l ( V ) l h i ( x ) = c},
any
III,
of
is o p e n
follows by
For
an o p e n m a p
fibres
~;
13. We p r o c e e d
W c U,
f(a).
the r e s u l t
theorem
map
being
that
that
a,
>~ d i m a S,
dimensional
is a n a l y t i c Y
X
irreducible
3, t h e r e
g o f : f-i (V) ~
Proposition and
3
(disjoint)
to p r o v e If
C p)
V
is
the p r o p o s i t i o n .
a pure
of
conversely
U
of
>~ k + d i m a
on
d i m x f-lf(x)
so that,
is a n e i g h b o u r h o o d open
a
= m - p
is a n e i g h b o u r h o o d
if
Q(fls.~) S~
a holomorphic
f(U)
is an
S' = d i m X, so t h a t W a (since X is i r r e d u c i b l e ) ' --a
of
be
by Proposition
= f(S)
g(a).
dim
a locally
that
-i
is an a n a l y t i c
2,
dim
proves
f-lf(x)
g
is a n e i g h b o u r h o o d
Hence
X
g(S)
of
Y(f(a) s
S W'
i.e.
This
Then,
if
(~
then
we h a v e
Let
and only
in
by Proposition
V.
p.
5,
point
any neighbourhood
is a n e i g h b o u r h o o d
Proposition
f(U)
be
= d i m a X.
in
sets
W
V
= g(S) nf(W)
W,
is a n a l y t i c
If
set
>~ d i m a S '
all
d i m a f-lf(S) f-lf(S)
Let
!
isolated
h
are
it s u f f i c e s
C p. from Chapter h =
III,
(h i ..... hp)
c = hl(Xo) , X o ~ f - l ( V )
I,
133
is m
a constant, - i;
Xi
and
onto
then
h (I)
an o p e n
X1
=
(h2,
set
dimxoh(1)-I but
clearly
h (I)-I
is a r b i t r a r y ,
3. E.E.
Levi's
We
give
in w h i c h of
our
4.
analytic
~(I{
subset
an
Proof.
Let
Chapter any is
IV,
as
A
X
a complex
Then
X - Y
and
hence
X ~ Y, Then,
is
by Chapter
of d i m e n s i o n
- Stein
of
in
we m a y
Bi
B,~U
g
set
of
has
(i = 1 ..... p)
A
Xo~f-l(V)
its
Proposition each
of
in
X.
many = O
on
Let is
i) , the Hence, of
a
of
have on
X - Y
X
X.
By
- 2
for
so t h a t X
X - Y
connected.
locally
irreducible
X = n.
an
analytic
closure for and
a zero
x~BnU.
on
dim
Hence,
irreducible for
an
X.
--a
is
points.
is n o w h e r e
finitely g(x)
U
we
Y
of
be m e r o m o r p h i c
P
its
case
function
X ~
8,
(Theorem
(which
a~X,
suppose
poles.
a proof
space and
~ dim
we may
Let
of
in the
Y ) A,
irreducible).
and
- p;
applied,
points
--a
(since
IV,
only
have
way
Levi
function
connected
analytic
be
any
that
Further,
set
U
for
dim
suppose
a neighbourhood in
can
singular
2 ' we have
the
is
choose
function that
X
the
any meromorphic
the
theorem
of
we
= m
since
of
E.E.
that
be
at
(p - I)
complex
a meromorphic
n - i
Remmert P
Then
also
Pe
to
to
globally
and
theorem
due
such
manifold.
map
induction,
-
example
a normal
we m a y
an o p e n
theorem
another
be
Theorem Hence
(m - i)
of d i m e n s i o n
is p r o v e d .
- Stein
X
extension
By
= h-lh(Xo ) ;
as
of
C p-I.
h (I) (Xo)
d i m _Ya 4 d i m _Xa - 2. has
in
=
theorem,
Let
is c l e a r l y
h(1) (Xo)
Remmert
following n d o m a i n s in C
dimensional,
.,hp)
contituation
the
Theorem
pure
proposition
next,
the
is
any
by
set
the
B = P point
aeY,
a holomorphic divisor)
such
components Let
ais
i - Y
134
(i = I, .... p). an integer
By the H i l b e r t
k
such
that
Nullstellensatz,
gk
is h o l o m o r p h i c
there
is
at the points
a i (i = 1 .... ,p). We assert in
U - Y.
Pf
dim Pf
IV,
nowhere ~
which
to
This
Remmert
- Stein
case
4. A n a l y t i c Let
a~X
X where
Chapter
theorem.
clearly
P.
in
that
Mk C
dimension
this
C n.
k-
(see e.g.
Let
Proposition
to
5.
is an analytic
f
then
P nU.
Hence
Pf = @
has
(by
a holomorphic in
U
(and
extension
without
using
the g e n e r a l
a direct
proof
is a n o w h e r e
case in the
functions of
f.
of
dense pl
on the c o m p l e x i.e.
the
We have analytic
b y itself
F~
• M k be
the
The closure
k times. set of
set
is a c o m p l e x
of
Ff = -CT,Ff of
X • M k.
Ff'
in
subset.
is an analytic
Ff' c (x - P)
set of
seen,
space
X - P.
subset
the
20).
P = i~l Pi"
{(x,fI(x) .... ,fk(x)) I x,X - P}. isomorphic
of
so
dependence
set of poles
Let
4,
theorem
and the c o m p l e m e n t I.
ai~P f,
unless
However,
be the p r o d u c t cMk
U - Y
theorem.
be m e r o m o r p h i c
P
in
is a m e r o m o r p h i c
the
long
the
f
is h o l o m o r p h i c
One r e d u c e s
(fi) a ~ a .
IV,
Let
F/g k
and a l g e b r a i c
and
Further
component
gA
to prove
fl ..... fk
of
Proposition
Since
is rather
of poles
8).
proves
to that of a d o m a i n
is h o l o m o r p h i c
(B i - Y).
a zero divisor), U.
Pf
k f = g ~
is not p o s s i b l e
VI,
U.
It is p o s s i b l e
space
set
no i r r e d u c i b l e
Proposition
F
to
latter
the
by C h a p t e r
extension
function
P nU = U
contains
Now,
of
in
~ n - 2,
Chapter
the
In fact,
is c o n t a i n e d that
that
in
X x
Then
135
The proof is almost proof of C h a p t e r
identical w i t h
VI, T h e o r e m
4
that given
(page 118)
in the
and is th er ef or e
omitted. If
~
denotes
of
X x ~
on
F~
composed with
Mk,
We say that the ma p of rank
< k
fl,...,fk
(Siegel,
and
fl,...,fk
Proof.
g iv en b y
Clearly
the nat u r a l
on
point of R(f)
< k).
We say that
if there exists
If
X
is a c om pa ct
are a n a l y t i c a l l y
R(flX - P) = R(~) By T h e o r e m
subset of
a
that
~ : Ff ~ M k
of d i m e n s i o n
Mk;
which vanishes
dependent.
2 and P r o p o s i t i o n
Mk
in particular,
complex
dependent meromorphic
< k,
algebraic
implies
by
X - P.
to the t h e o r e m of C h o w
by definition,
if
such that
the C o r o l l a r y
p(z I .... ,zk) ~ O
dependent
(which,
they are also a l q e b r a i c a l l y
is an analytic
into
has a jacobian
X - P
dependent
Thimm).
projection.
in
X - P
fl ..... fk
s i m p ly that
~ O
5.
X,
of the p r o j e c t i o n
are a n a l y t i c a l l y
p(z I ..... z k) ~ O
on
Ff
f.
are a l g e b r a i c a l l y
functions
~(Ff)
is
f l , . ~ . , fk
I, m e a n s
p(fl(x) ..... fk(x))
space,
~
at any regular
polynomial
to
then the na tu ra l m ap of
f : X - P ~ C
Proposition
Theorem
the r e s t r i c t i o n
(w 2),
there on
b ei ng
2,
< k.
~(Ff)
By
is
is a p o l y n o m i a l
~(Ff) nC k ) f(X - P). This,
fl .... 'fk
are a l g e b r a i c a l l y
dependent. Corollary
I.
dimension
If
n,
2.
meromorphic
If
algebraically
n + 1
complex
meromorphic
space of functions
on
X
dependent. X
functions
on a n o n - e m p t y
is a co mp ac t
then any
are a l g e b r a i c a l l y Corollary
X
is co mp ac t fl ..... fk
open subset of dependent.
X,
and i r r e d u c i b l e
and the
are a n a l y t i c a l l y then
fl,...,fk
dependent are
136
This follows
at once
The above proof
is due to Remmert
It can be proved, above,
from Theorem
b y methods
see Remmert
complex 29;
also
space 2,
to those used
functions
is an alqebraic 36,
37.
I.
29.
similar
that the field of m e r o m o r p h i c
(irreducible)
5 and Proposition
on a compact
function field;
137
BIBLIOGRAPHICAL
NOTES
The theory of analytic sets, e s p e c i a l l y of analytic n sets in C , is d e v e l o p e d in the books of M. Herv~ 19 and
S.S. A b h y a n k a r
H. Cartan,
1953/54
C. Houzel
12.
Rossi
14.
I I0
In less detail,
I.
in Herv~
19
in
theorem, proof to
II.
due to
sets over a r b i t r a r y
Theorem
J.P.
Serre.
theorem
I,
stated on page
i, which is one C. }{ouzel in
12
are
proof
Gunning
-
3,
is proved
differentiable
and
R. Remmert,
12;
he ascribes
remarks made
notes
apply,
theorem
- Remmert
w h i c h uses the n o r m a l i z a t i o n
theorem,
is due to
although the p r e s e n t a t i o n
the
for the the fourth, see
for
at the b e g i n n i n g
16.
24. of
to this chapter. Their proof,
is difficult.
The
L. Bungart - H. Rossi
is different.
17.
see M a l g r a n g e
above all,
Theorem 7 is due to Grauert
of the proof given here
24;
is unpublished,
and applications,
The general
in 25.
form of the p r e p a r a t i o n
ideas of M a l g r a n g e
functions
these b i b l i o g r a p h i c a l
e.g.
The third proof we have given
theorem uses
H. Grauert
III.
in
stated here w i t h o u t
For an analogue of the p r e p a r a t i o n
Chapter
(mostly a l g e b r a i c a l l y
of Houzel
as stated here,
is proved by
preparation
is
18.
The rank theorem, Chapter
That in A b h y a n k a r
(and/or A b h y a n k a r
F. Hartogs'
-
in algebraic geometry.
Most of the results
14).
is proved
work
in Gunning
although v e r y d i f f e r e n t
fields
The ideas in the treatment
are proved Rossi
in Herv6,
non-discrete
from G r o t h e n d i e e k ' s
18-21 by
it is treated
and treats analytic
c o m p l e t e l y valuated,
Chapter
Exposes
is based on the same ideas.
quite d i f f e r e n t
drawn
and 1960/61,
The treatment
in detail,
closed).
and in the seminar notes of
idea 7,
i38
Chapter
IV.
All known proofs of Oka's
are based on the ideas of Oka proof of the coherence
has a proof.
in
9,
Cartan's
26,
9,
although Oka
and Cartan h i m s e l f
that he understands
Oka's version,
from that of Cartan,
which
is given in
that Oka also
is not very d i f f e r e n t 27.
We have
in Grauert
- Remmert
a v e r y special case of a theorem of Grauert proofs
are, however,
given.
Another in
6.
v e r y different
proof,
16, 15.
and is These
from the one we have
quite different
This chapter
ii,
from all these,
is
follows c l o s e l y the papers of
Bruhat - Cartan
4,
5
and Bruhat - W h i t n e y
The unproved results concerning
in Cartan Whitney
II 6
(Proposition (Proposition
and functions, Malgrange
24.
17,
are given
Chapter VI.
The original
Cartan
where,
proof of
of Grauert - Remmert
27
33.
and K u h l m a n n
21.
see
2 given here
it is a direct g e n e r a l i z a t i o n
in algebraic geometry.
Oka
and
in the
is given
there is an error.
is unpublished,
The proof of T h e o r e m
are given
in Rossi
version of this proof is given I
properties
23.
however,
are due to A b h y a n k a r
sets
sets
in the book by
interesting metric
S. ~ o j a s i e w i c z
i0
2) and in Bruhat -
18).
the t r i a n g u l a b i l i t y of real analytic paper of
sets are
properties of real analytic
and applications Further
C-analytic
15, Example
16,
M a n y v e r y interesting
21;
followed
12.
Chapter V. Cartan
says,
presentation.
Theorem 7 is proved
given
3,
The first published
H. Cartan
to this theorem in
in a footnote
Theorem
of the ideal sheaf of an analytic
set, T h e o r e m 5, is that of refers
26.
theorem,
in
A complete
Other proofs The proof
17.
is due to Kuhlmann
of c o r r e s p o n d i n g
results
139
Theorem
2 can also be p r o v e d
a l r e a d y has T h e o r e m p r o o f of T h e o r e m Chapter also
VII.
3
(or T h e o r e m
if one
5). For a n o t h e r
algebraic
2, see A b h y a n k a r
The R e m m e r t
i0.
"geometrically"
The d e t a i l s
- Stein
are m o r e
general
form of the theorem.
We have g i v e n
in this
special
this
obtained
was
2 is due
to R e m m e r t
IV, P r o p o s i t i o n
formulation
suggested
by
The t h e o r e m H. K n e s e r
20)
been proved
given
28,
Rajwade.
due
to
E.E.
for d o m a i n s
in g e n e r a l
in
in the
be d e d u c e d
(in the case of d o m a i n s
of C h a p t e r
I, P r o p o s i t i o n
H. K n e s e r
20.
of m e r o m o r p h i c C.L.
Siegel
Theorem
36,
37
29;
who u s e d
functions. Grauert
See 3.
generalization functions
38.
see also
important
C n)
at the end,
from an a n a l o g u e functions;
see
on the
were
field
proved by
a very elementary method. is due
The p r o o f b y S i e g e l
has
led
to the t h e o r y of a u t o m o r p h i c
Andreotti
Borel has o b t a i n e d
2,
a very
of the e a r l y w o r k of S i e g e l
as an a p p l i c a t i o n
w o r k of Borel
31.
theorem
It can
The p r o o f g i v e n h e r e
applications
in p a r t i c u l a r A.
in
7).
it has
this reason.
5, and the t h e o r e m
stated
Theorem
seem to h a v e
although
12 for m e r o m o r p h i c
5 is due to T h i m m
to R e m m e r t to v e r y
functions
IV,
(see also
does not
a p r o o f for
cases of T h e o r e m
deduces
(just as we
22
literature,
We h a v e
proof
o f t e n used.
3 for R e m m e r t ' s
Levi
a more
simple
5 from C h a p t e r
Cn
with
Grauert
15
see
in the p r o o f
form m o s t
image
b e e n used.
Special
included
than
this
30;
in T h e o r e m
M.S.
32,
simultaneously
is the
t h e o r e m on the d i r e c t
Chapter
The
deal
case b e c a u s e
is in
complicated
given.
it from his
articles
theorem
we h a v e
Theorem
These
i.
of the m e t h o d s
is still u n p u b l i s h e d .
Andreotti
-
far-reaching on m o d u l a r of
3;
this
140
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Offsetdruck: Julius Beltz, Weinheim/Bergstr
Lecture Notes in Mathematics Bisher erschienen/Already published
Vol. 1: J. Wermer, Seminar i~ber Funktionen-AIgebren, IV, 30 Seiten. 1964. DM 3,80
Vol. 13: E. Thomas, Seminar on Fiber Spaces. VI, 45 pages. 1966. DM 4,80
Vol. 2: A. Borel, Cohomologie des espaces Iocalement compacts d'apr~s J. Leray. IV, 93 pages. 1964. DM 9,-
Vol. 14: H. Werner, Vorlesung fiber Approximationstheorie. IV, 184 Seiten und 10 Seiten Anhang 1966. DM 14,-
Vol. 3: J. F. Adams, Stable Homotopy Theory. 2nd. revised edition. IV, 78 pages. 1966. DM 7,80
Vol. 15: F. Oort, Commutative Group Schemes. Vl, 133 pages. 1966. DM 9,80
Vol. 4: M. Arkowitz and C. R. Curjel, Groups of Homotopy Classes. IV, 36 pages. 1964. DM 4,80
Vol. 16: J. Pfanzagl and W. Piedo, Compact Systems of Sets. IV, 48 pages. 1966. DM 5,80
Vol. 5: J.-P. Serre, Cohomologie Galoisienne. Troisi~me ~dition. VIII, 214 pages. 1965. DM 18,-
Vol. 17: C. MOiler, Spherical Harmonics. IV, 46 pages. 1966. DM 5,-
Vol. 6: H. Hermes, Eine Termlogik mit Auswahloperator. IV, 42 Seiten. 1965. DM 5,80
Vol. 18: H.-B. Brinkmann, Kategorien und Funktoren. Nach einer Vorlesung von D. Puppe. XII, 107 Seiten. 1966. DM 8,-
Vol. 7: Ph. Tondeur, Introduction to Lie Groups and Transformation Groups. VIII, 176 pages. 1965. DM 13,50 Vol. 8: G. Fichera, Linear Elliptic Differential Systems and Eigenvalue Problems. IV, 176 pages. 1965. DM 13.50
Vol. 19: G. Stolzenberg, Volumes, Limits and Extensions of Analytic Varieties. IV, 45 pages. 1966. DM 5,40 Vol. 20: R. Hartshorne, Residues and Duality. VIII, 423 pages. 1966. DM 20,-
Vol. 9: P. L. Iv~nescu, Pseudo-Boolean Programming and Applications. IV, 50 pages. 1965. DM 4,80
Vol. 21: Seminar on Complex Multiplication. By A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, J.-P. Serre. IV, 109 pages. 1966. DM 8,-
Vol. 10: H. LUneburg, Die Suzukigruppen und ihre Geometrien. VI, 111 Seiten. 1965. DM 8,-
Vol. 22: H. Bauer, Harmonische R~ume und ihre Potentialtheorie. IV, 175 Seiten. 1966. DM 14,-
Vol. 11: J.-P. Serre, AIg~bre Locale. Multiplicit~s. RL:,dig~ par P. Gabriel. Seconde ~dition. VIII, 192 pages. 1965. DM 12,-
Vol. 23: P. L. Iv~nescu and S. Rudeanu, Pseudo-Boolean Methods for Bivalent Programming. 120 pages. 1966. DM 10,-
VoL 12: A. Dold, Halbexakte Homotopiefunktoren. II, 157 Seiten. 1966. DM 12,-
Vol. 24: J. Lambek, Completion of Categories. IV, 72 pages. 1966. DM 6,80