Several Complex Variables - H. Grauert & K. Fritzsche

Graduate Texts in Mathematics
38
Editorial Board
F. W. Gehring
P. R. Halmos
Managing Editor
c. C. Moore
H. Grauert
K. Fritzsche
Several Complex
Variables
Springer-Verlag
New York Heidelberg Berlin
H. Grauert K. Fritzsche
Mathematischen Institut der Universitiit Mathematischen Institut der Universitat
Bunsenstrasse 3 - 5 Bunsenstrasse 3-5
34 Gottingen 34 Gottingen
Federal Republic of Germany Federal Republic of Germany
Editorial Board
P. R. Halmos F. W. Gehring C. C. Moore

Managing Editor University of Michigan University of California at Berkeley
University of California Department of Mathematics Department of Mathematics
Department of Mathematics Ann Arbor, Michigan 48104 Berkeley, California 94720
Santa Barbara, California 93106
AMS Subject Classifications: 32-01, 32A05, 32A07, 32AIO, 32A20, 32BIO, 32CIO, 32C35,
32D05, 32DlO, 32ElO
Library of Congress Cataloging in Publication Data
Grauert, Hans, 1930-

Several complex variables.
(Graduate texts in mathematics; 38)
Translation of Einftihrung in die Funktionentheorie mehrerer Veranderlicher.
Bibliography: p. 201
Includes index.
1. Functions of several complex variables.
I. Fritzsche, Klaus, joint author. II. Title. III. Series.
QA331.G69 515'.94 75-46503
All rights reserved.
No part of this book may be translated or reproduced in any form without written permission
from Springer-Verlag.
© 1976 by Springer-Verlag Inc.

Softcover reprint of the hardcover 1st edition 1976
ISBN-13: 978-1-4612-9876-2 e-ISBN-13: 978-1-4612-9874-8

DOl: 10.1007/978-1-4612-9874-8
Preface
The present book grew out of introductory lectures on the theory offunctions
of several variables. Its intent is to make the reader familiar, by the discussion
of examples and special cases, with the most important branches and methods
of this theory, among them, e.g., the problems of holomorphic continuation,
the algebraic treatment of power series, sheaf and cohomology theory, and
the real methods which stem from elliptic partial differential equations.
In the first chapter we begin with the definition of holomorphic functions
of several variables, their representation by the Cauchy integral, and their
power series expansion on Reinhardt domains. It turns out that, in l:ontrast
to the theory of a single variable, for n ~ 2 there exist domains G, G c en
with G c G and G "# G such that each function holomorphic in G has a
continuation on G. Domains G for which such a G does not exist are called
domains of holomorphy. In Chapter 2 we give several characterizations of
these domains of holomorphy (theorem of Cartan-Thullen, Levi's problem).
We finally construct the holomorphic hull H(G} for each domain G, that is
the largest (not necessarily schlicht) domain over en into which each function
holomorphic on G can be continued.
The third chapter presents the Weierstrass formula and the Weierstrass
preparation theorem with applications to the ring of convergent power
series. It is shown that this ring is a factorization, a Noetherian, and a Hensel
ring. Furthermore we indicate how the obtained algebraic theorems can be
applied to the local investigation of analytic sets. One achieves deep results
in this connection by using sheaf theory, the basic concepts of which are
discussed in the fourth chapter. In Chapter V we introduce complex manifolds
and give several examples. We also examine the different closures of en and
the effects of modifications on complex manifolds.
Cohomology theory with values in analytic sheaves connects sheaf theory
v
Preface
with the theory of functions on complex manifolds. It is treated and applied

in Chapter VI in order to express the main results for domains ofholomorphy
and Stein manifolds (for example, the solvability of the Cousin problems).
The seventh chapter is entirely devoted to the analysis of real differentia-
bility in complex notation, partial differentiation with respect to z, z, and
complex functional matrices, topics already mentioned in the first chapter.
We define tangential vectors, differential forms, and the operators d, d',
d". The theorems of Dolbeault and de Rham yield the connection with
cohomology theory.
The authors develop the theory in full detail and with the help of numerous
figures. They refer to the literature for theorems whose proofs exceed the
scope of the book. Presupposed are only a basic knowledge of differential
and integral calculus and the theory of functions of one variable, as well as a
few elements from vector analysis, algebra, and general topology. TheI book
is written as an introduction and should be of interest to the specialist and
the nonspecialist alike.
Gottingen, Spring 1976
H. Grauert
K. Fritzsche
vi
Contents
Chapter I
Holomorphic Functions 1
1 Power Series 2
2 Complex Differentiable Functions 8
3 The Cauchy Integral 10
4 Identity Theorems 15
5 Expansion in Reinhardt Domains 17
6 Real and Complex Differentiability 21
7 Holomorphic Mappings 26
Chapter II
Domains of Holomorphy 29
1 The Continuity Theorem 29
2 Pseudo convexity 35
3 Holomorphic Convexity 39
4 The Thullen Theorem 43
5 Holomorphically Convex Domains 46
6 Examples 51
7 Riemann Domains over en 54
8 Holomorphic Hulls 62
Chapter III
The Weierstrass Preparation Theorem 68
1 The Algebra of Power Series . 68
2 The Weierstrass Formula 71
vii
Contents
3 Convergent Power Series 74

4 Prime Factorization 78
5 Further Consequences (Hensel Rings, Noetherian Rings) 81
6 Analytic Sets 84
Chapter IV
Sheaf Theory 99
1 Sheaves of Sets 99
2 Sheaves with Algebraic Structure 105
3 Analytic Sheaf Morphisms 110
4 Coherent Sheaves 113
Chapter V
Complex Manifolds 119
Complex Ringed Spaces 119
2 Function Theory on Complex Manifolds 124
3 Examples of Complex Manifolds 128
4 Closures of en 144
Chapter VI
Cohomology Theory 150
1 Flabby Cohomology 150
2 The Cech Cohomology 158
3 Double Complexes 163
4 The Cohomology Sequence 167
5 Main Theorem on Stein Manifolds 174
Chapter VII
Real Methods 179
1 Tangential Vectors 179
2 Differential Forms on Complex Manifolds 185
3 Cauchy Integrals 188
4 Dolbeault's Lemma 191
5 Fine Sheaves (Theorems of Dolbeault and de Rham) 193
List of symbols 199

Bibliography 201
Index 203
Vlll
CHAPTER I
Holomorphic Functions
Preliminaries
Let e be the field of complex numbers. If n is a natural number we call the
set of ordered n-tuples of complex numbers the n-dimensional complex number
space:
Each component of a point 3 E en can be decomposed uniquely into real

and imaginary parts: Zv = Xv + iyv' This gives a unique 1-1 correspondence
between the elements (Zl' ... ,zn) of en and the elements (Xl' ... , Xn,
Yb' .. , Yn) of 1R 2n, the 2n-dimensional space of real numbers.
en is a vector space: addition of two elements as well as the multiplication
of an element of en by a (real or complex) scalar is defined componentwise.
As a complex vector space en is n-dimensional; as a real vector space it is
2n-dimensional. It is clear that the IR vector space isomorphism between
en and [R2n leads to a topology on en: For 3 = (Zl' ... , zn) = (Xl + iY1, ... ,
Xn + iYn) E en let
I I2
11311: = Ct1 ZkZkY 2 = Ct1 (xl + YDY ,
11311*: = max (lxkl,IYkl)·
k= 1, ... ,n
Norms are defined on en by 31-+11311 and 31-+11311*, with corresponding

metrics given by
dist(3b 32): = 1131 - 3211,
dist*(3b 32): = 1131 - 3211*·
I. Holomorphic Functions
In each case we obtain a topology on en which agrees with the usual topology
for ~2n. Another metric on en, defined by 131: = max IZkl and dist'(31) 32):
k= 1 •...• n
= 131 - 321, induces the usual topology too.
A region Been is an open set (with the usual topology) and a domain
an open, connected set. An open set G c en is called connected if one of the
following two equivalent conditions is satisfied:
a. For every two points 31, 32 E G there is a continuous mapping cp: [0, 1J -+
en with cp(o) = 31> cp(l) = 32, and cp([O, IJ) c G.
b. If B 1 , B2 C G are open sets with Bl u B2 = G, Bl n B2 = 0 and
Bl =F 0, then B2 = 0·
Definition. Let Been be a region, 30 E B a point. The set CB (30): =

{3 E B:3 and 30 can be joined by a path in B} is called the component of
30 in B.
Remark. Let Been be an open set. Then:

a. For each 3 E B, CB (3) and B - CB (3) are open sets.
b. For each 3 E B, CB (3) is connected.
c. From CB (3tl n CB (32) =F 0 it follows that CB (3i) = CB (32)'
d. B = U
3eB
CB (3)
e. If G is a domain with 3 E G c B, it follows that G c C B (3).
f. B has at most countably many components.
The proof is trivial.
Finally for 30 E en we define:
U.(30): = {3 E en:dist(3, 30) < e},
U:(30): = {3 E en:dist*(3, 30) < e},
U~(30): = {3 E en:dist'(3, 30) < e}.
1. Power Series
Let M be a subset of en. A mapping f from M to e is called a complex
function on M. The polynomials
ml, ... ,mn
P(3) = L av, ..... vn z'1' .... ' z~n,

Vl, ... ,Vn=O
are particularly simple examples, defined on all of en. In order to simplify

notation we introduce multi-indices: let Vi' 1 ~ i ~ n, be non-negative
integers and let 3 = (ZI,' .. ,zn) be a point of en. Then we define:
n Zii.
n n
Ivl: = LVi' 3v : =
i= 1 i= 1
m
With this notation a polynomial has the form P(3) = L a v3v•
v=O
2
1. Power Series
Def. 1.1. Let 30 E en be a point and for Ivi ~ 0, a v be a complex number.

Then the expression
00
I ai3 -
v=o
30V
is called a formal power series about 30'
Now such an expression has, as the name says, only a formal meaning.
For a particular 3 it does not necessarily represent a complex number. Since
the multi-indices can be ordered in several ways it is not clear how the
summation is to be performed. Therefore we must introduce a suitable
notion of convergence.
Def.1.2. Let~: = {v = (Vi"'" V"):V i

00
~ ofor 1 ~ i ~ n}, and 31 EC"befixed.
We say that I av (31 - 30)V converges to the complex number c if for
v=o
each B > 0 there exists a finite set 10 c: ~ such that for any finite set I
with 10 c: I c: ~
II
vel
av(31 - 30)V - cl < B.
00
One then writes I

av (3 - 30)V = c.
v=o
Convergence in this sense is synonymous with absolute convergence.
Def. 1.3. Let M be a subset of en, 30 E M, f a complex function on M. One

00
says that the power series I

av(3 - 30)V converges uniformly on M to
v=o
f(3) if for each B > 0 there is a finite set 10 c: ~ such that
II
vel
av (3 - 30)V - f(3) < I B
for each finite I with 10 c: I c: ~ and each 3 E M.

00
I av(3 - 30Y converges uniformly in the interior of a region B if the

v=o
series converges uniformly in each compact subset of B.
Def. 1.4. Let B c: en be a region and f be a complex function on B. f is

called holomorphic in B if for each 30 E B there is a neighborhood U =
00
U(30) in B and a power series I av (3 - 30)V which converges on U to

v=o
Note that uniform convergence on U is not required. We show now why

pointwise convergence suffices.
3
1. Holomorphic Functions
Def. l.5. The point set V = {r = (rl> . .. , rn) E ~n:rv ?= 0 for 1 ::S v :::::; n}
will be called absolute space. r:en -> V with r(3}: = (lz11, ... , IZnl) is the
natural projection of en onto V.
V is a subset of ~n and as such inherits the topology induced from to V ~n

(relative topology). Then r: en -> V is a continuous surjective mapping. If
B c V is open, then r- 1(B) c en is also open.
Def. l.6. Let r E V+: = {r = (r 1 , ... , rn) E ~n:rk > O}, 30 E en. Then P r (30):
{3Een:lzk - ziO) I < rk for 1:::::; k:::::; n} is called the polycylinder about
30 with (poly-)radius r. T = T(P): = {a E en: IZk - 4°)1 = rd is called
the distinguished boundary of P (see Fig. 1).
Figure 1. The image of a polycylinder in absolute space.
P = P r (30) is a convex domain in en, and its distinguished boundary is

a subset of the topological boundary JP of P. For n = 2 and ao = 0 the
situation is easily illustrated: V is then a quadrant in ~2, reP) is an open
rectangle, and reT) is a point on the boundary of T(P). Therefore
T = {a E e2:l z d = rl> IZ21 = r 2 }
= {a = (r 1 · ei81 , r2· ei82 )Ee 2:O:::::; 81 < 2n,O :;:;;: 82 < 2n}
is a 2-dimensional torus. Similarly in the n-dimensional case we get an
n-dimensional torus (the cartesian product of n circles).
If 31 E en:
= {a = (ZI> ... , zJ E en:Zk #- 0 for 1 :;:;;: k :::::; n}, then P3,: =
{a E en: IZkl < Izi1)1 = rkfor 1 :;:;;: k :;:;;: n} is a poly cylinder about 0 with radius
r = (r 1 , ••. ,rn ).
en. If the power series L:

00
Theorem l.1. Let 31 E ava v converges at 31, then

v=o
it converges uniformly in the interior of the polycylinder P 3,.
4
1. Power Series
PROOF
31' the set {av3~: Ivl ): o} is bounded.
1. Since the series converges at
Let M E IR be chosen so that lav3~1 < M for all v. If 31 E tn and < q < 1
then q' 31 E tn. Let P*: = Pnl . For 5 E P*, lavi = IZ1Iv1 .... 'lznlVn <
°
Iq· Z~111v1 .... ·Iq· z~111Vn = qV1 +"'+V n'lz~1)IV1 .... 'lz~l)IVn = qlvl '13~1, that
00
I I
~
is, lavl'13~1' qlvl is a majorant of avaV and therefore

v=O v=O
M· f
v=o
qV 1+"'+v n = M'( V1=0f qv 1) ..... ( f
vn=O
qv n ) = M.(~)n.
1 q
The set 3 of multi-indices is countable, so there exists a bijection tJ>: No --> 3.
00
Let bn(3): = a<1>(n1 . 5<1>(n1. Then I bn(3) is absolutely and uniformly con-
°
n=O
00
vergent on P*. Given a > there is an no EN such that I Ibn(a)1 < a

n=no+ 1
on P*. Let 10: = tJ>({O, 1,2, ... ,no}). If 1 is a finite set with 10 c 1 c 3,
then {a, 1, ... , no} c tJ>-1(l), so
IJo bn(a) - V~I av3vl = IJo bn(3) "E~l(ll bn(3)1

-
= InE~l(ll b (3)1 :;;; n=~+ 1Ib,,(3)1 < a for 3

n E P*.
00
But then I av 3V is uniformly convergent in P*.

v=O
°
2. Let K c P 1I be compact. {Pq. 'I : < q < 1} is an open covering of
POI' and thus of K. But then there is a finite subcovering {Pili' ,11' ... , P ql . ,l}
If we set q: = max(q1,"" q(), then K c Pqol ' and Pq ' ol is a P* such as
00
in 1). Therefore I av 3V is uniformly convergent on K, which was to be

v=o
shown. o
be brief we choose 30 = °
Next we shall examine on what sets power series converge. In order to
as our point of expansion. The corresponding
statements always hold in the general case.
Def. 1.7. An open set B c C" is called a Reinhardt domain if 31 E B => T oI : =

r- 1 r(3d c B.
Comments. Tli is the torus {3 E C": IZkl = IzP11}. The conditions of defi-
nition 1.7 mean that r- 1 r(B) = B; a Reinhardt domain is characterized by
its image reB) in absolute space.
Theorem 1.2. An open set B c C" is a Reinhardt domain if and only if there
exists an open set W c V with B = r -l(W).
5
PROOF
1. Let B = T- 1 (W), We V open. For 3 E B, '(3) E W; therefore
,-l(W) = B.
, - I T (3) c
2. Let B be a Reinhardt domain. Then B = ,-l,(B) and it suffices to
show that ,(B) is open in V. Assume that ,(B) is not open. Then there is a
point to E ,(B) which is not an interior point of T(B) and therefore is a cluster
point of V - ,(B). Let (tj) be a sequence in V - T(B) which converges to to.
There are points 3j E en with tj = T(3j), so that IzV)1 = rV) for all j and
1 ~ P ~ n. Since (tj ) is convergent there is an ME IR such that IrV)1 < M
for allj and p. Hence the sequence (3j) is also bounded. It must have a cluster
point 30, and a subsequence (3j) with lim 3jv = 30' Since T is continuous
v-oo
T(30) = lim T(3j) = lim tjv = to. B is a Reinhardt domain; it follows that
v-oo v-oo
30ET-1(to) c T- 1 T(B) = B. B is an open neighborhood of 30; therefore
almost all 3jv must lie in B, and then almost all tjv = T(3j) must lie in T(B).
This is a contradiction, and therefore T(B) is open. 0
The image of a Reinhardt domain in absolute space is always an open set

(of arbitrary form), and the inverse image of this set is again the domain.
Def. 1.8. Let G c en be a Reinhardt domain.

1. G is called proper if
a. G is connected, and
b. OE G.
2. G is called complete if
31 E G n en ==> P31 c G.
Figure 2 illustrates Def. 1.8. for the case n = 2 in absolute space.
IZ11
Figure 2. (a) Complete Reinhardt domain; (b) Proper Reinhardt domain.
6
1. Power Series
For n = 1 Reinhardt domains are the unions of open annuli. There is no

difference between complete and proper Reinhardt domains in this case; we
are dealing with open circular discs.
Clearly for n > 1 the polycylinders and balls K = {o: Iz 112 + ... +
IZnl2 < R2} are proper and complete Reinhardt domains. In general:
Theorem 1.3. Every complete Reinhardt domain is proper.
PROOF. Let G be a complete Reinhardt domain. There exists a point 31 E G n

tn, and by definition 0 E Po, c G. It remains to show that G is connected.
a. Let 31 E G be a point in a general position (i.e., 31 E G n en). Then the
connecting line segment between 31 and 0 lies entirely within Po, and hence
within G.
b. 31 lies on one of the "axes." Since G is open there exists a neighborhood
U.(31) c G, and we can find a point 32 E U.(31) n tn. Hence there is a path
in U e which connects 31 and 3z, and a path in G which connects 32 and o.
Together they give a path in G which joins 31 and o.
From (a) and (b) it follows that G is connected. D
00
Let ~(3) = L a3 v v be a power series about zero. The set M c en on

v=o
which ~(3) converges is called the convergence set of ~(3). ~(3) always con-
verges in M and diverges outside M. B(~(3)): = M is called the region of
convergence of the power series ~(3).
00
Theorem 1.4. Let ~(3) = L a v 3v be a formal power series in en. Then the
v=o
region of convergence B = B(~(3)) is a complete Reinhardt domain. ~(3)
converges uniformly in the interior of B.
PROOF
1. Let 31 E B. Then U~(31) = {3 E en:13 - 311 < 8} = Ue(zil)) x ... X
Ue(z~l)) is a polycylinder about 31 with radius (8, ... , 8). For a sufficiently
small 8, U~(31) lies in B. For k = 1, ... , n we can find a zk2 ) E Ue(zP») such
that Izf)1 > Izk1)1· Let 32: = (zi2), ... , Z~2»). Then 32 E Band 31 E P 32 • For
each point 31 E B choose such a fixed point 32.
2. If 31 E B, then there is a 32 E B with 31 E P 32 • ~(3) converges at 32, there-
fore in P 02 (from Theorem 1.1). Hence P 32 c B. Since P 3, c P 32 and To, c P 02 '
it follows that B is a complete Reinhardt domain.
3. Let P;,: = P 02 where 32 is chosen for 31 as in 1). Clearly B = UP;,.
h EB
Now for each 32 select a q with 0 < q < 1 and such that 33: = (1/q)52 lies
in B. This is possible and it follows that for each 51 E B ~(3) is uniformly
convergent in P;,. If K c B is compact, then K can be covered by a finite
number of sets P;,. Therefore ~(3) converges uniformly on K. D
7
The question arises whether every complete Reinhardt domain is the

region of convergence for some power series. This is not true; additional pro-
perties are necessary. However, we shall not pursue this matter here.
Since each complete Reinhardt domain is connected, we can speak of
the domain of convergence of a power series. We now return to the notion
of holomorphy.
Let f be a holomorphic function on a region B, 00 a point in B. Let the
00
power series L av (3 - oo)v converge to f(o) in a neighborhood U of 00'

v=O
Then there is a 01 E U with z~1) =I- z~O) for 1 ::::; v ::::; nand P'(31-30)(00) c u.
Now let 0 < e < min (lz~1) - z~O)I). From Theorem 1.1 the series con-
v= 1, ... ,n
verges uniformly on U~(30). For each v E ~ one can regard a.(3 - oo)v as a
complex-valued function on ~2n. This function is clearly continuous at 00
and consequently the limit function is continuous at 00' We have:
Theorem 1.5. Let Been be a region, and f a jitnction holomorphic on B.

Then f is continuous on B.
2. Complex Differentiable Functions
Def. 2.1. Let Been be a region, f: B -+ C a complex function. f is called

complex differentiable at 00 E B if there exist complex functions Ll1' ... ,
LIn on B which are all continuous at 00 and which satisfy the equality
n
f(o) = f(oo) + L (zv -
v=l
z~O») LI.(o) in B.
Differentiability is a local property. If there exists a neighborhood U =

U(OO) c B such that fl U is complex differentiable at 00, then fiB is complex
differentiable at 00 since the functions Llv(o) can be continued outside U in
such a way that the desired equation holds.
At 00 the following is true:
Theorem 2.1. Let Been be a region and f:B -+ C complex differentiable

at 30 E B. Then the values of the jitnctions LIb' .. , LIn at 30 are uniquely
determined.
PROOF. E.: = {o E en:z", = z~O) for A. =I- v} is a complex one-dimensional
plane. Let B.: = {( E C : (z\O), ... ,z~oll' (, z~oJ b ... ,z~O») E E. n B}.f~(z.): =
f(z\O), ... , z~oll' z., z~oJ b . . . , z~O») defines a complex function on B•. Since
f is differentiable at 00, we have on B.
f~(z.) = f(z\O), ... , z~~ 1, z., z~oJ b ... , z~O»)
= f(oo) + (z. - z~O») . LI.(z\o>, ... , zv, ... , z~O»)
= f~(z~O») + (z. - z~O») . LI~(z.).
8
2. Complex Differentiable Functions
Thus LI~(zv): = Llv(z\O), ... , Z~~ 1, z" Z~OJ 1, ... , z~O») is continuous at z~O).
Therefore f~(zv) is complex differentiable at z~O) E en, and LI~(z~O») = Llv(30)
is uniquely determined. This holds for each v. 0
Def. 2.2. Let the complex function f defined on the region Been be com-
n
plex differentiable at 30 E B. If f(5) = f(50) + I (z, - z~O») Llv(5), then
v= 1
we call Llv(50) the partial derivative of f with respect to z, at 50' and
write Llv(50) = oafZv (30) = 1,,,(30) = /v(30)'
Theorem 2.2. Let Been be a region and f complex differentiable at 30 E B.

Then f is continuous at 30'
n
PROOF. We have f(3) = f(30) + I (Zv - z~O») Llv(3); the right side of this
v=1
equation is clearly continuous at 30' o
Let Been be a region. f is called complex differentiable on B if f is
complex differentiable at each point of B.
Sums, products, and quotients (with nonvanishing denominators) of com-
plex differentiable functions are again complex differentiable. The proof is
analogous to the real case, and we do not present it here.
Theorem 2.3. Let Been be a region, f holomorphic in B. Then f is complex

differentiable in B.
PROOF. Let 30 E B. Then there is a neighborhood V = V(30) and a power
00
series I av (3 - 30)V which in V converges uniformly to f(3). Without loss

v=o
of generality let 30 = O. Then
00
I ava V = ao .. ·o + Z1' I aVI"'vnzI' 1. Z22 ••• Z~n

v=O vl~l
+ z 2 . '\'
~
aO • V2 . . . Vn ZV22 -1 . Z'3 3 ••• ZVnn + ... + zII . '\'
'-'
aO"'0"nZnVn-1
Vn~ 1
For now, this decomposition has only formal meaning. Choose a poly-
cylinder ofthe form P = Ve(O) X •.• X Ve(O) C V(O) and a point 31 E T "'"
{3 E en:IZkl = B}. Then POI = P and 31 E V (if Bis chosen sufficiently small).
00 00
I av3I converges, therefore I lav3I1 also converges. Since 31 E tn, IZ~l)l "# 0
v=o v=o
for all k. Therefore every sub series in the above representation at 31 also
converges absolutely and uniformly in the interior of P", The limit func-
tions are continuous and are denoted by Ll 1 , . . . , Lin. Since f(3) = f(30) +
z 1 . Ll1 (3) + ... + Zn • Ll n (3), it follows that f is complex differentiable at 30' 0
9
From this proof we obtain the values of the partial derivatives at a point
ao. For
L
CJ)
J(a) a v , ... vjZ1 - zlO))"' ... (zn - z~O))"n

Vl, .••• Vn =O
We obtain
fzJao) = ao .... , 0, l'
3. The Cauchy Integral

In this section we shall seek additional characterizations of holomorphic
fun,ctions.
Let r = (rlo ... , rn) be a point in absolute space with rv =1= 0 for all v.
Then P = {a E en: Izvl < r vfor all v} is a nondegenerate polycylinder about
the origin and T = {a E en:'t(a) = r} is the corresponding distinguished
boundary. It will turn out that the values of an arbitrary holomorphic
function on P are determined by its values on T.
First of all we must generalize the notion of a complex line integral.
Let K = {z E C:z = reiO, r > 0 fixed, 0 ~ e ~ 2n} be a circle in the com-
plex plane, J a function continuous on K. As usual one writes
SK J(z) dz = S027[ J(re iO ) • rie iO de.

The expression on the right is reduced to real integrals by
Lb q>(t) dt: = LbRe q>(t) dt + i' f 1m q>(t) dt.

Now let J = J(~) be continuous on the n-dimensional torus T =
g E en:'t(~) = r}. Then h:P x T --;. C with
h(3 ~). = J(~)

, . (~1 - Zl)"'(~n - zn)
is also continuous. We define
For each aE P, F is well defined and even continuous on P.
10
Def. 3.1. Let P be a polycylinder and T the corresponding n-dimensional

torus. Let f be a continuous function on T. Then the continuous function
ch(f):P -> e defined by
1 )n I(~) d~
ch(f)(5): =
(
2ni . IT (~1 - Z1)· ..(~n - zn)
is called the Cauchy integral oI f over T.
Theorem 3.1. Let Been be a region, P a polycylinder with PcB and T

the n-dimensional torus belonging to P. Iff is complex differentiable in B
then flP = ch(fl T).
PROOF. This theorem is a generalization of the familar I-dimensional

Cauchy integral formula.
The function I; with I;(zn): = I(~I' .. ·' ~n-I' zn) is complex differ-
entiable for fixed (~I' ... , ~n-I) E en-I in Bn: = {zn E 1C:(~b ... , ~n-I' zn) E
En n B}, where En is the plane {5 E en:z;. = ~;. for ), =f n}. But then f; is
hoI om orphic in Bn. Bn is an open set in IC. Kn: = {en E e: I~nl = rn} is
contained in B n , and the Cauchy integral formula for a single variable says
Therefore
Similarly for the penultimate variable we obtain
Theorem 3.2. Let Peen be a polycylinder, T the corresponding torus, and

h aIunction continuous on T. Then f: = ch(h) can be expanded in a power
series which converges in all of P.
PROOF.For simplicity we consider only the case of two variables. Let

T = {(~b~2)Ee2:1~11 = rb'I~21 = r2}, with fixed 3 = (ZbZ2)EP. Then
11
IZ11 < rl> IZ21 <

00
v~o
1'2
q? dominates v~o
00 (z .)Vj
and therefore qj: = (lzNrj) < 1 for j = 1,2. Hence
~~ for j = 1,2 and
is absolutely and uniformly convergent for (~1' ~2) E T. In particular arbitrary

substitutions are allowed, so,
also converges uniformly and absolutely on T. Since h is continuous on T

and T is compact, h is uniformly bounded on T: Ihl ~ M. Then, for fixed
(Zl' Z2) E P,
converges absolutely and uniformly on T, and we can interchange summation

and integration:
with
The series converges for each 3 = (Zl> Z2) E P. o
Theorem 3.3. Let Been be a region, I complex differentiable in B. Then I

is holomorphic in B.
PROOF. Let 30 E B. For the sake of simplicity we assume 30 = O. Then there

exists a polycylinder P about 30 such that PcB. Let T be the distinguished
boundary of P. From Theorem 3.1 liP = ch(fl T). II T is continuous so
from Theorem 3.2 I is holomorphic at 30' 0
12
Theorem 3.4. Let Been be a region, f holomorphic in Band 30 a point

in B. If PcB is a polycylinder about 30 with PcB, then there is a power
00
series 'l3(3) = L a.(3 - 30t which converges to f on all of P.

v=o
PROOF. IffisholomorphicinB, thenflP = ch(fIT), where the distinguished
boundary of P is denoted by T. From Theorem 3.2 flP can be expanded as
a power series in all of P. 0
Theorem 3.5. Let the sequence of functions (Iv) converge uniformly to f on

the region B with all Iv holomorphic in B. Then f is holomorphic in B.
PROOF. Let 30 E B. Again, we assume that 30 = O. Let P be a poly cylinder
about 30 with PcB. Let 3 = (Zb' .. , zn) E P. N(~): = (~l - Zl)··· .'
(~n - zn) is continuous and #0 on T; therefore, I/N(~) is also continuous
on T and· there exists an M E IR such that 11/N(~)1 < M on T. (fv) converges
uniformly on T to f so for every B > 0 there exists a Vo = Vo(B) such that
IIv - fl < BIM on all of T for v ~ Vo. But then
-
I~ ~I = I~I' IIv - fl < B.
Hence Ivl N converges uniformly on T to fiN and one can interchange the
integral and the limit.
flp = !~ (f.lp) = !~~ ch(fvI T) = ch (!~ (fvlT)) = ch(fIT).

f is continous on T since all the fv are continuous on T. From Theorem 3.2
it follows that f is holomorphic at the origin. 0
00
Theorem 3.6. Let 'l3(3) = L a 3 be a formal power series and G the domain
v v
v=o
of convergence for 'l3(3). Then f with f(3): = 'l3(3) is holomorphic in G.
PROOF. Let.3 be the set of all multi-indices v = (Vb' .. , vn ), 10 c .3 a finite
subset. Clearly the polynomial L a v3v is holomorphic on all of en.
velo
Let 30 E G be a point, P a poly cylinder about 30 with PeG. 'l3(3) con-
verges uniformly on P to the function f(3). If one sets Bk: = 11k for kEN
then in each case there is a finite set h c .3 such that 1 L a v3v - f(3)1 < Bk on
VEl
all of P for any finite set I with Ik c I c .3. For iT,: = Lava' we have iT,
velk
holomorphic and for each kEN, liT, - fl < 11k on all of P. Therefore (iT,)
converges uniformly on P to f. From Theorem 3.5 f is holomorphic in P
and in particular at 30' 0
13
Theorem 3.7. Let f be holomorphic on the region B. Then all the partial
derivatives h., 1 ~ fl ~ n, are also holomorphic in B. If PcB is a poly-
00
cyclinder with center at the origin and f(o) = I avo on P, then

V
v=o
00
{' (2)
J ZJl 0
= " '-'
a . v . zV, ... zV. -1
v I.l 1 I-l
... zV n
n
v=o
on P.
PROOF
1. Let PcB,
Xl
01 E P n tn. Then there is an M E IR such that la 3I1 v < M
for all v, where I avo is the power series expansion of f in P. If 0 < q < 1
v=o
00 00
and 32: = q' Ob then I av3z is dominated by M· I qlvl. Now 32 =

v=o v=o
(Z1> ••• , zn) with IZkl "# 0 for k = 1, ... , n. It follows that
Formally
00
For fl "# j, I qV. is a geometric series and therefore convergent. For fl = j

V,u =0
ro
the convergence of I VjqV j follows from the ratio test:
Hence the series
co
converges. By the comparison test the series I avVjZ,!' ... zjr 1 ... z~n is
v=o
also convergent at the point 32, and therefore in P'2' Since P is the union of all
the P')2 the series converges in all of P to a holomorphic function gj.
2. Let
1*(3): = f: j
gj(Zb . . . , Zj_ b ~, Zj+ b ... , zn) d~ + f(Zl, ... , 0, ... , zn)·
The path of integration can be chosen in such a way that it consists of the
line segments connecting 0 to Zj in the zrplane. Thus 1* is defined on P.
00 co
For hv(3): = av3", we have f(3) = I hv(o) and gh) = I (h v )z/3)· The path
v=o v=o
14
4. Identity Theorems
of integration is a compact subset of P and the series converges uniformly

there. Hence one may interchange summation and integration and obtains
1*(3) =v~o (f: i

(hv)ziZl>"" Zj_ b~' Zj+ 1, . . . ,Zn) d~ + hv(zb' .. ,0, ... ,Zn) )
OCJ
= I hv(3)
v=o
= f(3)
Hence fz/3) = f: j (3) = 9j(3)· o
We conclude this section with a summary of our results.
Theorem 3.8. Let Been be a region and f a complex function on B. The

following statements about f are equivalent:
a. f is complex differentiable in B
b. f is arbitrarily often complex differentiable in B
c. f is holomorphic in B. For every 30 E B there is a neighborhood U such
00
that f(3) = I av(3 - 30)" in U. Here the av are the "coefficients of

v=o
the Taylor series expansion":
1 aVI +"'+ v1
avl ... Vn = VI! ... Vn! . aZ~1 ... az~n (30)
d. For each polycylinder P with PcB, fiT is continuous and fiT =
ch(fIT).
PROOF. Nearly everything has already been proved, but we must still cal-
culate the coefficients avo For simplicity let 30 = and n = 2. In the proof
of Theorem 3.2 we obtained:
°
a
VIV2
= _1_
(2 m')2
ST
f(Zb Z2) dz dz .
vI+l. v2+1 1 2
ZI Z2
From the Cauchy integral formula for one variable it now follows that
a
VIV2
= _1 r _1_ [_1 r f(ZI, Z2) dZ 2] dZ 1
2ni JKI 2ni JK2 ZZ2+1
Z~I+l
1 1 aV2f dZ 1 1 aVI h 2f
= -- r -(ZI'
V2! 2ni JKI aZ z2
0)--
Z~I + 1
= -_.
VI !V2! az11 aZ z2
(0,0). 0
4. Identity Theorems
Different from the theory of one complex variable, the following theorem
does not hold in en: "Let G be a domain, MeG have a cluster point in
G and f1' f2 be holomorphic on G with fl = f2 on M. Then fl = f2 in G."
15
There is already a counter-example for n = 2. Let G: = 1[:2, M: =

{(Z1> Z2) E G:z 2 = O}, f1(Z1> Z2): = Z2 . g(Z1> Z2), f2(Z1> Z2): = Z2' h(Zl' Z2)
with g and h holomorphic on all of 1[:2. Then f1\M = f2\M, but f1 i= f2
for g i= h.
Theorem 4.1 (Identity theorem for holomorphic functions). Let G c en

be a domain and f1> j~ be holomorphic in G. Let BeG be a nonempty
regionwithf1\B = f2\B. Thenf1\G = f2\G.
PROOF. Let Bo be the interior of the set {3 E G:f1(3) = f2(3)} and Wo: =
G - Bo. Because B c B o, Bo i= 0. Since G is connected it suffices to show
that Wo is open, for then Bo = G follows. Let us assume Wo contains a
point 30 which is not an interior point. Then for every polycylinder P about
30 with PeG, P n Bo i= 0. Let rE!R and P: = {3:\Zj - z7\ < r} =
{3: dist'(3, 50) < r} be such a polycylinder. Let
P': = {3:dist'(3,30) < r/2} c P.
Then also P' n Bo i= 0. Choose an arbitrary point 31 E pI n Bo and set

P*: = {3:dist'(5, 31) < r/2}. Clearly 30 E P* and P* c P (triangle inequality).
Therefore P* c PeG. Let
00 00
f1(3) = I av(3 - 31t and f2(3) = I bv(3 - 31t

v=o v=o
be the Taylor series expansions of f1 and f2 in P*. Since f1 and f2 coincide

in the neighborhood of 31 E B o, av = b v for all v. (The coefficients are uniquely
determined by the function; cf. Theorem 3.8.) Therefore fdP* = f2\P* and
P* c Bo. It follows that 30 E B o, a contradiction. 0
Theorem 4.2 (Identity theorem for power series). Let G c en be a

00 00
domain with 0 E G, and I av 3V , I bv 3V two power series convergent in G.

v=o v=o
00 00
If there is an e > 0 such that I av3

V
= I bv3v in Vs(O) c G, then
v=o v=o
av = bv for all v.
00 00
PROOF. Let f(3): = I av3V , g(3): = I bv3v for 3 E G. By Theorem 3.6 f

v=o v=o
and g are holomorphic in G, and differentiation gives:
16
5_. Expansion in Reinhardt Domains
5. Expansion in Reinhardt Domains

In this section we shall study the properties of certain domains in en
in some detail.
Let r~, r~ be real numbers with 0 < r~ < r~ for 1 ~ v ~ n. Let r =
(1'1>' .. , rn) E Vbe chosen so that r~ < rv < <
for all v. Then T r : = {3: Izvl =
r v for all v} is an n-dimensional torus. We define
H: = {p.~ < Izvl < r~ for all v}

P: = {5: Izvl < r~ for all v}.
Clearly Hand P are Reinhardt domains.
Figure 3. Expansion in Reinhardt domains.
Let f be a holomorphic function in H. Then for all r E r(H), chUI7;) is a holo-

morphic function in Pr = {a: Izvl < rv for all v} (and therefore afortiori in P).
Proposition. g: P x r(H) ~ IC with g(3, r): = chUI7;)(3) is independent of r.

PROOF. We have
For eachj with 1 ~ j ~ n we have IZjl < rj = I~jl; therefore Zj =f ~j' Hence
the integrand is holomorphic on the annulus {Z{ rj < IZjl < r'j} and from
the Cauchy integral formula for one variable it follows that ifr = (r1> ... ,rn ) E
r(H) and r* = (r1, ... , r;) E r(H), then
r f(~l"'" ~n) d~. = r /(~b'''' ~n) d~ ..

JI~jl=rj J:. _
'oj
z.J J JI~jl=rj J:. _
'oj
z.J J
This yields the proposition. . o
17
Theorem 5.1. Let G c en be a domain and E: = {3 = (ZI' ... , zn) E en with

ZI = O}. Then the set G': = G - E is also a domain in en.
PROOF
1. E is closed, therefore en - E is open, and hence G' = G (\ (en - E)
is also open. Moreover, E contains no interior points.
2. We write the points 3 E en in the form 3 = (zr, 3*) with 3* E en-I. Now let
30 = (z\O), 3*(0») E G and let U~(30) = U,(ziO») x U~(3*(O») be an s-neighborhood
of 30' We show that U~ - E is still connected. Let 31 = (Z\I), 3*(1») and 3z =
(z\z>, 3*(Z») be two arbitrary points in U~ - E. Then we define 33: = (z\Z), 3*(1 »).
I
IL -31 _ _ _ _ _ _
___ ~
I
U~(30)
Figure 4. Proof of Theorem 5.1.
Clearly 33 E U~ - E. U,(ziO») is an open circular disk in the zl-plane, and

U,(z\O») - {O} is still connected. Hence there is a path cp which joins z\l) and
z\Z) and lies entirely within U,(z\O») - {O}; naturally there is also a path ljJ
which joins 3*(1) and 3*(Z) and which lies within U~(3*(O»).
We now define paths Wr, Wz by w 1 (t): = (cp(t), 3*(1») and wz(t): = (z\z>, ljJ(t)).
Then WI joins 31 and 33, Wz joins 33 and 3z, and the composition joins 31 and
3z in U~ - E. Therefore U~ - E is connected.
3. Let 3', 3" E G - E and let cp be an arbitrary path which joins 3' and 3"
in G. Since cp(I) is compact, one can cover it with finitely many polycylinders
U b . . . , Ue such that U). c G for A = 1, ... , t.
Lemma. There is a b > 0 such that for all t', t" E I with It' - t"l < (j, cp(t'),
cp(t") lie in the same polycylinder Uk'
PROOF. Let there be sequences (ti), (ti) E I with Iti - til ~ 0 such that
cp(ti), cp(ti) do not lie in the same poly cylinder Uk' There are convergent sub-
sequences (ti), (ti~) of(ti), (ti)· Letto : = lim ti" = lim ti'~. If cp(t o) E Ub then
v~oo IJ.-oo
there is an open neighborhood V = veto) c I with cp(V) c Uk' Then for
almost all v EN, ti, E V and tiv E V, so that cp(ti,) E Uk and cp(ti,) E Uk' This
is a contradiction, which proves the lemma.
Now let (j be suitably chosen and 0 = to < tl < ... < tk = 1 be a partition
of I with tj - t j- 1 < 0 for j = 1, ... , k. Let 3j: = cp(tj) and J;j be the poly-
18
5. Expansion in Reinhardt Domains
cylinder which contains OJ, OJ-1 (it can happen that Tj, = Tj2 for j1 "# jz). By
construction OJ-1 lies in Tj (\ Tj-1, so Tj (\ Tj-1 is always a non-empty open
set. Indeed, Tj (\ Tj _ 1 - E "# 0 for j = 1, ... , k.
We join 3' = 30 E VI - E and a point 01 E VI (\ Vz - E by a path 0/1
interior to VI - E. By (2) this is possible. Next we join 31 with a point
32 E Vz (\ V3 - E by a path O/Z interior to Vz - E, and so on.
Finally, let o/k be a path in Vk - E which joins 3~-1 with 3k = i3" E Vk - E.
The composition of the paths 0/)' ... , o/k connects 3' and i3" in G - E. D
Theorem 5.2. Let G be a domain in en, Eo: = {3 = (z), . .. ,zn) E en:z v = 0

for at least one v}. Then Go: = G - Eo is also a domain.
PROOF. For each J1 with 1 ~ J1 ~ n, G/l:G - E/l is connected, where
E/l: = {3 = (Zl' ... ,zn) E en:z/l = O}. This follows from Theorem 5.1 by
a simple permutation of the coordinates.
n
Clearly Eo = U E/l; therefore Go = (( (G - E 1 ) - E z ) ... ) - En. A
/l=1
trivial induction proof yields the proposition. D
Theorem 5.3. Let G c en be a proper Reinhardt domain, f holomorphic on

G, 30 E G (\ tn. Then chUI T,lO) coincides with f in a neighborhood of the
origin.
PROOF. We have Go: = ,(G (\ tn) c {r E V:rj "# 0 for j = 1, ... , n}.
1. Go is a domain:
a. G (\ tn is a Reinhardt domain; therefore Go = ,(G (\ tn)
is open by
Theorem 1.2.
b. Ifrb rz are points in Go, then there are points 3p E G (\ til with '(op) = rp
for p = 1,2. As shown above, G (\ tn is a domain, so that there is a
path 0/ in G (\ en which joins 01 and 3z. Then, 0/ is a path in Go which
0
joins r l and 1' z.

2. Let
B: = {1' EGo: ch U 1T) coincides with f in the vicinity of O}.
a. B is open: Ifro E B c Go, then there is a neighborhood U~(1'o) c Go which
can be written ,(H). This follows from the way we chose the set ,(H) at the
beginning of this section. Let P = P(O) be the corresponding polycylinder.
Then for 3 E P and l' E U~(ro) we have chUIT r)(3) = chUITro )(3). Moreover
g(3): = chUITro )(3) is a holomorphic function on P which coincides
with f near the origin because rEB. Therefore U~(1'o) c B.
b. W: = Go - B is open: The proof goes as in (a).

c. B "# 0: There is a polycylinder P.lo about 0 with P.10 c G. Then flp.10
chUI~lo)' and ro: = (IZ\O)I, ... , Iz~O)I) lies in B.
(1) and (2) imply B = Go. D
19
Theorem 5.4. Let G en be a proper Reinhardt domain, f holomorphic in G.

c
co
Then there is a power series 1.]3(3) = I a 3 which converges in G with
v
V
v=o
f(3) = 1.]3 (3) for 5 E G.
PROOF. If 30 E G then there is a 51 E G with Iz~O)1 < IZ~l)1 for j = 1, ... , n;

ro
therefore 30 E P31 • Let ch(fIT3 J(3) = I av 3" for 3E P 31 • The coefficients av

v=O
are those of the Taylor series about 0; they do not depend on 51. Since
30 was arbitrary it follows that the Taylor series of f about 0 converges in
all of G. It defines a holomorphic function g, which coincides with f near
the origin. By the uniqueness theorem, f = g on G. 0
Def. 5.1. If G c en is a proper Reinhardt domain, then G: = U P 3 is

3EGnC"
called the complete hull of G.
Remarks
l. G is open.
2. G c G. If 30 E G, then there is a 31 E G n en with 30 E G.
G n en with
P.ll C
3. G is a Reinhardt domain. Let 30 E G, 31 E 30 E P3 1" Then

Tao c Ph c G.
4. G is complete. Let 50 E G n 31 Een, G n en with
30 E 31 • Then P 30 c
P
P31 C G.
5. Gis minimal for the properties (1) through (4). LelLT c Gb G1 a complete
Reinhardt domain. If 3 E G n tn, then P a c G1 . Therefore G c G1 .
G is the smallest complete domain which contains G and we have the
following important theorem.
Theorem 5.5. Let G be a proper Reinhardt domain, f holomorphic in G.

Then there is exactly one holomorphic function F in G with FIG = f.
PROOF. By Theorem 5.4 we can write in G

00
f(-3) = I a v3 V
•
v=O
The series is still convergent on G, and actually converges to a holomorphic

function F. Clearly FIG = f. The uniqueness of the continuation follows
from the identity theorem. 0
For n ;:::, 2 we can choose sets G and Gin en so that G i= G. This constitutes
a vital difference from the theory of functions of a single complex variable,
where for each domain G there exists a function holomorphic on G which
cannot be continued to any proper superdomain.
20
6. Real and Complex Differentiability
We conclude this section with an important example of such a pair of sets

(G, G) with G #- G for n = 2.
Let P: = {3 E 1[2: 131 < 1} be the unit poly cylinder about the origin and
D: = {3 E 1[2:ql ~ IZll < 1, IZ21 ~ q} with 0 < ql < 1 and 0 < q < 1.
Then H: = P - D is a proper Reinhardt domain, and fj = U P3 = P.
3 EHO
The pair (P, H) is called a Euclidean Hartogs figure. Their image in abso-
lute space appears in Fig. 5.
Figure 5. Euclidean Hartogs figure in (:2.
The basis for the difference here between the theories of one and several
variables is that such a Hartogs figure does not exist in C. We already noted
that Reinhardt domains in I[ are open disks and annuli. Therefore a proper
Reinhardt domain in I[ is an open disk, i.e., a complete Reinhardt domain.
Hence G is not a proper superset of G.

Let M c en be a set, ! a complex function on M. At each point 30 EM
there is a unique representation !(30) = Re !(30) + i 1m f(30).
Therefore one can define real functions g and h on M by
g(x, t)) = Re !(3)

hex, t)) = 1m !(3)
where 3 = x + it). We then write:
! =g+ ih.
21
Def. 6.1. Let Been be a region, f = g + ih a complex function on B,

30 a point of B. f is called real differentiable at 30 if g and h are totally
(real) differentiable.
What does real differentiability mean? If g and h are differentiable, then

n n
g(x,1)) = g(x o,1)o) + I (Xv - X~o»)IX~(X, 1)) + I (Yv - y~o»)IX~*(X, 1)),
v=l v= 1
(1)
n n
h(x,1)) = h(xo, 1)0) + I (Xv - X~O»)f3~(X, 1)) + I (Yv - y~O»)f3~*(x, 1));
v=l v=l
where IX~, IX~*, f3~, f3~* are real functions on B which are continuous at (xo, 1)0)
and for which
1)0)
1X~(Xo, = gx/xo, 1)0)
1X~*(Xo, 1)0) = gy,(x o, 1)0)
f3~(xo, 1)0) = hx,(xo, 1)0)
f3~*(Io, 1)0) = hy,(x o, 1)0)·
We combine the equations:

n n
(2) f(3) = f(30) + I (Xv - x~o») L1~(3) + I (Yv - y~o») L1~*(3),
v=l v=l
where L1~ = IX~ + if3~ and L1~* = IX~* + if3~* are continuous at 30 and where
L1~(30) = gx,(30) + ihx,(30) = :fxJ30)
L1~*(30) = gy,(30) + ihyJ30) = :fy,(50)·
Theorem 6.1. Let Been be a region, 30 E B a point, f a complex function

on B. f is real differentiable at 30 if and only if there are functions L1~, L1:
on B which are continuous at 30 and satisfy in B the following equation:
n n
(3) f(3) = f(30) + I (Zv - z~o») L1~(3) + I (Zv - z~o») L1:(3)·
v=l v=l
PROOF
1. Let f be real differentiable at 30. We use the equations
Xv - x~o) = H(zv - z~o») + (zv - z~o»)]
and
1 [(z _ z(o») - (z _ z(o»)]
= _2i
Yv - yeo)
v v v v v'
Then
f(3) = f(30) + ±
v=l
(zv - z~o») L1~(3) -/L1~*(3) +
-
±
v=l
(zv _ ~o») L1~(3) +? iL1~*(3).
-
22
If we define
= Ll*v +2 iLl*'
A* . **
A
Ll' . = LJ v - ILJ v
and Ll'" v
V' 2 v •
then (3) is satisfied.

n n
2. Let f(3) = f(30) + L (zv - z~o») ,1~(3) + L (zv - ~o») Ll~(3), Ll~, Ll~
v= 1 v= 1
continuous at 30' The equations ,1~ = (Ll~ - iLl~*)j2, Ll~ = (Ll; + iLl~*)!2
appear in matrix form as
( Ll')
Ll ~ = 2' 1
1 (1 -i) i 0
(Ll*)
Ll;*
Let
A: = G -:}
Then det A = 2i =1= O. That means that the equations can be solved for
,1~ and LI~*. The solution functions satisfy equation (2); (1) follows from
decomposition into real and imaginary parts. Since the values of the func-
tions a~, a~*, f3~, f3~* are uniquely determined at the point 30, the same must
be true of the functions ,1~, Ll~. 0
We now write:
h,(30): = Ll~(30) = ~ [fx)30) - ifY )30)}
h,(oo): = ,1~(oo) = ~ [fxJoo) + ifyJoo)}

Theorem 6.2. Let Been be a region 30 E B, f a complex function on B. f
is complex differentiable at 00 if and only iff is real differentiable at 30 and
!z,(oo) = 0 for 1 ~ v ~ n. (This means that the Cauchy-Riemann differ-
ential equations must be satisfied:
gx, = hyv
for 1 ~ v ~ n.)
PROOF.
n
1. Let f(3) = f(oo) + L (zv - z~O») Llv(o), Llv(3) continuous at 00' Then
v= 1
n n
f(3) = f(oo) + L (Zv - Z~O») Ll~(3) + L (Zv - Z~O») Ll~(3) with ,1~(3) = ,1v(3)
v=1 v=1
and ,1~(3) = 0, so that fzJoo) = 0 for 1 ~ v ~ n.
23
2. Let / be real differentiable and hJ~o) = ° for 1 ~ v ~ n. Then /(3) =

/(30) + I
n
v=l
(Zv - z~o») .1~(3) + I
v=l
n
(Zv - ~o») .1~(3) with .1~(30) = ° for
V = 1, ... , n.
We define
o if Zv = z~O)
. _
1X.(3)· -
{ Zv-
-
-(0)
Zv • .1"( )
Z _ z(O) v 3
otherwise.
v v
Since
Zv - ~O)
Zv - z~O)
is bounded except at z~O) and lim .1~(3) = 0, it follows that IXv is continuous
3-30
at 30' But then
n n
/(3) = /(30) + I (Zv - Z~O») .1~(3) + I (Zv - Z~O») .1~(3)
v=l v=l
n
= /(30) + I (Zv - z~O»)(.1~ + IXv)(3)·

v= 1
Therefore / is complex differentiable at 30' o

We mention another differentiation formula.
1. If/is real differentiable at 30, we have at 30
for 1 ~ Jl ~ n.
for 1 ~ Jl ~ n.
2. Let / be twice real differentiable in a neighborhood of 30' Then at 30
for all v and Jl.
Theorem 6.3 (Chain rule). Let B), B2 be regions in en, respectively em.
9 = (g1> ... ,gm):B 1 ~ em be a mapping with g(B1) c B 2. Let 30 E B),
luo: = 9(30) and / a complex/unction on B 2. 1/ all gjl' 1 ~ Jl ~ m, are
real differentiable at 30 and / is real differentiable at luo, then /0 9 is real
differentiable at 30 and
m m
(f a 9)z)30) = I (fw)lUo))' ((gjl)z,(30)) + I (fwJlUo))' ((gjl)z.(30)),
jl=l jl=l
m m
(f 0 9)",(30) = I (fwJlUo))' ((gjl)", (30)) + jl=l
jl=l
I (fwJlUo))' ((gjl),,)30)).
PROOF. As in the real case, the proof follows from the definitions. D
24
Let B c C" be a region, I = (fb' .. ,fn):B ~ C" a real differentiable

mapping. Then we can define the complex functional matrix of I:
We assert that LI I: = det J I agrees with the usual functional determinant

as it is known for the real case. A series of row and column transformations
is necessary for the proof: We have
lv, z" = t(lv, x" - ifv, y),
Iv,,," = t(fv,x" + iiv,y,J
If we add the (n + .u)-th to the .u-th column, we obtain
LI I = det ~(j~~l_+_~if-,-xJ.;_-t.~,~"n),
\ CTv, x) ! (z-CTv, x" + ifv. y"))
therefore
LlI = 2- n det (~3J_~_~C:1'-~_!~!32).
\Uv,x) ! U:,x" + ifv,y)
Subtracting the .u-th from the (n + .u)-th column yields
Lli = 2- n det 6b~~_~_~xJ),
\(fv, x) ! (ifv, y)
therefore
LI I = 2 - nin det (i~_,xJ'L_~J{~~~\.
\Uv,x) ! UV,y);
{' 11 = g V'X fl
Jv,x + ih V'X/l'
.(,v, Yf.J. = g v. Yp + ih v, Y~l'
Subtraction of the v-th from the (n + v)-th row gives
LI I = in det (i(~i~L_)__~-((~i~~--)) = det (J(~~~))__ ~J(~~~~)\

\ v,x" I v,y" \ " XU I V,y";
This is precisely the functional determinant det J F of the real mapping
F = (gb ... , gn, hi> ... , hn).
25
7. Holomorphic Mappings
Def.7.1. Let Been be a region, gb"" gm complex functions on B.
g = (g1, ... ,gm):B --+ em is called a holomorphic mapping if all the com-
ponent functions gil are holomorphic in B.
Theorem 7.1. Let B1 c en, B z c em be regions, g = (gb ... ,gm):B 1 --+ B z

be a mapping. g is holomorphic if and only if for each holomorphic function
f on B z fog is a holomorphic function on B 1.
PROOF. Let g be a holomorphic mapping. Then all the component functions
gil are holomorphic, that is, (g/l)", = 0 for all v and fl. If f is holomorphic,
then fw. = 0 for all fl, fog is real differentiable, and from the chain rule it
follows that
m m
(f 0 g)z, = L fw• . (g/lh, + L fw• . (g/lh, = 0 for v = 1, ... , n.
/l=1 /l=1
Conversely, set f(w) == w/l' if the condition is satisfied. Then f 0 g(3) ==
g/l(3). 0
From this theorem it follows thatf g:B1 --+ e l is a holomorphicmapping

0
if g:B1 --+ B2 is a holomorphic mapping and f:B z --+ e l is a holomorphic

mapping.
Def.7.2. Let Been be a region, g = (gb ... , gm) a holomorphic mapping

from B into em. We call
9]( :
9
= ((
g/l.z, v
) fl = 1, ... ,
=
1, ... , n
m)
the holomorphic functional matrix of g.
Theorem 7.2. Let 30 E B, Wo = g(30), f and g as above. Then

9](Jog(30) = 9](J(wo) 0 9](g(30).
m
PROOF. (9)(Jo g)./l = (fv 0 g)z. = L Iv.

),=1
Wl • g)., z. = (9]( J 0 9](g)V/l" o
Def.7.3. Let Been be a region, g = (gb ... ,gn):B --+ en a holomorphic

mapping. M g: = det 9](g is called the holomorphicfunctional determinant
ofg·
Theorem 7.2 implies:
Theorem 7.3. Let the notation be as above and let m = n = 1. Then M Jog =
MJ·M g •
26
7. Holomorphic Mappings
Complex functional determinants of a holomorphic mapping have the

following form:
,1 = det (i~2)___ J[v-,-~2) = det (J.fh!~l_l ___O__)

9 (-)
\gv,~
(-
gv,~
_) 0 I
I
(-)
gv,~
= det( (gv,z)) . det( (~))
= det((gv,z))' det((gv,z)) = Idet((gv,z)W = IMgI2,
i.e., they are real and nonnegative. This means that holomorphic mappings
are orientation-preserving.
Def.7.4. Let Bb B2 be regions in e. A mapping g:Bl -+ B2 IS called

biholomorphic (resp. invertably holomorphic) if
a. g is bijective, and
b. g and g-l are holomorphic.
Theorem 7.4. Let Been be a region, g:B -+ e a holomorphic mapping.

Let 30 E B and roo = g(30). There are open neighborhoods U = U(30) c B
and V = V(roo) c e such that g: U -+ V is biholomorphic if and only if
My(30) =1= O.
PROOF
1. There are open neighborhoods U, V such that g: U -+ V is biholo-
morphic. Then 1 = M idu (30) = M 9 - 1 (roo) . M g(30), hence M g(30) =1= O.
2. g is continuously differentiable, and the functional determinant Mg is
continuous. If My(30) =1= 0, then there exists an open neighborhood W =
W(30) c B with (Mgl W) =1= O. So ,1gl W =1= 0 and g is regular (in the real sense)
at 30'
There are open neighborhoods U = U(30) c W, V = V(roo) such that
g: U -+ V is bijective and g-l = (gb' .. ,gn) is continuously differentiable.
gog -11V = id v is a holomorphic mapping. It follows that
n n n
0= (gv 0 g-lh" = L gv,z;.· gA,w" + L gv,z; . gA,w" = L gv,zJ.· gA,w"'
A=l A=l A=l
For each fl, 1 :( fl :( n, we obtain a system oflinear equations:
?JbW)
o ~ IDl. 0 ~
( 0_"
gn,w"
Since det 9)1g =1= 0 there is only the trivial solution: gA, w" = 0 for all A and
all fl. This holds in all of V. Therefore the Cauchy-Riemann differential
equations are satisfied and g-l is holomorphic. D
Theorem 7.5. Let Been be a region, g = (gb ... ,gn) holomorphic and one-
to-one in B. Then Mg =1= 0 throughout B.
27
This theorem is wrong in the real case: for example y = x 3 is one-to-one,

but the derivative y' = 3x 2 vanishes at the origin.
We shall not carry out the proof of Theorem 7.5 here. (It can be found as
Theorem 5 of Chapter 5 in R. Narasimhan: Several Complex Variables,
Chicago Lectures in Mathematics, 1971.)
Theorem 7.6. Let Bl c en be a region, g:Bl -+ en one-to-one and holomor-

phic. ThenB 2 : = g(B 1 )isalsoanopensetandg- 1 :B 2 -+ Bl isholomorphic.
PROOF
1. Let roo E B 2 . Then there exists a 30 E Bl with g(30) = roo. From
Theorem 7.5 Mg =1= 0 on Bb and therefore there are open neighborhoods
U(30) c Bb V(roo) c en such that g: U -+ V is biholomorphic. But then
V = g(U) C g(B 1 ) = B 2 ; that is, roo is an interior point.
2. From (1) for each roo E B2 there exists an open neighborhood V(roo) c
B 2 , such that g-llV is holomorphic. D
28
CHAPTER II
Domains of Holomorphy
1. The Continuity Theorem

In this and the following sections we shall systematically treat the problem
of analytic continuation of holomorphic functions.
Let P = {3 E en: 131 < 1} be the unit polycylinder, q10 ••• ,qn with
o < qv < 1 for 1 :::;; v :::;; n be real numbers. Then for 2 :::;; 11 :::;; n we define:
n
DI ,: = {3 EP :l z d:::;; ql andq!':::;; Iz!'1 < l},D: = U D!,andH: = P - D =
!,=2
H = {3EP:lzll > ql orlz!'1 < q!'for2:::;; 11:::;; n}

= {3EP:ql < IZll} u {3EP:lz!'1 < q!'for2:::;; 11:::;; n}.
(P, H) is called a "Euclidean Hartogs figure in en." H is a proper Reinhardt

domain, H = P its complete hull.
Def.1.1. Let (P, H) be a Euclidean Hartogs figure in en, g: = (gb ... ,gn):
P ---7 en be a biholomorphic mapping, and let P: = g(P), H: = g(H). Then
(p, H) is called a general Hartogs figure.
We shall try to illustrate this concept intuitively for n = 3. The Euclidean

Hartogs figure in absolute space appears in Fig. 6. In the future we shall use
the following symbolic representation in en. (Actually the situation is much
more complicated.)
29
II. Domains of Holomorphy
---t
I
IZ11
Figure 6. Euclidean Hartogs figure in C 3 .
g = (gb· .. , gn)
Figure 7. Symbolic representation of a general Hartogs figure.
Theorem 1.1. Let (P, B) be a general Hartogsfigure in 1[:", f holomorphic in

B. Then there is exactly one holomorphic function F on f5 with FIB = f.
PROOF. Let (f5, B) = (g(P), g(H}}, g: P - en be biholomorphic. Then fog
is holomorphic in H and by Theorem 5.5 of Chapter I there is exactly one
holomorphic function F* on P with F*IH = fog. Let F = F* g-l. Then 0
F is holomorphic in f5, FIB = f, and the uniqueness of the continuation

follows from the uniqueness of F*. D
Theorem 1.2 (Continuity theorem). Let Bel[:" be a region, (f5, B) a general

Hartogs figure with H c B, f a holomorphic function in B. If P n B is
connected, then f can be continued uniquely to B u P.
30
Figure 8. Illustration of the continuity theorem.

PROOF.f1: = fiR is holomorphic in R. Therefore there exists exactly one
holomorphic function f2 in P with fzlR = fl'
Let F( ). = { f(3) for 3E ~

3. f2(3) 3 E P.
Since B (\ P is a domain and fiR = f21R it follows (from the identity
theorem) that F is a well-defined holomorphic function on B u P. Clearly
FIB = f. The uniqueness of the continuation is a further consequence ofthe
identity theorem. D
The continuity theorem is fundamental to all further considerations.

Theorem 1.3. Letn? 2,P: = {3:131 < 1}betheunitpolycylinder,O::::; r~ < 1
for v = 1, ... , n, Pro: = {3:izvl : : ; r~ for all v} and G: = P - Pro' Then
every holomorphic function f on G can be extended uniquely to a function
holomorphic on P.
Figure 9. The proof of Theorem 1.3.
31
PROOF
1. Clearly G is a region. If 3l = (Z\l), . .. ,z~;'», A = 1, 2, are given,
then the points r(31), r(32) also lie in G. For A = 1,2 we can connect 3;. on
the torus T3A c G with r(3;} Define ({J;.:] --;. en by ({Jl(t): = (Z\ll(t), ... ,
z~}.)(t» with z~}.)(t): = IZ~l)1 + t· (max(!z~l)l, IZ~2)1) - IZ~l)D for A = 1,2,
v = 1, ... , n. Clearly Iz~;')(t)1 ~ IZ~l)1 > r? for v = 1, ... , n so that ({Jl(t) E G
for t E ] and A = 1,2.
({Jd2t) 0:::;t:::;1
Let {
({J(t): = ({J2(2 - 2t) 1 :::;
t :::; 1.
joins r(31) with r(32). Hence G is connected, and so is a domain.
({J
2. For v = 1, ... , n let E(,,): = {z" E 1[;:lz,,1 < 1}. Choose z~ E I[; with
r~ < Iz~1 < 1 :lnd set
g : P --;. P is a biholomorphic mapping with g(O, ... , 0, z~) = 0. If U =

U(z~) C {zn E I[;:r~ < IZnl < 1} is an open neighborhood, then E(l) x ... x
E(~ _ 1) X U c G, and therefore E(l) x ... x E(n -1) X T( U) c g( G). Choose
IZ11,···,IZ"-11
real numbers qb ... , qn with r? < q" < 1 for v = 1, ... , n - 1 and
{wn: Iwnl < qn} c T(U). Then
H: = {ro E P:q1 < IWl!} u {ro E p:lwJlI < qJl for J1 = 2, ... , n}
is contained in g(G) and (P, H) is a Euclidean Hartogs figure. (P, H) with
15: = g-l(p) = PandH: = g-l(H) is a general Hartogs figure with H c G.
32
Moreover, P n G = G is connected. The proposition now follows from the

continuity theorem. 0
The preceding theorem is a special case of the so-called Kugelsatz:
Let n ~ 2, G c en a domain, KeG a compact subset, G - K connected.

Then every function holomorphic in G - K can be uniquely extended to a
function holomorphic on G.
The proof of the Kugelsatz is substantially more difficult than that

of the preceding theorem. An important tool in its proof is the Bochner-
Martinelli integral formula, which is a generalization of the Cauchy integral
formula to a domain with piecewise smooth boundary.
Theorem 1.4. Let n ~ 2, Been be a region, and aD E B. Let f be holomorphic

in B': ~ B - {3D}. Then f has a unique holomorphic extension on B.
(For n ~ 2 there are no isolated singularities.)
PROOF. Without loss of generality we assume that aD = O. Let P be a poly-
cylinder about 30 with PcB, P': = P - {aD}. This is the situation of
Theorem 1.3; so there is a holomorphic function F' in P with F'IP' = flp'.
Let
F'(a)
{ f(a)
aE P
F(a): = 3 E B'.
F is the holomorphic continuation of f to B. o

Def.1.2. Let G c en - 1 be a domain, g:G --+ e a continuous function. Then
:F: = {3 E e x G:z 1 = g(Z2,"" zn)} is called a real (2n - 2)-dimensional
surface. If g is holomorphic, then :F is called an analytic surface.
Theorem 1.5. Let G c en - 1, G1 C e be domains, g: G --+ e be a continuous

function with g(G) c G1 and 30 E:F = graph(g). If V = U(ao) c G: =
G1 x G is an open neighborhood and f is a holomorphic function on S: =
(G - :F) u V, then f has a unique holomorphic extension to G.
PROOF. The uniqueness of the extension follows from the identity theorem
because G is a domain. For the proof of existence we treat only the case
G = {a* E en- 1 :la*1 < I}, G1 = E(l) (then G = P, the unit polycylinder in
en), and in addition assume that Ig(3*)1 < q < 1 for fixed q E IR and all
3* E G. The proof is in two steps:
1. S = (G - :F) u V is connected.
a. Let 31, 32 be points in G - :F. Then define
* ._
al . -
(1 +
2q' (1)
Z2 , . . . , Zn(1)) ' *. _
32. -
(1 + q
~'Z2"'"
(2) (2))
zn .
33
3;. and 31 lie in the punctured disk

(E(1) -
{g(7(l)
- 2 , .•• , -II J x {(z()')
7(A»)() 2 , ... , z().»)(
n J
and can therefore be connected by a path which does not cross :F. The line
segment connecting 3~ and 32 also lies in G - :F, so we can join 31 and 32
by a path in G - :F.
b. If 31 E U, 32 E G - :F, let U 1 be the connected component of 31 in U.
Since U 1 - :F is non-empty, we can join 31 in U 1 with a point 3~ E U 1 - :F.
In particular, 3~ then lies in G - :F and by case (a) we can join it with 32'
Ifal, 32 E U then both points can be connected with a point 30 E (; - :F and
therefore with one another.
2. Let n: e x en -1 -4 en -1 be the projection onto the second component.

Thennl:F::F -4 Gisa topological mapping with (nl:F)-1 = gandn(:F n U)
is an open neighborhood V of 30: = n(30).
Let h(zJ, ... , zn): = (idG,(ZI), h2(Z2),"" h~(zn)) with h~(zv): = (zv - z~)j
°
(z~ Zv - 1) for v = 2, ... ,n. h: P -4 P is a biholomorphic mapping with
h(O) = (0,30)' Set ql: = q and choose qv with < qv < 1 for v = 2, ... , n
so that h({(wJ,"" wn) E P:lwvl < qv for v = 2, ... , n}) is contained in
E(1) x V.
Let H: = {ro E P: Iwvl < qv for v = 2, ... , n} u {ro E P:ql < Iwd}. Then
(P, H) is a Euclidean Hartogs figure and (P, if) with if: = h(H) is a general
Hartogs figure. Clearly H c (E(I) x V) u {a E P:ql < IZ11} c S and by (1)
P n S = S is connected. The proposition follows from the continuity
theorem. D
Remark. If g is holomorphic, therefore :F an analytic surface, then there

is a holomorphic function f on (; - :F which does not permit a holomorphic
34
2. Pseudo convexity
extension beyond !F. For example set
PROOF. Assume there existed a point 30 E!F and an open neighborhood

U = U(30) c G such that f had a holomorphic extension F defined on
(G - !F) u U. Then there would be a sequence (3j) of points of (G - !F)
which converged to 30 and clearly as 3j -> 30, If(3j) I would tend to infinity.
But since F is continuous at 30 we would have lim f(3J = lim F(Oj) =
r--+ 00 j-+ co
F(30) and that would be a contradiction. D
With much more effort, one can prove the converse:

If !F eGis a real (2n -- 2)-dimensional surface and there is a holomorphic
fimction f in G - !F which is not holomorphically continuable to G, then !F
is an analytic surface.
2. Pseudoconvexity
Def.2.1. Let Been be a region. B is called pseudoconvex if for all general
Hartogs figures (P, H) with H c B, all of P lies in B.
Def. 2.2. Let Been be a region. f holomorphic in B, 30 E oB a point. f is

called completely singular at 30 if there exists a neighborhood V = V(30)
such that for any connected neighborhood U = U(Go) with U c V.
There is no holomorphic function F which in a non-empty open subset
of U n B coincides with f.
Def. 2.3. Let Been be a non-empty open set. B is called a region of

holomorphy if there is a function f holomorphic in B which is completely
singular at every point 30 E oB. If in addition B is connected, then B is
called a domain of holomorphy.
EXAMPLES
1. Since en has no boundary it trivially satisfies the requirements of
Def. 2.3. Therefore en is a domain of holomorphy.
2. The unit disk E(1) c C is a domain of holomorphy, as is shown in
1-dimensional theory.
3. The dicylinder E(l) x E(l) is a domain of holomorphy: If f: E(l) -> C
is a holomorphic function which is completely singular on oE(1), then
g:E(1) x E(1) -> C with g(Zb Z2): = f(Zl) + f(Z2) is a holomorphic func-
tion which is completely singular on o(E(1) x E(l))'
4. Let (P, H) be a Euclidean Hartogs figure, 30 E oH n P. For every
function f holomorphic in H there exists a function F hoi om orphic in P
with FIH = f. If V is an arbitrary open neighborhood of 30 which is entirely
35
contained in P and U is the connected component of 50 in V, then FI V is

holomorphic, Un H -=f. 0, and FlU n H = flU n H. Therefore H is not
a domain of holomorphy.
Theorem 2.1. Let Been be a region, G c en a domain with B n G -=f. 0

and (en - B) n G -=f. 0. Then for each connected component Q of B n G
G n 3Q n 3B -=f. 0.
PROOF. We have G = Q u (G - Q). Q is open and not empty, and because

(en - 0, G - Q is also non-empty. Since G is a domain it does
B) n G -=f.
not split into two non-empty open subsets. Hence G - Q is not open. Let
51 E G - Q not be an interior point. Then for every arbitrary neighborhood
U(51) c G it is true that U n Q -=f. 0. Therefore 51 lies in 3Q. If 51 E B then
there is a connected neighborhood V(51) c B n G (with V n Q -=f. 0 also).
But then Q u V is an open connected set in B n G which properly contains
Q. Since Q is a connected component this is a contradiction. Therefore ih
does not lie in B. Hence it follows that 51 E 3Q n 3B n G. 0
Theorem 2.2. Let G be a domain of holomorphy. Then G is pseudoconvex.

PROOF. Assume that G is not pseudoconvex. Then there is a Hartogs figure
(P, H) with H c G but P n G -=f. P. We choose an arbitrary 50 in Hand
set Q: = CPnG(30). Since H lies in P n G and is connected it follows that
H c Q. Furthermore, Q S P.
Since P n G -=f. 0 and (en - G) n P -=f. 0 there is by Theorem 2.1 a
point z 1 E 3Q n 3G n P.
Figure 12. The Proof of Theorem 2.2.
Let f be an arbitrary function holomorphic in G. Then flQ is also

holomorphic, and by the continuity theorem there is a function F holo-
morphic in P u Q = P with FIQ = flQ. Now if V = V(51) c P is an open
36
2. Pseudoconvexity
connected subset, then FI V is holomorphic, Q n V is open and non-empty,

and FIQ n V = flQ n V. Therefore G is not a domain of holomorphy.
This completes the proof by contradiction. 0
In 1910 the converse of the above theorem was proven in special cases
by E. E. Levi. The so-called Levi Conjecture, that this converse holds without
additional assumptions was first proved in 1942 by Oka for n = 2 and in
1954 for n > 2 simultaneously by Oka and by Norguet and Bremermann.
The proofis very deep and will not be presented here (see, for example, [7J).
To conclude this section, we will sketch the connection between the
pseudo convexity of a domain G and the curvature of its boundary.
Let Been be a region, 30 E Band cP: B --+ IR a twice continuously dif-
ferentiable function. One can regard B as a subset of 1R 2 n and consider the
tangent space T30 and the space T;o of the Pfaffian forms (see [21J, [22J).
The total differential of cP at the point 30 is the linear form
n n
(dcp)30 = L CPxv(30) dx. + .=1

.=1
L CPYv(30) dy. E T;o·
If f = g + ih is a complex-valued differentiable function, set df: = dg + idh.
Then dz. = dx. + idy., dz. = dx. - idy. and we can write the differential
in the form
n n
(dCP)30 = L CPzv(30) dz. + .=1

• =1
L CPl!.(30) dZ•.
Def. 2.4. A domain with C 2 boundary is a domain G c en with the following
properties:
1. G is bounded.
2. For each point 30 E 8G there is an open neighborhood U =
U(30) c en and a twice continuously differentiable function cP: U --+ IR
for which
a. Un G = {3E U:CP(3) < O}
b. (dCP)3 =1= 0 for all 3 E U.
Remark. Under the conditions of Oef. 2.4 the implicit function theorem
implies
1. 8G n U = {3 E U:CP(3) = O};
2. there is (after a reduction of U if necessary) a C 2 -diffeomorphism
(/>: U --+ B, where Been is a region such that (/>(U n G) = {3 E B:Xl < O}
and (/>(U n 8G) = {3 E B:Xl = O}.
We say that (G, 8G) is a differentiable manifold with boundary.
Theorem 2.3. Let G c en be a domain with C 2 boundary, U an open set with
U n 8G =1= 0. Let cP, t/J be two functions on U which satisfy the conditions
of De! 2.4. Then there is a uniquely determined positive differentiable func-
tion h on U such that cP ~ h . t/J.
37
PROOF. We only need to show that for each 00 E U n aG there is a neighbor-

hood V(oo) c U and in V exactly one differentiable function h with qJ\ V =
h· (t/tiV). Therefore let 30 E U n aG and W(oo) cUbe chosen so that there
is a CZ-diffeomorphism c]): W -> Been with C])(W n G) = {o E B:Xl < O},
C])(W naG) = {3 E B:Xl = O}. Then the functions (j): = qJ c])-1, lii: = 0
t/t 0 c]) -1 are twice differentiable in B. Without loss of generality we assume

that c])(oo) = 0 and B is convex (in the sense that for any two points in B
the connecting line segment lies in B). Define
Then (j) = hl . Xl and lii = hz . Xl' Since (dqJ)oo =I 0 and (dt/t)"O =I 0, near
a{j)jJXl and alii/ax 1 have no zeroes, 0 E B and the same holds for hb h z . Set
h: = (hdh2) c]) in a neighborhood of 00' Then
0
Here h is continuously differentiable and, near 00, has no zeroes. h is uniquely

determined, for outside aG we have h = qJ/t/t. 0
Def. 2.5. Let Been be a region, qJ: B -> ~ be twice continuously dif-
ferentiable, 00 E B. Then the quadratic form Lcp,3o with Lcp,oo(tIl): =
n
I qJz;Zj(oo)wiw j is called the Levi form of qJ at 30' qJ satisfies the Levi
i, j= 1
11
condition if the following holds: If tIl E en and I qJz;(oO)W i = 0, then

i= 1
Theorem 2.4. Let G en be a domain with C Z boundary, 00 E aG and U =

c
U(OO) an open neighborhood. Let qJ, t/t be two functions on U which satisfy
the conditions of Def. 2.4. If qJ satisfies the Levi condition at 30, so does tIt.
PROOF. We can find a twice continuously differentiable positive function h
n
on U with t/t = h· qJ. Now let tIl E en and I t/tZ;(OO)Wi = O. Then at 30
i= 1
n
o= I (h' qJz; + qJ . hz)wi

i=l
n 11
= h· I qJz; Wi (because of qJ\aG = 0), so I qJz;Wi = O.

i=l i= 1
38
3. Holomorphic Convexity
It follows that:
n n
LtjJ(ro) = L t/lziZjWiWj = L (hz/Pz j + ({Jzizjh + ({JZihz)WiWj
it Ctl
i,j=l i,j=l
= h· L<p(ro) + ({JZjW j ) hZiWi + jtl (tl ((JZiWi) hzjw j ,
where the last two terms vanish, as was shown above. Since h is positive, the
proposition follows. D
Def.2.6. For a domain G c en with C 2 boundary the Levi condition is

satisfied at a point 30 E 8G if there is an open neighborhood U = U(30)
and a function ({J on U which satisfies the conditions required by Def. 2.4
so that at 30 ({J satisfies the Levi condition.
Theorem 2.5. Let G c en

be a domain with C 2 boundary. Then G is pseudo-
convex if and only if the Levi condition is satisfied for every boundary point
ofG.
This theorem will not be proved here.
We will investigate whether there is a relationship between pseudocon-
vexity and the usual convexity of sets. We start with some observations
about convex domains in 1R2.
Let L be the set of linear mappings t : 1R2 ~ IR with
t(X) = aX l + bX 2 + c, a, b, CE IR.
A line g in 1R2 is a set of points x = Xo + to with t E IR and appropriate
fixed vectors xo, 0 E 1R2, 0 i= 0,
g = {x E 1R2:X = Xo + to, t E IR}.
Now let C E L with C(x) = aX l + bX2 + c and
(a, b) i= (0, 0). For b i=
let:ro: = (0, -c/b),o: = (1, -a/b); for b = Oanda i= Olet:ro: = (-c/a,O),
°
0: = (0,1). Then
t E IR} = g.
We therefore have two distinct ways of describing a straight line. We shall
use whichever description is most suitable.
Let g = {x E 1R2: x = Xo + to, t E IR} be a line. We denote the positive
ray {XE 1R2:X = Xo + to, t?= O} by g+ and the negative ray {XE 1R2:X =
Xo + to, t ~ O} by g -. If g is represented by the mapping t, then we define
H;: = {XEIR 2 :C(X) > O}, H;;: = {XEIR2: (x) < O}.
These are the two half-planes determined by g.
39
We shall use the following terminology: A set K lies relatively compact

in a set B (K c c B) if K is compact and contained in B.
Def. 3.1. Let M c ~2 be a subset. M is called geometrically convex if for

each point x E ~2 - M there is a line g with x E g and M c H;;.
Remark. The intersection of convex sets is again convex.
Def. 3.2. Let M c ~2 be an arbitrary subset. Then Me: = {x E ~2: t(x) :::;;
sup t(M) for all tEL} is called the geometrically convex hull of M.
Theorem 3.1 (The properties of the geometrically convex hull). Let M c ~2

be an arbitrary subset. Then:
1. Me Me.
2. Me is closed and geometrically convex.
3. Me = Me.
4. LetM l c M2 c ~2. Then (Ml)e c (M 2 )e'
5. If M is closed and geometrically convex, then M = Me.
6. If M is bounded, then Me is also bounded.
PROOF
1. Let x E M. Then for each tEL, t(x) :::;; sup l(M). Therefore x lies in Me.
2. Let Xo ¢ Me. Then there exists an ( E L with (xo) > sup (M). Since (
is continuous, it is also true that in an entire neighborhood of Xo we have
l(x) > sup l(M). Therefore Me is closed. t* with {*(x): = l(x) - t(xo) is in
Land t*(xo) = 0, sup t*(M e) = sup t*(M) = sup t(M) - sup l(M) = O.
Therefore g = {t E ~~: (*(x)",= O} is a line ~ith Xo E g and Me C H;;.
3. By (1) we have Me C Me. But fou E Me,t(x) :::;; sup (Me) :::;; sup (M)
for (E L. Hence it is also true that Me C Me.
4. sup t(M d :::;; sup t(M 2), for all ( E L, so (M l)e C (M 2)e'
5. Let Xo ¢ M. Since M is closed, there is an Xl E M with minimal distance
from Xo' If X2 is the midpoint of the line segment between Xo and Xl' then
X 2 ¢ M, and there is an tEL with (X2) = 0, tiM < O. Thus sup (M) :::;; 0,
but t(xo) > O. Therefore Xo ¢ Me and it follows that Me C M.
6. If M is bounded, then there is a closed rectangle Q with M c Q. For
each x E ~2 - Q there is a line g through x with Q c H;;, and therefore
an ( E L with t(x) = 0 and sup [(M) :::;; sup (Q) < O. That is, ~2 - Q C
2 ~ ~
~ - Me, there Me C Q. 0
Remark. Me is the smallest closed geometrically convex set which contains

M. (If M c K, K closed and geometrically convex, then Me C Ke = K.)
Theorem 3.2. Let B c ~2 be an open subset. B is geometrically convex if and

only if K c c B implies Ke c c B.
40
PROOF
1. Let B be convex. K c c B means that K is compact and lies in B.
Therefore K and also Ke is bounded. Since Ke is closed it follows that Ke is
compact. It remains to show that Ke lies in B.
We assume that there exists an Xo E Ke - B. Since B is convex there is
° °
and ( E L with ((xo) = and ((x) < for x E B. [ attains its supremum on
K so it is even true that C(xo) > sup ((K) ~ sup C(K). However, that con-
tradicts the fact that Xo lies in Ke. Hence Ke - B = 0.
2. Now we assume that Xo does not lie in B. First we show that for every
line 9 which contains Xo either g+ n B = 0 or g- n B = 0. From that we
shall deduce finally that there is a line go through Xo which does not intersect
B at all. We obtain go by rotating the above line 9 about Xo until the desired
effect occurs.
a. Assume that there exists a line 9 = {x E ~2: x = Xo + to, t E ~} with
g+ n B i= 0 and g- n B i= 0. Then let Xl = Xo + tID E g+ n Band
X2 = Xo + t2D E g- n B. The connecting line segment S between Xl and X2
is given by
S = {x = Xl + t(X2 - xl):t E [0, 1J}
= {x = t*Xl + t**X2 with t*, t** ~ 0, t* + t** = 1}.
Now let to: = -t 2/(tl - t 2) and to*: = 1 - to = tt/(tl - t2). Then xo: =
fOXl + to*X2 E Sand Xo = xo. Let C E L be arbitrary. We shall show that
((xo) ~ m = max(((x l ), ((X2)). Clearly, we can restrict ourselves to homo-
geneous functions (: ((x) = aXl + bx 2. Then ((xo) = ((taXI + to*X2) =
tof(xd + to*[(X2) ~ (to + to*)m = m.
Now let K: = {Xb x2 }. Then K c c B and therefore, Ke c c B. Because
f(xo) ~ max(((x l ), f(x 2)) = sup f(K) for each (E L it follows that Xo EKe.
That means Xo E B, which is a contradiction.
b. Now let such a 9 be given. If 9 + n B = 0 and 9 - n B = 0 we are
done. We assume that g+ n B i= 0. Let 80 be the angle between 9 and the
xl-axis, 8 1 : = sup{8:8 0 ~ 8 ~ 80 + 11:, g: n B i= 0}, where go denotes
the line which makes the angle 8 with the xl-axis.
Case 1. gt; n B i= 0. Then 8 1 < 80 + 11:. If Xl E gt; n B, then there
°
exists an e > such that Ve(Xl) lies in B. We can now find a 82 with 8 1 < 82 <
80 + 11: such that gt, still intersects Ve(Xl) and of course B as well. That
contradicts the definition of 8 1 , so Case 1 can be discarded.
Case 2. gOl n B i= 0. We proceed in exactly the same manner as above
to obtain a contradiction.
c. Let H+ and H- be the two half-planes belonging to gOl. From (b)
B c H+ U H-. But from (a) B must lie on exactly one side of go,. Suitable
choice of the orientation of go, yields that B lies in H- 0
One could use the conditions of Theorem 3.1 as the definition of convexity.
We now come to the notion of holomorphic convexity by replacing linear
functions by holomorphic fuhctions.
41
Def.3.3. LetB c enbearegion,K c Basubset. ThenK B : = {3EB:lf(3)1 ~

suplf(K)1 for every holomorphic function f in B} is called the holomor-
phically convex hull of K in B. When no misunderstanding can arise, we
write K instead of K B •
Theorem 3.3 (The properties of the holomorphically convex hull). Let

Been be a region, K c B a subset. Then:
l.KcK
2. K is closed in B.
3. K = K
4. Let K1 c K2 C B. Then K1 c K 2.
5. If K is bounded, then K is also bounded.
PROOF
l.Foq E K, If(3)1 ~ suplf(K)I.
2. Let 3 E B - 1(, Then there exists a holomorphic function f on B with
If(3) I > suplf(K)I· Since If I is continuous, these inequalities hold on an
entire neighborhood U(3) c B which is contained in B - 1(, Therefore
B - K is open.
3. suplf(K)1 = suplf(K)I·
4. The statement is trivial.
5. If K is bounded then there exists an R > 0 such that K is contained in
the set fa = (Zl,·'" zn):lzvl ~ R}. The coordinate functions fv(3) == Zv are
holomorphic in B, and therefore for 3 E K, Izvl = Ifv(3)1 ~ suplf.(K)I ~ R.
Hence K is also bounded. 0
Def.3.4. Let Been be a region. B is called holomorphically convex if

K c c B implies K c c B.
Remark. In e every domain is holomorphically convex.

PROOF. Let K c c G. Then K is bounded, and therefore K also. Hence K
is compact and it only remains to show that KeG. If there is a point
Zo E K - G, then Zo lies in aK naG. But then f(z) = 1/(z - zo) is holo-
morphic in G.
Now let (zv) be a sequence in K with lim Zv = zoo From the definition of K,
v~ro
If(zvll ~ suplf(K)1 ~ suplf(K)I, contradicting thefactthat {If(zv)l: v EN} is

unbounded. 0
By no means is every domain in en holomorphically convex. However,

we have
Theorem 3.4. Let Been be a region. If B is geometrically convex, then B

is also holomorphicaUy convex.
42
4. The Thullen Theorem
PROOF. We must first clarify when a region in en is geometrically convex.

Let t: en --+ ~ be a homogeneous linear mapping of the form
n n n n
e(3) = L
• =1
a.x. + L
.=1
b.y. = L
.=1
tX.z. + L P;z•.
.=1
Since we are supposed to have «3) = t(3) it follows that P. = a., and
therefore
B is geometrically convex if K c c B implies Ke c c B, where we define

Ke: = {3 E en:e(3) ~ sup t(K) for all homogeneous linear mappings t}.
Ke has the properties required by Theorem 3.l.
Now let K c c B. Then Ke C C B. Let 30 E B - Ke. Then there exists
n
a linear homogeneous mapping e with t(3) = 2 . Re L tX.z. and t(30) >
.=1
sup e(K).
Now we define a function f holomorphic on B by
f(3): = exp (2 ..t1 tX.z.)'

Then
therefore
If(30)1 = exp o e(30) > sup«expoe)(K» = suplf(K)I·
Thus 30 E B - KB, and we have shown KB C Ke C C B. This proves KB
cc B.
In general holomorphic convexity is a much weaker property than

geometric convexity.

Let M c en be an arbitrary non-empty subset. If 30 E en - M is a point,
then dist'(30, M): = inf 13 - 301 is a non-negative real number. If K c
3 EM
en - M is a compact set and M closed, then
dist'(K, M): = inf dist' (3, M)
3EK
is a positive number.
43
Def.4.1. Let Been be a region, e > 0. We define

BE: = {5 E B:dist'(5, en - B) ~ e}.
Remarks
1. {5} is compact, en - B is closed, so for 5 E B dist'(5, en - B) > 0.
2. If 5 E B, then 5 E BE for e: = dist'(5, en - B). Therefore B = BE' U
E> °
Theorem 4.1. BE is closed.
PROOF. Let 50 E en - BE' We define 15: = dist'(50, en - B). e > 15 ~ 0, so
e - 15 > 0. Let U: = U~-o(50) = kl5 - 501 < e - b}. For5E Uwehave
dist'(5, en - B) ~ dist'(3, 50) + dist'(30, en - B) ~ e - 15 + 15 = e. Therefore
U lies in en - BE> that is, en - BE is open. 0
We need the following terminology. Let M c en be an arbitrary

non-empty set. A function f is called holomorphic in M if f is defined and
holomorphic in an open set U = U(M) with U ::J M.
Theorem 4.2. Let B be a region, f holomorphic in 13, If(13) 1 ~ M, e > 0, and

30 E BE a point. In a neighborhood U = U(30) c B, let f have the power
00
series expansion f(3) = I av (3 - 30r. Then for all v,

v=o
PROOF. Let P: = {5 E e n :dist'(3, 30) < e}. Then for 3 E P, dist'(3, en - B) ~

en - B) - dist'(3, 50) > e - e = 0. Therefore P lies in B, that is
dist'(50,
PeE c VeE), where V is an open neighborhood of 13 and f is defined
and holomorphic on V. Then
1
av, ..... vn
1
= 1
1
(2nW
ST(~l -f(~b···'~n)d~l···d~n
z\O)y,+l ... (~n - z~O)rn+l
I
where T is the n-dimensional torus T: = {(~l' ... , ~n):~v

e v ~ 2n}. Because d~v = e . eiO , • ide v = i( ~v - z~O)) de.,
= z~O) + ee iO " °
~
holds. o
44
Theorem 4.3 Let B c: en be a region, f holomorphic in B, e > 0, and K c: Be

compact. Then for every () with 0 < () < e there exists an M > 0 such that
M
sup ia v(3) i ~ s:lvl'
3EK u
(We denote by a v (50) the coefficients a,. of the power series expansion
00
f(3) = L a (3 -
v 30)v.)
v=o
PROOF 0
1. Set B*: = (B.'-::).

We claim that K lies in (B*)a, that is, that for 30 E K
we have dist'(30, eN - B*) ~ (). Assume there is a 51 E eN - B* and a {y with
o < ()' < () such that dist'(30, 31) < {y'. Since 31 does not lie in B*, 31 is not
an interior point of Be-a. Therefore there still are points of en - Be-a arbi-
trarily close to 51' Now let e' > 0 be given. Then there is a 32 E eN - Be-a such
that dist'(31, 52) < e'. Since dist'(52, en - B) < e - () it follows that there
exists a 53 E en - B such that dist'(52, 33) < e - (). Therefore dist'(30, 33) ~
dist'(50, 31) + dist'(5b 32) + dist'(32, 53) < ()' + e' + e - (). This holds for
every e' > O. Therefore
dist'(50, eN - B) ~ (b' - ()) + e < e.

So 30 does not lie in B., contrary to our assumption. So K must lie in (B*)a'
2. K is bounded, so there exists a polycylinder P = P(O) with K c: P. We
can choose P in such a way that dist'(K, en - P) > (). Then let B': = P n
B*. B' is open and non-empty.
We shall apply Theorem 4.2 to the region B'. Clearly R' is compact. More-
over, R' c: P n R* c: P n Be-a c: B. Therefore f is holomorphic in R' and
can be bounded there by a constant M.
Because dist'(K, en -
P) > () and K c: (B*)(j, K c: B'". Therefore
supiUiB')i M
ia v(30)i ~ {)Ivl ~ ()Ivl
for every point 50 E K; in particular
M
sup ia v(5)i ~ 1vT'
3EK ()
o
Theorem 4.4 (Cartan-Thullen). If B c: en is a region of holomorphy, then B

is holomorphically convex.
PROOF. Let K c: c: B. We want to show that K c: c: B. Let e: =

dist'(K, en - B) ~ dist'(K, en -
B) > O. Clearly K lies in Be.
1. We assert that even the holomorphic convex hull K lies in Be. Suppose
this is not so. Then there is a 30 E K - Be. Since B is a region ofholomorphy,
there is a function f holomorphic in B which is completely singular at each
45
point 3 E aB. In a neighborhood V = V(30) c B f has the expansion

00
f(3) = I a v(3 - 30)V.

v=o
is holomorphic in B by the theorem on partial derivatives, and av(30) = avo

Because 30 E K, la v(30)1 ~ sup la v(3)1. And by Theorem 4.3 for every b with
3EK
o 0 with
M
sup lav (3)1 ~ ;5IvT'
3EK
Therefore
00
dominates I
lavC3 - 30YI· Now let Po(30) be the polycylinder about 30 with
v=o
radius b. For 3 E Po(30), I is a geometric series, and therefore convergent.
00
Hence I avC3 - 30Y converges on the interior of P0(30)'

v=o
Let p.: = {3:lzv - z~O)1 < e}. The sets Po with 0 < b < e exhaust P.,
00
hence I av (3 - 30Y is convergent in P., say to the holomorphic function]'

v=o
Near 30, f = ]. If Q: = Cp nB (30) is the component of 30 in p. n B" then
f = J in Q. There exists a p~int 31 E p. n aQ naB. If V = V(3d c p. is an
open connected neighborhood, then J is holomorphic in V, V n Q is open
in V n BandflV n Q = JIV n Q. Thatisacontradiction,forfissupposed
to be completely singular at 31' Therefore K c B•.
2. Since en - K = (B - K) u (Cn - B.) is open, it follows that K is
closed. Since K is compact, K is bounded, and by Theorem 3.3 K is also
bounded. Hence K is compact. This completes the proof. D
In the next section we shall show that the converse of this theorem also
holds.
5. Holomorphically Convex Domains

Theorem 5.1. Let B c c n be a region. Then there exists a sequence of
subsets Kv c B with the following properties.
1. Kv is compact for all v E N.
00
2. U Kv
v=l
= B.
3. Kv c KV+1 for. all v EN.
46
PROOF. It is clear how the Kv should be chosen: If Pv: = {3: Iz).1 ~ v for
all A}, we define Kv: = Pv ( l B 1/v . Obviously Kv is compact and lies in B.
Let 3 E B. Then e: = dist'(3, en - B) > 0 and there exists a Vo E N with
3 E P"o· Let v ~ max(vo, l/e). Then 3 E P v ( l B 1/v = Kv. Therefore B =
00
U Kv' Let 30 E B 1/v . Then

v=1
dist'(30, en - B) ~~> _1_,

v v+ 1
and
U = U(30): = {3 E en:dist'(3, 30) < 1_}
~v _v_
+1
is an open neighborhood of 30. For 3 E U, however,
dist'(3, en - B) ~ dist'(30, en - B)
_ dist'(3, 30) > ~
v
_ (~
v
__
v+ 1
1_) _1_.
+
=
v 1
Therefore U lies in B 1/(v+ 1)' H~nce 30 is an interior point of B 1/(v+ 1) and

B 1/v c B1/ (v+ 1)· Because P v c P v+1 it follows that Kv c Kv+ 1. 0
Remark. It is actually true that

cI) cI) cI) 00
B = UK v, since B = U Kv = U Kv - 1 C U Kv c B.
v=1 v=1 v=2 v=2
In the remainder of this section we shall call any sequence of compact

subsets of a region B which satisfy the conditions of Theorem 5.1 a normal
exhaustion of B. We define M 1: = K1 and Mv: = Kv - K v- 1 for v ~ 2.
Then:
1. M v (l MIL = 0 for v i= fl,

00
2. U Mv = B,
v= 1
Il
3. U Mv
v= 1
= Kil
Theorem 5.2. Let Been be a region and (K,,) a normal exhaustion of B.

Then there exists a strictly monotonic increasing subsequence (All) of the
natural numbers and a sequence (3 1l ) of points in B such that
1. 31l E M).,!
2. If G c en is a domain, G ( l B i= 0, G ( l (en - B) i= 0 and G1 a
connected component of G ( l B, then G1 contains irifinitely many points
of the sequence (3 1l )!' EN.
47
PROOF
1. A point 3 = (Zb ... , Zn) E en is called rational if
with Xv, Yv E Q for all v.
The set of U,(o) with rational 0 E en and I: E Q forms a countable basis for
the topology of en; we denote this basis by lID = {~: KEN}.
Now let m:={~EIID:~nB#-0 and ~n(lCn-B)#-0}. If
~ E m, then ~ n B has countably many components, as each contains
at least one rational point.
Let 58: = {BIl: There is a KEN such that ~ Em and Bil is a component
of~ n B}.
58 is now a countable system {BIl:p EN} of connected sets, and for each
pEN there is a K = K(p) such that Bil C ~ n B.
2. The sequences (Jell) and (Oil) are now constructed inductively: Let 01
00
be arbitrary in B l . Then Bl C ~(1) nBc Band B = U Kv' Therefore

v= 1
there exists a v(l) EN such that 01 lies in K v (l)' Since the system of Mv is a
decomposition of B, there is a Je(l) ~ v(l) such that 01 E M A(l)'
Now suppose 01, ... , Or 1 have been constructed so that 0, E K v (,) n B, and
Je(I) is chosen so that 0, E M A(,), I = 1, ... , p - 1. Choose 51l E Bil - Kv(r 1)
arbitrarily. That is possible since there is a point 0; E ~(Il) n oB n oBI'"
en - Kv(r 1) is an open neighborhood of 0; and contains points of BI'"
These points lie in Bil - KV(Il-l)' Now there is a v(p) E N with Oil E KV(Il)'
Therefore Oil E KV(Il) n B Il , and there exists exactly one Je(p) ~ v(p) with
OIlEMA(Il)'
3. If Je(p) ~ v(p - 1), we would have Oil E M;.(Il) c K V(Il-l) contrary
to construction. Therefore v(p - 1) < }e(p) < v(p); the sequences v(p) and
Je(p) are strictly monotone increasing.
4. Now let G c en be a domain, G n B #- 0, G n (en - B) #- 0, and
Gl a component of G n B. We assume that only finitely many 51l lie in Gb
say 01, ... ,3m' Then let
G*: = G - {01' ... ,3m},
Gi: = G1 - {Ob' .. 'Om}.
G* and Gi are again domains and Gi c G* n B. Let 51 E Gi, 0 E G* n B,
01 and 0 be connected by a path in G* n B. Then they can be connected by
a path in G n B, and 0 belongs to G1 n G* = Gi. It follows that Gi is a
component of G* n B.
Now let 00 E G* n oGi noB. There is a KEN such that ~ E m and
50 E ~ and such that ~ nBc G* n B. Moreover ~ n B must contain
points of Gi.
Now let 01 E ~ n B n Gi (and B*: = C w , n B(Ol)' Because Gi =
CGo n B(Ol) it follows that B* c Gi. B* is an element of 58 and therefore
contains a 31'" That is a contradiction. The assumption was false and we
have proved the theorem. 0
48
Theorem 5.3 Let B c 1C" be a region and (Kv) be a normal exhaustion of B.

In addition, suppose that, for each v E N, Kv = Kv.
Let (All) be a strictly monotone increasing sequence of natural numbers
and (3 11 ) a sequence of points with 311 EM)."
Then there exists a holomorphic function f in B such that If(3 1l )1 is
unbounded.
00
PROOF. We represent f as the limit function of an infinite series f = I f ll ;

11=1
we define the summands f/1 by induction.
1. Let fl: = 1. Now suppose fI> ... ,fll -l are constructed. Since 311 does
not lie in K}..(I1)-1 = K}..(/1)- I> there exists a function g holomorphic in B
such that Ig(31l )1 > q, where q: = supl(gIK}..(Il)-dl. By normalization one
/1-1
can make g(3 1l ) = 1. Hence q < 1. Now let a/1: = I
fv(3 1l ) and m be chosen
v=1
so that qll! < 1/(/1 + la ll !)' rll. This is possible since qll! tends to zero. We
set f/1: = (/1 + la/1I)' gil!. Then f/1 is holomorphic in B, f ll (3/1) = /1 + lalll and
supl(j~IKA(Il)-I)1 < r/1.
<Xl
2. We assert that I f/1 converges uniformly in the interior of B. Let

11=1
K c B be compact. There exists a Vo EN such that K c Kva - 1 ' Now let
/10 E N be chosen so that A(/1o) ): Vo. Then K}..(/1) :::J Kva for /1 ): /10, that is,
K}..(I1)-1 :::J K va - 1' By construction supIU/1IK}..(Il)-I)1 < 2- 1', therefore in
co 00
particular supl(J;,IK)1 < rll. As I r/1 dominates I fll in K, the series

/1=1 /1=1
OJ
converges uniformly in K. Therefore f = I f/1 is holomorphic in B.

11=1
3. If(31l) 1 = IV~1 h(3 JI 1 ): Ifll(31l )1 -I:t: h(3/l)1
co
I
we have
00
If(3/1) 1): /1 - I rv): /1 - 1.
Theorem 5.4. Let Been be a region. If B is holomorphically convex then

there exists a normal exhausting (Kv) of B with the property that Kv = Rv
for every v EN.
PROOF. Let (Kv) be any normal exhaustion of B. Then for all v, Kv c c B
and as B is holomorphically convex, it follows that Rv c c B. Rv is there-
fore a compact subset of B. We now construct a subsequence of the Rv.
LetK~: = R 1 • .
49
Suppose Ki, ... , K~-1 have been constructed (K~-1 compact and
K~-1 = K~-l). Then there exists a },(v) E N such that K~-l c K;.(v). Let
K~: = K;.(v) Clearly the K~ are compact subsets of B with K~ = K~. More-
00 00 CD
over U K~ = U K;.(v):::::J U K}.(v) = B and K~ c K}.(V+1) c K~+1.

v=1 v=1 v=1
We now come to the main theorem of this section.
Theorem 5.5. Let Been be a region. If B is holomorphically convex, then

B is a region of holomorphy.
PROOF. By the preceding theorem there is a normal exhausting (Kv) of B
with Kv = Kv for all v and hence sequences (}'IJ and (3 11 ) in the sense of
Theorem 5.2 and a function fholomorphic in B with If(311 )1--> co for /1--> co.
We now show that f is completely singular at every boundary point of

B. Assume that there exists a point 50 E aB at which f has no essential singu-
larity; that is, that there is an open connected neighborhood U = U(30)
and a function j holomorphic in U such that f = j in near some point
31 E Un B.
Let U 1: = C unB (51) be the connected component of31 in Un B. There
is a point 52 E U n au 1 n aB. Let V = V(52) be an open connected neigh-
borhood of 51 with Vee U.
V n U 1 contains a point 33. Let Vl : = CVn B(53). If 3 lies in Vl, then 3
can be joined to 33 in V nBc U n B, and 33 lies in U 1 so that it, too, can
be joined to 31 in Un B. Hence V1 c U 1.
Because "f = jin the region of 51", it follows thatf = Jin U 1 and from
this that f = j in V1 also. On the other hand, infinitely many points of the
sequence (5 11 ) lie in V1 • That is, Jis unbounded in V1 • That leads to a contra-
diction, since Jis holomorphic in U, Vis compact and therefore suplcJl V1 )1 :;::;
suplUiV)1 :;::; suplUI V")i < co. Therefore f is completely singular in aBo D
Def. 5.1. Let M c en be an arbitrary subset. D c M is called discrete if

D has no cluster points in M.
Theorem 5.6. Let Been be a region. B is holomorphically convex if and

only if for every infinite set M which is discrete in B there exists afunction
f holomorphic in B such that If I is unbounded on D.
(This theorem permits a simpler definition of holomorphically convex.
It holds in complete generality, both on complex manifolds and complex
spaces.)
PROOF
1. Let B be holomorphically convex, DeB infinite and discrete. More-
over, let (KJ be a normal exhaustion of B with K~ = Kv. Then Kv n D is
finite for every v E N.
50
6. Examples
We construct a sequence (31') of points in D by induction:

Let 01 ED be arbitrary, v(l) E N minimal with the property that 01 lies
in K v(1)'
Now suppose the points 31> ... ,01<-1 have been constructed. Then we
choose 01' E D - Kv(r 1) where v(J1. - 1) is to be the smallest number with
the property that 3r 1 lies in Kv(r 1)' That is possible, for Kv n D is finite,
so D - Kv contains infinitely many points.
Thus, v(J1.) is a strictly monotone increasing sequence of natural numbers
such that ol'lies in Mv(I')'
By Theorem 5.3 there is a function f holomorphic in B such that If(31')1
is unbounded. Therefore If I is unbounded on D.
2. Now let the criterion be satisfied. We assume B not to be holomor-
phically convex: that is, we assume there is a K c c B such that K is not
relatively compact in B. We construct an appropriate set D.
Let (Kv) be a normal exhaustion of B. Clearly K - Kv =I- 0 for all v,
otherwise we would have K c c B. We define D by induction as a point
sequence. Let 31 E K be arbitrary and v(l) minimal such that 31 E K v(l)'
Suppose 31>' .. ,31'-1 have been constructed, and for 1 ~ Il ~ J1. - 1 let
v(ll) always be the smallest number such that 3..1. E KV(A)' Then we choose
31' arbitrarily in K - Kv(r 1)' Then v(),) is strictly monotone increasing
and 31' E Kv(I')'
Let D be the set of points 31" J1. E N. If 30 E B, then there exists a J1. E N
such that 30 lies in KI' c ](1'+1 c ](V(I' + 1)' ](V(I<+l) is an open neighborhood
of 30, which contains only the points 31, ... ,31'+1' Therefore 30 is not a
cluster point of D. The set D is discrete in B. By assumption there is a function
f holomorphic in B which is unbounded on D. But then there exists a J1. E N
such that If(31')1 > suplf(K)I. That means that 31' does not lie in K, contrary
to construction. Therefore B must be holomorphically convex. 0
6. Examples
Theorem 6.1. Let Bee be a region. Then B is a region of holomorphy.
(H ence for every open set B in e there exists a holomorphic function which
cannot be extended to any proper open superset of B.)
PROOF. It was shown in Section 3, that every region in e is holomorphically
convex. From Theorem 5.4 it follows that B is a region of holomorphy. 0
In en we have the following theorem:

Theorem 6.2. Let Been be a region. Then the following statements are
are equivalent:
1. B is pseudo-convex.
2. B is a region of holomorphy.
3. B is holomorphically convex.
51
4. For every infinite set D discrete in B there exists afunctionfholomorphic

in B such thatfis unbounded on D.
PROOF. The statements have all been proved in the preceding paragraphs
(apart from the solution of the Levi problem: if B is pseudoconvex, then B is
a region ofholomorphy). ,
0
Theorem 6.3. If G c en is a geometrically convex domain then G is a domain
of holomorphy.
The n-fold cartesian product of open sets is a further example.
Theorem 6.4. Let VI, ... , v" c e be regions. Then V: = VI X ••• x v" c
en is a region of holomorphy.
PROOF. Let D c V be a discrete infinite set and (3/l) a sequence of distinct
points of D with 3/l = (z<t), ... ,zli»). If the sequence (z<t») in VI has a cluster
point z\O), then there exists a subsequence (z<t,(v») which converges to z\O).
If the sequence (z~,(v») in V2 has a cluster point z~O), then there exists a
subsequence (Z~2(V») which converges to z~O). Continuing this way until
n-th component (thus obtaining a subsequence (zlin(v») which converges to
a point z~O) E v,,), then the sequence (3/l n (v») converges to 3o: = (z\O), ... , z~O») E
V. This is a contradiction, because D is discrete in V.
Therefore there is a q E {l, ... ,n} and a subsequence z/l, = (z<t'), ... ,
zli'») of the sequence (3/l) such that the sequence (z~,») has no cluster point
in Vq.
By Theorems 6.1 and 6.2 there exists a functionfholomorphic in Vq for
which f(z~'») is unbounded. Now g(Zb . .. , tn ): = f(zq) is a holomorphic
function on V which is unbounded on D. Therefore V is holomorphically
convex. 0
Def. 6.1. Let Been and Vb ... , ~ c e be regions,fI' ... , fk holomorphic
functions in B, and U c B an open subset. The set P: = {3 E U :fh) E J.j
Figure 13. Analytic polyhedron in B.

52
6. Examples
for j = 1, ... , k} is called an analytic polyhedron in B if Pee U. If, in

addition, Vi = ... = ~ = {z E C: Izi < I}, then one speaks of a special
analytic polyhedron in B.
Theorem 6.5. Let Been be a region. Then every analytic polyhedron in B

is a region of holomorphy.
PROOF. Let U, Vb" . , ~, fb' .. , hand P be given as in Def. 6.1. Then
F:= U11U, ... ,fkl U): U -> Ck is a holomorphic mapping and P =
F-i(~ X ... x ~). By Theorem 6.4 V: = Vi x ... x ~ is a region of
holomorphy. Let D c P be an infinite discrete set. It suffices to show
that F(D) c V is infinite and discrete. For then there exists a function
holomorphic in V which is unbounded on F(D) and the function g: = f 0 F
satisfies the corresponding conditions in P.
Let (3j) be a subsequence of pairwise distinct points of D. F(3j) has a
cluster point too in V. Then there exists a subsequence F(3jJ which converges
to too. The points 3j, lie in P, and by assumption P is compact. Hence C3j)
has a cluster point 31 in P, and there is therefore a subsequence (3j,"(p)), which
converges to 31 E P c U. F(3j,"(p)) then converges to F(31) and to too simul-
taneously; that is, F(3d = too lies in V. This means that 31 lies in F- 1 (V) = P,
which is a contradiction to the assumption that D is discrete in P. Hence it
follows that F(a) has no cluster point in V, and we are done. 0
EXAMPLE. Let q < 1 be a positive real number and

P: = {3 E C2:l zd < 1, IZ21 < 1, IZ1 . z21 < q}.
IZ11
Figure 14. Example of a nontrivial analytic polyhedron.
P is clearly an analytic polyhedron, but neither a geometrically convex

region nor a cartesian product of regions. The analytic polyhedrons therefore
enrich our stock of examples of regions of holomorphy.
53
We shall show that every region of holomorphy is "almost" an analytic

polyhedron:
Theorem 6.6. Every region of holomorphy Been

can be exhausted by
analytic polyhedra in the sense that there exists a sequence (P j ) of special
CJ)
analytic polyhedra in B with P j c c P j + 1 and UP j = B.

j+ 1
PROOF
1. Let (K j ) be a normal exhaustion of B with K j = Kj • If 3 E oK j + 1 is an
arbitrary point, then 3 does not lie in K j C Kj + 10 and therefore not in Kj •
Hence there exists a function f holomorphic in B for which q: = supi f(K j) I <
If(3)1. By multiplication with a suitable constant we obtain q < 1 < If(3)1,
and then there is an entire neighborhood U = U(3) such that lUI U)I > 1.
Since the boundary oK j + 1 is compact, we can find finitely many open
sets U(3V)), . .. , U(3V))
J
and corresponding functions fV), ... ,fV)J holomor-
kj
phic in B such that oK j + 1 C U U(3~)) and IU~)IU(3~)))1 > 1. Let Pj: =
p=l
{3EKj+1:lf~)(3)1 < Iforp = l, ... ,kj }.
kj
2. ClearlyK j C Pj C Kj + 1 .ButbeyondthatM: = K j + 1 - U U(3~)) =
p=l
K+j 1 (\ (en - UU(3~))) is a compact set with P

p=l
j C M C K j + l' Con-
sequently Pj c M = M C Kj + 1; that is, Pj C C K j + l' Thus Pj is a special
apalytic polyhedron in B. It follows trivially from the relation "Kj C P j C C
K j + 1" that the sequence (P j ) exhausts the region B. D
In the theory of Stein manifolds one can prove the converse ofthis theorem.
7. Riemann Domains over en

If G is a domain in en,
we can ask if there exists a largest set M with GeM
over which every function holomorphic in G can be (holomorphically)
extended. It turns out that we cannot restrict ourselves to subsets of en.
We must consider domains covering en:
Def.7.1. A (Riemann) domain over en

is a pair (G, n) with the following
properties:
1. G is a connected topological space.
2. For every two points X10 X2 E G with Xl i= X2 there are open
neighborhoods U 1 = U 1(X1) C G, U 2 = U 2 (X 2 ) C Gwith U 1 (\ U 2 = 0
(that is, G is a Hausdorff space).
3. n:G ~ en
is a locally topological mapping (that is: If x E G and
3: = n(x) is the "base point of x", then there exist open neighborhoods
U = U(x) C G and V = V(3) C en
such that nl U: U ~ V is topological).
54
Remarks
a. The mapping n is in particular continuous.
b. G is path-connected.
Take Xo E G and let Z: = {x E G: x can be joined with Xo in G}.
1. Xo E Z, therefore Z =1= 0.
2. Z is open, since G is locally homeomorphic to en and therefore locally
pathwise connected.
3. Z is closed: If Xl E az, then there exists a neighborhood U(x l ) c G
homeomorphic to en with U n Z =1= 0. We can join Xl in U with a point
X2 E U n Z, and from the definition of Z we can join X2 with Xo. Therefore
Xl also belongs to Z.
It follows from statements 1,2, and 3 that Z = G.

c. If(G" nv) are domains over en
for v = 1, ... , £ and Xv E Gv are points
with nJx~) = 30 for v = 1, ... ,£, then there are open neighborhoods
Uv(xJ c Gv and a connected open neighborhood V(30) c en such that for
v = 1, ... ,£nvIU v: U v ~ V is topological.
PROOF. Choose neighborhoods Vv(x v ) c G" v,,(30) c en such that

nv IVv: Vv ~ v" is topological. Then let V be the component of 30 in V: =
n v" and U
t
v=l
v: = (nvIVv)-l(V) for v = 1, ... , t. 0
EXAMPLES
1. Domains in en. Let G c en be a domain. n: = id G the natural inclusion.
Clearly (G, n) is a domain over en in the sense of Def. 7.1.
2. The Riemann surface of JZ.Let G: = {(w, z) E e 2 :w 2 = z, z =1= O} be
provided with the relative topology induced from e 2 • G is a Hausdorff space.
The mapping <p:e - {O} ~ G defined by t 1-+ (t, t 2 ) is bijective and con-
tinuous. G is therefore connected.
Now let n:e 2 ~ e be defined by new, z): = z. Then n: = nIG:G ~ e is
continuous. If(wo, zo) EGis an arbitrary point, then Zo =1= 0, and we can find
a simply connected neighborhood V(zo) c e - {O}. From the theory of a
single variable we know that there exists a holomorphic function f in V with
f2(Z) == z and f(zo) = woo We denote f by JZ.
Then n-l(V) can be written
as the union of the disjoint open sets U +: = U = {(fez), z):z E V} and
U _: = {( - fez), z):z E V}. Let l(z): = (f(z), z). Then (nIU)-l = 1, that is
nl U is topological. Hence (G, n) is a domain over e, the so-called "Riemann
surface of JZ".
e
G can be visualized in the following manner: We cover with two additional
copies of e, cut both these "sheets" along the positive real axis and paste them
crosswise to one another (this is not possible in [R3 without self intersection,
but in higher dimensions, it is).
55
Figure 15. The Riemann surface of JZ.
Next we consider Riemann domains with a distinguished point.
Def.7.2. Let 30 E en be fixed. Then a (Riemann) domain over e" with base
point is a triple ffi = (G, n, xo) for which:
1. (G, n) is a domain over en.
2. n(xo} = 30.
The point Xo is called the base point.
Def.7.3. Let ffij = (G j, nj' xJ be domains with base point over en. We
say ffi l < ffi2 ("ffi l is contained in ffi 2") if there is a continuous mapping
rp: G 1 ~ G2 with the following properties:
1. nl = n2 0 rp ("rp preserves fibers")
2. rp(xd = X2.
Theorem 7.1 (Uniqueness of lifting). Let ffi = (G, n, xo) be a domain over
en with base point, Y a connected topological space and Yo E Ya point.
If t/lb t/l2: Y ~ G are continuous mappings with t/ll(YO) = t/l2(YO) = Xo
and n t/I 1 = n t/I 2, then t/I 1 = t/I 2.
0 0
PROOF. Let M: = {y E Y:t/ll(Y) = t/l2(Y)}. By assumption Yo E M, so

M i= 0. Since G is a Hausdorff space it follows immediately that M is
closed. Now let Yl EM be chosen arbitrarily, Xl: = t/ll(Yl) = t/l2(Yl) and
31: = n(xl)' There are open neighborhoods U(x 1), V(31) such that n[ U: U ~ V
is topological and there are open neighborhoods Qdyd, Q2(Yt) with
t/I;..{Q;J c U for A = 1,2. Let Q: = Ql n Q2. Then
t/ll[Q = (n[U)-l 0 n 0 t/ll[Q = (n[U)-l 0 n 0 t/l2[Q = t/l2[Q,
therefore Q c M. Hence M is also open, and since Y is connected, it follows
that M = Y. D
Theorem 7.2. Let ffij = (G j, nj' Xj) be domains with base point over en for
j = 1,2. Then there exists at most one continuous fiber-preserving mapping
rp:G 1 ~ G2 with cp(x 1 ) = X2.
56
PROOF. If there are two continuous mappings (fJ, I/I:G 1 -+ G2 with 1l:2 (fJ = 0
1l:1 =1l:2 0 1/1 and (fJ(Xl) = I/I(Xl) = X2, then it follows from Theorem 7.1
that (fJ = 1/1. 0
Theorem 7.3. The relation "<" is a weak ordering; that is:

1. 6> < 6>;
2. 6>1 < 6>2 and 6>2 < 6>3 => 6>1 < 6>3.
Def.7.4. Two domains 6>1> 6>2 with base point over Cn are called isomorphic
(symbolically 6>1 :::::: 6>2) if 6>1 < 6>2 and 6>2 < 6>1·
Theorem 7.4. Two domains 6>j = (G j , 1l:j' x), j = 1,2, are isomorphic if and
and if, there exists a topological fiber preserving mapping (fJ: G1 -+ G2
with (fJ(x 1) = X2.
PROOF. 6>1 :::::: 6>2 means that there exist continuous fiber-preserving
mappingS(fJl:Gl -+ G2 with(fJl(xl) = x2and(fJ2:G2 -+ G1 with (fJ2(X2) = Xl·
Then (fJ2 0 (fJl: G1 -+ G 1 is continuous and both 1l:1 0 «(fJ2 0 (fJl) = (1l:1 0 (fJ2) 0 (fJl =
1l:2° (fJl = 1l:1 and (fJ2 0 (fJl (Xl) = (fJ2(X2) = Xl. From the uniqueness theorem
(Theorem 7.2) it follows that (fJ2 (fJl = idGt • Analogously one shows
0
(fJl (fJ2 = id Gz • Hence (fJl is bijective and «(fJd- 1 = (fJ2. We set (fJ: = (fJ1·
0
To prove the converse we set (fJl: = (fJ and (fJ2: = (fJ -1. 0
Def.7.5. A domain 6> = (G, 1l:, xo) over Cn with base point is called schlicht
if:
1. G c en;
2. 1l: = idG is the natural inclusion. (In particular then Xo = 30.)
Theorem 7.5. 6>1 < 6>2 ~ G1 C G2 if 6>1> 6>2 are schlicht domains.
EXAMPLE. Let 6>1: = (G, 1l:, xo) be the Riemann surface of.JZ with the base
point Xo: = (1,1), 6>2: = (C, ide, 1). Then (fJ: = 1l::G -+ C is a continuous
mapping with ide (fJ = 1l: and (fJ(xo) = 1. Therefore 6>1 < 6>2·
0
Next we consider systems of domains over Cn• Let I be a set, (6),),eI

a family of domains over en with base point.
If Z E I, 6>, = (G" 1l:" x,), then 1l:,(x,) = 30.
Let X:
,eI
=G, = U
,eI
U
(G, X {z}) be the disjoint union of the spaces G,. Let
Kbe another set, (N,JKeK a family of sets. For each K E K, let
X~: = {X~:):VK E N K }
57
be a partition of X (that is:

1. xt) c X
2. U X~~) = X
v,
3. For v" -:/= J1,o X~:) n X~~ = 0).
Let X: = (X")"EK be the family of these partitions. We show that there
exists a partition r
of X which is finer than any X" with K E K. (That is, there
exists a partition X* = {Xv: v EN} for which, for any v E Nand K E K, there
exists a v" E N" with Xv c X~:).)
Let N: = n
NK and Xv: =
KEK
n
X~~) for v = (V")"EK E N, r: =
KEK
{Xv:v EN}. Then for each v E N, Xv c X, and
U Xv, = U
veN vellNK
(n X~:») n(u
KEK
=
KEK VKEN
X~:») = n
KEK
X = X
and
if v" -:/= J1K for some K E K. Therefore r is a partition, and clearly for fixed
v, Xv = n X~:) c X~:). That is, r is finer than any partition X"' K E K.
KEK
Definition. We say that the equivalence relation ~ on X has property (P)

if for all 1b 12 E I it is true that
1. (x"' 11) ~ (X ,2 , 12)
2. Ifl/!:[O, 1] ~ G", cp:[O, 1] ~ G'2 are paths (=continuous mappings)
with (l/!{O), 11) ~ (cp{O), 12 ) and n" 0 l/! = n,gcp then (l/!(1), 11) ~ (cp{l), 12)·
EXAMPLE. Let (y, 11) ~ (y', 12) ifn" (y) = n'2(y'). Clearly ~ is an equivalence
relation on X and ~ has property (P).
Now let K be the set of all equivalence relations on X which have property
(P). For K E K let X" be the partition of X corresponding to the equivalence
relation K, that is the set of equivalence classes.
For the partition system X = {X,,:K E K} one can construct a refinement
r = {Xv:v E N} as above.
Lemma 1. The equivalence relation ~ defined on X by r has property (P).

Furthermore, the equivalence classes Xv in each case contain only points
over the same fundamental point 5v E en.
PROOF. The equivalence relation K E K will also be denoted by "it'. Then

(x", 11)K(X'2' 12) holds for each K E K, 11' 12 E I. Therefore, for each K E K
58
set X~:). But then the points also lie in the set n
there exists a VIC E N /c' such that (x." 11)' (X. 2 , 12 ) simultaneously lie in the
X~:) = Xv for V =
nN
KEK
(VIC)/cEK E IC , that is (x." 11) ~ (x. 2, 12)' One shows similarly that the
/cEK
second requirement of (P) is satisfied..r is therefore the finest partition
of X which defines an equivalence relation with property (P). If two points
(x, 11), (x', 12 ) lie in Xv = n
X~:), then for every K E K (x, 11)i((X', 12), in
ICEK
particular for the equivalence relation given in the example..But then
n.,(x) = n. 2 (x'). The fundamental point uniquely determined by Xv will be
denoted by 3v' 0
Definition. Let X* = (Xv)vEN be the finest partition of X which defines an

equivalence relation on X with property (P). Then let G: = {Xv: v E N},
and let the mapping it: G . . . . en be defined by it(Xv): = 3v' Further, let
Xo = X Vo be the equivalence class which contains all the base points
(x" 1), 1 E I.
Subsequently it will be shown that <!> = (G, it, xo) can be given such a
topology that «; is a Riemann domain and (fj. < (fj for all 1 E I.
Definition. For 1 E I let a mapping ip.:G• ....... G be defined as follows: If

Y E G" then let ip'(y) = X v (.) E Gbe that equivalence class which contains
y. Clearly it 0 <P.(y) = n.(y) and ip.(x.) = xo.
It suffices, therefore, to give (G, it, xo) a Hausdorff topology so that all
mappings <PI are continuous.
Lemma 2. Let (Y1, 11), (Y2, 12) E X be equivalent, 31 E en the commonfimda-

mental point, V = V(31) c en a connected open neighborhood and Ui =
Ui(yJ C G' i open neighborhoods such that n.,IU i : U i ....... V is topological
(for i = 1,2). Then ((n lt iU1)-1(3),1 1) ~ ((n.2IU2)-1(3), 12)for every 3 E V.
PROOF. Let <P be a path in V which joins 31 with 3. Then t/Jl: = (n"IU.,)-1 0 <P
and t/J2: = (n.2IU.,)-1 ip are paths in U b resp. U 2, which connect Yl with
0
(n.,1 U.J -1(3), resp. Yz with (n.2IU.,) -1(3). The initial points are equivalent, and
therefore so are the endpoints. 0
Lemma 3. For all 110 12 E I it is true that: If MeG., is open, then

<P.~ 1(<p.,(M)) C G' 2 is open.
PROOF. <P.~ 1(<p,,(M)) = {x E G: There is ayE M with <P.,(y) = <P'2(X)} =
{x E G. 2 : There is ayE M with (y, 11) ~ (x, 12 )}.
Let x E ip.~ 1(ip., (M)) be given, y E M with (y, 11) ~ (x, 12) and 3: =
n.,(y) = n. 2 (x). There exist open neighborhoods U 1 = U 1{y), U 2 = U 2(x)
59
and a connected open neighborhood V = V(3) such that n"iU 1: U 1 --+ V,

n,21U2: U 2 --+ V are topological mappings. Let cp: = (n'2IU2)-1 0 (n"IU 1):
U 1 --+ U 2' From Lemma 2 it follows that for y' E U 1
(cp(y'), 12) '" (y', 11)'
M is open, and so are U1: = M nUl and U2: = cp(U1) c U 2'
But x E U2 c CP.~l(cp,,(M)). Hence x is an interior point, which was to be
shown. 0
Lemma 4. Let M" c G", M'2 C G'2 be arbitrary subsets. Then cp,,(M,.) n
cp'2(M'2) = cp'2(M'2 n cp,~l(cp,,(M,.))).
PROOF
1. Let y E cp,,(M,,) n cp'2(M'2)' Then there are points Y1 EM", Y2 E M'2
with cp,,(yd = CP'2(Y2) = y. Clearly Y2 E cp,~ l(cp.,(M,,)) n M'2'
2. Let Y E cp'2(M'2 n cp,~ l(cp'I(M,.))). Then there is a point Y2 E M'2 n
CP.~ l(cp'I(M'I)) with CP'2(Y2) = Y and furthermore there is also a point
Y1 EM" with CP'2(Y2) = cp,,(yd. Therefore Y E cp,.(M,,) n cp'2(M,J 0
Now we can introduce a topology on G:
Let l:': = {A c G: There exists an I E I, M, c G, open, such that
cp,(M,) = A} u {G}. Then:
1. 0 = cp,(0) for every Z E I, so 0 E l:'
2. GEl:' by definition
3. Ai> A2 E l:' => A1 n A2 E l:', from Lemmas 4 and 3.
l:' satisfies the axioms for the basis of a topology. Let l: be the corresponding
topology on G, that is, the set of arbitrary unions of elements of l:'.
Theorem 7.6. Let {ij). = (G" n" x,): I E I} be a family of domains over en
with base point, X = U G, the disjoint union of the spaces G" and X* =
,eI
(Xv)veN the finest partition of X which defines an equivalence relation with
(P). Let G: = {Xv: v E N} be the set of classes of X*. Let the point
Xo E G and the mappings ft: G --+ en, cp,: G. --+ G be defined as above, and
G be provided with the topology given above. Then:
1. ~ = (G, ft, xo) is a domain over en with base point.
2. Fpr every I E I, ij), < ~.
3. If ij)* = (G*, n*, xo) is a domain over en
and ij), < ij)* for all ZE I,
then also ~ < ij) *.
(~ is the smallest Riemann domain over en, which contains all domains Q)•. )
PROOF
1a. Gis a topological space and ft(xo) = 30 = n.(x,).
b. G is connected: If Y E G, then there is an I E I and a Y. E G, such that
Y = cp,(y.). Let 1/1 be a path in G, which connects Y. with x,. Then cp,O 1/1:
[0,1] --+ G is a mapping with CP. 0 1/1(0) = y, CP. 0 1/1(1) = xo. cp, (and hence
60
cp, 0 ljI also) is continuous: if MeG is open, then M = U cp, (M.), where
,El
M, c G, is open (possibly empty) for every l.
It follows that, for 10 E I, cp,~ 1(M) = U cp,~ 1(cp.(M) ) is open in G,o. We
'EI
can therefore connect every point to the base point by a path in G.
c. G is a Hausdorff space: Let Yl, Y2 E G with Y1 =1= Y2·
Case 1. n(Y1): = 31 =1= 32 = :n(Y2). Then there are open neighborhoods
V(3d, V'(32) with V n V' = 0, and n-l(V) n n- l (V') = 0. Therefore it
suffices to show that n is continuous. Let V c en be open, M: = n- l(V),
1 E I. Then cp,-l(M) = (n cp.)-l(V) = n,-l(V) is open in G" therefore
0
M = U cp,cp,-l(M) is open in G.
'EI
Case 2. Let 3: = n(Yl) = n(Y2). There are elements hE Gl[' Y2 E G'2 with
cpl[(h) = Yt and CP'2(Y2) = Yz. Furthermore we can find open neighborhoods
U 1 (Yd c Gq , U 2(Pz) C G'2 and a connected open neighborhood V(3) c en
such that n, ,I U 1: U 1 ~ V and n'21 U 2: U 2 ~ V are topological mappings. The
points CYb 11), (Y2, 12) are not equivalent, so by Lemma 2 it must be that
cp,,( U 1) n CP'2( U 2) = 0, and we have found disjoint neighborhoods.
d. n is locally topological. Let Y E G, 1 E I, y, E G, be such that cp,(y.) = y.
Let 3: = n(y) = n,(y,). Then there exist open neighborhoods U,(y,) and
V(3) such that n,iU,:U, ~ V is topological. U: = cp,(U.) is an open neigh-
borhood of y, nl U: U ~ V is continuous and surjective. From the equality
(nIU) (cp,IU,) = n,IU, it follows that nlU is also injective and (niU)-l is
0
continuous.
2. The mappings cp,: G, ~ G are fiber-preserving and by (1 b) are also
continuous. Therefore (£), < <b.
3. If (£)* is given, then there exists a fiber-preserving mapping cp;: G, ~ G*.
With the help ofthestatement "(y, 11) ~ (y', 12) ifand only ifcp:,(y) = CP;2(y')"
we can introduce an equivalence relation on X, which because of the unique-
ness lifting also has property (P): Namely, if ljI: [0, 1] ~ G", cp: [0, 1] ~ G'2
are two paths with (ljI(O), Id ~ (cp(O), 12) and n" ljI = n'2 cp, then Cp;, ljI(O) =
0 0 0
CP;2 o cp(O) and (because n* cp; = n,) also n* (cp;, ljI) = n* (CP;2 cp).
0 0 0 0 0
Hence Cp;, ljI = CP;2 cp, by Theorem 7.1.

0 0
In particular it follows that (ljI(I), ld ~ (cp(I), 12). But that means that
a mapping cp:G ~ G* is defined by cp cp, = cp;. cp is continuous and
0
fiber-preserving. 0
Def.7.6. The domain <b described in Theorem 7.6 is called the union of the
domains (£)" 1 E I, and we write
<b = U (£),.
'EI
Special Cases
1. From (£)1 < <fi and (£)2 < <fi it follows that <fil u (£)2 < <fi
2. From (£)1 < <fi2 it follows that (£)1 u <fi2 ~ (£)2
61
3. (fj u (fj ~ (fj

4. (fjl U (fjz ~ (fjz U (fjl
5. (fjl U (fjz u (fj3) ~ (fjl u (fjz) U (fj3
EXAMPLE. Let (fjl = (G b 11:b xd be the Riemann surface of w = jZ, with

Xl = (1, 1) as base point and with the canonical projection 11:1 :(w, z) H z. Let
(fjz = (G z , 11:z,x z)begivenbyGz : = {ZEC:t < Izi < 2},xz: = 11:l(Xl) = 1
and 11:z: = id G2 •
Then (fjl u (fjz = (G, if, xo), where G = (G l 0 Gz)/ ~ is the set of all
equivalence classes [(x, I)], IE {1, 2} with respect to the "finest" equivalence
relation with propery (P).
1. Let YE11:1 l (G z ) c G l . Then we can connect y with the point Xl by
a path IjJ in 11:1 1(G z ). The path 11:1 IjJ then connects 11:1 (y) in Gz with xz. But
0
now (Xl' 1) ~ (xz, 2), so (y, 1) ~ (11:1(y), 2) as well. On the other hand, the
equivalence classes contain only points over the same fundamental point, so
it follows that over each point of Gz there is exactly one equivalence class.
2. Let Z E C - {OJ be arbitrary. The line through Z and 0 contains a
segment IP:[O, 1] ~ C - {OJ which connects a point z* E Gz with z. Then
there existtwo paths IjJ 1> l/Iz in Gl with 11:1 0 IjJ 1 = 11:1 0 IjJz = IP and (1jJ 1(0), 1) ~
(ljJz(O), 1) ~ (z*, 2). Hence it follows that the points (ljJl(1), 2), (ljJz(1), 2) over
Z are equivalent From (1) and (2) we have:
(fjl u (fjz = (C - {OJ, idc-{oj, 1).
8. Holomorphic Hulls
Def.8.1. Let (G, 11:) be a domain over en, f:G ~ C a function. f is called
holomorphic at a point Xo E G ifthere exist open neighborhoods U = U(xo)
and V = V(11:(xo)) such that 11:1 U: U ~ V is topological and f (11:1 U)-l: 0
V ~ C is holomorphic. f is called holomorphic on G if f is holomorphic

at every point Xo E G.
Remarks
1. Holomorphy at a point does not depend on the neighborhood.
2. For schlicht domains the new notion of holomorphy agrees with the
prevIOUS one.
3. If f is holomorphic on G, then f is continuous.
Lemma 1. Let (G b 11:1> Yl), ... ,(G{, 11:/, Y(), (G, 11:, y) be domains with base
point over en and let 3 = n(y). ~r IPi: G ~ Gi are fiber-preserving mappings
with IPi(Y) = Yi for i = 1, ... ,e, then there exist open neighborhoods
U = U(y), v = V(3) and U i = Ui(yJ such that for every i all the mappings
in the following commutative diagram are topological
62
U CPi/ U ) U·
'I~v ;':I~;
PROOF. We can find open neighborhoods O(y), Y(o), and Oi(Y;) such that
the mappings n/O:O ~ Y and ni/Oi:Oi ~ V are topological. Since CPi is
continuous, there is an open neighborhood U(y) c O(y) with CPi(U) c Oi.
If we set V(o): = n( U) and Ui : = cp;( U), we obtain the desired result. 0
Def.8.2. Let 63; = (G i, ni' Xi), i = 1, 2 be domains with base points and
(£il < (£iz by virtue of a continuous mapping cP: G1 ~ Gz . If f is a com-
plex valued function on Gz , then we define f/Gl: = f 0 cpo
Theorem 8;1. Under the conditions of Def 8.2, f/Gl is holomorphic on G 1

whenever f is holomorphic on G z ·
PROOF. Let Yl E G 1 be arbitrary, yz: = CP(Yl)E Gz and 01: = n1(Yl) = nz(yz)·
By Lemma 1 we obtain a commutative diagram of topological mappings:
U1 cp/U1)U Z
"I~
V
J,W,
(with neighborhoods U 1 = U 1 (Yl), U z = Uz(yz) and V = V(ol)).Butthen
(f/G 1) 0 (nl/Ul)-l = f 0 (cp 0 (nl/Ul)-l) = f 0 (nz/U z )-l. 0
f is called a holomorphic extension or continuation of flGl to G z .

Def.8.3
1. Let (G, n) be a domain over en. If X EGis a point and f a holo-
morphic function defined near x, then the pair (f, x) is called a locally
holomorphic function at x.
2. Let (G 1 , nl), (G z , nz) be domains over en, Yl E G1 and Yz E Gz
points with n1(yd = nz(yz) = :0. Two local functions (ft. Yl), (fz, Yz)
are called equivalent (symbolically (fl)YI = (fZ)Y2) if there exist open
neighborhoods U 1 (Yl), Uz(Yz), V(o) and topological mappings nl/Ul:
U 1 ~ V, nzlU z : U z ~ V with flo (nlIU 1 )-1 = fz 0 (nzIUz)-l.
Remark. If (fd YI = (fZ)Y2' then clearly fl(yd = fz(yz). In particular if

G1 = Gz, n1 = nz and Yl = Yz then it follows that fl and fz coincide in
an open neighborhood of Yl = yz.
63
Theorem 8.2. Let (G b n1), (Gz, nz) be domains over 1[:", t/I).:[O, 1J --* G). be
paths with n1 0 t/l1 = n2 0 t/lz. Additionally, let fA be holomorphic on G).,
A = 1,2. If Ud"" (0) = (fZ)"'2(0), then also (f1)",,(I) = (f2)"'2(1)·
PROOF
1. Let Xl E G1, X2 E G2 be points with n1(x 1) = n2(xZ) = 30. Then there
are open neighborhoods U 1(X1), U Z(X2) and an open connected neighbor-
hood V(30) such that the mappings n).IU).: U;. --* V are topological.
If there exist points Xl E U b X~ E U 2 with n1(xI) = n2(x~) = 3 and
(f1)x\ = (fZ)X2 then f1 a (n1IU1)-1 = fz (nzIU z)-l near 3 E V and there-
0
fore, by the identity theorem, in all of V. It follows that if (f1)x 1 = (f2)X2'

then (f1)x\ = (fZ)X2 for all Xl E U b x~ E U 2 with n1(xI) = nz(x~). If(f1)x, #-
(fZ)X2' then (f1)x; #- (fZ)X2 for all Xl E U b x~ E U 2 with n 1(xI) = n2(x~).
°
2. Let W: = {t E [0, 1]:(f1)",,(t) = (fZ)"'2(t)}
a. By assumption W #- 0, as lies in W.
b. If t1 E W, then one sets X).: = t/I).(t 1). By (1) there exist open neigh-
borhoods U 1(X1), Uz(x z ) such that (f1)x\ = (fZ)X2 for all Xl E U b X~ E U z
with n1(x~) = nz(x~). Since the mapping t/I). are continuous, there exists a
neighborhood Q(t 1) c [0, 1J with t/I).(Q) c U)., A = 1,2. Therefore (f1)",(t) =
(f2)"'(t) for t E Q. This means that W is open.
c. One shows that [0, 1J - W is open in exactly the same way. Since
[0, 1J is connected, it follows that W = [0, 1]. D
Theorem 8.3. Let (fj). = (G)., n)., x).) be domains over Cn with n;.(x,.) = 30'
A = 1,2, and with (fj1 < (fjz.
Let f be a holomorphic function on Gb F a holomorphic extension of
f to G2 • Then F is uniquely determined by f.
PROOF. Let F b F Z be holomorphic extensions of f to Gz. By Lemma 1 there

exist neighborhoods U ;.(x).) such that the restriction ofthe canonical mapping
cp: G1 --* G2 to U 1 maps the set U 1 topologically onto U z. For v = 1,2 it is
true that Fv cplU 1 = flUb consequently F11Uz = F2IUz, and therefore
0
(F 1)X2 = (F 2)X2· Since each point X E Gz can be joined to X2, the equality
F 1 = F 2 follows from Theorem 8.2. D
For j = 1, ... , n let prj: I[:" --* C be the projection onto the j-th compo-
nent. If (G, n) is a domain over cn, then Zj: = prj n is a holomorphic
0
function on G, so the set A(G) of all holomorphic functions on G contains

more than the constant functions.
Def.8.4. Let (fj = (G, n, xo) be a domain over C" with base point fF a non-
empty set of holomotphic functions on G. Let {(fj" I E I} be the set of
domains over I[:" with the following properties:
1. (fj < (fj, for all I E I.
2. Iff E fF, then for every I E I there is an F, E A(G.) with F,IG = f.
64
8. Holomorphic HuIIs
Then Hg;(IJj): = U 1Jj, is called the holomorphic hull of IJj relative to :F

'EI
1f:F = A(G), then H(IJj): = HA(G)(IJj) is called the (absolute) holomorphic
hull of 1Jj. If :F = {f}, then HJ(IJj): = H(f)(IJj) is called the domain of
holomorphy of f.
Theorem 8.4. Let IJj = (G, n, xo) be a domain over en,:F a non-empty set of
functions holomorphic on G and Hg;(IJj) = (G, ii, x) the holomorphic hull of
G relative of :F. Then IJj < H g;( 1Jj), and for each function f E :F there exists
exactly one function FE A(G) with FIG = f. If IJjt = (G t , nt, xd is a
domain over en with IJj < IJjt and the property that every function f E :F
can be holomorphically extended to G t , then IJjt < Hg;(IJj).
PROOF
1. Let" ~" be the finest equivalence relation on X: = U G, with prop-
'EI
erty (P). Then G is the set of equivalence classes of X relative to ~. We now
define a new equivalence relation on X:(y, It) ~ (y', Iz), if and only if:
a. n,Jy) = n'2(y')·
b. Iff E:F and ft E A(G,,), fz E A(G'2) are holomorphic extensions of f,
then (ft)y = (fz)y. "~" is an equivalence relation and has property (P).
a. For each I E I there exists a continuous fiber-preserving mapping ({J,:
G -+ G, with ((J,(xo) = x,. We can find open neighborhoods U(x o), Ut(x,,),
U Z(X'2) and V(n(xo)) such that all mappings are topological in the two
commutative diagrams below.
U ({J'2I U ) U z
nl~ AIU,
v V
Thenfz 0 (n'2iUz)-t = fz 0 ({J'2 0 (nIU)-t = f 0 (nIU)-t = ft 0 ({J" 0 (nIU)-t =
ft 0 (n'lIUt)-t; that is the base points are equivalent.
b. If !/I A: [0, l] -+ G" are paths with (!/It (0), Id ~ (!/Iz(O), Iz) and n" o !/It =
n'2 0 !/Iz, then (ft)JjJl(O) = (fZ)JjJ2(O). It follows from Theorem 8.2 that: (fdJjJl(t) =
(fZ)JjJ2(t), and, therefore (!/It(l), It) ~ (!/Iz(l), IZ). Since "~" is the finest
partition with property (P), (y, It) ~ (y, Iz) implies (y, It) ~ (y, Iz).
2. For all I E I, IJj < 1Jj, < U 1Jj, = Hfj(IJj). Let rp,: G, -+ G and rp: G -+
'EI
G with rp = rp, 0 ({J, be the canonical mappings. Let f E:F be given. For
y E G there exists an I E I and a y, E G, with rp,(y,) = Y. Let F, E A(G.) be a
holomorphic extension of f. Then we set F(y): = F.(yJ If K E I, y" c G",
rp,,(y,,) = y, and if F" E A(G,,) is a holomorphic extension off, then (y" I) ~
(y", K). Hence (y" z) ~ (y", K) as well, so that (F.)y, = (F")YK. It follows that
F,(y.) = F,,(y,,). So F is well defined. Also, F 0 rp = F 0 rp, 0 ({J, = F,o ({J, =
f, so F is an extension of f. It remains to show that F is holomorphic:
65
Let Y E G. y = <p,(y,) and 3 = n(y). Then there exist open neighborhoods

U 1 (y,), U 2 {y), V(3) and a commutative diagram of topological mappings:
<p,IU1 U
U1 ) 2
n'l~
V
;lu,
It follows that F (nIU 2)-1 = F <p, (n,IU 1)-1 = F,o (n,IU 1)-1; the last is
0 0 0
a holomorphic function.
3. The "maximality" of H§'«fj) follows immediately from the construction.
The holomorphic hull H§'«fj) is therefore the largest domain into which
all functions f E :F can be holomorphically extended. D
Theorem 8.5. Let (fj). = (G)., n)., x), A. = 1,2 be domains over e with (fj1 u
(fj2 = (G, iE, X), and f1: G1 -4 C, f2: G2 -4 C be holomorphic functions. If
there is a domain (fj = (G, n, x o) with (fj < (fj). for A. = 1,2 and f11G =
f21 G, then there is a function 1 holomorphic on G with 11 G). = f;. for A. =
1,2.
PROOF. Let f: = f11G = j~IG, :F: = {J}. Then f1 is a holomorphic exten-
sion of f to G1 and f2 is a holomorphic extension of f to G2 . Therefore by
Theorem 8.4: (fj1 < H§'«fj) and (fj2 < H§'«fj). But then by Theorem 7.6
(fj1 u (fj2 < H§'«fj). Let !be the holomorphic extension of f to H§'«fj) and
1: = JIG. For A. = 1, 2,fIG = (JIG)IG = JIG = f = f).IG, thereforell G). =
h. D
Now let Pee be the unit polycylinder, (P, H) a Euclidean Hartogs
figure, tP: P -4 B e e a biholomorphic mapping. (B, tP(H)) is then a
generalized Hartogs figure. Since P and H are connected Hausdorff spaces
and tP is, in particular, locally topological, it follows that ~ = (P, tP, 0) and
~ = (H, tP, 0) are domains over e with base point and we have ~ < ~.
We regard the pair (~, ~) as a generalized Hartogs figure.
Theorem 8.6. Let (G, n) be a domain over e, (~,~) a generalized Hartogs

figure, and Xo EGa point for which ~ < (fj: = (G, n, xo).
Then every function f E A(G) can be extended holomorphically to
(fj u~.
PROOF. flH has a holomorphic extension FE A(G). Let (fj1: = (fj, (fj2: =
~, f1: = f, f2: = F. Because ~ < (fjb ~ < (fj2 and f11H = f21H, the pro-
position follows from Theorem 8.5. D
Def.8.S. A domain (G, n) over e is called pseudo convex if the fact that
(~, ~) is a generalized Hartogs figure and Xo EGa point with ~ < (fj: =
(G, n, xo) implies (fj u ~ ~ (fj.
66
Def.8.6. A domain ffj = (G, n, x o ) is called a domain of holomorphy if there

exists an f E A(G) with Hf(ffj) = ffj. In the schlicht case this definition
agrees with the old one.
Theorem 8.7.
1. If ffj = (G, n, xo) is a domain over en and F a non-empty set of
functions holomorphic on G, then H~(ffj) is a pseudoconvex domain.
2. Every domain of holomorphy is pseudoconvex.
The definition of holomorphic convexity can be extended from the

schlicht case; then we have
Theorem 8.8. (Oka, 1953). If ffj is pseudo convex then ffj is holomorphically
convex and is a domain of holomorphy.
The proof is tedious.
At present the concept of a holomorphic hull is only of theoretical interest,

although it is possible to construct the holomorphic hull by adjoining
Hartogs figures and it is conceivable that such a construction is realized
with the help of a computer. Quicker methods have been found in only a
few special cases, as for example, in connection with the Edge-of-the-Wedge
theorem which in quantum field theory serves as a proof of the PCT theo-
rem ("Under certain assumptions the product PCT of space reflection P,
time reversal T, and charge conjunction C is a symmetry in the sense of
field theory").
67
CHAPTER III
The Weierstrass Preparation Theorem
1. The Algebra of Power Series

In this chapter we shall deal more extensively than before with power
series in en. Our objective is to find a division algorithm for power
series which will facilitate our investigation of the zero sets of holomorphic
functions.
Let No: = N u {O}, No: = {(Vb' .. , Vn ) :V; E No}. We denote by C{a}
the integral domain of formal power series about 0 with variables Z 1, . . . , Zn
and coefficients in IC. Let Ih£~ be the set of n-tuples of positive real numbers.
00
An element f E C{a} can be written as f = I ava V •

v=o
co
Def.1.1. Let t = (tl' ... , tn ) E Ih£~ and f = I ava V
E C{a}. The norm of
v=o
f with respect to t is the "number"
00
Ilfll t: = L lavlt V
E Ih£+ u {O} u {oo}.
v=o
One can introduce a weak ordering on Ih£~ if one defines (tb ... , tn )
(t~, ... ,t~) if and only if t; ,,:; tt for i = 1, ... , n. The norm of f
,,:;
relative to t is then monotone in t: 1ft,,:; t*, then Ilfllt ,,:; Ilfllt"
Def. 1.2. A formal power series f E C{a} is called convergent if f(a) =

co
I ava is convergent in a polycylinder about O. (The definition of this
V
v=o
convergence was given in Chapter I.) We denote the ring of convergent
power series by H~.
68
1. The Algebra of Power Series
Theorem 1.1. f E 1C{3} is convergent if and only if there is atE 1R"t- with
Ilfll t < 00.
PROOF
co
1. Let f(3) = I a v 3V be convergent in the polycylinder P. Then there
v=o
exists atE 1R"t- with P t C P, and therefore Ilfllt < 00.
00
2. If Ilfllt = I lavlt V is convergent, then f(3) is convergent at the point

v=o
t, so f converges on all of Pt.
Der. 1.4. A set B is called a (complex) Ranach algebra if

1. There are operations + :B x B -+ B, . : C x B -+ Band 0: B x B -+
B such that
a. (B, +, .) is a C-vector space
b. (B, +, 0) is a commutative ring with 1
c. For all f, 9 E B and all c E C, c· (f 0 g) = (c' f) 0 9 =f 0 (c' g).
2. To every fEB a number Ilfll E IR+ u {O} is assigned with the
properties of a norm:
a'llc . fll = Icl'llfll for c E C,f E B.

b. If + gil :( Ilfll + Ilgll for f, 9 E B.
c. Ilfll = O<=> f = O.
3. Ilf gil :( Ilfll' Ilgll for f, g E B.
0
4. As a normed C-vector space, B is complete; that is, every Cauchy

sequence (fv) of elements of B converges to an element f of B.
Theorem 1.2. Bt is a Banach algebra for every t E 1R"t-.

PROOF. Clearly 1C{3} is a C-algebra. In order to show that B t is a C-algebra,
it suffices to show that B t is closed under the operations:
co 00
c· I av3V = I (c' a.)3 V ,

v=o v=o
00 00 co
I av 3V + I bv 3V = I (a v + bv)3 V
,
v=o v=o v=o
C~o av3v)oC~0 b#3#) A~O C+~=). a b#)3 = V

A
•
Straight-forward calculation shows that II' . '11t is a norm with properties

(2) and (3).
Now if c E C, f E B t , then c . f E B t because of (2a). If f and 9 are in B t ,
then f + 9 E B t because of (2b) and fog E B t because of (3).
69
III. The Weierstrass Preparation Theorem
It is clear that 1 lies in B t . All that remains is to show completeness:
L
00
Let (f)J be a Cauchy sequence in B t with fA3) = a~"l3v. Then for

,,=0
every B > 0 there is an n = n(B) E N such that for all A, f.1 ~ n
00
L la~"l - a\!,llt" = Ilf" - f"llt

v=o
< B.
Because tV # 0 it follows from this that
for every vENn.
For fixed v (a~"l) is therefore a Cauchy sequence in e which converges to

the complex number avo
rtJ
Let f(5): = L av 3". Let 6 > 0 be given. Then there exists an n = n(6)
such that
for A ~ nand f.1 EN.
Let I c No be an arbitrary finite set. There always exists a f.1 E N for A ~ n

such that
VEl
L
la~H"l - avlt v < 6/2, and then
L la~"l - avlt" ~ L la~"l -

vel vel
a~H"llt" + L la~H"l - avit" < 6
VEl
for A ~ n.
(jJ
In particular Ilf" - flit = v=o

L la~"l - a"lt ~ 6. Thus (f,,) converges v to f
Because Ilflll ~ Ilf - f"llt + Ilf"III' it follows that f lies in B t • 0
For the following we need some additional notation:

If v = (V.b"" vn ) E No, we set v': = (V2,"" vn ); if t = (t 1 , . . . , t n ) E IR~,
we set t': = (t 2 , . .. , t n); if 3 = (Zb' .. ,zn) E en, we set 3': = (Z2' ... ,zn).
Then v = (Vb v'), t = (tl' n,
3 = (Z1> 5'), and we can write an element
f E C{5} in the form
rtJ rtJ
f(3) = L f,,(5')z1
,,=0
where f,,(5') = L a",v'(5')"'.
,,'=0
This representation is called the expansion of f with respect to Z l' The
following assertions hold:
rtJ
1. f = L f"zi lies in B t if and only if every f" lies in B; = B t n

,,=0
C{3'} and
rtJ
L Ilf,,11r t1
,,=0
< 00.
2. For SE No, liz! .flit = tl . Ilfllt.

70
2. The Weierstrass Formula
3. Iff =
00
L avo v converges and ao ...

v~o
0 = 0, then for every 8 > °
there exists
atE 1R':c with Ilfllt < 8.
PROOF
(1) Ilflltv~o lavle = .Jo (%0 la",vWY) t1 = Jo Ilf"llt,tt.

=
(2) Ilz'1 . flit = 11,,~0 f),z1+t = ,,~o IIj~llrtt+s = t'1 . ,,~o Ilf"ll!' . t1
= t1 ·llflit.
00
3. If one sets };: = L ao, .. 0, ViVi + 1 .•• vnzii - 1 z;\\1 .. 'z~n, then z 1 . f1 +
Vi>O
... + zn . fn = f and Ilfllt = t1 '1lfdlt + ... + tn ·lIfnllt· Iff is convergent,
then there exists a to E 1R':c with Ilfllto < 00, and for t :( to
n n n
Ilflit = L till};ll! :( L tdl};llto :( max(t
i~l i~l
b · .. , t n )· L 11};llta'
i~l
which becomes arbitrarily small. o

Let a fixed element t E 1R':c be chosen. When no confusion is possible we
shall write B in place of B t , B' in place of B;, and Ilfll in place of Ilfllt.
00
Theorem 2.1 (Weierstrass formula). Let g = L g"z1 E B, let there be a s E No

,,~o
for which gs is a unit in B', and let there be an 8 with < 8 < 1 such that
11z'l - g' gs-lll < 8' t'1. Then for every fEB there exists exactly one q E B
°
and one l' E B'[Zl] with deg(r) < s such that f = q' g + l' ("Division with
remainder"). Furthermore,
1
(1) Ilgs' qll :( tIl ·llfll· ~
1
(2) 111'11 :( Ilfll· 1 - 8'
PROOF. Let h: = -(z'l - g' gs-l). Then Ilhll < 8' t1 and g' gs-l = z'1 + h.
Let us start with an arbitrary fEB and inductively construct sequences
(f,,), (q,,), and (1',,). We set fo: = f·
Suppose fo, ... , f" have been constructed. There exists a representation
00
ft, = L f",l(z~, and we define

1(~0
00 s-l
q,,: = L f",l(z~-s,
K=S
1',,: =, L
1(~0
f",l(z~ and
71
Then f;. = zi . q;. + r;. and fH1 = -h' q;. = f;. - r;. - gg;1. q;.. Clearly
the following estimates hold:
Ihll ~ Ilf;.ll,
Ilq;.11 ~ t 1S llf;.lI,
IIfH111 ~ Ilhll·llq;.11 < e ·llf;.ll, so
00 00
Letq: = L
;'=0
g;1· q;.andr: = L
;'=0
r;.. Then
By the comparison test the series converge. Since each r;. is a polynomial
with deg(r;.) ~ s - 1, it follows that r is a polynomial with deg(r) ~ s - 1.
00
Since the series L f;. also converges,

;'=0
, OJ co co 00
f=fo= L f;.- L fH1

;'=0 ;'=0
= L (f;.-fH1)= L (r;.+ggs-1 q;.)
;'=0 ;'=0
00 00
= g' L gs-1 . q;. + L r;. = g' q + r.

;'=0 ;'=0
The estimates now follow readily.
(1) Ilgsqll = II ;.~o q;.11 ~ ;.~o Ilq;.11 ~ t 1 llfll' ;.~o e;' = tiS '1Ifll' 1 ~ e'
S
CXl 00 1
(2) Ilrll ~ ;.~o Ihll ~ Ilfll' ;.~o e;' = Ilfll' 1 - e'
It still remains to show uniqueness.

Let there be two expressions of the form
f = q1g + r 1 = q2g + r2'
Then 0 = (q1 - q2)' g + (r1 - .'2)' From the representation g = gs(zi + h)
with Ilhll < e' t1 we obtain
o = (q1 - q2)gA + (q1 - q2)g.h + (r1 - r2)
and
II(q1 - q2)gszill ~ II(q1 - q2)gA + (r1 - r2)11
= II(q1 - q2)gs . hll ~ e . t1 . II(q1 - q2)gsll
= e '11(q1 - q2)gszi II·
Because e < 1, (q1 - q2)gsZ~ = O. Therefore q1 = q2 and r1 = r2' 0
Corollary. If the assumptions of Theorem 2.1 are satisfied andif in addition

f E B'[Z1J, g E B'[Z1J and deg(g) = s, then q E B'[Z1J and deg(q) =
max( -1, deg(f) - s) [with deg(O): = -1].
72
PROOF. Let d: = deg(f). For d < s one has the decomposition 1= 0· g + f,

let therefore d ~ s. Now -1::::; deg(q;J ::::; max( -1, deg(.f,J - s) and
deg(fo) ::::; d. If we assume that d v : = deg(Iv) ::::; d for v = 0, ... ,A, then
deg(q,,J ::::; d - s, therefore
deg(fHd = deg(f;. - r;. - ggs-l q;.)
::::; max(deg(f;.), deg(r;.), deg(q;.) + s)
::::; max(d, s - 1, (d - s) + s) ::::; d.
Hence deg(q;.) ::::; d - s for all A, and deg(q) ::::; d - s. On the other hand,
the representation I = q . g + r gives
deg(f) ::::; max(deg(q) + s, s - 1) = deg(q) + s,
therefore d - s ::::; deg(q). All together one obtains:deg(q) = max( -1, d - s).
D
Theorem 2.2. II B is a Banach algebra, IE Band 111 - III < 1, then I is a
unit in' Band 111-111 ::::; 1/(1 - 111 - liD.
PROOF. Let g: =
00
L
).=0
(1 - f);', 8: = 111 - III. Then °: : ; 8 < 1 and L
),=0
00
£;.
00
dominates g. Therefore the series L (1 - f)). converges and g is an element

;'=0
L L
00 00
of B. Moreover I' g = (1 - (1 - f)).g = (1 - f);' - (1 - I)Hl =

;'=0 ;'=0
00
(1 - f)0 = 1, and Ilgll::::; L 8;' = 1/(1 - 8). D

;'=0
L
00
Def.2.1. Let s E No. An element g = g;.zi E B satisfies the Weierstrass

),=0
condition (W-condition) at s if:
a. gs is a unit in B'.
b. liz! - gg;111 < tft·
Theorem 2.3 (Weierstrass preparation theorem). II g E B satisfies the W-

condition at s, then therp exists exactly one normalized polynomial W E B'[ ZI]
with deg(w) = s and one unit e E B such that g = e' w.
PROOF. We apply the Weierstrass formula to I = z'1: There are uniquely
determined elements q E Band r E B'[ z 1] with z! = q . g + rand deg(r) < s
("le take an 8 < t which satisfies the conditions of Theorem 2.1). But then
z! - ggs-1 = (q - gs-l)g + r is a decomposition of z! - ggs-1 in the sense
of Theorem 2.1; therefore we can employ formula (1):
That means that q . gs and hence q is unit in B.
73
Let e: = q-1 and w: = ~ - r. Then w is a normalized polynomial with

deg(w) = s, and e . w = q-1(~ - r) = g. If 9 = e1(~ - r1) = e 2(zl - r2),
then
but on the other hand, in the decomposition zi = q . 9 + r, the elements

q and r are uniquely determined. Therefore e 1 = e2 and r 1 = 7'2' 0
Corollary. If 9 is a polynomial in Zl> then e is also a polynomial in Z1'

PROOF. Ifwe use formula (2) in the decomposition ~ - gg; 1 = (q - gs-1). 9
+ r, we get
1 e
Ilrll ~ Ilzi - ggs-111· -1~ < ti . ~- < ti·
- e 1- e
Because Ws = 1 it is also true that
II~ - wws- 111 = II~ - wll = Ilrll < ti;

that is, w satisfies the conditions of Theorem 2.1. Since 9 = e' w is a decom-
position in the sense of the Weierstrass formula, the proposition follows
from the corollary of that theorem. 0
Comment. The Weierstrass preparation theorem serves as a "preparation

of the examination of the zeroes of holomorphic functions".
A function holomorphic in a polycylinder will be represented by a con-
vergent power series g. If there exists a decomposition 9 = e . w with a unit
e and "pseudo polynomial" w = ~ + A 1 (3')zi- 1 + ... + A s (3'), then 9 and
w have the same zeroes. However, the examination of w is simpler than that
of g.
3. Convergent Power Series

Def.3.1. 9 E C{3} is said to be regular in Z1 if g(Zl> 0, ... , 0) does not vanish
identically.
OCJ
If 9 = I g"z1 is regular in Z1' then ord(g) is that number S E No for

,,~o
which go(O) = ... = gs-1 (0) = 0, gs(O) =f. O.

We then say that 9 is regular of order s in Zl '
Theorem 3.1. For gl> g2 E C{3}

1. g1 . g2 is regular in Z1 if and only if g1 and g2 are regular in Z1>
2. ord(g1 . gz) = ord(g1) + ord(gz)·
PROOF. (gs' gz)(Zl> 0) = g1(Zl> 0)' g2(Z1, 0). Since C{zd is an integral do-
main, (1) holds; (2) is obtained by multiplying out. 0
74
°
Theorem 3.2. Let g E C{a} be convergent and regular of order sin Z1' Thenfor
every e > and every to E [R~ there exists a t ~ to such that g lies in B I ,
gs is a unit in B;, and Ilzt - gg; 111 I ~ e' tl..
00
PROOF. Let g = I g).z1 be the expansion of g with respect to Z1' Then

g).(O) = ° ),=0
for A. = 0, 1, ... , s - 1 and gs(O) oF 0.
1. Since g is convergent, there exists a t1 = (tl1), ... , t~1») E [R~ with
00
IlglII, = I Ilg).III,· tl »)' <1 co; therefore gA E B;,. In particular then

A=O
gs(a') _ 1 = 'f(-:>') E B'
gs(O) ." I"
and since f(O) = 0, there exists a t2 ~ t1 such that, for all t ~ t 2 , Ilflll < 1.
gs/gs(O) (and hence gs also) is therefore a unit in B;. Moreover, it is clear
that g lies in B t •
2. Let h: = zt - g' gs-l. Then hE BI for all t ~ t 2 , and we can write
= ° for
00
h =I d Az1 with ds 0, d A == - g).gs-1 for A. oF sand d;JO)

A=O
A. = 0, 1, ... , s - 1. For t ~ t2
f d).z1 1
I A=s+1 I
= f d 1- 11
Ilzt+ A=s+1 1
. AZ s
-
1
I
f dAZ1-s-111 ~t~+111 A=S+f

=tl.+ 1·11 A=S+ 1 I 1
d).z1- s- 111
12'
3. If t1 is sufficiently small, then
t1 f dAZ1-s-111
·11 A=s+1 12
< te;
therefore
Because d;.(O)
so small that
= °for A = 0, 1, ... , s - 1 we can choose for t1 a suitable t'
s-1
I Iid lkt1
),=0
A < te' tl..
For t = (t 1 , t') it then follows that
Remark. In a similar manner one can show that if gb ... ,gN E C{a} are
convergent power series and each gi is regular of order Si in Zb i = 1, ... , N,
75
then for every I> > 0 there is an arbitrarily small t E 1R"r for which
gi E Bb (gi)S, is a unit in B;
and
Ilzi' - gi(gi)S~ 111 ~ I> • tJ.'.
The problem of what to do if g is not regular in ZI now arises. We shall
show that if g does not vanish identically one can always find a biholomorphic
mapping which takes g into a power series g' regular in ZI'
Let A(O) be the set of all holomorphic functions defined in a (not fixed)
neighborhood of 0 E en, let cP:A(O) ~ Hn with cP(f) = (f)o be the mapping
which associates each local hoi om orphic function f with its Taylor series
expansion about the origin. cP is clearly surjective and commutes with
addition and multiplication in A(O). If U 1, U 2 are open neighborhoods of
oE en, 0': U 1 ~ U 2 a biholomorphic mapping with 0'(0) = 0, then for f,
g E A(O) with (f)o = (g)o we have
(f 0 0')0 = (g 0')0'
0
Therefore the mapping O'*:Hn ~ Hn with O'*((f)o) = (fo 0')0 is well defined
and moreover
1. 0'*( (fl)O + (fl)O) = 0'*( (fl)O) + 0'*( (fl)O)
2. 0'*( (fdo . (fl)O) = 0'*( (fl)O) . 0'*( (fl)O)
3. id*( (f)o) = (f)o
4. (0' 0 p)*( (f)o) = (p* 0 0'*)( (f)o)
5. 0'* is bijective, and (0'*)-1 = (0'-1)*.
0'* is therefore always a ring isomorphism. It is customary to write (f)o 0 0'
in place of 0'*( (f)o).
Def.3.2. Let c = (C2' ... ,cn) E en-I. Then 0',: en ~ en with 0',(w 1 , •.. , wn): =
(WI' Wl + C1 W1, . .. , Wn + cnW1) is called a shearing. Let the set of all
shearings be denoted by L'
Theorem 3.3. L is an abelian group of biholomorphic mappings of en onto

itself.
PROOF. Linear shearings are, of course, holomorphic. It follows from

the equalities
and
0'( 0 0' _( = 0'0 = id(n
that L is an abelian group and that shearings are biholomorphic. 0
Theorem 3.4. Let g E H m g #- O. Then there exists a shearing 0' E L such that
goO' is regular in ZI'
76
PROOF
00 00
1. Let g = L ayaV = L p•.(a) with p.l.(a) = L ava v be the expansion

v=o .1.=0 Ivl=.I.
of g into a series of homogeneous polynomials, Ao: = min{A. E No:p.l. =f. O}.
00
Then for every shearing u, go U = L (P.I. 0 u) is the expansion of g 0 U

.1.=.1.0
into a series of homogeneous polynomials.
2.
P.I. U(WI' 0, ... , 0) =
0 L ayw'J.I(c2WI)V2 ... (CnWI)Vn
Ivl=.I.
-
- "
L." acV2···cYnw.I._p-(c
v 2 n I - .I. 2,···, c)·w.I.
n I
Ivl=.I.
with 15.1. a polynomial in (n - 1) variables. Since by definition not all the

coefficients of 15.1. vanish, there are complex numbers c~O), ••• , c~O) such that
15.l.o(c~O), .. '. , c~O)) =f. O. Let
Then
L L
00 00
go uo(Wl> 0, ... ,0) = (P.I. uo)(Wl> 0, ... , 0) =

0 15.l.(c~O), ... , c~O))wt,
.1.=.1.0 .1.=.1.0
and it is clear that g 0 U0 is regular of order AO in WI' o

Remark. One can show that if gl" .. ,gN are non-vanishing convergent
power series, then there is a shearing U E L such that all gj 0 U are regular
in Zl'
Theorem 3.5 (Weierstrass formula for convergent power series). Let g E Hn
be regular of order s in Zl' Then for every f E Hn there is exactly one
q E Hn and one r E Hn-I[ZI] with deg(r) < s such that f = q' g + r.
PROOF
1. There is atE ~~ such that f and g lie in B t and gs is a unit in B t and
llzi - gg;llit :::; B' ti for an B with 0 q2, rl' r2 lie in B t and g satisfies the
W-condition in B t • From the Weierstrass formula for B t it follows that
ql = q2 and r l = r2. 0
We also have the
Coronary. If f and g are polynomials in Zl with deg(g) = s, then q is also a

polynomial. '
77
Theorem 3.6 (Weierstrass preparation theorem for power series). Let g E Hn

be regular of order sin t l . Then there exists a unit e E Hn and a normalized
polynomial WE Hn-I[ZI] of degree s with
g = e'w
PROOF
1. There exists atE IR~ such that g satisfies the W-condition in B t • The
existence of the decomposition "g = e' w" therefore follows directly from
the Weierstrass preparation theorem for B t •
2. whastheformw =z1- r,whererE Hn-I[ZI] anddeg(r) < s.Ifthere
exist two representations g = el(z1 - r l ) = ez(z1 - rz), then it follows that
ell. g + r l = z1 = ez
l . g + rz. But in this case the Weierstrass formula
says that ell = ezl and r l = rz. Therefore el = e z and WI = Wz (for
WJe: = Z1 - rJe)' 0
Corollary. If g is a polynomial in Zb then e is also a polynomial in Zl'
Theorem 3.7. f E Hn is a unit if and only if f(O) =I- O.

PROOF
1. Iff E H n is a unit, then there exists agE HII with f .g = 1. In particular
f(O) . g(O) = 1, so f(O) =I- O.
2. Iff E HII and f(O) =I- 0, then g: = [flf(O)] - 1 lies in HII and g(O) = O.
Therefore there is atE IR~ such that Ilgllt < 1, which means that !If (0) is a
unit in Bt • But then f is a unit in Hn also. 0
Remark. If the function g E A(O) does not vanish identically near 0,

then there is a shearing a such that (g 00')0 is regular in ZI' By the Weierstrass
preparation theorem we can find a decomposition (g 0')0 = e' W with0
e(O) =I- 0 and W = Z1 + A1 (3')z1- 1 + ... + A s(3') E Hn-1[ZI]. g then has a

zero at 0 if and only if w(O) = 0, and that is the case if and only if As(O) = O.
But As lies in H n _ I' The Weierstrass preparation theorem therefore allows
an inductive examination of the zeroes of holomorphic functions.
4. Prime Factorization
In the following let I always be an arbitrary integral domain and I*: =
I - {O}.
We quote some facts from elementary number theory (see for example,
v.d. Waerden, Vol. 1.).
Def.4.1. Let a E I*, bE I. We say that a divides b (symbolically alb) if there

exists aCE I such that b = a . c.
78
4. Prime Factorization
Def.4.2
1. Let a E 1*, a not a unit. a is called indecomposable if it follows
from a = a l . az with ab az E 1* that a l is a unit or az is a unit.
2. Let a E 1* not be a unit. a is called prime if alaI . a z implies that
ala l or alaz. It is true that a prime is always indecomposable. The converse
is not always the case, but does hold in some important special cases, such
as the ring of integers.
Def.4.3. I is called unique factorization domain (or UFD) if every a E 1*

which is not a unit can be written as a product of finitely many primes.
This decomposition is then determined uniquely up to order and mul-
tiplication by units.
In unique factorization domains every indecomposable element is prime.
Theorem 4.1. If k is afield, then k[ X] is a unique factorization domain.

PROOF. The euclidean algorithm is valid in k[X], hence k[X] is a principal
ideal domain. But every principal ideal domain is a unique factorization
domain. (Details are found in van der Waerden.) 0
Def.4.4
1. Let I be an integral domain. Then the quotient field of I, denoted
by Q(I), is defined by
Q(I): = {~:a,bEI,b # o}.

2. If I[X] is the polynomial ring over I, then we denote the set of
normalized polynomials of I[X] by IO[Xl
Remark. IO[X] is closed with respect to multiplication but not with

respect to addition. Therefore IO[ X] is not a ring. One can, however, consider
prime factorization in IO[ Xl
Theorem 4.2. Let I be a unique factorization domain, Q: = Q(I) the quotient

field. Furthermore, let Wb Wz E QO[X], WE IO[X], and W = Wl . Wz . Then
Wb Wz also lie in IO[Xl
PROOF. For A = 1,2, W;. = Xs), + A;.,lXS).-l+ ... + A;.,s with A;',IlEQ.
Therefore there exists a d;. E I such that d;. . W;. E I[ Xl In the coefficients of
d;. . W;, any common divisors are cancelled.
Now let d: = d l . dz . We assume that there exists a prime element p which
divides d. It follows that d,td;. . W;. for A = 1,2. Let Il;. be minimal with the
property that p,td;, A;., Il A' Now (dlw l ) . (dzw z ) = ... + Xlll+1l2 . [(d l • Al,lll) .
(d zAz'1l2) + terms divisible by p] + .... Therefore the coefficient of Xlll+1l2
is not divisible by p, hence (d l ' wl)(d z . wz) is not divisible by p, which clearly
79
is a contradiction of the fact that (d 1 . ( 1)(d z . wz) = d· W with ill E IO[X]

and pld.
Therefore d has no prime divisors, that is d = d 1 . d z is a unit. Hence
d;., }, = 1, 2 are units in I. It follows that W;. = d;: 1 . dA. WA E I[X] and
hence W;. E IO[ X]. D
Theorem 4.3 (Gauss' lemma). If I is a unique factorization domain, then so

is IO[X]; that is, every element of IO[X] is a product of finitely many
prime elements of IO[X]. (Only the multiplicative structure plays a role,
so one can employ the notion of "factorization" in IO[X].)
PROOF
1. LetwEIO[X] c Q[X].Thenw = W1 'wz ···wtwithw;.EQ[X]prime
(Theorem 4.1). In each case let a;. be the coefficient of the term of highest
degree in W;.. Then clearly 1 = a 1 ... at. Therefore
Without loss of generality we may assume, then, that the W;. are normalized.
2. By induction on e it follows from Theorem 4.2 that all W;. lie in 1° [X].
It still remains to be shown that the WAare also prime in IO[Xl Let wAlw' . w"
with w', w" E 1° [X].
This relation also holds in Q[ X] and there either w;.lw' or wAlw". Say
w;.lw'. Then w' = WA. w~ with w~ E Q[X] and hence QO[X]. By Theorem
4.2 it further follows that w~ E IO[ Xl Therefore W;. is prime in IO[ Xl D
We now apply these results to the special case I = Hn-
L PA be the expansion of f
00
Def. 4.5. Let f E H n , f = as a series of homo-

A=O
geneous polynomials. Then one defines the order of f by the number
ord(f): = min{A E No :p;. -# O}, ord(O): = co.
Then:
1. ord(f) ;:, O.
2. ord(f1 . fz) = ord(f1) + ord(f2)
3. ord(j~ + f2) ;:, min(ord(fd,ord(fz))'
4. f is a unit if and only if ord(f) = O.
Theorem 4.4. H n is a unique factorization domain.

PROOF. We proceed by induction on n. For n = 0, Hn = IC is a field, and
the statement is trivial. Suppose the proposition has been proved for n - l.
1. If f E Hn is not a unit, and f = f1 . fz a proper decomposition, then
ord(f) = ord(fd + ord(fz); therefore the orders of the factors are strictly
smaller. Consequently we can decompose f into a finite number of indecom-
posable terms: f = fl' .. ft·
80
5. Further Consequences (Hensel Rings, Noetherian Rings)
2. Now let f be indecomposable, f1, f2 arbitrary and #0, and flf1 . f2'
A shearing makes f1 0 a, f2 0 a, and f 0 a regular in Zl' Thus it follows that
there exists a decomposition f a = eo OJ and fv a = ev ' OJ v, v = 1, 2,

0 0
in the sense of Theorem 3.7. Since flf1 . f2 we have (f 0 0Wfl 0 a) . (f2 0 a);
therefore OJIOJ1 • OJ2 in H". There exists a q E Hn with q' OJ = Oh . OJ2' By
the Weierstrass formula (Theorem 3.6) q is uniquely determined, and by the
corollary q E H~ - 1 [ Z 1 J.
Since f is indecomposable, so is f a and thus OJ is indecomposable
0
(in H~-l[ZlJ). By the induction hypothesis H n - 1 is a unique factorization

domain, and by Gauss' lemma so is H~ _ 1 [z 1 J. Thus OJ is prime in H~ - 1 [z tJ.
Suppose OJIOJ1; then f 0 alj~ 0 a, so flf1 in H". Every indecomposable ele-
ment in H n is prime. D
5. Further Consequences
(Hen~el Rings, Noetherian Rings)
Hensel Rings
Let R be a commutative C-algebra with 1 in which the set m of all non-
units is an ideal. Let n:R --> Rim and I:C --> R be the canonical mappings.
Proposition
1. m is the only maximal ideal in R.
2. Rim is a field.
3. n 0 1: C --> Rim is an injective ring homorphism.
PROOF
1. Let a c R be an arbitrary maximal ideal. If a contains unit, then
a = R, and that cannot be. Therefore a em; that is, a = m.
2. If n(a) # 0, then a ~ m, and therefore is a unit in R. There exists an
a' ER with aa' = 1, and then n(a)' n(a') = n(a . a') = n(1) = 1 E Rim.
3. It is clear that n 0 1 is a ring homomorphism. If n 0 I(C) = 0, then
I(C) = c· 1 must lie in m, and that is possible only if c = O. Therefore no 1
is injective. D
Def. 5.1. Let R be a commutative IC:-algebra with 1. R is called a local

«:>algebra if:
1. The set m of all non-units of R is an ideal in R.
2. The canonical ring monomorphism n 0 I: C --> Rim is surjective.
Theorem 5.1. Hn is a local C-algebra.

PROOF
1. m = {J E Hn:f(O) = O} is clearly an ideal in H".
2. For f E H n , f = 1(f(0) ) + (f - 1(f(0)) ) with f - 1.(f(0)) E m; there-
fore n(f) = n 0 z(f(O)). Hence n 0 1 is surjective. Moreover, (n 0 /) -1 on(f) =
m D
81
Let R be a local C-algebra with maximal ideal m and the canonical

mappings n: R ~ Rim, I: C ~ R. Then there is a mapping p: R[ X] ~ q X]
with p eto rv xv ) = vto (n 0 Z)-l . n(rv)XV which is clearly surjective.
Def.5.2. Let R be a local C-algebra, p:R[X] ~ qX] the mapping given

above. R is called henselian if there exist normalized polynomials
f1, f2 E R[X] with P(f1) = gl, P(f2) = g2 and f = f1 . f2, whenever
f E R[X] is a normalized polynomial and p(f) = gl . g2 is a decomposi-
tion of p(f) into two relatively prime normalized polynomials gl>
g2 E qX].
Theorem 5.2. Hn is a henselian ring.
This theorem follows directly from Hensel's lemma:
Theorem 5.3 (Hensel's lemma). Let w(u, 3) E H~[ u] have the decomposition
w(u,O) = n (u -
t
;'=1
c,,)s). into linear factors (with Cv =1= Cll for v =1= p and
Sl + ... + St = :s = deg(w)). Then there are uniquely determined poly-
nomials Wl> ... , Wt E H~[u] with deg(w;.) = s;. and w;.(u, 0) = (u - c;.y)·
for A = 1, ... , C such that w = W1 ... Wt.
PROOF. We proceed by induction on C. The case C = 1 is trivial; we assume
that the theorem has been proved for C - 1.
1. First assume that w(O, 0) = o. Without loss of generality we can assume
that C1 = 0; thus w(u,O) = uS, . h(u) with deg(h) = s - Sl and h(O) =1= o.
This means that w is regular of order Sl in u and we can apply the Weierstrass
preparation theorem:
There is a unit e E Hn+l and a polynomial W1 E H~[u] with deg(w1) = Sl
such that w = e . W1. From the corollary it follows that e lies in H~ [u J.
W1(0,0) == 0, since w(O, 0) = 0 and e(O, 0) =1= 0; so W1(U, 0) = uS,. Therefore
e(u,O) = h(u) = n (u -
(
;'=2
c;.)s).. By induction there are elements W2, ... , Wt E
H~[u] with deg(w;.) = s;., w;.(u,O) = (u - c)Y). and e = W2 ... Wt. W =

Wi W2 ... w( is the desired decomposition.
2. If w(O, 0) =1= 0, then let W'(U,3): = w(u + Cl> 3). As in (1) we find a
decompsotion w' = wi ... w~ and with w;.(u, 3): = w~(u - cl> 3) obtain a
decomposition in the sense of the theorem.
The uniqueness of the decomposition is also proved by induction on C. In
Case 1 the induction step follows directly from the Weierstrass preparation
theorem, and Case 2 reduces to Case 1. 0
Noetherian Rings
Def. 5.3. Let R be a commutative ring with 1. An R-module M is called
finite if there exists a q E N and an R module epimorphism cp: Rq ~ M.
82
5. Further Consequences (Hensel Rings, Noetherian Rings)
This is equivalent to the existence of elements e10 ... , eq E M such that

q
every element x E M can be written in the form x = L rvev with rv E R.
v= 1
Def. 5.4. Let R be a commutative ring with 1. R is called noetherian if every

ideal J c R is finitely generated. An R-module M is called noetherian
if every submodule M' c M is finite.
Theorem 5.4. If R is a noetherian ring and q E N, then Rq is a noetherian

R-module.
PROOF. We proceed by induction on q.
The case q = 1 is trivial. Assume the theorem is proved for q - 1. Let
M c Rq be an R-submodule. Then J: = {rl E R: There exist r2, ... ' rq E R
with (rl' r2' . .. , rq) E M} is an ideal in R and as such is finitely generated by
elements r\;'>' A. = 1, ... , t. For every r\;') there are elements r~;'), ... , r~;') E R
such that t;.: = (r\;'), r~;'), ... , r~;'» lies in M for A. = 1, ... , t. M': = Mil
({O} x Rq-l) can be identified with an R-submodule of Rq-l and is there-
fore, by the induction assumption, finite.
Let t;. = (0, r~;'), ... , r~;'», A. = t + 1, ... , p be generators of M'. Ift E M,
e
we can write t = (r1o t') with rl E J, therefore rl = L a;.r\;'), a;. E R. But then
;'=1
t - L(
;'=1
a;.t;. -
-( tL a;.(r2 , ... ,rq))
0,'
t -
;'=1
(;.) (;.)
EM.,
That is, there are elements at+ 10 ... , ap E R such that

/ P
t - L a;.t;. = ;'=/+1
;'=1
L a;.t;.,
Hence
{t1o ... , t p } is a system of generators for M. o

Theorem 5.5 (RUckert basis theorem). Hn is a noetherian ring.
PROOF. We proceed by induction on n. For n = 0, Hn = IC and the state-
ment is trivial. We now assume that the theorem is proved for n - 1. Let
°
J c Hn be an ideal. We may assume that we are not dealing with the zero
ideal, so there exists an element g # in J. By application of a suitable
shearing (T, g': = goa is regular of order s in z 1. a induces an isomorphism
a*: Hn ~ Hn with a*(g) = g'. a*(J) is an ideal in Hn along with J, and if
a*(J) is finitely generated, then J = (a*)-l a*(J) is also finitely generated.
Without loss of generality we can then assume that g is already regular of
order s in Zl. Let tPg:Hn ~ (Hn - 1 )' be the Weierstrass homomorphism, which
83
will be defined in the following manner: For every f E Hn there are uniquely
defined elements q E Hn and r = 1"0 + r 1 z 1 + ... + r s _ 1 4- 1 E H,,-l[Zl],
such thatf = q' g + r. Let rpg(f): = (1'0, ... , rs -1)' rpg is an H n- 1-module
homomorphism. By the induction hypothesis H n - 1 is noetherian and so by
Theorem 5.4, (H n - 1 )' is a noetherian H n_ 1-module. M: = rp/J") is an
H n_ 1-submodule, and therefore finitely generated. Let r(}') = (rb.l.), ... , r~~ 1),
), = 1, ... ,t, be generators of M. If f E f is arbitrary, then f = q . g +
(ro + r 1 z 1 + ... + rs_1zt-1),andthereareelementsa1, ... ,atEHn_1sUch
(
that (ro, 1'1' ... , r s - 1) = I a.l.r(.I.). Hence we obtain the representation

.1.=1
t
f = q' g + I airb.l.) + ri.l.)Zl + ... + r~~lZt-1),
.1.=1
I.e.,
is a system of generators of f. o
Remark. We have up to now shown that Hn has a unique factorization,

and is a henselian and noetherian local C-algebra. If f c Hn is an arbitrary
ideal (with f i=- H n ), then A: = H,Jf is called an analytic algebra. A is like-
wise noetherian and henselian. Analytic algebras playa decisive role in the
local theory of complex spaces, a generalization of the theory of analytic sets
sketched in the following section.
6. Analytic Sets
Def. 6.1. Let Been be a region, M c B a subset and 30 E B a point. M is
called analytic at 30 if there exists an open neighborhood U = U(30) E B
and functions fI> ... , J" holomorphic in U such that
Un M = {3 E U:f1(3) = ... = '/;(3) = O}.
M is called analytic in B if M is analytic at every point of B.
Remark. If Been is a region and fI> ... ,J" are elements of A(B), then
we call the set
N(f1, ... ,J,,): = {3 E B:f1(a) = ... !t(a) = O}

the zero set of the functions f1' ... ,ff'
Theorem 6.1. If Been is a region and M c B is an analytic set in B, then

M is closed in B.
84
6. Analytic Sets
PROOF. We will show that B - M is open. If 30 E B - M, then there exists

an open neighborhood U = U(30) c B and functions f1> ... ,it E A(U) with
N(fb ... ,it) = Un M such that, say, fl(30) # o. Then there is an entire
neighborhood V = V(30) c U such that fllY vanishes nowhere, and hence
V is contained in B - M. Therefore 30 is an interior point, and B - M
is open. 0

be a domain. Then the ring A(G) of functions
holomorphic on G is an integral domain.
PROOF. We need only to show that A( G) has no zero divisors: Suppose f1' f2
are two elements of A(G) with f1 # 0 and j~ . f2 = o. Then there is a 30 E G
withf1(30) # 0, and hence an entire neighborhood V = V(30) c G such that
f1 never vanishes on V. But then we must have f21V = 0, and, by the identity
theorem, f2 = o. 0
We cannot conclude from this that A = A(G) is a unique factorization

domain. But we shall show that A o[u] is a unique factorization domain, as
a consequence of the following theorem:
Theorem 6.3. Let I be an integral domain, Q = Q(I) the quotient field of I.

IO[ X] is a unique factorization domain if I satisfies the condition:
PROOF. Although Gauss' lemma assumed that I was a unique factorization

domain, the proof only used the above property of I, which is satisfied for
every unique factorization domain. 0
We now show that I = A satisfies the hypothesis of Theorem 6.3. The

quotient field Q: = Q(A) is the field of "meromorphic functions" on G. The
elements h = fig can naturally be interpreted as functions only in a very
broad sense. Poles may occur, and more besides! If f and g vanish indepen-
dently at a point, then in general one cannot assign any reasonable value
to h at that point. Such indeterminate points only occur for merom orphic
functions of several variables. In what follows we confine ourselves to the
algebraic properties of Q.
For 3 E G let 13 = (Hn)~ be the ring of convergent power series at 3 and
Q3 = Q(I3) the quotient field of 13. Moreover, let A(3) be the set of all functions
defined and holomorphic on a neighborhood of 3. For f E A(3) let (f), denote
the power series of f at the point 3. Then for every 3 E G there exists a ring
homomorphism
e,:Q ~ Q3 • with e, (I) =

g
(f)3.
(g),
85
By the identity theorem, (g)o -# 0 and furthermore 80 is injective. Now if
h = £g E Q and (h)30: = 030(h) = ((f))30

g~
E 130 ,
then (g),o must be a unit in 130 and therefore g(30) -# O. But then there is an
open neighborhood V = V(30) c G such that g is nowhere vanishing on
V, and on V, h represents a holomorphic function. If (h)30 E 130 for every
point 30 E G, then h is a holomorphic function on G.
Theorem 6.4. If Wb W2 are elements of QO[u] with W1 . W2 E AO[u] then

Wb W2 E AO[u].
PROOF
1. If WE QO[u], then W has the form W = US + A 1u'-1 + ... + As with
A;E Q for i = 1, ... ,s. Let (W)3: = US + (A 1),u s- 1 + ... + (As)3E Q~[u].
If (w)3lies in Inu] for all 3 E G, then it follows from the above considerations
that A 10 ... , As are holomorphic functions; that is, W E A [u]. °
2. If W1, W2 are elements of QO[u] with W1 . W2 E AO[u], then for all
3 E G (W1)3' (W2)3 E Q~[u] and (W1)3 . (W2)3 E I~[u]. Since 13 = (Hn)3 is a UFD,
it follows that (W1)3' (W2)3 E 1~[u]. By (1) this means that Wb W2 E AO[u]. 0
Theorem 6.5. Let G c en be a domain, A = A(G). Then AO[u] is a unique

factorization domain.
The proof follows directly from Theorems 6.3 and 6.4.
Def. 6.2. Let I be an integral domain. 1 is called a euclidean ring if there

exists a mapping N: I ~ No with the following properties:
1. N(a' b) = N(a) . N(b).
2. a = 0 ~ N(a) = O.
3. For all a, b E 1 with a -# 0 there exists a q E 1 with N(b - q . a) < N(a).
EXAMPLES
a. 7!.. is a euclidean ring, with N:7!.. ~ No with N(a): = lal.
b. If k is a field, then k [X] is a euclidean ring, by virtue of the mapping
N:k[X] ~ No with
N(f): = 2deg (J) (and N(O): = 0).
Every euclidean ring is a principal ideal domain (and thus a unique

factorization domain). If a 1 , a2 are elements of a euclidean ring, then their
greatest common divisor can be written as a linear combination,
gcd(ab a2) = r 1 . a1 + r2 . a2,
where N(r1 . a1 + r2 . a2) is minimal. Of course, the greatest common
divisor is uniquely determined up to units only.
86
6. Analytic Sets
Again let G c en be a domain, A = A(G), Q = Q(A) the field of mero-

morphic functions on G. Q[u] is a euclidean ring. If W 1, W z are elements of
Q[u], consider all linear combinations W = P1 W1 + PzWz with Pb pz E
Q[u] and W =f. O. If W has minimal degree, then W is a greatest common
divisor of W1 and Wz. Let hE A be the product of the denominator of P1
and pz. The polynomials h . Pi lie in A [u] and (h . P1)Wl + (h . pz)wz = h· w.
But since h is a unit in Q[ u], we have
Theorem 6.6. If Wl, Wz are elements of Q[ u], then there exists a greatest
common divisor of Wl and Wz which can be written as a linear combination
of Wl and Wz over A[u].
°
Def. 6.3. An element WE A [u] is called a pseudopolynomial without multiple
factors if the factors Wi (by Theorem 6.5 uniquely determined) of the
prime decomposition W = Wl ... Wt are pairwise distinct.
Def. 6.4. Let a mapping D: A [u] ---+ A [u] be defined by
D eto Av(~)UV} vt = v' Av(~)uv-l.

If WE A[u], then one calls D(w) E A[u] the derivative of w.
Remark. The following formulas are readily verified:

1. D(wl + wz) = D(wd + D(wz)·
2. D(Wl . (2) = Wl . D(w2) + W2 . D(Wl)'
I
(
3. D(Wl ... Wt) = Wl ... Wv ... Wf' D(w.}. (Here, the hat on Wv in-
v= 1
dicates that this term is to be omitted.)
Now let W = Wl ... Wt = US + A l (;3)u S- 1 + ... + As(~) be a pseudo-
polynomial without multiple factors (in AO[u]). Then
I
(
D(w) = W2 ... Wt . D(wl) + Wl ... Wv ... w( . D(w.}.

v=2
Clearly Wl can only be divided by D(w) if Wl is a divisor of D(Wl)' How-

ever, since deg(D(wl)) < deg(wl), aWl E Q[ u] with Wl . wl = D(Wl) cannot
exist. Therefore Wl is not a divisor of D(w), and the same holds for W2' ... , Wt .
Hence wand D(w) have no common divisor.
Theorem 6.7. Let WE AO[u] be a pseudopolynomial without multiple factors.

Then there are elements qb qz E A[u] such that h: = ql . W + q2 . D(w)
lies in A and does not vanish identically.
PROOF. Wehaveshownabovethatgcd(w, D(w)) = 1,sothereexistelements
P1' P2 E Q[ u] with P1 W + P2 . D(w) = 1. If we multiply the equation by an
appropriate factor h E A (with h =f. 0), we obtain (Pl' h)' W + (P2 . h) . D(w) =
h,withpl·h,P2·hEA[U]. D
87
In the same way one proves:
Theorem 6.8. If Wb W2 E A[ u] are relatively prime, then there exist elements

qt. q2 E A[u] such that ql . WI + q2 . W2 lies in A and does not vanish
identically.
We must briefly entertain the notion of a symmetric polynomial.
Def.6.S. A polynomial p E E[X 1, ... , XsJ is called symmetric if for all v, p

p(X 1 , · · · , XV"'" XJl,"" X.) = p(Xt. ... , XJl,"" XV"'" Xs)·
The most important example are the elementary symmetric polynomials
at. ... , as where
al(Xt. ... ,Xs) = Xl + ... + X.,
a2(Xt. ... ,X.) = (X l 'X 2 + ... + XI·X.)
+ (X 2 ' X3 + ... + X 2 . Xs) + ... + X s- 1 . Xs
a,(Xt. ... ,XJ ~ Xl "'X s
(so in general
av(X b' .. , Xs): = I Xi! ... XiV)'

1 ~il < .. , <iv:::;s
In algebra (see van der Waerden I) one proves:
Theorem 6.9. Let P(Xb' .. , X.) be a symmetric polynomial with integer

coefficients. Then there is exactly one polynomial Q(YI , · · . , Y,) with
integer coefficients such that
P(Xb"" X.) = Q(al(X b "" Xs),"" as(X b ···, Xs)).
Another important example of a symmetric polynomial is the square of

the V andermonde determinant:
1, Xl, Xi,···, X~-ll

D(Xb"" X.): = det 2 [ .
•
.
•
= n (Xv -
V<J1
X,Y
1, X., X;, ... ,X~-l
Clearly D(X 1, ... , Xs) = 0 if and only if there exists a pair (v, p) with
v oF p and Xv = XJl"
Def.6.6. Let f(X) = Xs - alXs- 1 + a2XS-2 + ... + (-l)Sa s E qXJ be a

polynomial and let Q E E[ Xl, ... , X.] be that polynomial for which the
88
6. Analytic Sets
equation
D(X I ,··., Xs) = Q(CTI(Xb"" Xs),···, CTs(X I,···, Xs))
holds. Then A(f): = Q(al,' .. , as) is called the discriminant of f(X).
EXAMPLE. Let f(X) = X 2 - aX + b. For s = 2 we have

D(X I , X 2) = (Xl - X 2)2 = xi - 2X I . X 2 + X~,
CTI(X b X 2) = Xl + X 2, CT2(Xb X 2) = X I . X 2·
If we set Q(Yb Y2 ): = Yi - 4Y2 , then
Q(CTI(XI,X2),CT2(XbX2)) = (Xl + X 2)2 - 4,X I 'X 2 = D(Xb X 2)'
Therefore A(f) = Q(a, b) = a 2 - 4b.1f Cb C2 are both zeroes ofj(X), then:
f(X) = (X - c I ) . (X - C2) = X 2 - (CI + C2)X + CI . C2
= X 2 - CTI(CI, C2)X + CT2(CI, C2),
and therefore A (f) = D(Cb C2)'
Thus A(f) vanishes if and only if CI = C2'
s
Theorem 6.10. Let f(X) = TI (X - Xp) E C[X]. f has a multiple root if
p=l
and only if A (f) ;= O.
PROOF
f(X) = (X - Xd(X - X 2)··· (X - Xs)
= Xs - (Xl + ... + Xs)XS- 1 + (X I X 2 + .. ')X S- 2 + ...
+ (-I)'X I ·X2 ·"X.,
i.e.:f(X) = XS - alXs- I + a2XS-2 + ... + (-I)Sa s
with
av = CTv(X 10 ••• , X,,) for v = 1, ... , s.
Therefore
A(f) = Q(a1o···,as) = Q(CTI(X1o···,Xs),···,CTs(Xt,···,Xs))
= D(X1o""Xs) = TI (Xv - X/yo 0
V<Jl
Now let w(u, 3) = US - A 1(3)Us- 1 + ... + (-1)SAs(3) be a pseudopoly-

nomial over G. A holomorphic function on G is defined by A",(3): =
A (w(u, 3)) = Q(A 1(3), . .. , As(3)). Clearly A",(3) i= 0 if and only if w(u, 3)
has s distinct roots. But more is true:
Theorem 6.11. Let G c cn be a domain, w(u, 3) E AO[uJ a pseudopolynomial.

LI", does not vanish identically if and only if w has no multiple factors.
89
PROOF
1. Let W = wi . w with deg(w l ) > O. If 3 E G, then we can decompose
W l (u, 3) into linear factors,
w l (u,3) = (u - c l )· .. (u - ct ).
For W(ll, 3) we obtain a decomposition of the form
W(u,3) = (u - cd, .. (u - Ct )2(U - ct + l )· .. (u - cp ).
Hence
Since 3 was arbitrary, .1", = O.

2. Let W be a polynomial without multiple factors. Then there are elements
qb q2 E A[u] such that h: = ql . W + qz . D(w) E A does not vanish iden-
tically. We can find a 30 E G with h(30) =f. O. Let ai(u): = qi(U, 30) E C[u]
fori = 1, 2. Then
a: = al(u)' w(u, 30) + az(u)' D(w)(u, 30) =f. 0 (independent ofu).
If w(u, 30) = (u - cd Z • w(u), then
D(w)(u, 30) = 'D(w(u, 30)) = 2, (u - Cl) . w(u) + (u - Cl)Z . D(W(ll))
= (u - c l ) , (2w(u) + (u - Cl) , D(w(u))) = (u - Cl) . Wl(U),
and therefore
a = al(c l ) . w(c b 30) + aZ(cl) ' D(w)(Cb 30) = 0,
which cannot be, Hence all the roots Cb . , . , Cs of w(u, 30) must be distinct,
and .1",(30) = D(c b ' , , , cs ) =f. O. 0
Theorem 6.12. Let G c en be a domain, A = A(G),

w(u,3) = US - A l (3)uS- l + ... + (-1)SAs(3)EAO[u]
a pseudopolynomial without multiple factors,
Mro: = {(u, 3) E C x G:w(U,3) = O}, Dw : = {3 E G: .1,oC3) = O},
e
30
u-v-Dw en
G
Figure 16. Illustration for Theorem 6.12.
90
6. Analytic Sets
Then Moo and Doo are analytic sets and:

1. For 30 E G - Dro there exists an open neighborhood U (30) c G - D 00
and holomorphic functions fh ... ,I. on U with !v(3) -# f,J3) for v -# J1.
and 3 E U, such that w(u, 3) = (u - f1 (3» ... (u - 1.(3» for all 3 E U.
2. The points of Dro are "branch points," that is, above a point 3 E Doo
there always liefewer than s points of the set Moo.
PROOF. W(u,3) always has exactly s distinct roots above G - Dro; above
Doo multiple roots appear. Now let 30 E G - Doo , w(u, 30) = (u - cd··· (u - cs)·
w30 is a polynomial over the ring (Hn)w and by the Hensel lemma there
are polynomials (Wi)30' i = 1, ... , s, with the following properties:
1. (Wi)30 (u, 30) = u - Ci for i = 1, ... , s
2. (Wl)ao ... (ws)ao = wao
3. deg( (Wi)30) = 1 for i = 1, ... , s.
In particular we can write
(Wi)30 = u - ri with ri E (Hn),IO for i = 1, ... , s.
Then there exist a connected open neighborhood U (30) c G - D00 and
holomorphic functions fl' ... ,I. on U.;such that the power series ri converge
to k If we set w(u, 3): = (u - fl (3» ... (u - 1.(3», we obtain
W30 = (u - (fd ao ) ... (u - (!.)ao) = (u - r1) ... (u - rs) = wao'
Therefore, near 30-and by the identity theorem in all of U - wand
W must coincide. Hence W(u,3) = (u - f1 (3» ... (u - 1.(3» on U, and
because U c G - Doo , fv(3) -# fi3) for v -# J1.. 0
We now can continue with the study of analytic sets. We begin with
hypersurfaces:
Let G c en be a domain, f be holomorphic and not identically zero on
G and N: = {3 E G:f(3) = O}. Let 30 E N be a fixed point. Since a shearing
does not change an analytic set essentially, we can assume without loss of
generality that (f)30 is regular in Zl ' By the Weierstrass preparation theorem
there exists a unit (e)ao and a pseudopolynomial (w)ao such that (f)30 =
(e)ao . (W)3o' We can find a neighborhood U(30) c G on which (e)30 resp. (w)ao
converge to a holomorphic function e and a pseudopolynomial W such that
fl U = e' w.lfwe choose U sufficiently small then e(3) -# 0 for all 3 E U, and
therefore
{3 E U:f(3) = O} = {3 E U: W(Zl' 3') = O}.
Now let W = Wl ... Wt be the prime decomposition of w. Then
(
{3 E U:f(3) = O} = U {3 E U:wi(3) = O}.

i=l
Ifmultiple factors appear, then the corresponding components of the analytic
set are equal; it is sufficient therefore to restrict our attention to pseudopoly-
nomials without multiple factors. Let 30 = (z~o), 30), G1 be an open neighbor-
hood of z~o) E C and G' be a connected open neighborhood of 30 E en - 1 such
91
III. The Weierstrass PreparatIOn Theorem
that
G 1 x G' c U and
Moreover, let
DO) = {a' E G':LlO)(a') = OJ.
N n (G 1 x G') represents a branched covering of G1 whose branch points
lie over D", (see Theorem 6.12); over G1 - DO) the covering is unbranched.
One knows the analytic set N c cn once we know the analytic set D", c Cn- 1
Figure 17. Representation of an analytic set as a branched covering.
and the branching behavior of N. Inductively one obtains such an overview

of the construction of N. We will consider special cases:
(A) n = l. Let G c C be a domain, f: G --+ C a holomorphic function
which vanishes identically nowhere. The local pseudopolynomials corre-
sponding to f are polynomials over C, each having finitely many zeroes. The
analytic set N = {z E G:f(z) = O} therefore consists of isolated points which
may cluster at the boundary of G.
(B) n = 2. It suffices to consider pseudopolynomials.
l. Let G = C 2 , w(u, z): = u2 - z, N: = {(u, z) E G:w(u, z) = OJ. N -
{CO, O)} is the Riemann surface of)2. The discriminant is LlO)(z) = 4z. Clearly
D", = {zEC:Ll",(z) = O} = {OJ. For ZoEC - DO) there is a neighborhood
V(zo) c C - DO) and above V there is a decomposition WeLl, z) =
(u - )Z)(u + )Z). This yields a 2-sheeted covering above C - DO) and a
branch point above DO) = {OJ. N as well as N - {CO, O)} are connected topo-
logical spaces.
2. Let G = C 2 , w(u, z): = u 2 - Z2 = (u - z) . (u + z). Then
N: = {(U,Z)EC 2 :W(U,z) = O}
= {(u; z) E C 2 :u = z} u {(u, z) E C 2 :u = -z}.
92
6. Analytic Sets
The discriminant is .dw(z) = 4Z2 with zero set

Dw = {z E C:.dw(z) = O} = {O}.
In this case globally N consists of two distinct schlicht sheets which intersect
only above the origin. N is connected but N - {(O, O)} is no longer. In such
a case one speaks of pseudo-branching.
(1) and (2) are the two characteristic cases which can occur. One inductively
reduces cases of higher dimension-as described above-to cases A and B.
There still remains the question how to proceed in the case of analytic sets
which are described by several equations.
Let there be given a domain G c en and holomorphic functions f1> f2 on
G. Both fl and f2 vanish nowhere identically. Then let M: = {3 E G:fl (3) =
f2(3) = O} and 30 E M. A shearing makes (fl),o and (f2)30 simultaneously
regular in z 1> and then there are a connected neighborhood U = U 1 X U' of
30 and pseudopolynomials WI' W2 E A(U,)O[ZIJ with
for i = 1,2
and
We can assume that the polynomials Wi contain no multiple factors; but in

general they are not relatively prime. There are polynomials ill, W'1> w~ E
A(U')O[ZIJ with
and gcd(w'I' w~) = 1.
Hence M (') U = Ml U M2 with
and
M 1 is a "hypersurface" such as we have already considered. M 2 is given by

two relatively prime pseudo polynomials. By Theorem 6.8 there exist poly-
nomials q1> q2 E A(U')[ZIJ such that h: = ql . W'1 + q2 . w~ is a nowhere
identically vanishing holomorphic function on U'. Let
M': = {3 E U':h(3') = O}.
If n: U ~ U' is the projection with n(z b 3') = 3', then it is clear that n(M 2)
lies in M'. Naturally above each point 3' E M' there lie only finitely many
points of M 2 • "M2 lies discretely over M'."
By means of sheaf theory one can show that n(M 2) is itself an analytic
hypersurface in U' and that there exists a nowhere dense analytic subset N
in n(M2) such that M2 - n- 1 (N) is a smooth several sheeted covering of
n(M2) - N.
Similar considerations apply to analytic sets which are given by several
functions. At this point we want to give one example showing that in general
93
analytic sets cannot be defined by global equations. Let

Q1: = {3 = (Zl,Zz)EC 2:j Z11 < t, IZzl < 1},
Q2: = {3 = (Zl,Z2)EC 2:lz11 < 1,t < IZ21 < 1},
Q: = Q1 U Q2·
Furthermore let
IZll
Figure 18. An analytic set which cannot be defined globally.
1. QI> Qz are open subsets of Q and M n Q1 = 0, M n Q2 = {(Zl' Z2) E

Q2 :Zl - Zz = O}. M is therefore an analytic subset of Q.
2. We assume that there exist holomorphic functions fI> ... ,ft on Q
such that M = {3 E Q:f1(a) = ... = fAa) = O}. But then there exist holomor-
phic extensions F 1, . . . , Fe on P (with FdQ = h for i = 1, ... , t), and holo-
morphic functions Fi: {z E C: Izi < 1} ~ C are defined by Fi (z): = Fi(z, z).
For t < Izl < 1 (z, z) lies in M and therefore Ft(z) = Fi(z, z) = .t;(z, z) = O.
By the identity theorem it then follows that Fi = 0 for i = 1, ... , t. There-
fore h(O, 0) = Fi(O, 0) = Fi(O) = 0 for i = 1, ... , t; that is, (0. 0) lies in M.
That is a contradiction, and the analytic set M cannot be defined globally.
Nevertheless, by means of sheaf theory one can prove the following
theorem:

be a domain of holomorphy, MeG analytic.
Then there exist holomorphic functions f1' ... ,fn+ 1 on G such that
M = {3 E G:f1(3) = ... = fn+1(3) = O}.
Next we present a short survey of further results from the theory of

analytic sets.
94
6. Analytic Sets
Theorem 6.14. Let G c en be a domain. Then:

1. 0 and G are analytic subsets of G.
e
2. If M 1> ••• , Me are analytic in G, so is U Mi'
i= 1
3. If M 1, ... , Me are analytic in G, so is n Mi'

(
3'. If(M,),eI is a system of analytic sets, n

i= 1
,eI
M, is analytic in G.
PROOF
1. 0 = {3 E G: 1 = O}, G = {3 E G:O = O}.
(
2. Let 30 EM: = U Mi' Then there exists an open neighborhood

i= 1
U(30) c G and holomorphic functions h,j' j = 1, ... , di such that
Mi n U = {3 E U:h,1(3) = ... = h,d,(3) = O}.
Letj(i1, ... ,j,): = f1,i1 ... ft,i/' Then
M n U = {3 E U:fu" ... ,j,l3) = 0 for all indices (j1>'" ,j,)}.
n Mi' Then
(
3. Let 30 EM': =
i=l
Un M' = {3 E U:f;j3) = 0 for i = 1, ... , e, j = 1, ... , d;}.

3'. is more difficult to prove. The proof will be omitted here. 0
Comment. (1), (2), and (3') are the axiomatic properties of closed sets in a
topology. In fact, we get the so-called Zariski topology on G by defining
U c G to be open if and only if there exists an analytic set M in G with
U = G - M.
DeC. 6.7. Let G c en be a domain, M analytic in G. A point 30 E M is called

a regular point (ordinary smooth point) of M (of dimension 2k) if there
exists an open neighborhood U(30) c G and functions f1,'" ,fn-k
holomorphic on U such that
(1) Un M = {3 E U:f1(3) = ... = fn-k(3) = O}.
(2) Of; (30

r k (( -
»)i. = 1, ... ,n - k) -_ n - k.
OZj } = 1, ... , n
A point 30 E M is called singular (a singularity of M) if it is not regular.

One denotes the set of singular points of M by SCM). Let 30 be a regular
point of M. Without loss of generality we can assume that
det ((Of; (30») ~ : 1, ... , n -

OZj } - 1, ... , n - k
k) =1= O.
95
Now let F: U --+ en be defined by

F(Zb ... ,zn): = (f1(zb ... , zn), ... ,fn-k(zi> ... , zn),
Zn-k+ 1 - z~Olk+ i> ... , Zn - z~O»).
Let
OJ; (30)) ~ :
((( OZj 1, ... , n - k) i * )
a-- ] - 1, ... , n - k I
$': = ----------------------~-1-----
o I ~1
be the functional matrix of F at the point 30. Then clearly det :J' i= 0 and
there exist open neighborhoods V(30) c u, W(O) c en such that F[ V:
V --+ W is biholomorphic. But F(V n M) = W n {(Wi> . . . , wn ) E en:
W 1 = ... = W n - k = O} is a real 2k-dimensional plane segment.
Theorem 6.15. Let G c en be a domain, M analytic in G and 30 E M a

regular point of M of dimension 2k. Then there exists an open neighbor-
hood V(30) c G such that M n V is biholomorphically equivalent to a
plane segment of real dimension 2k.
Theorem 6.16. Let G c en be a domain, M analytic in G. Then the set S(M)

of singular points of M is a nowhere dense analytic subset of G.
Der. 6.8. An analytic set M is called reducible if there exist analytic subsets
Mi c G, i = 1,2, such that:
1. M = M1 U M 2 •
2. Mi i= M, i = 1, 2.
If M is not reducible, it is called irreducible.
Theorem 6.17. Let G c en be a domain, M analytic. Then there is a countable

system (MJ of irreducible analytic subsets of G such that
1. U Mi = M.
iEN
2. The system (MdiEN is locally finite in G.
3. If Mil i= M i2 , then Mil ¢ M i2 .
We speak of a decomposition of M into irreducible components. This
decomposition is unique up to the order in which the components appear.
The proof is lengthy and requires the help of sheaf theory.
Remark. Let M be irreducible. Then:

1. M - S(M) is connected. (This condition is equivalent to irreducibility.)
2. The dimension dim 3(M) of the point 3 E M - S(M) is independent of
96
6. Analytic Sets
3. The number thus obtained is denoted by dim;;!(M). The complex dimension

of Mis dimdM): = t dime«M).
If M = U Mi is the decomposition of an arbitrary analytic set into
ieN
irreducible components, then we define
dimdM): = max dimdMJ
iEN
dimdM) ~ n always. In particular, if dime(Mi) = k for alI i E N, then we

say that M is of pure dimension k.
Theorem 6.18. If M is an irreducible analytic set in G and f a holomorphic

function on G with flM "# 0, then dimdM (1 {3 :f(3) = O}) = dimdM) - l.
For every irreducible component N c M (1 {3 :f(3) = O} we have
dimdN) = dimdM) - 1, hence:
Theorem 6.19. Let G c IC" be a domain, let f1>' .. ,fn-k be holomorphic

fimctions in G, M: = {3 E G:fl(3) = ... = !,,-k(3) = O}, M' c M an irre-
ducible component. Then dimdM') ~ k.
PROOF. G itself is an irreducible analytic set. Then, by Theorem 6.18,
dimd{3EG:fl(3) = OJ) ~ n - 1, and the set Ml = {3EG:fd3) = O} is
pure dimensional. Let M 1 = U M 1 ) be the decomposition of M 1 into
iEN
irreducible components.
Then dimdN~l) (1 {3EG:f2(3) = OJ) ~ n - 2 and we obtain the
same value for each i EN. Therefore dimd {3 E G:fl (3) = f2(3) = O}) =
dime (.U M1)
lEN
(1 {3 E G:f2(3) = OJ) ~n- 2. We are finished after finitely
m~~~. 0
In conclusion we consider one more example of an analytic set:

Let f: IC" -> C be defined by
with
if and only if Zi = O. Thus only the origin could be a singularity. It can be

shown that S(M) = {OJ. In this case we say that M has an isolated singularity
at the origin.
Clearly M belongs to the family (MJt E C of analytic sets which are given by
Mt = {(Zl"",Zn)EIC":Zl' + ... + r"n = t}.
M = M 0 is an analytic set with an isolated singularity at the origin, while
alI sets M t with t "# 0 are regular. The family (Mt)tEC is calIed a deformation
ofM.
97
-------
Figure 19. Deformation of an analytic set.
One can consider corresponding situations in the real analytic case.

Suppose a, b are real numbers with a < 0 < b and let (MI)IE[a,bl be a family
of real analytic sets which are free of singularities for t "# 0 and which have
a singularity at the origin. It can then occur that for t = 0 the topological
structure jumps, that is:
All sets MIl' MIl with tb t2 < 0 are homeomorphic, all sets MIl' MI2 with
tb t2 > 0 are homeomorphic, but for tl < 0 and t2 > 0, MIl and Mt2 are
not homeomorphic.
R. Thorn recently applied this theory to the developmental processes
in biology for example. One can call the jumping ofthe structure a revolution.
Thorn speaks instead of a catastrophe!
98
CHAPTER IV
Sheaf Theory
If 30 E en is a point, then (930 = (Hn) denotes the if-algebra of convergent

power series convergent at 30. An arbitrary element of (930 has the form ho =
C()
L a (3 -
v=o
v 30r·
Therefore there is a if-algebra (93 for each point 3 E en. The disjoint union
(9: = U(9,1 of these algebras is a set over en with a natural projection
3 E C'
n: (9 ~ en taking a power series h onto the point of expansion 3. There exists
a natural topology on (9 which makes n a continuous mapping and induces
the discrete topology on every stalk (9,1' derived as follows.
If 100 E (9, then there exists an open neighborhood U(30) c en and a holo-
morphic function f on U such that the series 1;0 converges uniformly to f
in U. Therefore, the function f can also be expanded in a convergent power
series at each point 3 E U. Hence f induces a mapping s: U ~ (9 with the
following properties:
1. nos = idu
2. S(30) = 100 E s(U) c (9.
All such sets s(U) form a system of neighborhoods of foo in (9. If we give (9
the topology induced in this way, then the topological space (9 is called the
sheaf of convergent power seies. The if-algebras (93 = n - 1 (3) are called stalks
of the sheaf. n is locally topological and the algebraic operations in (9 are con-
tinuous in this topology.
1. Sheaves of Sets
Def.1.1. Let Been be a region, Y' a topological space, and n:Y' ~ B a
locally topological mapping. Then 6 = (Y', n) is called a sheaf of sets over
B. If 3 E B, then we call Y'3: = n - 1(3) the stalk of 6 over 3.
99
IV. Sheaf Theory
Remark. In exactly the same manner we define sheaves over arbitrary

topological spaces. If it is clear how the mapping n is defined, we shall also
write!/' in place of 6.
Def. 1.2. Let (!/', n) be a sheaf over B,!/'* c !/' open and n*: = nl!/'*. Then
(!/'*, n*) is called a subsheaf of!/'o
Remark. Each subsheaf(!/,*, n*) ofa sheaf(!/" n) is a sheaf. We need only

show that n*:!/'* ~ B is locally topological.
For every element a E !/'* there are open neighborhoods U(a) c !/' and
V(n(a)) c B such that nlU: U ~ V is topological. But then U*: = Un!/'*
is an open neighborhood of a in !/'*, V*: = n(U*) is an open neighborhood
ofn(a) in Band n*IU* = nIU*: U* ~ V* is a topological mapping. 0
If WeB is open, !/'IW: = n- 1(W), then (!/'IW, nl(!/'IW)) is also a sheaf,

the restriction of!/' to W.
Def.1.3. Let (!/', n) be a sheaf over B, WeB open and s: W ~ !/' a con-
tinuous mapping with nos = id w. Then s is called a section of!/' over W.
We denote the set of all sections of!/' over W by T( W, !/').
s(W)~
..--------. I s
I I
I I
IsS I
I J I
II I
I
:
I
In
I ~----~I---------------B
\. 3 }
W
Figure 20. The definition of sheaves and sections.
Theorem 1.1. Let (!/', n) be a sheaf over B, WeB open and s E T(W, 9').
Then n:s(W) ~ Wis topological and s = (nls(W))-1.
PROOF. By definition nos = id w. For 3 E W

so (nls(W) )(S(3)) = son 0 S(3) = S(3).
Therefore s (nls(W)) = ids(w).
0 o
Remark. The equation s = (nls(W)) -1 holds even if s is not continuous.
100
1. Sheaves of Sets
Theorem 1.2. Let (Sf', n) be a sheaf over B, WeB open and s: W ---+ Sf' a
mapping with nos = id w . Then s E r(W, Sf') if and only if s(W) is open
in Sf'.
PROOF
1. Let s be continuous, 0'0 E s(W), and 30: = n(ao). Then S(30) = 0'0 and
there are open neighborhoods V(30) c Wand U(ao) c Sf' such that
nl U: U ---+ V n W is topological. Moreover, there exists an open neighbor-
hood V'(30) c V with s(V') c U. Therefore (nIU) (sjV') = (n s)jV' = id v'.
0 0
But then (nIU)-1(V') = s(V') c s(W) is an open neighborhood of 0'0; that

is, 0'0 is an interior point of s(W).
2. Let s(W) be open, 30 E W, and 0'0: = S(30). Then there are open neigh-
borhoods V(30) c W, U(ao) c s(W) such that nlU: U ---+ V is topological.
s = (nls(W)) -1, so sjV = (njU) - \ and this mapping is continuous at 30' 0
Theorem 1.3. Let (Sf', n) be a sheaf over B, a E Sf'. Then there exists an open
set V c B and a section s E r(V, Sf') with a E s(V).
PROOF. Let 3: = n(a). Let open neighborhoods U(a) c Sf' and V(3) c B be
chosen so that nl U: U ---+ V is topological. Then V and s: = (nl U) - 1 satisfy
the conditions. 0
Theorem 1.4. Let (Sf', n) be a sheaf over B, WeB open. If for two sections
S1' S2 E r(W, Sf') there is a point 3 E W with S1(3) = S2(3), then there is an
open neighborhood V(3) c W with sdV = S2jV.
PROOF. Let a: = Sd3) = S2(3). Then U: = Sl(W) n S2(W) is an open neigh-
borhood of a and nl U: U ---+ V: = n( U) c W is a topological mapping of
U onto the (consequently) open set V. Hence sdV = (nIU)-1 = S2jV. 0
Def. 1.4. Let (Sf'1' nl), (Sf' 2, n2) be sheaves over B.

1. A mapping cp: Sf'1 ---+ Sf' 2 is called stalk preserving if n2 cp = n 1
0
(therefore cp( (Sf'1)3) C (Sf'2)3 for all 3 E B).

2. A sheaf morphism is a continuous stalk preserving mapping
cp: Sf'1 ---+ Sf' 2'
3. A sheaf isomorphism is a topological stalk preserving mapping
cp: Sf'1 ---+ Sf' 2' The sheaves Sf' b Sf' 2 are called isomorphic if there exists
a sheaf isomorphism between them.
Theorem 1.5. Let (Sf'b n1), (Sf'2' n2) be sheaves over B, CP:Sf'1
---+ Sf'2 a stalk
preserving mapping. Then the following statements are equivalent:
1. cp is a sheaf morphism.
2. For every open set WeB and every section s E r(W, Sf'd cp 0 s E
r(W, Sf'2)'
3. For every element a E Sf'1 there exists an open set WeB and a
section s E r(W, Sf'1) with a E s(W) and cp s E r(W, Sf' 2)'
0
101
IV. Sheaf Theory
PROOF
a. If cp is continuous, WeB open and s E r(W, [1'1) then cp s is also
0
continuous. Moreover: nz (cp s) = (nz cp) S = n 1 s = id w. Therefore

0 0 0. 0 0
cp s lies in r(W, [I' z).

0
b. If (F E [I' b then there exists an open set WeB and an s E r(W, [I'd
with (F E s(W). If the conditions of (2) are also satisfied, then cp s lies in
0
r(W, [1'2)'
c. If for a given (F E [1'1> aWe B and a s E T(W, [1'1) with (J E s(W)
and cp s E r(W, [1'2) are chosen according to condition (3), then s: W ~
0
s(W) is topological. Therefore cpls(W) = (cp s) S-1 :s(W) ~ [1'2 is con-

0 0
tinuous, and therefore cp is continuous at (J. D
Remark. For every open subset WeB a sheaf morphism cp: [I' 1 ~ [1'2
defines a mapping cp*:r(W, [1'1) ~ r(W, [1'2) by cp*(s): = cp s. 0
Def.1.5. Let Been be a region. For every open set WeB let there be
given a set M wand for every pair (V, W) of open subsets of B with V c W
let there be given a mapping df :M w ~ M v such that:
1. r~ = id Mw for every open set WeB.
2. If U eVe W, then r~ rV' = r~.
0
Then the system {M w, rV'} is called a pre-sheaf (of sets) and the map-
pings rV' are called restriction mappings.
With every sheaf (9', n) over B a pre-sheaf is associated in a natural
manner:
If V, W are open subsets ofB,then we set Mw: = r(W, [I')andrV'(s): =
slY for s E Mw. Clearly {r(w, [1'), rV'} is a pre-sheaf; it is called the
canonical pre-sheaf of the sheaf [1'.
Conversely a sheaf can be constructed for each pre-sheaf:
Let the system {Mw, rV'} be given, 3 E B fixed. On the sets {(W, s): W is
an open neighborhood of 3, s E M w} the following equivalence relation is in-
troduced: (Wi> S1) 1"' (W2' sz) if and only if there exists an open neighborhood
V of 3 with V c W1 n W2 and rV"(sd = rV'2(sz). Let the equivalence class
of (W, s) be denoted by (W, S)3' and let [1'3 be the set of all classes (W, S)3'
Finally, let [1': = U [1'3 and n:[I' ~ B be the canonical projection. [I' will
3 EB
now be provided with a topology such that n becomes locally topological:
If WeB is open and s E M w , then define rs: W ~ [I' by rS(3): = (W, S)3'
Let!B: = {rs(W): WeB is open, s E Mw} U {[I'}. If WI, W2 C B are open
sets, SI E M W" S2 E M W2 , then let W: = {3 E WI n Wz :rsl(3) = rSz(3)}.
a. W is open: If 30 E W, then (WI' SI ).10 = (W2' S2)30; therefore there exists
an open neighborhood V(30) c WI n W2 with rV"(sl) = rV'2(sz). But then
for every 3 E V also (WI' sd3 = (W2' SZ)3' therefore rSI(3) = rS z(3). Hence
V lies in Wand 30 is. an interior point of W.
102
1. Sheaves of Sets
b. Let s: = r:;'(sl) E Mw. Then rs 1(W1) n rS2(W2) = rs(W): An element

(J Y lies in rs 1(W1) n rS2(W2) if and only if there exists a 3 E W1 n W2
E
with rS 1(5) = (J = rS 2(3), that is, a 3 E W with rS 1(3) = (J. This holds if and
only if (J = (WI> Sl)3 = (W, S)3 E rs(W).
For two sets rS1(Wd, rs 2(W2) E ~ the intersection rS1(~) n rs 2 (J-t;) also
lies in ~. Hence ~ is a basis for a topology on Y whose open sets are arbitrary
unions of elements of ~. It remains to show that n is locally topological.
a. Let (J E Y, 3 = n(o-). Then there is an open set WeB with 3 E Wand
an s E Mw with (J = (W, s)j = rS(3). We set U: = rs(W). U is an open
neighborhood of (J in Y and n 0 rs = id w, rs 0 (nl U) = id u . Therefore
nlU: U -> Wis bijective, (nIU)-l = rs.
b. Ev y open set U' c U is of the form U' = U rsl(Tv,) where in every
U Tv, is open in W.
lEI
case Tv, is open in Wand Sl E M w ,. Therefore n(U') =
IE I
n/ U then takes open sets onto open sets and hence rs is continuous.
c. If W' c W is open, then (W, s) 1" (W', r:;,s) for every 3 E W', so that
rslW' = r(r:;,s). Hence rs(W') = r(r:;,s)(W'), which is an open set. Also
rs maps open sets onto open sets, and thus nl U is continuous.
We now have proved the following theorem:
Theorem 1.6. Every pre-sheaf {Mw, r~n defines a sheaf Y over B in the
above manner (forming the inductive limit), Every element s E Mw is
associated with a section rs E T(W, Y). If 3 E Band (J E Y j , then there is
an open neighborhood W(3) c B and an s E M w such that (J = rS(3).
Theorem 1.7. If Y is a sheaf over B, then the sheaf defined by the canonical
pre-sheaf {T(W, Y), df} is canonically isomorphic to Y.
PROOF, Let (g, iC) be the sheaf defined by the canonical pre-sheaf.
a. If (WI> Sl) 1" (W2' S2) then Sl(3) = s2(3) and the converse also holds.
Therefore cp: (W, s)j I-> sC) defines an injective mapping cp: g -> Y which
is stalk preserving, If (J E Yo, then there exists a neighborhood W(3) and an
s E T(W, Y) with S(3) = (J. rS(3) = (W, S)3 then lies in g3' and cp(rs(3)) = (J.
Hence cp is also surjective.
b, If (J E gj, then there exists an open set WeB and an s E T(W, Y) with
(J = (W, S)3 = rS(3), Then rs E T(W, g3)' (J E rs(W) and cp (rs) = s E T(W, Y),
0
Therefore cp is continuous at (J,

c. If WeB is open and s E T(W, Y), then cp-1(S) = rs E T(W, g).
Therefore cp -1 is also continuous. 0
Theorem 1.8. Every sheaf morphism is an open mapping.
PROOF. Let cp: Y 1 -> Y 2 be a sheaf morphism. Since Y 1 is canonically iso-

morphic to the sheaf g 1 defined by the canonical pre-sheaf {T(W, Y d, rn,
the sets s(W) with s E T(W, ff'd form a basis of the topology of Y l' If s
103
IV. Sheaf Theory
lies in r(W, .9' d, then cp s lies in r(W, Y' 2) and hence cp (s(W)) = (cp s)( W)
0 0
is open in .9'2' The proposition follows. 0
Def. 1.6. Let (.9' 10 1td, ... , (.9'l' 1tt ) be sheaves over B. For open sets WeB
let Mw: = r(W, .9'1) x ... x r(W, .9't), for S = (SI,"" Sf) E Mw and
open subsets V c W let rV's: = (sliV, . .. , seiV) E Mv. Then {Mw, r:r}
is a pre-sheaf and the corresponding sheaf.9' = Y'1 EB . . . EB .9'( is called
the Whitney sum of the sheaves .9' 1, ... , .9'r.
Theorem 1.9. Let (.9'10 1td, ... , (.9'l' 1t() be sheaves over B, and let .9' =
.9' 1 EB ... EB Y't be their Whitney sum. Then for every 3 E B there is a
bijection cp:.9'a--> (.9'd a x " ' , X (.9't)a defined by (W,(Sb ... ,St))af-+
(SI (3), ... , St(3))·
PROOF
'a. Let s). = (s\).), ... , s~).») E r(W;., .9') for A = 1,2,3 E WI n W2.
(Wb sd 3' (W2' S2) if and only if there exists a neighborhood V(3) c WI n
Wz such that
(SI(1)IV, ... , S,(1)IV) -- (SI(2)IV, ... , St(2)IV) .
This is equivalent to SP)(3) = SF)(3) for i = 1, ... , t. Therefore an injective
mapping is defined by (W, (S1o ... , sl))a f-+ (SI (3), ... , S,(3)).
b. If a = (a 10 ..• , at) E (.9' 1)3 x ... X (.9't)3 and, say, ai = S;(3) with
s; E r(W;, .9'i), then W: = n W; is an open neighborhood of 3 and Si:
t
i=1
=
s;IW E r(w, .9'i)' Consequently s: = (SI,' .. , St) lies in M w , and rs is a

section of the sheaf.9' with rS(3) = (W, S)3 f-+ S(3) = a. The mapping defined
above is therefore also surjective. 0
Theorem 1.10. Let (.9' 10 1tl),' .. , (Y't, 1t() be sheaves over B. Then the canonical
projections Pi:.9' 1 EB ... EB .9't f-+.9'i (with Pi(al,' .. , at): = ai) are sheaf
morphisms.
PROOF. The mappings Pi are stalk preserving, by definition. If a E (Y'1 EB ...
EB .9")3 = (.9' 1)3 x ... x (.9'( )3' then there exists sections Si in .9'i with
Si(3) = pM) and rS(3) = a for s: = (SI' ... ,St). Therefore Pi rs = Si is 0
continuous, Pi a sheaf morphism. 0
For Si E r(W, .9';), i = 1, ... , t, defined a mapping SI EB ... EB Sf: W -->

.9' 1 EB ... EB .9'l by (SI EB ... EB St)(3): = (SI(3), ... , S (3)). Clearly, (S1o' .. , St)
lies in M w, and r(sb"" St)(3) = (W, (SI' ... , St))3 = (SI (3), ... , S,(3)) =
(SI EB ... EB St)(3); therefore SI EB ... EB Sf = r(s1o ... , St) E r(W, .9' 1 EB ...
EB .9'(). Hence we can identify the sets r(W,.9'1 EB ... EB .9't) and r(W,
.9' d x ... x r(W, g().
104
2. Sheaves with Algebraic Structure
If global sections Si E 1 ~D, Sf'J for i = 1, ... , t are given, then we can
define corresponding injectionsk = NSb ... , Si' ... , Sf): Sf'i ~ Sf'1 EB ... EB
Sf't where
ji(a): = (S1 (3), ... , Si-1 (3), a, si+ d3), ... , S((3)), for a E (Sf'J.
Clearly ji is stalk preserving and for s E r(W, Sf'J
ji 0 S = (S1!W, ... , Si-1!W, s, Si+1!W, ... , s !W)
lies in r(W, Sf'1) x ... x r(W, Sf'r) = r(W, Sf'1 EB··· EB Sf't) that is, ji is
continuous. Pi 0 ji = ids; holds for i = 1, ... , t.

Let B c cn be an open set.
Def. 2.1. A sheaf (Sf', n) over B is called a C-algebra sheaf if:
1. Every stalk Sf'a is a commutative C algebra with 1.
2. Sf' EB Sf' ~ Sf' (with (ab (2) ~ a1 + az) is continuous.
3. Sf' EB Sf' ~ Sf' (with (O"b O"z) ~ 0"1 . O"z) is continuous.
4. For every c E C, Sf' ~ Sf' (with 0" ~ c . 0") is continuous.
S. The mapping 1: 5 ~ I, E Sf'a lies in r(B, Sf').
Consequences
1. 0:3 ~ OJ E Sf', lies in r(B, Sf').
2. Sf' ~ Sf' (with 0" ~ - 0") is continuous.
3. If WeB is open, then r(W, Sf') is also a C-algebra.
°
PROOF
1. Because 0 . I, = 0,),0· I = 0, and the zero section is continuous.
2. It follows from the definition that the mapping 0" ~ - 0" = ( - 1) . 0" is
continuous.
3. Addition, multiplication, and multiplication by a complex number
are defined pointwise, so the axioms of a C-algebra are satisfied since they
hold in every stalk. Continuous sections go into continuous sections. 0
Theorem 2.1. Let Sf' b ... , Sf'r, Sf' be sheaves over B given by pre-sheaves
{MW, rX}, i = 1, ... , f, and {Mw, dr}. Suppose that for every open set
WeB there exists a mapping CPw: M~P x ... x M\V ~ M w (for ex-
ample, an algebraic operation) with r~ CPW(Sb ... , Sf) = CPV(rfVsb . .. , r~s()
for arbitrary elements Si E MW, i = 1, ... , e, and open sets V c W. Then
there exists exactly one sheaf morphism cP: Sf' 1 EB ... EB Sf't ~ Sf' with
CP(rsb· .. , rs t ) = rCPW(s1' ... , St)·
PROOF
1. Let W, Wbe open in B, 3 E W n Wand (W,s;),.,.., (W,s;) for i = 1, ... , {.
Then there exists a neighborhood V(3) c WnW with rXsi = rftsi for
105
IV. Sheaf Theory
therefore
(W, ipW(Sl, ... , Sf)) J (lV, ipW(Sl> ... , St))·
Hence a mapping ip: Y 1 EB ... EB Y ( -+ Y is defined by
(rs1(3), ... , rS((3)) H (W, ipw(s1o ... , Sc) h = ripW(Sl, ... , St)(3);
It IS stalk preserving and ip(rS1,"" rs t ) = ripw(s1o' .. , Sf). Hence ip IS
uniquely determined.
2. For a = (a b ... , at) E (Y d.l X ... x (Yt ) there is a neighborhood
W(3) and elements Si E MW for i = 1, ... , t such that ai = (W, S;).l' Then s: =
(rs1o ... , rs t ) E r(W, Y 1 EB ... EB Y t ), a E s(W) and ip 0 S = ripw(s1o ... ,Sf) E
r(W, Y). Therefore ip is continuous. D
Def. 2.2. Let {Mw, rn

be a pre-sheaf with the following properties:
1. Every M w is a I[>algebra.
2. rtf: M w -+ My is always a homomorphism of C-algebras. Then
{M w, rn
is called a pre-sheaf of C-algebras.
Theorem 2.2. Let {Mw, rtf} be a pre-sheaf of C-algebras, Y the corresponding

sheaf. Then Y is a sheaf of C-algebras and for every open set WeB
r:Mw -+ r(W, Y) is a homomorphism of C-algebras.
PROOF. For WeB let ipw:Mw x Mw -+ Mw be defined by ipW(Sb S2): =
Sl + S2' Then
rtfipW(sl' S2) = rtf(sl + S2) = rtfs1 + rtfs2 = ipy(rtfs1' rtfs2)'
By Theorem 2.1 there is exactly one sheaf morphism ip:Y EB Y -+ Y with
ip(rs 1, rS2) = ripW(sl, S2) = r(sl + S2)'
An addition Y EB Y ~ Y is defined by a1 + a2: = ip(a1' (2), so
rS1(3) + rS 2(3) = ip(rs 1(3), rS2(3)) = [ip(rs1o rS2)J(3) = r(sl + S2)(3);
therefore
rS 1 + rS 2 = r(sl + S2)'
The remaining operations are defined analogously; r transfers them to the
stalks, and it is clear that r is a homomorphism of C-algebras. D
Def. 2.3. Let d be a sheaf of C-algebras over Band Y some sheaf over B.
Y is called a sheaf of d -modules if:
1. For every 3 E B, Y. is a unitary d.-module.
2. Y EB Y -2+ Y is continuous.
3. dEB Y ~ Y is continuous.
106
Remarks
1. Let 0 3 be the zero element of Sf' 3' Then 0:31---+ 0 3 defines the zero section
° 2. For every
E r(B, Sf').
W, r(W, Sf') is a r(W, d)-module.
Def. 2.4. Let {M w, rf} be a pre-sheaf of I[:>algebras, {M w, rf} a pre-sheaf

of abelian groups, and d resp. !7 the corresponding sheaves. If for every
open set WeB, Mw is a (unitary) Mw-module and for every s E Mw
and every SE M w , r:'(s' s) = rf(s)· rf(s), then ({Mw, rf}, {Mw, rf})
is called a pre-sheaf of modules.
Analogous to Theorem 2.2 it can be shown that every pre-sheaf of modules

defines a sheaf of d-modules. r:Mw ~ r(W, Sf') is then a homomorphism
of abelian groups with r(s . s) = rs . rs.
Let M w be the set of hoI om orphic functions in Wand let rf:

EXAMPLE.
Mw ~ Mv be defined by rf(f): = f/V. Clearly {Mw, rf} is a pre-sheaf of
I[>algebras. The corresponding sheaf (f) is a sheaf of IC-algebras and called
the sheaf of germs of holomorphic functions on B.
An element (W, f)3 of the stalk (f)3 is an equivalence class of pairs F,),(w.,
where W. is an open neighborhood of 3 and f. a holomorphic function on w..
Two pairs (Wb f1) and (W2' f2) are equivalent if there exists a neighborhood
V(3) c W 1 n W2 with f11 V = f21 V, that is, if and only if f1 and f2 have the
same power series expansion about 3. Hence we can identify the stalk (f)3
with the IC-algebra of convergent power series, so that nothing new has been
added to (f)3 as introduced above. In particular the power series J;, and the
equivalence class (W, f), coincide.
For every open set WeB r:Mw ~ r(W, (f)) is a homomorphism of
IC-algebras and r(fl V) = rfl v.
Proposition. r is bijective.
PROOF
1. If rf = 0, then for every 3 E W we have rf(3) = 0" therefore (W,f)3 =
0,; that is, there exists a neighborhood V(3) c W withflV = 0, in particular
f(3) = O. Therefore f = O.
2. If s E r(W, (f)) then for every 3 E W there exists a neighborhood
U(3) c Wand a holomorphic function f on U with (U, f)o = S(3). Then
there is a neighborhood V(3) c U with rfl V = sl v.
Now let (U')'EI be an open covering of W such that there is a holomorphic
function f. on each U, with rf. = sl U,. Then a holomorphic function f on
Wis given by flU,: = f., for which
rfl U, = r(fl U,) = rf. = sl u,.
Therefore f E Mw and rf =s. D
107
IV. Sheaf Theory
Hence the following theorem holds:
Theorem 2.3. r:Mw ~ r(W, (I)) is an isomorphism of C-algebras.

Henceforth we shall identify the functions holomorphic on W with the
elements of r(W, (I)).
EXAMPLE. Let Mw = C and r~ = ide for all V, W. Then {Mw, rn is a

pre-sheaf of C-algebras, indeed, of fields. Let d be the corresponding sheaf.
(Wb c l ) '3 (W2' c 2) if and only if C l = c 2, that is d, = C for all 3 E B.
If s E r(W, A) and 3 E W, then c: = S(3) lies in d, = C = M w , and
rC(3) = c = s(3). Then there exists a neighborhood V(3) c W with V = sl
rcl V, that is S(3) = c for 3 E V. One can thus regard s as a locally constant
complex function.
We call d the constant sheaf of the complex numbers. Clearly d is a sub-
sheaf of (I).
Def. 2.5. An analytic sheaf over B is a sheaf:7 of (I)-modules over B.
EXAMPLES
1. (I) is an analytic sheaf.
2. Let :7 be an analytic sheaf, :7* c :7 a subsheaf. If for every 3 E B,
:7; C :73 is a submodule, then :7* is likewise an analytic sheaf: If, say,
(Sb S2)Er(W,:7* EB :7*) c r(W,:7 EB :7), then Sl + S2 belongs to r(W,:7),
and therefore to r(W, :7*). This shows addition is continuous. Multiplication
by scalars is treated similarly. Note that if :7* c :7 is an analytic sub sheaf,
then r(W, :7*) c r(W, :7) is a r(W, (I))-submodule.
3. If Jf c (I) is an analytic subsheaf, then Jf 3 c (1)3 is an ideal. Hence we
also call Jf an ideal sheaf.
Def. 2.6. Let Jf c (I) be an ideal sheaf. Then we call N(Jf): = {3 E B: (1)3 =f. Jf J
the zero set of Jf.
For J; E (1)3' f3 converges near 3 to a holomorphic function which we

denote by f.
Theorem 2.4. Let Jf c (I) be an ideal sheaf over B. Then N(Jf) = {3 E B: For
all f3 E Jf3, f(3) = O}.
PROOF
1. Let 3 E N(Jf), J;, E Jf" but f(3) =f. O. Then on a neighborhood of W(3),
Ilf is holomorphic and 10 = rl(3) = r(1If)r(f)(3) E Jf 3 ; therefore Jfo = (1)0·
That is a contradiction, so f(3) must be zero.
2. If 3 ¢ N (Jf), then Jf 3 = (I) a' thereforel o E Jf 3; on the other hand, 1(3) =f. O.
. D
108
EXAMPLE. Let 0: = U {OJ and let n:O --+ B be the canonical mapping.
3 EB
If we give 0 the topology of B, then n is a topological mapping. In this way
o becomes an analytic sheaf, the zero sheaf
Theorem 2.S. Let Y'1,"" Y't be analytic sheaves over B. Then Y': =
Y'1 EB ... EEl Y'( is analytic.
PROOF. Clearly Y'3 = (Y'1)3 x ... X (Y'J3 is always an {D3 -module. It
remains to show that the operations are continuous. We only carry out the
proof for addition:
Let
(s, s) E r(W, Y' EB Y') = r(W, Y') x r(W, Y')
(Si' Si): = (Pi S, Pi S)E r(W, Y'J
0 0 x r(W, Y'J
= r(W, Y'i EB Y'J for i = 1, ... J
Then
Si + Si E r(W, Y'J for i = 1, ... , C,
therefore
Def.2.7. For q E N let q{D: = (9 EB .; . EB q. (In the literature (Dq is the most
q tImes
common notation for this.) q{D is always an analytic sheaf.
To conclude this section, we consider quotient sheaves. Let Y' be an

analytic sheaf over B, Y'* c Y' an analytic subsheaf. For open sets WeB
we define N w : = r(W,{D) and Mw: = r(W, Y')/r(W, Y'*) interpreted as N w-
module. There is a canonical projection q:r(W, Y') --+ Mw. For s E r(W, S)
let <s>: = q(s). Then (for V eWe B) we can define
= <siV>
r~«s»: for <s> E Mw.
Clearlyr~fiswell-defined:<s1> = <s2>ifandonlyifs 1 - s2Iiesinr(W,Y'*).
But then (S1 - s2)iV E r(V, Y'*), so <sliV> = <s2iV>. Hence {Mw, rn is a
pre-sheaf of abelian groups and for <s> E Mw and f E N w,
r~(f' <s» = r~«f' s» = <(f' s)iV> = <(fiV)' (siV»

= (fiV)' <siV> = (fiV) . r~ «s».
({ N w, r~}, {Mw, r~}) is a pre-sheaf of modules whose associated sheaf

f2 is an analytic sheaf. We call f2 the quotient sheaf of Y' by Y'* and write
f2 = Y'IY'*.
Theorem 2.6. Let Y' be an analytic sheaf over B, Y'* c Y' an analytic sub-
sheaf, f2 = Y' I Y'* the quotient sheaf Then for every 5 E B there is an
isomorphism ljJ :f23 --+ Y')Y'; (of (D3-modules) defined by (W, <s» ~ S(5).
(0' denotes the image of (J E" Y'3 in Y'olY'n
109
IV. Sheaf Theory
PROOF
1. (Wi> <Sl») '3 (Wz , <sz») if and only if there is a neighborhood V(3) c
WI n Wz such that
<sllV) = r~l«sl») = r~2«sz») = <szlV)
and that is exactly the case when (Sl - sz)1V lies in T(V, 9"*). But by the con-
tinuity of Sl - Sz, this is equivalent to having Sl(3) - Sz(3) E 9";; therefore
sl(3) = sz(3)· Hence 1/1 is well-defined and injective.
2. Since
Moreover
1/I(f3' (W, <s) ),) = 1/1 ( (W, <f . s) ),) = (f. S)(3) = f,' S(3) = f,' 1/1 ( (W, <s) ),).
1/1 is therefore an {D,-module homomorphism.
3.· If (Y E 9")9"~, then there exists a neighborhood W(3) c B and an
s E T(W, 9") with S(3) = (J. But then (W, <S»)3 is in 22 3, and I/I((W, <S»)3) =
S(3) = (Y. Therefore 1/1 is also surjective. D
Henceforth we can identify 223 with 9",/9";.

Remark. If sET( W, 9"), then r< s) lies in T( W, 22). If we define s: W -+ 22
by 8(3): = S(3), then 1/1 r<s) = S. Hence we can identify r<s) and
0 s by
means of 1/1.
3. Analytic Sheaf Morphisms

Def. 3.1. Let 9" b 9" z be analytic sheaves over B, cp: 9" 1 -+ 9" z a sheaf mor-
phism. cp is called an analytic sheaf morphism (or a sheaf homomorphism)
if for every 3 E B, cp :(9" d3 -+ (9" Z)3 is an (D3-module homomorphism.
EXAMPLES
1. Let 9" be an analytic sheaf over Band 9"* c 9" an analytic subsheaf.
Let q: 9" -+ 9"/9"* be the canonical projection with q ((J) = (Y. Then q: 9"3 -+
9",/9": is always an (D3-module homomorphism and for s E r(W, 9"), q s = 0
S = r<s) E r(W, 9"/9"*). Therefore q is a surjective sheaf homomorphism

(a sheaf epimorphism).
2. If 9" is an analytic sheaf, then there is exactly one sheaf morphism
9" -+ 0, and it is clearly a sheaf epimorphism.
3. Conversely, though there can be several sheafmorphisms 0 -+ 9", there
is only one analytic sheaf morphism (mapping 0 3 onto 0 3), This homo-
morphism is injective (a sheaf monomorphism).
4. If 9"* c 9" is an analytic subsheaf, then the canonical injection I =
id.9'j9" *: 9"* 4 9" is a sheaf morphism.
Remark. (2) is a special case of (1), with /7* = /7; and (3) is a special case of (4),
with /7* = O.
110
3. Analytic Sheaf Morphisms
5. If [/1, ... , [/( are analytic sheaves, then the canonical projections
P;:[/1 $ ... $ [/( ~ [/; are sheafepimorphisms.
6. If 0; is the zero section in [/;, then the canonical injection j; =
j;(Ob.·., 0;, ... , Ot):[/; '+ [/1 $ ... $ [/t are sheaf monomorphisms.
7. Let j;:(9 '+ q{9 be the canonical injections. If 1 E r(B, (9) is the "unit
i-section", then we define the unit sections in q{9 by
e;: = j; 01 = (0, ... , I, ... ,0).
Now let cp:q{9 ~ [/ be an analytic sheaf morphism and let S;: = cp e; E
0
r(B, [/). Then for (a1' ... , aq ) E q{93'
cp(a1o ... , aq ) = cp Ct1 a;e;(3») = J1 a;s;(3)·
So the sections S10 ••• , Sq determine the homomorphism completely, and

conversely we can define an analytic sheaf morphism cp = cp(. h ••• , Sq) by the
above equations for S1' •.• , Sq.
8. If cp: [/ 1 ~ [/2 and ifJ: [/ 2 ~ [/3 are analytic morphisms, then so is
ifJ a CP:[/1 ~ [/3·
Def. 3.2. Let cp: [/ 1 ~ [/2 be an analytic sheaf morphism. Then we define
Imcp: = cp([/d c [/2; Kercp: = cp-1(o) c [/1.
Theorem 3.1. If cp: [/ 1 ~ [/2 is an analytic sheaf morphism, then 1m cp and

Ker cp are analytic sheaves.
PROOF
1. Since every sheaf morphism is an open mapping, 1m cp = CP([/1) c [/2
is open in [/2, and is therefore a subsheaf. Since (1m CP)3 = cp( ([/1)3), 1m cp
is analytic.
2. Because cp is continuous and 0 c [/2 is open, Kercp = cp-1(O) c [/1
is open and therefore a subsheaf. Because (Ker CP)3 = {CT E ([/ d3: cp(CT) =
0 3 E ([/2)a} = Ker(cpl([/ d a), Ker cp is analytic. 0
Def.3.3. Let [/1, [/2 be analytic sheaves over B. A mapping CP:[/1 ~ [/2
is called an analytic sheaf isomorphism if (1) cp is stalk preserving; (2) cp is
topological; and (3) cpl([/ d3:([/ 1)3 ~ ([/2)3 is an {93-module isomorphism
for every 3 E B.
We write [/ 1 ~ [/2 if there exists a sheaf isomorphism cp: [/ 1 ~ [/2·

Remark. If a mapping cp: [/ 1 ~ [/2 is a bijective sheaf homomorphism
then it is an analytic sheaf isomorphism. Namely, cp is stalk preserving and
continuous, and for every 3 E B, cpl([/ 1)3:([/ 1)3 ~ ([/2)3 is an {93-module iso-
morphism. Since every sheaf morphism is open, it now also follows that
cp - 1 is continuous; cp is therefore topological.
111
IV. Sheaf Theory
Theorem 3.2. If (p :!/' 1 ---> !/' 2 is an analytic sheafmorphism, then!/' dKer qJ ~

1m qJ.
PROOF. ip(O'): = qJ(O") defines a stalk preserving bijective mapping ip:!/'d

Ker qJ ---> 1m qJ which induces a @,-module isomorphism in every stalk. If 0' E
(!/' dKer qJ)" then there exists a neighborhood W(3) and an s E r(W,!/' 1) with
5(3) = 0' and ip 05= qJ s E r(W, 1m qJ). Therefore ip is also continuous.
0
Hence by the above remark ip is a sheaf isomorphism. 0
Remark. If qJ:!/' 1 ---> !/' 2 is an analytic sheaf morphism and if q:!/' 1 --->
!/' dKer qJ and I: 1m qJ 4 !/' 2 are the canonical mappings, then one has the
canonical decomposition of qJ:
qJ = I a ip 0 q:!/' 1 ~>!/' dKer qJ ~ 1m qJ 4 !/' 2'
Def. 3.4. Let!/' 10 ••• , !/'( be analytic sheaves over B, and let qJi:!/' i ---> !/' i + 1
be analytic sheaf morphisms for i = 1, ... , t - 1. Then we call the
sequence
an analytic sequence of sheaves.

The sequence is called exact at !/'i ifIm qJi-l = Ker qJi' The sequence
is exact if it is exact at each !/'i'
Remarks. The sheaf homomorphism which maps every element stalkwise

onto zero will be denoted by O.
1. 1m qJi-l = Ker qJi means:

a. qJi 0 qJi-l = 0
b. If qJ;(O") = 0, then there is a ff with qJi-l (ff) = 0".
2. 0 ---> !/" ~ !/' is exact if and only if qJ is injective.
3. !/' ~ !/''' ---> 0 is exact if and only if t/J is surjective.
4. If qJ:!/' 1 ---> !/' 2 is an analytic sheaf morphism, then we have a canonical
exact sequence:
o ---> Ker qJ ---> !/' 1 ---> 1m qJ ---> O.
If qJ is injective, then Ker qJ = 0 and!/' 1 ~ 1m qJ; if qJ is surjective, then
!/' dKer qJ ~ !/' 2 .
Def.3.5. Let (!/' 10 1!1), (!/' 2, 1!2) be analytic sheaves over B. Hom(!1(!/' 10 !/' 2)
is the set of all analytic sheaf morphisms (p:!/' 1 ---> !/' 2'
If we set (qJl + q(2)(0"): = qJl(O") + qJ2(0") and (f. qJ)(O"): = f"1(")' qJ(O")
for qJ, qJ10 qJz E !/' 1, and f E F(B, @), then Hom(!1(!/' 10 !/' 2) becomes a r(B, @)-
module.
((qJ! + qJz) s = qJl S + qJz sand (f. qJ) s = f· (qJ s) are also
0 0 0 0 0
sections, and hence qJl + qJ2 and f . qJ are continuous.)
112
4. Coherent Sheaves
4. Coherent Sheaves
B will always be a region in en.
Def.4.1. An analytic sheaf Y over B is called finitely generated if for every
point 3 E B there exist an open neighborhood W(3) c B, a natural
number q and a sheaf epimorphism q>:q(9IW ......> YIW.
Let Ci be the i-th unit section of q(9, Si: = q> (cd W) the images under q>.
0
If (J E Y 3 then (J comes from an element (at. . .. , aq ) E q(9; that is, (J =

q
q>(at. ... , aq ) = I ai si (3)· The sections S1" .. ,Sq therefore generate the
i=1
(9o-module Yo simultaneously over all of W.
Def.4.2. If Y is analytic over B, then one calls the set Supp(Y): = {3 E B:

Yo =1= 03} the support of Y.
Theorem 4.1. If Y is finitely generated, then Supp(Y) is closed in B.

PROOF. We show that B - Supp(Y) is open in en. Let 30 E B - Supp(Y) be
chosen arbitrarily, and let W(30) c B be an open neighborhood over which
a sheaf epimorphism q>:q(9 ......> YIW exists. Let S1' . . . , Sq be the images of
the unit sections over W. Then s1(30) = ... = Si30) = 0 30 = 0(30) E Y 30 •
Hence there exists a neighborhood V(30) c W with S1jV = ... = SqjV =
OjV; therefore YjV = 0, so V c B-Supp(Y). D
EXAMPLES
1. q(9 is finitely generated, since id: q(9 ...... q(9 is a sheaf epimorphism.
2. Let e: Y 1 ...... Y 2 be a sheaf epimorphism with Y 1 finitely generated.
Then trivially Y 2 is also finitely generated.
3. Let Y' c Y be an analytic subsheaf with Y finitely generated. Then
2, applied to the canonical projection q:Y ...... YjY', shows that YjY' is
finitely generated.
4. Let A c B be analytic in B. The ideal sheaf .Y(A) is defined as follows:
Let .Yoo: = {(J E (930: There exists a U(30) c B and a holomoFphic fin U
with flU n A = 0 and rf(30) = (J} for 30 E B; then 'y(A): = U .Y3•
3EB
a . .Y(A) is a subset of (9, and for (J E .Y.l there exists a neighborhood
U(3) c B and an f such that rf(3) = (J. But then the set Ij(U), open in (9,
lies in .Y and contains the element (J. Therefore (J is an interior point and
.Y is open in (9.
b. That every stalk .Yo is an ideal in the ring (93 follows immediately
from the definition. Hence .Y c (9 is an analytic subsheaf and an ideal
sheaf.
113
IV. Sheaf Theory
By (3) the quotient sheaf J'l' = (!J/f (a sheaf of C-algebras!) is finitely

generated. We show that Supp(J'l') = A. For 30 E B - A, f30 = (!J,o' there-
fore J'l' 30 = 0 30 , For 30 E A, f 30 i= (!J 30' since otherwise 130 E f 30 and there
would be a holomorphic functionf on a neighborhood U(50) withfl U ! l A =
o and rf(30) = 1,10 = r1(30). But then rf and r1 would agree on a neighbor-
hood V(30) c U. Since in this case r is bijective", it follows that flY = llY,
in particular 0 = f(30) = 1.
Remark. Clearly
N("?(A)) = Supp (i)j..?(A) = A.
Yet for an arbitrary ideal sheaf j' c (i), the equation ..?(N(j')) = j' is false.
5. Let B be connected, B' c B open, B' i= 0 and B' i= B. An open subset
Y = n- 1 (B') u O(B) of (!J is defined by YIB': = (!JIB' and YI(B - B') = o.
It is a sub sheaf. Since Y 3 c (!J3 is always an ideal, Y is an ideal sheaf; but
Supp(Y) = B' is not closed. Hence it follows that Y is not finitely generated.
Def. 4.3. Let Y be an analytic sheaf over B. Y is called coherent if:

1. Y is finite (that is, finitely generated).
2. Y is relation finite (that is, if U c B is open and q>: q(!J IU ~ YI U
is an analytic sheaf morphism, then Ker q> is finitely generated).
Let Si E r(U, Y) be the images ofthe i-th unit section ei E r(U, q(!J) under
q>:q(!JIU ~ YIU· Then an element (ab' .. , aq ) E q(!J)smapped onto 03ifand
q
only if the "relation" L: ai si (3) = 0 is satisfied. We call Ker q> the relation
i=1
sheaf of Sb ... , Sq.
Consequences
1. Coherence theorem of 0 ka: (!J is coherent.
2. Coherence theorem of Cartan: The ideal sheaf f (A) of an analytic set is
coherent.
These two results are very deep and cannot be proved here.
3. 0 is coherent. (This is trivial.)
4. If Y is coherent and Y* c Y a finitely generated subsheaf, then Y*
is also coherent.
PROOF. Let WeB be open, q>:q(!JIW ~ Y*IW be an analytic sheaf mor-
phism, I: Y*I W ~ YI W the canonical injection. Then I 0 q>: q(!J1 W ~ YI W
is also an analytic sheaf morphism, and Ker q> = Ker(lo q» is finitely
generated. D
Theorem 4.2 (Existence of liftings). Let q>: Y ~ Y* be a sheaf epimorphism,

e*: q(!J ~ Y* an arbitrary sheaf homomorphism. Then for every 50 E B there
is a neighborhood U(30) c B and a (non-canonical) sheaf homomorphism
e:q(!JIU ~ YIU such that q> 0 e = e* (one calls any e with these properties
a lifting of e*).
114
4. Coherent Sheaves
PROOF. Let si: = 1',* Ci E r(B, 9'*) for i = 1, ... , q. Then for 30 E B there
0
are elements ai E 9'30 with cp(ai) = s;(30). We can find a neighborhood W(30) c
B and sections Si E r(W, 9') with Si(30) = ai; therefore cp S;(30) = si(30). 0
cp Si and si coincide on a neighborhood U(30) c W. 1',: = cp(S! ..... Sq):

0
qlDl U ---* 9'1 U is an analytic sheaf morphism with

q
l',(a 1, ... , aq ) = I aiSi(3) for
i= 1
therefore
q
cp 0 l',(a1, ... , aq ) = I aisi(3) = I',*(a b ... , aq ). D
i= 1
Theorem 4.3. Let 0 ---* 9'* .:!..,. 9' !... 9'** ---* 0 be an exact sequence of ana-
lytic sheaves over B. If 9'* and 9'** are coherent, 9' is also coherent.
PROOF
1. 9' is finitely generated: Since 9'* and 9'** are finitely generated,
there are for every 30 E B a neighborhood W(30) c B and sheaf epimorphisms
1',*:q*lD->} 9'*, 1',**:q**lD ---*} 9'** over W. Since p:9' ~ 9'** is surjective, there
is (w. I. o. g. also over W) a lifting of 1',**
l',:q**lD ~ 9' with pol', = 1',**
If pr 1: q*lD E:B q**lD ---*> q*lD and pr 2: q*lD EEl q**lD ~> q**lD are the canonical
projections, then
1/1: (q* + q**)lD ~ 9'
with
1/1(0'): = j 0 1',* 0 pr1(a) + 1',0 pr2(a)
is an analytic sheaf morphism. It remains to show that 1/1 is surjective:
Let a E 9',,3 E W. Then there is a 0'1 E q**lD3 with 1',**(0'1) = pea). Clearly
0'- 1',(0'1) lies in Ker p = Imj, therefore there is a 0'2 E 9'; with j(a2) =
0'- 1',(0'1)' Furthermore, we can find a 0'3 E q*lD3 with 1',*(0'3) = 0'2' Now
1/1(0'3,0'1) = j 0 1',*(0'3) + 1',(0'1) = j(a2) + 1',(0'1) = a.
2. 9' is relation finite: Let WeB be open, cp: qlDl W ~ 9'1 W an analytic
sheaf morphism and 30 E W an arbitrary point. Since 9'** is relation finite
there is a neighborhood V(30) c Wand, over V, a sheaf morphism 1/11:
rlDlV ~> Ker(p cp)W This gives the exact sequence:
0
rlDlV ~ qlDlV ~ 9'**.

Because Ker p = 1m j ~ 9'*, we can regard cp 1/1 1: rlD ~ Ker p as a map- 0
ping cp 0 1/1 1: rlD ~ 9'*, and since 9'* is relation finite, there is a neighborhood
U(30) c V and a sheaf epimorphism 1/12:slD1U ~> Ker(cp I/IdlU. This yields 0
the following exact sequence:

slDlU ti rlDl U ~ 9'.
115
IV. Sheaf Theory
Hence we obtain (over U):

a. cp 0 (1/11 0 1/12) = (cp 0 I/Id 01/12 = 0
b. If a E q(!) and cp(a) = 0, then p 0 cp(a) = 0 also and there is a 0'1 E r(!)
with 1/11(0'1) = a. Then cp 0 1/11(0'1) = 0 and there is a 0'2 E s(!) with 1/12(0'2) =
0'1' Then 1/11 1/12(0'2) = a.
0
(a) and (b) imply that the following sequence is exact:

s(!)1 U ~ q(!)1 U ~ 9'1 U
Therefore Ker cp is finitely generated. D
Theorem 4.4. Let 9'* .L,. 9' ~ 9'** --+ 0 be an exact sequence of sheaves
over B. If 9'* and 9' are coherent, then 9'** is also coherent.
PROOF
1. Since p is surjective, it follows immediately that 9'** is finitely generated.
. 2. Let e**: q**(!) --+ 9'** be an arbitrary sheaf homomorphism on an open
set WeB, e:q**(!) --+ 9' a lifting (with poe = e**). Since 9'* is finitely
generated, we can find a neighborhood V(30) c Wand a sheaf epimorphism
e*: q*(!) --+> 9'* on V for every point 30 E W. Now let 1/1: q*(!) EEl q**(!) --+ 9'
be a sheaf morphism on V defined by
1/1(0'1,0'2): =j 0 e*(ad + e(a2)'
Since 9' is coherent there exists an exact sequence q(!) ~ q*(!) EEl q**(!) .t 9'
on a neighborhood U(30) c V. Let r:x:q(!) --+ q**(!) be defined by r:x: = pr2 0 cpo
The theorem will be proved once we show the exactness of the sequence
q(!) ~ q**(!) ~ 9'**. For 3 E U and a E q**(!)3 the following statements are
equivalent:
1. a E Ker(e**)
2. e(a) E Ker p = Imj
3. There is a 0'1 E q*(!J3 withj 0 e*(ad = e(a)
4. 1/1(-0'1, a) = 0 for a 0'1 E q*(!)3
5. There is a 0'2 E q(!),l with cp(a2) = (-0'10 a)
6. a = pr2° cp(a2) = r:x(a2) Elm r:x. D
Theorem 4.5. Let 0 --+ 9'* .L,. 9' ~ 9'** be an exact sequence of analytic
sheaves over B. If 9',9'** are coherent, then 9'* is also coherent.
PROOF. We may regard 9'* as an analytic subsheaf of 9', so it suffices to
show that 9'* is finitely generated. Let 30 E B be chosen arbitrarily. Since 9'
and 9'** are coherent there is a neighborhood W(30) c B, and over W, a
sheaf epimorphism e: q(!) --+ 9' and a sheaf epimorphism cp: q*(!) --+ q(!) such
that the sequence
is exact.
Then eo cp(q*(!)) = e(Im cp) = e(Ker(p 0 e)) = Ker p = Imj, so eo cp(q*(!)) ~
9'*. Hence 9'* is finitely generated. D
116
4. Coherent Sheaves
Theorem 4.6
1. If g is a coherent sheaf over Band g* ega coherent analytic
subsheaf, then g / g* is also coherent.
2. If g 1>' •• ,gt are coherent analytic sheaves over B then g 1 EB ... EB gt
is also coherent.
PROOF
1. There exists a canonical exact sequence 0 -> g* -> g -> g/g* -> o.
The result follows by Theorem 4.4.
2. For t = 2, apply Theorem 4.3 to the exact sequence 0 -> g 1 4 g 1 EB
g 2 ~ g 2 -> O. The result follows by induction from the isomorphism
gl EB ... EB g( ~ (gl EB'" EB gt-1) EB Yr. 0
Theorem 4.7. Let cp: g 1 -> g 2 be a homomorphism of coherent sheaves over

B. Then Ker cp and 1m cp are also coherent.
PROOF. The sequence 0 -> Ker cp -> g 1 -> g 2 is exact, so Ker cp is coherent.
Since 1m cp ~ g dKer cp, the coherence ofIm cp follows from Theorem 4.6. 0
Theorem 4.8 (Serre's five lemma). Let g' 4 g" 4 g 4 g* ~ g** be an

exact sequence of sheaves. If g', g", g*, g** are coherent, so also is g.
PROOF. The sequence 0 -> g"/Imj1 -> g -> Ker P2 -> 0 is exact and the
sheaves g"/Imj1 and Ker P2 are coherent. Hence the result follows from
Theorem 4.3. 0
Remark. With Serre's five lemma, we can deduce the other theorems:
For example, if the sequence 0 -> g* -> g -> g** is exact, then so is
the sequence 0 -> 0 -> g* -> g -> g**. If g and g** are coherent, then
the coherence of g* follows from the five lemma.
EXAMPLE. Let A c B be an analytic set, §(A) its ideal sheaf and Yl'(A) =
CQ/J1(A). Since J1(A) is coherent, the sheaf Yl'(A) is also coherent.
If we choose, for example, A = {O} c en,
then J1(A)o = {fo:f(O) = O}
is the maximal ideal in (1)0, §(A)3 = (1)3 for 3 #- O. Therefore
Yl'(A)3 = {OC for a = 0

otherwise.
In conclusion we note that the general theorems and constructions of this

chapter carryover word for word if one admits as the base space an arbitrary
topological space instead of a region of en.
We define: Let X be a topological space, &I a sheaf of C-algebras over X.
A sheaf of &I-modules over X is called coherent if it is a finite and relation
finite sheaf of 9I!-modules.
In particular &I itself is coherent if for every open set U c X and each
9I!-homomorphism cp: q9l!1 U -4 9I!1 U the sheaf Ker cp is finite over 9I!.
117
IV. Sheaf Theory
Theorem 4.9. Let fYl be a coherent sheaf of C-algebras over X, ~ c fYl a

coherent ideal sheaf. Then as a sheaf of C-algebras fYl/~ is coherent.
PROOF. We already know that as a sheaf of fYl-modules fYl/~ is coherent.
Now let n: fYl ---7 fYl/~ the canonical projection, V c X open, rp: q(fYl/~)i V ---7
(fYl/~)iU a given (fYl/~)-homomorphism.
n induces an fYl-homomorphism nq: qfYl ---7 q(fYl/~) and we set ljI: =
rp nq:qfYliU ---7 (fYl/~)iU. ljI is an fYl-homomorphism and hence for every
0
30 E V there is an open neighborhood V(50) c V and sections S10 . . . , sp, Si E

r(V, Ker ljI) which generate Ker ljI over V. For the sections
81: = nq(sJ E r(V, q(fYl/~))
we have
hence Si E r( V, Ker rp) .

.It is easily verified that the 81> ... , sp generate the sheaf Ker rp over V as
an (fYl/~)-module.
Corollary. If Been is a region, A c B an analytic subset, then £,(A) and

hence J'l'(A)iA is a coherent sheaf of C-algebras.
118
CHAPTER V
Complex Manifolds
1. Complex Ringed Spaces

Let R be a local q::::-algebra with maximal ideal m, (see Def. 5.1 in
Chapter III), and let n:R ~ Rim ~ C be the canonical projection. If J E R,
the value of J is the complex number [fJ: = n(f).
EXAMPLE. Let J be a holomorphic function on a region B, 30 E B a point.

Then 100 = (W, f)30 is an element of the local C-algebra (!J30 and rJ E reB, (!J)
with rJ(ao) = ho·
We introduce the complex valued function [rfJ on B by setting

[rfJ(30): = [rJ(30)].
Then [rfJ(ao) = Lho] = n(J;o) = J(30), so that [rfJ = J. Consequently the
inverse of the isomorphism r:Mw ~ r(W, (!J) is given by r-1(s): = [sJ.
Def. 1.1. A pair (X, ~) is called a complex ringed space if:

1. X is a topological space;
2. ~ is a sheaf of local C-algebras over X.
If W c X is an open set, then the set of all complex valued functions

on W will be denoted by ST(W, C).
If J E r(W, ~), then there is an element [J] E ST(W, C) defined by
[J](x): = [J(x)] E ~xlmx ~ C. The correspondence r(W,~) ~ ST(W, C)
given by J f-* [1] is a homomorphism of C-algebras but, in general, is
neither surjective nor injective.
119
v. Complex Manifolds
Comment. If (X, £) is a complex ringed space and W c X open, then

naturally (W, £!W) is also a complex ringed space.
Def. 1.2. Let (X l> £ 1), (X 2, £2) be complex ringed spaces. An isomorphism
between (Xl> £1) and (X2' £2) is a pair CfJ = (cp, CfJ.) with the following
properties:
1. cp: Xl -? X 2 is a topological mapping.
2. CfJ.: £ 1 -? £2 is a topological mapping.
3. CfJ. is stalk-preserving with respect to cp; that is, the following diagram
is commutative:
£'1~£2
nIl _ ln2
Xl -L X2
4. For every x E Xl, CfJ.I(£1)X:(£1)X -? (£2)iji(x) is a homomorphism of
IC-algebras.
The existence of an isomorphism CfJ between (X 1, £ d and (X 2, £2)

is expressed briefly by
Theorem 1.1. Let CfJ = (cp, CfJ.):(X 1, £1) -? (X2' £2) be an isomorphism of
complex ringed spaces. Then for every open set V c X 2 there is a IC-algebra
isomorphism
cp:r(V, £2) -? r(cp-1(V), £d
defined by cp(s): = CfJ; loS a cp.
PROOF. cp(S):cp-1(V) -? £1 is clearly continuous, and

n 1 a (CfJ;l a S a cp) = (nl 0 CfJ;l) a (s 0 cp) = (cp-1 a n2) a (s 0 cp)
= cp-1 0 (n2 0 s) 0 cp = idiji-l(V).
It is clear that cp is a homomorphism of IC-algebras. An inverse is given by

cp-1(t) = CfJ.otocp-1. 0
Lemma 1. If R is a local IC-algebra with the maximal ideal m, then a E m

ifand only ifa - c· 1 <t mfor all c E IC - {O}.
PROOF
1. Let a E m, a - c· 1 Em. Then also c· 1 = a - (a - c· 1) E m. That
is, c cannot lie in IC - {O}.
2. For all c E IC - {O} let a - c· 1 <t m. We set c: = neal. Then
n(a - c· 1) = O. Therefore a - c· 1 Em, hence c = 0 and a E m. 0
120
Lemma 2. Let P:(Rb md ~ (R2' m 2 ) be a <r:>algebra homomorphism between

local C-algebras. Then p(m 1) c m 2 . If in particular p is an isomorphism,
then p(ml) = m2.
PROOF
1. If (J E m 1, then (J - c· 1 ¢ m 1 for all c E C - {O}. Therefore for every
c E C - {O} there is a (Je with (Je . «(J - c . 1) = 1, and then
p«(JJ . (p«(J) - c· 1) = 1.
Hence p«(J) - c . 1 ¢ m 2, so p«(J) lies in m2.
2. If p is an isomorphism, then p-l(m 2) c mI. Therefore m2 =
pp-l(m 2 ) c p(ml) c m2, and p(ml) = m2. 0
Theorem 1.2. Let 
%(cp-l(V), q be defined by <p*(f): = fa cp. Then for every S E r(V, Yl' 2)'
[<Ii(s)] = <p*([s]) (therefore [<p*-1 So cp] = [s] cp).
0 0
PROOF. Let V c X 2 be open, y E V, x: = cp-l(y) and s E r(V, Yl' 2). Then

s(y) = ([s](y))· 1 + (J* with (J* E (m2)y, therefore
<p; I(S(y)) = ([s](y))· 1 + <p; 1«(J*),
with <p;: 1«(J*) E (mlk Hence
[<p;:1 osocp](x) = [<p;:I(S(Y))] = [s](y) = [s] ocp(x)
follows. o
We thus obtain the following commutative diagram:
t E r(ip-l(V), Yl'd ( (j5 r(V, Yl' 2) :3 s
I 1
[t]E %(cp-l(V), C) ~%(V, q
1 I :3 [s]
Since <Ii and <p* are isomorphisms, t f---* [t] is injective if and only if s f---* [s]
is injective.
Def. 1.3. A (reduced) complex space is a complex ringed space (X, Yl') with
the following properties:
1. X is a Hausdorff space.
2. For every point Xo E X there is an open neighborhood U(xo) c X and
an analytic set A such that (U, Yl'1 U) ~ (A, Yl'(A)).
(A lies in an open set Been and Yl'(A): = (@/JE(A))IA, where JE(A) is
the ideal sheaf of A. Yl'(A) is a coherent sheaf of local IC-algebras and
hence Yl' is also coherent.)
121
V. Complex Manifolds
A reduced complex space therefore looks locally like an analytic set. If

this analytic set has no singularities, then we call the complex space a complex
manifold:
Der. 1.4. A complex manifold is a complex ringed space (X, J'f) with the
1. X is a Hausdorff space
2. For every point Xo E X there exists an open neighborhood U(xo) c X
and a region Been such that (U, J'fIU) ~ (B, (D).
Theorem 1.3. Let (X, J'f) be a complex manifold, We X an open subset.

Then the mapping T{W, J'f) --> ST(W, C) given by f I--> [f] is injective, and
for every f E T{W, J'f), [f] is continuous.
PROOF. There is an open covering (U'),ei of Wand a system (B,),ei of open

sets such that (U" J'fIU,) ~ (B" (DIB,). For f E T{W, J'f)
[flU'] = [f]IU,.
Hence it suffices to prove the proposition for the sets U,. It follows im-
mediately from Theorem 1.2 and the equation r- 1 (s) = [s] that the mapping
T{U" J'f) --> ST(U" C) is injective. If f E T{U" J'f), then Cf'*0 f cp-1 =
0
tp-1(f) lies in r(B,,(D); [tp-1(f)] is therefore continuous. Hence
is also continuous. o
Der. 1.5. Let (X, J'f) be a complex manifold, W c X open. A holomorphic

function over W is an element of the set [T{W, J'f)] = {[f] E ST(W, C):j E
T{W, J'f)}.
Remarks
1. The mapping f I--> [f] defines an isomorphism from T{W, J'f) onto
the set of holomorphic functions over W.
2. Every hoi om orphic function is continuous.
3. If U c X is open, Been a region and Cf': (U, J'f) --> (B, (D) an iso-
morphism, then for every open subset V c U a function fE ST(V, C) is
holomorphic if and only if f cp - 1 is holomorphic.
0
4. If U c X is open, Been a region and Cf': (U, J'f) --> (B, (D) an isomor-
phism, then the pair (U, {ji) is called a complex coordinate system for X. If
(U 1, {ji1), (U 2, C(2) are two complex coordinate systems with U 1 n U 2 =I 0,
then {ji12: = {jil {ji:;1:{ji2(U 1 n U 2 ) --> {ji1(U 1 n U 2 ) is a homeomorphism
0
of open sets in I[:n.
122
Figure 21. Compatibility of complex coordinate systems.
If f is holomorphic on iiJ I(U I n U 2), then f 0 iiJ I is holomorphic on

U 1 n U 2 (by 3). Therefore, f 0 fPl2 = (f 0 fPI) 0 fPil is holomorphic on
fP2(U I n U 2). Then iiJ12 is a holomorphic mapping (see Theorem 7.1,
Chapter I).
To conclude this section we shall show conversely that we can also
define a complex manifold with the help of a suitable system of complex
coordinates.
Let X be a Hausdorff space, (U,),eI an open covering of X. For every U,
let there be given a homeomorphism fP, from U, onto a region B, c Cn
such that every coordinate transformation
fP'l 0 fP,~ I: fP 12 (U'l n U'2) --+ fP'l (U'l n U,J
is holomorphic. One calls the system {(U" fP,):l E I} a complex atlas for X.
Now let W c X be open, f E ff(W, q and Xo E W. f is called holomorphic
at Xo if there exists an lo E I and a neighborhood U(xo) c W n U'o such
that f 0 fP,~ I is holomorphic in fP'o(U) c B,o. f is called holomorphic on W
if f is holomorphic at every point x E W.
Because ofthe compatibility condition for the coordinate systems f fP,-1 0
is holomorphic at fP,(xo) for every l with Xo E U" whenever f is holomorphic

at Xo.
the usual restriction mapping. Then {Mw, rn

Let M w be the set of holomorphic functions on W, and r~: M w --+ M v
is a pre-sheaf. The corre-
sponding sheaf Ye is called the sheaf of germs of holomorphic functions over X.
Ifxo E U, n W andf E M w , then
(w, f)xo = (W n U" flW n U,)xo.
The system M w n u, together with the corresponding restriction mappings
form a pre-sheaf for the sheaf Yel U,. An isomorphism between the pre-sheaf
of J'l'IU, and the pre-sheaf of <!JIB, is defined by f 1--+ f fP,-I, and this iso-
0
morphism induces an isomorphism (cp,)*:YeIU, --+ <!JIB,. Ye is thus a sheaf of

local C-algebras, cp,: = (fP" (cp,)*):(U" Ye) --+ (B" <!J) is an isomorphism of
complex ringed spaces, and (K, Ye) is a complex manifold.
123
2. Function Theory on Complex Manifolds

Let (X, JIP) be a complex manifold, Xo E X. Then there is a neighborhood
U(xo) c X and a homeomorphism q> of U onto a region B c cn. The natural
number n is independent of the particular choice of q> and one defines
dimxo(X): = n.
Henceforth we always assume that dimAX) = n = constant on all of X.
Then (X, JIP) is called an n-dimensional complex manifold.
Theorem 2.1. Let (X, JIP) be an n-dimensional complex manifold, W c X

open. Then (W, JlPIW) is also an n-dimensional complex manifold.
PROOF. It is clear that W is a Hausdorff space and JlPIW a sheaf of local
C-algebras. For every point Xo E W there is a neighborhood U(xo) c X and
an isomorphism q>:(U, JlPIU) -4 (B, (1)). Then W n U is a neighborhood of
Xo in Wand (W n U, JlPIW n U) ~ (q5(W n U),(1)). D
Der. 2.1. A complex manifold (X, JIP) is connected if the underlying topo-
logical space is connected (so there is no decomposition X = Xl U X 2
into two disjoint non-empty open subsets).
Theorem 2.2 (Identity theorem). Let (X, JIP) be a connected complex

manifold, f1' f2 holomorphic functions on X and V c X a non-empty open
subset with f11 V = f21 V. Then f1 = f2·
PROOF. LetW1: = {xEX:rf1(x) = rf2(x)}, W2 : = X - W1. W1 isnotempty
since V is contained in WI> and W1 is open since the set where two sections
coincide is always open. Let Xo E W2 be an arbitrary point. Then in X there
exists an open neighborhood U(xo) and an isomorphism (U, JIP) ~ (B, (1))
where B denotes a domain in cn. If Xo is not an interior point of W2 , then
U n W1 i= 0 is an open neighborhood and rf11 U n W1 = rf21 U n W1. By
the identity theorem in C n it now follows that rf1 and rf2 coincide on U,
and so in particular, that Xo lies in W1 • That is a contradiction. Every point
of W2 is an interior point of W2 , so Wz is open. Since X is connected, it follows
that W1 = X and therefore f1 = f2' D
Theorem 2.3 (Maximum principle). Let (X, JIP) be a connected complex mani-
fold, f holomorphic on X, Xo E X a point. If If I has a local maximum at
xo, then f is constant.
PROOF. There is a neighborhood U(xo) c X and an isomorphism q>:
(U, JIP) ~ (B, (1)). Without loss of generality we may assume that q5(xo) = 0
and B is a polycylinder about the origin. For 3 E Band 3 i= 0 let E3: =
{t3:t E C}. Then E3 n B is a circular disk in the complex t-plane, and
IU 0 q5-1IE3 n B)I has a local maximum at the origin. By the maximum
124
principle of one dimensional complex analysis this means that f 0 Ep -11E3 n B

is constant. In particular f(Ep-1(3)) = f(xo). Since 3 E B was chosen arbi-
fl
trarily it follows that U is constant, and by the previously proved identity
theorem that holds only if f is constant. 0
Theorem 2.4. Let (X, Jf') be a connected compact complex manifold. Then
every function holomorphic on X is constant.
PROOF. If
Iff is holomorphic on X, then I is continuous on X and therefore
attains a maximum on the compact manifold X. By the maximum principle,
f is constant. 0
EXAMPLE. If we give the Riemann sphere X: = IC U {oo} the usual topology,

one obtains a Hausdorff space.
cp:X - {oo} ~ IC with cp(x): = x and 1/1: = X - {O} ~ IC with I/I(x): =

l/x are topological mappings, and the transformations
are holomorphic.
Hence {(X - {oo}, cp), (X - {O}, I/I)} is a covering of X by compatible
complex coordinates which induces a sheaf Jf' on X. X is a one-dimensional
complex manifold.
1. X is compact:
Let E1 : = {ZEX - {oo}:lzl ~ I},
E z: = {zEX:lzl ~ I}.
Then E1 is compact and (1/IIEz):E z ~ E1 is a homeomorphism. Therefore E z
is also compact. The proposition follows because X = E1 U E z .
2. X is connected, since the sets Eb E z are connected and El n E z =F 0.
By Theorem 2.4 it follows that every function holomorphic on the whole
Riemann sphere is constant.
Def. 2.2. An abstract Riemann surface is a connected one-dimensional

complex manifold.
The Riemann sphere is an abstract Riemann 'iurface. In the next section

we shall examine the so-called "concrete Riemann surfaces."
Note. When no confusion can arise, we shall denote a complex manifold
simply by X.
Def. 2.3. Let X l' X z be complex manifolds. A continuous mapping cp:

X 1 ~ Xz is called holomorphic if for every open set U c X z, 9 0 cp is
holomorphic over cp -1( U) whenever 9 is hoI om orphic over U. If cp is
topological and cp and cp - 1 are holomorphic, then cp is called biholo-
morphic.
125
Remarks
1. idx:X ~ X is always biholomorphic.
2. If qJ: X 1 ~ X 2 and ljI: X 2 ~ X 3 are holomorphic mappings, then
ljI 0 qJ: X 1 ~ X 3 is also holomorphic.
3. Let f:X ~ IC be a continuous mapping. f is holomorphic (in the
sense of Def. 2.3) if and only if f is a holomorphic function.
PROOF
a. If f is a holomorphic mapping, then f = ide 0 f is a holomorphic
function over f-1(C) = X.
b. Let f be a holomorphic function, U c IC open and 9 holomorphic
over U. Then for every point Xo E f-1(U) C X there exists a neighborhood
V(xo) c X and an isomorphism qJ:(V, £') ~ (B, (9). Since by definition
f 0 ip - 1 is holomorphic over B, (g 0 .f) 0 ip - 1 = 9 0 (f 0 ip - 1) is holomorphic
over B, and that means that 9 0 f is holomorphic at Xo. D
Theorem 2.5. A mapping ljI: X 1 ~ X 2 is biholomorphic if and only if there

exists an isomorphism qJ:(X 1, £'1) ~ (X2' £'2) with ip = ljI.
PROOF
1. If ljI: X 1 ~ X 2 is a biholomorphic mapping, then ljI and ljI - 1 carry
holomorphic functions into holomorphic functions and hence induce an
isomorphism between the canonical pre-sheaves. Naturally ip = ljI for the
corresponding isomorphism qJ:(Xb £'1) ~ (X2' £'2)'
2. If qJ = (ip, qJ*):(X 1, £' 1) ~ (X 2, £'2) is an isomorphism, then ip is a
topological mapping. If U c X 2 is open and 9 holomorphic over U, then
there is an s E r(U, £'2) with 9 = [s], and 9 ip = [s] ip = [qJ;l S ip]
0 0 0 0
with qJ;l 0 so ip E r(ip-1(U), £'2)' Therefore go ip is holomorphic over

ip -leU). Hence ip is holomorphic. One shows similarly that ip -1 is holo-
~~~ D
Def. 2.4. Let X be a complex manifold. A subset A c X is called ana-

lytic if for every Xo E X there is an open neighborhood U(xo) c X and
holomorphic functions f1' ... ,it over U such that UnA = {x E U:
f1(X) = ... = f(x) = O}.
Theorem 2.6. Let X be a connected complex manifold and M #- X an ana-

lytic subset of X. Then M = 0.
PROOF. If M #- 0, there exists a point Xo E aM such that for every open
neighborhood U(xo) c X the set U n M is open and non-empty. We could
choose a connected U such that there exist holomorphic functions fb ... , fd
on U with Un M = {x E U:f1(X) = ... = ..ft(x) = O}. ThenilU n M = 0
for i = 1, ... ,d. By the identity theorem it follows that i IU = 0 for
i = 1, ... , d and therefore U c M, a contradiction. D
126
Def. 2.S. A complex manifold X is called holomorphically separable if for

every Xo E X there are holomorphic functions f1' ... , ft on X such that Xo
lies isolated in the set {x E X:f1(X) = ... = ft(x) = O}.
Remark. One can show that always t ~ dim X.
EXAMPLE. Let (X, ljJ) be a domain over en. Then for every Xo E X there are
open neighborhoods U(xo) c X and V(ljJ(xo» c en such that ljJiU: U ~ V
is topological. (U, ljJ) is therefore a complex coordinate system, and since the
identity is always the coordinate transformation, X becomes a complex
manifold. The mapping ljJ: X ~ en is holomorphic (in the sense of Def. 2.3).
A continuous mapping ljJ: X ~ Y between topological spaces is called

discrete if for every y E Y, ljJ -l(y) is empty or a discrete set in X.
ljJ is a discrete mapping.
PROOF. Let Xo E X, 50: = ljJ(x o) and Xl E ljJ-1(50)' Then there is a neigh-

borhood U(x 1) c X and a neighborhood V(50) c en such that ljJiU: U ~ V
is topological. But that can only be if ljJ-1(50) n U = {xd. Therefore the
fiber ljJ - 1(50) is a discrete set. 0
Hence it follows that every domain over en is holomorphically separable:
PROOF. Let gi(Zl"'" zn): = Zi - z? and J;: = gi 0 ljJ:X ~ e for i =

1, ... , n. fb ... ,J,. are holomorphic functions on X, and Xo lies isolated in
{x E X:f1(X) = ... = J,.(x) = O} = ljJ-1(50)' 0
In particular, every domain G c en is holomorphically separable. One

can generalize the above results in the following manner:
Theorem 2.7. An n-dimensional complex manifold X is holomorphically

separable if and only if there exists a holomorphic discrete mapping ljJ:
X~ en.
(One direction is clear, for the other see: H. Grauert: "Charakterisierung
der holomorph-vollstandigen Raume", Math. Ann., 129: 233-259, 1955.)
Def. 2.6. Let X be a complex manifold.

1. If K c X is an arbitrary subset, then
K: = {xEX:if(x)i ~ supif(K)i}
for every holomorphic function f on X} is called the holomorphically
convex hull of K.
2. X is called holomorphically convex if K c X is compact whenever
K c X is compact.
127
Def.2.7. X is called a Stein manifold if

1. X is holomorphically separable.
2. X is holomorphically convex.
Theorem 2.8. For a domain (X, t/J) over en the following properties are
equivalent:
1. X is a Stein manifold.
2. X is holomorphically convex.
3. X is a domain of holomorphy.
The non-trivial equivalence of (2) and (3) was proved in 1953 by Oka.
Theorem 2.8 leads us to regard the Stein manifolds as generalizations of
domains of holomorphy.
EXAMPLE. If X is a compact complex manifold and dim X > 0, then X is

holomorphically convex but not a Stein manifold.
PROOF. If K c X is compact, then K c X is always closed. If X is compact

it follows that K is also compact. Therefore X is holomorphically convex.
Since X is compact there exists a decomposition of X into finitely many
connected components, which are also all compact: X = Xl U··· U Xc.
If Xo E Xi> f is holomorphic on X and f(x o) = 0, then by Theorem 2.4 f
vanishes identically on Xi. Therefore each set of the form
{x E X:fl(x) = ... = fm(x) = O}
contains, in addition to the point xo, the open subset Xi eX; Xo is therefore
not an isolated point and X cannot be holomorphically separable. 0
3. Examples of Complex Manifolds

Concrete Riemann Surfaces
Def.3.1. A (concrete) Riemann surface over e is a pair (X, cp) with the
1. X is a Hausdorff space.
2. cp: X -+ e is a continuous mapping.
3. For every Xfl E X there is an open neighborhood U(xo) c X, a con-
nected open set Vee and a topological mapping t/J: V -+ U such that
a. cp t/J: V -+ e is holomorphic.
0
b. (cp t/J)' does not vanish on any open subset of V.

0
One also calls the mapping t/J a local uniformization of the Riemann surface
(X, cp).
128
Theorem 3.1. Let (X, cp) be a Riemann surface over C. Then X has a canonical
one-dimensional complex manifold structure, and cp: X -+ C is a holomorphic
mapping.
PROOF
1. Let Xo EX, zo: = <p(Xo) E C. Then there is a neighborhood V(xo) c X
and a connected neighborhood V(zo) c C as well as a topological mapping
l/!:V -+ V with the local uniformization property. (V, l/!-1) is therefore a
complex coordinate system for X at Xo.
Now two such coordinate systems (V 1, l/!11), (V 2, l/!:;l) may be given. Then
l/!: = l/!11 l/!2:l/!:;I(V 1 ( l V 2) -+ l/!ll(Vl ( l V 2) is a topological mapping.
0
If we set fA(t): = cp l/! A(t) for t E v,., ), = 1,2, then fA is a holomorphic

0
function on V;. whose derivative does not vanish on any open subset of V;..
Let to E l/!11(Vl ( l V 2) be chosen so thatf'1(to) # O. Then there is a neigh-
borhood V(to) c l/!11(Vl ( l V 2) and an open set We C such that flIV:
V -+ W is biholomorphic. Let
gl: = (fIIV)-1 = (cp l/!1IV)-1:W -+ V.
0
The mapping
l/!IIV:V -+ V: = l/!1(V) c VI (l V2
is topological, and so is
cplY = gIl 0 (l/!liV)-1 = «l/!dU) 0 gl)-l: V -+ W.
1/1-1/11 10 1/12
w= cp(V)

It now follows that
l/!1l/!-1(V) = l/!11 0 l/!21l/!:;1(V) = l/!11 0 (cplV)-l 0 (cplY) 0 l/!21l/!:;1(V)
= l/!11 0 l/!1 0 g1 0 cp 0 l/!21l/!:;1(V)
= g1 0 (cp 0 l/!2)1l/!:;1(V) = gl 0 f21l/!:;1(V),
129
which is a holomorphic function. By the identity theorem D: = {t E V1 :

1'l(t) = O} is a discrete set, and so is
D': = 1/I-1(1/I1 1(U 1 n U 2 ) n D).

If
then
therefore
f~ (to) i= O.
As we just showed, there is a neighborhood U(to) such that 1/I11/I-1(U) is

holomorphic. In particular 1/1 is holomorphic at So = 1/1 -l(t O).
A continuous mapping which is holomorphic outside a discrete set must,
however, be everywhere holomorphic by the Riemann extension theorem.
Hence the coordinate transformations are holomorphic, so X is canon-
ically a complex manifold.
2. Let Bee be open, g holomorphic on B. Then W: = cp - l(B) is open
in X and go (cpIW): W -+ Cis continuous.If(U, 1/1-1) is a coordinate system,
then f: = cp 1/1: 1/1 - 1( U) -+ C is holomorphic, and so is (g cp) 1/1 = g f,
0 0 0 0
which means that g cp is holomorphic on W.

0 0
Now let (X, cp) be a Riemann surface, 1/1: V -+ U a local uniformization

and f: = cp 1/1. By assumption the set D: = {t E V:1'(t} = O} is discrete in
0
V. Let to E V and xo: = I/I(to} E U.

Case 1. 1'(to} i= o.
Then there are neighborhoods V1(to) c V and W(f(to}} c C such that
f1Y1: V1 -+ W is biholomorphic. U 1: = I/I(Vd is open in U, and 1/11: =
1/1 (fl V1) -1: W -+ U 1 is topological. Moreover, cp 1/1 1 = id w . Therefore
0 0
there exists a local uniformization 1/1 1: W -+ U 1 with Xo E Uland cp 1/1 1 = 0
id w . Thus, via cp, U 1 is a sheet over W.

In this situation we say that X is unbranched at Xo. Clearly X is unbranched
everywhere outside a discrete set.
Case 2. 1'(to) = O.
Let zo: = f(to}. Then at to, f - Zo has a zero of order k ;:, 2, that is, on V
f(t} = Zo + (t - to}k. h(t),
where h is a holomorphic function with h(to ) i= o.

There exists a neighborhood V1 (t O } c V and a holomorphic function g
on V1 with gk = h. In particular g(to) i= O. Let r: V1 -+ C be defined by
r(t}: = (t - to) . g(t). Then r'(t o} = g(t o} i= 0, so there is a neighborhood
V2 (t O ) c V1 and an open set W2 c C such that r1Y2: V2 -+ W2 is biholo-
morphic. Clearly
130
is topological and
cp 0 (1/1 0 (r!V2)-1) = f 0 (r!V2)-1
holomorphic, and (f 0 (rl V2 ) -1)' vanishes at most on a discrete set. Moreover
f 0 (r!V2)-1(s) = f (r!V2)-1(r(t)) = f(t) =
0 zo + ((t - to)' g(t))k
= zo + r(t)k = zo + Sk.
Therefore there exists a local uniformization 1/1 2 : tV; -+ U 2 with Xo E U 2 and
cp 1/12(t) = cp(xo) + tk.
0
Since the coordinate transformation has a non-vanishing derivative, the

order k of a zero is not affected by a change of chart, that is, k depends only
on the point Xo. Wesay that Xo is a branch point of order k. 1/12, where
cp 1/12(t) = cp(xo) + t, is called the distinguished uniformization.
0
X then locally represents a branched k-fold covering over cp(x o), in the
sense that there lies exactly one point of X over cp(xo), while over every
point z =1= cp(xo) in some neighborhood of cp(xo) there lie exactly k points
ofX.
EXAMPLE. Let X: = {(w, z) E C 2 :W2 = Z3}. With the topology induced by

C 2 , X becomes a Hausdorff space, and the mapping cp: X -+ C with cp( w, z): =
z is continuous.
In order to show that X is a Riemann surface over C, we must specify
the local uniformization. Let I/I:C -+ X be defined by I/I(t): = (t 3, t 2).
a. 1/1 is injective. If l/I(t 1) = l/I(t 2), then ti = t~ and t~ = t~. If I/I(t) = 0
then t = O. If t1 =1= 0, then also t2 =1= 0, so we can divide and then t1 = t 2.
b. 1/1 is surjective. If 0 =1= (w, z) E X, then z =1= 0 so there exist two complex
numbers t 1, t2 such that {tb t 2 } = {t:t 2 = z}. Then t1 = -t2 so that
w2 = Z3 = (t~f = (t~)2; therefore WE {d, tn, so either 1/I(t1) = (w, z) or
l/I(t 2) = (w, z).
c. 1/1: C -+ X is topological. The continuity of 1/1 is clear. Because of the
continuity of the roots, 1/1 -1 is continuous.
Hence there is a global uniformization for X, given by 1/1. (cp I/I(t) = t 2 0
is hoi omorphic, and has a derivative which does not vanish identically
anywhere.)
Let r:X -+ C be defined by r: = 1/1-1. Since r 1/1 = ide, r is a holo- 0
morphic function on X. r cannot be holomorphically extended into C 2 •

Otherwise, suppose, g(w, z) = L avl'wVzl' is a holomorphic function on C 2
V,I'
(for example, in a neighborhood of 0 E ( 2 ). Then
(gIX)(w, z) = g(l/I(t)) = L aVl't 3v + 21'
V,I'
= aoo + a01t2 + a10t3 + a02t4 + allt5 + a20 t6 + ....

If i were a holomorphic extension of r, then we would have t = r 0 I/I(t) =
(iIX)(I/I(t)). But that cannot be.
131
Complex Submanifolds
Let X be a complex manifold, (U, <p) a coordinate system on X. If Xo E U
and f holomorphic on a neighborhood V(xo) c: U, then we define the
partial derivatives of f at Xo with respect to (xo): =
au a(xo) = L avl"
1'=1
(DJ)q>'(xo),
Therefore, if fl' ... ,!<J are holomorphic functions on V, the natural number
is independent of <p.
Def. 3.2. Let X be an n-dimensional complex manifold, A c: X analytic.

A is called free of singularities of the codimension d if for every point
Xo E A there exists a neighborhood U(xo) c: X and holomorphic func-
tions f10 ... ,fd on U such that:
1. An U = {XE U:fl(x) = ... = fd(x) = O}
2. rkAf1o ... ,!<J) = d for all x E U.
Theorem 3.2. An analytic set A c: X is free of singularities of codimension d

if and only iffor every point Xo E A there exists a neighborhood U(xo) c: X
and an isomorphism <p = ([P, <p*): (U, JIt') --+ (B, (9) such that [P( UnA) =
{(W1o" ., wn ) E B:Wl = ... = Wd = O}.
Figure 2 . Illustration for Theorem 3.2.
132
PROOF. Let Xo E A be given, U(xo) c: X a neighborhood and qJ: (U, £) --"

(13, (0) an isomorphism. A II U is singularity free of co dimension d if and
only if cp(A II U) is a regular analytic set 13 of the complex dimension n - d.
That, however, is equivalent to the existence, for every 3 E cp(A II U) of a
neighborhood V(3) c: 13 and an isomorphism tjJ:(V, (0) --" (B, (0) such that
iJI(cp(A II U) II V) = {to E B:W1 = ... = Wd = O}
(see Theorem 6.15 in Chapter III).
qJl: = tjJ 0 qJ:(cp-l(V(cp(xo))), £) --" (B, (0)
is such a function. o
Theorem 3.3. Let A c: X be an analytic set,free of singularities of codimension
d. Then X induces a canonical (n - d)-dimensional manifold structure on A,
and the natural imbedding jA:A ~ X is holomorphic.
PROOF. A, with the relative topology induced by X, is clearly a Hausdorff
space. A function f defined on an open set W c: A will be called holo-
morphic if for every x E W there is an open neighborhood U(x) c: X and
a holomorphic function J on U such that U II A c: Wand JI U II A =
fl U II A. If X is an open set in en and A is a part of a (n - d)-dimensional
plane, then this new notion of holomorphy on A agrees with the earlier
notion. The set of holomorphic functions defines a pre-sheaf on A.
If Xo E A, then there exists a neighborhood U(xo) c: X and an isomor-
phism qJ:(U, £) --" (B, (0) such that
cp(U II A) = {to E B:W1 = ... = Wd = O} = B II ({O} X en-d)
The pre-sheaf of hoI om orphic functions on U II A is mapped isomorphically
by qJ onto the pre-sheaf of locally holomorphically continuable functions on
B II ({O} X en-d) = B', and the latter coincides with the pre-sheaf ofholo-
morphic functions on the region B' c: en-d. For the sheaf £' of germs
of holomorphic functions on A it is then true that (U II A, £') ~ (B', (0).
Hence A is a complex manifold.
If U c: X is open and f holomorphic on U, then by definition f 0 jA =
flA II U is also holomorphic, and hence jA:A ~ X is a holomorphic
mapping. 0
Remark. An analytic set A c: X, free of singularities, is also called a

complex submanifold of X. The example X = {(w, z) E e 2 :w2 = z3} con-
sidered in part (a) is not a submanifold of e 2 .
Cartesian Products
Theorem 3.4. Let Xl, ... , Xl be complex manifolds, ni: = dim Xi for
i = 1, ... , t and n: = n 1 + ... + nc. Then there is an n-dimensional
manifold structure on X: = Xl X .•. x X (, such that all projections
Pi: X --" Xi are holomorphic.
133
PROOF. With the sets W = WI X .•. x ~, W; C Xi open as the basis for

the topology of X, X becomes a Hausdorff space.
If Xo = (Xb ... , Xe) E X, then there exist neighborhoods Ui(X i ) C Xi and
isomorphisms ({Ji :(U;, Jt";) -4 (B;, (I);). Let
U: = U I x ... X Ut , <PI X ... x <Pe: U -4 B: = BI x ... x Bt

be defined by
«PI x ... x <PtHX /I , ... , x~): = (<PI (XiI), ... , <P1(X~)).
Then (U, (<PI x ... x <Pe)) is a complex coordinate system at Xo. If (v,
(ifr I x ... x ifrt)) is another coordinate system, then the transformation
(<PI x .. · x <Pt) ° (i(J1 X ... X ifre)-I = (<PI ° i(JI I X .. , X <p( ° i(J;I)
is holomorphic. Therefore X is an n-dimensional complex manifold. Suppose

W .C X 1 is open, and g is holomorphic on W. Then
Let Xo E V and
be a coordinate system for X at Xo' Then
(g ° PI) ° (<PI x ... X <P1)-I(Zb"" ZI) = go <PI 1(zd

= (g <pI1) ° prl(zt> ... , zn),
0
and
is holomorphic. Therefore g ° Pl:X -4 C is also holomorphic; that is, PI is

a holomorphic mapping. The prooffor P2, .... , Pe is similar. 0
Theorem 3.5. Let X be an n-dimensional complex manifold. Then the diagonal

D: = {(x, x):x E X} C X x X is an analytic subset free of singularities
of codimension n.
PROOF
1. Since X is a Hausdorff space, the diagonal D C X x X is closed.
Therefore D is analytic at each point (x, y) E X X X-D.
2. Let (xo, x o) E D. Then there is a neighborhood U(xo) C X and an
isomorphism ({J:(U, Jt") c:::: (B, (I)) and then U: = U x U is a neighborhood
of (xo, xo) in X x X, which is biholomorphically equivalent to B x B.
Therefore there exist coordinates ZI, ... , Zn, W b ·· ., Wn (with Zv: = prv ° <P,
W v: = prn+v° <p)inUsuchthatD n U = {(X,X)EX x X:(Zi - w;)(x,x) =
ofor i = 1, ... , n}. Moreover
134
rk(X,x)(ZI - WI, ..• , Zn - Wn)
) ),p (X, x)) i = 1, ... , n )

= r k ((
(Dv (Zi - Wi
v
= 1, ... , 2n
1 0 i -1 0)
= rk ( 0~1 i 0~-1 = n
which was to be proved. o
Theorem 3.6. Let X be a complex manifold, D c X x X the diagonal. Then
the diagonal mapping d:X -;. D by d(x): = (x, x) is biholomorphic.
PROOF. d is bijective, and the inverse mapping r
l = Pl/D is holomorphic.
It remains to be shown that d is holomorphic. Let WeD be open, g holo-
morphic on W, (xo, xo) E W. Then there exists a neighborhood U(xo) c X
and a holomorphic function {J on U x U such that (U x U) n DeW
and (J/(Ux U) n D = g/(U x U) n D. Without loss of generality we may
assume that there is an isomorphism q>: (U, .no) -;. (B, (()). The mapping
d*: B -;. B x B with d*(3): = (3, 3) is holomorphic and
(g 0 iP - 1(3) = {J d iP - 1(3) = {J (iP x iP) - 1 d* (3).
d) 0 0 0 0 0
Therefore (g d) iP - 1 and hence god is holomorphic.

0 0 o
Complex Projective Spaces
We define a relation on en+ 1 - {O} by setting 31 ~ 32 if and only ifthere
existsatEiC - {0}with32 = t'31'
It is clear that "~" is an equivalence relation, and we denote the equi-
valence class of 30 by G(30) = {3 = t30:t E iC - {O}}. G(30) is simply the
complex line through 0 and 30 with the origin removed.
Def. 3.3. The set [pn: = {G(3): 3 E en + 1 - {O} } is called the n-dimensional
complex projective space and the mapping n: en + 1 - {O} -;. [P" with
n(3): = G(3) is called the natural projection.
n is a surjective mapping, and we give [pn the finest topology in which

n is continuous. A set U c [pn is therefore open if and only if n-l(U) c
iC n+ 1 - {O} is open.
Let W;: = {3 = (Zll""Zn+l)EiC,,+I: Zj = I}, for i = 1, ... ,n + 1. Then
W; is an affine hyperplane in en+ l - {O}, and in particular, it is an n-
dimensional complex submanifold. Let
Wi: = {3 = (Zll"" Zn+l) E en+ l :Zj =1= O}.
A holomorphic mapping exj : Wi -;. en is defined by
135
Then ailVVi: VVi --+ en is biholomorphic with

(adVVi)-l(Z1>"" zn) = (Z1>"" Zi-l, 1, Zi"'" zn)·
If W c VVi is open, then for 3 E Wi, 3 E ai- 1(alW)) if and only if there is
a 3' E W with ai(3) = aM), and therefore with 1/(z;)3 = 3'- That happens if
and only if 3En-1n(W). Therefore n- 1n(W) = ai- 1alW) is open, which
means that:
1. The system of sets U i: = n( VVi) forms an open covering of [pn.
2. niVVi: VVi --+ U i is an open mapping.
If n(3) = n(3') for 3, 3' E VVi, then there is atE e with 3' = t· 3; therefore
1 = z; = tZ i = t, so 3 = 3'. Hence niVVi: VVi --+ Ui is injective, so with the
preceding considerations, it is topological. Hence for each i, ({Ji: = ai 0
(nl VVi) -1: Ui --+ en is a complex coordinate system for [pn. Moreover
(nIVVi)-1 0 n(z1>"" Zn+1) = (nIHii)-1 0 n (~i (Z1>"" Zn+1))
1
= -(Z1>"" zn+d
Zi
= (adVVi)-1 (~(Z1>"" Zb"" Zn+d)
= (aiIVVi)-1 oai(Z1>""Zn+1),
and hence the coordinate transformations
have the form

({Jj 0 ({Ji- 1(Z1,"" zn) = aj 0 (nlH-j)-1 0 n 0 (adVVi)-1(Z1>"" zn)
= a/z1>"" Zi-1> 1, Zb"" zn)
which is a holomorphic mapping.

We still must show that [pn is a Hausdorff space. Let Xl' Xz E [pn, Xl i= Xz.
1. If both points lie in the same coordinate neighborhood Ui> it is trivial
to find disjoint neighborhoods.
2. Suppose Xl' X2 are not elements ofthe same coordinate neighborhood.
Then for arbitrary points 3i E n-l(xi) we have
Z<'1) • Z<'2)
J J
= °'
j = 1, ... , n + 1.
Without loss of generality, then, we may assume that
31 = (1, z~l), ... , z~1), 0, ... ,0), with Z)l) i= °for j = 2, ... , S.
32 = (0, ... , 0, z~l! 1> ••• , Z~2), 1).
136
Let
V1 : = n({(1,W2, ... ,Wn+1)Ecn+1:IWn+11 < 1}),
V2 : = n({(W1, ... , Wn , 1) E e+ 1 < 1}). :lw11
V1 is an open neighborhood of Xb V2 is an open neighborhood of X2, and
VI n V2 = 0.
Theorem 3.7. The n-dimensional complex projective space is an n-dimensional

complex manifold, and the natural projection n: e + 1 - {O} -> /pn is
holomorphic.
PROOF. In order to complete the proof we have to demonstrate the holo-

morphy of n. Let W c X be open, g holomorphic in W. Without loss of
generality we may assume that W c U 1. Then g 0 qJl 1 = go no (0: 11W1)-1:
cn -> Cis holomorphic, and so is g 0 n = (g 0 qJl1) 0 (0:1IW1). 0
Theorem 3.8. /pn is compact.
PROOF. Let S: = {3Ecn+1:11311 = 1} = S2n+1. For 3Ee+ 1 - {O}, 3: =

(1/11311) . 3 lies in Sand n(3) = n(3). Therefore nlS: S -> /pn is a surjective con-
tinuous mapping. Since S is compact and /pn is separated, it follows that /pn
is also compact. 0
The I-dimensional complex projective space /p 1 is covered by two coor-

dinate neighborhoods U b U 2 • Here U 1 = n({3 = (1, Z2):Z2 E C}), and
U2 - U 1 = n({3 = (0, Z2):Z2 E C - {O}}) = {G(O, I)} consists of a single
point. Hence /p 1 = U 1 U {G(O, I)}.
Theorem 3.9. Let X = C u {oo} be the Riemann sphere. A biholomorphic

mapping qJ:X -> /p 1 is defined by qJ(oo): = G(O, 1) and qJ(z): = qJl 1(z) =
n(l, z).
PROOF. It is clear that qJ is bijective. On X one has two coordinate systems

IjJ 1 : X - {oo} -> C, and IjJ 2 : X - {O} -> C. Let Xl: = X - {oo}, X 2: =
X - {O}. Then
A. = 1,
for
A. = 2.
Therefore qJ IX1: X 1 -> U 1 is biholomorphic for ), = 1, 2, and so qJ IS
biholomorphic. 0
The n-dimensional Complex Torus

Let Cb ... , C2n E cn be linearly independent as real vectors. Then
r: = {3 = I 1=1
k1C1:k1 E Zfor A. = 1, ... , 2n}
137
is a subgroup of the additive group of en (a translation group). Two points

of en will be called equivalent if there is a translation of r carrying one into
another; that is,
This is in fact an equivalence relation, and we give the set Tn of all equivalence
classes the finest topology in which the canonical projection TCT: en -+ Tn is
continuous. We call the topological space Tn an n-dimensional complex torus.
Any two n-dimensional tori are homeomorphic. For 30 E rand U c en
let U + 30: = {3 + 30:3 E U}. If U is open, then U + 30 is open for every
30 E r, and so is TCT 1TCT ( U) = {5 E en :5 - 5' E r for a 3' E U} = U (U + 30)'
30Er
Thus TCT is an open mapping. Let 30 E en be an arbitrary point. Then the set
1 1
and --2 < r v <-2 v= 1, ... ,2n}
is open in en.
2n
For two points 3, 3' E F3o' 5 - 3' = I (rv - r~)cv with Irv - r~1 < 1 for
v= 1
V = 1, ... , 2n. Therefore 3 and 3' can only be equivalent if they are equal,
that is
is injective. Hence
is a complex coordinate system for the torus, and the set of all U30 covers
the entire torus.
Theorem 3.10 The n-dimensional complex torus Tn is a compact n-dimen-

sional complex manifold and the canonical projection TCT: en -+ yn is
holomorphic. [Since one can show that the complex structure on Til depends
on the vectors c1> ... , c2n' we also write: Til = T n (c 1 , · · · , C2n).J
PROOF
1. Any two complex charts for yn are holomorphically compatible.
is a topological mapping, where

2n
CfJ 31 0 CfJ3~ 1(3) = 3 + I kv(3)C v
v=l
and the functions kv are integer valued. Since {C1> ... , c2n } is a (real) basis
of en, the kv must be continuous, and therefore locally constant. But then
CfJ 31 CfJ3~ 1 is holomorphic.
0
138
2. Tn is a Hausdorff space: Let Xl = 1rT(31) i= 1rT(a2) = X2. Then we can

write:
2n 2n
al - 32 = L
v= 1
kvcv +
v
L
= 1
rvcv,
with kv E 7l.. and 0 ~ rv < 1 for v = 1, ... , 2n. Moreover not all rv can
vanish simultaneously. Suppose 1'1 i= 0 and let e > 0 be chosen so that
2e < 1'1 < 1 - 2e.
is open, and hence

u l(al): = u + al and U 2(32): = U + a2
are open neighborhoods. Suppose 1rT(Ud n 1rT(U 2) i= 0, so there are
points 3' E U b a" E U 2 with a' ~ a". But then we have
2n 2n
3' = 51 + L r~cv and 3" = 52 + L r~cv with Ir~1 < e and Ir~1 < 8,
v=l v=l
therefore
2n 2n 2n
5' - a" = (51 - a2) + L (r~ -
v=l
<)cv = L kvcv + L (rv + (r~ -
v=l v=l
r~))cv.
Since
1> 1'1 + 28 > 11'1 + (r'l - r'DI > 1'1 - 28 > 0,
1'1 + (1"1 - I'D cannot be an integer. That is a contradiction, so 1rT(U l ) and
1r T (U 2) are disjoint.
3. If 5 E en, then 3 is equivalent to a point
5' E F: = {5 v~/vcv: - ~ ~ rv ~
= u·
F is compact, 1rT is continuous, Tn is a Hausdorff space, and: 1rT(F) = Tn.
Hence it follows that Tn is compact.
4. 1rT:en ~ yn is holomorphic. If We Tn is open, if g is holomorphic in
Wand if 50 E V: = 1ryl(W), then go 7rTIV n Foo = g 0 CPo~llV n Foo IS
holomorphic. 0
H apf M anifalds
Let p > 1 be a real number, r H:= {l: k E 7l..}. r H is a subgroup of the mul-
tiplicative group of the positive real numbers. Two elements 31> 52 E en - {O}
are considered equivalent if there is a pk E r H with 32 = l31. The set H of
all equivalence classes will be given the finest topology in which the canonical
projection 1rH: en - {O} ~ H is continuous. We obtain complex coordinate
systems for H in the following manner.
l39
Let
Fr : = {oEen - {O}:r < 11011 < pr}
for arbitrary real numbers l' > O. Then U Fr = en - {O}, and we can
re lR+
show that
nHIFr:Fr ~ Ur : = n(F r } c H
is topological. (Un qJr) is therefore a complex chart. In a manner similar to
that of the preceding examples we can prove:
Theorem 3.11. H is a compact n-dimensional complex manifold (the so-called

Hopfmanifold), and nH:en - {O} ~ H is holomorphic.
If, for 31' 32 E en - {O}, nH(31} = nH(3z}, then there is a k E7L with 02 =
pk 01' But then G(02) = G(Ol)' Therefore there is a mapping h:H ~ [pn defined
by h(nH(O)): = G(o). We obtain the following commutative diagram.
en - {O} n ) [pn-1
~J H
Since nH is locally biholomorphic, it follows that h is hoI om orphic. D
Meromorphic Functions and Projective-Algebraic Manifolds

Let X be an arbitrary complex manifold.
Theorem 3.12. Let U c X be open, Xo E U. Let g, h be holomorphic functions

on U with g(xo) = h(xo) = O. If the germs gxo' hxo are relatively prime,
then for every complex number c there exists a point x arbitrarily close to
Xo with hex) =1= 0 and g(x)/h(x) = c.
PROOF. Without loss of generality we can assume that U is a polycylinder

in en and Xo = O. By the Weierstrass preparation theorem one can further
assume that gxo' hxo are elements of(D~o[ Zl]. If we denote the quotient field
of (D~o by Q~o' then it follows from Theorem 4.2 of Chapter III that gxo' hxo
are already relatively prime in Q~o[ Zl]. By Theorem 6.6 of Chapter III
there exists a greatest common divisor of gxo' hxo which can be written as
a linear combination of gxo' hxo with coefficients in (D~oC ZlJ, and that greatest
common divisor clearly must be a unit in Q~o [Z 1]. Thus there exists a neigh-
borhood YeO) c U and there are holomorphic functions gl, h1 on Vas well
as a nowhere vanishing function d independent of Z 1 such that on V
d = glg + h1h.
Now suppose that the theorem is false for c = O. Then there is a neighbor-
hood W(O} c V such that for every 0 E W, g(o) = 0 implies h(o) = O. Since
140
the zeroes of a polynomial depend continuously on the coefficients and

since the polynomial g(z b 0) has a zero at z1 = 0, for suitably small
°
3' E W n ({O} x en-I) there is always a ZI with (ZI' a') E Wand g(ZI' 3') = 0.
But then h(Zb a') = also and consequently d(a') = 0. Therefore d vanishes
identically near Xo = 0, which is a contradiction. The assertion is thus proved
for c = 0, and if we replace 9 by 9 - c· hand g/h by (g - c· h)/h, we obtain
the theorem for arbitrary c. 0
Def. 3.4. A meromorphic fimction on X is a pair (A, f) with the following

properties:
1. A is a subset of X.
2. f is a hoI om orphic function on X-A.
3. For every point Xo E A there is a neighborhood U(xo) c X and holo-
morphic functions g, h on U such that:
a. An U = {x E Ulh(x) = o}
b. The germs gxo' hxo E (9xo are relatively prime.
c. f(x) = g(x)/h(x) for every x E U - A.
Remark. If (A, f) is a merom orphic function on X it follows immediately
from the definition that A is either empty or a l-codimensional analytic set.
We call A the set of poles of the meromorphic function (A, f).
Theorem 3.13. Let Y c X be an open dense subset, f a holomorphic function

on Y. For every point Xo E X - Y let there be a neighborhood U(xo) c X
and holomorphic functions h, 9 on U such that gxo and hxo are relatively prime
and for every x E Y, g(x) = f(x) . h(x).
°
Finally, let A be the set of all points Xo E X - Y such that given a real
number r > and a neighborhood V(xo) c X, there is an x E V n Y with
If(x) I > r.
Then there exists a uniquely determined holomorphic extension] of f to
X - A such that (A, ]) is a meromorphic function.
PROOF. Let Xo E X - Y. By assumption there exists a neighborhood
U(xo) c X and holomorphic functions g, h on U which are relatively prime
at xo, such that g(x) = f(x) . h(x) for x E U n Y.
If h(x o) =F 0, then g/h is bounded in a neighborhood of Xo. Therefore Xo
°
does not lie in A.
If h(xo) = and g(xo) =F 0, then f = g/h assumes arbitrarily large values
near Xo. Furthermore, if h(xo) = g(x o) = 0, f is also not bounded near x o,
by Theorem 3.12. Thus Xo lies in A.
Hence An U = {x E U:h(x) = a}.
(g/h) is a continuation of f on U - A. We can carry out this construction
at every point of X - Y. Y is dense in X, so by the identity theorem the
local continuation is already uniquely determined by f, and so we obtain
a global holomorphic continuation] of f to X-A. It follows directly from
the construction that (A,]) is a merom orphic function. 0
141
Theorem 3.13 allows us to define the sum and product of merom orphic
functions:
If (A, f), (A',f') are meromorphic functions on X, then Y: = X -
(A u A') is open and dense in X, and at every point of A u A' we can write
f + I' and f . I' as the reduced fraction of two holomorphic functions.
There are analytic sets AI, A2 c X and merom orphic functions (Ar. fd,
(A 2,f2) on X with AI, A2 c A u A' andflly = f + f',f2l y = I-f'.
One sets
(A, f) + (A', 1'): = (AI,fI)
(A, f) . (A', 1'): = (A 2, f2)'
If X is connected, then the meromorphic functions on X form a field. We can
think of any holomorphic functionf on X as a meromorphic function (0, f).
EXAMPLES
1. Let X = C u {oo}, the Riemann sphere with the canonical coordinates
t/J I: X I
-l- C, t/J 2 : X 2 -l- C (see Theorem 3.9). Let p and q be two relatively
prime polynomials in C[z], and let N q : = {x E Xl :q(x) = o}. Then Y: =

X I - N q is a dense open subset of X and f(x): = p(x)lq(x) defines a holo-
morphic function f on y. Let
p.={{oo} if deg(q) < deg(p)

. 0 A:=NquP.
if deg(q) ?= deg(p),
We want to show that there is a holomorphic function J on X - A with
JI y = f, such that (A, J) is meromorphic. It suffices to show that there
exists a neighborhood U (00) c X and holomorphic functions g, h on U with
~ IU n y = flU n Y.
For then, since A = {x E X:f is bounded in no neighborhood of x}, the

existence of an J with the desired properties follows from Theorem 3.13.
Now let U: = {XEX 2 :p(X) #- 0, q(x) #- O}, set g: = llq and h: = lip
on U - {oo}, and g(oo) = h(oo): = O. Then g, h are continuous functions on
U. p: = go t/J21 and q: = h a t/J21 are continuous on t/J2(U) andholomorphic
except at the origin. By the Riemann extension theorem p, q are actually
holomorphic on the whole set t/J2(U), hence also g, h on U. Moreover, glh
and f concide on U n Y. This finishes the proof. 0
The meromorphic function (A, .1) is written in short as plq, since both
the values and the poles of f are uniquely determined by p and q.
2. Let X = C 2, A: = {(ZI, Z2) E i[:2:Z2 = O}. Thenf(zI' Z2): = Zr/Z2 is a

holomorphic function on X - A and (A, f) is a meromorphic function on X.
For 30 = (z\o>, 0) E A set 3n: = (z\O) + (lin), lln 2 ). Then the sequence of
points 3n (outside A) tends towards 30; the values f(3n) = z\O) . n2 + n are
unbounded (for arbitrary z\O»). At the point 30 = 0, the case of an "indeter-
142
minate point" (which cannot occur in the one-dimensional case) arises, as

numerator and denominator vanish simultaneously. The function assumes
every possible value in an arbitrary neighborhood of the indeterminate point.
3. Let 30 E en - {O} be a fixed vector, G

G(30). If 3 E G, then pk 3 E G
=
also. Hence one can also divide G by r H. T = GjrH is a I-dimensional
complex torus and at the same time a submanifold of H. Iff is a meromorphic
function on H, then f 7tH is meromorphic on en - {O}. For n ~ 2 there
0
is a continuity theorem for merom orphic functions, which in the present case
says that f 7tH can be continued to a meromorphic function J on en.
0
Naturally JIG is then also meromorphic. If 30 were a pole of JIG, then all the
points p k 30 E G would also be poles of JIG, and these points cluster about the
origin. Since this cannot be, we must either have JIG == 00 or JIG holo-
morphic. If JIG is holomorphic, then fl T is also holomorphic and therefore
constant (since T is compact). The submanifolds T c H are precisely the
fibers of the holomorphic mapping h: H -+ [lJ'n -1. Hence we can show that
there exists a meromorphic function g on [lJ'1! - 1 with g h = f. In other words,
0
on the n-dimensional manifold H there are no "more" merom orphic functions

than on the (n - 1)-dimensional manifold [lJ'n - l.
Def. 3.5. An n-dimensional compact complex manifold is called projective-

algebraic if there exists an N E N and an analytic subset A c [lJ'N which
is free of singularities and of codimension N - n such that X ~ A.
By a theorem of Chow every projective-algebraic manifold is already

"algebraic" in the sense that it can be described by polynomial equations.
Furthermore:
Theorem 3.14. Let X be a projective-algebraic manifold. Then for arbitrary

points xl> X2 E X with Xl =P X2 there is always a meromorphicfunctionf on
X which is holomorphic at Xl and X2 with f(xd =P f(x 2 ).
This means that there are "many" merom orphic functions on projective-
algebraic manifolds. The Hopf manifold is not projective-algebraic. One can
also interpret this topologically.
For a topological space X let Hi(X, IR) be the i-th homology group of X
with coefficients in IR. If X is a 2n-dimensional compact real manifold, then
for i = 0, ... , 2n
for i> 2n
We call Bi(X) the ith Betti number and associate with X the Betti polynomial
211
P(X): = L B;(XW· For cartesian products there is the formula
i=O
P(Xx Y) = P(X)· P(Y).
143
Theorem 3.15. If X is a projective-algebraic manifold, then the Betti numbers

satisfy
B 2i + l(X) E 271.,
B 2i (X) =F O.
This theorem is proved within the framework of the theory of "Kahler
manifolds." It constitutes a necessary condition which is not fulfilled for a
Hopf manifold. We can easily convince ourselves that H is homeomorphic
to s2n-l X Sl. But for spheres S\ p(Sk) = 1 + tk. Hence it follows that
P(H) = p(s2n-l). P(Sl) = (1 + t 2n - 1). (1 + t) = 1 +t+ t 2n - 1 + t2n.
For n ?:: 2 therefore Bo(H) = 1, B1(H) = 1, and B 2(H) = O. On the other
hand, the n-dimensional complex torus satisfies this necessary condition for
projective-algebraic manifolds:
Tn ~ Sl X ..• X Sl (topological)
'----v----'
21l-times
therefore
p(r) = (1 + t)2n = i~O Cin) {

Hence Bi(rn) = (2in} and the condition is satisfied. Nevertheless not every
torus is projective-algebraic. This property depends very critically on the
vectors Cb . . . , C2n which define the torus. It can be shown that the so-called
"period relations" (which involve only the vectors Cb . . . , C2n) furnish a
sufficient condition.
4. Closures of en
Def. 4.1. Let X and Y be connected n-dimensional complex manifolds. If
Y is compact and X c Y is open, then we call Y a closure of X.
EXAMPLE. The coordinate neighborhood U 1 C [pn is isomorphic to en.

Hence [pn is a closure of en. (We use the notation of Section 3, above.)
A holomorphic function f is defined on U 1 = en by

p
f(n(l, Z2, . . . ,Zn+ 1)): == I aV2 ••• vn + IZ22 ... Z~+\l
Ivl =0
(with QV2 . . . Vn+ 1 E e and Ivl: = V2 + ... + vn+d. If
x = n(l, Z2, ... , Zn+ d= n(wb ... , 1, ... , W n + dE U 1 n U;,
then
144
4. Closures of IC"
Hence there is a meromorphic function /; with /; IU i nUl = fl U i nUl

given by
1 with II U i: = /; is then a merom orphic function on [fl'/J with llC" = f·
Def.4.2. Let Y be a closure of C". Y is called a regular closure ofC" if every

polynomial defined on en extends to a me rom orphic function on Y.
Clearly [fl'/J is a regular closure of C".
Theorem 4.1. If Y is a regular closure ~fC", then Y - C" is an analytic set of

codimension 1.
PROOF. Let Z1> ••• , Zn be the coordinates of en. By hypothesis they can be
continued to meromorphic functions f1' ... ,in on Y.
The set Pi of poles of /; is an analytic set of co dimension 1, and so is
n
P: = U Pi' Hence it suffices to show that Y - en = P.
i= 1
Let 30 E (lc" c Y. Then there is a sequence (3J in C" with lim 3i = 30'
i-+ 00
This means that (z~)) is unbounded for at least one k E {l, ... , n}. We can
find a subsequence (z~v;)) with lim Iz~vi)1 = 00. Hence, for i --> 00, .h(3v') tends
i-+ 00
to infinity, so 30 is a pole of k Thus 30 lies in P, and since 30 E (lc" was chosen
arbitrarily, (lc" c P.
An analytic set of co dimension 1 cannot separate a manifold; that is,
°
Y - P is connected. Hence for every point 30 E (Y - C") - P there is a path
qJ: [0, 1] --> Y - P with qJ(O) = and qJ(l) = 30' Since such a path always
meets the boundary (lc", we must have (Y - C") - P = 0. 0
Remark. For n ): 2 Bieberbach has constructed an injective holomorphic

mapping [3: C" --> C" whose functional determinant equals 1 everywhere and
whose image U: = [3(C") has the property that there exist interior points in
en - U. We can regard U as an open subset of [fl'n. Then [fl'/J is a closure of
C" ::::: U, but this closure is not regular since C" - U contains interior points.
A 1-codimensional analytic set cannot have interior points (Theorem 2.6)!
As a further example we consider the Osgood closure of C".

Let
n-times
145
Each factor !pI is isomorphic to the Riemann sphere, which has the canonical
coordinates 1/11,1/12. We obtain coordinates on en by letting U = Vj . • • Vn :
U V1 x··· x UVn and
(with VA E {I, 2}). en is compact, and I[:" ~ U 1 ... 1 C en is an open subset.

Therefore en is a closure of en and we can see directly that this closure is
regular.
If Y is a closure of 1[:", we call the elements of Y - I[:" infinitely distant

points. In special cases one can describe the infinitely distant points more
exactly.
1. Let Y = !p n be the usual projective closure of 1[:". Then
I[:" ~ U 1 = n({(1,z2, ... ,Zn+l)EI[:"+I}) = n({(ZI, ... ,Zn+l)EI[:"+I: ZI =1= O}),
Therefore
!pn - I[:" = n({(O, Z2, ... , Z,,+l) E 1[:"+1 - {O} n,
and this set is isomorphic to !p 1 .n-
2. Let Y = en be the Osgood closure of 1[:". Then I[:" ~ U 1 ... 1

U 1 X ... X U 1 and
en - I[:" = {(Xl> ... , xn) E!P 1 X ... X !pI there is an i with Xi ¢ Ud

= {(x 1, ... ,X n)E!pl X ... X !pI: there is an iwithxi = oo}
= ({oo} X !pI X ... X !PI) u··· U (!PI X ... X !pI X roo}).
In the first case the set of infinitely distant points is free of singularities of
codimension 1 ; and in the second case it is the finite union of analytic subsets
of codimension 1 which are free of singularities and each of which is iso-
morphic to 1[:"-1. For n = 1, e = !pl. One can prove that only this closure
exists in the I-dimensional case. For n = 2, e 2 - e 2 = ({ oo} X !PI) U
(!PI X {oo}) with ({oo} x !PI) n (!PI x {oo}) = {(oo, oo)}. The analytic set
e2 - e2 has a singularity at the point (00, 00) as can easily be demonstrated.
Def.4.3. Let X and Y be connected n-dimensional complex manifolds, let
M c X and N c Y be closed proper subsets, and let n: X - M --"* Y - N
be a biholomorphic mapping. Then (X, M, n, N, Y) is called a modification.
For example (!P n, !pn-l, iden, e" - en, en) is a modification. We can
therefore use modifications to describe transformations between distinct
closures of cn.
Def. 4.4. Let cp: X --"* Y be a holomorphic mapping between connected

complex manifolds, dim X = n and dim Y = m. Then
E(cp): = {x E X:dimAcp-l(cp(x))) > n - m}
is called the set ofdegeneracy of cpo
146
4. Closures of C"
If dim X = dim Y, then as can be shown, E(tp) = {x E X:x is not an

isolated point of tp-l(tp(X))}.
Both the following theorems were proved by Remmert:
Theorem 4.2. If tp: X --+ Y is a holomorphic mapping between connected com-

plex manifolds, then E(tp) is an analytic subset of x.
Theorem 4.3 (Projection theorem). If tp: X --+ Y is a proper holomorphic map-

ping between complex manifolds and M e X is an analytic subset, then
tp(M) e Y is also analytic.
Def. 4.5. A modification (X, M, n, N, Y) is called proper if n can be con-

tinued to a proper holomorphic mapping ft:X --+ Y such that M = E(ft).
Theorem 4.4. Let (X, M, n, N, Y) be a proper modification, ft:X --+ Y a con-

tinuation of n in the sense of Def 4.5. Then M and N are analytic sets, and
ft(M) = N.
PROOF. By Theorem 4.2, M = E(ft) is analytic, and by Theorem 4.3, N*: =

ft(M) is analytic. It remains to show that N = N*:
1. Suppose there is a Yo E N* - N. We set xo: = n-1(yo) E X - M and
choose an Xo E M with ft(xo) = Yo. Then we can find open neighborhoods
U(xo), V(xo) and W(Yo) such that:
a. Un V=0
b. We Y-N
c. n(U) = W
d. ft(V) e W.
But from this it follows that V - M e X - M is qpen and non-empty, and
n(V - M) = ft(V - M) lies in W. Therefore
V - M = n-1n(V - M) c n-1(W) = U.
That is a contradiction; and so N* e N.
2. Y - N is open and non-empty, so for every point Yo E 8(Y - N) there
is a sequence (Yi) in Y - N with lim Yi = Yo. The set K: = {Yo, Yl' Yz, ... }
i-+co
is compact, and since ft is proper, K*: = ft-l(K) is also compact. In par-
ticular, K* contains the uniquely determined points Xi E X - M with n(xi) =
Yi. We can find a subsequence (xv,) of (Xi) which converges to a point Xo E K*.
Since ft is continuous, we must have ft(xo) = Yo; therefore Xo E M and
Yo E ft(M) = N*. Hence we have shown that 8(Y - N) lies in N*.
Suppose there is a point Yo E N - N*. Then since N* is analytic, we can
connect Yo with a point Yo E Y - N by a path running entirely in Y - N*.
Each such path, however, intersects 8(Y - N), and hence N*. That is a
contradiction; so N e N*, and thus N = N*. 0
147
The most important special case is the Hopf (I-process:
Theorem 4.5. Let G c C" be a domain with 0 E G, n:C" - {O} -+ IFD n- 1 the
natural projection. Then X: = {(3, x) E (G - {O}) x IFD n- 1:x = n(3)} u
({O} x IFD n- 1) is an analytic set of codimension n - 1 in G x IFD n- 1 which is
free of singularities, therefore an n-dimensional complex manifold.
PROOF. Let CPi: U i -+ Cn- 1 be the canonical coordinate system of IFDn-l. If
3= (ZI,"" zn) E G - {O} and x = n(3) E U 1 , then
X = n( 1,-,"
Z2 . ,Zn)
- .
ZI ZI
Therefore w;.(x) = ZH dZI for A. = 1, ... , n - 1, where we denote the

coordinates on U 1 by W;.. Hence it follows that
Xn(GxU 1 )
= {(3,X)EG x U 1:Z 1 #Q,w;'(X)=Z~:1 for A.= 1, ... ,n-l}U({O} x U 1)

= {(ZI"",Zn;X)EG X U 1 :Z 1 'W 1(X)-Z2
= ... = ZI . wn - 1(x) - Zn = O}.
'there is an analogous representation for U 2 , ••• , Un. Therefore X is an
analytic set in G x IFDn - 1.
Since clearly rk(3.X)(ZI • WI - Z2,'"'' ZI . W n - 1 - Zn) = n - 1 on all of
U 1 and an analogous statement can be made for U 2 , •• • , U m X is free of
singularities of codimension n - 1. D
Theorem 4.6. Let X c G x IFD n- 1 be the analytic set described in Theorem

4.5, cp:X -+ G the holomorphic mapping induced by the product projection
pr 1:G x IFD n- 1 -+ G, ljJ: = cpl(X - ({O} x IFD n- 1)). Then (X, {O} x IFD n-1,
ljJ, {O}, G) is a proper modification. It is called the "(I-process."
PROOF
1. ljJ':G - {O} -+ X - ({O} x IFD n- 1) with ljJ'(3): = (3, n(3)) is clearly
holomorphic, and:
ljJ 0 ljJ'(3) = pr 1(3, n(3)) = 3,
ljJ' 0 ljJ(3, x) = ljJ' ljJ(3, n(3)) = ljJ'(3) = (3, n(3) ).
0
Therefore ljJ' = ljJ -1, and ljJ: X - ({ O} x IFDn - 1) -+ G - {O} is biholomorphic.

2. cP is a holomorphic continuation of ljJ, and cP -1(3) = {(3, x) E G x
IFDn-I:(3,X)EX} = {(3,n(3))}foqEG- {O},cp-l(O) = {O} x IFD n- 1.There-
fore E(cp) = {O} x IFD n- 1 ::= IFD n- l .
3. If KeG is compact, then K x IFD n- 1 is compact, and so is cp-l(K) =
(K x IFD n- 1) n X. Therefore cp is proper. D
148
4. Closures of C"
To be identified
'------_....
Figure 24. The Hopf a-process.
Remark. Clearly we can regard [pn-l as the set of all directions in en.
By the O"-process these directions are separated in the following sense:
If one approaches the origin in G - {o} from the direction Xo E [pn - 1,
say along a path w, then one approaches the point (0, xo) along the directly
lifted path l/J - loW in X _ [pn - 1.
It can be shown that the O"-process is invariant under biholomorphic
mappings. Hence it can also be performed on complex manifolds.
149
CHAPTER VI
Cohomology Theory
1. Flabby Cohomology
In this chapter we apply, with the help of cohomology groups, the methods
and results of sheaf theory to complex manifolds.
X will always be an n-dimensional complex manifold and R a commutative
ring with 1. If :/ is a sheaf of R-modules over X and U c X is open, then
we let t(U, :/) denote the set of all functions s: U -+ : / with nos = id u
(where n::/ -+ X is the sheaf projection), and we call these not necessarily
continuous functions generalized sections. Clearly r( U, :/) is an R -submodule
of t(U, :/).
If <p::/ 1 -+ : / 2 is a homomorphism of R-module sheaves, then
<P. :t(U, :/ 1) -+ t(u,:/ 2) with (P.(s): = <P.) is an exact covariant functor from
the category of R-module sheaves over X to the category of R-modules.
Therefore:
1. if:/ is an R-module sheaf, then t(U, :/) is an R-module;

2. if <p::/ 1 -+ : / 2 is a homomorphism of R -module sheaves, then <P.:
t(U,:/d -+ t(u, :/2) is a homomorphism of R-modules;
3. a. (id y '), = idt(U. y);
b. (<p 0 l/tt = <P. 0 l/t.;
4. if :/ 1 .:!!.. :/ 2 .!. :/ 3 is an exact sequence of R-module sheaves, then
oj! <p
r(U, :/ 1) ---* r(U, :/2) -+ r(U, :/ 3) is an exact sequence of R-modules.
~ ~ ~
The proof is completely trivial.
150
For U c X open, we set Mu: = t(U, Sf'); if U, V c X are open with

V c U, then we define r~:Mu -> Mv by r~(s): = slY. Then {Mu, rn IS
a pre-sheaf and we denote the corresponding sheaf by W(Sf').
Theorem 1.2.
1. The canonical mapping r:M u -> r(U, W(Sf')) is an R-module
isomorphism.
2. The canonical injection iu: r( U, Sf') c.. t( u, Sf') induces an injective
sheaf homomorphism c:S -> W(Sf') with c.Ir(U, Sf') = r iu. 0
PROOF. (1) is proved exactly as is Theorem 2.3 in Chapter IV. To prove (2):
Clearly iu(s)1V = iv(slV) for s E r(U, Sf'). If we identify the sheaf induced
by {r(U, Sf'), r~} with the sheaf Sf', then it follows from Theorem 2.1 of
Chapter IV that there exists exactly one sheaf morphism c:Sf' -> W(Sf')
with c.(s) = riu(s) for s E r(U, Sf'). If 0" E Sf'x and 13(0") = Ox, then there
exists a neighborhood U(x) c X and an s E r(U, Sf') with sex) = 0". Therefore
Ox = 13(0") = 13 0 sex) = c*(s)(x) = riu(s)(x), with riu(s) E r(u, W(Sf')). Then
IV
there exists a neighborhood Vex) c U with riu(S) = 0; therefore iu(s)1V =
o by (1), and then clearly slV = O. Hence 0" = sex) = Ox. D
Let <p: Sf'1 -> Sf' 2 be a sheaf homomorphism. Then for open sets U, V c X
with U c V and s E t(U, Sf'1) we have <p.(s)1V = <p.(slV). By Theorem 2.1
of Chapter IV 
W(Sf' 2) with (W<p)*(rs) = r(<p*(s)).
Let s E r(U, Sf'd. If c).:Sf'). c.. W(Sf';J are the canonical injections (for
A. = 1, 2), then
(W<p) 0 1310 S = (W<p).(ri~)(s)) = r(<p*(i~)(s))) = r(i~)(<p*s)) = 1320 r(U, Sf') is surjective.
Theorem 1.3. If Sf' is a sheaf of R-modules over X, then W(Sf') is aflabby sheaf.
PROOF. We can identify r(U, W(Sf')) with t(U, Sf'). If s E t(U, Sf') then we
define s· E t(X, Sf') by
s*(x): = {sex) for x E U;

o for x E X - U.
Clearly r~s* = s. D
Theorem 1.4. (W:Sf' N'> W(Sf'), 9", <P: 9" ---> 9"2 be sheafhomomorphisms and s E r(V, 9" 1)·
Then Weep ljI) rs = r((
•"
9" --->
(P
9"2 be exact.
a. Then W<p 0 WljI = W(<p 0 ljI) = W(O) = O.
b. Let (J E W(9")x and W <p((J) = Ox. Then there exists a neighborhood
Vex) c XandansEF(V,9")withrs(x) = (J,thereforeW<p°rs(x) =
Ox·
Hence there exists a neighborhood Vex) c V with 0 = W<p 0 rslY =
r(<p 0 s)lY; therefore (<p 0 s)jV = O. We can construct an SI E F(V, 9" 1) with
ljI 0 SI = slY pointwise. Then WljI 0 rS I = r(ljI 0 SI) = rslY, and therefore
WljI(rsl (x)) = (J. 0
Def. 1.2. Let 9" be a sheaf of R-modules. A resolution of 9" is an exact

sequence of sheaves of R-modules:
o ---> 9" ---> 9" 0 ---> 9" 1 ---> 9" 2 ---> •. . .
If the sheaves 9"0,9" 1,9" 2, ... are all flabby, then we speak of a flabby
resolution.
We now show how to assign a canonical flabby resolution to any sheaf 9".
1. The sequence 0 ---> 9" .:. W(9") is exact. Let Wo(9"): = W(9").
2. Suppose we have constructed an exact sequence 0 ---> 9" ---> Wo(9") ~ ... ~
~(9"), with flabby sheaves Wo(9"), Wl (9"), ... , ~(9").
Then there is an exact sequence
J;Y((9") .'i. ~(9")jlm(d(_ 1) !.. W(Tt;(,'I')jlm(dr - 1))

Let ~+1(9"): = W(~(9")jlm(d{_I))'
el t : = j 0 q.
Clearly Ker(d{) = Ker(q) = Im(d t _ d; that is, the extended sequence
o ---> 9" --->
Wo(9") ---> ••• ---> ~(9") ---> ~ + 1(9") remains exact. Thus we con-
struct an exact sequence Wo(9") ---> WI (9") ---> W2 (9") ---> ••• by induction. We
write W(9") as an abbreviation. The exact sequence 0 ---> 9" .:. W(9") is
called the canonical flabby resolution of 9".
Theorem 1.5. Let <p: 9" 1 ---> 9"2 be a homomorphism of sheaves of R -modules
over X. Then there are canonical homomorphisms Tt;<p: W;(9" 1) ---> W;(9" 2)
with (W;+I<P) 0 di = di 0 (W;<p) for i E No and (Wo<p) 0 /; = /; 0 <po
PROOF. We proceed by induction. Let Wo<P: = W<p. If Wo<P, Wl <p, ... , ~ ~(S" z)!lm(dt - 1) with
(If ql(a) = 0, then there exists a cr* with dt-1(cr*) = cr; therefore Weq>(cr) =
dt - 1 ° (We-lq»(cr*), so qz 0 Weq>(cr) = 0.) We define We+lq>: = WIj;. All
diagrams remain commutative. 0
The system of homomorphisms Vli;q> is denoted by W(q». We can regard

W(q»:W(S"l) -> W(S"z) as a "homomorphism between flabby resolutions."
Clearly W(id'l') = id 'lll(<I') , W(lp ° q» = W(Ij;) 0 W(q».
Therefore (W: S" Iv-> W( q») is a covariant functor. We need the next two
lemmas in order to show that W is also an exact functor.
Lemma 1. Let the following diagram of sheaves of R -modules be commutative,

have exact rows and columns, and moreover let the mapping q>o be surjective:
If cr E S" 6 and ljJ 6 0 q>3 (cr) = 0, then there exists a fi E S" 3 with
q>3(cr - ljJ3(fi)) = o.
PROOF. Letcrl: = q>3(cr)ES"7.
1. Because 1j;6(crl) = 0 there exists a crz E S"4 with ljJ4(crZ) = crl.
2. Ij;s(q>z(crz)) = q>4(1j;4(crZ)) = q>4(q>3(cr)) = 0; therefore there exists a
cr3 E S" 2 with Ij;Z(cr3) = q>z(crz), and there is a cr4 E S" 1 with q>0(cr4) = cr3.
3. q>2 0 Ij;l(cr4) = Ij;z 0 q>0(cr4) = q>Z(cr2); therefore q>2(crZ - Ij;l(cr4)) = o.
Hence there is a cr s E S" 3 with q> 1(cr s) = cr 2 - Ij; 1(cr 4).
4. Letfi: = cr s.
Then
q>3(cr -1j;3(fi)) = q>3(cr) -1j;4°q>l(crS) = q>3(cr) -1j;4(crZ) = q>3(cr) - crt = O. 0
153
Lemma 2. In the sequence
let qJl be surjective, qJ4 injective and Ker qJ3 = 1m qJz.

Then
is exact.
o
Theorem 1.6. \.ill is an exact Junctor.
PROOF
1. Let 0 ~ [I" ~ /7 ~ /7" -> 0 be exact. We show by induction that
o~ Jt;(/7') -> Jt;(/7) -> Jt;(/7") ~ 0
is exact. For C = 0 this has already been proved in Theorem 1.4. Therefore
let C ): 1. We consider the case C = 1; the general case is handled entirely
analogously.
The following diagram is commutative:
(with f2: = Wo(/7)j/7, f2' and f2" similarly). All columns and the three top
rows are exact.
a. Since qJ'2 and ljJ'2 are surjective qJ'i, is also surjective.
b. Since ljJ~ is surjective and qJ'2 qJ~ = 0, also qJ'i, qJ':, = O.
0 0
c. Let (J E f2 with qJ'i,((J) = o. Then there exists a (J* E Wo(/7) with ljJz((J*) =
(J; therefore ljJ'2 0 qJ'2((J*) = O.
By Lemma 1 there is a BE/7 with qJ'2((J* -ljJl(B)) = O. Therefore there
exists a (J' E Wo(/7') with qJ~((J') = (J* - ljJl(B). It follows that ljJ~((J') E f2' and
qJ':, 0 ljJ~((J') = ljJz((J* - ljJ 1 (B)) = (J.
d. Let (J' E f2' with qJ':,((J') = O. Then there is a (J* E Wo(/7') with ljJ~((J*) = (J'.
Hence ljJ2 qJ~((J*)= O.
0
154
By Lemma 1 there is a 8 E Y" with rp'z((J* - ljJ'1(8)) = O. Since rp'z is in-

jective it follows that (J* - ljJ'l (8) = 0, hence
o = ljJ'z((J* - ljJ'1(8)) = ljJ'z((J*) = (J'.
Thus, the last row of the diagram is exact, and by Theorem 1.4 we now have
the exactness of the sequence 0 ~ W1(Y") ~ W1(Y') ~ W1(Y''') ~ o.
2. Now let Y" ~ Y' ~ Y''' be exact.
Then we obtain the following exact sequences:
o ~ Ker rp ~ Y" ~ fiL' ~ 0 (with fiL': = Y"jKer rp)
o ~ fiL' ~ Y' ~ 1m ljJ ~ 0
o ~ 1m ljJ ~ Y''' ~ fiL" ~ 0 (with fiL": = Y'''lIm ljJ).
Applying (1), we obtain an exact sequence of the form
rt;(Y") -+> rt;( fiL') ~ rt;(Y') -+> rt;(Im ljJ) ~ rt;(Y'''),
where the first mapping is surjective, the last mapping is injective and the
sequence in the center is exact. By Lemma 2 it follows that vr;(Y") ~ vr;(Y') ~
vr;(Y''') is exact. But that means that lID is exact. D
Def. 1.3. A co chain complex over R is a sequence of R-module homo-

morphisms
withdiod i - 1 = OforiE N.
zn(Me): = Ker dn is called the n-th group of the cocycles,
Bn(Me): = 1m dn- 1 is called the n-th group of the co boundaries.
We set BO(Me ): = O. Then clearly Bn(Me) c zn(Mo), and Hn(Me): =

zn(Me)jBn(Me) is called the n-th cohomology group of the complex Me.
Remark. Clearly M· is exact at (the location) n > 0 if and only if Hn(Me) =
O. In this sense, one says that the cohomology groups measure the deviation
of the complex Me from exactness.
DeC. 1.4. An augmented cochain complex is a triple (E, e, Me) with the
1. E is an R-module.
2. M· is a cochain complex.
3. e:E ~ MO is an R-module monomorphism with 1m e = Ker dO.
Remark. If (E, e, Me) is an augmented complex, then

E ::::: 1m e = Ker dO = Z°(1\1 9 ) ::::: HO(MO).
If H((Me) = 0 for f, ;;::: 1, we call the complex acyclic.
155
Theorem 1.7. (T:fJ' ~ r(X, fJ'), ({J ~ ((J*) is a left-exact functor; that is, if
o ---> fJ" .!!.,. fJ' ~ fJ'" ---> 0
is exact, then so is
o ---> r(X, fJ") !; r(X, fJ') ~ r(X, fJ'")

exact.
PROOF. Since f is exact (see Theorem 1.1), it is clear that ({J* is injective and
tjJ * ({J* = O. Now let s E r(X, fJ') with 0 = tjJ *(s) = tjJ s. Then there exists
0 0
a generalized section s' E reX, fJ") with ((J.(s') = s. We must still show that
s' is continuous. For every point x E X there is a neighborhood U(x) and an
s* E r(U, fJ") with ({J s*)(x) = sex). Therefore there is a neighborhood
0
Vex) c U with ({J s*1V = sl V.

0
Since ({J is injective, it follows from ({J 0 s*1V = ({J 0 s'lV that s* = s'lV, IV
so that s' is continuous at x. D
Theorem 1.8. Let fJ' be a sheaf of R-modules over X,

W8(fJ'):r(X, Wo(fJ')) ---> r(X, ~(fJ')) ---> r(X, W2 (fJ')) ---> ••••
Then (r(X, fJ'), s*, W·(fJ')) is an augmented co chain complex.
PROOF. Clearly W·(fJ') is a complex, s* :r(X, fJ') ---> r(X, Wo(fJ')) an R-

module monomorphism, and (d o)* 0 s* = o.
Consider the mapping
do:Wo(fJ') --t> Wo(fJ')/Im(s) ~ W(WoCfJ')jIm(s)) = Wl(fJ').
Let s E r(X, Wo(fJ')) and 0 = do 0 s = j 0 q 0 s. Then q 0 s = 0; therefore

sex) E Im(s) for every x E X. Since Im(s) : : : : fJ', T(X, Im(s)) : : : : r(X, fJ'); so
there is an s* E T(X, fJ') with s*(s*) = s. D
Def. 1.5. Let fJ' be a sheaf of R-modules over X. Then we define
We call
Hf(X, fJ'): = Zf(X, fJ')/W(X, fJ') = Ht(W8(fJ'))
the i-th cohomology group of X with values in fJ'.
Remark. Clearly HO(X, fJ') : : : : r(X, fJ').
Theorem 1.9. If 0 ---> fJ" .!!.,. fJ' ~ fJ'" ---> 0 is an exact sequence of sheaves
of R-modules, and if fJ" is ajiabby sheaf, then
o ---> T(X, fJ") ~ r(X, fJ') 'A r(X, fJ'") ---> 0

is exact.
156
I. Flabby Cohomology
PROOF. We need only show that Ij;* is surjective. Let s" E T(X, g") be given.
1. If Xl> X z are points in X, then there are neighborhoods U(x 1 ), V(xz) c X
and sections sET( U, g), s* E T( V, g) with Ij; 0 s = s" IU and Ij; 0 s* = s" W
If U (\ V = 0, then this defines a section over U u V, whose image is
s"IU u v.
Suppose U (\ V =1= 0. The sequence
o -+ T(U (\ V, g') -+ T(U (\ V, g) -+ T(U (\ V, g")
is exact, and since Ij; 0 (s - s*) IU (\ V = 0, there is an s' E T( U (\ V, .'I")

with (P s' = (s - s*)iU (\ V.
0
Since g' is flabby, we can extend s' to an element sET( V, g'). Let
SeX) for XEU
Sl(X): = {( ~ *)()
qJ0s+s x for X E V.
Then s1 lies in T( U u V, g) and Ij; s 1 = s" IU u V. In this case there is

0
also a section over U u V whose image is s" IU u V.

2. We consider the system Wl of all pairs CO, S) with the following
properties:
a. V c X is open with U c V
b. SE T(V, S) with slU = sand Ij; S = s"IO. 0
In Wl we consider all subsystems (V" S;),El with the following properties:

For (ll> lz) E I x I either V" c V'2 and s'210'! = S'2' or V, c V, and
S"IV'2 = S'2' For each such system the pair (V, S) with V: ~ V: and U
,EI
~ V,: = S, is again an element of Wl. Zorn's lemma 1 implies that there exists
a "maximal element" (U 0, so) in Wl. By (1) U 0 cannot be a proper subset of X.
This completes the proof. 0
As a consequence we have:
Theorem 1.10. Let g be afiabby sheaf of R-modules over X and 0 -+ g -+

go -+ g 1 -+ ... a flabby resolution of g. Then the sequence
is exact.
1 Let X be a non-empty set with a relation,,; such that:

I. x ,,; x for all x E X.
2. if x ,,; y and y ,,; Z, then x ,,; z for all x, y, Z E X.
3. if x ,,; y and y ,,; x, then x = y for all x, y E X.
A chain in X is a set K c X with the property that for any two elements x, y E K either
x ,,; y or y ,,; x. Zorn's lemma says that if there exists an upper bound for every chain K c X
(an element SEX with x ,,; s for all x E K), then there exists a maximal element in X (an element
Xo E X such that for x E X it always follows from Xo ,,; x that x = xol.
157
PROOF. Let B8).: = Im(cp).:9',l.-l -> 9'.d for A = 0,1,2, ... and 9' -1: = 9'.
1. We show by induction that all fJ9,l. are flabby: For B8 0 ~ 9' this is true
by assumption; suppose we have proved that fJ9 o, B8 b . . . , fJ9 t - 1 are flabby
sheaves.
For V c X open, the exactness of the sequence -> T( V, .%'( - tl ->
T( V, 9'( _ 1) -> T( V, B8() -> °
°
follows from the exactness of the sequence
0-> .?4t - 1 c.. 9't -1 -> fJ9( -> 0 by Theorem 1.9. Let s E T(V, .%'().
Then there is an s' E T(V, 9't-1) with CPt s' = s. Since 9'(-1 is flabby
0
there is an s* E T(X, 9'( -1) with s*jU = s'. But now CPt s* E T(X, .%'() and
0
CPt 0 s* IV = s. Therefore, fJ9t is flabby.

2. The following sequences are exact.
o -> fJ9t - 1 -> 9'( -1 -> .%'t -> 0
o -> fJ9 ( -> 9't -> fJ9e+ 1 -> 0
0-> .%'(+1 -> 9'(+1 -> fJ9(+2 -> O.
By Theorem 1.9 the associated sequences of the modules of sections are exact.
We can combine these into a sequence which satisfies the conditions of
Lemma 2:
T(X, 9'(-1) ......, T(X, fJ9{) -> T(X, 9'r) -> T(X, B8(+ 1) c.. rex, 9'( + 1)'
Then the sequence T(X, 9'1-1) -> T(X, 9'() -> T(X, 9't + 1) is exact, as was
to be shown. 0
Thus we have obtained
°
Theorem 1.11. If 9' is a flabby sheaf over X, then the complex W·(9') is
acyclic; therefore Ht(X, 9') = for t :;: : 1.
EXAMPLE. Let ~(A) be the ideal sheaf of the analytic set A = {Of E en.
Then Jff(A) = @/§(A) is a coherent analytic sheaf over en, in particular, a
sheaf of C-modules. Clearly .Yt'(A) is flabby, and
HO(en, Jff(A)) ~ C, H1(en, Jff(A)) = ° for t :;:::: 1.
2. The Cech Cohomology

Let X be a complex manifold, R a commutative ring with 1, and 9' a
sheaf of R-modules. Moreover, let U = (V.),El be an open covering of X,
with V, =f 0 for every 1 E I. We define
V'O'" 'I: = V'O (I ... (I V'I'

1(: = {(lo, ... ,I():V,o··"1 =f 0}.
Let 6" be the set of permutations of the set {a, 1,2, ... , n - 1}. For 0" E 6", let
+1 if 0" is the product of an even number of transpositions
sgn(O"): = { -1
otherwise.
158
2. The tech Cohomology
Def.2.1. An {-dimensional (alternating) cochain over H with values in [I' is

a mapping
(la, ... ,Iil
with the following properties:

1. ((10, ... , I() E r(U'a''''[' [1').
2. ((1,,(0), ... , I,,(t)) = sgn(O')((lo, ... , It) for
The set of all (-dimensional alternating co chains over H with values in
is denoted by Ct(H, [1'). Ct(H, [1') becomes an R-module by setting
[I'
((1 + (2)(1 0, ... , It): = (1(1 0, ... , I{) + ~2(IO" .. , It)

and
(r' ~)(Io, ... , It): = r' ~(IO" .. , I().
Theorem 2.1. ot:C'(H, [1') -+ C H1 (U, [1') with

t+ 1
W~)(lo,···, It): = L (_I)Hl(~(lo,···, I)., ... , 1(+l)!U,a''''[+I)
;.=0
is an R-module homomorphism with Of 0 0(-1 = O.

PROOF
1. First we show that o(~ is alternating. It suffices to consider transposi-
tions.
(O(~)(lo,···, lv, Iv+1>' .. , 1(+ d
L (_l)Hl~(lo,···,I)., ... ,lt+l) + (_l)V+l~(lo,···,I" ... ,I(+l)
Atv,v+l
+ (-I)'+2~(lo,···, I v+1>"" It+d
L (-1))'+ 1~(lo, ... , lA' ... , Iv+ l' I" ... , It+ 1)
;.tv,v+l
+ (-I)'+1~(lo,· .. , IV+1> Iv,' .. ,1(+1)
+ (_1)v+2. ~(IO"'" (,+1' IV' ... , Ie+d
= - 0 ~(Io, ... , IV+1> 1" . . . ,1(+1)
2. It is clear that Of is a homomorphism. Moreover,
(0[+1 0 O(~)(lo, ... , 1e+2)
(+2
= L (_I)H1(Ot~)(lo, ... ,IA, ... ,Zt+2)
=
),=0
(+2
L (_I)H1
[).-1L (_I)'I+l~(lo, ... ,I~, ... ,IA, .. ·,1r+2)
),=0 ~=o
+
q
tf (-I)~~(lo, ... , I A, . .. , I~, ... , Zt+2)]
=).+ 1
L (-I)Hq~(lo, ... , Iq, ... , I ,1£+2) A ,· ••
q<A
+ L (-1)Hq+1~(lo,···, I A, . .. , Iq, ... , Zt+2) = o. 0
A<~
159
Def. 2.2. t5: = t5(: Ct(U, 5") ~ Ct + l(U, 5") is called the coboundary operator.
We denote by C(U, 5") the Cech complex
CO(U, 5") ~ C 1 (U, 5") ~ C 2 (U, 5") ~ ....
8: r(X, 5") ~ CO(U, 5") is defined by (8S)(/): = sIU,.
Theorem 2.2. (r(X, 5"), 8, C 8 (U, 5")) is an augmented co chain complex.

PROOF. Clearly 8 is an R-module homomorphism. If 8S = 0, then sl U, = 0
for every I E I; therefore s = O. Hence 8 is injective.
Let e E CO(U, 5") and t5e = 0. Since
(t5e}(/ o, 11) = (-e(/1) + e(/ o))IU,o',
this is equivalent to e(lo)IU,o', = e(/ 1)IU,o',. Therefore there is a section
s E r(X, 5") with 8S = e
defined by siU,: = e(/). 0
Def.2.3. The elements of zt(U, 5"): = zt(C(U,5")), resp. Bt(U, 5"): =

B/'( C 8 (U, 5") ) are called (alternating) C-dimensional coc ycles, resp. cobound-
aries, over U with values in 5". H((U, 5"): = Z!(U, 5")jBt(U, 5") =
Ht( C 8 (U, 5")) is the t -th Cech cohomology group of U with values in 5".
In particular HO(U, 5") ~ r(X,5").
Ifwe choose the covering ofU too coarse, then all the higher cohomology
groups vanish:
Theorem 2.3. If X itself belongs to the elements of the covering U, then
H((U,5") = 0 for t ~ 1.
PROOF. IfU = (U,),eI, then there is apE I with X = Up. Let eE zt(U, 5"),
t ~ 1. There is an element I] E C t - 1 (U, 5") defined by
1](/0' ... , 1(-1): = e(p, 10' ... , It - 1)·

Since t5e = 0 we have
I
o= t5e(p, 10, ... , It) = - e(l o, ... , If) + I (-l)'''e(p, 1o, ... , I)., ... , If);
),=0
therefore
(
t5(-I]}(/o,···, It) = I (_I)H11](/o,···, I)., ... , Ie)

),=0
(
I (-I)).e(p, 10, ... , I)., ... , I() = e(/ o, ... , IJ.

),=0
In other words, t5( -I]) = e, so e E Bt(U, 5"). o

Theorem 2.4. Let U be an arbitrary covering of X and 5" a flabby sheaf.
Then H((U, 5") = 0 fort ~ 1.
160
2. The Cech Cohomology
PROOF. We proceed by induction onf: Let ~ E zt(U, ,<;J),f ~ 1. If U c X is

open, then we set U n U: = {U n U, =1= 0: U, E U} and
(~IU)(IO' ... , It): = ~(IO'· .. , It)IU n U'O···'I

With this notation we have ~I U E zt( U n U, ,<;J).
For arbitrary Xo E X, there is an 10 E I and an open neighborhood
U(xo) c U'o. But then U E U n U, so Ht(U n U, ,<;J) = 0 for f ~ 1, and
there is an IJ E Ct - 1(U n U, ,<;J) with (jIJ = ~I U.
If V c X is an open set for which there is an IJ' E C t - 1(V n U, ,<;J) with
(jIJ' = ~IV, we set
s: = (IJ - IJ')IU n V E zt-1(U n V n U, ,<;J).
If f = 1, then s lies in r(U n V, ,<;J), and since ,<;J is flabby, we can extend s
to an S E r( V, ,<;J). Then set
IJ(X) XE U
{
s*(x): = IJ'(x) + s(x) XE V.
Clearly s* E r(U U V, ,<;J) and (js* = ~I U u V (because (js = 0).
If f > 1, then by the induction hypothesis there is ayE C t - 2 (U n V n-U,
,<;J) with (jy = s. Since ,<;J is flabby,
Y(lo, .. . ,1/ -2) E r(U n V n U'O·· .'f_2',<;J)

can be extended to an element
')1(1 0 , . . . ,1(-2) E r(V n U'O·· .'1-2' ,<;J).
Let
*( )( ). _ {I](IO" .. ' It - 1)(X) for x E U n U'O ... 'I_l

I] 10,···, 1(-1 x. - , ~
(I] + (jy)(zo,···, It -1)(X) for x E V n U'O ... 'I_ 1
Then 1]* E C t -- 1((U U V) n U,,<;J) and (jIJ* = ~IU u v.
By Zorn's lemma there must be a "maximal element" (U 0, so) for f = 1,
resp.(U 0,1]0) for f > 1 with So E r(U 0,,<;J) and (jso = ~I U 0, resp.l]o E Ct(U,,<;J)
and (jl]o = ~I U o· But an element is only maximal if U 0 = X; therefore
~ E W(U, ,<;J). 0
Remark. Let U be a covering of X and ,<;J a sheaf of R-modules, ~ E

C 1(U, ,<;J). It is worth noting the following criteria:
1. ~ E Zl(U, ,<;J) if and only if
~(IO' 12) = ~(Io, 11) + ~(Ib 12)

on U'O"'2.
2. ~ E B1(U, ,<;J) if and only if for all 1 there exists an p(l) E r(U" ,<;J) with
~(Io, 11) = p(lo) - p(11)
The first condition is also called the compatibility condition.
161
Def.2.4. A system (V" f.),El is called a Cousin I distribution on X if

1. U: = (V.),El is an open covering of X;
2. I. is meromorphic on V,;
3. 1.0 - I., is holomorphic on V,o" for all 10, 11·
A solution of a Cousin I distribution is a merom orphic function f on X

such that I. - f is holomorphic on V,.
Theorem 2.5. Let (V" 1.),El be a Cousin I distribution on X, Yf the structure

sheaf of X, U: = (V.),El. Then
1. Y(lo, 11): = (1.0 - I.JIV,o', defines an element Y E ZI(U, Yf).
2. (V" 1.),El is solvable if and only if Y lies in Bl(U, Yf).
PROOF
1. Clearly
Y(lo, 11) + Y(Ir. 12) = (1.0 - I.J + (I., - I.') = 1.0 - 1.2 = Y(lo, 12)
on V'01l'2.
2. a. Let (V" J.)'EI be solvable. Then there is a meromorphic function f
on X such that (I. - f)IV, is holomorphic. Let
p(I): = (I. - f)IV, E r(V" Yf).
P lies in CO(U, Yf) and
p(l o ) - p(l l ) = (1.0 - f) - U;, - f) = 1.0 - I., = Y(lo, 11)
on V'o".
b. If Y lies in Bl(U, Yf), then for every I E I there is a p(l) E r(V" Yf)
such that p(lo) - p(ll) = Y(lo, Id on V'OIl. Then
1.0 - I., = Y(lo, 11) = p(IO) - p(ll),
so 1.0 - p(lo) = I., - p(ll) on V'o',. Then there is a meromorphic function
f on X defined by fIV,: = I. - p(l) with
(I. - f)iV, = p(l) E r(V" Yf). 0
Corollary. If Hl(U, Yf) = 0, then every Cousin I distribution belonging to the

covering U is solvable.
EXAMPLE. Let X = IC. A Mittag-Leffler distribution on C is a discrete point

sequence (zv) together with principal parts Iv which define a meromorphic
function in C.
°
Now let V o : = C - {zv:v E N},fo: = and Vv be an open neighborhood
of Zv which contains no point zJl with f1 v. Then IvI(Vv - {zv}) is"*
holomorphic.
162
3. Double Complexes
Hence U = (Vv)"EN is an open covering ofC and (Iv - f/1)IVv/1 is always

holomorphic. Therefore (VV' Iv)VEN is a Cousin I distribution. Each solution
of this Cousin I distribution is a solution of the Mittag-Leffler problem.
3. Double Complexes
Def.3.1. A double complex is a system (Ci ) of R-modules (with i, j E No)
and R-module homomorphisms, d':Cij ~ Ci+l,j and d":Cij ~ Ci,j+l,
such that
1. d'd' = 0
2. d"d" = 0
3. d'd" = -d"d'
(thus
d: = d' + d": EB Cij ~ EB C ij with dod = 0).
i+j=n i+j=n+l
A double complex is therefore an (anticommutative) diagram of the
following form:
d" d" d"
COO~COI~C02~'"
ld' d"
ld' d"
ld' d"
CIO~ C II ~CI2~'"
ld' d"
C20~C21~C22~'"
ld' d"
ld' d"
ld l
ld' ld l
Def.3.2.
Zij: = g E Cij with d'~ = 0 and d"~ = O}
BOj: = d"(g E CO,j-1 with d'~ = O}) for j ?; 1,
B;o: = d'(g E Ci-I,O with d"~ = O}) for i ?; 1,
Boo: = 0 and Bij: = d'd"Ci-l,j-1 for i, j ?; 1.
We call the elements of Zij cycles ofbidegree (i,j); the elements of Bij are
called boundaries of bidegree (i, j).
Clearly Bij is an R-submodule of Zij for all i, j and we define the homology
group of the double complex of bidegree (i,j) by Hij: = ZdBij. Let the
canonical projection be denoted by %:Zij ~ Hij'
163
Theorem 3.1. Let (M, Ae ), (M, £2, Be) be two augmented cochain com-
£1'
plexes. Let there be given a double complex (C VIL ' d', d") and homomorphisms
dj:Aj ~ C Oj and d;':B i ~ C;o such that
and d' d;'
0 = d;'+ 1 d,
0
where d denotes the operation in A e and Be;

(2) (Aj, dj, Cej) and (B i, d;', Cie) are augmented cochain complexes. Then
Hj(A e ) ::::: HOj and H i(B 8 ) ::::: H;o.
PROOF
1. ZOj = g E COj:d'~ = 0 and d"~ = O} = g E C o{ There is an I] E Aj
with djl] = ~, d" ~ = O} = g E C Oj : There is an I] E Aj with djl] = ~ and
dj+l(dl]) = O} = g E C Oj : There is an I] E Aj with djl] = ~ and dry = O} =
dj(Zj(Ae) ).
2. BOj = {d"~:~ E C O. j- 1 with d'~ = O} = {d"~: There is an I] E Aj-l
with dj-ll] = ~} = dj(Bj(A e )) for j ~ 1 and Boo = 0 = do(BO(A e )).
3. Since dj is always injective, it follows that
H Oj = ZOj/B oj ::::: Zj(A 8 VBj(A8) = Hj(Ae).
One shows that H;o ~ H i (B 8 ) analogously. o
EXAMPLE. Let X be a complex manifold, Y a sheaf of R-modules over X,

U an open covering of X. If W(Y): Y ° ~ Y 1 --+ Y 2 --+ •.. is the canonical
flabby resolution of Y, then let
d d d
we(y): r(X, Y 0) ~ r(X, Y 1) ~ r(X, Y 2) --+ . . . .
(r(X, Y), £., we(y)) is an augmented cochain complex.

If one sets
C(U, Y): CO(U, Y) ~ C 1 (U, Y) ~ C2 (U, Y) --+ ... ,
then (r(X, Y), £, C(U, Y)) is also an augmented co chain complex.
Now let
Cij: = Ci(U, Y j), d': = b(j) = b:Ci(U, Y j ) ~ Ci + 1(U, Y j),
d": = (- l)id.: Ci(U, Y j) --+ Ci(U, Y j+d (with d*~(lo, ... , I;): = d.(~(lo, ... , I;})).
Clearly d'd' = 0 and d" d" = O. Moreover it is true that
i+l )
(d.bO(l o,···, li+ d = d. ( I
),=0
(_l)H 1~(10,···, lie,"', li+ d
i+ 1
= I (_l)Hld.~(lo,···,lA, ... ,li+l)=(bd.~)(lo,···,li+d;
ie=O
therefore
d'd" + d"d' = b(j+ 1)( ~ l)id. + (_1)i+ Id.b(j) = ( _1)i. (b(j+ 1)d. - dAj)) = O.
164
3. Double Complexes
Thus (C ij , d', d") is a double complex which we shall henceforth describe

as the canonical double complex of (X, Y, U). We obtain the following
diagram.
Since all the hypotheses of Theorem 3.1 are fulfilled,

for all i, j.
We can therefore use the homology groups Hij of the canonical double
complex to compute the flabby and Cech cohomology groups of X with
coefficients in Y. Homomorphisms ((J{ : H((1.l, Y) --? H((X, Y) will be con-
structed with the help of these double complexes.
Theorem 3.2. Let (Cij' d', d") be a double complex.

1. Let the d' -sequences be exact at the locations (i, j) and (i - 1, j).
Then there are homomorphisms
((Jij:Hij --? H i - 1 ,j+l for i ~ 1, with ((Jij 0 qij d'
0 = qi-l,j+l 0 d".
2. Let the d"-sequences be exact at the locations(i - 1,) + 1),(i - I,}).
Then there are homomorphisms
t/!ij:H i - 1 ,j+ 1 --? Hij for i ~ 1, with t/!ij 0 qi-l,j+ 1 0 d" = qij 0 d'.
3. If hypotheses (1) and (2) are satisfied simultaneously, then ((Jij is an
isomorphism with ((JiJ 1 = t/!ij'
PROOF
1. If ~ij E Zij, then d' ~ij = O. Therefore there is an 1]i-l,j E Ci-1,j with
d'1]i-l,j = ~ij' We set ((Jii%(~ij)): = qi-l,j+l(d"1]i-l,j)'
a. Let ~ij = d'1] = d'r/*. Then d'(11 - 1]*) = 0; therefore there is ayE C i - 2 ,j
withd'y = 1] -l]*(fori ~ 2), and it follows that d"1] - d"1]* = d"d'YEBi-1,j+l'
165
Therefore
qi-1,j+ l(d"l]) = qi-l,j+ l(d"I]*),
If i = 1, set y*: = I] - 1]*, Then d'y* = 0, Therefore d"y* E B o, j+ 1> and

furthermore qO,j+ l(d"l]) = qO,j+ l(d"I]*), The definition does not depend
on the choice of l]i-1,j'
b. Let ~ij E Bij' If i ;:, 1 and j ;:, 1, then ~ij = d'd"y with y E Ci - 1, j - 1
and d"(d"y) = O. If j = 0, then ~ij = d'y* with d"y* = O. Therefore the
definition depends only on the cohomology class of ~ij'
(c) d"(d"l]i-1,j) = 0, d' (d"l]i-1,j) = - d"(d'1J;-l,j) = d"( - ~ij) = O.
Therefore d"l]i-1,j lies in Zi-1,j+1'
Because of (a), (b), and (c), ({Jij actually defines a mapping from Hij to
Hi-l,j+l' It is clear that the map is a homomorphism.
2. The existence of lj;ij follows exactly like that of ({Jij' If ({Jij and lj;ij both
exist, then
Hence it follows that ({JiJ 1 = lj;ij. o
Theorem 3.3. Let X be a complex man(fold, !? a sheaf of R-modules over X,

and U an open covering of X. Then there is a (canonical) R-module homo-
morphism
for t ;:, 1.
({J 1 is, in particular, injective.
PROOF
1. Let (Cij' d', d") be the canonical double complex of (X, !?, U). Then
Hj(X,!?) ~ H oj , Hi(U,!?) ~ HiQ, and we can define
({Jt: = ({J1,f-1 0 ••• 0 ({Jr-1,l 0 ({Jr,O
[Since all sheaves !?j,j ;:, 0 are flabby, we have Hi(U, !?j) = 0 for i ;:, 1,
j ;:, O. Therefore the d' -sequences are exact!J
2. ({J1 = ({J10:H10 -> H01 is given by ({JlO 0 ql0 0 d' = q01 0 d". If 0 =
({J10(Q10 0 d'l]) = Q01 0 dill], then d"l] lies in BOb therefore there is an If E Coo
with d'l]* = 0 and d"l]* = d"l], Then d"(1] - 1]*) = 0 and d'(1] - 1]*) = d' 11,
therefore d'l] E B 10 ; that is, Q10 0 d'l] = O. 0
Def.3.3. Let!? be a sheaf of R-modules over X and U = (U.)IEl an open

covering of X. U is called a Leray covering of !? if Ht(UIO"'li'!?) = 0
for C ;:, 1 and all la, ' .. , Ii'
Theorem 3.4. If U is a Leray covering of !?, then ({Jr: HI (U, !?) -> H{(X, !?)
is an isomorphism for everye ;:, 1.
166
4. The Cohomology Sequence
PROOF. If Ht(U,o' "'i' 9") = 0, then by the definition of flabby cohomology

the following sequence is exact:
r(U,O'" 'i' 9"j_ d ~ r(u,o'" 'i' 9"j) ~ r(U'Q'" 'i' 9"j+ d

If d*~ = 0, then 0 = (d*~)(/o, ... , IJ = d*(~(/O"'" Ii)) for all (/0"" I;).
Therefore there are elements 1](/0, ... , li)withd*(1](/o, ... , Ii)) = ~(IO"'" IJ-
In each case it suffices to determine one ordering 1](/0' ... , IJ of the indices,
since the values for other orderings are determined by the antisymmetric
rule.
In this way a cochain 11 with d*11 = ~ is determined.
The d"-sequences in the canonical double complex are therefore exact
and the proposition follows. 0

Let X be a complex manifold, 9"*, 9", 9"** sheaves of R-modules over X.
(A) Let cp:9"* ---+ 9" be a homomorphism. Then \fi(cp):\fi(9"*) ---+ \fi(9")
is a homomorphism between canonical flabby resolutions, defined by the
mappings ~cp:9"i ---+ 9"i' These mappings induce mappings
(~(p)*: r(X, 9"t) ---+ r(X, <'l'J,
Theorem 4.1
1. If ~ E Zi(X, 9"*), then (~cp)*~ E Zi(X, 9").
2. If ~ E Bi(X, 9"*), then (~cp)*~ E Bi(X, 9").
PROOF. The following diagram is commutative:
r(X, 9"1-1) Lr(x, 9"n-Lr(X, 9"1+1)

l(~-lcp). l(~cp). l(~+lcp)*
d r(X, 9";) ~
r(X, 9"i- d ~ d r(X, 9"i+ 1)
1. Ifd~ = 0, thend«~cp)*~) = (~+lcp)*(d~) = O.
2. If ~ = d1], then (~cp)*~ = (~cp)*dl1 = d«~-lCP)*I1)· o
Corollary. Let
qi:Zi(X, 9"*) ---+ Hi(X, 9"*), qi:Zi(X, 9") ---+ Hi(X, 9")
be the canonical residue class mappings. Then there exists a homomorphism
cp:Hi(X, 9"*) ---+ Hi(X, 9"), given by cp qi = qi (~cp)..
0 0
Theorem 4.2. (Hi:9" ~Hi(X, 9"), cp \fV"'>cp) is a covariant junctor, that is:
1. Idy = idHi(x.y).
2. l/J 0 cp = iii 0 cp.
167
(B) Let 0 -+ Y* ~ Y ~ Y** -+ 0 be exact. Then we obtain the following

commutative diagram with exact columns:
o 000
... --+ r(X, Yi-1)

I d Id
r(X, Yi)
---=--+ ---=--+
I d j
r(X, Yi+ 1)---=--+r(X, Yi+2) ----+ ...
lO'V;-1IPL l(W;IP) * 1(W;+1IP)* 1(W;+2IP)*

... ----+r(X, Y i - 1) L r(X, Y i) L r(X, y i + 1)Lr(x, Y i +2)--+ ...
1(W;-1t/1)* 1(W;t/I)* 1(W; +1 t/I)* 1(W;+2t/1)*

... ----+r(X, Yi~1) Lr(x, Yi*)L r(X, Yi~1)Lr(x, Yi~2)--+'"
1o 1o 1o 1
o
Theorem 4.3
1. If ~ E Zi(X, Y**), then there exists an '10 E r(X, Y i ) and an '1 E
Zi+1(X, Y*) with ~ = (W;t/I)*'10 and d'1o = (W;+1IP)*'1. '1 is determined up
to an element '1* E Bi+ 1(X, Y*).
2. There exists a homomorphism, canonically induced by (1),
a:Hi(X, Y**) -+ Hi+1(X, Y*) with ip 0 a =0 and a 0 Ifi = O.
PROOF
1. If ~ E Zi(X, Y**), then d~ = 0, and there exists an 110 E r(X, Y i ) with
(W;t/I)*'10 = ~. Clearly then 0 = d( (W;t/I)*'10) = (W;+ 1 t/I) *d'1 0 , that is, there
exists an '1 E r(X, Yi+1) with (W;+1IP)*'1 = d'1o. The element '1 is a cycle,
because 0 = dd'1o = d«W;+1IP)*'1) = (W;+2IP)*d'1 and therefore d'1 = O. '1 is
uniquely determined by '10' If ~ = (W;t/I)*'1o = (W;t/I)*'10, then there exists a
P E r(X, (1) with (W;IP)*P = '10 - '10 and we have
d'1o - dllo = d«W;IP)*p) = (W;+1IP)*dp;
therefore '1' - '1" = dp.
2. A homomorphism O:Zi(X, Y**) -+ Hi+1(X, Y*) is defined by o(~): =
qi+1('1) such that
ip 0 00 (W;t/I)*'10 = = qi+1 (W;+1IP)*1I = qi+1(d'10) = o.
ip 0 qi+1 '1 0
If~ = d~*, then there is a O'Er(X,Y i - 1) with (W;-1t/1)*O' = C; therefore

= d«W;-1t/1)*O') = ~. We can choose '10 = dO' and from the con-
(W;t/I)*dO'
struction we obtain that o(~) = O. Therefore 0 induces a homomorphism
with a qi*
0 = O.
In particular (a) qi*, (W;t/I)* are surjective and
ip 0 a qt*
0 0 (W;t/I)*('10) = ip 0 00 (W;t/I)*('10) = o.
168
(b) qi is surjective, (W;cpL is injective, and for ~ E Zi(X, Sf') we have

with (W;cpLI] = d~ = 0
therefore I] = O. Hence ?f5 0 = 0 and 0 0 0 lfI = O. o
Theorem 4.4. Let 0 -+ Sf'* ~ Sf' .i. Sf'** -+ 0 be an exact sequence of sheaves
of R-modules. Then the following long cohomology seqllence is also exact
o -+ r(X, 8'*) ~ r(X, Sf') "4 r(X, Sf'**) ~ H1(X, Sf'*) -+ . . .
. . . -+ H i - 1(X, Sf'**) ~ Hi(X, Sf'*) ~ Hi(X, Sf')! Hi(X, Sf'**) -+ ...
PROOF
a. The sequence 0 -+ r(X, Sf'*) -+ r(X, Sf') -+ r(X, Sf'**) is exact, since
r is a left exact functor.
b. The cohomology sequence is exact at Hi(X, Sf'*), i ~ 1:
1. ?f5 0 = 0 by Theorem 4.3.
0
2. If ~ E Zi(X, Sf'*) and

o= ?f5 0 qi(~) = qi 0 (W;cp)*~,
then (W;cp)'~ = dl] with I] E r(X, Sf'i-d and

d((W;-l!fr).I]) = (W;!frLdl] = (W;!fr)'(W;cp)'~ = O.
(W;-l!frLIJ thus lies in Zi-1(X, Sf'**) and
0 qi::1 0 Q (W;-1!fr).1] = O(W;-l!fr).1] = qi~·
Therefore Ker?f5 c 1m o.
c. Exactness at Hi(X, Sf'), i ~ 1:
1. By Theorem 4.2, lfI " ?f5 = O.

2. Let ~ E Zi(X, Sf') and 0 = lfI q;(~) 0 = qi* 0 (W;!fr)'~. Then
(W;!fr)*~ = de with ~* = (W;-1 !frLI] E r(X, Sf'i:: 1)

and hence d(~ - dl]) = 0 and (W;!fr)*(~ - dl]) = O. Thus there is a
a E r(X, Sf';) with (W;cp).a = ~ - dl]. Clearly da = 0 also, and
?f5 0 qi(a) = qi (W;cp)*a = 0 qi(~ - dl]) = qi(~)·
Therefore Ker lfI c 1m ?f5.

d. Exactness at Hi(X, Sf'**), i ~ 1:
1. 0 lfI = 0 by Theorem 4.3.
0
2. Let d~ = 0 and
o= 0 0 qi*(~) = a~ = qi+11J with ~ = (W;!fr)*l]o
and
169
Then I] = dO', and

d(l]o - (W;<p).O') = 0, (W;t/t).(l]o - (W;<p)*O') = ~;
therefore
l[i qi(I]O - (W;<p).a) = qi* (W;t/t).(l]o - (W;<p)*a) =
0 0 qi*~.
Hence Ker ac 1m l[i, and the proof is complete. o

(C) Let X be an n-dimensional complex manifold with structure sheaf (!). For
every open set U c X there is an associated multiplicative abelian group
Mu = {f:f is holomorphic on U and f(x) #- 0 for x E U}. Mu becomes a
Z-module (with n . f: = f n ), and together with the usual restriction mappings
r~:Mu -> Mv yields a pre-sheaf of Z-modules. The corresponding sheaf of
Z-modules (!)* is called the sheaf of germs of non-vanishing holomorphic
functions. We write the group operation in (!)* and in the derived modules
adtlitively. If Nu is the additive abelian group of holomorphic functions,
then there exists a Z-module homomorphism eXPu:Nu -> Mu defined by
f f--> e21tiJ • For V c U the commutative law expv 0 r~ = r~ 0 expu holds.
This defines a sheaf homomorphism exp: (!) -> (!)* with exp(rf) = r(e 21tiJ ).
Theorem 4.5. 0 -> Z -> (!) ~ (!)* -> 0 is an exact sequence of sheaves of Z-
modules (where Z also denotes the sheaf of genns of continuous Z -valued
functions).
PROOF. Continuous Z-valued functions are locally constant, in particular,
locally holomorphic. Hence we can regard Z as a subsheaf of (!), and we need
only show that Ker(exp) = Z and Im(exp) = (!)*.
1. Let a = (rf)(x) E (!) x, fEN u, exp(a) = o. Then 0 = exp(rf)(x) =
(r(e 2niJ ) )(x). There exists a connected neighborhood Vex) c U with
rV
r(e 21tiJ )1Y = 0; that is, e 21tiJ = 1. Then there is an n E Z with flY = n.
Conversely if a E Zx c (!)x, it follows that exp(O') = O.
2. Let p = (rf)(x) E (!);, f EMu, x E U. Without loss of generality we
may assume that U is an open set in en, so that log(f) is holomorphically
definable on U. Let
1
h: = -2.. log(f), a: = (rh)(x) E (!)x·
m
Then
exp(a) = exp( (rh)(x)) = (r(e 21tih ) )(x) = (rf)(x) = p. 0
Theorem 4.6
1. Let f reX, (!)*). Then there is an hE reX, (!)) with f = e21tih if and
E
and only if a(f) = O.
2. If Ht(X, (!)) = 0 fort ~ 1, then Ht(X, (!)*) ~ H(+ leX, Z) for C ~ 1.
PROOF. Look at the, long exact cohomology sequence of the short exact
sequence 0 -> Z -> (!) -> (!)* -> o. 0
170
Def. 4.1. A system (U" .f.), El is called the Cousin II distribution on X if

1. U = (U,),El is an open covering of X.
2. I. is holomorphic on U, and vanishes identically nowhere.
3. On U,o', there is a nowhere vanishing holomorphic function h,O" such
that .f.o = h,o" . f;, on U,o,,·
A solution of this Cousin 11 distribution is a holomorphic function f
on X such that I. = h,:f with nowhere vanishing holomorphic functions
h,on U,.
Remark. The functions h,o', are uniquely determined by the distribution
(U" !.),El :
If!.o = h,O" .1., = h'ott . 1." then 0 = (h,O" - h,o'.) .1.,.
If it were true that (h,O" - h,Oq)(xo) =f. 0 for an Xo E U'ott' then it would
also hold that (h,O" - h,ott)(x) =f. 0 for x E V, Van open neighborhood of
Xo in U,O". Therefore 1.11 V = 0 which is a contradiction.
Theorem 4.7. Let (U" i.),El be a Cousin II distribution on X, U = (U.)'EI.

Then:
1. h(/o, 11): = rh'o'l defines an element h E Z1(U, (1)*).
2. (U" i.)'EI is solvable if and only if h lies in B1(U, (1)*).
PROOF.
la. Because
it follows that h( It> 10 ) = - h( 10 , I d.

b. Because
it follows that h(lo, ld + h(11' l2) = h(lo, 12)·

2a. Let (U" i.)'Ei be solvable. Then I. = h, . f with nowhere vanishing
functions h, andf;o = h,o', .1. 1 , therefore h,o . f = h'o'l . h'l . f. Let p(I): = h,.
Then
p(/O) - p(11) = r(h,o· h,~1) = r(h'o,J = h(lo, 11);
therefore fJp = h.
b. If h lies in B1(U, (1)*), then for every 1 E 1 there exists a p(l) E T(U" (1)*)
such that p(lo) - p(/1) = h(10' 11) on U,o". Then h,: = [p(I)] is a nowhere
vanishing holomorphic function, and on U'o" we have h(lO' Id = r(h'OII) =
r(h,o . h,~ 1); therefore h,o = h,ott . h".
Similarly we have!.o = h'o'l . I.,. Hence it follows that f;o . h,~ 1 = 1.1 . h,~ 1.
Thus there is a holomorphic function f on X with!. = h, . f defined by
flU,: = I. . h,-l. 0
Remark. The question of the solvability of a Cousin II distribution is a

generalization of the Weierstrass problem.
171
Corollary. If Hl(U, (9*) = 0, then every Cousin II distribution belonging to

the covering U is solvable.
Theorem 4.8. Let X be an n-dimensional complex manifold with structure

sheaf (9.
1. If Hl(X, (9) = 0, then every Cousin I distribution on X is solvable.
2. If Hl(X, (9*) = 0, then every Cousin II distribution on X is solvable.
PROOF. The canonical homomorphisms Hl(U, (9) ~ Hl(X, (9) and Hl(U,
(9*) ~ Hl(X, (9*) are injective for every covering U (See Theorem 3.3). D
Def. 4.2. Let hE Zl(U, (9*) be the cocycle of a Cousin II distribution

(U" J.),El' h. the corresponding cohomology class in Hl(X, (9*), and
8:Hl(X, (9*) ~ H2(X, Z) the "boundary homomorphism" of the long
exact cohomology sequence of 0 ~ Z ~ (9 ~ (9* ~ O. Then c(h): =
8(h) E H2(X, Z) is called the Chern class of h (resp. of (U" J.)'EJ.
Theorem 4.9. If H{(X, (9) = 0 for e ;::, 1, then the Cousin II distribution
(U" D'E! (with the corresponding cocycle h) is solvable if and only if
c(h) = 0 (and that is a purely topological condition !).
PROOF. By Theorem 4.6 Hl(X, (9*) ::,;: H2(X, Z), under 8. h is thus solvable
if and only if h. = 0, and that is the case if and only if c(h) = 8@ = O. D
EXAMPLE. There exist very simple domains of holomorphy on which not

every Cousin II distribution is solvable. Suppose,
X: = {(z, w) E C 2 :[[z[ - 1[ < e, [[w[ - 1[ < e}.
X is a Reinhardt domain and, as one can readily see, is logarithmically
convex, therefore a domain ofholomorphy.
The "center of X"
T: = {(z, w) E C 2 :[z[ = 1, [w[ = I} c X
is the real torus.

g: = {(Z,W)EC 2:W = z - I}
is a complex line, and therefore a real 2-dimensional plane.
For (z, w) E g
[W[2 = W. tv = (z - 1)(z - 1) = zz + 1 - (z + z) = [Z[2 + 1 - 2x
(with z = x + iy)
If [z[ = 1, then in particular we have [W[2 = 2 - 2x, so [w[ = 1 if and only
if x = 1/2. Let
Zl: = t(1 + i.j3),z2: = t(1- i.j3),w 1 : = Zl - l,w2: = Z2 - 1.
172
Hence it follows that

Tn g = {(ZI' WI), (Z2' w2)}
The mapping cI>: g -> C with cI>(z, w): = Z is topological with cI> -1 (z) =
(z, Z - 1). Let
Re: = {zEC:l - s < Izi < 1 + e} = {zEC:llzl- 11 < s},

Re: = {ZEC:Z - 1 ERe} = {zEC:llz - 11- 11 < s}.
Re, Re are two congruent annuli displaced from one another with
Re n Re = cI>(g n X) :::> cI>(g n T) = {Zb Z2}'
Figure 25. Illustration for the example.
Ifwe choose e sufficiently small, then Re n Re decomposes into two connected

components Yl' Y2 •
LetF",: = cI>-I(Y;) for A = 1,2.ThengnX = F1 uF 2 withF 1 nF2 =

o and the sets FA are analytic in X. Let
U 1 : = X - F 2 , U 2 : = X - Fb g(z, w): = w - Z + 1,
as well as
11: = gIU 1,f2: = 11 U 2'
g has no zeroes in U 12 = X - (F1 u F 2) = X - g, and we haveI11U12 =
g . 121 U 12' Therefore ((U 1,11), (U 2,12)) is a Cousin II distribution on X.
We can introduce real coordinates on T:
(z, w) = (eiq>, eiB ) 1-+ (cp, 8).
Then glT = eW - eiq> + 1 = (cos 8 - cos cp + 1) + i(sin 8 - sin cp), and

173
the mapping.
r:V 1 n T -> [R2 with r(cp, e): = (cos e - cos cP + 1, sin e - sin cp)
is real analytic and has exactly one zero (CPo, eo) ~ (Zl' Zl - 1).
For the functional determinant we have
sin CPo - sin eo) _ det ( sin CPo -sin CPo )
det J,(cpo, eo) = det (
-cos CPo cos eo - cos CPo cos CPo - 1
= d et ( sin
-cos
CPo
CPo -1 0) -_ -sin CPo = -1m Zl = -tJ3 =F O.
Hence we can find a neighborhood V = V(cpo, eo) c V 1 n T which is
mapped by r biholomorphically onto a domain of [R2. Let V*: =
V - {(CPo, eo)}·
We can regard r as a complex valued function. Then on V* the differential
form 0) = dr/r is defined and clearly dO) = 0.
We now choose an open subset Bee V which relative to r IV is the
inverse image ofa circular disc{z E C:lzi ~ s}. Let H: = aB. Then
1 iH
ill =
dr
-
1,I=s r
=F o.
Now suppose there is a solution f of the above Cousin II problem. Then
fl V 1 =g . h, with a nowhere vanishing holomorphic function h in V 1, and
fiT has a zero only at (CPo, eo). Therefore ill: = dhlh is a differential form
°
on V 1 n T, IX: = dflf a differential form on T - {(CPo, eo)} and dill = 0,
dlX = and IXI V* = 0) + w. Thus it follows that
SH ill = 1B ill = SB dw = 0,
SH IX = - 1(T-B) IX = - ST-B dlX = 0,
but
SH IX = SH + SH ill = SH =F O.
0) 0)
That is a contradiction. A solution f cannot exist. o
5. Main Theorem on Stein Manifolds

The two following theorems of Cartan-Serre are the basis for the theory
of Stein manifolds. The proofs are difficult and cannot be included here.
Theorem 5.1 (Theorem A). Let (X, (I) be a Stein manifold, !f a coherent ana-
lytic sheaf over X. Then for every point Xo E X there are finitely many
global sections Sb . . . , Sk E T(X, !f) which generate !f Xo over (i7 x o.
sheaf over X. Then H((X, !f) =

manifold, see Chapter V, Section 2.)
°
Theorem 5.2 (Theorem B). Let X be a Stein manifold, S a coherent analytic
for e ~ 1. (For the definition of a Stein
174
Theorem 5.3. Let X be a Stein manifold, and 0 --* 9"* --* 9" --* 9"** --* 0 an
exact sequence of coherent analytic sheaves over X. Then
o --* r(X, 9"*) --* r(X, 9") --* r(X, 9"**) --* 0
is exact and
rex 9"**) ~ r(X, 9") .
, r(X,9"*)
PROOF. The cohomology sequence

o --* r(X, 9"*) --* r(X, 9") --* r(X, 9"**) --* H1(X, 9"*) --* .•.
is exact and by Theorem B, HI(X, 9"*) = O. o
Theorem 5.4. Let X be a complex manifold and V 1, V 2 C X open Stein
manifolds. Then V: = V 1 (} V 2 is also a Stein manifold.
PROOF
1. If Xo E V, then there are holomorphic functions fI' ... , ff on VI such
that Xo is an isolated point in
{XE V I :f1(X) = ... = fe(x) = O}.
Then the functions fII V, ... , J; IV are holomorphic and Xo is also isolated in
{x E V:f1(x) = ... = J;(x) = O}.
Therefore V is holomorphically separable.
2. Let K c V be compact. Then K is also compact in Vi' and so, for the
holomorphically convex hulls, K c Kb Ki compact. Clearly K is contained
in K1 (} K2. V - K is open; therefore K1 (} K2 - K = K1 (} K2 (}
(V - K) is open in KI (} K2. Since KI (} K2 is compact, it follows that
K is compact. 0
Def. 5.1. Let X be a complex manifold. An open covering U = (V.),El of

X is called Stein if all the sets V, are Stein.
Theorem 5.5 (Leray). Lex X be a complex manifold, 9" a coherent analytic

sheaf on X, U a Stein covering of X. Then U is a Leray covering of X and
for all t, He(U, 9") ~ Hf(X, 9").
PROOF. If U = (V.),El is Stein, then by Theorem 5.4 all sets V'O""i are
Stein, and by Theorem B, Hf( V,o" "i' 9") = 0 for t ~ 1. Therefore U is a
Leray covering and CPt: H((U, 9") --* Hf(X, 9") is an isomorphism. 0
Theorem 5.6. If X is a complex manifold, then there are arbitrarily fine Stein
coverings of X. If 9" is coherent analytic on X, then for every open covering
U of X there exists a refinement m such that Hf(m, 9") ~ Hf(X, 9") for
all t ~ O.
175
PROOF. Let Yf be the structure sheaf of X. If Xo E X, then there is an

open neighborhood U(xo) c: X, a domain G c: en and an isomorphism
qJ:(U, Yf) -+ (G, (9). If V(xo) is an arbitrary neighborhood, then there exists a
polycylinder P with ip - l(P) c: c: V (") U. Then ip - l(P) is Stein. Therefore
there exist arbitrarily small Stein neighborhoods and hence arbitrarily small
Stein coverings. 0
Theorem 5.7. Let X be a Stein manifold,!J? a coherent analytic sheaf over X,

U an arbitrary open covering of X. Then Hl(U, !J?) = O. In particular,
every Cousin I distribution over X is solvable.
PROOF. Hl(X,!J?) = 0 and qJl :Hl(U, !J?) -+ Hl(X,!J?) is injective. 0
Theorem 5.8. If X is Stein, then for all t' ~ 1, HI(X, (9*) :::::: Ht+l(X, Z).
PROOF. Theorem B and Theorem 4.6. o
Theorem 5.9. Let X be Stein, (Up J.),el a Cousin II distribution on X,
hE Zl(U, (9*) the corresponding cocycle. Then (U" f.),el is solvable if and
only if c(h) = O.
PROOF. Theorem B and Theorem 4.9. o
At the end of the last section we gave an example of a Stein manifold on
which not every Cousin II problem is solvable. Let us assume the following
two (topological) results without proof:
1. If X is a connected non-compact Riemann surface (X is then Stein by a
theorem of Behnke-Stein), then H2(X, Z) = O.
2. If X is a Stein manifold which is continuously contractible to a point,
then H2(X, Z) = o.
Theorem 5.10. If X is a Stein manifold and H2(X, Z) = 0, then every Cousin I I

problem on X is solvable.
PROOF. Immediate corollary of Theorem 5.9. o
Therefore, every Cousin II problem on X is solvable if X is a non-compact
connected Riemann surface or an arbitrary contractible Stein manifold.
Specifically it follows that if G c: e is a domain, then all Mittag-Leffler
and Weierstrass problems on G are solvable. So far we have only used
Theorem B. Interesting possibilities for applications of Theorem A are found
primarily in the area of analytic subsets of Stein manifolds.
Def. 5.2. Let A be an analytic subset of a complex manifold X. A complex

valued function f on A is called holomorphic if for every point Xo E A
there is a neighborhood U(xo) c: X and a holomorphic function f on
U with flU (") A ;", flU (") A.
176
For analytic sets which are free of singularities (therefore sub manifolds)
this coincides with the old notion of holomorphy.
Theorem 5.11. Let (X, (!)) be a Stein manifold, A c X an analytic subset and
f a function holomorphic on A. Then there is a holomorphic function J on
X with JIA = f. (Global continuation!)
PROOF. We assign to every point x E A a neighborhood U x c X and a
holomorphic function Ix such that IxIA nUx = flA nUx. To every point
x E X - A let there be assigned a neighborhood U x c X with U x n A =
o and the function Ix: = 0IU x' Let
U: = (Ux)xEX' 1](x): = Ix E T(U x , (!)).
Then 1] E CO(U, (!)) and ~: = blJ E Zl(U, (!)). Moreover, for all Xo, Xl E X
~(Xo, xdl A n U xox • = LolA n U xox • -L.IA n U xox • = O.
Therefore ~ E Zl(U, J(A)), where we denote the ideal sheaf of A by J(A).
By Theorem B, H1(X, J(A)) = 0 and hence also H1(U, J(A)) = O. There-
fore there is apE CO(U, J(A)) with bp = ~, that is, b(1J - p) = O. There is
a holomorphic function J E T(X, (!)) defined by
JI U x: = lJ(x) - p(x) = L - p(x)
and
That is, JIA = f. D
Theorem 5.12. Let (X, (!)) be Stein, X' c c X open, Y a coherent analytic
sheaf over X. Then there are sections Sb ... , s( E T(X, Y) which at each
point x E X' generate the stalk Y x over (!)x'
PROOF
1. Let Xo E X'. Then there exists an open neighborhood U(xo) c X and
sections tb ... , tq E T(U, Y) such that for every point x E U the stalk //'x
over (!)x is generated by t1 (x), ... , tq(x). Now, by Theorem A there are global
sections 8 1 , •.. , sp E T(X, Y) and elements aij E (!)xo such that
p
ti(xo) = L aijsj(xo)
j= 1
for i = 1, ... , q.
There exists an open neighborhood V(x o) c U and sections aij E T(V, (!))
with aiixo) = aij for all i,j. Hence it follows that there exists an open neigh-
borhood W(xo) c V with tilw = Ct1 aijSj)lw for i = 1, ... ,q; that is,
sp generate each stalk Y x , x E W.
Sb' .. ,
2. Since X' is compact, we can find finitely many points Xl, ... , Xr EX',
open neighborhoods JV;(x;) and global sections
(i)
Sl , ... ,
(i)
Sp(i)' i = 1, ... , r
177
such that Wi u ... u Tv, covers X, ;

(i) (i)
Sl , . . . , Sp(i)' i = 1, ... , r,
generate g' on W;. Then
(i) (i)
Sl , ... , Sp(i)' i = 1, ... , r,
generate the sheaf g' on X'. o
Theorem 5.13. Let (X, lP) be Stein, X' c c X open, A c X analytic. Then
there are holomorphic functions flo ... , ft on X such that
A n X' = {x E X':fl(X) = ... = ft(x) = O}.
PROOF. Since J(A) is a coherent analytic sheaf on X, by Theorem 5.12
there exist global sectionsfl, ... ,ft E r(X, J(A)) c r(X, lP)which generate
each stalk of J(A) over X'. Clearly
A n X' c {x E X': [fl(X)] = ... = [fAx)] = O},
so we need only show the converse. (Recall that for an element f E r(X, lP)
the corresponding holomorphic function is denoted by [f].)
l
If Xo E X' - A, then there are elements av E lP"o with I avfv(xo) =
v=l
1 E lP"o. Then in a neighborhood V(xo) c X' - A the function 1 has the
I avE.f.], where the av are holomorphic functions in
l
representation 1 =
v=l
V. But then not all the [Iv] can vanish at Xo.
Therefore
{x E X': [fl(X)] = ... = [ft(x)] = O} cAn X'. o
We record the following sharpened version of Theorem 5.13 without proof.
Theorem 5.14. Let X be an n-dimensional Stein manifold, A c X an analytic

subset. Then there exist holomorphic functions fl' ... ,fn+ 1 on X such that
A = {xEX:fl(X) = ... = fn+l(X) = O}.
We note that the theorem does not imply that J(A) is globally finitely
generated. Indeed, there is an example due to Cartan which shows that this
is not possible, in general.
178
CHAPTER VII
Real Methods
1. Tangential Vectors
In this section X is always an n-dimensional complex manifold.
DeC. 1.1. Let kENo. A k-times differentiable local function at Xo E X is a

pair (U,f) such that:
1. U is an open neighborhood of Xo in X;
2. f is real-valued function on U continuous at Xo; and
3. there exist a neighborhood V(x o) c U and a biholomorphic mapping
t/I: V -+ G c en such that f t/I-l at t/I(x o) is k-times differentiable.
0
Complex valued local functions can be defined correspondingly.

Let the set of all k-times differentiable functions at Xo be denoted by
E»~o' Instead of (U, f) we usually write f.
Remark. Since the coordinate transformations are biholomorphic, so in

particular k-times differentiable for every k, Definition 1.1 is independent of
the choice ofthe coordinate system (V, t/I). The elements of E»~o can be added
and multiplied by real or complex scalars in the obvious fashion. (For
example (U, f) + (U', 1'): = (U n U', f + 1').)
A well-known theorem says that if f E E»!o' g E E»~o and f(xo) = g(x o) =
0, then f . g E E»~o (see [21 ]).
Def. 1.2. A (real) tangent vector at Xo is a mapping D:E»!o -+ IR such that:

1. Dis IR-linear;
2. D(l) = 0; and
3. D(f· g) = 0 for f E E»!~ and g E E»~o such that f(x o) = g(x o) = O.
179
VII. Real Methods
We call (2) and (3) the derivation properties. The set of all tangent vectors
at Xo is denoted by Txo.
Remark. T Xo forms a real vector space. The partial derivatives

a a
OX1'···' oXn' OY1'· .. , 0Yn
which depend on the choice of the coordinate system form a basis for T Xo
(see [21]). Therefore dimij;l Txo = 2n. For complex valued local functions
f = g + ih at Xo and DE Txo we set D(f): = D(g) + iD(h). D remains IR-
linear!
Theorem 1.1. If C1> ••• , Cn are arbitrary complex numbers then there exists
n a
.exactly one tangent vector D with D(f) = V~l Cv oXv (f) for each function f
holomorphic at xo. In particular, a given tangent vector D is already uniquely
determined by its values on the holomorphic functions. In local coordinates
D has the representation
nan a
D = V~l Re(D(zv» oX v + V~l Im(D(zv» 0Yv
PROOF. If cv = a v + ib v for v = 1, ... , n, we set

nan a
D:= I a -+
v=l
I b
v
oX v=l
v -·
0Yv
v
Then for each function f holomorphic at Xo (because h, = ifxJ
Hence Cv = D(zv) for v = 1, ... , n. It is clear that D is uniquely determined

by its values on the holomorphic functions as well as by the numbers c 1 , . . . ,
~. 0
Theorem 1.2. If c E C and DE T xo ' then there exists exactly one tangent
vector c· D E T Xo such that (c . D)(f) = c· (D(f» for every function f
holomorphic at Xo.
PROOF. There exist complex numbers c1 , ••• , Cn such that
180
for every function f holomorphic at Xo; and by Theorem 1.1 there is exactly
n 0
one tangent vector D* with D*(f) = V~l (ccJ oX v (f) = c· (D(f)) for holo-
morphic f. We set c· D = D*. 0
Theorem 1.3. Let

. 0 0 . 0
r·-=- and r·-= for v = 1, ... , n.
oX v oYv oYv
Then Txo is an n-dimensional complex vector space with basis {0/ox 1 , ••• ,
%xn}, and this new complex structure is compatible with the given real
structure on T Xo.
PROOF. Iff is holomorphic at xo, then
( i . ~) (f) = i (~(f)) = i· fzv = ~ (f).

oXv oXv °Yv
The axioms of a Q:>vector space are clearly satisfied; in particular
o
i. O~v = i· (i . o~J = (i. i) . O~v = ox;
Therefore
{0~1'···' o~J
forms a system of generators of Txo over C.
n 0
If I Cv • = 0 with cv = av + ib v for v = 1, ... , n, then
~
v= 1 UXv
nOn 0 non 0
0= I av~ + i· I bv~ = I av-" + I bv -;-,
v=l UXv v=l UXv v=1 oXv v=1 UYv
therefore av = bv = 0 for v = 1, ... , n. That is, {%x b ... , O/ox n} is a basis

for T Xo over C. 0
Remark. A complex tangent vector at Xo is a C-linear mapping D:~!o ~ C

with the derivation properties (2) and (3) of Def. 1.2. Let the set of all complex
tangent vectors at Xo be denoted by T~o. Then we set
T~o: = {D E T~o:D(J) = 0 if f is hoI om orphic at x o},
T~o: = {D E T~o: D(f) = 0 if f is holomorphic at xo}·
We call the elements of T~o holomorphic tangent vectors, the elements of T~o
antiholomorphic tangent vectors. The partial derivatives 0/OZ1, ... , %zn resp.
0/OZ1, ... , %zn form a basis for T~o resp. T~o' and T~o = T~o EB T~o·
181
VII. Real Methods
We can now assign to every element DE Txo complex tangent vectors

D' and D" E T~o such that D = D' + D". If
E T~o
a a
Iv=1 avaxv + I b -,
II II
D = -
v=1 Vayv
we set
1 a
I
II
D': = -2 (a v + ibv)-a
v= 1 Zv
1 a
I a-
II
D": = -2 (a v - ib v )
v= 1 Zv
Clearly D'(f) + D"(f) = D(f) for every f E f0,!o. Hence we can write every
real tangent vector D E T Xo in the form
a a
I + I cv - ·
II II
D = cv -
v= 1 az v az..= 1
If c E C, then
a a
I CC. -a + I a- .
II II
c.D = CC.
• =1 Z. .=1 Z.
Def. 1.3. An r-dimensional complex differential form at xo is an alternating

IR-multilinear mapping
({J : !' Xo x . ~. x T x~ ~ C
r-times
The set of all r-dimensional complex differential forms at xo is denoted

by Ft~.
Remarks
1. Ft~ is a complex vector space. We can represent an element ({J E Ft~
uniquely in the form ({J = Re( ({J) + i Im( ({J), where Re( ({J) and Im( ({J) are
real-valued differential forms (cf. [22]). It follows directly that
dim~ Ft~ = ern) + ern).

so that
·
d Ime Xo
F(r)
= (2n)
r .
2. BY conven t IOn
· xo -- i0
F(O) 'G.
F or r -- 1 we 0 bt· Xo -- T*xo d7
am F(l) CJ:\ ·T*
I xo'
with T~o = Hom~(T Xo' IR). F~~) is the complexification of the real dual space
of Txo.
3. We associate with each element ({J E Ft~ a complex-conjugate element
lp E Ft~ by setting
182
We have
a. cp = cp.
b. (cp + ljJ) = cp + l/i, ccp = c· cp.
c. cp is real if and only if cp = cp.
If we define the element dz. E F~~) by dz.(~): = ~(z.), then we obtain an
additional element dZ. E F~~ from
dZv(~): = az.(~) = ~ = ~(zv) = ~(zv)'
{dz 1 , ••• , dz n, dZ1 , .•• , dZn} is a basis of F~~). In general cp = Re(cp) -
i Im( cp ); as a special case
dz v = dx. + i dy., dZ. = dx v - i dy•.
4. Let cp E F~~, ljJ E F~~. The wedge product cp /\ ljJ E F~o+S) is defined as
in [22]:
cp /\ ljJ(~l>""~" ~r+ 1>' •• , ~r+s): =
1
-'-I
r.s.
I
ae ,sr+s
(sgn O')CP(~O'(l)' ... , ~O'(r») . ljJ(~O'(r+ 1), ... , ~O'(r+s»)'
Then:
a. cp /\ ljJ = (_I)'·sljJ /\ cp (anticommutative property);
b. (cp /\ ljJ) /\ W = cp /\ (ljJ /\ w) (associative property).
In particular
00
With the multiplication " /\" F xo:. = EB F~~ becomes a graded associative
r=O
(non-commutative) ring with 1.
5. For j = 1, ... , n let dz n + j: = dZj • Then F~~ is generated by the elements
dZ v1 /\ ••• /\ dZ vr with 1 ~ V1 < ... < Vr ~ 2n. The number of these ele-
ments is exactly (2rn) ; so they form a basis.
Theorem 1.4. If Zl' ... , Zn are coordinates of X near xo and if cp E F~~, then
there is a uniquely determined representation
cp=
(normal form of cp with respect to Zl" •. ,zn). In particular cp = 0 for

r > 2n; therefore F~~ = 0 for r > 2n.
Def. 1.4. Let p, q E ~ 0 and p + q = r. cp E F~~ is called a form of type (p, q)

if
for all c E C.
183
VII. Real Methods
Theorem 1.5. If <p E Fr~, ... , ~r) 1= O. Then
<p(c~1> ... ,c~r)= { cpcq<P(~ 1, . . . , ~r)

p'-q' (I' 1')
C C <P<"1"",<"r
Therefore cp'c q'= cp'cq' for each c E C. Let c = eie with arbitrary E IR. Then e
ei9 (p-q) = ei9 (p'-q'). That can hold for all only when p - q = pi - q'. Since e
p + q = pi + q' = r by assumption, it follows that p = pi, q = q'. 0
Theorem 1.6.
1. If <p is of type (p, q), then lp is of type (q, p).
2. If <p, l/J are of type (p, q), c E C, then ... ,c~r) = <p(c~l> ... ,c~r) = cPcq<P(~1""'~r) =cpcqlp(~l>"" ~r)'
(2) Trivial.
(3) <p(c~ 1, ... , c~r)l/J( c~r + 1> ••• , c~r+.) = cpcqcp'c q' <p( ~ 1>' •• , ~r )l/J( ~r + 1> ... , ~r+s)'
Therefore
 ••• ,
1')
<"r+s . o
Theorem 1.7. If <p E Fr~, then <p has a uniquely determined representation
<p = L <p(p, q)
p+q=r
where <p(P, q) E Fr~ are forms of the type (p, q).
PROOF. Clearly dz v is of type (1,0), rlzv of type (0, 1). Hence it follows that
monomials dZ il /\ ... /\ dz ip /\ dZh /\ ... /\ dZjq (with 1 ~ i1 < ... < ip ~
nand 1 ~ j1 < ... < jq ~ n) are forms of type (p, q).
<p = L <p(P, q)
p+q=r
with
<p(P,q): =
L
1 ~ i l < ... < ip ~ 11
ail'" ip,lI+ h,'" ,11+ jq dZil /\ ... /\ dz ip /\ dZh /\ ... /\ rlzjq
1 :::;;. it < .. < jq ::;; 11
is therefore a representation of the desired sort. Let

<p = L <p(P,q) = L 0(P,q).
p+q=r p+q=r
184
2. Differential Forms on Complex Manifolds
Then
L ljJ(P, q) =0 for ljJ(P,q) = cp(p,q) - ip(p,q)
p+q=r
It follows that
o= L ljJ(p,q)(C~b"" C~r) = L CPcqljJ(p,q)(~l"'" ~r)
p+q=r p+q=r
For fixed (~l> ... , ~r) we obtain a polynomial equation in the polynomial
ring iC[ c, c]. Then the coefficients ljJ(P, q)( ~ b . . . , ~r) also vanish for all p, q.
Since we can choose ~b . . . , ~r arbitrarily, we have cp(P,q) = ip(p,q) for all
~q. 0

Def.2.1. Let X be a complex manifold. An ({arm on X is a mapping
cp:X -- U F~)
XEX
with the property that cp(x) E F~) for every x E X. If Zl' ... , Zn are coor-
dinates on an open subset U c X, then for x E U
X f--+a" ... ,ix) defines a complex valued function a"", 'f on U. We call
cp Xo E U if all functions a" ... If are k-times differ-
k-times differentiable at
entiable at Xo. This definition is independent of the choice of coordinates.
cp is called k-times differentiable (on X) if cp is k-times differentiable at
every point of X.
Henceforth the set of all arbitrarily often differentiable (-forms will be

denoted by AU) = A(!)(X), by A(p,q) the set of all arbitrarily often differ-
entiable forms of the type (p, q).
Def.2.2. If f is an arbitrarily often differentiable function on X (therefore

an element of A(O»), then we define an element df E A(l) by (df)x(~): =
~ (f) for ~ E T x (total differential of f).
Remarks
1. For the basis elements dz v , dz v the definition does not change anything.
2. In local coordinates
n n
df = L h, dz
v=l
v + L fzv dz
v=l
v'
PROOF. We write
n n
df = L a v dz v + L b v dz v ·
v=l v=l
185
VII. Real Methods
and hence
n n
df(~) = ~(f) = I Cv!ZV + I Cv!ZV·
v=l v=l
In particular it follows that
We can also define a total differential d:A(I) --+ A(l+ 1) on manifolds. It

has the following properties:
1. 4- is IC-linear.
2. d(f) = df (in the sense of Def. 2.2) for f e A (0).
3. d(<pIU) = (d<p) IU.
4. If <p e A(r), ljI e A(s), then d(<p A ljI) = d<p A ljI + (-1)'<p A dljl.
5. If
<pIU = I all ••• , ( dz" A ..• A dz,(,
1::S;Jt<"'<u::S;2n
then
d<P1 U = I da" ... It A dz" A ••• A dZI{"
1 ::S;ll < ... <1(::E;2n
6. dod = O.
7. d is a real operator; that is, dip = ([(p. In particular then d<p = d(Re <p) +
id(Im <pl.
Theorem 2.1. If <peA(p,q), then d<p = d'<p + d"<p with d'<peA(p+l,q) and
d"<p e A(p,q+1).
PROOF. One usually abbreviates the normal form of <p(p,q) as
<p(p,q) = I aI,Jd?H A aoJ·

I,J
Then
d<p(p,q) = I daI,J A d31 A aoJ

I,J
which is a decomposition of d<p(P,q) into a form d'<p(P,q) of type (p + 1, q)

and a form d"<p(P,q) of type (p, q + 1). 0
186
If <p = L <p(p, q) is an arbitrary t-form, then we call d' <p: = L d' <p(P, q)
p+q=l p+q=l
the total derivative of <p with respect to 3 and d" <p: = L d" <p(P, q) the
p+q=l
total derivative with respect to 3. (In the English literature one generally
writes (} instead of d' and "0 instead of d".)
Theorem 2.2.
1. d' and d" are ~>linear operators with d' + d" = d.
2. d'd' = 0, d"d" = 0 and d'd" + d"d' = O.
3. d', d" are not real. Moreover d'<p = d"ip and d"<p = d'ip.
4. If <p is an t-form, ljJ arbitrary, then
d'(<p /\ ljJ) = d'<p /\ ljJ + (-Ir<p /\ d'ljJ,
d"(<p /\ ljJ) = d"<p /\ ljJ + (_I)I<p /\ d"ljJ.
PROOF. It suffices to prove this for pure forms:

(1) is trivial. For (2):
o= dd<p = (d' + d") (d' + d")<p = d'd'<p + d'd"<p + d"d'<p + d"d"<p.
0
If <p has type (p, q), then d'd'<p has type (p + 2, q), (d'd"<p + d"d'<p) has type
(p + 1, q + 1), and d"d"<p has type (p, q + 2). Since the decomposition into
forms of pure type is uniquely determined, the proposition follows.
For (3), since d<p = dip it follows that
d'<p + d"<p = d'ip + d"ip; therefore (d' <p - d"ip) + (d" <p - d'ip) = o.
Hence d'<p - d"ip has type (q, p + 1) and d"<p - d'ip has type (q + I, p).
Therefore both terms must vanish.
For (4), both formulas follow from Rule (4) for the total derivative d by
comparing types as in (2) and (3). D
Remark. A real differentiable function f is holomorphic if and only if

f7., = 0 for v = 1, ... , n, that is if d'j = O. Correspondingly it follows for
<p = <p(P,O) = L ai ! ••. ip dz i ! /\ ••• /\ dz ip
l~il<···<ip~n
that d" <p = 0 if and only if ai, ... ip is always holomorphic. Hence we make
the following definitions.
Def. 2.3. <p E A(t) is called holomorphic if
1. <p is of type (p, 0), and
2. d"<p = O.
<p E A(t) is called antiholomorphic if
1. <p is of type (0, q), and
2. d'<p = O.
Remark. Clearly <p is antiholomorphic if and only if ip is holomorphic.
187
VII. Real Methods
3. Cauchy Integrals
The Poincare Lemma from real analysis (see, for example, [22J) can be
formulated as follows:
Let Been be a star-shaped region (for example, a polycylinder), cP E A(t),
.e > 0, dcplB = O. Then there exists a t/I E AU-I) with dt/l = cp.
We will below prove a similar theorem for the d" operator. In order to
do this, we must first generalize the Cauchy integral formula.
If Bee e is a region and J a complex valued, continuous, bounded
function on B, then there exists a continuous function Cht) on C defined by
1. f
cht)(w): = -2
1tl JB z
J(z)
- W
dz 1\ az.
Specifically, let el>: [0, (0) x [0, 2n) --I- e be defined by el>(r, 0): = re i8 + w,
and let B* be the region el> - I(B). Then
( J(z) dz
z - w
1\ az) 0 el> = J(iP(r, 0» del>
el>(r, 0)
1\ d~
= 2i . J(r· e i8 + w) . e- i8 dr 1\ dO
is a continuous, bounded differential form on B*. Hence
z- w
J(z) dz 1\ az
is integrable over B, the integral is continuously dependent on w, and
Cht)(w) = ±~ fBo J(re i9 + w)e- i8 dr 1\ dO.
If the real numbers R, k > 0 are chosen so that IZI - z21 ~ R for Zl> Z2 E B
and IJ(z) I ~ k for z E B, then we get the following estimate:
Icht) (w) I ~ ~ fBo dr 1\ dO ~ 2kR.
Now let Pee be a circular disk (therefore a polycylinder), and T: =

OP. If g is holomorphic on P, then the Cauchy integral formula holds:
1. f
g(w) = ch(gIT)(w) = -2 g(z) dz, for WE P.
1tl JT Z - W
As a generalization we obtain
188
3. Cauchy Integrals
Theorem 3.1. Let g be continuously differentiable on P, J: = gz bounded.

Then Jar w E P
g(w) = ch(glT)(w) + Chr)(w).
PROOF. Let w E P, Hr a small circular disk about w with Hr C C P, and

T,.: = oHr· If T and T,. are given the usual orientation, then it follows from
Stokes' theorem (see [22]) that:
Chr)(w) = _~
2m JP-Hr
r
d (g(Z) dZ) + ~
Z - w
J(z) dz /\ dZ
2m JHr Z - W
r
1 r g(z)
= - 2'Tti Ja(p-H r ) z _ w dz + Ch~Hr)(w)
= _~
2m JT z - w
r
g(z) dz + ~ r
g(z) dz + Ch1r)(w)
2m Jrr z - w
= -ch(gIT)(w) + ch(gITr)(w) + Ch1r) (w).
Hence the function p(r): = ch(gITr)(w) + Ch~Hr)(w) has the constant value
ch(glT)(w) + Chr)(w), and it suffices to consider the limit for r --+ 0:
p(r) = a(r) + b(r) + c(r)
with
a(r):
1
= -2'
'Ttl
i --
Tr
g(w)
Z- W
dz
1
= g(w) . -2'
'Ttl
i -- =
Tr
dz
Z- w
g(w),
b(r): =~ r g(Z) - g(W) dz and c(r): = Ch1r) (w).

2m JTr Z - W
Since g is continuously differentiable as a function of w there exist functions

A', A" which depend continuously on w such that
g(z) = g(w) + (z - w) . A'(z) + (z - w)' A"(z).
If we choose ro and M such that IA'(z)l, 1,1 "(Z) 1 < M for z E Hr and r ~ ro,
then we get
- g(W)1 ~ IA'(z)1 + IA"(z)I' ~ - wi ~ 2M

Ig(Z)z-w Iz-w for z E Tr and r < ro;
therefore
1
Ib(r)1 ~ -2
'Tt
iTr
Ig(Z) - g(W)1 dz ~ 2M· r
Z - W
for r < roo
Hence
Ib(r) + c(r) 1 ~ 2Mr + ICh~Hr)(w)1 ~ 2r' (M + 2· sup 1J(P) i),
and this expression becomes arbitrarily small. Hence it follows that
p(r) == g(w). 0
189
VII. Real Methods
Theorem 3.2. Let f be continuously differentiable on C, Supp(f) c c C,

Pee a circular disk with Supp(f) c P.
Then g: = Chr) is continuously differentiable on C, with gz = f.
PROOF. Let
Pc: = {ZEC:Z + CEP},
y(w, c): = Chr)(w + c) = ~ r f(z) dz !\ dz
2m Jp Z - W - C
= _1_ r f(z + c) dz !\ dz.

2ni Jpc Z - W
Because
Supp(f) c P, y(w, c) = -2.

1 ~ f(z + c) dz !\ dz.
m C z - w
By known theorems on parametric integrals (see [22]), y is continuously
differentiable with respect to c and c. Since y(O, c) = g(c), g is differentiable.
Applying formulas for the derivative of parametric integrals and the chain
rule gives
gz (c) = _1
2'
mC
~ fiz + c) dZ!\ d-Z = _1
z
fz(z) d
mPz-c z
2'
l .:r.:
!\ uZ
= Ch(P)(
J c,
)
l
Z
1
gz(c) = -2' -h(z)
- - dz !\ dz = Chr)(c).
m Pz-c z
Since f vanishes on T: = ap, it also follows from Theorem 3.1 that gz =

Ch)!i = f - ch(flT) = f. 0
Theorem 3.3. Let Bee C be a region, f continuously differentiable and

bounded on B.
Theng: = Chj!l) is continuously differentiable on Band gz = f.
PROOF. Let Wo E B be given, H an open circular disk about Wo with H c c B.

We can then find an arbitrarily often differentiable function p: C ~ IR for
which
1. 0 ~ p ~ 1,
2. plH = 1,
3. Supp(p) c c B.
Then let f1: = p . f, f2: = f - fl' Clearly
f1 + f2 = f and Chj!l! + Chj!l; = Chj!l).
Moreover, f11H = flH and f21H = O. f1 is even continuously differentiable
on all of C and if P is a circular disk with B c P, then Ch~) = Chr,). Hence
it follows from Theorem 3.2 that Ch~) is continuously differentiable on C
and (ChW)z = fl'
190
4. Dolbeault's Lemma
For w E H we also have
Ch(B)(w) = _1_ r f2(z) dz!\ dz = _1_ r f2(z) dz !\ dZ

h 2ni JB z - w 2ni JB-H z - W '
the integrand is continuous and bounded on B - H, as well as holomorphic

with respect to w. From the theory of parametric integrals it follows that
Ch(1!IH is continuously differentiable and
(Ch~lIH)w = °
Therefore glH is continuously differentiable and (gIH)z = flH. D
Remark. If Bee, B* c [Rn are regions, Bee B open and f : B x B* ~

C arbitrarily often differentiable, then it follows from the theory of para-
metnc
<
h C h(B)'
. .Integra Istat f wIth Chf(B) (w, x): = -.
1 f(z,-
- x) dZ!\ d'
2m
L
Z IS arb'1-
B Z - w
trarily often differentiable on B x B*, and
(Ch(B») (w x)
f x,,,
= _1_
2ni
r f~,(z,
JB z _
x) dz
W
!\ dZ (Ch(B»)_
'f w
= f.
4. Dolbeault's Lemma
Theorem 4.1 (Dolbeault's lemma): Let Kv c C be compact sets for v =
1, ... ,n, Uv open neighborhoods of K v , K: = KI x ... K n , U: =
U I x ... X Un.
Moreover, let 0. Then there
exist an open set U' with K c U' c U and a ljJ E A(O, q-l)(U') with
d"ljJ = <pIU'.
If <p is arbitrarily often differentiable as a function of real parameters,
then ljJ is also arbitrarily often differentiable as a function of these parameters.
PROOF. By induction on n.
= 1, then also q = 1 and <p has the form 1. The
operators d; and O/OZI are defined by
d; (IJ
aJd3J ): = I f ~~J
J v=2 UZ v
Mv !\ d3J
so that
191
VII. Real Methods
If we write cP in the form cP = dZI /\ CPl + CP2, where CPb CP2 no longer
contain azb then
O = d" cP = uZ1
-1= /\ (d"
- .CPl + OCP2)
OzI + d".CP2·
Since d;CP2 contains no az b it follows that d;CP2 = O.

Now regard ZI as an additional parameter and apply the induction
hypothesis.
Let
K.: = K2 x .. , x K n, V.: = V 2 x ... X Vn.
There is an open set V: with K. c V: c V. and a Ij; = Ij;(O.Q-l) in which
ZI appears as a parameter, such that
d 'I'I'IV I x V'• = CP2

l
•' • IV 1 X V'•.
On V': = V 1 X V: we have
".I,
cP - d 'I' = cP -.'1' -
d".I, -1=
uZ1 olj; = d-ZI
/\ OZI /\
( CPl - OZI
Olj;) '
where CPl - (OIj;/OZI) contains no dz1. On the other hand
o= d"(cp - d"lj;) = dZI /\ d; (CPl - :~}

therefore
For the case q ~ 2 by the induction hypothesis there are an open set
V" with K c V" c V' and a·7. = .7.(O,Q-1) on V" such that
* * * * tp ljI *
d"'7.
.'1' = (m _ Olj;)\v"
't'1 Cl-
UZI
••
Hence on V": = VI x V;
d"(dz1 /\ ijJ) = -dz1 /\ d;ijJ = -dz1 /\ (CP1 - :~) = d"lj; - cP,
thereforecp = d"(1j; - az 1 /\ ijJ).

For q = 1, CP1 - (01j;/OZ1) is a function a = a(zb Z2,' .. , zn) which is
holomorphic in Z2, ... , Zn' We regard Z2, ... , Zn as additional parameters
and determine a region V'I with K c V'I C cUI by (1), and a function
f = ChiU'!l with d''f = a az1 = cP - d"lj;. Then cP = d"(f + Ij;). This com-
pletes the proof. D
We immediately obtain the following result for manifolds X.

If cP E A(O,q)(V), q ~ 1, V c X open and d"cp = 0, then for every x E V
there exists an open neighborhood V(x) c V and a Ij; = Ij;(O,q-l) on V with
d"lj; = cpl V. (Let K = {x}.) We present the following theorem without proof.
It provides us with some more precise information.
192
5. Fine Sheaves (Theorems of Dolbeault and de Rbam)
Theorem 4.2 (Lieb). Let Gee be a domain with smooth boundary 8G

(see De! 2.5 in Chapt. II; the function <p defining the boundary are arbi-
trarily often differentiable). Let the Levi form of the defining functions be
positive definite everywhere. (In such a case one calls G strongly pseudo-
convex.)
Let w = L:
aJ aoJ be an arbitrarily often differentiable form of type
J
(0, q) on G with d" w = 0. Moreover let there be given a real constant M with
Ilwll: = max suplaAG)1 ~ M.
J
Then there exists a constant k independent of wand a form !/J of type

(0, q - 1) on G with d"!/J = wand II!/JII ~ k· M.
According to Siu and Range there exists a generalization of this theorem

for domains with piece-wise smooth boundaries (See R. M. Range, and
Y. -T. Siu: Uniform estimates for the a-equation on intersections of strictly
pseudo-convex domains. Bull. Amer. Math. Soc., 78(5): 721-722, 1972).
5. Fine Sheaves (Theorems of Dolbeault and de Rham)

In this section X is always a paracompact complex manifold.
Def. 5.1. A test function on X is an arbitrarily often differentiable function

t:X ~ IR1 with compact support.
Let the ring (without I) of all test functions be denoted by T. Let:!l be the
sheaf of germs of test functions on X.
Remark. Let U = (U,},El be an open covering of X. Since X is para-

compact, there exists for U a subordinate partition of unity, that is, a system
(t,), El of test functions with the following properties:
1. ° ~ t, ~ 1 for every 1 E I;
2. Supp(t,) c U, for every 1 E I;
3. the system of sets Supp(t,) is locally finite;
4. I t, = 1 [by (3) the sum is finite at each point].
,eI
Def. 5.2. Let !I' be a sheaf of T-modules over X. !I' is called fine if for all
x E X, (j E !I' x and t E T
1. t· (j = 0 if x ¢ Supp(t)
2. t· (j = (j if x ¢ Supp(l - t).
Remarks
1. If !I' 1, . . . ,!I'r are fine sheaves, then !l'1 EB ... EB 9'; is also fine.
2. The sheaf Jdp.q of germs of (arbitrarily often differentiable) forms of
type (p, q) defined by the pre-sheaf {Ap.q(U), rn is clearly a fine sheaf.
193
VII. Real Methods
The sheaf
p+q=l
is fine, by (1). Here
reU, dt) = EB reU, dp.q) = EB Ap.q(U) = At (U),
p+q=l p+q=l
that is, d t is the sheaf of germs of arbitrarily often differentiable i-forms.
Theorem 5.1. Let ff', ff" be fine sheaves over X, <p: ff' -> ff" an epimorphism
of sheaves of T-modules. Then <P* :reX, ff') -> r(X, ff") is surjective.
PROOF
1. Let s' E reX, ff"), x E X. Then there exist a (J E ff'x with <p((J) = s'(x),
a neighborhood W(x) c X and a section s* E reW, ff') with s*(x) = (J, so
that <p 0 s*(x) = s'(x). We can find a neighborhood Ux(x) c W with
<p 0 .s*IUx = s'IU x ' Let sex): = s*IU x'
2. u = {Ux:x E X} is an open covering of X. Let (t(X»XEX be a subordinate
partition of unity. For x E X t(x) . sex) is an element of reX, ff'). Since the
system of sets Supp(t(X» is locally finite, for fixed Xo we have t(x) . s(X)(xo) = 0
for almost all x E X. Therefore
s: = L t(x)' sex)
XEX
is also an element of reX, ff') and
(<p 0 s)(xo) = <p (L xeX

t(x)' S(X)(x o») = L t(x)' <p(s(x)(xo»
xeX
= L t(x)' s'(xo)
xeX
L
xeE
(t(x)' s'(xo» = (L xeE
t(X»)' s'(x o) = s'(xo),
where E is a finite set and L t(x) ==

XEE
1 near xo' 0
Theorem 5.2. If ff' is fine, then Hf(X, ff') = 0 for t ~ 1.

PROOF. Let 0 -> ff' -> ff' 0 -> ff'1 -> ff' 2 -> ... be the canonical flabby res-
olution of ff'. ff' and all the ff'v are sheaves of T-modules. By induction on
v we show that the ff'v are all fine.
ff' 0 = W(ff') is defined by the pre-sheaf {f(U, ff'), r~}. f(U, ff') is a
T-module with t· s = 0 ifSupp(t) n U = 0 and t· s = s ifSupp(1 - t) n
U = 0. Therefore ff' 0 is fine.
Now let ff' 0, . . . , ff't be fine and t ~ O. The homomorphisms which
appear take into account the T-module structure. Therefore the subsheaves
f!lJi: = Im(ff'i-1 -> ff'i)
are fine for i = 0, ... , t and ff' _ 1: = ff'; hence ff't + 1 = W(ff' tl f!lJ() is fine.
Since all the sheaves Xi: = Ker(ff'i -> ff'i + 1) are fine, we obtain epi-
morphisms offine sheaves: ff'i-1 -» Xi = f!lJi' By Theorem 5.l,r(X, ff'i-1) ->
194
5. Fine Sheaves (Theorems of Dolbeault and de Rbam)
r(X, X;) is also surjective, and therefore

Im(r(X, :7 i - 1 ) --+ r(X,:7J) = Ker(r(X,:7;) --+ r(X, :7 i + 1)),
i.e., Hi(X, :7) = 0 for i ;;: 1. o
Def. 5.3. The sheaf of germs of holomorphic (p, O)-forms on X will be
denoted by QP. A holomorphic (p, O)-form qJ = qJ(P'O) has a local
representation
qJ = L ail' .. ip dZ il !\ . . . !\ dz ip '
1 ::::;il < ... <ip~n
with holomorphic coefficients ai I ... i p '
Thus the sheaf QP is locally isomorphic to the (free) sheaf (;) . (JJ. We
also call QP a locally free sheaf. In particular QP is coherent.
There is a canonical injection 1>: QP ~ .sip. and the differential °
d":Ap,q(V) --+ AP,q+1(V)
induces homomorphisms of sheaves of abelian groups:
Theorem 5.3. The following sheaf-sequence is exact:

o --+ QP t. .siP' ° --+
d" d"
.siP' 1 --+ .siP' 2 --+ •..
PROOF
1. It is clear that d" 0 I> = 0 and d" 0 d" = O.
2. Let x E X, V be a coordinate neighborhood of x in X. An element
qJ E A p, q( V) has the form
qJ = L dZ il !\ ... !\ dz ip !\ qJil " ip'

l::Sil<"'<ip~n
with qJi l '" ip E A 0, q(V). Therefore

d"qJ = L (-1)P dZ il !\ ... !\ dz ip !\ d"qJi1"'i p'
l::::;il<"'<ip~n
and so d"qJ = 0 implies that d"qJil"'i p = 0 for all i 1, ... , ip.

According to Dolbeault there are neighborhoods Vii'" ip of x with
Vii'" ip C V, as well as forms t/!il'" ip of type (0, q - 1) on Vii'" ip such
that d"t/!il"'i p = qJil"'ipIVil ... ip' Let V' be the intersection of all sets
Vil " ' ip and
t/!: = L
1 ::::;i 1 < ... <ip::::;n
(-1)Pdz i1 !\'''!\ dZ ip !\ t/!il"·ipIV'.
Then d"t/! = qJl V'.

3. Let (J E .sIP'o, X E V, V an open neighborhood of x and qJ E AP' O(V)
such that (J = rqJ(x). 0 = d" (J = r(d" qJ )(x) if and only if d" qJ = 0 near x,
195
VII. Real Methods
if therefore cp is holomorphic near x. That means (J E QP. The sequence is

exact at Slip· 0.
4. Let q ;? 1, (J E Slif,q, U a neighborhood of x and cp E AM(U) be such
that (J = rcp(x). 0 = d"(J = r(d"cp)(x) if and only if d"cp = 0 near x, and
without loss of generality, on U.
By (2) this is equivalent to the existence of a neighborhood U'(x) C U
andaIjlEAP,q-l(U')withd"ljI = cp!U',andthatisequivalentto(J = rcp(x) =
r(d"ljI )(x) = d"(rljl )(x). Therefore the sequence is exact at SliM. 0
Def.5.4. The induced sequence

o ~ r(X, QP)':" r(X,SliP' 0) ~ r(X,SliP' 1) ~ ...
is called the Dolbeault sequence. Clearly we have an augmented cochain
complex (of (:-vector spaces). The associated cohomology groups
Ker(r(X, SliM) ~ r(X, SliP,Q+1))
HM(X): = Im(r(X, SliP,q-l) ~ r(X, SliM))
are called the Dolbeault groups.
Theorem 5.4 (Dolbeault)

Hp,q(X) ~ Hq(X, QP) for
PROOF. Let 0 ~ SliM -? Slib ~ Slit -? SIi~ -? ... be the canonical flabby
resolutions of the sheaves SliM (all SIi~ are fine !). Let CVIl : = r(X, SIi~) for
V,f,lE No· Let6':Cvll ~ Cv+l,lland6":C"Il-? C V ,Il+1 be the homomorphisms
induced by the flabby solution 0 -? QP -? Yo ~ Y 1 ~ . . . and the
Dolbeault sequence, with signs so that (C VIl ' (j', 6") is a double complex. We
obtain the following diagram:
196
5. Fine Sheaves (Theorems of Dolbeault and de Rham)
All the hypotheses of Theorem 3.1 of Chapter VI are satisfied, so

Hp,q(X) ~ H qo , Hq(X, QP) ~ Hoq.
Since Hi(X, dp,q) = 0 (for i ;:: 1) the c)"-sequences are exact. Since the
sequence 0 ~ QP ~ dP'o ~ d P' 1 ~ . . . is exact and I.ID is an exact functor,
all sequences 0 ~ !/'y ~ d? ~ dt ~ ... are exact. Since all sheaves are
flabby, the c)'-sequences are exact. Therefore Hoq ~ H qo , and the theorem
is proved. 0
Theorem 5.5. Let X be a Stein manifold, q ;:: 1. If cp is aform of the type (p, q)
on X with d" cp = 0, then on X there exists a form l/J of the type (p, q - 1),
with d"l/J = cpo
PROOF. By Theorem B Hq(X, QP) = 0 for q ;:: 1; therefore Hp,q(X) = 0 for
q;::1. 0
Remarks. With the help of Poincare's Lemma one shows that the sequence
o~c ~ dO ~ d l ~ d 2 ~ . .• is exact. The associated cohomology
groups
W(X): = Ker(r(X, d r) ~ r(X, d'+l))jIm(r(X, dr-l) ~ r(X, d r ))
are called the de Rham groups. As above one shows
Theorem 5.6. W(X) ~ H'(X, C)for r ;:: O.
Since
d t = EB dp,q
p+q=f
we would expect that a connection between the topological cohomology

groups H'(X, C) and the analytically defined cohomology groups Hq(X, QP)
exists. That is in fact the case. If, for example, X is a Kahler manifold (for
example, a projective-algebraic manifold), then according to Kodaira,
H'(X, C) ~ EB Hq(X, QP).
p+q=r
As a consequence one obtains:

Bl(X) = 2p = even;
on X there exist p linearly independent differentials of the first kind, that is,
elements of r(X, Ql).
197
List of Symbols
cn, 11311, 11311*, dist, dist* 1 [J], (X, JIt'), ff(W, C) 119
131, dist', C B (3o), U., Ui, U~ 2 cp = (i{J, CPJ 120
Pr(3o) 4 dimxo (X) 124
en 4 (Dvf)<p(x o), rk xo (f1, ' , , ,f.t) 132
[pn 135
T(U, Y), W(Y) 150
W(Y) 152
M', zn(M'), Bn(M'), Hn(M') 155
ch(f) 11
Zt'(X, Y), Bt(X, Y), Ht(X, Y) 156
fx" f y,. 22
U io · .. it' 6 n , sgn(o-) 158
JJ' D. J 25
Ct(U, Y) 159
W g, Mg 26
zt(U, Y), Bt(U, Y), Ht(U, Y) 160
K c c B 40
H1j(f)), A(G) 65 f0~0 179
No, No, C{3}, 1R':r, Hm Ilfllt 68 a a
B t 69 T xo ';:;--';:;-- 180
uX1 UY1
]O[X], QO[X] 79
N(fl> ' , , ,It) 84 TC
Xo'
T'
xo,
T" a a
xo, OZ1' OZ1
181
gcd(a 1 , a 2 ) 86
D(X 1, ' , , , X s), D.(f) 88 F(r) T* 182
Xo' Xo
(9ao' (9, Y a 99
cp 1\ tjJ, dx y , dy y , dz y , dzy 182
T(W, Y) 100 A(p, q), A(l), df, d 185
r:Mw ~ T(W, Y), {Mw:rn 102
Y 1 EB ' , , EB Y t 104 L aI,J d31 1\ d5J 186
I,J
1,0, la' 0a 105
N(§), q(9 108 d', d" 187
y/y* 109 Ch<f)(w) 188
1m cp, Ker cp 111 d\ dp,q 194
Hom(1lY l' Y 2) 112 QP 195
Supp(§), §(A) 113 HP,q(X) 196
JIt'(A} 117 H'(X) 197
199
Bibliography
Textbooks
1. Abhyankar, S. S.: Local Analytic Geometry. New York: Academic Press, 1964.
2. Behnke, H., Thullen, P.: Theorie der Funktionen mehrerer komplexer Veriinder-
lichen. Ergebn. d. Math., Bd. 51,2. erw. Aufiage. Berlin-Heidelberg-New York:
Springer 1970. .
3. Cartan, H. : Elementary Theory of Analytic Functions of One or Several Complex
Variables. New York: Addison & Wesley, 1963.
4. Fuks, B. A.: Introduction to the Theory of Analytic Functions of Several Complex
Variables. Trans!. of Math. Monogr., 8. Providence, Rhode Island: American
Mathematical Society, 1963.
5. Fuks, B. A.: Special Chapters in the Theory of Analytic Functions of Several Complex
Variables. Trans!. of Math. Monogr., 14. Providence, Rhode Island: American
Mathematical Society, 1965.
6. Grauert, H., Remmert, R.: Analytische Stellenalgebren. Grund!. d. math. Wiss.,
Bd. 176. Berlin-Heidelberg-New York: Springer, 1971.
7. Gunning, R. c., Rossi, H.: Analytic Functions of Several Complex Variables.
Englewood Cliffs, N.J.: Prentice-Hall, 1965.
8. Hormander, L.: An Introduction to Complex Analysis in Several Variables. Prince-
ton, N.J.: Van Nostrand, 1966.
9. Vladimirov, V. S.: Les Fonctions de Plusieurs Variables Complexes(et leur applica-
tion Ii la tMorie quantique des champs). Paris: Dunod, 1967.
Older (Classical) Books

10. Bochner, S., Martin, W. T.: Several Complex Variables. Princeton: Princeton
University Press, 1948.
11. Osgood, W. F.: Lehrbuch der Funktionentheorie, 2. Bd., 1. Lieferung. Leipzig-
Berlin: Teubner, 1924.
201
Bibliography
Lecture Notes
12. Bers, L.: Introduction to Several Complex Variables. New York: Courant Institute
of Mathematical Sciences, 1964.
13. Cartan, H.: Semina ire Ecole Normale Superieure 1951/52, 1953/54, 1960/6l. Paris.
14. Herve, M.: Several Complex Variables. Tata Institute of Fundamental Research
Studies in Math., l. London. Oxford University Press, 1963.
15. Malgrange, B.: Lectures on the Theory of Functions of Several Complex Variables.
Bombay: Tata Institute of Fundamental Research, 1958.
16. Narasimhan, R.: Introduction to the Theory of Analytic Spaces. Lecture Notes in
Mathematics, Vol. 25. Berlin-Heidelberg-New York: Springer, 1966.
17. Schwartz, L.: Lectures on Complex Analytic Manifolds. Bombay: Tata Institute of
Fundamental Research, 1955.
More Advanced and Supplementary Books

18. Colloquium uber Kiihlersche Mannigfaltigkeiten. G6ttingen: Ausarbeitung des
Mathemat. Inst., 1961.
19. Ehrenpreis, L.: Fourier Analysis in Several Complex Variables. Pure and Applied
Mathematics, Vol. 17. New York: Wiley-Inter science Publishers, 1970.
20. Godement, R.: Toplologie algebriqueet theoriedesfaisceaux. Paris: Hermann, 1964.
2!. Grauert, H., Fischer, W.: Differential- und Integralrechnung II. Heidelberger
Taschenbiicher, 36. Berlin-Heidelberg-New York: Springer, 1968.
22. Grauert, H., Lieb, I.: Differential- und Integralrechnung III. Heidelberger Taschen-
biicher, Bd. 43. Berlin-Heidelberg-New York: Springer, 1968.
23. Hirzebruch, F., Scheja, G.: Garben- und Cohomologietheorie. Ausarb. math. und
physik. Vorl., 20. MUnster, Westf.: Aschendorffsche Verlagsbuchhandlung, 1957.
24. De Rham, G.: Varietes differentiables. Paris: Hermann, 1960.
25. Weil, A.: Introduction d l' etude des varietes kiihleriennes. Paris: Hermann, 1958.
202
Index
A Cartan-Thullen (Theorems of), 45

Absolute space, 4 Catastrophe, 98
Acyclic complex, 155 Cauchy integral, 11
Alternating cochain, 159 Cauchy-Riemann differential equations,
35
Analytic algebra, 84
polyhedron, 53 C 2-boundary, 37
sequence of sheaves, 112 Cech cohomology groups, 160
set 84, 126 Chain rule, 24
sheaf, 108 Chern class, 172
sheaf isomorphism, 111 Chow's theorem, 143
surface, 33 Closure of a manifold, 144
Antiholomorph, 181, 187 Coboundary, 155
Augmented cochain complex, 155 operator, 160
Cochain complex, 155
Co cycle, 155
B
Coherence theorem of Cartan, 114
Banach algebra, 69 of Ok a, 114
Base point, 56 Coherent, 114, 117
Behnke-Stein Theorem, 176 Cohomology group, 155, 156
Betti number, 143 Cohomology sequence, 169
polynomial, 143 Complete, 69
Bieberbach ('S example), 145 Complete hull, 20
Biholomorphic, 27, 126 Reinhardt domain, 6
Branch points, 91, 131 Completely singular, 35
Complex atlas, 123
c coordinate system, 122
C-algebra sheet, 105 differen tial be, 8
Cartan-Serre (Theorems of), 174 function, 2
203
Index
Complex atlas [cant.] Exact functor, 150

functional matrix, 25 sequence, 112
manifold, 122 Expansion in a series of homogeneous
ringed space, 119 polynomials, 77
space, 121 in Zl, 70
submanifold, 133
torus, 138 F
Connected, 2, 124 Fiber preserving, 56
component, 2 Fine sheaf, 193
Continuity theorem, 30 Finite analytic sheaf, 113
Convergence of power series, 3, 68 module, 82
Coordinate transformation, 123 Finitely generated, 113
Cousin I distribution, 162 Five lemma, 117
Cousin II distribution, 171 Flabby resolution, 152
Covariant functor, 150 sheaf, 151
Form of type (p, q), 183
D Formal power series, 3
Deformation, 97 Functional determinant, 25
de Rham groups, 197 holomorphic, 26
Derivation properties, 180 Functional matrix, holomorphic, 26
Derivative of a polynomial, 87 Functor, 150
Diagonal, 135 exact, 150
Differential form, antiholomorphic, 182 covariant, 150
holomorphic, 187
normal form of, 183 G
Dimension of an analytic set, 97 Gauss lemma, 80
of a complex manifold, 124 Geometrically convex, 40, 43
Discrete mapping, 127 set, 50 hull, 40
Discriminant, 89 Greatest common division (GCD), 86
Distinguished boundary, 4 Group, Dolbeault, 196
Divisibility, 78 de Rham, 197
Dolbeault groups, 196
lemma, 191 H
sequence, 196
Hartogs figure, 29, 66
theorem, 196
Hensel's lemma, 82
Domain 2
Henselian ring, 82
of convergence, 8
Holomorphic differential form, 187
ofholomorphy, 35, 65
domain, 35, 64
with C 2 boundary, 37
function, 3,44, 62, 63, 122, 176
Double complex, 163
functional determinant, 26
functional matrix, 26
E hull, 65, 127
Edge-of-the-Wedge theorem 67 mapping, 26, 125
Euclidean Hartogs figure, 21, 29 region, 35
ring, 86 tangent vector, 181
204
Index
Holomorphically convex, 42, 127 discrete, 127

separable, 127 holomorphic, 26, 125
Hopfmanifold,140 Maximum principle, 124
CT-process, 148 Mittag-Leffler problem, 163
Hypersurface,91 Meromorphic function, 85,141
Modification, 146
Multi-index, 2
I
Ideal sheaf, 108
Identity theorem, 15, 124 N
Indecomposable, 79 Noetherian, 83
Indeterminate point, 85 Norm of a power series, 68
Inductive limit, 103 Normal exhaustion, 47
Infinitely distant point, 146 form of a differential form, 183
Integral formula of Bochner-Martinelli, 33
Irreducible analytic set, 96
Isolated singularity, 97
o
Order of a convergent power series, 80
Isomorphism of complex ringed spaces,
Osgood closure, 145
119
of Riemann domains, 57
p
K Partial derivative, 9, 132
Kahler manifold, 144, 197 Partition of unity, 193
Kodaira's theorem, 197 Period relations, 144
Kugelsatz, 33 Permutation, 158
Poincare lemma, 188
Pole, 85, 141
L Polycylinder,4
Leray covering, 166 Power series, formal, 3, 68
Leray's theorem, 175 convergent, 3, 68
Levi conjecture, 37 norm of, 68
condition, 37 order of, 80
form, 38 Presheaf, 102
Lieb's theorem, 193 of modules, 107
Lifting, 56, 114 of IC-algebras, 106
Locall[:-algebra, 81 Prime, 79
analytic, 84 decomposition, 79
uniformizing,128 Projection theorem, 147
Locally differentiable function, 179 Projective algebraic manifold, 143
free sheaf, 195 space, 135
topological, 58 Proper modification, 147
Reinhardt domain, 6
Pseudo branching, 93
M Pseudo polynomial, 87
Manifold, 122 Pseudoconvex, 35, 66
Mapping, biholomorphic, 27, 126 Pure dimension, 97
205
Index
monomorphism, 110
Q morphism, 101
Quotient field, 79 morphism, analytic, 110
sheaf, 109 of A-modules, 106
of (:-algebras, 106
R of sets, 99
Real differentiable, 22 Shearing, 76
Reducible analytic set, 96 Singularity (singular point), 95
Region, 2 free, 132
of convergence, 7 isolated, 33, 97
of holomorphy, 35 Stalk, 99
Regular closure, 145 Stalk preserving (mapping), 101
in ZI, 74 Stein manifold, 128
point, 95 covering, 175
Reinhardt domain, 5 Strongly pseudo convex, 193
complete, 5 Submanifold, 133
proper, 5 Subsheaf, 100
Relation finite, 114 Support of an analytic sheaf, 113
sheaf, 114 Symmetric polynomial, 88
Relatively compact, 40
Resolution of a sheaf, 152 T
Riemann domain, 54 Tangent space, 37, 180
with distinguished point, 56
Tangent vector, antiholomorphic, 181
Riemann surface, abstract, 125
holomorphic, 181
concrete, 128
real, 179
of )Z, 55, 62, 92
Taylor series expansion, 15
Riemann sphere, 125
Test function, 193
RUckert basis theorem, 83
Theorem A, 174
B,174
s Topological mapping, 54
Schlicht domain, 57 Torus, complex, 137
Section, 100 real, 4
Sequence, exact, 112 Total differential, 37, 185, 186
of sheaves, analytic, 112 Type (p, q), 183
Serre's five lemma, 117
Set of degeneracy, 146
Sheaf, analytic, 108
u
conherent, 114, 117 Unbranched, 130
constant, 108 Union of domains, 61
epimorphism, 110 Unique factorization domain, 79
fine, 193 Unit section, 111
finitely generated, 113
homomorphism, 110 w
isomorphism, analytic, 111 Wedge product, 183
locally free, 195 Weierstrass condition, 73
206
Index
formula, 71, 77
homomorphism, 83
preparation theorem, 73, 78
problem, 171
Whitney sum, 104
z
Zariski topology, 95
Zero section, 105
set, 108
set offunctions, 84, 109
sheaf, 109
Zorn's lemma, 157
207

Several Complex Variables - H. Grauert & K. Fritzsche

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Graduate Texts in Mathematics

P. R. Halmos F. W. Gehring C. C. Moore

Library of Congress Cataloging in Publication Data

Grauert, Hans, 1930-

All rights reserved.

© 1976 by Springer-Verlag Inc.

ISBN-13: 978-1-4612-9876-2 e-ISBN-13: 978-1-4612-9874-8

with the theory of functions on complex manifolds. It is treated and applied

Gottingen, Spring 1976

3 Convergent Power Series 74

List of symbols 199

Each component of a point 3 E en can be decomposed uniquely into real

Norms are defined on en by 31-+11311 and 31-+11311*, with corresponding

Definition. Let Been be a region, 30 E B a point. The set CB (30): =

Remark. Let Been be an open set. Then:

P(3) = L av, ..... vn z'1' .... ' z~n,

are particularly simple examples, defined on all of en. In order to simplify

Def. 1.1. Let 30 E en be a point and for Ivi ~ 0, a v be a complex number.

is called a formal power series about 30'

Def.1.2. Let~: = {v = (Vi"'" V"):V i

One then writes I

Def. 1.3. Let M be a subset of en, 30 E M, f a complex function on M. One

says that the power series I

for each finite I with 10 c: I c: ~ and each 3 E M.

I av(3 - 30Y converges uniformly in the interior of a region B if the

Def. 1.4. Let B c: en be a region and f be a complex function on B. f is

U(30) in B and a power series I av (3 - 30)V which converges on U to

Note that uniform convergence on U is not required. We show now why

V is a subset of ~n and as such inherits the topology induced from to V ~n

Figure 1. The image of a polycylinder in absolute space.

P = P r (30) is a convex domain in en, and its distinguished boundary is

en. If the power series L:

Theorem l.1. Let 31 E ava v converges at 31, then

is, lavl'13~1' qlvl is a majorant of avaV and therefore

vergent on P*. Given a > there is an no EN such that I Ibn(a)1 < a

IJo bn(a) - V~I av3vl = IJo bn(3) "E~l(ll bn(3)1

= InE~l(ll b (3)1 :;;; n=~+ 1Ib,,(3)1 < a for 3

But then I av 3V is uniformly convergent in P*.

in 1). Therefore I av 3V is uniformly convergent on K, which was to be

Def. 1.7. An open set B c C" is called a Reinhardt domain if 31 E B => T oI : =

The image of a Reinhardt domain in absolute space is always an open set

Def. 1.8. Let G c en be a Reinhardt domain.

Figure 2 illustrates Def. 1.8. for the case n = 2 in absolute space.

For n = 1 Reinhardt domains are the unions of open annuli. There is no

PROOF. Let G be a complete Reinhardt domain. There exists a point 31 E G n

Let ~(3) = L a3 v v be a power series about zero. The set M c en on

The question arises whether every complete Reinhardt domain is the

power series L av (3 - oo)v converge to f(o) in a neighborhood U of 00'

Theorem 1.5. Let Been be a region, and f a jitnction holomorphic on B.

2. Complex Differentiable Functions

Def. 2.1. Let Been be a region, f: B -+ C a complex function. f is called

Differentiability is a local property. If there exists a neighborhood U =

Theorem 2.1. Let Been be a region and f:B -+ C complex differentiable

Theorem 2.2. Let Been be a region and f complex differentiable at 30 E B.

Theorem 2.3. Let Been be a region, f holomorphic in B. Then f is complex

series I av (3 - 30)V which in V converges uniformly to f(3). Without loss

I ava V = ao .. ·o + Z1' I aVI"'vnzI' 1. Z22 ••• Z~n

J(a) a v , ... vjZ1 - zlO))"' ... (zn - z~O))"n