From Holomorphic Functions to Complex Manifolds (Graduate Texts in Mathematics)

Klaus Fritzsche Hans Grauert

From Holomorphic Functions to Complex Manifolds With 27 Illustrations

Klaus F n t z s c h e Be rg i sch e Universitat Wup pertal GauBstraBe 20 D-42 1 1 9 Wuppertal

H a ns Grauert Mathematisches Institut

Geor g-August-Univer sitatGottingen BunsenstraBe 3-5 .

G ermany

D-37073

[email protected]

Germany

Gottingen

Editorial Board S. Axler Mathematics Department

San Francisco

State

University San Francisco, CA

94 132

F.W. G e h r i n g

K.A.

Mathematics Department East Hall

Mathematics Department

University of Michigan Ann Arb o r , MI 48 109

USA

USA

[email protected]

fgehring ® ma th.lsa.

Ribet

University of California, Berkeley Berkeley, CA

94720-3840

USA rib et@ math. berkeley .edu

umich.edu

Mathematics Subject Classification (2000): 32-01, 32Axx, 32D05, 32Bxx, 32Qxx, 32E35 Library of Co ngress Cataloging-in-Publication Data Fritzsche, Klaus. From holomorphic functions to complex manifolds p. em. -(Graduate texts in mathematics ; 213)

I

ISBN 0-387-95395-7 (alk paper) I. Complex manifolds. 2. Holomorphic functions.

Klaus Fritzsche, Hans Grauert.

Includes bibliographical references and indexes,

I. Grauert, Hans, 1930-

III. Series.

II. Title.

QA331.7 .F75 2002 2001057673

5!5'.98---dc21 Printed on acid-free paper.

©

2002 Springer-Verlag New York, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use i n connection with any form of information storage and retrieval, electronic adaptation, computer soft­ ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Production managed by Michael Koy; manufacturing supervised by Erica Bresler. Photocomposed from the authors' LaTeX files. Printed and bound by Maple-Vail Book Manufacturing Group, York, PA. Printed in the United States of America. 9 8 7 6 5 4 3 2 I ISBN 0-387-95395-7

SPIN 10857970

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Media GmbH

Preface The aim of this book is to give an understandable introduction to the the­ ory of complex manifolds. With very few exceptions we give complete proofs. Many examples and figures along with quite a few exercises are included. Our intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involved with sheaves, coherence, and higher-dimensional cohomology are avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional co­ cycles are used. Nevertheless, deep results can be proved, for example the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem. The first chapter deals with holomorphic functions defined in open sub­ sets of the space @". Many of the well-known properties of holomorphic functions of one variable, such as the Cauchy integral formula or the maxi­ mum principle, can be applied directly to obtain corresponding properties of holomorphic functions of several variables. Furthermore, certain properties of differentiable functions of several variables, such as the implicit and inverse function theorems, extend easily to holomorphic functions. In Chapter II the following phenomenon is considered: For n > 2, there are pairs of open subsets H c P c en such that every function holomorphic in H extends to a holomorphic function in P. Special emphasis is put on domains G c en for which there is no such extension to a bigger domain. They are called domains l for v = l , .. . ,n . The n T(e;e .zo) = T(zo), but l e;e . z0 - z0 I = le;e - 11 . lzo I > l e;e - 11, and for suitable () this expression may be greater tha n E. So pn (zo, c ) is not a Rei nhardt domain.

Definition. Let G c en be a Rei nhardt domain. 1 . G is called proper if 0 E G. 2. G is called complete if V z E G n (C*t : Pz C G (see Figure 1 . 2).

Later o n we shall see that for any proper Rei nhardt domain G there is a smallest comp lete Rei nhardt domain G containing G. �

(a)

Figu r e 1.2.

(b)

(a) Complete and (b) noncomplete Rei nhardt domain

Exercises 1 . Show that there is a n ope n set B C e2 that is not co nnected but whose image T(B) is a domai n in absolute space. 2. Which of the following domains is Rei nhardt, proper Rei nhardt, complete Rei nhardt? (a) G1 := { z E e 2: 1 > lz1l > lz2l}, (b) G2 : = {z E e2 : lz1 l < l a nd lz2 l < 1 - l zii}, (c) G3 is a domain in e2 with the property z E G ===;. eit · z E G for t E JR. 3 . Let G c en be an arbitrary set. Show that G is a Rei nhardt domain 1 -¢:=? 3 Q C "f/ ope n and connected such that G = T- (Q). 4. A domai n G c en is called conwex, if for each pair of poi nts z,w E G the line segment from z to w is also co ntai ned in G. Show that a n arbitrary domain G is co nvex if and o nly if for every z E 8G there is a n affine linear fu nctio n >. :en --+ lR with .X (z) = 0 a nd >.ic < 0.

2.

2.

Power Series

9

Power S erie s

Polynomials . In order to simplify notatio n, we i ntroduce multi-i ndices. For v = (v1, ... ,vn) E zn and z E en define n

a nd The notatio n v > 0 (respectively v > 0) means that vi > 0 for each (respectively v > 0 and vi > 0 for at least one i).

�

A fu nction of the form z r-+ p (z) =

L

lv l<m

av z v , with

a,

E C for lvl < m,

is called a polynomial (of degree less than or equal to m ) . If there is a v with lvl = m a nd a, 1- 0, the n p (z) is said to have degree m . For the zero polynomial no degree is defined. An expression of the form a v zv with a, 1- 0 is called a monomial of degree m == lvl. A polynomial p (z) is called homogen eous of degree m if it consists only of monomials of degree m . 2. 1 Propositi on. and only if

PROOF :

A

polynomial p (z) 1- 0 of degree m i s homogeneous if

p (Az) = Am · p (z),

fo r all A E C.

Let p (z) = avz v be a monomial of degree m . The n

The same is true for finite sums of monomials. O n the other hand, let p (z) = 'I:I io c, and discuss the m atter of co nvergence. The trouble is that there is no cano nical order o n N3.

I.

10

Holomorphic Functions

Definition. The series :Z:::: v >o cv is called convergent if there is a bi­ jective map t.p : N --+ N3 such that "'£': 1\cO c, is convergent ! cv ! {L vEI

:

I

if

and only

C N� finite

if

}

is a bounded set. The proof is trivial. 2 . 3 Pro p osit io n . If the series :Z::: v�o Cv converges to the complex number c, then for each E > 0 there exists a finite set Io C N� such that: 1.

L: ! cv ! < E,fo r anyfin ite

2.

L c,

vEI

set K C N3 with K nio = 0 .

- c < E, fo r any fin ite set I with Io C I C N3

PROOF : We choose a bijective map t.p : N --+ N�. Then I:� 1 c 0 there exists an io E N o such that I:� ;0 jco fv is called normally convergent on M if the series of positive real numbers :Z::::v>ollfviiM is convergent.

2.4 Prop ositi o n . Let the series :Z:::v>O fv be normally convergent o n M. Then it is convergent for any z E M, and for any bijective map t.p : N--+ N0 the series 2::� 1 f : o l fv (z) I is convergent for any z E M. But then 2::v>o fv(z) is also convergent, and there is a complex number f(z) such that I:� 1 f'P(i)(z) converges to f(z), for every bijective map t.p :N--+ N0. If an

E

> 0 is given, there is an io such that I:� io II !'P(i) liM < E. Then m

L f'P(i) (z) i=k

'!!!


O a v wv is convergent. Then there is a constant c such that lav wv I < c for all v E N0. PROOF:

We choose real numbers q; with 0 < q; < 1 such that lz;l < q;l w;l for any z = (z1, . . . , Zn ) E P' and i = 1 ... , n. It follows that ,

iav z v l
O a v z v is normally convergent on P'. • Definition. We say that a power series :Z:::: v >o a v (z - zo) v converges compactly in a domain G if it converges normallyon every compact subset K c G. 2.6 Corollary. Let P c e n he a polydisk around the origin and w be a point of the distinguished bo undary of P. If the power series :Z::: v 7o a v zv converges at w, then it converges compactly on P. Let K C P be a compact set. Then there is a q with 0 < q < 1 such that K c q · P cc P. Therefore, the series is normally convergent on • K.

PROOF:

Let S (z) = :Z::: v >O av z v be a formal power series about the origin, and B :={Z E e n

:

S (z) convergent} .

2.7 Propositi on. The interior BO ts a complete Reinhardt domain, and S (z) converges compactly in B0• Let w be a point of B0• There is a polydisk p n (w ,c) C B0 and a point v E pn (w, c) n (C*r' such that w E Pv(O) . Then Tw C B , and if wE (C*) n , then also Pw(O) C B".

PROOF:

"

To see that B0 is a complete Reinhardt domain, it remains to show that it is connected. But this is very simple. Every point of B0 can be connected to a point in B0 n (C* ) n , and then within a suitable polydisk to the origin. 3

The notation U CC V means that U lies relatively compact in compact set which is contained in V.

V;

i.e., U is a

2. Power Series

13

From these considerations it follows that B0 is the union of relatively compact polydisks around the origin. Therefore, S ( z ) converges compactly o n B . • "

The set B0 is called the domain of convergence of S(z ). 2.8 Propositi on. Let G be the domain of convergence of the power series S ( z ) = :Z::: v >O av z v . Then

L av

v >O v1>o

·

VI

VJ· z1

V

J· · · z.

J

1

· · · z

V

n

n

also converges compactly on G.

PROOF: Let w be any point of (C*) n n G, and lavwv l < c for every v E N�. If 0 < q < 1 and z = q .w, then

Now,

is convergent. Therefore, SzJ ( z ) is convergent, and it follows that SzJ is normally convergent on Pz(O). Since every compact set K C G can be covered by finitely many polydisks of this kind, Sz. is compactly convergent on P. • J

Definition. Let B c en be an open set. A function f : B --+ C is called holomorphic if for every z0 E B there is a neighborhood U = U (z 0 ) c B and a power series S (z ) := :Z::::v>o a v (z - z0 t that converges on U to ! ( z). -

The set of holomorphic functions on B is denoted by 0( B).

It is immediately clear that every holomorphic function is continuous. Exercises 1 . Let j, g be two nonzero polynomials. Prove that

deg(f ·g)

=

deg(f) + deg(g) .

2. Let f = h · · .!k be a homogeneous nonzero polynomial. Show that fi is homogeneous, for i = 1 .. . , k. ,

4

I. Holomorphic Functions

3. Fi nd the domain of co nvergence for the following power series:

4.

Determine the limit and the domain of convergence of the series v z'' w . (2zt + F(z, w) = tL>O v >O

L )

L(

5. A polyradius r (r1, . . ., r,) E "f/ is called a radius cf convergence for the power series f(z) l:: v >O a v zv if f(z) is co nvergent i n P pn (O, r), but not co nvergent in any po lydisk P' pn (O, r') with P cc P' . =

=

=

=

Prove the following generalizatio n of the root test: r is a radius of convergence for f (z) if a nd o nly if lmnl 1'\/lavlr v

3.

=

1.

C om plex Differ entiable Functions

The C omp lex Gr adient Definit ion. Let B c en be an ope n set, Zo E B a poi nt. A fu nctio n f : B --+ e is called complex differentiable at zo if there exists a map A : B --+ en such that the following hold: 1 . � is co nti nuous at zo . 2. f(z) = f(zo) + (z - zo) . � (z) t for z E B .

Complex differe ntiability is a local property: For f to be complex differen­ tiable at z0 it is sufficie ntthat there is a small neighborhood U U(zo) C B such that the restrictio n flu is complex differentiable at zo . =

3 . 1 Proposit ion. Iff is complex differentiable at zo, then the val ue f unction A at zo is uniq uely determined.

cf

the

Assume that there are two maps �� and � 2 satisfying the condi­ tio ns of the defi nition. The n PROOF:

In particular, there is a n E > 0 such that the equ atio n holds for z zo +te, 1 , ... , n. 1f �.\ ( �i-'>, ... , ��>->) , the n with t E e, It! < E, and i =

=

t· (��'> (z) - ��2>(z ))

=

=

0 for It! ,

< E,

z = Zo +te, a nd i

=

1 , . ..,n .

Because the � �>-> are co ntinuous at z0 , it follows that ��(zo) = � 2 (zo ) .

•

3. Complex Differentiable Functions

D efi n i t i on . Let f : B exists a representatio n

---+

e be complex differentiable

at

15

z0. If there

f(z) = f(z o) + (z -zo) . �(z) t ,

with A continuous at z0, the n the uniquely determi ned numbers M (zo) Bzv

= fzv (zo) : = ev . � (zo) t

are called the partial derivatives o ff at z0. The vector

is called the complex gradient of f at z0. Rem ark s 1 . If f is complex differentiable at z0 , the n f is continuous there as well. 2 . A fu nctio n f is called complex differentiable in a n ope n set B if it is complex differentiable at each poi nt of B. The n the partial derivatives of f define fu nctio ns fzv o n B. 1f each of these partial derivatives is again complex differentiableat zo , the n f is called twice complex differentiable at z0, a nd o ne obtai ns second derivatives

[)2j OZvOZJL

(zo)

= fzvz, (zo) ·

By inductio n, partial derivatives of arbitrary order may be defined. 3. Sums, products, and quotients (with no nvanishi ng denominators) of com­ plex differe ntiable fu nctio ns are again complex differentiable.

Weakly Holomorphic Functions . Let B c en be an ope n set, z0 E B a poi nt, a nd f a complex-valued fu nction o n B. For w -=!= 0 let 'P w : e ---+ en be defined by 'Pw(()

:= z o + (w.

The n f o t.pw ( () is defined for sufficie ntly small (. Iff is complex differentiable at zo , the n we have a representatio n f(z) = j(zo ) + (z -zo) . � (z) t, with A co ntinuous at z0. It follows that

and f o 'P w is complex differe ntiable at ( = 0, with

16

I. Holomorphic Functions

This is the complex directional derivative of f at z0 in the directio n w . We de note it by Dwf (zo) · In particular, fzv (zo) = Dev f (zo) for v = 1 .. . , n. ,

A n arbitrary fu nctio n f is called partially diflerentiable at z0 if all partial derivatives Dev f ( zo ) exist for v = 1 . . . , n. ,

A fu nctio n f is called weakly holomorphic on B if it is continuous and partially differe ntiable o n B. The n for z = (z1 , . . . , zn ) E B and v = l , . . . , n the fu nctio ns ( r-+ f(zl, . . . , Zv- 1 , C, zv+ l 1 , zn ) •

•

•

are holomorphic functions of one variable. I ff is complex differentiable o n B, the n f is also weakly holomorphic o n B. Later o n we shall see that weakly holomorphic functions are always complex differe ntiable, in co ntrast to the behavior of real differentiable fu nctio ns.

H olomor phic Funct ion s 3 . 2 Prop o sit io n. Let P C en be a polydisk around the origin, and S(z) = :Z::: v >o av z v a power series tha t converges compactly on P to a function f . Th en f is complex differentiable a t 0 , with fz1 (0 )

= a1, 0 , . . , 0 , · . . , fzn (0) = ao, . . . ,O, l ·

We choose a small polydisk Pc: cc P arou nd the origin such that S(z) is normally co nvergent o n P,. But the n the series obtai ned by any rearrangement of the terms is also normally co nvergent, and it co nverges to the same limit. We write PROOF:

f (z)

L:> v zv

v>O a o , o . . . . , o + z1

+ Z2 ·

v1 > 0 vz , .. . ,vn > O

+

· · ·

+ Zn ·

vl = . . . = Vn-1=0 Vn > O

Since the series ll 1 ( z) , . . . ,ll n (z) converge normally o n Pc: to conti nuous fu nc­ • tio ns, f is complex differentiable at 0 , with fzv (O) = flv ( O ) . 3 . 3 Cor ollary. If B c en is an open set, and f : B function, then f is complex diflerentiable on B.

-+

e

a holomo rphic

4.

The Cauchy Integral

17

Let z0 E B be a n arbitrary poi nt. There is a power series S (w) co nverging compactly near 0 to a holomorphic fu nctio n g such that

PROOF:

f ( zo + w )

=

g( w)

=

g(O) + WI . �I (w) + ·..+ Wn .�n (w) ,

with conti nuous fu nctio ns �I , .. . , A,. It follows that f is complex differen• tiable at zo . Exercises 1 . Show that there is a fu nctio n f : e n --+ e that is complex differe ntiable at every poi nt z = ( zi , . . . , zn ) with Zn = but is nowhere holomorphic. 2. Prove the following chain rule: If G c en is a domain, J : G --+ e a complex differentiable fu nctio n, and t.p = ( 'P I , . . . ,t.p n ) : A --+ G a map with holomorphic compo nents t.p ; , the n f o t.p : A --+ e is a holomorphic fu nctio n, with (! o t.p ) ( () = V f ( t.p ( () ) . t.p' ( () t .

0,

'

4.

The C auch y I ntegr a l

The Integr al Formula. Let r = (ri , .. . , rn ) be a n element of iR+. , P = p n ( o, r ), T = Tn ( o, r ) , a nd f a co ntinuous fu nctio n o n T. The n kt (z , ()

:=

f( () ( ( z) (I, . . ,I) _

defines a co nti nuous functio n kf : P x T --+ C. Defi nit ion.

(2:i) J kt (z , () d( J n

T

((n - Z n )

is called the Cauchy integral o ff over T. Obviously, Cf is a co nti nuous fu nctio n on P. 4 . 1 Theorem (C a u chy integral formula). Let P, T be as above, and U = U(P) be an open neighborhood cf the closure of P. Iff is weakly holomorphic on U, then Ctlr (z) = f(z ) fo r an y z E P.

Drn (0), we may assume that U = UI x · . . x Un , with ope n neighborhoods U; = U; (Dr, ( 0)) for i = 1 , .. . PROOF:

If P

=

0

(O ) x . . . x

,

, n.

18

I.

Holomorphic Functions

Since f is weakly holomorphic, we can fix z ' = ( z1 , . . . , Zn - d E U1 x · . · x Un -1 and apply the Cauchy i ntegral formula in o ne variable to (n r-+ f(z', (n ) · For Zn E Drn ( 0) it follows that f(z 1 , Zn )

=

1 21r i

J \(n\= rn

f (z' , (n ) d(n (n - Zn

Similarly, for the penultimate variable Zn - 1 and z" = (zl , . . . , Zn - 2 ) E ul . . . x Un - 2 we obtain f(z11 , Zn - l , Zn )

�

=

2 i

and after n steps, f(z)

j

=

f(z" , (n- 1 , Zn ) dl"n - 1 (n - 1 - Zn - 1 '>

\(n- l l =r n-1

f(z " , (n - 1 , (n ) dl" ":,n - 1 , ":,n dl" ) ((n - 1 - Zn - I ) ( (n - Zn

J

J

X

CJ I T(z) , for z E P.

•

4.2 Theorem (power series exp ansion). Let P = pn (o, r) c en be a polydisk and T its distinguished boundary. Iff : T -+ C is a continuous function, then there is a power series l:: v 7 o av z v that converges to Ct (z) in all of P. Th e coefficients a, of this series are given by

Q,V1 · · · Vn

PROOF:

=

( -1 ) n j 27r l

"

J((l , · · · , (n ) dl"l · · · d�" + v1 + V '> n · l l '> n . . · (n I" T '> 1

Setti ng 1 := ( 1 , .. . , 1 ) E N�, for z E P a nd (

1

( ( -z) l

1 ( (I - zl) · · · ((n - Zn )

2:

v, = O

•

T it follows that

1 Zl · · · · 1 (n (I (I

( �� ) VI l: ( �: ) Vn •

E

(

) · · · ( 1 - Z(n ) n

•

If r = (r 1 , . . . , rn ) , the n for fixed z E P and arbitrary ( E T we have

Zj (j

-

=

Since T is compact and l f(() l

<M

z - l jl qj ·.rj

--

f

.1 , . . . , n. or J f 1 ,
O q, , where q = ( ql ' . . . , qn )' and therefore it is normally convergent as a series of functions on T with limit f ( ( ) j ( ( -z) 1 . We can interchange summation and integration:

( 1 )n r Ct (z) = 1r

"' v, ! (() = z av � l Jr ( ( z) d( v >O

2 i

with av .

·

_

- ( 27r1 i )n lrr ! (() d(.

The series converges for each

z E

c+ l

P.

•

4.3

Osgood's theorem. Let B c en be an open set. Th e following state­ ments about a function f : B ---+ e are equivalent:

f is holomorphic. 2. f is complex differentiable. 3. f is weakly holomorphic. 1.

We already know that a holomorphic function f is complex differ­ entiable, and it is trivial that then f is weakly holomorphic. PROOF:

On the other hand, let f : B ---+ e be weakly holomorphic, and z0 E B an arbitrary point. There is a small polydisk P around z0 that is relatively compact in B . If T is its distinguished boundary, then J I P = Cf/ T, and the Cauchy integral is the limit of a power series. So f is holomorphic. •

In addition, if f is weakly holomorphic on B , z 0 E B a point, and P CC B a polydisk around z o , then there is a power series S(z) = :Z::: v > o av (z - z0) v that converges to f on all of P.

-

Holomorphy of the Derivatives

4.4 Weierstrass ' s convergence theorem. Let G c en be a domain, and (ik ) a sequence of holomorphicfunctions on G that converges unzformly to a fu nction f . Th en f is holomorphic. The limit function is continuous. Let zo E G be a point, P cc G a polydisk around z0, and T its distinguished boundary. Then PROOF:

Since T is compact, we can interchange integral and limit. Thus, for any fixed z E P, lim Cfk/T (z) C lim fk / T (z) Cf/T (z) .

k--+CXJ

=

k -+ =

=

20

I . Holomorphic Functions

Since f is continuous on T, the Cauchy integral Cf iT has a power senes • expansion in P . Therefore, f is holomorphic at z0 .

4 . 5 Prop o sition . Let S(z) = 2:: v >o av z v be a power series and G its do­ m a in of convergence. Then the limitfunction f of S(z) is holomorphic on G, and the formal derivative Szi (z)

=

L av · Vj Z�1

v >O v1> o

v· -1 V ZJ J . . znn •

converges to fz1 In particular, all partial derzvatzves of f are likewise holo­ morphzc. •

Since S(z) converges compactly on G, f is locally the uniform limit of a sequence of polynomials. Then it follows from Weierstrass' theorem that f is holomorphic. But also S'z (z) converges compactly on G, and its limit function g must be holomorphic on G. PROOF:

.1

Now let z0 be an arbitrary point of G. Since G is a complete Reinhardt domain, there is a polydisk P around the origin with z0 E P C C G. We define f* (z)

=

r] 9j (Zl , · . . , Zj-I , (, Zj+l , · . . , zn ) d( + f( zl , · . . , 0, . . . , z n)· Jo

For the path of integration we take the connecting segment between 0 and Zj in the zFpl ane. Then f* is defined on P .

Let S(z) = I:� 0 p; (z ) be the expansion into a series of homogeneous polyno­ mials. Then Bz1 ( z) = I:� 0 (p;)zj (z) , and this series converges uniformly on the compact path of integration we used above. Therefore, we can interchange summation and integration, and consequently,

00

L p; (z)

=

f(z) ,

i =O

for z E P. Hence fz1 (zo) = J;j (zo)

=

9j (zo) .

•

4 . 6 Cor ollary. Let G c en be a dom a in and f : B --+ C a holomorphic function. Th en f is infinitely complex differentiable in G. Let v = ( v1 , . . . , vn) be a multi-index. Then we use the following abbrevia­ tions:

4 . The Cauchy Integral

21

1 . VI. '. = VI I. . . · Vn .I .

2 . If f is sufficiently often complex differentiable at z 0 , then 81 v l j v D f(zo) : = 8 v1 8 vn ( z o). · · Zn z1 ·

4. 7 Iden tity theorem (for power series). Let f(z) = :Z::: v >O avz v and g (z) = l: v >O bv z v be two convergent power series in a neighborhood U = U(O) c en . Ifthere is a neigborhood V(O) c U with f l v = 9 l v, then a, = b, fo r all v . PROOF: We know that f and g are holomorphic. Then D v f (0) = D v g ( O ) for all v , and successive differentiation gives Dv f ( O )= v! . av and Dv g ( O ) = v! . b,. •

4 . 8 Cor ollary. Let G c e n be a domain, Zo E G a point, and f : B --+ e a holomorphic fu nction. If f ( z) = :Z::: v >o av (z - zo) v is the (uniquely deter­ m ined) power series expansion near z0 -E G, then a, =

� . Dv f(zo ) , fo r each

v.

v

E N�

4 . 9 Cor ollary (C auchy ' s in equ alities). Let G c e n be a doma in, f : G --+ e holomorphic, z0 E G a point, and P = p n (z 0 , r ) C C G a polydisk with distinguished boundary T. Th en

Let f (z) = :Z::: v >O a v (z - z0 ) v be the power series expansion of f at z 0 . Then Dvf(zo) = v!av and 1 n !( ( ) r d av Jr ( ( zo) v+l ( , = 27r i and therefore PROOF:

( )

I Dv f(zo )l

-

< v'

(27r ) n v'·

J [0 ,27r] n

' f(z1< o) + r 1 c i t t ' . . . ' zn + rn ci t n )l rV

dt 1 · · · dtn

< r_v · sup ! fl . T •

22

I.

Holomorphic Functions

The Identity Theor em . Let G c e n be always a domain. The con­ nectedness of G will be decisive in the following. 4. 1 0 Identity theorem (for h olomorphic fu nctions).

Let !I, h be two holomorphic functions on G. If there subset U C G with !1 l u = h l u , then !I = h · PROOF:

is

a nonemp ty open

We consider f := !I - h and the set N := {z E G : D" f ( z ) = 0 for all v}.

Then N 1- 0, since U c N. Let zo E G be an arbitrary point, and f(z)

=

"""' _!_ Dv f(zo ) (z - zot � v! v >O

the power series expansion of f in a neighborhood V = V(z0) C G. 1f zo belongs to N , then flv 0, and also V C N . This shows that N is open. Because all derivatives D" f are continuous, N is closed. Since G is a domain, • we get N = G and !I = h -

Remark. In contrast to the theory of one complex variable, it is not suf­ ficient that !I and h coincide on a set M that has a cluster point in G. Consider for example, G = e2 and M = { ( z1 , z2) : z2 = 0). The holomor­ phic functions !I ( z1 , z2 ) := z2 ( z1 - z2 ) and h ( z1 , z2 ) := z2 ( z1 + 2 2 ) are equal on M, but fi (O, 1) = - 1 and /2 (0, 1) = 1 . 4. 1 1 Theorem (maximum p r in ciple).

Let f : G ---+ e be a holomorphic function. If there is a point zo E G such that IJ I has a local m aximum at zo , then f is constant. We consider the map t.p w : e ---+ en with 'Pw(( ) = zo + (w, for an arbitrary w 1- 0. Then f o 'Pw is a holomorphic function of one complex variable, defined near ( = 0. Now, since I f o 'P w l has a local maximum at the origin, this function must be constant in a neighborhood of the origin. But the direction w was chosen arbitrarily, so f also has to be constant in a neighborhood of 0 E en . The identity theorem implies that f is constant on • G. PROOF:

Exer cises 1 . Prove Liouville's theorem : Every bounded holomorphic function on en is constant. 2. Prove that if f E O(en ) and l f(z) l < C · llz v ll for some C > 0 and some v ENS , then f is a polynomial of degree at most l v l .

5.

The Hartogs Figure

23

3 . Let G c en be a domain and f E O( G) not constant. Prove that then !( U) C 2 Pn (z ) with pairs Pn ( z ) = (P�n ) ( z ) , p�n ) (z) ) z. Hint: Use 2. A typical example is the Euclidean Hartogsfigure (P", H), where pn = p n ( O , 1 ) is the unit polydisk, and H = { z E P"

:

1 z1 1

> q1 or

l zvl < qv for v = 2, . . . , n} .

Here q1 , . . . , qn are real numbers with 0 < qv < 1 for v = 1 , . . . , n. Every holomorphic function f on H has a holomorphic extension jon P".

general Hartogs figure. We use the symbolic picture that appears as Figure IL l Z2 , . . .

, Zn

'-------'-- - Figur e Il. l . General Hartogs figure 1 . 1 C ontinuity

theorem. Let G c (51 " be doma in, (P, H) a general Har­ togs figure with ii C G, f a holomo rphicfTf_n ction on G. lf GnP is connected, then f can be continued uniquely to G u P.

44

II. Domains of Holomorphy

PROOF:

Let g : pn ---+ en be an injective holomorphic mapping such that P == g(Pn) and H == g ( H ) . The function h = = f o g is holomorphic in H . Therefore, there exists--exactly one holomorphic function h on P" with h i H = h. Si!2 ce g : P" ---+ P is biholomorphic, the function fo : = h o g - 1 is defined on P, and it is a holomorphic extension of fi H - We define -

-

�

f(z) :=

{

f(z) fo(z)

for z E G, for z E 2;.

Since G n P is connected and f = fo on H, it follows from the identity theorem that f is a well-defined holomorphic function on G U P. This is the desired extension of f . • -

�

-

Example

Let n > 2 and P' c c P be polydiscs around the origin in en . Then every holomorphic function f on P - P' can be extended uniquely to a holomorphic function on P. For a proof we may assume that P = pn is the unit polydisk, and P' = pn(O, r), with r = (r1 , . . . , r ) and 0 < rv < 1 for v = l, . . . , n . It is clear n that G := P P' is a domain. Given a point zo = ( zio > , . . . , z�o) ) E G with I z;;'> I > rn , we choose real num­ bers q1 , . . . , qn as follows: For v = 1 , .. . , n - 1 , let qv be arbitrary numbers, with r, < qv < 1. To obtain a suitable q , we define an automorphism T of n the unit disk D by ( - z;;'> T (() : = z;:'>( - 1 This automorphism maps z�0> onto 0 and a small disk D C { ( E e : rn < ICI < 1} around z;:'> onto a disk K c D with 0 E K. Notice that 0 need not be the center of K. We choose qn > 0 such that Dqn (0) C K •

If we define H : = {z E P" : l z1 1 > q1 or l zv l < qv for v = 2 , . . . , n } , then (P", H) is a Euclidean Hartogs figure. The mapping g : pn ---+ pn defined by g(z1 , . . . , zn ) : = ( z1 , . . . , Zn - 1 , T- 1 ( zn ))

is biholomorphic, and (P, ii) = (P", g(H)) is a general Hartogs figure, with

-

Since P n G = G is connected, the continuity theorem may be applied. The preceding example is a special case of the so-called Kugelsatz which we shall prove in Chapter VI.

1.

45

The Continuity Theorem

(0 ) Zn g

K Figur e 1 1 . 2. A

Hartogs figure for concentric polydiscs

Removable Singularities . Let G c ccn be a domain. If A c G is an analytic set and f a holomorphic function on G - A that is locally bounded along A, then by Riemann's extension theorem f has a holomorphic extension to G. If n > 2 and A is a complex linear subspace of codimension greater than or equal to 2, then every function holomorphic on G - A has such an extension.

1 . 2 Theorem. Let P" = pn (o, 1 ) be th e unit polydisk in ccn , n > 2, k > 2, and E := {Z = (z l , . . . , zn ) E ccn : Zn - k+l = = Z n = 0}. Th e n every holomorphic function f on pn - E can be holomorphically ex­ tended to pn . .

.

.

PRooF:

Set P' := {z' := (z1 , . . . , zn - k) : l z' l < 1 } , and for 0 < r < 1 define ?;' := {z" = ( Zn - k + l , . . . , zn ) : l z" l < r } . Let P" : = P{' and fix an E with 0 < E � 1 . Then P" n E c P' x P�', and for w E P' the function fw (z") == f (w, z") is holomorphic on P" - Pf:l From the example above we know that fw has a holomorphic extension fw to P" . Now define f : pn ---+ CC by f(w , z " ) := fw (z"). On P" - E, f is equal to f and is therefore holomorphic. �

_......._

_......._

_......._

_......._

For w E P' take a small open neighborhood U = U(w ) c c P ' .Then U x 8P�' is compact. By the maximum principle we conclude that

I j(z', z" ) I

=

l iz, (z " ) I < l l fz" ll aP;' < llfiiK
2, G0 C c_n - 1 a domain, g : Go ---+ C a contin­ uous function, and f := {z = (z' , Zn ) E Go X c : Zn = g (z' ) } the graph of g in G := G0 x C . In addition, let z0 be a point of f and U = U ( z 0 ) C G a small neighborhood.

If f is a holomorphic function on ( G- f) UU, then f has a unique holomorphic extension to G.

PROOF: The uniqueness of the extension follows from the identity theorem. For the proof of existence ( which is only a local problem ) we may assume that Go = {z' E c_n - l : l z' l < 1 } and that there is a q with 0 < q < 1 such that l g (z') I < q for z' E G0. It also may be assumed that U is connected. Then it is clear that G' : = ( G - f) U U c pn pn ( 0 , 1) is connected. Since g : z' r--+ (z' , g (z')) is continuous, U' g - 1 (U ) is an open neighborhood of z� with (U' X D) n f c u and therefore U' X D c G' . For v = 1 , . . . ' n - 1 let Tv be the automorphism of D defined by =

:=

Then h : pn ---+ pn with h(z1 , . . . , zn ) := (T1 (z2) , . . . , Tn - 1 ( zn ) , z! ) is holo­ morphic, h(O) (z�, O) , and h({z E pn : ! z1 l > q}) C {w E pn : lwn l > q } . =

q '

r,, '

/

I

zo

lz' l

Figure 11.3. Extending a holomorphic function across a hypersurface

1.

We define q1

:=

The Continuity Theorem

47

q, and for v = 2, . . . , n choose qv such that h (D

X

Dq2 (0)

X · · · X

Dq, (O)) C U'

X

D.

2, . . . , n} Then ( Pn , H) with H {z E pn : l z1 l > ql or l zv l < qv for V is a Euclidean Hartogs figure, and (P, H) ( Pn , h(H)) is a general Hartogs figure, with H C G' (see Figure II.3). Since P n G' G' is connected, the proposition follows from the continuity theorem. • :=

=

-

=

-

=

Sometimes we wish to use a family of analytic disks instead of a Hartogs figure. The Continuity Principle.

Definition. A family of analytic disks is given by a continuous map t.p : D X [0, 1] ---+ en such that 'Pt (() : = t.p((, t ) is holomorphic in D, for every t E [0, 1] . The set St : = 'Pt (D) is called an analytic disk, and bSt := t.pt (8D) its boundary.

Observe that in general bSt is not the topological boundary of St . Definition. A domain G c en is said to satisfy the continuity prin­ ciple if for any family { St , t E [0, 1] } of analytic disks in en with U o < t s ( z) for every z E G. As in the case of functions that have the weak mean value property it suffices to show that s is constant in a neighborhood of a . If this is not the case, there is a small disk D = Dr(a ) CC G and b E aD with s(a) > s (b) . Since s is upper semicontin­ uous, there is a continuous function h on aD with s < h < c and h(b) < c. Solving Dirichlet' s problem we can construct a harmonic continuation of h on D. Now

h ( a)

r = __!__ 21r J

o

h(a + reit ) dt < c = s(a).

This is a contradiction

•

For later use we give the following criterion for a function to be sub harmonic: 2.5

Theorem. Let G c C be a doma in and s : G ---+ lR U { -co} an upper semicontinuous fun ction. Suppose that fo r every disk D C C G and every fun ction f E O(D) with s < Re(f) o n aD it follows that s < Re(f) o n D . Th en s is subharmonic. PROOF: Let D = Dr ( a ) c c G, h : D ---+ lR continuous and harmonic on D , and s < h on aD. For simplicity we assume a = 0. For v E N, a harmonic function h, on D, := D ( v /( v - l ))r(O)

::J

D is given by

Then (h, ) converges on D uniformly, increasing monotonically to h. Further­ more, for every v there is a holomorphic function fv on D, with Re(fv) = h,. Let E > 0 be given. Then there is a vo such that / h - hv I < E on D for v > v0. Therefore, s < h v + c- = Re(fv + c) on aD for v > Vo . By definition it follows that s < h, + E on D. Since (h,) is increasing, it follows that s < h + c and therefore s < h on D. •

Differentiable S ubharmonic Functions 2.6 Lemma. Let s : G ---+ lR be a �2 fun ction such that Szz > 0 o n G. Then s is subharmonic.

II. Domains

56

of

Holomorphy

PROOF: Let D Dr (a) c c G and let a continuous function h : D -+ lR be given such that h is harmonic on D and s < h on 8D. We define r.p := s - h. =

Assume that r.p takes its maximum at some interior point zo of D . Then we look at the Taylor expansion of r.p at zo in a small neighborhood about zo:

r.p(zo + z)

= r.p (zo ) + 2 Re Q(z) + 'Pz-z(zo)zz + R(z) ,

where Q(z) : = 'Pz (z0)z + � r.pzz (zo)z2 is holomorphic and R(z) / l z l 2 -+ 0 for z -+ 0. The function 'ij;(z ) := 2 Re Q(z) is harmonic, with '1/J(O) = 0. Since it cannot assume a maximum or a minimum, it must have zeros arbitrarily close to but not equal to 0. On the other hand, r.p(zo + z) - r.p ( zo) < 0 and 'Pz-z (zo)zz > 0 outside z = 0. This is a contradiction. Thus r.p must assume • its maximum on the boundary of D , and s < h on D. 2. 7

Theorem. Let s : G -+ lR be and only if Szz > 0 on G .

a

�2 function. Then s is subhamwnic if

PROOF: (a) Let Sz-z(z) > 0 for every z E G. Then we define s" on G by setting s" := s + ( 1 /v)zz. Obviously, (sv)zz = Szz + ( 1 /v) > 0. Then s" is subharmonic by the above lemma. Since (s, ) converges, monotonically decreasing, to s , it follows that s is subharmonic. (b) Let s be subharmonic on G. We assume that Sz-z (a ) < 0 for some a E G. Then there is a connected open neighborhood U = U(a) c G such that Sz-z < 0 on U. By the lemma it follows that -s is subharmonic on U. Then • s must be harmonic on U. So Sz-z(a) = 0, contrary to assumption.

Pluri subharmonic Functions. We return to the study of domains in arbitrary dimensions. Let G c e n be a domain and ( a,w) a tangent vector at a E G. We use the holomorphic mappmg cta,w : e -+ e n defined by cta, w( ( ) := a + ( w. Definition. Let G c e n be a domain. An upper semicontinuous func­ tion p : G -+ lR U { oo} is called plurisubhannonic on G if for every tangent vector ( a,w ) in G the function -

Pa,w ( ( ) := P o cta, w( ( ) = p(a + (w ) is subharmonic on the connected component G (a , w ) of the set a a, !... (G) e containing 0. R e m arks 1 . Plurisubharmonicity is

a local property.

c

57

2. Plurisubharmonic Functions

2. If f E O(G), then log l f l is plurisubharmonic. 3 . If PI , P2 are plurisubharmonic, then so is PI + P2 . 4. 1f p is plurisubharmonic and c > 0, then c . p is plurisubharmonic. 5 . If (Pv) is a monotonically decreasing sequence of plurisubharmonic func­ tions, then p : = limv--too p , is also plurisubharmonic. 6. Let (Pa )a EA be a family of plurisubharmonic functions. If p := sup (pa ) is upper semiconlinuous and finile, Lhen il is also plurisubham10nic. 7. 1f a plurisubharmonic function p takes its maximum at a point of the domain G, then p is constant on G.

The Levi Form Definition. Let U C e n be an open set, f The quadratic form2 Lev( !) : Ta --+ lR with

2 ( U ; JR), and a

E 9&'

E

U.

is called the Levi form of f at a. Obviously, Lev(!) is linear in f . Examples

1. In the case n = 1 we have Lev(s) (a, w) = S z-z ( a) ww . So s is subharmonic if and only if Lev(s) (a, w ) > 0 for every a E G and w E C. 2. Let f( z) : = llz ll 2 = 2::: 7 I Zi Zi . Then Lev(f) (a, w ) = llwl/ 2 for every a. 3 . If f E 9&'2 ( U; lR) and g : lR --+ lR is twice continuously differentiable, then

Lev ( g o !) (a, w) = g11 ( f (a) ) · l (8f)a( w) l 2 + g' ( f (a) ) Lev(f)(a, w) . ·

4. If F : U --+ V c em is a holomorphic map and g

E

9&'2 ( V; JR), then

L ev (g o F) (a, w) = Lev(g ) (F(a) , F'(a) ( w))

5 . For

f E 9&'2 ( U ; JR) the Taylor expansion at a E U gives f ( z ) = f (a) + 2 Re (QJ (z -

where Q J ( w ) = I:�= l fz (a) w v quadratic polynomial, and v

2

If H :

T

x

T --+ C

a))t L ev ( f ) (a, z - a) + R(z - a),

+ � I: v fzv z (a) w ,J.L

. R(z - a) hm = Z--Ta II Z - a 11 2 o .

"

"

w J.L

is a holomorphic

is a Hermitian form on a complex vextor space, then the associated quadratic fo rm Q : V --+ lE. is given by Q(v) :=H(v, v).

58

11.

Domains of Holomorphy

2.8

Theorem. Lev(f) (a, w) >

A function ! E 1&'2 (U; JR) is plurisubhamwnic if and only if Ofor every a E U and every w E T,.

PROOF: Let (a,w) be a tangent vector in G and Q := cta, w · Then f oa(O) f(a) and (fo a)(((O) = Lev ( f o a )( O , l) = Lev ( f) ( a w ) .

=

,

Now, f is plurisubharmonic if and only if f o a is subharmonic near 0 for any a = aa, w · Equivalently, (t o a)(((O) > 0 for any such a. But this is true if • and only if Lev(f) (a, w ) > 0 for any tangent vector ( a,w) in G. 2.9 a

Corollary. Let G1 c en and G2 c em be domains, F : G1 -+ G2 holommphic map, and g E 1&'2 (G1 ; JR) plurisubharmonic. Then g o F zs

plurisubharmonic on G1 .

PROOF:

This is trivial, because of the formula in Example 4 above.

•

Exhau stion Function s . For every domain G c e the function log 8c is subharmonic. In higher dimensions it is in general not true that this func­ tion is plurisubharmonic for every domain G. -

Definition. Let G C en be a domain. A nonconstant continuous func­ tion f : G -+ lR is called an exhaustionfunction for G if for c < sup c ( f ) all sublevel sets G ( f) := {Z E G : f (z ) < C) c

are relatively compact in G.

Example

For G = en , the function f (z) := llzll 2 is an exhaustion function. For G # en , we define the boundary distance 8c by 8c(z) : = dist (z, en - G).

Then -8c is a bounded, and log 8c an unbounded, exhaustion function. We only have to show that 8c is continuous: -

For every point z E G there is a point r(z) E en 8c (z)

-

G such that

= dist(z, r(z)) < dist(z , w) for every w E en

Then for two arbitrary points u, v E G we have

-

G

.

2. Plurisubharmonic Functions

8c (u) = ll u - r (u) ll < and in the same way 8c (v )
0 such that f(z ) - c:((zll 2 is plurisubharmonic on U. Pl (z) P2 (z) P3 (z )

log( 1 + _ - log( 1 -

60

3.

1 1.

Domains of Holomorphy

P seu d ocon vexity

P seudoconvexity A domain G c en is called pseudoconvex Defin ition . strictly plurisubharmonic �= exhaustion function on G.

if there is a

Remarks

1. By the smoothing lemma the following is clear: 1f - log 8c is plurisub­ harmonic, then G is pseudoconvex. 2. Pseudoconvexity is invariant under biholomorphic transformations.

3 . 1 Theorem. If G c en is a pseudoconvex domain, then G satisfies the continuity principle. Let p : G --+ lR be a strictly plurisubharmonic exhaustion function. Suppose that there exists a family { St : 0 < t < 1} of analytic disks given by a continuous mapping ip : D X [0, 1] --+ en such that So c G and bSt c G for every t E [0, 1] , but not all St are contained in G. PROOF:

The functions p o 'Pt : D --+ G are subharmonic for every t with St c G. It follows by the maximum principle that p / St < maxbs, p for all those t.

We define to := inf{t E [0, 1] : St C/- G}. Then to > 0, St0 C G, and St0 meets aG in at least one point z0. We can find an increasing sequence (tv) converging to to and a sequence of points z v E Stv converging to zo. So p ( z v) --+ c0 := sup c (p ) , but there is a c < co such that P/bs, < c for every • t E [0, 1] . This is a contradiction. 3 . 2 Corollar y.

If G is pseudoconvex, then G is Hartogs convex.

The Boundary Distance

3 . 3 Theorem. If G c en is a Hartogs convex domain, then - log 8c plurisubharmonic on G. PROOF:

For z E G and

u

E en with

8c , u ( z ) :=sup{t > 0 :

is

/lui/ = 1 we define z

+ Tu E G for /T/ < t } .

Then 8c (z ) = inf { 8c , u ( z ) : // u// = 1}, and it is sufficient to show that - log 8c , u is plurisubharmonic for fixed u . (a) Unfortunately, 8c,u does not need to be continuous, but it continuous:

is

lower semi­

3. Pseudoconvexity

61

Let zo E G be an arbitrary point and c < 8c, u (z0 ) . Then the compact set K := {Z = zo + Tu : ITI < c} is contained in G, and there is a 6 > 0 such that {z : dist(K, z) < 8 } C G. For z E Bo (zo) and ITI < c we have ll (z + T U)

-

(zo + Tu) ll

= l i z - zo ll

< S, and therefore 8c,u (z) > c.

(b) The function - log 8c ,u is upper semicontinuous, and we have to show that s(() : = - log 8c, u (zo + (b) is subharmonic for fixed u ,z0, b . First consider the case that u and b are linearly dependent: b = X u , >. -=/=- 0. Let Go be the connected component of 0 in { ( E C

:

z0 + (b E G}. Then

sup{t > 0 : zo + (b + Tu E G for ITI < t } sup{t > 0 : ( + T/A E Go for I T I < t } 1>-1 . sup{r > 0 : ( + J E Go for IJI < T J 1>-1 · 8c o (() ,

8c , u (zo + (b)

and this function is in fact subharmonic. (c) Now assume that u and b are linearly independent. Since these vectors are fixed, we can restrict ourselves to the following special situation:

Zo = O, h = e1 ,

n = 2,

and

u = e2 .

Then s(() = - log sup{t > 0 : ((, T) E G for ITI < t}. We use holomorphic functions to show that s is subharmonic. Let R > r > 0 be real numbers such that ( (, 0) E G for I( I < R, and let f : DR (O) -+ C be a holomorphic function such that s < h :=Re f on 8Dr (O) . We have to show that s < h on Dr (O ) . We have

the following equivalences:

s (() < h(()

e - h(() ( ( , c e- f(( ) ) E G for c E D. ·

(d) Define a holomorphic map F by F (z1 , z2 )

:= (rz1 , z2 e- f(r z, ) ) .

Then F is well defined on a neighborhood of the unit polydisk P 2 = P 2 (0, 1 ) . It must be shown that F (P 2 ) C G. We already know the following: 1 . F (z1 , z2) E G for l z1 l = 1 and lz2 l < 1 , because s (t) < h (t) on 8Dr (O ) . 2 . F (z1 , 0 ) E G for l z1 l < 1 , because ( (, 0) E G for 1(1 < T.

62

1 1.

Domains of Holomorphy

These facts will be used to construct an appropriate Hartogs figure. First, note that

JF ( z1 , z2 ) =

(:

� )

e-f rzl )

,

SO

det JF(ZI , Z2 ) � 0.

By the inverse function theorem it follows that F is biholomorphic. For 0 < 6 < 1 we define ho : C 2 --+ C 2 by h0 ( z1 , z2 ) to the compact set

c

:=

:=

(z1 , 8 z2 ) and apply h0

{ ( zl , z2 ) E C 2 : (/zl l < 1 , Z2 = 0) or ( /zi / = 1 , /z2 /

I

1)} c P 2 .

Consequently,

Co := ho (C) = { (zl , Z2) E C2 : (/zl / < 1 , Z2 = 0) or ( /zl / = 1, / z2 / < 8)} . Then F (C0) C G, as we saw above, and therefore Co C F - 1 ( G) . For 0 < c < min(8, 1 - S) we define a neighborhood U6 of C0 by U6

{ ( zi , Z2 ) E C2 : ( /zi / < 1 + c, / z2 / < c) or ( 1 - c < /zi / < 1 + c, /z2 / < D + c ) } . If we choose small enough, then U6 c F- 1 (G ) . 1 Finally, we define He; :=hi (U6 n P 2 ) n P 2 (see Figure 1 1 .6). Then c

H e;

{ (z1 , z2 ) E P 2 : (z1 , 8z2 ) E U6 n P2 } = ( z1 , z2 ) E C2 : (/ zi / < 1 , / z2 / < ) or ( 1 - · E< / z1 / < I , / z2 / < 1 )

{

�

c

b .

.

}.

. .. . . . ..

Figure 1 1 .6. Construction of the Hartogs figure Since (P2 , H 6) is a Euclidean Hartogs figure, (F o h0 ( P2 ) , F o ho (H c; )) is a general Hartogs figure with Fo h0 (H6) c F (U n P 2 ) c G. Since G is Hartogs c;

3. Pseudoconvexity

63

pseudoconvex, it follows that F o h6(P 2} c G. This is valid for every 8 < 1 . But P2 = Uo 0, it follows that g o a has a minimum at t = 0. Then (g o a)" (O) > 0, and Hess(g) is positive semidefinite on T. Now let the criterion be fulfilled, assume that 0 E G, and define g6 by

For small E and large N the set G, := {x : g6(x) < 0} is a domain. We have G, C C6 C G for E' < c:, and Uc;> O C6 = G. Therefore, it is sufficient to show that G, is convex. ,

The Hessian of {!c; is positive definite on Tx (ac) for every X E ac. Thus this also holds in a neighborhood U of aC. If E is small enough, then aC6 C U. We consider S := { (x, y) E C6

x

G, : txt ( 1 - t ) y E G, for O < t < 1 ) .

Then S is an open subset cf the connected set G, x G,. Suppose that S is not a closed subset. Then there exist points x0 , y0 E G, and a to E (0, 1) with toxo+ ( 1 - to)Yo E ace; . So the function t H {!c; 0 a(t) , with a (t) :=txo + ( 1 - t)yo , has amaximum at to. Then (g6 oa) " (t0) < 0 and Hess(g6) (a(t0 ) , x0 - y0) < 0. This is a contradiction. •

A domain G = {g < 0) is called strictly convex at xo E ac if Hess(g) is positive definite at x0. This property is independent of g and invariant under affine transformations. Now we return to Levi convexity. 4.4

Lemma.

Let U C en be open and r.p E � 2 (U; �) . Then

Lev( r.p) (z, w)

=

1

4

(Hess( r.p ) (z , w ) + Hess( r.p ) (z, iw ))

68

II.

Domains

oE

Holomorphy

This is a simple calculation!

PROOF:

•

4.5

Theor em. Let G c c en be a dom ain with smooth boundary. Then the following statements are equivalent: 1.

G is strictly T�evi convex. 2. The re is an open neighborhood U = U( EJG) and a strictly plurisubhar­ monic function g E 'l!f= (u; JR) such that U n G = {z E U : g(z) < 0) and (dg)z -=/:- 0 fo r z E U. 3. For eve ry z E EJG there is an open neighborhood W = W(z) C en , an open set V c en, and a biholomorphic map F : W --+ V such that F(WnG) is convex and even strictly convex at every point of F(W n EJG) . PROOF:

( 2) : We choose a global defining function g for G, and an open neighborhood U = U ( EJG) such that g is defined on U with (dg)z -=1- 0 for z E U. Let A > 0 be a real constant, and {!A : = eA e - 1. Then {!A is also a (1 )

===?

global defining function, and

Lev(gA)(z , w) The set K : = EJG

X

= Ae

A e (z ) [Lev(g) (z, w) + A l (8g)z (w) l 2 ]

•

S2n- l is compact, and

K0 : = {(z, w) E K : Lev(g) (z, w) < 0 } is a closed subset. Since Lev(g) is positive definite on Hz ( EJG) , we have ( ag)z (w) -=1- 0 for (z,w) E K0 . Therefore,

min Lev(g) (z , w) > - oo ,

M

K

min l (8g)z (w) l 2 > 0 .

C

Ko

We choose A so large that A ·

C +M

> 0. Then

Lev(gA )(z , w) = A · [Lev (g) (z, w) + A l (8g)z(w )l 2 ] > A · (M + AC) > 0 for (z, w)

E

K0 , and

Lev (gA)(z , w) > A 2 · 1 (8g)z(w)l 2 > 0 for (z, w) E K - Ko. So Lev(gA)(z, w) > 0 for every z E EJG and every w E en - {0}. By conti­ nuity, {!A is strictly plurisubharmonic in a neighborhood of EJG. (2 )

===?

( 3 ) : We consider

transformations:

a

point zo E EJG and make some simple coordinate

4.

Levi Convex Boundaries

69

By the translation z H w = z zo we replace zo by the origin, and a permutation of coordinates ensures that (!w1 (0) -=/=- 0. -

The linear transformation gives u1 = w . V' g(O) t, and therefore g( u )

-

2 Re (u . V'(g o w) (O) t ) + term sofdegree > 2 2 Re (u . Jw (O) t . V' g(O) t ) + terms of degree > 2

2 Re (w . V' g(O) t ) + terms of degree > 2 - 2 Re( u1) + terms of degree > 2. -

Finally, we write g( u) = 2Re(u1 + Q(u)) + Lev(g) (O, u) + . . . , where Q is a quadratic holomorphic polynomial, and make the biholomorphic transforma­ tion It

follows that

+ Lev(g) (O, v) + terms of order

g(v) = 2Re(v1)

> 3.

By the uniqueness of the Taylor expansion g(v)

=

Do(O) (v)

+ �Hess(g) (O, v) + terms of order

> 3,

and therefore Hess(g)(O, v) = 2 . Lev(g) (O, v) > 0 for v -=/=- 0 (in the new coordinates). Everything works in a neighborhood that may be chosen to be convex. (3)

===?

( 1 ) : This follows from Lemma 4.4: Hess(g) > 0 on Tz (8G)

===?

Lev(g) > 0 on Hz (8G) .

The latter property is invariant under biholomorphic transformations.

•

A Theorem of Levi. Let G cc en be a domain with smooth boundary. If G is strictly Levi convex, then it is easy to see that G is pseudoconvex.

We wish to demonstrate that even the weaker Levi convexity is equivalent to pseudoconvexity. For that purpose we extend the boundary distance to a function on en. dc(z)

:=

8c(z) 0 -8cn _c (z)

for z E G, for z E 8G, for z !f. G.

70

II.

Domains of Holomorphy

4.6 Lemma.

-de is

a

smooth defining function fo r G

PROOF: We use real coordinates x = ( x 1 , . . . , X N ) with N = 2n. It is clear that G = { x : -de( x) < 0} . Let Xo E ac be an arbitrary point and (! : U (xo) -+ JR. a local defining function. We may assume that Qx N (xo) -=/=- 0. Then by the implicit function theorem there is a product neighborhood U' x U" of xo in U and a smooth function h : U' -+ JR. such that { (x' xN ) E U' x U" : g(x' , xN ) = 0 } = { (x jl,( x')) : x' E U' } ,

'

It follows that O = "Vx' Q(x', h(x')) + gxN (x' , h (x')) . Vh(x ') . At the point (x' , h(x')) E 8G the gradient "Vg(x', h(x')) is normal to 8G and directed outward from G. Every point y in a small neighborhood of the boundary has a unique representation y = x + t . V g(x) , where t = -de( Y ) and X is the point where the perpendicular from y tO 80 meetS the boundary. Therefore, we define the smooth map F : U' x JR. -+ JR.N by y

F (x ' , t )

=

=

(x', h(x' )) + t . V g(x' , h (x')) .

Then there are smooth functions =

JJR,F (x', t) and therefore

A

and b such that

( VEhN(x') - I + t A(x') + t b(x') ·

.

E det ( N - I "V h (x')

)

h (x')) V h (x ') t ' (x gx , N det JJR,F(x' , 0) Qx N (x' , h(x')) "V h (x') t EN - 1 - (!x N (x' , h (x )) . de t O' 1 + II "V h (x' ) l l 2 - (!xN (x', h(x'))( l + 11Vh( x') l l 2 ) -=/=- 0. 1

·

(

)

It follows that there exists an c > 0 such that F maps U' x ( - s, s) diffeo­ morphically onto a neighborhood W = W (x0 ) , and U' x {0} onto ac n W . Moreover, since de (x + t V g(x) ) = -t for I t I < c and c small enough, it follows that de = ( t)J y - 1 is a smooth function near ac. If p ' is defined by p' (x', t ) := (x' ,0), then the projection ·

-

p = p'

o

y- 1 : X + t \l g(x) f-+ X, for X E 80, ·

is a smooth map, and de is given by de( Y ) = CJ . IIY - P( Y ) II , where CJ = 1 for y E G and CJ = - 1 elsewhere. For y # 8G we have

4.

Levi Convex Boundaries

71

N

- Pk ( y )) (8k v - (Pk )yv (Y)) ( · Yk : � II IIY (Y) p ( y ) j p (Y)) N J Y) IIY -:( y )ll · [Yv - Pv ( - (y _

Yv

-

and therefore

'

a . [y - p ( y) - Dp ( y ) ( y - p (y )) ] . IIY - P ( Y ) II Since g(p( y )) 0, it follows that Dp( y )("V g( p ( y ))) = 0. But y - p ( y ) is a multiple of \1 g( p ( y )). Together this gives . "Vg( p ( y)) y - p ( y) "V de(Y ) = a · II Y - (Y) II = ± II g( ( ))ll . P "V p y If y tends to 8G, we obtain that "Vde( Y ) of. 0. • "Vde(Y)

=

E.E. Levi showed that every domain of holomorphy with smooth boundary is Levi convex, and locally the boundary of a strictly Levi convex domain G is the "natural boundary" for some holomorphic function in G. Here we prove the following result, which is sometimes called "Levi's theorem". 4.7

Theorem. A domain G with smooth boundary is pseudoconvex if and only if it is Levi convex.

PROOF :

( 1 )Let G be pseudoconvex. The function -de is a smooth boundary function for G, and - log de = log 8e is plurisubharmonic on G, because of the pseudoconvexity. We calculate -

1

1

Lev( -de )(z , w) + I ( 8(de )) z (w) 1 2 · · de (z)2 de(z ) This is nonnegative in G. If z E G, w E T,, and ( 8 (de)) z (w) = 0, it follows that Lev(-de) (z , w) > 0. This remains true for z -+ 8G, so - de satisfies Lev( - log de )(z, w)

=

the Levi condition.

(2) Let G be Levi convex, and suppose that G is not pseudoconvex. Then in any neighborhood U of the boundary there exists a point z0 where the Levi form of - log 8e has a negative eigenvalue. This means that there is a vector wo such that

'P(((O) = Lev(log 8e) (z0, wo) > 0, for cp(() := log 8e (zo t (wo) . Consider the Taylor expansion

cp(()

) cp(O) + 2 Re(cp((O)( + 2 cp(((0 )( 2 ) + 'P(((O) I ( I 2 + - cp(O) t Re(A( + B(2 ) + >-1( 1 2 + . . . ,

72

II.

Domains of Holomorphy

with complex constants A , B and

a

real constant >. > 0.

We choose a point Po E 8G with 8e (zo) = IIPo - zo l l , and an arbitrary c > 0. Then an analytic disk 'ljJ : Dc:(O ) --+ en can be defined by

'lj;(() := Zo + ( wo + exp (A( + B(2) (Po - zo). We have 'lj; (O) = p0 , and we wish to show that 'ljJ (() E G, for 0 < ICI < c and E sufficiently small.

Since cp(( ) > cp(O) + R e(A( + B( 2 ) + (>. / 2 ) 1(1 2 near ( 8e (zo + (wo)

=

0, it follows th at

exp(cp( ( )) > exp ( cp (O)) I exp (A( + B( 2 ) I exp > 8e(zo) I exp (A( + B( 2 ) I - ll exp (A( + B( 2 ) (Po - zo) II , ·

·

·

C 1(1 2)

for ( small and -=f. 0 . This means that we can choose the E in such a way that 't/J(() E G, for 0 < ICI < c. The analytic disc is tangent to ac from the interior of G. Now f( () = de ( 't/J ( ( )) is a smooth function with a local minimum at ( Therefore ( 8de)p0 ('t/J'(O)) = (8! )0 (1) = 0 , and

f( ( )

=

Re (!((( 0 ) ( 2 ) + !((1(1 2 +terms of order >

Since Re ( f 0 for every t, it follows that Lev( de )(po , 't/J' (0 ))

=

0.

3.

f(((O) > 0 . This is a contradiction to the Levi condition at p0, because -de is a defining function for G.

=

•

Exer cises 1 . Prove Lemma 4.4. 2. Assume that G c c C2 has a smooth boundary that is Levi convex outside a point a that is not isolated in 8G. Show that G is pseudoconvex. 3. Assume that G C C2 is an arbitrary domain and that S c G is a smooth real surface with the following property: In every point of S the tangent to S is not a complex line. Prove that for every compact set K c G there are arbitrarily small pseudoconvex neighborhoods of S n K. 4. Assume that G c c C2 is a domain with smooth boundary. Then G is strictly Levi convex at a point Zo E ac if and only if the following condition is satisfied: There is a neighborhood U = U (zo), a holomorphic function cp : D --+ U with cp (O) = zo and cp'(O) -=f. 0, and a local defining function g on U such that (g o cp)(( ) > 0 on D - {0} and (g o cp) (((O ) > 0 .

5.

Holomorphic Convexity

73

5. Let G c c en be a domain with smooth boundary. If G satisfies the strict Levi condition at zo E 8G, then prove that the following hold: (a) There is no analytic disk r.p : D --+ en with

r.p (O) = z0

and

lim (-+ o

bc( r.p (()) llr.p ( ( ) - r.p (O) II 2

=

0

(b) There are a neighborhood U = U(z0) and a holomorphic function f in U with G n { z E U : f (z) = O} = {zo } . 6. A bounded domain G c en is called strongly pseudoconvex if there are a neighborhood u = u (ac) and a strictly plurisubharmonic function g E '6'2 (U) such that G n U = { z E U : g(z) < 0} . Notice that a strongly pseudoconvex domain does not necessarily have a smooth boundary ! Prove the following results about a strongly pseudoconvex bounded do­ main G: (a) G is pseudoconvex. (b) If G has a smooth boundary, then G is strictly Levi convex. (c) For every z E 8G there is a neighborhood U = U (z) such that U nG is a weak domain of holomorphy. 7. Let G c en be a pseudoconvex domain. Then prove that there is a family of domains G, C G such that the following hold: (a) G, CC Gv+l for every v. ( b) U:;" 1 Gv = G. (c) For every v there is a strictly plurisubharmonic function fv E '6'00 (Gv+I ) such that G, is a connected component of the set

{z

5.

E Cv+ I

:

fv (z) < 0} .

Holomorp h ic C on vexity

Affine C onvexityWe will investigate relationships between pseudocon­ vexity and affine convexity. Let us begin with some observations about convex domains in lRN . Let 2 be the set of affine linear functions f : JRN --+ lR with

f(x) = a 1x1 + . . . + aN X N t b,

a 1 , . . . , aN , b E lR.

If Jvf is a convex set and x0 a point not contained in M, then there exists a function f E 2 with f(x0) = 0 and J I M < 0. For any c E JR, the set {x E JRN : f (x) < c} is a convex half-space.

Definition .

Let

H( M)

M

c JRN be an arbitrary subset. Then the set

:= {x E JRN : f (x) < sup f, for all f E 2

is called the affine convex hull of M.

M

}

74

11.

Domains of Holomorphy

5. 1 Pr oposition. Let M, M1 , M2 c lRN be arbitrary subsets. Then 1.

M c H (M) . 2. H ( M) is closed and convex. 3.

H ( H ( M ))

4. If M1

c

=

H ( M) .

M2 , then H(M1 )

c

H(M2). 5. If M is closed and convex, then H(M) = M . 6. If M is bounded, then H ( M ) is also bounded.

PROOF:

( l ) is trivial.

5e

with f (xo) > supM f . By conti­ nuity, f(x) > sup, f in a neighborhood of xo. Therefore, H(M) is closed.

(2) If x0 rf_ H ( M ) , then there is an f E

If x0 , y0 are two points in H ( M ) , then they are contained in every convex half-space E = { x : f ( x) < sup M f } , and also the closed line segment from x0 to y0 is contained in each of these half-spaces. This shows that H(M) is convex.

( 3) We have to show that H (H( M )) C H(M) . If x

E H(H(M)) is an arbi­

trary point and f an element of 2', then f (x) < supH ( M ) f < supM f , by the definition of H(M). (4) is trivial.

( 5) Let M be closed and convex. If xo rf_ M, then there is a point Yo

E M

such that d ist (xo , M) = dist(x0, Yo ) (because M is closed). Let zo be a point in the open line segment from x0 to yo. Then z0 rf_ M , and there is a function f E 5e with f(zo) = 0 and JI M < 0. Since t H f(txo + ( 1 - t)yo) is a monotone function, it follows that f(x0 ) > 0 and therefore x0 rf_ H ( M ). (6) If M is bounded, there is an R > 0 such that M is contained in the closed • convex set BR (O) . Thus H( M ) C BR (O). Remark.

H(M) is the smallest closed convex set that contains M .

5 . 2 Theor em . A domain that H(K) c c G.

G c JRN is convex if and only if K c c G implies

PROOF: Let G be a convex domain, and M c c G a subset. Then H(M) is closed and contained in the bounded set H( M ). Therefore, H(M) is compact, and it remains to show that H(M) C G. If there is a point xo E H(M) - G, then there is a function f E 5e with f (x0) = 0 and fi e < 0. It follows that sup M f < 0, and f(xo) > supM f . This is a contradiction to xo E H( M ).

5.

Holomorphic Convexity

75

On the other hand, let the criterion be fulfilled. If xo , Yo are two points of G, then K := { xo, Yo } is a relatively compact subset of G. It follows that H(K) is contained in G. Since H(K) is closed and convex, the closed line segment • from xo to Yo is also contained in G. Therefore G is convex.

H olomor phic C on vexity. Now we replace affine linear functions by holomorphic functions. Definition.

Let G C en be a domain and K c G a subset. The set

K = Kc :

{z E G

:

l f(z) l < s� I J I , for all f E

O(G) }

is called the holomorphically convex hull of K in G.

5 . 3 Pr oposition. Then

Let G c en be a dom ain, and K, K1 , K2 subsets of G.

1. K c K . 2. K is closed in G. 3. K = K . 4. If K1 C K2 , then K1 C K;. 5. If K is bounded, then K is also bounded. � �

�

�

PROOF:

( 1 )is trivial. �

(2) Let z0 be a point of G - K. Then there exists a holomorphic function f on G with l f(zo ) l > su pK i f l . By continuity, this inequality holds on an entire neighborhood U = U(zo) c G. So G - K is open. �

(3 ) SUPx l f l

tmPK i f l .

(4) is trivial.

(5) If K is bounded, it is contained in a closed polydisk pn (O, r ) . The coordi­ nate functions Zv are holomorphic in G. For z E K we have l zv l < supK i zv l I • r. Hence K is also bounded. �

Definition . A domain G c en is called holomorphically convex K cc G implies that K CC G. �

if

II.

76

Domains of Holomorphy

Example

In C every domain is holomorphically convex: Let K cc G be an arbitrary subset . Then K is bounded, and it remains to show that the closure of K is contained in G. If there is a point z0 E R - G, then Zo lies in aR n aG. We consider the holomorphic function f(z ) : = 1 /(z - z0) in G. If (z,_, ) is a sequence in K converging to zo, then l f ( z,_, ) l < supKifl < supKI ! I < oo . This is a contradiction. For n > 2, we will show that there are domains that are not holomorphically convex. But we have the following result. �

�

5.4

Pr op osition . If G morphically convex.

C

C" is an affine convex domain, then it is holo­

PROOF: Let K be relatively compact in G. Then H ( K ) C C G. If zo is a point of G - H ( K ) , then there exists an affine linear function >. E £ with >.( zo) > supK >. . Replacing >. by >. - >. ( 0 ) we may assume that >. is a homo­ geneous linear function of the form >.(z)

=

2 Re(a 1 z1 +

·

·

·

+ a , z, ) .

Then f(z) := exp ( 2 . (a 1 z1 + . . . + a, z, )) is holomorphic in G, and

lf (-3 ) 1

=

exp ( >. ( z)) . Therefore, l f (zo )l > supKifl, and zo E G - k .This proves K c c • G. In general, holomorphic convexity is a much weaker property than affine convexity.

The C artan-Thullen Theor em . Let G c C" be a domain, and E > 0 a small real number. We define G, : =

{Z

E G : 8c (z) > s}.

Here are some properties of the set G,: 1 . If z E G, then there is an E > 0 such that 8c(z) > c . Therefore, G = U c: >O G,. 2. If c1 < c 2 , then G, => Gc: 2 • 3. G, is a closed subset of C" . In fact, if zo E C" - G, then 8c (zo) < c or z0 rf. G. In the latter case, the ball B, (zo) is contained in C" - Gc:. If z0 E G - Gc: and 8 := 8c(zo ) , then Bc:-o (zo) C C" - G,. So C" - G, is open.

f

a holomorphic function in G. If K C G, thenfo r any 8 with 0 < 8 < E there exists a constant C > 0 such that the following inequality holds:

5 . 5 Lemma. Let G c C" be a dom ain, K C G a compact subset, and

5.

77

Holomorphic Convexity

PROOF: For 0 < 8 < E, G := { z E G : dist(K, z) < 8} is open and relatively compact in G, and for any z E K the closed polydisk Pn (z , 6) is contained in G' c G. If T is the distinguished boundary of the polydisk and l f l < C on G', then the Cauchy inequalities yield a! . a! a f s < p < I D (Z ) I 8 1al � l f l 8 1al . C. •

5.6 Theor em (Car tan-Thullen). If G is a weak dom ain of holomorphy, then G is holomorphically convex. �

Let K CC G. We want to show that K c c G. Let dist(K, en - G) > dist (K, en - G) > 0. Clearly, K lies in G,. PROOF:

E

We assert that the holomorphically �onvex hull R lies even in G,. Suppose this is not so. Then there is a z0 E K - G,. Now let f be any holomorphic function in G. In a neighborhood U = U(z0 ) C G, f has a Taylor expansion

f(z)

= L a,_, (z

v>O

- zo) " , with

a,

=

1D" f(z0 ). v!

The function z H a,_, ( z) := ;, D " f ( z) is holomorphic in G. Therefore, l a,_, (zo) l < supK i a,_, (z) l . By the lemma, for any 6 with 0 < 8 < E there exists a C > 0 such that supK ia,_, (z) l < C/8 1 '-' l , and then

On any polydisk pn (z0 , 6) the Taylor series IS dominated by a geometric series. Therefore, it converges on P = pn ( z0 , E) to a holomorphic function f. We have f = f near z0 , and then on the connected component Q of zo in P n G. Since P meets G and en - G, it follows from Lemma 1..9 that there is a point Z l E p n 8Q n ac. Then f cannot be completely singular at z l . This is a contradiction, because f is an arbitrary holomorphic function in G, and G is a weak domain of holomorphy. • �

�

E xer ci se s

1 . Let G1 c G2 c en be domains. Assume that for every f E O(G1 ) there is a sequence of functions f,_, E O(G2 ) converging compactly on G1 to f. Show that for every compact set K c Gl it follows that Kc2 n cl = Kc l '

78

II.

Domains of Holomorphy

2. Let F : G 1 --+ G2 be a proper holomorphic map between domains in en ,

respectively em . Show that if G2 is holomorphically convex, then so is G1 . 3. Let G c en be a domain and S c G be a closed analytic disk with boundary bS. Show that S C (bS ) c· 4. Define the domain G C e 2 by�G := P 2 (0, 1 ) - P 2 (0, 1/2). Construct the holomorphically convex hull Kc for K : = { ( z1 , z2 ) : z1 = 0 and l z 2 l = 3/4}. Is Kc a relatively compact subset of G ? 5. Let F be a family of functions in the domain G. For a compact subset K c G we define -

{

K;: : = z E

}

G : lf(z ) l < s� l f l for all f E F . �

The domain G is called convex with respect to F, provided that K;: is relatively compact in G whenever K is. Prove: (a) Every bounded domain is convex with respect to the family 'l!f0 (G) of all continuous functions. (b) The unit ball B = B1 (0) is convex with respect to the family of holomorphic functions zi . z� with v, fJ, = 1 , . . . , n and k, l E No .

S in gu lar Fun ction s

6.

Nor m al Exh au stion s .Let G c en be a domain. lf G is holomorphically convex, we want to construct a holomorphic function in G that is completely singular at every boundary point. For that we use "normal exhaustions." Definition . A normal exhaustion of G is a sequence ( K, ) of compact subsets of G such that: 1 . K, C C ( Kv+ l t , for every v. 2. U::" 1 Kv = G .

6 . 1 Theor em. Any domain G in en admits a normal exhaustion. ff-G I S holomorphically convex, then there is a normal exhaustion (K, ) with K, K, fo r eve ry v. In the general case, K, := Pn (o, v) n G 1 ;v gives a normal exhaustion. 1f G is holomorphically convex, K" c c G for every v. We construct a new exhaustion by induction. PROOF:

Let Ki := K 1 . Suppose that compact sets Ki , . . . , K�-1 have been con­ structed, with Kj = Kj for j = 1, . . . , v - 1 , and Kj CC (JC;*+ 1 t . Then there exists a >.(v) E N such that K�_ 1 c ( K;.. ( v) t . Let K� : = K;.,(v) ·

6. Singular Functions �

It is clear that (K� ) is a normal exhaustion with K�

=

79

K� .

•

Unb ounded Holomor phic Function s . Again let G c en be a domain. �

6 . 2 Theor em. Let ( K, ) be a normal exhaustion of G with Kv = K, >. (J.L) a strictly monotonic increasing sequence of natural numbers, and (zJ.L) a sequence of points with zJ.L E K;.. ( J.L) + l - K;.. c,"l .

Then there exists a holomorphic function bounded.

f

in G such that

l f (z J.L ) I

is un­

PROOF: The function f is constructed as the limit function of an infinite series f = 2:;:" 1 fw By induction we define holomorphic functions fJ.L in G such that:

2 - J.L for J.L > 1. J.L - 1 2 . I JJ.L(zJ.L) I > J.L + 1 + Ll fj (zJ.L) I for J.L > 2 . j=l 1.

I JJ.L I K>--.(fl,) and (zJ.L) are constructed. Let z1 be arbitrary in CI - KI . Then there is a unique number >. (I) such that z i E K>-(1)+1 - K>-( I ) · Now suppose z i , . . . , zJ.L - I have been constructed such that Zj E cj n (KA(j)+l - K,A(j ) ), for j = 1 , . . . , f-L - 1.

We choose zJ.L E CJ.L - K>-(J.L - IJ+l and >-.(J-L) as usual. That is possible, since there is a point w E BV(J.L) n acfl. n ac if CJ.l. is a connected component of B v (J.L ) n G. Then en - KA(J.L-I ) +I is an open neighborhood of w and contains points of C,. (3) Now we show that property (2) of the theorem is satisfied, Let z0 be a point of ac, U = U(zo ) an open connected neighborhood, and Q a connected component of U n G. We assume that only finitely many zJ.L lie in Q , say ZI , . . . ,z m . Then U * := U - {z i , . . . , z m }

and

Q* : = Q - {z 1 , . . . , zm }

are open connected sets that contain no z,. Obviously, Q* is a connected component of G n U * . There is a point wo in U* n 8Q* n 8G, and a ball B, C U* with wo E B,. Then B, n G c U* n G. Moreover, B, n G must contain a point WI E Q*. The connected component C* of w1 in B, n G is a subset of the connected component of WI in U* n G. But C* is an element CJ.Lo of C. By construction it contains the point zJ.Lo . That is a contradiction. Infinitely many members of the sequence belong to Q. • 6.5 Theor em . If G is holomorphically convex, then it is a dom ain

morphy.

d

holo­

�

PROOF: Let ( K, ) be a normal exhaustion of G with K = K, and choose " sequences A(J-L) E N and (z, ) in G such that z, E K>-(J.Ll+l - K>-(J.L) · We may assume that for every point z0 E ac, every open connected neighborhood U = U(z0) , and every connected component Q of U n G there are infinitely many z, , in Q.

Now let f be holomorphic in G and unbounded on D := {zJ.L clear that f is completely singular at every point Zo E 80.

:

f-L

E N} . It is •

82

II. Domains of Holomorphy

Remark. It is not necessary that a completely singular holomorphic func­ tion is unbounded. In 1978, D. Catlin showed in his dissertation that if G cc en is a holomorphically convex domain with smooth boundary, then there exists a function holomorphic in G and smooth in a neighborhood of G that is completely singular at every point of the boundary of G. Exer cises 1 . A domain G c c en is holomorphically convex if and only if for every z E 8G there is a neighborhood U(z) such that U n G is a domain of holomorphy. 2 . Let G1 c en and G2 c em be domains of holomorphy. Iff : G1 --+ e m is a holomorphic �pping then j - 1 ( G2) n G1 is a domain of holomorphy. 3 . Find a bounded holomorphic function on the unit disk D that is singular at every boundary point. ,

7.

E xamp les and Ap p lication s

Domains of Holomorphy 7 . 1 Pr op osition. holomorphy.

Every doma in in the complex plane e zs a domain of

PROOF: We have already shown that every domain in e is holomorphically • convex. Therefore, such a domain is also a domain of holomorphy.

7 . 2 Theor em. alent:

Th efollowing statements about domains G E en are equzv­

1 . G is a weak doma in of holomorphy. 2. G is holomorphically convex. 3. For every infinite discrete subset D C G there exists a holomorphicfunc­ tion f in G such that I ! I is unbounded on D . 4. G is a doma in of holomorphy. The equivalences have all been proved in the preceding paragraphs. Fur­ thermore, we know that every domain of holomorphy is pseudoconvex. Still missing here is the proof of the Levi problem: Every pseudoconvex domain is holomorphically convex. We say more about this in Chapter V. Every affine convex open subset of en is a domain of holomorphy. The n-fold Cartesian product of plane domains is a further example. 7 . 3 Pr op osition. If G1 , . . . , Gn C e are arb itrary domains, then G G 1 x · · · x G is a domain d holomorphy. n

:=

7 . Examples and Applications

83

PROOF: Let D = {zJ.L = (zf , . . . , zf:: ) : f-L E N} be an infinite discrete subset of G. Then there is an i such that (zf) has no cluster point in Gi , and there is a holomorphic function f in Gi with limJ.L--> = I f ( zf) I = = · The function f in G, defined by [(z1 , . . . , z ) : = f (zi ) , is holomorphic in G and unbounded n on D. •

Remar k. The same proof shows that every Cartesian product of domains of holomorphy is again a domain of holomorphy.

C omplete Reinhar dt Domains . Let G c en be a complete Rein­ hardt domain (see Section 1 . 1 ) . We will give criteria for G to be a domain of holomorphy. For that purpose we define a map log from the absolute value space "f! to JRn by

Definition . A Reinhardt domain G is called logarithmically convex if log r(G n (C* )n) is an affine convex domain in JRn . Remark. For z = (z1 , . . . , zn ) E G we have lo g r (z ) = (loglz1 l , . . . , logiz i ) . n If z E (C* )n, then i zil > 0 for each i , and log r(z) is in fact an element of JRn . 7.4 Pr oposition. The doma in d convergence of a power series S(z) Ev > o av z " is logarithmically convex. PROOF: Let G be the domain of convergence of S(z) , and M : = log r(G n (C* )n) c JRn . We consider two points x , y E M and points z , w E G n (C* )n with log r (z) = x and log r(w) = y . If >. > 1 is small enough, >.z and :XW still belong to G n (C* )n. Since S(z) is convergent in >.z, Xw, there is a constant C > 0 such that l av l . >. 1 " 1 - l z" l < C

Thus

and

l av l . >. 1 " 1 . l w" l < C, for every v E N8

lav l . ;. l v l . l z" l t - l w" l l - t < C, for every v and 0 < t < 1 .

It follows from A bel's lemma that S(z) is convergent in a neighborhood of

This means that Zt E G and tx + ( 1 - t) y = log r (z t ) E M , for 0 < t < 1 . Therefore, M is convex. •

84

I I . Domains of Holomorphy

7 . 5 Pr op osition. Let G be a complete Reinhardt domain. If G is logarith­ mically convex, then it is holomorphically convex. PROOF: Let K be a relatively compact subset of G. Since G is a complete Reinhardt domain and K a compact subset of G, there are points z1 , . . . , z�,; E G n (C* )n such that

K c G'

:=

k

U P n ( O , qi )

i=l

C G,

where qi := T(z i ) ·

We consider the set Jlt = { m (z) = zv : v E N0} of monomials, which is a subset of O(G). For z E pn (o, Qi) and m E A we have Let Z

:=

{z 1 ,

•

•

,

•

l m(z) l

�

z �,; } . Then for z E K it follows that


1 on Uv ,j , and 8Kv+l c U�v 1 Uv ,j · We define P" == {z E (Kv +l t : lfv,j (z)l < l for j = l , .. . , kv } · Clearly, K, C Pv C (Kv+ l t · Furthermore, M :=Kv+l - (Uv,l U . . . U Uv,kJ is a compact set with P" C M C (Kv+l t · Consequently, P" CC Kv+l · Thus P" is a special analytic polyhedron in G. It follows trivially that the sequence • (Pv) exhausts the domain G.

8. Riemann Domains over en

87

In the theory of Stein manifolds one proves the converse of this theorem. Exercises 1. If R is a domain in the real number space JRn , then

TR

= R + ilRn := {z E en : (Re(zi ), . . . , Re(z )) E R }

n is called the tube doma in associated with R . Prove that the following properties are equivalent: (a) R is convex. (b) TR is (affine) convex. (c) TR is holomorphically convex. (d) TR is pseudoconvex. Hint: To show (d) ::::::::} ( a) choose x0 , yo E R. Then the function cp ( ( ) - In 8rR (x0 + ((Yo - x0 )) is subharmonic in D. Since 8rR ( x +iy ) = 8R(x) , one concludes that t � - ln 8R ( x0 + t (y0 - x0 )) assumes its maximum at t = 0 or t = 1 . 2 . Let G c en be a domain. A domain G c en is called the envelope of holomorphy of G if every holomorphic function f in G has a holomorphic extension to G. Prove: (a) If R c JRn is a domain and H(R) its affine convex hull, then G := H(R) + ilRn is the envelope of holomorphy of the tube domain G = R + i JRn. (b) If G c en is a Reinhardt domain and G the smallest logarithmically convex complete Reinhardt domain containing G, then G is the envelope of holomorphy of G . Hint: Use the convex hull of log T(G) . 3. Construct the envelope of holomorphy of the domain Gq P2 ( 0, ( 1 , q)) u P2 (0, (q, 1)) . 4. A domain G c en is called a Runge do main if for every holomorphic function f in G there is a sequence (P v ) of polynomials converging com­ pactly in G to f. �

�

�

�

:=

Prove that the Cartesian product of n simply connected subdomains of e is a Runge domain in en. 5. A domain G C en is called polynomially convex if it is convex with respect to the family of all polynomials (cf. Exercise 5.5). Prove that every polynomially convex domain is a holomorphically convex Runge domain.

8.

R iem ann Dom a in s

over

en

Riemann Dom ains. It turns out that for general domains in en the envelope of holomorphy (cf. Exercise 7 .2) cannot exist in en . Therefore, we have to consider domains covering en .

II.

88

Domains of Holomorphy

Definition . A (Rzemann) domain over e n is a pair (G, 1r) with the following properties: 1. G is a connected Hausdorff space. 4 2. 7r : G -+ e n is a local homeomorphism (that is, for each point X E G and its "base point" Z := 1r( X ) E e n there exist open neighborhoods U = U(x) c X and V = V(z) c en such that 1r : U -+ V is a homeomorphism). Remar ks

1. Let ( G, 1r) be a Riemann domain. Then G is pathwise connected, and

the map 1r : G -+ e n is continuous and open. The latter means that the images ci open sets are again open. 2. If (G, 1r") are domains over e n for v = 1 , .. . , l, and x, E G, are points over the same base point z0 , then there are open neighborhoods U" = Uv (xv) c G, and a connected open neighborhood V = V(zo ) C e n such that 1l"v luv : U" -+ V is a homeomorphism for v = l , .. . , l . Examples

1.

If G is a domain in en , then (G,idc) is a Riemann domain.

2. The Riemann surface of vz (without the branch point) is the set G := { (z, w) E C* x e : w 2 = z} .

If G is provided with the topology induced from e* x e , then it is a Hausdorff space. The mapping r.p : C* -+ G defined by ( � ( ( 2 , () is continuous and bijective. Therefore, G is connected. The mapping r.p is called a uniformization ci G.

Now let 1r : G -+ e be defined by 1r(z, w ) := z . Clearly, 1r is continuous. If (z0, w0) E G is an arbitrary point, then z0 -=/=- 0, and we can find a simply connected neighborhood V(z0) c @*. Then there exists a holomorphic function f in V with f 2 (z) - z and f(zo) = wo. We denote f (z) by vz. 1 The image W == f (V) is open, and the set 1r- (V) can be written as the union ci two disjoint open sets U± := {(z, ± f (z)) : z E V } = (V x ( ± W)) n G. Let f (z) == (z, f(z)). Then f : V -+ G is continuous, and_:rr o f(z) = z. The open set U == U+ i s a neighborhood ci (z0 , w0 ) , with f(V) = U and 1r(z, w) = (z, w) on U; that is, 1rlu : U -+ V is topological. Hence (G,1r) is a Riemann domain over C . �

4

�

general topological space X is said t o be connected if it is not the union of two disjoint nonempty open sets. A space X is called pathwise connected if each two points cf X can be joined by a continuous path in X . For open sets in en these two notions are equivalent. A

8 . Riemann Domains over en

89

The space G can be visualized in the following manner: We cover e 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 !R 3 without self intersection, but in higher dimensions, it is). This is illu strated in Figure 1 1.8. 0

Figure

1 1 . 8 . The Riemann surface of

v'z

8. 1 Pr op osition (on the uniqueness of lifting). Let ( G 1r) be a domain over en and Y a connected topological space. Let y0 E Y be a point and 'l/.;1 , $2 : Y --+ G continuous mappings with 'l/J1 (y0) = 'l/J2 ( y0) and 1r o 'l/.;1 = 1r o 'l/.;2 . Then 'l/.;1 = 'l/.;2 . ,

Let M := {y E Y : 'lj.;1 (y) = 'lj.;2 (y) } . By assumption, y0 E M , so M -=1- 0. Since G is a Hausdorff space, it follows immediately that M is closed. Now let y E Y be chosen arbitrarily, and set x := 'lj.;1 (y) = 'l/.;2 (y) and z := 1r(x) . There are open neighborhoods U = U(x) C G and V = V(z) C en such that 7r : u --+ v is topological, and there is an open neigborhood W = W(y) with '1/J>- (W) c U for >. = 1 , 2. Then PROOF:

'1/Jd w = (7r I u) -

1

0 7r 0

'l/.;1 I w = (7r I u) -

1

0 7r 0

'l/J2 1 w = 'l/J2 1 w '

and therefore W C M. Hence M is open, and since Y is connected, it follows that M = Y. • Definition. Let Z o E en be fixed. A ( R iemann) domain over en with distinguished point is a triple Q = (G,1r, x0) for which: 1. (G,1r) is a domain over en. 2. x0 is a point of G with 1r(x0) :=z0.

90

II.

Domains of Holomorphy

Let Yj = (Gj , 1rj , Xj) be domains over e n with distin­ guished point. We say that YI is contained in Y2 (denoted by YI -< Y2 ) if there is a continuous map r.p GI --+ G2 with the following properties: 1. 1r2 o r.p 7ri (called "r.p preserves fibers" ). 2. r.p(x! ) x 2 .

Definition.

:

=

=

8.2 Proposition. with r.p( XI) = x2 is

If YI Y2 , then the fiber preserving map r.p : GI uniquely determined. -
-, 1r>-, x>-) over en with distinguished point. An equivalence relation on X is said to have property (P) if the following hold: 1. X>- "' xll, for >., g E L. 2. If [0, 1] --+ G>- and (3 : [0 , 1] --+ Gil are continuous paths with a(O) rv (3(0) and 7r.A o a = 1rll o (3, then a(1 ) rv (3(1). We consider the family of all equivalence relations on X with property (P) . It is not empty, as seen above in the case of two domains. Therefore we can construct an equivalence relation (as above) that is finer than any equivalence relation with property (P) . We denote it by rv p . It is clear that 7r,A (X) = 7rll(y) if x E G.A , y E Gil, and x "' P y. The relation "'P also has property (P), and the elements of an equivalence class X" all lie over the same point z = z(Xv ) · We define G := Xj rvp and ?r(Xv ) := z(Xv ) · The equivalence class of all X,A will be denoted by x. z. If 8.6 Lemma. Let y E G.A and x E Gil be given with 1rll(x) = 7r>-(Y) a :

=:

we choose open neighborhoods U = U(y) C G.A, V = V(x) C Gil, and an open connected neighborhood W = W(z) such that 1r>-I : U --+ W and 1rll : V --+ W are topological mappings, then for r.p := (1rIl l v ) - o 7r.A U --+ V the following hold: 1 . r.p(y) = X. y, then r.p(y') y' for every y' E U. 2. If x :

"'P

"'P

PROOF: The first statement is trivial. Now let a [0, 1] --+ W be a con­ tinuous path with a(O) = z and a(1) = 7r>-(Y') for some y' E U. Then (3 := (7r>- lu ) - 1 o a and "( := r.p o (3 are continuous paths in U and V with (3(0) y rv p X = r.p(y) = "f(O) and 7r,A 0 (3 = 7rll 0 r.p 0 (3 = 7rll 0 "!· Therefore, y1 = {3 ( 1) rv p "( (1) = r.p(y1 ). • :

=

8. 7 Theorem.

There is a topology on G such that

is a Ri�mann domain over e n with distinguished point x, and all maps 'P>- : G>- --+ G with 'P>-(x) := equivalence class of x are continuous and fiber preserving.

8.

Riemann Domains over en

93

PROOF: (1) Sets of the form 'P>. (M) for M open in G>. together with G constitute a base of a topology for G. To see this it remains to show that the intersection of two such sets is again of this form. Let M C G>. and N C G12 be open subsets. Then 'P >. (M) n cp12 ( N)

=

cp12 ( N n cp 12 I ('P >. (M))).

But cp12 I ('P>. (M)) is open in G12 • In fact, let x E cp12 I ('P>.(M)) be given, and y E M be chosen such that 'P>. (Y) = cp12 (x) (and therefore y "'P x). Let z : = 7r>. ( Y) = 1r12 (x) . Then there exist open neighborhoods U = U(y) and V V(x) and an open connected neighborhood W = W(z) such that 7r>. : U --+ W and 1r12 : V --+ W are topological mappings. Let 'P : = ( 1r12 1 v ) -I o 7r>. : U --+ V. By the lemma, cp(y) x and cp(y') "' p y' for every y' E U. So V' := cp(MnU) is a neighborhood of x in G12, and since cp12 (cp(y')) = 'P>.( Y') for every y' E U, it follows that V' C 'P 12 I ( 'P>. (M) ) . =

=

Consequently, every 'P>. is a continuous map. (2 ) Remark: Since every y E G is an equivalence class 'P>.(x), we have M = U 'P>. ( 'P>. I ( M)) for any subset M C G. AEL (3) 7r : G --+ e n is continuous: Let V c e n be an arbitrary open set, and M : = :;r-I (V). Then 'P >. I (M) = 1r>. I (V) is open in G>. , and therefore M = u ).. E L 'P>. ('P ).. I (M )) is open in G. ( 4) G is a Hausdorff space: Let YI , Y2 E G with YI -=/=- Y2 , and Z I ir(y! ) , Z 2 : = ir (y2 ) . There are two cases. If z i -=/=- z2, then there are open neighborhoods VI (z i ) and V2 (z2) with VI n V2 = 0 . Then :;r- I (VI ) and :;r- I (V2) are disjoint open neighborhoods of YI and Y2 · If Z I = z2 , then we choose elements XI E G>. , x2 E G12 with 'P>. (x! ) = YI and cp12(x2 ) = y2 , and since X I and x2 are not equivalent , the above lemma implies that there are disjoint neigborhoods of YI and Y2 · (5) G is connected: Let y = 'P>. (x) be an arbitrary point of G. Then there is a continuous path a : [0 , 1] --+ G>. that connects the distinguished point X>. to x. Then 'P >. o a connects x to y. (6) 7r is locally topological: Let y = 'P>. (x) be a point of G, and z = ir(y) 7r>. (x) . Then there exist open neighborhoods U = U(x) C G>. and W = W(z) c e n such that 7r>. : U --+ W is a topological mapping. U := 'P>. (U) _ is an open neighborho od of y, with ir(U) 7r>. (U) = W. In addition, ir l u is injective, since 7r o 'P>. = 7r>. and 7r>. l u is injective. :=

=

=

94

II.

Domains of Holomorphy -

( 7) Clearly, the maps 'P>.. : G >.. --+ G are fiber preserving, and it was already • shown that they are continuous. Now g has the following properties:

1. 2.

9>-.

-
-.

PROOF:

_

Q* for every >., then g

-
.. by x N x1

: -.. , x' E G12 and r.p)_ (x) = 'P� (x').

It follows from the uniqueness of lifting that N has the property (P).Now we define a map r.p : G --+ G* by r.p(r.p>.. (x))

:=

r.p)_ (x) .

Since "' p is the finest equivalence relation with property ( P ) ,r.p is well defined. • Also it is clear that r.p is continuous and fiber preserving. Therefore g is the smallest Riemann domain over en that contains all do­ mains Y>.. · -

Definition. The domain g constructed as above is called the union of the domains Y).. , and we write g = u ).. E L Y).. · Special cases:

1. 2.

From YI -< g and Y2 -< 9 it follows that YI U Y2 From YI -< Y2 it follows that YI U Y2 "' Y2 · 3. g u g "' g . 4 . YI u 92 "' 92 u YI . 5. YI u (92 u 93 ) "' ( QI u 92 ) u Y3 ·

-
. we can find open neighborhoods U = U(xo ) , V = V(x;.. ) , and W = W(1r(x0)) such that all mappings in the following commutative dia­ gram are homeomorphisms:

Then for f E .Jk and its holomorphic extension F;.. on G;.. we have that 1 F;.. o (7r;.. / v ) - 1 = F;.. o r.p;.. o (1r/ u ) - 1 = f o (1r/ u ) - is independent of >.. Therefore, all distinguished points x;.. are equivalent. (iqf a : [0, 1] --+ G;.. and (3 : [0, 1] --+ G, are continuous paths with a(O) "' (3(0) and 1r;.. o a = 1re o (3 , then (F:.. ) a(O) = (Fe)f3(0) · It follows that (F:.. ) a(l) = (Fe)f3( I ) as well, and therefore a(1) c::: (3(1). Since g --< 9:.. and 9:.. --< Hg; (Q) , it follows that Q --< HF (Q) . Furthermore, the fiber preserving map rp : = cp;.. o r.p;.. does not depend on >.. Now let a function f E .Jk be given. We construct a holomorphic extension F on G as follows: �

If y E 8 is an arbitrary point, then there is a >. E L and a point Y>. E G>. such that y = cp;.. (y;.. ), and we define

1f y = 'Pe(Ye) as well, then y;.. "'P Ye , and therefore y;.. "' Ye as well. It follows that F;.. (y;.. ) = Fe (Ye), and F is well defined. 6

For the definition of property (P) see page

92.

9 . The Envelope of Holomorphy

99

We have F o cp = F o VJ>-. o 'P>-. = F>-. o 'P>-. = on G. This shows that F is an extension of and from the equation F o $>-. = F>-. it follows that F is holomorphic (since $>-. is locally topological).

f

f,

•

The maximality of H,y; (Q) follows by construction.

The 9-hull Hg; (Q ) is therefore the largest domain into which all functions E § can ce holomorphically extended.

f

9. 5

Iden_!ity theorem. Let YJ = (GJ , 1rj , XJ ) , j = 1 , 2, be domains over e n , and Q = (G, ?r, x) the union of Q1 and G2. Let fJ : GJ --+ e be holomor­ phic functions and Q = ( G,1r, x) a domain with 9 --< Yj fo r j = 1 , 2-such that Then there is a holomorphic function on G with fJ , JfO f' J = 1 , 2 .

h i e. = hi e ·

f

f

h /e

f ie = J

h/ e ,

PROOF: Let and § == { ! } . Since Y1 --< Hg; ( Q ) and = == Y2 --< Hg; ( Q ) , it follows that 91 U Y2 --< Hg; ( Q ) . �

�

�

Let be a holomorphic extension o ff to G (where Hg; (Q) = G, 1?, x) ), and = = ]115. Then

1

f

(

(fleJ i e = 11e = (J10) l e = J1e = f. fl e1

Therefore, is a holomorphic extension o ff to Gj · Due to the uniqueness = fJ for j = 1 , 2. of holomorphic extension, •

J/ ej

P seu doconvexit y. Let P" c e n be the unit polydisk, (P n , H ) a Eu­ clidean Hartogs figure, and : p n --+ e n an injective holomorphic map­ ping. Then ((P n ) , ( H )) is a generalized Hartogs figure. P = (P", @, 0 ) and 1i = (H, , 0 ) are Riemann domains with 1i --< P. We regard the pair (P.'H) as a generalized Hartogs figure. 9.6 Proposition. Let ( G, 1r) be a dom ain over e n , (P, 'H) a generalized Hartogs figure, and xo E G a point fo r which 1i --< g := (G,1r, x0 ) .

Then every holomorphic function G UP.

f o n G can be extended holomorphically to

The proposition follows immediately from the identity theorem. Definition. A domain ( G, 1r) over e n is called Hartogs convex if the fact that (P.'H) is a generalized Hartogs figure and x0 E G a point with 1i --< Q := (G,1r, xo) implies g u p "" r;; .

A domain Q = (G, 1r, xo) over en is called a dom ain of holomorphy if there exists a holomorphic function on G such that its domain ci existence is equal to Q.

f

1 00

II. Domains of Holomorphy

Remark. If G c e n is a Schlicht domain, then the new definition agrees with the old one.

9 . 7 Theor em. 1.

If Q = ( G,n, x0 ) is a dom ain over e n and § a nonempty set of holo­ morphic functions on G, then Hg; (Q) is Hartogs convex. 2. Every dom ain of holomorphy is Hartogs convex.

PROOF: Let (P, 'H) be a generalized Hartogs figure with 1i --< Hg; (Q). Then every function f E § has a holomorphic continuation to Hg; (Q) UP .There­ fore, Hg; (Q) U P + H.� ((]). On the other hand, we also have Hg; (Q) < • Hg; (Q) U P .s o Hg; (Q) U P "' Hg; (Q) . A Riemann domain (G, n) is called holomorphically convex if for every infinite discrete subset D C G there exists a holomorphic function f on G that is unbounded on D.

9.8 Theor em (Oka, 1 953). If a R iem ann dom ain ( G,n) is Hartogs pseu­ doconuex, it is holomorphically convex (and therefore a dom ain qf holornor­ phy) .

This is the solution of Levi's problem for Riemann domains over en . We cannot give the proof here. It seems possible to construct the holomorphic hull by adjoining Hartogs figures (cf. H . Langmaak, [La60] ) . It is conceivable that such a construction may be realized with the help of a computer, but until now (spring 2002) no successful attempt is known. We assume that parallel computer methods are necessary.

Boundary Point s . In the literature other notions of pseudoconvexity are used. We want to give a rough idea of these methods. Definition . Let X be a topological space. A filter (basis) on X IS a nonempty set R of subsets of X with the following properties: 1. 0 rf_ R. 2. The intersection of two elements of R contains again an element of the set R .

Example

1. If xo is a point of X , then every fundamental system of neighborhoods of xo in X is a filter, called a neighborhood filter of xo .

9 . The Envelope of Holomorphy

101

2 . Let (Xn ) be a sequence of points of X . If we define SN == { Xn n > N) , then R := {SN : N E N} is the so-called elementary filter of the sequence (x n ) . A filter is therefore the generalization of a sequence. Definition . A point x0 E X is called a cluster point of the filter R if x0 E A, for every A E R. The point x0 is called a limit of the filter R if every element of a fundamental system of neighborhoods of x0 contains an element of R. For sequences the new notions agree with the old ones.

If f : X --+ Y is a continuous map, then the image of any filter on X filter on Y , the so-called direct image.

IS

a

Definition . Let (G, 1r ) be a Riemann domain over e n . An accessible boundary point of ( G,n ) is a filter R on G with the following properties: 1 . R has no cluster point in G. 2. The direct image 1r ( R) has a limit zo E e n . 3. For every connected open neighborhood V = V(zo ) C e n there is exactly one connected component of 1r - 1 (V) that belongs to R. 4. For every element U E R there is a neighborhood V = V ( z0 ) such that U is a connected component of 1r - 1 ( V) .

Remark.

For a Hausdorff space X the following hold:

1 . A filter in X has at most one limit. 2. If a filter in X has the limit x0 , then filter .

x0

is the only cluster point of this

(for a proof see Bourbaki, [Bou66] , § 8 . 1 ) Therefore, the limit z0 in the definition above is uniquely determined. There is an equivalent description of accessible boundary points that avoids the filter concept. For this consider sequences (x , ) of points of G with the following properties: 1 . ( x , ) has nu cluster point in G. 2. The sequence of the images 1r ( x, ) has a limit zo E e n . 3. For every connected open neighborhood V = V(z0) C en there is an m E N such that for n, m > n0 the points x, and Xm can be joined by a continuous path a : [0, 1] --+ G with 1r o a([O, 1]) c V .

Two such sequences ( x, ) , (y,) are called equivalent if

7r(Yv) = Zo . 1 . limv--.oo 7r(Xv) = li mv 2. For every connected open neighborhood V = V ( z0 ) there is an n0 such that for n , m > m the points x, and Ym can be joined by a continuous path a : [0, 1] --+ G with a([O, 1]) c V. --; 00

n o

102

I I . Domains of Holomorphy

An accessible boundary point is an equivalence class of such sequences. Let 8 G be the set of all accessible boundary points of G . Even if G is schlicht, this set may be different from the topological boundary 8G. There may be points in ac that are not accessible, and it may be happen that an accessible boundary point is the limit of two inequivalent sequences. We define 6 == Gui!JG. If r0 = [xn] is an accessible boundary point, we define a neighborhood of r0 in G as follows: Take a connected open set U C G such that almost all Xn lie in U and 1r ( U) is contained in a neighborhood of z0 := limn---+ oo 1r(xn)· Then add all boundary points r = [yn] such that almost all Yn lie in U and limn-too 7r(Yn) is a cluster point of 1r ( U) . With this neighborhood definition G becomes a Hausdorff space, and * : G --+ e n with v

1r(x) :=

{

if x E G, 7r( x) nlim ---t oo 7r(xn) if x = [xn]

E i!JG,

is a continuous mapping. Definition . A boundary point r E 8G is called removable if there is a connected open neighborhood U = U(r) C G such that ( U, *) is a Schlicht Riemann domain over e n and 8 G n u is locally contained in a proper analytic subset of U. A subset M C 8G is called thin if for every ro E M there is an open neighborhood U = U(r0) C G and a nowhere identically vanishing holo­ morphic function f on U n G such that for every r E M n U there exists a sequence (xn) in U n G converging to such that limn-too f (x n ) = 0. v

r

Example Let G C e n be a (schlicht) domain and A C G a nowhere dense analytic subset. Then every point of A is a removable boundary point of G' := G - A. The points of the boundary of the hyperball Br (0) C e n are all not removable. Let B be a ball in the affine hyperplane H = { (zo , . . . zn ) E en+l : Zo = 1 } , and G c e n+ l - {0} the cone over B. Then every boundary point of G is not removable, since locally the boundary has real dimension 2 n + 1 . The set M := {0} is thin in the boundary, as is seen by choosing J (zo , . . . zn ) : = zo . ,

,

(G,1r) be a Riemann domain over e n . If r.p : D --+ G is a continuous mapping, 7r o r.p : D --+ en holomor phic , and (7r o r.p )' ( () -=/=- 0 for ( E D, then S := r.p(D) is called an analytic disk in G. The set bS := r.p(8D) is called its boundary.

Analytic Disk s . Let

v

9 . The Envelope of Holomorphy

l 03

Let I == [0, 1] be the unit interval. A family (St)tEI of analytic disks 'Pt (D) in G is called continuous if the mapping (z , t) H 'Pt (z ) is continuous. It is called distinguished if St c G for 0 < t < 1 and bSt c G for 0 < t < 1 . Definition . The domain G is called pseudoconvex if for every distin­ guished continuous family (St)tEI of analytic disks in G it follows that s1 c G.

The domain G is called pseudoconvex at r E 80 if there is a neighborhood U = U(r) C G and an c > 0 such that for every distinguished continuous family (St)tEI of analytic disks in G with 7r(St) C B10(7r(r) ) it follows that St n U c G for t E /. v

As in en one can show that a Riemann domain is pseudoconvex if and only if it is Hartogs pseudoconvex. 9.9 Theorem (Ok a). A R iemann dom ain (G, 1r) is pseudoconvex if and only if it is pseudoconvex at every point r E 80. 9. 10 Corollary. If (G, 1r) is a dom ain of holomorphy, then convex at every accessible boundary point.

0

is pseudo·

The converse theorem is Oka's solution of Levi's problem. Finally, we mention the following result: 9. 1 1 Theorem. Let (0,1r) be a Riemann dom ain over en , and

a thin set of non removable boundary points. point of 80 - M , then 0 is pseudoconvex. PROOF:

See [GrRe56] , §3, Satz 4.

If 0

M C 80 is pseudoconvex at every •

E xer cises 1 . Prove that a Reinhardt domain Q over e n must be Schlicht if it is a domain of holomorphy. 2. Prove that if ( G, 1r) is a Reinhardt domain, then for every f E 0( G) there is a power series S(z) at the origin such that f(x) = S(1r(x)) for E G. 3. Prove that the envelope of holomorphy of a Reinhardt domain is again a Reinhardt domain. 4. Prove that the Riemann surface of the function f(z) = log(z) has just one boundary point over 0 E C. 5 . Find a Schlicht Riemann domain in e 2 whose envelope of holomorphy is not schlicht . 6. Construct a Riemann domain g = (G, 1r , x 0) over e2 such that for all x,y E 1r-1 (1r(xo )) and every f E O (G) the equality f(x) = f(y) holds. X

104

II. Domains of Holomorphy �

7. Let (G, 1r ) be a Riemann domain and ( G, 7?) its envelope of holomorphy. If f is a holomorphic function on G and F its holomorphic extension to G, then f(G) F (G) 8. Consider =

G

.

{ (z, w) � < l z l < l, lwl < 1} :

ei t , w) : : < r < l , t > O and lw

Determine the envelope of holomorphy of G. 9. Let G c e n be a domain and p G --+ JR a plurisubharmonic function. {z E G p ( z ) < c} c c G, 1f zo is an accessible boundary point of B then B is pseudoconvex at zo , in the sense of the last paragraph. :

==

:

C h ap t er III

Analyt ic S ets 1.

T h e Algeb r a of P ower Ser ies

T he Banach Algebr a Bt . In this chapter we shall deal more exten­ sively with power series in e n . Our objective is to find a division algorithm

for power series that will facilitate our investigation of analytic sets. We denote by C[z] the ring of formal power series I: v>o avz" about the origin. Let JR.+ be the set of n-tuples of positive real numbers. Definition . Let t = (t1 , . . . , tn ) E JR.+ and f = L: v> O avz" define the "number" I I J il t by

I I J ilt : =

{ Ev>ol av lt"

E

C [zl We

if this series converges,

otherwise.

Let Bt : = { ! E C [z� : II f l it < oo }. Remark.

One can introduce a weak ordering on JR.+ if one defines

For fixed f, the function t 1---t II J il t is monotone: If t < t * ,then II J i lt < I / f li t · · Definition . A set B is called a comp lex Banach algebra if the following conditions are satisfied: 1 . There are operations +

:

B B --+ B, · : C B --+ B x

x

and

o B :

x

B --+ B

such that (a) ( B , +, · ) is a complex vector space, (b) ( B ,+,o) is a commutative ring with 1 , (c) c . ( ! g) = ( c . f) g = f ( c . g) for all f, g E B and c E C. 2. To every f E B a real number 11!11 > 0 is assigned that has the properties ci a norm: o

( a) ( b)

o

o

li e · !II = lei . 1/!11 , for c E C and f E B , 1/ f + g /1 < 1/ f /1 + 1/ g ll , for f, g E B,

106

1 1 1 . Analytic Sets

IIIII = o � J = o . II! o g il < IIIII . ll g l l , for J, g E B. (c)

3.

is complete; i.e., every sequence in B that is Cauchy with respect to the norm has a limit in B.

4. B

Bt

1 . 1 Theor em. for any t E JR+. .

= {! E

C[z] II J ilt :

< oo} is a complex Banach algeb ra

Clearly, C[z] is a commutative C-algebra with 1. Straightforward calculations show that II· . . li t satisfies the properties ( 2a) , ( 2b) , (2c) and (3). It follows that Bt is closed under the algebraic operations, and all that remains to be shown is completeness. a >-J z" . Then for every Let ( f>.. ) be a Cauchy sequence in Bt with f>.. = L v>o - � c > 0 there is an n = n (c) E N such that for all A, f-L > n ,

PROOF:

Z::: l a �>-J - a �l l t" = ll f>.. - !M ilt v> O Since t" = t�1

•

•

•

t�n #

<E

0, it follows that E

for every v E Nn0 .

I ac>-J " ] < t" " - aC>-J ) is therefore a Cauchy sequence in C which converges to a complex number a,. Let f = L: v> O avz". This is an element of C[z]. Given S > 0, it follows that there exists an n = n ( 8) such that Y,

=

:2::: I a�>-) - a�>-+ >< l - a I t" < 8/2, and then "

:2::: I a�>- J v El

- a,

It"

< 8,

for >. > n .

In particular, ll f>.. - Jilt < 6. Thus f>.. - f (and then also and ( f>.. ) converges to f . Expan sion with Respect t o

additional notation: If v E N0 , t E JR+. , and E e n , write z

z1 .

= J-L( >.) E

f)

belongs to Bt,

•

For the following we require some

1 . The Algebr a of Power Series

107

(t1 , t' ) , and z = ( z1 , z1 ) . An element f = L v> O avz" E C[z] can be written in the form v = ( v1 , v1 ) ,

t

00

"""' f>. z A1 , f =�

=

with

I

f>. (z )

""

= �"'

I

I

a( A , v '){Z }" .

The series fA are formal power series in the variables z2 , . . . , zn. We call this representation of f the expansion off with respect to z1. Now the following assertions hold: 1.

2.

00

/>. E Bt ' for all A, and l: II!A IIt ' � < oo . ll zf . Jilt = t� . II J i l t , for s E No.

f E Bt

¢==;>

PROOF:

( l ) Since we are dealing with absolute convergence, it is clear that

ll!ll t

=

00

2: 1 1 />. ll t 't� . A=O

(2) We have z . f = I:r 0 fA z; + s . The right side is the unique expansion z . f with respect to z1 . Now the formula can be easily derived.

f

f

of •

Let B be a complex Banach algebra and (!>.) a sequence of elements of B. The series I::r 1 fA converges to an element f E B if the sequence Fn I:;�= 1 fA converges to f with respect to the given norm. 1 .2 Proposition. Eve ry f E B with Il l - !II < 1 is a unit in B with C onver gent Series in Banach Algebr a s . ==

f- 1

00

- """' (1

-6

-

f) A and ll f - 1 1 1

1 < - 1 - 11 1 -

111 ·

E :=

Ill - f 11 . Then 0 < c < 1 , and the convergent geometric series I::r 0 sA dominates the series I: r 0 ( 1 - ! ) A . As usual, it follows that this series converges to an element g E B. We have PROOF:

Let

n

n

( 1 - ( 1 - f)) . 2: ( 1 X=O

Jf'

t (l - !) A - I: (l - f f' A=O

As n tends to infinity we obtain

Xl

1 - ( 1 - J) n+ l f . g = 1 and 1/ g //
O avz" is called convergent if it is convergent in some polydisk P around the-origi n. In that case there exists a point t E P n JR+. , and since f converges absolutely

at t . it follows that f E Bt . On the other hand, if f E Bt , then by Abel's lemma f converges in P = pn ( o , t ) . 1 . 3 Theorem. :3 t E JR+. with II f li t < oo} is the set d conve rgent { f E C [z] 1. Hn :=

power serzes. 2. Hn is a C-algebra. 3. There is no zero divisor in H.: If f . g = 0 in H, then f = 0 or g = 0.

We have already proved the first part, and then the second part follows easily.

The last part is trivial, since C[z] contains no zero divisors. Remark. If f is convergent and f(O) = 0, then for every c > 0 there is a t E JR+. with II Jil t < c. In fact, since f (O) = 0, we have a representation f = Z1 h + · · · + zn fn . If 11 /ll t < oo, then also 11/i ll t < oo for i = 1 , _ . , n, and n

n

II Jilt = 2::.:: ti ll/ill t < max( t1 , · · · t n ) · 2::.:: 1 1/i ll t i=1 ,

This expression becomes arbitrarily small as t

--+

i=1

0.

When we go from Bt to Hn , we lose the norm and the Banach algebra structure, but we gain new algebraic properties: 1 . f E Hn is a unit ¢==;> f (O) -=1- 0. PROOF: One direction is trivial. For the other one suppose that f(O) -=1- 0. Then g == f . f(0) -1 1 is an element of Hn with g(O) = 0. So there exists a t with 1/ g//t < 1 , and f . f (0) 1 is a unit in Bt . Thus f is a unit • in Hn . 2 . The set m := {f E Hn f(O) = 0} of all nonunits in Hn is an ideal: -

-

:

(a) /I , h E m ===? h + h E m. (b) f E m and h E Hn ===? h f E m. ·

An ideal a in a ring R is called maximal if for every ideal b with a c b c R it follows that a = b or b = R. One can show that in any commutative ring with 1 -=/:- 0 there exists a maximal ideal. If a C R is maximal, then Rfa is a field. 1 .4 Theorem. The set m of nonunits is the unique maximal ideal in H, and Hn /m "'

C.

If a C Hn

is a proper ideal, then it cannot contain a unit. Therefore, it is contained in m . The homomorphism r.p Hn --+ C given by f H f( O ) is • surjective and has m as kernel. PROOF:

:

1 . The Algebra of Power Series

1 09

Distinguished Dir ection s . An element f E Hn is called zl- regular order k if there exists a power series fo ( z! ) in one variable such that:

of

1. f(z1 , 0, . . . , 0) = z f . fo (z1 ) . 2. fo (O) -1- 0. If f is zrregular of some order, f is called z1 -regular. Let f (z) = I:� 0 f>.. zr be the expansion of f with respect to z 1 . Then f is zrregular of order k if and only if fo(O' ) = . . . = fk_1 ( O') = 0 and fk (O') -1- 0. f is zl-regular if and only if f (z1 , 0, . . . , 0 ) � 0. We often need the following properties: 1. f is a unit in Hn -¢==? f is 21-regular of order 0. 2. If f>. is zl-regular of order k>. , for >. = 1 , 2 , then h . h is z1-regular of order k1 + k2 . There are elements f -1- 0 of Hn that f (0) = 0 which are not zrregular, even after exchanging the coordinates. Definition. map ac : en

Let c = ( c2 , . . . , en ) be an element of en - 1 . The linear --+ en with

is called a shear. The set I; of all shears is a subgroup of the group of linear automorphisms of en ' wilh ao = idcn . We can write ac (z) := z + z1 . (O , c ) . ln particular, we have ac (ei) = ( l ,c ) . Let f E Hn be a nonze ro e lement. Then there exists a shear a such that f o a is z1 -regular. 1 . 5 Theorem.

Assume that f converges in the polydisk P. If we had f(z1 , z') = 0 for every point (z1 , z') E P with z1 -1- 0, then by continuity we would have f = 0, which can be excluded. Therefore, there exists a point a = (a1 , a ' ) E P with a1 -1- 0 and f ( a) -1- 0. We define c := (a i ) - 1 . a ' and a : = ac . Now, PROOF:

So f o a(z1 , 01) � 0, and f o a is 21-regular.

•

Rem ark. If JI , . . . , /z are nonzero elements in Hn , then f == h . . fz -1- 0 converges on a polydisc P, and there exists a point a E P with f ( a ) -1- 0 and a1 -1- 0. As in the proof above we obtain a shear a such that h o a, . . . , fz o a are zl-regular. ·

110

1 1 1. Analytic Sets

Exercises

f(z) L: v>O avz" be a formal power series. (a) P r ve the "Cauchy estimates": f E Bt

1 . Let

=

===?

o

l av l
o av , nz" be a sequence of power series with //!n i l < C. If every sequence (a,,) converges in C to a number a, then show that (fn) is a Cauchy sequence in Bt converging to f(z) = L: v> O avz" , for every t < s . s

2. The Krull topology on Hn is defined as follows: A sequence Un) converges in Hn to f if for every k E N there is an no with f - fn E mk for n > no . What are the open sets in H,? Is Hn with the Krull topology a Hausdorff space? 3. Let B be a complex Banach algebra with 1. Show that for every f E B the series exp(f) = I::� o r fn! is convergent, and that exp(f) is a unit in B. 4. If f is a formal power series and f I:;� 0 P>. its expansion into homo­ geneous polynomials, then the order of f is defined to be the number =

ord(f) := min{s E No

: p8 -=f- 0).

Now let (fn) be a sequence of formal power series such that for every k E N there is an no with ord(fn) > k for n > no. Show that I:� 1 fn is a formal power series. Use this technique also for the following: If 91 , . . . , gm are elements of Hn with ord(gi) > 1 , then

:2::: afl. wfl. H :2::: afl. (g 1 (z) , . . . , 9m ( z))fl. J.L > O J.L > O defines a homomorphism rp Hm --+ Hn of complex algebras. :

2.

The Preparati o n The orem

Let a fixed element t E JR+. be chosen. When no confusion is possible we write B in place of Bt , B' in place of Bt' , and II ! I/ in place of II f 1 / t . The ring of polynomials in z 1 with coefficients in B' is denoted by B' [z1 ]. 2. 1 Weierstrass Formula in Bt . Let f and g = I:;� 0 g;.. z� be two elements of B. Assume that there exists an E No and a real number E with Division w ith Remainder in Bt .

0 < E < 1 such that

g8 is

a

s

unit in B' and ll z f

- g g-; 1 11 ·

< c t� . ·

111

2. The Preparation Theorem

Then there that

exists exactly one q E B f

=

and one r E

B'[z1 ]

with deg ( r )
.. + I ) < s. If we define q : I:: r 0 g_; 1 q>.. and r : 2:r 0 r>.. , then =

=

00

00

00

00

l::: (gg_; 1 q>.. + r>.. )

X=O

=

g · q + r.

112

1 1 1. Analytic Sets

When using this algorithm we do not need the abstract transformation T and the B anach theory of such transformations. However, it is necessary to prove the convergence of all of the series that were used. - ( zf - gg-;; 1 ) . Then ll h ll < c: . tf and gg-;; 1 = zf + h. From f>.. = zf q;.. + r;.. it follows that li rA II lll f>.. ll and 1/ q;.. /l . ll f>.. ll . Furthermore, from f>.. + I = - h · q;.. it follows that

For this let h

==

lt1 8

ll f>..+ I II

;..

.l h ll . ll q;.. ll < E .

II!AII­

0 !A converges. Th us II /A II < E . II I II and Since 1 q < >-. s g 1 ll . II I II and li rA / / < c:;.. IIJII, g ll ; ;.. ll c: rt ll ; the series q = I: 0 g; 1 q;.. and r 0 r;.. also converge.

I;�

�

= I;�

The estimates for 1/ gsq/1 and IIl ii follow readily: < ll r ll


s. PROOF:

=

0 · g +f.

2. The Preparation Theorem

We assume that therefore

deg (JJ.L ) < d fo r f-L

=

1 13

0 , . . . , A . Then deg(q>-) < d - s , and

deg( f>- +d = deg (JA - r>- - gg; 1 q>-) lnax (d, s - l ,s + (d - s ) ) = d. Hence deg ( fA) < d and deg(q>-) < d - s for all A. It follows that deg( q) < d - s, and from f = g . q + r we can conclude that deg ( q) = d - s. •

The Weierstras s Condition . We use the notation from above. Definition. Let s E N0 . An element g = I:; 0 g>-z� E B satisfies the Weierstrass condition (or W-condition) at position s if 1 . gs is a unit in B'. 2. l l zf - gg; 1 ll < � tf .

�

Let R be an integral domain.' A polynomial f ( u) = fs u + fs - 1 u s- 1 + . . . + h u + fo E R[ u] is called monic or normalized if fs = 1 . A polynomial f E B'[zi] is normalized if and only if it is z 1 -regular of some order k < s. 8

lfg E B satisfies the W­ 2.3 Weierstrass preparation theorem in Bt · condition at position s, then there exists exactly one normalized polynomial w E B'[z 1 ] of degree s and one unit e E B such that g = e . w. PRooF: We apply the Weierstrass formula to f = zf . There are uniquely determined elements q E B and r E B'[zd with zf = q g + r and deg(r) < s (we choose an c < i such that ll zf - gg; 1 l l < stf ) . ·

But then zf - gg;1 = ( q - g;1 )g + r is a decomposition in the sense of the Weierstrass formula. Therefore, we have the estimate

llgs q - 1 11 < t� s ll zf - gg; 1 ll ·

1

� c < 1 � c < 1.

That means that g8 q and hence q is a unit in B . Let e := q - 1 and w == zf - r. Then w is a normalized polynomial of degree s, and e . w = q- 1 (zf - r) = g .

If there are two decompositions g = e 1 (zf - r!) = e2 (zf - r2 ) , then

z 1 = e1 1 . g + r1 = e2 I . g + r2 . 8

From the uniqueness condition in the Weierstrass formula it follows that e1 = e2 and r1 = r2 . • 2.4

z1 . 1

Corollary.

If g is

a

polynomial in

z1 ,

then e is also a polynomial in

An integral domain is a commutative nonzero ring in which the product of two nonzero elements is nonzero.

1 14

III. Analytic Sets

PROOF: We use the decomposition zf - gg; 1 = (q - g; 1 ) g + r . From the Weierstrass formula it follows that // r // < // z f - gg; 1 // · 1

Since w,

=

� E: < t� - 1 � c < tf .

1 , it is also true that // z f - ww; 1 // = // z f - w// = // r// < t� .

Therefore, g = e . w + 0 is a decomposition in the sense of the Weierstrass • formula, and the proposition follows from Corollary 2.2. The Weierstrass preparation theorem serves as a "preparation for the exami­ nation of the zeros of a holomorphic function." If the function is represented by a convergent power series g , and there exists a decomposition g = e· w with a unit e and a "pseudopolynomial" w (z1, z') = zf + A 1 ( z')z� - 1 + · · · + As (z') , then g and w have the same zeros. However, the examination of w is simpler than that of g .

Weier str ass P olyn omials. Now we turn to the proof of the Weierstrass formula and the preparation theorem for convergent power series. The ring Hn is an integral domain with l . If f E Hn and f (z) = I; 0 f>.. (z')z£ with h. = 0 for >. > s , then f is an element of the polynomial ring Hn - dz!] . If fs -=1- 0, then deg( f ) = s. lf f is normalized and f>.. (O') = 0 for >. < s, then f is zrregular exactly of order s, and f(z1 , O') = zf.

�

Definition. A normalized polynomial w E Hn - dz1 ] with and w ( z1 , O') = zf is called a Weierst rass polynomial.

deg(w)

=s

We have seen that a normalized polynomial w E Hn d z1 ] with deg(w) = s is a Weierstrass polynomial if and only if it is z1 -regular of order s . It follows easily that the product of two Weierstrass polynomials is again a Weierstrass polynomial. -

If g = e . w is the product of a unit and a Weierstrass polynomial of degree s,

then we also have that g is z 1 -regular of order s, since the unit e is z1-regular of order 0. We now show that conversely, every z 1 -regular convergent power series is the product of a unit and a Weierstrass polynomial.

2.5 Th eor em . Let g E Hn be z 1 -regular d orde r s . Then fo r every c > 0 and e v e y to E JR+. the re exists a t < to such that g lies in Bt, gs is a unit in Bt' , and //zf - gg_;-- 1 //t < c . tf .

PROOF: Let g = 0 gJ.. Z� be the expansion of g with respect to z . Then 1 g>. (O') = 0 for >. = 0 , 1 , . . . , s - 1 and g8(0') -=1- 0.

I;�

Since g is convergent, there exists a t1 < to with l / g / / t 1 < oo . Then g>. E Be1 for all >., and in particular,

2 . The Preparation Theorem

/(z ')

=

1 15

g8 ( 0') - 1 gs (z' ) - 1 E Bti .

Now, f(O') = 0. Thus there exists a t2 < t1 such that 1 1 ! / l t ' < 1 for all t < t2 . Therefore, g8 is a unit in Bt ' , and g an element of Bt . Let h := zf - gg; 1 . Then h E Bt for all t < t 2 , and we have an expansion h = 2::: � 0 h;.. z � with h. = 0 , h;.. = - g;.. g; 1 for >. -=/:- s, and h;.. (O') = 0 for >. = 0, 1 , . . . , s - 1 . 1f t1 > 0 is sufficiently small, then 00

E t1 . < t2 2' s

for all t = ( t 1 , t ' ) < t2. And since h;.. (O') = 0 for >. = 0 , . . . , s - 1 . for every small t 1 there exists a suitable t ' such that

Consequently, 1/ h//t < E . tf.

•

Remar k . In a similar manner one can show that if g 1 , . . . ,gN E C [z] are convergent power series and each gi is zrregular of order si , then for every E > 0 there is an arbitrary small t E JR+ for which gi E Bt , (g; )s, is a unit in Bt' and llzt' - gi (gi ) s/ 11 < E . t�'

Weier str ass Pr ep ar ation Theor em

2.6 Theorem (Weier str ass division for mula) . Let g E Hn be z 1 - regula r of order s. Thenfo r every f E Hn the re are uniquely dete rmined e lements q E Hn and r E Hn - dz1 ] with deg(r) < s such that f=

q · g + r.

Iff and g are polynomials in z1 with deg(g)

=

s, then q is also a polynomial.

PROOF: There exists a t E JR� and an E with 0 < E < 1 such that f and g lie in Bt., g8 is a unit in Bt ' , and ll z f - gg; 1 ll t < E . tf . It then follows from the division formula in Bt that there exist q and r with f = q . g + r .

Let two decompositions of f be given:

1 16

III. Analytic Sets

We can find a t E JR� such that f, q1 , q2 , r1 , r2 lie in Bt and g satisfies the vV-condition in Bt . From the Weierstrass formula in Bt it follows that q1 = q2 • and r 1 = r2 .

Let g E Hn be z1 regular of orde r s . Then the re exists a uniquely dete rmined unit e E Hn and a We ierst rass polynomial w E Hn - dzi] of degree s such that

2.7

Theorem (Weierstrass preparation theorem) .

g=e

·

w.

If g is a polynomial in z1 , then e is also a polynomial in z 1 . PRooF: There exists a t E JR� such that g satisfies the W-condition in Bt . The existence of the decomposition g = e . w with a unit e and a normalized polynomial w of degree s therefore follows directly from the preparation the­ orem in Bt . Since g is z1-regular of order s, the same is true for w. So w is a Weierstrass polynomial.

Now, w has the form w = zf - r, where r E Hn - 1 [z1] and deg ( r) < s. Thus, if there exist two representations g = e1 ( zf - r1 ) = e2 ( zf r2 ), it follows that 1 1 zf = e;- · g + r1 = e2 .g t r2 . The Weierstrass formula implies that e1 = e2 , • r1 = r2 and therefore w 1 = w2 -

.

Exercises

1 . Write a computer program to do the following: Given two polynomials f (w, x,y) (of degree n in w and degree m in x and y) and g(w, x, y) with g(w, 0, 0) = w 8 , the program uses the Weierstrass algorithm to determine y and r (up to order m in x and y) such that f = y . g + r. 2. Let f : p n- I x D --+ C be a holomorphic function andn 01< r < 1 be a real number such that ( H f(z' , () has no zero for z' E p - and r < I CI < 1 . 1 Then prove that there is a number k such that for every z' E p n the function ( H f(z' , () has exactly k zeros (with multiplicity) in D . Use this statement to give an alternative proof for the uniqueness in the Weierstrass preparation theorem. 3. Show that the implicit function theorem for a holomorphic function f : n e X (1: --+ (1: with f (O) = 0 and fzn (0) # 0 follows from the Weierstrass preparation theorem.

3.

Prim e Factorization

Unique Factorization. Let I be an arbitrary integral domain with 1 . Then I * : = I { 0} is a commutative monoid with respect to the ring multiplication, and the set JX of units of I is an abelian group . -

3.

Prime Factorization

1 17

Let a, b be elements of I*. We say that a divides b (symbolically a I b) if there exists c E I * with b = a . c. We can also allow the case b = 0. Then every element of I* divides 0, and a unit divides every element of I . Definition. Consider an element a E I * - Ix . 1 . a is called irreducible (or indecomposable) if from a = a1 . a2 (with a 1 , a2 E / * ) it follows that a 1 E / x or a2 E J x . 2. a is called p rime if a I a1a2 implies that a I a1 or a I a2 . Irreducible and prime elements can be defined in an arbitrary commutative monoid. In / * every prime element is irreducible, and in some rings (for example, in Z or in JR[X]) it is also the case that every irreducible element is prime. In Z[H] one can find irreducible elements that are not prime. Definition. I is called a unique facto rization domain (UFD) if every element a E J X can be written as a product of finitely many primes. One can show that the decomposition into primes is uniquely determined up to order and multiplication by units. In a UFD every irreducible element is prime and any two elements have a greatest common divisor (gcd). Every principal ideal domain2 is a UFD, and in this case the greatest common divisor of two elements a and b can be written as a linear combination of a and b. For example, Z and K[X] (with an arbitrary field K ) are principal ideal domains. So in particular, C [X] is a UFD .

Gaus s ' s Lemma. Let I be an integral domain. Two pairs ( a ,b), ( c ,d) E I x I * are called equivalent if ad = be. The equivalence class of a pair ( a,b) is called a fraction and is denoted by a fb. The set of all fractions has the structure of a field and is denoted by Q(I). We call it the quotient field of I. The set of polynomials f ( u) = a +a 1 u+ . + an un in u with coefficientsai E I 0 constitutes the polynomial ring J [ u] . The set J0[u] of monic polynomials in J[u] is a commutative monoid. Therefore, we can speak of factorization and irreducibility in J 0 [u] . 3 . 1 Gauss' s lemma) . Let I be a unique facto rization domain and Q = Q(I) . lf w 1 , w2 are ele ments of Q0 [u] with w1 w2 E J0 [u] , then W>, E JO[u] fo r >. = 1 , 2. PROOF: For >. = 1, 2, W>, = a>,,o + a;>,, I u + . . . + a>,,s,_ _ 1 us ,.. - l + u8" with ax,, E Q . Therefore, there exist elements d>- E I such that d>, . w>, E J [u] . We can choose d>, in such a way that the coefficients of d>, . W>, have no common divisor (such polynomials are called p rimitive). 2 A principal ideal domain is an integral domain in which every ideal is generated

by a single element.

1 1 1. Analytic Sets

1 18

We define d := d1d2 and assume that there is a prime element p with p I d. Then p doesn't divide all coefficients d>.a:>.,v of d>. . W>. . Let J-l>. be minimal such that p f d>.a>.,JL>- . Then JL ( d 1 wi) ( d2w2 ) = . . . + u �' ' + 2 ( da1 ,JL, a2,JL2 + something divisible by p ) + . . . .

Since I is a UFD, p doesn't divide (d1a1,JL, ) (d2a2,JL2 ) . So the coefficient of u�' ' + JL2 is not divisible by p. But since w1w2 has coefficients in I, every divi­ sor of d must divide every coefficient of d . w1 w2 = ( d1wi ) ( d2w2 ) . This is a contradiction ! 3 When d has no prime divisor, it must be a unit. But then d1 and d2 are also 1 units, and W>. = d>. (d>.w>.) belongs to I0 [u] . • 3 .2 Corollary. 1.

2.

3.

4.

Let I

re

a unique factorization domain.

If a E J0 [u] is prime in Q[u] , then it is also prime in I0 [u] . If a E J0 [u] is reducible in Q[u] , then it is reducible in J0 [u] . Every element of I0 [u] is a product offinitely many prime elements. If E J0 [u] is irreducible, it is also prime. a

PROOF: l . Let a E J0 [u] be a prime element in Q[u] . 1f a divides a product a' a" in I0[u] , then it does so in Q [u]. Therefore, it divides one of the factors in Q[u] . Assume that there is an element b E Q[u] with a' = ab. By Gauss's lemma b E J0 [u]. This shows that a is prime in J0 [u] .

E J0 [u] be a product of nonunits a 1 , a2 E Q[u) . If ci E Q is the 1 1 highest coefficient of ai , then c1 c2 = l ci 1 ai E Q0 [u) and a = ( c1 a1 ) ( c;- a2) . 1 By Gauss c; ai E I0 [u] , and these elements cannot be units there. So a is reducible in I0 [u] .

2. Let

a

,

3 . Every element a E J0 [u) is a finite product a = a 1 . . . az of prime elements of Q[u]. One can choose the ai monic, as in (2). Using Gauss's lemma several times one shows that the ai belong to I0 [u] . By (1) they arc also prime in JO [u].

4. Let a E J0 [u] be irreducible. Since it is a product of prime elements, it • must be prime itself. Remark. In the proof we didn't use that I is a unique factorization do­ main. We needed only the fact that Q[u] is a UFD (since Q is a field) and the statement of Gauss's lemma: If a 1 , a2 E Q0 [u) and a1 a2 E I0 [u] , then ai E I0 [u) for i = 1, 2. 3

The original version of Gauss's lemma states that the product of primitive poly­ nomials is again primitive. The reader may convince himself that this fact can be derived from our proof.

3. Prime Factorization

Factorization in Hn . I = H, .

1 19

Now the above results will be applied to the case

Definition. Let f E Hn, f = L� o P>. be the expansion of f as a series of homogeneous polynomials. One defines the order of f by the number

ord(f) : = min{>. E N0

:

P>. -=1- 0}

and

ord(O) := oo.

(See also Exercise 1 .4 in this chapter) Then the following hold:

ord(f) > 0 for every f E H,. 2. ord(f) = 0 -. E (Hn )* for >. = 1 , 2. There exists a shear a such that h o a, h o a and f o a are zi-regular. If we can show that f o a divides one of the f>. o a , then the same is true for f and f>. . Therefore, we may assume that h, /2, and f are ZI-regular. By the preparation theorem there are units ei, e2, e and Weierstrass polyno­ mials WI , w2 , w such that h = e i -WI , h = e2 ·w2, and f = e · w. Then w divides wi w2. If wi w2 = q . w with q E Hn , then the division formula says that q is uniquely determined and a polynomial in zi . So w divides wiw2 in H�_ dzi] . Since w is irreducible in H,, it must also be irreducible in H�- I [zi] · By the induction hypothesis Hn- I is a UFD, and therefore w is prime in H�- I [zi] . It follows that w I WI or w I w2 in H�_dzi] and consequently in Hn. This means • that f I h or f I h in Hn ·

Hensel' s Lemma. Let w E Hn[u] be a monic polynomial of degree s. There is a polydisk P around 0 E en where all the coefficients of w converge to

1 20

III.

Analytic Sets

holomorphic functions. Therefore, we can look at w as a parametrized family of polynomials in one variable u . By the fundamental theorem of algebra w(O, u) splits into linear factors, and the same is true for every w(z, u) with z E P. We now show that such splittings are coherently induced by some splitting of w in H� [u] , at least in a neighborhood of 0 . 3.4

Let w(O, u) = TI�= I ( u - C>.)8" be the decomposition into linearfacto rs (with c, -=/=- cJL for v -=/=- f-L and si + · . . + s z = s). Then there are uniquely determined polynomials WI, . . . ,wz E H� [u] with the following prope rties: Hensel's lemma.

deg(w>.) = S>., fo r ).. = 2. W>. (O, u) = (u - C>.)8" . 3 . W = WI · · · · · Wz . 1.

1 , .. . , l .

PROOF: We proceed by induction on the number l. The case l We assume that the theorem has been proved for l - 1 .

=

1 is trivial.

First consider the case w(O, 0) = 0 . Without loss of generality we can assume that c1 = 0. Then w(O, u) = u8' h( u ) , where h is a polynomial over C with deg(h) = s - SI and h(O) # 0. So w is u-regular of order SI , and there exists a unit e E H� [u] and a Weierstrass polynomial wi with w = e . WI · Since wi (O, u) = u8' , it follows that •

l

e(O, u) = h(u) = IT (u - c>.)8" >.=2 By induction there are elements w2, . . . , wz E H�[u] with deg(w>.) = S>., W>. (O, u) = (u - C>.) 8" and e = w2 · · · w1 . Then w = wiw2 · · · wz is the de­ sired decomposition.

If w(O, O) -=/=- 0, then we replace w by w'(z, u) := w(z, u + ci ) and obtain a

decomposition w'

= wi . . . wf

as above. Define

This gives a decomposition w = w1 . . . wz in the sense of the theorem. The uniqueness statement also follows by induction. •

The Noetherian Property. Let R be a commutative ring with 1 . An R-module is an abelian group M (additively written) together with a composition R x M -+ M that satisfies the following rules:

r(xi + 2 2 ) = rxi + rx2 for r E R and xr, x2 E M . 2. (ri + r2)x = ri x + r2x for ri , r2 E R and x E M . 3 . ri (r2x) = (rir2)x for ri , r2 E R and x E M . 1.

3. Prime Factorization

4. 1

· x

121

= x for x E M.

These are the same rules as those for vector spaces (and the elements of a module are sometimes also called vectors). However, it may happen that rx = 0 even if r -=!=- 0 and x -=1=- 0. Therefore, in general, an R-module has no basis. So-called free modules have bases by definition. An example is the free module RQ := R x . . . x R (q times), with a basis of unit vectors. An example of a nonfree module is the Z-module M : = Z/6Z, where 2 · 3 = 2 · 3 = 0.

If M is an R-module, then a submodule of M is a subset following properties: 1. 2.

N

C M with the

x,

r

y E N =? x + y E N . E R and x E N =? r x E N .

A submodule of an R-module is itself an R-module. Example The ring R is also an R-module. The composition is the ordinary ring mul­ tiplication. In this case the submodules of R are exactly the ideals in R. An R-module M is called finite if there is a finite set { x 1 , . . . , xn } C M such that every x E M is a linear combination of the xi with coefficients in R. The free module Rq is obviously finite. But Z/6Z is also finite, being generated by the class I. Definition. An R-module M is called noethe rian if every submodule N C M is finite. A ring R is called noetherian if it is a noetherian R-module. This means that every ideal in R is finitely generated (in the sense of a module). 3.5

of

Proposition. ideals

becomes stationary,

Let R be a noetherian 1ing. Then any ascending chain Io

i.e.,

c

/1

c Jl c · · · c R

there is a ko such that h

=

ho fo r k > ko .

PROOF: The set J : = u� 0 h is obviously an ideal. Since R is noetherian, J is generated by finitely many elements h , . . . , fN . Each f" lies in an ideal h., . If ko = max ( k1 , . . . , kN ) , then all f" are elements of ho . So h = ho for

k > ko.

•

3.6 Theorem. PROOF:

IfR is

a

noethe rian ring, then Rq is a noetherian R-module.

We proceed by induction on q.

122

III.

Analytic Sets

The case q = 1 is trivial. Assume that q > 2 and the theorem has been proved for q - 1 . Let M c Rq be an R-submodule. Then I : = { r E R : 3 r' E Rq- I with (r, r' ) E M ) is an ideal in R and as such is finitely generated by elements ri , . . . , rz . For every r;.. there is an element r� E Rq I such that r;.. : = (r;.. , r� ) lies in M . The set M ' := M n ( {0} x Rq - I ) can be identified with an R-submodule of Rq - r , and by the induction assumption it is finite. Let r ;.. = ( 0, r�. ) , >. = I + 1 , .. . , p, be generators of M '. An arbitrary element x E M can be written in the form x = (xi , x' ) with J

XI E I. Then XI = I:>-. = l a;.. r;.. , a;.. E R, and

x-

l

2.:: a;.. r ;..

X=l

=

(0, x' -

l

2.:: a;.. r �)

X1

E M' .

That is, there are elements az + I , . . . , ap E R such that x

1

- 2.:: a;.. r ;.. = X1

p

2.::

>-.= 1 + 1

a;.. r;.. .

Hence { ri , . . . ,rp} is a system of generators for M . 3 .7 Riickert b a sis theorem.

noetherian.

•

The ring Hn of convergent power series is

PROOF: We proceed by induction on n. For n = 0, Hn = C, and the statement is trivial. We now assume that n > 1 and that the theorem has been proved for n - 1 . Let I C Hn be a nonzero ideal and g -=/=- 0 an element of I. Without loss of generality we can further assume that g is 2 1-regular of order s .

Let = 9 : Hn -+ ( Hn- I ) 8 be the Weierstrass homomorphism, which is defined in the following manner: For every f E Hn there are uniquely defined elements q E Hn and r = ro + r1zi + · · · + rs - I zf I E Hn- dzi] such that f = q · g + r. Let (f) : = (ro , . . . , rs- d ·

Now, is an Hn- rmodule homomorphism. By the induction hypothesis Hn- I is noetherian, and so (Hn- ! ) 8 is a noetherian Hn- rmodule. Since M : = @ ( Ijs an Hn-I -submodule, it is finitely generated. Let r;.. = (r6"l , . . . , r�?:) I ) , >. = 1 , .. . , l , be generators of M . If f E I is arbitrary, th en J = q · g + r with r = ro + ri zi + · · · + rs -I zf - I , and there are elements ai , · . . , az E Hn- I such that (ro, ri , · . . , rs - 1 ) = 9 ( ! ) L�= l a;.. r;.. . Hence we obtain the representation

4.

f

Branched Coverings

1 23

l

= q.

g + z:::a ).. . (rg'' ) + riA) Z1 + · + r�A\ zf- 1 ) , .>..= 1

The set {g, rymmetric polynomials a1 , . . . , a, defined as follows: a 1 (u 1 , . . . , us ) - U1 + · · · + Us , a2 (u 1 , . . . , us ) - u l (u2 + . . . + us ) + u 2 (u3 + . . . + us) + . . · + us - 1Us ,

The following result is proved, e . g . , in the book [vdW66] 4.5 Theorem. lfp E Z[u1 , . . . , us] is symmetric, then there is exactly one polynomial Q(y 1 , . . . , Y s) E Z[y 1 , . . . , Ys ] such that p = Q( a 1 , . . . , a, ). Consider the special symmetric polynomial pv(u1 , . . . , us) = IJ (u i - Uj ) 2

The Discriminant.

i <j

(square of the Vandermonde determinant). Since it is symmetric, there is a uniquely determined polynomial Qv ( y1 , . . . , y,) with integral coefficients such that

4. Branched Coverings

1 27

Definition . If w u8 +h1 (z)u s - l + · . + hs(z) is a pseudopolynomial in 0° [u] , then A, = Q v (- h1 , h2 , . . . ( - 1 ) 8 hs ) is called the discriminant of w. It is a holomorphic function in G, and we denote its zero set by D,. .

=

,

It is well known from the theory of polynomials that where w 1 , . . . , w , are the zeros of the polynomial if and only if there is a pair i # j with w; = w1 .

u

f-t

w(u, z) . So �w (z)

=

0

Assume now that w is without multiple factors. Then there is a linear combi­ nation of w and D(w) that is a function h E 0* . We restrict to a point z E G with h(z) # 0. Then the greatest common divisor of w(u, z) and D(w) (u, z) is 1 . This means that w( u, z) has no multiple factors; i.e., the zeros of w( u, z) are all distinct. So �w (z) # 0, and D, is nowhere dense. Example Let G c en be a domain, a, b holomorphic functions in G, and u2 - a(z) . u + b(z) . In this case

w(u, z)

pv(u 1 , u 2 ) = IJ (u ; - u1 ) 2 = (u1 - u 2 ) 2 = (u1 + u2 ) 2 - 4 · u l u 2 . i <j

·

�w (z) = Q v ( a(z) , b(z)) = a(z) 2 - 4b(z). If z E G and �w (z) # 0, there are two different solutions of w(u, z) = 0. Hyper sur fa ces. We use the theory of pseudopolynomials to study ana­ lytic hypersurfaces. Such analytic sets are locally the zero set of one holomor­ phic function. Assume that f is a holomorphic function in a connected neigh­ borhood of the origin in en+ I that is not identically 0. Without loss of gener­ ality we may assume that A = N( f) contains the origin. Then a generic com­ plex line £ through 0 meets A in a neighborhood of 0 only at the origin. After a linear coordinate transformation, £ = { ( u ;z) : z = 0 } is the first coordinate axis. By the Weierstrass preparation theorem fco,o) = e (o,o) · wco,o) in the ring Hn+I , where e co, O ) is a unit in Hn+l , and W(o,o) E Hn [u] a Weierstrass poly­ nomial. We can represent the germs locally by holomorphic functions. Thus there is a domain G C e n containing 0 , and a disk D = { u E e : lui < r} such that in U : = D x G there are a holomorphic function e that does not vanish in U and a pseudopolynomial w over G with f = e . w in U. We may assume that A n (8 D x G) = 0 . Therefore, the zero set of f in U is that of W.

128

1 1 1. Analytic Sets

We can decompose w into prime factors. Using the fact that any power of a prime factor vanishes at the same points as the prime factor does, we may assume that w is without multiple factors. Then the discriminant A, is not identically zero in G. We set D, = {z E G : 6w (z) = 0 } .

4 . 6 Theorem (on branched cover ings). If z o E G - D, there are a neigborhood W = W(zo) C G - D, and holornorphic functions ft , . . . ,fs zn W with fi ( z ) -=/= J1 (z) fo r i -=/= j and z E W such that

w(n, z ) = (n - JI (z) ) · . . (n - fs (z)) in C W There are fewer than s points over any point zo E D, (see Figure III. l ) . X

·u

A

G

Figure 1 1 1. 1 . A branched covering over G

A point z E G above which there are fewer than s points is called a branch point. All points of the discriminant set Dw are branch points. Over all other points our set A is locally the union of disjoint graphs of holomorphic func­ tions, and is therefore regular. PROOF: For zo E G - D, the polynomial w(n, zo) has s distinct roots. We write w(n, zo) = (n - ci ) · · · (n - c8 ), where the ci all are distinct. If w(n, z ) = '1L3 + h1(z)n 8 - 1 + . . . + hs (z ) , then the germ

Wz0 := n 2 + (hi )z01L 8 - 1 + + (hs)z0 Hn- By Hensel's lemma it has is a polynomial over Oz0 w, , = w1 .zo . . . ws ,zo with the following properties: ·

"'

·

·

a

decomposition

4.

1 . w,,z0 (u, zo ) = U 2. deg (wi,zo ) = 1 .

- C;

Branched Coverings

1 29

for i = 1 , . . . , s .

We have w;,z0 u - ri , with ri E H,. There are a connected neighborhood W(zo) C G - Dw and holomorphic functions h , . . . , is in W such that the power series ri converge to j;. Since the germs of w and ( u - h) . . . ( u - is ) coincide at z0 , it follows from the identity theorem that =

wkxw

=

(u - h ) . . . ( u - is ) ,

and since W c G- D, it also follows that ii (z) -=!= i1 ( )for i -=!= j and z

z

E W.

•

Examples 1. Let G = e and w = zf - z2 . Then the discriminant is given by A, (z2) = 422, and Dw = {0} c C. If z2 E @*, there is a neighborhood W C e - D, where VZ2 is well defined. There we have w = (z1 - VZ2) . (z1 + VZ2) · This gives a surface above e that is a connected unbranched 2-sheeted covering over e- {0}. The point 0 is a branch point. This is the (branched) R iemann surface of y'Z. The unbranched part was already discussed in Section 1 1.8. 2. A completely different situation is obtained if we take w = zf - z� (z1 - z2 ) (z1 + z2 ) . The discriminant is 4z� in this case, and the discrim­ inant set D, is again the origin in C. The set A consists of two distinct sheets, which intersect above 0, and both are projected biholomorphically onto C. The set A - { 0 } , i.e., A without the branch point, is no longer connected. 3. In higher dimensions the situation is even more complicated. Let us con­ sider the analytic set A = N (f) , where f ( z1 , . . . , Zn ) = z�' + . ' . + z�n with si > 2 for i = 1 , . . . , n. This is a very simple holomorphic function. The derivatives are iz, = si . zf'- 1 , and their joint zero set consists only of 0 E e n . So all other points of A are regular. Every line £ through the origin lies completely in A, or i has a zero of order s with s > min( s 1 , . . . , s,) on £ at the origin. Therefore, there is no line that intersects A in 0 transversally. From this one can conclude that 0 is in fact a singular point of A (see, for example, Exercise 8.2 in Chapter I). Now we look on f : en -+ e as a fibration with general fiber ·

At = {z E e n : z�' + · · · + z�n = t} .

Then A = Ao has an isolated singularity, while all other sets At are regular everywhere. We call the family (At)tEIC a deformation of A.

130

III. Analytic Sets

The Unbranched Part . We assume that

w(u, z) a pseudopolynomial over A = { (u, z)

G

c en is a domain and

G of degree s. We set G' = G - Dw , E e x G : w(u, z) = 0} ,

A' = AIG' the part of A over G'. Then A' is an unbranched covering of G'. It is an n-dimensional submanifold of e x G', and we have the canonical projection 1r A --+ G. If ( u0 , z0 ) E A' is a point , there is a small neighbor hood and

:

B = B ( u0 , z0) c A' that is mapped by 1r holomorphically and topologically onto a ball around z0 in G'. We also call B a ball. The holomorphic map (7r i B ) - 1 : 7r(B) --+ en+l is a local parametrization of A'. A complex function f in B is called holomorphic if f o (7r i B ) - 1 is holomorphic. In particular, the components of 1r itself are holomorphic functions on B.

For holomorphic functions in B we have the same properties as for holomor­ phic functions in a domain of en . For example, the identity theorem remains valid, and we obtain the following results: 4. 7 Proposition. Assume that A1 is a connected component of A' and that M is an analytic subset of A1 . Then M = A1, or M is nowhere dense in A1 . 4.8 Proposition. If f is a holomorphic function on A', and A1 a connected component of A', then either f vanishes identically on A1 or its zero set is nowhere dense in A1 . 4.9 Proposition. Let A1 again be a connected component of A'. Assume that M is a nowhere dense analytic set in A1 and that f is a holomorphic function in A1 - M that is bounded along M . Then f has a unique holomor­ phic extension to A 1 . Decompositions. We consider the interaction between the decomposi­

tion of a pseudopolynomial into irreducible factors and the decomposition of its zero set into "irreducible" components. 4.1 0 Proposition. Let G c en be a domain and w(u, z) a pseudopolyno­ mial over G without multiple factors. Then w is irreducible if and only if the intersection of its zero set A with e x ( G - E) is connected for every nowhere dense analytic subset E C G which contains the discriminant set Dw . Since locally over G' = G - Dw the set A' = AIG' looks like a domain in en , a nowhere dense analytic set does not disconnect A' , locally and globally. Therefore, we may assume that E = Dw·

PROOF:

If w is not irreducible, every factor wi defines an analytic set Ai over G - Dw . The intersection of different Ai is empty. So AI(G - Dw) is not connected.

4.

Branched Coverings

131

If, on the other hand, AI G1 has the connected components A;, i = 1 , . . . , s , with s > 0, then for any point z E G-Dw there is a ball B C G - Dw around z such that Ai lE splits into graphs of holomorphic functions f1 : j = 1 , . . . , si. In each case we form the pseudopolynomial Wi = ( u - h ) · · · (u - is , ) . The zero set of this wi is exactly Ai I B, and it determines w; and vice versa. So over the intersection of two different balls the pseudopolynomials must be the same, and thus we obtain global holomorphic functions wi in G - Dw . If z E Dw , then there is a neighborhood W of z such that Ai i (W - Dw ) C AIW is a bounded set. S o the coefficients of w; are bounded over this neighborhood and extend holomorphically to G. We also denote this extension by Wi , and • for reasons of continuity it follows that w = WI · · · Ws. If the w; are the irreducible factors of w, we call their zero sets A; the irre­ ducible components of A. The sets A� = A; IG' are the connected components of AIG'. 4.11 Proposition. Assume that w * , w are pseudopolynomials without mul­ tiple factors over a domain G and that A * = { w * = 0} C A = { w = 0}. Then w * is a factor of w. Let D denote the union of the discriminants of w * and w. It is a nowhere dense analytic set in G. Over G - D we decompose the two unbranched coverings into connected components. There we have A * = AI U · · · U As• and A = AI U · · · U As with s * < s. This yields pseudopolyno­ mials over G - D that extend to pseudopolynomials WI , . . . , Ws over G, with • w * = WI · · · Ws• and w = WI · · · Ws. This implies the result.

PROOF:

The following result is proved analogously. 4.12 Proposition. Assume that w is free of multiple factors and that A = {w = 0} is the disjoint union of two nonempty sets M', M" that are closed in C x G. Then there are pseudopolynomials w', w" over G with M' = {w' = 0}, M" = { w" = 0} , and w' · w" = w. The construction is first carried out outside Dw . \Ve set G' = G - Dw and use the fact that every nonempty open subset of A' = AIG' must be a union of connected components of A'. If w = WI · · · ws is the decomposition into irreducible factors, then we may assume that there is an s * with 0 < s * < s such that for w' = WI · · · Ws • and w" = Ws• + I · · · Ws we have M' IG' = { (u, z) E C x G' : w '( u , z) = 0} and M" IG' = {( u , z) E C x G' : w" ( u z) = 0} .

PROOF:

,

It is now essential that in a continuous family f ( u, z) of holomorphic functions of one variable u the zeros depend continuously on the family parameter z ( "continuity of roots" ) . We omit the proof here. If we apply this fact ( and the equations A = M' U M". w = w' · w" ) , we get that the sets M'IG' and

132

III.

Analytic Sets

M " I G' are not empty (i.e., 0 < s * < s) and that their closures in e x G are M', respectively M". • Project ions. In the next section we will investigate zero sets of several

holomorphic functions. Here we begin with the simplest case, the common zero set N of a pseudopolynomial w over G C en and an additional holomor­ phic function f in a neighborhood of A = { (u, z) E e x G : w(u, z) = 0}. Our method involves the projection of N to G.

4.13 Proposition. Assume that f = f(u, z) is a continuous function on A that is holomorphic outside of e x Dw and does not vanish identically in a neighborhood of any point of A. Then the projection of N = {! = w = 0} to G is an analytic set N' = {! = 0}, where f is a holomorphic function in G that does not vanish identically.

PROOF: If z E G - Dw , we have a ball B C G - Dw around it such that

over B our w has the form w(u, z) = (u - fi (z)) · · · (u - fs (z) ) . The function f does not vanish identically on any graph u = fi · Consequently,

f(z)

:=

f( JI (z) , z) · · · f( Js (z) , z)

does not vanish identically. In the usual way we obtain the holomorphic function f in the entire set G - Dw · It is bounded along Dw . So it extends to a holomorphic function in G. • Now consider the following situation: Assume that G is a domain in e n and that w is a pseudopolynomial over G without multiple factors. Let f be a holomorphic function in a neighborhood of A = {w(u, z) = 0} C e x G that does not vanish identically on any open subset of A and define

N

:=

{ (u, z) E e x G : w(u, z)

=

f(u, z)

=

0}.

Denote by N' the projection of N to G. We want to give a definition for "unbranched points" of N. The difficulty is that there may exist such unbranched points of N lying in the set of branch points of w.

4.14 Proposition. For any point z0 E N' there is an arbitrarily small linear coordinate change in z1 , . . . , Zn such that thereafter the line parallel to the z1 -axis through zo intersects N' in zo as an isolated point. In such coordinates there is a neighborhood U(zo) C G, a domain G' in the space en - l of the Variables Z1 = (z2 , . . . , Zn ) , and a pseudopolynomial w'(z1 , z') over G' such that { (z1 , z') E e x G' : w'(z1 , z') = 0} = N' n U. A "small" linear change of the coordinates z1 , . . . , Zn means here that the transformation is very near to the identity. Since f does not vanish

PROOF:

4.

Branched Coverings

133

identically, after a small generic coordinate transformation the line parallel to the zra.xis through zo intersects N' in z0 as an isolated point . And then • it is also clear that U, G' , and w ' with the desired properties exist. Let us now assume that we have chosen a point zo E N' and suitable coordi­ nates as above, and that U, G' , and w ' have also been chosen.

Definition. In the given situation, a point (u, z ) E N n (C x U) is called an unbranched point of N if z E N' - C x Dw' and there is a neighborhood V = V(z) C N' - C x Dw' with a holomorphic function g on V such that N n (C x V) is the graph { u = g(w) : w E V}. (Figure III.2 shows the situation. )

u

A

..... . :--...

o

examples of unbranched points �

Figure 111.2.

C

X

Dw'

Branched and unbranched p oints of N

4.15 Theorem. In the given situation, in every neighborhood of an arbi­ trary point ( u 1 z i ) of N n (C x U) there are unbranched points of N. ,

PROOF: We may assume that

z1

E N' - (C x Dw' ) . Then we take a small

neighborhood w = E X ul of (u l , z i ) such that the following hold: ul c u - (C X Dw' ) . 2 . ( u � , zi ) is the only point of A above

1.

z1

in W.

134

3.

III. Analytic Sets

The set A n W is defined by a pseudopolynomial w * over

Setting N� c N' n projection 1r : W --+

U� to be the image set of W n N under U1 , take Ut so small that N� = N' n U1 .

U1 . the canonical

We restrict w * to N� and replace possible multiple factors by one factor at a time. So we obtain a new pseudopolynomial w1 without multiple factors over N� such that

{ (u, z) E E x N� : w1 (u, z) = 0 } , E a suitable disk. near to (u1 , z1) we can find points (u z , z2) E N n W lying over

NnW=

Arbitrarily N{ - Dw, . All of these points are unbranched points of N, since we can find neighborhoods W2 (u 2 , z2 ) c W and U2 (z 2 ) C U1 with the same properties as W and U1 • Now choose U2 so small that it contains no point of Dw, and that Wt i N' n u2 decomposes into linear factors. Then every sheet of A I (N' n U2) • with the property that it contains points of N is a graph over N' n U2 . Exercises 1 . Prove that every symmetric polynomial in u1, . . . , Us can be written as a polynomial in the power SUmS Sk : = U � + . . . + u�. 2 . Calculate the discriminant of a cubic polynomial. 3. Let D : = Dr (O) C C and f be a holomorphic function in an open neigh­ borhood of D without zeros in 8D. 1f f has in D the zeros c1, . . . ,c8 (some of them may be equal), then

_ 1 r 21r i len

f u, z1 ,

:

4. Let

D

Zn -

("'' ! '(( ) d( = t em . f(( ) j=l 1

x P" --+ C be a holomorphic function in the variables

Assume that f is u-regular of order s and that for every z E P" the function u f--t J(u, z) has exactly s zeros u1 (z) , . . . , u 8 (z) (with multiplicity) in D. Show that the coefficients of the "pseudopoly­ nomial" w(u, z) : = n;=l (u - Uj (z)) are holomorphic. 5. Prove the "continuity of roots": Let f(u, z) be u-regular at the origin. Show that there is an r > 0 such that if g(z) is a function defined in a neighborhood of 0 with lg(z) I < r and f(g(z), z ) = 0 ,then g is continuous at 0. 6. A complex function f on an analytic set A in a domain G C en is called holomorphic if it is locally the restriction of a holomorphic function in the ambient space. (a) Let A : = {(w, z) E C 2 : w2 - z3 } be the Neil parabola. There is a bijective holomorphic parametrization of A given by w = t3 and z = t2 Describe the holomorphic functions on A as functions of the parameter t. Is there a meromorphic function on A that has a pole at oo with main part tm ? •

•

•

,

•

5.

135

Irreducible Components

(b) Show that the analytic set A = {(w, z 1 , z2 ) E C 3 : w2 = z1 z2 } is not regular at the origin. Consider the holomorphic map (t 1 , t2 ) �--+ (t 1 t2 , ti, t§). It is a "two-to-one" map. Describe the local holomorphic functions on A in t1 , t2 • 7. Prove that there is a topological holomorphic map from @* onto A := { (w;) E C2 : W 2 = Z1 z2 }. 8. Prove that there is no topological holomorphic map from @* onto the "elliptic surface" A := { (� ) E C 2 : w 2 = (z 2 - 1)(z 2 - 4)}. 9. Define the pseudopolynomial w E O(C 2 )[u] by w(u, z 1 , z2 ) := u 2 - u . z 1 and determine the discriminant set and the irreducible components of A := {w = 0}. Let f on A be defined by f(u, z 1 , z2 ) := z2 . (u 1 ) . Consider N := {f = w = 0} c C 3 and determine the projection N' c C 2 and the set of unbranched points of N. -

5.

I r r edu cib le C omp on en t s

Embedded -Analytic S et s . We wish to study general analytic sets. Since it is easier to work with pseudopolynomials than with arbitrary holo­ morphic functions, we introduce the notion of "embedded-analytic sets." These are subsets of the common zero set of finitely many pseudopolyno­ mials, and they are not a priori analytic by definition. But later on, it will turn out that they are indeed analytic. Assume that G c e n- d = { z ' = (zd+ I , . . . , zn )} is a domain and that wi (zi; z ' ) ,i = 1 , .. . , d, are pseudopolynomials over G without multiple fac­ tors. The zero sets of the single Wi intersect transversally4 in cd X G. We denote by D c G the union of the d discriminant sets belonging to the wi. We call it the union discriminant set. We put

A := {(z1 , . . . , zd , z ' ) : wi (zi; z') = O, for i =

1 , .. . , d and

z' E G}.

�

Over any ball B C G - D the set A lB consists of finitely many disjoint holomorphic graphs. Every graph is contained in a connected component Z of AI ( G - D ) . We call the closure of Z in A an irreducible embedded-analytic component of A. �

�

�

�

Definition . If A is defined as above, any union of finitely many irreducible embedded-analytic components of A is called an embeddedanalytic set of dimension n - d. Two submanifolds M, N C en intersect transversally at a point z E M n N if the entire space is spanned by vectors that are tangent to M or N at z. fu our case the common zero set of the pseudopolynomials w; contains enough vectors to span en. Therefore, � say that these zero sets intersect transversally. �

4

136

1 1 1. Analytic Sets

By definition any embedded-analytic set A can be decomposed into finitely many irreducible components.

The surrounding set A :J A is not uniquely determined. Sometimes we can make A smaller by throwing away those irreducible factors of w; that do not vanish identically on A . Then the wi are uniquely determined by A . �

Assume that A is an embedded-analytic set in A c r,cd xG and that f is a holomorphic function in a neighborhood of A that does not vanish identically on any open subset of A. If N = {Z E A : f(z) = 0 } , then fo r any point z0 E N there is an arbitrarily small linear change cl the coordinates z' such that the affine space parallel to the (z1 , . . . , zd+l)- "axis " through z0 intersects N in a n isolated po int. If z1 E N is any point near z0, then there are unbranched po ints qf N arbitrarily near z1 . A t all of these points N is a submanifold of dimension n - d - 1 . 5 . 1 P r opo sition .

PROOF:

We proceed as in the proof for the last theorem of the previous section. The procedure to find unbranched points will be denoted by ( * ) .

First we construct the projection f of j, which is holomorphic in G. For this observe that if D is the union discriminant set of the polynomials w 1 , . . . ,wd and z' E G - D, we always have the same number of points z�, . . . ,zs in A over z'. We set f(z') = f(z1) . . . f(zs) and obtain j, which is holomorphic on G - D. Since it is bounded along D , we can extend it holomorphically to G. Therefore, the projection set is N' = {J-(z') = 0}. Assume now that z� E N' . Then, after an arbitrarily small linear coordinate change in the variables z', the line L parallel to the Zd+ 1-axis through z� intersects N' in an isolated point. Then by Weierstrass's theorem we can find a neighborhood U = U(z�) C G, a domain G', and a pseudopolynomial w'(zd+ l , z") over G' without multiple factors such that U n N ' is equal to the set {z' = (zd+ l • z") E C x G' : w'(zd+ I , z") = 0}, with z" = (zd+2 • . . . , zn ) ·

Since the space r,c d x L intersects N at z0 in an isolated point, it remains to prove the existence of unbranched points. We apply ( * ) to N n (Cd x U) and prove in the same way as before that for points z1 E N n (C d x U) there are unbranched points of N arbitrarily near to z1 . Of course, at these points N • is a submanifold of dimension n - d - 1 . For the following we use the same notation and hypotheses as above. 5 . 2 Theorem.

Let N' C G be the projection cl N . For every point zo x G, after a suitable linear change of the coordinates z' (z0, z�) E N there is a neigborhood U U(z�) C G, a domain G' in the space of the variables Zd+2 , . . . , Zn , and a pseudopolynomial w' over G' without multiple factors such that:

C r,c d

I. N'

n

U=

{w' = 0} .

=

5.

2. N n (e d

x

Irreducible Components

U) is an embedded-analytic set of dimension

n -

d

-

137

1.

PROOF: We use the notation and results from the proof above. Thus the first statement is clear. We set �+ I :=w'. Restricting the Wi to ed X (N' n U) and projecting them down to e d X G' we get pseudopolynomials w i ( Zi; z") ' over G'. Then N n (e d x U) is in the joint zero set of w 1 , . . . , w d+I · Let A be the union of those irreducible components of this set that contain the unbranched points of N. Since N is the closure of unbranched points, it +I follows that N n (e d x G) C A . By the mapping theorem that we prove in the next paragraph, every irreducible component of A is in N. So we have the desired equality. •

Images of Embedded -Analytic Set s . Assume that G = e d X G' c en is a domain and A c G an irr educible embedded-analytic set over G'.

Let G1 = e dl X Gi c en, be a domain, A I c G 1 an embedded-analytic set over Gi , F a holomorphic mapping from a neigh­ borhood of A C G into GI such that - C A I for some nonempty open subset U c A. Then F(A) c 4 5 . 3 M apping theo r em .

We denote by D C G the union of the discriminant sets of the wi (zi, z' ) that define the surrounding set A for A . It is sufficient to prove that F (A n (ed x (G ' - D ) ) ) C A I . Since we can connect two points of A n (ed x (G' - D ) )by a chain of arbitrarily small balls, it is enough to give the proof for such a ball. So we may replace A by a ball in en- d and may assume that F is defined in a neighborhood of B . PROOF:

�

Let A I be an embedded-analytic set in

� (WI , W' ) = · · · = W � d 1 ( Wd 1 , W ) = 0 } A 1 -- { WI 0 . So - c 4 Then w; o Flu 0, and by the identity theorem wi o FJB �

I

_

For an arbitrary point v E G� we choose a small transformation of the coordinates in ed , such that AI is also embedded in a set Ay = { wY = . . . = wd, = 0} . The transformation can be made arbitrarily small, and we can do it so that

A1 n .Ar n (ed,

x

{v}) = A I n (ed ,

x

{v}) .

Then AI is given by the infinite set of holomorphic equations wi = 0, w'[ = 0, v E G� . If F maps a nonempty open part of B into A 1 , then by the identity theorem wi o F = w; o II 0, and consequently - c A I · This completes the proof. • _

Remark. Assume that A is an analytic set in a domain G c en and that z0 is a point of A . If zo is a regular point of dimension n - d, then there is a neighborhood U = U(zo) c G such that A n U is an embedded-analytic set.

138

llL Analytic Sets

In fact, there is a neighborhood U with holomorphic functions JI , . . . , jd such that N(fi , . . . , fd ) = A n u and the rank of the Jacobian is d everywhere. We may assume that i_

1, . . . , d J - 1, . . . , d

) # 0.

Then the transformation F ( z 1 , . . . , zn ) = (h (z) , . . . !d ( z) , zd+ l , . . . , zn ) maps a neighborhood of z0 biholomorphically onto a neighborhood of the image point. If the inverse is given by ,

then A is given by the equations

Zd

=

gd ( O, . . . , 0, Zd+ l , . . . , Zn ) ·

So A n U is an embedded-analytic set.

Local Decomposition. We use embedded-analytic sets to show that an arbitrary intersection of analytic sets is again an analytic set. First we consider the following situation. Let G c e n be a domain and Zo E G a point. Assume that at z0 a set 9of local analytic functions f is given such that 1 . For every f E 9 there is a connected open neighborhood U ( z0) C G with f E O(U) and f =/=- 0. 2. f(zo) = 0. We want to construct a "maximal" analytic set S* in a neighborhood U* ( zo ) c G such that for each zero set N of finitely many elements f E :7 there is a neighborhood V = V(z0 ) c G with S* n V c N n V. Then S* is uniquely determined near z0 and can be considered as the common zero set of the functions f E 9 Lt may be nontrivial even if the domains of definition of the functions f tend to the point z0 . For example, if :7 is the set of the func­ tions fn (z) : = z!/ (1 - nzl ) , defined on Un := {z E C n : Re(z l ) < 1/n}, then S* is the analytic set { z1 = 0} in an arbitrary neighborhood of 0 , whereas the intersection of the Un does not contain any neighborhood of the origin. We employ the results from the beginning of this section several times and carry out an induction on the codimension of the embedded-analytic sets obtained from the functions f E Y' .

5.

Irreducible Components

139

(A) We begin with one arbitrarily chosen function f E Y'. The equation f 0 gives an analytic set5 S of codimension 1 . We decompose S into irreducible components si in a neighborhood U(zo) (given by pseudopolynomials in c X G'), and we choose the neighborhood U so small that the Si stay irreducible in the whole neighborhood. =

(B) Next we try to obtain codimension 2. If every function f

E .'/'

vanishes identically near z0 on Si, we leave Si unchanged (and have it as a codimension ! component for our S* ) . Otherwise, there is an f' E :7 that does not vanish identically in any small neighborhood of z0 on Si. We apply Theorem 5.2: After an arbitarily small linear change of the coordinates z' the set Si n { f' 0} is a finite union of irreducible embedded-analytic sets Sij of codimension two, which stay irreducible if we pass to some smaller neighborhood of z0 . The Sij are embedded in the zero set of two pseudopolynomials wi1 ( z1 ; z3 , . . . , z,) and w�(z2 ; z3 , . . . , zn )· =

(C) Now codimension 3 follows. For this we need consider only the Sij . Leave Sij unchanged if every f vanishes on Sij (and get codimension 2 components for S* ). Otherwise, find an f" E :7 not vanishing identically on Sij , and (after an arbitrarily small coordinate change of the variables z" (z3 , . . . , z,)) the set Sij n { ! " 0} is the union of a finite set of irreducible embedded-analytic sets Sij k , given in the zero set of three pseu­ dopolynomials w� k (z;.. ; z4 , . . . , zn ), >. 1 , 2 , 3. =

=

=

(D) Continuing, it is possible to obtain components of codimension 1 , 2, . . . , n 1 , and finally one reaches dimension 0. If there is a ! -dimensional component S Si, .. . in - l such that not every f E :7 vanishes on S , we have to replace S by the one-point set {z0 } . Then the procedure stops. Only finitely mar,.y steps were necessary. -

=

We obtained a finite system :70 of local holomorphic functions j, f ', J " , . . . and a finite system A of irreducible embedded-analytic sets Si, Si1, Sij k , . . . and may assume that they all are defined in one neighborhood U(z0) C G, that every s E A of dimension d is embedded in a set e n- d X Gd, and that the union discriminant sets Ds C Gd C <e d belong to the embedding of S. The necessary linear coordinate change in z' can be made at the beginning of the procedure, i.e., once for all steps of the procedure. If S E A is an irreducible embedded-analytic set that has an open part in the union of the other sets of A , then it also has an open part in an irreducible S' E A, S' i= S . It follows by the mapping theorem that it is completely contained in S '. Then we simply throw it away and denote the new system again by A . After finitely many steps we have that the intersection of every 5

An exact definition of dimension and codimension of analytic sets will be given later. Here we use embedded-analytic sets, for which the dimension has already been defined. An analytic hypersurface is obviously embedded-analytic.

140

111.

Analytic Sets

S with the union of the rest of A is nowhere dense in S . Moreover, the points of S over Ds are nowhere dense in S . We denote by S* Usu t S the union of all remaining components. Then S* is given by the finitely many holomorphic functions f E Y0 . Therefore, it is an analytic set. 1f Y c Y is an arbitrary finite subset, then in a small neighborhood of z0 every f E Y vanishes at every z E S * .So S* c N(Y') . Obviously, S* is uniquely determined by this property. =

A

A

Finally, we want to show that the decomposition into irreducible embedded­ analytic sets is unique. For that we use the notion of regularity for points of embedded-analytic sets just as in the analytic case. Clearly, the intersection of two different S, E A contains no regular point. So the points of every S E A are regular if they are not in such an intersection and not over DA . We denote the set of regular points of S* by S* and set S S n S* for S E A . Then for S E A the sets S are the connected components of S * . Since the set S* is uniquely determined in a neighborhood of zo , this is also true for its connected components. And since the closure of S is S, the irreducible embedded-analytic components S are also uniquely determined near zo . .

.

.

=

.

.

5 .4 Theor em.

The intersection of (even infinitely m any) analytic sets is an analytic set and is locally afinite union of components that are irreducible embedded-analytic sets. This decomposition is locally uniquely determined.

Let {A ' : L E I } be a family of analytic sets in a domain G c e n , and Zo E A == n,E/ A ' an arbitrary point. We consider the system y of all local holomorphic functions f such that:

PROOF:

1 . f is defined in an open neighborhood U of z0 (depending on f). 2. f =j:. 0 near zo. 3. There is an L E I such that f vanishes near z0 on A , . As above, z0 is contained in an analytic set S* that is the union of irreducible embedded-analytic sets S and that is given by a finite subsystem Yo c Y.

If z is a point of A that is sufficiently near z0 , then every f E Yo is defined at z and vanishes on some A , and consequently at z . This shows that A c S * in a neighborhood of z0 . On the other hand, let z be a point in the intersection of S* with a small neighborhood of z0. Any analytic set A , is given by finitely many holomorphic functions fL . . . , J'N that belong to the system Y. Then by construction every ft vanishes on every embedded-analytic component S of S * ,in particular at z . Therefore, S* C A , for all L. Thus S * is contained S* near zo . in the intersection A of the A , and we have the equality A Since S* is an analytic set that has a unique decomposition into irreducible s embedded-analytic sets, this completes the proof. =

5.

Irreducible Components

141

Analyticit y . Now we are able to prove the following result, which we announced at the beginning of the section: 5.5 P ropo sition . Every embedded-analytic set A in a domain red X G c e n is an analytic set. PROOF: As in the last part of the proof of the mapping theorem, it follows that the embedded-analytic set A is given as the joint zero set of infinitely many holomorphic functions. Theorem 5 .4 shows that A is an analytic set. • Consequently, every analytic set has locally a unique decomposition into ir­ reducible analytic components.

The Zariski T op ology. We prove that the system of all analytic sets has the properties of the system of closed sets of a topology. 5 . 6 Theor em.

The system A the following properties:

ci

all analytic sets in a domain G

c en

and the empty set belong to A . 2. If A 1 , . . . , At E A, then also A = Ut= l A i E A . 3. If I is an index set and {A , : L E I } a collection cl analytic sets in then A = n, E J A, is also an analytic set in G .

has

1. G

G,

PROOF:

( 1 ) G is defined by the zero function, and 0 by the constant function 1 .

(2) Let z E A = A 1 U · · · U At . Then in a neighborhood U(z) there are holomorphic functions fi,J : i = 1 , . . . , l j = 1 , .. . , d, such that for all i we have U n Ai = N ( fi , l , . . . , !i, dJ · It follows that U n A = N ( h ,1, . . . ft,J1 :ji = 1 , . . . , di) ·

3) This is Theorem 5 .4.

8

So the analytic sets are the closed sets of a topology in G. We call this topology the (analytic) Zariski topology of G. It plays an important rule in complex algebraic geometry.

Global Decomp ositions . Assume that G c e n is a domain and A c G . an analytic subset. We call A irreducible if the set of regular points A c A is connected. It follows that A has the same dimension d at all regular points. This number d is called the dimension of A and is denoted by dim( A ). Every irreducible embedded-analytic set is also an irreducible analytic set. 5 . 7 Theor em.

Every analytic set A has a unique decomposition into count­ ably m any irreducible analytic subsets A, . The covering .9/ = { A i : i = 1 , 2, 3, . . . } is locally finite.

142

III. Analytic Sets

We decompose A into connected components. Let A' be such a component. It has dimension d in all of its points. PROOF:

We consider a point z 0 E A that lies in A' . In a neighborhood U = U(zo) C G we have a decomposition of A into finitely many irreducible embedded­ A analytic components 1 ' . . . , A,. By Af we denote the set of points of Ai that are not over the union discriminant set. Some d-dimensional Af meet A '. Their union A* is contained in A' and dense in A' n U . Hence, the closure of A* in U is equal to A' n U. But A * is an analytic set. From this it follows that A' is an analy!ic set, that only finitely many A' locally). Since the intersect U and that the union of all A ' is A (as it istopology of G is countable, it follows that the set of the A' is countable. • '

If A is irreducible and A = A1 arbitrary analytic sets, then A = A 1 or A = A2.

5 . 8 Corollary.

Sometimes this condition is used

as

U

A2 , where A1 , A2 are

the definition of irreducibility.

5 . 9 Proposition. Let A, B C G be irreducible analytic sets. If there is an open set U C G such that A n U -=/= 0 and A n U C B n U, then A C B .

P ROOF:

This is an immediate consequence of the mapping theorem.

•

Another corollary is the following: 5 . 10 Identity theorem (for analytic set s ) . Let A, B C G be irreducible analytic sets. If there is a point zo E A n B and an open neighborhood U = U(zo) C G with A n U = B n U, then A = B . 5 . 11 Prop osition. Let A , B c G be analytic subsets with A C B. If A is irreducible, then A is contained in some irreducible component of B . U .x. EA B.x. be the unique decomposition into irreducible Let B components. We can choose an open set U C G and a finite set {>..1 , . . . , >- t } C A such that U n A i= 0 is irreducible and U n B = (U n B.x., ) U · · · U (U n B.x.1 ) ­ Then U n A = U n A n B = (U n A n u . . . U ( U n A n B.x.1 ) . Thus there is an index j such that U n A = U n A n Aj , so U n A c U n B.x.j . It follows • that A is contained in B.x.j .

PROOF:

=

B.x.tJ

Now we can generalize the notion of the dimension to arbitrary analytic sets. Definition . If A C G is an analytic set with irreducible components Ai, then dim( A) := supi dim( A i) is called the (complex) dimension of A.

6. Regular

and

Singular Points

143

In general, the dimension of an analytic set can be oo. But if C1 cc G is a rel­ atively compact subdomain, then only finitely many irreducible components intersect C1 . So the dimension of A n C 1 is finite. An analytic set is called pure-dimensional of dimension d if all its irreducible components have the same dimension d. Exercises 1. Let A be an analytic set near the origin in e n . Assume that every ir­ reducible component of A has dimension > 1. Show that there exists a neighborhood U = U(O) such that A n U is the union of irreducible one-dimensional analytic sets containing 0. 2 . Consider A := {(z1 , z2 ) E e 2 : z� = zr + zD. Show that A is irreducible, but A has a nontrivial decomposition into irreducible components in a small neighborhood of the origin. 3. Let A1 1 A2 c e n be analytic sets. Prove that A1--=-A2· is analytic. 4. Let {A; : i E N} be a locally finite family of irreducible analytic sets in a domain G c en . Suppose that Ai � A . for i # j and prove that the Ai are the irreducible components of theif union. Regu lar and Sin gu lar P oin t s

6.

Comp a ct Analytic Set s .

Our goal is to prove the following simple

proposition. 6 . 1 Prop osition. If G c e n is a domain and A c G a n irreducible compact analytic set, then A consists of a single point. We first prove a lemma. 6 . 2 Lemma. Iff is a holomorphic function in a neighborhood of A, then

f !A

is constant.

PROOF: We assume that the dimension of A is n - d. Since A is compact, there is a point z0 E A where Iii takes its maximum. After a linear coordinate change there is a neighborhood U = U(z0) c G and a domain G' c e n-d such that A n U is an embedded-analytic set over G'. Denote by D c G' the union discriminant set. Over every z" E G' - D our A n U has points. The point z0 lies over some z0 E C' and we may assume that it is the only point of A n U over z0. It remains to construct the elementary symmetric functions associated with f !Anu over G' - D. For this, if z1, . . . ,z 8 are mapped onto z", we define fi(z") = ai(J(zi ) , . . . ,f(zs)). These functions are holomorphic on G' - D and bounded along D. So they extend to holomorphic functions in G'. The s

:

144

III.

Analytic Sets

absolute value of every extension takes on its maximum at z�. By the max­ imum principle each such is constant in G'. Since the values of f can be reconstructed from the values of the f; , it follows that f is constant over G' , in particular in some open subset of A. Since A is connected, it follows that • f is constant on A and then by continuity also on A. PROOF of the proposition: All coordinate functions z; must be constant on • A . So A is a single point. A consequence is that every compact analytic subset A C G consists of only finitely many points.

Embeddin g of Analytic S et s . Assume that A c G is an analytic set in a domain G c e n , that 0 E A, and that the plane P = {zd+l = · · · = zn = 0} intersects A in an isolated point.

In a neighborhood U(O) the set A is an analytic subset d a n embedded-analytic set of dimension n - d that is defined over a domain G' in the space d variables z' = (zd+l , . . . , zn ) . If the set d ( n- d)- dimensional regular po ints is dense in A , then A is itself a n embedded-analytic set.

6 . 3 Theorem .

Remark.

No coordinate transformation is necessary for this statement !

PROOF: By definition, A is the zero set N(JI , . . . , fN ) of finitely many holomorphic functions in a neighborhood of 0. Since A n P = { 0} is isolated, there is an i such that j; does not vanish identically in any neighborhood of 0 on the z1-axis. Consequently, f; is z1-regular, and we can apply the Weierstrass preparation theorem, which implies that A is locally contained in the zero set of a pseudopolynomial w(z1 ; zz , . . . , zn ) · Now we proceed by induction on d. In the case d = 1 there is nothing t o prove. 1f d > 1 we consider the projection 7r : e n -+ en -l with Z f--t Z1 = (zz , · . . , Zn ) · To h , . . . , fN there are associated as usual holomorphic functions f 2 , . . . , f N of z' such that

1r({w = h = . . · = fN = 0} ) = A := {z' : j 2 (z') = . . . = J N (z') = 0) in some neighborhood of 0 . The intersection of P' = {z' : Zd+ I = . . . = zn = 0} with A contains 0 as an isolated point. So we have for A the same situation, but with one dimension fewer. By the induction hypothesis it follows that there are pseudopolynomi­ als wz , . . . , wd such that A locally is contained in the set {wz (zz ; z ") = . . . = wd (zd ; z") = 0} . By exchanging z1 with Zz, we obtain a pseudopolynomial w1 (z1 ; z") such that A is contained in {w 1 = Wz = . . . = Wd = 0} .

6 . Regular and Singular Points

1 45

- d are dense in A , we can take the union A* of those irreducible components of the embedded analytic set that contain such a regular point. Then A * and A are identical. This completes the proof. If the regular points of dimension

n

•

Remark. This result is also known as the "embeddingtheorem of Remmert and S Lein."

Again we consider a domain G c en, an analytic set A c G, and a domain G' c en- d such that 7r(G) c G' (where 1f : z M z' (zd+ I , . . . , z )). Suppose n that there exists a domain G* C G such that for every z' E G' there exists a neighborhood U U(z') c c G' with (e d x U) n G * c c (e d x U) n G and (ed X U) n A c G * . Then the following holds. =

=

If the set d regular (n - d) - dimensional po ints of A is dense in A , then A is a n embedded-analytic set over G'.

6 .4 Pr op osition .

PROOF: We take an arbitrary point zb E G'. It follows from the hypotheses that the set (ed x {z6}) n A is compact and analytic. Therefore, it consists of finitely many points. Each of these points has a neighborhood such that the restriction of the set A to this neighborhood is an embedded-analytic set over a neighborhood of zb . By multiplying the pseudopolynomials by the same distinguished variable belonging to our various points we obtain d uniquely determined pseudopolynomials over a neighborhood U'(zb) C G' such that A n (e d x U') is just their joint zero set. But these pseudopolynomials glue together to form global pseudopolynomials over G' . •

Regular P oints of an Analytic Set . Assume again that G c en is a domain, and A C G an analytic set. 6.5 Theorem .

For any z0 E A there is a fixed neighborhood U(z0 ) c G with finitely m a ny holomorphic functions JI , . . . , JN whose joint zero set is A n U such that fo r all d at every regular point z E A n U of dimension n - d the rank of their Jacobian at z is equal to d. It is remarkable that this statement is also true for a singular point z0 of A . PROOF: After applying a linear coordinate transformation in en we can find a neighborhood U U(zo) C G such that A n U is a finite union of irreducible embedded-analytic components. To give these in the canonical form a further coordinate transformation is not necessary. We denote by A' the union of all ( n - d)-dimensional irreducible components of A n U and choose pseudopolynomials w1 , . . . , wd of minimal degree such that A' is contained in the common zero set of the wi . =

�

Let D c A' be the set of points that lie over the union discriminant set DA' . Its dimension is equal to n - d - 1 . Now we carry out the proof in

146

III.

Analytic Sets

several steps and construct sets D t , D2 ,.... . . . , D n- d+ I = 0 with D i + I c Di and dim (Di ) = n - 4- i . We begin with DI :=D. The Jacobian of w1 , . . . , Wrt has rank d in A' - D1 (implicit function theorem). �

�

Next we decompose D1 into irreducible components C>.. and choose, if possi­ ble, in each C>.. a point z ;-, where A' is regular. We can make U so small that only finitely many C>.. occur in U. Then we apply another linear transforma­ tion in en that is near the identity such that in a small neighborhood of any Z>.. the set A' can be written as an ( n - d)-dimensional holomorphic graph {z

:

Zi

=

J>.. , i (Zd+l , . . . , zn )

for i = 1, . . . , d} .

Finally, we apply a linear transformation that is very near the identity to the variables ZI , . . . , Zrt such that for every >. and every point z in A' above (z�+ l ' . . . ,z�) the first d coordinates of z are distinct. Now we use Proposition 6.4. In the new coordinates (and in a slightly smaller neighborhood, which we again denote by U) the set A' is again an embedded-analytic set contained in the common zero set of pseudopolynomi­ als WI , . . . ,wd. We choose wi with minimal degree. We can assume that the components of D1 are still irreducible in U, and that the points z ;-, are still in U. In a neighborhood of z;-. we have a decomposition �

with w;_ i (z>.. ) -=/= 0. So the Jacobian determinant of WI , . . . ,wd with respect to the �ariables .z t ' . . . , z,t does not vanish at any Z>.. - We denote the zero set of this Jacobian in DI n U by D2 . It-has dimension n - d - 2, and WI ' . . . , Wrt , WI ' . . . ,wd have rank d on A' - D2 · �

�

Now apply the same procedure to D2 and ob.t ain an ( n- d D3 and continue in this way until reaching D n- d+ I = 0. �

- 3)-dimensional

By putting all of the pseudopolynomials together, in a small neighborhood U of zo we get holomorphic functions Jr , . . . , jd, fd+ I , . . . . fN (with N = ( n - d + 1) . d ) whose rank is d in every regular point of A' n U. Since the pseudopolynomials always were chosen with minimal degree, it follows that A' = N(h , . . . , JN ) near zo .

Now set A + = A n U - A' . It is the union of the remaining irreducible com­ ponents of A n U. We may assume that U is so small that A + is the common zero set of finitely many holomorphic functions gi , . . . , gs in U. Multiplying the fi by the g1 yields finitely many holomorphic functions in U that de­ scribe the set A n U. No point of A' n A + is a regular point of A. For every z E A' - A + there is a gj that does not vanish there. So the rank of the Jacobian of the fi . gj is equal to d at every nonsingular point of A' - A + .

6.

Regular and Singular Points

147

The same procedure can be used for every d and the corresponding A', and consequently, in finitely many steps we obtain a neighborhood of z0 and holomorphic functions in this neighborhood with the desired properties. •

The Singular L o cus. From the preceding theorem we conclude the following: 6 . 6 Theorem.

Th e set Sing(A) again an analytic set.



singular points

d

an analytic set A is

The intersection of two irreducible components of A belongs to Sing( A) . The union S of all these intersections is an analytic set.

PROOF:

Assume that z0 is a point of an irreducible component A' of A and dim(A') = n - d. Then there is a neighborhood U = U (z0 ) C G with holomorphic functions h , . . . N vanishing exactly on A' n U such that their Jacobian has rank d in each of the regular points. Let S* be the analytic set of all points of A' n U where all d x d minors of the Jacobian vanish. Clearly, S* is contained in Sing( A') n U. On the other hand, at any point of Sing( A') n U the Jacobian of h , . . . N cannot have rank d. So Sing(A') n U = S *, and Sing( A') is analytic in G.

,f

,f

The union of S and the sets Sing(A') for all irreducible components set Sing(A) . It is analytic, since the union is locally finite.

A'

is the •

The set Sing(A) is called the singular locus of A.

Extending Analytic Set s . Let G c

cc n

be a domain.

6 . 7 Lemma. Let Zo = (ziO} ' . . . ' z�0)) c e n be a n arbitrary point and E = { z : Zi = zi0l for i = 1 , .. . , d} a n affine plane of codimension d containing zo . IfA C G is a n irreducible analytic set of positive dimension that is n o t a

subset

of

E, then there is a n open dense subset C C ed such that

fc( zl , . . . , zn ) : = c1 ( z1

- zi0> ) + . . . + cd (zd - zA0> )

does not vanish identically o n Afo r every c = (c1 , . . . ,cd ) E C . In particular, fo r any hyperplane Ho C en containing E there is a hyperplane H arbitrarily close t o H0 and also containing E such that dim(Ai ) < dim( A) - l fo r every irreducible component A, of A n H . We define cp : )!A contradiction, and consequently, V must be a proper subspace of ed. For any does not vanish identically on A. c in the open dense subset C := e d - V, 0} is a hyperplane containing E . Thus He := { z : • PROOF:

:= fc·

: !ciA

fc(z)

fc

=

148

llL Analytic Sets

Our main tool for extending analytic sets is the following.

If E = {z E en : Z l = . . . = Zd = 0} is a n ( n - d) ­ dimensional plane and A an analytic set in G - E , whose irreducible compo­ nents all have d imens ion n - d + l with 0 < 1 < d , then the closure A of A in G is an analytic set in G . 6. 8 P r op osition .

The proposition is of a local nature. We may assume that A is irreducible and 0 E E n G. It is enough to construct a continuation of A into a neighborhood of 0 . PROOF:

Let c = (O, . . . , O, ed+l , . . . , c,) be an arbitrary point of E n G. We consider the following family of ( d - 1 )-dimensional planes through c : For

i = 1, . . . , n - d + l j = I, . . . , d - l we have the linear map LA : e d - l -+ e n - d+ l and define P (c, A) := c t { (w', LA (w' )) : w' E So P(c, A) consists of vectors w

ed - 1}.

= (WI , · . . , wd - t , Wd - l +l • · . . , wn ) with

d-l for i = 1 , . . . , 1, :2::: aij . wj d- l ed -l + i + Z::: aij . w, for i = l + 1 , . . . , n - d + l . ,=1 Then every P( c, A ) meets E exactly in c = ( 0 , . . . ,O,ed+l , . . . ,en )· If 0 is the zero matrix, then P( c) == P( c, 0) is the plane ed - l { ( 0 , . . . , 0, ed+l ' . . . ,en )} ' �1

X

In the next chapter we will introduce Grassmannian manifolds and a topology on the set of linear subspaces (with fixed dimension) of a given vector space. In our case it follows that a neighborhood of c + Po is given by the set of all planes P = c + P with P ffi (Ox e n - d+ l ) = e n . This shows that every ( d - l ) ­ dimensional plane through c that is near P(c, O)is of the form P ( c, A ) . We choose real numbers 0 < r1 < r2 and r > 0 so small that the "shell" S = {z = (

'

z ,z

"

) E ed - l e n- d+ l . r1 x

" ' < li z II < r2 and i z l < r }

is a relatively compact open subset of G. Only finitely many irreducible com­ ponents A , of A enter S . We can find a hyperplane H1 containing E that intersects the A, not at all or in codimension 1. Let A , be the finitely many irreducible components of H1 n A , that enter S. We can find another hyper­ plane H2 containing E that intersects all A , at most in codimension 2. We continue this procedure running through the irreducible components of the A, n H2 . After finitely many steps we have n - d + l hyperplanes such that their intersection P is a plane P( c, A ) that meets A n S in at most finitely

6. Regular and Singular Points

149

many points. By the above lemma we can choose this plane arbitrarily near

Po := P(c).

Now apply a linear coordinate transformation very near to the identity that leaves E invariant and maps our plane P (c, A) onto Po and replace the transformed shell S by a new shell S' (in the new coordinates) that is a little bit smaller such that S' is contained in the old (tran sformed) S. Thi s can be done so that 8S' n P0 n A 0. =

Earlier we proved that A is an embedded-analytic set in a neighborhood of the points of the intersection A n P0 n S' over a domain G' in the space of variables Zd- l + I , . . . , Zn . So there is a small closed ball B c cn- d+ l around the origin such that (cd - l x B) n 8S' n A remains empty, every irreducible component of (<e d- l x B) n S' n A enters P0 n S', and every plane through a point of B and parallel to P0 intersects A n S' in at most finitely many points.

Zd+ I , . . .

,

Z

n

S'

E

c

I I . . . .

0

A n Bo

Figure

11 1.3. Intersecting A

with B0 and Po

Zl , . . . , Zd-l

Now, the set B0 {z' E <e d- l : ll z' ll < r2} x B is a neighborhood of the origin in e n , and each of our parallel planes through points of B E meets A n B0 in a compact analytic set and therefore in at most finitely many points (cf. Figure 111.3). At all of these points the set A is locally an embedded-analytic set over B. By multiplying the pseudopolynomials by the same distinguished variable that we obtained for the different intersection points over the same base point in B - E, we have that An ( cd-l X (B-E) ) is -

150

Ill.

Analytic Sets

an embedded-analytic set over B - E . The coefficients of the corresponding pseudopolynomials over B - E are bounded along E . Hence, they can be analytically extended to B. This means that A n (B0 - E ) has a unique analytic continuation to B0 • • The proposition just proved is also true if A c G - E is an analytic set, whose irreducible components all have dimension greater than n - d, since we can write A as a finite union of pure-dimensional analytic sets. As a consequence we have the following theorem.

Assume that G c en is a domain, K c G an ( n - d)- dimensional analytic subset, and A an analytic subset of G - K all components of which have d imension > n - d. Th en the closure A of A in G is a n analytic set in G. 6 . 9 Theorem of Remmert-Stein .

PROOF : If zo E K is a regular point, then K can be transformed in a neighborhood of z o to a plane E. So A is analytic in a neighborhood of zo and therefore in all regular points of K. We can replace K by the set K1 of singular points of K, which is analytic again and has lower dimension. By the same argument we show that A is analytic at all regular points of K1 . • Continuing in this way we prove that A is analytic in G.

This theorem first was proved by

R.

Remmert and

K.

Stein; see [ReSt53] .

The Local Dimen sion . We show how our results are linked with the classical dimension theory of analytic sets.

Let G c en he a domain, A c G an analytic set, and Zo E A a point. There is an open neighborhood U = U(z0) c G such that U n A is a finite union of irreducible analytic components A 1 , . . . , At . If we choose U small enough, then the A, are uniquely determined. Definition .

In the given situation the uniquely determined number dimz0 (A) := max dim(A.>-) .\ = 1 ,

.

.. , 1

is called the (local) dimension of A at z 0. The set A has dimension 0 at z0 E A if and only if there is an open neigh­ borhood U = U(z0) c G such that U n A = {z0}.

Let k : = dimz0 (A) be positive. Th en k is the smallest number with the property that there are holomo rphic functions . . . , ik in a small neighborhood U of z0 such that z0 is isolated in A n N(JI , . . . , !k ) ·

6 . 1 0 Pr oposition .

h,

PROOF: If dimz0 (A) = k, then there must be at least one k-dimensional irreducible component Pt of A at z0 .

6 . Regular and Singular Points

151

If f is any holomorphic function near z0 , then either !l A' 0 (and therefore A n N ( f) still k-dimensional) or A' n N(J) is ( k - I)-dimensional. So at least k functions are required. _

On the other hand, by Lemma 6 . 7 we can find a holomorphic function h near z0 that does not vanish identically on any irreducible component A' of dimension k at z0• It follows that A' n N(h ) has dimension k - 1 for all such components A '. We can repeat this process, and after k steps we reach dimension zero, so that z0 is isolated in A n N ( h , . . . , fk ) . • Definition . If A has dimension k at z0 , then any system {h , . . . , fk } of holomorphic functions with AnN( h , . . . , fk ) = { z0} is called a parameter system for A in z0 .

6. 1 1 Ritt' s lemma. Let B C A be closed analytic sets in a domain G C en . Th en B is nowhere dense in A zj and only if dimz (B) < dimz (A) for every z E B . PROOF:

Let the criterion be fulfilled, and z0 be an arbitrary point of B. Then there exists an open neighborhood U of z0 in G and a parameter system { it , . . . , fk } on U for B at z0• Since dimz0 (A) > k, it is not possible that z0 is isolated in AnN( h , . . . , fk ) · This means that (A-B) nN(h , . . . , fk )nW -=1= 0 for every neighborhood W = W ( z0) . So B is nowhere dense in A. On the other hand, let B be nowhere dense in A, and z0 a point of B . In a small neighborhood U of z0 we have unique decompositions into irreducible components: BnU

=

B1

U

...

U Bm

and

A = A1

u

· · .

u

A1 .

Every component B; is contained in a component Aj(i ) , and for any open neighborhood W = W(zo ) we have (Aj (i) B; ) n W -=1= 0, because otherwis e, there would exist points z E W n B; where B is dense in A. So dim( B;) < dim(Aj(i ) ) for all i . It follows that dimz0 (B) < dimz0 (A) . • _

Let G c en be a domain, A c G an analytic set, and Z o E A a point. 1f dimz0 (A) = k , then the number n - k is called the codimension of A at z0. 6 . 1 2 Second Riemann exten sion theorem . Suppose that n > 2 and that the analytic set A C G has everywhere at least codimension 2 . Th en any holomo rphic function f on G - A has a holomorphic extension to G . PROOF:

We may assume that A is irreducible of codimension d > 2. If z0 is a regular point of A, then there is a neighborhood U of z0 such that U n A

152

III.

Analytic Sets

is biholomorphic ally equivalent to an open subset of a linear subspace E of codimension d. By the theorem on removable singularities (see Section II. l) f can be holomorphically extended to zo . We repeat this procedure. Beginning with the set Sing(A), which has codi­ mension d + 1 , after finitely many steps only a set of isolated points remains. Since f can also be extended to these points, we obtain the desired result. • Exercises 1 . Let A 1 , A2 c en be analytic sets. Show that dim(A 1 n A2 ) > dim( A! ) +

dim(A2 ) - n. 2. Assume that G c en is a domain and j a nonconstant holomorphic function on G. Prove that there is an at most countably infinite set Z c e such that A, := {z E G : f(z) = c ) is regular for c E e - Z. 3. Let G c en be a domain and A c G an analytic set. Prove that for any k > 0 the closure of the set A k : {z E A : dimz (A) = k } is either empty or is a pure k-dimensional analytic subset of G. 4. Let G c en be a domain and fl , . . . , fm holomorphic functions on G. Denote by N the common zero set N(fi , . . . , fm ) · Show that if the rank of the Jacobian of h , . . . , fm at some point zo E N is equal to r, then there is a neigborhood U U(z0) c G and a closed submanifold M c U of dimension less than or equal to n - r such that U n A c M . 5 . Let A be an analytic set in a domain G c en. For every point Zo E A the set lz0 (A) := { (J)z0 E Oz0 : f \ A 0} is an ideal in Oza · Show that there are a neighborhood U of z0 and holomorphic functions h , . . . , fk on U such that: (a) A n U = N ( h , . . . , fk ) · (b) lz0 (A) is generated by the germs (JI )z0 , • • • • (Jk )z0 • Show that the vector space n Tzo (A) = w E en l::: w vfzv (zo ) = 0 for every f E lz0 (A) =

=

{

v=l

has dimension n - rkz ( h ' . . . , fk ) . It is called the Zariski tangential space of A at zo. Let A be defined as in Exercise 5, and suppose that z0 is a regular point of A. Show that Tz0 (A) is the set of tangent vectors a(O) (see Section I. 7) , where a : I -+ en is differentiable, a(I) c A, and a ( O ) Zo . Let A be an analytic set in a domain G c en ' and let Zo E A be an arbitrary point. The embeddinf? dimension of A at zo is the smallest integer e such that there is an open neighborhood U = U(zo) and a closed submanifold M c U of dimension e with A n U C M . It is denoted by embdimz0 (A). Prove that z f--t embdimz (A) is upper semicontinuous on A, and that embdimz0 (A) dimc (Tzo (A)). Consider the analytic set A { w exp ( l /z ) } c {( w , z ) E e 2 : z -=!= 0}. Determine the closure of A in e 2 . 0

6.

}

=

7.

a.

=

=

=

C h ap ter IV

C omp lex M an ifold s 1.

T h e C omp lex Str u ctu r e

C omplex C oor din ates . Let X be a Hausdorff space, i. e . a topological space satisfying the Hausdorff separation axiom. Sometimes such a space is also called a separated space or a 2'2-space. Hausdorff spaces are the most common in topology (for example, every metric space is a Hausdorff space), but non-Hausdorff spaces do arise, in particular in algebraic geometry. The space en with the Zariski topology is not Hausdorff. ,

We think that a space X is too big if there exists a discrete subset with the cardinality of the continuum. Therefore, we demand that the topology of X have a countable base. In this case X is said to satisfy the second axiom
_

(x , g;� ( x)h ,. (x)p).

Since G is acting faithfully, it follows that ( C)

Two systems of transition functions (g;,_) and (g:�) with respect to the same covering are called topologically (respectively analytically) equivalent or cohomologous if there are continuous (respectively holomorphic) maps h< : U< --+ G satisfying condition ( C ) . Equivalent fiber bundles have equivalent systems of transition functions. On the other hand, it is easy to see that fiber bundles constructed from equiv­ alent systems of transition functions are themselves equivalent. Now we will demonstrate that the latter remains valid in passing to finer coverings.

2.

Complex Fiber Bundles

1 75

Let us assume that there are given systems of transition functions g' = (g:,.J and g" = (g:� ) with respect to a covering fl = { U< : L E I} . We call them equivalent as well if there is a refinement Y = { V, : v E N } of fl (with refinement map T : N -+ I ) and a collection of maps h, : Vv -+ G such that

To show that the systems are equivalent in the old sense, we define h< : U< -+ G by In fact, h, is well defined, since on U< n V,, we have

an d therefore g :� (v ) hv (g ;T (v ) ) - l get

The bundles are equivalent! If two bundles are given with respect to two different coverings, then they are

called equivalent if they are equivalent with respect to a common refinement, for example, the intersection of the coverings. Then everything works as above.

C omplex Vector Bundles . manifold.

Let X be an n-dimensional complex

Defin ition . A complex topological (respectively holomorphic) vector bundle of rank r over X is a topological (respectively holomorphic) fiber bundle V over X with
can be defined by

If g
For v E Vx and e E Fx the tensor product

v

®e

E ( V ® F) x is defined by

(v ® e) (.\) := .\ (e) . v.

180

IV.

Complex Manifolds

Lifting of Bundles. Let f : X -+ Y be a holomorphic map between complex manifolds and p : V -+ Y a vector bundle of rank T. Definition .

The

j* V : = X

lifted

bundle. or pullback. f*V over X is defined by

x y V :=

{(x , v ) E X

x V :

J (x) = p (v)}.

The bundle projection p : f* V -+ X is given by p(x, v ) : = x. The fiber of j * V over x E X is given by (f* V)x = Vf(x) · Therefore, the lifted bundle is trivial over the preimage sets f- 1 (y). One has the following commutative diagram: v

j*V -1-

p

X

�

-1-

y

p

If f/ = { U, E I } is an open covering of Y such that V is trivial over U, , then f/ := { U, = rl (U, ) : E I } is an open covering of X such that !* v �

: L �

L

�

is trivial over U, . If - := U, n U,. n U>- .

3.

Cohomology

1 83

A 0-dimensional cochain with values in V (with respect to � ) is a function s that assigns to every L E I a section s, E r(U" V ) . The set of 0-dimensional cochains is denoted by C0 ( � , V ) . Defin ition .

A !-dimensional cochain (with values in V )is a function � that assigns to every pair 1 L , K) E I x I a section �"' E r(Uu, , V ) .The set of !-dimensional cochains is denoted by C 1 (0/o, V ) . Finally, a 2-dimensional cochain (with values in V ) is a function A that assigns to every triple 1 L, K, v) E I x I x I a section A,, E r(U uw , V ) , and the set of all these 2 -dimensional cochains is denoted by C2 (� ' v ) . Assume that a 0-dimensional cochain s is given. One may ask whether the sections s, E r ( U, , V ) can be glued together to a global section s E r(X, V ) . For that it is necessary and sufficient that s, = s" on U," . This can be expressed in another way: If we assign to every 0-cochain s a 1-cochain 6s by ( Os ) , " := s" - s,, then s defines a global section if and only if 6s = 0. Defin ition .

The coboundary operators

are defined by · ­ .

(on u,l< ) '

A cochain s E C0 (� , V ) (respectively � E C 1 ( � , V ) ) is called a cocycle if 6s = 0 (respectively o� = 0). The sets of cocycles are denoted by Z0 (� , V) (respectively Z 1 (� , V) ) . Remark s

The sets of cochains and the sets of cocycles are all complex vector spaces. 2. We can identify Z0 (�, V ) with r( X , V ) . 3. A 1-cochain � is a cocycle if and only if the following compatibility con­ dition holds: �w = �'" + �� 2 and nilni+l for i 1 , . . . , s 1 such that =

-

The number r is called the rank of G, and zr the free part of G. The sum of the ZjniZ is called the torsion part of G . In many cases cohomology can be computed from homology:

If X is a compact good topological space (for example, a compact connected complex manifold), then Hq (X) is finitely generated for every q, and for q > 1 there is an isomorphism

3.5 Theorem.

H q (X) '"" (free part of Hq (X)) EB (torsion part of Hq _ 1 (X) ) .

The rank of H 1 (X) is called the first b1 (X).

Betti number of X , and is denoted by

Now we give an application of Cech cohomology methods to singular coho­ mology. Let X and Y be good topological spaces. 3.6 Theorem (Kiinneth formula) .

We choose open coverings 02/ = {U, : L E I} of X and )/ = {Vn n E N} of Y such that all the U,, Vv and all their pairwise intersections are connected. We write Wm := U, X Vn and Wm,l<m : = Wm n Wl<m = u,l< X Vnm · A cocycle '1/! E Z 1 (%' X 1/ , Z) is given by constant maps '1/Jm , l<m : Wm,l<m --+ z.

P ROOF:

We identify any cocycle � E Z 1 (%' , Z) with a cocycle [ E Z 1 (%' x 1/, Z) by [n ,�<m = �'" and also any rJ E Z 1 (1/, Z) with an fj E Z 1 (%' x 1/, Z). This ' induces natural injections

with Im(j1 ) n Im(j 2 ) =

{0}, and therefore an injective map

j : H 1 (%' , Z)

x

H 1 (1/, Z) --+ H 1 (%'

x

1/, Z)

by j ( a , b) := J 1 (a) + jz (b). Given a cocycle '1/J E Z 1 (%' x 1/, Z), for any L E I we define '1/J, E Z 1 (1/, Z) by ('1/J ,) n m : = '1/J m,,m · The cohomology class of '1/J, in H 1 (1/, Z) is independent of the i n dex We see this as follows. L

3.

Cohomology

191

(a) If Uu, f. 0 , then by the co cycle property we have 1/JLn,K,n + 1/JKn,Km + 1/JK,m,Lm - 1/'K,n ,Km 1/JLn,Kn - 1/Jim,Km ·

Setting ('Pu,)n := '1/!m .>i Zi . ·

i=l In addition, there is a neighborhood V = V(p) c U such that there exist minimal defining holomorphic functions fi for Zi in V. Then

di v

N

N

=l

i =l

(iIJ it'') = L ni . zi n

v

= Dlv

In this way every divisor is locally the divisor of a meromorphic function.

200

IV.

Complex Manifolds

Associated Line B undle s . Let X be a connected n-dimensional com­ plex manifold. If Z c X is an analytic hypersurface, then there is an open covering %' = { U, : L E I } of X with the following property: If U, n Z -=/= 0 , then there is a minimal defining function f, E O(U,) for Z. Setting f, == 1 if U, n Z = 0 , in Uu, we get the two relations

with suitable holomorphic functions

gu,

and g,"'" Then

Since f, does not vanish identically, it follows that g,"g'" 1 on U," . This shows that gu, E 0* (Uu-,) and g"' = g,"1 . Furthermore, on U,">-. we have the compatibility condition =

The system of the nowhere vanishing functions g," = f,j f" defines a holo­ morphic line bundle on X, which we denote by [Z}.It is easy to show that this definition does not depend on the covering and the functions k 4.8

Proposition.

1. Th ere is a section sz E f(X, [ Z] }vith Z = {x 2. [Z]is trivial over X Z .

EX

: sz (x) = 0} .

-

PROOF: The system of holomorphic functions f, defines a global section sz with {x E U, : sz (x) = 0) = {x E U, : f, (x) = 0) = U, n z . Then it is clear • that [ 41 - z is trivial.

We can generalize the concept of associated line bundles to the case of divi­ sors. If D is a divisor on X , then there is an open covering %' = { U, : L E I } of X , and meromorphic functions m, on U, with Di u, = div ( m, ) . It follows that the functions are nowhere vanishing holomorphic functions on U," . They define a line bun­ dle, which we denote by [D]Jf D = k . Z, then [D) [Z] k . If D = D1 + D2, then [ D ]= [D1] Q9 [D2] . Thu s the map 6 : 51( X) --+ Pic( X ) , is a homomorphism of groups. 4.9

Theorem.

Th e sequence v¢!(X ) *

o(D) d

==

isomorphy class of [D] ,

group homomorphisms

� 51(X) � Pic(X)

4.

Meromorphic Functions and Divisors

201

is exact. PROOF: ( l ) Let m f- 0 be a meromorphic function. Then [div(m)] is given by only one transition function mjm = 1 . Therefore, o o div(m) = 1 .

( 2 ) Let D be a divisor on X with J(D) = 1 . We assume that D i u, = div(m,) and [D] is represented by g,"' = m,jm"' . Since [ D ]ls trivial, there are nowhere vanishing holomorphic functions h, with h, . g,"' = 1 . h, on U,"' . Then h, . m, = h,

·

m,

Therefore, a meromorphic function m on X can be defined by ml u, : = h, . m, . Obviously, div(m) l u, = div(m,) = D i u, , and therefore div (m) = D . •

Meromorphic Sections . Let X be a connected n-dimensional complex manifold. Any analytic hypersurface Z c X leads to a line bundle [ Z jq,­ gether with a global holomorphic section sz that vanishes exactly on Z. The construction of sz fails in the case of an arbitrary divisor D and its associ­ ated line bundle [ D] Therefore, we introduce the notion of a meromorphic section. Definition. Let 1r : L --+ X be an analytic line bundle and A c X an analytic hypersurface. A holomorphic section s E f(X A L ) is called a meromorphic section over X in L if for every point x E X there is an open neighborhood U = U(x) c X , a function h E O(U), and a holomorphic section t E f(U, L ) such that: 1. h · s = t over U - A . 2. N(h) c A n U. -

,

If we have a system of trivializations rp, : 1r - 1 (U,) --+ U, x C and transition functions g,"' , then we have the following description of s .

For x E U, - A we define s,(x) by rp, o s(x) = (x, s, (x) ). Then s, = g,"' · s"' on U,"' - A. If we choose the U, small enough, there are holomorphic functions h, t, on U, such that h, . s, = t, over U, - A and N ( h, ) c A n U, . This means that s, is a meromorphic function on U, . We could have as well said that a meromorphic section is a system (s,) of meromorphic functions with s, = g"" . s"' . If Z is an irreducible hypersurface, then ordz(s,) = ordz (s"') , and we denote this number by ordz(s). The sum

div(s)

:= ZC X irreducible hypersurface

ordz (s) . Z

is called the divisor of the meromorphic section s.

202

IV. Complex Manifolds

Now we have the solution for our problem: Let D be the divisor on X given by D \ u div(m, ) . Then [D] is described by the transition functions 9u< = m,m"' 1 , and the system of the m, defines a global meromorphic section sv of [D]with div(sv) = D. ,

=

So far our definitions seem to be purely tautological. But for example, they allow us to determine the space of holomorphic sections of [D ]in terms of meromorphic functions on X . For that we need the following notation: If D�, D2 are two divisors on X , then D 1 > Dz if and only if D1 - Dz is a positive divisor. 4 . 1 0 Theo r em. Let D = 2: z n z . Z be a divisor on X . Th en there is a natural is omorphism {m E v¢!(X ) * : div(m) > - D) ----+ r(X, [D]) . ""

PROOF: Let sv be the global meromorphic section of [D]with div( sv) D . Then for any meromorphic function m on X also t : = m · s v is a meromorphic section of [ D}. =

If

m

is a meromorphic function with div(m) > - D, then div(t) = div(m) + div(sv) = div(m) + D > -D + D = 0.

This means that t is a holomorphic section. The map m H m · s v is obviously injective. Let t E r(X, [D] ) be given. If D \ u

,

=

div(mJ , then

Hence t,m-; 1 = t "' m"' 1 on U,"' ' and there exists a meromorphic function m on X with m\u, = t,m-; 1 . Therefore, div(m) \ u, = div(tJ - div(m,) > - D \ u, • So the map is an isomorphism. ·

Example

Let X

=

C be the Riemannian sphere and D

r(x, [D] )

=

=

n · 00 · Then

{ m E v¢!(X)* : ord00 (m) > - n and ordp(m) > 0 otherwise } .

The holomorphic sections in [D] are just the meromorphic functions on X that have a pole of order at most n at oo . Exer ci ses

1 . Let x be a point in a complex manifold X and let fx , 9x be nonunits in Ox . Prove that fx and 9x are relatively prime if and only if dim(N (f, g)) < n -

2.

5.

Quotients and Submanifolds

203

2 . Let G c en be a domain, A c G an analytic hypersurface, U c G an open subset with U n A -=!= 0 , and f : U --+ C a minimal defining function for A. Prove that Sing(A) n U = {z E U : f(z) = 0 and \7 f(z) = 0}. 3. Consider the meromorphic function m(z 1 , z2) := 2 212 1 on C2 . Show that the closure X of the graph { (z1 , zz , w) : z1 -=/= 0 and w = m(z 1 , z2 ) } in C 2 x C is an analytic hypersurface. Determine a minimal defining function for X at (0, 1 , oo ) . 4. Classify the singularities of m(z1 , zz) :=sin(z1 )1 sin(z1 z2 ) . 5. Let L --+ X be a holomorphic line bundle. Prove that L = [D ]for some divisor D on M if and only if L has a global meromorphic section s -=!= 0. 6. Let X be a compact Riemann surface. Show that for every nonconstant meromorphic function f on X the numbers of zeros and poles are equal (counted with multiplicity) .

5.

Qu otien t s a n d Sub manifold s

T op ological Qu otients. Let X be an n-dimensional complex manifold and '"" an equivalence relation on X . If x,y E X are equivalent, we write x '"" y or R( x, y ) . For x E X let X (x) := { y E X : y rv x} = { y E X : R(y , x ) }

be the equivalence class of x in X . These classes give a decomposition of X into pairwise disjoint sets. The set X1R of all equivalence classes is called the topological quotient of X modulo R. Let 1r : X --+ XI R be the canonical projection given by 1r : x H X (x) . Then XI R will be endowed with the finest topology such that 1r is continuous. This means that U c XI R is open if and only if 1r� 1 (U) c X is open. We call this topology the quotient topology.

A set A c X is called saturated with respect to the relation R if

5 . 1 P r op o si tio n . 1.

2. 3.

A saturated -{==} A = UxEA X ( x ) . If U C XIR is open, the-n 0 such that f i Bc ( D) nX - 0, then Pv l x = 0 fa r ev ery

5. 1 2 Lemma.

l/ .

PROOF:

Let

z

f. 0 be an arbitrary point of B6 (0) n X. Then 00

u=o

vanishes identically for I A I < I . So Pv ( z) = 0 for every v , and since X 1s • conical, P v I x 0 for every v . Now let F'r , . . . ,Fk be homogeneous polynomials in the variables zo , Then the analytic set �

X

:=

{z

�

E

en +

is a cone. If we set X ' := X set

X In U;

by

1

:

F1 (z)

= . . . = Fk (z) = 0 }

- {0 ) , then the image X

:= 1r (X ' ) c ]pm is the

= { (ZQ : . . . : Zn ) : F ( zo , . . . , Zn ) = = Fk (ZQ ,

= {(zo

1

.. . . . : z,)

=

. . . , z,.

·

•

·

· ··

,

Zn )

= 0}

•

z; f. 0}, we can define holomorphic functions hv zo Zn . , . . . Fv = ' ) J; ,v (zo : . . . : Zn : z, z, {x E U; : fi , 1 (x) = . . . = J; , k (x) = 0}, and consequently X

Then X n U; is an analytic set.

:

(

)

5. Quotients and Submanifolds

217

Defin it ion . An analytic set X c Jpm that is the zero set of finitely many homogeneous polynomials is called a (p rojective) algebraic set. The subsets X n Ui are called (affine) algebraic.

E

A complex manifold X is called projective algebraic if there is an N N and a holomorphic embedding j : X --+ lP'N such that j (X) is a regular algebraic set.

Every analytic set X in projective space is the zero set of finitely m any homogeneous polynomials F1 , . . . ,F8 such that if x X is a regular point of codimension d, then rkz ( F1 , . . . , F, ) = d fa r every z 1r - 1 (x) . 5 . 1 3 Theorem of Chow.

E E

PROOF: If X c lP'n is a nonempty analytic set, then X' = 1r - 1 (X) is also analytic. Since dimz (X') > J for all z c n +l _ { 0}' by the extension theorem of Remmert-Stein its closure X = X' u { 0} is analytic in cn + I .

E

By Theorem 6.5 in Chapter III we can find an open neighborhood U = U(O) c C n + I and finitely many holomorphic functions ft , . . . ,fm on U with N(ft , . . . , f ) = U n and rkz ( !I , . . . , fm) = d at any regular point z of m dimension n+1 -d in U n . Now we expand fi into homogeneous polynomials Pi ,v· Then Pi ,v l x = 0 for all i , v.

X

X

Let h c (')0 Hn + I be the ideal that is generated by all i = 1 , . . . , m . Since 11 C I, C . . . C 00 '""

Pi, v,

v < k,

is an ascending chain of ideals in a noetherian ring, it must become station­ ary. Thus there are homogeneous polynomials F1 , . . . , F8 such that every Pi v is a finite linear combination of the Fa . But then every fi is also a linear combination of the F,: ,

a=l It is clear that N ( ft , . . . , fm ) = N (F1 , . . . , F8) near the origin. But since X is a cone, even X = N ( F1 , . . . , F, ) . Setting �

A (z) = ( ai, a(z) we have

i = 1, . . . , m CY = 1 , . . . , s

)'

A (z) . J(F, , . . . , Fs ) (z) for z E X near 0. Therefore, d = rk, (!I , . . . ,Jm ) < rk, (F1 , . . . , Fs) . If X is regular of codimen­ sion d at x, then X is also regular of codimension d at every z E 1r- 1 (x) , J(f, , ... ,Jm) (z) =

218

IV. Complex Manifolds

because 1r is a submersion. It follows that rkz (F1 , . . . , Fs) cannot be greater than d at these points. Since the rank is constant along 1r - 1 ( x) , it must be • equal to d at every z E 1r - 1 (x) . Examples

1 . Let L C e n+ I be a complex linear subspace of codimension q . Then there are linear forms y l , . . . , yq On en + l SUCh that L

=

{z E e n+ I

:

tp 1 (z)

=

...

=

tpq (z)

=

0}.

Since the linear forms are homogeneous polynomials of degree 1 ,

IP' ( L)

:=

{ ( zo : . . . : z,) E IP'n

:

tpf.t ( zo , . . . , Zn )

=

0 for f..L

=

L

. . . , q}

is a regular algebraic set. We call !P'( L) a (projective) linea r subspace. It is isomorphic to IP'n - q . 2. We now show that the Grassmannian manifolds are projective �lgebraic. Let 0 < k < n be given, and N : = G) - 1 . We identify 1\ e n with eN+ l and define the Plucker embedding pl : G k . n --+ IP'N as follows: 1f a subspace v c e n has the basis { al , . . . , ak } , then

It is clear that this is a well-defined injective map . To see that it is a holomorphic immersion, we choose another descripton. As above, when we introduced the Grassmannian, we use the set of multi-indices

To any I E Y'k, n there corresponds a permutation matrix P r E GL n (e) such that for A E St(k, n) we get A . P I = (Ar iAr) . We define p : St(k, n) --+ e N + l - { 0} by p(A)

:=

(det Ar I I E Y'k , n ) ·

Then p(G . A) = det G . p(A) , so p induces a map p : G k, n --+ IP'N such that the following diagram commutes: p

-

St(k, n)

cl'k, n

p

5.

Quotients and Submanifolds

L

a 1 /\ · · · /\ ak =

I E..Yfk ,n

219

det AI - ei ,

and therefore p = pl. Let 'P I : UI --+ Mk , n - k (C) be a complex coordinate system for Gk , n · Then rp i 1 (B) = 7rk,n ((Ek iB) - P[ 1 ) and 1 1 p o rp [ ( B ) = Jr o p ((Ek i B ) · P[ ) = 1r (det ((Ek i B) · P[ 1 ) J :

J E

Y'k , n) ·

Obviously, p is a holomorphic map. Every matrix

E

B E Mk ,n - k (q can be written in the form B = (b� 1 , . . . , b�) +

with b, C k for p, = k + 1 , .. . , n . We have det ((Ek iB) . P/) I = 1 , and if J = (I- {iv }) U {p,} for some 11 { 1 , . . . , k } and p, {k + 1 , .. . , n } , then

E

E

and therefore det ((En iB) . P[ 1 ) J = bvw So p o rp / (B) contains the components 1 , b, for all 11, p, and some other components. It follows that pl is an immersion. Since Gk,n is compact, pl is an embedding.

Pr oj ective H yper sur fa ces . The simplest example of an analytic hy­ persurface in Jpm is the hyperplane H0 { z0 0). =

=

If S E Mn+I (q is asymmetric matrix, then qs (z) := z·S·z t is a homogeneous

polynomial of degree 2. The hypersurface Q s : = { ( zo : . . . : z,)

: qg ( zo , . . . , Zn) = 0}

is also called a hyperquadric. It follows easily from the classification of sym­ metric matrices that Q 8 "" QT (biholomorphic) if and only if rk(S) = rk(T) . In particular, every quadric of rank n + 1 is biholomorphically equivalent to the standard hyperquadric Qn -1 = { (zo : . . . : z, ) : z5 + · · . + z� = 0). Since Qn - 1 n Uo = (( 1 : t 1 : . . . : tn) : ti t . . . + t � = - 1 }, Qn - 1 has no singularity in Uo . The same works in every Ui , so Qn _ 1 is a projective algebraic manifold. Now consider the Grassmannian G 2 ,4 '-+ lP'5 . Since dim (G 2 4) = 4, it is a , hypersurface. From multilinear algebra one knows that a 2-vector w E f\ 2C 4 is decomposable (i.e., of the form w = a A b ) if and only if w 1\ w = 0. So G2 ,4 = {1r(w) : w12W34 - w13 W24 + W14 W2 3 = 0}

220

IV.

Complex Manifolds

is a hyperquadric that is isomorphic to Q4 . This means that there is a 1 to 1 correspondence between the set of proj ective lines in lP'3 (which is the same as the set G2 ,4 of planes in C4) and the set of points of a 4-dimensional quadric.

Every analytic hypersurface Z C lP'n single homogeneous polynomial. 5 . 14 Theor em .

a�

the zero set of a

�

PROOF: If Z c lP'n is an analytic hypersurface, then Z = 1r -1 (Z) U ( 0 ) c c n+ 1 is also an analytic hypersurface. Therefore, there exists an open neigh­ borhood U = U(O) c c n + 1 , a point z E U n Z ,and a holomorphic function f : U -+ C with \7 f(z) f. 0 and Z n U c N(f). Making U smaller if necessary, we can find a holomorphic function g dividing f such that U n Z = N(g) . Without loss of generality we may assume that u U' X U" c c X e n and g (z 1 , z') = Z � + ak- 1 (z')z� - 1 + · . . + ao (z') �

�

=

is a Weierstrass polynomial. If a�..z) = A • • F ( z ) . This means that F is a homogeneous polynomial of degree k, with sF = s . Consequently, we have the following result.

EN

For k the vector space f(lP'n , O(k) ) is isomorphic to the space of homogeneous polynomials of degree k in the variables z0 , . . . , zn . Analytic hypersurfaces in lP'n are exactly the zero sets of global holomorphic sections cf O(k) . there is a homogeneous polynomial For any Zo c n + l - ( 0 ) and any k F of degree k with F(z0) f. 0. Therefore O(k) is generated by global sections. 5 . 1 5 Pr oposition .

EN

E

The bundle 0(1) can be described in purely geometric terms. For this we note that if -1 Po := (0 : . . . : 0 : 1 ) lP'n 1 , then a projection 1r0 : JP'n + 1 - {Po} --+ lP'n is given by

E

7ro (zo : . : zn : Zn + l ) : = ( zo : . . . : zn ) We obtain local trivializations Yi : Jr() 1 (Ui) --+ ui X c by ·

·

(

·

)

lf!i ( Zo : . . . : Zn : Zn+ 1 ) : = ( Zo : . . . : Zn ) , Zn+ l Zi with lf!i 1 ( (zo : . . . : zn ) , c) = (zo : . . . : Zn : czi) Obviously this gives the 1 + !ran_sition functions Zj zi _If F . is a linear form on e n ! ' then the section sF IS g1ven at x = 1r ( z ) = �zo . . . . . z ) by n SF (x) = 'Pi 1 (x, z;- 1 F(z) ) = (zo : . . . : Zn : F(z) ) . ·

.

,

222

IV. Complex Manifolds

The Euler Sequence. Here we discuss the tangent bundle of the pro­ jective space in more detail. Let 1r : c n+ l - {0} ---+ ]pm be the canonical projection. Then for any point z E c n + l - {0} we have a linear map lf!z : c n+ l ---+ T7r (z) (lP'n ) defined by

lf!z (w ) [ j]

:=

d dt

0

(f o 1r(z + tw) ) .

The vector w can be interpreted as tangent vector 0:(0) = a. , o ( (djdt)o) with a(t) := z + tw. It follows that

7f, ,z (w ) [ f]

= ( 1r o

a) .,o(d j dt)o [f]

=

(dj dt)o [f o 7f o a] = lf!z (w ) [ f] .

Since 1r is a submersion and lf!z ( z ) = 0, the linear map lf!z is surjective with Ker( lf!z ) = Cz. Ifwe use the local coordinates tv : = zv / zi in Ui = { (zo : z1 : . . . : zn ) l zi f- 0} , then for any z with zi f- 0 it follows that

ZiWv - WiZv

�(

)

0

.

lf!z (w ) = i atv Zi2 'v# Thus a trivialization 'l/Ji : T(lP'n ) l u, ---+ ui X en is given by 'l/Ji ( 'Pz (w ))

:=

(

1r ( z )

,

with a , = w, - ( wiz;,- 1 )zv for 11 f- i.

� (aa , . . .

, iii , .

)

. . , an ) ,

Let 0(1 ) Eil ( n +l) be the direct sum of n + 1 copies of the hyperplane bundle. Recall that every linear form F on cn + l defines a global section SF of 0(1) by ( sF )i = z; 1 F. Then two vector bundle homomorphisms j : OJI'n = lP'n x C ---+ 0( 1 ) Eil (n+ l ) and q : 0 ( 1 ) Eil ( n+ l ) ---+ T(lP'n ) can be defined by

j ( X, ) C

:= C '

( S zo ( X ) , . · , S Zn ( X )) ·

and

q( sF0 (7r (z) ) , . . . , sFn (1r (z) )) : = lf!z (Fo(z) , . . . , Fn ( z ) ) , where Fa , . . . , Fn are linear forms on c n + l .

Using the canonical trivializations, j and q can be given over Ui by

j : ((zo : . . . : zn ) , c) H ((zo : . . . : zn ) , c · (zo z; 1 , . . . , zn zi 1 )) and

� . , a , , . . . , a, )) ,

q : ( ( zo : . . . : Zn ) , (Wo , . . . , w n )) H ( ( zo : . . . : Zn ) , ( ao , . with av = wv - (wiz;,- 1 )zv. This shows that both of these homomorphisms have constant rank. Since there is always an index i such that sz, ( x ) f- O, j is injective. Since 0(1) is globally generated, q is well defined and surjective. ·

5.

Quotients and Submanifolds

223

Defin ition . A sequence 0 ---+ V' � V 4 V" ---+ 0 of vector bundles called exact if the following hold: 1. j and q have constant rank. 2 . j is injective and q is surjective. 3 . Im (j ) = Ker(q).

5 . 1 6 Theorem .

is

We have a n exact sequence of vector bundles

The sequence is called the Euler sequence. PROOF: It is clear that q o j (7!' ( z ) , c ) = 'Pz ( c z) = 0. If, on the other hand, x = 7l'(z) and q(sF0 (x), . . . , sFn (x)) = 0, then (Fo (z) , . . . , Fn ( z )) must be a multiple of z . So there exists a c E C such that (sF0 (x), . . . , sFn (x)) (X ) , . . . , S Zn ( X )) . C ' (S • ·

zo

Ration al Fun ctions. We consider some connections with complex alge­ braic geometry. A meromorphic function m on ]pm is called rational if m = 0, or if there are homogeneous polynomials F and G of the same degree such that F f- 0 and

) m ( za .. . . . .. zn ) _ - F ( ZQ , , Zn . G(zo, . . . , Zn ) ·

5. 1 7 Theorem .

PROOF:

If

.

.

Every meromorphac functson on ]pm

ai

rational.

m E .4'(1P'n ) , then its divisor has a finite representation

with n, E Z and irreducible hypersurfaces Zi C lP'n . By the theorem of Chow every Zi is the zero set of a homogeneous polynomial F; of degree d, > 1 . Then F = = Il FJ'i is a rational function on cn+ l that is homogeneous of degree d := 2:: i n i di E Z . Since div(m o 7!' ) = div(F) on cn + l - {0 } , the function f := ( m o 1l' ) . F- 1 is

holom_!?rphic there witho_Et zeros. It has a holomorphic extension j to cn + l with f( O) f- 0, because f cannot have isolated zeros.

For c E C* , the equation f( c z ) = c - d . [( z) is valid on an open subset of c n+ l , and then (by the identity theorem) everywhere on c n + 1 . Thus d < 0 ·

224

IV. Complex Manifolds �

�

and c- d = l. From [( c . z ) = f (z) it follows that f ( z ) is a constant wo , and • m o 7l' ( z ) = w 0 . F ( z ) a rational function. submanifold X c Jpm is the restriction m i x of a rational function m on P". We have already seen that every polynomial p on U0 en can be extended to a rational function m on Jpm . Therefore, on every algebraic manifold there is a large number of rational functions.

A rational function on

a

rv

]pm

is a projective algebraic set, then A; := A n U; is called an affine algebraic set. A complex manifold is called an affine algebraic manifold if it is biholomorphically equivalent to an affine algebraic set. A regular function on an affine algebraic manifold j : X '---+ U; is a holomorphic function f : X ---+ e such that there exists a polynomial p on U; e n with f = p o j . It can be shown that a rational function on an affine algebraic manifold is always a quotient of regular functions. If A c

'""

In algebraic geometry there is a more general definition for regular functions, which coincides with our notation in the case of affine algebraic manifolds. On projective algebraic manifolds all regular functions are constant, whereas there are many rational functions. In the affine case the field of rational functions is exactly the quotient field of the ring of regular functions. If Z c ]pm is a hypersurface of degree k , then X : = algebraic manifold. We can see this as follows:

]pm

- Z is an affine

Let I be the set of multi-indices 11 = ( 110 , . . . 11n ) with 110 + · . . +11n = k. Then # ( I ) = (nk k) is the number of monomials z v = z�0 z�n , 11 E I. We set N : = # ( I ) - 1 and define the Veronese m ap vk, n : pn ---+ pN by ,

•

Vk, n ( zo : . . . : Zn )

:=

•

•

( Zv 1 11 E I) .

One can show that vk, n is an embedding, and so its image is Vk, n = vk, n (JPn ) an algebraic submanifold of pN . If p is a homogeneous polynomial of degree k with zero set Z, then there are complex numbers a, 11 E I, such that p = � vEl av z v . It follows that Vk,n (Z) = Vk,n n

{ (wv ) vEl : I:> vwv = 0 } vEl

is the intersection of Vk, n with a hyperplane H c pN . Therefore, pn - Z '"" Vk, n ( Pn - Z) = Vk, n n ( JPN - H ) is affine algebraic. Exercises

0}

1. Show that M : = { ( zo : z1 : zz : z3 ) E 1P3 : zS + · · · + z� = is a projective manifold. Consider the group G := {gm : 0 < m < 4}, where g : JP 3 ---+ JP3 is defined by

5.

Quotients and Submanifolds

Q

g(zo : ZI : zz : z3) : = (Qzo : Q2ZI : Q3z2 : Q4z3 ) ,

=

225

e2 1l' i /5

Prove that X := M/G is a complex manifold (called a Godeaux surface). 2. Let F : C 3 ---+ C 3 be defined by F(t, ZI, zz) := (t, azi

3.

+ tzz , azz) ,

a E JR, 0 < Ia I < 1 .

Set G = {F m : m E Z} and prove that X : = (C x (C 2 - {0}))/G is a corn­ pact complex manifold. Consider the holomorphic map 1r : X ---+ C that is induced by the projection onto the first component. Then Xt : = 7r - I ( t) is a 2-dimensional complex manifold. Show that Xt is biholomorphically equivalent to XI for every t f- 0, but that there is no biholomorphic map between Xo and XI . Show that Xo is a Hopf manifold. For WI , . . . , Wzn E e n define n := (wf , . . . , win ) E Mn ,2n (C) and prove that

WI , . . . ,wzn linearly independent

-{==}

det

(�) f- 0 .

Let To be the torus defined by the lattice r := Zwi t . . . + Zwzn · Prove that To is biholomorphically equivalent to To' if and only if there are matrices G E GL n (C) and M E GLzn (Z) such that f!' = G . f! · M . 4. The Segre m ap (J'n : lP'I x lP'n ---+ lP'2n + I is defined by

(J'n ( (xo : xr ) , (Yo : . . · : y)) := (XoYo : . . · : xoYn : x l Yo : . . · : xi Yn ) · Show that � n : = (J'n (lP' I x lP'n ) is a complex manifold and (J'n an injective

holomorphic immersion. 5. Show that for n > 2, two irreducible hypersurfaces in nonempty intersection. 6. Decompose A := { (zo : ZI : zz) : z6 + 3z6zi

lP'"

always have a

+ 2z6 + 4z6z� + 5z?z� + 3zi = 0}

into irreducible components. 7. Let d > 0 and n > 0. Prove the following theorem of Bertini. If Hn + I (d) is the vector space of homogeneous polynomials of degree d in z0 , . . . , zn , then there is a dense open subset U c Hn +l (d) such that N (F) = {(zo : . . . : Zn ) E lP'n : F(zo , . . . , Zn )

=

0)

is a projective manifold for every F E U. 8. Use the notation from the introduction to Grassmannian manifolds. For I E Y'k , n set Vr : = {A E St (k, n) : det Ar =/= 0) and Ur : = 1fk, n ( Vr) C Gk n · Then '

give the transition functions of a vector bundle Uk on Gk ,n · there are n independent global holomorphic sections in uk .

Show

that

226

IV. Complex Manifolds

9. Show that H 1 (ll1m , 0) 0 for n > 1 and H 1 (1P'1 , 0 ( k)) = 0 for k > 0. Prove that there is an exact sequence of vector spaces =

for n > 2 and k > 1 . Conclude that H1 (1P'n , O(k)) = 0 for n > 1 and

k > 0.

10. Let X be a compact complex manifold and 1r : V ---+ X a holomorphic vector bundle of rank r > 2 . Show that there is a holomorphic fiber bundle P ---+ X with typical fiber lP'r-1 and a line bundle L ---+ P such that Px lP' (Vx) and LIPx rv OpJ 1 ) . 1 1. Prove that for every automorphism f of lP'n (i.e., every biholomorphic m a p f : ]pm ---+ P") there is a n A E GL n +1 (C) with j(1r(z)) 1r(z · A t) . This is formulated as Aut( lP'n ) P G L n+ l (C) := GL n + 1 (C)/C* . 12. Let r = Zw1 + Zwz be a lattice in C and p the Weierstrass function for this lattice. Show that the map z H ( 1 : p(z) : p' (z)) induces an embedding rp of the torus T C/f into lP'2 . Determine the equation of the image rp (T) in lP'2 . Determine transition functions for rp*O( 1 ) . =

=

=

=

6.

Br anched Riem ann Dom ain s

Br anched Analytic C over ings. Let f : X ---+ Y be a continuous map between Hausdorff spaces. Then the image of compact sets is compact, and the preimage of closed sets is closed. Definition. The map f is called closed if the image of closed subsets of X is closed in Y. It is called proper if the preimage of compact subsets of Y is compact.

6 . 1 Pr op osition.

Iff : X ---+ Y is closed and A C X a closed subset, then f l A : A ---+ Y is closed. 2. Iff is closed, y a point off ( X ) , and U a n open neighborhood of the fiber f - 1 (y) , then there is a n open neighborhood W = W(y) C Y such that f - 1 ( W ) c U. 3. Let X and Y be additionally locally compact. (a) If every point y E Y has a neighborhood W such that f- 1 (W) ts compact, then f is proper. (b) Iff is proper, then f is closed. ( c ) Iff is closed and every fiber f - 1 (y) compact, then f is proper. 1.

PROOF:

(1) is trivial.

6. Branched Riemann Domains

227

(2) Let f be closed and U be an open neighborhood of some nonempty fiber f - 1 (y) . Then W := Y - f ( X - U) is an open subset of Y. If y were not in W , then there would be a point x E f - 1 (y) - U. This is impossible, and therefore W is an open neighborhood of y with f - 1 (W) C U. (3a) Let Y be locally compact and K c Y compact. It follows from the criterion that there are finitely many sets W1 , . . . , Wm such that f- 1 (K) is contained in the compact set f - 1 (WI ) U · · · U f - 1 (Wm) · Since f - 1 (K) is closed, it must itself be compact.

(3b) Let f be proper and A c X be a closed set. For every y E j(A) there is a sequence of points xv E A with lim, f(xv) = Y · The set N : = { y } U {f(xv) : v E is compact (using the local compactness of Y) , and consequently f- 1 (N) is compact. Thus there is a subsequence (xvf.t) converging to some point X E f - 1 (N) n A . It follows that y = f(x) E f(A) .

N}

(3c) Let f be closed and assume that every fiber is compact. For y E Y choose a compact neighborhood U of f- 1 (y) . This is possible, since X is locally compact. By ( 2) there exists an open neighborhood W of y in Y with f - 1 (W) c U. Then f - 1 (W) c U as well. As a closed subset of a compact set it is itself compact, and from (3a) it follows that f is proper.

•

Definition . A continuous map f : X ---+ Y between locally compact Hausdorff spaces is called finite if it is closed and if each fiber has only finitely many elements. Obviously, every finite map is proper. Conversely, if a proper map has only discrete fibers, then it must be finite. Definition . A holomorphic map 1r : X ---+ Y between n-dimensional complex manifolds is called a branched (analytzc) covenng ifthe following hold: 1 . The map 1r is open, finite, and surjective. 2 . There is a closed subset D c Y with the following properties: (a) For every y E Y there is an open neighborhood U = U(y) c Y and a nowhere dense analytic subset A c U with D n U c A. (b) 1r : X - 1r - 1 (D) ---+ Y - D is locally biholomorphic. The set D is called the critical locus. A point x E X is called a branch poznt if 1r is not locally biholomorphic at x. The set B of branch points is called the branch locus . The covering is called unbranched if B is empty. The branch locus B is nowhere dense in X . Now we consider a domain G c en and d pseudopolynomials w; (wi , z ; over G and define

228

IV.

Complex Manifolds A == {(w, z) E red X G

Let

7f :

:

w; (w;, z) = 0 for i = l , _ . , d} .

A ---+ G be the restriction of the projection

6 . 2 Pr oposition.

prz

: red X G ---+ G.

The map 1r is surjective, finite, and open.

PRooF:

( 1 ) The surjectivity follows from the fact that, for fixed z E G, each w; (w; , z) has at least one zero. ( 2) Let M c A be a closed subset and z0 a point of 1r( M) . Then there is a sequence (wv, zv ) in M with z, ---+ z0 . Since the coefficients of the wi are bounded near z0 , the components of the zeros w, are also bounded. Consequently, there is a subsequence (wv M ) of (wv) that converges in red to some w0 . It is clear that (w0 , zo) lies in M . Thu s zo belongs to 1r ( M ) and 1r is closed. ,

Since 1r has finite fibers, it is a finite (and in particular a proper) map. (3) It remains to show that 1r is open. Let (w0 , z0 ) E A be an arbitrary point. If V(w0) c red and W(z0 ) c G are open neighborhoods, we have to find an open neighborhood W'(zo) c 1r(A n ( V x W) ) . �

By Hensel 's lemma there are pseudopolynomials w:; and w:;* such that

Define

A : = {(w , z) E V x W : w; (w 1 , z) = · · · = w'd (wd , z) = 0} . Then A c A n (V x W ) ,and (7riA) -1 (z 0 ) = {(wo, z0 ) } . Since 1riA is a closed map, there is a neighborhood W'(z 0 ) c W such that A n (Cd x W ' ) = (n iA) - 1 (W ') c A n (V x W) . Since also KIA : A ---+ W is surjective, it follows that W' = 7r(A n (cd x W') ) c 1r (A n (V x W)) . • Thus 1r is open. Remark. If X c A is an irreducible component that is everywhere regular of dimension n, then the same proof shows that 1r l x i�also surjective, finite, and open. If D c G is the union discriminant set for A , then 1r : X ---+ G is a branched analytic covering with critical locus D.

Br anched Dom ain s. A continuous map f : X ---+ Y between Hausdorff spaces is called discrete at x E X if the fiber f - 1 (f(x) ) is a discrete subset of X . The map f is called discrete if it is discrete at every point x E X . A holomorphic map f : X ---+ Y between complex manifolds is discrete at x0 if and only if there is an open neighborhood U = U ( xo ) c c X with

f(xo) � f(8 U) .

6. Branched Riemann Domains

229

Now let M be an n-dimensional complex manifold. Definition . A branched domain over M is a pair (X, 1r) with the fol­ lowing properties: 1 . X is an n-dimensional connected complex manifold. 2 . 1r : X ---+ M is a discrete open holomorphic mapping. Let 1r : X ---+ M be a branched domain and x0 E X an arbitrary point. We can choose a coordinate neighborhood B of 1r (x0 ) in M such that B is biholomorphically equivalent to a ball in en and 7r ( Xo ) = 0 . Since 7r is discrete and open, there is a connected open neighborhood U = U(x0 ) cc X such that 1r(U) c B is open and Jr - 1 (0) n U = {x0 } . We assume that 1r(U) = B . Define j : U ---+ U x B by j(x) : = (x, 1r (x)). Then j is a holomorphic embed­ ding, and A : = j ( U) = {(x , z ) E U x B : z = Jr(x) }

is a regular analytic set in U x B . We have a factorization 1r = 7r o j , with 7r := pr2 IA · Let w1 , . . . , w, be complex coordinates near x0 in U. It follows from the results of Section 111.6that over a small neighborhood of 0 in B there are pseudopolynomials wi (wi , z ) such that A is an irreducible component of the embedded analytic set �

A = { (w , z ) : wi (wi , z ) = 0 for i = 1 , . . .

, n}.

�

If D c B is the union of the discriminant sets of the wi , then A (and therefore also A ) is unbranched over the complement of D. Therefore, if 1r : X ---+ M is a branched domain, then for every x E X there is a neighborhood U ( x) c X and a neighborhood W(f(x) ) c M with 1r(U) = W such that 1r l u : U ---+ W is a branched analytic covering. If M = en or M = Jpm , then a branched domain over M is also called a

branched R ie mann domain.

Definition . Two branched domains (XI,1rl ) (over M ) and (X2, 1r2 ) (over N ) are called equivalent if there are holomorphic maps rp : M ---+ N and ip : X 1 ---+ X2 such that the following diagram commutes: 7r l

x1 -1 M

�

N

Tor sion P oints . As an example we consider a polydisk P = pn (o, r ) c en ' a positive integer b, and the set

Xo : = { (z, w ) E P

x

e : wb - z 1 = 0 } ,

230

N.

Complex Manifolds

together with the projection 1l'o : (z , w ) H z. Then X0 is a regular analytic hypersurface in P x C and consequently an n-dimensional complex manifold, which is the Cartesian product of a polydisk P' c c n -1 and a neighborhood of the origin in the Riemann surface of w = ifZ. It is unbranched outside {z : z 1 = 0) . Definit ion . Let ( X ,1l' ) be a branched domain over some manifold M . A point x0 E X is called a torsion point (or winding p o int) of order b if there is an open neighborhood U = U(xo) c X such that 1l' : U ---+ 1l'(U) is equivalent to 1l'o : Xo ---+ P. In this case we say that 1!'1 u is a winding covenng. We consider the local case, i.e., a branched analytic covering 1l' : X ---+ Q over some polydisk Q c en around the origin, and we assume that it is branched of order b over 0 and unbranched outside { z E Q : z 1 = 0). Then X ' : = X-1!'-1 ( { z1 = 0}) is still connected, and the covering 1l' : X ---+ Q is equivalent to 1l'o : Xo ---+ P. For a proof see [GrRe58] , §2.5, Hilfssatz 2. One shows that the two coverings outside the branching locus have the same fundamental group. Therefore, they must be equivalent there. Using the theorem of Remmert­ Stein it follows that X and X0 are equivalent. The following is now immediate. 6 . 3 Pr op osition. Let (X, 1l') be a branched analytic covering over M , with b ranching locus D . If x0 E X is a point such that z0 = 1l'(xo) is a regular point of D , th en the re are neigborhoods W = W(xo) C X and U = U (z0) C M such that 1l' : W ---+ U is a winding covering. The proof follows simply from the fact that one can find coordinates such that locally the situation above is at hand, and then X is locally equivalent to Xo .

concrete Riemann surface is a branched Riemann domain 1l' : X ---+ C or 1l' : X ---+ lP'1 . It is an abstract connected Riemann surface, and in the latter case it may be compact. The infinitely sheeted Riemann surface of w = logz is not a concrete Riemann surface in our sense. Concrete Riem ann Sur faces. A

Here is a method for constructing concrete Riemann surfaces. Let X be a Hausdorff space, M = C or M = lP' 1 , and 1l' : X ---+ M a continuous mapping such that for every x0 E X there is an open neighborhood U = U(x0) C X , a domain V c C (in the case M = C ) , respectively a domain V in an affine part rv C of M (in the case M = lP' 1 ) and a topological map 'l/J : V ---+ U with the following properties: 1 . 1l' 'lj; : V ---+ C is holomorphic. 2. (1!' o 'lj;)' does not vanish identically. o

6. Branched Riemann Domains

23 1

The map '1/J is called a local uniformization. Obviously, (U, 'I/J - 1 ) is a complex coordinate system. We have to show that two such systems ( U1 , '1/!1 1 ) and (U2, '1/!2 1 ) are holomorphically compatible. We consider only the case M = C . The map

'1/!1 1 o 1/!z : 1/J2 1 (U1 n Uz) ---+ '1/J1 1 (U1 n Uz) is a homeomorphism. We denote the set W>-. 1 (U1 n Uz) by v; . Then D 1 := { t E Vt : (7r o '1/J I )' (t) = 0} is discrete in Vt , and therefore Dz := '1/J - 1 (D I ) C V2* is also discrete. Let t0 be a point of V2* - D 2 . Then '1/!

:=

(7r o '1/J I ) ' ( '1/J( t0 )) f- 0. Therefore, there are open neighborhoods U = U('lj!(t0 )) c Vt and W = W(z0 ) c C (for z 0 : = 7r('I/J2(t 0 ))) such that 7r o '1/J 1 : U ---+ W is biholomorphic. Then 7r l ,p, ( U) : '1/! 1 (U) ---+ W is a homeomorphism, and

'1/J = 'l,61 1 o (7r l ,p, (U) ) - 1 o 7r o 'I/Jz

=

(n Oij;l ) - 1 o (7r o '1/Jz) , on

V

:=

'1/J - 1 (U) .

Since V is an open neighborhood of to, we see that '1/J is holomorphic on V2* -D2. As a continuous map it is bounded at the points of D2 , and therefore it must be holomorphic everywhere in V2* . From the above it follows that X is a Riemann surface, and obviously 7r is a holomorphic map. We still want to see that it is a finite branched covering. If x 0 E X , then there is an open neighborhood U = U(xo) C X and a local uniformization '1/J : V ---+ U. We define : U ---+ V x 7r(U) C C 2 by

(x) := ('1/J - 1 (x) , 7r(x)). Then (U) = {(z, w ) E Vx 7r(U) : 7r o 'lj!(z) = w }. Restricting the projection (z, w) H w, we obtain a branched covering p : (U) ---+ 7r(U) that is equiva­ lent to 7r l u : u ---+ 7r(U). Let be f := 11' 0 1/J and Zo == w -1 (xo ) . If f'(zo) i- 0, then p is unbranched. If f' has an isolated zero of order k at z 0 , then

f(z)

=

7r(x0 ) + (z

_

z0) k · h(z),

where h is a holomorphic function with h(zo) f- 0. It is clear that then p is branched of order k, equivalent to the Riemann surface of ifZ.

Hyperelliptic Riem ann Sur faces .For g > 2 choose 2g + 2 different points z 1 , zz, . . . , zz9+I , z2g+ 2 in Xo := 1P' 1 . We assume that they are all real, in natural order, and not equal to oo .

Now we take two copies X1 , X2 of lP'1 regarded as lying above Xo and cut them along the lines £i between z2,+ 1 and z2i + 2 , i = 0 , . . . , g . We define

X.\ := {f E X>-. : lm(t) < 0 }

and

Xt := {t E X>-.

:

Im(t) > O}

232

IV. Complex Manifolds x2+

-

-

-

-
2 and ordp (m) > 0 otherwise } .

234

IV. Complex Manifolds

But such a function m is holomorphic and vanishes at oo , and conse­ quently is identically 0. Thus H 0 ( 1P' I , Kll'' ) = 0. 3 . Let X be a compact Riemann surface, and W I , wz E H 0 (X, Kx) two holomorphic 1-forms. In local coordinates w.\ l u, = wf l dt for ,\ = 1 , 2. Then f l u, := wi2l jwi ' l defines a global meromorphic function f on X such that wz = f . WI · '

"

Now we return to the hyperelliptic surface X . A holomorphic map j : X ---+ X is defined by (s,t )H (-s, t ) . Clearly, j permutes the two leaves, and j 2 = id. This map is called the hyperelliptic involution, and it induces a linear map j* : H0 (X, Kx ) ---+ H0 (X, Kx ) with (j* ) 2 = id. We get a decomposition of H0 (X, Kx ) into the eigenspaces associated with the eigenvalues ± 1 . But if j*w = w , then w = 7!' * ( cp) , where 'P is a holomorphic 1-form on lP'I . Since H0 (1P'I , Kll'' ) = 0, it follows that w = 0. Thus j*w = -w for every w E

H0 (X, Kx ).

The equation s 2 = F(t) implies that 2s ds = F' (t) dt. If s (t) = 0, then t = zk for some k. But F' (zk ) = IT;# (zk - z; ) =/= 0. Therefore, 2ds wo := s = F' (t) dt

is a holomorphic 1-form on X ' . Near oo we have s ( t) � t9 + I and therefore w0 � ( 1 /t) 9 - I d( 1 /t) . This shows that w0 is a holomorphic 1-form on X that has a zero of order g - 1 over oo . The same argument shows that the 1-forms w, : = t11 ( dt / s) are holomorphic on X , for v = 1 .. . , g - 1 . ,

•

Now let w be an arbitrary element of H 0 (X, Kx ) . Then there is a meromor­ phic function f on X with w = f · w0 . Since w0 has no zero over lP' I - {oo } , f is holomorphic there . And since j *w = -w and j*w0 = -wo , we have j* f = f . This means that f = h o 1l' for some meromorphic function h on lP'I that is holomorphic outside oo . Consequently, h is a polynomial, say of degree d. It has d zeros in C and a pole of order d at oo . It follows that f also has a pole of order d at every point above oo . Since w f . w0 is holomorphic, d < g - 1 . Therefore, w is a linear combination of the 1-forms =

dt s

dt

ts

'

...

'

It follows that dime H0 (X, Kx ) = g. 6. 5

Pr op osition. If X is a hyperelliptic R i emann surface, then the genus g = g (X) is equal to dim H0 (X, Kx ) . We will later see that the same result is true for arbitrary compact Riemann surfaces.

7. Modifications and Toric Closures

235

Exer c ises : X ---+ M be a branched analytic covering with critical locus D c M . Show that if there is a point y0 E D such that D has codimension 2 at y0 , then there is no branch point in 1r- 1 ( y0 ). 2. Let 1r : X ---+ M , and 1r1 : X ' ---+ M , b e branched analytic coverings and D, respectively D ', the critical locus. Prove that if X - 1r- 1 (D) is equivalent to X ' - (1r' ) - 1 (D' ) , then X is equivalent to X ' . 3. Consider the holomorphic map F : C 2 ---+ C 3 defined by 1 . Let

1r

(w, z t , zz) = F (h , tz) = ( h · tz, t i , t � ) . It is the uniformization of a 2-dimensional analytic set A c C3 . Show that A - {0} is a branched domain over C 2 - {0}. What arc the torsion points? Describe A as the zero set of a pseudopolynomial w (w, z ) . Is A also a branched domain in our sense? 4. Let F be a homogeneous polynomial of degree k and

{ (zo : . . . Zn Zn + ) E lP'n +l Z/{ + 1 - F(zo , . . . , zn ) = 0 } . Assume that D = {(zo : . . . : zn ) E lP'n : F(zo , . . . , zn ) = 0 } is regular of codimension 1 and prove that the canonical projection 1r : X ---+ lP'n X

=

:

:

:

t

:

is a branched Riemann domain. Determine the torsion points and their orders. 5. Define rp : 1P' 1 ---+ lP'n by rp(z0 : zt ) := (z0 : Z(t - lz 1 : . . . : zf) and prove that the image C : = rp(lP' 1 ) is a regular complex curve in lP'n . Define

{(p, H) E C Gn , n +l P E H}, where an element H E Gn , n + I is identified with a projective hyperplane in P". Show that the projection 1r = pr, : X ---+ Gn , n + I is an n-sheeted X :=

x

:

branched covering. 6. Let X c lP'n be a d-dimensional connected complex submanifold. Prove that there is an ( n - d - 1 )-dimensional complex projective plane E c lP'n not intersecting X , a d-dimensional complex projective plane F c lP'n with E n F = 0 , and a canonical projection 1r : lP'n - E ---+ F such that 1r l x : X ---+ F is a branched domain.

7.

M odification s and Tor ic C losur es

Pr oper M odifications . Assume that X is an n-dimensional connected complex manifold and A C X a compact submanifold. Is it possible to cut

out A and replace it by another submanifold A' such that X ' = ( X - A ) u A' is again a complex manifold? In general, the answer is no, but sometimes such

complex surgery is possible. Then we call the new manifold X ' a modification of X .

236

IV. Complex Manifolds Definition . Let f : X ---+ Y be a proper surjective holomorphic map between two n-dimensional connected complex manifolds. The map f is called a (proper) modification of Y into X if there are nowhere dense analytic subsets E c X and S c Y such that the following hold: 1 . f(E) c S. 2. f maps X - E biholomorphically onto Y - S. 3 . Every fiber f- 1 (y) , y E S, consists of more than one point. The set S is called the cente r of the modification and E = f- 1 (8) the exceptional set.

Assume that we have a proper modification define X

f

: X ---+ Y with center S and

:= {(y, x) E Y x X : y = f(x)}

Then X is an n-dimensional connected closed submanifold of Y x X. If K c Y is compact, then 1r- 1 (K)� = ( K x X) nX is a closed subset of K x f- 1 ( K) and therefore compact; 1r : X - 1r- 1 (S) ---+ Y - S is biholomorphic, with inverse y H (y, f-1 (y) ) ; and for Yo E S we have 7r- 1 ( Yo) = {(yo , x) : f (x) = Yo} rv f-1 (y0) . This is a set with more than one element. Therefore, we can generalize the notion of a proper modification in the fol­ lowing way. Let X and Y be two connected complex manifolds. A generalized (proper) modification of Y in X with center S is given by an irreducible analytic subset � X of the Cartesian product Y x X and a nowhere dense analytic subset S C Y such that 1.

2. 3. 4.

.5 .

�

1r :=pr 1 lx : X ---+ Y is proper. 1r- 1 (S) is nowhere dense in X . f - 1r- 1 tS) is a complex manifold. � maps X - 1r-1 (S) biholomorphically onto Y - S . X has more than one point over each point of S.

The center S is sometimes also called the set of indeterminacy. If X is a manifold, then with center S .

7 . 1 Pr op osition.

1r : X ---+ X

1r- 1 (S)

is an ordinary modification of X into

X

�

has codimension 1 in X .

PRooF: We consider only the case where X is regular. A proof of the general case can be found in [GrRe.55] .

Assume that x 0 E 1r-1 (S) is isolated in the fiber of 1r over Yo�:= 7r(x0 ) . Then there is a neighborhood U of xo in Y x X such that U n X is a branched covering over Y. Since yo lies in S , there must be at least one additional

7. Modifications and Toric Closures

237

point x E X in the fiber over y0 . But then in any small neighborhood 1 of y0 there are points y E Y - S with several points in 1r-1 (y) . This is a contradiction. Therefore, dim, Jr-1 (1r(x 0 )) > 0. This is possible if and only if rkx0 (7r) < n :=dim(Y), and this shows that 1r- 1 (S) in local coordinates is given as the zero set of det In: . Therefore, it has codimension 1 . •

Blowing Up . Let U = U(O) be a small convex neighborhood around the origin in Cn + l . We want to replace the origin in U by an n-dimensional complex projective space. If 1r : cn + l - {0} ---+ Jpm is the canonical projection, then every line Cv through the origin determines an element x = 1r(v) in the projective space, and x determines the line £(x) = 1r- 1 (x) U (0) such that Cv = £(1r(v)). Now we insert ]pm in such a way that we reach the point x by approaching the origin along £(x). We define

X := {(w, x) E U X lP'n

:wE

£(x) }.

This is a so-called incidence set (another example is considered in Exercise 6.5) . We first show that it is an (n + I )-dimensional complex manifold. In fact. we have

( w,1r(z)) E X

z f- 0 , and 3 ,\ E C with w = Xz 3 i with zi f- 0 and Wj = '114- · Zj for j f- i z f- 0 and ZiWj - wizj

U x Jpm , with X n (U Uo ) rv {(w, t) E u X en : Wj

Zi

= 0 for all i, j.

So X is an analytic subset of X

= Wo tj for j = l , .. . , n }.

In u X ui (for i > 0) there is a similar representation. It follows that X is a submanifold of codimension n in the (2n + I )-dimensional manifold U x lP'n . The map q : = pr 1 l x : X ---+ U is holomorphic, mapping X - q - 1 (0) biholomorphically onto U - {0} by q : ( w ,x) H w and q- 1 : w H ( w ,1r(w) ) . Obviously, q is a proper map. The preimage q - 1 ( 0) is the exceptional set { ( 0, x) : 0 E £( x)} = { 0} x lP'n . So q : X ---+ U is a proper modification. It is called Hopf's 0'-process or the blowup of U at the origin. If w # 0 is a point of U and (>- n ) a sequence of nonzero complex numbers converging to 0, then q- 1 (>-n w) = (>- nw , 1r(w)) converges to (0, 1r (w)). This is the desired property.

The Tautological Bun dle . We consider the case U = manifold F := { (w, x) E c n+ l x lP'n : w E £(x)}

c n+ I ,

i.e., the

238

IV.

Complex Manifolds

and the second projection

Then p - 1 (x) = £(x) is always the complex line determined by x. The manifold F looks like a line bundle over the projective space. In fact, we have local trivializations 'Pi : p - 1 (Ui) ---+ ui X c defined by 'Pi(w, x) : = (x, wi)· Clearly, 'Pi is holomorphic. In fact, it is biholomorphic with inverse map

So over UiJ ,

Hence F has the transition functions giJ = zi/ ZJ , and consequently, it is the dual bundle of the hyperplane bundle 0 ( 1 ) . It is denoted by 0( - 1 ) and is called the tautologzcal bundle, because the fiber over x E pn is the line £(x), which is more or less the same as x. Sometimes F is also called the Hopf bundle, because it lies in c n + l X pn and the projection onto the first component is Hopfs a-process (see Figure IV.3).

j : F ---+ pn c n+ l Ui X c n+ l by

Let

X

be defined by

j(w, x) := (x, w ) , and Ji : Ui X C ---+

J,(1r(z) , c) := ( 1r(z), : . z ) .

Then we have the following commutative diagram :

F / u; jt

Ui X (: n +l

�

ui c x

-1 Ji

rv

Ui X (: n +l

This shows that F is a subbundle of the trivial vector bun< � p n X c n+ l. In particular, it follows that r(lP'n , 0 ( - 1 )) = 0; i. e . , there is no global holomor­ phic section in the tautological bundle. There is an interesting geometrical connection between F and F'. Consider the point x0 : = (0 : . . . : 0 : 1 ) E pn+l and the hyperplane

Ho : = {7r(z) E pn+l : Zn + l = 0}. Then the hyperplane bundle F' = 0 ( 1) is given by the Jpm+ l - {xo} ---+ Ho rv lPm, with 7r + ( ZQ : . . . : Zn : Zn+ 1 ) : = ( ZQ : . . . : Zn : 0).

projection

1r+

7.

Modifications and Toric Closures

�

.

Figure I V .3 .

.

239

£(x )

'

Hopf's a-process

Here H0 is the zero section of 0(1). Removing the zero section gives a man­ ifold isomorphic to @"+' - { 0}. and the fibers of the hyperplane bundle correspond to the lines through the origin. Blowing up the point x0 gives us an additional point to each fiber and so a projective bundle F' over Jpm with fiber lP' 1 . But looking in the other direction, F' - H0 is nothing but the tautological bundle 0( -1) over the exceptional set of the blowing up.

Quadratic Transformatio ns . We consider the case n = 1. Let M be a 2-dimensional connected complex manifold and p E M a point. Let U = U(p) c M be a small neigborhood with complex coordinates z, w such that (z(p), w(p)) = (0, 0) . Let X c U x lP' 1 be the blowup of U at the origin. Then

Qp (M) := ( M - U ) U X = ( M - {p ) ) U lP' 1

is again a 2-dimensional complex manifold, with a nonsingular compact an­ alytic subset N = 1P'1 and a proper holomorphic map qP : Qp(M) + M such that q; 1 (P ) = N and qP : Qp(M) - N ---+ M - { p } is biholomorphic. We call Qp(M) the quadratic tran5fo rmation of M at p. Let F : M1 ---+ M2 be a biholomorphic map between 2-dimensional complex manifolds and F(p l ) = P2 · Let q1 : Qp1 (M1 ) ---+ M1 and q2 : Qp2 (M2 ) ---+ M2

240

IV. Complex Manifolds

be the quadratic transformations. Then there exists a biholomorphic map F : Qp1 (Ml ) ---+ Qp2 (M2 ) such that q2 o F = F o q 1 . This follows directly from the construction, and it shows that the quadratic transformation is a canonical process. �

In [Ho55] Hopf proved that every proper modification with a single point as center is a finite sequence of quadratic transformations. 5 We consider the quadratic transformation U

Then Q0(U)

= { ( z , w ) E C2

q

:

Q0(U) ---+ U, where

: l z l < r 1 and

lwl

< r2 } .

1P' 1 : ZTJ - w( = 0) = Ua U U1 , with {(TJ , z) E C2 : l zrJI < r2 and l z l < r l }, {((, w) E C2 lw(l < r 1 and lwl < r2 }

= {(z, w )(

: rJ)) E U

x

:

and W

Thus

q

=

ZTJ

1

( = TJ

and

-

on Ua n

U1

is given by

and

((, w) H (w(, w). Consider, for example, the curve C := {(z, w) E U : w 2 = z3 }, which has (TJ, z) H (z, zrJ)

singularity at the origin. Then

q - 1 (C - {o}) n Ua q - 1 (C - {O}) n U1

{(TJ, z) E Ua {((, w ) E U1

a

: z # 0 and z - rJ2 = 0}, : w # 0 and l - w(3 = 0}.

In U1 - Ua we have ( = 0. So q - 1 (C - {0}) lies completely in Ua , and its closure in Q0 (U) is the singularity-free curve C' := {(TJ, z) : z = TJ 2 }. We call it the strict transform of C. The map q : C' ---+ C is called a resolution of the sing ularities of C. One can show that for any curve in a 2-dimensional manifold the resolution of the singularities is obtained by successive quadratic transformations. A proof can be found, for example, in [Lau71] . If a 2-dimensional manifold M contains a compact submanifold Y rv lP' 1 , one can ask whether Y is the exceptional set of a blowup process (then one says that Y can be "blown down" to a point) . Here we give a necessary condition.

We determine the transition functions for the normal bundle of the fiber Fa := q - 1 (0) in Q0 (U) . The divisor Fa is the zero set of the functions fa : Ua ---+ C with fa (rJ, z) : = z and h : U1 ---+ C with fl ((, w) := w . Therefore, the associated line bundle [Fa] on Qa is given by the transition function 5

In algebraic geometry this was already known

century, for example by the italiens

at the beginning of the twentieth G. Castelnuovo and F. Enriques.

7.

Modifications and Toric Closures

( z 9oi ( z, w, (( : TJ )) = -:-: = -:-:TJ

on

241

Uo n U1 .

But ( are the homogeneous coordinates on Fa 1P'1 . S o NQo (Fa ) = [Fo] I Fo is described by the same transition function the tautological bundle 0 ( -1) It follows that a submanifold Y lP'1 in a complex surface M can be excep­ tional only NM(Y ) 0(-1). We give a topological interpretation of this criterion. We set so := 1 on Uo and which defines a meromorphic section s in 0 ( - 1 ) that has no zero and only one pole at = (0 1).We can use this section to calculate the self­ intersection number Y . Y. Since more knowledge about topology is needed for the complete calculation, we can give only a sketch. The self-intersection n umber of a compact submanifold Y c M can be de­ scribed follows. Take a copy Y' of Y (which need be only a topological or piecewise differentiable submanifold of M ) such that it intersects Y tr n ver­ sally. Then the intersection Y n Y' is a zero-dimensional submanifold, i.e., a finite set. Now count the number of these intersection points respecting orien­ tation. The result is the self-intersection number. Furthermore, one can show that if Z c NM ( Y ) is the zero section, then Y . Y = Z Z, and the number Z Z can be calculated by intersecting Z with some not identically vanishing (continuous) section Z' . We obtain such a Z' in the following way. Let s be the meromorphic section in N = NM ( Y) mentioned above. Choosing appro­ priate coordinates near we can there write N = { (z, w) C2 : lzl < 1) and s ( z) = (z , 1 / z) (since s has a pole of order one). For 0 < E < 1 and lzl = E we have s ( z) = ( z,z/c2 ) . Then we define s by for lzl > E, s ( z) s(z) : = lzl < E. (z, 2: ) So s is a piecewise differentiable section, homologous and transversal to the zero section with one zero of order 1. Since gives an orientation opposite to that defined by z, we have the self-intersection Y . Y = - 1 . If y lP' 1 is exceptional, then y . y = -1. It is a deeper result that the converse of this statement is also true (cf. [Gr62]). M onoidal Tr ansfor mations. We consider a domain G c en and holomorphic functions fo , . . . , fk on G such that A := N(J0, . . . , fk) G is a singularity-free analytic subset of codimension k + 1. Then we define k+ f (fo, . . . , fk ) : G ---+ C 1 and ,

rv

as

rv

if

rv

oo

:

as

a s

·

·

oo,

€

for

z

rv

c

:=

242

IV.

Complex Manifolds

{(w, x) E G x lP'k : f(w) E £(x)} {(w, 7r ( z)) E G x lP'k : zdj (w ) - Zj ji(w)

X

= 0 for i , ]= 0, . . .

,k }.

The map p : = pr 1 l x : X --+ G is holomorphic and proper. For wo E G - A we have p -1 (wo) = { wo} x lP'k , whereas p : X -p- 1 ( A) --+ G- A is biholomorphic with p- 1 : w H (w, (fo(w) : . . . : fk (w))) . The set X is called the monoidal transformation of G with center A. The exceptional set E( f) : = p - 1 ( A) is the Cartesian product A x lP' k . In the case k = n - 1 and fi = zi for i = 0, . . . , k this is the a-process. One can show that the monoidal transformation is also a canonical process. Therefore, it can be generalized to an n-dimensional connected complex man­ ifold M and a submanifold A c M of codimension k + 1 . Then the monoidal transformation of M with aner--A is an n-dimensional manifold M with a hypersurface E c M such that M - E rv M - A. The exceptional set E is a fiber bundle over A with fiber lP'k . The fiber over x E A is the projective space lP'(Nx), where N = NM ( A) is the normal bundle of A in X . The divisor E determines a line bundle L : = [E] on M with L !JP'( Nx) = OIP'(Nx ) (- 1) . We leave the details to the reader (see also Exercise 5.10). -

Mer omor phic M ap s . Let X be an n-dimensional complex manifold and f a meromorphic function on X with polar set P and S c P its set of indeterminacy points. Then f is holomorphic on X - P , and we can extend it to a holomorphic map f : X - S --+ 1P'1 by setting f( x ) : = oo for x E P - S. The set Gt : = { (z,y) E (X - S ) x 1P' 1 : y = f(x) } is called the graph of X X lP' 1 •

f.

Let Gf c X

x 1P'1

be the closure of the graph in

7.2

Prop o sition. Th e graph Gt is an irreducible analytic subset of X x 1P'1 and defines a proper modification of X in 1P'1 with S as set of indeterminacy. PROOF: Let xo E P be an arbitrary point. Then there is a neighborhood U = U(x0 ) c X and holomorphic functions g , h : U --+ C such that f( x ) = g(x)jh(x) for x E U - P . We can assume that 9x and h, are relatively prime for x E U. Then P ll U = { Z E U : h(x) = 0 } and S n U = { x E U : g ( x ) = h(x) = 0} . Hence P is a hypersurface and S has codimension 2 in X . Then dim( S x lP'1 ) < n and Gf c (X x lP' 1) - ( S x lP' 1 ) is an n-dimensional analytic set. B y the theorem of Remmert and Stein its closure G f in X x 1P' 1 is also analytic. In fact, we have The map 1l' : = pr, : Gf --+ X is obviously a proper holomorphic map, with 7l' - 1 (x) {x} x JID1 for 3: E S and =

7. Modifications and Toric Closures

1!'- l (x) =

{ (X,(x, (( 1 : f1 (z) ) ) ) ) Q

243

for x E X - P, for X E p - S.

:

So 1l' : G1 - 1!'- 1 (S) ---+ X - S is biholomorphic, and G1 defines a proper modification of X in lP' 1 with indeterminacy set S. As usual, X is assumed to be connected. Then Reg ( G1) is also connected, and G1 irreducible. •

Now we consider the notion of a meromorphic map. To develop the correct point of view, we start with a very special case. Let

I(Jim

:=

JP' l

X . . . X m

JP'l

times

be the so-called Osgood space. If h , . . . ,fm are meromorphic functions on X without points of indeterminacy, then they define a holomorphic map f = ( !I , . . . , fm ) : X ---+ I(Jim . 1f fi has a set Si as set of indeterminacy, we define S := 81 U . . U Sm . Then f is a holomorphic map from X - S into I(Jim . Again we have its graph: ·

Gr = { (x ; yl , · . . , ym ) E ( X - S ) x I(Jim : Yi = fi(x) for i = l , . . , m}. For x0 E S we define Tr(xo) := {y E I(Jim : 3 xv E X - S with x, ---+ x0 and f(xv) ---+ y } . Then Tr(x0 ) is a nonempty compact subset of I(Jim . Setting Tr(x) := {f(x) } for x E X - S , we get a map Tf from X into the power set P ( I(Jim) .6 We define Gr { (x, y) E X i(Jim : y E Tr(x)}. One sees easily that Gr is the topological closure of Gr in X x I(Jim . We cannot apply the Remmert-Stein theorem to show that Gr is an analytic set, but .

:=

x

locally it is an irreducible component of the analytic set A with

{(x ; (z6 : z i ), . . . , (z[J" : z;:")) E U x i(Jim : z(tg 1 (x) - z�h 1 (x) . . zlJ"gm (x) - zfhm (x) 0 } , where fi = gi/ hi on U - Pi, for i = 1 , . , m. We could carry out the simple =

·

=

=

. .

proof here, but we refer to [Re57] , Satz 33.

Definition. A meromorphic map between complex manifolds X and Y is given by a nowhere dense analytic subset S c X , a holomorphic map f : X - S ---+ Y, and an irreducible analytic set X c X x Y such that

f o pr 1 = pr2

6

�

on X n ((X - S) x Y)

and 1l' : = pr 1 1 x : X ---+ X is proper and surjective. The set S is called the set cf indeterminacy of f. We also write f : X ---' Y. A s usual, the

p o w e r set

P(M) of a set M is the set

of all subsets of M .

IV. Complex Manifolds

244

We define TJ : X --+ P(Y) by nonempty compact set, and

T1 (x) := pr 2 (n- 1 (x)) .

Then

TJ

(x)

is always a

X = {(x, y) E X x Y : y E TJ (x) } . For x E X - S we have n- 1 (x) = {(x, f(x))}, and for x E S we have n - 1 (x) = X n ( { x} x Y) .This is a compact analytic set. If S is minimal, then n : X ---+ X is a proper modification with center S. �

�

So alternatively we could have defined a meromorphic map between X and Y to be a map T : X --+ P(Y) such that:

1. 2.

X := { (x, y) X x Y.

n :=pr 1 lx

E X

x

Y :

y

E

T

(x) }

is an irreducible analytic subset of

�

: X ---+ X is a proper modification.

This definition is due to Remmert. A meromorphic map is called surjective if pr 2 (X) = Y . If f is a surjective meromorphic map between X and Y and g a meromorphic map between Y and Z, then it is possible to define the composition g o f as a meromorphic map between X and Z (cf. [Ku60] ) . But it is not a composition of maps in the set-theoretic sense ! Any meromorphic function f on X defines a meromorphic map between X and 1P' 1 , any m- t uple ( h , . . , fm ) of meromorphic functions a meromorphic map between X and I(Jim . .

7.3 Proposition. S. Then f - 1 : Y indeterminacy. PROOF:

Let f : X -+ Y re a proper modification of Y with center X is a surjective meromorphic map with S as set cl

--"

We use the analytic set y

= f(x)} = { (y, x) E Y X : x E r 1 (y)} . Then f - 1 : Y - S --+ X is holomorphic with f - 1 o pr 1 (y, x) = j- 1 (y) = x = pr 2 (y, x) for (y, x) E Y n ( ( Y- S ) x X ) . Obviously, n := pr 1 1¥ : Y ---+ Y is a proper holomorphic map. It is surjective, since f is surjective. Finally, we • have pr2 (Y) = X . So f- 1 defines a surjective meromorphic map. �

Y : = {(y, x) E Y X

�

X

:

X

�

Toric Clo sures. Let X be an n-dimensional complex manifold. A clo­ sure of X is an n-dimensional compact complex manifold M such that X is biholomorphically equivalent to an open subset of M . We are interested in closures of X = e n . In the case n = 1 the situation is simple. Assume that we have a compact Riemann surface M that is a closure of X = C. Then we define f : M - {0} ---+

c by

7. f(z) : =

{

Modifications and Toric

Closures

245

for z E C - { 0}, otherwise.

1/z 0

The function f is continuous and is holomorphic at every point z with f(z) f0 . By the theorem of Rad6-Behnke-Stein-Cartan (cf. [Hei56] ) f is everywhere holomorphic. But then the zero set N(f) is discrete. We can consider f as a meromorphic function on M with exactly one pole of order one at z = 0. So f can have only one zero (cf. [Fo81 ] , Section 1 .4, Proposition 4.24 and corollary), and M must be the Riemann sphere lP' 1 , the well-known one-point compactification of C. In higher dimensions one can find several closures of e n . In [Bie 33] an injective holomorphic map (3 : e2 ---+ e2 is constructed whose functional determinant equals 1 everywhere and whose image U = (3 (e 2) has the property that there exist interior points in e 2 - U. We can regard U as an open subset of lP'2 . Then lP'2 is a closure of e 2 rv U, but this closure contains interior points in its boundary. We want to avoid such "pathological" situations, even if they may be interesting for those working in dynamical systems. Definition. A closure of e n is a triple (X,U, ) where: 1 . X is an n-dimensional connected compact complex manifold. 2. U c X is an open subset with U = X . 3. : e n --+ u is a biholomorphic map. The closure (X,U, ) is called regular if for any polynomial p the holo­ morphic function p o - 1 extends to a meromorphic function on X .

7.4 Proposition. If (X, U, ) is a regular closure ct en , then Xoo : = X - U is an analytic hypersurface in X . PROOF: Let fi be the meromorphic extension of Zi o - 1 , for i = 1 , . . . , n . The polar set Si of f; is a hypersurface in X . We set P : = P1 U . . . U Pn and claim that P = X - U. One inclusion being trivial, we consider a point x0 E X - U. There is a sequence of points xv E U with limv oo x, = xa . If we write 4 := - 1 (xv ) as 4 = (z iv1 , . . . , zi;'l ) , there must be an index k with l zk"l l ---+ oo . So l fk (xv) l ---+ oo for v + oo . This means that x0 E Pk C P. • --+

--

Example The manifold X = lP'n is a closure of e n . As open subset U with U = X we can take the set U : = Uo = {(zo : . . . : zn ) I zo f- 0}. Then 0 : en --+ U with o(h , . . . , tn ) : = ( 1 : h : . . . : t n ) is biholomorphic. We have seen in Section 5 that any polynomial on U extends to a meromorphic function f on Jpm with

246

IV.

Complex Manifolds

polar set H0 = { (z0 : . . . : zn ) I zo = 0 }. So (lPm , Uo , o) is a regular closure of e n , with Xoo = Ha . Another example is the Osgood space I(Jin , which contains e n via (a :

(h , . . . , t n ) H ( ( 1 : h ) , . . . , ( 1 : tn ) ) .

Also, (I(Jin , e n , (a) is a regular closure, but here

� = ({ m } x lP'1 x . . . x lP' 1 ) U · · . U (lP' 1 x . . . x lP' 1 x {m}) is an analytic hypersurface with singularities. The complex Lie group Gn @* X . . . X e* (n times) acts in a natural way on e n and is contained in e n as an open orbit of this action. Definition . A complex manifold X is called a toric variety or to rus embedding if G n acts holomorphically on X and there is an open dense subset U c X biholomorphically equivalent to Gn by g H gx for each x E U. Remark. In algebraic geometry the group G n is called a torus. This is the reason for the notion "toric variety." A "variety" is an irreducible algebraic subset of en or lP'n . It may have singularities. Here we do not allow singular­ ities, but we allow arbitrary manifolds that do not have to be algebraic. More information about toric varieties can be found in [Oda88] , [Fu93] , and [Ew96] . We wish to restrict the closures of en as much as possible. If the axes of en can be extended to the closure, then the action of Gn should also extend to the closure. Definition . A closure ( X ,U, (a) of en is called a to ric closure if the action of G n on e n extends to an action on X such that (g · z) = g · ( z ) for g E Gn and z E en . Examples 1 . The projective space Jpm is a toric variety with g ( zo : . . . : Zn ) := ( zo : gz 1 : . . . : gzn ) ·

and G = (e* )n c en = U0 c lP'n. Since the embedding en = U0 '---+ lP'n n is "equivariant," lP'n is a toric closure of en . 2 . A further simple example is the Osgood space I(Jin. Here the to ric action is given by 1 . . ) ) .. = ( (z01 .. gz 1 ) , . . . , (zon .. gzn ) ) . g . ( ( z0 . z 11 ) , . . . , ( z0n . z n 1 1 1

7.

Modifications and Toric Closures

247

3 . Let Ad c N0 be the set of multi-indices 11 with !11! :::; d and define d : Gn ---+ lP'N by d(z) := (z" : 11 E A d ) . Then Xd := d(Gn ) C lP'N is the image of lP'n under the Veronese map vd. n with

(

)

Vd , n (zo : z 1 : . . . : zn ) = zg - lv l z v : 11 E Ad . Obviously, Xd is a toric closure of C'

7.5 Prop osition. Let ( X ,U, (a) be a regular closure oj en and f : Z ---+ X a proper modification with center S c X,. Th en (Z, f- 1 (U) , f - 1 o (a) is again a regular closure. PROOF: It is clear that (Z, f -1 (U), f - 1 (a) is a closure. Now let p be a polynomial. Then the holomorphic function p o - 1 : U ---+ e has a mero­ morphic extension fi on X . But then fi o f is a meromorphic extension of p u- 1 0 )- 1 = (p o - 1 ) 0 f. • o

0

Now let ( X ,U, (a) be an arbitrary regular toric closure of en. The meromor­ phic extensions fi of zi o- 1 define a meromorphic m!lpping f = (h , . . . , fn ) : X ----" on and therefore a generalized modification X of X in on . We know that X is an irreducible component of the analytic set �

A = { (x; z1 , ·

. .

, z,) E X X

where we write zi E fi = (!& : !1) .

lP' 1

o n : zb f: (x) - zUMx) = 0 for i = 1 , . . . , n } ,

in the form zi = ( zb

z� ) and fi in the form

X

The two projections 1r := pr 1 l x : X ---+ X and q : = pr 2 l x : ---+ 0" are proper holomorphic maps. It may be that X has singularities over the indeterminacy set S of f. Nevertheless, we consider X as a closure of en. We take fJ : = 1r-1 ( U) as dense open set, and the biholomorphic map ( 0 and div (f) = D . A divisor D on a complex manifold X is called a principal div isor if there is a meromorphic function m on X with div(m) = D. If D > 0, then m is holomorphic. So we are interested in conditions on X such that every divisor is principal. From the exact sequence .4(X)* � 5?(X) � Pic(X) it fol­ lows that Ker ( 6) = 51( X) is a necessary and sufficient condition. This is, for example, the case if every analytic line bundle on X is trivial. Due to Oka's principle, if X is Stein, it is sufficient that every topological line bundle be trivial. One can prove that the topological line bundles on X are classified by elements of H 2 (X, Z). Here we will show directly that the vanishing of H2 (X, Z) is sufficient for the solvability of the generalized Weierstrass prob­ lem. Definition. Let X be a complex manifold. A Cousin-1 1 distribution on X consists of an open covering 'P£ = { U, : L E I } together with meromorphic functions f, f- 0 on U, such that on U"" there are nowhere vanishing holomorphic functions g,"' with f, = g,"' . f"' . A solution of the Cousin-Il distribution is a meromorphic function f on X such that flu, = h-;: 1 . f, with a nowhere vanishing holomorphic function h, on U, . A Cousin-Il distribution is therefore a meromorphic section in an analytic line bundle L that is defined by the transition functions g,"'. If L is trivial, then there are nowhere vanishing holomorphic functions h, on U, with g,"' = h,h"' 1 , and fl u, : = h-;: 1 . f, defines a solution. The Cousin-Il distribution uniquely determines the cocycle Z 1 ('P£ , 0* ). If f is a solution with flu, = h-;: 1 . j, , then

'Y

=

(g,"')

E

1 . Stein Manifolds

255

The following shows that the converse also holds.

Let 'Y E Z 1 (%' , 0 * ) be the cocycle of a Cousin-11 distri­ distribution is solvable if and only if 'Y E B 1 (%' , 0 * ) .

1 .7 Propo sition.

bution.

The

If H1 ( X, 0 * ) = 0, every Cousin- II distribution on X has

a

solution.

Unfortunately, Theorem B cannot be applied to H1 (X, 0 * ) . So even for Stein manifolds we need an additional topological condition.

Chern Class and Exp onential Sequence . Let X be an arbitrary n-dimensional connected complex manifold. We choose an open covering %' = { U, : L E I } of X such that the U, and all intersections U"" are simply connected and construct an exact sequence of abelian groups:

H 1 (%', Z) � H 1 (%', 0 ) � H1 (%' , 0 * ) � H2 ( %', Z ) 1 . If � = (�u '-+ (z1 , . . . z . The restriction wi e may contain multiple factors. We throw away such superfluous factors, so we can assume that every wi e is free of multiple factors. =

l

Let us first assume that the union of the discriminant sets of the pseudopoly­ nomials over Ji consists only of the point 0' and that A' is irreducible and has s sheets over £. Then A' is the Riemann surface of V'z. We write A' in the form A ' � ( ( t ;a l z, . . .

,an-d Z) : z = t8 }. ,an-dt8 ) we have a localt parametrization of A '. Then By F(t) == (tll 1 t8 , (p o F)a (t) = H(p o F, t) is given again by J H (p, F(t)) . J, where = ( 1 ;a1 , . . , an- d) t denotes the Jacobian of the holomorphic map F . The proof is the same as in the case of a biholomorphic map F . So ( p F)a > and •

•

J

•

.

.

:= PI A'

p o F is a subharmonic function of t. We get p'

-

o

c on A '.

0,

The same is true if A' is not irreducible but has 0' as the joint discriminant set, since 0 is the only point of A ' over O'. Now assume that the union of the discriminant sets is general. Every point in A' can be connected with 0 . We introduce the subset K of all points x E A ' with = c. If K0 # A ', there is a boundary point of K in A '. We know that A' is an embedded-analytic set. Then there is a neighborhood U ! C A' that is embedded-analytic over a disk B' (zr ) C Ji around a point z1 E B such that over z1 the only point of U is x1 and the union discriminant set consists of z1 only. Then we get c (by the same argument as above), which is a contradiction to the property "boundary point." So p' - c follows.

p' (x) (x )

x1

p' l u

This holds for all £, and therefore p c over the whole ball B ,which is in a full open neighborhood of in A . So we have * ) . •

x0

(

_

Exercises 1 . Assume that p is a real-valued smooth function on the complex manifold X . If � � e is a tangent vector and r.p a complex coordinate system at E X , then define

xo

n

(8p) xo ( � ) : =

L (P

v=l

0

r.p- 1 ) zv (r.p(x o ))

·

�v .

Show that (8p)x0 : Txo --+ C is a complex-valued real linear form that does not depend on the local coordinates. Prove the following formulas: (a)

Lev(p q)(x, �) ·

p(x) Lev(q)(x, �) + q(x) . Lev(p)(x, �) + 2 Re ( (8p) x ( � ) (8q)x ( � ) ) . ·

·

266

V. Stein Theory (b)

Lev ( h o p)(x, �) == h" (p(x)) . I (8P)x (�) 1 2 + h' (p(x)) Lev (p) (x, �). ·

2. Let G cc X be a relatively compact domain with smooth boundary. Show that there is an open neighborhood U of 8G in X and a real­ valued smooth function cp on U such that G n U = { x € U : < 0} and (8cp)x -=/= 0 for X E ac. Show that

cp (x)

Hx (8G) = {� E Tx (X) : (8cp) x ( � ) = 0 } is a well-defined subspace of Tx (X) that does not depend on the boundary function cp. Show that if for every X E ac there is a local boundary function 7/J such that is positive definite on Hx (8G), then cp can be chosen as a strictly plurisubharmonic function. 3 . Let G cc X be a relatively compact domain with smooth boundary, and cp : U (8G) --+ l!l1. a global boundary function. If ( cp) has for every X E ac at least one negative eigenvalue on Hx (8G) , G is called pseu d o­ concave.

Lev ('I/J)

Lev

Show that if X is connected and there is a nonempty pseudoconcave domain in X , then every global holomorphic function on X is constant.

3.

P seu docon v exity

Pseudoconvex C omplex M an ifold s. If X is an arbitrary complex manifold, then there exists a sequence of compact subsets Ki E X with the following properties: 1. The set Ki- l is always contained in the open kernel (Kit of Ki . 00

i=l If X is holomorphically convex, then the Ki can be chosen in such a way that the holomorphically convex hull Ri =

{ x E X : I J (x) l < s�; I J I for all f E O(X) }

always equals Ki . (One uses the same proof as for domains in en . )

x I !I 1! 1

f

Therefore, for every point E X - K, there is a holomorphic function in > 1 and < 1 on K, . By passing over to a multiple and X such that a power of we can make arbitrarily small on K, and arbitrarily big in a fixed neighborhood of

If (x) I f,

x.

Since Ki + - (Ki + I t is compact, there are finitely many holomorphic func­ 2 tions , tr;) in X such that for Pi : = L,;:'i 1 2 the following hold:

J? ), . .

.

1 1 Jt)

3.

Pseudoconvexity

2fj7

1 . supKi Pi < 2 - " . 2. Pi > i on Ki +2 - (Ki+l t . Of course, p, is a nonnegative plurisubharmonic function in X .

The sum 2:::: � 1 Pi converges compactly on X to a nonnegative function p , with < c } CC X for every c > 0. EX :

{x

p(x)

If f is a holomorphic function on a domain G c e n , and G' c c G a subdo­ main, then we have the Cauchy estimate · sup l fl , for z E G ' . c' Using this estimate in the intersection of a local coordinate system for X with (Ki t , one shows that all derivatives of Li Pi converge compactly in X to the corresponding derivatives ofp . So p is 'tf'00 and again a plurisubharmonic function. One can even show that p is real-analytic (see [DoGr60] ) . Definition . A complex manifold X is called pseudoconvex if there exists a nonnegative smooth plurisubharmonic exhaustion function p on X ( i. e . a 'ff'= function p with EX : < r CC X for all r > 0 such that the Levi form of p is everywhere positive semidefinite) .

{x

,

p(x)

}

If we can find for p a strictly plurisubharmonic function in X , then X is called stricly pseudoconvex or l-comp lete 1 . If p is strictly plurisubhar­ monic only outside a compact set K c X , then X is called ] -convex or strongly pseudoconvex (at infinity). In the literature a pseudoconvex manifold is often called weakly complete.

1-

Above it was shown that

every holomorphically convex comp lex manifold is pseudoconvex. We shall prove later on that every 1-complete complex manifold is holomor­ phically convex, and even Stein (solution of the Levi problem) . Also, strongly pseudoconvex manifolds are holomorphically convex. But in general this is not true for weakly 1-complete i . e. , pseudoconvex) manifolds.

(

Examples. Strict pseudoconvexity ( ! -completeness) is one of the most important notions in the analysis of complex manifolds. Many constructions can be carried out only in the strict pseudoconvex case. Let us consider some examples. Example 1 : The theory was inaugurated by the following result: 1 An n-dimensional complex manifold X is called q - complete if it has an exhaustion function p such that at every point of X the Levi form of p has at least n - q + 1 positive eigenvalues.

268

v. Stein Theory An unbranched doma in G over en is strictly pseudoconvex it is a domain of holomorphy.

if

and only if

For proofs see [Oka53] , [Br54] , [No54] . E xam p le 2: Every unbranched Hartogs convex domain over en is strictly pseudoconvex. See [Ri68] , where a smoo lhing procedure for slriclly pseudo­ convex functions is introduced. E xamp le 3 : The proofs

of

the following statements are elementary.

Every compact complex manifold is pseudoconvex. The Cartesian product of finitely many (strictly) pseudoconvex complex manifolds is (strictly) pseudo­ convex. Any submanifold of a (strictly) pseudoconvex complex manifold is (strictly) pseudovonvex . E x amp le 4: Assume that G c en is a domain of holomorphy and that A c G is an analytic hypersurface. Then G - A is Hartogs convex (and therefore strictly pseudoconvex) . For the proof we just take a Hartogs figure

in the unit poly disk P " = { t : lti I < 1 for i = 1 , . . . , n} and a biholomorphic mapping F : H --+ G - A. The mapping F extends to a holomorphic mapping pn --+ G. If A F- 1 (A) c P" is not empty, then it is an analytic hypersurface in P" - H . Some lines L(t') { t (t 1 , t') : t1 E e } will intersect it in a compact subset of P " - H ; for other t' the intersection is empty (see Figure v. 1 ). =

=

<E-

Figure

=

L(t')

V . 1 . Hartogs convexity of G - A

There is a limit tb with c < ltb l < 1 such that the intersection of L(tb) with A is not empty but in arbitrarily small neighborhoods of tb there are points t' for which the int<M"section is empty. If t E A n L(tb ) , then there is a neighborhood U where A is given by a holomorphic equation f = 0. The function tt >-+ j(t1 , t0) has isolated zeros in U n L(tb ) , and by the theorem

3.

Pseudoconvexity

269

t1 f(t1 , t') Thus F(P n ) c G - A, and the domain G - A is Hartogs pseudoconvex. Example 5: Let A c ]pm be an analytic hypersurface. We know that there is a homogeneous polynomial such that A = { (zo : Z1 : . . . : Zn ) E lP'n : w(zo , . . . , Zn ) = 0}. of Rouche >-+ also has to vanish at some points of U n L (t' ) for t' near to t � . This is a contradiction.

w

Let

w

be homogeneous of order s. Then

p(zo : . . . : Zn ) := log

l (zo · . . , znW } w

,

lP'n - A.

is a well-defined smooth exhaustion function for the affine algebraic manifold We calculate the Levi form in local coordinates tv Zv / 0 v 1 , . . . , n using the properties of the logarithm. Since the Levi form of + vanishes for any holomorphic function it follows that =

z, f f

=

f, Lev(p) (to, e) s . Lev (log(1 + l i t 1 2 )) (to, e) S · ( ( 1 + ;L I 2 ) 2 · l (to , e) l 2 + 1 + l�tol l 2 · l e 11 2 ) 2 · l l + ( l t o l 2 · l e l 2 - l (to , e) l 2 )) , e ( 1 + l :o ll 2 ) 2 ( and this expression is positive for e cft 0 . So p is strictly plurisubharmonic everywhere, and X = lP'n -A is 1-complete. In this case we can show directly _

that X is holomorphically convex:

Every function f

(zo

:

. . . : z . .)

:

zo80 · · · zn _ w(zo, . . , Zn )

n

Sn

= -

.

'

with

l: si = s,

a=O

values of A.

all these is holomorphic in X , and the maximum of the absolute functions tends to infinity as approaches So K c c X for any subset K c c Consequently, X is holomorphically convex, and it is even Stein, as one can see from the following theorem.

(zo : . . . : zn )

�

X. Let X be a holomorphically convex connected complex man­

3 . 1 Theorem . ifold that contains n o compact analytic subset is a Stein manifold. PROOF:

of positive

dimension. Then X

E X be an arbitrary point. Then the set A { E X : f ( ) f ( o) for every f E O(X)}

Let

x0 :

=

x

x

=

x

270

V.

Stein

Theory

is a closed a�ic subset of X . Clearly, it is contained in the holomorphi� < convex hull Since for all has to be compact, is likewise compact. This is possible only consists of isolated points. Then there exists an open neighborhood and in such that holomorphic functions h ,

{xo} f E O(X)}. {xo} = {x E X : lf(x) l lf(xo )l A if A U = U(xo) . . . JN U {xo} = A n U = rx E U : h(x) = . . . = fN (x) = O}.

So X is holomorphically spreadable and therefore Stein.

3 . 2 Proposition .

•

A ]-complete complex manifold cannot contain compact

analytic subsets of positive dimension. PROOF: Let p be a strictly plurisubharmonic exhaustion function in the manifold X . Then p is plurisubharmonic, and by the maximum principle it must be constant on any compact connected analytic subset A c X . If A has positive dimension, then there is a point and an open neighborhood c X such that A n is a submanifold of of positive dimen­ �ion. Tpe function PlAnu is strictly plurisubharmonic and constant. That is Impossible. •

U = U(x)

U

xEA

U

3 . 3 Corollary. Let X be a ] -comp lete manifold that is holomorphically convex. Th e n X is a Stein manifold.

At the end of this chapter we will see that the condition "holomorphically convex" is not necessary. 3 .4

Corollary. Let

A C ]pm be an analytic hypersurface. Then ]pm - A is a

Stein manifold. Every analytic subset B C ]pm of positive dimension meets A in at least one point.

Example 6: There is a famous theorem by H. Cartan:

A domain G c C2 is a domain of holomorphy if and only if the first

Cousin problem is always solvable.

The solvability is also true for higher-dimensional domains of holomorphy, but there is a greater class of domains with this property. Take, e.g . , the domain G c pn c en (with n > 3) that is the union of the three open sets

. .

ui - {z u2 - { z {z u3 . · -

E pn : l zii > EJ , E P" : hi > E) , E P" : I (Z3 , . . , Zn ) I < c}. ·

3. Pseudoconvexity

Figure

V.2.

Cousin-I in

a

27 1

domain which is not Hartogs convex

The domain G = U1 u U2 U U3 is not Hartogs pseudoconvex (see Figure V.2). But all three covering elements Ui are domains of holomorphy. Therefore, we have H 1 (Ui , 0) = 0 for i = 1 , 2, 3, as will be proved in Section V.5. So '2/ = { U1 , U2 , u3) is an acyclic covering for 0, and every Cousin-! distribution in G can be given by a cocycle f E H 1 ('2/, 0 ) pp to a coboundary. Such an f consists of holomorphic functions fiJ in Uij with f1 2 + f2 3 + !3 1 = 0 on U1 2 3 · This implies that the Laurent series of /1 2 (around the origin) in U1 23 contains no powers zf z; with i < 0, j < 0. By the identity theorem this is true on the whole set U1 2 . Therefore, we can subtract a coboundary J{g 1 , g2 , 0} from our cocycle such that thereafter f1 2 0. Then the new !2 3 , fi 3 coincide on U1 23 . Together they give a holomorphic function h in (U1 U U2) n U3 that extends holomorphically to U3 . Then the new cocycle f is equal to the coboundary 6(0, 0, h ), and hence the old cocycle also cobounds (i.e. , is a coboundary) . Consequently the Cousin-! problem is solvable. Example 7: Assume that X is an n-dimensional complex manifold and that x0 E X is a point. We can blow up X in x0. Then we obtain an n-dimensional complex manifold X , an ( n - I )-dimensional complex submanifold A c X that is isomorphic to ]pm - l , and a proper holomorphic map 1r : X --+ X that maps A to x0 and X Abiholomorphically onto X - { x0 }. We have a strictly pseudoconvex neighborhood U around x 0 . We can lift the strictly plurisub­ harmonic exhaustion function p on U by 1r to 1r- 1 (U). This is a strongly pseudoconvex neighborhood of A that is not, however, strictly pseudocon­ vex. �

�

�

-

Example 8: For a similar example consider the analytic set

2 A := {z E C n : ZI Zn+j - Zj Zn+ l = 0 for j = 2 , . . n }. ·

,

O,

Outside of the origin A is regular of dimension n + 1. For example, if zi # then Zn+ J = zi . Zn +d zi for j = 1 , . . . , n . So A is there parameterized by z 1 , . . . , zn and zi + l · It follows that dim( A) = n + 1.

272

V.

Stein Theory 0"

We have a meromorphic map from C 2" to the Osgood space wi = zi / Z n + i , i = 1 , .. . , n. Its graph is the set

X = { (z, w)

E

C 2" X ((])"

:

given by

WiZn +i - Zi = 0 for i = 1, . . . , n } .

A

We denote by X the closure of the part of X that lies over - {0}. Let be the restriction of the canonical projection from x 0" 1r : X --+ to Then 1r maps a ! -dimensional projective space onto 0 and the rest biholomorphically onto - {0}.

A

A.

A

A

The space X is an (n + I)-dimensional complex manifold, locally given by the equations Zi = Wi Zn+i for i = 1 , . . . , n, for i = 2, . . . , n . Wi = W 1

A

{

zr /zn+ l = · · · - Zn /z2n = s } is an The set E ( s ) : = (zr , . . . , 2n) E n-dimensional plane for every s E JID1 , and is the union of all these planes. Since E ( s! ) n E ( s 2 ) = {0} for s 1 -=/= s2 , is singular at the origin. It follows that X is a vector bundle of rank n over 1r - 1 (0) "" lP'1 .

A

A

Now we use the function p(z) = Li ZiZi on C2" . It induces a strictly plurisub­ harmonic function on the complex manifold - {0}. The (n+ I )-dimensional complex manifold X is strongly pseudoconvex by p o 1r, but not 1-complete, since it contains the compact analytic subset 1r - 1 (0) .

A

Examp le 9: Consider the covariant tangent bundle T' of JID" . If � is a global holomorphic vector field on lP'" , then a plurisubharmonic function Pr; on T' is defined by We consider local coordinates in a set Ui c JID" , for example tv : = zv / zo for v = 1 , . . . , n in the case i = 0. Then every w over Ui can be written uniquely in the form w = L v wvclfv. So t 1 , . . . , t,, w 1 , . . . , w, are local coordinates in T' over ui · If � = Lv �!/a;azv , then

We have the following n + n2 holomorphic vector fields over Ui that extend to lP'" :

a atv

- q(O,

a . . . , � , . . . , ) and ll v at

O

uth place

t

= q(O, . . . ,

�

, . . . , 0) ,

uth place

where q : O( l ) ffi · · · ffi O ( l ) --+ TlP'" is the canonical bundle epimorphism in the Euler sequence. Plurisubharmonic functions p� , respectively p�ll' are defined in local coordinates by WvWv , respectively WvWv · t!ltW By adding them all we obtain the plurisubharmonic function

3. Pseudoconvexity

273

0,

if

Finally, we add the Pi constructed for the various Ui , i = . . . , n , we obtain a function p. If Z denotes the zero section in T', then p vanishes on Z and is strictly plurisubharmonic and positive outside of Z. It tends to oo for llwll --+ oo . So the complex manifold T' is strongly pseudovonvex, but not 1-complete. Example 10: This example is probably due to J.P. Serre.

Assume that E is an elliptic curve (i.e. , a compact Riemann surface of genus 1 ). Then E is a !-dimensional torus, given by a lattice of periods (1, O, where the imaginary part of � is positive. We may write the elements of E as real linear combinations = s . 1 + . �' with s, 1]. The first cohomology group H 1 (E , 0) is equal to C (this will follow from results of the next chapter, but it is also a very classical result in the theory of Riemann surfaces) .

t E [0,

t

z

We have a covering '2/ of E consisting of the two elements

u, u2

{ z = s + t� 0 < s < 1/2, O < t < l}, {z = s + t� I /2 < s < 1 , O < t < l}. Denote by C c E the circle {z = 1 + t � 0 < t < 1 ) and define there the function f 1 (and 0 on the other component of U1 n U2 ). Then f 1 -

.

:

is a cocycle in Z ('2/, 0). It yields a nonvanishing cohomology class, since otherwise we would obtain a nonconstant bounded holomorphic function on C. We construct a fiber bundle above E with fibers = C by gluing above C the point X C. We denote this X C with point in the bundle by [ z,

(z, w) E U2 w]

A

A

A,

(z,w - l)E U1

•

Now, is topologically trivial (since we can find a continuous function g on that coincides with f on C such that g} = f) ,but it is not analytically trivial, because it defines a nontrivial cohomology class. It follows from the construction that the notions of real lines, planes, and convexity are well defined in

U2

6{0,

A.

A

Let A be the bundle with typical fiber JID1 that is obtained from by adding the point at infinity to each fiber. Then A is compact and has the infinite cross section D = D= over E. We put X = A - D. Then X does not contain any compact analytic set of dimension 1. Otherwise, there would be a number b such that the analytic set meets every fiber in exactly b points. We could pass over to their barycenters and would obtain an ordinary holomorphic cross section in This would imply the triviality of �

:=A

A.

A.

We consider the real 3-dimensional surface

S = Sr = {[z, w]E A z = s + t� , W = s + r exp ( 27ri :

.

B

) and

O < < l }. B

274

V. Stein Theory

>0

Here r is a very big number. The surface S bounds a tube G, which converges to A as r tends to oo . We look at the holomorphic tangent H in any point of S . We can consider H as a real plane in A . By subtracting s in the fiber above s + we pass over from A to E x C. The hypersurface S is transformed diffeomorphically into a convex cylinder over E and H into a contacting plane that docs not enter the interior of the cylinder. So H does not enter the interior of the tube G. If H were contained in S , then the transformed H would lie in the boundary of the cylinder and therefore be compact. So H itself would be a compact analytic set. We saw that such a set does not exist. So there remains only the possibility that H contacts S of first order; i.e. , the intersection H n S is a real line. We can choose a complex coordinate ( = x + i y in H such that our real line is exactly the z-axis.

t�

If(! is a smooth defining function for S, then (! behaves on H like the function y2 . It follows that (QIH ) (( This means that the Levi form of (! is always positive definite on the holomorphic tangent H. Now we can construct a strictly plurisubharmonic function {j in U D (where U is a neighborhood of D in A) whose level sets are the manifolds Sr such that {j converges to oo It follows that X is strongly pseudoconvex. when approaching

> 0.

�

-

D. 2

Later on we shall prove that a strongly pseudoconvex manifold is holomorphi­ cally convex, and since X contains no positive-dimensional compact analytic subset, it is a Stein manifold. So there are many holomorphic functions in X . Assume that there is a meromorphic function on A that has poles of order m on D only. Then the coefficient of the highest polar part off is a holomorphic cross section ry in the mth tensor power of the normal bundle of D . Because this normal bundle is topologically trivial, ry cannot have zeros on D. So f = const} meets D. tends to infinity approaching D, and no analytic set This is a contradiction, since A is not analytically trivial. Every holomorphic function on X must have essential singularities on D. �

{!

Analytic Tangents . Let G be a domain in en with n > 2 and p a strictly plurisubharmonic function in G. Denote by X the set {z E G : p ( z) < 0}. Let B c c G be an open subset and w E 8X n B an arbitrary point. The expansion of p in

w

is given by

p(z) = p (w) + 2 Re Q(w, z - w) + Lev(p)(w, z - w} + R(w, z - w}, where

1 t w) + L h · Hess (p) (w) . h t, Q(w, h ) = h . \lp (

2 We may assume that g is globally defined and

{!

=

unbounded strictly monotonic smooth function h very large. The function {i := h o g will do the job.

r

:

on

Sr .

Then we choose an � -+ � such that h" /h' is

3 . Pseudoconvexity

with the complex Hessian Hess (p) ( w) =

.

��

(Pz, z1

R (w, h ) llhll 2

275

)

( w ) i , J = 1 , . . . , n , and

= o.

The map h >-+ Q(w, h ) is a holomorphic polynomial of degree The assignment w >-+ Q(w, h ) is smooth.

2,

for every w .

Let G' c c G be an open neighborhood of B. We can find constants c, k > 0 such that

Lev (p) (w, z - w ) + R(w , z - w ) > kllz - w ll 2 - cllz - w l l 3 for z E G' and arbitrary w .

B if

We say that a real number c > 0 is suffici ently small with respect to every ball U(w) with center w E ax n B and radius c belongs to G', and

is valid on these U(w) . Assuming that this is the case, for w E ax n B we define the analytic set

A(w)

:�

{z

E U(w) : Q(w, z - w ) = 0 }

in U ( w) . On A(w) we have p( z) > 0. Therefore, A(w) - { w } is outside of X . Since w belongs to the boundary of X , this implies that A = A(w) has codimension 1 . We call A an analytic tangent (or, in German, a Stiitzfiache) for X at w . 3.5

Prop o s ition. Let G, p, X, B , G' be as above. There exists a differen­ tiable family A(w) qf analytic tangents to ax at the points w E B n ax with A(w) n X { w} . =

Here "differentiable" means that the defining quadratic polynomials for A(w) depend smoothly on w •

Exercises

1. Let X be a complex manifold and Y c X a closed complex submanifold. Construct a nonnegative smooth function f : X --+ l!l1. with the following properties: (a) Lev ( f) ( x, �) > 0 for every x E X , � E Tx (X). (b) For every x E Y there is a linear subspace Px C Tx (X) such that Px + Tx (Y) = Tx (X) and Lev ( f) ( x, �) > 0 for � E Px , � # 0 . 2 . Let X be a 1-complete complex manifold and fl , . . . , fq E O(X). Show that X - N(fi , . . . , fq) is q-complete. 3. Let G cc e n be a strictly convex domain with smooth boundary. Con­ struct the differentiable family A(w) of analytic tangents t o ac.

276

V.

Stein Theory

4 . Let cp be a strictly plurisubharmonic smooth function defined in a neigh­

borhood U U (O) c @" with cp(O) = 0 , and 7/J an arbitrary smooth function in U. Prove that there are r > 0 and > 0 sufficiently small such that cp + . 7/J is strictly plurisubharmonic in Br (O) and =

c

c

{

z

E Br ( 0) : ( cp +

c · 7/J) ( ) < 0} z

is a Stein manifold.

4.

Cuboids

Distinguished Cuboids. If Q is a closed subset in en , then in this section we say that something is defined on Q if it is defined in a small neighborhood of Q. Two objects are called equal on Q if they coincide in a small neighborhood of Q. So actually we consider "germs" along Q. Definition .

A cuboid is a closed domain

where ai < bi are real numbers for i = 1, . . . , 2n. If there are partitions

for

i

=

1 , .. . , 2n,

then we denote by A the system of sequences

a. 1• .1

'

=

1 , . . . , 2n,

j = 0, . . . , nzi.

A closed covering 'PEA of Q is defined by the system of cuboids Qj . . . ,hn = 1 ,

{

z

: ai i

1- < Xi < a{i , for i

=

1 , .. . , 2n}.

For any open covering of Q there is a closed cuboid covering which is finer. In the following it will be not enough to have a covering of Q. Additionally, we need a system of complex submanifolds Q = Xo � X1 �

· · · � Xi- 1 � Xi � · · · � Xs , s < n, - such that there is a holomorphic function fi

where xi has dimension n i ' in Xi _ 1 vanishing everywhere on Xi to first order (and maybe also vanishing at points of Xi - 1 - Xi) · Definition . A distinguished cuboid is a cuboid that is equipped with a system { ( Xi , fi) : i = 1 , . . . , s } . The number n - s is called the manifold dimension of the distinguished cuboid.

4 . Cuboids

277

Vanishing of Cohomology. As described above, everything on a closed cuboid is assumed to be defined on an open neighborhood of the cuboid. Therefore, we can consider cocycles and coboundaries with respect to closed cuboid coverings.

4. 1 P r op osition. Assume that Q is a cuboid and %'A a closed cuboid cov­ ering d Q. Th e n every cocycle .; E Z 1 ( %'A, 0 ) cobounds.

PROOF: First we consider a very simple system A. We just take the case where m 1 = 2 and all other mi = 1 . Then

has the minimal possible number of elements. The cocycle .; is given by one holomorphic function .;0 1 on Qo n Q l = { z E Q : Zl = a n . Let c be a small positive number. Then in the z1-plane we can choose two continuous paths aj from a� + i (a�+l - r:: ) to a� + i (a;+ l + E) (for j = 0 on the left side and for j = 1 on the right side of the line x 1 = ai, see Figure V . 3 ) and define rt = {ry0 , ry 1 } by the Cauchy integral .;o r (w , Z2 , . . . , Zn ) W - Z1

6

d

Then we get ry = .; .

z1-plane

Figu r e V . 3 . Cohomology of

a

cuboid

The next step is an induction on the number m = L:i mi . We just han­ dled the case m = 2 n + 1 . Now we assume that m > 2 n + 1 and that the proposition already has been proved for any number smaller than m. We put io : = max{i : 1 < i < 2 n and mi > 2} and define

278

V. Stein Theory

By the induction hypothesis, we have � � %' ' = J(r/) and � � %'" = J(r/') with cuboid coverings C1/ ' for Q' and C1/ " for Q". Hence, � - 17', 17" } is a cocycle with respect to the covering C1/ that is given by a holomorphic function g on the surface Q' n Q" { z E Q : Xi0 a:·o - 1 } ·

6{

=

=

As above, by integration we obtain holomorphic functions 170 on Q' and 171 on Q" with g = "11 - 11o . So we have a cochain (}' = 17' + TJo, 17" + "11 } on Q with J(J' = � (since the functions 1]j are already cocycles) . This completes the • induction.

{

V anishing on the Embedded M anifolds. We consider a distin­ guished cuboid Q in e n with manifold dimension n - s, and for i = 0, . . . , s we prove the following:

Every cocycle of Z 1 (Cf/.A n Xi, 0) cobounds, and every holomorphicfunction on Xi+ I can be extended to a holomorphicfunction on

4.2 Proposition.

xi .

PROOF: We carry out an induction on i. In the case i = 0 the proposition has been proved already. The induction hypothesis now states that it holds for some i, 0 < i < s - 1 , and we prove the extension property for i + 1 . Assume that g is the holomorphic function on Xi + I · Define g ( z) = 0 on the other connected components of N(fi +1 ) c Xi . If the covering Cf/.A is sufficiently small, we can extend this g to a cochain 17 E C0 ( 0&'.A n Xi , 0) . The coboundary 6 (17) vanishes on N (fi+ 1 ) . Therefore, there is a cochain a E C1 (Cf/.A n Xi , 0) with 6(17) = a · fi +1 . It is clear that a is a cocycle, and by the induction hypothesis there is a 1 E C0 (Cf/.A n Xi, 0) with a = J (r) . Since J(ry - fi + I . y) = 0, we get g = TJ - fi+ 1 . 1 as a holomorphic extension of g.

Now we prove that any � E Z 1 (Cf/.A n Xi + l , 0) co bounds . As in the proof of the preceeding theorem we have to show this or.ly in the case where m 1 = 2 and mi = 1 for i > 1. So � is simply a holomorphic function g on (Q 1 , 1 , ... , 1 n Q2, 1 , . . . , 1 ) n Xi + l · We have to find holomorphic functions f' on Q 1 , 1 , ... , 1 n Xi+ l and !" on Q2, 1 , ... , 1 n Xi+l with !" - f' = g. For that we first extend g to ( Ql , l , . , 1 n Q2,l, . . . , r n Xi , construct f', !" for Xi (induction hypothesis), and • then restrict them to Xi+ l · That completes the proof.

)

..

Cuboids in a Complex M anifold. We assume that X is an n ­ dimensional complex manifold and p a smooth real-valued function on X with p(x) ---+ 2 when X ---+ aX . 3 We assume further that p is strictly plurisub3

This means t h at for every E > 0 there is a compact subset p(x) > 2 E for E X K.

-

-

K CX

such t h at

4.

Cuboids

279

harmonic on {p(x) 1 ) and define Y : = {x E X : p(x) < 1 ) . Then Y is a relatively compact strongly pseudoconvex open subset of X . =

For an n-tuple z

=

(z1 , . . . ,zn) we use the norm l z l supmax(IRe(zi )l , l lm(zil). :=

1

For an open neighborhood G c X of 8Y we want to apply the results from the end of V . 3 . (It does not matter that there are no global coordinates on G.) We consider a relatively compact open neighborhood B B(8Y ) C C G and choose a G' with B C C G' CC G. =

For any point x0 E 8Y there is a compact cuboid U* with center x0 in a coordinate neighborhood S around x0 . If a real number c: > 0 is given, we can choose every U* so small that U* c U(w) c c S for every point w E U * , where U(w) is the ball with center w and radius c: (with respect to the local coordinates) . Then for every w E U* n 8Y we have the analytic tangent A(w) given by a quadratic polynomial fw in U(w) . It follows that A(w) n U* n Y w , and the function p i U* n (A(w) z) is positive. This is illustrated by Figure V.4. =

-

For an open subset Y' c c Y we have the following proposition.

There is a distinguished cuboid Q* the following property: 4 .3 Propositi on.

=

U*

x

Qc

eN with

For s n the submanifold Xs is projected biholomorphically onto a N compact set U' c U* with Y' n U* C U' C Y . =

-

U*

F igur e V .4. Projection of the distinguished cuboid

en

and assume that x0 is the origin. We use the PROOF: We may work in differentiable family of analytic tangents A(w) N(fw) with w E 8Y n U* . =

280

V. Stein Theory

Take a positive number r so big that we always have 1 1 /fw(z) l < r for w E 8Y n U* and z E Y' n U * . In a small neighborhood of w we have 1 1 /fw(z) l > r. Therefore, by compactness there are finitely many of these fw , say /I , . . . , fs , with maxi l 1 / fi (z) l > r for all z E 8Y n U * . For a fixed i we denote by Q� the cuboid {z' = (zn+l , . . . ,zn+i) : lz' l < r} and by Q�' the cuboid {z" = (zn+i+l , . . . , zn+s) : lz" l < r}. We put Q* = U* x Q: x Qr The submanifolds Xi c Q* are obtained in the following way: Consider the graph of the i-tuple ( 1 //I , . . . , 1 /fi) in U* x QL take the union of those connected components that contain points over Y , and multiply this union by Q�' - The manifold Xi has dimension N - i. In Xi- l we have the holomorphic function Zn+i . fi (z) - 1 , which vanishes on Xi to order 1 . Finally, the projection U ' of X , contains no point of 8Y. Hence, it is contained • in Y. Since U* n Y' is contained in U', the proposition is proved. 4.4 C or ollar y.

In the above notation, H1 (U' , O) = 0 .

E � nlarging U' . We use the same notation as before and construct a set U' c U* n Y that is bigger than U' where the vanishing theorem still holds. For that purpose we take an open set Y ' instead of Y' with Y' CC 8' C C Y � and U' c Y'. We need a bigger r > r such that we still have 1 1 / fw (z) l < r on 8' for w E 8Y n U* . But now ffiaKi 1 1 / fi (z) l > r no longer is true on 8Y n U * . So we add some functions to the old ones, !s+l , . . . , fs , such that we get the old situation again.

� ' We get the Q� and the Q� , for a = 1 , .. . , s. But we� write � � Q� instead. of Q� , Q�' � � ,.., Jii Wit h N < N . for Q:' ' and xi for xi . Then we have Q* = U* X Qi' X Qi" c "" The projection of X-s is a compact set D' c Y n U* with Y ' n U* again we have the vanishing of the cohomology.

c

D'.

So

The following statement is proved in the next paragraph. ( * ) Every holomorphzc functzon o n U' can be approxzmated arbitrarily well by holomorphic functzons on U'. Suppose all cochains are given with respect to a covering '2/ of U' X-s. If .; E Z1 ( '2/ , 0) is a cocycle over D', we have .; = ( ry) over U' and .; = (if) over D', with cochains ry and if with respect to the coverings UnU' and unD'. Then ij - ry is a holomorphic function over�U' , which we approximate by a sequence of holomorphic functions [}j E O (U'). We may replace the cochain ij by ij1 : = ij - [}j and obtain that ij3 approximates TJ over U' .

6

Now we can prove the following theorem.

""

6

4.

Cuboids

281

If x0 E 8Y and U* is a small cuboid in a local coordinate system around xo, then H 1 (Y n U* , O) = 0 .

4.5 Theorem.

PROOF: Take a fixed open covering '2/ of Y n U* and exhaust Y by a sequence of domains Yi with Yi c c Yi+ I c c Y . We get a sequence of compact sets UJ c U* n Y with Yi n U* c UJ.

u;

If � E Z1 ( '2/ , 0) is a cocycle, for every i we get a cochain 'r/i on such that � = J(ryi ) on As we have seen above, we may assume that the difference of 17·i + I and 'r/i on is as small as we want. Then the sequence 'r/i converges to some ry E C 0 ( 'P/ , O) with J(ry) = � . •

u;,.

u;

Ap pr oxima tion . We first prove the following simple result.

Assume that Q is a cuboid in CN. Th en any holomorphic function ! o n Q can be approximated arbitrarily well by polynomials. 4.6 P r op osition .

PROOF: We make the induction hypothesis that f is already a polynomial in the variables z' = ( z 1 , . . . , zs) . Let /c; be the boundary of the rectangle with

corners (as + l - c, a N +s + I - c) , (bs+ l + c, aN + s+l - c) , (bs+ l + c, b N + s + l + c ) , and ( as+ I - c, bN +s + l + c) in the (xs + l , X N + s+ l )-plane. We have the integral representation

f(z ' , zs + I , z " )

=

� 2Jrl

j

rs

f(z ' , �, z " ) d� � - Zs+ l

It is necessary to approximate 1/(� - Zs +I) with a fixed � E Ire I by a poly­ nomial. We consider only the case where Re( 0 ; i.e., the topology is Hausdorff. 2. The space V is complete in this topology: Every Cauchy sequence converges. A Fre 'chet space is a complex vector space with a Frechet topology. Every Frkchet space possesses a metric that defines its topology. The vector space O(X) of holomorphic functions on a complex manifold X is a Fnkhet space. We just take an increasing sequence of compact subsets Ki with Ki C (Ki +I) 0 and U i Ki = X , and we define pi (f) := sup/f(K ) / . If (!k) is a Cauchy sequence of holomorphic functions on X , then fk / ( Ki) o converges uniformely to a holomorphic function on (Ki) o . Hence (fk ) con­ verges compactly to a holomorphic function f on X . This means that (!k) converges to f in the topology of O(X). Since every Frechet space V is a metric space, a subset M c V is compact if and only if every sequence (fk ) in M has a convergent subsequence with limit in M. A linear map v : E -+ F between two Frechet spaces Definition. is called compact or completely continuous if there is a neighborhood c E such that is relatively compact in F .

U (O)

v(U)

Obviously, every completely continuous map is continuous. We have the following famous theorem of Schwartz. 5 .3

T heorem of L. Schwartz . Assume that E , F are two Fre'chet spaces and that u, v : E -+ F are two continuous linear maps with the following properties:

1 . u is surjective. 2. v is completely continuous. Then the quotient Fj (u + v) (E) has finite dimension. A typical example of a completely continuous map is given as follows: Let X be a complex manifold and Y c c X an open subset. Using the increasing sequence of compact subsets from above, we have Y c Ki for some i. Now, : Pi (!) < 1} is a neighborhood of 0 in : = { f E O(X) E := O (X ) Let v : E -+ F := O(Y) be defined by v ( f ) : = f / y . Then

U

.

v(U)

c .'JlJ

: = {g E

O (Y) : snp/g/ < 1).

286

V. Stein Theory

Since !!& is closed and bounded, it follows from Montel's theorem that it is compact. This shows that the restriction map v is completely continuous. Remark. It is clear that we can replace the space of holomorphic functions by the space of holomorphic cross sections in an analytic vector bundle.

Finiteness of C ohomology. We wish to apply Frkchet space theory to our standard situation. Let X be an n-dimensional complex manifold and Y = { x E X : p ( x ) < 1 } , where p is a smooth exhaustion function on X with sup(p) = 2. Assume, as usual, that p is strictly plurisubharmonic on a neighborhood of 8Y. Let a small real number c > 0 be fixed such that Y"' is strongly pseudoconvex for 0 < a < c . We have shown that the restriction map H1 (Ya, 0) -+ H1 ( Y, 0) is surjective for sufficiently small a. We start with a special pair ('2/ , % * ) of cuboid coverings for (Y, Y ) , choose numbers � with ti < � < ri and define Ui = { z : lzl < � } - This gives a covering c¥/ = { Vi : i = 1, . . . , m } of Y a · a

�

The spaces C0 ('P/ n Y, 0), Z1 ('P/ n Y, O ) , and Z1 ('P/ n Y" 0) are Frkchet spaces, as is Z1 ( '2/ n Ya , 0) ffi C0 ( '2/ n Y, 0) . Since the cohomology can be extended from Y to Ya , it follows that the map �

�

u : E = Z1 ('P/ n Ya; O)

ffi

C0 ('P/ n Y, O) -+ F = Z 1 ('PE n Y, 0),

with u(�, rt) : = � l %' nY + o(ry) , is surjective. The map v : E -+ F is defined as the negative of the restriction map from Z1 ( '2/ n Ya , 0) to Z1 ( '2/ n Y, 0) , i.e. , by v(�, rt) := -�l%'nY· Then v is completely continuous. �

The map u + v is the coboundary map r5 : C0 ('P/ n Y, O) -+ Z1 ('P/ n Y, O ) , since the first summand goes to 0. So the quotient Fj(u+v) (E) is the first co­ homology group of Y. Its complex dimension is finite by Schwartz's theorem. Thus we have proved the following result.

If Y is a strongly pseudoconvex relatively compact subset of a complex manifold X , then dime H1 (Y, 0) < oo.

5 . 4 Theor em .

H olomorphic C onvexity. We consider the same situation as above and choose an a > 0 such that Y cc Ya and: 1 . dime H1 (Ya, 0) < 00 . 2. For every x E 8Y we have an analytic tangent A(x) c Ya to 8Y, with A (x) n Y = { x } , given by a holomorphic function f in a neighborhood U of A(x). x

5.

Special Coverings

287

Let x E 8Y be an arbitrary point, f = fx, A = A(x) , and U a suitable neighborhood of A . Then 'llx = {UnYa, Ya - A} is a covering of Ya consisting of only two elements. So Z1 ('llx, 0) = O(U n Ya - A ) ,and every function f - k 1 belongs to Z ( 'llx , 0) .Since H 1 ( Ya , 0) is finite-dimensional, for m ::;» 0 there are complex numbers a 1 , . . . , � not all zero, such that the cocycle given by the function m ) ( Un Y A ) g = ( at . r 1 + · + a, .

.

.

J I -

"

-

is cohomologous to 0. Therefore, we find holomorphic functions h 1 in Ya - A and h2 in U n Ya such that h 1 - h2 = g. The function h

..

_

{ hht + 2

g

on Ya - A , on U n Ya ,

is meromorphic in Ya with poles in A . So hi y is a holomorphic function, and for y -+ x it tends to oo . If K is compact in Y, then the holomorphic convex hull K does not approach x . This is true for every x E 8Y. So K is compact. As a consequence we have the following result. �

If Y is a strongly pseudoconvex relatively compact subset of a complex manifold X , then Y is holomorphically convex.

5 . 5 Theor em.

Moreover, we have a solution of the Levi problem in this special case. 5 . 6 Theor em. Assume that Y = { x E X : p (x ) < 1 } , where p(x) is a smooth function in X with p (x ) -+ 2 tor X -+ ax that is strictly plurisubhar­ monic o n 8Y. IfY contains no higher-dimensional compact analytic subsets, then Y is a Ste in manifold. PROOF: In Section V . 3 we showed that every holomorphically convex man­ ifold that contains no compact analytic subset of positive dimension is a Stein • manifold. 5 . 7 Corollary. If p is a strictly plurisubharmonic exhaustion function of X , then Y = { x E X : p (x ) < 1} is a Ste in manifold.

This is clear, since under the assumption there are no compact analytic sub­ sets of positive dimension in X .

Negative Line B undies. Assume now that Z is a compact n-dimensional complex manifold and that F is a holomorphic line bundle on Z. Let 1r : F -+ Z be the canonical projection. Definition . The line bundle F is called negative if there is a strongly pseudoconvex neighborhood Y c c F of the zero section of F .

288

V. Stein Theory

We identify Z with the zero section in F , and we call Y a tube around Z. For an arbitrary holomorphic vector bundle V of rank s on Z we consider the pullback bundle 1r* V on F. If U c Z is an open set, then the holomorphic cross sections in 1r * V over 1r - 1 (U) are holomorphic maps f : 1r- 1 (U) -+ V such that for any z E U the restriction of f to the fiber Fz has values in V� The trivial line bundle on F is denoted by Op . Its local holomorphic cross sections are the local holomorphic functions on F . If VI u rv u X c s ' then the holomorphic cross sections of 1r* V over 1r - 1 (U) are s-tuples of local holomorphic functions. Since the cocycles in our theory are given by lo­ cal holomorphic functions, we may replace OF by 1r* V on F and get the same results for X = F and the strongly pseudoconvex set Y . So we have dime H 1 (Y, 1r* V) < oo. Among the holomorphic maps f : 1r - 1 (U) -+ V that correspond to sections in 1r*V we have the maps f that on every fiber Fz are homogeneous of degree m (with values in Vz) · We denote the space of these maps by Om ( 1r- 1 (U) , V). It can be identified with the space of holomorphic cross sections in the tensor product V ® p- m (see Section IV.2) .

1,

}

�

We choose a finite open covering '2/ = { Ui i = 2, . . . , l of Z. Let '2/ be � the covering of F given by the sets Ui = 1r- 1 (Ui) We denote by z;, ('2/, 1r*V) the vector space of cocycles �m = with E Om (7r- 1 (Uij ), V ) . These are homogeneous of degree m on the fibers of F. We also call them cocycles of degree m. Every finite sum � = Li �i of cocycles of distinct degrees mi is con­ tained in Z 1 (F, 1r* V) and can be restricted to Z 1 (Y, 1r*V). It is a coboundary if and only if all i are coboundaries. Since the cohomology H 1 (Y, 1r*V) is �

{�Ij}

:

�0

�

finite, we obtain the following theorem: 5 . 8 Theor em. Assume that

F

is a negative line bundle and V an arbi­ trary holomorphic vector bundle o n the compact manifold Z. Th en there is an integer mo such that H 1 (Z, V ® p- m ) = Ofo r all m > mo . Remark. A line bundle F on Z is called positive if its complex dual F' is negative. Since (F' ) ' = F , the above result can be reformulated as follows: If F is a positive line bundle on Z, then H 1 (Z, V ® F ) = O fo r m > mo . "

Bundles over Stein M an ifolds. Now we again consider a general n-dimensional complex manifold X and assume that we have a strictly plurisubharmonic function in X with -+ 2 when X -+ ax. Then is called a special Ste in manifold. < Z ={ EX :

x

p( x) 1}

p

p(x)

We use similar methods to those in the last paragraph. Now let F be the trivial line bundle X x C on X and the strictly plurisubharmonic function p

6.

The Levi

289

Problem

on F be defined by

p(x, w ) :=p(x) + w w . The manifold Y = { (x , w) E F : x E Z and p(x, w ) < 1 ) is a strictly pseudoconvex tube around the zero section in F l z. However, Z is not compact now; it is only relatively compact in X . The tube extends to a neighborhood of Z. The boundary of the restriction to Z has an "edge" 8Y n (8Z x C).

But the proof of finiteness of cohomology goes through in the same way as if there were no edge. We have to apply the method of analytic tangents to 8Y and 8Z simultaneously. If V is a holomorphic vector bundle on X, we denote the pullback of V to F by 8. It then follows that dime H 1 (Yi z , 0) < oo. But in this situation all the tensor powers of F are again the trivial line bundle. We have V 0 p- m V for every m. From this we conclude the following: ""'

5 . 9 Theorem. If Z is a special Stein manifold (as described above) and V a holomorphic vector bundle on Z, then H 1 (Z, V ) = 0.

Exer cises 1 . Let E be a Frcchet space and let U U(O) c E be a neighborhood such _ that U is compact. Prove that E is finite-dimensional. 2. Let u : E --+ F be a continuous linear map between Frechet spaces. Prove that if u (E) is finite-dimensional, then u is compact. 3 . Let E be a Frkchet space. A subset M c E is called bounded if for every neighborhood U = U(O) c E there is an r0 E l!l1. such that M c r U for every r > r0 . Show that a compact set K c E is closed and bounded. Let X be a complex manifold. Show that every closed and bounded subset K c O (X) is compact. 4. Let X be an n-dimensional Stein manifold and Y c X a closed subman­ ifold of dimension n - 1 . Prove that the line bundle Nx (Y) is positive. 5. Let F = 0( - 1) be the tautological bundle over JID2 . Prove that F is negative and that =

·

6. Let X c lP'3 be a regular hypersurface and i : X injection. Prove that H 1 (X, i * O(k)) = 0 for k E Z.

6.

'--7

JID3 the canonical

T h e Levi P r ob lem

Enlarging: The I dea of the Pr oof. We wish to show that every strongly pseudoconvex ( i.e. , 1-convex) manifold is holomorphically convex, and that every strictly pseudoconvex (i.e. , 1-complete) manifold is Stein. This solves the Levi problem for complex manifolds.

V.

290

Stein Theory

Recall that an n-dimensional manifold X is called strongly pseudoconvex if there is a smooth exhaustion function p on X that is strictly plurisubharmonic > 1}. on EX

: p(x)

{x

Xs = { x

: p( x)

For s > 1 the open sublevel set < s} is holomorphically EX convex, and its cohomology-with coefficients in 0 is finite. We will construct an open set with X, c c X by extending X, in several steps such that every holomorphic function on X, can be approximated by holomorphic functions on

X

�

X. �

{x

: p(x) xi, Ui Ui (Uit UiA(xo) x0 Cri By disturbing p a little bit in a neighborhood of xi, we get a new function P2 and a bigger set X2 = {P2 2 < s} with Xs c X2 and E X2. We do this carefully enough to get X strongly pseudoconvex again. Since everything happens in a cuboid, we can apply the theory of Section V.4 and obtain the desired approximation property for functions in X = X, and X 2 . Then we repeat this process and construct a bigger set X 2 , and so on. N already cover Since X s is compact, finitely many set X i ' X 2 ' X 3 ' ' X Xs, and we order them such tha.!_Xi+ i is obtained bJ: enlarging Xi in a neighborhood of some point i E X 8 • Finally, X, CC X :=XN, and every holomorphic function on X, can be approximated by holomorphic functions In the first step we choose a point Xi E s } , a local coordinate EX system around a sufficiently small compact cuboid around x i in that coordinate system, and an open concentric cuboid cc such that every analytic tangent with E - X, is defined in =

Xi

i

.

•

•

X

�

on X .

Enlar ging: The Fir st Step . We choose an s > 1 . Then p is strictly and also on X - X,. We put plurisubharmonic in a neighborhood of s } and get EX

8Xs = {x

8Xs, 8Xs C 8Xs.

:p(x) = Assume that E axs is a point that is the origin in a local coordinate system, and that Ui = {z : lzl < r} is a compact cuboid in that coordinate system. Furthermore, let U1 Ui be a concentric open cuboid such that every analytic tangent A(x0) with x0 E Ui - X, is defined in Ui- On A(x0) we have p(x) > s outside x0, and there is an c > 0 with p (x) > s + c on 8Ui n A(xo). We assume that p is strictly plurisubharmonic on Ui6 . 1 P r op osition . We can add to p a function < 0 with support U i such Xi

cc

h

that h(xl) < 0, p + h is still strictly p lurisubha rmonic, and the following approximation property is sati5fied:

{x

: p(x)

+ h(:�;) < s}, then every holomorphic function (or If X : = E X cross section in a holomorphic vectorbundle) over X, can be approximated over (i. e . , on every compact subset of X, ) arbitrarily well by holomorphic functions (respectively holomorphic cross sections) over

Xs

X. �

6. The Levi Problem

29 1

PROOF: Assume that K c X, is compact. We choose an R » 0 and con­ sider the family A(x ) of analytic tangents to the points of U 1 n axs . There is a finite set of points yi , . . . , y:r,1 E U 1 n axs such that for the defining functions ft of A( yi ) and the corresponding vector g* = = ( 1/ fi , . . . , 1/ /:;. 1 ) we have the estimate

lg* ( x ) l := maxl 1/ft (x) l > R on aXs n U 1 . Now we replace the points Yi by points Yi very near to Yi , with p( yi ) > s, and the functions ft by fi (the defining function for A(yi)) such that we still have l g (x ) l > R on aXs n U l , for g := ( 1//I , . . . , 1 / /m 1 ) . We choose R and

m1

so big that

l g (x ) l

< R in K n Ui .

Now we add-a smooth function h with support U 1 that is strictly negative on U1 . We let X := {x E X : p(x) + h(x ) < s} and may assume that the points � � Yi are not contained in X and that l g (x )l > R on X - X . We can make h and its derivatives so small that p + h is still strictly plurisubharmonic in Ui (see Figure V .6). ,

�

X A (yJ )

u; Figur e

V. 6 .

The enlarging process

Next we take an R > R and add further functions fx with X E ax , sax fm1 + 1 , . . . , !m , such�that for g = ( 1 / /1 , . ;..: , 1 / fm ) it follows that l g (x) l > R at every point of ax n Ui , and l g (x ) l < R in K n Ui . Then we define

fJ = {x E X n Ui : lg (x) I < R }.

292

V.

We define

Stein Theory

U'

�

by replacing the number R by R in the first

m,

inequalities.

We use the enlarging method of Section V.4 and prove in the same '!ay that every holomorphic function in U' can be approximated by those in U. �

Now we let U and U' increase by making the number m biggcr (i .e., adding new functions) and, simultaneously, by also making R and ..R bigger with R < R. So we can construct sequences Uj , UJ where the U1 converge to Ui n X and the Uj to Ui n X, . Hence, we have the approximation of the holomorphic functions in X, n Ui by those in X n Ui . �

�

�

Let 1f/ be the open covering of X consisting of the two elements W, = X , and W2 = Ui n X . On the intersection W1 2 we can approximate every holomorphic function by those on W2 . Since the cohomology of X is finite, there are cocycles 6 , . . . , �k E Z1 (11/, 0) such that the corresponding cohomology classes � 1 , . . . , � k form a basis of the image of Z 1 (11/, 0) in H 1 (X, 0) . The mapping cp : c k X C0 (1f/, 0) -+ Z 1 ( 1f/ , 0) defined by �

�

cp(a , , . . . ,a k , rt)

:=

a , 6 + · · · + ak � k + J(ry)

is a surjective mapping of Frechet spaces. By the open mapping theorem (see [Ru74] ) cp is an open map.

If g is a holomorphic function in X, , then g can be approximated on W1 2 by a sequence of holomorphic functions fj E O(W2). Since cp is open, the cocycle g - !1 (which is a small function on W1 2) can be written as a sum fj +6(rtj ) , with a small function fj E O(W12) and a (small) cochain 'r/j · Replacing fj - fj by a suitable approximation f, and 'r/j by a suitable (and also small) cochain { g, , Q2 } , we get g - f = Qz - el, with Q1 E O (WI ) and Q2 E O(W2 ) - The function g ( x ) + Q1 (x ) �

g (x ) . .

_

{ f (x ) + Q2 (x )

is holomorphic on X and approximates g on X , as well as we want. �

•

Enlarging: The W.llole Pr oc ess . Let X 1 = X8 , p, = p , and h, = h. Then we define X 2 :=X and p2 := p + h. Repeating the process from above with a suitable point X2 E 8X 2 , cuboids U2 C C U2 , and a suitable function h2 we obtain a new set X = X3 that is defined by a new function p3 = p2 +h2 , and every holomorphic function on X 2 can be approximated arbitrarily well by holomorphic functions on X 3 . Continuing in this way we can always choose the correction functions hi so small that the strict plurisubharmonicity is never disturbed. After finitely many steps we obtain the following:

6.

The Levi Problem

293

There is a strictly pseudoconvex complex manifold X with X, c c X such that every holomorphic function o n X , can be approximated over X , arbitrarily well by holomorphic functions o n X . 6.2 Pr op o sition .

�

We have the analoguous result for cochains.

Assume that 1f/ is an open covering of X and that � E Z1 (1f/ , 0) is a cocycle with � X = 6(rts), � = J(ry) . Th en, by changing ry, the difference ry - rts can be made arbitrarily small o n X, .

6 . 3 Pr opo sition .

I

s

�

Remark. We can replace X by a set Xs +c:, with an c > 0, since such a manifold is contained in X. �

Solution of the Levi Pr ob lem . Let X be an n-dimensional strongly pseudoconvex manifold that as above is endowed with a smooth exhaustion function p that is strictly plurisubharmonic on {p > 1 ) . The solution of the Levi problem will be given in three steps. First we prove an approximation theorem for X , and X .

Eve ry holomorphic func tion in X, s > 1 , can be approxi­ mated by holomorphic functions in X . 6 . 4 Theorem.

PROOF: There is a maximal s with s < s < 00 such that every holomorphic function on X , can be approximated by those on Xs. We have to prove 8 = oo . If this were not the case, we could approximate each holomorphic function on X8 by those on Xs+c: with r:: > 0. This would imply the approximation of the holomorphic functions on X, by those on X8+c: and contradict maximality. • Assume now that K is a compact set in X, s > 1 . Then K is also contained in Xt for t > s. We denote the holomorphically convex hull of K in Xt by Kt. In the second step of our proof we show that Kt = K, for every t > s. �

�

�

�

It is always the case that Ks C Kt , and, due to the approximation results, Kt n X, = K8 • Suppose that there exists a t > s such that Kt is bigger than K8 • Then we can find a minimal t' > s with Kt c X t' · It follows that t' < t (since Xt is holomorphically convex), and there is at least one point xo E Kt n 8Xt' · We choose a very small number c > 0 and consider the set �

�

�

�

�

-

�

Xt' + c:-

Let A(x o) c Xt' + c: be the analytic tangent to Xt' at x0. We may assume that there is a neighborhood U = U(x0) and a holomorphic function g in U such that A(x0) = {x E U n Xt' + c: : g(x) = 0}, and we find a meromorphic function f on Xt' +c: with poles only on A(xo ).

294

V.

Stein Theory

xo

Then we construct a sequence of points � outside of X t' converging to and meromorphic functions on Xt ' + c: with poles only on This can be done in such a way that converges to f with respect to the Fubini metric in C (which will be described in Section V l . 3 ) . Then is holomorphic in a neighborhood of Xt' , takes arbitrarily large values, and supK i f stays bounded. By approximation we can fin supK i f* l , i.e. ,

A(xv)·

fv fv l!v(xo) l

fv

l f*(xo) l

vl

xo

Let us give some more details for the construction of f and fv · To begin with J , we use the open covering '11 = {U1 , U2 } of Xt'+c: given by U1 = U n Xt'+c: and 1 u2 = Xt'+e - A(xo ) . Since dim H (Xt'+e) 0) < oo , there are complex numbers a! ' . . . ' am such that the function a == alg� 1 + . . . + arng- rn is a cocycle on ul 2 that is cohomologous to zero . Then a = h - !I , with !I E O(UI ) and h E O(U2 ) . So we get the meromorphic function f that is equal to a + f1 on U1 , and equal to h on U2 . It has A(xo) as po lar set. We can find a holomorphic family of analytic tangents A(z) , always given by one holomorphic equation 9z = 0, with z arbitrarily near to xo. If m is big enough, then 1 for every z we have a vector space Pz of principal p arts p = a1g-; + . . . + arng-;=

1

and a linear map 'Pz : Pz -+ H (Xt'+c: , O) . Let E, C Pz be the kernel of t.pz . Then dim(Ez) > 1 , and there is a minimal dimension mo < m for Ez that will occur at generic points. Taking E, only at generic points and forming the closure for z -+ xo , we get a regular holomorphic family of vector spaces. If we always take merom orphic functions fz with principal p art in Ez , we get a continuous family (fz) and the desired convergence fz -+ f for z -+ xo.

The last step of our proof is simple: Let K c X be any compact set. Then K is contained in some K, s > l , and from above we know that Kt = Ks for every t > s. Let x be an arbitrary point of the holomorphically convex hull K . Then E Xt , for some t > s . I ff is a holomorphic function in Xt , then there is a holomorphic approximation f E O(X ). Since we have < < supK i f l , we can conclude that supK as well. So x E Kt , and K = K, . This proves the following theorem. �

�

�

I !I

�

l[(x)l

�

x

l f(x) l

�

6 . 5 Theorem . If X is strongly pseudoconvex, then X is also holomorphi­ cally convex. It is a Ste in manifold if and only if there are n o compact analytic subsets in X ofpositive dimension.

In addition, we have the following result. If X conta ins a compact analytic subset A of positive dimension, then already A c X 1 .

6 . 6 Proposition .

�

�

The PROOF is the same as that for K = K1 . We also o b t ain t h e following

theorem.

6.

The Levi Problem

295

6 . 7 Theorem . If X is a strictly pseudoconvex (i. e., ]-complete) manifold and V a holomorphic vector bundle o n X , then V ) 0.

H 1 (X,

=

PROOF: This has already been proved for every distinguished Stein mani­ fold X, s > 1 . By approximation of coboundaries we get the result for X .

•

The C ompact C a se. Let X be a compact complex manifold, and V a holomorphic vector bundle on X . We can find a pair of finite open cc coverings '2/ , % * ) for X consisting of concentric cuboids such that V) 0 for all i. Then C0 ('P/, V), Z 1 ('P/, V) and V) Z ( '2/ * , V )are Fnkhet spaces. The finiteness of the coverings is essential here!

( 1 (Ui , H 1

=

Ui

H 1 (Ut,

Ut

=

Again we consider the linear mappings

such that u is surjective, v is completely continuous, and Im( u + v ) = B 1 ('P/, V). Since the coverings are acyclic, it follows from the theorem of Schwartz that dime ( X, V ) < oo.

H1 All restriction maps f(Ut, V) f(U, V) are completely continuous. From this it also follows that id f(X, V) f(X, V) is completely continuous, and f ( X, V ) is finite-dimensional. So we have the following theorem. :

-+

-+

6 . 8 Theorem . If X is a compact complex manifold, then dime

f(X, V )
--+

a i .. . ir ( x ) are smooth functions. ,

Henceforth, the set of all (smooth) r-forms on U will be denoted by szfr (U) , and the subset of all forms of type ( p , q) by szf (p, q) (U). If f is a smooth function on U, then its differential df E s4' 1 (U) is given by x >--+ ( df )x . In local coordinates we have

dJ =

n

n

v=l

v= l

2:: !zv dzv + 2:: hv azv .

A (smooth) vector field is a smooth section of the tangent bundle T(X) . So in local coordinates it can be written in the form

1

The tangent bundle T(X) is a complex-analytic vector bundle of rank n . Here we denote its r ea l dual bundle by T* (X) . It is a real-analytic bundle of rank 2n (over �). The complex dual bundle T' (X) of T(X) (with fiber T� (X) = Homc (Tx ( X ) IC) ) is a complex-analytic bundle of rank n over IC. ,

1 . Differential Forms

301

where the coefficients �v are smooth functions. Then we can apply df to such a vector field and obtain n

n

v=l

v=l

dJ( � ) = L �vfzv + L �Jzv · For any open set U the differential can be generalized to the Poincare map d = du : Jt1r (U) ---+ Jt1r+ I (U) in the following way:

If w =

""' /

a . · w· A �1

"'

. .

1-r

'L l

1 0 for every tangent vector v -1- 0. Prove that i88llzll 2 is a positive form on e n . 3 . Consider the (n, n - I)-form ry0 on e n defined by

1]o

:=

n k ( - 1) n ( n- 1 )/ 2 L(-1) z k dz1 A · . . A dz, A dz 1 A · · .A di;., A · . · A az n k=l

and calculate 8ry0 . Let f be a holomorphic function on an open neigh­ borhood of the closed ball Br (O) , and w : (f(z)/ llzll 2n ) · ry0 . Show that dw 0 4. Let X = = lPm . On Uo : {(z0 : z1 : . . . : z,) E ]pm : zo -1- 0} we use the holomorphic coordinates t, zv / z0 . Then w0 dt 1 A · · · A dt, is a holomorphic n-form on U0 . Show that a holomorphic n-form on an open subset U c X is the same as a holomorphic section over U in the canonical bundle Kx . Prove that there is a meromorphic section s in Kx with s lua w0 and div(s) - (n + 1)Ho , where H0 {z0 = 0} is the hyperplane at infinity. =

=

.

=

=

=

2.

=

=

=

Dolb ea u lt Theor y

Integr ation of Differ ential Forms . We recall some results from real analysis. Let G c e n "' lll1. 2n be a domain. A closed subset M c G is called an r-dimensional submanifold of class � k if for each z E M there exists a neigh­ borhood U of z in G and a � k map f : U + lll1. 2n - r such that: 1. u n M r - 1 (0). 2. The rank of the (real) Jacobi matrix o ff is equal to 2n - r. =

VI. Kahler Manifolds

304

It follows from the (real) implicit function theorem that for any z E M there is a neighborhood U of z , an open set W c Jll1.r , and a � k map cp : W ---+ G such that:

1. cp maps W homeomorphically onto U. 2. The rank of the Jacobian matrix of cp is equal to r.

Then

cp

is called a local parametrization of M.

An r-dimensional differential form w = Lp+ q=r L I , J afJ dz1 A az J is called an r-form of class � k if all coefficients are functions of class k. Then dw is an ( r + 1 )-form of class �k - 1 . We always assume that k > 1 . (a

---+ B2 is a �k map between domains B1 c em and B2 c en , then every r-form w of class �k on B2 can be lifled back Lo B1 by

If

: B1

( * w)z ( v1 , . . . , vr )

W ( z) ( * v1 , . . . , * vr ) · If u 1 , . . . ,u 2n are real coordinates in B2 then * (L ai, ... ir dui , A . . · A duir ) L(a i, .. . ir o ) d(u i, o ) A " · A d(uir o ). :=

=

* w is again an r-form of class �k . We have: * (w1 A w2 ) = * w1 A * w2 . * ( dw) = d( * w) , in particular * (df) = d(f o (a).

Therefore,

1. 2.

Now let w = a dz 1 A . . . A dz , A az 1 A . . . A d:Zn be an arbitrary (2n)-form. Since dzi A azi = -2i dxi A dyi, it follows that

a · ( -l) n( n - 1 )/ 2 dz1 A az 1 A . . . A dzn A dz n n n 1 2 n dx 1 A dy 1 A · · · A dx, A dy n -2) i . ( )/ . ( a ( - l ) n +1 a · ( - lt ( n ) 1 2 (2i) n dx 1 A dy 1 A . . . A dx n A dy,. We write a := a · (-l) n( n+ 1 ll 2 (2�r · If B c en is a bounded domain and a continuous in a neighborhood of B, then w

l w l a(x1 + iy1 , :=

0 ° 0 ,

Xn + iyn ) dx 1 dY1

° 0 0

dx n dYn·

M c G is an r-dimensional submanifold of class �k , w an r-form of class �k , and K c M a compact subset that is contained in the range of a local parametrization cp : W ---+ M , then we define

If

r w jK

·=

1 0,

·

8 operator. For this we start in

f E 'i&' 1 (G) . Let f E 'i&' 1 (C) be

for bounded functions

a func tion with compact support. Then there is a contin uous ly diffe rentiable function u on C with U z- = f .

2. 3 Theorem.

PROOF: Choose an R > 0 such that supp(f) cc D := DR(O) . Then u : Ch f is a continuously differentiable function on D, and we can assume that the integral is taken over the whole plane. It follows that

=

( ) = 21 '

_

U Z

7r

I

1 f (( + z) dl 2, K c G a compact subset. If G - K is connected, any holomorphicfunc tion on G - K has a holomorphic extension ! o n G .

f

�

P ROOF : Choose an open neighborhood U = U(K) cc G and a smooth function (! on e n with supp(Q) cc G and (! lu = 1 . Define

9k : = f · UZlc ;;; , We have For fixed

z"

= ( z2 ,

ogk azz .

•

.

,

=f.

for k

EJ2(! 8zk azz

Zn ) we define

=

1,

. . . , n.

VI. K ahler Manifolds

308

Then U-z1 = 91 (we can apply the previous corollary, since support) , and for k > 2 we get

91

(9 1 ) zk (( , Z2 , . . . , Zn ) d( 1 2 7rl ( - Z1 � 1 (9k ) z1 (( , Z2 , . . . , Zn ) d ( 21r1 (-z

�

IC

1

c

has compact

1\

d(

1\

d(

9k (z1 , z2 , . . . , Zn ) · Therefore, u is a holomor­ For z rt supp (Q) we have U-zk ( z ) = 0, k = 1 phic function outside s upp (Q) . Since 91 ( (, z' ) = 0 for arbitrary ( and large II z' II, it follows that u ( z1 , z ') = 0 for large II z' I I , and so u is defined everywhere ,

.. . , n .

and is identically zero on the unbounded connected component Z of G - K.

There exists a nonempty open subset V of Z n (G - supp (Q)) C G - K . The function f := ( 1 - Q) . f + u is defined on G and coincides with f on V. In addition, we have �

in G - K, in So f

is

U.

holomorphic in G.

•

Dolb eault' s Lemma. Before provmg the 8 analogue of Poincare's lemma we need the following result.

Let P c c Q be polydisks around the origin in en . Let be a smooth June tion in Q that is holomorphic in zk + 1 , . . . , Zn , fo r some

2.6 P r op osition.

g

k>

1.

Then there exists a smooth function u Zk+ 1 , . . . , Zn such that u-zk = g on P .

in

Q that is again holomorphic

in

PROOF: There are disks E cc D around 0 in the zk -plane such that P = P' x E x P" and Q = Q' x D x Q". Let Q be a smooth nonnegative function on C with QI E = 1 and Q k - D = 0. Then we define

U ( Z1 , . . . , Zn ) . -_ _1_ .

.

2 7rl

1 Q(() · 9(z1 , · · · , Zk-- 1 , (, zk+ 1 , · · · , zn ) d1 C

o.,

( Zk

1\

dl o.,

The function u 1s defined on Q , and the integrand has compact support. Therefore, U-zk (z) = (Q . 9) ( z) = 9(z) on P, and by the rules for parametric integration we get u-z So u is holomorphic in these variables.

,

= 0 for i = k + 1 , . . .

,n.

•

2.

2. 7

en .

Dolbeault Theory

309

Dolb eault' s lemma. Let P c c Q be poly disks around the origin in For each :p E ( Q) with q > 0 and 8:p = 0 there exists a 7/J E with 87jJ = :P IP ·

a'p,q

JL1P. q-1 (P)

PROOF : Without loss of generality we may assume that p = 0. Let k < n be the smallest number such that :p does not involve the differen­ tials dzk + 1 , . . . , d:Zn . We carry out the proof by induction on k .

For k

= 0 there is nothing to prove,

since

q

> 0.

Now we assume that k > 1 and the theorem has already been proved for k - 1 . Then we write

:p = dzk A + (3, o

where If a = then

a

E J4'0 ·q- 1 (Q) and (3 E !4'0 •q (Q) are forms not involving azk ,

I: jJ I=q-1 aJazJ 0

--

(where

I:* means that azk > . . . , d:Zn do not occur),

-azk A 8a + 8(3 - L azk A azv A

= D:p

v =#

. . . , d:Zn .

(aJ)zv dzJ) + 8(3. ( I JL* I=q-1

= 0 for v > k. By the above proposition there Then it follows that are smooth functions in Q that are holomorphic in Z k +1 , . . . , Zn such that = on P . We define y : = Then

(aJ)zv AJ

(AJ )zk aJ

a,

=

I:jJ I=q-1 AJ azJ. L* L (AJ )zv azv A dzJ I J I=q-1 v !f-J L * aj azk A dzJ + L (AJ)zv azv A cLzJ v . . . ,d:Zn . We see that also (3 - ry does not involve azk , · · · , dzn · Since (3 - ry = (:p -azk Aa) -(81-azk Aa) = :p -81 , we have 8((3-ry) = 8:p = 0, and the induction hypothesis implies that there is a form (}' E J4'0·q- 1 (P) with {)(}' = (/3 - rt ) IP · Setting 7/J := (}' + y , it follows that

87/J = ((3 - rt ) + (az k A a + rt ) = :p

on P.

This completes the proof. We immediately obtain the following result for manifolds X :

•

3 10

VI. Kahler

Manifolds

If U C X is an open set and cp E pf0·q (U) a smooth fo rm with q > 1 and acp = 0, then fo r every x E U there exists an open neighborhood V = V (x) C U and a fo rm 7/J E JZ1D,q - l (V) with a'ljJ = cp l v . Without proof we mention the following result, which provides more precise information (see [GrLi70] and [Li70] ) . 2.8 Theorem (1 . L ieb). Let G c c e n be a Levi convex domain with smooth boundary and w = L J aJ cLzJ a smooth (0, q) -form on G with aw = 0. Moreover, suppose that there is a real constant M > 0 with

llw ll

:=

ma J x suGp l aJ I < M .

The n there exists a constant k independent on G with a'ljJ = W and 11¢11 < k . M .

ci

w and afo rm 7/J

of

type ( O,q - 1)

Siu and Range generalized this theorem to domains with piecewise smooth boundary (see [RaSi] ) . L. Hormander has given a quite different proof for the solution of the 8-equation with L2 -estimates on pseudoconvex domains using methods of functional analysis (see [Hoe66] ) . In the meantime there exists a big industry of solving 8-equations, but we do not want to go here into detail.

Dolbeault Gr oup s . Let X be an n-dimensional complex manifold and U c X an open subset. Then we have natural inclusions2

Together with the Poincare differential d and the a-operator we get the fol­ lowing sequences of C-linear maps:

1 . The de R h a m sequence

If U is diffeomorphic to a star-shaped domain, then the sequence is exact, i.e., {f E d0 (U) : df = 0) = C and zr (U) := Ker ( d : Jd'r (U) ---+ Jt1r + 1 ( U)) is equal to Br(U) := Im ( d : dr- 1 (U) ---+ dr(U) ). In general, we have only Br(U) c zr ( U) (because d o d = 0) . 2 . The Dolbeault sequence

We define { cp E dp,q ( U) : acp = 0} for 0 < q < n , - a(dp,q - 1 (U)) for 1 < q < n and BP · 0 (U) Then Bp,q C Zp,q, because a o a = 0 (respectively 0 C SlP(U) ) . 2 Here C stands for the set of locally constant functions.

.·

0.

2.

Dolbeault Theory

311

Definition. The complex vector space Hp,q (U) := Zp,q(U)j Bp ,q(U) is called the Dolbeault (cohomology) group of type (p, q ) of the set U, and the complex vector space Hr(U) := zr (U)j Br(U) is called the rth de R ham (cohomology) group of U.

We have

> 1 the condition Hp, q (U) = 0 means that for every cp E dp , q (U) with flcp = 0 there is a 7/J E dp , q - l (U) with 87/J = cp. From the solution of the 6'-equation (using, for example, [Hoe66] , chapter IV , or Dolbeault's lemma and an approximation theorem due to Oka) we obtain that Hp,q (G) = 0 for every strictly pseudoconvex domain G c en and q > 1. Eor

q

If U is connected, then H 0 (U) = C. If U is star-shaped, then Hr (U) = 0 for r > 0. Now let n� == /'tT' (X) be the holomorphic vector bundle of holomorphic p-forms on X (in particular, 0� = Ox , 01: = T' (X) and flx = Kx ) . Then flP (U) = r ( U, n� ) for every open set U c X. We construct a linear map

For every � E Z1 (X , n� ) there is an open covering CZt = { Ui : i E I } of X such that � is given with respect to this covering. Then we can pass to any refinemenl. So we may assume Lhal CZt is as fine as we wanl.

Let � = ( �ij ) E Z1 ( ezt , 0� ) be given. Then �ij E flP ( Uij ) and �ij +�jk = �ik on Uijk· Let ( ek ) be a partition of unity for the covering CZt . We define differential forms 'Pi := L ek�ki E sz1P·0 (U; ) , for i E /.

kEf

'PJ - 'Pi = I > k ( �kj - �ki ) = I > k�ij = �ij · k

In

UiJ we have

acp1 - acpi = a�;1

k

= o.

Therefore, a global (p, 1 )-form w on X can be defined by w \ u, : = acpi. Now, dw = o. s o w E ZP·1 (X) .

If � is a coboundary, � = Jry for some ry E C0 (CZ? , O� ) , then 'rfJ - 'r/i = �,1 = 'PJ - 'Pi on Uij , and global (p,O)-form 7/J on X can be defined by 7/J\u, : = 'r/i - 'Pi· We get a

3 12

VI.

Kahler Manifolds

8( -7,/J) I ui = 8cpi - 8rti = w I u, ,

since 'Tfi is holomorphic.

D

So the assignment � >--+ w induces a map : H 1 (X , n� ) ---+ HP· 1 (X) by [�] >--+ {w], where [ . . . ] denotes the residue class. One can show that this definition is independent of the covering CZt .

2.9

D ol beau lt ' s

theorem .

isomo rphism.

The map D

:

H 1 (X, O�) ---+ HP· 1 (X ) is an

D

is injective, we PROOF: We use the notation from above. To show that suppose that there is a (p,O)-form (! on X with 8 (! = w. Then 8cpi = 8Qi ui , for every i E I. It follows that 8( 'Pi - (!) = 0 and therefore Ti := 'Pi - (! 1s holomorphic on ui 0

Then T = (7;) is a cochain of holomorphic pform s, with

So the class of � vanishes. Now let an element w E ZP· 1 (X ) be given. Since 8w = 0, we can find a covering CZt = {Ui : i E I } of small polydisks (in local coordinates) with (p, 0)-forms 'Pi such that 8cpi = wl ui . Then 8( 'Pi - 'Pi ) = 0 for every pair (i, j), and �ij := 'Pi - 'Pi lies in flP (Uij ) · Clearly, � = (�ij ) is an element of Z 1 (ezt , n� ) whose cohomology class is mapped by onto the class of w. This shows that is surjective. •

D

D

Dolbeault's theorem has an interesting consequence:

Let X be a Stein manifold. Ifcp is a (p, l ) -form on X with = 0, then there exists a fo rm 7,/J o n X of type (p ,O) with 87,/J = cp.

2. 1 0 Theorem.

8cp

PROOF: By Theorem B for Stein manifolds we have H 1 (X, n� ) = 0 for every p, and the result follows from Dolbeault's theorem. • Remark. Dolbeault's theorem is also true for forms of type (p, q) with q > 1 . For the proof one has to introduce higher cohomology groups H q (X, 0� ), to generalize Theorem B for those cohomology groups and to show that H q (X, 0�) "' Hp,q (X ) .

In an analogous way one can prove de Rham' s t h e or em :

where Hr (X, q denotes the cohomology group with values in the set of locally constant functions (isomorphic to the singular cohomology group with values in C ) . Let CZt be an open covering of X such that all intersections tractible. In the case r = 2 the de Rham isomorphism

u.o

. .

·•k

are con­

2.

Dolbeault Theory

3 13

is given as follows: Let w E d2 (X) be given, with f3v E d1 (Uv) with

dw = 0 . On every Uv wluv = df3v ·

Then d(f3�" - f3v) = 0 on Uv w Therefore, on every (complex-valued) function fvl" with

Now,

there exists an element

Uv�" there exists a smooth

d(fl">. - fv>. + fvl") = 0 on Uvl">. · It follows that av J"A : = ( !!").. - !v>. + !vl") l uv,>.

is constant. So we have a = (av�">.) E Z2 ('1f , C) , and the de Rham class {w]E H2 (X) will be mapped onto the cohomology class {a] E H2 ('1£ , q. Exercises

1 . Let ry0 be the

( n ,n - 1 )-form defined in Exercise 1 .3, and

function on an open neighborhood of the closed ball

f a holomorphic B r (O). Prove that

(n - 1 ) ! r f(z) J (O) = (2 7ri ) n Ja sr ( O ) ll zll2 n . 'Tfo (integral formula of Bochner�Martinelli) . 2. Let (rv) be a monotone increasing sequence tending to r > 0, and let cp be a smooth (0, q )-form on p n (o, r) with q > 0 and 8cp = 0. Construct a sequence ( 7/Jv) of smooth (0, q - 1 ) -forms on p n (o, r) such that (a) 87/Jv = cp on p n (o, rv)· (b) 17/Jv +l - 7/Jv l < 2 � v on p n (O , rv � !) in the case q = l, and 7/Jv + l = 7/Jv On p n (O , Tv� I ) for q > 2. Show that Hp,q (P) = 0 for q > 1 and every polydisk P in en . 3. Let X be a complex manifold and 1r : E ---+ X a holomorphic vector bundle of rank q over X . If cp : Elu ---+ U x Cq is a trivialization, there are sections �v : U ---+ E defined by cp o �v(x) = (x, ev) · The q-tuple � = (6 , . , �q ) is called a frame for E . Use the concept of frames to define differential forms of type (p, q ) with values in E locally as q-tuples of forms. Let dp,q(U, E) be the space of (p, q ) -forms with values in E over U. Show that there is an analogue of the Dolbeault sequence for forms with values in E. .

.

3 14

3.

VI.

Kahler

Manifolds

Ka hler

Metrics

Her mitian metrics . Assume that X is an n-dimensional complex man­ ifold. Definition . A He rmitian metric H on X is an assignment of a positive definite Hermitian form Hx to each tangent space Tx (X ) such that locally in complex coordinates it can be written in the form n

Hx = L 9ij (x) dziazj ,

i ,j =l

with smooth coefficients 9·iJ

If a : [0, 1 ] ---+ X is a differentiable arc, we can assign to it a length

LH (a ) : =

1

1 JHa(t) ( a (t) , a (t)) dt,

where a (t) = a * a;at = 2::: " a� (t)a;azv+ I:" a� (t)a;az" and a = (a1 , . in local coordinates. So

. . , an)

The length is positive if a is not constant

K

ow, for a tangent vector � E Tx (X ) a norm can be defined by

1

Then LH (a) = f0 lla * 8j8tii H dt. There is a uniquely determined differen­ tiable 1-form a along a defined by

(

(J'a( t) (�) = Ha( t) � lla '

����� )

· H : It followsthat a * a = a d t, with a = a(a * 8j8t) = lla * 8j8t11H · Traditionally, a is called the line element and is denoted by ds, though it is not the differential of a function. We have

LH (a) a * Ha(t ) = lla* 8/8tl17:r dt 2 tian metric H by ds 2 . and

=

e a * ds

fo

=

I. ds, ,.

= (ds) 2 . Therefore, we also denote the Hermi­

30 Kahler Metrics

315

Let CZt = (U,),E I be a (countable) open covering of X by local coordinate charts with coordinates (zL 0 0 0 , z� ) Then there exists a partition of unity (Q,) for CZt , and o

ds 2

n

L Q, 0 L dzt dzi

:=

LE I

i=l

is a Hermitian metric on X 0 Thus we have the following result.

3 0 1 Pr oposition 0 nco

0n

every complex manifold there exists a Hermiti an m e t­

The Fund amental For m . In a local coordinate system we associate to our Hermitian form H = ds2 = Li ,J. 9ij dziazj the differential form t ,J

(In the literature one often uses w = i /2 0 L 9ij dZi A - i 0 Li ,j 9ijazi A dz, = W So w is a real ( 1 , 1)-formo

azj o)

We have w -

o

Let G = (% I i) = 1 , 0 0 0 ,n) be the (Hermitian) matrix of the coefficients of H, and e, rJ the coordinate vectors of two tangent vectors �, 'rf Then H ( �, rt ) = e c ry t o

o

.

o

If we apply a holomorphic coordinate transformation z = F (w) = (fr (w) , . , 0 , fn (w)) ,

we obtain

This can be written in matrix form as t G (w) = JF (w) 0 G (F (w)) 0 JF (w) ,

which is the transformation formula for Hermitian forms. Thus F*wH = WF· H 0 So the correspondence H +-+ w is independent of the coordinates. The form w is globally defined. In fact,

j _

o

-

(e

-

c

o

ry

t

2 1m ( H ( � ry)) ,

ry

-

o

c

o

-t

e

)

o

The form WH is called the fundamental fo rm of the metric H

VI.

316

Kahler Manifolds

Definition . A Kahle r me tric on X is a Hermitian metric H with dw H = 0. If X possesses a Kahler metric H, the pair ( X ,wH ) is called a Kahler manifold. In this case we also say that X is a Kahler manifold. In general, a Hermitian metric H does not satisfy the condition dwH = 0 . There are many complex manifolds that cannot have a Kahler metric. How­ ever, on a Riemann surface X every 3-form vanishes. So every Riemann sur­ face is a Kahler manifold.

Geodesic C oor dina tes. Assume that H is a Hermitian metric on the manifold X and that z = ( z l , . . . , zn ) are holomorphic coordinates around some point x0 E X . Definition . The coordinates z are called geodesic in x0 (with respect to H ) if H = "' L.t,J g'·J· dztazJ· and 1. 9ij ( x0 ) = 6ij , i.e. G ( xo ) is the unit matrix, 2. (9ij)zv ( xo) = O and (gij);Zv( xo) = O for v = l , .. . , n, i .e., ( dgij )x0 = 0. ·

·

We will show that the existence of geodesic coordinates is a characteristic property of Kahler metrics. For that we need a lemma. 3 .2 Lemma. H = Li ,J. 9ij dziazj is a Kahler me tric (9ij ) z,, = (9vj )zi

and

if

and only

(9ij hv = (giv )zJ , for all i, j,

if

v.

( The firs t equation implies the second one.) PROOF: dw

Let

w

= w H be the fundamental form of H. Then we have

( + ?= (9ij )z v azv A dzi A azj ) I · ( 2:) 2:) (9ij ) zv - (9uj)zJ dzv A dzi) A azj + 2::) 2:) (9iv )z1 - (9ij )zJ azv A d:Zj ) A dz; ) . v J I · ?:= (9ijk dzv A dzi A azj �,),V

't 1 ) 1 V

v 0, and (r.p , r.p) = 0 if and only if r.p = 0 . 1.

('PI + 2. (c 3. ('t/J ' ·

PROOF:

( 1 ) and (2) are trivial.

7/J A * r.p 7/J 1\ * r.p = r.p 1\ * 7/J . 0 always. If r.p "I 0, then there is a point x0 (4) It is clear that ( r.p, r.p) X with 'Px o "I 0. If x o E Uv and Qv ( x0 ) > 0, then we can find a small neighborhood V = V ( x0 ) C C Uv such that Qv'P = c dV on V, with c > 0 everywhere in V . It follows easily that ( r.p , r.p) > 0. Since ( . . . , ) is a Hermitian scalar product on dr(X) , a norm on this space is defined by II r.p II : ( r.p , r.p) I 1 2 . 4 . 6 P r op osition . Twojo rms different type are orthogonal to each other. PROOF: Let r.p, 7/J be two forms of type (p, q) , respectively ( t ) , such that + n - t). If Then r.p 1\ * 7/J is a form of type ( p + n +t p +q s > p , then q > t and therefore q + n - t > n. If s < p , then p + n n. So r.p 1\ * 7/J 0 in these cases. =

(3) We have

E

>

•

. . .

=

s

=

=

=

r.

of

s, s, q

s >

•

328

VI. Kahler Manifolds

Unfortunately, dr (X) is not complete in the induced topology, so that we cannot apply Hilbert space methods directly. We will show how to construct a closure of dr(X).

Curr ents. Let X be a compact Hermitian manifold. A sequence 'P v = :E r a 1,v dn1 of (smooth) r-forms on X is said to be convergent to zero in dr (X) if for every a all sequences Dcxar ,v are uniformly convergent to zero. Definition. A current of degree 2n - r is an JR-linear map T : dr(X) --+ C such that if ( 'Pv) is a sequence of r-forms converging to zero, then T ( 'Pv) converges to zero in C.

The set of all currents of degree 2 n

-

T

Examples

7/J

is denoted by dr(X)'.

7/J

1 . A current of degree 2n is a distribution in the sense of L. Schwartz. 2. Let be a differential form of degree 2 n - r. Then defines a current T,p of degree 2 n - r by

T,p [r.p]

:= lx 7/J I\ r.p. r

It is easy to see that T,p is, in fact, a current. So .0"2n- r (X) C dr (X)' . 3. Let M C X be an r-dimensional differential submanifold. Then a current TM is defined by TM [r.p] := JM r.p. Therefore, we say that a current of degree 2 n - r has dimension r .

Let ( 'Pv) be a sequence in dr (X) converging to some r-form r.p (i.e., 'Pv - r.p --+ 0 ) . Then also *('Pv - r.p) --+ 0, and therefore T,p [*('P v - r.p)] --+ 0 in C for every 7j; E dr (X). It follows that

v-+ oo 7/J lim (

Definition.

,

'P v)

* 'Pv = lim T,p [ * 'Pv] 7/J 1 T,p [ * r.p] = 7/J * r.p = (7/J r.p) . 1 ' lim

V---7 00

{

X

For T E dr (X)' and

(7/J ,

1\

V ---7 00

1\

r.p E .0"2n - r(X)

r.p) . If In particular, we have (T,p , r.p) = limv-+oo ( 7/Jv , r.p). This motivates the following.

7/Jv --+

we define

7/J , then (T,p , r.p)

5. Hodge

Definition. limv-+ oo 7/Jv E

Decomposition

329

If (7/Jv) is a Cauchy sequence in ,0" 2n -r(X), then T = dr(X)' is defined by

(T , cp)

:= lim (7/Jv , V -+ 00

cp)

The limit T is called a square integrable current. We omit the proof that T is actually a current.

Before defining the notion of a limit of a sequence of currents we note that ( T, cp) - (7/Jv , cp )

T = vlim 7/Jv -+ oo

T[cp] - T,p Jcp]

....-.+

....-.+

0 for all

0 for all

cp E .0"2n-r (X)

cp E dr (X) .

Definition. A sequence (T, ) of currents in d r (X)' is called convergent to a current T E dr (X)' (written as T = limv-+oo T, ) if

lim

V -+ 00

(T[cp] - Tv [cp])

=

0 for all

cp E dr (X) .

We state the following without proof.

Th e space dr (X)' of currents of degree 2n - r is complete; i. e . , every Cauchy sequence converges. Th e set of square integrable currents -2n-r fo rms a complete subspace of dr (X)', which we call the closure d (X) of the space of (2n - r)-forms. 4 . 7 Theorem.

Exer cises 1 . Compute the volume of the complex projective space JIDn with respect to the Fubini metric, and the volume of a p-dimensional linear subspace. 2. Let (X, w ) be a compact Kahler manifold and Y c X a closed subman­ ifold with dim(Y) = m. Then vol(Y) = c . JY wm for some constant c. Calculate this constant! 3. Try to define the product of a current and a differential form! 4. For T E dr (X)' define dT E ,0" r+ l (X)' by dT[cp] : = ( - l y+1T[d!t?] · Show that dT is in fact a current with d(dT) = 0. Calculate dT for the case that T is a form or a submanifold.

5.

H od ge Decomp osition

Adjoint Op er ator s . A complex vector space with a Hermitian scalar product is called a unitary vector space. Let V1 , V2 be two unitary vector

330

VI.

Kahler Manifolds

spaces, and T : V1 ----+ V2 an operator (i.e., a (>linear mapping). If there exists an operator T* : V2 ----+ V1 with

(T ( ) , w) = ( , T* ( )) v

v

w

for v E

V1 , w E V2,

then T* is called an adjoint operator for T. It is clear that T* is uniquely determined, but in general it may not exist.

If V1 , V2 are unitary vector spaces and T : V1 ----+ V2 is a continuous operator, then it follows from the Cauchy-Schwarz inequality that v f--+ (T( v ) , w) is continuous for any fixed w E V2. If V1 , V2 are Hilbert spaces, then it follows by the Riesz representation theorem that there exists an element T* ( w ) with (T(v) , w) = ( v , T* (w) ) , and that w f--+ T*(w) is continuous. Now let X be an n-dimensional complex manifold

5.1

Lemma.

is compact and r.p a (2n- 1)-form on X , then fx dr.p = 0 .

If X

PROOF: Let %' = {Uv : v = 1 , . . , N} be a finite open covering of X by local coordinates and ( f2v) a subordinate partition of unity. We choose open subsets U� cc Uv with piecewise smooth boundary and supp(f2v) c U� . Then .

by Stokes's theorem. 5.2

The operator o := - Hi* dr+l (X) ----+ dr(X) is the d : dr(X) ----+ ,0"r+l (X). :

Propo sition.

adjoint of

PROOF:

r.p 1\

•

* 7/J

Let r.p E dr (X) , 7/J E is a (2n - 1)-form, and

,0"r+l (X)

be two arbitrary forms. Then

Therefore ,

(dr.p ' 7/J)

L dr.p i\ *7/J = L d(r.p i\ * 7/J ) - ( - 1 Y L r.p l\ rl* 'ljJ ( - 1 y+ I + 2n-r L r.p i\ * * M 'l/J = L r.p l\ *(* rf* 'l/J) ( r.p ' -* ri* 'l/J) . -

•

5 . Hodge Decomposition

331

5.3 Propo sition . _ Th e operator 6 := - * ch : ,0"P+ l . q (X) ....-.+ ,0"P. q (X) , re­ spectively {) := - * crt. : ,0"p,q+1 (X) + .0"p,q (X), is the adjoint of d , respec­ tively 8.

PROOF: We consider only the first case. Let r.p be a form of type (p ,q) with P + q = r and 7/J a form of type ( p + 1 , q) . Then r.p 1\ * 7/J is a form of type (n- 1,n) .Therefore, 8(r.p A * 'tP ) = 0 and

It follows that (8r.p ' 7/J)

fx ar.p A * 7/J = l d(r.p A * 7/J) - ( - 1 Y l r.p A 8(* 7/J) ( - 1r+ I +2n -r l r.p i\ * * 8 ( * 7/J ) = (r.p , - * fn 'l/J ) . •

5.4

l. 2. 3. 4-

We have

Propo sition.

o = rJ + rJ , oo = 0, {){) = 0 and {){) = 0, {){) + {){) = 0, rJr.p = rJ( r.p) and rJcp = rJ( r.p) .

The proofs are trivial.

The Kahlerian Case . Now let X be an n-dimensional Kahler manifold. For the moment X is not assumed to be compact. A canonical operator L : dr(X) ....-.+ ,0" r+2 (X) is defined by Lr.p :=

l 2 r.p 1\

w,

where w is the fundamental form. If cp is a (p,q)-form,then Lcp is a (p+ 1, q+ 1 ) form. Since dw = 0 and w is of type ( 1 ,1), we also have dw = ow = 0. It follows that 8Lr.p = -1 8(r.p 1\ w) = � 8r.p 1\ w = L8cp z ? ­

and

2

[)Lr.p =

In addition, we have

! o(r.p 1\ w) = � {)r.p 1\ w = Lor.p . 2

Lcp = Lip.

332

VI. Kahler Manifolds

5. 5

Proposition. If X is a compact (Kah ler) manifo ld, then the operator A : dr+2 (X) ....-.+ dr (X) ot L is given by

If '1/J is an ( r + 2)-form, then w A

PROOF:

( Lcp , '1/J )

* 'ljJ is a

(272

- r)-form, and

� k cp (\ w (\ * '1/J � fx ( - 1r cp (\ ** w =

_

adjoint

(\

* '1/J

(cp , (-lr * L * 'I/J) .

•

Note also that A is a real operator: Acp 5. 6

Proposition. hold.

=

Acp.

Th e commutativ ity relations {)A

=

A{) and {)A

=

A{)

We leave the proof to the reader.

Br acket Relations. Lel k, l be inlegers, and S : dr (X) ....-.+ dr+k (X), T : dr(X) ....-.+ ,0"r+ 1 (X) linear operators that are defined for every r. Then [S .T] := S o T - T o S : .0"r ( X )

....-.+

dr+ k+ l

is called the bracket or the commutator of S and T. Example

As we have seen above, 6, {) : dr ....-.+ dr- l and A : dr ....-.+ dr -2 have the brackets [rJ, A] = [rJ, A] = 0 (as operators from d ' to dr- 3 ). Before we calculate more brackets, we have to calculate {)cp very explicitely for a form cp of type (p, q) . For that we need geodesic coordinates. So everything that follows is valid only on Kahler manifolds. Let cp = I: 1 J aiJ dz1 1\ azJ be an arbitrary (p ,q)-form. We consider geodesic coordinates a t a point Xo E X such that w = i I:v dzv (\ azv . Then

* cp = L aiJ *(dz1 1\ azJ ) = a (p , q) · L aiJ dzl' 1\ azp , I,J !,1

where a (p , q) = 2p+q-n i n ( -l ) q (n-p)+n(n+ l ) / 2 , and C J , J' = C J ',J' = 1. In a neighborhood of x0 the formula is more complicated, because the coefficients 9iJ of the metric are involved. We have to apply a in this neighborhood, but because we are working with geodesic coordinates, we obtain the exact value at Xo by simply applying a tO OUr representation Of * cp at Xo . It follOWS that

a *

-. 1\ dz1 1\ dz J =

i acp . •

5. B C or ollary.

[ �] = - i a,

[a, A] = - i rJ

[a, A] = i rJ .

and

The proof is an easy exercise.

T he Laplacian . Let X be an n-dimensional compact complex manifold. Definition .

The (real) Laplacian A : dr(X ) -+ dr(X ) is defined by

The complex Laplacian

D

: dp,q (X) -+ dp,q(X) is defined by u p : = (o rJ + rJ IJ) cp .

Since d and 6 are real operators, A is also real. 5.9

Proposition.

On a compact Kahler manifold,

0

= D.

5.

PROOF: We have a {) = - {} a and a {) bracket relations. Therefore,

-i o

=

Hodge Decomposition

335

- {} a, as can be seen from the

- i (a rJ + rJ a) a( - i rJ) + ( - i rJ)a a(aA - Aa) + (aA - Aa)a.

From this one easily derives that

-

i D = i D.

•

A further consequence is

Therefore, operator.

D

io

= a(a A - Ad) + (a A - Aa)a = i (arJ + rJa) .

= arJ + {)a and

Dp

= (a{) + rJa)r.p

= D

5 . 10 P r op osition. On a compact Kahler manifold, PROOF:

r.p;

D =

i.e., D

is a real

� A.

We have A

do + od - (a + a)(rJ + rJ) + (6 + rJ)(a + a) (a rJ + rJ a) + (a rJ + rJ a) 2 0. •

This formula is not valid on general compact manifolds !

Har monic Forms. Let X be a compact Hermitian (not necessarily Kahler) complex manifold. We define

: - d.'Zf r - l (X), : - oJZfr+ I (X), : - {r.p E .'Zfr(X) : �r.p = 0}. The elements of yrer (X) are called harmonic forms. Example For calculating the star operator on (p , q)-forms one needs the number (p , q) = 2P+q-nin ( - 1) q (n-p)+n (n+l ) /2 . 1f n = 1 , then a(1, 0) = 1) = and a(1, 1) = - 2i. It follows that in geodesic coordinates

a

-i, a(O,

��

=

odf

- * d * Uz dz + !z az) - * d( ifz az - i fz dz) - *( 2i fzz dz A az) - 4fzz = - fxx - Jyy ·

i

336

VI. Kahler Manifolds

This motivates the name "harmonic. " 5. 1 1 P r op o sition . 1.

r.p E £r 2 . Then b 1 = 1 , as we have shown in Chapter IV. So H cannot be a Kahler manifold!

Exercises

1. Prove the commutativity relations 19A = A19 and 19A = A19. 2. Prove the bracket relations [19, L] = - i a, [a, A] = - i 19, and i 19'

[a, A]

=

6.

Hodge Manifolds

341

3. Let X be an n-dimensional compact complex manifold. Use the norm 1 11 4? 11 = (cp, cp) 1 2 on dr (X) and prove for a given cohomology class u E Hr(X) that among all representatives cp E dr (X) of u a representative cp0 has minimal norm if and only if J cp0 = 0. 4. Calculate Hp,q (X) for X = JIDn and all p, q .

6.

Hodge Manifolds

Assume that X is an n-dimensional compact complex manifold and that F is a holomorphic line bundle on X. We define a new notion of negativity for F. The new definition is formally more restrictive, and if we would generalize it to arbitrary vector bundles, it would, in fact, be different from our old notion. But it turns out that for line bundles the new negativity is equivalent to the old one. Negat ive Line Bundles .

Let a system of trivializations cp, : F l u ---+ U, x C ( with transition functions guJ be given. A fiber metric on F is given by a system h of positive smooth functions h, such that ,

Then a smooth function

X.h

:

F ---+ lR can be defined by

Definition. A line bundle F over a compact manifold X is called Griffiths negative if there exists a system of trivializations cp, : F l u ---+ U, x C (with transition functions guJ, and a fiber metric h = (h,) on F such that X.h is strictly plurisubharmonic on F - Zp (where ZF denotes ,

the zero section in F).

The line bundle F is Griffiths negative if and only if there exists a system of positive smooth functions [h on U, such that:

6. 1 Proposition. 1.

2.

- log [h is strictly plurisubharmonic on U, .

[h =

l gu, l · f2>< on U, " .

PROOF: Let h be any fiber metric on F and consider a point x0 E X. There is a trivialization cp at x0 such that h'P(x0) = 1 and all first derivatives of h'P vanish at x0 . Then h'P(x) · l w l 2 is strictly plurisubharmonic at every (x0, w ) with w =f. 0 if and only if h'P is strictly plurisubharmonic at xo . Taking such trivializations cp, we have only to set g, : = ( h J - 1 12 . •

VI.

342

Kahler Manifolds

The tube T : = { v E F : Xh ( v) < 1} is a strongly pseudoconvex neighborhood of the zero section in F. Locally, for T, := T n n-1 (U, ), we have The boundary of T is given by the equation ww / f2Z ( x)

=

1.

6.2 Proposition. The line bundle F is Griffiths negative if and only if F is negative (in the old sense).

PROOF: One direction is trivial. To show that every negative line bundle is Griffiths negative one needs rotations in the fibers and a smoothing proce­ • dure. For details see [Gr62] .

The line bundle F on X is called Griffiths positive if the complex dual F' is negative. Definition.

It is clear that F is Griffiths positive if and only it is positive in the old sense. We assume that F is a positive line bundle on a compact manifold X . Then F' is negative. We always denote the coordinate on the fibers of F' by w . Of course, the variable w depends on the local trivialization. Special Holomorphic Cross Sections.

The strongly pseudoconvex tube around the zero section of F' is locally given by T = {(z , w) E U x C l w l < Q (z ) } , where U is a domain in e n . From the last chapter we know that dime H1 (T, 0) oo . In this situation we prove the following result. :

6.3 Proposition. are valid:

If m is sufficiently large, the following two statements

1.

If x 1 , x 2 E X are two different points, then there are holomorphic cross sections s0, s1 E f(X, Fm ) with s0 ( x i ) -/= 0 and s 0 (x 2 ) -/= 0 such that the quotient sd s0 has different values at these points. 2. If x0 E X is a point, then there are holomorphic cross sections so , S l , .

·

·

,

S

m n E f(X, F )

with so (xo ) -/= 0 and det J( s i fs0 , . . . ,s n /so ) (xo ) -/= 0 .

PROOF: If s o , s1 are two sections in a line bundle with so ( xo ) i= 0, then there is a holomorphic function f in a neighborhood of xo such that s1 (x) = f ( x) s0 ( x) in that neighborhood. The quotient sd s o is defined to be equal t o f near xo. ·

6. Hodge Manifolds

343

We operate in several steps:

(1) Let x0

X be a point. We assume that in local coordinates x0 corresponds to the origin. We call a trivialization of F' around x0 usable if E

i,j in a neighborhood of 0. Here a is a smooth function and ative definite.

:E i . 9iJZdij is neg,J

The existence of a usable trivialization follows easily. We take an arbitrary trivialization and write log(g(z))

= ao + Re ( Q(z )) + L 9iJ ( z )zi Zj + llzll:� · a( z) ,

with a0 E lR and a quadratic polynomial Q(z ) with Q(z ) the trivialization by dividing w by exp (a0 + Q(z )).

= 0. Then we change

(2) Now let x1 , x 2 be two different points in X . We choose usable trivializa­ tions simultaneously over neighborhoods U1 (xi) and U2 (x2 ) , and, moreover, strictly increasing sequences 0 < r1 < r2 < · · · < 1 and 0 < s1 < s2 < · · · < 1 such that I s i/ri I of. 1 and { w = ri } n T, { w = si } n T are analytic sets in T, contained in F' I U1 (respectively in F' I U2 , cf. Figure VI . l ). We get

W=

Ti

-�>

s� �==s :�� / \

F'

W=

Si

X Figure VI. l .

Construction of special cross sections

principal parts of meromorphic functions 9i = 1/(1 - wjri) in T n (F' I UI ) and gi = 1/(1 - w/ri) in T n (F' I U2 ) , and hi = 1/(1 - wjsi) in T n (F' I U2 ) as well. Because of the finiteness of cohomology there are complex numbers ai that are not all zero such that there are meromorphic functions g, h in T with principal parts I;i ai(9i + gi ) and I; i ai (9i + hi)- We may assume that the first nonvanishing coefficient ai is equal to 1. The functions g, h are holomorphic in a neighborhood of the zero section , and locally they have power series expansions

344

VI.

Kahler Manifolds 00

00

g(x, w ) = L 9rn (x)wrn and h(x, w) = L hrn (x)wrn . m =O m =O The coefficients 9m (respectively hm ) define global holomorphic sections so,m (respectively s1,m ) in the tensor bundle F ". If over x1 the first nonvanishing part of g is aigi = 1/(1 - wjri), then we have

g(x1, w)

=

1 ( r� + Cm ) wm ,

with

Cm

-+

0 for m

-+ oo .

So for large m we have 9m(x1) >:::J 1/(rf'), and also 9m (x2) >:::J 1/(rf'). Analo­ gously, hm (xl) M 1/(rf') and hm (x2) >:::J 1/(sf' ). We take so = so, m and s1 = s1, m for such a large m. Then the quotient sl/ so at x1 has a value that is nearly l, and at x2 nearly (rdsi) m . These values are different for large m. ( 3) Now we work at some point x0 , and over a neighborhood U of xo we define g, = 1/(1 - w/ri) and hu (z , w) = 1/[1 - (w/ri) · (1 + Ezj)], for j = 1 ,. . . , n . If U and c are sufficiently small, the polar sets of 9i and hi,J are always analytic sets in T. As above, we obtain meromorphic functions g and hj in T, for j = 1 . , n, which have power series expansions g ( x, w) = :Em 9m ( x) w m and hj (x, w ) = :E m hj,m (x)w m in a neighborhood of the zero section. Again let g; be the first principal part with nonvanishing coefficient a, . For large m we have 9m (x0) >:::J (1/ri) m and hj,m(x0) >:::J (1/r;) m (l + m . Ezj)· As above, 9m and hj . m define sections so and Sj in p m with (sj/so) (xo) >:::J 1+ mEZj . So the Jacobian Of (sl/so, . . . , s n /so) at Xo is approximately mn cn and hence is • not zero for m large enough. ,

_

Pr ojective Em bed dings . Let X be an n-dimensional compact complex manifold and F a positive line bundle on X . Then for large m we have many global holomorphic cross sections in pm . If there are sections s0, . . . , s N that do not vanish simultaneously, then

0, are holomorphic '

7. Applications

353

functions on L that are homogeneous polynomials of order m on the fibers on L. We add a point at infinity to each fiber of L and obtain a compact algebraic surface X . We denote by D the divisor of all points at infinity, and by F( m) the line bundle over X , associated to the divisor mD. Then all cross sections in L - m are sections in F(m) over X that vanish on the zero section Z = ZL c L c X of order m. These sections are holomorphic functions on the dual bundle F ( - m ) : = F(m ) ' over X (associated to the divisor -mD) that are linear on the fibers of F ( - m) ,but vanish over Z. Let 9 be the set of all these functions. The function f(x) - 1 on X is also a cross section in F(m) that vanishes on D to mth order, but it does not vanish on Z. It gives a holomorphic function on F( - m )that is linear on the fibers, vanishes over D , but does not vanish over Z. We denote it by h and add it to the system !7. Now, 9 is a set of holomorphic functions on F ( - m )that are linear on the fibers and whose joint zero set is the zero cross section of F ( - m ) . Forming the sum of several products f f, f E 8 .,we construct a plurisubhar­ monic function ' 0 is a constant that depends only on '<j;. Then - , ' 0 SUCh that for every Xo E an there is a COOrdinate neighborhood U(x0 ) and a constant c > 0 such that

2. Subelliptic Estimates

359

1 1 4?11 ; < c . Q ( cp , cp) -* ) . for all cp E d'rf• q (U n 0) n dom(o

-

-

81P) +( o*cp, o*1P) is the so-called energy form Here Q (cp , ' 0 . The proof of this theorem is rather difficult, of course. The so-called Kohn Morrey inequality is involved. See [DL8 1], for instance.

The Neumann Oper a tor . There is an old theory on elliptic bound­ ary problems, beginning with Dirichlet's problem (see the old papers of L . Bers and L . Nirenberg, for instance). Here the Gdrding inequality plays an important role. If it holds for a problem, existence and regularity of solutions of the problem can be proved. In our terminology it can be expressed in the form

ll4?lli < c . Q (cp, cp ) .

If c < 1, then the inequality for subellipticity is much weaker than the Gdrding inequality. But nevertheless, in the papers of Kohn and Nirenberg it was proved that there are results on the smoothness of the solution of the Neumann problem that are similar to those in the old theory and that follow from subellipticity alone. We state a first important result (see [DL81] ): 2 . 2 P r op o sition . Assume that 0 is strongly pseudoconvex and th at the Ne um ann problem is subellipticfo r (p,q)-forms on R . Th en there is a unique operator N : L; , -+ dom(D) with the following properties: q 1 . N(Yt"p,q ) = 0. 2. D o N = id . 3. N(L; ) is orthogonal to Yep , q ,q This operator N is called the Ne umann operator. Under the above conditions the following theorem is true: 2 . 3 Theorem.

1. Th e harmonic space Yep, q isfinite-dimensional. It is contained in d'p, q (0) and fo r cp E £p, q we have or 1\ *

1 if and only if all germs of local complex-analytic sets A c an have dimension less than q . Moreover, we have the following result: 2.5 P r o p osition. Assume that n c c X = en . Th e n there is no germ of a local complex-analytic set A c aG ci positive dimension. For a proof see [DiFo78] and [Ko79] .

E xam ples . We will construct two 2-dimensional examples n cc X ,where n contains just one compact irreducible curve Y. From the existence of Y it will follow that the cohomology group H 1 (!1, 0) is different from zero. The cohomology classes are given by unique harmonic forms that satisfy the boundary conditions. We can study these as if they were functions, which may be useful for our investigations. Unfortunately, the forms in £0 • 1 are not in general d-closed. So there is no connection between £0 • 1 and the cohomology with coefficients in e as in the case of compact Kahler manifolds. We will see this more precisely in the examples.

2. Subelliptic Estimates

36 1

Example 1

We take a compact Riemann surface Y of genus g > 1 and a line bundle X on Y with Chern class - 1 . This line bundle is negative. There is a real­ analytic metric on the fibers of X such that n = { E X : ll x ll < 1 } is a strongly pseudoconvex relatively compact subdomain of X with real­ analytically smooth boundary. We get a uniquely determined Kahler metric on X by using the metric of constant curvature on Y and the metric on the fibers and requiring that the surfaces { x E X : ll x ll < r} be orthogonal to the fibers.

Over Y we have the vector bundle Oi of homogeneous polynomials cf degree i on the fibers of X . If i is sufficiently large, then H 1 (Y, Oi) vanishes. So the finile-dimensional cohomology group H1 (!1, 0) is generaled by finilely many of these groups H 1 (Y, Oi). The cohomology classes are given by differential forms r.p E m"0 ,1 (Y, Oi) = r(Y, A0 ' 1 (Y) Q9 Oi) . They all may be viewed as differential forms on n whose coefficients are holomorphic polynomials of degree i on the fibers. They vanish on Y to order i. We denote the set of these forms by 2t� ' 1 . Examples for forms of this type are given by the derivatives af' where f is a differentiable function in n whose restriction to the fibers is a holomorphic polynomial of degree i. We take the orthogonal complement of these forms and get £0' 1 = EB :/f;0 '1 , where :/f;0 ' 1 C 2t� ' 1 . This follows by a i direct computation. Thus we obtain a basis of finite length, and the restriction of these forms to an satisfies the boundary conditions of harmonic forms.

Since g > 2, the space of harmonic forms �0 ' 1 has positive dimension. We can obtain a basis by explicit computation. The elements are harmonic forms in n vanishing on Y c R. If there were an isomorphism onto the cohomology with coefficients in t, then for any c > 0 there is a neighborhood V (0) such that II J i l t < ell f i l s for all f with compact support in V . Use this to prove the same estimate for the tangential Sobolev norm. Assume that the Neumann problem is solvable for (p, q)-forms. Let a form r.p E L�, q with ar.p = 0 be given that is orthogonal to Jt'P ,q . Prove that 'ljJ := a N r.p is the only solution cf the equation fhjJ = r.p that is orthogonal to every 5-closed form. Let n c c en be a strongly pseudoconvex domain. Prove that for every positive number C there is a smooth plurisubharmonic function f2 on n with 0 < f2 < 1 such that Lev( g) (z, t) > C l l t l l 2 for every z E a n . Take Example 2 and prove that that the derivative d( r.p l y ) is different from 0 and that this form is not harmonic. Take Example 1 and prove that £1,0 has infinite dimension. The following problem requires some knowledge about sheaves and com­ plex spaces. Take integers r < s such that r, s are free of common divisors. Consider at 0 E e the holomorphic functions that are given by conver­ gent power series in wr and w 8 • Then replace the ring of holomorphic functions in 0 E IP\ by these holomorphic functions and repeat the pro­ cedure of Example 2. Since now the "structure sheaf' 0 of JID 1 is smaller, a new curve Y is obtained. Construct an X around Y such that there is a 2-dimensional strongly pseudoconvex tube n around Y. Compute the first cohomology of Y. . ,

2.

3.

"

-

-*

4. 5.

6.

7.

3.

Nebenhullen

General Domains . Assume that n c en is a general domain. We take the open kernel of the intersection of all domains of holomorphy n :::) n and its connected component N(f!) that contains R . (More precisely, we work in the category of unbranched domains over en as in the case of the construction of the hull of holomorphy). �

Definition.

The domain N(f!) is called the Nebenhiille of !1.

We can see very easily:

The Nebenhulle is a dom a in of holomorphy. The Nebenhulle contains the enve lope of holomorphy of !1 .

-

3. Nebenhiillen

365

If n cc e n is a domain of holomorphy with (smooth) real-analytic boundary, then an does not contain any germ of an analytic set of positive dimension. Moreover, one can show that N(n) n. That implies that one can approxi­ mate n by Stein neighborhoods. See [DiFo77] for details.

=

2 : l w l l zl
0 if for all points w , z E n the estimate IIF ( w ) - F ( z ) l l < M . ll w - z l l € holds with a constant M > 0. Here we state a special case of a theorem that was independently proved by Henkin and by Vormoor (see [He73] and [Vo73] ) .

368

VII.

Boundary Behavior

4.2 Theorem. Assume that !1 1 , !1 2 E 0} e { 0} -

imaginary unit A V* V' F(V)

real dual space of a real (or complex) vector space complex dual space of a complex vector space C-valued real linear forms on a complex vector space

Mp,q(k) Mn (k) GLn (C) A B ( or A · B)

k-valued matrices with p rows and q columns k-valued square matrices of order n general linear group product of matrices

z zt (z l w) m ll z l l , dist (z , w ) \ z l w) lzl

vector z = ( z 1 , . . . , Zn ) and transposed vector standard Euclidean scalar product in lRm Euclidean norm and distance standard Hermitian scalar product sup-norm (or maximum-norm)

D r ( a) D

Br (z) pn (z, r) P� (z) pn T : en --+ "f/ Pz Tn (z , r) Tz

open disk (in q around a with radius r unit disk D 1 (0) open ball with radius r around z polydisk in e n around z with polyradius r polydisk with polyradius r ( r, . . . , r ) unit polydisk P� (O) natural projection onto absolute value space polydisk pn (0, T (z)) distinguished boundary (torus) distinguished boundary Tn ( o , T(z) )

u cc v

U lies relatively compact in V

Ct (z)

Cauchy integral of f

V'f ( z) V'f ( z) V,J(z) V'yf (z)

holomorphic gradient ( fz , ( z ) , . . . Jzn (z)) antiholomorphic gradient (fz, (z), . . . , fzn (z))

'

=

= =

:

( fx, (z) , . . . ,fxn (z) ) ( fy, (z) , · . . , fYn (z) )

T

--+

e at z E P"

382

Index of Notation

Dv J (z) DJ (z) (of)z (w ) (of)z( w ) (df)z Jr (z) JIR,r (z) 6"(G) O(G) t

=

(z , w )

Tz a(O)

N(JI , . . . , Jq )

A (or Reg( A )) Sing( A) rkz (fr , . . . , fq ) (P" , H) (P, H) �

�

bSt

higher partial derivative total real derivative of f at z = fz, (z)wr + . . · + fzn (z)wn = !z1 (z)wr + . . · + fzn (z)wn (of)z + (of)z complex (or holomorphic) Jacobian matrix real Jacobian matrix space of smooth functions on G space of holomorphic functions on G =

tangent vector at z with direction w tangent space at z tangent vector (a(O) , a' (O)) common zero set of fr , . . . , fq set of regular points of A set of singular points of A rank of the Jacobian J(/1 , . . . , /q ) Euclidean Hartogs figure general Harto gs figure boundary of an analytic disk St

=

4?t (D)

connected component of A1 containing boundary distance (for a domain G) Lev (f) ( z, w ) Hz (oG) Hess( f ) (z, w )

z

Levi form I:: v /-L fzvz, (z)wvw ,_. ,

complex (or holomorphic) tangent space of oG Hessian 'E v, ,_. fxvx, (z)wvw,_. holomorphically convex hull of K in G

61 -< 52

H,� (Q) , H( g )

a

qz] Bt Hn

m

I * ' JX JD [u] fz

the Riemann domain r;;1 is contained in r;;2 9 - h u 11 of g and envelope of holomorphy set of accessible boundary points of G ring of formal power series = { ! E qz] : ll f l l t < oo } ring of convergent power series maximal ideal in H, set of nonzero element, set of units in an ideal I set of monic polynomials with coefficients in I germ of the holomorphic function f at z

fudex of Notation w

(u , ) z

D (w ) .6-w , Dw

pseudopolynomial

dimz ( A ) , dim ( A )

local and global dimension of an analytic set