Techniques of extension of analytic objects 0824761685

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TECHNIQUES OF EXTENSION OF ANALYTIC OBJECTS Yum-Tong Siu DEPARTMENT OF MATHEMATICS YALE UNIVERSITY NEW HAVEN, CONNECTICUT

MARCEL DEKKER, INC.

NewYork

1974

COPYRIGHT◎ 1974 by MARCEL DEKKER, INC.

ALL RIGHTS RESERVED

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.

MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York

LIBRARY OF CONGRESS CATALOG CARD NUMBER: ISBN:

0-8247-6168-5

Current printing (last digit): 10 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERI CA

10016

74-83962

To Sau-Fong and Brian

\

TABLE OF CONTENTS PREFACE

iii

INTRODUCTION

1

CHAPTER 1 EXTENSION OF HOLOMORPHIC AND MEROMORPHIC FUNCTIONS Levi 's Theorem

13

Rothstein's Theorem

16

APPENDIX The i-Theorem of Koebe-Bieberbach

31

CHAPTER 2 EXTENSION OF SUBVARIETIES AND HOLOMORPHIC MAPS The Theorem of Thullen-Remmert-Stein Bishop's Theorem

34 51

Rothstein's Theorem

67

Ext ens ion across IR n

76

Extension of Holomorphic Maps

85

APPENDIX Jensen Measures

94

Ranks of Holomorphic Maps

96

Thick Sets

101

Special Analytic Polyhedra

104

A Special Case of the Lemma of DolbeaultGrothendieck

117

CHAPTER 3 HOMOLOGICAL CODIMENSION, LOCAL COHOMOLOGY, AND GAP-SHEAVES

123

i

APPENDIX 147

M-sequences CHAPTER 4 EXTENSION OF COHERENT ANALYTIC SUBSHEAVES

152

CHAPTER 5 EXTENSION OF LOCALLY FREE SHEAVES ON RING DOMAINS

163

APPENDIX Topology of Sheaf Cohomology Groups

184

Triviality of Holomorphic Vector Bundles

187

CHAPTER 6 EXTENSION OF LOCALLY FREE SHEAVES ON HARTOG' DOMAINS

202

APPENDIX Duality

229

CHAPTER 7 EXTENSION OF COHERENT ANALYTIC SHEAVES

2.'.35

REFERENCES

250

INDEX

254

INDEX OF SYMBOLS

256

ii

PREFACE

This set of lecture notes is a reproduction (with minor modifications) of the notes I distributed to the students in the "troisieme cycle" course "Techniques of Extension of Analytic Objects" I gave at the University of Paris VII in the second semester of 1971-1972 while I was on a leave of absence from Yale University. The results presented here are known.

However, some of

the proofs (for example, the proof of the extension theorem for coherent analytic sheaves from a Hartogs' figure) are new. Most of the results appear here in book form for the first time. The reader is assumed to have some familiarity with the basic theory of several complex variables, as given, for example, in Gunning-Rossi's "Analytic Functions of Several Complex Variables". The appendices at the end of the chapters contain background material and are mainly for the convenience of the reader.

Some of the material there are slightly different

from the standard formulations usually found in the literature and some are special cases of well-known results for which direct proofs are supplied which are much easier than those of the general cases. I wish to thank the University of Paris VII and Professor F. Norguet for their hospitality during my stay in

iii

Paris.

I would like also to thank the Alfred P. Sloan Found-

ation and the National Science Foundation f·or providing part of the financial support respectively during my leave of absence from Yale University and during periods in which this set of lecture notes was prepared and organized.

Finally I

wish to thank Betsy Buslovitz for her excellent typing.

iv

TECHNIQUES OF EXTENSION OF ANALYTIC OBJECTS

INTRODUCTION

We are concerned with the theory of extension of what we call analytic objects. an analytic object is.

We will not define abstractly what

By an analytic object, we simply mean

one of the following entities whose extension we will con. sider:

holomorphic and meromorphic functions, analytic sub-

varieties, coherent analytic subsheaves and sheaves, holomorphic maps, etc.

In this introduction, we will describe the

main results, point out their interrelationships, and give a ge~eral

idea of their proofs. First, we want to say something about the domains on

which the analytic objects are defined

~

the domains before

extension and the domains after extension.

We consider the

following types: l)n

Hartogs domains of order In ~ 2

consider the following set

n' = {O~x 1) n , because we can exhaust ·the associated

polydisc "' D

P •

U(J,

c

D

when

)

such that

a

is small,

ii)

oUa

(\ ( D-D)

iii)

Ua

is strictly

oUa n

ua

n

by an in-

(0 J ®

to a fiber

F

d

is a coherent

The analytic

means the image of

is the structure sheaf of

7[n](U)

=

ind. lim. f(U-A,']) A

A runs through the set of all subvarieties of

11

U of

dimension ~ n • 7[n] When

= ']

Always we have a canonical map "}

-->

7[n] •

means that this canonical map is an isomorphism.

n = l ,

2)n-l

is due to Trautmann [33].

Siu [22,24]

used Grauert•s power series method to obtain 2)n

and

O)n;

and Frisch-Guenot [5] provided an elegant proof of 4)n-l Douady's method of privileged sets.

by

In these .lecture notes,

we will give a new nroof of O) n which uses neither the series . method nor the method of privileged sets. The idea is to· use projections

to reduce it to a special case and for this spe-

cial case the method of power series and the method of privileged sets are unnecessary.

This proof is more in line With

the proofs used in the extension of functions, subvarieties, and subsheaves.

O)n

for sheaves generalizes

O)n

for sub-

sheaves. V.

Holomorphic Maps A holomoprhic map from a punctured ball of dimension

~

2

to a compact Kahler manifold can be extended to a mero-

morphic map defined on the whole ball.

This depends on

Bishop's theorem applied to the graph and also on the fact that a closed nonnegative

(1,1)

current on the punctured

ball can be extended to the whole ball. Griffiths [6] and Shiffman [20].

12

This theory is due to

CHAPTER 1 EXTENSION OF HOLOMORPHIC AND MEROMORPHIC FUNCTIONS

The main results in this Chapter are a theorem of E. E. Levi and a theorem of W. Rothstein.

(1.1)

Let .6(r) = {z E

a:j I zl

Theorem (Levi).

O)n

i. e. Suppose Qx

< r}

and

L::::.

=Li.(l) •

is true for meromorphic functions,

f( zl' ••• , zn,w)

is a meromorphic function on

(b.(r) -LS), where r>l and .QCC[n is open and connected.

Suppose

A is a thick set in

.Q

and suppose, for every

(z 1 , ••• , zn) EA , f(z 1 , ••• , zn,w) function on .6(r) •

Then

f

extends to a meromorphic

extends to a meromorphic func-

tion on 52 x .6(r) • Proof.

l" Special case.

Qx (.6(r)-.6).

all

f

is holomorphic on

We have a Laurent series expansion 00

=

f(z,w)

where

Assume

~ c ( z )wv v=...oov

(1 < lwl

is holomorphic on Sl •

c)z)

Let

A

p

< r) be the set of

z E A such that the meromorphic extension of

L:::.(r)

has at most

p

f(z,w)

on

poles (with multiplicities counted).

00

Since thick.

A= VA

there exists some

p=O p

such that

p

A p

is

Consider the vectors

... '

=

over the field

F

c

-v-p

)

(v ~ 1)

of meromorphic functions on 52 •

13

We claim

that the vector subspace \.(' c Fp+l

spanned by

}°

{V

0

\!

dimension

~

p c c

= det

D:

for all

1

~

over

c

-\12

v1

has

F , i.e. c

-\11

v=l

c

-\11-l

c

-vrl

< ... < vp+l < oo.

-\11-P -vrp

on .Q

:O

We know that, for every

a0 , ••• , ap E

0 •

Proof. Since on

Let g

H.

C = f(z)

and let

z = g(()

H

is nowhere zero on

be the inverse of f.

a branch of

It extends continuously to

log g

exists

(-1,1) •

log g(it 0 ) - log g(O) log _l_

!al

=

I

jRe(log g(it 0 ) - log g(O))

t=t o _l_ f.t=O lzl

~ /log g(it 0 ) - log g(O) I ~

14-theorem

Now we have to use the

to estimate

1

II dz

dt

dC C=it

•

ldzl

I z I de

'=it •

Let

'

'

=

c-

=

h(O

it

1 + itC

Since 1

=

1 - t

2 ,

we have

'~~'

C=it

=

~ ld~1 1 C' =O

1

t2

By Schwarz reflection principle, g

can be extended to a biho-

lomorphic map ~ g:

where

C'

.6

-> ( -

is the reflection of

19

'

(C V C ) , C

With respect to

oA .

Let -

a

(dz) d

(gh-l>' •

=

c' c' =O

Define

cp

= ~(gh- 1

cp is a univalent function By the

A

- g(it})

with

•

=0

cp( O}

~-theorem of Koebe-Bieberbach (see cp(~} JA(t}.

of Chap. l}, we have

the point

-~g(it}

and

'

qi ( 0} = 1 •

(l.A.3) oftheAppendbc

g

Since

is not in the image of

is never zero,

cp •

Hence

1

4,

~

i.e.

1:z,,

j c'=0

4jg(it>I •

~

It follows that

_l_ jdzj lzl

~

dCC=it

4 1 - t2

Consequently,

log _l_ !al

~

J

t=to 4 dt t=O 1 - t2

i

2 log

+ t0

1 - t0

and

Q.E.D.

20

(1.7)

Lemma.

~(z)

Suppose

C and

G are as in (1.6).

is the harmonic measure of

(,) is harmonic on

G

Suppose

G , i.e.

C with respect to

and approaches

l

.Q!l

C and

at

0

> 0 , there exists 8 = 8(e) > 0 , independent of C , such that w(z) > 1-e for G•

other points of

lzl




corresponds to

corresponds to the upper half unit circle. because we can map both

H

(where

[-1,l]

is

H

and

C

This is possible,

and H biholomorphically onto .6

G

and find a fractional linear transformation mapping any prescribed triple of distinct points to another prescribed triple of distinct points. a E G•

Take

dicular to both 1R Let

b

~..ci..

Let

a

0•T •

such that

o.ll.

oA

and

•

(-1,b,l) Then

oa(a)

oA

T(a) •

intersects the upper half of

y

to the triple

(-1,i,l) •

is a biholomorphic map from

= ita

with

[-1,l]

and

ta> 0

and such that

Let

G to

H

31:::..

C corresponds to the upper half

By ( 1. 6 ) , t

Let

and which passes through

be the fractional linear transformation mapping

corresponds to of

be the circle which is perpen-

y

be the point where

the triple of aa =

Let

w

'

a

:: a:

f

is continuous on

YP = the

(where (ii)

such that

= f(z,·)

fz

denotes the f

with

interior of M), and

sup

\lfzH

z E N

(where

n llu

M x U and holomorphic on

i' x

norm on

>

1 •

Expand

f(z,w)

f

av

Since

llfzn ~A, .6(r) -

is continuous on

M and holomorphic on

!a .. (z)j ([t x U t+l, ••• ,n

(a(x),T) •

Since

a EA - x Ut+l, ••• ,n

--1~( ) dimaa a a

it follows that

n

L: "-.c,·Z· j=l J J

~

B ()(AX U.t+l, ••• ,n)

such that

k-.C.+l

(Ax U.t+l, ••• ,n>

is a subvariety of

Ax U

t+l, ••• ,n (see the Appendix of Chapter 2 for this and also for statements concerning the properties of rank in this proof). ( b)

Hence

B is a subvariety of A x Gn_.c,((n) •

Let

and

37

be the natural projections.

R = T(B) •

We have

Since the

holomorphic image of a complex space is always a countable union of subvarieties of open subsets, to finish the proof we

R has no interior in

need only show that (c)

We are going to reduce the problem to the special situ-

at ion of hood

k

= .(, .

u of

r' C T II

T' (\ T in

Take

in

T

=

T R

T' E Gn-k((n)

Fix

T" E Gk-t(

U , then the coefficients of the polynomial

P

(z·Z)

in

Z associated to the function

n ~- 1 (1 n u) ' rjx n ~- 1 (L n U) on the analytic cover X n ~- 1 (L n U) ~:> L n U are precisely rlx

L

nU (ii)

the restrictions to

of the coefficients of the polynomial If

f

and

g

Pr(z;Z)

Z •

X and f

are holomorphic functions on

assumes ?I. distinct values on X n ~- 1 cz')

in

for some

z

'

EU,

then there exists uniquely

= (where

is a holomorphic function on

bi(z)

g(x)Pr for every Pr(z;Z)

?1.-1 . ~ b.(z)Z 1 i=O 1

' (~(x);f(x)) = (where

x EX

are uniformly bounded on X

ponent of ?I.

L

nU

is the derivative of

The functions

formly bounded on an open subset

dimensional plane in

and

C can be extended to a holomorphic

ii

is proper, every element of

,..,

C is

U and therefore can be extended to a

locally bounded on

,...,

holomorphic function on every element of ,.., U •

on

n

'1T- 1 (x}

Let D x

When (b) is satisfied, clearly

C can be extended to a holomorphic function

Suppose (c} is satisfied.

x E U such that X

U •

•

Y

f

Let

assumes less than

be the set of all

Y

distinct values on

~

is a subvariety of pure codimension

Y ~-> D be induced by the natural projection

~=

(.6(~) -A(a})

->

D.

x E Y such that

of all thin in

D

Y1

Let rankx~

be the closure of the set

< k-1

Then

,

t E D-cr-(Y }

For

let

set of all the coefficients of the polynomials PwjlXt (z;Z}

TfjXt,wjlXt (z;Z}

holomorphic functions

~

j

~

,

cr-(Y)

is

(see the Appendix of Chapter 2 for the definition

and properties of rank) •

(1

in U.

1

N) •

f!Xt ,

Elements of

in wjlxt

Ct

z

ct

be the

pf Ixt ( z ; z > ,

associated to the

on the analytic cover

are precisely the restrictions

45

of elements of

{t} x (.6(fl) - .6(a))

t E D-o-(Y ' ) •

for

t E A the analytic cover

Since for

-> {t}

Xt

C to

can be extended to the analytic

X ( .6 (fl) - .6( a) )

,.,

cover

Xt

~->

{t} x

.6(~)

every element of

tended to a holomorphic function on t E A-o-(Y ' ) •

{t}

x

By (1.2) every element of

Ct

can be exfor

D.(S)

C can be extended

,.,

to a holomorphic function on

U •

By using the extensions of ,.,

C , we obtain new polynomials

elements of

TfjX,w.jx(z;Z) J

EU

U•

Z are

The set

X of all

satisfying

x [N

=

{Pfix(z;f) ( t)

whose coefficients in

~

holomorphic functions on (z;w1 , ••• , wN)

PfjX(z;Z) ,

Pwjlx(z;wj)

0 =

0

=

TfjX,w.IX(z;wj)

(1

~

j

~ N)

,.,, wjPf!X(z;f)

J

(where

is the derivative of

respect to

"'"1x

that

Z)

'"" N U x

D

of a branch of

A

of dimension

k , it is clear that we can confine ourselves to the special

case

D =Ak

taking

and

n'

=(A(£)) k

for some

that

0

< £ < 1 •

By

1-dimensional linear subspaces of ([ k , we reduce the

general case to the special case

k = 1 •

We can also assume

A is irreducible. Take

€

Q.E.D.

From considering a boundary point of the image under

the projection ~

and

is a subvariety of

Suppose

diction.

.

'

f.1 (x) =O

such that

a compact subset of

.

(i = 1,2)

is

is an open subset of

zm(x)

=

=

lz(x*)

is not constant on

There exists

J.

zm(x*)

jz(x0 >1 < B

are constant on

f.

>

* fi(x)

A and

I A

.

We can choose otherwise both

A is a compact subvariety

x E A n (Li(B)

x

G)

such that

sup x E A (\ (.6.(B) x G) is equal to the

because

on

4g

sup

of

which is compact and because

>

sup x EA() ((d~(5))

~

x

G)

This contradicts the maximum modulus principle for the funcf. A tion _.!. Q.E.D. m • z

( 2. 5) and

Corollary.

Suppose

G is a Stein open subset of ctn

E is a proper subvariety of

are open subsets of that Proof.

II

AC G

and

G

G•

t

II

Suppose

G CC G

G' -E

is a subvariety of

A

such

dim A ~ 0 •

Then

Identify .Q •

in

0

in ([ n-k , and

L

K

Let

Q

with

SL

x [ O}

'1r- 1 (D)

•

is an

D •

"1., ••• ,

Choose

W = Q x {O} ,

be induced by the natural projection

We identify

analytic cover over

x

is a connected open subset

.Q

0

through

, we can assume

L ,

.Q x

is an open neighborhood of

is a compact neighborhood of 'IT":

G=

V (\ (Q.x (L-K)) = ¢ , where

p~ssing

k

µn-k E D as defined in the proof of

(2.3).

l a'V} c

We claim that, i f 'V --->

oo ,

contrary. that

then

g(a) --> 0

By replacing

Ig(av> I 6

€

{a'V}

for all

'V

as

D and

a'V---> a E () D as

'V -->

Suppose the

oo •

by a subsequence, we can assume and for some

€

be the number of sheets of the analytic cover Choose

c

6 l

such that

>

0 •

50

u of

0

in

,.,.

'IT"-l(D) - > D.

c 6, the sup of If I on K •

there exist an open neighborhood

Let

L

By (2.5) and a

connected open neighborhood

.

sup of I f I on U jg(av)j 6

V r\

and

E

nD

wise, since

n (E

k

for

'

av

E'

(E '

Vn

•

av E E

such that

v 6Vo.

--> E '

x U)

-

, V

n

(E

Vn ({a VO } ¢.

x U) =

Since

be the compo-

¢ , other-

x U) =

0

pure dimension

V

such that the

Q.

v, there exists

containing

V (\ (E

contradicting

in

c

({av} x U) =¢

nent of

a

< -,;:::r €' and

is

for all

€'

E of

'

x U)

is proper and

->

x U) = ¢ .

E

'

is an analytic cover,

By (2.4),

E C D , contradicting

Hence

is of

V

a E

an .

The

claim is proved.

g

Define a function

g=

and

holomorphic function on Sl proper subvariety of subvariety in

G •

/"'

g=g

by setting

/"'

From (2.1) we conclude that

on Q.-D •

0

on S2

g(b) -/= 0 ,

Since

By (2 • 3) '

.Q •

Hence

D = .Q ,

~-l(D) "

D

on

is a

g

is a

Q. -D

extends to a

contradicting

aD

x E

•

Q.E.D. (II). (2.7) f: X

Theorem of Bishop Lemma (Federer).

-->

Suppose ~

Y is a map and

d ( f ( x) , f ( x ' ) ) ~ ~d ( x , x ' ) is the distance between ~ ~

0 , and

(where Let

d(B)

a 6 0 •

c

> 0

for all a

and

X,Y

are metric spaces and

such that x,x ' E X a' ) •

Suppose

> 0 such that, if B c f(X) is the diameter of If

B ) , then

ha+~(X) < oo , then

51

(where

and

'

d(a,a )

B> 0 , d(B)

~

B

h~(B) ~ cd(B)~ •

f*

where Proof.

If

denotes the upper integral. Ac X and

d(A) ~~,then ~ ~d(A)

d(f(A))

~

8

and


0 ,

52

Q.E.D. Lemma.

(2.8)

Suppose

is an interval in lR

I

and

is a

X

Let R x X be given the metric such that the

metric space.

. distance between

(a ' ,x ' ) E 1R x X

(a,x) E 1R. x X and

( d(a,a ' ) 2 + d(x,x ' ) 2)1/2 •

Let

p

>0 •

is

Then, for any

p+l A

c

X ,

Proof.

hp+l(I x A) ~ 2 2 Let

>

€

of subsets of 1

~

v

0 For

by a minimum number of disjoint intervals 1

We have for

v E E •

Since

p

>

0 ,

finite number of intervals

and the length of each

for

v E F

tv {Ivj} j=l

r,, j

v-~

we can cover

.

Clearly

.(,

v

00

I x A

c

u U

v=l j=l

We have

53

~ h"'J.11 +

such that

J2e

is

t

I . x Gv VJ

1

I

by a

Hence

Taking

inf

on both sides for all choices of

{Gv], we obtain

l!t!

;;;; 2 2 (h1 (I) + e)h~(G) + s • Let

€

-:> 0 •

It follows that

J!tl

~ 2 2 (2.9)

Lemma.

Suppose

'IT":

h1 (I)hP(A) •

Q.E.D.

B ~-:> X is a differentiable fiber

bundle whose base and fiber are both differentiable manifolds. Let

B and

X be metrized by Riemannian metrics.

the fiber is compact and has real dimension relatively compact open subset of exists a constant

c

depending only on

hp+q('IT"- 1 (A)) ~ chP(A) Proof. (1

~ j

Cover ~

of

Let

p > 0 • Q and

p

for all Borel subsets A of

Q be a

Then there such that Q.

Q by a finite number of disjoint sets

w.J

k) , each of which is the difference of two open sub'IT"- 1 (G.)

sets, such that Gj

X and

q •

Suppose

Wj

(l

J

~

j

~

k) •

is trivial for some open neighborhood Since every Borel set in X is

54

k

~ hP(A II W.) j=l J

hP(A) =

hP-measurable, we have

•

This reduces

the general case to the special case of a trivial bundle and the case of a trivial bundle follows from (2.8).

(2.10) Proposition. cr:n-{O}

give

The fiber of

( M induced by ,,,. ,

independent of

55

A such that

Since

~- 1 (P)

maps

CT

n rr- 1 (A)

bijectively onto

proposition follows.

(2.11)

Lemma.

centered at

in

Bc

~n

U-P with

U , and

0 E A•

Proof. Let

X

A (\

Suppose

A

oB

U •

n dB-H

H of

is at least

Let J4. be the uniform algebra on

complex manifold

=

=

(An:B)v (Pf\B)

R X

=1

•

con-

(Ar1B)U (An'dB)V (PnB)

measure on

A (\ B , the Silov boundary

(An dB) u (P

contained in er

n B) .

Let

1-dimensional

A

rr of

is

be a Jensen

µ

for the uniform algebra consisting of restric-

tions of functions of

J4

defined by the point

O

to

rr

and for the maximal ideal

(for the definition and the existence

see the Appendix of Chapter 2, (2.A.l) and (2.A.2)).

Since we can find a holomorphic function on

R

2·

X of holomorphic functions

because of the maximum modulus principle on the

µ

is

Since

"An:B c (AVP)n:B

of

P

has the structure

We can assume without loss of generality that

sisting of all uniform limits on on

R

is a subvariety of pure dimension

A

Then the arc length of

=A (\ B •

.

B,

of an analytic arc except perha.ps at a countable set points.

' the

is the open ball of radius

U is a Stein open neighborhood of

a subvariety of 1

Q.E. D.

Suppose

0

nA

p

P and nonzero at

f

on

O and for such a function

56

U vanishing f

we have

- oo

< logtf(o) I ~

f

loglfl ctµ ,

()

we conclude that single point

B) = 0.

µ(crn P n

n oB ,

x E A

holomorphic function on

to the arc length

U vanishing at

'"': ([n

and let

v

-> ([ s

We are

of

s

A (I ClB-H

£1!:.s dd ~ 2 • Without x = (l,O, ••• ' 0)

and

.

be the projection onto the first coordinate 7T

.

> 0 let Ks

=

=

Re

{ C E {::;;,, C-1 l

~ 5}

and let hs(C)

1

=

' - ( l+S) The function

hs

Hence

~

-1

=

s

lim + -> 0

0

'

le -

- (l+S) ( l+S) 12

f

on

Ks ,

h(C)dv(c)

7r(

hs ~ ~

Since

l+S

8

Re

is harmonic on an open neighborhood of

and negative on ~ •

-1

O.

is absolutely continuous With respect

µ

be the direct image of µ under

For

and nonzero at

x

A n aB-H •

x of

loss of generality we can assume that Let

because we can find a

µ( {x}) = 0 ,

Take an arbitrary point going to show that

Likewise, for every

28

~ 2

-

Since

arc-length of '"'-l (K8 ) 28

57

n

A

n aB

~

1 ,

~

it follows that s

!J:l!:. S. 2 •

and

is absolutely continuous with respect to

µ

Finally the lemma follows from

ds -

Jcr dµ

1

s

=

2ds

An aB-H

Q.E.D.

In his original proof, Bishop uses the Poisson kernel

(2.12)

Proposition.

ing only on

n

A is a pure h 2k(A)

Proof.

0

21rR

instead of

in

B

depend-

is the open ball of radius

a: n and P is a subvariety of B and

k-dimensional subvariety of

6

R

2 . c > 0

There exists a constant

such that, if

centered at

then

f

~

2 (arc-length of A nd B-H) •

to get the better lower bound

R

ds

An ;rn-H

= Remark.

!J:1!:. ds

B-P

With

0 E A ,

cR 2k •

We can assume without loss of generality that

R

=1

We need only show that

for some universal constant and

z1 , ••• , zn S ince

I Zl 2

c 1 , where

J

z!

are the coordinates of [n • is strictly plurisubharmonic and hence

cannot be constant on any subvariety of positive dimension,

k = 1 follows from (2.11) and from applying (2.7)

the case with

X

=

A

i

n { ~ Iz I ~

l} ,

f (z)

= Iz I ,

and

Y

= IR •

The general case follows from applying (2.10) with

58

.

s = [~ ~

~

lzl

l}

p

'

=

2

and

'

k

replaced by

k-1

Q.E.D.

(2.13)

Pro:12osition.

and

p

U-P

of pure dimension

Proof. r

Denote by

0

... '

A

Then

the open ball in ~n

B(a,r)

a •

such that €

z E

s

to JR

mapping

(lz 1 j 2 + ••• + lzn! 2 ) 1/ 2 , we conclude that

to

there exists

There exists

p (\ A n

0

such that

and

61

D: W = Bk(s)

Let

,..,

... , zn)

is proper.

7T"

Let sure

0

Fix

proper.

x

=

'TT

*E

rr!Ann

be the component of

holomorphic function on

'

Ph(z ;Z)

=g

that

f

G and

such that

7T"- 1 (G)

=0

f

on

W-G •

G

be the

~-> G as defined in f

on

W

by setting

We are going to prove

> 0 and a sequence

€

jf(xµ)

I

6

for all

€

rr- 1 (x 0 )

We claim that pose the contrary.

Then

y1 , ••• , Yt

0-dimensional.

and

Let

g{ z ' )

Let

x 0 E ';JG ,

ber of sheets of the analytic cover

hoods

*

•

is continuous.

Then there exist

points

x

Define a function

Suppose, at some point

x0

7T"-1( x *)

Z for the holomorphic function

on the analytic cover

on

U which vanishes

on

G which is the constant term of the

in

the proof of (2.3). f

h

containing

W-F

is

7T"

is a finite set,

E and is nowhere zero on

identically on

h!7r-l(G)

by (2.13) and

= O

there exists a holomorphic function

polynomial

.

Since

W-F •

.

zk)

is a closed subset of mea-

F

h 2 k(An E)

W , because

... ,

(zl'

71-(A n En D)

F =

in

Let

.

A.n D -> W .·be defined by

"' 7T":

and let 71-( zl'

Then

Bk( s) x Bn-k(r) cc u

=

H x Vj

( H x av

t

fxµ} c G approaching

/t •

>A..

A.

Let

7T"- 1 (G)

has at most

rr- 1 (x0 )

is not continuous.

be the num-

~-> G • A.

contains By (2.5),

points. t

Sup-

distinct

~- 1 cx0 )

is

We can choose disjoint open polydisc neighborof

. )· n A

J

with

f

=

yj

(l~j;;;t)

such that

r/;

(l;;;j~t)

Since

62

H x Vj CC D-E

( Hx V . ) J

7r ((

n

H x V j)

A --> H is proper and

n A)

-->

7r- 1 (G)

G

~-sheeted analytic cover.

is a

~

11-l (XO) = { y :.

that, for· µ

:t •'

The claim is

where

y -e,J

We claim

sufficientl~l)rge, 1 7r-

(x) µ

Suppose the contrary.

.(,

c

UHXV . • j=l J

f.x µ J by a subse-

After we replace

quence, we can assume that there exists

x'µ E

7r- 1

(xµ )

such

that .(,

UH xv. j=l J

and x'·

µ

->

')

x'

W x Bn-k(r)

E

lh(x'll >

Since 7r(X

= x 0 , contradicting

x'

0.

E

x ' E A and

Hence

t UHXV .• j=l J

The claim is

proved. Choose

x µ EH

such that .(,

c

UH x V . • j=l J

Let

cp:

k,

( 1 ~ j ~ ·t) , contradicting that

= H

proved. Let

A is of pure dimension

7r- 1

t (Gf\Hl _

UHxv. j=l J

63

->

GnH

be induced by is proper.

xµ ~ Im

'Tr •

Let

xµ

Since

xµ

i. Im cp that

A

(Hx(W.}(lA=r/J

Since

Cf •

G·n H containing

Q be the component of

is of pure dimension Q

n Im

cp = ¢

.

it follows from

k

11'-l(H)

The set

-

t

\JHxV. j=l J

is empty, otherWise it is a subvariety of pure dimension k H x ( Bn- (r) -

t -

,,

'

J

k in

\

UV ·I - E j=l J

and is disjoint from both t

v .))

H x (aBn-k(r) U ( U j=l

J

Q x Bn-k(r} , contradicting (2.4).

and

The continuity of

f

follows, because the emptiness of

'Tr-

contradicts

1

(H} -

t

-

U H x V. j=l J

x0 E F •

From (2.1} we conclude that Since of D •

W.

f(x*} :/= 0 , By (2.3),

An U

W-G

is holomorphic on

W•

is contained in a proper subvariety

7r- 1 (G)

can be extended to a subvariety in

is a subvariety at

(2.15)

Theorem (Shiffman).

(n , E

is a closed subset of

o•

Suppose U with

is a subvariety of pure dimension is a subvariety in

f

U •

64

k

Q.E.D.

U is an open subset in h 2k-l(E} = O , and in

U-E •

Then A

A

nu

An u

We need only prove that

Proof.

arbitrary point

x

generality that

x =O

En A.

of

.

is a subvariety at an

We can assume without loss of

,

h2k-l(E) =O

Since

h2k+l(A) = 0.

As in the first paragraph of the proQf of (2.14), after a linear coordinates transformation we can assume that there exist

s

> 0 and r > 0 such that

and

w =Bk( s)

Let

, let 'IT (

zl'

'Tr:

... '

and

F='lf(AnEnD)

.

W-F

is connected.

Since

W-F

,

An D -> w be defined by =

zn)

(zl,

h2k-l(F) = 0 '7f-

1 (W-F)

... '

zk)

'

By the lemma below, is an analytic cover over

the theorem follows from (2.3) and the lemma below.

Q.E.D.

(2.16)

Lemma.

Suppose

is a closed subset of morphic function

f

U is an open subset of ~n h 2n-l(E)

U with on

U-E

=O

We need only show that

f

E

Then any holo-

which is locally bounded on

can be extended to a holomorpl1ic function on Proof.

•

and

U

U •

can be extended to a holo-

morphic function in an open neighborhood of an arbitrary point x

of

x

=O

E • •

We can assume without loss of generality that

Since

h 2n-l(E) = 0 , as in the first paragraph of

the proof of (2.14), after a linear coordinates transformation

65

s > 0

we conclude that there exist

r > 0

and

such that

and

First we consider the case distance between 1}

>

0

.

G.1

of open discs

k I:

i=l

k

c l.J

n B1 (r)

G.

i=l

(1

=

i

~

< 7 • Let

= a(

1

z

1 €

f

21Ti oBl(r)

Since

f


z1 * E IR n A(r) as v - > oo. •

Fix

1

Take E

6(r) - IR - A

such that

< 81

1

v

Im a.].

It suffices to show that

and approaches

on some subsequence of

0

Let

rc(v)J

By replacing (v) ' k 'Ir(

r

>

"'k

*

Ck * ) = zl *

of all

as

by a subsequence, we can assume that

-->

v

(1 ~ k ~A.) •

oo

(1 ~ k ~A.) •

For

(z 1 , ••• , zn) E ([n

Fix

(1 ~ .t ~ n) •

* ... ,

Cm+l'

'1 '

>

such that

... '

Cm*

T

let

0

be the set

€

Jim z.tl


*J D2

is an open

and

is disjoint from

D1 induced by

'Ir

1 ,

$2

D2

••• , Cm

is an open subset of ([ ,

'

n1 x

is pro per and

X

Since

X has pure

For

v

for

1

sufficiently large, C(v) E Dl ~

k

~

and

A. , contradicting r (v) (v)l tC • 1 , ••• , CA.

The claim is proved. Since conjugate of

z1* E IR

{'l

*'

x' -

and

Cm* l

•· · '

R(Z):

T€

is itself.

z .(ck )) J

k=l

~ 2A. •

are all real and have absolute values

Ck* E T€

for

m< k

~ ?I. ,

The coefficients of

*

nm (z -

=

is self-conjugate, the

Since

the imaginary parts of the coeffi-

cients .of s ( z):

have absolute values

(0

)Im a/I 0 as z1 -> some point of 1R n 6 (r) (0 ~ i + ai (zl)

=

ai(zl),

where

Hence

ai(z 1 )

ai (z1 )

on

can be extended to a holomorphic function

.6 ( r) •

Let A.

z

+

A.-1

. a . ( zl) z i=O 1

2:

1

~

Let

X be the set of all

Pj(z1 ;zj) = 0 dimension

Remark.

l

for in

(z1 , ••• , zn) E .6n(r)

2 ~ j ~ n • An(r)

Then

containing

X

X

such that

is a subvariety of

n ~n(r)

•

Q.E.D.

In the proof of (2.22) the step of proving (*) can be

avoided if we use directly the projection defined by

-> and if we define

X as the common zeros of functions corre-

sponding to those in (t) of the proof of (2.3).

84

Then we can

conclude (*) of the proof of (2.22) as a consequence of (2.22).

V.

Extension of Holomorphic Maps Suppose

is the open unit ball in ~n

Bn

is a compact Kahler manifold and

= Bn

f

-{O}~->

and

M

M is a

We will consider the problem of extending

holomorphic map.

to a meromorphic map from

Bn

to

M•

f

Extension to a holo-

morphic map is impossible as is shown by the counter-example where

M=

IPn-1

and

f

is the restriction of the canonical map

IPn-l (n~2). The problem for Griffiths [6].

n;:::, 3

was solved by

Shiffman [20] improved the result to

by proving the following lemma.

n ;:::, 2

The proof of this lemma given

below is new and is very elementary.

(2.23)

Lemma.

Suppose

C00 (1,1)-form on matrix a

(aij)

e:.2 - {O}

.r:I

2 L:

a . ..-dz.,..dz.

i, j=l lJ

l

J

u

on

is a closed

which is nonnegative (i.e. the Then there exists

is semipositive hermitian).

function

Proof.

cu=

A2 -{O} such that

W

d~U =

W •

First we prove

f

(*)

a 11 J-1 dz 1A dz1 " }:Idz 2 Adz2

=

2 0(€ ) •

i(€)-{O}

Take a

cf'° function cp

such that

cp= 1

=

on .C.(~)

cp( z1 ) and

function

$5

on .. - {O}

.

Likewise

for

Z2 E D. -

{O}

.

Since

Therefore

df dz 2

2.sQ.

Jz 1

and

f t are both Jz1

fr a - and f t a - are d 21 d- 12 zl zl

D.(l) it follows that 2 '

.

d azI

0 Zz

for

D.2

=

and

As a consequence , f(z 2 ) 1 For

df

admit

~

0

on

~

on

extensions to

/:::,,,

;jz2

is uniformly bounded on

e

0 .

0

By Stokes's theorem,

=O.

It follows that

dw= O

on .D.

2

•

By Poincare 1 s lemma for currents, there exists a 1-current

~ on D.2

such that

W=

d~

Write

where

cr1 0

is a

'

rent.

Since

(1,0)-current and

c.u = dcr and

w

cr0 1

is a

(0,1)-cur-

'

is

(1,1) , it follows that =

0

=

0 •

By using the lemma of Dolbeault-Grothendieck (see (2.A.19) of the Appendix of Chapter 2), one obtains distributions v2

2 ti

on

{ on

.0.2

w.

-fO}

=

(n

~

(v2-vl)

.

J:i 'iJu is d'° on

Then w=

i t follows that

'

Theorem.

2)

and

Suppose

u

Bn

to

variety of Proof.

Let

Bn

Since w .6.2 - f 0}

(see

Q.E.D.

is the open unit ball in {n

M is a compact Kahler manifold.

Then every

f: Bn-{O}~> M extends to a meromorphic map

holomorphic map from

coo

dV2

'

(2.A.20) of the Appendix of Chapter 2).

(2.24)

is

dv 1

'

cro l u =

and

such that CTl 0

Let

v1

M ,

i.e. the graph

G of

f

extends to a sub-

Bn x M •

?

be the Kahler form of

M and let

• • • + dz n A dzn ) • Let

Bn(€)

radius

be the open ball in ~n

€ > 0 •

Then the volume

90

V

with center €

of

0

and

is equal to

To finish the proof, by Bishop's theorem (2.14) it suffices to

v€

prove that

is finite for some

€

>0 .

We are going to prove that there exists a

C00

function

on

u

D.n(_!_)Jn

Q

{OJ

such that

W The case Then

n=2

coo

"1 0

1-form on

'

is and

=

(1,0)

"1

and

'

o + "o

O"l J 0 = 'dv 1

"o

J

1

= av2 u:

n ?: 3

cf. ( 3. 2) ) •

w =de;

'

1 •

.

'

vl, v2 Then

=

satisfies

on

Now assume

Since 1 is (0,1) it follows from Hl(S2,n(O) = 0

"o

that there exist and

.

Write

.Q •

"o 1 = 0 ' coo' functions

"1 J 0 = 0

W

H1 (Q, n

~

By

(2.A~7)

71"( s

.)

J

x

is

E Bj thin in

We are going to show that 00

A

Since each

Bj

C

u 71"( s J. ) j=l

.

is a countable union of compact subsets, 00

A where trarily

Ci

X

G C

U Ci ,

i=l

is a compact subset of some t EA•

Bj(i) •

Fix arbi-

Since ft}

X

G

by Baire category theorem there exist empty open subset

U of

G such that

101

1 < i
D

factor.

V c W•

Then

We use the following notation. and

A is a thick

and

Ll.d

Ip

'II

T.

-2:_>

Ll.k

are biholomorphic maps and

Let

1 < i
r(u' ,(!))

f(U , I'(U ' ,(!))

'

, we

is

conclude that there exist

II hi_< II v

e

such that

and the fibers of the·. map

$ i $ k)

(2

h 2 , ••• , hk E F

U

'

~>I + Ih.1 ( x >I




B(ci,?) ci).

Since

denotes the open

X is compact,

f (X-B( c., ~ l) l.

Since

0

such that

such that, for

110

k

~

k0 , we have




(cs)N - 1

and 1 ?: CN - SN > c1 N

jfl(x)jN The claim is proved. Consider the map

92:V 0 (){jf1 i

defined by ( f 2 , ••• , f k ) • fl fl For

?: cl} -::> ([ k-1 Now assume that

x F TN

By (2.A.14), for any component

N is so large

n D n Vo

and

A of

n n n v0

TN

2

< i
r(U,~)

and (2.A.12) and {2.A.15).

frame

F

f 1 , ••• , fk E F

subset of

that

is a complex space of

D is a polyhedral region in X with

and

with a continuous monomorphism is

Q.E.D.

X with frame

U,

g 1 , ••• , gn E F, and

S C QC D • Proof.

By {2.A.16) there exists a polyhedral region

with frame

(W,gl,

... '

s ("

Q

c D

compact open neighborhood of

oQ

gl,

••• J

gn E F

'

and

w cu

such that

gn)

116

. in

Let

w

G

Q in X

'

be a relatively Since the boundary

of the compact set

n

Q i)G

for

1 < i

·principle that

Q-G

is

Q () aG

Igi I
( g+h) IG

~ uUppose D'

u0 =n'xb.N(b),

0

Q.E.D.

•

< _ a


aj]J

( 1 $ j $ N) , and 7J/. = {U1. J N • Then for 1 $ v $ N-1 there i=O exists a continuous linear map ~v= Zv(L'l,n+NV) --> cv-l(u,n-INV) such that cular, for

B~v = the identity map of

1 $ v $ N-1 ,

124

zv(Zil,n+N~) •

In parti-

(where

Moreover, 'flv

~

such that

~-1

=

0··· i v

zv 2 ('2'l,n+~)

{~i

o· · · 2

is

L

i }

v

on

-

E cV('Zl,n+NCO)

n ui , and

ui n 0

v

denotes the intersection of

L

zv(Vl,n+NV)

with

Cv 2 (Vl, n+N(Q) ) • L

For a holomorphic function

Proof.

f

on

u. n .•• n ui 10

with Laurent series expansion

=

f

00

in

zj , we denote by

v

ej(f)

the function

L le={)

ckz .k • J

Take

= with

1

$ v $ N-1 •

{~.

1

i}

0··· v

(1-e.)~. J 1

zv (VC, n+N ~ 1 (X,7) -> ~(X,7) -> H~ 11 (X,7) ->

127

Ht 1 (X,'])

->

... .

~)Suppose

ring

and

is an, isomorphism for

M is a finitely generated module over a local

(R,llM) •

A sequence

j

~

k •

f 1 , ••• , fk

Any permutation of an

M-sequence.

An

in"""' is called an

. not a zero- d"1v1sor . f or is

f J.

M-seguence if ~

q~k

for

k+l • Suppose

l

HP(X,~q:n=o

If

HAP( X, ']) - > r( X, 9i_ P;t)

p ;::: 1 , then p ~

k;:::o.

M/j;lf·iM f or ~ i=l M-sequence is still an

M-sequence is called maximal if it is not

contained in a longer

M-sequence.

have the same length.

This common length is called the homo-

logical codimension of simply by

codh M •

local ring

=

codhRM

S-module. 0

~>

K

If

M over

All maximal

R , denoted by

M-sequences

codhRM

or

R is a quotient ring of another

S , then it is trivial to see that

codh8M when M is reparded naturally as an If

R

= n~o

and if

~> RPt-1 ~> RPt-2 ~>

~> RP1~> RPO ~> is exact, then

K is free if and only if

~

M

~>

0

+ codh M;::: n

(All the above statements will be proved in the Appendix of Chapter 3 for the case where

R

is a quotient ring of some

nVO , which is the only case used in what follows.) If '1 (X , k-n}

k=O

Z is a subvariety of

x E Sk(7)

such that

136

X •

Let

Tk

• be the set of

Since 00

V Tk ,

Z C rank

'Ir

IZ
n([J~ -> nf()ql -> ... -> v~-d-3 ~> (Qqn-d-2 n

138

n

which is exact on

X-A.

Since by (3.3)

RAkn((J =O

for

1 < k < n-d-1 , it follows from (#) that

Since

R,0 (!)= ('.) A n n '

(Im a )[A]

q .D..d+l x [O} •

defined by multiplication by Since

Lf.

direct image of

under the inclusion map

-§,=

t:.d x {O}C

is nonzero, contradicting

139

40

= 0 •

s

*

of

-ffo

Q.E.D.

defined

(J.13)

Proposition.

?

Suppose

on a complex space X and

d

is a coherent analytic sheaf

is a nonnegative integer.

Then

-> 7[d] is a sheaf-isomorphism if and only if dim Sk+ 2 (7) S k for k < d •

the natural sheaf-homomorphism "}

Proof. (a)

The "i.f" part.

Suppose

is an open subset of

U

X and A is a subvariety of dimension k are going to prove by induction on

re u, 1 > Since

codh 7

~ k+2

on

k

in

U • We

that

re u-A ,7 >

~

Sd

•

U-(A n Sk+l ("])) , by ( J.J)

r(u-(Ansk+l