Volumes, limits, and extensions of analytic varieties

Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich

19 Gabriel Stolzenberg Brown University Providence, Rhode Island

1966

Volumes, Limits,, and Extensions of Analytic Varieties

Springer-Verlag. Berlin. Heidelberg. New York

I!

All rights, especi,lly th-t of translat/on into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechan/cal means (photostat, microflhn and/or m/crocard)or by other procedure without written permission from Springer Verlas. @ by Sprb~er-Verlag Berlin- Heklelberg 1966. Librat7 of Congress Cstalog Card Number 66- 2~792.~Printed in Germany. Title No. 7339.

Contents

Introduction

Chapter I Analytic Varieties Minimize Volume C h a p t e r II A L o c a l L o w e r B o u n d f o r the V o l u m e of an A n a l y t i c V a r i e t y

16

C h a p t e r III H a u s d o r f f M e a s u r e and the H a u s d o r f f M e t r i c

21

C h a p t e r IV The Use of the P r o p e r

31

Mapping

Appendix .

39

Bibliography

44

Introduction

T h i s e x p o s i t i o n is b a s e d o n l e c t u r e s I g a v e i n N o v e m b e r the Brown Analysis Seminar.

M y m a i n s o u r c e w a s E. B i s h o p t s

" C o n d i t i o n s f o r t h e a n a l F t i c i t y of c e r t a i n s e t s " [1] . references

1"965 a t

w e r e H. F e d e r e r ' s

" Some theorems

My other principal

on i n t e g r a l c u r r e n t s " [Z]

,!

a n d H. R u t i s h a u s e r : s meromorphen

" U b e r F o l g e n und S c h a r e n yon a n a l y t i s c h e n und

Funktionen mehrerer

Variablen,

sowie yon analytischen

A b b i l d u n g e n " [7] . H e r e a r e t he t h e o r e m s

(A) ( W i r t i n g e r ) minimizes

(B)

t h a t w i l l be c o v e r e d .

An a n a l y t i c s u b v a r i e t y of a d o m a i n in

n

volume (globally) .

If a p u r e l y

k-dimensional

s u b v a r i e t y of a n

R - b a l l in

~n

p a s s e s t h r o u g h t he c e n t e r of t h e bal_l_then i t s Z k - v o l u m e i s a t l e a s t ~(Zk) R 2k

( where , for any positive integer

d ,

~(d)

is the v o l u m e

of t h e u n i t b a l l i n E u c l i d e a n d - s p a c e . )

(C) ( B i s h o p )

T h e limi_t of a s e q u e n c e of p u r e l 7 k - d i m e n s i o n a l

_analytic v a r i e t i e s w h o s e purely

k-dimensional

Z k - v o l u m e s a r e u n i f o r m l y b o u n d e d is a g a i n a

variety.

-Z-

(D) (Stoll) A purely k-dimensional ~lobal subvariety of ~u whose intersection O ( R zk)

with every

R=ball about

O has

Zk-volume

is a l g e b r a i c .

These theorems

w i l l be p r o v e d in C h a p t e r s I - I V .

Then in the

A p p e n d i x I w i l l d i s c u s s b r i e f l y t w o m o r e r e s u l t s of B i s h o p w h i c h a r e c l o s e l y r e l a t e d to (C) a n d (D) .

(E) ( B i s h o p )

Let

Namely ,

W b e a s u b v a r i e t y of a d o m a i n

U .

If V i s

m

a purely

k-dimensional

s u b v a r i e t ~ of U - W

{ c l o s u r e in u ) has z e r o

such that

Vfl W

Zk-dimensional Hausdorff measure

then

V

i s a n a n a l y t i c s u b v a r i e t Y of U .

(F) ( B i s h o p ) a purely

Let

k-dimensional

finite then

W b e a s u b v a r i e t y of a d o m a i n s u b v a r i e t y of U - W

.

If V i__ss

Z k - v o l u m e is

whose

V i s a n a n a l y t i c s u b v a r i e t y of U .

For nonsingular varieties

(A) i s W i r t i n g e r ' s

g e n e r a l v a r i e t i e s i t is p r o v e d b y F e d e r e r Theorem

U

(A),most naturally~in terms

T h e o r e m [11] .

in [Z] ~ H e f o r m u l a t e s

of c u r r e n t s ,

and also gives a

v e r y s i m p l e p r o o f of t he b a s i c W i r t i n g e r I n e q u a l i t y . his approach here,

but without currents.

I will present

For

-3-

Federer

a l s o s h o w e d m e h o w t o d e r i v e (B) f r o m (A) , b y a

t e c h n i q u e w h i c h h a s b e e n u s e d i n r e c e n t y e a r s in w o r k on t h e P l a t e a u problem

.

I do n o t k n o w i t s o r i g i n .

( B r i e f l y , t h e m e t h o d is t h i s .

C o m p a r e t h e v o l u m e of t h e v a r i e t y w i t h t h a t of t h e c o n e t h r o u g h

O

o v e r t h e i n t e r s e c t i o n of t h e v a r i e t y w i t h t h e

, for

each

r < R .

r-sphere

about

O

Then integrate.)

Theorems

(C), (E) a n d (F)

w e r e a l l p r o v e d b y B i s h o p i n [1] .

I w i l l p r e s e n t h i s p r o o f of (C) i n C h a p t e r s III a n d IV ; a n d i n t h e A p p e n d i x I w i l l d i s c u s s t h e r e d u c t i o n of Theorem

(F)

to

(E) .

(D) is S t o l l ' s c r i t e r i o n f o r a g l o b a l s u b v a r i e t y of

to b e a l g e b r a i c [8] .

( It i s a l s o n e c e s s a r y . )

In C h a p t e r IV I w i l l

d e r i v e (D) f r o m (C) b y f o l l o w i n g a n o l d i d e a of R u t i s h a u s e r y i e l d s t h e s i m p l e s t p r o o f of S t o l l ' s c r i t e r i o n . now a p p e a r s ,

take

U = ~pn

hyperplane at infinity. in

(D) i m p l i e s t h a t

This

( A nd C h o w ' s T h e o r e m

consequence of

complex projective

n-space and

W =the

V h a s finite v o l u m e when v i e w e d in

g i v e s ~ " a s a s u b v a r i e t y of

~pn

w h i c h is a l s o a c o r o l l a r y of (F) , s a y s t h a t The earliest

(F) .

T h e g r o w t h c o n d i t i o n on t h e v o l u m e of V c ~n

= ~;nu W w i t h t h e s t a n d a r d IC~hler m e t r i c T h e n (F)

[7] .

where it belongs, as a corollary.)

T h e o r e m (D) i s a l s o a n i m m e d i a t e ( Essentially,

~n

s t a t e m e n t of

on p r o j e c t i v e s p a c e .

. and Chow's Theorem, V is a l g e b r a i c .

)

(C) t h a t I h a v e s e e n - f o r c u r v e s i n ~

-4-

is t h a t of K. O k a in h i s p r e - R o m a n n u m e r a l s 1934 n o t e [5] .

In 1950

i n [7] R u t i s h a u s e r p r e s e n t e d a c o n t r o v e r s i a l p r o o f of t h i s c a s e .

It

d i d n o t w i n g e n e r a l a c c e p t a n c e , p r o b a b l 7 b e c a u s e of h i s u n c o n v i n c i n g statement " Nach ~ I,h) kann jetzt

J

noch aus analytischen FlRchen

o d e r a n a l y t i s c h e n H T p e r f l R c h e n b e s t e h e n " on p. Z64 , on w h i c h h i s argument apparently depended.

In t h i s s a m e p a p e r R u t i s h a u s e r s h o w e d

h o w to g e t (D) f r o m (C) f o r c u r v e s in

~

.

In C h a p t e r IV I w i l l e x t e n d

t h i s to t h e g e n e r a l c a s e . F i n a l l T , in 1962, T h e o r e m (C) w a s p r o v e d f o r c u r v e s in

~

by

N i s h i n o [4] a n d Oka [6] ~ a n d in 1964, b y a s o m e w h a t s i m i l a r m e t h o d , S t o l l [9] s e t t l e d i t f o r h y p e r s u r f a c e s . hypersurface integral

{ f = 0 } in a d o m a i n

In t h i s a p p r o a c h t h e v o l u m e of a U is r e l a t e d to t h e

~ u l o g If l d U .

Rutishauser's

a t t a c k on (C) w a s d i f f e r e n t .

He r e l i e d on a l o c a l

l o w e r e s t i m a t e l i k e (B) to be p l a y e d off a g a i n s t t h e g i v e n u p p e r b o u n d . T h i s is t h e s a m e a p p r o a c h t h a t B i s h o p [1] f o l l o w e d w i t h g r e a t s u c c e s s in (C) a n d (F) - a h d w h i c h I w i l l p r e s e n t in C h a p t e r s HI a n d IV . (Bishop's proof of ( C ) s h o w s ,

incidentally, that Rutishauser's m e t h o d

works. )

T h e r e d u c t i o n of (F) to (E) a l s o r e q u i r e s a l o c a l l o w e r b o u n d . Namely, suppose that

B is a n R - b a l l in

~ja a n d V i s a p u r e l y

k-dimensional subvariety of B - { z = 0 } w h o s e closure ~ ..... n

passes

-5-

through the center

of

indePendent of V

and R ,

(#)

B .

Then~ for some constant

VOlzk(V) >

This estimate

c ( k , n ) 9 R Zk

c(k,n) ,

.

c a n a l s o b e u s e d i n p l a c e o f (B) t o p r o v e (C) , b u t (B)

gives the best value of the constant,

c ( k , n ) = G(Zk) .

worked

only for

out this local lower estimate

the general

case would follow by induction on

s k e t c h of h i s a r g u m e n t

( My own interest constructed

a sequence

through the center, analytic disks. construction

will be presented

In [ I ] B i s h o p

k = 1 and asserted k .

A more

detailed

in the Appendix.

in these problems

stems

of analytic curves

in a ball in

from the time [I0] I ~2 , all passing

such that the limit set did not even contain

The areas

contradicted

that

grew (inevitably)

tremendously,

KutishauserWs assertion

about

" J"

any

and the o n p . Z64

of [?]. [ believe all these matters are n o w firmly settled, )

.%11 t h e b a c k g r o u n d

material

on analytic varieties

needed is contained in the first three of Several

Complex Variables"

chapters

by R.C.

that will be

o f ,, A ~ a l y t i c F u n c t i o n s

G u n n i n g a n d Ho R o s s i [3] .

CHAPTER

I

A n a l y t i c Vari.eties Min'm~.ize Volume

The e x a c t s t a t e m e n t will c o m e l a t e r . the possible presence Wirtinger's

of s i n g u l a r i t i e s . ) The p r o o f h a s t w o i n g r e d i e n t s ,

Inequality and Stokes' T h e o r e m .

Wirtinger's let

Inec~uality.

Let

L be a complex linear space and

M be a real even-dimensional

d e f i n i t e H e r m e t l a n f o r m on symmetric

L

subspace.

.

Then

a n d A is a l t e r n a t i n g .

Let

M w h i c h is o r t h o n o r m a l w i t h

with equality holding precisely when s u b s p a c e of L .

([2])

(Here

Ak is the

Firstly,

restriction

of A to M

representation.

Namely,

,

H be a positive

( ml,...,mZk

S is

} b e a b a s i s of

Then

[ (Xil'

~

e l= dXilA...AdX.ld_l

Also define

" r [ l I ( HI {6 )) 9 T h e r e s t r i c t i o n s

any smooth function

f

and any positive integer

~i = 5

on

- ~ I ( H I ( 5 ))

0 imply that for

j < e

I

and

~e) WlAd.x. = ~ J For each

I

such that

and

projection

9 Xld_l) "

"'',

andlet

~

I m

Then set

I

f(~IAC~" . J

c h o o s e a s e q u e n c e of s m o o t h f u n c t i o n s 0
r

T w o propositions will be proved.

PROPOSITION V

V-S

of a d o m a i n in

S i s a c o u n t a b l e u n i o n of s m o o t h m a n i f o l d s

For any Euclidean space

subvariety

subvariety

r < R

let

2k

-17-

PROPOSITION

smooth

2.

Let

d-dimensio_nal submanifold of a neighborhood of O

such that

0 E M

, and define

1 lira r---~ O ~ ( d ) r d

From (B)

Let M be a

d < e be positive integers.

Mr : { m

is true whenever

B u t e v e n i f O ES a sequence

such that

V O l z k ( y N B (vii r i ) ) V N B (v i~ r i )

O EV-S

( where

R e

Then

1

that Theorem

S i s t h e s i n g u l a r l o c u s of V . ) v.

I

B(v i ~ ri)=

~ VOlzk(V ) ,

implies also

=

it follows immediately

t h e r e is a s e q u e n c e

r.1 ~ R

IImll < r }

9 Vol d ( M r )

these two propositions

in

in

V-S

with

B(O ~ R)

so Theorem

.

v.

> O

I

, and

Then

(B) f o r e a c h

V o l z k ( V ) > ~(Zk) 9 R Zk .

Therefore

, it

s u f f i c e s to p r o v e t h e t w o p r o p o s i t i o n s .

P r o o f of 2.

>I.

T b e t h e t a n g e n t s p a c e to

on

Xl''''IXe

where

lim

is

) Let

near

( N o t i c e t h a t a l l w e n e e d f o r (B) i s t h a t t h e

O

,

M

5(O)=

IRe so that

T

is defined by equations

0

and

at

O and choose coordinates

= { Xd+ 1 = 0 , ...,

dhj(O)= O.

h(x) = (hd+l(X),..., he(X)) . T h e n if we define

M

xe = 0 } .

x. = h . ( X l , . . . s X d ) j J

Write

Then,

j=d+l,...,e

x = (Xl,...,x d) and

Uh(x) ll = o( llxll ) . Therefore,

-18-

re(r)

then

-- m i n { IIx[I : II( x , h ( x ) )

m(r)/r

> 1 as

H: ]Re-----> T For each

r

> 0 .

be the projection

s > 0 let T

= { p ET

II :

r }

To use this information

(Xl,...,Xe)

> (Xl,...,x

: llPll < s } .

Then

S

and, for all small

rl ( M r )

c~(d)-(m(r)) d


1 .

~(d)r

the proof of Proposition ,x d

enough, U

V O l d ( T s ) = ~{d). s

~(d)r

VOld(FI(Mr) )

VOld(Mr) ~

0 ) 9

"

1

lira r--~ 0

But

d, 0 . . . . .

r ,

Tm(r) C

Hence,

let

and set

Z view

U r = II (M r ) .

an upper bound for

of the Jacobian

as

]R d w i t h

Then, by linear

VOld(Mr)

of t h e s u m o f t h e a b s o l u t e

submatrices

T

algebra,

is given by the

v a l u e s of t h e d e t e r m i n a n t s

matrix

of the mapping

-19-

(x I , . . . .

) (Xl' .... Xd' hd+l(X)'''''he(X))

x d)

integrates

out to

of at least one Ur = •(Mr)

.

One

surnmand

V O l d ( U r ) :, a n d e v e r y o t h e r o n e i n v o l v e s t h e g r a d i e n t

h. , s o t h a t i t s i n t e g r a l j

over

U

r

is

o(VOld(Ur) ) .

But

so we have

1

lira r

d >0

" V~

)

~(d)r

1 lim r )0

Proof

of 1 .

F(r) = V O l z k ( V r )

9 VOld(r[ (Mr))

This will be an application for

r < 1% .

integrating a n d exponentiating,

Zk r

which I will rewrite

expression

vertex all

r

0 )

< --

F(r)


O

of t h e f o r m

co

~

( diam S.)d J

j=l where as

S =

8 ~ 0

co U S. j=l J

and each

, and we define the Hausdorff

Hd(S)

( The

i

d i a r n S. < ~ . J

is purely

=

I

Z'~

9 lira

8

for aesthetic

9 O

I(S,8 ) increases d-measure

I(S,

8

) .

reasons.)

2d

Here

are some

important

elementary

properties,

of

(weakly)

S to be

-ZZ-

PROPERTY

I. if Hd(S ) < co and

PROPERTY

Z.

another metric

x I, x 2 s ScX

Suppose

space

d > 0 ,

then

f is a Lipschitz

Y with Lipschitz

dist(f(Xl),f(Xz))
oo

S f]K for every

Note that it does not follow that S, 0 K

compact

K.

i

(For

example,

s.n{p}

if p ES

=

but

p

i s n o t in a n y

S.

1

then

sn{p}.)

I

If X

can be expressed as a countable union of compact sets then,

by a diagonal process, every sequence

S. yields a convergent 1

subsequence.

This is the sense in which we will discuss

analytic subvarieties

of a d o m a i n i n

The next proposition

relates

of

~,n .

the metric

and the measure,

shows how a local lower bound like that of Theorem off against an upper bound.

limits

(B)

and

can be played

-24-

P R O P O S I T I O N 3. ( B i s h o p ) L e t S. ~

Hd(Si) < M

a n d , fO r e a c h . c o m p a c t

a n y S.10K a n d r_< r(K) , B(p; r) compact

d> 0 .

S a s c l o s e d s u b s e t s of X a n d s u p p o s e ther_e_ is a c o n s t a n t

M > 0 such that every N >

L e t X be a m e t r i c s p a c e a n d

KcX

9 S u p p o s e als~ Iso t h e r e is a c o n s t a n t

r

p i.~n

( where

Hd(S i (] B(p; r)) _> N . r d

is the open metric ball.of radius KcX

such that, for

an r(K) > 0

about p . ) Then, for a n y

,

H d (lira (S i O K ) ) < 4 d M / N i---> co

.

If X i% G - c o m p a c t t h e n Hd(S) < 4d/vi/N.

Proof:

Let

S K = lira (S i f l K ) . If p ES K t h e r e is a s e q u e n c e i - - ~ co Pi E S., N K w i t h P i ' - - - > p " By t h e a s s u m e d l o c a l l o w e r b o u n d it m u s t be that for 8 < r(K)

r_~m Hd( S i n B(p; 8 )) _> N 9 8 i--~ oo

Also, for any 8> 0 there are finitely many points i n SK s u c h t h a t t h e b a l l s

B(pi(8 ) ; 8 / 2 )

sKcu B(pi(8); r i

d

pl(8),...

are all disjoint and

,pn(8)(8)

-ZS-

This can be demonstrated

as follows.

Since

S K is c o m p a c t a n d i s

c o v e r e d b y { B (p; S / Z ) : p s K } t h e r e is a f i n i t e s u b c o v e r -

B(q i; S/2~,..., B(qm~ E / Z ) . Let PI(S) = ql B(qj~ E/Z) which meet Let

Remove

say

all

B(PI(S)~ E/Z) . They all lie in B(PI(S)% S ) .

B(qj2; S/2 ) be the next ball that remains.

Let

pz(S) = qJ2 and

continue . Combining these remarks, M > lira

we now have, for

0 < S < r(K) ,

Hd(Si)

i---> co n(S ) _> i i m i - - ~ co

~

H d (S i n B ( p j ( s )

; S/Z))

j=l

n(S) > ~,

N(S/Z) d

;

j=l a n d , f r o m t h e d e f i n i t i o n of H a u s d o r f f m e a s u r e ,

Hd(SK) < lira S---> 0

Therefore,

Hd(S K)


I as

Therefore,

But F(S)

(a, b) , and d e f i n e a m a p

o z(p)

'

and, by assumption,

by Property 2 ,

contains

s p a c e in

(~kxRby

Z(p----)-'''"

This is a Lipschitz m a p

the set

~k.

o >

andwe express

o Zk+ WkZ = 0 } is an n - k

= 0,.,.,

W ~ .

But

~k f o r w h i c h

in

>a}

P

6 ~ (I, Z )

o

0 } for suitable coordinates then

Z ) has no i n t e r i o r in Call this set

L

.

W x I. o

F(S)

has m e a s u r e

Therefore

W

o

ReZlp)

.

H2k+l(S)

= 0 .

0 in

~kx~

= •Zk+l

cannot h a v e any i n t e r i o r

in c k . Q.E.D.

.

CHAPTER

IV

T h e U s e of th.e. P r o p e . r M a p p i n g

We will now prove

THEOREM subvarieties limit set

(C) .

Theorem

Let

U .

is again a purely

Proof:

be a sequence

I f VOl2k(Vi)

V is a subvariety

of N

.

U wema7

p

is the origin.

coordinates Zl'''''Zn and Nn. k of O

rI: ~n

~

rl to V D N n

k-dimensional.

n

on @;

3 ~ and the local lower so that every non-

such that

V nN

is a subvariety

as well take

p ~V

Then , by Proposition

and neighborhoods,

maps

VONn

; and

4 , there

N k of O

n-k in ~; , such that if N n = N.KX nN-k

~;k by (Zl,...pZn)

For,

of U I will prove that for each

NcU

so that

V

Zk-measure.

there is aneighborhood V is closed in

to a (.non-e~

U .

over in the limit,

p EU

translate

of

of Proposition

empty open subset of V has positive

Since

k-dimensional

bounded above then

it must be purely

HZk+I(V) = 0 by the application (B) c a r r i e s

is uniformly subvariety

If V i s a v a r i e t y

To show that

of purely

G;n w h i c h c o n v e r g e

k-dimensional

bound of Theorem

(D) .

1

U in

of a domain

V in

V.

(C) a n d T h e o r e m

in

are k

~;

and

~ (Zlp...,Zk)~thenthe restriction of

properly to N k . Also N n C

U

. Therefore,

-31

since

V.

compact

> V

in

U

open subset

restriction

Let

, it follows directly DkCN k

o f 11 t o

-

and all

i

sufficiently large,

V.1 O ( D k X Nn_ k ) i s a p r o p e r

D k be a p o l y d i s k a n d d e f i n e

the restriction

o f II

to

V. O D

.

Since

~;

k

variety

and

Dk

Let H i be

V. O D i s a p u r e l y 1

H.

1

is a proper

mapping to a polydisk in

it follows that

ViA D

is a finite-sheeted

branched

discussion

of these matters

especially

Theorem

Then,

l~i >

the

mapping to

D = Dk;(Nn_ k.

1

k-dimensional

that, for any relatively

if M

cgvering

Dk

>

of

s(i)

are also uniformly

V.

V.

1

have the same

, so that the sheet n u m b e r s

Therefore

we can extract

a

have the same limit set

sheet number

of a p r o p e r

s .

analytic mapping

C h a p t e r I I I . o f [3] ) g i v e s t h e f o l l o w i n g i n f o r m a t i o n

such that

.

Kelabeling,

V ) call

again.

The local analysis

For each

sheet number

VOlzk(V i ) w e have

( which will naturally

1

a l l of w h o s e m e m b e r s this subsequence

s(i) 9 VOlzk(Dk)

bounded above.

of t h e

( For a complete

be the associated

is an upper bound for all the

) >

Dk .

r e a d C h a p t e r H I , S e c t i o n B o f [3] ,

Zl. ) Let

VOlzk(V lAD

subsequence

11i

i there is an open dense connected

( as described

. subset

D ki

of

Dk

in

-3Z-

a) over

D ki

e v e r y b o u n d e d a n a l y t i c f u n c t i o n on

Dk

(in fact,

b) II"I i

( D ki )

c) II'1{ i

Dki

i Dk

Dk -

extends analytically

i s a s u b v a r i e t y of

Dk )

i s d i s j o i n t f r o m t h e s i n g u l a r l o c u s of V. i

)

~

D ki

is a n o n - s i n g u l a r

analytic even

s- sheeted

c o v e r i n g of D k . Also, for each

q E V. n D

,

once

G is a s m a l l enough n e i g h b o r -

1

h o o d o f r[(q) i n

Dk

t h e n t h e s h e e t n u m b e r of t h e r e s t r i c t i o n

t h a t c o m p o n e n t of I1:1 (G)

which contains

1

this number xED

k

mi(q)

q

remains

, t h e m u l t i p l i c i t y of H i a t

q .

of ]I1 t o

the same.

Call

Then , for all

,

mi(q)

=

s

.

Hi(q) = x

Therefore, u

in D

of points

since

its associates q

in V. N D

II: D

~ Dk

, we can define for each point

[ Ai (u),..., A is (u) ] as t h e u n o r d e r e d

s-tple

( counted with multiplicity ~ for w h i c h

Ili(q) = If(u). Now I w i l l c o m p l e t e t h e p r o o f of T h e o r e m for each point

v ED - ( V ND )

a function

(C) b y c o n s t r u c t i n g a n a l y t i c on

h

D such

V

that

h

V

= 0 on V ND

First,

fix

v

but

hv ( v ~ ~

0.

a n d p a s s t o a s u b s e q u e n c e of t h e

Vi

,

s o that.

-33-

(after

relabeling)

point AC~ (v)

in

J

Next~

for each

let

j = 1,...,s

,

A~(v ) converges

to some

VAD

f

be any bounded analytic function on

D with the

v

property

that

fv (v) ~

( For example,

fv (Ajc~

we could take

f

j : l,...,s .

combination

to be a suitable linear

v

of Zl,...,z n. ) Then

define functions

h

i v

on D by

s

hiv (z) : H (fv (z) - fv (A~(z)) ) j=l

These functions are polynomials

of degree

s

in f

whose coefficients v

are uniformly

bounded

on

i Dk ~ Therefore,

Dk

,

and

{ h iv

functions on uniformity

D

( with respect these coefficients

} is a uniformly .

Hence,

to

i ) on

bounded sequence

h

and are analytic

must be analytic everywhere

some subsequence

to an analytic function

Dk

on

of a n a l y t i c

converges

D

.

with local

Evidently,

h

v

i f a n d o n l y i f f v (u)

is the limit,

of { f v ( A ~ ( u ) ) } .

Therefore,

(u) = o v

for some hv

on

j

= 0 on

, of a certain VAD

but

subsequence

hv (v} ~ 0 .

Q.E.D.

-34-

THEOREM of

~n

.

(D) . L e t V be a p u r e l y

K > 0 such that

S u p p o s e t h e r e is a c o n s t a n t

VOlzk ( V N B ( O ; R))


(zl,oo.pZ k)

s-sheeted branched covering

~~ Dk

-35-

T h e n the r e s t r i c t i o n

of • to e a c h @R ( V R A D ) 1

a proper

is again

1

s-sheeted branched covering

V f] PR.(D)

> PR.(Dk)

1

Since the V ApR.(D)

"

1

exhaust

V

and the p R (Dk) e x h a u s t

i

i

= VApR.(D)

1

~;k a s

I

~- oo it follows that

v

is a l s o p r o p e r a n d

nlv

~ ck

s-sheeted.

This by itseH will not insure that c a n b e c o m p l e t e d in t h e f o l l o w i n g w a y .

V is a l g e b r a i c , For each

c o n s t r u c t a p o l y n o m i a l v a n i s h i n g on V b u t n o t a t To do t h i s I f i r s t c o n s i d e r , [A l(z),..., Aj(z) 6 V

for each point

As(Z) ] w i t h r e s p e c t to r [ I v : v and II(Aj(z)) = If(z).) Also,

i , there are the associates I]i: YR. liD

p 6 •n_

~;k

, its a s s o c i a t e s ( determined by

for each point w s

[ A i1( w ) , . . . ,

A is ( w ) ]

V Iwill

p .

z 6 (I; n )

but the p r o o f

and each

w i t h r e s p e c t to

> Dk .

I

There is a relation.

p I/R.(~)~D, 1

Namely,

for z 6 ~;n and all i so large that

-36i i Pl/R.(As(Z))] = [Al0tl/R.(Z)),...,As(Pl/R.(Z))]

(~)

[Pl/R.(AI(Z)},-.., 1

( because

1

Pl/R.

1

commutes with

1

r[ ) .

1

Now let

n

Zp(Z)

=

~, c . z .

33

j=l be a n y l i n e a r c o m b i n a t i o n of t h e c o o r d i n a t e s s u c h t h a t Zp(p) ~ Zp(Aj(p)) r

n

j = 1,...,

s .

D e f i n e an a n a l y t i c f u n c t i o n

hp

on

by

S

h (z)

P

=

[I j=l

(Zp(-) - Zp(Aj(-)))

Then S

h p (z)

= Z

H ~ ( z 1 ...,z k) 9 (Zp(Z)) ~

~=0 where each

H.. is a n a l y t i c on

is clear that

ck

.

h (p) ~ 0 a n d h = 0 on V . P P

linear,

i t w i l l s u f f i c e to p r o v e t h a t e a c h

degree

< s-~

all

( w 1 leeo

By t h e f i r s t f o r m u l a f o r

.

,Wk)

Therefore,

since

H~ is a p o l y n o m i a l ,

h Z

P

P

it is

in f a c t of

B 7 C a u c h y ~ s E s t i m a t e it is e n o u g h to s h o w t h a t , f o r in t h e n e i g h b o r h o o d

D k of O in

~;k

(Ri)~-s " H~(R:wI'z "'''lliWk ) is bounded as R.I~ co . To this end

-37-

gpi on D by

define a n a l y t i c f u n c t i o n s

Sip(W)

=

s11

(Zp(W) - Zp(A~(w)))

.

j=l

Then s

~(wl,...,wk)

"

(

a n d the f u n c t i o n s

G~& a r e e v i d e n t l y u n i f o r m l y b o u n d e d ( w i t h r e s p e c t

to i ) on / 9 . .

But s i n c e

K

Z

P

is l i n e a r the r e l a t i o n

(:)

yields

i

hp(Riw) = (Ri)s. gp(W) ; so each (Ri)~-s

9

H~&IRiw I . . . . ' R i w k )

= G~(w 1 , . . . , w

k)

is b o u n d e d as

R. ~K co a n d w e a r e done. I

Q.E.D.

Appendix

Let

W be a subvariety

k-dimensional

subvariety

t h a t i f V O l z k ( V ) < co

of a domain

of U - W .

then

V

The proof that Proposition

( closure

U

U

in

to b e a ball.

embedding

U

replace

HI .

However

In that case W

by some

( z I , . .. , z n , f(z I ,..., z n )) w e

plane

{z

n

W

of Theorem

U ) is a subvariety

that

Hzk (V nw)

(F) i s of

U .

= 0 and

from

it is convenient to make the Firstly, w e m a y

suitable

{f = O } .

s p a c e via

( z I,

can a r r a n g e that W

n o w on w e

that of

localize a n d

is cut out by global equations on

in a h i g h e r - d i m e n s i o n a l

Therefore,

V be apurely

Hzk (V n W) = O closely parallels

3 of Chapter

so w e m a y

Let

(E) .

following reductions before proceeding. take

.

The assertion

This can be proved by first demonstrating then proving Theorem

U

shall a s s u m e

= O } . In proving that H z k ( V n w )

that : 0

V O I z k ( V ) < co plays the role of the u p p e r bound.

Finally, by ..., z n )

is a hyperplane. W

is the h y p e r -

, the a s s u m p t i o n

But besides that one

n e e d s a local l o w e r b o u n d of the following type.

(~)

There is a constant

is so small that

c(k,n)

B (p~ R ) c U t h e n

> 0

such that if p EV

and

R > 0

VOl2k( V A B (p ~ R )) ~ c ( k , n ) 9 R 2 k .

-39-

The case an argument

k = 1 is Theorem

of Bishop for the general

Remark.

By integrating,

V O l 2 k _ l ( V fl S ( p ; R ) ) By translation and

2 o f [1] .

R = I .

case,

it suffices to find

> d(k,n)

" R zk-I

where

Let

S = S(O;

d(k,n) > 0

I.

For each

2 )

__in ~ n

k .

so that

S ( p ; R ) = { p . IIp[I = R } . to the case

p = O

I) .

0< r < I

let

L

l

Jl-r

by induction on

and a change of scale we can restrict

LEMMA { zI =

Here is a sketch of

andfor

each

r

be the hyperplane

w = (w I ....

,Wn) ES

let

Pw

n

the hyperplane

{ ~ w.z. = 0 } i=l 1 1

.

There

is a constant

c(n) > 0

that

(1)

VOlzn_3(L r flS)

Proof:

(2)

_> c(n) I V O l z n . 5 ( L r N S f l P w . ) d S(w) ,

We have

Vo12no3 ( L r A S )

wh~.re

-

cl(n)- r

cl(n ) is the volume of the unit L

AS A P r

is a w

2n- 3

2n-3

svhere.

2n-5 sphere with equations

such

be

-40-

zI =

Izj I g = r 2

.

.

j=2 Its radius for

is therefore

c(n)

For a proper

~Vol 2n-5 (U NPw)

c h o i c e os c o o r d i n a t e s

dS(w} .

z I ,...

Bz n

and

-41 -

r > 0

L is the

large number

L

of L e m m a l

r

.

Wemay

IV[ of t h e s e p i e c e s .

~/N

LNS

into a

N of d i s j o i n t p i e c e s w h i c h a r e n e a r l y c o n g r u e n t u n d e r

the u n i t a r y group, in such a way that some

decompose

U i s v e r y n e a r l y t h e u n i o n of

T h e n t h e two s i d e s of (3) a r e v e r y n e a r l y

t i m e s t h e c o r r e s p o n d i n g s i d e s of (I) ,

T h u s (3) f o l l o w s f r o m

(1) i n the l i m i t .

LEM~ 1

c(k+1) ~d(k-l,

n-1)dS(w)

=

c(k+l)d(k-l,

n-l) VOIzn_l(S ) .

P a s s i n g to t h e l i m i t g i v e s

VOl2k-l(VnS)

with

> d(k.n)

d(k,n) = c(k+l) d(k-1, n-l)Vol2n_l(S)

.

T h i s c o m p l e t e s o u r s k e t c h of B i s h o p ' s a r g u m e n t f o r n o w a s i m p l e m a t t e r to m o d i f y t h e p r o o s of P r o p o s i t i o n

(~) .

It is

3 to g e t

H2k(V f]W) = 0 . A s f o r t h e p r o o f of T h e o r e m comment.

(E) I w i l l m a k e o n l y t h e f o l l o w i n g

The condition Hzk ( ~ N W ) yields

4) ~ l o c a l l y , a p r o j e c t i o n

.~n(Nk•

)

H > Nk = c k

( similar to Proposition

-43-

that is proper closed

, a n d s u c h t h a t t h e i m a g e of V N W N

a n d of m e a s u r e

zero

a n a l y s i s of t h i s p r o j e c t i o n elementary properties

Nk

.

k)

is

Bishop than makes a close

p u s i n g RadoSs T h e o r e m a n d s o m e

of r e p r e s e n t i n g

to get analytic equations for of [1 ] .

[l

in

(NkXNn.

measures

~r on N k R N n . k

for uniform algebras, .

T h i s is L e m m a

T h e r e a d e r i s n o w i n v i t e d to t u r n t o t h a t p a p e r .

?

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2.

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3.

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10. G. S t o l z e n b e r g ,

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