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

[ <_ k !

M is a complex

k-dimensional

k - t h e x t e r i o r p o w e r of A .)

t h e v a l u e of

d e p e n d on t h e c h o i c e of o r t h o n o r m a l

Let

H : S + iA w h e r e

r e s p e c t to S .

IAk(ml,...,rnzk)

Proof:

( It is c o m p l i c a t e d b y

IAk(ml ,...,m2k)

b a s i s of M

.

l does not

S e c o n d i y , the

AM , has a certain canonical t h e r e is an o r t h o n o r m a l

basis

m~,...

,m~ k

-7-

of M

with dual basis

~I''''' ~~

such that

k

AM j=1 . ) . aj = A ( m ZI j _ l , m 'Zj

where

Ak

H e n c e , on M ,

k = klAa,(~Zj_iA~zj j=l J

),

so t h a t

IA k ( m

nk

[aj

I

j=l

Therefore,

t h e g e n e r a l r e s u l t will f o l l o w d i r e c t l y f r o m t h e c a s e

applied to the real span of e a c h p a i r But w h e n H(ml,ml)

{ m ~ j _ l , m~j } .

k = I w e have A ( m l,m2) = - iH(m l,mT)

= 1 = H(mz,m2)

alternating. ) Therefore,

( because

and

S ( m ~ , m l j ) -- 6 ~ I j a n d A is

b y S c h w a r z l s Lnequality a p p l i e d to H ,

IA(m 1 , m Z ) [ <_I w i t h e q u a l i t y if a n d o n l y i ~ m Z = c m I f o r s o m e complex

number

c ~ 0 . Q.E.D.

Note.

B y a s u i t a b l e s h u f f l i n g of t h e b a s i s i t c a n b e a r r a n g e d t h a t

Ak(ml ,...,mZk

)_> 0 .

k =1

-8-

To a p p l y W i r t i n g e r t s on c o m p l e x

n-space

d~n

Inequality let

z I, ...,z n be coordinates

and set

n

n-= ~

dzj^d

.

j=l T h i s is t h e f u n d a m e n t a l ? - - f o r m of t h e s t a n d a r d K ~ h l e r s t r u c t u r e ~ja a n d j a t e a c h p o i n t

on

p E ~n s &~ is t h e a l t e r n a t i n g p a r t of t h e P

positive definite Hermetian form

n

~

d~j(p) 9 d~j(p)

j=l on t h e t a n g e n t s p a c e to

d~n a t

p .

2k-dimensional manifold immersed

Therefore in

Cn

, if %

is a n y s m o o t h

, Wirtinger~s Inequality

implies immediately that

~

~Gk

_< ~ I d ~

= Volume2k('~)

w i t h e q u a l i t y is a n d o n l y if " ~

is a c o m p l e x

~I.o, o~o~ ~ o~ ~ . .

~ ~o~

Zk-I

form

~k

~o,

which can be computed.

we can now deduce the non-singular

~

k-dimensional

~:

~~

manifold.

~o~ . o ~ ,

Hence, by Stokes ~ Theorem

v e r s i o n of T h e o r e m ( A ) .

Namely ,

-9-

Let

('~ , b ) and (~ , ~) be smooth compact

Zk-manifolds

with boundary i m m e r s e d in ~n and having the s a m e boundary b . _If "b~

is a complex

Proof:

VOlzk( ~

k-manifold then VOlzk( ~

) =

. ~k

= 13r

) < VOlzk(~ ) .

=

~

Now I w i l l s t a t e t h e m o r e g e n e r a l v e r s i o n of

_ VOlzk( h

(A) t h a t w i l l b e

proved here.

T H E O R E M (A) . v a r i e t y of a d o m a i n i n

Let

W

be a purely

~n a n d l e t

W ,, a n d s u c h t h a t

with boundary and and

bv = b - sG~

Z k - m a n i f o l d in

.

Let ~

Let

is a s m o o t h

has finite 2k-I volume.

Let

Zk-manifold ~V

d~n w i t h t h e p r o p e r t i e s

that

(~,

bY ) i s a ( n o t .

c o m p a c t ) m a n i f o l d with b o u n d a r y and that the set

5 =~-(~U

~V ) ( c l o s u r e in

~n)

is a c o m p a c t c o u n t a b l e u n i o n of

smooth manifolds of dimensions < Zk - 2 . Then

"~V

= V-SQV

be an), s m o o t h r e l a t i v e l y c o m p a c t

necessarily

(Note.

V be

, with topological boundary, b

( V - Sf~V , b - S G 5 )

b - SQb

analytic sub-

S be its s i n g u l a r l o c u s .

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

k-dimensional

The inequality is strict unless ~

V O l g k ( ~ V ) < VOlzk( ~ ) .

is an o p e n d e n s e s u b s e t of

but that will not be proved here. ) T h e p r o o f is b a s i c a l l y t h e s a m e a s i n t h e n o n - s i n g u l a r

case.

It

-10-

is only a matter of justifying the step

1 :

This w i U foUow directly from the fact that VOlzk( " ~ V ) < oo ( t o b e proved) and the following extension of Stokes' Theorem.

Stokes Extended.

Suppose

is a smooth relatively compact

~

d-manifold in IR e with finite d-volume and such that ~ ( closure in

-'~

IRe ) can be d e c o m p o s e d into b U 6 w h e r e

(i) ( ' ~ , b) is a ( not necessarily compact)

manifold-with-

boundary (ii) 5 has finite d-I volume (iii) 6 is compact (iv) the image of 8 under each coordinate projection d-I (Xl,...,Xe)

"i (Xil,..., x. ) has m e a s u r e Id_ 1

Then , for any d-1 form

~T Proof:

([2])

=

T on

~

dT

R%

e

0 in

IR

,

.

F o r each I = ( i l , . . . ,

id.l)

(where

-11-

I_< i I < . . . HI :

Re

< i d . I_< e ) s e t

) •d-I

(Xl,...,

x e)

and ~I

=

be the associated

> (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 <

<_1 ,

m= 0

m

=

~I

0 II

m

I'

express

on

o n a n e i g h b o r h o o d of

~ 1 u n i f o r m l y on c o m p a c t s u b s e t s of ~I

Tm

T as

R ~T

d-1

I

e I

(5) ,

- II i(5) . '

and define

'rm = I;lrIOI~t" Since these forms

T

v a n i s h i d e n t i c a l l y on ( v a r y i n g ) n e i g h b o r h o o d s m

of 6

, the standard version

~

dT

rn

of S t o k e s ~ T h e o r e m

=

~ T ~ m

applies -

-12-

Now c o m p u t i n g

)

Also,

-- ] ~ L d ( T l ~ )

ar m

The.reforep if we e x p r e s s

d( T l ~

because each

d

o%,,tOI

oo

9

) as

~ crI ,J.wiAd.x.J

j=l then

e

~111 ar

e

m

=

z z s j=l

~ZF j=l I ")~I

3

I

I,j~I ^d~'J - ~

dT

.

Hence

fal" - J'bldr . Q.E.D.

Note.

The f i ~ t e v o l u m e a s s u m p t i o n s w e r e u s e d in taking l i m i t s .

W h a t n e e d s t o b e c h e c k e d i n o r d e r t o a p p l y t h i s e x t e n s i o n of

-13-

S t o k e s ~ T h e o r e m to ~

w~V

and

V' 8V ?

Conditions (i)-(iii)

m

are evident for

, bv

and 8 =

~F)- ( ~ U 8 v ) k a n d

(iv)obtains

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

because <2k-2.

Therefore,

if V O l z k ( ~ ) < co t h e n t h e e x t e n s i o n a p p l i e s

and ~

1

f~k__ ~

~-[

@k

"

~v

On the o t h e r h a n d j w h e n VOl2k ( ~ )

= co t h e c o n c l u s i o n of (A) is

immediate . Similarly,

(i)- (iii) a r e d i r e c t f o r

Vfl s = ~-~v " ( ~ v

~V

9 bV and

U 8v ) " Condition (iv) holds because Vfl S is

contained in the singular locus S w h i c h , b e i n g a n a n a l y t i c v a r i e t y of dimension manifolds

k - I , is ( b y i n d u c t i o n ) of d i m e n s i o n s

< k-I .

VOlzk(I

Since

~-nV = ~

a c o u n t a b l e u n i o n of c o m p l e x

Therefore,

V ) < co

t h e o n l y m i s s i n g p i e c e is

9

is a compact subset of the variety W it s u f f i c e s

to prove the following local resttlt.

LEMMA. such that

F o r e a c h p o i n t p E W t h e r e is a n e i g h b o r h o o d N

VOlzk ( N f ] ( w

- S )) < co .

-14-

Proof: problem

W e m a y take p to be the origin O

is,

of course

Since

W

of coordinates Z.

=

11 { near

0

,

Chapter

is a purely z 1,...,z

9

= 0

Z.

)

Then,

O s

on

: 1

D about

neighborhood

NI

~n

( Almost

O

in

< i k 5_ n }

( Z i l , 9 . . ,.Zik ) m a p s

branched

covering

of

9 Let

s(I) h

I there is a hypersurface

systems

W

have this

of a v a r i e t y

( see

I = (i 1,...,i

k)

a

I

NI N W as a finite-sheeted

be the sheet number. of

~,

D

A/so,

for

such that

-1 lq I ( h I ) a n d

NINSC

Nin w

is a regu/ar Let of the

of the form

4~,such that the projection " , 4 ~ ~

zn )

each

choice

9 of Section C ) there is a

Corollary

( z 1 9 ....

D

for a suitable

will meet

local analysis

~k and for each

of o in

)

( n-k)-plane

all coordinate

by the standard

locus.

variety,

, each

<_ i 1 < . . .

H I o f [3] , e s p e c i a l l y

polydisk

the singular

k-dimensional

n

1k only at O .

O)

property.

, when

. ( The only

s(I)-sheeted

in

with the

W

D

hI

covering.

N be the intersection

HII(AI)

intersection

n~(~i~ '~

of the

Then

Zk-manifold

T

N I and let

T be the union

is a hypersurface W - S has measure

in

W 0 .

, so its Therefore

-15-

VOI2k(Nn(W-S))

= ~

~

~s(I).

VOlzk(D)

.

Nn (W-T) Q.E.D.

This justifies

a n d c o m p l e t e s t h e p r o o f of T h e o r e m

( Note .

(A) .

T h e m o s t g e n e r a l a n d n a t u r a l v e r s i o n of (A) s a y s t h a t a n

o p e n r e l a t i v e l y c o m p a c t s u b s e t of a n a n a l y t i c v a r i e t y i s a m i n i m a l c u r r e n t [2]. )

H

CHAPTER

A Local L o w e r B o u n d for the V o l u m e of an Analytic Variety

If V i s a p u r e l y define

k-dimensional

VOlzk(V ) = VOlzk(V-S ) .

Zk-manifold dimension.

and

( Note that

~n

is a smooth of l o w e r

)

a point p to O

THEOREM

let

liP II denote the distance f r o m

be the openball of radius

(B)

Let

R > 0

o f B( O; R) i__nn

Proof:

~

and let ~(d) be the v o l u m e of the unit ball. For each

r > 0 let B(p~r)

n

and let

such t h a t

V

r about the point p .

be a purely

= V[IB(O;r

1.

For e a c h p o s i t i v e

) . Then

r

1

Zk " v~ Y

is (weakly) increasing.

Zk (Vr)

k-dimensional

0 6 V. T h e n VO~k(V) > 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 <

c

Tr

V O i d ( H ( M r ) ) < ~(d). r d

d "V~

li~m r > 0

To complete coordinates if

r

is small

integral of the

over dx d

x 1 ....

r

,

d

SO

>

( rl ( M z ))

1

1

so that

d " V~

(Mr)

>

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)

<

2k

Theorem

of (A) .

This is a m o n o t o n e to p r o v e

r

9F'(r)

1

it is e n o u g h

a.e.

dr

a.e.

dr

Let

function of

r .

to s h o w

that

;

an inequality between volumes.

on the right is the volume

over

1 .

as

F(r)

so as to express

=

~(d)r d

~r = ( p ~V-S: (A) a p p l i e s ,

I claim that the

of t h e o p e n c o n e

lip II : r } .

and that

yielding the desired

C

r

with

~ ~or almost

inequality.

By

-20-

To justify real analytic

this,

p

)

lip II z on V - S . It is

so , e x c e p t ffor isolated v a l u e s of

analytic manifold. VOlzk_l(

consider the map

Then we can express

Bt)dt

r

,

VOlzk(Vr)

r

is a real

as

so t h a t

o

F'(r)

Therefore,

= VOl2k_l (~r)

2kr F t ( r )

8r = ~r - ( ~r U ~r ) "

since

(,%) i t r e m a i n s

( Everything

S N { p:

r e a s o n i n g as a b o v e , t h a t

lip II = r } ,

( for almost all

f i n i t e u n i o n of r e a l a n a l y t i c m a n i f o l d s 8 r

dimensions

dr.

Cr

. ) But

so it is compact. < k-I r )

8 r is

Mso.

we have, by the same S O { p:

of d i m e n s i o n s

lip

II = r }

< 2k-3 .

i s c o n t a i n e d i n a f i n i t e u n i o n of s m o o t h m a n i / o l d s <_2k-2 , s o t h a t T h e o x e m

"

only to examine

else is in order

S i s i t s e l f a v a r i e t y of d i m e n s i o n

the cone

a.e.

i s i n d e e d t h e v o l u m e of t h e c o n e

T o b e a b l e to a p p l y T h e o r e m

the c l o s e d c o n e o v e r

< co

is a

Therefore, of

(A) a p p l i e s a n d w e a r e d o n e .

CHAPTER Hausdors

Measure

m

and the Hausdorff

Metric

T h e m e t r i c will be used to take a limit of analytic varieties. The measure

i s i n l i e u of a v o l u m e

DEFINITION negative let

.

Let

real number.

X

be a metric

Let

I(S, 8 ) be the infimum

element

on the limit

space

S be any subset of all sums

of

and let X

.

set.

d be any nonFor

each

8> 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)) <

and

d< e

He(S ) = 0 .

map from

constant

K .

X

t_~o

( For all

9Ddist (Xl,Xz). ) Then, for an),

Hd(f(S)) < KdHd(S) . In particular, if Hd(S ) = 0

then Hd(f(S))= 0 .

PROPERTY

e

3. If X =

and

S is a smooth

d-manifold

in

e

]R

then VOld(S ) = ~ (d). Hd(S) .

The first two properties

are direct

The third can be quickly verified reasoning here. positive

by means

used to prove Proposition

( But I remark constants

Ad , Bd

Comp(K) metric

.

such that

Let

of Property

I will not carry

Z and the out the argument

A ~ H d ( S ) < VOld(S) < B ~ H d ( S ) . )

metric.

K be a compact

be the set of all compact

on Comp(K)

Z.

of the definitions.

that all we will need is that there are some

Next I will define the Hausdorff

DEFINITION

consequences

is defined by

subsets

metric of

K

space and let .

The Hausdorff

-Z3-

dist (S,T)

= maxftmin s s

dist(s,t)}+ m a x { r a i n

E T

t ET

This is indeed a metric and with it C o m p ( K ) For any metric subsets

space

X

, if

S.

1

dist(t,s)~.

s ES

is itself a compact space.

i = 1,Z,...

and

S are closed

I will say that

S.

~

S

I

provided

that for every compact

sequence

Lu. C o m p (K)

K c X # S. fl K i s a c o n v e r g e n t 1

and

s = u

lira

(si o x ) .

K i--> 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) <

If X is

)d

j=l

4d M/N

.

(Y - c o m p a c t e x p r e s s it a s t h e u n i o n of a s e q u e n c e of

compact subsets decreasing

n•S) (2S

Kt

t = lsZ,..,

and let

s e q u e n c e of p o s i t i v e n u m b e r s

SKt w e h a v e a s s o c i a t e d p o i n t s

pl(St),...

r t = r(Kt) .

such that

L e t St b e a

St < r t

.

For each

9 Pn (S t) (%) " If w e a r r a n g e

-26-

t h e s e p o i n t s i n the o r d e r

P1(81 ) . . . . .

and repeat sequence

Pn(81)(81)'

the elimination

P l (~Z)'" . . ' P n ( ~ z ) ( ~ 2 ) ' " " "

proceedure

used above we obtain a

of p o i n t s

Xl (~1)'" " ' ' Xm(~ 1)(~1 )'

Xl (~Z)' " ' ' '

Xm(~2)(~2 ) ' ' ' '

such that the balls B(xi(~t) ; ~/2) are all disjoint and

S

=

tU SKtC i, tU B(xi(~t ) ; ~t ) "

Then, as above, we get

M

> ~ No(VZ) d

and

Hd(S) < lira

~(Z~t }d

-

i,t

1

i,t

so that Hd(S) < 4dhf/N.

APPLICATION

a sec~uence of p u r e l y to

V

.

Let

U be a domain in

k-dimensional

s o m e c l o s e d s u b s e t of

bounded above then

HZk+I(V)

U .

subvarieties If

= O.

the

~u and let V.1 -be of U which c o n v e r g e s

VOlzk(V i) a r e u n i f o r m l y

-Z7-

F o r VOlzk(Vi) of V.

= VOlzk(V i - Si )

where

S.1 is t h e s i n g u l a r l o c u s

~ a n d i t f o l l o w s d i r e c t l y f r o m t h e s u b a d d i t i v i t y of H a u s d o r f f

1

measure

and from Property

3 that

when there are singularities

VOlzk(V i)

present.

H z k ( V ) < oo ~ so b y P r o p e r t y

1

= ~(Zk)- Hzk(Vi)

Therefore,

H2k+I(V)

even

by Proposition

3 ,

= 0 .

T h i s c a n be u s e d in t h e f o l l o w i n g w a y to g e t a c e r t a i n v a l u a b l e proper

mapping.

P R O P O S I T I O N 4. ( B i s h o p ) L e t Let

contains

O

.

HZk+I(S)

=

0 . T h e n there are coordinates

nieghborhoods

,

that S O ( N k X N n _

Nk

S be a c l o s e d s u b s e t of U

of O

in

11>

Nk

_ _

k)

~k by

n

U be a domain in

and

@;

which

such that

z I .... , z n

Nn_ k

of O

( z I ..... z n )

on

~n

and , such

i__nn ( n - k

> ( Zl,...,z k )

is_

a proper mapping.

Proof: n

, L AS

Suppose that for s o m e is totally-disconnected.

{ z I = 0 , ..., z k = 0 } . with

L = {O}

n-k

X ~ n-k

dimensional subspace

of

C h o o s e coordinates so that L

This d e c o m p o s e s

. Since

L

~n

into

S is closed in U

is

~k x ~n-k

and

L AS

is totally

d i s c o n n e c t e d t h e r e m u s t be a r e l a t i v e l y

c o m p a c t n e i g h b o r h o o d of O in

L N U w h o s e b o u n d a r y is d i s j o i n t f r o m

L ns

b o u n d a r y a r e of t h e f o r m

{O} XNn_ k and

.

This neighborhood and

{O}Xbn_

k where

Nn_ k

-28-

is a b o u n d e d neighborhood of 0

in

~

n-k

with boundary

bn. k .

m

Thus

({O}Xbn_k)

f~S= ~

and

compact and

S is c l o s e d in U ,

about

r

O in

NkXNn_kCU

Therefore

(~kXbn-

Sn(NkXNn_k)

is

C o n s e q u e n t l y , if [q: S n ( N k X N n . k ) K i s a n y c o m p a c t s e t in N k t h e n

=

~ and

N k x bn_ k . > Nk by p r o j e c t i o n and

[I " I ( K )

a n d is a t a p o s i t i v e d i s t a n c e f r o m b o t h

k) A S

is c l o s e d i n

N k x Nn_ k a n d is at a p o s i t i v e d i s t a n c e f r o m

(where

b n-k

if N k is a s m a l l e n o u g h p o l y d i s k

it w i l l s t i l l b e t r u e t h a t .

. Since

{O}XNn_kCU

is c l o s e d in S n (NkX Nn_k)

N k X bn_ k a n d

b k = b o u n d a r y of Nko ) T h e r e f o r e

II-I(K)

bkX Nn-k

is c o m p a c t ; a n d

this s h o w s that II is a proper mapping. It r e m a i n s to locate an

L

such that L [Is is totally-disconnected.

This can be done by a category argument The space

~

n-k

d i m e n s i o n a l s u b s p a c e s of

by taking the Hausdorff metric

- as follows. ~

n

form a complete metric

on t h e i r i n t e r s e c t i o n s

with

the c l o s e d u n i t b a l l . N o w f i x one c o o r d i n a t e s y s t e m

z 1, . . . ,

zn for reference

and let

Z v a r y o v e r a l l l i n e a r c o m b i n a t i o n s of z 1 , . . . ,

Zn w i t h c o e f f i c i e n t s

in ~ ( i ) .

in

Let

I v a r y o v e r all i n t e r v a l s

lie in e 9 Define

~(I,Z) : { L E ~ :

(a,b)

Re Z(L

~

whose end-points

A S ) DI } . T h e relevant

observation is that if L N S is not totally-disconnected then to s o m e

(I,Z) . ( T a k e

a

Z

such that R e Z

L

belongs

is not constant on s o m e

-Z9-

non-trivial c o m p o n e n t of L D S . ) Therefore, e

aust

it suffices to prove that the

r

. Since each

z)

~

( I, Z ) do not and the

is clearly closed

(I, Z ) can be e n u m e r a t e d it is enough ( by the B a i r e C a t e g o r y Theorem)

to p r o v e that e a c h

~{I,Z)

has no i n t e r i o r in ~

f o r that i t w i l l b e

enough to show that if

L ~ as

..., z k =

{ z I = 0,

of all w = ( w 1 , . . . , w k ) Lw = { e l + WlZ ~(I,

F: {p 6r

Express '>

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))

<

K"

R Zk

k-dimensional subvariet~

for all R > 0 . Then

V

i_~s

algebraic.

Proof:

R > 0 l e t PR : ~;n

For each

m u i t i p l i c a t i o n by R

.

be s c a l a r

Since all our coordinate changes and projections

a r e l i n e a r t h e y w i l l c o m m u t e w i t h PR Then each

> ~n

VR i s a p u r e l y

" D e f i n e V R = p l / R ( V [ ~ B(O~R)).

k - d i m e n s i o n a l s u b v a r i e t y of t h e u n i t b a l i

B( O~ 1 ) ~ and the condition that VOlzk ( V N B ( O ~ R )) < K 9 R Zk is equivalent to V O l z k ( V R ) <

K.

subsequence of the family { V R }

Note also ( R ~

that every convergent

co ) c o n t a i n s

O in its l i m i t

set . By t h e p r o o f of T h e o r e m (C) , f o r s o m e s e q u e n c e

( with R. ~ 1

co ) there exist coordinates

a neighborhood

Zl,.-.,z n on

vR.} I!;

n

1

,

D = D k X Dn_ k of O , and an integer s > 0 s u c h t h a t

the projection I~: ~;n ~

~ k by

induces , for each i , a p r o p e r

VR.N D 1

(Zlp...jz

n)

> (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

<_ r

cz(n ) = the volume

VOlzm.5

= - w 1.

wj.j

j=2 if

[w 1 I _< r

of t h e u n i t

and

0 otherwise.

Zn-5 sphere,

( L r n s r ] p w ) dS(w)

<

cz(n) r

c3(n ) > 0

Hence,

we have

Zn-5

dS(w) Iw 1

But for some

- rz

r

,

d s (w)

< c3(n)

Z 9

r

Iw I I < r

T h e n (1) h o l d s w i t h

DEFINITION

c(n) = c 1 (n)/(cz{n) * c3(n) ) .

.

A

subset of

L AS w h e r e

(complex)

dimension

LEA~/~4A Z. If U

2k-i volume

element

U

of S is an open

L is a complex linear variety

in

~n

of

k .

isa

Zn-S dimensional

volume

element

of S

then

(3) V O l z n _ 3 ( U )

Proof:

_> 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
(4)

3 .

H

Zk-i d i m e n s i o n a l v o l u m e e l e m e n t of S 9

, then

Vol::,k_l(U) _> c ( k + l ) J'VOlzk__-3 (U N P w ) dS(w)

Proof:

n

.

In an a p p r o p r i a t e c o o r d i n a t e s y s t e m

L = { Zl= 2l-~rZ, . Zk+ with

U _i s_a

= 0, ..., z

r e p l a c e d by

n

(4)

= O } . Therefore

r e d u c e s to (3)

k+l.

Now, b y t h e r e m a r k a t t h e b e g i n n i n g of t h i s d i s c u s s i o n w h a t we require is a constant

d(k,n) > 0

By T h e o r e m Z of [ I ] w e m a y t a k e have

d(k-1 p n - I ) .

such that V o l Z k _ I ( V

d(l,n)

Then approximate

= Z~

.

nS)>d(k,n)

.

S u p p o s e we a l r e a d y

V G S by t h e u n i o n of f i n i t e l y

m a n y Zk- 1 d i m e n s i o n a l v o l u m e e l e m e n t s

UI,...,U

m

of S .

B y (4)

-42m V~

VOizk_1(Uj) _>

~

1 ( V r] s )

j=l 111

V~

3 ( Uj N pw, ) dS(w)

j=l

c(k+l) ~'VOlzk 3 ( VNS Np ,) dS(w) >

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.

H. F e d e r e r ,

S o m e T h e o r e m s on i n t e g r a l c u r r e n t s ,

Trans. A.M.S.

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3.

R . C . Gunning and H. R o s s i , " A n a l y t i c F u n c t i o n s of S e v e r a l Complex Variables", Prentice-Hall,

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.

H. R u t i s h a u s e r ,

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8.

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W. StoU, The growt!~ o f the a r e a of a t r a n s c e n d e n t a l a n a l y t i c s e t . I. and I I , Math. Ann. 156 (1964) 4 7 - 7 8 and 1 4 4 - 1 7 0 .

9.

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

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11.

W. W i r ~ g e r ,

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Eine Determinantenidentit~t

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