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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 .
?
Bibliography
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E. B i s h o p , Conditions f o r the a n a l y t i c i t y of c e r t a i n s e t s I Mich. Math. J o u r . 1 1 ( 1 9 6 4 ) 2 8 9 - 3 0 4 .
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,
4.
1965.
T. N i s h i n o , Sur l e s f a m i l i e s de s u r f a c e s a n a l y t i q u e s ,
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Kyoto Univ. 1(1962) 357-377.
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K. Oka, N o t e s u r l e s f a m i l i e s de f o n c t i o n s a n a l y t i q u e s m u l t i f o r m e s etc.,
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J. Sci. H i r o s h i r n a Univ. A 4 ( 1 9 3 4 ) 9 4 - 9 8 .
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H. R u t i s h a u s e r ,
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meromorphen Funktionen mehrerer a n a l y t i s c h e n Abbildungen~
8.
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