Holomorphic mappings into smooth projective varieties

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Holomorphic mappings into smooth projective varieties

BJ()RN DAHLBERG University of G6teborg, Sweden

1. Introduction

Let M and N be two complex manifolds of complex dimensions m and n respectively. A holomorphic map f: M - + N is called nondegenerate if in each component of M, there exists a point p, such that the induced linear map of the holomorphic tangent spaces f , : Te(M)--> Tf(p)(N) is surjective. This can only happen if m _ ~ n .

I f L is a holomorphic line bundle on a complex manifold N and R is a positive integer, then we say t h a t R has property P with respect to L if there exists a positive holomorphic line bundle T and a positive integer s, such that

H ~ R Q K N ) 8 | -1) # 0

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where KN is the canonical bundle of N.

The main result of this paper is the following generalization of the big Picard Theorem.

THEOREM ] .2. Let N be a smooth projective algebraic variety and L a holomorphic line bundle on N. Suppose further that F 1 .. . . , F n are sections of L in normal position, where R has property P with respect to L, and put F = F 1 • . . . Q F ~.

Let M be a complex manifold and f: M ~ S --> N ~ IDFI a nondegenerate holo- morphic mapping, where S is an analytic subvariety of M. Then f can be extended to a meromorphic map from M into N.

COROLLARY 1.3. Let f: M ~ S --> Pn be a nondegenerate holomorphic mapping, where S is an analytic subvariety of M and Pn is the n-dimensional complex projective space. I f f fails to meet R d-dimensional hypersurfaces in normal position and R ~_ (n ~- 1)/d, then f can be extended to a meromorphic mapping of M into Pn.

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60 B J O R N DAHLBE:RG

L e t us b r i e f l y outline t h e idea o f t h e p r o o f o f T h e o r e m 1.2. A f t e r a suitable i m b e d d i n g o f N into a Pq has been chosen, f can be r e p r e s e n t e d b y c e r t a i n a n a l y t i c fimctions gl . . . gq. T o show t h a t f has a m e r o m o r p h i c extension, it is sufficient to show t h a t t h e g r o w t h o f gl . . . . , gq is n o t too wild n e a r S. W e control t h e g r o w t h w i t h p o t e n t i a l - t h e o r e t i c m e t h o d s , which are o u t l i n e d in section 3.

I t m a y be p o i n t e d o u t t h a t t h e R i e m a n n e x t e n s i o n t h e o r e m implies t h a t i f codim (S) > 2, t h e n a n y h o l o m o r p h i c m a p p i n g f: M ~ S - ~ N can be e x t e n d e d t o m e r o m o r p h i c m a p p i n g of M into N , w h e n e v e r N is a s m o o t h p r o j e c t i v e v a r i e t y . F o r a b a c k g r o u n d t o this p a p e r we refer t o t h e s u r v e y article b y G r i f f i t h s [3]

a n d Carlson-Griffiths [2].

T h e p l a n o f t h e p a p e r is t h e following. I n section 2 we give t h e n e c e s s a r y b a c k - g r o u n d material. W e p r o v e in section 3 some technical results, a n d in section 4 we p r o v e T h e o r e m 1.2.

2. Notations and terminology

L e t M be a c o m p l e x manifold. R e l a t i v e t o a suitable open covering { Ui} o f M, a h o l o m o r p h i c line b u n d l e L is g i v e n b y h o l o m o r p h i c t r a n s i t i o n f u n c t i o n s fq: Ui A Uj --> C* = C ~ {0}, satisfying t h e cocycle condition

f,k = f,,fik in U, n u j n u k .

T h e t e n s o r p r o d u c t L 1 @ L 2 o f t w o line b u n d l e s L 1 a n d L 2 is g i v e n b y m u l t i - plication o f t h e c o r r e s p o n d i n g t r a n s i t i o n functions. A h o l o m o r p h i c section F = {F~) of L is given b y h o l o m o r p h i c f u n c t i o n s Fi: Ui - ~ C, satisfying t h e c o m p a t i b i l i t y r e l a t i o n

F , = f q F j in U, n u j .

T h e v e c t o r space o f all h o l o m o r p h i c sections o f L is d e n o t e d b y H ~ L ) . W e o b s e r v e t h a t a section F o f L in a n a t u r a l w a y gives rise t o a divisor D F on M. W e d e n o t e b y [DF[ t h e s u p p o r t of DF, which is t h e set {z C M: F~(z) = 0 for some i}. Suppose F 1 , . . . , F P are s~ctions o f L a n d p u t

F = F 1 @ . . . @ F ~ E H ~ Lp).

W e s a y t h a t F 1, . . . , F P are in n o r m a l p o s i t i o n if for eac21 p o i n t x C ]DFI we can fired an i, such t h a t x E Ui a n d {F~: s C Z(x)} can bz t a k e n as p a r t o f a c o o r d i n a t e s y s t e m a r o u n d x, where Z ( x ) - - {s: F~(x) = 0}. I~ is e a s y to see tha*~ a collection o f hype~planes m e e t s this c o n d i t i o n i f a n d o n l y if t h e y are in general position.

A m e t r i c in L is given b y positive C ~ f u n c t i o n s a~ in Ui, sa+Asfying in n

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]-[OLO1VfO~I~:KIC Ir INTO S~/fOOTI~ PI~OJECTIVE VARIETIES 61 T h u s for a holomorph~e section F = {F~} E H ~

L),

t h e l e n g t h

]]Fll 2 = ]Fi 12a~ 1 is well defined.

I n particalay, i f we let { U~} be a covering o f M w i t h c o o r d i n a t e n e i g h b o u r - hoods U i w i t h h o l o m o r p h i c coordinates Z i : (z~ . . . . , zm), t h e n t h e canonical b u n d l e K M of M is g i v e n b y t h e t r a n s i t i o n ~unctions

g~j -= det ~zjl) in

u i n u ~ .

Thus, if F = {F~} is a section o f

KM,

t h e n t h e locally g i v e n h o l o m o r p h i c (m, 0)- f o r m s

F~dz~ A . . . A dz7

p a t c h t o g e t h e r to a h o l o m o r p h i c (m, 0)-form on M . I n t h e same way, we o b s e r v e t h a t t h e metrics on KM are in o n e - t o - o n e c o r r e s p o n d e n c e w i t h t h e positive (m, m)-forms on M.

I f h is a C ~ real form of t y p e (1, 1), t h e n h is g i v e n locally b y h = V -- l ~ h,jdz, A dSj (hij = hji).

W e Shall s a y t h a t h is positive if (h~j) is a positive d e f i n i t e H e r m i t i a n m a t r i x . W e are n o w in a position to define w h a t is m e a n t b y a positive line bundle.

Definition

2.1. L e t L be a h o l o m o r p h i c line b u n d l e on t h e c o m p l e x m a n i f o l d M.

T h e n L is said to be

positive,

if t h e r e exists a m e t r i c {ai} on L, such t h a t t h e real (1, 1)-form h on M , given in U~ b y

h -- ~ r 1 a~ log a~

is positive. This m e t r i c is t h e n said t o be positive.

3. Some technical results

Since we are going to w o r k in C m, w e s t a r t b y collecting some n o t a t i o n .

][Zll 2 = [Z!] 2 -~- . . . Jr- [Zm[ 2 for

z = ( z l , . . . ,

Zm) E Ca m,

- 2 - 2 {Za JAd J} j = l

V k ~-- V 1 A V 1 A . . . A V 1, V 0 = 1,

A = 4 ~- . . . @ , t h e L a p l a c e o p e r a t o r .

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6 2 B J O R N D A H L B E R G

W e shall f r o m n o w on assume t h a t N is a s m o o t h c o n n e c t e d n - d i m e n s i o n a l p r o j e c t i v e v a r i e t y a n d t h a t L is a h o l o m o r p h i c line b u n d l e on N a n d t h a t R has p r o p e r t y P w i t h r e s p e c t to L.

L e t F 1 , . . . , F R E H ~ L) be in n o r m a l position a n d p u t F = F 1 | . . . | F R C H ~ LR).

H e n c e , we can f i n d a positive b u n d l e T, an integer s > 0 a n d an H c H ~ (L R | KN)" | T -~)

such t h a t H ~ 0 . L e t { a t } = a be a positive m e t r i c on T, a n d let e be a n y positive n u m b e r . W e o b s e r v e t h a t

R

cr • (alHI2)" 11F]-2 1-1" [log (ellFJll2)] -2 j=l

is a section o f K N Q R ~ in N ~ IDFI, a n d t h e r e f o r e gives rise to a n o n - n e g a t i v e (n, n ) - f o r m W in N ~ IDFI, g i v e n locally b y

W = \ 2 / (a, iH, l:)'-t]F,I-2 ]-[ [log (eliFJH2)]-2dwl A dffh A . . . A dw~A dff;,.

W e are now in a position to f o r m u l a t e t h e m a i n r e s u l t o f this section. F o r a r e l a t e d r e s u l t see Carlson-Griffiths [2].

THEOREM 3.3. There exist two positive numbers e and c with the following property: For any domain M in G '~ (open, connected set) and any nondegenerate holomorphic function f: M --> N ~ IDF] the nonnegative function uf in M defined by

f * W A V " - " = u f V m is such that log u I is plurisubharmonic and

A log uf ~ c(uy) 1!~ in M ~ {z E M: uy(z) : 0}. (3.4) Proof. L e t W 1 be a n y (n, n ) - f o r m on N given locally b y

W1 ~ \ 2 / hdwl A d ~ A . . . A dw~ A d ~ ,

a n d let Q b e t h e class o f subsets o f { 1 , . . . , m} c o n t a i n i n g e x a c t l y n elements.

I f we w r i t e f = ( f ~ , . . . , f , ) in local coordinates, t h e n one f i n d s easily t h a t f * W ~ A V " - ~ = h o f d e t \ o z j / j e s

Since

f

is a s s u m e d t o be n o n d e g e n e r a t e , we h a v e t h a t

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HOLOMOR1)HIC MA1)iPINGS II~TO SMOOTI-[ PROJECTIVE VARIETIES 63

is n o t identically zero. B y using Corollary 1.6.8 in H S r m a n d e r [5], we f i n d t h a t log Jf is p h r i s u b h a r m o n i c . T o show (3.4) it is t h e r e f o r e sufficient to p r o v e t h a t t h e r e exists a c > 0 such t h a t t h e i n e q u a l i t y

R

C(~f) 1In ~ A l o g {(ai o f l H i

oflZ)s-llFi

o f [ - 2 ]7- l o g ( e l l F " o fl12) -2}

r = l

holds in M 1 = M ~ . {z e M: uf(z) = 0}. Since b o t h log [Hi o f]~ a n d log IFi o f]2 are h a r m o n i c in M 1 it is e n o u g h t o show t h a t

C(Uf) x/n < UO, (3.6)

w h e r e we h a v e p u t u0 = s-l{A log (ai o f ) } - - ~ , ~ 1 A log (log (eHF o f H2)) 2.

W e s t a r t b y choosing e so small t h a t cHFI] 2 < 1 in N for 1 < r < R . This is possible b y t h e c o m p a c t n e s s o f N. W e localize a r o u n d ]DFI. F r o m t h e a s s u m p - t i o n s a b o u t F a n d t h e c o m p a c t n e s s o f N, it follows t h a t we can c o v e r ]DFI w i t h f i n i t e l y m a n y c o o r d i n a t e n e i g h b o u r h o o d s {U}, w i t h local c o o r d i n a t e s (wl . . . . , w , ) , such t h a t for some p, 1 < p < n , t h e r e exist kl . . . . , k v 1 < k 1 < k S < . . . < k~ _< R, w i t h w 1 = • k l , . . . , W p = F ~ , a n d if k q { k , , . . . , kp};

t h e n irrf~euIIFk(z)ll > 0. F u r t h e r m o r e , t h e r e exists a positive b E C k ( U ) , such t h a t if G E H~ L), t h e n IIGII2 = b]Ggl 2. W i t h o u t loss of g e n e r a l i t y we m a y a s s u m e t h a t wj = F [ for 1 _< j < p. Since we h a v e a s s u m e d t h a t T is positive, t h e r e exists a n u m b e r c > 0, such t h a t

s-lA log (a~ o f ) > c - - . (3.7)

-- i = I k = l OZk

I f v > 0, t h e n A log v = v-iAv -- 4 v - 2 ~ = 1 IOv/Ozkl 2. W e a p p l y this r e l a t i o n t o v = (log (e]]F of[j2)) 2. T h e n we h a v e in f - l ( U ) [3 M 1

A log (log (~IIF ~ o f i t 2 ) ) ~ = 2 ( l o g ( ~ i l f ~ o f l i 2 ) ) - l A log II F~ o f i l ~ - -

O log liE" ~ 2 2 (3.8)

8 (log of[J2)) -2

k=l/" azk "

I t is o b v i o u s t h a t t h e r e exists a n u m b e r C > 0, such t h a t if 1 < r < R, t h e n IA log ]IF" o fit 2 ] _< C ~

i=1"= k=l a n d if p + 1 < r < R , t h e n

0 log IIF" ofll 2 < C Z Ofi2

- - O z ~ "

k=l OZlt i=1 k=l

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6 4 B J O R ~ D A H L B E R G

T h e r e f o r e , b y combining t h e e s t i m a t e s a b o v e w i t h (3.7) a n d b y choosing s u f f i c i e n t l y small, we get t h e following e s t i m a t e of %, w i t h possibly a different c > 0

Ofi 2 0 l o g ] i F , ofl)-~ 2

l/ roJll ))

- - i = 1 k = l r = l k = l OZk "

N o w we h a v e t h a t

0 log IIF ~ o fll 2 0 log b o f 0fi

0z~ - 0zk + (w~ o f ) - I 0z~"

W e recall t h e following e l e m e n t a r y i n e q u a l i t y . I f ~1 a n d ~2 are two c o m p l e x n u m b e r s , t h e n ]~1 + ~2[ 2 ~> 1/21~1]2 - - r~212. B y using this i n e q u a l i t y w i t h

~ = Ofl/Ozk a n d ~2 ~ 0 l o g b of/Ozk we f i n d f r o m

(3.9)

t h a t

n & Ofi 2 l, ~ 2 _ _ O f f

U o > C ~ k~__li~Zk l + 4 Z Iw~ofl-2(log(ellF~ofll2)) -2

- - i = 1 r = l k = l I OZk

- - 8 Z (log (~IIF r ofl12)) -~ I 0 log b

,=l k=i 0zk "

W e m a y as before absorb t h e t h i r d t e r m of t h e r i g h t h a n d side of t h e i n e q u a l i t y a b o v e into t h e f i r s t t e r m b y choosing e s u f f i c i e n t l y small, a n d we are left w i t h u~ > c ~ Of, p

- - i = 1 k = l 0Z~k @ 4 ~ ~:l k=~ Iw~of[-2(log(elIF'of[I2))-2 Off2 Ozk > c ~(i) 2 ,

- - i = 1 k = l ~ k

w h e r e we h a v e p u t

~(i) = { ^I+ ]w~ of]_2 (log (e]]F ~ of]l=))-= if I < i < p 1 i f

p + l < i < n .

H e n c e we can f i n d a c > 0, such t h a t

0fl 2 SEQ i = 1 kES

(3.10) F r o m t h e i n e q u a l i t y b e t w e e n t h e a r i t h m e t i c a l a n d t h e g e o m e t r i c a l m e a n we h a v e

T

k e s OZk - - \ i = 1

i = 1 = kfiS i = 1

N o w we h a v e f r o m H a d a m a r d ' s i n e q u a l i t y t h a t

of, 2 i I~-t/Ofi\l<-i<n 2

fr :

i = I k6S

Since inf{]lF~(z)l]: z e U} > 0 for p + 1 < r < R, it follows t h a t

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H O L O 1 K O R P H I C 1 K A P P I N G S I I ~ T O SI~OOTI-I P R O J E C T I V E V A I % I E T I E S 65

R

~(i) ~ cI[F ofl/-~ T-[ (log (~IIF" o f[12))-%

i = 1 r ~ l

for some c > 0 in f I(U). B y c o m b i n i n g this w i t h (3.10) we get, w i t h some c > 0

Uo >--c(llF~ (l~ ~ ))- ) / Z tdet~-~/

^I ⁹

2in

r = l S ~ Q ~ \ U ~ ' k / k E S i

F r o m this i n e q u a l i t y (3.6) follows b y observing t h a t t h e t e r m s a~ a n d 1H~] can be t a k e n to be b o u n d e d a n d b y a p p l y i n g t h e i n e q u a l i t y ~ x ~ / " ~ (~. x;) 1/~ t o t h e last f a c t o r o f t h e i n e q u a l i t y above. T o c o m p l e t e t h e p r o o f we h a v e t o show t h a t we can choose e such t h a t log uf is also p l u r i s u b h a r m o n i c , b u t t h a t is done in t h e same way, a n d is t h e r e f o r e o m i t t e d .

W e shall n o w t u r n t o some consequences of T h e o r e m 3.3. S u p p o s e P0 C N ~ ID~[

~nd t h a t t h e r e exists a n H E H ~ R Q K N ) ~ | -1) w i t h H(po) r F i x a c o o r d i n a t e n e i g h b o u r h o o d U a r o u n d P0 w i t h local c o o r d i n a t e s (wl . . . . , w~).

I f f: Bm(r) -~ N is a h o l o m o r p h i c m a p p i n g , w h e r e B"(r) ~- {z e C,m: Hzll < r}, w i t h f ( 0 ) = Po, t h e n we p u t

/af~\,<,<o2

t J j i - -

s~ e d e t t ~ ) for z E f - ' ( U ) .

g l ( ~ ) = Q, ~ '

H e r e Q is t h e set of all subsets o f {1 . . . m} c o n t a i n i n g n e l e m e n t s a n d we h a v e w r i t t e n f ~ (fl, 9 9 fn) in local c o o r d i n a t e s (Wl . . . . , wn).

As a n a p p l i c a t i o n of T h e o r e m 3.3 we shall p r o v e t h e following g e n e r a l i z a t i o n o f L a n d a u ' s t h e o r e m (cf. K o d a i r a [7] a n d Carlson-Griffiths [2]).

THEOREM 3.11. To each m >_ n there exists a constant r, > 0 with the following properties: For any holomorphic mapping f: Bin(@) ----> N Nx 1DFI with f(O) ~ Po and Jf(O) > 1, the intequality @ <_ r o holds.

F r o m this r e s u l t we d e d u c e t h e following v a r i a n t o f t h e (small) P i c a r d t h e o r e m . COROLLARY 3.12. A n y holomorphic mapping f: C'~--> N "~ IDF] is degenerate.

Proof of Corollary 3.12. P i c k a G C H ~ (L R Q KN) s (~ T - I ) w i t h G r 0.

L e t M be t h e set o f all points z C {3 m such t h a t f , : T~(C')-+Tf(z)(N ) is surjective. S u p p o s e t h a t M is n o t e m p t y . T h e n M is o p e n a n d h e n c e f(M) is o p e n in N. T h e r e f o r e f ( M ) contains a p o i n t P0 such t h a t G(po) r 0. W e m a y a s s u m e f(0)--~ P0 a n d 0 C M. W e can choose a c o o r d i n a t e s y s t e m a r o u n d Po such t h a t Jf(0) - - 1. T h e o r e m 3.11 implies t h a t

f

can be d e f i n e d o n l y in a b o u n d e d d o m a i n a n d this c o n t r a d i c t i o n finishes t h e proof.

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66 B J O R ~ D A H L B E R G

Proof of Theorem 3.11. W e wish to each t > 0 construct a positive infinitely differentiable f u n c t i o n V, in B"(t) w i t h t h e following properties:

(a) A log V, < V~/~,

(b) tim V,(z) = oo for all zo 60Bm(t),

z - + z o

(c) lira Vr(z) = V,(z) for each fixed z e Bin(r),

r ~ t

(d) lim Vt(O) = O.

t-+o0

I f t > 0 is given, t h e n we p u t g t ( r ) = n l o g ( 8 n ( m + 1 ) ) + l o g ( t 2 ~ ( t 2 - r 2 ) - 2 " ) , 0 < r < t. We construct t h e f a m i l y { E } b y p u t t i n g

W e n o w have

V,(~) = e x p g,(r), I l z l / = r.

)

A log Vdz) = r -2"+1 - - dr (g&)) = d

r r 2 m - - 1

2m(t ~ - r 2) @ 2r u

--- 4n -4 8n(m + 1)te(t 2 -- r2) -2 = (Vt(z)) l/n, (t 2 ^-^- r2)2

a n d hence the f a m i l y { Vt} satisfies (a). The rest is s t r M g h t f o r w a r d verification.

N e x t we consider t h e f u n c t i o n uf as c o n s t r u c t e d in T h e o r e m 3.3. F r o m the assump- tions a b o u t f, it follows t h a t there exists a n u m b e r B > 0, such t h a t

uf(O) > B, for all f E ~e (3.13)

where T is the set of all holomorphie m a p p i n g s f: Bm(~) --~ N ~ [ D / , satisfying f(0) ----P0 a n d Jr(O) _> 1. L e t 9 ( be t h e set of all n o n d e g e n e r a t e holomorphie mappings f: B"(~) - ~ N ~ IDFI. W e w a n t t o show, t h a t there exists a n u m b e r A > 0, such t h a t for all Q > 0 a n d all f C g ~ we have

uf < AV~. (3.14)

W e prove relation (3.14) w i t h the aid of a classical trick due to Ahlfors [1]. F r o m p r o p e r t y (c) of Vt a n d the c o n t i n u i t y of uf, it follows t h a t to each t < ~, there exists a point ~ E B'~(t), such t h a t

uf(~)/Vt(~) -- sup {uf(z)/V,(z): z e Bin(t)}.

Since f is nondegenerate, we m u s t h a v e t h a t uf(~) > O, a n d hence uf is infinitely differentiable in a neighbourhood of ~. Moreover, since ~ is a local m a x i m u m for log (uf/V,), we have

A log (ugV,)(~) <_ O.

Now we have f r o m T h e o r e m 3.3 t h a t

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I t O L O M O R P H I C l V [ A P P I N G S I N T O S M O O T H 1 ) I ~ O J E C T I V E V A R I E T I E S 67 (u:(~)) :/" <_ CA log uf(~) <_ CA log V,(~) _< C(V,(~)) ,/~,

and hence uf(~) < A V~(~). Therefore, we have for all z E B~(~) uf(z) <_ A lim V,(z) = A V~(z).

This proves relation (3.14). I n particular, we have from (3.13) t h a t if f C ~o, then B < uf(O) <_ AV(O) = A{8n(m + 1)}~ -2~,

and this gives

~_ (A/B):/2~8n(m ~- 1), which completes the proof of Theorem 3.11.

From the proof of Theorem 3.11 we have the following corollary.

COROLLARY 3.15. To each m >_ n there exists a constant K > 0 with the following property: For all domains M in C m and all holomorphic mappings f: M --> N ~ IDoL the inequality

uf(z) < K(d(z, M)) -2~ (3.15)

holds, where d(z, M) = dist {z, aM} and f is nondegenerate.

Proof. We continue using the notation of the proof of Theorem 3.11. I t is sufficient to treat the case z = 0. For each Q < d(0, M) we have f C 9(~ and from (3.14) we have

u2(0 ) < A V~(O) = A{Sn(m + 1 ) } n ( 2n.

Letting ~ --~ d(0, M) we have (3.16), and this finishes the proof.

I t is possible to prove Theorem 3.11 along different lines.

Second proof of Theorem 3.11. We continue using the notation of the previous proof of Theorem 3.11. Let G be the Green function of B'~(r) with pole at 0.

This function is given by

log if m = 1

G(z)

= g(llzLI) = I ~

{[lz[] 2 - : ~ - r : - : ~ if m > 2 .

We have from the Ricsz representation formula applied to the subharmonic function log uf t h a t there exists a positive measure /~f on Bm(~), such t h a t for all r, 0 < r < ~,

u (o) = <, f l o g - .fa(z)d j(z). ^(a.16)

log

0B"(:) Bin(r)

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6 8 B J O R N D A H L B E R G

H e r e da is t h e area m e a s u r e o n 0B'~(1), ~,~ t h e area o f 0Bin(l), fil = (271;) - 1 a n d fin --~ { % ( 2 m - - 2)} -1 if m > 2. F r o m T h e o r e m 3.3 we h a v e dlq > c(uf)~/nV m.

P u t Pf(r) = fB~(0 (uf)l/'~V~" H e n c e we h a v e

r

Bin(r) 0

A p a r t i a l i n t e g r a t i o n gives t h a t

r

~., f P)(t)g(t)dt

0

= ~,., fP+(t)t~-:"~dt,

0

w h e r e ~m is a c o n s t a n t . Recalling (3.13) a n d (3.16) we h a v e

r

o OB~O)

Arguing as in K o d a i r a [7], one can f r o m i n e q u a l i t y (3.17) finish t h e proof. W e o m i t t h e a r g u m e n t .

4. The generalized Pieard Theorem L e t B = {z e C: [zl < 3}, B* = B ~ {0},

D - - B • 2 1 5 W e n e x t p r o v e T h e o r e m 1.2.

D* = B* x B • . . . • B a n d

m ~ l

Proof of Theorem 1.2. Since t h e set o f singular p o i n t s o f S are o f c o d i m c n s i o n

> 2, we m a y a s s u m e t h a t S is nonsingular. Localizing, we can a s s u m e t h a t f is d e f i n e d in D*. W e h a v e to show t h a t t h e pull b a c k o f t h e r a t i o n a l f u n c t i o n s o n N c a n be e x t e n d e d t o m e r o m o r p h i c f u n c t i o n s on D. T h e r e exists b y K o d a i r a [6]

a positive line b u n d l e E on N , such t h a t if {s ~ . . . , s q} is a basis o f H~ E), t h e n t h e m a p p i n g F : N - + Pq, d e f i n e d b y

v : z - ~ [s~ . .: sq(z)] ( 4 . 1 3

is a n i m b e d d i n g . L e t s

eH~ E),

s r

0,

a n d p u t G =f-l(JDs]). W e s t a r t b y d e m o n s t r a t i n g t h a t 0 is an a n a l y t i c s u b s e t of D. W e m a y assume t h a t IIs]] < 1.

I t follows f r o m B i s h o p ' s T h e o r e m ( T h e o r e m F o f S t o l z e n b e r g [8]), t h a t it is s u f f i c i e n t t o show t h a t t h e r e exists a n e i g h b o u r h o o d U o f {z E D: zj = 0}, such t h a t

j V real < o9. (4.2)

* s N U

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I I O L O M O R P I I I C M A P P I N G S I N T O S M O O T K P R O J E C T I V E v A R I E T I E S 69 I t follows f r o m t h e p r o o f o f T h e o r e m 3.3, t h a t t h e r e exists a ($ > 0, such t h a t t h e f u n c t i o n

vf =

log II s o

fllsuf

is p l u r i s u b h a r m o n i c in D*. W i t h o u t loss o f general- ity, we m a y a s s u m e t h a t t h e r e exists a zl, such t h a t

vf(zl,

0 . . . 0) > - - oo, other~

wise we change o u r c o o r d i n a t e system. W e can also assume, possibly a f t e r a change of t h e c o o r d i n a t e system, t h a t if IzlL = l, lz21 _< 1 , . . . , Iz,n[ <__ 1, t h e n

vf(zl, z2,...,z~)

> - - ~ . L e t Bin_ 1--{~6Gm-1:11~11 < 1} a n d

n(r,~)

be t h e n u m b e r of roots, c o u n t e d w i t h multiplicity, of t h e e q u a t i o n

8(f(%, ~))= 0 in

{ Z l : r < [Zl] < 1 } . F i x ~ 6 B .... 1 a n d p u t

v~(z)

= v f ( z ,

~).

W e can f i n d a sequence {v/}~l of twice c o n t i n u o u s l y differentiable s u b h a r m o n i c f u n c t i o n s in B*, such t h a t vj \ %. F o r each j, an a p p l i c a t i o n of G r e e n ' s f o r m u l a gives

2 z 2:r

fo 3rr v/(e/-~'~)dv~ -- r ~r vj(re/~)dvQ 4 f 0z105~ vj(zl)dV(zl) .

0 0 r < [ % ] < l

H e r e d V is t h e p l a n a r Lebesgue m e a s u r e on G. Dividing b y r a n d t h e n i n t e g r a t i n g , we h a v e

2 z 2~

f vj(re/-~i<>) d o - f v j ( e / - ~ ) dv~ +

0 0

2~ 1 (4.3)

fo f

@ (log

r 1) 3r vj(eV-~'~)dO ~ 4t-1 aZl~l vj(zx)dV(zl).

0 r t < l z ~ l < l

I t is clear t h a t t h e r e exists a c o n s t a n t K , such t h a t if lZll = 1, t h e n [vj(zl) I < K for all j . Moreover, this c o n s t a n t can be chosen i n d e p e n d e n t of ~ 6 B~_~. T h e f u n c t i o n s

vj

can be t a k e n as i t e r a t e d m e a n values. F r o m t h e explicit r e p r e s e n t a t i o n of t h e d e r i v a t i v e s o f such functions, see H e l m s [4, p. 20], it is clear t h a t t h e r e exists a c o n s t a n t K , also i n d e p e n d e n t of ~ E B b _ l , such t h a t if ]Zll = 1, t h e n

iOvj(zi)/~rl < K.

L e t t i n g j t e n d to oo in such a w a y t h a t t h e measures

(a2/OziOgl)VflV

t e n d w e a k l y t o t h e Riez m e a s u r e o f v~, we f i n d t h a t

1 2~ 2~ ( 4 . 4 )

f t-ln(t,.~)dt- f

^log

Hsof(re/-~,9, $),,dO ~_ f

^log

uf(re 1/-~, ~)d~ + K

log r - k

r 0 0

One c o n s e q u e n c e of (4.4) s h o u l d be n o t e d here. I f we t a k e ~ = 0, t h e n an i n t e g r a t i o n gives

I t follows f r o m Corollary 3.15, t h a t i n d e p e n d e n t of ~ 6 Bin_l,

1

f v+(=l, 0) v+ ^{>_ -} ^f t l o g t - l d t ~ -- K.

(4.5)

r < ]%1 < 1 r

log

us(ref-~'~, ~) < K

log r - t , w i t h K a n d h e n c e we h a v e f r o m (4.4)

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70 B J O R N D A H L B E E G

1 2 ~

f t-in(t, ~)dt -- ~ f

log [[so

f(re l/-~'~,

~)][dv ~ < K log r -1. (4.6)

r 0

W e g e t f r o m (4.6) t h a t

n(t, ~) < K

f o r all ~ E

B,n_ 1.

(4.7) W e m a y t h e r e f o r e a s s u m e t h a t

vf(z 1,0)

> - - oo f o r 0 < ]zl[ _< 1. I f z C B * , t h e n w e d e f i n e vz(~ ) =-

v/(z, ~), ~ E Bin_ 1.

L e t A,n_ 1 b e t h e L a p l a c e o p e r a t o r o n C ~-1. Since v~ is s u b h a r m o n i c in Bin_l, w e h a v e b y t h e R i e s z r e p r e s e n t a t i o n f o r m u l a

vo(o) = f vo(t)d m_l(t) -- f

( 4 . 8 )

0 B m _ 1 B m - - 1

w h e r e d % _ 1 is t h e s u r f a c e m e a s u r e o n ~B ~-1, n o r m a l i z e d so

fos,~_da.._l = 1, dVm_l

is t h e L e b e s g u e m e a s u r e o n

cm--l,g,,(~)

⁼ [~l 4 - 2 ' n - i , n ~ >__ 3,92(~ ) =

= log l~i 1 a n d c% = ( 2 m - - 4 ) ( 2 m - - 2)

fB,=_l d V ~ - I

f o r m >_ 3 a n d ~2 =

= (2~) -1. L e t /t(z) b e t h e 2 ( m - - 2 ) - v o l u m e o f G [3 {z} X {~ E C~-1: II~ll < 1/2}.

S i n c e g~(~) > ~ 1 > 0 if ]l~]] < 1/2, w e h a v e f r o m (4.8) t h a t /~(z) < K l o g Iz1-1 - -

~m2mV(Z, 0).

I f w e n o w m a k e u s e o f (4.5), t h e n a n i n t e g r a t i o n g i v e s

f # ( z ) d V ( z )

<

K

f o r all _T, 0 < < 1. (4.9) r<!~[<l

NOW t h e r e l a t i o n s (4.7) a n d . (4.9) e s t a b l i s h B i s h o p ' s c o n d i t i o n (4.2).

P i c k a n y t w o s e c t i o n s s I a n d s 2 o f E , s 2 r 0. T o c o m p l e t e t h e p r o o f , it is s u f f i c i e n t t o s h o w t h a t t h e m e r o m o r p h i c f u n c t i o n

h = (s 1 ofl)/(s2 o f)

h a s a m e r o - m o r p h i e e x t e n s i o n t o D . I n v i e w o f (4.2), w e m a y a s s u m e t h a t

f-~(ID,~I) c {z e D: z~ - - 0}.

H e n c e w e m a y a s s u m e t h a t h is a n a l y t i c in D * . L e t g h a v e t h e L a u r e n t e x p a n s i o n

g(z, z2 . . . . , zm) = zlAj(z2, 9 9 zm).

j ~ - c o

F i x ~ e B ~ _ 1. W e n o t e t h a t ]h I = l[sl

~ ~

a n d since w e m a y a s s u m e t h a t IIs~l I _< 1, i = 1, 2, w e h a v e t h a t (log+ = m a x (0, log))

2zr 2,~

flog+ tg(re d ~ , <

^-

[ log IIs o f(re ~ 1 ~ , )ll ,a _< log K

^{r - 1 ,}

(4.10)

0 0

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H O L O M O I ~ P H I C ~ A : P I ~ I N G S II~ITO S M O O T H I ~ I ~ O J E O T I V E V A R I ~ 3 T I E S 71

where t h e last i n e q u a l i t y follows f r o m i n e q u a l i t y (4.6). F r o m t h e N e v a n l i n n a t h e o r y it n o w follows t h a t t h e r e exists a n u m b e r k ~ 0 such t h a t A j ( ~ ) = 0 for all j ~ - - k . I f we for r C Z + p u t H ( r ) = { ~ C B m _ ~ : A j ( $ ) = 0 for all j G - - r } , t h e n we h a v e j u s t s h o w n t h a t [J, H(r) - - B,~_I. Since t h e sets H ( r ) are closed, a t least one, H(r2) say, c o n t a i n s a n o p e n subset o f Bm_~. Since t h e f u n c t i o n s Aj are analytic, we h a v e t h a t A i ~ 0 for j ~ r 0 a n d this p r o v e s T h e o r e m 1.2.

P r o o f of Corollary 1.3. L e t H be t h e h y p e r p l a n e b u n d l e of P~ a n d K t h e canonical b u n d l e o f P~. Since K = H , - 1 it is easily seen t h a t if R C Z + a n d R > ( n + 1)/d, t h e n R has p r o p e r t y P w i t h r e s p e c t to H 4.

References

1. AHLFORS, L., An extension of Schwartz's lemma. Trans. Amer. Math. Soc. 43 (1938), 359-- 364.

2. CARLSOn, J., & GRI~FI~HS, P., A defect relation for equidimensional holomorphie mappings between algebraic varieties. Ann. of Math. 95 (1972), 557--584.

3. GRIFFITItS, P., I-Iolomorphie mappings: Survey of some results and discussion of open problems. Bull. Amer. Math. Soc. 78 (1972), 374--382.

4. I-IE~MS, L. L., Introduction to potential theory. Wiley-Interscienee, 1969.

5. I-IoRMANDER, L., A n introduction to complex analysis in several variables. Van Nostrand, 1966.

6. ](ODAmA, K., On ]~iihler varieties of restricted type. Ann. of Math. 60 (1954), 28--48.

7. -- ~-- I-Iolomorphie mappings of polydiscs into compact complex manifolds. J. Differential Geometry 6 (1971), 33--46.

8. STOLZE~BERG, G., Volumes, limits, and extensions of analytic varieties. Lecture Notes in Math, ematics, Vol. 19 (1966), Springer-Verlag.

Received November 10, 1972;

in revised form June 30, 1973

Bj6rn Dahlberg

Department of Mathematics University of G6teborg G6teborg, Sweden

Holomorphic mappings into smooth projective varieties