Introduction to the carlson — Griffiths equidistribution ...shiffman/publications/22 carlson-griffiths...INTRODUCTION TO THE CARLSON- GRIFFITHS EQUIDISTRIBUTION THEORY Bernard Shiffman

INTRODUCTION TO THE CARLSON- GRIFFITHS EQUIDISTRIBUTION THEORY

Bernard Shiffman

Preface.

Nevanlinna's theory ofthe distribution of values of a meromorphic

function f on C was cast in a geometric form by L. Ahlfors in 1941.

In Ahlfors' theory, f can be viewed as a mapping into the complex pro-

jective line C~ I (which is also known as the Riemann sphere). The

Nevanlinna-Ahlfors theory was generalized to several complex variables

by W. Stoll in 1953. In 1973, J. Carlson and P. Griffiths transformed

equidistribution theory to a differential-geometric setting and obtained

new defect relations for holomorphic mappings in terms of the geometry

of line bundles on an algebraic manifold. In the Carlson- Griffiths per-

spective, Nevanlinna's upper bound 2 on the total deficiency of f

equals the first Chern class of C~ I.

In these lecture notes, we present the equidistribution theory of

meromorphic mappings from the Carlson - Griffiths viewpoint. The notes

begin with a complete proof of Nevanlinna's First and Second Main Theo-

rems using the ideas of Carlson and Griffiths. These methods are then

extended to provide proofs of the First and Second Main Theorem, defect

relations, and the lemma of the logarithmic derivative for meromorphic

mappings in several complex variables.

I. The Nevanlinna Defect Relations.

We begin these notes by giving the classical Nevanlinna theory for

meromorphic functions on C as described geometrically by Ahlfors. In

this chapter we give complete proofs of the First and Second Main Theo-

rems. These proofs are specializations of the proofs given in Chapter

II for the Carlson- Griffiths - King theory of mappings in several com-

plex variables. Our proofs of both the one and several variable theorems

differ from those in the literature in that we use the differential

calculus of currents instead of the classical method of "integrating

twice".

We can regard meromorphic functions on ~ as holomorphic mappings

from C into C~ I by identifying the complex projective line ~I

with { U {~}, where the point (z :z I) 6 CP I corresponds to Zl/Z O 6 o

C U {~}. The projective line C~ I is a l-dimensional complex manifold.

Recalling that the exterior derivative d has the decomposition d =

45

+ ~ on a complex manifold, we let

d c i ~ - 2) = ~ (i .I)

Thus

d d c = ~ ~ (1.2) 2~

(The operator d c coincides with the operator d c in Carlson- Griffiths

[5], Griffiths- King [14], and Stoll [33] and equals I/4n times the

classical conjugate differential, which is also commonly denoted d c.

For a C1-function ~ on ~,

I ~ ~ dx), dC~ = ~ (~ dy -

where z = x + iy.) On CP I we have the 2-form

= d d C ( l o g (IZo 12 + Iz112)). (1 .3 )

Writing Zl/Z ° = z = x + iy we have

= i ~log(1 + Izl 2) = I x 2 y2 ~-~ ~(I + + )-2dx ^ dy (1 .4)

and thus by elementary calculus

C~ I_ ~ = I .

(Here the deRham class of ~ is integral and generates H2(C~I,z).)

We shall use the notation

{[r] = {z 6 ~: Izl s r}, C<r> : {z 6 ~: Izl : r]. (1 .5)

We let [z o] denote the Dirac 6-measure at the point z ° 6 C. Thus

([Zo],~) = ~(z o) for a continuous function ~ on C.

For the remainder of this chapter, we let f : ~ + ~ U {~} be a non-

constant meromorphic function on C. (Soon we shall regard f as a

mapping into ~I.) We define the 0-divisor ~o(f) as the measure on

Uo(f) = [ nj[zj], (1.6)

where the z. are the zeroes of f and the n. are the multiplicities 3 3

of the corresponding zeroes z.. We also define the divisors 3

Va(f) = ~o(f - a), a 6 ~, and ~ (f) = ~o(I/f). (1.7)

For a divisor @ on C ( e is a measure given as in (1.6) or (1.7)),

we have the unintegrated counting function

n(@,r) = 0(f[r]), for r > 0, (1.8)

and the counting function

46

r N(@,r) = ~ n(8,t)t-ldt,

s

(1 .9)

for r > s, where s is a fixed positive constant. Since we shall not

be "integrating twice", we shall instead use the following alternate

description of the counting function. Let

log(r/s) for [z I f s

/r(Z) = log(r/[z[) for s S ]z[ S r

0 for Izl ~ r

r s : l o g + i z - - [ - l o g + ~ - ~ . ( 1 . 1 0 )

(Recall that log+a = max(log a,0).) By Fubini's Theorem, one easily

obtains

N(@,r) = (@,/r) . (1.11)

For our meromorphic function f, we define the counting functions

n(a,r) = nf(a,r) = n(Va(f),r), (1.12)

N(a,r) = Nf(a,r) = N(ma(f) ,r) , (I .13)

for a 6 C U {~}. Thus, by (1.11),

Nf(a,r) : ~ nj/r(Wj) , (I .14)

where f takes the value a at the points w with corresponding 3

multiplicities n . 3

We now regard f as a holomorphic mapping into CP I . We define the

Nevanlinna- Ahlfors characteristic function of f,

T(r) = Tf(r) = S lrf*~ : ~ Nf(a,r)~(a), (1.15) Cp I

where the last equality of (1.15) follows from (1.14). For points A =

(ao:al), B = (bo:b I) in C~ I, we define the chordal distance

laob I - albol d(A,B) [[A]I ]IS[ [ S 1 , (1.16)

2 + a 2 ! where IIAI[ = (ao I )2. We let

(z) = - logd(z,a) > 0 , (1.17) a

and we define the proximity term

2~ I

l

m(a,r) = mf(a,r) - 2~ S ~a (f(rel0))d8 • 0

(I .18)

The classical First Main Theorem (F.M.T.) can now be stated:

Theorem 1.1. (F.M.T.) N(a,r) + m(a,r) - m(a,s) : T(r).

47

We shall prove Theorem 1.1 later. Note that since m(a,r) ~ 0, it

follows from Theorem 1.1 that

N(a,r) ~ T(r) + O(I). (1.19)

Let {z.} denote the set of points at which the map f : { ÷ C~ I 3

is ramified. Then f is m.-to-1 at z., where the m. are integers 3 3 3

2. We define the ramification divisor of f,

Rf = [(mj - I) [zj].

Regarding f : { ÷ • U {~}, we have

Rf = ~o(f') + 2~ (f) - ~ (f').

(I .20)

(1 .21)

We write

N1(r) = N(Rf,r) = Nf, (0,r) + 2Nf(~,r) - Nf, (~,r) . (I .22)

The fundamental result of Nevanlinna is the Second Main Theorem (S.M.T.) :

Theorem 1.2. (S.M.T.) Given distinct points al,...,a

[I (q- 2)T(r) <-

6 ~]p1 , q

q N(aj,r) - N1(r) + O(logT(r)) + o(logr).

j=1

The notation

II A(r) < B(r)

means that there exists an open subset E c [0,+~) such that Sdr < E

+ ~ and A(r) < B(r) for all r C [0,+~) - E.

We now describe the Defect Relations that follow from Theorem I .2.

Since by (1.15) ,

T(r) = frt-ldt f f~w > c(logr - logs) (1.23)

s cEt]

where

c = f f*~ > 0, ¢[s]

it follows that T(r) + + ~ as r ÷ +

O(logT(r)) + o(logr) -< o(T(r)).

We define the defect

and

6(a) = I - lim sup N(a,r) = lim inf m(a,r) T(r) T(r)

r ÷ +~ r ~ +~

(1 .24)

where the second equality is by the F.M.T. 1.1. We write

N(a,r) = N({f -I (a) },r) , (I .25)

48

where

the points of f-1(a) ; i.e., each point of f-1(a)

plicity I. We let

N(a,r) @(a) = I lim sup T(r) '

r ÷ +~

{f-1(a) } is the divisor given by the sum of the 6-measures at

is given the multi-

for a 6 C37 I. We have

(I .26)

0 -< 6(a) -< @(a) _< I . (1.27)

are distinct points in C3? I . One easily sees that Suppose a I , . . . ,aq

q ~ ~ (f) < ~ {f-1 (aj) } + R= (I .28)

9= I aj 9= I

(I .29)

and hence

< 2 .

(I .30)

q q

J:1[ N(aj,r) $ j!IN(a3'r) + N1(r).

Thus it follows from (I .29) and the S.M.T. I .2 that

q q ~ (T(r) - N(aj,r))

@(aj) ~ lim inf j=1 j=1 r ÷ +~ T(r)

q qT(r) - [ N(aj,r) + N I (r)

< lim inf j=1 r + +~ T(r)

Since the points a. are arbitrary, we have the following consequence 3

of the Second Main Theorem:

Theorem 1.3. (Defect Relations) a6fP[ 1 6(a) d a6 ~I 8(a) =< 2.

We now develop the machinerv needed to prove Theorems 1.1 and

1.2. We begin by reviewing the basics of the calculus of currents. We

consider currents on an n-dimensional complex manifold 9, in antici-

pation of the methods of Chapter 2. For details, see L. Schwartz [25].

We let Dk(~) denote the space of (complex valued) C~-k-forms on

with compact support. (The space Dk(Q) is an inductive limit of

Frechet spaces.) We let

D,k(~) = D2n-k(~) ,

denote the space of currents of degree

of R is 2n.) The divisors Ua(f) given by (1.6) and (1.7) are thus

elements of D'2({) = D°({) ' If u 6 D'k(~), then du 6 D' (k+1) (~)

and u A ~ 6 D' (k+J) (9), for a j-form ~ on ~, are given by

(I .31)

k. (Note that the real dimension

49

(du,~) = (-1)k+1(u,d~), (1.32)

(u ^ ~,~) = (u,~ ^ ~). (1.33)

We also have the decomposition

D'k(Q) = @ D'P'q(~), (1.34)

p+q=k

where D'P'q(~) is the dual space to the space Dn-p'n-q(~) of C ~-

forms of type (n-p,n-q) on Q with compact support. The operators

2, ~, and d c (recall 1.1)) are defined on the spaces of currents by

using equation (1.32) with d replaced by the appropriate operator.

Lemma 1.4. (Green's Formula) Let U be a bounded domain with smooth

boundary in C, and let ~,~ ~ C~(U). Then

f(~ddC~ - ~ddC~) = ~ (~dC~ - ~dC~).

U ~U

Proof. Use Stokes' Theorem and the easily verified identity

d(~dC~ - ~dC~) = ~ddC~ - ~ddC~ . (1.35)D

The following lemma is a restatement of the Poincar~ formula in the

language of currents.

Lemma 1.5. Let g be a meromorphic function on a domain 9 in C.

Then loglgl 2 6 ~oc(~) c D'°(~) and

ddCloglgl 2 = ~o(g) - ~ (g).

Proof. Since loglgl 2 is harmonic where g ~ 0,~, and ddCh = 0

for harmonic h, it suffices to show that

ddCloglzl 2 = [0]. (1.36)

just a restatement of the fact that ~ loglzl 2 is (Formula (1.36) is

a fundamental solution of the Laplacian on C.) For ~ 6 g°(C), we have

(ddeloglzl2,~) = (loglzl2,ddC~) : lim floglzr2ddC~ e÷0 U £

where U e = {z 6 • : £ < ]z I < A} where A is chosen so that supp~ c

C[A - I]. By Green's Formula (Lemma 1.4),

ll°glzl2ddC~ = I (loglzl2dC~- ~dCloglzl 2 U ~U £ c

= f (~dCloglzl2 _ log lzI2dC<0) ! 12~ ~<e> = 2~0 ~(eelS)d0 + O(e log~) ÷ ~(0)

as e ÷ 0, which verifies (1.36). []

50

Recall that f : C ÷ C~ I is a non-constant meromorphic function.

I Lemma 1.6. For a 6 {]P ,

2ddC(la of) = f*~ - Va(f) .

Proof. From (1.3), (1.16) and (1.17), we obtain

2ddCl = w on fp1 {a}. (I .37) a

Applying f~ to (1.37) and noting that supp Ua(f) = f-1(a), we con-

clude that Lemma 1.6 is valid on f - f-1(a) . Let a = (I:~) and re-

gard f as a map into C U {~}, identifyinq a with

~ ~. (If e = ~, replace f by I/f and note that

~ (f) = ~o(I/f), and f~ = (I/f)~.) Since

la(Z) = i (z) - loglz - ~I (1.38)

(identifying z 6 C with (1:z)) we have

I a of = I of - loglf - el- (1.39)

Using (1.37) with a replaced by ~ and Lemma 1.4, we then obtain by

differentiating (1.39)

2ddC(la o f) = f~w - ~o(f - e) on f _ f-1 (~). (1.40)

Since ~ (f - e) = ~ (f), we can now conclude that the lemma is valid o a

on all of ~. []

For r > 0, we define the current ~ 6 D'2(f) by r

2~ i [2

(Or,Q) = ~0 ~ ~(reiS)dO : ~ ~dClog[z , (1.41) ~<r>

for ~ 6 D°(C).

e. We can assume

1 o f = I 0 (I/f), o

Lemma 1.7. 2ddCl = o - @ . r r s

Proof. By (1.10), it suffices to show that

2ddClog+~ = o r

Let ~ 6 D°(~ - {0})

on C - {0}.

be arbitrary. Then by Lemma 1.4,

(I .42)

(2ddClog+~,~) = (21og+~,ddC~) = 2 ~ log~ddC<0 C[r]

: 2 s llog dC - dClog ) : 2 dClog[zl ~<r> ~<r>

(Ur,~),

which verifies (1.42).

We note that Jensen's Formula (given below) is a consequence of

51

Lemma 1.7. Our proofs of the First and Second Main Theorems in both one

and several variables use the method of the proof of Jensen's Formula

given below.

Corollary 1.8. (Jensen's Formula) Let f be a meromorphic function

on fIR] with f(0) # 0,~. Then for 0 < r < R,

2~ loglflrei lld loglfI011 + Imjlog - Injlo

2~ 0

where [aj} and {bj} are the zeroes and poles of f in f[r] with

respective multiplicities {mj} and {nj}.

Proof. Let u s denote the convolution of log lfl with an approxi-

mate identity ~s on ~. (For s > 0, u s = (l°glfl) ~ ~s' where ~s 6

C~({), supp ~s c {[s], ~s > 0, ~ ~s = I.) By Lemma 1.7

(o r - @s,Us) = (2ddC/r,Us) (/r,2ddCus). (1.43)

By Lemma 1.4

2ddCus = (2ddCl°glfl) * ~s = [~o (f) - ~(f) ] ~ ~s ÷ Vo (f) - ~(f)

on the space C(~[r]) ' of measures on C[r]. Furthermore

(Or,U E) ÷ (Or,loglfl)

as s ÷ 0, since loglf I is the difference of two subharmonic functions.

Thus, letting s ÷ 0 in (1.43) we obtain

(o r - Os,loglfl) (/r,~o(f) - ~ (f)) = ~mj/r(a j) - [nj/r(bj). (1.44)

Finally we let s ÷ 0 in (1.44) to obtain Jensen's Formula. []

The First Main Theorem 1.1 can now be proven using the method of the

above proof of Jensen's Formula: Let a be a point in {~I, and let

v s : (I a 0 f) * ~s'

where ~s is an approximate identity on C

(o r - Os,Vs) = (/r,2ddCvs).

as above. By Lemma 1.7,

(I .45)

By Lemma I .6,

2ddCvE = [f~ - Va(f)] ~ ~s ÷ f~ - ~a(f)

in C(C[r]) ', as s ÷ 0. Letting s ÷ 0 in (1.45) and recalling the

definitions (1.13), (1.15), and (1.18), we then obtain

m(a,r) - m(a,s) = (o r - Os,la 0 f) = (/r,f~ - la(f)) = T(r) - N(a,r) . o

The proof of the Second Main Theorem uses the following analytical

52

lemma:

Lemma 1.9. Let ~ be a domain in C, let a 6 Q and let u be a

real-valued subharmonic C~-function on ~ - {a}. If

lul + ~ = o( ),

then u extends to a subharmonic function ~ on ~ and

(ddC~,~) = f ~ddCu 9-{a}

for all ~ 6 D°(Q).

Remark. Note that (ddC~,~) = S uddC~ by definition. If u has ~-{a}

a subharmonic extension ~ on ~, then ddC~ must be a measure, and

thus

ddC~ = c[a] + {ddCu[ (~- {a}) }

where the term in brackets is ~I near a. The equality in the conclu- I

sion of the lemma states that c = 0; we then say that ddC~ 6 ~loc"

Proof. Assume without loss of generality that a = 0. Clearly u

extends to ~ 6 ~Iloc(~) " Let U s = ~ - C[c] for s > 0. Then for

6 ~o(~),

(ddC~,~) = (~,ddC~) = lim ~ uddC~ = lim[ ~ddCu + ~ (~dCu - udC~) ]

e+0 U ~÷0 U ~<s> C £

where the last equality above is by Lemma 1.4. By hypothesis the coef-

ficients of the boundary integrand are o(I/Izl), and hence

lim f~ddCu = (dde~,~) . (1.46) 6÷0 U

£

By considering ~ ~ 0 in (1.46), we conclude that ddCu is ~I near

0 and ~ is subharmonic. The conclusion then follows from (1.46). []

Suppose ~ is a positive 2-form on any Riemann surface (l-dimen-

sional complex manifold), in particular on ~I. In local holomorphic

coordinates z = x + iy, we can write

= g(½dz ^d~)= gdxAdy,

where g 6 C , g > 0. We define the Ricci form

Ric~ = ddClogg . (1.47)

One easily sees that Rice is independent of the choice of coordinates

and that

K Ric a = - 2--~e (1.48)

53

where K is the Gaussian curvature of the metric g(dx 2 + dy2) . On

C~ I with w given by (1.3), we have using (1.4)

I izi2)-2 g = ~(I +

and hence

Ric ~ = - 2w . (1.49)

Let D and D ~ denote the unit disk {Izl < I} and the punctured

unit disk {0 < Izl < I} in f respectively. The Poincar@ metric on

D (which is invariant under the automorphisms of D) is given by the

2-form

~D (I iz12)-2 i = - (2dr A d~) . (1.50)

We compute

I Ric qD = ~ qD' (1.51)

which says that the Poincar6 metric has constant curvature -I. We can

use (1.50) to obtain a natural metric qD ~ on D ~ as follows: The

universal covering map T : D ÷ D ~ can be given by

z - I T(z) = exp(~-~-~) . (1.52)

Writing w = T(z), we have

-I _ I + log w z = T (w) I - log w

(a multivalued function). We let

(i .53)

-1)~qD~: I -2 i qD ~ = (~ ~(]wlloglwl) (~dwAdw) . 1.54)

(The metric qD ~ is the Kobayashi metric on D ~; see [17].) Let p > 0

be arbitrary; replacing w by Q-lz in (I .54) we obtain a metric

I q ~(izl (log p loglzl))-2 i = - (ydz ^ d~) (I .55)

on {0 < Izl < p} with constant Gaussian curvature -I. Equation (1.55)

provides the motivation for the Carlson-Griffiths metric T on

~]p1 _ {a I, .... aq} given by

q = ( N d(z,aj) (2q - logd(z,aj)))-2~ , (1.56)

j=1

for distinct points al,...,a q in ~]pl. Recall that d(z,0) =

Izl/(1 + IzI2) ½, SO if a I = 0, then ~ N q near 0. We shall later show

that if q > 3, then ~ has strictly negative Gaussian curvature

(Lemma 1.13) .

54

We write

f*~ = ~(~dz ^ d~) .

Then ~ is a non-negative

more

(I .57)

on C - f-1{a 1,...,aq}. C~-function Further-

f*Ric ~ = Ric f*~ = ddClog ~ on C - f-1{a I ..... aq} - supp Rf. (1.58)

I ~,o Lemma 1.10. log~ 6 ~]ioc(C) c (C)

and

ddClog~ = f*Ric ~ - ~ ~a (f) + Rf

3

on all of C.

Proof. Write ~ = h'~, where

h = H(2q - logd(z,aj)) -2, (1.59)

? = ( ~d(z,aj)-2)~ . (1.60)

Let

f,~ ~ i = ~(~-~ dz ^dz), (1.61)

so that ~ = (h 0f)~ and thus

log ~ = logh of + log~ . (1.62)

By (I .60) , (I .61) , and (I .4) , we have

log~ : 2[ la. of + log((1 + If12)-2]f' I 2) 6 ~I loc (C) . (I .63) 3

Thus by Lemma 1.6 and (1.49)

ddClog~ = ~(f*al - ~ (f)) + f*Ric w + Rf = (q-2)f*w - ~ ~ (f) + Rf. a. a.

3 3 (I .64)

By (1.58) and (1.51)

f * R i c ~ = d d C l o g ~ = d d C l o g h 0 f + d d C l o g ' ~ o n 112 - f - l { a 1 . . . . . a q } - R f .

(I .65)

To complete the proof, we must show that log h 0 f 6 ~I and loc

ddClog h of 6 ~I (as in the remark following Lemma I .9). Let loc

u = - 21og(2q - logd(z,0)) . (1.66)

It suffices to show that u 0 f and ddCu 0 f are in ~I . We can loc

write

u = - 2 log (<0- log Izl) (1.67)

where <0 = 2q + ½ log (I + [zl 2) is a positive subharmonic C~-function

on C. Thus

55

1 u : o(logT~)

and

(I .68)

~u -2 ~<0 2~ 8~ = ~ - log Izl (~-~ - ) '

(I .69) ~2u _ 2 I ~<0 I 2 2 ~2<0

~z~ (~ _ log[zl)2 [~z 2z - ~ - l°glz! ~zS~

on ~ - {0}. Let ~ = u + <0. Recalling that <9 - loglzl -> 2q and that

is subharmonic on {, it follows from (I .69) that ~ is subharmonic

on ~ - {0}. Let a 6 f-1 (0) be arbitrary. Thus by (I .68),

~ 0f = O(iog~_~) < o ( ~ )

near a. By (1.69), 8~/3z = o(I/Iz - a[) and hence ~(~ 0 f)/~z =

o(I/Iz - a I) near a. Therefore by Lemma 1.9, ~o f and dde(~ 0 f) are

i n ~ l o c (~:) a n d t h u s h o f a n d dd c (h o f ) a r e i n ' ~ oc (~:) " Thus by

( 1 . 6 2 ) a n d ( 1 . 6 3 ) l o g ~ 6 ~ 1 • l e c ( £ ) ; by ( 1 . 6 4 ) a n d ( 1 . 6 5 ) , d d C l o g ~ i s a

m e a s u r e . By (1 . 5 8 ) , t h e i d e n t i t y o f t h e lemma i s v a l i d on

C - f - l { a 1 . . . . . aq} - s u p p R f . By ( 1 . 6 4 ) a n d t h e f a c t t h a t d d C ( h o f )

contains no 6-measures, it follows that the identity is valid as an

equality of measures on the discrete set {a 1,...,aq} U supp Rf. []

The following useful identity is an immediate consequence of the above

proof. (See (1.64) and (1.65).)

Lemma 1.11. f*Ric ~ = (q - 2)f*~ + ddClogh 0 f 6 ~oc(C) .

Lemma 1.11 allows us to make the definition (from Carlson-Griffiths

[5])

flrf~Ric ~ . (I .70) T~(r)

Since ddClog ~ is a measure by Lemma I .10 and 1.11, log ~ is locally

the difference of subharmonic functions. (This fact can also be seen

explicitly from the proof of Lemma 1.10.) Thus log ~ 6 ~I (~<r>) for

all r > 0, and we can define

2w u(r) = ½(Or,log ~) = 4~ f log ~(rei~)d~ . (1.71)

0

Corollary 1.12. T~(r) + N I (r) = ~ N ,r) + ~(r) - ~(s). j=1 (aj

Proof. This corollary is proven using Lemma 1.10 in the same manner

as the proofs of Corollary I .8 and Theorem 1.1. We begin with equation

(I .43) with u s = (log ~ ) • ~8" By Lemma 1.10,

= - (f) + Rf) • ~s (I .72) ddCug (f*Ric ~ ~ ~a~

56

Substituting (1.72) in (1.43) and letting s ÷ 0, we obtain the desired

identity. []

Lemma 1.13. If q k 3, there exists a constant c = c(al,...,a q) > 0

such that Ric y ~ c~ on ~p1 {al,...,aq } m

Proof. Let

u. = - 2 logv. 3 J

where

vj = 2q - logd(z,aj) = 2q + la. ,

so that logh = ~ u. J

any local coordinate

~2uj ~ 2 ~2vj

~z~[ vj ~z~

and hence by (1.37)

> I ~ ddCl = - ½w ddClogh = ~ ddCuj = - q a.

J

on fp1 _ {al, .... aq}. By the proof of Lemma 1.10,

Ric ~ = (q - 2)~ + ddClog h

(Lemma 1.11 without f~), it follows that

5 Ric ~ ~ (q - ~)~ ~ ½ ~ > 0

(I .73)

(I .74)

3

for the function h given by (1.59). Then (using

z)

221

I a. J (1.75)

q ~z~z

(I .76)

(I .77)

(I .78)

for q ~ 3. To complete the proof, we must look at (Ric ~)/~ as z ÷ a. 3

Assume without loss of generality that a = 0 6 { U {~}. Then u. co- 3 3

incides with u given by (1.66). By (1.69),

Ric ~ ~ ddCu ~ (Izlloglzl)-2(i dz A d~) ~ Y (1.79)

as z ÷ 0. (In (1.79), A(i dz Adz) ~ B(i dz Adz) means A = O(B) and

B = O(A) as z ÷ 0.) The lemma then follows from (1.78) and (1.79). []

Remark. Lemma 1.13 is equivalent to the bound: Gaussian curvature

of ~ ~ - 2~c. The Big Picard Theorem can be obtained from Lemma 1.13

and the Schwarz - Pick- Ahlfors Lemma. (See Kobayashi [17].)

Lemma 1.14. T~(r) = (q - 2)T(r) + O(logT(r)) .

Proof. By Lemma 1.11, (1.15), and (1.70)

(q - 2)T(r) + SfrddClog(h o f) . (I .80) T~(r)

By smoothing logh 0 f and applying Lemma 1.7 as in the proofs of Theo-

rem I .I, Corollary I .8, and Lemma I .11 , we obtain

57

£rddClogh 0 f = ½(o r Os,log h o f).

Since

logh 0 f = - 2 [ log(2q + I 0 f) < 0 a.

3

it suffices to show that

(I .81)

I .82)

({r,log(2q + la. o f)) -< O(logT(r)) . 3

By the concavity of the logarithm and the F.M.T.

1 .83)

(~r,log(2q + la. 0 f)) ~ log(Or,2 q + I 0 f) a.

3 3

= log(2q + m(aj,r)) -< logT(r) + O(I) ,

which verifies (1.83).

r d7 T t e 2 ~ ( t ) Lemma 1.15. S -~- ~ dt -< =Lzc T~(r)

s 0

where c is the constant from Lemma 1.13.

Proof. By concavity of the logarithm

2u(r) = (Or,log ~) ~ log(Jr, <) . 1.84)

(One can show that (Ur,~) is finite, but we do not need this fact.)

Thus by (1.57), (1.84) and Lemma 1.13,

T T I te2U(t)dt =< ; (dt'~)tdt = --2~ ; ~ dx A dy = ½ S f*~ = < 12c ~ f*Ric ~.

0 0 ~[~] ~[~] ~[T]

By (1.70) (1.85)

r dT

f T f f*Ric~ : f~ f*Ric~ = T~(r). (1.86) s {[T] { r

The lemma follows from (1.85) and (1.88).

We use the followin~ elementary fact to differentiate the left-hand

side of the inequality of Lemma 1.15:

Lemma 1.16. Let F

Then for all e > 0

I+E II F' (r) -_< F(r)

Proof. Let E denote the open subset of

Then

Sdr < ; F'_qq_j.~ f+~ F'~ dr _< ____I E F '' dr -<

0 ~ EF (0)

be a positive, increasing C1-function on [0,+~).

, F 1+e . [0 +~) where F' >

< + ~ n

Lemma 1.17. II ~(r) -< O(logT~(r)) + o(logr).

Proof. Let 0 < t < I be arbitrary. By Lemma 1.16 with F(r)

to the left-hand side of Lemma 1.15, we obtain

equal

58

rte2P (t) (I .87) II ~1 f dt ~ c'Ty(r) I + s 0

where c' is a constant independent of s. (We may take c' =

max(I/2c, (I/2c)2).) Multiplying (1.87) by r and again applying Lemma

1.17, we obtain

2 II re2p(r) -< c''rl + (I + s) STy(r) (I .88)

where c" is independent of s. Thus

I[ 2~(r) ~ 4 log T~(r) + s log r + log c" (1.89)

It remains to show that s log r can be replaced by o(log r) . (If f

is transcendental, then the following argument is unnecessary. Although

we are studying transcendental functions, we nonetheless include the

argument for use in Chapter II.) For m = 1,2,3,..., we let E m denote

the exceptional set (where the inequality is not satisfied) for (1.89)

with 6 = I/m. Since f dr < + ~, for each m we can choose r m s E

such that m

f dr ~ I-- .

E m n (rm,+~) 2 m

Let

E = U E n (rm,+~). m=1 m

Then ~ dr ! I and E

I log r + logc" ~(r) ~ 4 log T~(r) + for r 6 (rm,+~) - E.

I .90)

1.91)

I .92)

The len~a follows immediately from (1.92). []

We now complete the proof of the Second Main Theorem 1.2: By Lemmas

1.14 and 1.17 we conclude that

II ~(r) ~ O(logT(r)) + o(logr).

By Lemma 1.14 and Corollary 1.12,

(q - 2)T(r) = T~(r) + O(logT(r)) =

The Second Main Theorem then follows from (1.93) and (1.94).

(I .93)

q N(aj,r)-N I (r) + ~(r) + O(logT(r))

j:1 (I .94)

II. The First Main Theorem.

Chapters II and III give the Carlson- Griffiths [5] generalization of

Nevanlinna theory to meromorphic maps from C n into a projective-

59

algebraic manifold V. We give complete proofs following the methods

of [5], [14] and [27]. The greater part of this chapter is concerned

with developing the necessary notation and machinery.

We write

z,2½ Ilzll = (Izl 12 + "'" + I nl )

for z = (Zl, .... Zn) 6 c n, and as in (1.5) we let

cn[r] {z 6 cn = : [Izll ~ r}, {n<r> = {z 6 {n : llzl I = r}. (2.1)

we let

i : ddCllz[l 2 = ~ ~ dzj ^ dzj (2.2)

on C n, so that

Bn 2n =r

cn[r]

We let

; ddClogllzll 2

on ~n _ {0}. Thus ~ = p~w, for a unique

p : C n - {0} ÷ c~n-1 is the usual map. Then

w n-1 : I.

n-1 C~

The invariant probability measure on the (2n - 1)-sphere cn<r> is

(2.3)

{0 6 01 'I (c]pn-1) , where

given by the form

dClogl[z[l 2 A~ n-1 [ cn<r>.

We let u 6 D'2n(cn) be given by r

~n- I dr,q0) = ; h0dClogl[zl]2 A w -

cn<r>

I ^ , (2.4) 2n-I f ~de II z LI 2 Bn-1 r cn<r >

for <0 6 ~°(cn), so that (~r,~) is the average of q0 over cn<r>.

The following generalization of Green's Formula (Lemma 1.4) to ~n is

verified by an elementary computation:

Lemma 2.1. Let <0 6 ~,p-l,p-1(~n), ~ 6 •n-p'n-P(Cn). Then

<0A ddC~ - ~ ^ ddC~ = d(~ A dC~ - ~ A dC<0) . []

Suppose A is an analytic hypersurface (a complex analytic set of

codimension I) in a complex manifold ~. Then A defines a current

[A] 6 D,I,1(~) given by

60

([A],~) = ~ ~ (2.5) A reg

for ~ 6 ?n-l,n-1(~), where A denotes the set of regular points reg

of A. (See [19] or [35].) Suppose g is a meromorphic function on ~.

If A = A I UA 2 UA 3 U .-. is the irreducible decomposition of the zero

set of g, then we write

~o(g) = [ nj[Aj] 6 ~,I,1(9), (2.6)

where nj is the order of vanishing of g on A D A . (The sum in 3 reg

(2.6) is locally finite.) We define the divisor of g

Div g = ~o(g) - ~o(I/g). (2.7)

A divisor on 9 is a current D 6 p,I,1(~) of the form D = [mj[Aj]

where m. 6 ~ and the A. are the irreducible components of an ana- 3 3

lytic hypersurface in Q; D is a positive divisor if the m. are 3

positive.

We have the following generalization of Lemma 1.5:

Theorem 2.2. (Poincar6- Lelong Formula) Let g be a meromorphic func

tion on Q. Then loglgl 2 6 ~loc(Q) c ~,o(9) and

ddClog!gl 2 = Div g .

Proof. We can assume without loss of generality that g is holo-

morphic and ~ c fn. Then log]gl 2 is plurisubharmonic and hence is

in ~oc(~). Write

n = i UjkdZj ^ d~ k . (2.8) Divg - ddCloglgl 2 ~ 3

j,k=1

We must show that Ujk = 0 for all j,k. By considering several linear

coordinate systems, one sees that it suffices to show that u = 0. Let nn

i i i A d~n) - i' (~dz I Adz I) ^ "'" A (3dZn-1A dZn_1), i = I' ^ (3dZn

Then for ~0 6 ~o(9),

(Unn,~l) = (Divg,~l') - (ddCloglgl2,~l ') . (2.9)

For z' = (z I ..... Zn_1) 6 {n-l, we write gz' (z n) = g(z',Zn) ' ~z' (Zn) =

~(z',Zn). By Lemma 1.5

(Divg,~l') = f (Vo(gz,),~z,)l' = f (ddCloglgz, 12,~z,)l ' {n-1 cn-1

61

= (ddCloglgI2,~l ') (2.10)

and thus u = 0. (The integrals in (2.10) are well-defined, since nn cn_l

the set of z' such that gz' m 0 has Lebesgue measure 0 in .) []

It follows from Theorem 2.1 that if A is an analytic hypersurface

in Q, then

d[A] = 0 . (2.11)

Equation (2.11) is a special case of "Stokes' Theorem with singular-

ities" (see [35]) and also is given by Lelong [19].

We now describe the growth measures or "counting functions" for an

analytic hypersurface A in C n. The normalized area function

n(A,r) - 1 Bn-1 2n-2 f (2.12)

r A N cn[r]

generalizes the unintegrated counting function (1.8). The counting

function of Stoll [31] and Carlson- Griffiths [5], which we denote

Ns(A,r), is given by

r r dt 6n-I

Ns(A,r) = f n(A,t) = f t2n_ I f (2.13) ~n s s A n [r]

where s is a fixed positive constant. We shall modify the definition

(2.13) slightly to simplify the analysis: Choose a real-valued function

6 C~(C n) such that

T(Z) : log [Izl[ for lizll ~ I, T(z) 0 for llz[l < I . (2.14)

Define

= max(log r - T,0), (2.15) r

and let

= 2ddCT. (2.16)

Thus

Tr(Z ) = log+ r 7~7 ' ~ = ~' for llz I > I. {2.17)

We shall define a counting function for a general class of currents

that includes divisors. The characteristic function will be a special

case of this counting function.

A current in D'k(~) is said to be of order 0 if its coefficients

are measures on ~, or equivalently if it can be extended to a linear

functional on the space of (2n - k)-forms with continuous coefficients.

(For example, divisors are currents of order 0.) A current ~ is said

to be real if ~ = ~. Recall that ~ is closed if d~ = 0.

82

Definition. If % is a real, closed current of order

D' I, I (fn), then we define the counting function for

N(¢,r) = (% m n-l) ' r

0 in

(2.18)

Although definition (2.18) apparently depends on the choice of T,

we shall see in (2.21) that modulo a constant term, N(~,r) is indepen-

dent of ~. For example, if A is an analytic hyDersurfRce in {n, then

N([A],r) = f (logr - 7)&n-l,

A N {n[r]

for r > I.

The counting function can also be described "classically" as follows:

Let } be as in the above definition. Then for r > s > 0

r ~ n-1 n-1 f f ~ A ~ = (%,l r ) = N(%,r) + c (2.19)

s {nit ]

where c is independent of r (but depends on %, m, and s) and I r

is as in (1.10). If % has C~-coefficients, then for t > I,

= n-2 ~n-2 f ¢ A n-1 f ¢ A 2dC~ ^ : f e ^ dClog[iz[[ 2 ^

{n[t] cn<t> cn<t>

Thus n-1

f ~ ^ a {n[t]

,2 Bn-2 I Bn-1 I f ~ ^ d C ] l z l i A - f ¢ ^ .

t 2n-2 ~n<t > t 2n-2 ~n[t ]

= I f ¢ A B n-1 (2.20) t 2n-2 ~n[t ]

for t > I. We verify (2.20) for non-smooth ~ by replacing % by

~ = % * ~ in (2.20) and then letting s ÷ 0. Thus

r N(¢,r) = f dt [ ~ A B n-1 - c . (2.21)

S t 2n-I {n[t ]

In particular, by (2.13) and (2.21)

N([A],r) = N (A,r) - c . s

Henceforth, we shall use (2.18) to define the counting function.

i Suppose ~ = ~ ~ ~jkdZj ^ dz k 6 D'I'I(~), where ~ is a domain in

C n. We say that ~ is positive if ~ is real and the hermitian matrix

[ (~jk,~n) ]1~j,kSn

is positive semi-definite for all ~ ~ 0 in D°(cn). If ~ is positive,

83

it follows by the Riesz representation theorem that ~ is of order 0

and ~ ^ B n-1 is a positive measure.

Suppose ~ 6 D'I'I({ n) is real, closed, of order 0, positive and

not identically 0 on {n. Then there exists r > I such that o

~ ^ n-1 I Bn-1 = 2n-2 ~ ~ i = a > 0.

cnEro ] r O {n[ro]

Thus by (2.19) and (2.20), for r > r , o

r~ n-1 r~ n-1 r N(%,r) + c = f -- f ~ ^ e ~ f ~ ~ ^ e : a log~--- ,

I cn[t] r O ~n[ro] o

and hence

N(%,r) ~ a log r + O(I). (2.22)

Recall that a current u 6 D'°(C n) is given by a plurisubharmonic

function if and only if ddCu is positive. (See Lelong [19].)

I n Definition. A pluripotential is a real-valued function u 6 ~loc({ )

such that ddCu is of order 0.

For example, a plurisubharmonic function is a pluripotential. The

pluripotentials that we shall consider are differences of plurisubhar-

monic functions, but not every pluripotential can be expressed as such

a difference. If u is a pluripotential, then the Laplacian of u

(which equals a constant times the trace of the coefficient matrix of

ddCu) is a measure and hence u is locally integrable on all real

hypersurfaces in C n.

The following lemma is a generalization of Jensen's Formula:

Lemma 2.3. Let u be a pluripotential on C n such that u is con-

tinuous on C n - S, where S is a complex analytic set in {n. Then

I N(ddCu,r) = ~(Or,U) + c

for r > I, where c is independent of r.

Proof. It suffices to consider u 6 C~({n), since for general u,

we replace u by u s = u • ~s and let ~ ÷ 0 as in the proof of Co-

rollary 1.8. By Lemma 2.1, for r > I,

N(ddCu,r) = (ddCu,Tr an-l) = ~ (log r - T)e n-1 ^ ddCu

cn[r]

= - S uddC7 ^ n-1 + S ((log r -T)~ n-1 ^ dCu + udCT ^ n-l)

cn[r] fn<r>

84

U n + i udCT ^ n-1 (2.23)

cn[r] {n<r>

For r > I,

udCT ^ n-1 =

{n<r> fn<r>

by (2.4). Since n : ~n = 0 on

and (2.24) that

N(ddCu,r) = ½(Or,U) - ~ ue n,

cn[1]

udClogliz]l ^ ~n-1 = ½(Or,U ) (2.24)

C n - {n[1], we conclude from (2.23)

(2.25)

which completes the proof of Lemma 2.3. []

i Suppose ~ = ~ ~ ~jkdZj ^ dz k is a current of order 0 in ~,I,1 (~)

. . . n

is a domaln in C . We define the positive measure II~II on where

by

n

l~jk , (2.26) j,k=1

where l~jk I is the total variation measure of ~k'~ The following

lemma is a consequence of a result of H. Federer on locally flat cur-

rents [9, 4.1.20]. An elementary proof of this lemma is given in Appen-

dix I.

Lemma 2.4. Let S be a complex analytic set of codimension > 2 in

a domain ~ in C n. If u is a pluripotential on ~, then IIddCull (S)

= 0. []

The following lemma is the generalization of Lemma 1.9 to {n:

Lemma 2.5. Let A be an analytic hvpersurface in a domain

{n, and suppose f is a holomorphic function on

[A]. If u is a plurisubharmonic C~-function on

n lul + [ IDu/Dzjl = o(I/]fl) ,

j=1

then u extends to a plurisubharmonic function ~ on ~ and

Eldde~li (n) : 0.

Remark. The identity (2.28) is equivalent to

(ddCj,~) = f ddCu ^ Q-A

for ~ 6 ~n-l,n-l(Q). (Recall the remark following Lemma 1.9.)

in

such that Div f :

- A such that

(2.27)

(2.28)

65

Proof. We first suppose A is smooth. Since the estimate (2.27) is

independent of coordinates, we may assume that f = z I and ~ = ~1 x W,

where ~I c ~ and W c ~n-1. Since lul = o(I/Izi[) , u extends to

6 ~Iloc(~) " Let U = (~I - C[£]) x W. By Lemma 2.1, as in the proof

of Lemma 1.9,

(ddC~,~) = (~,ddC~) = lim ~ u ddC~

~÷0 U s (2.29)

= lim( ~ ~ ^ ddCu + ~ (~ ^ dCu - udC~)), s÷0 U C<c>xV

6

for ~ 6 pn-l,n-1(~). By (2.27), the coefficients of the boundary integ-

ral (over C<£> x V) in (2.29) are o(I/e) and hence

lim f ~ A ddCu = (ddC~,~). E÷0 U

(2.30)

Equation (2.30) is equivalent to:

lim f __~2u ~ n = ( ~2~ ,~ n) (2.31)

~+0 U ~zj~ k ~zj~z k

for ~ 6 D°(~), I ~ j,k ~ n. Since u is C 2 and plurisubharmonic on

(~I - {0}) × W, it follows from (2.31} that the matrix

~2u n [( ,~ ]

Dzj~k 1~j,k~n

is positive semi-definite for all £0 > 0. Thus, ~ is plurisubharmonic

on ~, and (2.28) then follows from (2.30) and the above remark.

We now consider the case where A is not smooth. Let S be the set

of singular points of A. By the above, u has a plurisubharmonic exten-

sion u' to ~ - S and 11 ddcu' II (A - S) = 0. By the Grauert-Remmert

extension theorem [10] (see also [15] or [26]), u' has a plurisubharmonic

extension ~ to ~. By Lemma 2.4, ]IddCull (S) = 0 and thus IlddCull (A)

= 0. []

We now briefly review the theory of holomorphic line bundles, upon

which the Carlson- Griffiths theory is based. For details, see Chern

[8] or R. O. Wells [37]. Let V be a complex manifold. A holomorphic

line bundle L on V is given by an open covering {Ua} of V and

holomorphic "transition functions" gab : Ua nu B ÷ ~ - {0} with

gaBgBy = get on U a N U B NUT. A holomorphic section s of L is given

by a collection of holomorphic functions s a 60(U ) such that s a =

gaBsB on U nu B. We let F(V,L) denote the space of holomorDhic

66

sections of L. If s 6 F(V,L), the divisor of s is the positive di-

visor Div s 6 D'I'I(v) given by

(Div s) ]U a = Div s a. (2.32)

We let ILl denote the set of positive divisors of the form Div s for

s 6 F(V,L). If D is an arbitrary divisor on V, then there exist an

open cover {U a} of V and meromorphic functions f~ on U a such that

Div fe = DIUa" The transition functions ge8 = fe/f8 define a line

bundle, which we denote by L D. If D is positive, then the collection

{fa} defines a section s 6 F(V,L D) with Div s = D. Thus a positive

divisor D is in ILl if and only if L = L D-

A hermitian metric on a holomorphic line bundle L with transition

{ge6 + (0,+~) functions } is a collection of C -functions h e : U

such that

h e = Ige@l-2h8. (2.33)

A hermitian line bundle is a holomorphic line bundle L with a hermitian

metric {ha}. For s 6 F(V,L), the norm l[sil 6 C~(V) is given by

llsEl 2 : helsal 2, (2 341

which is well-defined on V by (2.33). The curvature form ~L of L

(with the metric {h }) is given by e

~L = - ddCl°ghe ' (2.35)

which is a global (1,1)-form on V by (2.33). (The deRham cohomology

class of rl L is the Chern class c1(L); see [8], [37].)

Suppose ~ is a real (1,1)-form on V. For x 6 V, we can write O

i (x O) = ~ [ ajkdW j A dw k

where [ajk] is a hermitian matrix and the wj are local coordinates

at x O. We say that n(x o) > 0 (resp. ~(x o) ~ 0) if [ajk] is positive

definite (resp. positive semi-definite). We say that ~ > 0 (resp.

q ~ 0) if q(x o) > 0 (resp. q(x o) ~ 0) for all x ° 6 V. We shall also

write ~I > ~2 (resp. ~I ~ ~2 ) if nl - q2 > 0 (resp. n I - n 2 ~ 0) .

A holomorphic line bundle L on V is said to be positive if L has

a hermitian metric with DL > 0.

The set of holomorphic line bundles on V has a natural group struc-

ture. Let L, L' be holomorphic line bundles with transition functions

{gaB} and {ga6' } respectively. The product L ~ L' has transition

functions { ' with ga@gaB}. The identity element is the trivial bundle L °

transition functions gab ~ I. (Thus F(V,Lo) = O(V).) The inverse L -I

67

of L has transition functions {g~} (L -I is called the dual of L

and is often written L~). Thus for p 6 Z, L p has transition functions

{g~B}. If D, D' are divisors on V, then

LDOLD' = LD + D' (2.36)

and thus the map D ÷ L D is a group homomorphism. The set of hermitian

line bundles on V also forms a group, where LSL' is given the prod-

uct metric {heh~}. By (2.35) we have

~L ® L' = ~L + DL'" (2.37)

Example I. Let V = C~ k . The sets

U e = {(Wo:W1:''':w k) 6 {~k : we ~ 0}, (2.38)

for 0 $ e ~ k, form an open covering of {~k. The transition functions

ge~ = wB/w~ (2.39)

define a holomorDhic line bundle on CP k, which we denote by H. The

sections of H are described as follows: Let I : ~k+1 ÷ ~ be a linear

function. Then

s e = l(w)/we : U ÷

defines a secdion s £ F(f~k,H) with

Div s = ~(Ker I) .

All sections of H are of this form and thus H is called the hyper-

plane section bundle. Thus H = L A where A is any projective hyper-

plane (of the form P(Ker I)) in fpk. The standard hermitian metric

on H is given by

h = Iw [2/11wlL 2 (2 .40)

(where w = (w ° ..... Wk)). By (2.35) and (2.40) we have

n H = ddClogllwll 2 = ~. (2.41)

Thus H is positive.

Example 2. Again let V = ~]pk. For p 6 ~., the line bundle H p has

~ P p transition functions g = wg/w e. If p > 0, the holomorphic sections

s of H p are of the form

Q(w O ..... w k ) s = e D '

w- e

where Q is a homogeneous polynomial of degp. If p < 0, then H p

68

has no holomorphic sections. We have

n = p~. (2.42) H p

One can show by cohomology theory, that all holomorphic line bundles on

f~k are of the form H p.

Example 3. Let V be a submanifold of ~N. (By Chow's Theorem,

V is the zero set of homogeneous polynomials; V is thus called a

projective-algebraic manifold.) Let j : V ÷ C~ N denote the imbedding.

We let U = V N {w # 0}, for 0 ~ ~ ~ N. The hyperplane section bundle

on V, denoted j H, is given by transition functions geB o j and

metric h e o j where gas and h a are given by (2.39) and (2.40). We

have

~j*H = J*~H = J*~' (2.43)

and thus j*H is positive.

Example 4. Let V be a k-dimensional complex manifold. The canonical

bundle K V of V is the holomorphic line bundle whose sections are

holomorphic (k,0)-forms on V. Choose a covering {U a} of V with

holomorphic coordinates w~ ~)" ,...,w~ e)" on U a. If ~ is a holomorDhic

k-form on V, we can write

0 = s dw (~) (a) a -I A "'" A dw k

on U e, where sa E 0(Ue)" Then s~ = gdBsB on Ua N U B where

• (2.44) gab = det[~w~ ~ 1~l,m~k

Thus (2.44) can be taken as the definition of K V. The reader should

verify that

Kf~ k = H -(k+1) (2.45)

(An easy method is to find t 6 r(c~k,K -I) with Div t = (k + I) [A]

where A = {w ° = 0}.) A volume form ~ on V is given by

i ~a) ~--(e) i ~ (e) --(a) = ha( ~dw A aWI ) A "'" A (~aW k ^ dw k ) (2.46)

> 0 and h = I 12h~ on U n u~ for given by where h a e gab a gab

(2.44). Thus by (2.43), Q can be regarded as a metric on the dual

bundle . We define the Ricci form Ric Q by

Ric ~ = ddClog h (2.47) a

on U . Recalling (2.35), we see that Ric ~ is a global (1,1)-form e

69

on V and

Ric ~ = - q -I " (2.48) KV = qKv

In contrast to the case of meromorphic functions on ~I , a meromorphic

map of complex manifolds is not necessarily well-defined. Suppose M

and V are complex manifolds and V is compact. A meromorphic map f

from M into V is given by an analytic set G c M x'V (G is called

the graph of f) such that there exists a dense open set M O of M

such that

i) the set G o = G N(M ° x V) is the graph of a holomorphic map from

M into V, o

ii) G is the closure in M x V of G o

We write f : M ~ V to indicate that f is a meromorphic map. The in-

determinacy set If of f is the set of points at which f is not

holomorphic. It is a classical result that If is an analytic set in

M of codimension ~ 2; thus we can choose M ° = M - If in the above

definition. If fo,...,fk are holomorphic functions on M, then f =

(fo:...:fk) defines a meromorphic map from M into ~k. Conversely,

if f : {n ~ c~k is a meromorDhic map, then it follows by Cartan's

Theorem B that f can be written in the form f = (fo:.-.:fk) where

-1{0 } fj 6 0(~ n) and If = n fj

Henceforth we assume that f : C n ~ V is a meromorphie map, where

V is a projective-algebraic manifold. (In Chapter III, we must assume

that dimV ~ n for the Second Main Theorem.) Let L be a positive

line bundle on V with curvature form q > 0. We shall define the

Carlson- Griffiths characteristic function of f with respect to the

line bundle L. If f is holomorphic, this characteristic function is

given by

T(L,r) = Tf(L,r) = N(f*q,r) = f (log r - T)~ n-1 ^ f*q . 2.49)

fn[r]

(Alternately, we can consider V c ~N and f = (fo:..-:fN) and de-

fine the characteristic function T(r) = (Ur,logllf]i) as in Ahlfors

[I] and Weyl [38] when n = I. As was seen in Example 3, every imbedding

j : V ÷ {~N determines a positive line bundle j*H. One can then show

that T(r) = T(j*H,r) + c.) For meromorphic f we also use (2.49) to

define T(L,r) except we define f*q as follows: Let G c C n x V be

the graph of f and let ~I : G ÷ ~n and 72 : G ÷ V be the projections.

We define

* 11 f*q = ~I,([G] A ~2n) 6 D' ' (on). (2.50)

70

Dk(M I ~k(M 2 ) (If ~ : M I ÷ M 2 is a proper C -map, then z, : ) ' + ' denotes

the dual of the map ~* : ~k(M 2) ÷ Dk(MI). Note that ~, changes the

degree if dimM I ~ dimM2.) It is easy to verify that the current f*~

is real, closed, and of order 0. We then define Tf(L,r) for mero-

morphic f by (2.49) and (2.50).

We note two more properties of f*~ : Let M = {n _ If, f = flM o, -I o o

and Go = z1 (Mo). First, f*~IM ° is the ordinary pull-back f~ of

the form ~. To verify this fact, we let ~ 6 vn-1'n-1(Mo ) be arbitrary.

Then ~2 = fo o nl on G O , and

* ~ ) f ~ A ~ = I f *~ ,~ l : I [ a ] ^ ~ 2 n , ~ a 2n* "1~ GJ ~T(f~n ̂~I o o (2.51)

= ~ f~n A ~ = (f~n,~). M O

Secondly, f*~ is absolutely continuous, which by (2.51) means that

C ~ llf*~jl (If) = 0. To show this, choose a sequence Xj E ({n) decreasing

pointwise to I on If and to 0 on C n - If. It suffices to show

that (f*~,Xj~) ÷ 0 for arbitrary ~ 6 ~n-l,n-1({n). We have

( f * n , X j ~ ) = f (X j o ~ 7 ) ~ n A * ~i ~ + 0 G rag

by the Lebesgue dominated convergence theorem. These two facts say that I

the coefficients of f*n are in ~loc(f n) and are given by the ordinary

pull-back f*~o on {n _ If.

The characteristic function T(L,r) does not depend on the choice

of metric on L, modulo a bounded term. To see this, let {h } and ! {h~} be two metrics on L. Then h = eUh , where u 6 C~(V). Thus

~' = ~ - ddCu and hence by (2.50),

f*~' = f*~ - ddC(u 0 f)

where u o f is a bounded measurable function on

Lemma 2.3

(2.52)

{n. Therefore by

N(f*~,r) - N(f*~',r) : N(ddC(u o f),r) : ½(~r,U 0 f) + c : 0(I)

If L and L' are line bundles on V and L is positive, then

~L' = < c~ L for some constant c and hence

T(L',r) ~ cT(L,r). (2.53)

In particular, if both L and L' are positive, then the orders of

the functions T(L,r) and T(L',r) are equal.

Let D be a positive divisor on V such that f({n) ~ suppD. We

71

define the proximity term m(D,r) as follows: Choose a metric on L D

and choose s 6 F(V,LD) such that Div s = D and llsI] £ I on V.

The proximity term is given by

m i D , r } = m f ( D , r } = ( a r , - l o g l l s o f l l ) ~ 0 . (2.54)

One easily verifies that miD,r) does not depend on the choice of the

metric or of s, modulo a bounded term. The pull-back divisor f~D is

defined as follows: Choose a covering {U } of V, and choose s 6

0(U ) representing the section s given above. Letting fo = f lCn - If

~D on ¢n _ I{ by as before, we first define the divisor fo

fo*Dlfo l{U} : Div s Ofo

Since codim If ~ 2, then by the Remmert--Stein Theorem, we can define

f~D as the unique extension of f~D to ~n. Finally, we define the o

counting function

N(m,r) = Nf(D,r) = N(f~D,r). (2.55)

Note that (2.55) makes sense only if f(C n) ~ supp D.

The following lentma is an "unintegrated" First Main Theorem:

Lemma 2.6. Let D be a positive divisor on V such that f(cn)

suppD. Let s 6 P(V,L D) such that Div s = D. Then

ddCloglls o fll 2 = f*D - f*~,

where q is the curvature form of L D-

Proof. Using the notation from above, we write Ilsl] 2 = h Is ] 2 on

U and hence

loglls0fl] 2 : logh of ÷ logls 0f[ 2 (2.56)

-I on f (U e) - If. It then follows from Theorem 2.2 and (2.35) that the

identity of the lemma is valid on C n - If. Since llf~Dll (If) :

Ilf~ll (If) = 0, it suffices to show that loglls 0 fl] 2 is a pluri-

potential on fn, so we can apply Lemma 2.4 to conclude that the iden-

tity is valid on all of C n. Choose an imbedding V c C~ N and choose

a positive constant c such that c~IV - q ~ 0, where e = WC~ N .

Choose fo ..... fN 6 0(C n) such that f = (fo:.-.:fN) and If = n f71(0). 3

Let u = c log [ Ifj[ 2 Then

adC{log[[s0fll 2 ÷ u) : f*n - f*n + cf*~ ~ 0

on C n - If. Thus log[Is o fll 2 + u is plurisubharmonic on C n - If and

extends to a plurisubharmonic function on {n by the Grauert- Remmert

72

extension theorem. Thus loglIs o fl[ 2 is the difference of plurisub-

harmonic functions on {n, completing the proof.

~n

Proof. Let s 6 F(V,LD) such that

Lemma 2.6,

[]

Theorem 2.7. (First Main Theorem) Let D be a positive divisor on

such that f(C n) ¢ supp D. Then

N(D,r) + m(D,r) = T(LD,r) + c.

Div s = D and lls[l ~ 1. By

N(ddClogll s o fll2,r) = N(f~D,r) - N(f~n,r) = N(D,r) - T(LD,r). (2.57)

By Lemma 2.3,

N(ddCloglls 0 fl[2,r) = ½(Or,logIls 0 fll 2) + c = - m(D,r) + c. (2.58)

The conclusion follows from (2.57) and (2.58). []

Corollary 2.8. If D is as in Theorem 2.7, then

N(D,r) $ T(LD,r) + 0(I).

Example 5. (This is actually a continuation of Example I.) Let V =

C~ k. Writing f = (fo:.--:fk) and ilf[l 2 = ~ Ifj[ 2, we have

f~n H : f~ : ddClogllfE1 2

Thus by Lemma 2.3,

T(H,r) = (~r,logIlfll) + c.

Let a 6 C k+1 with flail = I

D = {w 6 {]pk : w.a = 0} 6 iHI.

The section s 6 F({]pk,H) with

s = (w-a)/w ,

where U e = {w ~ 0}

(2.40) , we have

llsl] : lw'al/LEw]1

and thus

(Or 'lOg~l~ai ) . m(D,r)

(2.59)

and consider the hyperplane

(2.60)

Div s = D can be given by

(2.61)

as before. Then using the metric on H given by

(2.62)

73

III. The Second Main Theorem

Throughout this chapter, we let V be a k-dimensional projective-

algebraic manifold, with k $ n, and we let f : C n ÷ V be a non-degen-

erate meromorphic map. By non-degenerate, we mean that the image of f

contains an open subset of V. In order to state the Second Main Theorem,

we must extend the definition of the ramification divisor Rf given in

(1.20). Let z ° be an arbitrary point of ~n at which f is holomorphic

and let Wl,...,w k be local coordinates in a neighborhood U of f(z O)

in V. Since f is non-degenerate, the matrix

[~(wj of)]

Sz m 1$jSk,1~m!n

has rank k somewhere (in fact, almost everywhere) on f-1(U). By per-

muting the coordinates {Zl,...,z n} if necessary, we shall assume that

the determinant

A = det (wj 1~j,m~k ~z m

does not vanish identically on f-1(U). The ramification divisor Rf

is defined to be the unique divisor on {n such that

Rf I f -I (U) = Div A. (3.1)

Note that (3.1) gives a well-defined divisor on C n - If by the chain-

rule, and this divisor has a unique extension (which we call Rf) to

C n by the Re~mert-Stein Theorem. (Of course, if k < n, then Rf de-

pends on the choice of coordinates Zl,...,z n in C n. If k = n, then

Rf is independent of coordinates.)

Recall that an analytic hypersurface D in V is smooth if and only

if for each a 6 D there exist holomorphic coordinates Wl,...,w k at

a such that Div w I [D] near a. Suppose DI,...,D q are smooth hyper-

surfaces in V. We say that DI,...,D q have normal crossings if for

each a 6 U Dj, there exist holomorphic coordinates Wl,...,w k at a such

that

m q

Div(3~lW j) = ~ [Dj] (3.2) "= j=1

on a neighborhood of a, where m is the number of the D. that con- ]

tain a. Note that if the D have normal crossings, then m ! k (for 3

all a) and the D must be distinct. ]

We shall prove the following version of the Second Main Theorem:

Theorem 3.1. Let D I,...,D be smooth hypersurfaces with normal

crossings in V, and let A = [ [D ]. Then j=1 3

74

II T(LAeKV,r) s N(A,r) - N(Rf,r) + O(logT(L,r)) + o(logr)

where L is a positive line bundle on V.

Note that by (2.53), the choice of the positive line bundle L is

immaterial. Theorem 3.1 for f holomorphic and k = n was proven by

Carlson and Griffiths [5]. The generalization to k < n with ~n re-

placed by an affine algebraic manifold (see Theorem 3.10) was given by

Griffiths and King [14]. The generalization to meromorphic f was given

in Shiffman [27] where a generalization to singular divisors without

normal crossings is stated. (See appendix II for an errata to [27].)

Definition. Let L be a positive line bundle on V, and let

The defect 6(D) is given by

N(D,r) 6(D) = lim inf [I T(L-L-I].,~, (3.3)

By the First Main Theorem 2.7, we have 0 ~ 6(D) < I.

Corollary 3.2. (Defect Relation) Let L be a positive line bundle

on V and suppose DI,...,D q are smooth hypersurfaces with normal

crossings in ILl . Suppose I 6 ~ such that

- < lnL- nK v

Then

q 6(Dj) < I.

j=1

Proof. (assuming Theorem 3.1)

(3.4)

(3.5)

Let A = U D-3 so that L A = L q. Then

@(Dj) ~ lira inf [ [T(L,r) - N(Dj,r) ]/T(L,r)

(3.6) = lira inf [T(LA,r) - N(A,r)]/T(L,r).

By (3.4)

-T(Kv,r) = N(-f*~KV,r) $ N(If*nL,r) = IT(L,r). (3.7)

Thus by Theorem 3.1, (2.22) and (3.7)

II T(LA,r) - N(A,r) ~ -T(Kv,r) + o(T(L,r)) ~ [l + o(I) ]T(L,r). (3.8)

Equation (3.5) then follows from (3.6) and (3.8). o

Example 6. Let V = CP k and let L = H p where p > 0. (Recall

Example 2.) By (2.45), K V = H -(k+1) and thus giving L and K V the

metrics induced by the metric on H, we have

m C I L l .

75

~K V

and thus

k + I - - q (3.9)

P

6(Dj) $ k P+~ (3.10)

for smooth hypersurfaces D of degree p in fpk with normal cross- 3

ings. If p = I, then the D. are hyperplanes (normal crossings then 3

means "in general position") and (3.10) remains valid for k > n by

a result of W. Stoll [31]. It remains an open problem to find conditions

on f so that (3.9) is valid for k > n, p > I. (See [2], [28], [29].)

Remark. The reader may check that

Ric ~ = -(k + 1)w : qK{~k '

which together with (2.42) explicitly verifies (3.9).

Corollary 3.3. If K V is positive, then there are no non-degenerate

meromorphic maps f : {n m V.

Proof. (again assuming Theorem 3.1) Let L be a positive line bundle

on V and let D I be a smooth hypersurface in ]LI . Since qK > 0

there exists a constant c such that n L 5 CqKv and thus

I <

-qK v = - C ~L"

If there exists a non-degenerate f : {n m V, then by Corollary 3.2,

6(D I) ~ -I/c < 0; contradiction. []

We begin the proof of Theorem 3.1 by constructing the Carlson-Griffiths

volume form on V - A. Let DI,...,D q and A be as in Theorem 3.1.

Choose metrics on the LD . These metrics induce a metric on the line

= ® -.- ®LDI. Choose a volume form 9 on V; ~ induces bundle L A LD1 metrics on K~ I_ and on qK v. Choose sections sj 6 F(V,LDj) such that

Div sj = Dj J and SUPvllSj] I < I, for 1 S j $ q. The Carlson-Griffiths

volume form ~ is then defined by

q : ( A Ilsj[llogllsjll)-2d. (3.11)

j=1

The metric ~ is C on V - A and becomes infinite along A. The

reader should check that (1.56) is a special case of (3.11) with s. = 3

e-2qs(aj), where s(aj)j is given as in (2.61) with a replaced by aj.

Suppose @ is a real 2n-form on an n-dimensional complex manifold

M. We say that @ > 0 [@ ~ 0] if on each coordinate neighborhood[ U

of M we can write

76

i dz I i dZn) @ = u(~dz I ^ ) ^ -.- A (~dz A , n

C ~ where u > 0 [u > 0] on U. (Thus, a volume fo~m on M is a 2n-

form ~ with Q > 0.) We say that 01 > 02 [01 -> 02 ] if 01 - 02 > 0

[01 - @2 -> 0]. Suppose ~j is a real (1,1)-form on M with ~j > 0

for I < j _< n. Then one easily checks that ~I ^ "'" ^ nn -> 0. Now

suppose that @'3 is a real (1,1)-form with @.3 > nj > 0 for I < j < n.

Then @j ~j + pj where pj -> 0 and therefore

@I A "'" ^ On = (~I + Pl ) ^ "'" ^ (~n + Pn ) >- ~I ^ "'" A ~n" (3.12)

Lemma 3.4. If LA®K V is positive and SUPv,j]Isjl I is sufficiently

small, then Ric ~ > 0 and there exists c > 0 such that

(Ric ~)k ~ c~ (3.13)

on V - A.

Proof. On V - A, we have by definition

2 Ric ~ : ~ - ~ ddClog(logllsjIl)

where

(3.14)

n : Ric ~ - X ddCl°gllsjE12 13.151

By (2.35) and (2.48), ~ is the curvature form of LAeK V and hence

> 0 on all of V. We compute

_ddClog (logll sj Ii ) 2 : 2 ddClog ii sj I]

-l°gll sj II (3.16)

+ i ~(logiisjil)2 ~ logllsjll ^ ~ logllsjll.

Since ddClogllsjll is C ~ on v, if SUPvllSjl I is sufficiently small,

then the first term on the right hand side of (3.16) is small and

I i Ric ~ > ~ + ~ [ (logllsj]l)-2 ~ logllsj] I ^ [ log]Isjl I > 0. (3.17)

Thus (Ric ~)k ~ 2-k k > 0 on V - A. Thus, to verify (3.13) it suf-

fices to show that (Ric ~)k/~ is bounded away from 0 near A. Let

a 6 A be arbitrary, and choose coordinates Wl,...,w k at a satisfying

(3.2). After permuting the sj, we can write

llsjll 2 = e~Jlwjl 2 (3.18)

for I ~ j ~ m, where ~j is C ~ near a, and sj(a) ~ 0 for j > m.

77

Hence

dw. J 2~ logllsjl I = w. + ~j J

for I ~ j ~ m. Thus by (3.17) and (3.19),

Rie ~ > ~1(iog,lsj,,)-2(~J + ~j)^ (Cw~ +. ~j)

3 J

near a, for I ~ j ~ m. Since ~ > 0, we also have

i > e' ~ ~dwj ^ dwj

near a, for some c' > 0, and hence

(3.19)

(3.20)

ic I Ric ~ > ½ ~ > Tdwj A dwj (3.21)

near a, for I -< j _< k. Thus by (3.12), (3.20), (3.21) (for m+ I < j < k),

and (3.18) ,

dwj + ] ^ (Rie ~)k -> Hm [~(logi llsj II -2) (dWJw. + S~j) ^ (~. ~j)

9=1 j J

ic ' - ic ' --~- dWm+ I A dWm+1) ^ .-- A (TdWk A dw k) (3.22)

m ~ (llsj log][sjll) -2 i i (2dWl ^ dWl) A "'" ^ (~dw k ^ dw k) .

J

If follows from (3.11) and (3.22) that

lim inf (Ric ~)k/~ 0. [] w ÷ a

Write E = f-1 (A) U If. We let ~ 6 C~(C n - E) be given by

i - i ^ d-n)Z ~Bn = f*~ ^ (~dZk+ I A dZk+ 1) ^ ... ^ (~dz n (3.23)

Let ~ be given by (3.15) and let

q )-2 h = ( H logllsjl I g

j=1

Lemma 3.5.

i) log ~ and log h 0 f are pluripotentials on fn ,

ii) ddClog ~ = f*Ric ~ - f*A + Rf ,

iii) f*Ric ~ = f*~ + ddClogh 0 f .

Proof. Fix j and let

(3.24)

u = -2 log (-iogllsjll). (3.25)

I n We must show that u o f 6 ~oc(~ ) and that dd c u 0 f is an absolutely

continuous current on ~n. The lemma then follows by the arguments of

78

the proof of Lemma 1.10. Let

v : -logllsj o fll (3.261

so that u 0 f = -2 logv, and thus

2 dd c v = I f,j dd c u o f ~ -~ -Q (3.27)

on fn _ E, where nj is the curvature form of LDj. Consider an im-

bedding V c ~N and let f = (fo : "'" : fN )' so that

f*~ : dd c logllfll 2 13.281

on C n - If. By Lemma 2.4, (3.28) is valid on all of fn. Since e > 0

on V and v is bounded from below by a positive constant, there exists

c > 0 such that

I f, U ~j ~ cf*~ (3.29)

on {n _ If. Let

y : u0f ÷ clogl]fll 2 (3.30)

By (3.27) and (3.29), y is plurisubharmonic on {n _ E. By Lemma 2.5

(with u replaced by y) it follows that y is plurisubharmonic on

C n - If and ddCy is absolutely continuous on {n _ If. Hence by Lemma

2.4 and the Grauert-Remmert extension theorem [10], y is plurisubhar-

monic and ddCy is absolutely continuous on all of C n. By (3.30)

C n dd c u o f is absolutely continuous on . D

By Lemma 3.5 (ii) , f*Ric ~ is a closed current of order 0 on {n,

and thus we can define

T~(r) = N(f*Ric ~,r)

as in Chapter I. By Ler~na 3.5 (i) , we also define

(3.31)

~(r) : ½(Ur,log <). (3.32)

The following identity is an immediate consequence of Lemma 2.3 and

3.5 (ii):

Lemma 3.6. T~(r) : N(A,r) - N(Rf,r) + ~(r) + c.

Next we estimate T~(r).

Lemma 3.7. T~(r) = T(L A ®Kv,r) + O(Iog+T(LA,r) + I).

Proof. By Lemma 3.5. (iii),

79

"Ty(r) = N(f*~,r) + N(dd c logh o f,r)

= T(L A®KV,r) + ½(Or,log h 0 f) + c.

(3.33)

Since log h o f $ O(I), the following estimate completes the proof of the

lemma:

- ½(Or,logh 0 f) = [ (Or,log log I ) $ ~ log(Or,log I ) llsjl] llsj]l 1 = ~ l o g m ( D j , r ) ~ q l o g l ~ m ( D j , r ) ) ~ q Z o g T ( L A , r ) + 0 ( 1 ) .

[]

We now c o m p l e t e t h e p r o o f o f T h e o r e m 3 . 1 . Add s m o o t h h y p e r s u r f a c e s

t o t h e c o l l e c t i o n { D 1 , . . . , D q} s o t h a t L A ® K V i s p o s i t i v e a n d t h e

D. s t i l l h a v e n o r m a l c r o s s i n g s . ( T h e s e a d d i t i o n a l h y p e r s u r f a c e s c a n 3

b e t a k e n t o b e g e n e r i c h y p e r p l a n e s e c t i o n s o f a n i m b e d d i n g V c c ~ N . )

By C o r o l l a r y 2 . 8 , i t s u f f i c e s t o p r o v e T h e o r e m 3 . 1 f o r t h e e n l a r g e d

c o l l e c t i o n { D . ] . T he c o n c l u s i o n o f T h e o r e m 3 . 1 f o l l o w s f r o m Lemmas 3 . 6 3

a n d 3 . 7 t o g e t h e r w i t h Lemraa 3 . 8 b e l o w :

Lemma 3.8. Suppose L A~K V is positive. Then

II ~(r) ! O(IogT(LA®Kv,r)) + o(logr).

Proof. Let

i - i ^ din) (3.34) I = (2dZk+1 A dZk+1) ^ "'- ^ (SdZn

By Lemma 3.4

~B n = f*~ A 1 < c(f*Ric ~) k ^ I, (3.35)

where throughout this proof, we let c be the symbol for a positive

constant. Write

n

i l,~=lalmdZl ^ d ~ m ' ( 3 . 3 6 ) f*Ric ~ =

and consider the n × n and k × k matrices

M = [alm]1~l,m~n, M k = [alm]1~l,m~ k.

By Lemma 3.4, M is positive semi-definite at each point of C n - E.

By (3.34) and (3.36),

i . n (f*Rie ~)k A i = (det M k) (~z I ^ d~ I) ^ .-. ^ (gZn ^ dz n) =~-~(det M k) ~n.(3.37)

Thus by (3.35) and (3.37)

80

< c det M k. (3.38)

Hence

(I/k ~ c Trace M k ~ c TraceM. (3.39)

Noting that

(f*Ric ~) A B n-1 ~ B n = (~ Trace M) , (3.40)

we thus conclude that

(I/k6n ~ cf*Ric ~ A 8n-I (3.41)

Thus by (2.21) and (3.41),

r dt I/kBn r t2n_ I n~ ~ _< c

I C [t] I dt ~ f*Ric ~ ^ B n-1

t 2n-I cn[t]

= cT$(r) + O(I).

(3.42)

By the concavity of the logarithm,

k ~I/k) k /k ~(r) = ~(Or,log ~ ~log(Or,~ I ) (3.43)

and thus

t I/k) s2n-lds I/k~n t (2/k)~(S)s2n_ids S (as,~ c nf ~ . (3.44) f e < = 0 0 C [t]

Combining (3.42) and (3.44), we obtain

r dt Ste (2/k) ~ (s) 2n-I t2n_ I s ds < cT$(r) + O(I). (3.45)

I 0

We conclude from (3.45) exactly as in the proof of Lemma 1.17 that

lJ ~(r) ~ O(log T~(r)) + o(log r). (3.46)

By Lemma 3.7 and (2.53) we have

T~(r) = [I + o(I) ]T(LA®Kv,r). (3.47)

Lemma 3.8 follows from (3.46) and (3.47). This completes the proof of

the Second Main Theorem. D

We give one more application of Theorem 3.1:

Corollary 3.9. Let D 1,...,Dq be smooth hypersurfaces with normal k

crossings in C]P . If [ degree (D.) > k + 2, then there are no non- k u

degenerate holomorphic maps f : C ÷ C]P k - (D I U ... U Dq) .

Proof. Let A = D I U-.-U Dq and let p = degA = ~ degDj.

81

Let

L : L A®K = H p OH -(k+1) : H p-k+1 (3.48)

which is positive since p > k + I. Suppose on the contrary that there

exists a non-degenerate holomorphic map f : fk ÷ f~k _ A. Then

N(A,r) = 0 and thus by the Second Main Theorem 3.1,

II T(L,r) _< O(logT(L,r)) + o(logr) < o(T(L,r)),

which is a contradiction. D

For other "Picard theorems" see M. Green [11].

Both smoothness of the D. and normal crossings are essential in 3

Theorem 3.1, as the following example (from [27]) shows:

Example 7. For p ~ 4, let

Ap : {(w o:w 1:w 2)}E~2:w~- WoW~-1 : 0}.

The hypersurface A is singular at the point (I : 0 : 0) and is smooth P

everywhere else. However there exists a non-degenerate holomorphic map

f : {2 + ~2 _ A given by P z 2 (1-P)Z 1 z I

f(zl,z 2) = (e + e : I : e ).

We now state without proof the generalization of Theorem 3.1 to mapp-

ings from an affine algebraic manifold. Let M be an n-dimensional

algebraic submanifold of C N. By considering the closure of M in {pN,

we can find a linear map ~' : C N ÷ C n such that the map

: z'IM :M + C n

is proper. The counting functions N(%,r) for % 6 D'I'I(M) (where

is real, closed, and of order 0) are defined as before with 7 re-

placed by T 0 7. Recalling (2.4), we define o r 6 D'2n(M) by

~n-1 (ar'~) : I f n ~*(dClog[Iz[l 2 A ~ ). (3.49)

~- (e <r>)

(The measure u given by (3.49) is not a probability measure. In fact r

(Or,l) = degM .) Let f : M ~ V be a non-degenerate meromorphic map

where V is as above. The First Main Theorem is valid as before. We

define the ramification divisor of an equidimensional mapping on M

as in (3.1), except the z. are replaced by local holomorphic coordi- ]

nates in M. We define Rf to be the ramification divisor of the equi-

dimensional map

82

(f,zk+ I 0 ~,...,z n 0 ~) : M ÷ V x fn-k.

By the methods of these lecture notes, one can prove the following form

of the Second Main Theorem:

Theorem 3.10. Let V, A, L be given as in Theorem 3.1 and let

f : M ~ V be as above. Then

]I T(L A ®Kv,r) ~ N(A,r) + N(R ,r) -N(Rf,r) + O(log T(L,r)) + o(logr).

Theorem 3.10 is given in Shiffman [27, Theorem 3.2] and is given for

holomorphic maps in Griffiths-King [14]. For another form of the Second

Main Theorem, see Stoll [34]. Note the extra term N(R~,r) in Theorem

3.10. However, one has N(R~,r) = O(log r). Thus, if f : M ~ V is tran-

scendental, then the defect relation of Corollary 3.2 is valid for this

f. For details, see [27].

We complete these notes by using the above proof to derive the "lemma

of the logarithmic derivative" on fn given by Vitter [36]:

Theorem 3.11. Let g be a meromorphic function on fn. Then

II (°r,l°g+l~ ~I) $ O(logT(g,r)) + o(logr), for I $ j ~ n.

3

Our proof below is a simplification of the proofs given by Griffiths

[13] and Vitter [36]. (For a proof by entirely different methods, see

[4] or [34].)

To prove Theorem 3.11, we first return to the general setting of

Theorem 3.1. Let A = [ [Dj] be as in Theorem 3.1 and suppose that

L A is positive and that the curvature of L A®K V vanishes identically

(as it does, for example, when L A = ). Choose a volume form ~ on

V and a metric on L A such that qLA is positive and

Ric 2 + ~LA = qLA®K V 0. (3 .50 )

We use the notation from the proof of the Second Main Theorem. Also,

we let

@ = Ric T + (3.51) qL A"

By ( 2 . 3 5 ) , ( 3 . 1 4 ) , (3 .15 ) and ( 3 . 5 0 ) , we have

@ = Ric ~ + 2qLA + ddClogh : ~LA + ddClog h (3.52)

on V - A. By repeating the proof of Lemma 3.4, we then obtain the fol-

lowing result:

83

Lemma 3.12. If suPllSjl I is sufficiently small, then @ > 0 and

there exists a constant c > 0 such that @k ~ c~.

Next, we have the estimate:

Lemma 3.13.

II (°r,log+<) $ O(iog T(LA,r)) + o(log r).

Proof: By concavity of the logarithm,

(Or,log+~) ~ k(Or,log(~I/k + I)) ~ k log (o r ,~I/k + I). (3.53)

Thus

(I/k) (Or, log+~) e - I ~ (Ur,61/k). (3.54)

By (3.54), Lemma 3.12, and the proof of Lemma 3.8

r ft[e(1 dt /k) (~r'l°g+~) - 1]s2n-lds $ N(f*@,r) + 0(I) 1 t 2n-I 0

and thus

(3.55)

I/ (~r,l°g+~) $ O(iogN(f*@,r)) + o(log r). (3.56)

By (3.52) and the proofs of Lemmas 3.5 and 3.7 we obtain the estimate

N(f*@,r) = T(LA,r) + O(IogT(LA,r)). (3.57)

Lemma 3.13 follows immediately from (3.56) and (3.57). []

In order to complete the proof of Theorem 3.11, we now let V = ~I,

D I = {(I : 0) }, D 2 = {(0 : I) }, and A : [D I] + [D2]. Let g be a mero-

morphic function on {n, and let f : (fo : fl ) : {n ÷ V such that g =

fl/fo and f-1{0}o n f11{0}- = If. For a (1,0)-form y on V, we shall

use the notation

I ~12 = i ~ ^ T .

We have

tWll I%1 I ] s 111 = ~ , IIs 211 = ~

[Iwll I lwl l where I lwlE2 = 1%12+ [W112, and ~

s e t

= ddClogllw11 2

(3.58)

is a small positive constant. We

I lWodWl - WldWo 12 2~ Elwii4

(3.59)

(The reader should verify (3.50).) Thus

84

82h lWodW I -WldWo 12

= I l s l l l - 2 1 1 s 2 1 1 - 2 h ~ - 2~ iWoWl l2

Hence

2hl dlwW--ll I 2 13601 2z I o "

1 3g 2 = c(h o f)1~ ~--~] I ,

where c > 0. We thus have

(3.61)

~-~- l + 1 (°r,log+l I) = < 2(@r,log <) + ½(5r,log+h--~) + 0(I). 1

By the proof of Lentma 3.7,

I ½(Qr,lOg+h__~) $ ½(Jr,log I ~-~) + ½(Or,log(h 0 f + I))

< 21ogT(LA,r) + 0(I).

(3.62)

(3.63)

Note that T(LA,r) = 2T(g,r). The conclusion of Theorem 3.11 then follows

from (3.62), Lemma 3.13 and (3.63). []

Appendix I.

In this appendix, we give an elementary proof of Lemma 2.4. We let

be a domain in {n, and we identify C n with ~2n with coordinates

Xl,.--,X2n-

Lemma A.I. Let u 6 9' (~). If ~u/~x is of order 0 for I < i < 2n, 1

u £ __~oe (~)" then

Proof. (See also [25].) Let < denote the fundamental solution for

the Laplacian on IR 2n ; i.e, X ~2</~x2 equals the Dirac delta measure

at 0. (If n > I, < = -[4~n/(n - 2) !] IIxII-2n+2.) Then

32 8K 8u u = • 2 (<*u) = X ~x *~x

( A . 1 )

~X. 1 1

I The lemma follows from (A.I) and the fact that ~K/~x i 6 __~]oc" []

We let h p denote Hausdorff p-measure on Euclidean space, for p-> 0

see [9] or [35]). If

V = ~ Vil ..... ikdXil A "'' A dXik (I ~ i I < "'" < i k ~ 2n)

is a current of order 0 on ~, we define the measure llvl] on ~ by

]Ivll = I Ivi I ..... ikl.

Lamina A.2. Let 0 < p ~ 2n and let E be a closed subset of

such that hP(E) = 0. If v 6 D'2n-p(d) is closed and of order 0, then

j[vllIEl = 0.

85

Proof. Let E, v satisfy the hypotheses of the lemma, and let

y = XEV, (A.2)

where XE is the characteristic function of E. (In (A.2), we regard

the coefficients of v as measures. If ~ is a measure, then

< 2n (XE~) (A) : ~(A n E).) We must show that ~ = 0. Let I ~ i I < ..- < ip =

and let ~ £ D°(~). If suffices to show that

Let

(y,<0dx ^ ..- A dx ) = 0. 11 lp

: ]R 2n ÷ ]R p be given by

(A.3)

~(x I ..... X2n) = (Xil ..... x i )- P

To verify (A.3) , we must show that

~.(~y) = 0. (A.4)

Choose a sequence pj 6 C~(~) such that 0 _< pj < I

~j = ~.(pj~v) 6 D'°(zRP),

for j = 1,2,3 ..... Then ~j ÷ ~,(~0y), and

d~j = ~,(d(pj<0) ^ v) ,

which is of order 0. Thus, by Lemma A.I, ~j 6 ~I(]RP). Let

and let K = support <0. For k > j,

and pj ~ XE. Let

(i.5)

(A.6)

c : sup I~I

[[~j - ~k~J[ (]Rp) < II (Pj - Pk)<°vll (K) < c f P~d[Ivll ÷ 0, (A.7) K-E J

as j ÷ ~. Since llgj - ~k][ (]RP) equals the ~I norm of ~J - ~k' it

follows from (A.7) that {~j} is a Cauchy sequence in ~I (]RP) and

thus converges in ~I (]Rp) . Therefore, z,(~y) 6 ~I (jR n) . But supp ~,(<~y)

m z(E) and hP(~(E)) -< hP(E) = 0. Since h P is Lebesgue measure on

]R p, (A.4) follows. []

Lemma A.2 is a special case of Federer [9,4.1.20]. The above proof

shows that Lemma A.2 is valid for v 6 D'2n-p(Q) such that v and dv

are of order 0. (Such a current is said to be "locally normal.")

Lemma 2.4 is a special case of Lemma A.2: Let u, S be given as in

Lemma 2.4. Since S is a countable union of submanifolds of real di-

mension -< 2n - 4, it follows that h2n-2(S) = 0. (In fact h2n-4+S(S)

= 0.) The conclusion of Lemma 2.4 then follows from Lemma A.2 with

p = 2n - 2, E = S, and v = ddCu.

86

Appendix II.

We include here an errata for [27].

Page 174: Replace lines 4-7 by the following:

II T~(LA,r) - N(A,r) + T~(Kv,r) < N(R ,r) - N(Rg,r)

(4.14) + O(logT~(LA,r) + o(logr)

A

where g = ~^ = (~,z I o ~ ..... Zn_ k o 7) . Thus f = p 0 g, where

A

p : p x idcn_k: ~ × C n-k ~ V × ~n-k.

By the chain rule, we have

= : + Rg ~ --*[Sj] + R . R~ g'R; + Rg ~*R : qjf g

Thus,

N(Rg,r) = N(R~,r) - ~ qjN(Sj,r).

By the First Main Theorem 2.3,

Replace lines 12-21 by the followinq:

II T~(L~ + K~) - N(D,R) < N(R ,r) - N(R~,r)

(4.15) + [ qjN(~j,r) + O(logT~(LA,r))+ o(logr).

Note that

N(D,r) = N~(p*D,r) = N(~,r) + ~ (pj - I)N(Sj,r). (4.16)

Therefore by (4.13) and the F.M.T.,

Tf(L D + Kv,r) - N(D,r) = T~(L~ + K~,r)

+ [ ~oj -qj - I)T (n~ ,r) -N(~,r) - [ (pj - I)N(Sj, r) 3

= T~(L~ + K~,r) - N(~,r) + [ (pj - qj - 1)m(Sj,r) - [ qjN(~j,r) +O(I).

Thus by (4.15)

II Tf(L D + Kv,r) - N(D,r) -< N(R ,r) - N(R~,r) + [ (pj - qj - 1)m(~j,r)

+ O(logT~(LA,r) ) + o(logr).

87

Replace lines 24-27 by the following:

Let D' = p(A) c V. Note that for a positive divisor B, the order

function T(LB,r) = N(B,r) + m(B,r) + O(I) is bounded below. Since

p*D' - A is positive, we thus have

• ~(LA,r) S ~(Lp,m,,r ) + 0(I) = Tf(LD,,r) + 0(I). (4.18)

Page 175: Delete lines I-2.

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Department of Mathematics The Johns Hopkins University Baltimore, MD 21218 U.S.A.