Entire Holomorphic Mappings in One and Several Complex Variables. (AM-85) (Annals of Mathematics Studies)

ENTIRE HOLOMORPHIC MAPPINGS IN ONE AND SEVERAL COMPLEX VARIABLES BY

PHILLIP A. GRIFFTTHS

Herman Weyl Lectures The Institute for Advanced Study

ANNALS OF MATHEMATICS STUDIES

PRINCETON UNIVERSITY PRESS

Annals of Mathematics Studies

Number 85

ENTIRE HOLOMORPHIC MAPPINGS IN ONE AND SEVERAL COMPLEX VARIABLES BY

PHILLIP A. GRIFFITHS

Hermann Weyl Lectures The Institute for Advanced Study

PRINCETON UNIVERSITY PRESS AND

UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1976

Copyright © 1976 by Princeton University Press ALL RIGHTS RESERVED

Published in Japan exclusively by

University of Tokyo Press; In other parts of the world by

Princeton University Press

Printed in the United States of America

by Princeton University Press. Princeton. New Jersey

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

HERMANN WEYL LECTURES

The Hermann Weyl lectures are organized and sponsored by the School

of Mathematics of the Institute for Advanced Study. Their aim is to provide

broad surveys of various topics in mathematics, accessible to nonspecialists, to be eventually published in the Annals of Mathematics Studies. The present monograph is the second in is series. It is an outgrowth of the fifth set of Hermann Weyl Lectures, which consisted of five lectures given by Professor Phillip Griffiths at the Institute for Advanced Study on October 31, November 1, 7, 8, 11, 1974.

ARMAND BOREL JOHN W. MILNOR

TABLE OF CONTENTS INDEX OF NOTATIONS

ix

INTRODUCTION

(a) Some general remarks

3

(b) General references and background material The prerequisites and references for these notes, some notations and terminology, and the Wirtinger theorem.

5

CHAPTER 1: ORDERS OF GROWTH (a)

Some heuristic comments Emile Borel's proof, based on the concept of growth, of the

Picard theorem, and the classical Jensen theorem are discussed. (b) Order of growth of entire analytic sets The counting function, which measures the growth of an analytic set V C Cn, is defined and some elementary properties are derived. Stoll's theorem characterizing algebraic hypersurfaces in terms of growth is proved. (c) Order functions for entire holomorphic mappings The order function Tf(L,r), which measures the growth of an entire holomorphic mapping f : Cn -> M relative to a positive line bundle L - M, is defined and the first main theorem (1.15) proved. Crofton's formula is used to give a geometric interpretation of Tf(L,r), and is then combined with Stoll's theorem to give a characterization of rational

8

11

17

maps.

(d) Classical indicators of orders of growth The Ahlfors-Shimizu and Nevanlinna characteristic functions, together with the maximum modulus indicator, are defined and compared. Borel's proof of the Picard theorem is completed. (e) Entire functions and varieties of finite order Weierstrass products and the Hadamard factorization theorem are discussed. The Lelong-Stoll generalization to divisors

in Cn is then proved.

vii

25

30

viii

TABLE OF CONTENTS

CHAPTER 2: THE APPEARANCE OF CURVATURE

(a) Heuristic reasoning It is shown that curvature considerations arise naturally in trying to measure the ramification of an entire meromorphic function. The use of negative curvature is illustrated.

40

(b) Volume forms Volume forms and their Ricci forms are introduced and some examples given. The main construction of singular volume forms is presented. (c) The Ahlfors lemma The ubiquitous Ahlfors lemma is proved and applications to Schottky-Landau type theorems are given, following which appears a value distribution proof of the Big Picard Theorem.

46

(d) The Second Main Theorem The main integral formula (2.29) and subsequent basic estimate (2.30) concerning singular volume forms on Cn are derived.

59

53

CHAPTER 3: THE DEFECT RELATIONS

(a) Proof of the defect relations The principal theorem 3.4 and some corollaries are proved. (b) The lemma on the logarithmic derivative A generalization of R. Nevanlinna's main technical estimate is demonstrated by curvature methods. (c) R. Nevanlinna's proof of the defect relation 3.10 Nevanlinna's original argument and an illustrative example are discussed. (d) Ahlfors' proof of the defect relation 3.10 This is the second of the classical proofs of the Nevanlinna defect relation for an entire meromorphic function. (e) Refinements in the classical case There are many refinements of the general theory in the special case of entire meromorphic functions of finite order. We have chosen one such, due to Erdrei and Fuchs, to illustrate the flavor of some of these results.

65

BIBLIOGRAPHY

97

70

73

79

81

INDEX OF NOTATIONS

Cn is complex Euclidean space with coordinates z = (zl, , zn); n

ziwi and

(z,w)=

11z!,12

= (z,z);

i=1

Pn is complex projective space with homogeneous coordinates Z= lin] ; P1 = C U I-l is the Riemannian sphere;

B[r] = 1zECn : 1Izil < rl is the ball of radius r in Cn;

S[r] = S fl B[r] for any set S C Cn; A(r) = 1zfC : JzJ < rl is the disc of radius r in C; A = A(1) is the unit disc; A* = 1CEC :0< 1} is the punctured disc; is a polycylinder in A*(R)= 1zfCn: zi!1 < Ri and R =

x (,)n -k is a punctured polycylinder;

A* n =

,lY,

C n;

denote volume forms;

denote (1,1) forms; do =

4 (a - a);

On C with z = rei9

1 T 70 ®dr dc = 1- , j ®d6 - 477 (b = ddc;1zIl2 =

2 C±

T?7

19r

dzindzcl is the standard Kahler form on Cn 1

f,, = ddclogllz112 is the pull-back to Cn - 101 of the Fubini-Study Kahler

metric on

Pn-1;

A holomorphic line bundle is denoted by L -. M;

H - Pn is the hyperplane line bundle; ix

INDEX OF NOTATIONS

x

D is a divisor and [D] the corresponding line bundle,c1(L) is the Chern form (curvature form) of a Hermitian line bundle; Q(M,L) is the space of holomorphic sections of L -. M; ILI is the projective space of divisors of sections s c (9(M,L))HDR(M,R) is the 2nd deRham cohomology of M; A real (1,1) form cu on M is positive in case locally zj = E-1: 2 c/iijdziAd I

where (i/iij) is a positive definite Hermitian matrix; [c/fl

denotes the class in HDR(M,R) of a closed form % on M;

A class X in HDR(M,R) is positive in case X = [u] for some positive form

cjr;

KM is the canonical line bundle of M;

L* is the dual line bundle to L - M; G(Cn) are the entire holomorphic functions on Cn.

Entire Holomorphic Mappings in One and Several Complex Variables

ENTIRE HOLOMORPHIC MAPPINGS IN ONE AND SEVERAL COMPLEX VARIABLES

Phillip A. Griffiths* INTRODUCTION

(a) Some general remarks

These talks will be concerned with the value distribution theory of an entire holomorphic mapping

where M is a compact, complex manifold. The theory began with R. Nevanlinna's quantitative refinement of the Picard theorem concerning a non-constant entire meromorphic function

f:C-P1. If we let nf(a, r) be the number of solutions to the equation

f(z) = a,

Izj
then Picard's theorem says that the sum nf(a, r) + nf(b, r) + nf(c, r)

is eventually positive, where a, b, c are three distinct points on P1. Roughly speaking, Nevanlinna's refinement states that the above sum is eventually larger than the average

t f(r) = f nf(a, r) da aEP1 *This work was partially supported by NSF Grant GP38886. 3

4

ENTIRE HOLOMORPHIC MAPPINGS

number of solutions to the equation in question. Since the appearance of Nevanlinna's book [31], there has been considerable attention to the subject of value distribution theory, first in the classical case of an entire meromorphic function [25], then in the study of entire holomorphic curves in projective space ([44] and [45]), and, in recent years, in the general theory of holomorphic mappings between arbitrary complex manifolds (cf. [37] for a survey). We shall concentrate on the equidimensional case where dimCM = n and where f is non-degenerate in the sense that the Jacobian determinant

of f is non-identically zero. Aside from the Ahlfors theory of holomorphic curves in Pn, it is here that the defect relations of R. Nevanlinna have been most directly generalized. For example, Picard's theorem becomes the assertion that no such f can omit a divisor on M which has simple normal crossings and whose Chern class is larger than that of the anticanonical divisor on M. The corresponding defect relation will be proved in Chapter 3. Aside from proving this theorem, there are two main purposes of these lectures. The first is to attempt to integrate more closely the deeper analytic aspects of the classical one variable theory with the formalism

and algebro-geometric flavor in the several variable case. The second is to try to isolate the analytic concept of growth and differentialgeometric notion of negative curvature as being perhaps most basic to the theory. A glance at the table of contents should make it pretty clear how our discussion has been centered around these two purposes. One other aspect of these notes is that we have tried to give the heuristic reasoning which historically led to the recognition that growth and curvature were central to theory. Aside from the original proof of Picard's theorem using the modular function, we have included three additional proofs, among them the "elementary proof" of Emile Borel in which the central importance of growth was first clearly exhibited. Similarly, aside from the negative curvature derivation of the general defect relation,

INTRODUCTION

5

we have included the two classical proofs due to R. Nevanlinna and Ahlfors, each of which has its own distinct merit. (b) General references and background material

The classic book in the subject is [31] by R. Nevanlinna. A second book of his [32] and the more recent monograph by Hayman [25] contain

further discussion of the value distribution theory of an entire meromorphic function. All that is required to read any of these books is standard basic

knowledge of complex function theory.

In several variables, we shall use the formalism of complex manifold theory, especially that centered around divisors, line bundles, and their Chern forms and subsequent Chern classes. The basic references here are the books by Chern [11] and Wells [43]. In practice all that we shall really require is fully explained in the introduction (pp. 151-155) of Griffiths-King [22].

From several complex variables, one needs to know a little about the local structure of an analytic hypersurface, especially as regards integration over an analytic hypersurface and Stokes' theorem in this situation. For the former the second chapter of Gunning and Rossi [24] is more than sufficient, and for integration we suggest the notes by Stolzenberg [41]. Aside from the facts that integration of a smooth differential form over a possibly singular analytic variety is possible and that Stokes' formula is valid, the fundamental result we shall use concerning integration on complex varieties is the Wirtinger theorem:

Let ds2 =

hij dzi dzj and V be respectively a Hermitian metric and

k-dimensional analytic variety, both defined in some neighborhood of the

closure of an open set U C Cn. Let

_

2

I hidzi A di] be the J

i,j

exterior (1,1) form associated to the Hermitian metric and vol (V) the 2k-dimensional volume of v Il u computed with respect to the Riemannian metric associated to ds2. Then

6

ENTIRE HOLOMORPHIC MAPPINGS

vol (V) =

(0.1)

J 0k vnu

1

k!

where ok= k-times

PROOF. If V* is the set of non-singular points of V, then by definition vol(V) = vol(V*) and f Ok = f Ok. Thus it suffices to prove v*nu vnu (0.1) in case V is a complex submanifold of U, and this obviously reduces to establishing (0.1) in case V=U. By applying the Gram-Schmidt process to the differentials dzl, ,dzn relative to the Hermitian metric (hid), we may write

n

ds2

i=1 where the (ki =

aijdzj give a Cm basis for the (1,0) forms over U.

Expanding

C)i = ai + \ Pi in real and imaginary parts, the associated Riemannian metric is n

ds2

ai + /3i

=

i=1 The volume form for this metric is, by definition, dp =

alA13lA...AanAOn

.

On the other hand,

2 \j ()n l Ann =

so that

on =

i= 1

n! aIAt3l A...AanAlen

n!dp.

Q.E.D.

INTRODUCTION

7

The principle that the volume of an analytic variety V C Cn is computed by integration of a differential form defined on all of Cn, as opposed to the computation of arclength, surface area, etc. in the real case, is of fundamental importance.

CHAPTERI ORDERS OF GROWTH

(a) Some heuristic comments

In its simplest terms, the study of entire holomorphic mappings is concerned with growth. For example, when the well-known growth behavior of a polynomial is plugged into the Cauchy integral formula

n (0 r) =

r p'(z) dz

1 -

=1 J

p(z)

one obtains the fundamental theorem of algebra. Emile Borel realized that growth was also essential to an understanding of Picard's theorem, viewed as a transcendental analogue of the fundamental theorem of algebra. Here is the heuristic reasoning behind his proof of that theorem: An entire holomorphic mapping f : C P1 may be written in homogeneous coordinates as f(z) = [f0(z), f1(z)] where f0 and f1 are entire holomorphic functions having no common zeroes. We assume that f is nonconstant and omits three points, which may be taken to be [1, 0], [0, 11,

and [1, -1]. Then ho

r

ht

and

eh0/ehi + 1 = e h2

where h0, h1 and h2 are entire holomorphic functions. Multiplying the second equation by e

-h 2

gives a linear relation

8

ORDERS OF GROWTH

9

,g(z) + eh(z) = 1 between exponentials of entire holomorphic functions. Taking derivatives yields

g'(z) eg(z) + h'(z)eh(z) = 0 ,

and these equations may be solved to obtain

In the simplest situation when g and h are polynomials, such a relation can hold only if g' = h', in which case f is constant. In general, it seems at least plausible that the left hand side should in some sense grow exponentially faster than the right hand side. Making this precise necessitates a careful discussion of growth, a discussion which should include growth of entire meromorphic as well as holomorphic functions, and which

should also relate the growth of a function to that of its derivatives. This will be the object of the first chapter, with the completion of Borel's argument appearing at the end of Section (d). Before going on to do this, we want to derive Jensen's theorem, which together with its variants, the First and Second Main Theorems, constitutes the main tool in the subject. Given an entire holomorphic function f(z), the number of solutions n(a, r) to the equation f(z) = a

in the disc IzI < r is given by the Cauchy formula 1

n(a, r) = 2 77

f' z dz

J f(z)-a

Izl=r

10

ENTIRE HOLOMORPHIC MAPPINGS

The difficulty in using this to estimate n(a, r) is that the integratd is complex while n(a, r) is real, which suggests adding to it the conjugate formula. To facilitate doing this, we write

f' z dz = a log [f(z) - a]

f(z) - a

= a loglf(z)-a12 since d log(f(z)-a) = 0 by the Cauchy-Riemann equations. It follows that

n(a,r) =

1

J

((9-d) loglf(z)-a12

4rr

Izl=r

f dc logIf(z)-a12

=

where

dc = 4 (a-a) = 4n ra ®d9- -147r t

0®dr.

Plugging this in gives n(a, r) =

I

dr

log If(z)- a12 d0

,

(41'-

Izl =r

an equation whose form suggests that we integrate with respect to Setting N(a, r) =

f

dr

r

I n(a, p) - n(a, 0)} dp + n(a, 0) log r

0

we obtain Jensen's theorem: 2r,-

(1. 1)

logIf(0)-aI + N(a,r) = 2n f logIf(rei0)-aI

d4

,

0

valid for an entire holomorphic function f(z) with f(0) / a. In the general case when f(z) is meromorphic and f(0) = a or oo, the same proof gives

f

ORDERS OF GROWTH

(1.2)

N(a,r) =

11

N(,r)+C

where 2n

f

2n

f

log I f(e e10) - al dO

.

0

The formula (1.1) gives an upper bound on N(a, r), and hence on

n(a,p) for p < r. It fails, however, to give a lower bound unless we know something about the mean proximity 2n

2 r log f(relo)- al dO 0

of f(z) - a to infinity on large circles. For example, if f(z) = ez and a = 0, then f(rei0) tends to - for -n/2 < 0 < n/2 and symmetrically to 0 for n/2 < 0 < 31r/2, and thus the mean proximity is zero. In some sense, understanding integrals of the general form

2rr

r

log I f(rei0)I dO

(a <

is what the theory is all about. What is not so clear is how the concept of negative curvature appears naturally in this mean proximity problem, and this will be taken up in the second chapter. (b) Order of growth of entire analytic sets We now turn to a precise analytical description of growth, first for analytic varieties and then for holomorphic mappings.

Suppose that V C Cn is an entire analytic set of pure dimension k. Denote by B[r] = Iz c Cn : I!zI! < r) the ball of radius r in Cn and V[r] = V fl B[r] that part of V in B[r]. (Exception: When n = 1, we let L(r) denote the disc of radius r in C.) The Euclidean volume vol (V[r])

12

ENTIRE HOLOMORPHIC MAPPINGS

of V[r] is finite, and we set n(V, r) = Ck vol (V[r])

v(V,r) = n V r 2k

r

where the constant Ck is determined so that v(Ck, r) = 1. (1.3) PROPOSITION: The function v(V, r) is increasing in r, and lim v(V, r) = v(V, 0) r -. O

is the multiplicity of V at the origin. PROOF. We consider the (1,1) forms = ddcllz112 =

2(

dzi n dzil

=

(dz,dz)

(1.4)

(z,z)(dz,dz)-(dz,z)(z,dz)

= ddc log11z112 =-1 277

112112

The first one cb is the standard Kahler form on Cn, and by Wirtinger's theorem (0.1) (omitting the factorials)

(' vol (V[r]) = J (bk v[r]

The second form rv is the pullback to Cn - 101 of the standard Kahler

form on the Pn-I of lines through the origin in Cn. We note that on the tangent space to the sphere llzll = r

2 dcllzll2 = dc

log1Iz112

r

2 ddcllzll2 = ddc log11z 112 r

(since d1IzIl2 = 0 there)

.

ORDERS OF GROWTH

13

Now use these remarks together with (1.4) and Sta#es' theorem"to catc..late, for r < R,

v(V,R)-v(V,r)=

R2

ok f ¢k rI f V[r]

V[R]

2k f Ik

dcIIzII2

A0k-12k r2k

R

= f v[R] do log lI

I2 A ok-1

-

f

f

d`IIzII2A¢k-1

do log 11 2112 A ak-i

aV [r]

Wk

fV[f,Rl where V[r,R] is the intersection of V with the annular ring r < IIzII < R. This proves that v(V,r) is non-decreasing in r. To evaluate lim v(V,r), r-O we write for r < R

v(V,r) = J

dc log IIzII2 n c0k-1 +

v(V,R)-

aV [r]

f

dc log,: z 2 n wk-1

aV [R]

For very small r, V is approximately its tangent cone Vo whose equa-

tions are the homogeneous terms of lowest degree in the defining equations of V. The corresponding projective variety P(VO) C pn-1 satisfies degree (P(VO)) = multo(V) =

r

J P(V0)

o k-1

by the Wirtinger theorem again. In each line 6 e Pn-1, do log I! zII2 is

the angular measure d(Be) where

reei

'

9

are polar coordinates in

Thus lim r-0

do log 11.112

A

Wk-1 = degree(VO) ,

aV [ r ]

which proves our contention.

Q.E.D.

ENTIRE HOLOMORPHIC MAPPINGS

14

The above proof gives the beautiful formula vol (V[r])

C

k

(1.5)

= v(V,r)

r2 k

where

v(V,r) = multo(V) + (volume in Pn-I of the set of lines Oz, 0 A z e V[r]I The function v(V,r) is the basic growth indicator of an entire analytic set. For reasons arising from Jensen's theorem, we shall also have occasion to use the counting function

rr I-(V,p)-v(V,0)) dp + v(V,O) log r

N(V,r) =

.

n

is zero-dimensional, v(V,r) = n(V,r) is the number of points in B[r] and an integration by parts gives, assuming In case V = ( aI , a2 ,

1

101/ V, N(V,r) = fo, n(V, p)

log

LP

P

IIa I < =r

r

I!aIi F<

IL

In general, from (1.4) and (1.5) we observe the properties:

If B[z, r] is the ball of radius r and center z, then the lower bound (1.6)

vol (V fl B[z, r]) > Ck r2 k

(z ( V)

on the volume of V is valid;

If V ci Cn is a hypersurface (k=n-1), 01 / V, and if for each line e e Pn-I passing through the origin we let n(V,6, r) be the number of points of V[r] fl 6, then (1.7)

v(V, r) =

fn(ver)d

is the average of n(V,1~, r) over all lines 6.

ORDERS OF GROWTH

15

(PROOF. n)"-1 = de is the invariant density on f,n-1)

If V C" is an algebraic set of degree d, then lim v(V, r) = d

r-.c

(PROOF: We may assume that 104 / V. The projection of V on the

hyperplane P"-1 at infinity is the set of lines Oz (z'V), and is an algebraic variety of degree d in Pn-1.) The converse of (1.8) is the important

THEOREM OF STOLL: V is algebraic if, and only if, v(V,r) = 0(1)

(1.9)

We shall prove that

"v(V, r) < d implies V algebraic" in case V is a hypersurface. As with most questions involving higher codimension, the general situation is at present considerably more difficult.

For simplicity we assume that the origin does not lie on V. The residual mapping

n: V-,Pn-1

i P(Z) = Oz is equidimensional and v(V, r) is the volume of n(V[rj). By (1.7), on the

average V meets each line 6 through the origin in at most d points. What we shall prove is that this is true for every line e. If this has been done, then an easy argument (cf. the proof of (1.27) for divisors in Cn)

Shows that V is algebraic of degree < d. For each point z A 0 in Cn we denote by Lz the line Oz, and by

n(V,z,r) = #(VflLzflB[!IzIlrj)

16

ENTIRE HOLOMORPHIC MAPPINGS

the number of points of intersection of V with the line Lz in the ball of radius IIzIlr. We consider the logarithmic means N(V, r) = J

c

v(V, P) dP

0

n(V, z, p) dp

N(V, z, r) =

P

JO

It is standard (Cousin II problem in Cn) that there is an entire holomorphic function f c (9(Cn) whose zero set is V and with f(O) = 1. By Jensen's formula (1.1),

N(V, z, r) = 21

f

log f(rel0z)dO

It follows that N(V, z, r) is a plurisubharmonic function of z e Cn. Such functions Vi(z) satisfy the sub-mean-value principle (z) <

'3[z,a 0(w) (w)

Q

J

nn

where b(w) is Euclidean measure. If furthermore Vf > 0, then V, (z) 2C

11z,1

n

Vf(w)4D(w)

fB[2!lzlll

Applying this to N(V, z, r) and taking IIzjj = 1 and using (1.7) gives the estimate (1.10)

N(V, z, r) < C(N(V, 2r))

valid for general analytic hypersurfaces V in Cn. Now it is clear that n(V, z, r) is 0(1) if, and only if, N(V, z, r) is 0(log r) and similarly for v(V, r) and N(V, r). When combined with (1.10), we are done.

ORDERS OF GROWTH

17

(c) Order functions for entire holomorphic mappings By definition, an entire holomorphic mapping is a holomorphic mapping

f:Cn*M into a compact, complex manifold M. If dimCM = n, we shall say that f

is equi-dimensional, and in this case f is non-degenerate, when the Jacobian determinant is not identically zero. The special case

is just an entire meromorphic function, frequently written as w = f(z). We shall measure growth of an entire holomorphic mapping relative to

a positive holomorphic line bundle L M. Let q be the Chern form of this line bundle relative to some fiber metric. If s is a non-vanishing holomorphic section of L over an open set on M, then by definition

q = ddc log

1

Isle

A different choice of metric Is1'2 gives a new Chern form rl' related to i7

by

rl' = q + ddco

(1.12)

where a= log(Is1'2/Is12) is a C°° function on M. In particular, the deRham cohomology class

[71] E HDR (M, R)

is well-defined. The potential theory on M which we shall require is contained in the converse to (1. 12): be a real, closed (1,1) form such that [&1 = [rlJ in H2DR(M,R). Then Vf is the Chern form of some metric in L -> M. (1.13) LEMMA. Let

1,

PROOF. Since M has a positive line bundle, it carries a Kahler metric and we let Ad and dc be the Laplace-Beltrami operators of d and

18

ENTIRE HOLOMORPHIC MAPPINGS

dc relative to this metric. By a standard Kahler identity Ad =

Adc

Thus the Green's operators also satisfy

Gd=Gdc Since 77-ilr is zero in cohomology,

77-0 = AdGd(r1,0) as follows from the Hodge decomposition. Similarly

17-0 = AdcGdc(71-VG) .

Combining these with d(>)-0) = 0 = dc(rl-c/i) gives the relation

rf-0 = ddc(d*dc

GdGdc(71-r/r)) = ddca

M by e-a, then the new Chern

If we multiply the given metric in L form is

.

Q.E.D.

i/r.

For an entire holomorphic mapping f : Cn - M, we define

tf(L, r) = J

f*r7 A CO n-1

B[ r]

Tf(L, r) = J tf(L, p) p o

Tf(L, r) is called the order function of f relative to the line bundle

L -. M. Interpreting wn-I as the standard measure de on the projective Pn-1 of lines e through the origin in Cn, space

tf(L, r) = f(f ( f*rl) de(e[r] =f1B[r). [r

ORDERS OF GROWTH

19

It follows that (1.14)

Tf(L, r) =

r

Tf(L, C, r) de

,J P P n-1

is the average of the order functions of f restricted to lines Using (1. 12) and Stokes' theorem together with do =

1 r

a 0dO-4nrae 111 0dr

a different choice of metrics gives a new order function

fr(f[P] Tf(L,e,r) = Tf(L,6,r)+

= Tf(L, e, r) +

ddcaf

f

r

r

LP

P

dcadp P

Jae[P] =T

f(L,

', r) +

(af=f*a)

4n P aP

a f dB

(faelpl f

= Tf(L,6, r) +

dp

P

2 af(0)

41- £e[r] = Tf(L, f, r) + 0(1) ,

because a is bounded on M. Since L - M has been assumed to be positive, there is some metric for which

77

is positive, and consequently

Tf(L, r) > C log r unless f is constant, a case which we exclude. It follows from these two observations that the growth of the order function is intrinsically defined up to a relatively insignificant 0(1) term. Suppose now that D E I LI is the divisor of a holomorphic section s E C(M, L), and f: Cn -. M is an entire holomorphic mapping whose

20

ENTIRE HOLOMORPHIC MAPPINGS

image does not lie in D. Then Df = f -I (D) is an effective divisor in Cn, which may be described as the zero locus of the section f*s of f*L -a Cn. The First Main Theorem (F.M.T.) gives the basic relation between the growth functions N f(D, r) = N(Df, r)

and

Tf(L, r)

.

To state it, we may multiply s by a non-zero constant to assume that the length Is12 < 1, and then define the proximity form mf(D, r) = 2

f

l s[rJ

(og

-

0

1 '2!

where 1z112)n-1

E = dc log 11z112 A (ddc log

is the unique, closed 2n-1 form on Cn - {O which is invariant under unitary transformations and has integral one over a sphere of any radius.

When n = 1 and z = rely, E

21, d0

and in general

F = 2 dOs A de where doe is angular measure in the line 6 e Pn-I. With the obvious notation,

mf(D, r) _

fmf(Der)dC. 6

The F.M.T. is the formula (1.15)

Nf(D,r) + mf(D,r) = Tf(L,r) + 0(1)

where the 0(1) is independent of D e ILl.

ORDERS OF GROWTH

21

PROOF. Since all terms appearing in (1.15) are averages over the lines Pn-1 of the corresponding 1-variable quantities, it will suffice to eE prove the result in the special case n = 1. If f*s has no zeroes, then by the usual computation Tf(L, r) =

f0

J rr .J

a(p)

r ,J

ddc log * z If s )

f

do log

J

1

f*sI2

d

dP P

(Stokes')

dp P

(formula for dc)

= mf(D, r) + C.

In the general case, we may write f*s =

on L(r+ E) where s' is a non-zero holomorphic section of f*L and h is a holomorphic function having the same zeroes as f* s there. The F.M.T. (1.15) is then the sum of the formula just proved, applied this time

to s' instead of f*s, and the Jensen theorem (1.1) for h.

Q.E.D.

COROLLARY (Nevalinna inequality): (1.16)

Nf(D, r) < Tf(L, r) + 0(1)

In general, the proximity form mf(D, r) is large when f(aB[r]) is on

the mean close to D, so that in some sense the left hand side of (1.15) measures the total attraction of f(z) to the divisor D in the ball of radius r. The F.M.T. says that this total attraction is essentially independent of the particular divisor. An important special case of an entire holomorphic mapping is when

M = Pm is complex projective space and L is the hyperplane line

22

ENTIRE HOLOMORPHIC MAPPINGS

bundle H. Denoting by Z = 60, , m] the homogeneous coordinates of a point in Pm and by A = [a0, , am] points in the dual projective space Pm* of hyperplanes in Pm, the divisors A ( CHI = Pm* are linear spaces given by an equation

=aoip+ +am3.m= 0. The length function of a section defining A may be taken to be 1< A, Z>I

u(A, Z) =

IIAII IIZI!

The subsequent Chern form -q = ddc log

= ddc log II Z112

1

u(A, Z)2

is the standard Kahler form (Fubini-Study metric) on projective space. The unitary group Um+1 acts on Pm and Pm*, and we denote by dA the unique invariant measure on Pm* with total volume one. Equiva-

lently, if v(A) is an integrable function on Pm*, then

Im*dr ,J

P

TE U m+1

where dT is the invariant measure on the unitary group and AO is a fixed point in Pm* (by invariance, it doesn't matter which AO we select). Perhaps the best interpretation of the order function of f : Cn -. pm is that furnished by Crofton's formula: (1.17)

Tf(H,r)=

f

Nf(A,r)dA A(Prn*

expressing the order function as the average of the counting functions Nf(A, r).

23

ORDERS OF GROWTH

PROOF. Using unitary invariance of the inner product and of the measure dT,

log u(TA0, Z)dT

log u(A, Z)dA =

log u(STA0, SZ)dT TE Um+l

log u(STA0, SZ)d(ST) TE Um+l

log u(A, SZ) dA AEPm*

for any fixed S E Um+1

It follows that

I

mf(A, r) dA

AEP m*

is a constant Co. Averaging (1.15) gives

f

Nf(A, r) dA = Tf(H, r) + C ,

AEP

and C = 0 ' by letting r-0.

Q.E.D.

The tension between the simultaneous relations Nf(A, r) < Tf(H, r) + 0(1)

fNf(A.r)dA = Tf(H, r)

suggests the deeper aspects of the theory. For example, it follows immediately that the image f(Cn) cannot omit an open set U of hyperplanes, since such a U would have positive measure. This is the Liouville theorem, derived here by purely integral methods.

24

ENTIRE HOLOMORPHIC MAPPINGS

As an application of (1.16) and (1.17), we shall prove: The mapping f is rational if, and only if, Tf(L, r) = O(log r)

.

Before giving the argument, some explanation may be in order. Since

the line bundle L M is assumed to be positive, M has uniquely the structure of a projective algebraic variety. More precisely, by a famous theorem of Kodaira, some high power Lk = L 0 ® L of the bundle L k-times

will have enough holomorphic sections to induce a projective embedding

j:M -pN with

j*(H)=Lk and where JHI is the linear system of hyperplanes in PN. Since Tf(Lk, r) = kTf(L, r) ,

we may as well assume that M = Pm and L = H is the standard line bundle. Then f is rational if, and only if, the functions f*(,. i/3 o) (i= 1, , m) are rational functions on Cn. Now an entire meromorphic function g(z) (z EC) is rational of degree d if, and only if, the equation g(z) = a

has at most d solutions for all points a e PI. Similarly, an entire meromorphic function g(z) (z a Cn) is rational of degree d if, and only if, all level sets g(z) = a are algebraic hypersurfaces of degree < d. In the question at hand, these level sets are inverse images f (A) of linear hyperplanes in Pm. Thus, if f is rational of degree d, then Nf(A, r) < d log r + C for all A e Pm*, and by Crofton's formula (1.17)

ORDERS OF GROWTH

25

Tf(H, r) < d log r + C.

Conversely, if this estimate on the order function is valid, then by (1.16)

Nf(A, r) < d log r + C

for all A and by Stoll's theorem (1.9) the mapping f is rational of degree < d. Q.E.D. (d) Classical indicators of orders of growth Working backwards from a historical viewpoint, we shall discuss some classical measurements of growth and give a few applications. For an entire meromorphic function f

: C - P1

and hyperplane (= point) line bundle, our order function reduces to the

Ahlfors-Shimizu characteristic function Tf(H, r) =

rr r o

f*??

(P)

d

=

rrAf(P) dP

where

r

A(p) _

f (z)l2 dz

A (p)

A 2dz

2 (1+If(z)l

)

is the spherical area of the image f(A(p)). This order function was used as an alternative to the Nevanlinna characteristic function Tf(r) = Nf(oo, r) +

2n

r

2rr

log+ I f(rei0)j d9

0

where

log+a = max (O, log a)

.

Thus, Tf(r) measures the total attraction of f on A(r) to the point at infinity on P1 = C U (-}. Using

26

ENTIRE HOLOMORPHIC MAPPINGS

log a = log+a - log+ a Jensen's formula (1.2) may be written as

f97, (1 .18)

Nf(a, r) + I 277

log+

dO = Tf(r) + 0(1)

1 If(rei0)

- al

expressing the symmetry of total attraction of

f

to all points a e P1. The

formula (1.18) was the F.M.T. in Nevanlinna's original work. This seemingly innocuous splitting of Jensen's formula to give (1.18) and subsequent interpretation provided the key to the theory. If we use the inequalities log+If(reiO)I < log (1+(f(reiO)I2)2 < log+if(re10)I + log 2

in the F.M.T. (1.15) 2n

Tf(H, r) = Nf(-, r) + 4n fio(1+If(rei0)1z)dO

for the Ahlfors-Shimizu characteristic function and in the definition of Tf(r), we find that T f(r) < T f(H, r) < T f(r) + log 2 ,

so that the two order functions are essentially equivalent. The Nevanlinna characteristic function has the substantial advantage that algebraic relations such as Tf+g(r) < Tf(r) + T9(r) + 0(1) (1.19)

Tfg(r)

Tf(r) + Tg(r)

T1/f(r) = Tf(r) + 0(1) are apparent.

The relation between Tf(r) and Tfg(r) is considerably more subtle and in some sense may be said to hold the key to the defect relations. Indeed, R. Nevanlinna's main technical tool was an estimate on

ORDERS OF GROWTH

f' rei6

27

dO

f(rei6)

in terms of log Tf(r). We shall return to these questions in Chapters 2 and 3.

For the moment we wish to discuss the relation between Tf(r) and the maximum modulus indicator

Mf(r) = max log If(z)I IzI
of an entire holomorphic function f(z). Certainly Mf(r) is the most ob-

vious, and also most classical, growth indicator. We shall show that Mf(r) and Tf(r) grow at essentially the same rate by using the PoissonJensen-Nevanlinna formula for meromorphic functions 277

log If(z)I = 2>r fo log If(reie)I P(rei0, z)dO (1.20) +

L.

g(z, at)

g(z, b,) -

Ialµ
bv I
where 2

2

r

r2 + p2 - 2r p cos (0 - 9)

is the Poisson kernel, 2

-wz g(z, w) = log rr(z-w)

>0 =

is a Green's function, and I aµ1, f bv1 are the respective zeroes and poles of f(z) in A(r). To prove (1.19) we write r

Nf(0, r) = J nf(0, P)

=

log Ia

µ

by an integration by parts (assuming f(0) A 0, ne), and similarly for Nf(oo, r). Then (1.20) results from the usual Jensen's theorem (1.2) by

28

ENTIRE HOLOMORPHIC MAPPINGS

making a standard linear fractional transformation in A(r) sending z

to 0. If case f(z) is entire holomorphic, we have (1.21)

Tf(r) < Mf(r) < (Q ± it Tf(ar)

(a> 1)

PROOF. The first inequality is obvious, and the second follows from (1.20) and I P(a rei0, z)I < a

1

valid for Izl < r and a > 1.

Thus, Tf(r) gives a measure of growth for meromorphic functions generalizing the maximum modulus of a holomorphic function.

As a non-obvious (at least to me) application of (1.21), we have the result: If f(z), g(z) are entire holomorphic functions such that f/g is also holomorphic, then (1.22)

Mf/g(r) < (a +

(a> 1)

1)1mf(ar) + Mg(ar)} + C

We shall use (1.22) to complete the proof of Picard's theorem begun in the introduction to this chapter. First we observe that if f(z) is a holomorphic function, then Cauchy's formula

f'(z) =

1

r Iwl=R

t w dw (w-z)2

(I zj
gives an estimate (1.23)

Mf'(r) < CMf(ar)

(a> 1) .

Next, we recall the Borel-Caratheodory lemma (1.24)

f(z)l < Ca

max

IRef(w)I

(IzI
Iwl <ar

which is proved as follows: Applying (1.20) to F = of gives for lzl < r

ORDERS OF GROWTH

29

Ref(w) Re (w +Z1 d(argw)

Ref(z) = 277 J IwI = r

,

/J

Taking harmonic conjugates we find Im f(z) = Zn fIWI

=r

Re f(w) Im (w + it d(arg w)

The Borel-Caratheodory lemma follows by adding these and making the

obvious estimate.

Now to the Picard theorem. As discussed in (a) of this chapter, an entire meromorphic function omitting three values on P1 leads to a linear relation eg + eh = 1

between exponentials of entire holomorphic functions g and h. Differentiating and solving gives h,

eg =

eh

=

h,-g, g

g'-h'

This holds unless h' = g', in which case the original map is constant. Setting

S(r) = Mg(r) + Mh(r) ,

if we combine (1.22)-(1.24) we obtain

1

S(r) < log S(ar) + O (log a

1

)+C.

To show that this leads to a contradiction, we use the following lemma of Borel which was devised for just this purpose: LEMMA. Let S(r) > 0 be continuous and monotonically increasing for

ro
30

ENTIRE HOLOMORPHIC MAPPINGS

S ( + S(r) < 2S(r) outside an exceptional set E of finite total measure.

PROOF. Assuming E is non-empty, set r1 c E, SI = S(r1)> 0 and Al = S . Inductively define rk, Sk = S(rk), Ak = Sl by k

1

rk+1 = min{rcE: rk+L\k
k

Sk = S1

(L 2k)

SI

and E is covered by the Ak.

Q.E.D.

To complete the proof of Picard's theorem, we determine a by

ar = r+

S(r)

or

a-1 Then the inequality on S(r) becomes S(r) < log S (r + RI ) + C' log S(r) + C

.

Comparing this with Borel's lemma yields the desired contradiction.

(e) Entire functions and varieties of finite order We shall conclude this first chapter by discussing analytic functions and varieties of finite order. Thus far, these are the most important entire transcendental quantities. Given a positive increasing function ¢(r), if the upper limit lim

log ¢(r) = A r

ORDERS OF GROWTH

31

is finite, then 0(r) is said to have finite order A. This is equivalent to either of the following assertions:

Given e > 0, ¢i(r) < rA+ E for all large r, while 0(rn) > rn- E for some sequence rn

r 0(t+ dt is convergent for µ > A and

The integral

1

divergent for

p. < A .

PROOF. If ¢(r) < rA+E for large r, then f

1

(dt

evidently converges

t

for p > A. If conversely this integral converges, then for large

1 > r - O(t)dt

Jr

r

sufficiently

>

prA

tp

If we set

f r 0(t)dt t 1

and use integration by parts r

Ir)d=

i

tp+ t

rA

1

q(t) dt tp+1

+c,

we find that:

fi(r) has finite order

r

if, and only if, this is true of 0(r) .

The following definitions are the obvious ones: An entire analytic set V C Cn has finite order A if this is true for v(V, r), or equivalently for the counting function N(V, r). An entire holomorphic mapping f : Cn M has finite order k if this is true for the order function Tf(L, r). (Remark: Since for any two positive line bundles L and L',

Tf(L, r) < Tf(L', r) 5 CTf(L, r)

32

ENTIRE HOLOMORPHIC MAPPINGS

for a suitable C > 0, the finite order condition is independent of the positive line bundle.) For entire meromorphic functions we may use either the AhlforsShimizu or Nevanlinna characteristic functions. For entire holomorphic functions we may additionally use the maximum modulus indicator. The

order is the same in all cases. For an entire holomorphic function f(z) of finite order A, the zero locus D = 1a1, a2, ... l has finite order < A. This follows from Jensen's theorem (1.1), and here is a proof based on the maximum principle: To begin with, assuming 1011 D

fr N(D, r) =

n(D, P)

O

n(D,r)

dP p

=

µ_l

loglapr

using an integration by parts. Consider the analytic function g(z) =

f(z)

n(D,r)

(z - a µ=1

in the larger disc Izi < ar. By the maximum principle, max

f(0)!

[Ia

If(z)l

IzI=ar

= 1g(0)1 <

M f(a r) <

iar ITIz - aAI

e

(a - 1)rn(D,r)

lZm

µ

Taking logarithms gives n(D,r)

log l f(0)l + I log µ=1

r I

< Mf(ar) + log (a 1 1)

.

iP

Conversely, given D = 1aµ) of finite order A in C, we may find an entire function f of order A whose zero set is D. For this one considers the Weierstrass primary factors

ORDERS OF GROWTH

33

E(u q) = Elementary considerations give the inequalities log 1E(u,q)l S log(1+(u)) < lul (1.25)

q+1

(q> 1)

< C lul q+1

log IE(u, q)l < C I1+ lul

Choosing a smallest

q

such that the series

converges, the

infinite product

II E (a , q) µ

µ

converges normally to an entire function f(z) having D as its zero locus By what was proved earlier, f(z) has order at least A. On the other hand, using integration by parts =

1

dnD tJ dt

frtP

rp

la lp

la/I
P

tp+

0

µ

Taking p = q+ 1 it follows that f(z) has finite order A < q+ 1. To obtain a more precise estimate, in case q > 1 using (1.25), we have Mf(r) < C

rq+1

p

=

(1.26)

do

= Crq+1

laµlq(lapl+r)

t

tq(t+r) q+(qr+(q+l)t)n(D,t)dt

Cr'

tq+1 (t+r)2

p

< C(q+1)rq+l

r

n D t dt

J0

tq+1 (t+r)

or

n tq+ dt +

Mf(r) < C rq

[fT

D+ dtl

34

ENTIRE HOLOMORPHIC MAPPINGS

A similar argument gives the same estimate when q = 0. Taking any µ > A so that n(D, t) < C"tu, we find from (1.26) that Mf(r) < C""rµ

.

Thus f(z) has finite order exactly equal to A, and we have proved onehalf of the Hadamard factorization theorem:

(1.27) Given D =I aA a divisor of finite order A in C, there exists an entire holomorphic function f(z) of order A having D as its zero locus. If g(z) is another such function, then I

g(z) = eP(z)f(z) where p(z) is a polynomial of degree < [A]. The uniqueness follows by writing g/f = eP for some entire function h, using (1.22) to deduce that max Re p(z) = O(rA+e) for every s > 0, zl
Using the one variable result and semi-continuity principle (1.10), we shall prove the same theorem (1.27) for divisors of finite order in Cn, this result being due to Lelong and Stoll. To begin with, if f(z) is an entire holomorphic function of z = (zi, ..., zn) ( Cn such that Mf(r) =

max log If(z)I IIZI1
has finite order A, then the zero locus D of f(z) has finite order < A by the one-variable result and averaging formula (1.7). If D has order exactly equal to A and g(z) also defines D, then g = ePf where p(z) is a polynomial of degree < [A] by the one variable result applied to the lines through the origin. It remains to prove existence. For convenience we assume that 101 / D. By (1.10) the intersections De = D fl a all have finite order < A. We shall use the Weierstrass

ORDERS OF GROWTH

35

products on each line a and patch together by uniqueness. Here is the proof in detail for n = 2, the general case being similar. We cover C2 - 10, 01 by the two open sets Ui = 1(z1, z2) : zi >< 01

(i=1, 2). The picture is something like 6= 1z2/zi =w}

z 1

D

The holomorphic mapping (1.28)

(v, w) - (v, vw)

maps C2 onto (C2 -1 z1= 0 }) U 1(0, 0)1, and has the inverse

(z1,z2) -. (z1,z2/zl) on U1. For fixed w, the mapping (1.28) gives a parametric representation of a line LK,. The boundary of the ball of radius r meets Lam, in the circle IvI =

r

(1+Iwl2)2

The intersection D Lw = v1(w) + v2(w)+

36

ENTIRE HOLOMORPHIC MAPPINGS

is a divisor in the v-plane of finite order < A. The points vi(w) are algebroid functions of w; i.e. they locally vary in a holomorphic manner with w, except for branching. Setting q = [Al + 1, the product Tj µ

E

v q

C°u(w)

)

converges normally in (v, w) to a holomorphic function having finite order < A on each line w = constant. In fact, if we assume for simplicity that N(D, r) < CrA ,

then by (1.10) C'(RAIwI;')

where C' is independent of w and N(D Lw, R) is the logarithmically averaged counting function for the number of vi(w) satisfying lvi(w)l < R. It follows from (1.26) that

max log If(v, w)l < C"(RkIwIx) Ivl
.

The function

fl(z1,z2) = f(zl,z2/z1) is defined in UI and satisfies max IzI l
// loglfl(zl,z2)I < C~(RX

Izz1) "

)

On each fixed line ;z2/z1 = constant, fl has finite order < A and extends holomorphically across the origin. Moreover, for fixed z2

log Ifl(z1, z2)I = 0(lZ1I ")

as Iz11

0. The picture is something like

37

ORDERS OF GROWTH

z2

2/zl = constant

z2 = constant

zi

Similarly, we may construct f2 on U2. On the intersection Ul fl U21

fl = h12f2 where h12 is non-vanishing. Now if h(z) is non-vanishing on the punc-

tured plane 0 < 'zl < -, then z-kh(z) has a global logarithm where

k=

J1dlogh.

1

2rr .

IZI =r

Similarly, h12 z11z22 has a global logarithm on U1 n U2. Multiplying fl by zl

kl

and f2 by z22, we may assume that

fl

=

ep12 f2

on U1 U U2. We now expand p12 in a Laurent series P12 =

I

m,n= --

.

mn z2

38

ENTIRE HOLOMORPHIC MAPPINGS

For fixed z2, Re P12 = 0(zi I-A)

as Iz1 - 0, and thus amn = 0 for m < - [A]. Similarly amn = 0 for m < - [A]. On any line z1 = 6 z2(6 A 0), f1 and f2 have at most a pole at the origin. Thus amn = 0 for m+n < 0. Similarly, amn = 0 for m > [A], n > [A], or m+ n > W. Thus p12 is a finite Laurent series of total degree at most [A] in both positive and negative powers of z1 and z2. Setting p1

mn a mnz1 z2 jr n<0

P12 = -PI + P2 where p1 is holomorphic in Ul and p2 is holomorphic in U2. Thus eP1f1 = eP2f2

=f

gives a holomorphic function in U1 fl U2 = C2 - 10, 01, and hence by Hartogs' theorem on all of C2, a function which has zero divisor D. Moreover, f has finite order < A by the growth properties of f1, f2, p1, and p2.

Q.E.D.

Bibliographical remarks for Chapter 1

The original proof of Stoll's theorem (1.9) appears in [38]. Interestingly enough, his argument uses Nevanlinna theory, but applied to the analytic variety V. A proof of (1.9) in local form was given by Bishop [5] with refine:aents by Shiffman [34]. The proof presented here for the case of hypersurfaces is one of the two found in [22]; the other one given there also uses value distribution theory on V and is similar in spirit to Stoll's original argument. Proposition (1.3) may be found in Lelong [30] and also in the paper by Stoll [38]. The multiplicity statement is due to Thie and Draper. A detailed discussion using blowing up, together with the original

ORDERS OF GROWTH

39

references, appears in Griffiths-King [22]. The general technique of reducing questions about an analytic function or analytic hypersurface in

Cn to the one variable case seems to date back to Kneser [27], and has been used by Stoll [39] and Kujala [26] among others. The potential-theoretic lemma 1.13 is due to Kodaira. The argument we have given assumes Hodge theory for compact manifolds and the standard KAhler identities, both of which are proved in Wells [43] or in the set of notes [14]. The order function Tf(L, r) is a special case of that used by Bott and Chern [6]. Our First Main Theorem is a special case of the formalism in Stoll [37]. A general discussion of order functions appears in [10]. Crofton-type formulae expressing growth indicators as averages of

counting functions are ubiquitous in the theory. For example, (1.7), (1.14), and (1.17) are all of this kind. The general philosophy comes from integral geometry, for which we refer to the book by Santalo [33]. A proof of Crofton's formula for holomorphic curves in projective space, together with an interpretation vis-a-vis the Wirtinger theorem, is given in [14]. The classical order functions are extensively discussed in Nevanlinna [32], to which we also refer for manifold uses of the Poisson-JensenNevanlinna formula. Theorem 1.27 is due to Hadamard; a very elegant proof using the Poisson-Jensen-Nevanlinna formula is in [32], where more detailed properties of Weierstrass primary factors are presented. The n-variable version of (1.27) is due to Lelong [30] and Stoll [39]; our argument appears to be similar to that of Stoll. The problem of growth properties of entire analytic sets of higher codimension is in an unclear state. On the one hand there is the Bezout counterexample of Cornalba-Shiffman [15], and on the other the existence theorems of Pan and Skoda [36]. Further discussion is given in [20], in the book [40] by Stoll, and in Carlson [7].

CHAPTER 2

THE APPEARANCE OF CURVATURE

(a) Heuristic reasoning In the first chapter we discussed various ways of measuring the growth of entire analytic sets and entire holomorphic mappings. The First Main Theorem provides the basic relation between growth of f : Cn -, M and the growth of divisors Df = f-1(D). For an entire meromorphic function f(z), a consequence of this result is that the number nf(a, r) of solutions to the equation f(z) = a in the disc IzI < r can grow no faster than the

average number of solutions to this equation. In general, because of the somewhat intractible character of the proximity form mf(D, r), the F.M.T. has thus far been only used to estimate the counting function Nf(D, r) from above by the order function Tf(L, r) (D( ILI ), and not to discuss the deeper Picard-type question of how large a divisor f(Cn) can miss. An exception was E. Borel's proof of the little Picard theorem given in Chapter 1. For this argument we not only needed a good formalism for measuring the growth of entire meromorphic functions, but also required information on the growth of the derivative of an entire holomorphic function. In order to discuss the more profound question of obtaining a lower bound on

nf(a, r) + nf(b, r) + nf(c, r)

(a, b, c being three distinct points on the sphere), one needs to bound the growth of f'(z) for an entire meromorphic function f(z). A related problem is to estimate the number of ramification points of f : C P1 in terms of the growth of f. Questions of this sort lead naturally into curvature considerations, as we hope to now make evident. 40

THE APPEARANCE OF CURVATURE

41

In a somewhat general fashion, we consider a holomorphic mapping f

:C-S

into a compact Riemann surface on which one is given a conformal metric ds2. In terms of a local holomorphic coordinate w on S ds2 = h(w)Idwl2

,

and the Gaussian curvature is

a2

K(w) _ _-4 h(w)

log h(w)

9w ,9;7v

Writing ds2 = f*(ds2) = k(z)jdz12

where

k(z) = lw,(z)I2 h(w(z))

,

we see that dsf is a pseudo-metric on C having isolated zeroes on the ramification divisor Rf of f and Gaussian curvature K(z) = K(f(z)). Denoting by dAf = 2 k dz A d2- the pullback to C of the area element on S, ddc log k = -KdAf

.

Applying the proof of the F.M.T. (1.15) to the function k(z) then gives

fr( (2.1)

N(Rf, r) =

f

A(p)

KdAf pP + 21

r2

log k(reie) d9 + C .

o

This is the Second Main Theorem (S.M.T.) in the present context, and formulae of this general type will be systematically used to give inequalities going in the opposite direction to (1.16). To explain how (2.1) may be used to estimate N(Rf, r), we assume the upper bound

42

ENTIRE HOLOMORPHIC MAPPINGS

on the Gaussian curvature and denote by

fr( Tf(r) =

the Ahifors-Shimizu order function for the metric ds2. The growth of Tf(r) is determined up to the multiplicative constant fdA, for the same S

reason that the growth of Tf(L, r) was seen to be intrinsic in Chapter 1. We shall prove that, given E > 0, (2.2)

N(Rf, r) < (K0 + E ) Tf(r)

where the notation //E means that "the stated inequality holds outside an open

set I, C R+ with

at <

The derivation of (2.2) from (2.1) proceeds by two steps which are ubiquitous in Nevanlinna theory. The first is the concavity of the logarithm: (2.3)

f 2-log u(9) ae < log 2n r 2-

u(9) ae

27-7

o

for a function u(O) > 0 such that log u is integrable. This is proved from the finite form: 1

(u1 ... un)° < n (uI + ... + un)

F (log ul +

+ log un) < log (n (ul

by passing to the limit. Applying (2.3),

+

+

un))

THE APPEARANCE OF CURVATURE

f277 2n

log k(reie)dO < log

43

2'i

(i_f

k(rei0)d9 = log

)

[r2 (d log r)2]J

so that from the S.M.T. (2.1) d2Tf(rr)

N(Rf, r) <_ KOTf(r) + log

I(2.4)

rI2 (d log

)2

I

If Tf(r) were a nice growth function, such as logar rR erY, then d2 Tf(r) is evidently O(Tf(r)), and this proves (2.2) (without exr2 d log r2 ceptional intervals). In general, we shall use the case n = 1 and S(r) _ Tf(r) from the following calculus lemma:

(2.5) LEMMA. Let S(r) be a positive function such that

r2n-1 dS r

dr

2

is positive and increasing and d SZ is continuous. Then, given positive dr

Eand6, r2n-1 d (r2n_1 dr

<

r4n-2(S(r))1drdr)=

Es

PROOF. We shall use twice the assertion: Let f(r), g(r), a(r) be positive increasing functions of r such that g'(r) is continuous, f'(r) W is piecewise continuous, and f Ac) < -. Then (2.6)

f'(r) < g'(r)a(f(r))

outside an open set I C R+ such that

fdg
(Proof. On the set I where (2.6) fails to hold, g '(r) < a consequently

fdg<J(' f'rdr
xf'rdrr a(f(r))

J

dr

a(r)

and

44

ENTIRE HOLOMORPHIC MAPPINGS

Applying (2.6) with f(r) = S(r), g(r) = rµ , a(r) = r1' and 1.c,v > 1 gives ru-1 (S(r))"

S'(r)

//

.

Keeping the same g and a and setting f = r2n-1S'(r) gives

dd (r21 dS(r) <

r µ-1 r(2 n-1 v(S'(r))v

//

Combining these and readjusting constants yields

r2n-1 d (r211_1 dS r) < r4n-2+e(S(r))1+S

dr J

dr as was to be proved.

Proof of (2.2). By the calculus lemma, log

r2 log (r) r)2 ] d2Tf C1 (d

On the other hand, for large

<_ E

-

log r+ (1+E) log Tf(r)

E

Tf(r) > C log r (C> 0), and the result follows from this together with (2.4). r

In case S = P1 is the Riemann sphere with coordinate w, the standard metric

ds2=V1 277

Idw12

(1+Iw12)2

has constant Gaussian curvature +2, and so

(2.7)

lim r) < 2 . r -m N(Rf, Tf(r) _

The Weierstrass p-function associated to a lattice A = I mwl + nw2 : m,n f Z I in the plane is a doubly-periodic entire meromorphic function. In each

fundamental parallelogram, p(z) is a 2-to-1 mapping onto P1 and has 4 points of ramification. Thus

THE APPEARANCE OF CURVATURE

45

N(Rf, r) = 2Tf(r) + o(Tf(r))

so that (2.7) is sharp. We next wish to illustrate how negative curvature may be used. Suppose that S is a compact Riemann surface of genus g(S) > 1, which is the same as saying that the canonical line bundle KS is positive. Then, by linear potential-theoretic methods (cf. Proposition 2.17 below) we may find a metric on S whose Gaussian curvature satisfies K < -1. The S.M.T. (2.1) then gives

Tf(r) = J

J dAf

pP

<J

(P)

J - KdAf pP < C log T{(r)

//

(P)

under the assumption that f is non-constant. Since Tf(r) -+ o, this is clearly impossible, and we have proved the following result also due to Picard: A holomorphic mapping f : C

S with g(S) > 2 is necessarily constant.

Of course this theorem is trivial if one assumes the uniformization theorem, which is equivalent to finding a metric of constant Gaussian curvature K = -1. This latter is, however, a deep theorem which basically involves solving a non-linear equation of the type

Au=eu. Moreover, the method will not work in several variables. Finding a metric with K < -1 boils down to solving a linear equation of the type (cf. Lemma 1.13)

Au = v

on S, and this will generalize to compact Kahler manifolds. At this stage, however, we mainly wish to point out that curvature appears quite naturally in the theory, and that the presence of a negatively curved metric has very strong function-theoretic implications.

46

ENTIRE HOLOMORPHIC MAPPINGS

(b) Volume forms

On a complex manifold M of dimension n, a volume form tY is a smooth positive (n, n) form. In a local holomorphic coordinate system

z = (z1, ... zn),

T(z) = h(z)(D(z)

where

(D(z) = H j=1

2

(dzjAdzj)

is the Euclidean volume form and h(z) is a positive Co function. A pseudo-volume form is a smooth non-negative (n,n) form T having a similar local expression, but whose coefficient function is h(z) = Ig(z)I2 k(z)

with g(z) A 0 being a holomorphic function and k(z) being positive. Pseudo-volume forms arise by pulling back volume forms under equidimensional, non-degenerate holomorphic mappings f. The ramification

divisor Rf of such a mapping is given locally by g(z) = 0. The Ricci form of a pseudo-volume form 'P is the smooth (1,1) form given locally by

Ric T = ddc log k(z) . Since ddc log I j(z)I2 = 0

for a non-zero holomorphic function j(z), it follows that Ric T is globally defined on M and that Ric (f*V) = f*(Ric 'Y)

for an equi-dimensional, non-degenerate holomorphic mapping f. Use of the formula Ric (euW) = Ric W + ddcu (2.8) affords easy manipulation of Ricci forms. In particular,

THE APPEARANCE OF CURVATURE

(2.9)

Ric (CT) = T

(C > 0 a constant)

47

,

so that we may always adjust constants. Volume forms which satisfy the curvature estimates 5

(2.10)

Ric'Y>0 (Ric'Y)n>IF

play a crucial role in the theory. These inequalities are evidently invariant under non-degenerate holomorphic mappings. Here are some examples. (i) When n = 1 so that M is a Riemann surface, there is a natural

1-1 correspondence between conformal metrics and volume forms given by ds2 =

h(z)IdzI2 " 21 h(z)dz A dz = 'Y

For such a 'Y, Ric tY = (-K)'Y

where K is the Gaussian curvature of ds2. Hence, in this case (2.10) says that ds2 has Gaussian curvature K < -1. In particular, on the disc L(R) = 1z cC: Izi < R}, the Poincare metric (2.11)

R2 dz A dz

1(R) _ IT

(R_Iz12)2

satisfies (2.12)

Ric ®(R) = ®(R)

We write A = A(1) and ® = 0(1). The punctured disc A* 9 cC : 0 < IC I < 11 has A as universal covering, and the Poincare metric there induces _ (2.13)

dC A dl'

®* _ 17

ICI 2(log ICI 2

on A* such that (2.12) is satisfied for W.

2

)

48

ENTIRE HOLOMORPHIC MAPPINGS

(ii) On the polycylinder

On(R)= I(zl,...,zn)E Cn Izjl < Rjl :

R = (RI,..., Rn) with Rj > 0

,

the product of Poincare metrics induces a volume form ®n(R) such that Ric ©n(R) > 0 (2.14) I Ric ©n(R)Vn = On(R).

As above, On = ©n(1, , 1) on the unit polycylinder induces a volume form satisfying (2.14) on the punctured polycylinders A*kn =

(A*)k X An-k

(iii) On a general complex manifold M, a volume T is the same as a metric for the dual KM of the canonical line bundle, and Ric T is the Chern form of KM for this metric. When M is compact, we may find a volume form satisfying the curvature conditions (2.10) exactly when KM

is positive in the sense of Kodaira. (Proof: If u is a positive (1.1) form representing cl(KM) in H2(M, R), then by Lemma 1.13 we may find a metric in KM whose Chern form is u. Letting 'P' be the corresponding volume form, Ric'Y' _ u and so (Ric'Y')n > C'! " (C> 0) by the compactness of M. Now adjust constants to obtain (2.10).) This suggests that on a general compact, complex manifold M, we should look for such "negatively curved" volume forms on Zariski open sets

M-D

where D is an effective divisor such that KM 0 [D] is positive. Some restrictions on the singularities of D are necessary, and so we assume that D has simple normal crossings in the sense that D = DI

++DN

THE APPEARANCE OF CURVATURE

49

where the Dj are smooth divisors meeting transversely. In a neighborhood 11 of a point p e M through which exactly k of the divisors pass, we may choose local holomorphic coordinates (z1, , zn) such that D fl `U is given by the equation zl zk = 0 . Thus (M-D) fl `U = Ak n is a punctured polycylinder, and this suggests the following global version of the Poincare volume form on 4k n (cf. (2.13)): Choose a volume form TM on M and metrics in the line bundles

[Dj] such that the inequality of Chern forms (2.15)

cl(KM) + c1([D]) > 0

is satisfied (recall that [D]

N

j=1

[D ] and

j

cl([D]) = cl([D1]) + - + cl([DN])) . Let sj a O(M, [Dj]) be a holomorphic section which defines Dj , and set (2.16)

TM(D) = N

TM

sj12(log(a1sj12))2 II j=1

(2.17) PROPOSITION. Given c > 0, there exists 0 > 0 such that for 0 < a < 13 the volume form TM(D) is C°° on M - D and satisfies

Ric'I'M(D)_ (1-e)1cl(KM)+ cl([D])i+0 , V'> 0 and ifin > WM(D)

In particular, the curvature conditions (2.10) are satisfied. REMARK. The reason we have replaced (2.10) by the excess relation above is to remove the ambiguity caused by the possibility of adjusting constants according to (2.9).

50

ENTIRE HOLOMORPHIC MAPPINGS

PROOF. Set m = cI(KM) + cl([D]) > 0. Then Ric GYM = cl(KM)

ddc log

1

aIsjj2

= c1([Di 1)

N

I

1([Di 1) = cl([D])

.

j=1

Using these relations in (2.8) gives ddc log (log aIs)I2)

Ric'YM(D)

2

j=1

= (1- 2 e)c, + & where c / i = 2t". co -

ddc log (logaIsj 12)z

Now then

ddclog(logalsi 2)2 =

-2ddclog aIsI2

4d log

A do log al

+

(logaIs)12)

1

(log aI sjl2)2

The first term on the right is

log al sj12

which tends to zero as a for a sufficiently small N

0 and may thus be absorbed in F(v. Thus,

d log (al,s]I2) A d log (als]l2) 2

j=1

(log alsj(2)

since each term in the sum is non-negative.

THE APPEARANCE OF CURVATURE

51

We now localize around a point p e D. We may choose a coordinate neighborhood 11

of p in which D has the equation z1

zk = 0. In

fact, we may assume that Dj fl `u is given by zj = 0 for j= 1, , k, and that Dk+l DN do not meet 11. Then

,,

a log (alsjl2) n a log (alsjl2)

k

(2.18) 77

(logalsjI2)2

j=1

Now alsjl2 = ayj(z)lzjl2 where yj is a positive C°° function. Consequently

dzj n dz I+ pj 2

a log (alsjl2) A a log (alsjl2)

(2.19)

Izjl

where aYj n ayj + ayj n dzj (2.20)

P

=

Y

+

dzj n 7yj

zj yj

zj Yj

has the property that lzjl2pj is C°° and vanishes on Dj . Since dzj n dzj ) for some C > 0, it follows from (2.18)-(2.20)

> C j

(D(z) + A(z) Izj12

((z))n > c'

(log a1sjl2)2

i

where I(z) is the Euclidean volume form and A(z) is C°° and vanishing on D. By (2.16) on > C"'YM(D)

holds in a perhaps smaller neighborhood `U' of p. By compactness, we

(C'" > 0) on all of M, and then we may make C"" = 1 by adjusting constants. Q.E.D. may assume that i/i° > C""'YM(D)

As a special case, we consider complex projective space Pn with homogeneous coordinates Z = [30, , 3n] and standard Kahler form

52

ENTIRE HOLOMORPHIC MAPPINGS

w = ddc log 11Z112

Then (,)n = T is a volume form and Ric W _ - (n+1)C

/

(Proof: In affine coordinates Z = [1,w1, ,wn], Cu =

2 djlog(1+11w112)

- v1 (dw, w) (w, dw)

dw, dw 217

2rr

1 + IIwI12

(1+11w1!2)2

= a+0 where a is the first term and /3 the second. Since /3 A (3 = 0, wn

= an+ na n-1

v /3

n(n-1)!

n!

11w112

1 + 11w112

_(w)(1+1,wl!2)n

n ! 'I(w) llw112)n+l

(1+

It follows that

Ric'P = -(n+1)ddc log (1+11w112) = -(n+1)C ./

We assume that the divisors Dj are linear hyperplanes given by equations

= 0 .

j,

The condition that D = DI + + DN have simple normal crossings is that these hyperplanes should be in general position. Since ddc log

11A 112112112

I

(2)

THE APPEARANCE OF CURVATURE

53

the Chern form cl([D]) = N w and the positivity condition (2.15) is

N > n+1 . In summary: On Pn - IN > n+2 hyperplanes in general position#, there is a volume form which satisfies (2.10). In particular, this is true on Pl - IN > 3 distinct pointsl. As another example, if M is a compact Riemann surface of genus g then (2.16) gives a metric ds2 on with distinct marked points

(N+2g-2> 0)

M-

having Gaussian curvature K < -1. We may use this metric W on Pt - Ia,b,cl, together with the S.M.T. (2.1), to give another proof of the Picard theorem as follows: Using Proposition (2.17), write Ric 'P = (1 -e)w + vi

where

0 >1Y

and w is the standard metric on Pl. If a non-constant holomorphic mappint; f : C -. Pl - I a, b, cl exists, then we set S(r) =

fr(f

o(p)

and apply (2.1) just as at the end of (a) in Chapter 2 to obtain an inequality

(1-e)Tf(r)+ S(r) < log (S(r))

// ,

which is a contradiction. (c) The Ahlfors lemma Among the most subtle and far-reaching tools in the study of holomorphic mappings is Ahlfors' generalization of the Schwarz lemma:

54

ENTIRE HOLOMORPHIC MAPPINGS

(2.21) AHLFORS LEMMA. Let T be a pseudo-volume form on the polycylinder An (R) which satisfies the curvature estimate (2.10). Then 'P < ©n(R).

PROOF. For r < R we write 'f'(z) = ur(z)©n(r)

(z c An(r))

It will suffice to prove that ur < 1 for r < R, because lim ur(z) = uR(z) r-+R

for fixed z c An(R). For r < R, ur(z) tends to zero as z - ddn(r) since ©n(r)(z) goes to infinity there. Thus ur(z) has an interior maximum at some point zo, and by (2.8) and the maximum principle 0

v217

dc3 log ur(zo)

=

Ric '(zo) - Ric ©(r)(zo) .

Using (2.10) and (2.14) and taking determinants in this inequality gives

'P(zo) < Ric'(zo)n < Ric Q(r)(zo)n = ©(r)(zo) which exactly says that ur(zo) < 1.

Q.E.D.

For a holomorphic mapping

f:A .A we may apply this lemma to the pseudo-metric f*(©) and obtain Pick's invariant form of the Schwarz lemma If'(z)I 11 = 1

I - If(z)12

_

In case f(0) = 0 this reduces to f'(0)I < which is the usual Schwarz lemma.

1. ,

1 Iz,2

THE APPEARANCE OF CURVATURE

55

An application of the Ahlfors lemma is the (2.22) SCHOTTKY-LANDAU THEOREM. Let f : An(R) - M be a non-

degenerate, equidimensional holomorphic mapping into a complex manifold M having a volume form tY satisfying (2.10). Writing f"1(z) = Jf(z)(D(z)

where c is the Euclidean volume, i

R=(R1,... Re).

R1 ...Rn < CJf(0)-z

PROOF. By (2.11) and the Ahlfors lemma at z = 0

(1 °

2

1

R2 ... R2 1

n

Applying Proposition 2.7 gives the

(2.23) COROLLARY. Let M be a compact, complex manifold and D a divisor with simple normal crossings such that

cl(KM)+ cl([D)) > 0 . Then any non-degenerate, equidimensional holomorphic mapping

fAn(R)- M-D with non-zero Jacobian at z = 0 satisfies R1 ... Rn < C

where the constant C depends only on the magnitude of the Jacobian of f at z = 0. In particular, an entire holomorphic mapping

f:Cn -,M-D is necessarily degenerate.

56

ENTIRE HOLOMORPHIC MAPPINGS

Taking M = pn and D to be a set of n+2 hyperplanes Aj in

general position, we find the theorem of A. Bloch, reproved independently by Mark Green, that an entire holomorphic mapping

f:

Cn_ Pn-1AI+...+An+2#

is degenerate. For n = 1 this is the Picard theorem. The SchottkyLandau theorem gives both results in finite form. It is perhaps of interest to compare the Ahlfors lemma with the methods based on the integral formula (2.1). Both imply that if ds2 = h(z)Idz12 is

a metric on the disc A(r) whose Gaussian curvature satisfies K < -1, then r < =. From the Ahlfors lemma, one finds the bound

r < Ch(0)-

(2.24)

J. .

On the other hand, setting fr

S(r) =

r h(z) 2r- (dz A dz)pP A(p)

S(r) < log

d25(r) (d log r)2 Ir2 1

Then the calculus lemma (2.5) implies that r +-. Being more careful in the proof of this lemma, one may also prove (2.24) by this method. The advantage of integral formulae is that this technique also works for singular volume forms obtained by pulling back (2.16) under a non-degenerate, equidimensional holomorphic mapping

f:Cn - M, and leads to a lower bound on the size of Df = f -I(D). Actually, the Ahlfors lemma implies, at least in principle, such a lower bound on the size of Df as follows: First, for a point z r Cn we let

THE APPEARANCE OF CURVATURE

57

r centered at z. If V C Cn is a k-dimensional analytic set, then (1.6) gives the estimate B[z, r] be the ball of radius

voi (V n B[z, r]) > cr2 k

(2.25)

for any point z c V. Next, for an entire holomorphic mapping f : Cn -. M

as above, we set

f*TM(D)

= Jf4)

where 'PM(D) is the singular volume form (2.16) and $ is the Euclidean volume form on Cn. If An(z, R) is the polycylinder Iwj - zjj < Rj , then for any such r\n(z, R) contained entirely in Cn - Df , (2.23) gives RI ... Rn < C Jf(z)- 2

.

Thus, for any point z e Cn - Df there is a bound on what size and shape polycylinder centered at z may be put in Cn - Df . This bound depends only on the size of Jf(z), and so if Jf(z) bounded from below off a relatively small subset of Cn, we obtain a lower bound on the size of Df using (2.25). By the remarks in Section (a) of this chapter, we may expect that Jf(z) is small only on a subset of Cn which is small relative to the growth of f, thus putting a lower bound on the size of D f . It seems difficult to make this heuristic reasoning precise, and so we shall rely on the more analytical integral formulae. Before going to the general Second Main Theorem, we shall combine the Ahlfors lemma with the integral formula method to prove the BIG PICARD THEOREM. A holomorphic mapping

f : A* . PI - I

cl

extends across the origin to a holomorphic mapping into PI. PROOF. We let A *(r) be the punctured disc 10 < j z 1 < r} and A(r) the annulus 1 < zj < 1 . By change of scale, we may assume that f is r =

defined on A*(1+=) for some e > 0.

ENTIRE HOLOMORPHIC MAPPINGS

58

Let u(z) be a non-negative C°° function on A*(1+e) having the local form

u(z) = z-zol uz0 > 0

uzo(z)

µ(zo) > 0

and

around any point zo. We let D be the divisor

Yu(zo) - zo and n(D, r) the counting function gives (2.26)

N(D,r)+

21 77

I

µ(z0). Then the proof of (1.15)

z0(A(r)

ft(

J

log u- dB=

J

IZI=T where N(D, r) =

ff

n(D, p)

ddc log u

LP

+ O(log r)

A(p)

d

.

Let A c PI be a point and Af = f -1(A) the inverse image. Then the estimate n(Af, r)

-=

N(Af, r) = O(log r)

0(1)

implies that Af is finite; and if this is true for any A F P1 the CasoratiWeierstrass theorem implie:a that f extends as desired. To derive this estimate on N(Af, r), we use homogeneous coordinates and write f(z) = Z(z) _

o(z),51(z)]

A = Lao, all U(Z)

11Z(Z)11z 11Aflz

>1 iIz =

.

Then ddc log u(z) = f*co= mf is the pull-back of the standard Kahler form on P 1, and (2.26) gives

THE APPEARANCE OF CURVATURE

N(Af,r) < f

(2.27)

( f fl pP f

\A(p)

+ 0(log r)

59

.

J

Now then by inspection

co < C'!' 1(la,b,cl) P

for the singular volume form (2.16) in this case. Moreover,

T 1(la,b,cl) < v'-1 =

P

2n

dz A dz iz12(log (1+E)%z12)2

by the Ahlfors lemma. Since this Poincare volume form on A*(1+E) has finite integral over A*(1), it follows that

J ()f

J O,f = 0(1) > J A*(1)

dp

= 0(log r) ,

A(p)

1

which when combined with (2.27) proves our result.

REMARK. The same proof works for an equi-dimensional, non-degenerate holomorphic map

f : Ak,n

M-D

where M and D satisfy the hypotheses in (2.23), and gives a meromorphic extension of f to n. (d) The Second Main Theorem

On Cn we consider a singular volume form A(z) = A(z)c(z)

Where '(z) is the Euclidean volume and where the coefficient function has the local form (2.28)

A(z) _

Ig(z)j2h(z) k

lf)(z)I2 log(al(z)(f)(z)l 2)2

j=1

60

ENTIRE HOLOMORPHIC MAPPINGS

with g(z), fe(z) being non-identically zero holomorphic functions and h(z), a)(z) being positive C°` functions. We also assume that A(z) is C around the origin. Of course, we have in mind A's of the type f*('Y) where f : Cn -. M is a non-degenerate, equi-dimensional holomorphic map-

ping and T is given by (2.16). The ramification divisor R and singular divisor D are respectively defined by

g(z)=0 k

f(z) = 0

f(z) = 11 f)(z)

where

j=1

The general Second Main Theorem (S.M.T.) is (2.29)

J0

r

J

Ric AA

,,n-1

PP + N(R, r) = N(D, r) + f log ,1 1 + C

B[p]

dB[r]

PROOF. By averaging over the lines through the origin as in the proof of (1.5), it will suffice to prove the case n = 1. Were it not for the 2 (log aj ifj12) terms, (2.29) would be the same as (1.5), with the trivial modification that Jensen's theorem for meromorphic functions must be utilized. What must therefore be proved is

J

r

J ddc log (log ajf!2)

PP

log (log aIf12)2de + C ,

217

0

I=I=P

(P)

where f is holomorphic and a is positive and C°°. Setting 0 = alf'2, the equations

a log(log f3) =

as log (log o3)

a log a log a

as log 13

log

+

df f log f3

a log [3 A a log f3 (log f3)2

.

THE APPEARANCE OF CURVATURE

61

show first that ddc log (log a f'2)2 = Zn ad log (log f3) is integrable. Next, Stokes' theorem may be applied to d(d log (log /3)) = ad log (log 13)

by cutting out =-discs around the zeroes of f, then applying Stokes' theorem to the complement of these discs, and finally observing that dz

lim

eyo

IZ,_,

log'izl

r J

= 0 = lim -,o

dz

Ii log!zI

IZI

Using this and an elementary interchange of limits argument gives r o

ddc log(logalfl2)2 Pp =

J

rr f

,f

dc log(log aIfl2)2

fr(J

(P)

0

dA (p)

1) p

1

n p

=

1

dp

log(log alf2)2d0

dP

aA(p)

j log(log a!f,2)2dO + C dA(p)

Q.E.D.

To apply the S.M,T., we assume that A satisfies the curvature conditions where

r]

Ric A = 77+0

is a closed, positive C°° (1,1) form and

A

.

The motivation for writing Ric A in this manner is given by Proposition 2.17. We shall prove the following Basic Estimate: (2.30)

7)n(on-1

pP + N(R, r)

N(D, r)

//.

62

ENTIRE HOLOMORPHIC MAPPINGS

PROOF. Following the same calculation as in the proof of (1.2), p2n-2

J tbA6) n-1

= P2n-2 5 d(OAdc logjjZ112 A (ddc

B[p]

B[p]

p2n-2

J V1Adc log1'zj, 2A(ddc

log!izji2)n-2

9B[p] OAdclizl2A(ddcllzii2)n-2

J dB[p] °

0Aon-1

1

B[p]

where 0 is the standard Kahler form on Cn. Writing ij dzi A dzj)

2r

,

the non-negative Hermitian matrix (0i)) satisfies 1

1

F (Trace (O i])) > (Det (w ij))" > a"

j

the second step being the assumption on > A. Combining these two steps gives G A (on-I

dp

r

J

P

A¢n-1

.B[p] =

J

dP

zn-1 P

Trace (0 i))4) > 2n r

B[p]

>f, =f r

1

P 1

xn

dp

2n-I

BIP] n X

[5{3B]

tzn-1 dt

21

THE APPEARANCE OF CURVATURE

63

Using Ric A = rj+yri and setting S(r) =

Jan

dp P

B p]

fr{f

ff

d

Ric AAwn-1

(2.31)

77 Aw

L P

B[p]

B[p]

On the other hand, concavity of the logarithm gives

J log A E< n log

i

xi s

dB[r]

aB[r]

[r2n-i d (r2n-i dS r )1 1 = n log r4n-2 dr I J dr L //'.

< n e log r + n(1+ =) log S(r)

by the calculus lemma 2.5. Since

S(r) > C log r

(C> 0)

for large r, we find that (2.32)

S(r) -

f

log A s < 0

F

aB[r]

Combining (2.31) and (2.32) with the S.M.T. (2.29) gives the basic estimate (2.30).

Q.E.D.

Bibliographical remarks

The importance of negative Gaussian curvature in value distribution theory seems to have been first recognized by F. Nevanlinna (cf. the references cited in [32]). Extensive use of differential-geometric methods

64

ENTIRE HOLOMORPHIC MAPPINGS

was made by Ahlfors in [7] and [2]. It was he who first realized that the curvature condition K -1 instead of K = -1 was all that was necessary

The ubiquitous Ahlfors lemma appears in [2] for n - 1, with the extension to volume forms being due to Chern [12] and Kobayashi [28]. In a certain sense, this lemma founded the whole subject of hyperbolic complex analys15: cf. Kobayashi [29]. It also has extensive applications to algebraic geometry especially to variation of Hodge structure; cf. [23] and the references cited there. The proposition 2.16 on volume forms appears in Carlson-Griffiths [9]. The existence of such negatively curved volume forms was suggested by algebro-geometric considerations given in the thesis of Carlson [8]. The proof of the Big Picard Theorem using Nevanlinna was proved in a more general setting in Griffiths-King [22]. The Second Main Theorem is given in [9]. A general discussion of the role of negative curvature in the study of holomorphic mappings appears in [21]. The use of these methods in the geometrically somewhat more subtle non-equidimensional theory of entire holomorphic curves in Pn is given in Cowen-Griffiths [16], where references to this subject together with some additional heuristic discussion of negative curvature also appears. Regarding Picard-type theorems for holomorphic curves in general algebraic varieties, the recent compendium of examples by Green [19] is quite interesting. It was he who, together with Fujimoto, first proved the sharp Picard-type theorems for non-degenerate maps from C:m to Pn -- 1hyperplanesf -- cf. the references given in [19].

CHAPTER 3

THE DEFECT RELATIONS (a) Proof of the defect relations

Let L M be a positive line bundle over a compact, complex manifold M. We consider the projective space I LI of divisors of sections s ( O(M, L). An entire holomorphic mapping f : Cn M is said to be algebraically non-degenerate in case the image does not lie in any D e I LI . In this case the F.M.T. (1.15) Nf(D, r) + mf(D, r) = Tf(L, r) + C

and subsequent Nevanlinna inequality (1.16) Nf(D, r)

Tf(L, r) + C

are valid. We may then define the defect (3.1)

5f(D) = 1

-

l

[!ELr)] = !L

mf(D, r)1

ro

Tf(L, r)J

Tf(L, r)

r

with the properties:

0
8f(D1)+...+3f(Dk) < k5f(D1 +...+Dk) 8f(D) = 1

if

f(Cn)

misses

(Proofs: The first follows from

0<m/T<1+o(1) 65

D.

66

ENTIRE HOLOMORPHIC MAPPINGS

and the third from Nf - 0 in case f(Cn) fl D = 0. As for the second inequality, from Chapter 1 r) m f(DI , r) Tf(L, + ... + mf(Dk, rf(L r) = r)

where D = DI + + Dk a I01. Now use

mf(D, r)

k ,rf(Lk,

r)

(lim) ( lim(E). l

In general, if the defect 8f(D) is positive, then the image f(Cn) meets D less often than is the case for an average divisor. For example, suppose that M is projective space Pm and L is the hyperplane line bundle H. Then 1HI is the dual projective space Pm* of linear hyperplanes A, and Crofton's formula (1.7) together with "f lim < lim f " implies that

f

8f(A)dA = 0

.

pm *

To state the main result, we define the quantity (3.3)

8(M; L) = inf la > 0 : act(L) + cl(KM) >

More precisely, we consider all a

0 for which there are real closed (1,1) forms 6L and OK representing cI(L) and cl(KM) in HDR(M, R) such that a 61L + rbK > 0 ,

and then 8(M, L) is the infimum of such a. For example, if M = Pin and

L = Hd is the dth power of H, then 8(Pm, Hd) =

(m-d/ 1.

(Proof: We consider the unitary group G = ;Um+1 acting on Pin, and

denote by dT the invariant measure on G. Given any differential form

0 on Pm, the average

THE DEFECT RELATIONS

* = J T*z(i

67

dT

G

is well-defined, and gives an invariant form tli*. If. (o is the standard Kahler form on Pm, then any invariant form is a multiple Acoq of some

power of w. Thus, if then by averaging

acb Hd + OK -1 0

But

* d = d w and chK = - (m- 1)w; so that a > M+1

.

H

Q.E.D.)

Given a non-degenerate, equidimensional holomorphic mapping

f : Cn - M, we denote by Rf the ramification divisor of f and set Rf(L) = lira

N(Rf, r)

Tf(L, r)

Then Rf(L) measures the amount of ramification of growth. The main result is

f

relative to its

THEOREM (Defect Relation). Suppose that f : Cn -+ M is a non-degenerate, equidimensional holomorphic mapping, L M is a positive holomorphic line bundle with

cl(L)+ cl(KM) > 0 ,

and D e ILI is a divisor with simple normal crossings. Then Sf(D) + Rf(L) < S(M, L)

.

PROOF. Under the assumption cl(L) + cl(KM) > 0 we may construct the volume form TM(D) given by (2.16) and having the curvature properties as stated in Proposition 2.17. By the basic estimate (2.30), given F > 0 we have an inequality

68

ENTIRE HOLOMORPHIC MAPPINGS

(1-e)Tf(L®KM,r)+ N(Rf,r) < N(Df,r)

//E

'

Rewriting this using the F.M.T. (1.15) and Tf(L®KM, r) == Tf(L, r) Tf(KM, r) leads to (3.5)

m(Df, r)

N(Rf, r)

Tf(L,

Tf(L,

r) +

Tf(KM, r)

r) < (1- `) Tf(L, r) +

Noting that 8(M, L)cl(L) > cl(KM) implies Tf(KM, r) < l (M, L) Tf(L, r)

we find lim r -. no

[m(Dfr) N(Rf, r) Tf(L r) + Tf(L ]

Since 1 (lim) < lim (1) and

<

(1-E)S(M, L) + E .

> 0 was arbitrary, we have proved (3.4). Q.E.D.

We now give some variants and special cases of (3.4). From (3.3) we note that S(M, L) <:

(since L , M is positive)

(3.6)

S(M, LN) = N S(M, L) (since cI (LN) = Ncl (L))

The condition

cl(L) - cl(KM) > 0 is equivalent to S(M, L) <

1 .

If, on the contrary 5(M, L) > 1, then trivially Sf(D) < 1 < 6(M, Q. In summary: Under the assumptions that L -> M is positive, D e L'' has simple normal crossings, and f : Cn M is non-degenerate and equidimen-

sional (3.7)

Sf(D) < S(M,L) .

THE DEFECT RELATIONS

69

Here are some special cases:

(i) Assume that D1, , DN e 'LI are divisors such that D = DI

DN has simple normal crossings. Then N

Sf(Di) < S(M, L)

.

i=1

PROOF. Using the second property in (3.2), (3.6), and (3.7) N

N

Sf(Di) < NSf

Di

< NS(M,LN)

= 6(M, L) .

(ii) Taking M = Pn and L = Hd, we find that if D1,

, DN are

smooth hypersurf aces of degree d meeting transversely, then

+Lr

N

Sf(DI) < (!J).

(3.9)

N

af(Ai) <. n+ 1

i-1 Where AI, , AN are hyperplanes in general position in Pn. For n = 1,

the Ar are distinct points on the Riemann sphere, and (3.7) reduces to the beautiful Nevanlinna defect relation -

(3.10)

Sf(Ai) < 2

Which began this whole business.

70

ENTIRE HOLOMORPHIC MAPPINGS

(b) The lemma on the logarithmic derivative In the preceding section we have used what might be termed the "method of negative curvature bounded away from zero on M -

D" to

obtain a lower bound on the size of Df = f -I(D) for non-degenerate, equidimensional holomorphic mappings f : Cn M. Now, in R. Nevanlinna's original proof of the defect relation (3.10) for an entire meromorphic function f(z), the crucial analytical step was the: LEMMA ON THE LOGARITHMIC DERIVATIVE:

(3.11) 277

fiog+

f' rein) dO 1- C (log T f(r) + log r) f(rei )

//

.

It was F. Nevanlinna who pointed out that (3.11) could also be proved by a negative curvature argument, but where the curvature is allowed to tend toward zero as one approaches D. What we shall do here is show that the same effect may be achieved by putting a metric on M - D which is strongly negatively curved in a neighborhood of D, but which may have "positive curvature" outside such a neighborhood. We shall be interested in the situation where the dual canonical bundle KM -. M is positive and where A is a meromorphic n-form on M with no zeroes and whose polar divisor D e IKMi has simple normal crossings. The prototype example is M = Pn so that KM = H(n+l) and A is the differential form given in affine coordinates by the Cauchy kernel dw1 dwr A ... A wI Wn

f : Cn - M is a non-degenerate, equidimensional holomorphic mapping, then the pullback If

f*A = Af=Af(z)dz1 A...A dzn

where Af(z) is an entire meromorphic function whose zero divisor is the

THE DEFECT RELATIONS

71

ramification divisor Rf of f and whose polar locus is Df -= f -I(D). In case n = 1, A f(z) = f( is the logarithmic derivative of the meromorphic function f(z). We shall prove the estimate (3.12)

I

log+ kkf !, I = 0 (log T f(KM, r) + log r)

as [r]

which reduces to (3.11) in the case of an entire meromorphic function

w = f(z) and A=. Step one in the proof. Let rI be a closed positive (1,1) form on M which represents c1(KM). As usual, we write D = D1 + + DN where the Di are smooth divisors meeting transversely, and we choose metrics in the line bundles [Di] -, M and sections si c ON, [Di]) such that

'sit < 1 and

1 = 17. Letting I1 M be a volume form on M i=1 ddc logIsll2

with Ric TM = - 71, we consider the singular volume form TM(D) _

TM N

Is. 2 (log alsi,2)2

i=1

The proof of Proposition 2.17 applied to this singular volume form gives: the conditions Given e > 0, there is 0 > 0 such that for 0 < a < Ric TM(D)

FYI

(3.13) Vi > 0

and

on > TM(D)

are valid. From the definition of TM(D), if we write (f*WM(D)) (z) = h(z)t(z)

,

72

ENTIRE HOLOMORPHIC MAPPINGS

then for some constant C > 0 1,\f(z)I2

h(z) < (3.14)

C

N

(log

< Ch(z)

.

lsil2)2

i=1

In particular, if vf(r) =

log+Ixf1E

J aB[r]

is the quantity we wish to estimate, then (3.15)

log hE < 2vf(r) + C'

I

aB[r]

Step two in the proof. We apply the S.M.T. (2.29) to the singular volume form h(z)(D(z), ignore the ramification term, and use (3.15) to obtain n con-1

PP < Nf(D, r) + E Tf(KM, r) + 2vf(r) + C

.

Now, arguing as in the proof of the basic estimate (2.30) we have 0

1

Li

B[p]

f nn-1

2n1 dP

J hn

dP P

B[p]

P

Combining these inequalities with (3.14) and the F.M.T. (1.15) gives fr

J

2nP 1 P

fpt2n-1

dt

J aB[t]

j

f2/n I

2/n

< C(Tf(K , r) + vf r)) M

11(log 'siI2)

Applying the calculus lemma twice to the left-hand side of this estimate gives

THE DEFECT RELATIONS

73

IA I2/n f

(3.16)

(log5j2)2/fl)

< C(Tf(K, r)1+

ra

+ vf(r))

//,,a

.

N

We now take a = JA f'2/n and 0 = II in the inequality 1 i=1 (loglsij2)2/n elog+a+log t3 < af3 + 1

valid for a > 0 and 0 <

r

2 log+IAf

1,

to obtain from (3.16)

N

- n i=1 log ( tog `\

e

1

Isil

2

2

< C(Tf(KM,r)1+E+ra+vf(r))//s.

Md Applying concavity of the logarithm, together with

('

J

log

log

)

2

logIsIl2 1s

< log

71.1i

aB [r]

aB[r] = log (m f(Di, r))

< log (Tf (K*,

to the integral estimate yields vf(r) < C (log Tf(KM, r) + log r + log vf(r))

//

.

The last term on the right-hand side may be absorbed in the left-hand side, thereby proving (3.12).

(c) R. Nevanlinna's proof of the defect relation 3.10

For historical reasons, and also because it is very pretty, we shall use the lemma on the logarithmic derivative 3.11 to derive the defect relation for a non-rational entire meromorphic function f(z). In this section we let

74

ENTIRE HOLOMORPHIC MAPPINGS

mf(a, r) =

1 27r

log+

If(1er

al

(a A

dO

ti mf(-, r) = ;5L 1

log+ If(reiO)j dO

rr

,

0

and shall use the Nevanlinna characteristic function Tf(r) = Nf(=, r) + m f(-, r)

By Jensen's theorem, Tf(r) is essentially independent of a (this is the F.M.T. for Tf(r)). We denote by r(r) any positive function of r satisfying t-(rn) = o(Tf(rn))

for a sequence of values rn -. = (3.17)

According to Lemma 3.11,

.

m f7f(°°, r)

` (r)

,

and log r is also c(r) since f(z) was assumed non-rational. Given distinct points al, ,aq on the sphere F1, which may each be assumed finite, obtaining a lower bound on the counting function sum q

I Nf(a,,, r) µ=I is, by the F.M.T., equivalent to finding an upper bound on the proximity function sum q

I mf (aµ' r)

µ=1

If we define the auxiliary function q

g(z)

k=1

1 f(z)-aµ

THE DEFECT RELATIONS

75

(1.19) and (3.17) together with

log'- ja+(31 < log' jaI + log+ f3j + log 2 implies that (3.18)

mf,g(o, r) = E(r)

On the other hand, the two expressions mf(aµ, r) =

2n

J log+

1

jf-a11l

dO

and

mf g(c, r) + mf-(0, r))

1

=2r

(

f'

log

-a

+log+jI i

dO l1

q

I mf(a,`, r) - mf"(0

ii = 1

r)

t- (r)

where the right-hand side is independent of the points (ateThis is the idea behind the argument.

More precisely, for fixed S > 0 the points on the circle jz = r .where f(re'O)-auj 5 contribute only an 0(1) term to mf(au, r). Choosing fi small relative to min I aP - al,at the points where f(re1O) - ail < S the IL

contributions to mf(ap, r) and mg(-, r) are the same, up to an 0(1) term. Repeatedly using the inequality

log+la-/31 < log+ ai + log+log 2 we thus find an estimate q

I mf(au, r) < mg(«>, r) + 0(1) . p=1

76

ENTIRE HOLOMORPHIC MAPPINGS

On the other hand, from 1og+ Iaol < log+ (al + log+ 1/31

we obtain mg(°°, r) < mf,g(°°, r) + mf'(0, r)

which, when combined with (3.18) and (3.19), yields q

mf(aµ, r) < mf'(0, r) + E(r)

(3.20)

.

µ=I

Now add Nf'(0, r) to both sides to find 9

mf(aµ, r) + Nf'(0, r) < Tf'(r) + s(r) µ=1 Using the F.M.T. for f'(z),

Tf'(r) = Nf-(c, r) =

+

mf'(', r)

Nf,(oc, r) + mf(co, r) + 6(r)

by (3.17) again. Adding 2Nf(oo,r) to both sides we obtain

mf(aµ, r) + NI(r) < 2Tf(r) + 6(r)

µ=1 (3.21)

where

NI (r) = Nf'(0, r) + (2N f(°°, r)- Nf'(o, r))

The point here is that the term NI(r) is non-negative and measures the ramification of f (the points in f -I(o) where f is ramified are counted in a funny way). Thus

THE DEFECT RELATIONS

77

q

mf(ap, r) < 2Tf(r) + E(r)

,

µ=1

and this implies the defect relation.

Note that the "2" appeared essentially because on the sphere the differential dw has a pole of order two at w = w, which is the same reason as in the algebro-differential-geometric proof. Here is an example of an entire holomorphic function f(z) having p + 1 deficient values b0, ,bp_I, by = oo with corresponding defects given by (Ii0,...,p_l)

f(bp) = P (3.22)

8 f(bp) = Sf(°°) = 1

For this example, the Nevanlinna defect relation 3.10 is sharp. The general question of finding an entire meromorphic function f(z) with preI Sf(b) < 2 is quite interesting, scribed defects Sf(b) subject only to b, P1

especially when there are restrictions on the growth of f, and is a problem which has seen considerable progress in recent years. The function in which we are interested is defined by f(z) =

ize_tPdt.

Setting t = ael(k,

le t= e-apcosp, It follows that in the angular sector

w ={z:largz-

I<-c}(=0,1,..,2p-1) ))

the integrand tends uniformly to zero for v even and to infinity for v odd. We shall correspondingly find that f(z) tends to

ENTIRE HOLOMORPHIC MAPPINGS

78

27rµ \/=1 f

bu=e

2r7µ \/=1

e tpdt=e

p

p

F(1+P1

in A2jt (jc=0, , p-1), and to by=W in A2µ+I (je=0,... p-1). Indeed, in ',2j,+1 a double integration by parts gives e -z

p

(8 .0 as Izj-«)

pzp-I (1+6)

f(z)

Thus in ,2,+I log

I

rp cos pc(1+8)

(z= re

from which it follows that

(8-.0 as r-.oo)

mf(00,r) = rp (1+S)

Since Nftoo, r) = 0, we have for the characteristic function

Tf(r) _ L- (I+&) 27

In the sector AO'

f(z) - bo = iZe_tPdt

- retp dt 0

= - J Ae-tp dt

=

zp -e pzp -I (1+S)

,

where the last step is again integration by parts. Thus

THE DEFECT RELATIONS

log

1

Is(z) - b01

79

= rp cos pq(1+&) ,

which gives mf(b0, r) = Pn (1+8)

.

A similar result holds for bI, ,bp-I in the sectors A2' ... , 12p-2' This implies the result (3.22) on the defects of

f.

(d) Ahlfors' proof of the defect relation 3.10 As mentioned previously, it may be useful to give the two "classical" proofs of (3.10). The Ahlfors proof, which is based on the tension between the simultaneous relations Nf(a,r) < Tf(r) + 0(1) (3.23)

f Nf(a, r)da = Tf(r) , a(P 1

is the quickest. Here, Tf(r) is the Ahifors-Shimizu characteristic function and da is the standard invariant measure on P normalized so that the total area is one. As mentioned in the discussion following equation (1.17), integration of the inequality in (3.23) over the range f((:) gives a proof of the Liouville theorem that f(C) is dense in PI. To obtain finer Picard-type theorems, one might try and apply the same method, but with a density having singularities at the points al, , aq in which we are interested. This is the idea. Denoting by w the coordinate on PI = C U 1-1, the standard metric is dw A dw

(w) P

27r

(1+Iw12)2

Assuming that the aµ are finite, we consider

80

ENTIRE HOLOMORPHIC MAPPINGS

l

w-aµ

u(aµ, w) =

1+1w

which the reader will recognize as the length of the section sµ e C(P1, H) which defines aµ. The density CIF I(w) P

'I (w) = -q

jT u(aµ w)2A µ=1

has singularities at the I aµl and is integrable for k < 1. We choose the constant C so that the total area is one. Writing the mapping f : C P1

as w(z) and setting

\T1-

f*(T 1) _

h(z)dz A dz , 277

P

we now integrate the inequality in (3.23) with respect to 'I'(a) and obtain

f

r

r

J

V 27r

h(z) dz A ddp <

Tf(r) + 0(1)

q

H u(aµ, w(z)) 2X

gy(p)

µ=1

Applying the calculus lemma 2.5 twice just as in step two of the proof of (3.12) gives 2r,

h(rele) dO

1

2n

fo

< C(Tf(r)1+ F-

q

2X

u(aµ, w(reie))

µ=1

The integrand may be written as

e

log h+2AIq

µ=I

log(1/u(a ,w)) µ

a

+r )

//F 8 .

THE DEFECT RELATIONS

81

and concavity of the logarithm gives 2n (3.25)

Zn

q

r h(rel©) dB + 1 1 o

mf(aµ, r) < C(log Tf(r) + log r)

//

.

µ=I

The Gaussian curvature of

is constant +2, so that the S.M.T. (2.1)

'I'

may be used in (3.25) to yield 4

mf(aµ, r) + N(Rf, r) < 2Tf(r) + E(r)

(3.26)

µ=1 where

lim IT f(r)1

J

Dividing (3.26) by the order function and ignoring the ramification gives q

µ1 1=

Letting A

2

mf(aµ,r)

lim r - W

<

L Tf(r)

X

1 yields the defect relation (3.10).

Q.E.D.

(e) A refinement in the classical case The reader may have noticed that much of the analysis in the theory as developed above is essentially of a one-variable character, and is centered around the distributional equation A logifj = 1f=01 for a holomorphic function f. As might be expected, there are considerable

refinements of the Nevanlinna defect relation in the classical situation of an entire meromorphic function of finite order. Generally speaking, these refinements appear to be based on the following two principles: (i) Such a function has an infinite product representation given by the h:adamard factorization theorem (1.27). For example, if f(z) is an entire

82

ENTIRE HOLOMORPHIC MAPPINGS

holomorphic function of finite non-integral order p, then the equation (3.27)

f(z) = a

has an infinite number of solutions for every a

a. (Proof: The function

f(z) - a has the same order p, and by (1.27)

f(z) - a = zkeP(z) JJ E(z/ap, q) µ

where q = [p], p(z) is a polynomial, and the aµ are the roots of (3.27). If the product were finite, then f(z) - a would evidently have integral order.)

(ii) Entire functions of finite order whose zeros and poles are on the negative and positive real axes respectively are frequently extremal for problems in value distribution theory. This principle seems to be originally due to Gol'berg and has recently been put in beautiful conceptual form by Baernstein (cf. the bibliographical remarks below). It is of a different character from what we have encountered thus far, in that the angular - as opposed to the radial - distribution of the roots is involved. In order to illustrate the flavor of these results, we shall prove a result of Edrei and Fuchs concerning an entire meromorphic function of finite order p with 0 < p < 1. To state their theorem, we let a, b be two distinct points on the Riemann sphere and set

A = 1 - bf(a) B = 1 - bf(b) On general grounds, all we can say is that (3.28)

0 < A,B < 1,

i.e. (A, B) lies in the unit square in the first quadrant.

THE DEFECT RELATIONS

83

THEOREM (Edrei-Fuchs): In addition to (3.28), (3.29)

A2 + B2 - 2AB cos ap > sine rrp

Moreover, it either A < cos rip or B < cos rip, then B = 1 or A = 1

respectively. COROLLARY (Valiron). If p = 0, then f(z) can have at most one defi-

cient value.

Thus functions of order zero have a striking similarity to rational

functions. The sort of restrictions imposed on (A, B) by this theorem are illustrated by the shaded region in the following figure

0
suppose that a = 0, b = -. Finally, multiplying f(z) by Czk does not change the problem, and allows us to assume that f(0) = 1.

84

ENTIRE HOLOMORPHIC MAPPINGS

By the Hadamard theorem,

f(z) =

11 (1 - z/an) n

IT (1- z/bm) M

where the an, bm are the respective zeros and poles of f(z). We associate to f(z) the symmetrized function

I(1+z/lanD f(z) =

n

ll(1-z/Ibm1)

,

m

obtained by moving the zeros and poles of f(z) to the negative and positive real axes along circular arcs. The first step is to prove the result, due to Gol'berg, that f(z) is extremal for the present problem in the sense that 8f(0) < 8f(0) (3.30)

Then, with the obvious notations and using (1.26),

A
p = inf A:1la n

< W, " -

Ian

<

I

` m

m

so that it will suffice to prove (3.29) for f(z). The proof of (3.30) is based on the

(3.31) PROPOSITION. Given f(z) and f(z) as above, and th(t) a convex function of log t for t > 0, then for r 1 Ianl, ';bml ,0(if(rei0)I)d0 < 2zn fo

2n 0

O(If(rer©)!)dO

THE DEFECT RELATIONS

85

Proof of (3.30) from (3.31): Taking 0(t) = log+ t,

mf-, r) < m F(-, r) for r # janl, Jbm'', and then for all r by continuity. Since obviously

Nf(°°, r) = Nf (a, r)

,

$f( °) = lim rmF r)(+ Nf(oo, r)] < lim [mf(oom)(+ Nf

(°°,r)]

and similarly for 8f(0) and 8f(0).

Q.E.D.

The proof of (3.31) is, in turn, based on the following technical

(3.32) LEMMA: Suppose that av(0) are even, periodic with period 27r, monotone increasing for 0 < 0 < n, and satisfy 0 < m av(0) < M. Then

J

c6( nR

f7 r

f U,0+0') dO < V=1

I

J

av(0))

`v-1

dO

.

l

Proof of (3.32): By assumption the o' (0) are decreasing functions of cos 0. Since

cos(O-k)- cos(O+k) = 2 sin0 sin k implies that, for 0 < 0 < rr, 5 cos (0 - k) > cos (0 + k)

0
lcos (0+ k) > cos (0 - k)

n < k < 277

,

we infer the inequalities av(0 - k) <_ av(B + k)

0<0
0
k)
0 <0
77
(3.33) or,(0

.

86

ENTIRE HOLOMORPHIC MAPPINGS

We next need to derive a consequence of the convexity of c(et). If q(t) is any convex function of t, then

77(t+s)+77(t-s) is an increasing function of s > 0. Indeed, if

77(t)

is smooth, then

d (77(t+s)+ q(t-s)) = 71'(t+s) - 77'(t-s) > 0 . The general case follows by writing r1(t) as the limit of smooth convex functions. From this we deduce that, for t1, t2, s1, s2 non-negative,

77(t1+t2+s1-s2) + 77(t1+t2+s2-s1) G 77(t1+t2+s1+s2)+ i7(t1+t2-s1-s2).

Taking ra(t) = 0(t) and setting

t1-s1

e e

t2-s2

tl+sl
t2+s2

we find the inequality O(Ab) + cb(aB) < q(AB) + q (ab) (3.34)

0
To prove the lemma, we may suppose that 0 < 01, < 2u and some

0 > 0. Define

k.

by

kt, = 0, - 2 0 (2n), 0 < kt, < 2n, and set

11

il

ai,(O+kL)

0
II

77< kt,< 27r

av(-0+ k.) .

THE DEFECT RELATIONS

87

By periodicity and a,(0) = av(-0), n

77

0E 7;

Cv=

('n

ov(0+Ov)) d0 = J

i

II

n

b( n

av(0+kv)) d0

\v=1

J

rrr 0(E1(0)Y-2(-0))d0

J

,7

d0

)+ 0

IT

= f q5(11(O)E2(O))d0 n

where the next to the last step follows from (3,33) and (3.34). Combining gives

f7l-

rl ( av(O+Ov)) d0 < v=i

/

(n

ov(O+Oi )) d0

v=1

/

where

Ov=kv-20µ(27r),

0<0'v<2n,

if

0
0;,

0<0'<2n,

if

rr
kv - 2 0µ(2O,

.

If Ov = 0, then kv = 2n - 2Oµ > n so that 0, = 0. Moreover, k _ < n so that Oµ = 0. Proceeding in this way, after at most n steps 2 we reach a situation where all Ov = 0. Proof of (3.31): We may assume that

Q.E.D.

ENTIRE HOLOMORPHIC MAPPINGS

88

N

rl f( z )

_n

(1 - z/a n)

M I

(1- z/bm) m

=1

and deduce the case of infinite products by passing to the limit. Setting

an = Ianl la e° , Qv(0) = 1 _ re

i9

bm = Jbml e l m

(v=1,...,N)

©v=_av

lavl

0

°N+lc(O) =

N+µ=77-a11

(µ=1,...,M)

in (3.32) gives the proposition. Step two: We may assume that

an, bm > 0 .

f(z) = =I

The theorem is trivial unless S f(0) > 0 ,

Sf(oc) > 0 .

Assuming this, we may suppose that for large

r

log+ f(1ei0)I d©> 0

fmfr) = Zrr (3.35)

log+

dB > 0 .

mf(0, r) =

fo

I f(re

)I

89

THE DEFECT RELATIONS

The point in using the symmetrized function f(z) above is that one may get some handle on the proximity functions as follows: First note that

is an even function of 0 which decreases as 0 goes from 0 to rr - indeed, this is true for each of the factors If(rei0)I

I1+re"I an

1 - re

i6

bm

By (3.35)

logif(-r)I < 0 < logif(r)I

,

and thus there is a unique /3(r) with 0 < 0(r) < it such that log 1f(relj6(r))I = 0 .

From this we deduce

f

/3(r)

lgf(reie)dB

mf(or) _ 1 (3.36)

mf(0, r) = n

log (3(r)

1

If(r!

dO

.

LEMMA. The integral representation

Tf(r) = nf Nf(0,t)

(3.37)

JO

r sin /3 dt Z + t +2tr cos /3+r 2

f

Nf(-

t)

r sin /3 dt t2-2tr cos /3+r 2

is valid.

PROOF. For z not on the real axis,

log f(z) = I log(1+z/an)+I log(1-z/bm) n

f =

m

log(1+z/t)dnf(0,t)+ fo, log(1-z/t)dnf

90

ENTIRE HOLOMORPHIC MAPPINGS

Since at [log(l+z/t)] = t(t+z)' itd (

1)

-1

(t + z)

2 , and nf(O,t) = 0(tp+e)

where p + e < 1, we may integrate by parts twice to obtain

Cl 1

flog(l+z/t)dnf(0,t) = z

0

0

r

dt

Nf(O, t)

t

t+z

Doing the same for the polar terms gives

log f(z) = z

t+z

fdN(ot)

=

(3.38)

nf(0,t)dt

f +z

(t+z)2

0

NO, t)dt (t + z)2

r

dt

Nf(oo, t)

2

(t - z)

0

We will prove the lemma by combining (3.36) and (3.38). Thus the proximity function 13 (r)

mf(x, r) _

log If(reiO); df3

lim

re 1

dz (t+z)2

Nf(0, t) dt J

Im J

lim

re

ij3 re

Nf(-,t)dt

i Im

dz

2

(t - z)

0

re

We set

P(t, r, y) = 1 Im (

Q(t, r, y) = 1 n and deduce that

Im(-

1

t -rely

)

1

t+rery =

r sin y

=

t2+2tr cos y + r 2

r sin y t2 - 2tr cos y +

= P(t, r, 77 - y) , r2

THE DEFECT RELATIONS

mf(,-,r) = J

Nf(0,t)P(t,r,s)dt + J

0

91

Nf(oo,t)Q(t,r,f3)dt

0

lim J Nf(0,t)P(t,r,8)dt - lim -0

8-.0

0

J

a Nf(oo,t)Q(t,r,8)dt

.

0

The first limit is obviously zero. As for the second, Q(t, r,&) -, 0 as S 0 uniformly in t outside any fixed interval [r-E, r+E]. Also, Nf(oo, t) is approximately equal to Nf(oo, r) inside such an interval. Thus,

as 8 - 0, f'x

J

T+E

Q(t, r, 8) dt + 0(1)

Nf(oo, t) Q(t, r, 8) dt = Nf(oo, r) J

r-E

0

= Nf(o, r) fo Q(t, r,8)dt + 0(1)

= Nf(oo, r) + 0(1)

since fo

Q(t, r, S) dt = 1 for 3 > 0 by a residue calculation. Combining

we find

Tf(r) = mf(o,r) + Nf(-,r) =fo

Nf(O,t) P(t,r,T) + Nf(°°,t) Q(t,rd0)I dt

as was to be proved.

Step three: The idea is that, according to (3.37) in the form x x Tf(r) = fo Nf(0,t) P(t,r,f)dt + fo Nf(x,t) P(t,r,r7 -(3) dt ( 3.39)

P(t,r,y) = t2 + 2tr cos y + r2

the assumptions S f(0) > 0,

b f(-) > 0

92

ENTIRE HOLOMORPHIC MAPPINGS

imply that both Nf(0, r), Nf(o, r) grow at a slower rate than Tf(r), and we

might hope to use this in (3.39) to force some sort of inequality. In order to carry this out, it is pretty clear that we will need some method for dealing with the possibly irregular growth of Tf(r) - cf. Borel's lemma on page 29 and the calculus lemma 2.5. This is provided by the following LEMMA ON POLYA PEAKS. Given a(t), 13(t) continuous positive func-

tions of t > to with the assumptions /3(t) is non-decreasing

lim a(t) _ + o

t - 00

lim t-7 -Ut =0

t - a NO then there exist rn - w such that

to < t < rn

a(t) < a(rn) , (3.40)

/3(t)

rn
(Such rn are called Polya peaks.) PROOF. Suppose t1 > t 0 is given. Then there is t2 > t1 such that a(t2) =

sup

to
a(t)

Next, there is t3 > t2 such that a(t3) NR(t 3)

1,*

s

t>3

a/,(,t)

Finally, there is r with t2 < r < t3 such that

THE DEFECT RELATIONS

a(r) =

sup

t2
a(t)

93

sup

t0
a(t)

This implies (3.4) for rn = r since /3(t) is non-decreasing.

Q.E.D.

To apply this lemma, we choose E > 0 sufficiently small, set A _ p + E < 1, and take Tf(r) a(r) A-2F. r

/3(r) =

rzE

We may then find arbitrarily large r such that

t0
Tf(t)

<

Tf(r)

tx

=

rA

r
.

Now to the proof of theorem. Choose

51>AO>A=1-5f(0)

1> Bo>B= 1-8f(-), so that for t > to Nf(O, t) < A0T f(t)

Nf(', t) < BOTf(t) Plugging these inequalities into (4.39) and using (4.41) gives, for arbitrarily large Polya peaks r, r

Tf(r) < Tf(r) AO

_2 P(t,r,a)(r)

r P(t,r,f3)(L)

dt + AO

+ Bo

rr 0

dt

r

0

dt +

BOJ-P(t,r,n-/3)( r

j)

A

dt - E (r),

ENTIRE HOLOMORPHIC MAPPINGS

94

where the remainder term

c (r) < Nf(0, to) J tOP(t, r, Q)dt + Nf(-, t0)

`OP(t, r, n-13)dt

Jo

o

Clearly E. (r) =, and by making A0, B0 slightly larger may be ab-

sorbed in the term

above.

To estimate the integrals, we set t = rs to have 0 < s < 1, and then r

J

(' t A-2S t P(t,r,y)( i) dt + J P(t,r,y)(r) dt / r

=f' SA-2e P(s,1,y)ds

+

f sAP(s,l,y)ds I

= J sAP(s,l,y)ds + J (sA-25-sA)P(s,1,y)ds 0

0

An elementary contour integration gives 1

s"P(s,l,y)ds

=

sin(yA)

(0
sin (17 A)

fo

Moreover, the other integral is 0(c) for A > A0 > 0. We deduce that 1<

max 0< p< 77 L

A0 sin (/3A) + B0 s in (n-/3)A sin r7A

+ Cc .

Choose y = y(a0, B0, A) to maximize A0 sin yA + B0 sin (ii-y)A .

For this value of y, (1-Cc)sin nA < (A0-B0cos 77,k) sin yA + B0 sin riAyA

[(A0-B0cos

7,,A)2

+ (B0 sin rA)21Isin2yA + cos2yA]

Ao + Bo - 2AOB0 cos nA

THE DEFECT RELATIONS

using the Cauchy-Schwarz inequality. Letting E B0

95

0 and then A0

A,

B gives (3.29).

To prove the remainder of the theorem, we suppose that A < cos rrp. Then A < cos n(p+E) for sufficiently small E. Taking A = p+ E we

choose Bo with B < Bp < 1 and A < B0 cos nA, and then set A0 = B0 cos nA in (3.42) to obtain

(1-CE) < Bo .

We let B0 decrease until either B 0 cos riA = A or B 0 = B. Thus either

1-CE < A cos nA

or

(1-CE) < B . Letting E -. 0 so that A desired.

p, the first cannot hold, so that B = 1 as Q.E.D.

Bibliographical remarks

The defect relations, viewed as a quantitative form of Picard's theorem for an entire meromorphic function, are given in R. Nevanlinna ([31] and [321). Subsequent proofs were found by F. Nevanlinna [32] and Ahlfors [1]. A modern exposition of Ahlfors' proof in a differential-geometric setting appears in Chern [13]. The equidimensional defect relation (3.4) is taken from CarlsonGriffiths [9]. In spirit, the proof is similar to that of F. Nevanlinna only

with "curvature < -1" replacing "curvature = -1" as suggested by the Ahlfors lemma. In [22], an extension to equidimensional maps from an

arbitrary algebraic variety to M is presented. Defect relations when D has worse singularities than simple normal crossings have been found by Shiffman [35], which also contains a somewhat different derivation of the First and Second Main Theorems. The thesis of Drouhillet [18] contains extensions to f : Cn Mn of R. Nevanlinna's theorem on ramified values

96

ENTIRE HOLOMORPHIC MAPPINGS

and unicity of holomorphic mappings. Defect relations for an entire holomorphic curve in Pn are given in [16], where further references and some historical discussion also appears. The lemma on the logarithmic derivative is proved using the PoissonJensen-Nevanlinna formula in [31], and another differential-geometric proof by F. Nevanlinna appears in [32]. A weaker equidimensional version of

(3.7) is given in [221. The proof of the classical defect relation from the lemma on the logarithmic derivative follows the argument in [321, from which the example f(z) = f0 a-tpdt is also taken. The refinements of Nevanlinna theory in the classical case are due to very many people. The book [25] by Hayman contains an extensive bibliography up through 1964. Our proof of the Edrei-Fuchs theorem closely follows his. In recent years, there has been substantial progress on many long-standing problems. The papers by Weitsman [42] and Drasin [17] contain a beautiful refinement of the Nevanlinna defect relation for functions of finite order and a solution to the inverse problem (finding f(z) with

pre-assigned defects 5(a) subject only to 1:8(a) < 2) respectively. Finally, the papers [3] and [4] of Baernstein give an entirely new interpretation of the Nevanlinna characteristic function Tf(r), probably leading to conceptual proofs for the extremal behavior of symmetrized functions f(z) and ultimately a much deeper understanding of the proximity integrals

f log+ If(reie)! dO a

(a
.

BIBLIOGRAPHY [1] [2] [3]

Ahlfors, L., Uber eine Methode in der Theorie der meromorphen Funktionen. Soc. Sci. Fennica, vol. VIII, no. 10, pp. 1-14. , An extension of Schwarz's lemma. Trans. Amer. Math. Soc., 43 (1938), pp. 359-364.

Baernstein, A., Proof of Edrei's spread conjecture. Proc. London Math. Soc., 26 (1973), pp. 418-434.

[4] [5]

[6]

[7]

, Integral means, univalent functions and circular symmetrization, to appear in Acta Math. Bishop, E., Conditions for the analyticity of certain sets, Mich. Math. J., 11 (1964), pp. 289-304.

Bott, R., and Chern, S. S., Hermitian vector bundles and the equidistribution-of the zeros of their holomorphic sections. Acta Math., Vol. 114 (1966), pp. 71-112. Carlson, J., A remark on the transcendental Bezout problem, ValueDistribution Theory (Part A), Marcel Dekker, New York (1974), pp. 133-143.

[8]

, Some degeneracy theorems for entire functions with values in an algebraic variety. Trans. Amer. Math. Soc., 168 (1972), pp. 273-301.

Carlson, J., and Griffiths, P., A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. of Math., vol. 95 (1972), pp. 557-584. [10] , The order functions for entire holomorphic mappings. Value-Distribution Theory (Part A), Marcel-Dekker, [9]

New York, (1974), pp. 225-248.

[11] Chern, S. S., Complex manifolds without potential theory, van Nostrand, Princeton, N. J. (1967). [12] , On holomorphic mappings of Hermitian manifolds of the same dimension. Proc. Symp. Pure Math., vol. 11 (1958), pp. 157-170. [13] , Complex analytic mappings of Riemann surfaces 1. Amer. Jour. Math., 82 (1960), pp. 323-337. 97

98

BIBLIOGRAPHY

[14] Cornalba, M., and Griffiths, P., Some transcendental aspects of algebraic geometry. Proc. A.M.S. Summer Institute on Algebraic Geometry (1974).

[15] Cornalba, M., and Shiffman, B., A counterexample to the "Transcendental Bezout Problem." Ann. of Math., vol. 96 (1972), pp. 402406.

[16] Cowen, M., and Griffiths, P., Holomorphic curves and metrics of negative curvature, to appear in jour. d'Analyse. [17] Drasin, D., A meromorphic function with assigned Nevanlinna deficiencies. Bull. Amer. Math. Soc., 80 (1974), pp. 766-768. [18] Drouhillet, J., Ramification and unicity of equidimensional holomorphic maps. Thesis at Rice University, Houston, Texas (1974). [19] Green, M., Some examples and counterexamples in value distribution theory for several variables, to appear in Composito Math. [20] Griffiths, P., On the Bezout problem for entire analytic sets, appeared in latest Annals. [21] , Differential geometry and complex analysis. Proc. Symp. Pure Math., vol. 27 (1974), pp. 127-148. [22] Griffiths, P., and King, J., Nevanlinna theory and holomorphic mappings between algebraic varieties. Acta Math., 130 (1973), pp. 145220.

[23] Griffiths, P., and Schmid, W., Variation of Hodge structure (a discussion of recent results and methods of proof), to appear in Proc. Tata Institute Conference on Discrete Groups and Moduli. [24] Gunning, R., and Rossi, H., Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs (1965). [25] Hayman, W., Meromorphic Functions. Oxford Math. Monographs, Oxford University Press (1964). [26] Kujala, R., Functions of finite A-type in several complex variables. Trans. Amer. Math. Soc., 161 (1971), pp. 327-358. [27] Kneser, H., Zur theorie der gebrocherien funktionen mehrerer veranderlichen, Iber. deutsch Math. Ver., 48 (1938), pp. 1-38. [28] Kobayashi, S., Volume elements, holomorphic mappings and the Schwarz lemma. Proc. Symp. Pure Math., vol. 11 (1968), pp. 253-260. _ , Hyperbolic manifolds and holomorphic mappings. [29] Marcel Dekker, New York (1970).

BIBLIOGRAPHY

99

[30] Lelong, P., Fonctions entieres (n variables) et functions plurisousharmoniques d'ordre fini dans Cn, J. d'Analyse 12 (1964), pp. 365-407.

[31] Nevanlinna, R., Le theorem de Picard-Borel et la theorie des fonctions meromorphes. Gauthier-Villars, Paris (1929). , Analytic functions. Springer-Verlag, Berlin and New [32] York (1970).

[33] Santalo, L., An introduction to integral geometry. Hermann, Paris (1953).

[34] Shiffman, B., On the removal of singularities of analytic sets. Mich. Math. J., 15 (1968), pp. 111-120. , Nevanlinna defect relations for singular divisors, to [35] appear in Inventiones. [36] Skoda, H., Sous-ensembles analytiques d'ordre fini ou infini daps Cn, Bull. Soc. Math. de France, vol. 100 (1972), pp. 353-408. [37] Stoll, W., Value distribution of holomorphic maps into compact complex manifolds. Lecture Notes in Math, no. 135, Springer-Verlag (1970). [38] , The growth of the area of a transcendental analytic set (1 and 11). Math. Ann. 156 (1964), pp. 47-98; and 156 (1964), pp. 144170. , About entire and meromorphic functions of exponential type. Proc. Symp. Pure Math., 11 (1968), pp. 392-430. Value Distribution Theory: Part B (Deficit and Bezout [40] estimates), Marcel-Dekker, New York (1974). [41] Stolzenberg, G., Volumes, limits, and extensions of analytic varieties. Lecture Notes in Math. no. 19, Springer-Verlag, New York

[39]

(1966).

[42] Weitsman, A., A theorem on Nevanlinna deficiencies. Acta Math. 128 (1972), pp. 41-51. [43] Wells, R., Differential analysis on real and complex manifolds. Prentice-Hall, Englewood Cliffs, New Jersey (1973). [44] Weyl, H., and Weyl, J., Meromorphic functions and analytic curves. Princeton University Press, Princeton (1943). [45] Wu, H., The equidistribution theory of holomorphic curves. Princeton University Press, Princeton (1970).

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72. Introduction to Algebraic K-Theory, by JOHN Mn.NoR 71. Normal Two-Dimensional Singularities, by HENRY B. LAUFER 70. Prospects in Mathematics, by F. HIRZEBRUCH, LARS HBRMANDER, JOHN MILNOR, JEANPIERRE SERRE, and I. M. SINGER

69. Symposium on Infinite Dimensional Topology, edited by R. D. ANDERSON 68. On Group-Theoretic Decision Problems and Their Classification, by CHARLES F. MILLER, III 67. Profinite Groups, Arithmetic, and Geometry, by STEPHEN S. SHATZ 66. Advances in the Theory of Riemann Surfaces, edited by L. V. AHLFORS, L. BERS, H. M. FARKAS, R. C. GUNNING, I. KRA, and H. E. RAUCH

65. Lectures on Boundary Theory for Markov Chains, by KAI LAI CHUNO 64. The Equidistribution Theory of Holomorphic Curves, by HUNG-Hsi Wu 63. Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, by ELIAS M. STEIN 62. Generalized Feynman Amplitudes, by E. R. SPEER 61. Singular Points of Complex Hypersurfaces, by JOHN MILNOR 60. Topology Seminar, Wisconsin, 1965, edited by R. H. BIND and R. J. BEAN 59. Lectures on Curves on an Algebraic Surface, by DAVID MUMFORD 58. Continuous Model Theory, by C. C. CHANO and H. J. KEISLER 56. Knot Groups, by L. P. NEUWIRTH 55. Degrees of Unsolvability, by G. E. SACKS 54. Elementary Differential Topology (Rev. edn., 1966), by J. R. MwN®iEs 52. Advances in Game Theory, edited by M. DRESHER, L. SHAPLEY, and A. W. TUCKER

51. Morse Theory, by J. W. MILNOR

50. Cohomology Operations, lectures by N. E. STEP.NRoD, written and revised by D,B.A. EPSTEIN

48. Lectures on Modular Forms, by R. C. GUNNING 47. Theory of Formal Systems, by R. SMULLYAN 46. Seminar on Transformation Groups, by A. BOREL et at. 44. Stationary Processes and Prediction Theory, by H. FURSTENBERo 43. Ramification Theoretic Methods in Algebraic Geometry, by S. AaHYANKAR

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Library of Congress Cataloging in Publication Data Griffiths, Phillip, 1938Entire holomorphic mappings in one and several complex variables. (Annals of mathematics studies ; no. 85) Holomorphic mappings. I. Title. II. 1. 75-31631 515'.9 QA331.G753 ISBN o-691-08171-9 ISBN 0-691-08172-7 pbk.

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