Entire Holomorphic Mappings in One and Several Complex Variables. (AM-85), Volume 85 9781400881482

Table of contents :
TABLE OF CONTENTS
INDEX OF NOTATIONS
INTRODUCTION
(a) Some general remarks
(b) General references and background material
CHAPTER 1: ORDERS OF GROWTH
(a) Some heuristic comments
(b) Order of growth of entire analytic sets
(c) Order functions for entire holomorphic mappings
(d) Classical indicators of orders of growth
(e) Entire functions and varieties of finite order
CHAPTER 2: THE APPEARANCE OF CURVATURE
(a) Heuristic reasoning
(b) Volume forms
(c) The Ahlfors lemma
(d) The Second Main Theorem
CHAPTER 3: THE DEFECT RELATIONS
(a) Proof of the defect relations
(b) The lemma on the logarithmic derivative
(c) R. Nevanlinna’s proof of the defect relation 3.10
(d) Ahlfors’ proof of the defect relation 3.10
(e) Refinements in the classical case
BIBLIOGRAPHY

Citation preview

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

PR IN CE TON

UNIV ERSITY PRESS AN D

U NIV ERSITY OF TOKYO

PR INCETO N,

NEW 1976

PRESS

JERSEY

C o p yrigh t ©

1976 by Prin ceton U n iversity Press A L L RIGH TS RESERV ED

Published in Japan exclusively by U n iversity o f T o k y o Press; In other parts o f the w o rld by Princeton U n iversity Press

Printed in the United States o f A m erica by Princeton U n iversity Press, Princeton, N e w Jersey

L ib ra ry o f Congress C a talogin g in Publication data w ill be found on the last printed page o f this book

HERM ANN W E Y L L E C T U R E S

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, ac c e ssib le to nonspecial­ ists, to be eventually published in the Annals of Mathematics Studies. The present monograph is the second in this series.

It is an outgrowth

of the fifth set of Hermann Weyl Lectures, which consisted of five lectures given by Professor P h illip Griffiths at the Institute for Advanced Study on October 31, November 1, 7,

8,

11, 1974.

ARMAND BOREL JOHN W. MILNOR

T A B L E OF CONTENTS

IN D EX OF N O T A T IO N S

ix

IN T R O D U C T IO N (a ) (b )

Some general remarks

3

General references and backgroundmaterial

5

The prerequisites and references for these notes, some notations and terminology, and the Wirtinger theorem. C H A P T E R 1: (a )

O RDERS O F GROWTH

Some heuristic comments

8

Emile B o re l’s proof, based on the concept of growth, of the Picard theorem, and the c la s s ic a l Jensen theorem are discussed. (b )

Order of growth of entire analytic sets

11

The counting function, which measures the growth of an analytic set V C C n , 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

17

The order function T f(L ,r ), which measures the growth of an entire holomorphic mapping f : C n -* 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 T f(L ,r ), and is then combined with Stoll’s theorem to give a characterization of rational maps. (d )

C la ssica l indicators of orders of growth

25

The Ahlfors-Shimizu and Nevanlinna characteristic functions, together with the maximum modulus indicator, are defined and compared. B o re l’s proof of the Picard theorem is completed. (e )

Entire functions and varieties of finite order W eierstrass products and the Hadamard factorization theorem are discussed. The Lelong-Stoll generalization to divisors in C n is then proved.

30

viii

T A B LE OF CONTENTS

C H A P T E R 2:

T H E A P P E A R A N C E OF C U R V A T U R E

(a ) Heuristic reasoning

40

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. (b )

Volume forms

46

Volume forms and their R icci forms are introduced and some examples given. The main construction of singular volume forms is presented. (c )

The Ahlfors lemma

53

The ubiquitous Ahlfors lemma is proved and applications to Schottky-Landau type theorems are given, following which appears a value distribution proof of the B ig Picard Theorem. (d )

The Second Main Theorem

59

The main integral formula (2.29) and subsequent basic estimate (2.30) concerning singular volume forms on C n are derived. C H A P T E R 3: (a )

T H E D E F E C T R E L A T IO N S

Proof of the defect relations

65

The principal theorem 3.4 and some corollaries are proved. (b )

The lemma on the logarithmic derivative

70

A generalization of R. N evanlinna’s main technical estimate is demonstrated by curvature methods. (c )

R. Nevanlinna’s proof of the defect relation 3.10

73

Nevanlinna’s original argument and an illustrative example are discussed. (d )

Ahlfors’ proof of the defect relation 3.10

79

T his is the second of the c la s s ic a l proofs of the Nevanlinna defect relation for an entire meromorphic function. (e )

Refinements in the classical case

81

There are many refinements of the general theory in the sp ecial 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. B IB L IO G R A P H Y

97

IN D E X O F N O T A T IO N S

C n is complex Euclidean space with coordinates

z = ( z lf •••, zn);

n

(z,w )= 2

Zi ^ i and IUII2 = (z.z);

i= l P n is complex projective space with homogeneous coordinates

Z = UoP 1 = C U {~ j

is the Riemannian sphere;

B [r] = i z e C n : ||z|| < r! S[r] = S H Btr]

is the b a ll of radius

for any set

A (r ) = { z c C : |z| < r! A = A (l)

’

r in C n;

S C C n;

is the disc of radius

r in C;

is the unit disc;

A * = i £ e C :0 < \C\ < l i

is the punctured disc;

A * (R ) = ! z £C n : |zj < R • and R = ( R lf

is a polycylinder in C n;

A £ n = (A * )^ x ( A ) n” k is a punctured poly cylinder; denote volume forms; co,

rj, • •• denote (1 ,1 ) forms;

4C_V=I« (d-d); 477

On C

with

z = re^,

d C

4> = ddc ||z||2 = ^? ~r^

*

dz^Adz^j

is the standard Kahler form on C n;

co = ddc log||z||2 is the pull-back to C n — {Oi

of the Fubini-Study Kahler

metric on P n~ I ; A holomorphic line bundle is denoted by L -> M; H -» P n is the hyperplane line bundle;

X

D

INDEX OF NOTATIONS

is a divisor and

CjCL)

the corresponding line bundle;

is the Chern form (curvature form)

G (M ,L ) iLi

[D ]

of a Hermitian line bundle;

is the space of holomorphic sections of L -» M;

is the projective space of divisors of sections

4 r (M ,R )

is the 2nc* deRham cohomology of M;

A real (1,1)

form ^

where

is a positive definite Hermitian matrix;

[if/]

(^ y )

on M is positive in case locally

denotes the c la ss in H ^ r (M ,R )

A cla ss x

in

of a closed form ijj

on M;

*s Po si ^ ve *n case X = tAJ f ° r some positive

form if/) Km

is the canonical line bundle of M;

L*

is the dual line bundle to L -> M;

0 ( C n)

s € 0 (M ,L );

are the entire holomorphic functions on Cn.

Entire Holomorphic Mappings in One and Several Complex Variables

E N T IR E H O LO M O R PH IC M A P P IN G S IN ONE AN D S E V E R A L C O M P L E X V A R IA B L E S P h illip A. G riffith s*

IN T R O D U C T IO N (a ) Some general remarks T hese talks w ill be concerned with the value distribution theory of an entire holomorphic mapping f : C*1

M

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 If we let

P1 .

nf(a, r) be the number of solutions to the equation

f(z ) = a ,

\z\ < r and

a eP1 ,

then P ic a rd ’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 P 1.

Roughly speaking, Nevanlinna’s refinement states that the above sum is eventually larger than the average

tf(r) =

I

nf(a ,r )d a

a eP1 * T h is w ork w a s p a r tia lly su p ported by N S F Grant G P38 88 6 .

4

ENTIRE HOLOMORPHIC MAPPINGS

number of solutions to the equation in question.

Since the appearance of

N evanlinna’s book [31], there has been considerable attention to the su b­ ject of value distribution theory, first in the c la s s ic a l 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 sh all concentrate on the equidimensional case where dimc M = n and where

f is non-degenerate in the sense that the Jacobian determinant

of f is non-identically zero. A side from the Ahlfors theory of holomorphic curves in P n, it is here that the defect relations of R. Nevanlinna have been most directly generalized. For example, P icard ’s theorem becomes the assertion that no such

f can omit a divisor on M which has simple

normal crossings and whose Chern c la ss is larger than that of the anticanonical divisor on M.

The corresponding defect relation w ill be proved

in Chapter 3. A side 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 c la s s ic a l 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.

A side from the original proof of

P icard ’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 c la s s ic a l proofs due to R. Nevanlinna and Ahlfors, each of which has its own distinct merit. (b ) General references and background material The c la s s ic 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.

A ll that is required to read any of these books is standard basic

knowledge of complex function theory. In several variables, we sh all use the formalism of complex manifold theory, esp ecially that centered around divisors, line bundles, and their Chern forms and subsequent Chern c la sse s.

The basic references here

are the books by Chern [11] and W ells [43].

In practice a ll that we sh all

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 integra­ tion over an analytic hypersurface and Stokes’ theorem in this situation. For the former the second chapter of Gunning and R o ssi [24] is more than sufficient, and for integration we suggest the notes by Stolzenberg [41]. A side 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 sh all use concerning integration on com­ plex varieties is the Wirtinger theorem: Let

ds 2 = 2 h^: d z .d z - 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 C n .

Let

cf> =

2

I

exterior

i,j

h -d z - a d z-

J

be the

J

(1,1) form associated to the Hermitian metric and vol (V )

2k-dimensional volume of V fl U

the

computed with respect to the Riemann-

ian metric associated to d s2 . Then

6

ENTIRE HOLOMORPHIC MAPPINGS

(0.1)

vol(V) = i

if

I