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

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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)-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 r fo log If(reie)I P(rei0, z)dO (1.20) +

L.

g(z, at)

g(z, b,) -

Ialµ 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 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
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

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 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


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) 0 is equivalent to S(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) 0 and 0
, 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 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 t 0 is given. Then there is t2 > t1 such that a(t2) =

sup

to3

a/,(,t)

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

THE DEFECT RELATIONS

a(r) =

sup

t2