Vector valued Nevanlinna theory

Research Notes in Mathematics

H J W Ziegler

Vector valued Nevanlinna Theory

Pitman Advanced Publishing Program MELBOURNE

73

H J W Ziegler University of Siegen

Vector valued Nevanlinna Theory

Pitman Advanced Publishing Program BOSTON LONDON MELBOURNE

PITMAN BOOKS LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto First published 1982

© H J W Ziegler 1982 AMS Subject Classifications: (main) 30, 30C, 30D (subsidiary) 31, 32, 53

Library of Congress Cataloging in Publication Data

Ziegler, H. J. W. (Hans J. W.) Vector valued Nevanlinna theory. (Research notes in mathematics; 73) Bibliography: p. Includes index. 1. Functions, Meromorphic. 2. Value distribution theory. 3. Nevanlinna theory. I. Title. II. Series. QA331.Z53 1982 515.9'82 82-13202 ISBN 0-273-08530-1

British Library Cataloguing in Publication Data

Ziegler, H. J. W. Vector valued Nevanlinna theory.-(Research notes in mathematics; v. 73) 1. Functional analysis 2. Vector-valued measures II. Series 1. Title 515.7 OA320 ISBN 0-273-08530-1

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Contents

Preface

Extension of the First Main Theorem of Nevanlinna Theory and Interpretation by Hermitian Geometry 1

§1 §2 §3 §4

Generalization of the Formula of Poisson-Jensen-Nevanlinna Interpretation by Hermitian Geometry The Generalized First Main Theorem The Example of Rational Vector Functions

Some Quantities arising in the Vector Valued Theory and their Relation to Nevanlinna Theory

1

1

9

16

27

2

§5 §6 § 7

Properties of V(r,a) Properties of T(r,f) The Connection of T(r,f),m(r,f)

T(r,f .),m(r,f .) §8

]

]

34

39

and

N(r,f)

with

and N(r,f .)

The Order of Growth

34

45

J

51

Generalization of the Ahlfors-Shimizu Characteristic and its Connection with Hermitian Geometry 3

§9 §10

§11 §12

0

The Generalized Ahlfors-Shimizu Characteristic T(r,f) 58 The Generalized Riemann Sphere 68 The Spherical Normal Form of the Generalized First Main Theorem 73 0 The Mean Value Representation of T(r,f) 75

4 Additional Results of the Elementary Theory §13 §14

58

The Genus of a Meromorphic Vector Function Some Relations between M,m ; N,n ; V,v and

82

82 A

89 vii

5

Extension of the Second Main Theorem of Nevanlinna Theory

110

110

§16

The Generalized Second Main Theorem The Generalized Deficiency Relation

§17

Further Results about Deficiencies

141

§15

129

Appendix: Rudiments of Complex Manifolds and Hermitian Geometry

168

Bibliography

189

Table of Symbols

197

Index

199

viii

Preface

w] = f1(z) be a meromorphic function in the Gaussian complex plane C. Nevanlinna theory or the theory of value distribution gives answers to the question of how densely the solutions of the equation Let

f1(z)

=

a1

(z E C , a1 E Cu{-})

are distributed over C ; it also studies the mean approximation of the function f 1( z) to the value a1 along large concentric circles around the origin z = 0 , a problem which turns out to be equivalent to the former. Nevanlinna theory originates from a general formula of F. and R.Ncvanlinna [451, by which they were developing a general method for the investigation of meromorphic functions. This formula includes both the Poisson formula and the Jensen formula as special cases, and in its most important form it expresses the logarithm of the modulus of an arbitrary meromorphic function by the boundary values of the function along a concentric circle around the origin and the zeros and poles of the function inside this circle. Nevanlinna theory was created at the moment when Rolf Nevanlinna gave the formula an ingenious interpretation, This happened about 1924. The most general result of Nevanlinna theory can be summarized by saying that the distribution of the solutions to the equation f1(z) = a1 is extremely uniform for almost all values of aI there can only exist a small minority of values which the function takes relatively rarely. The investigation of these exceptional values constitutes the main task of value distribution theory in the sense of Nevanlinna. The earlier value distribution theory before Nevanlinna can be traced back to the year 1876, when K . Weierstrass [571 showed that in the vicinity of an isolated essential singularity a meromorphic function f 1(z) approaches every given value a1 arbitrarily closely. In 1879 E.Picard [50] even proved the surprising fact that a meromorphic function takes in the vicinity of an isolated essential singularity every finite or infinite value a1 with 2 exceptions at the most. Points which are not taken are now called Picard exceptional values of the function. The results which were found after by the mathematicians E.

Laguerre, H.Poincare, J.Hadamard, E.Borel and others revealed that in spite ix

of the possible existence of Picard exceptional values the distribution of zeros or, more generally, the distribution of a -points of an entire function is controlled, at least in some sense, by the growth behaviour of the maximum modulus function M(r,f1)

max

IzI= r

if (z) 1

which has the function of a transcendental analogue of the degree of a polynomial. This approach of early value distribution theory breaks down, however, if f 1(z) is meromorphic, since then M(r, f 1) becomes infinite if f 1(z) has a pole on the circle IzI = r . An attempt by E.Borel [3] himself of including meromorphic functions in this framework was not very successful. In Nevanlinna theory the role of logM(r,f1) is taken by ar; increasing real valued function T(r,f1) , the "Nevanlinna characteristic function" which is associated to the given meromorphic function f 1(z) . A great deal of work had been done in establishing the relationship between distribution of values and growth when Rolf Nevanlinna created his epoque making theory. This theory, which applies to entire functions, as well as to meromorphic functions, even improved tremendously the earlier value distribution theory of entire functions. There have been many attempts to extend the Nevanlinna theory in several directions. Besides older investigations of E.Borel, A.Bloch, R.Nevanlinna and H.Cartan the most important of these, known as the theory of holomorphic or meromorphic curves, was initiated by H. and J.Weyl [431 in 1938; the most difficult problem of this extension, the proof of the defect relation for holomorphic curves, was solved by L.Ahlfors [2] ; recently a very modern treatment of this theory was given by H . Wu [461 In its most simple form this theory investigates the distribution of the zeros of linear combinations .

A0fo(z) +

+ Anfn (z)

of finitely many integral functions w. = f.(z) for different systems of con] in other words, this theory analyzes stant multipliers A = (A0, ... ,An) Pn the position of a non-degenerate meromorphic curve C relative to in complex projective space P. +A w the hyperplanes A 0 w o + = 0 n n The theory of holomorphic curves by Weyl-Ahlfors was further extended in a very general way to a higher dimensional theory first by W.Stoll [561 and then in a different direction, stressing Hermitian differential geometric aspects, by x

H.I.Levine [431, S.S.Chern [101 , R.Bott and S.S.Chern [7] and other au thors. In 1972, introducing once more fascinating new ideas, the Ahlfors-Weyl theory was extended in a different direction, more regarding to algebraic ge ometry, by J.Carlson and P.Griffiths [15] to equidimensional holomorphic mapCm_ V , where V pings is a projective algebraic variety and where m m you are interested in how the image meets the divisors on V m . This theory was further generalized in the same direction by P.Griffiths and J.King [30] to the study of holomorphic mappings

is an algebraic, V a projective algebraic variety. Given an algebraic subvariety Z C V , the 2 basic questions which are treated in this setting are in analogy to Nevanlinna theory: (A) can you find an upper bound on the size of f 1(Z) in terms of Z and the "growth" of the mapping f (B) can you find a lower bound on the size of f(Z) , again in terms of Z and the growth of the mapping. The most important special case of this problem is when A = Cm and V = Pn , the complex projective space. Then f may be given by n meromorphic functions where

A

f(z)

_

(f1(z),...,fn(z))

The subvarieties pa(`°1' *,wn) the equations

,

z = (z1,. .. , zm) E Cm

will be the zero sets of collections of polynomials and so the questions amount to globally studying solutions to Z

pa(fl(z) ...,fn(z))

=

0

Concerning the extensions of Nevanlinna's theory for holomorphic mappings between Riemann surfaces, we refer to L.Sario and K.Noshiro [30] and to the more Hermitian differential geometrical versions of S.S.Chern [11] and H.Wu [62] . We disregard here the several extensions of Nevanlinna theory to certain classes of non-holomorphic functions, note however the extension of E.F.Beckenbach and G.A.Hutchison [4] to triples of conjugate real harmonic functions, as it rests on a similar underlying idea as our developments. In this Research Note I am presenting an extension of the formalism of Nevanlinna theory to systems of n '_ 1 meromorphic functions f 1(z), f 2(z) , ..,fn(z) in a way, which is fundamentally different from the theory of holomorphic or meromorphic curves of Weyl-Ahlfors and its higher dimensional ai

generalizations. As in Nevanlinna theory again the starting point is a formula, this time a generalization of the formula of Poisson-Jensen-Nevanlinna, which I discovered in 1964, when I was trying to extend the Nevanlinna formalism to the simultaneous solutions of systems of n equations

where

f1(z)

=

a1

f2(z)

=

az

f11(z)

=

an

wj = fj(z)

,

zEC

,

j=1,

,n

are

aIE C

,

n

1

,

, anE C

meromorphic functions.

We note that already G.Polya [52] and R.Nevanlinna [47] have studied functions with respect to values they assume at the same points; contrary to the present study, however, these values were taken to he one and the same complex number a1 , and they investigated the condition under which necessarily f1(z) = f2(z) I succeeded in extending formally both the main theorems of Nevanlinna theory, together with the Nevanlinna deficiency relation. Although the above system of equations has only solutions for points a = (a1, ,an) E f(C) , a set which is rather thin for n > 1 , these results seemed to be quite inter esting. However, one difficult main problem was still to solve; the problem of finding the true geometric meaning of the extended quantities, a problem which was proposed to me by Helmut Grunsky and by Rolf Nevanlinna in 1967/68. It took me several years to find its solution, and was finally achieved partly in my doctoral thesis and partly in my Habilitationsschrift, on which the present Research Note is based. in addition The main difficulty was the appearence of a new term V(r,a) to the generalized Nevanlinna value distribution quantities m(r,a), N(r,a) T(r,f) in both the extended First and Second Main Theorems. I then and tried to compare my result with the totally different theory of Weyl-Ahlfors and its extensions and with the more recent developments of complex manifolds and Hermitian geometry. So I gradually found out that the notions of the m dimensional complex projective space Pm , its Fubini-Study Kahler metric and the complex differential geometry of Hermitian line bundles, which are now a central fact in the theory of holomorphic or meromorphic curves and in generalized multidimensional Nevanlinna-Weyl-Ahlfors theories, are also the key for the xii

proper understanding of the geometric meaning of the main new term V(r,a) , and even for the interpretation of my "generalized Ahlfors-Shimizu characteristic function" T(r,f) . A fundamental role is played by the curvature form, whose cohomology class represents the characteristic Chern class of the hyperplane section bundle over complex projective space. While the theory of holomorphic curves by Weyl-Ahlfors and its generalization give results for the problem of how often the image of the mapping meets a set of hyperplanes or subvarieties in the image space, the theory of the present Research Note does not render results in this direction when n > 2 In contrast to this we study the growth of the projection of the curve Pn-1 C - Cn into f(z) - a .

and the connection of this growth with the distribution in to the system of equations f(z)

=

C

of the solutions

a

whilst the point a varies over Cn , a problem which has no effective counterpart for n = 1 . The theory reduces to Nevanlinna theory if n = 1 and stays in close contact with the original Nevanlinna formalism if n >_ 1 The readers of this book must have hardly any prerequisites from Nevanlinna theory, but only a good lesson in function theory. An advantage, however - even a necessity in some places - would be a little familiarity with the sources R.Nevanlinna [46] , [27] or [28] . In Chapter 1, §2 , Chapter 3, §9 and in Chapter 5, §15 some background knowledge of fundamental ideas from Hermitian geometry and algebraic geometry might aid the deeper understanding of certain formulas. The interested reader will find some hints on these matters in the Appendix and further details in the literature of the Bibliography. With pleasure I thank W.Helmrath and Professor R.Schark for correcting part of the manuscript, but above all I want to thank my wife for helping me with the literature and having all the patience during the time of writing. Last but not least, I would like to thank Pitman Publishing for their excellent cooperation.

Siegen, West Germany April 1982

Hans J. W. Ziegler

xiii

1 Extension of the first main theorem of Nevanlinna Theory and interpretation by Hermitian geometry §1

Generalization of the formula of Poisson-Jensen-Nevanlinna

We denote by

the usual

Cn

the coordinates

n dimensional complex Euclidean space with (w 1, ... , wn) , the Hermitian scalar product

w



v1w1+ ... +vnwn

(v,w E Cn)

and the distance 1

liv - wM

2

+

_

Let w1

f1(z),

=

.

.

.

wn

.

fn(z)

=

complex valued functions of the complex variable meromorphic and not all constant in the Gaussian plane C1 finite disc be

na1

CR

{*

=

Thus in CR , 0 < R 5 valued meromorphic function (f1(z),

f(z)

c

R}




E

m GE

-

G

=

U dE

U(z)

,

q=0 q

=

log If(z)II

Green 's formula aU

ag

g - ds

CU - -

(UAg

-

g AU ) dx Ady

an

an a GE

GE

}

is valid, where the derivatives under the integral of the left hand side are with respect to the inner normal of the boundary D G E of G F- . Because of Ag(z,z0) = 0 (z E GE) and g(z,z0) = 0 (z E IG) this can be written m

Ig(z,z0) ds

U

an

ag(z,z0) U

+

an

- g(z,z0)

au \ an

/Ids

a dE aq

aG

g(z,z0) AU(z) dx A dy

(1.2)

GE

Since

the circle

is harmonic in

g (z, z0) + log I z-z0I

C

-

g

the integral over

ds

an

3n

aU\

alogIz-z0I lim E-> 0

,

au --

ag

U

=

G

satisfies

a dE

U

(-

+

an

loglz-z0

) ds

an

Here the integral over the second term on the right side tends to zero as E->0 The counter-clockwise traversed circle a do has the parametric representation

z

z0 + Ee -1

=

increases from

as

0

to

. From

27r

alog Iz-z01 and

1

ds

E

an

we obtain

2n

r

slog Iz-z0

U(z0+Ee io ) do

ds

- U

(1.3)

,

an 0

ad E

so that 1

lim

9g

au

an

an

U - - g - ds

E+ 0

(1.4)

- 27rU(z0)

ado

In the punctured vicinity of

z

q

(q=1, ...

m)

we have the develop -

ment x

_

Ilf(z)M

1Z-z

q

I

q v q (z)

v

q(Zq)

X

0,+m

where Aq is a positive or negative integer of absolute value equal to the is Cm with respect to x,y multiplicity of z , and where V (z) q q z = x + iy around q q q Now, for q = 1,...,m ag

log V (z) - q

g

an

alog Vq (z)

ds

0

(1.5)

an

ad E

q

since the integrand is continuous in

zq

. We conclude that for

q=1, ... , m

-

ag

lim

E-0

an

g-aU

ds

an

adE Fq

alogIz-z

ag lim

X log z-z I - - gA q q an q

E+ 0

I

q

(1.6)

ds

an

adE

Fq

Since the integral of the first expression on the right hand side tends to zero, we get, repeating the argument which led to (1.4), for (1.6) the limit -

2,r a

q

g(z ,z0) q

.

Using this we obtain from (1. Z) and (1.4), letting

E+0

Ig(z,z 0) log If(z)II

log IIf(z0) II

an

ds

DG

m F, Agg(zq'z0) q=1

g(z,z0) A1ogIIf(z)II dx A dy

(1.7)

G

(1.7) holds even if

z0 E {z1, ... ,z}

.

since then both sides become infi= +m . nite; here, if zq is a pole of f (z) , we understand Il f (zq) II Also, (1. 7) remains valid if there are zeros or poles of f (z) on a G , this can be easily seen, modifying the proof by indenting the boundary suitably at the singular points and by performing a limit process, taking into account the mild logarithmic nature of the singularities. Substituting z for z0 and r for z , and denoting by zi (0) and z. (-) respectively the zeros and poles of f (z) in G , each counted according to its multiplicity, we have for any zEG the basic formula ,

log If(z)II

=

ds

log IIf(c)Ij -

2,r

an

3G

E g(z0),z)

z

E

+

g(zj(W),z)

z.(") E G

.(0) E G J

7

1

g(r;,z)

27T

A

G

If in particular we choose Green's function with pole in log

In order to compute a(E,z)

z

is

rZ

-z

:

_

{

z


- 2

Because of this geometric meaning the

V(r,a,f)

will henceforward be called the "v o 1 u m e

f u n c t i o n" associated with

the meromorphic vector function f(z) . Of importance will also be the "little volume function" v(r,a) = v(r,a,f) .

Now the "counting function" of finite or infinite

a - points of Nevanlinna theory will be generalized to vector valued meromorphic functions by the following definition:

r

N(r,a) = N(r,a,f)

=

n(t,a) - n(0,a)

n(O,a)logr

dt

+ t

r

N(r,f) = N(r,W)

=

n(O,f)logr

n(t,f) - n(O,f) +

dt t

0

Applying this notation and (3.4),(3.5),(3.6), we now rewrite (3. 3) , observing that in (3.3) the number q = n(0,0) - n(0,f) is equal to the difference of the number of zeros and the number of poles of f(z) at the origin. The result is the following identity, which because of its importance we formulate as a theorem. Theorem

3.1

:

Let

w = f(z)

=

(f1(z) , ... , fn(z))

vector function on CR (0 < R 5 + cc) . Then for 0 < r < R the following identity holds

be a meromorphic a E Cn , f (z) it a

2n

alldq + N(r,f) = V(r,a) + N(r,a)

1

2r

+

logllcq(a)lI (3.9)

0

here cq(a) is given by the development (3. 2) ; in particular, if z = 0 is not a pole for f(z) and if f(0) x a , then simply c (a) = f(0) - a q

Note that this identity gives a differential geometric interpretation of the integral mean in the left hand side of (3.9). Remark 1 : In the special case n = 1 and a = 0 formula (3.9) boils

down to a formula of Nevanlinna theory, for which H. and J. Weyl use the name "condenser formula" t, refering to the electric potential of a circular condenser. Remark 2 From formula (3.9) follows immediately that the mean value 2 7r

1

11(r,a)

.

log IIf(rei ) - all d¢

2ir 0

is a continuous function of r for 0 < r < R , since all other expressions in (3. 9) are continuous in r Remark 3 Up to the r - values such that on D C r lie poles or a-points of f , this mean value is differentiable with respect to r and .

2n

r

a

1

ar

2 rr

all d + n(r,f) = v(r,a) + n(r,a)

log j 0

f(z) _ (f1(z),...,f(z))

is a vector valued integral function such that f (z) a E Cn . Then it follows from (3. 9) and from remarks 2,3 that the mean value ' (r, a) is a continuous, non-decreasing and convex function of log r for 0 < r < R f (z) = (f 1(z) , ... , fn (z)) is a vector valued integral Remark 5 If function without common zeros of f 1(z) , ... , fn (z) , then up to an additive constant the integral mean Assume that

Remark 4

:

2,r r

(r, 0)

=

dO

2n d

0

arising in (3.9) for a = 0 equals the characteristic order function of H. and J. Weyl for the corresponding meromorphic curve in P n-1 tt . If f 1(z), f 2(z) , ...,f n(z) are arbitrary meromorphic but not all identically vanishing, then

Weyl [43],p.73

t*Weyl [43]p.81 21'

in view of (2.4), (2.5) and (3.8) the characteristic order function of H. and J. Weyl t equals V (r, 0) . With the usual abbreviation

max (log x ,

log x

(x > 0)

0)

we have the decomposition log X

logx -

=

log X

so that we can substitute in (3. 9) 1

log !If - a I

=

- log

logIlf - a 11

We introduce the following generalized quantities of Nevanlinna theory: 2,r

m(r,f)

m(r,W,f)

=

1

_

d*

2n 0 27r

f

m(r,a)

1

1

m(r,a,f)

__

d

log

2,r

,

(a

all 0

T(r,f)

m(r,f) + N(r,f)

=

For finite or infinite

a

is thegeneralized Nevan-

m(r,a)

linn a proximity f unction" ; vergence of the vector function

it measures the asymptotic conT(r,f) is the to the point a

f

.

"generalized Nevanlinna characteristic funct i o n " . Further, from the general inequality log (x1 + x2)


0)

we obtain logllfll

=

tWeyl [431,p.142 22

logll (f-a) +all

logllf-all

+

loglIall

+ loge

and

logIIf-all


0

dt + n(0,f)logr0

0

and the same holds if f is replaced by hand inequality of (7.8) we conclude that

48

. Then we can write for

f

r0

f. J

(j=1, ... n)

.

Using the right

r

n

N(r,f)

n(t,f.)

Z


r0 , such that for

r > r1

dt

s(r)

s(t)

to +1 dt

s(r)

u ru

r which shows that

r . Now assume that

pp

.

In this case we put

such that

tp+E

s(t)

t o +1 dt


0) to be of order p (0 < p < +c) is that the integral

Proposition 8.3 s(r)

:

s(t)

J +1 dt

(8.1)

r0

00

is of order .

s(r)

(r > r0) is a positive function of finite order p , then the p=p In the first case integral (8. 1) can be convergent or divergent for we say that s(r) belongstotheconvergence classoforder p, If

.

in the second case it belongs to the divergence classtoforder p. is increasing and of convergence class of order 0 < p < +m then the first part of the reasoning above Lemma 8.3 shows that s(r) must If

s(r)

T=0 . be of minimumtype i.e., For meromorphic vector functions in the plane

C

or in a finite disc

CR

tNevanlinna [27] 53

it is natural to define order and class exactly as in Nevanlinna theory case of the plane e.g., we give Definition 8.4 : The " o r d e r "

= (f 1(z),...,fn(z))

in

i.e.,

log r

r-> +m

o r d e r"

The " 1 o w e r

T(r,f)

f(z)

log T(r,f)

lim

P

In the

of a meromorphic vector function

p

is the order of

C

.

A

of

f (z)

is the limit inferior

log T(r,f)

lim

log r

r-> +m

0 < p < +W

If

, then the "type " and the " c l a -s s

of

f(z)

are

respectively defined to be the type and the class of T(r,f) A=p If , then f ( z ) is called of " r e g u l a r growth IT in the 0<E< sense of Borel If there exist constants E , F , such that

0 +m

then f ( z ) will be called of " v e r y sense of Valiront. If the limit lim r->+m

exists, then

lim

T(r,f) rP

f(z)

It' is clear that

54

F

rp

r e g u l a r g r o w t h " in the

will be called of "perfectly regular

rowth" in the sense of Valiron t

tValiron [37]

T(r,f)

0 p > max p. such that j inequality (8.4) and Proposition 8.3 we conclude Now choose a number

n

E k=1

p

T(t,fk) p+1

_

+ m

to >

0

,

t

t0

which is impossible, since by the same Proposition each integral 55

T(t, fk) k=1,

dt

.

n

to +1 t0 < is convergent. Thus we must have p max pj and observing (8.4) this proves (8.2). J (8.3) follows immediately from the left hand inequality (8. 4) .

Let

f

_ lim

r *+W

be of order

0 < p < +W

. Then from inequality (8.4) we have

_

--

T(r,f.)

'T(r,f)

rp

T(r,fk)

n

lim

0

/1 :

ii) q=0 :

)

I

=

(1)

f/

0

(1)

(c1

f

=

0 zq cq

+

z

+

\O1/

+

q+1

+

z2 ('2)

+,\

0

iii) q< 0

:

(f )

zq

(c0

q)J

+

+...

zq+1 q+1

+ 1

z \Q

1/

In these three cases respectively the expression

59

1

log 1 + If(r0eie)II2

r0 + 0

behaves as

log II

like i)

logVI

+

II

f

or

o(r0)or

ii) log

1If(r0eie)II2

+ 1110

1V

iii) -n(0,f)logr0

1I`

+

+

o(r0)

or

+ o(r0)

logllcgil

In the cases i) and ii) the point z = 0 is not a pole of f ; in the case iii) the point z = 0 is a pole of f In case i) we have f (O) = 0 = (0'. ..10) ; in case ii) we have . i) and ii) can be IIc0II = IIf(0)II considered together so that we have .

log 1 + IIf(0)ll2

finite at 1og"1

+

z=0

+ o(r0)

if

f

is

(cases i) and ii))

II f (roeie) II 2

-n(0,f)logr0 + logllcgll + o(r0) z = 0 is a pole of f (case iii) ) Thus, if r

dt (AlogVfl r0

is not a pole of

z=0

+

IIf(z)II2 dx A dy

=

Ct

r

2n

n(t,f)-n(0,f) t dt r

we obtain from (9. 3)

f

0

so that, letting r0

+

1

log 1 +

2n 0

tend to zero,

I

f(reie)I2 de - log V1 + IIf(0) II2 -o(r0)

if

r r

dt

AlogN/1

2 Tr tt

+ IIf(z)II2 dx - dy

J

J

Ct

0

2 Tr

N(r,f)

1

+

2,r

1logVl

+

IIf(reie)II2de - log 1 + I1f(0 )II2

(9.4)

0

If on the other hand r dt

f

, then we get from (9. 3)

1 + If(z)II2 dx A dy

Alog

2,r tt

is a pole of

z=0

0t

r0

2n

r r

n(t,f)-n(O,f) dt + n(O,f)logr

+

logN/l

2n

+

If(reie)I2 de

0

r0

-log IIcgII - o(r0)

and letting r

r0-r 0

,

r dt

0

1 + IIf(z)I2 dx A dy

Olog

27T t

0t

2n

N(r,f)

+

1°g /1 -L 2n I

+

If(reie)II2 de

-

log IIcgII

(9.5)

1

0

Now the function

61

271

r

1

logV/i + IIf(rei0)II2 d0

2n 0

which appears in (9.4) and (9.5), behaves asymptotically very similar to m(r,f) ; it can serve equally well as m(r,f) as a measure for the mean approximation of f to infinity on circles ac r . We can therefore introduce the following modified proximity function 0

0

m(r,f)

=

m(r,W,f)

with respect to infinity: Definition 9. 1 27T

0

1

m(r,f)

log1 + IIf(Yeie)II2d0 - logV1 + 11f(0)II'

2 it

(9.6)

0

if

z=0

is not a pole of

f

, and

2,r r

0

1

m(r,f)

1 + Iif(rel0)112de - log IIcgII

log

2ir

(9.7)

0

is a pole of f . Here cq is the first non-vanishing coefficient vector in the Laurent development of f (z) at z = 0 if

z=0

.

With this definition (9.4) and (9.5) can be written in this unified form: r r

r

0

m(r,f)

+

dt

N(r,f)

elog

2nt 0

1 +

IIf(z)II2dx A dy

Ct

Now we already observed on p.26 that in the scalar case

62

.

f (z) = f 1(z)

(9.8)

4 If(z)12 nlog(1 + If1(z)12)

_

(1 + Ifl(z)12)2

and that 1

slog /1-1 Ifl(z)12 dx A dy

2

j Ct

is the spherical area of the Riemannian image of Ct under the mapping f 1; and in this case the sum in the left hand side of (9. 8) is called the characteristic function of Ahlfors and Shimizu or the spherical characteristic of fI We now try to find in the present vector valued case the correct geometric meaning of the expression 1

AlogV1

2

+

If(z)II2 dx A dy

Ct

occurring in equation (9. 8). We compute

2 nlog% 1 + IIf(z) I2 dx A dy a

Z.4.

i

aa

2

log

1

+ Ilf(z)I

azaz log/1 + Iif(z)II2

i

2

Z dz A dz

a a log(1

+

Efi fJ

=

dz

a

+ Ekfkfk E_'k1kk

i

(1 + Ekfkfk)( Ekfkfk) - ( Ekfkfk)( Ekfkfk) 1 + Ekfkfk)'

dz A dz

(9.9)

tAhlfors [11, Shimizu [541 63

Here each sum is extended from

is sitting in

Cn

Now

Cnc Pn

k=n as an open set, and we have the inclusion map k=1

Pn

to

.

,wn) -- (1,w1.

(w1,

Pulling back by the inclusion

.wn)

ti

Cn C Pn

we obtain the " F u b i n i

the Fubini-Study metric on Pn -Study metric on Cn " t. It is given

by

1 + Ekwkwk) (Ekdwk QQ dwk ) - (Ekwkdwk ® (Ekwkdwk ) ds2

=

1+

Ekwkwk

)2

(9.10)

Its Kahler form is i w0

Z

(1 + E k w k w k ) ( Ekdwk A dwk )- ( Ekwkdwk ) A (Ekwkdwk (1 + )2 E w w k k k

4 ddclog (1 + Ekwkwk )

If ds2

(9.11)

for example, then P1 is the usual spherical metric on

is the Riemannian 2-sphere

n=1

ds

2

dw

C

,

x( dw 1

(1 +

1

w

1"`1) 2

which is a conformal (Hermitian) metric of constant Gaussian curvature and the associated Kahler form

t

S2, and

dw 1 A dw1 (1 + w1w1)2

4

dul A dv1

(I

+w1w1)2

(wl=u1+iv1)

is the spherical volume form.

We now return to the general case is the pull-back of w0 to CR

tWu [63] 64

n?1

.

(9.11) and (9. 9) show that

Alog

1'V

If(z)

+

1

2dxA dy

(9.12)

This shows that in the general vector valued case

1 + 11f(zo dx A dy

Alog

2

the integral

n?1

f*u

=

(9.13)

J

Ct

Ct

is nothing but the v o I u m e of the image of

z -+

Ct

in

Pn

under the map

(lrfl,...1fn)11,

(If (9. 13) is divided by it we obtain the normalized volume ) . We now return to equation (9. 8). At this point, in view of equation (9.8) and the given geometric interpretation (9.13) of its right0 hand side, it is natural to introduce a new modified characteristic function T(r,f) by

T(r,f)

:

0

where

0

0

Definition 9.2

m(r,f)

We will call

.

=

m(r,f) + N(r,f)

is given in Definition 9. 1 0

T(r,f) the"generalized Ahlfors -Shimi-

zu characteristic "orthe "generalized spherical c h a r a c t e r i s t i c ", since it agrees with the characteristic of Ahlfors Shimizu in the scalar case. Summarizing, we can formulate this result: be a vector valued meromorphic function on CR Then denoting by w0 the Kahler form (9.11) of the Fubini-0Study metric on Cn and defining the generalized spherical characteristic T(r,f) by Defi nition 9. 2 , we have the formula : Theorem 9.3

:

Let

f

.

65

r f

0

dt t

1

T(r,f)

2,r

Olog%/1

+ IIf(z)II2 dx A dy

6t

0

r r

1

dt

71

t

(f x const.)

f WO

(9.15)

.

Ct

0

0

This geometric interpretation of T (r , f) gives also a quasi-geometric interpretation of the generalized characteristic T(r,f) of Nevanlinna because of the following

Proposition 9.4

T(r,f)

:

0

T(r,f) as

r-R

T(r,f)

=

0

T(r,f) only by a bounded term:

differs from +

0(1)

(9.16)

,

.

0

Thus, in many investigations T can be used instead of T and vice versa, without any changes of formulas. The estimate (9.16) can be seen as follows. We have 2 Tr

0

1

m(r,f)

log V1 + Ilf(reie)II2 do

27T

-

d

0

where

log'1l

d

if

IIf(0)II2

is not a pole for

z=0

pole for

+

f

.

f

, and

d

=

z=0

if

log IIcg1I

Since logv1 + X11112

log lIfII

log IIf1I

_

+

1ogV_

it follows that

m(r,f)




0

We summarize:

Theorem 11.2 (First Main Theorem in Spherical Formulation):

(fI(z) , ... ,fn(z)) identity 0

T(r,f) holds for

is a non-constant meromorphic function in 0

=

0R

S 2n

Since the integrand here is non-negative, we conclude that almost everym on Stn lim

r>R an

1

-

V(r,a)

V(r,a) =

T(r,f)

1

- lim r-rR T(r,f)

=

0

i.e., we have

Corollary 12.3

:

V(r,a) lim

(12.9)

0 r-R T(r,f)

holds almost everywhere, unless 0

T(r,f)


0 such that the series .

q+1

E

is convergent. Let

E(u,p)

_

(1-u) e

u+ u2 2

+

UP

p

E(u,0) = 1-u

denote the prime factor of Weierstrass. If the sequence ite,it can be shown that the infinite product

(13.1)

z1, z2, ...

is infin-

TTE(zz q) V

converges uniformly in each bounded region of the plane C and represents If the thus an integral function which vanishes exactly in the points zv sequence is infinite and the series .

E

1

z

q

v

is divergent, then q >_ 0 is called the g e n u s of this infinite product. If the sequence is finite, the genus of the corresponding finite product with q=0, which is a polynomial, is defined to be zero. Now let f1(z) be a (scalar) non-constant meromorphic function in the

plane. Let z11(0),z21(0),... and zll(m),z21(-),... respectively denote the zeros and poles of fI(z) outside the origin z = 0 . Let q >_ 0 be an integer such that the integral

T(r,f1) (13.2)

dr

r

q+2

is convergent. Then the series q+1

1

2:

E

and z)j 1(0)

both converge. Let

zV 1(c0)

be the smallest integer

k1

q+1

I

such that the series

>0

k1+1

1

zu 1(0) I

converges, and let k2 be the smallest integer ? 0 such that the analogue series for the poles z\; 1(') converges. Then clearly k1 < q , k2 < q Nevanlinnat proved the following fundamental representation of f 1(z) . Under the above conditions the meromorphic function has the representation Theorem 13.1

:

z

z

Pm e

1

(z)

,kl)

1

TTE

r1

f1(z)

zU 1(0) (13.3) z

TTE (

where

r1 > 0

is an integer and

PM (z)

,k2

zV 1(-)

is a polynomial of degree m1

1



0)

(13.6)

r0

is convergent or divergent. If this integral is convergent, the genus is If this integral is divergent, the genus is g1 = pl 1 , provided that the series gl 11 - 1 .

84

PI

p1

1

1

E zp 1(0)

Z zvl(x)

and

both converge; and the genus is series is divergent.

gl

p1

if at least one of these

Thus the genus g 1 of f 1 is determined uniquely by the order p 1 alone, with the exception of the case where T(r,f1) belongs to the minimum type of the integer order p 1 > 1 and the integral (13.6) diverges. We can now apply these results of Nevanlinna to vector valued meromorphic functions in the following way. First we propose to define the genus of a vector function by

,.

be a meromorphic vector function of finite order in the plane C ; we assume that f 1, ... Ifn are all zero . Let gj (j=1,...,n) denote the genus of fj(z) . Then the " g e n u s g o f f(z) " is defined to be the integer Definition 13.2

f (z) = ( f 1(z) (z)

Lpt

:

fn (z))

max g

g

1

From result (i) above we have the inequalities g.

g. + 1

P.

Since the order of

f (z)

is

, p

(j=1,...,n) max P. J

13.2

we obtain from Definition

I

Proposition 13.3 : The order p and the genus g of a meromorphic vec tor function f (z) (f,, ... I fn Ft zero ) of finite order in the plane satisfy the inequality g

p


>

T(r,f)

T(r,f) lim r->+m

=

and

0

rp

r P+1

dr

=

r0

Applying inequality (13.8) we get

T(r,f.) lim r->+m

0

rp

and there is an integer

j0

for which

(j=1,.. ,n)

p.

=

0

p

and

+00

T(r,fj

)

(13.10)

dr

0

r0

From the result iii)b) of Nevanlinna we conclude that least one of the two series p

I

E

=

p. JO

if at

p =

p

1

E zv.

and

zuj0 (0)

g. JO

(oo) 0

zuj (0) and z vj (co) respectively denote the zeros diverges, where and poles of f ].0 outside the origin. In Othis case we conclude from the general inequality g

max g

=

that

p

g

p

J

and (13. 10) is Now assume that for all indices jo , for which p. = p J satisfied, both series (13.11) converge. Then by result iii)b) of Nevanlinna we have gJ . = p - 1 for these indices j0 . For indices j l , for (13.10) is not satisfied for jo replaced by j 1 , we have either p is of convergence class. In the or we have p. = p and f. J1 J1 case, since f, is of minimum type, we p lf iii)b) of Nevan linna. In the former case have
0 the

r

1

logM(t,f) dt

r

C(k) T(kr,f)


1

Again the proof rests on the generalized formula of Poisson-Jensen-Nevanz = tell linna. Similarly to the previous reasoning we have for 27T F

t2 d6 2 + t - 2st cos(6-4) 2

logllf(Seie)II

2n

2

s 0

Iz

E .(-)I< s

log

J

Here for

t 1 there exists a sefor x ' x0 such that if h(x) is any other positive increasing and x such that h(x) < g(x) for x>x0 , we have

Suppose that

vex function of xj -r +quence convex function of

x

h'(xj)

Here

h'(x)

derivative of

g(x)

eQ g'(xj)


1

.

109

5 Extension of the second main theorem of Nevanlinna Theory §15

The generalized second main theorem

f(z) = (f 1(z),...,fn(z))

be a meromorphic vector function in was shown in §9, Proposition 9.4 that the characteristic functions

Let

T(r,f) = m(r,f) + N(r,f)

and

CR . It

0

T(r,f)

_

Ct

differ by a bounded term only; here c,b denotes the Kahler form of the Fubi0 ni - Study metric on Cn . So T(r,f) as well as T(r,f) can be thought of as measuring the volume of the image of the disc C r under the mapping f (z) in Cr' , equipped with the Fubini - Study metric. Associated with the curve f(z) is its Gauss map ti

f'

CR

P n-1

which is a holomorphic curve in

(15.1)

Pn-1

and is defined by projecting the de-

rived curve f'

=

(f...... fn

.CR

Cn

Pn 1 by natural projection, and by extending the result holomorphically into the poles of f and into the common zeros of f...... fn . In the case of the original Nevanlinna theory n = I of course,the Gauss map is 0 useless, since P is a point. The Gauss map associates in particular with each tangent plane at each point f(z) E Cn of the complex curve f the point, which this plane defines in Pn-I ; this notion of Gauss map generalizes the corresponding notion of differential geometry in R3 , where to each tangent plane of a surface,is associated the point, which the normal defines on into

the unit sphere. According to Chapter 1 (formulas (3.8) and (2.4)) the function

110

r

v(t,0,£)

V(r,0,f)

dt t

with r 1

v(t,0.f)

1

f w

n

Ct

at

where Pn-1

1

w

.77

Alog IIf(C)11 do A dT

2 7T

is the curvature form of the hyperplane section bundle H over of the image of the disc

measures the volume 7rv(t,0,f) in Pn-1 ti Ct under the mapping f So, if we define the function G(r) = G(r,f) r r

G(r)

=

V(r,O,f')

dt

=

0

dt

if*w

t

r

r

r

r

by putting

Clog 11f'(E )1I do A dT

2-Wt

IT

Ct

Ct

0

(15.2)

pn-1 G(r) measures the volume of the image of the disc Cr in ti under the Gauss map f' , if n a 2 , and vanishes if n = 1 ;(this volume corresponds in differential geometry of R3 to the area of the spherical image under the Gauss map). A second geometric interpretation of the function G(r) can be obtained

then

as follows.

D CC

In general, if on a domain h where

=

g dw(2) dw

=

g

(du2+ dv2)

is a positive C - function on

g =

a Hermitian metric assumes the form (w = u + iv) D

, then its volume form is

g duAdv

and its Kahler form is n h

=

gdwndw

so the volume and Kahler forms are equal. The Gaussian curvature K of the metric

h

is defined by 111

K

g Alog g

where A

a2

a2

au2

3v2

_

a2

4

aw aw

is the usual Laplacian. Hence _

a 2log g

Ki

- 2 ddclog g

dw A dw

i

aw aw

We now define the Ricci formt Ric i Rich

s log g

a

27T

of the volume form

t by putting

4 ddclog g

=

(15.3)

Thus the formula Ric

K

1

_

(15.4)

2

is valid. We will now apply this. By the map f

Cn

'CR

the flat metric ds2

of

Cn

=

induces on

dsf

=

dz1 Q dz1 + Co

CR

.

.

.

+ dzn Q dzn

the pseudohermitian metrictt

IIf'(w)II2 dw®dw

,

A

which is Hermitian on

OCR

. According to the above its volume form dsf =dAf

is

dAf

on

2

If'(w)II2 dw A dw

OCR

tCarlson and Griffiths [151 112

By (15.3) , (15.4) the Ricci form of this volume form is Ric dA f

2-' K dAf

2r ddclog IIf'(w)I

(15.5)

where K

- IIf'(w)II-2Alog IIf'(w)II

=

is the Gaussian curvature of the metric dsf2 . Since log IIf'(w)II is sub0 as harmonic on OCR we remark that on OCR , K 2 (v = 1,...,p) and put

distinct finite points

av

=

(av,...,an)

E

1

F(z)

_

pp

L. IIf(z) - av v=1

From the inequality log (x1x2) we get for

log x1

+

1IgF(re')d j 0

(15.14)

0 < r < R

2n

2,r

(xl,x2 ? 0)

log x2

2 Tr

m(r,O,f') +

log{F(re'o) Ijf'(re1')jj1d

r

.

(15.15)

j

0 115

Put =

6

min I I al - a1 1

1

ixJ

p E (1,2 , ... ,p}

Let for the moment where

IIf(z) - a"II


- 2)

(15.16)

the inequality IIf(z) - av II

='

6

Ila" - av II - IIf(z) - a" II

>

6

- 2p

36

4

(since p> 2)

. Therefore the set of points on ac r which is determined by (15.16) is either empty or any two such sets for different " have empty intersection. In any case for

vxu

2n

P 1

1ogF(rei$)do

2n

Zn

E u =1

1og F (rel$) d$ J

0

I l f -a"

< 2p

P >

1

1

log

2n

Ilf(rel$ ) - au II IIf-a

P I
0)

,

(15.22)

]

J

j=1

j=1

gives 2,r

2-T

r 1

p

log{F(rel") IIf -(rel') II)do

2n

5

log

2,r

k=1

j

j

i IIf(re

)

- a

kII

do + log p

0

0 so that

2,r

2 7r

+

S1(r)

log

5

II f' (rein ) II

IIf'(reio)I) log

+

If(reil)I!

k=1

I 0

0

+ log p

+ p log 6 -

log IIci.

do II f (re 10)

ak

(15.23)

I

We now need the following important Theorem of Nevanlinna theory.

Suppose that the complex scalar valued function 4(z) is is finite and not zero, then for all r c0 = 4(0) meromorphic in CR . If and the inequality s (0 < r < s < R) Theorem 15.2t:

m(r,

)

24 + 3log


0} is at most countable and summing over all such points we have 2] [6(a) + 0(a)]

+

6

1 8(a)



3

and in particular at most 2 points with Corollary 16.2 we see that there is at most d (a)

6 (a) 1

>

;

3

finite point

analogously from

such that

a

1-dG >

2

and in particular at most 1 One can pose the general

138

finite point such that

d (a)

>

1 2

{a Given sequences (a 6 G > 0 , such that and a number

Problem

:


n(+o,a) Example 1. As an example we consider the entire rational vector function (z3

f(z)

_

z5)

,

5 3 Here every point a E C2 not of the form z0 cc , is a = (z0 , z0) z0 * 0 not assumed. In every point the function f(z) has an a = 3 5 (z0 , z0) - point of multiplicity 1 , and in z0 = 0 it has an a = (0, 0) point of multiplicity 3 . Let us first examine the deficiency relation (17. 10) We have *(o) = 5 n(r,a) = n(r.a) up to the point a = (0,0) where n(+',a) = 3 n(+m,a) = 1 so that the deficiency relation ,

,

(17.10) is

2

5

+

d

i.e.

1

G

On the other hand we can compute

f'(z)

(3z2

_

N(r,0,f')

=

,

,

m(r,0,f') = o(1)

G(r,f) = V(r,0,f') = 2logr + O(1) On the other hand,

144

=

3 5

directly as follows. From

we obtain using the first main theorem,

T(r,f)

G

T(r,f') = 4logr + O(1)

5z4)

2logr

dG

d

5logr + 0(1)

so that in fact

aG

For completeness let us compute the other equidistribution quantities for this example. We have T(r,f) = m(r,f) = 5logr + 0(1) , so that of d p(-) = 1 . For finite a we distinguish the following course cases. ae

(i) Here

V(r,a)

a 4 (z03

CZ

,

5

zQ)

m(r,a) = o(1) , N(r,a) = = 5logr + 0(1) , so that b V(a)

(ii)

=

a

=

0

=

(z0

,

6(a)

0

d (a)

=

(z0 E C)

,

, and by the first main theorem =

0(a)

z0) * (0.0)

Here m(r,a) = 0(1) , N(r,a) = log r + theorem V(r,a) = 4logr + O(1) so that

V(a) O(a)

(iii) Here

1

=

4

-

4 + 1 =

, and by the first main

5 =

1

6(a)

=

0

0(a)

=

5

a

(0,0)

=

m(r,a)

=

V(a)

1-

=

=

1

n(0,a) = 3 , N(r,a) = 3logr V(r, a) = 2logr + 0(1) , so that

0(1)

first main theorem

5 (a)

=

0(1)

and by the

,

2

3

=

5

©(a)

1-

=

5

2+3

_

5

0

,

0(a)

=

2

1

2

=

5

3-1 5

=

2 5

As compared to scalar Nevanlinna theory, where we have to consider the value distribution quantities 6(a)

,

0(a)

,

©(a)

we have in the vector valued theory the additional quantities 6V(a) and SG. The relations between these quantities in the vector valued theory are more 145

complicated than in the original Nevanlinna theory, and it is useful to examine a little the interdependence of some of these quantities, in particular under special assumptions. For the point a we have from the definitions and from V(r,°°) = 0 6(W)

=

- lim

1

r-+R

p(er)

- lim

1

=

N(r,f) T(r,f) N(r,f)

r4R T(r,f)

N(r,f) - N(r,f)

6V(-)

e(W)

1

=

lim r-+R

T(r,f)

and f (z) = (f 1(z) , ... , fn (z)) be a meromorphic vector Let Proposition 17.1 function in CR . Then the following conclusions hold.

N(r,f)

N(r,f) 1

lira

0

=

lim

,

G(-)

1

=

e(-)

,

©(°°)

lira

1

=

e(m)

=

r+R T(r,f) 1

lim r-+R

.

r-+R T(r,f) =

0(m)

,

1

T(r,f) =0

lim

,

=

1

N(r,f)

N(r,f) 0

5

0

lim

0

=

N(r,f) - N(r,f) (iii)

=

N(r,f)

N(r,f) (ii)

0

=

r-+R T(r,f)

r-+R T(r,f)

lira

,

=1

r-+R T(r,f)

r-+R T(r,f)

N(r,f) (iv)

(v)

146

6(Cc)

e(o°)

0

=

lira

r-+R T(r,f) =

0

_ N(r,f) lim

r-+R T(r,f)

=

1

,

1 (co)

=

0

0.

_ N(r,a)

N (r,a) (vi)

O(W)

lira

0

=

urn

r+R T(r,f)

r+R T(r,f)

We also note

Proposition 17.2 Let f(z) = (f1(z),...,f(z)) be a non-constant meromorphic vector function in CR . Then the following inequalities are valid for a E CnU{-} :

_ N(r,a)

N(r,a) - 6 (a)

S V(a)

lim

rr}R T(r,f)

N(r,a) lim

r+R T(r,f) N(r,a)

SV(a) - 0(a)

5

r+R T(r,f)

lim

5

lim

(17.12)

r+R T(r,f)

Proof of (17.11). m(r,a)

m(r,a) + N(r,a)

SV(a) - S (a)

=

lim

lira

T(r,f)

r+R

r+R T(r,f) - m(r,a)

m(r,a) + N(r,a)

lim

T(r,f)

r+R

urn

+

r+R

T(r,f)

The right side is

N(r,a) lira

r+R T(r,f)

N(r,a)

and

?

lim

r+R T(r,f)

This shows (17.11) ; (17. 12) is shown analogously.

From Proposition 17.2, from the inequality the definitions we deduce Proposition 17.3:

Let

morphic vector function in

8(a) + S (a)

CR

S (a)

=

1

or from

. Then the following conclusions hold.

V(r,a) (i)

0(a)

be a non-constant mero-

(f1(z),...,fn(z))

f(z)

5

N(r,a)

lim

=

r+R T(r,f) S V(a)

=

1

,

0

lira

,

0(a)

=

0

r+R T(r,f) =

1. ,

©(a)

=

0

147

(ii)

N(r,a) 6 (a)

N(r,a)

lira

0

=

6 V(a)

5

r-*R T(r,f)


R T(r,f)

_ N(r,a)

1-lim r--R T(r,f)

N(r,a)

` 1-lim r-*R T(r,f)

6(a)

_
_R

?

lim

r+RT(r,f)

+

6(a)

It shows that we must have

V(r,a) lira

r->RT(r,f)

if there are sufficiently many a - points so that the expression N(r,a) 6N(a)

lim

assumes the maximum possible value

1

.

In the case R = +ro we assume in the rest of this are non-constant. It follows then that f n

T(r,f.)

--b

152

as

0 < R < +m

In the case

T(r,fj)

+m

-->

+m

r-*+w

,

§

that all

j = 1,...,n

we assume for simplicity always that as

r-+R

,

j = I, ... n

f 1, ... ,

Using the estimate

T(r,f)

T(r,fj)

?

j = 1,...,n

0(1)

+

and the first main theorem of Nevanlinna theory we obtain the following 2 estimates for the same quantity SN(a) for a = (a1....,an) ECn j = 1,...,n ; k=1, ,n . N(r,ak}

N(r,a)

(i)

SN(a)

=

T(r,fk)

< Jim < Jim rr->R T(r,f) r-R

Jim

r7>RT(r,f)

T(r,f) T(r,fk)

lim

RT(r,fj)

N(r,a)

SN(a)

(ii)

=

lim

N(r,ak)

N(r,a) lim


RT(r,f)

5

lim

r->RT(r,f.)

r-*RT(r,f.)

T(r,fk) lira

(17.15)

Replacing N by N we obtain the analogous estimates for replacing lim by lim we obtain the analogous estimates for for A -(a) For we obtain a=m

N(r,f)

6N(-)

lim

r-;RT(r,f)

= lim

5 lim

(17.16)

AN(m) and

,

0(a)

,

N(r,ak) - N(r,ak) + 0(1)

< lim r+R

T(r,f) N(r,ak) + 0(1)

T(r,f)

0(a)

=

lim r-->R

o lim

T(r,f)

T(r,fk)

T(r,fk) lim

r-R T(r,f) lim

r+R

(17.17)

rr-;RT(r,f.)

N(r,a)
+oT(r,f) What can be said about the generalized Nevanlinna deficiencies 6 (a) = 6(a,f) beyond the general deficiency relation? Applying a selection of known results on Nevanlinna deficiencies to the component functions fe(z) of f(z) we will now give a few rather simple conclusions about generalized Nevanlinna deficiencies .

As was shown by R.Nevanlinna a meromorphic function

r(z)

of order

p

159

distinct values ar , b E Cu{-} such that 6(ar,r) = d (br, r,) = 1 only if PC is a positive integer or if p = +m Assume now that for a meromorphic vector function f(z) = (f1(z),...,In (z)) for 2 distinct points a = (al., ... an) E Cn, b = (bl, ... ,b n) E Cn 6(a) = 6(b) = 1 . Then each component we have maximum deficiency function f.(z) must have the same order p = p as f(z) by Propoj d (a) < 6 (a.,f.) 1 , 6 (b) 6 (b., sition 17.9. Because of the inequalities can have

2

0 . Theorem 17. 11 shows ] J that for each j = 1 , ... ,n the value a. is the only deficient point of is the only f.(z) . Thus inequality (17.27) shows that a = (a 1, ... , an) finite deficient point of f(z) Proof. Since

a

J

J

>_

J

{ak )R

k = 1, 2, ... be the set of finite deficient points a (a1,...,any ) E Cn for the vector valued meromorphic function f(z) This set is either finite or is countably infinite. For each j = 1, ... ,n and each 2, we have by (17.27) the inequality Now let

0


av Using these results, Proposition 17.19, Proposition 17.13 and the deficiency relation we deduce that if C

A

C,

C,

,

-

Y

6 (a)

-

=

1

aECn t

tt

Hayman [161 ,p. 104

Pfluger [49] -'-ttEdrei and Fuchs [201, [211 165

is an entire component function of finite order p and then p = AA. is a positive integer, which is equal to and all the deficiencies a. of f.(z) are integral J vJ -1 multiples of p Further, all the deficient values a. of f .(z) are J ] asymptotic values of f .(z) and the inequality holds and if f .(z) J of lower order A. J the order of f(z)

.

J

Y

0(a)

aEC

N(r,f)

6,

+




are differential forms in U m From the orthogonality relations (2) we get by differentiation 174

(4)

< dWA , WB >

CAB

+

SBA

< WA, dWB >

+

0

=

(5)

0

=

for any fixed T , the give OAB a basis for the left invariant Maurer-Cartan forms in U m (3) and (5) say that under infinitesimal displacement, the frame -F undergoes an infinitesimal transformation with coefficient matrix 0AB . They are the structure equations of a moving frame. Taking the exterior derivative of (4) we get using (3) Since

F)

WA(T

T WA(F)

=

.

dOAB

=

- < dWA,dWB >

C OACWC '

CD

< WC'WD >

D BBD

AC ^ ABC

0AC ^ BBD

'

so that by (5) (6)

X SAC A GCB

C

which are the Maurer-Cartan equations of the unitary group (3) and (2) we get < dW0'WO>

< dW0,dW0 >

U

m

. From (4),

X00


M-1

_ (7)

C

where the multiplication of differential forms is understood in the sense of ordinary commutative multiplication. From (7) we get m-1

dW0 , dW0 > - < dW0 , W > < WO , dW0 >

C1

0

(8)

175

. The vector

w c Cm-0

Now let WO

- wE Cm-0

=

(9)

IIwII

has length 1 . From (9) we compute IIwII dw - w dIIwII dW0

so that

-

1

< dW0 , W0 >

dllwll

,

IIwII2

< WO , dW0 >

1

dw > -

<w

1

IwII2 1

< dWO , dW0 >

< dw

,

dw> -

1

< wdliwli




+

IIwII4

w d IIwII

>

1

< wdliwNI

4

,

wd1Iw1I >

IIwII

dIIwI!

1

< dw , dw> -

_

IIwIIdw

IIwII

2 T, -WI?

I

dIIwII

IIwII

IIwII2

dIIwII

< dw , w > -

< w, , dw>

IIwII3

IIwII3

(dIIwHI)2

This gives for (8) M-1 C=1

_

1

2

1

< dw , dw >

< dw , w > < w , dw >

IIwII4

IIwII

(10)

Remark. The last expression could be abbreviated by IIWII2IIdwiI2 IIw1114 {

176

1< w , dw >1 2

}

1 11W

IwII4

A

dwll2

1

2

jwJdwk - wkdw

4

j1k

From this calculation we conclude that we can define a Hermitian metric in Pm-1 by the formula M-1 ds2 C =1

OC'OC

2 3

Lkwkw k)( 'k dwk O dw

'kwkw k)

3

k

Ekwkdw k)

xQ

Ekwkdw

J

(12)

In fact, the right hand side shows that the metric is Hermitian and the left side shows that it is positive definite. This is the Fubini-Study metric of PM 1 In terms of the left hand side of (12) the associated Kahler form can be written m-1

m-1

c

C 1 HOC A HOC

1

00C A 0CO

where for the latter expression we have used (5). Now from (6), using 0 , we have 4,00 A 000 - X00 A X00 m-1

C0

m-1 HOC A

C1

`NCO

HOC 11

¢CO

Thus the Kahler form can be written m-1 =

c

2i

1 HOC

1 A

(13)

2i

NCO

so that the Kahler form is closed and the metric Kahler. As was shown in Chapter 1, §2 we can also find an expression for the Kahler form from a calculation in terms of the right side of (12) ; we obtained that w can be written as i

a

a log jwjl

d do log jjw

=

(14)

Z

In the special case m = 2 (12) is the Fubini-Study metric of Explicitly, we obtain from the right side of (12)

P1

=

S2

177

ds2

2)-Z{(w

(w w +w w

=

1

2

1

1

w +w w 2

1

) (dw

2

1

(D dw

1

+ dw

2

(E dw 2)

- (dw1w1 + dw2w2) ® (w1dw1 + w2dw2) } (w1dw2 - w2dw1) Q (w1dw2 - w2dw1) (wlw1 +

w2w'2)2

We will now express this in the inhomogeneous coordinate w2 (15)

w1

With

w1dw2 - w2dw1

d =

w1dw2 - w2dw1

_ do

2

-2 wI

_

w1

we can write w? wZ 1

.

ds2

dOd

do

(1 + C

(w1w1 + w2w2)Z

2

Thus the Fubini-Study metric on P1 is just the natural metric of the Rie mann sphere S2 of constant curvature 4 Its Kahler form is .

w

2

d4 A d

da A dT

(1 + )2

(1 +

Cr + i T

,

(16)

which is the spherical volume element. This can also more quickly be obtained as follows. is the coordinate on the open set U1 = (w1 x 0) in P1 By using on U1 the lifting w = (1 , d we obtain from (14) a a log(1 + C

w

_

The volume of

i

)

=

4

dAd

2(1+CO is

S2

2n +m

tdtdO

(1+ )Z 0

178

0

=

n

Olog(1 + 4 c) d A d

Differential forms. Let AP(M,R) denote the space of differential forms of degree p on M , and ZP(M,R) the subspace of closed pforms. Since d2 = 0 , d(AP 1(M,R)) C ZP(M,R) . The quotient groups D.

ZP(M,R)

HDR(M,R) dAP-1(M,R)

of closed forms modulo exact ones are called the de Rham cohomology groups of ZP(M) M . Similarly if AP (M) and denote respectively the spaces of complex valued p-forms and of closed complex valued p-forms on M , we have the corresponding quotient ZP(M)

HDR(M)

If

HDR(M,R) 0 C

dAP-1(M)

is a complex- manifold the decomposition

M

TC w(M)

T

of the cotangent space to

w

(M)

+Q

®

=

w

at each point

M

r

A TC,w(M)

T

(

A

wEM

gives a decomposition

Tw (M) 0 A Tw, (M))

p+q=r

Therefore the space of r-forms can be written Ar(M)

Q APq(M)

=

p+q=r where

AP'q(M)

=

{

E Ar(M)

P

is the space of r-forms of type (p, q) . For day

E

q

: (w) E A Tw (M) 0 A T( M) for all w E M} E AP' q(M)

we have

AP+1.q(M) O APq+I(M)

and we can define operators APq(M)

AP'q+I(M)

APq(M)

AP+1'q(M) 179

where

d

a

+

a

=

In terms of local coordinates

.

a form is of type (p,q) if it can be written

(w)

w = (w 1... 1w m )

''Ii (w) dwI A dwJ #Irp

#J =q

where for each multiindex dwl

dwi

=

A

I

...

a

p

and

are given by

a

_

i(w)

a

,

}

P

A dwi

1

The operators

{ il, ... ,i

=

(w)

lJ(w) dwj A dwI A dwJ

I,J,j

awj

I,J,i

aw

IJ(w) dwi A dwl A dwJ i

E. Vector bundles. Let M be a C differentiable manifold. A complex vector bundle E over M consists of a space E and a projection map n E - . M , such that: {U, V, ...1 of M with n 1(U) i) There is an open covering 7 1(U) -+ U X Ck equivalent to U X Ck by a C- map : U ou(n _1(x)) _ (x} x Ck oU must preserve fibers ii) On we require Y n 1(U V) ) :

:

U0 0u (x, V) where

(x

_

,

gUV(x) &V)

U nV -C k - 0

gUV

_

(x

, U)

,

are C functions, called the

transition functions. The transition functions necessarily satisfy the identities gUV(x)

-

gVU(x)

gUV(x)

-

gvW(x)

=

I

gWU(x)

for all =

I

xEUnV for all x c UnVnW

(17) .

(18)

is called a trivialization of E over U . A complex vector bundle on M is called trivial if it is of the form M X Ck . E is called a line bundle if oU

180

k = 1 . E is called holomorphic if M is a complex the fiber dimension manifold and if the transition functions are holomorphic. We give a few examples without details. 1. Let M be a complex manifold, and let T (M) be the complex tanx gent space to M at x . For x e U C M and U-+Cm a OU coordinate chart, we have maps

Tx(M) for each

xEU

spanC {

a ,

3 V.

= C 2m

, hence a map U

U

a

3 U.

UC 2m

Tx(M)

xEU

giving T(M)

U

=

Tx(M)

xEM the structure of a complex vector bundle, called the complex tangent bundle. 2. For each xEM we have a decomposition Tx(M)

T'(M)

=

) T' (M)

T'(M) C T(M) {T'x (M) C Tx(M)} form a subbundle The subspaces called the holomorphic tangent bundle ; it has the structure of a holomorphic vector bundle. 3. If E -o M is a complex vector bundle, then the dual bundle E is the complex vector bundle with fiber Ex = (E x ) ; the M trivialization s EU ------ I- U X C k

U

,

EU = it 1(U)

induce maps

EU --UxCk =

U

*

*

which give

E

has transition functions tion functions lUV

=

t

UxCk

-1 gUV

M the structure* of a manifold. If E then E -. M is given by the transi{gUV } ,

U Ex

t 9VU 181

Similarly, if E -- M , F - M are complex vector bundles of fiber dimension k and Z and with transition functions {gUV } and respectively, then you can define the following bundles: {h UV } , 4.

EO F

,

given by the transition functions E

lUV

E©F

gUV

*

T (M) T

Ck )

+

given by the transition functions

,

lUV 5.

GL(Ck

T(M)

=

(M)

,

r*(P'q)(M)

© hUV E GL(Ck X CQ)

T

*

(M)

the complex cotangent bundle

.

the holomorphic and the antiholomorphic cotangent bundle ;

.

(M) © A T

T

=

A

(M)

Aholomorphic vector bundle with fiber dimension 1 is called a holomorphic line bundle. We will now give an example of this in detail. Pn-1 F. The universal bundle J on . Let w 1 , ... , wn denote Euclidean coordinates on Cn and also the corresponding homogeneous coorPn-1 Pn-1 Pn-1 dinates on , all . Let x Cn be the trivial bundle on fibers being identified with Cn . We will define a holomorphic line bundle Pn-1 Cn J J is a subbundle of Pn-1 Pn-1 ; its fiber Jw over each point wE is the line {Xw}X C Cn represented by w , i.e., 6.

-

{ A(w1.. .wn)

Jw

,

XeC}

We show now that there exists in fact a line bundle with these fibers. Let J denote the disjoint union of all J w . Then any point v E J can be represented (not in a unique manner) in the form v

where it

:

182

=

(Awl,...IXwn)

=

(w1, ...Iwn) E Cn - 0

J - Pn-1

is given by

A(w11....wn) E C n ,

and

AEC

.

Moreover, the projection

7r(a(wl,...,wn)) Putting

Ui

Now if

v

v

A(wl,...,wn) E 7r 1(Ui)

=

1 W.

Aw

.EC

71-1(U

1

1

.

x 01

Pn-1

, we see that

AEC

wi

,

, then we can write

0}

v

.

in the form

1

is uniquely determined by

1

i

w

,

E

W.

ith

1

=

(wl,...,w11)rk,

{v = A(wi,...,wn) E Cn

_

=

and A. 1 mapping

E Pn-1

,w n)

{(wl'

=

1(Ui)

r

n(wi,...,wn)

) -+ U

. 1

. We can define the

v

xC

by setting

Yv)

4i

the fibers of

A(wi,...,wn) E

=

((w1.....wn)"

is bijective and is linear from the fibers of U xC U. . Suppose now that 1

The mapping v

i(A(w1,...,wn))

_

n

1(Ui)

to

1(U. f Uj)

then we have 2 different representations for relationship. We have $i(v)

A.)

((wl, .... wn)"

,

A.)

((w1..... wn)"

,

A.)

v

and we want to compute the

,

where A. 1

=

Aw.

A.

1

J

=

Aw. 1

Therefore A

=

A. 1

=

wi

A.

wj

i.e.

a. 1

=

-

W.

-wja. 1

j

Thus if we put w. 1

gij

w.

183

then it follows that gij gjk ' gki = 1 . We deduce that J given the structure of a line bundle by means of the trivializations the transition functions

can be

*

w.

{oi}

and

U1 .nU. - GL(1,C) =C - 0

1

w.

.

n

of a vector bundle E- M over

Sections. A section s

G.

is a C

UCM

map

U- E

s

s(x) E E x

such that

for all x E U . A frame for E over U CM is a collection s1 , ..., sk of sections of M over U such that is I (x),. .. , sk(x)} is a basis for Ex for all xEU . A frame for E over U is essentially the same thing as a trivialization of E over U : Given E

U

U

------- 0-

UXCk

a trivialization, the sections

Ul(x,ei

si(x)

the canonical basis of

{ei}

si, ... , sk , we can define a

form a frame, and conversely, given a frame trivialization A

=

U(A)

(x

_

in

G A .1 s 1.(x)

Given a trivialization

of

0U

we can represent every section a

=

s

U , it is important to note that over U uniquely as a C vec-

over

of

E

by writing

Ul(x,ei)

0V

ai(x) '

X

E

is a trivialization of E corresponding reoresentation of If

E

(al,...,ak)

ai(x)

Y

for

(A 1, ...' Ak))

,

Ck

over s I V n i7

ai(x)

U1(x,ei)

and

V

a'

(a,, ... , ok)

the

, then '

V1(x,ei)

so

ai(x)

184

ei

=

of (x)

U4 j(x,ei)

,

i.e.

o

gUV a

Thus, in terms of trivializations

EU -+ Ua X Ck }

{a

a

sections of

over

E

{ as

U Ua

correspond to collections

,...,C.. , aka) }a

=

of vector valued C functions such that a

for all {oa}

gas

a

'

as

gas are transition functions of

where the

a

E

relative to

.

A section s of the holomorphic bundle E over U C M is said to be U E holomorphic if s : is a holomorphic map, a frame s = (s1,... is called holomorphic if each s i sk is; in terms of a holomorphic frame {s. } a section 1 ai(x)

s(x)

.

si(x)

is holomorphic if and only if the functions a,i are. H. The hyperplane section bundle H - Pn-1 is the dual H = J of the universal bundle J , i.e., it is the holomorphic line bundle whose fiber over we Pn-1 corresponds to the space of linear functionals on the {Aw}A . It has global sections line given by the linear forms (P n-1 ,H) A(w) + + anwn C . Such a form a w on A(w) determines a 1 I divisor, which is given by the hyperplane A(w) = 0 . In more detail, let al, ... an be constants and n the projection Cn - 0 --- Pn-1 . The linear form A (w) in Cn - 0 has in the local coordinates in 1(U ) i the expression A(w)

wi(a1 ill + ... + I + ... + an

=

inn)

ith

where

w. =

ill

wi

j

,n

Denoting the expression in parentheses, which is essentially the linear form at the left hand side in "non-homogeneous" coordinates in Ui , by ai

=

(a 1 ic

l ++1++a nn ) ith 185

we see that in

n-1(Ui n u.) W.

w. a. 1

so

{o. } 1

--' G. wi J

w. a.

=

1

J

.

J

1

defines a section

s

i

C. J

in the line bundle whose transition functions

are

j

W.

gij

w=

=

1

i

-1

Because of this origin the latter bundle is called the hyperplane section bundle H of Pn 1 It is the dual J of the universal bundle J I. Divisors and Chern class. In the last example the hyperplane A (w) = 0 is defined by ai = 0 in Ui = (wi x 0) , and the transition functions in can be written Ui n u .

.

.

J

More generally, a divisor D on a complex manifold locally finite formal linear combination D

Ia

=

M

is defined to be a

V. 1

1

of irreducible analytic hypersurfaces of M. lection of holomorphic functions

It may be thought of as a colU. - C such that

M. .

1

1

m.

1

gij

mj

are non zero holomorphic functions in Ui n Uj for all i , j . D to be the zeros of the functions mi in U. . In U. in Uj n Uk gij

gjk

'

gki

is defined we have

mk

M.

M.

m

mk mi

It follows that m. 1

gij

M.

are transition functions of a line bundle associated to the divisor D . 186

[Dl

.

It is called the line bundle

In view of the above then we can say that the hyperplane section bundle H Pn-I is the line bundle which is associated to the divisor of a hyperplane in It can be shown that the line bundle (D] associated to a divisor D on M is trivial if and only if D is the divisor of a meromorphic function. If M is compact we have Poincare'duality between H (M, Z) and q H2m q(M, Z) In particular, a divisor D on M carries a fundamental .

homology class {D}

H2(M,Z)

E H2m-2(M,Z)

as an element in the de Rham group HDR(M,R) Then the divisor D is said to be positive if {D} is represented by a closed positive (1, 1) form m . This means that locally We' may consider

{D}

2

i,j

hij dwi

,

dwj

where the Hermitian matrix (hij) is positive definite. It can be shown that collections {g..} and {g'.} of transition funca. E tions define the same line bundle if and only if there exist functions (Ui) satisfying 'i

gij

gij

The transition functions sent a Cech 1-cochain on (18) mean that d ( {gij})

(19)

{ gij E 0(Ui n U.) }

of

E -M repre-

the relations (17), with coefficients in = 0 , i.e. , {gis a Cech cocycle. Moreover, by 'define the same line bundle if and only if and {gij}

(19) two cocycles {gij}

their difference

]

M

{g.. gis a Cech coboundary. Consequently the set of

holomorphic line bundles on The coboundary map

-0

M

is the Cech cohomology group

H (M, VVV

d

HI (M,(0 *)

H2(M,Z)

arising from the cohomology sequence of the exponential sheaf sequence 0

defines the Chern class c($) = 6 ( {gij }) of a line bundle. If the bundle E carries an Hermitian metric in its fibers, with the curvature matrix 0 , Chern has shown the important theorem that c(E) is represented in the de 187

Rham cohomology group

i 27r

188

HD R( M

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Characteristic classes of hermitian manifolds, Ann. of Math. 47, No 1, 1946, 85-121.

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Chern S.S.

The integrated form of the first main theorem for complex analytic mappings in several complex variables, Ann. of Math. 71, No 3, 1960, 536-551.

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Complex analytic mappings of Riemann surfaces I, Amer. J. Math. 82, 1960, 323-337.

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Holomorphic curves in the plane, Differential Geometry, in Honor of K. Yano Kinokuiya, Tokyo, 1972, 73-94.

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Holomorphic curves and metrics of negative

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Deligne P.

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Address of author Mathematics Department

University of Siegen D-5900 Siegen West Germany 196

Table of symbols

Cn

1

w

H2(Pn 1 Z)

1

OCR


1

li

1

f

C

1

v(r,0)

14

CR

1

a

16

f(z) - a

16

z.(a)

17

II

C +,n

1 2

ac

2

z.(0)

8

z(-)

8

Pn-1

9

*

13

J

n(r,a,f) = n(r,a) V(r,a) = V(r,a,f) v(r,a) = v(r,a,f) N(r,a) = N(r,a,f) N(r,f) = N(r,o)

18

18,19 19,20 20

20

u (r, a)

21

log

22

m(r,f) = m(r,f)

22

m(r,a)

22

T(r,f)

22

10

CR

24

F

10

V(r,W)

24

F

10

0

ti

w

9

a

10

a

10

d

10

ac

10

T(r, A)

* n

28 10

*(a)

28

11

n(+oo,co) = n(+or,f)

28

11

n(+W,a)

29

*

C J

25

*

*

J

12

n(a)

29

H

12

v(+-,a)

30

c1(H)

12

v(r,m)

30

197

m(r,c,f

45

Ric j N1(r)

114,118

45

nl(r)

114,118

45

N(r , a)

126 130

n(r,a)

126, 130

45

n(r,f)) = n(r,oo,fi)

N(r,-,f T(r, f.) J

m(r,ai) = m(r,aj,fi)

46

n(r,a.) = n(r,a.,f.)

46

J

7

N(r,a.) = N(r,a.,f.) J

46

3

d (a)

112

,

(a, f)

d(co) = d(c",f) N(r , f) = N(r , w) O(a)

O(a,f)

130

130 130 131

51,54 0(a) = 0(a,f) 52

131

(r,a)

132

N1(r,a)

132

n

1

54 57

m(r,f) =

m0

(r,-,f)

62

64

65 68 69

70, 71, 72 72

74

S 2n

75

A(r,f)

77

E(u,p)

82

g

85

M(r,a)

89

M(r,f) = M(r,-)

89

M(r,a.)

97

J

98

A

198

= 6 G (f)

133

6V(a) = 6V(a,f)

139

6G

Index

admissible, 133

a -point, 2, 16 a -point at infinity, 28 associated 2 -form, 10

characteristic class, 12 Chern class, 12 Chern form, 11, 12 class, 54

completely multiple point, 140 complex Euclidean space, 1 complex projective space, 9 convergence class, 53 counting deficiency, 150 counting function, generalized, 20 counting function of multiple a -points, generalized, 132 counting function of multiple points, generalized, 118

curvature form, 11, 12, 111 curvature function, 113

deficiency, 131, 132 deficiency, counting, 150 deficiency relation, generalized, 134 deficiency, volume, 139 deficient point or value, 132 degree, 29 divergence class, 53

exceptional value, see deficient point, 132 and volume deficie exceptional value, generalized Borel, 129 exceptional value, generalized Picard, 127

first main theorem, generalized, 23, 26, 75 Fubini-Study metric, 10 Fubini-Study metric on Cn , 64 function, meromorphic vector valued, 1 Gaussian curvature, 111 Gauss map, 110, 111, 133 Gauss map, index of the, 133 generalized Ahlfors-Shimizu characteristic, 65

generalized Ahlfors-Shimizu proximity function, 74, 75 generalized Borel exceptional value, 129 generalized chordal distance, 72 generalized counting function of multiple a -points, 131 generalized counting function of multiple points, 118 generalized first main theorem, 23, 26, 75 generalized genus, 85 generalized Nevanlinna characteristic function, 22 generalized Nevanlinna deficiency relation, 134 generalized Nevanlinna proximity function, 22 generalized Picard exceptional value, 127 generalized Poisson-Jensen-Nevanlinna formula, 8 generalized Riemann sphere, 73 generalized second main theorem, 114 generalized spherical characteristic, 65 generalized spherical distance, 72 generalized spherical proximity function, 74 generalized theorem of Picard, 127, 138 generalized theorem of Picard-Borel, 129 genus, 82, 83, 85 Hermitian geometry, 9 Hermitian metric, 11, 64, 111 holomorphic curve, 12 holomorphic line bundle, 11 homogeneous coordinates, 9 hyperplane section bundle, 12

index of multiplicity, 132 index of the Gauss map, 133, 137 index, Ricci, 133 inhomogencous coordinates, 9

Kahler form, 11, 64 Kahler metric, 10, 11

local affine coordinates, 9 lower order, 54 meromorphic function, vector valued, 1 multiple point, 118 multiplicity, 1, 2, 16, 28, 139 multiplicity, index of, 132 normalized volume, 14, 19,

order of growth, 51, 54 200

65

perfectly regular growth, 54 Picard, generalized theorem of, 127 Picard-Borel, generalized theorem of, 129 plurisubharmonic function, 37 point, completely multiple, 140 point, multiple, 118 pole, 1, 2, 28 pole at infinity, 28 pseudohermitian metric, 112

rational vector function, 27, 141, 143 regular growth, 54 Ricci form, 112 Ricci function, 113, 114 Ricci-index, 133

second main theorem, generalized, 114 stereographic projection, 70

total curvature, 113 transcendent, 27 type, 52, 54 universal bundle, 11 vector valued meromorphic function, 1 very regular growth, 54 volume, 14, 19, 65 volume deficiency, 139 volume deficient point, 139 volume element, 13 volume function, 19 volume, normalized, 14, 19, 65 zero, 1,

2