Value Distribution Theory for Meromorphic Maps (Aspects of Mathematics)

Wihem Sto Value Distribution Theory for Meromorphic Maps

Aspects of Mathematics Aspekte der Mathematik Editor: Klas Diederich

Vol. E1:

G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part A

Vol. E2:

M. Knebusch/M. Kolster, Wittrings

Vol. E3. G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part B Vol. E4: M. Laska, Elliptic Curves over Number Fields with Prescribed Reduction Type Vol. E5: P. Stiller, Automorphic Forms and the Picard Number of an Elliptic Surface Vol. E6: G. Faltings, G. Wustholz et aI., Rational Points (A Publication of the Max-Planck Institut fur Mathematik, Bonn)

Vol. E7: W. Stoll, Value Distribution Theory for Meromorphic Maps Vol. 01: H. Kraft, Geometrische Methoden in der I nvariantentheorie

The texts published in this series are intended for graduate students and all mathematicians who wish to broaden their research horizons or who simply want to get a better idea of what is going on in a given field. They are introductions to areas close to modern research at a high level and prepare the reader for a better understanding of research papers. Many of the books can also be used to supplement graduate course programs. The series comprises two sub-series, one with English texts only and the other in German

Wilhelm Stoll

Value Distribution Theory for Meromorphic Maps

Friedr. Vieweg & Sohn

Braunschweig I Wiesbaden

Prof. Dr Wilhelm Stoll is Professor of Mathematics at the University of Notre Dame, Notre Dame,.ln"cliana 46556, USA

AMS Subject Classification: 32 H 30, 32 A 22

1985 All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1985 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means. electronic, mechanical, photOCOPYing. recording or otherwise. Without prior permission of the copyright holder Produced by Lengericher Handelsdruckerei. Lengerich Pronted in Germany

ISBN

3-528-08906-7

D dicated to the memory of Yozo Matsushima

CONTENTS Page Preface

VIII

Letters

XI

Introduction

1

A.

Value Distributl n Theory for Fixed Targets

B.

Value DistributIOn Theory for Moving Targets

56

1.

Hermitian Geomptry

92

2.

Meromorphic MdPs on Parabolic Manifolds

115

3.

The First Main T eorem

134

4.

Associated Maps

151

5.

Frenet Frames

163

6.

The Ahlfors Estimates

191

7.

General Position

116

8

The Second Main Theorem

245

9.

Value Distribution over a Function Field

275

10.

An Example

:310

11.

The Theorem of Nevanlinna-Mori

317

12.

References

334

\3.

Index

344

Preface Value distribution theory studies the behavior of mermorphic maps. Let

f: M -

target family A

N

be a meromorphic map between complex manifolds.

£I - (E) a aE A

of analytic subsets

is a connected, compact complex manifold.

family

f*(£I1 ~ (f-1(EallaEA

created by many contributors.

is investigated.

Fa

IJ

g : M -- A

f(zl E Eg(z)

g E

Ea

stay fixed.

of meromorphic maps for

z E M

and

IJ.

Here we investigate this situation: dimension

is given where

A substantial theory has been

However we can consider a finite set

some

N

The behavior of the inverse

Usually the targets

and study the incidence

of

A

m and

N - IP n

family of hyperplanes in projective space

IP*n

IP n

M

is a parabolic manifold of

is the n-dimensional pro ective space.

The

is the target fami y parameterized by the dual

We obtain a Nevanlinna theory consisting of several

First Main Theorems, Second Main Theorems and Defect Relations and extend recent work by B. Shiffman and by S. MorL

We use the

Ahlfors-Weyl theory modified by the curvature method of Cowen and Griffiths. The Introduction consists of two parts.

In Part A, we sketch the

theory for fixed targets to provide background for those who are familar with complex analysis but are not acquainted with value distribution theory. In Part B, we outline the results of this monograph.

A specialist can read

Part B directly, and has only occasionally to look up some definitions, notations or facts in Part A. The main part consists of 11 Sections, which can be read independently from the introduction.

In Section I, a new contraction

operator is introduced, which is most helpful.

After necessary preparations

in Section 2, the First Main Theorem for an abstract operation is proved in Section 3.

The general form of the First Main Theorem proves to be most

helpful in its various and sometimes surprising applications.

In Section 4,

the definition of associated maps is recapitulated and convenient notations for the interaction of the associated maps of the given map

f: M -

IPn

IX

and the target map studied in Section 5.

g: M -

1P~

are introduced.

This interaction is

Here the stress curvature formula together with the

stress gradient formula is most likely the deepest result of these investigations.

These identities should be of considerable interest to

complex differential geometry quite independently from value distribution They enable us to prove the Alfors Estimates for moving targets in Section 6.

The treatment of general position in Section 7 is another highpomt in

this research. exposition.

The new contraction operator simplifies and clarifies the

A gauge mea ure and the First Main Theorem for the exterior

product permit us to solve the problem of general position for moving targets.

The Second Mam Theorem and the defect relation are established

in Section 8.

They are

pplied in Section 9 to create a value distribution

theory over a function fl d of rank

m - 1.

As a special case and up to

minor modifications in the assumptions, Shiffman's defect relation for functions is obtained

The particular choice of the holomorphic form

B

defining the different! I operator in the associated maps is surprising and sheds new light upon thiS method which I introduced thirty years ago. example is calculated in Section 10.

An

It shows that deriving the Second

Main Theorem is like walking a tight rope.

In Section 11, the defect

relation of Mori is improved and established on parabolic covering manifolds References and an index conclude the monoeraph. This research was carried out from January 1982 to July 1982 at the University of Notre Dame and completed at the Research Institute for Mathematical Sciences, Kyoto UniverSity, Kyoto, Japan where I was from May 13 to August 16 1983.

In the Summer of 1984, Part A of the

introduction was written on the request of the editor and Part B revised accordingly.

Also Section 11 was added.

In March 1982, I was invited to the Centro de Investicacion y de estudios a vanzados del lPN, Mexico, for a week, where I lectured on a very preliminarY version of the Second Main Theorem.

In thE' Spring of

1982, [ learned about Shiffman's result, which was available to me, when developed the theory now written up in Section 9.

The Research Institute

for Mathematical Sciences, Kyoto, held a conference on value distribution theory July 4 to July 8, 1983 where I reported on the results of this monograph in a series of lectures. conference.

I learned about Mori's results at this

x This research was supported by the University of Notre Dame through a leave during the Academic Yedr 1981 82. by the National Science Foundation Grants MCS 82 01158 and DMS 84-04921. by the Japan Society for the Promotion of Science and by the Research Institute for Mathematical Science. Kyoto University.

I thank all these agencies and

institutions for their help and support without which this work would have been impossible Spring Dawn Redder designed the non-standard characters for this monogrdph and typeset the entire manuscript on the IBM Personal Computer using the Radio Shack TRS-80 DMI' 2100 printer dn for mdthemalicdl mdnuscripts written by R. J University.

a program designed

Milgrdm of Stanford

I thdnk here for her great energy. remarkable expertice and

dedication to this project. On April 9. 1983. Professor Yozo Matsushima di(.d in Osaka.

He was

d gredt mdthematician. a fine gentleman. a long time colleague and a good fl'icrlll

I dedicatE' thl'se pages til him. Wilhelm Stoll

Letters Latin Capital

Latin Small

A

a

CI

B

h

C

German Small

Greek Capital

Greek Small

.£A

A

a

£,

I)

B

8

c

1:

1-

r

~

D

d

.,'}

.'J

6

Ii

E

e

~

'I\.

E

E

F

f

~

of

Z

1"

G

g

QJ

g

H

T/

H

h

~

f

e

I)

I

i

J

1-

I

l

J

j

it

~

K

IC

K

k

~

If!

A

A

L

I

;t

-e

M

...

M

m

'IUIt

'Woo

N

))

N

n

Ul

1A

-

~

0

0

6

0

0

p

p

":p

H

D

If

Q

q

L1

(It

P

P

R

r

~

1('

L

o,c.

s

s

It

1

T

T

t

"'I-

.

T

U

u

11

iii.

V

v

10

'fJ

W

w

lru)

-.0

~

X

x

X

~

0

y

y

~

Lg

Z

z

;f

it

Gcrmdn Capital

Y

u

4>

cp

X

X

'" w

Introdution A.

V lue Distribution Theory for Fixed Targets

First we sket h some features of classical value distribution theory for fixed targets to p ovide the necessary background. At.

Functions of 2.....! variable.

The degree of a polynomial measures its

growth, counts the number of points in its fibers and provides a valuation of the ring of polynomlals.

A polynomial splits into linear factors.

Within

50 years, 1875-1925, st kingly similar features were discovered in the theory of transcendental entire functions.

Weierstrass [116] provided the

factorization.

Picard [74J showed, that all values are assumed except

perhaps one.

Jensen's formula connects the number of zeros with the

growth.

The order of an entire function is an analog to the degree of

a polynomial. However, these results were difficult to extend to meromorphic functions since the maximum modulus could no longer serve as a growth measure.

Within three years, 1922-1925, R. Nevanlinna created value

distribution, which revolutiOnized the theory of entire functions and extendec it to meromorphic functions solving the problem distribution has remained Nevanlinna theory,

Ever since, value

We will outline his two main

theorems.

A2.

Classical value distribution of

~

variable.

(A.t)

I[[r I - Ir

E:

I[ I I z I , r}

I[lr)

IA.2)

I[ - Iz

E:

I[ I I z I - r)

II:.

For - Iz

r > 0 E:

define

I[ I I z I < r)

II: - {OJ.

On any complex manifold, the exterior derivative splits

a

d

+

a

and

twists to

(A.3)

An exhaustion

of

T

G:

is a Kaehler metric and such that

a - d C log

da

ThE'n

'T.

measure on each circle

o.

=

lI:

I

(A.4)

T(z) ~ 1z 12

is defined by log

is harmonic.

'T

a

The form

such that

dd C .,

Define

mduces a rotation invariant

with

a-I

for

O
G: The Riemann sphere diameter

in

1

with total measure

1'1

between the points w

1Pl - (a)

E:

1.

Let

wand

a

n(w)

a - 00,

0 w;a 0

in

1P 1.

be the chordal distance

If

a

E

1'1

and if

=

-

1'1 - G: V (00)

dd c log 0 w;a 0 2 . is identified with the compactified plane

CA.6)

on

Let f : G: -

f 1'1

be a non - constant. meromorphic function on

G:.

II:

Then

is an open, holomorphic map which spreads each disk

over an open subset of

CA.7)

is fixed and if

is variable, we have

(A.5)

If

n be the rotation invariant surface measure on

Let

1Ji13.

is realized as an euclidean sphere of

11'1

1'1.

The spread volume is

Af(t) -

J lI:(t)

f*(n) > 0 .

G:(t)

-3-

The spherical image function

o~

s < r,

Af

increases and is continuous

the Ahlfors-Shimizu Characteristic function

Tf

For is defined by

(A.B)

For fixed

s,

the chdracteristic serves as a growth medsure for

increases,

is of class

el

and is convex in

is a holomorphic isometry, then

Take

a

E

11'1.

T JJ of - Tf

log r.

f.

J.I: 1P 1 -

If

11' 1

in particular

T e compensation function

mf

of

f

for

a

is

defined by

(A.9)

mflr;a) -

J

log 0

f, a

r > 0 .

for all

0 a > 0

It(r>

v

A divisor is an integral valued function (A.10)

v -

SUpp

is a closed set of isolated points. valence function

Nv

v

of

(z E I[ I vIz) 'I:- 0)

The counting function

nv

and the

are defined by

(A.10)

Nv(r,s) - Jr nvlt)

(A.Il)

whose support

Z

It -

dtt

~

t

E ~

for

0

for

0 < s < r

E

~ •

s

The divisors form a module

v

E ~

[f

non-negative.

V

~

0,

For

then ZEit,

~.

and

nJl

and

nv

and

N v'

for fix!'d

let

JJ~lz)

be the

NJI

arE' additive in s,

increase and are

~-multiplicity

of

f

at

-4-

a

U:

Then

is a non- negative divisor on

and valence function are denoted by

whose countinl function

11:,

nf(t;a)

and

Nf(r,s;a)

nr(t,O) - nf(t,oo) -

J

respectively.

The argument principle

(A.12)

df

T

I[

ir

n

lI:

supp [u~ +

u;] - 0

implies the Jensen Formula for

O<s
J

Nr(r,s,O) - Nr(r,s;oo) -

(A.131

log I f I a -

M For

0 < s < r,

J

log If I a .

M<s>

we have Nevanlinna's First MalO Theorem (FMTH)

(A.141

The theorem is a conservation principle.

The measure for the fiber

over a plus the compensation for the outflow over the boundaries of the annulus a.

lI:(r)

lI:[s)

balances to a quantity independent of the fiber over

All terms in (A.l4) are non-negative permitting estimates:

(A.15)

Up to an additive constant, the growth measure fiber measurE'

Nr

(A.16)

for all

or

r-l(a).

E:

!PI

of

We easily calculate

log

W

Tf

=D~w":.:"a--:D= (lCa) =

An exchange of integration yields

2'1

f

majorizes the

·5·

J

(A.I7)

mr(r;a)O(a) -

t.

8EIPl

Now (A.14) and differentiation yield the Mean Value Theorem

(A.I8)

Tf(r,s) -

J

Nf(r,s;a)O(a)

8EIPl

(A.I9)

Af(t) -

J

nf(t;a)o(a) .

8EIPl

For a boundary integral representation of the characteristic some preparations are needed.

For

1l -

(zl,z2)

~ =

and

(wl'w 2) define

the (A.20)

inner

prod ct



- zlw 1 + z2w 2

(A.21)

exterior

prod ct

It '"

- zlw2 - z2 w l

(A.22)

hermitian

product

(A.23)

the rum!

a:! - a: 2

- (O).

a:,..2 Ie. 1Pl - e V

Let

Put line:

"I -

identify and

w

E:

{oo},

'\10

(It 1...0) - zlwl + z2 w2

IIltll -

~ C~·llt)

-

j

I z 1 12 +

Iz2 I2

Identify the Riemann sphere with the projective 2

F: 1:,.. then

"I

be the residual map.

1P(z,w) - z/w

and

1P(w,O) -

00

[f we for

z

E

1:• .

A holomorphic vector function

C)

-

(g,h) :

a: -

a:!

is said

to be 8 reduced representation of f if f - 1P 0 10 ~ g/h. Stokes Theorelll :fields the boundary inteKl'al representation of the characteristic

II:

·6·

(A.24)

Tf(r,s)

J

=

J

log 1110 1100 -

log 1110 1100

1[<5>

I[

which is most helpful to gain estimates of the characteristic. and (A.24) prove the FMTH (A.14) easily. functions on

1[,

f1

If

and

f2

Also (A.13)

are meromorphi(

then (A.24) yields

(A.25)

T f +f (r,s) :!l; T f (r,s) + T f (r,s) + const. 1 2 1 2

(A.26)

T f f (r,s) :!l; T f (r,s) + T f (r,s) + const. 1 2 1 2

which corresponds to the valuation property of the d gree of polynomials. If

At -

r

Tf -

(A.27)

is constant, the definitions (A.7) and (A 8) still hold with

o.

If

f

is not constant, then

T f(r,s)/log r

--+

Tr(r,s) _ . .

Af (..) - 1 i m Af(t)

for

and even

r

--+ 00

t ......

Ar

where

If

f

<.. if and only if f

is an entire function, the maximum modulus Mf(r) = Max( I fez) I I z

(A.28)

is a natural growth measure of 10

-

(A.29)

is rational.

(l,f)

f.

[Ir])

If we use the reduced representation

in (A.24), we obtain 1

T fer,s) :!l; log Mr(r) + 2" log 2

Take 0 < (] < 1 and 0 < s < r. Since subharmonic, the Poisson formula implies (A.30)

E

log II

10

II > 0

is

+_ (] IJ (T f (r,s) + mf(s,oo))

"/"

Again,

Tf

is justified as a growth measure.

In the FMTH, the valence function usually dominates, whilE' the compensation function remains small.

The proper measure is the defect

(A.31) Nf(r.s;a)

1 -

which does not depend on

If

!;C(a) = 1. times, then

f

!;C(a)

If

f

1.

=

=

~

1

a,

then

assumes a only finitely man~

f

Fatou's Lemma and the Mean Value Theorem easily

Cor almost all

0

Tf(r.s)

does not assume the value

is transcendental, and if

!;f(a)

show that

s.

lim sup r->oo

a

E

!PI .

Nevanlinna's Second Main Theorem (SMTH) improves this result tremendously: Let

f

Let

10.

p

of

Take a finite subset cS > 0

there is a constant

f.

be a reduced representation of

1 0 : (C --- (C;

(or Wronski) divisor

It.

be a non-constant meromorphic function on

10

A

S '#. t2l

does not depend on the choice of

10'

of

The ramification

!P 1

and a subset

Take

0 < s

E

IR.

Then

of finite measure in

such that (A.32)

for all

r

E

IR - Es

with

r > s.

This Second Main Theorem implies the

defect relation

(A.33)

A value

a E!P1

is deficient for

f

if

!;f(a) > O.

Thus there are at

most coontably many deficient points and we can write

·8·

(A.34)

Picard's Theorem follows, but the defect relation is stronger and deeper. The deficiencies of meromorphic functions have been studied extensively. points:

We cannot outline the vast theory here, but mention these

Baerenstein's spread relation [3] has helped greatly with functions

of finite lower order. Umkehr problem:

Most strikingly, Drasin [24) solved Nevanlinna's

"Given a non-negative function

II

on

lPl

positive

at at most a countable number of points such that the sum of all function values of on

II

a: with

does not exceed II f - II."

2,

there exists a meromorphic function

A meromorphic function

f

f

of finite lower order

has additional restrictions on its deficiencies, since

(A.35)

converges for a =

j.

I,

a

where

~

1 '3

(Hayman [39) for 1

a - 3'

1

a > '3'

Weitsman [117) for

is optimal.

Ahlfors created a value distribution theory of covering surfaces with differentiated versions of the two Main Theorems.

Wittich and his school

studied the value distribution of meromorphic solutions of complex differential equations.

Value distribution was essential in the composition

0

meromorphic functions, in product representation of functions, the study of function classes, the existence of boundary values and many other topics. The theory was lifted to Riemann surfaces from the study of algebroid functions to the study of holomorphic maps between arbitrary Riemann surfaces. The reader can find a present day view in the memorial volume 7 for Rolf Nevanlinna, Annales Academiae Scientiarum Fennicae Series AI Mathematics 1982, in particular the contributions of O. Lehto [481. W. H. J. Fuchs (28), W. K HaYman (40), S. Rickman [761 and H. Wittich [1211

Also

·9·

we recommend the books by G. Valiron (113). R. Nevanlinna (66). (67). M. Tsuij (111). W. K. Hayman (39). B. Ja. Levin (53). L. Sario and K. Noshiro (78). H. Wittich (121). F. Gross (37) and A. Dinghas (23). Value distribution has been extended outside the district of dimensional complex analysis. to subharmonic functions in

IPn.

ne

see IR n

W. K. Hayman and P. B. Kennedy [41). to quasi-regular maps from

into itself. see Rickman [771 and of course to several complex variables which will be the topic of the rest of this monograph. A3.

The beginninl of value distribution in several variables.

hope for a good Ne anI nna theory for holomorphic maps of maximal rank.

Naturally. we

f: I[m -

Most fibers are discrete and easy to count.

IPm

Yet in 1922

Bieberbach (6). extending a result of Fatou [271 constructed a biholomorphic map

f: I[m -

D.

where

D is open but not dense in

I[m.

Our hopes

are shattered.

For meromorphic functions on

I[m.

Picard's Theorem is trivial; yet

the extension of Nevanhnna theory is not obvious. achieved the break through. f

In 1938. H. Kneser [461

He restricted the given meromorphic function

to the complex lines through the origin and took the integral average of

the value distribution functions over the parameter space these lines.

A FMTH like (A.14) emerges.

result would be of little consequence.

P n-1

of

Had Kneser stopped here. his

However. he expressed the terms in

the FMTH in geometrically meaningful ways. which today remain basic to value distribution of functions and maps of several. independent complex variables.

Therefore we shall formulate his result here.

I[m - M

Abbreviate M is defined by given by dd c ?, a - d C log

T

A

T (11-) U

-

IaI

> O.

m- I .

M. - M - (0).

and I ht 112.

On Then

An exhaustion

The standard Kaehler metric on

M.

da -

define

o.

IaI -

Thus

dd c log .,

T

~ 0

of

T

M is and

is strictly parabolic.

For anY S ~ M. let S[rl. SIr). S be the intersection of S with. respectively. the closed ball. the open ball. the sphere of radius r > 0 centered at 0 IE: M. Then a induces the rotation invariant measure of

·10·

total measure

1

on each sphere

meromorphic function on for

t > 0

M.

M.

Let

f

be a non -constant,

The spherical image

Af

of

f

is defined

by

J

Af(t) - t 2 - 2m

(A.36)

I

M[ t

J

f*(n) " urn 1

f*(n) "

101

m. 1 + Af(O)

M[ll

where

Af(O) =

11m

Af(t)

is an integer.

If

f

is determined at

0,

f

is

t-oO

which Kneser dssumed, then defined for

0 < s < r

Af(O)

~

0

The cha a teristic of

f

is determined at

by

(A.37)

where

T f(r,O)

Take ,!.E

M.

a

Let

exists if and only if

E

lP1

Fa

=

Let

u~('!)

be the a multiplicity of

(it E MIU~(iI) > 0)

be the support of

is empty or a pure (m-I)-dimensional analytic subset of function of

(A.38)

is defined for

nr(t;a) = t 2 - 2m

u m- 1

U~

at

u~

which

The countinl

by

I

is the euclidean volume element of the variety

valence function of

(A.39)

t > 0

M

f

J Fa [t

where

a.

is defined for

0 < s < r

by

Fa'

The

·11·

The compensation function of

(A.40)

f

for

a

I

mf(r;a) -

r > 0

is defined for

by

log 0 f ; a 0 a

M As said before. the functions are the integral averages over

1Pn-1

over

the corresponding functIOns on the lines through the origin which gives the First Main Theore (A.41)

Tf(r.s) - Nf(r.s;a) + ~(r;a) - ~(s;a) .

Rneser adds two Important theorems: of

M,

then

Tf I L

majorizes

T f .n

1)

"If

L

is a linear subspace

He proves this theorem for

holomorphic fUnctions. but it is true for meromorphic functions. be an entire function of finite order on fI}}) ~ 0

maximal with polynomial and in

M(s)

h

for all

}}

E

M

M(s).

2)

"Let

with

flO) - 1.

Take

Then

f - he g

where

s > 0 g

is

Fa

of

f."

This is the

Later Stoll [91] showed that

such an integral over the given zero set

in fact constructs an entire

h

Fa

whose growth is controlled by the growth of

[491. [511 constructed the same canonical function different integral representation. integral representation. created.

Later.

h

FO'

P. Lelong

independently using a

Ronkin [77] gave still another

A theory of functions of finite order on

M

was

In [921. Stoll used the canonical function to construct theta

functions to given 2m-periodic divisors. the first outside application of value distribution theory.

All these results were surveyed by Stoll [l01].

The technical difficulties at the time should not be under estimated. Several complex variables was in its infant state. were not available.

i

is an entire function whose logarithm can be expressed

as an integral over the zero set

analogon to the Weierstrass product theorem.

function

f

Many technical devices

For instance. Rneser used integrals and the Stokes

Theorem for bordered domains on analytic sets. later by Lelong [50] and Tung [112\.

This

WdS

established only

-12-

Kneser did not prove the Second Main Theorem, which was proved by Stoll [93] in a much more general setting, but contains the above setting, see Satz 24.2. There is another corner stone of value distribution in several complex variables.

In 1897, E. Borel [7] reformulated and proved Picard's Theorem

in the form: "If such that

F

and

F + G - 1,

he shows, if

G are entire functions on then

F 1, ... ,Fn

F1 + ... + Fn - 1,

F

and

a:

without zeros

G are constant."

More generally

are zero free, entire functions with

than at least one

Fj

is constant.

We would expect,

that the beautiful result of Borel would be mentioned in most standard textbooks on basic complex analysis in conjunction With Picard's Theorem. This is not the case.

Thus we should not be surprised by D_ J. Newman's

question in the 1984 Mathematical Intelligencer [68]: "Do there exist three non-constant, non-vanishing, entire functions F + G + H - 1?"

F, G, H on

a:

such that

Borel's Theorem does not extend to infinitely manY

non-constant, zero free functions: 00 ~ L

(A.42)

_1_ pz-e pie

p=o

z _ 1

.

.

In modern, geometrical language, Borel's Theorem reads: "A holomorphic map

f: a: -

11' n

which misses

general position is linearly degenerated_"

n + 2 hyperplanes in

A local Borel Theorem in several

variables was proved by M. Green [33], (34). In 1932, H. Cartan (12) proved a Second Main Theorem for a holomorphic map

f:

a: -

• n'

Let

be a holomorphic vector function with 1P : a:~+ 1

(A.43)

_

11'n

10

f - F

is the residual map.

T fIr) -

J

-

log II

(vO' ... ,vn ) : 0

10

a: -

a:~+ 1

where

Then the characteristic

10

lIa - 10g11

10

(0)11

a: is an increasing function. convex in choice of 10. Assume that v·(O) J

'I:

log r, which does not depend on the 0 for j - 0, ... ,n and that

·13·

a:.

vO' ... ,v n are linearly independent over
(
has rank

J.t~

Let by

a: n+1

in

J.tj(z) - Min(n,J.t~ (z))

for

n + 1.

Take

j - 1, ... ,q. Define

q > n + 1.

Take

Assume that the

P j - ajOvO + '" + ajnVn

be the zero divisor of

Pj .

Define

J.tj

j

for

Z E

a:.

Then there exists a constant

J

c > 0

and a set

(A.44)

E of finite measure in

(q

for all

r

E

IR+

such that

n

IR+ - E.

We kept close to Cartan's formulation, which does not contain much geometry. He does not mtroduce a compensation function and the FMTH appears only in the form of an estimate (A.15). He uses the maximum norm instead of the euclidean norm to define implies Borel's Theorem.

Tf .

He observes that his SMTH

HiS proof rests upon the Lemma of the logarithmi(

derivative, which much later was extended to meromorphic functions on

a: m maps

by Vitter [115], who proved the Second Main Theorem for meromorphic f:

a: m -

1P n by Cartan's me hod.

Also see Biancofiore and Stoll

[5] and Stoll [107]. H. Weyl and J. Weyl [118], [119] and L. Ahlfors [1] pioneered the theory of value distribution of holomorphic curves. They established the two Main theorems and the Defect relation for the intersection of a holomorphic map

f: M -

1Pn

with the hyperplanes in

1Pn'

where

M

is a Riemann surface. In a stunmng performance, Ahlfors obtained a defect relation for associated maps. H. Wu [126] reorganized this theory in a modern fashion. In 1953-54 Stoll [93] united the Ahlfors-Weyl Theory and the Kneser Theory into a value distribution theory of meromorphic maps f : M-

1Pn'

where

M is a non-compact Kaehler manifold.

The two

Main Theorems and the Defect relation were established for the intersectiOI1

·14· of the image of

f

with the hyperplanes in

11'n'

We will sketch this

theory now. In subsequent years many different extension and versions of value distribution theory were given.

They cannot be sketched here.

For an

dccount see Stoll [1021. The First Main Theorem fur meromorphic maps.

A4.

a'

Hermitian geometry.

needed in Section B. n ~ 1.

Define

Let

We introduce some concepts which are also

V

be a complex vector space of dimension

V. - V - (OJ.

Let

V·

be the dual vector space.

< t:) ,'lID>

be the inner product between

t:)

Let

be the exterior product space.

For

"V

E

V a d

"V p-q

the interior product 'lID

"va.

E

Since

V - V... ,

1,

If

is defined.

the exterior product

"V

£

t:)

p

'lID

E

Let

V·.

and

'WJ

E

"V q

p

q ,

is defined for

p , n + I, €

t:)

"V

and

p

the roles of

V and

va

can be

q

interchanged. Let I.at

1P{VI = V./([;.

1P: Va -

»{V,

IP{AI - 1P(A " Val.

be the projective space associated to

be the residual map.

The Grassmann

~

For

A

of order

p

~

V.

£

V.

define

Z[O,nl

(A.45'

and the Grassmann manifold If

x -

(A.46,

1P(~1 €

Gp{VI.

Gp{VI - 1P{G p{VII

of order

p

are defined.

the {p+ I)-dimensional linear subspace

E(xl -

(t:)

E

V I ~ '"

10

-

O}

is assigned. where E(x) - 1P(E(x)) is a projective ~ of dimension p in 1P(V). Put q - n - p - 1. For a - 1P(-u) E Gq(V·I the (p+ 1)- dimensional linear subspace

·15·

(A.47)

Ela] -

is assigned, where p in »(V).

E[a] - "(Ela))

V I «L

E:

(10

~he=r,--_..::n

is defined.

OJ

-

is a projective plane of dimension

A positive definite hermitian form

( I I on

V

is called a

V,

and

V

together with

vector space.

On

V

a norm

hermitian product (or metric) on said to be a

10

lit-II

Natural hermitian products are induced on

( I I is =

J(~I ~)

V·, AV, ®V p

p

ev (YDlmetric tensor productl. If V and Ware

(tensor product),

p

hermitian vector spaces, so are the direct sum product

V $ Wand the tensor

V ® W.

Define

T

A V _

p:

by

IR+

p+l

Fubini-Stud,y form

on

.,. p(~1 = 1I~1I2.

The

A VI is the unique positive form '\, of

1P(

p+l

bidegree

(1,11

Take

x -

... (~) - dd c log., p.

with

I'(~) E: 1'(

A V)

and

Also write

a = 1P(<<)

I:

p+l

o, and to

p 'n.

q ,

1P(~L«1

xLa a

If

x

then

1P( A V·)

with

q+l

~ L« ;I!:

0,

exists.

In any case the projective distance from

then

(x,al

is said to be free for

L

is defmed by

IItL«1I [] xLIl [] = II~II 11«11

(A.48)

If

D - DO

I:

Gp(VI

~L«

-

and

a

<~,«> E:

It.

if

a

I:

then

0

~

0 xLa 0 '1.

If

p

~ q,

We write

o ~ []

(A.491

If p - q - 0, varlable,

Gq(V*I,

I:

I <1:..()\> I

x;a 0 - 0 xLa 0 - II~II 11«11 :s: 1 .

1P(V*)

is fixed and if w

I:

1P(V) - Ela)

is

l

-16-

new)

If

x

dd c log 0 w;a []2

-

we have (Weyl [119])

1P(V),

E:

=

(A.51)

f l o g 0 x.. a aE:1P(V*) b)

Hermitian v

tor bundles.

extend to vector bundles.

The operations

ED, 8, ., L, A

The concept of a hermitian metric along the

fibers of a vector bundle is defined.

A holomorphic vector bundle together

with a hermitian metric along its fibers is called a hermitian vector bundle. Let

re: W

-+

N be a holomorphic vector bundle over a pure

k-dimensional complex manifold sections of let

Z(s)

W over

Let

M

pulls back (re ,f).

x

E:

For

Z(s) , k - 2,

f'"(W)

s E: r(N,W),

then

then

s

s

is reduced.

A holomorphic map over

M,

is said to

f: M

-+

N

the relative product

The standard model is defined by f'"(W) - «x,w)

K: f'"(W)

where

s

r(N,W).

dim x Z(s) :E; k - 2,

be a complex manifold.

W to the bundle

(A.52)

f:

If

If dim

N.

I:

The vector space of holomorphic

N is denoted by

be the zero set.

be reduced at

N.

f*(W)

-+

r(N,W)

f(s)(x)

=

Let

W with

Y

M with

If s

M X WI fIx) - lI'(w»

K(X,W) - x

f(x,w) - w

a lifted section

(x,s(f(x))).

11 : W -+ Y

-+

E:

[Is)

is a vector bundle and where

is a bundle map over E:

r(M,C'"(W))

is reduced,

fis)

f.

If

is defined by

may not be reduced.

be another holomorphic vector bundle over

N.

be a surjective bundle homomorph'sm with kernel

Let X.

Let

be a hermitian metric along the fibers of Wand let Xl. be the differentiable subbundle of W perpendicular to X. Then one and only one hermitian metric along the fibers of y, also denoted by x and called the quotient metric, exists such that isometry,

11:

Xl. -

Y is an

·17·

c) N

Hermitian line bundles.

with hermitian metric

associated Chern form. Z(s) - 0,

with

be a hoi om orphic line bundle over along its fibers. Let c(L,K) be the

K

Let

U ~ 0

If

L

is open in

N

and if

s

E:

r(U,L)

then

(A.53)

Let

be a I near subspace of

V*

o < dim V* = n : N X V* -

n + 1 <

n(x,~)

E:

- ~(x)

spans

L

2

Let ELla)

a

=

with

A bundle map

V - V**.

L,

c led the evaluation, is defined by

Lx

f r all

x

n

if and only 1f

Take

Put

00.

r(N,L)

1P(~) E:

E:

~ E V*.

N and

By definition

V*

is sudective. then

V*),

be a herm1tian metric on

ELla]

V*.

=

Z(~)

only depends on

The distance from

x

E

N

a. to

is defined by

(A.54)

where all

is said to be distingui hed if

(.(,2)

x

E

N

and

a

E

1P(V*).

If

0 ~ 0 x,a 0.( ~ 1

for

t

can be

Then we can take for

.(

the induced

x

V· -

N 1S compact,

or

K

multiplied by a constant to make the pair distinguished.

V·

Assume that 2.

We have

tlJ(x)

E

tlJ : N (A.55)

0 x,a O 2 ,2 nx(~)

defined by 1P(V)

spans

-

~(x)

For

is uniquely defined.

1P(V)

E

N,

the map

is linear with kernel

nx

EltlJ(x))

Lx

where

This dual classification map

is holomorphic with

o :!ii:

0 x,a 0 1 ,1 - 0 lP(x';a 0 :!li: 1 .

Thus (1,1) is distinguished. tautological bundle (A. 56)

L.

The name is explained as follows.

So(V, - {(x,!;.,

E

F(V, X V I!;.

E

E(x',

The

-18-

is a subbundle of the trivial bundle bundle

QO(V)

1P(V) X V

and defined the g,uotient

by the exact sequence

(A_57)

o -

SO(V) -

1P(V) X V -

QO(V) -

0,

whose dual sequence is given by (A.58)

o -

where

QO(V)'" -

H - 0(1) - So (V)'"

with

v'" -

E(x).

If


t

metric

r(1P(V),H).

1P(V) X V*

E:

V·,

on

V'"

SO(V)* -

0,

is the hYperplane section bundle on

If

x

then

n

1P(V) X V'"

£

1P(V),

then

Hx - E(x)'"

11 (x,
is the dual of

The hermitian

induces a hermitian metric along the fibers of

t

which defined a quotient metric

alo g the fibers of

L

=

CP"'(H)

is the pullback of

H

under

cP

H

n - c(L,t).

where the Fubini-Study form coincides with the Chern form: Then

1P(V)

such that

c(L,t) - CP*(c(H,t)) - cp"'(n) .

(A.59)

d)

Meromorphic maps.

manifolds with such that

dim M - m.

Let

is analytic.

M-A

M and N be connected, complex

Let

A "t: '" be an open subset of Then A is dense in M. Let

M

f: A M be a holomorphic map. The closure rtf) of the graph ((x,f(x)) I x E: A) in M X N is said to be the closed mEh of f. The map

f

is said to be meromorphic on

and if the projection If

=

(x

E:

Obviously

M 11*1(-l(x) > 1) If (\ A

f : M - If -

N.

not identic to

00.

Let

1(: rtf) -

V

=

0.

M

M

is analytic in

The map

f

if

rtf)

is proper. M

is analytic in

M X l'

The indeterminacy

with

dim If

:!O:

m - 2.

continues to a holomorphic map

A meromorphic function is a meromorphic map into

be a complex vector space of dimension

n + 1 > 1.

1P1

Take

N - 1P(V) and take A as above. Let f: A 1P(V) be a holomorphic map. Let U;t 0 be an open, connected subset of M. A holomorphic map ~: U V is said to be a representation of f if ~ iI!! 0 and if fIx) - 11'( ~ (x)) for a\1 x £ U "A with ~ (x) ;t O. If a £ U

·19·

then

is a representation at

1:)

a.

The representation is reduced if

dim 1:) -1(0) :E; m - 2. The map f is meromorphic if and only if there is a representation at every point of M. A meromorphic map admits a reduced representation on every Cousin II domain in M. If 1:) : V _ V is a reduced representation, then Let

VM 8 Lf

over

"Let

Z(

Ir -

-1(0).

1:)

1:)

Lf

Vp to isomorphism, there

and a holomorphic section

F - Ff

M umquely determined by the property:

1:):

V

V -

~ r(V,VM) by

with

n

VM - M X V be the trivial bundle.

exists a holomorphlc I e bundle

1:)

U

b.) - 0

be a reduced representation of f.

;; ( ) - (x, such that

Mostly we write bundle on 1P(V). Then Therefore we call Lf

1:)

(x)).

F IV -

Then ;;

8

1:)

b.

E

r(V,L f ) exists

b.."

1:)

Let H be the hyperplane section I (M - If) is isomorphic to f*(H) on M - If

-

Lr

1:)

1:).

the hyperplane section bundle of f

representation section of

Define

f.

(See Proposition 2.1).

and

Ff

The section

the

Ff

overcomes the possible absense of global reduced representations.

e) Divisors. Since M is a manifold. we can identify a divisor with its multiplicity function. Let f,a 0 be a holomorphic function on M. Take x E M. Let U be an open neighborhood of x and let

a: m(1)

11 : V -

Integer

q

degree

q

~

be a biholomorphic map with I1(X) - O. For each there exists uniquely a homogeneous polynomial Pq

0

a: m

on

of

such that 00

flV -

(A.60)

!

q-O

Pq

a

converges uniformly on every compact subset of is a number depending on

J.l -

f

J.l~ (x), and

x

II

U.

Since

called the zero multiplicity of f only. such that

f:ll! 0, at

x

there and

of

·20·

v.~: M -

The function

f.

called the zero divisor of is said to be a divisor on connected neighborhood h it: 0

such that

An integral valued function

M

if for every

U of

x

supp v -

(A.61'

E

M

module

"'M.

Z

there is an open. g iii 0

and

The support of a divisor

<x

E

M

I ,,( x) '1:

O}

is either empty or an analytic set of pure dimension v ~ O.

is said to be non negative if

is

,,: M -

and holomorphic functions

v.~ - v.~.

v IU

x

Z

m - 1.

A divisor

The set of all divisors on

M

"

is a

Now we shall introduce specific d VIsors which will be

needed.

a' m - 1

Simple divisors. of

M.

S - supp

Vs

"s

The divisor

~ O.

Let

S

be an analytic subset of pure dimension

"s

Then there exists one and only one divisor and

"S(x' - 1

Vs

for every regular point

If

is called simple.

S

x

of

with

S.

Then

"s

is irreducible.

is

called prime. 8'

Decomposition.

finite set

f,

and a family

If

" it: 0

of irreducible. {kB'Bd.

is a divisor. there is

8

unique. locally

(m-l,-dimensional analytic subsets

B

of

IV

of non-zero integers such that

(A.62,

where

,,~O

if

kB > 0

and only if

set

f,

and family

p

z.

the p-truncated divisor

E

(kB'BEf,'

for 811

B

E

f,.

Given such a

then (A.62) defines a divisor. ,,(p,

of

"

Given

is defined by

(A.63)

y) Divi.ors of sections in vector bundle!. Let W be 8 holomofphic vector bundle over M and let s it: 0 be a holomofPhic section in W

-21-

over

M.

Then one and only one (zero)-divisor

Us

of

s

is defined

by the property:

"Let

U;t 0

be an open, connected subset of

be a reduced section.

Let

h: U -

II:

U,

then

Us I U

U~

that

s

ht

=

If and

on

is a Cousin II domain,

U

supp Us

Z(s)

!;;

~

M

--+

~

A section

E:

V

r(M,M X V)

U

N

--+

divisor

U

C 1(U)

;t

v IU If

=

on

M

g

0

0

u g - Uh

Let

N fOO(v)

't 0

Then

U~

=

Ilfis) .

is a

v

exist with

=

Ul

~

~

;; (x) = (x, ~ (x)).

0_

thus

is defined.

v: M -

be a divisor.

Z

f(M) g; supp v.

and

h 't 0

Let

The pullback

f*(v)IU

=

is defined on

N

with

0

0

u gof - Uhof f: M - If -

M - If.

f(M) g; supp Us' exists with

with

be holomorphic functions on

is meromorphic, then

fis) e: r(M,f*(W))

Ufisl ~ f"(~(s)'

~ 0

v

If

be a complex vector space

V

f*(v)

on

M.

N

Take

then

[·(U s )

fis) 't 0

if

where equality holds for line bundles.

. N

is

dim If ~ m - 2,

Since

be a hoI om orphic vector bundle on

s e: r(N,W)

section

r(M,L)

E:

Us ~ 0

Obviously,

is defined by the property:

Let

0.

f: M -

W

s

is defined by

the divisor continues uniquelY to a divisor Let

exist.

be an open, connected subset of

holomorphic and

o~

h

Let

be a hoi om orphic map with

f*(v)

"Let

with

Land

of

Pullback divisor.

€)

r(U,W)

E

"

be a holomorphic vector function with

the divisor

t

be a holomorphic functton such

and

Divisor of ~ vector function.

6)

f : M

t

Let

For line bundles we have equality.

divisor, a holomorphic line bundle

Let

=

M.

exists.

Also a

f(M) g; Z(s). Abbreviate

then

l

·22·

If

V"

a - P(u)

0

t:)

f iii! 0,

n f a

iii! 0

t:)

- u't depends on a and f

u~

then

E: 1P(V"),

If

is a holomorphic vector function on

then

.. .... t:)

~

"ft:)

of - ....

The a-divisor of ! meromorphic function.

Then

Pl.

)la - vIa)

U~ - f·(v a )

a-divisor

connected subset of

g iii! 0

if

holomorphic functions on

If

the divisor of

f.

If

fl

f: M -

and

If

E:).

f iii! 0,

f2

N with

Take

PI

If

PI

U;t 0

f

~

the

a,

is an open,

h iii! 0 are co-prime

and

then

hf - g,

U with

only.

A meromorphic function

is a prime divisor on

is defined by

M,

if

f"(u _ ). ...,

can be regarded as a meromorphic map E:

r(N.W).

is a finite dimensional linear subspace of

a

0

uf I U -

o

then

if

Ug-ah 00

uf - Uf - Uf

is called

are meromorphic functions on

M then

Ur1f Z - UrI + Ur 2 . The intersection divisor.

1/)

dimension

n + 1.

a

with

P(V·)

£:

F(V),

U~

u

:

£:

f: M -

f(M) ~ Efa].

Let

If

U V:,

e)

V

U;t 0

be a complex vector space of

V

be a meromorphic map.

1P(V)

is defined by

1.1: IU

-

U~

t:)

if

on

M,

a - 1P(u)

if

with

.u> .

The basic integral theorems. m

and

holomorphic vector bundle over

N.

Take

f(M) ~

M

Let

complex manifolds of dimension

be a meromorphic map with

f,

H

Also

£:).

is an open, connected subset of

is a reduced representaton of

then

Take

Using the hyperplane section bundle

the intersection divisor f"'(vEla).

t:)

Let

Z(s).

be a hermitian metric along the fibers of

n

and

N

be connected

respectively. s

£:

r(N,W).

Define W.

S -

Let Let

supp

Abbreviate

W

f

1.1:

be a M -

Let

N K

·23·

(A.64)

Let

A - 10g1lSIl;

G;t 121

boundary

aG -

cO.

real. class E'

be an open, relative compact subset of

G-

'Iubmanifold of

C1

I

(A

aGo

fidE' +

0

G

Let

d(A

0

S" supp(x I aG)

dx '" dC(A

0

G

then

Suppose

(A

D

fiE' .

(m-I.m-I)

has measure zero on

x '" ddC(A

E' - dCx.

I

fI '" E' -

0

dx '" dCA

S.

on

M.

Assume that

Then the Residue Theorem holds

f) +

G

If we take

M.

and bidegree

COO

J

f) +

on

Let

aG

fI is integrable over aGo

0

2m - 1

G.

Then we have the Singular Stokes

G

(A.66)

I

I

be a form of class

X

Assume that x " dC(A

M oriented to the exterior of

and degree

is inte rable over

(A.65)

Assume that the

G is either empty or a pure (2m-I)-dimensIonal.

be a form of class

(A D fIE' Theorem

M.

0

r

S~G

f - dA

0

f "dcx.

We obtain the

Green-Residue Theorem

J G

x " ddc(A

D

f) -

I G

(A

0

f)dd c X +

r

S~G

(A.67)

Proofs for even more general situations can be found in Stoll [103]. Also see Stoll [93]. [96]. [97]. [98]. [108] and Griffiths and King [35]. Before we proceed a number of important applications shall be listed.

-24-

APPLICATION:

1_

w - «:

Theorem of Poincar':-Lelon.l [51].

X It. s E r(<<:.W)

with

M. Then

dd c " a f ::

O. Assume that

ood choice of

divisor of

J

J

M

S

0

is the current of integration over the zero

f. The Argument Principle (Wirtinger [120]. Stoll [93]).

N. W. sand =

as in the 1. Application.

f

J x"

APPLICATION.

f

that the boundary

X

submanifold of

> 0

on

Then

G - g.

dC log I f I 2 .

The Jensen Formula (Stoll [93]).

as in the 1. Application.

COO

dx - O.

aG

S(\G

3.

Assume

and

J

(A.69)

We have

Then

ddc[log I f I 2]

APPLICATION:

dCx

be a holomorphic

has compact support which is contained ir

log If 12 dd c X -

Thus the current

2.

X

f ~ 0

Let

«:.

N -

is the zero-divisor of f.

J1f

G.

(A.68)

Take

0

s J1 f -

function on

G for a

s(z) - (z.z).

Take

C)g

~

Let

g _"

Take

be open with

N. W. sand

i C G.

Assume

i - g is a pure (2m-l'-dimensional, real. class

M oriented to the exterior of Take

0 < RElit

Let

continuous function on G which is of class solves the Dirichlet problem

(A.10,

A function If, of class COO exists on The Green-Residue Theorem implies

0

~

COO

g.

'"

~

on

"'Ii - 0 M such that

Assume that R

G-

be the unique g

which

",laG - R.

If, I (G

-

g) - ",.

-25-

J U~(R

- if)x

J

~

snG

J

(log If 12)d c ", " X -

(log If 12)dd c (if x)

aG

g

J

sng which adds up to the Jensen formula (A.71)

J

(log IfI 2)d c ",,, X -

(log IfI 2 )dc ",,, x

ag where 4.

d C" , , , x

APPLICATION.

Stoll [108]).

Let

defines a non-negative measure on

aG

and

ago

The unintecrated First Main Theorem (Levine [541, W - L

be a line bundle.

dd c A - -c(L1,x).

Then

Assume dx - O. Then d C X - O. The Green-Residue Theorem becomes the unintecrated First Main Theorem

(A.72)

J

f-(c(L,x)) "

x-

J u:

x -

SnG

G

rG X"

~(

This is a balancing statement, the integral over and is non-negative if

X ~

0

and

c(L,x)

~

O.

dC log lis

D

£II ~

G does not depend on

s,

The integral over

S "G measures the intersection of f with the zeros of the section s and the difference of these two integrals is compensated by a boundary integral.

Unfortunately nothmg can be said about the sign of the boundary

integral, which makes estimates impossible. The existence of the integral has even to be assumed. To correct those short comings (A.72) will be "integrated. "

, 5. Af,PLICATIO,H. Poincare dual. In addition to the assumptions in the 4. Application we require that M - G - N is compact and that f is the identity· Then

·26·

!

(A.73)

for every closed form

X

c(L,x) '" x -

of bidegree (m-l.m-I)

and

class

Coo.

thus the Chern form c(L,x) is the Poincar:-dual to the zero divisor of any s E: r(M,L). If M is a compact Riemann surface, we take x - 1. Then (A.73) means, that the degree of the divisor of a holomorphic sectior in a line bundle equals the Chern number of the line bundle. r) The First Main Theorem. Let M and N be connected complex manifolds of dimension m and n respectively. Let L be a holomorphil line bundle over N. Let IC be a hermitian metric along the fibers of L

Let X ~ 0 be a form of class COO and bidegree (m-l,m-l) on M with dx = O. Let f: M N be a meromorphic map. Let (G,g,y,) be a condensor on M. This means: g and G ~ 0 are open, relative compact subsets of ag

~

M with

ii c G. The boundartes aG - G - G and

g - g are either empty or are pure (2m-l)-dimesional. real,

submanifolds of class

COO

of

M oriented to the exterior of

G

respectively g. The function y,: G - IR is continuous on G and of 2 class C on G - g and y, I g - 0 and y, I aG - R > 0 are constant On

G we have

0 ~ y, 'R.

For

t

~

0

define

Gt - (x

E:

G I y,(x) < t

Let V· be a linear subspace of r(N,L) with 0 < dim V* < Let I be a hermitian metric on V*. Assume that (x ,I.) is distinguished. Take a E: 1P(V·) with f(M) g; ELla). Abbreviate Fa - sUPP

I..l:'

counting function

the spherical image nf

~

compensation functions

(A.74)

Ar(G.L,K)

-

the characteristic

0,

the valence function

Nf

mr

~

Df

I G

(A.7S)

Ar,

0

and the deficit

f*(c(L,IC) '"

X

~

0,

Tf'

00.

the

the

are defined by

·27·

(A.76)

nf(G,a,L)

r

-

G~Fa (A.77)

J

Nf(G,a,L) -

J.l~(R

R

- ¥-)x -

J0

nr(Gt,a,L)dt

GnF a

(A.7B)

IDt(aG,a,L,K) -

(A.79)

mf(C>g,a,L,K)

1

J C>G

log 0 f,a 0x,l 2dc ¥- '"

X

J

log 0 f.a 2d c ¥- '" °x,l

X

,a °K,l 2dd c ¥- '"

X

ag

Df(aG,a,L,K)

(A.BO)

-J

log 0

f

G

If the line bundle

Ar(G,L,x)

~ 0

L is non-negative, that is if

and

Tf(G,L,K)

~

o.

C(L,K)

~

If we apply the same

0,

then

~

trick as in

the 3rd application we obtain the First Main Theorem of line bundles and condensors. (A.BI) Tt(G,L,K) • Nf(G,L,a) + mf(aG,a,L,K) - mf(as,a,L,K) - Dt(G,a.L,x)

We have a balancing statement. term

Df(G,a,L,K)

does not depend on

appears called the deficit.

nicely on the line bundles. bundles then

Tf

If

(L 1,x 1)

and

a.

A new

The characteristic depends (L 2,x 2'

are hermitian line

-28-

(A.82)

The characteristic introduced by Nevanlinna turns out to be the integrated version of the characteristic of a line bundle introduced by Chern, a relation which holds in much more general situations. If we take

bundle on

N - F(V)

!P(V·),

Study form.

Now

B[a) - EL[d)

and

then

and

v· -

L - H

r(IP(V),H)

=

0 f,a Dt,t - 0 f,a 0

as the hyperplane section

(.')(1)

and

cIH,I) = 0

is the Fubini

is the distance from

fIx)

u ~ is the intersection divisor with the hyperplane

defined by

a.

appreviate

Af(G) - Af(G,H,t), Tf(G) - Tf(G,H,I), nc(G;a) - nr(G;a,H),

Nr(G,a) and

=

We substitute all this in the definitions (A. 74) and

NC(G,a H), mc(oG,a) - me(oG,a,H,t), me(og,a) - me(Oi,a,H,t)

DC(G,a) - DC(G,a,H,L).

We have the First Mam Theorem

(A.83)

Exchange oC integration and (A.S!) imply

(A.84)

n

f

mf(oG,a)On(a) -

J

mC(oi,a)On(a) =

u:1P(V*)

(A.85)

J

U

u- 1

f

de", "

X

d C ",

X

oG n

aE1P(V*)

(A.86)

.1

~

~ JJ-1

*J

A

Og

n

DC(G,a)On(a) -

&E1P(V*)

while Stokes Theorem implies

~

.1

u"'l JJ

L

dd c ", "

G-g

X

to

·29·

J

(A.87)

G-g We obtain the Mean Value Theroem

(A.88)

T rIG)

=

J

Nr(G.a)On(a)

a£1P(V tc )

which does not hold in general for

Let Nh(G)

=

h

jI!

0

be a meromerphic function on

Nh(G.O) - Nh(G.oo).

FMTH implies the G

(A.89)

NUh (G)

Tf(G.L.K).

Since

M.

Put

I h I - 0 h;O 0/0 h;oo

o.

the

r I Jensen Formula

~

J

log I h I dCof "

X -

aG

J

log I h I dCof "

X

ag

I_

log I h I ddC ",

"

x .

G-g

If there is a reduced representation a

II

'0

'0:

M -

V of

f.

take

< '0 ,w jI! O. Define a - lI'(a). Then II lIall 0 f;a 0 - < 10 ,w. If we apply (A.89) to h - < '0 ,a>,

E:

V·

such that

the FMTH implies

(A.90)

-I

(log II

'0

lI)ddcof "

X •

G-g g) Remarks to the First Main Theorem. We rediscover many useful features of the classical FMTH as well as the close connection to the Chern forms. However there are substantial disadvantages. We have a new

-30-

deficit term and our set up depends on the arbitrary choices of These problems can be resolved.

If

given. choose

X

problem and

M

ddCo{I

o{Il ClG

X

>0

with

o.

A

X

For on

= 0

G - g

is

o{Il g - 0

with boundary conditions R(G) > 0

is chosen such that

(A.9I)

1

The deficit is eliminated. for

0

=

we take the solution of the Dirichlet

o{I

where the constant

R(G) > 0

=

dx

v, we can

is a Kaehler manifold with exterior form u m- 1 >

=

We assume

R, X

o{I,

We say that

(G,g,o{I)

is a harmonic condensor

x.

If

M

is a non-compact Riemann surface, Weyl (119) took

X

=

1

and succeeded to establish a Second Main Theorem and Defect relation by exhausting

M

with harmonic condensors.

relation is the question if the capacity zero.

Decisive for the type of defect

J:

inf(l/R(G))

=

Wu [126] recast and enlarged this theory.

theory to

dim M > 1

by introducing the form

is positive or

Stoll [93] extended the x.

Thus Nevanlinna

theory was fully established for meromorphic maps from Kaehler manifolds into projective space with the family of hyperplanes as targets.

The

exhaustion by condensors remains cumbersome, and we will see how this can be rectified. h)

Higher Codimension.

target fdmilies?

Could value distribution be extended to other

The breakthrough came with the papers of Levine [54] and

Chern (14) where the family of p{:planes in projective space as the target family.

In general we have this situation:

complex manifolds of dimension be a meromorphic map.

m

Let

and

n

A target family

II

M

Define

Fa

=

assume (respectiVely construct) a closed form and class bidegree

COO

on

(p-l,p-l)

N

and for each

and class

COO

a on

E

N

respectively. =

{Ea)aEA

In-p)-dimensional analytic subsets is given, where is a compact complex manifold.

and

m

elm) a Cl

~

0

A a form

N - Ea

be connected

Let

f: M --- N

of pure

~

p

and where

for

a E A.

of bidegree Aa ~ 0

such that

A We

(p,p)

of Aa

has

-Jl-

residue

1

on

N - Ea

and such that x ~ 0

take a closed form and a condensor

(G,g.",)

of class

COO

and bidegree

We

1m P.rn p)

as before. define the value distribution functions

and obtain a FMTH containing a deficit term is impossible to eliminate the deficit. since

DfIG.a;CI).

Now. however. it

dd c ",,, x _ 0

is an over

determined system and in general the Dirichlet problem cannot be solved. The existence of the deficit accounts for the accurance of Bieberbach rnap5 and other phenomena as the transcendental Bezout problem of Ph. Griffiths. Still we can consider the pseudoconcave case where pseudaconvex case wh re

dd c ", ~ O.

dd c ", ~ 0

and the

Integral averaging and capacity

methods still allow C so ati-Weierstrass type theorems under reasonable assumptions.

For these problems and the case of higher codimensions in

general the reader

IS

ref rred to:

Hirschfelder [45]. Wu [125] Bott-Chern

[8], Stoll [961. [97], [981. [99]. [104], [105]. Tung [112]. Cowen [20]. Shiffman [791, Molzan [561, Molzon-Shiffman-Sibony [59]. Dektjarev [221. Griffiths King (36] and of course the papers of Levine and Chern_ Application of the First Main Theorem.

i)

Internal applications of

the value distribution functions and the FMTH are common and shall not be listed here.

There are some interesting applications outside value

distribution theory. u) with

The Theorem of Fatou.

d", '1' 0

on

aGo

bidegree

Im-1.m-1)

Such

and

'"

X

on

f

M.

X

T fiG)

IG.g.",)

be a condensor on

If

f: G -

\PIV)

x

condition does not depend on the choice of

is a meromarphic

x

and

",.

bounded holomorphic function is in the Nevanlinna class.

f : G -

1P(V)

is closed.

still can be defined, but may be infinite.

is said to belong to the Nevanlinna class if

result of L. Lempert [52] in

M

> 0 be a form of class COO and of

We do not require that

always exist.

map, the characteristic The map

Let

Let

a: n ,

T fiG) <

00.

The

Naturally. a In extension of a

Patrizio (73) proved: "A meromorphic map

in the Nevanlinna class admits admissible boundary values in

the sense of Korany-Stein at almm,t every point of aGo If flM) is not contained in any hyperplane in 1PIV). if a E 1PIV·). then the set of all

x

E

oG

where

f

has an admissible boundary value at

x

belonging

-32-

to E[a) has measure zero in aG." This is a beautiful extension of the classical theorem of Fatou-Nevanlinna for the unit disc. Without the apparatus of value distribution thoery. Lempert's theorem could not have been reformulated, localized and extended to manifolds. 8) 1I ~

0

Blaschke products. Take '" and )( as in a). A divisor on G satisfies the Blaschke condition if N pIG) < 00. The

condition does not depend on the choice of '" and )(. The zero divisor of a holomorphIC function in the Nevanlinna class satisfies the Blaschke condition by the FMTH. On the unit disc a divisor, which satisfies the Blaschke condition. is the zero divisor of a bounded holomorphic function. which is not true on the unit ball in a;m with m > 1. but Skoda [86). and Henkin [43) proved independently: A principal divisor satisfying the Blaschke condition is the zero divisor of a holomorphic function in the Nevanlinna class. Thus a difficult problem was solved. The classification of the zero divisors of functions in y)

Mfine algebraic varieties.

Let

A

HP

is still unresolved.

be a pure p-dimensional

analytic subset in a;m. Adopt the notations of A.3 around (A.36) and (A.37). The counting function nA of A is defined by (Kneser [46). Lelong [50))

J

(A.92)

A [r)

where the Lelong number

n A(0)

A

J

[r)

is an integer (Thie [110)).

Stoll [94) extended the previous theory to the analytic set. in applying the FMTH to the map 1': A IPn-1 and showed that A is affine algebraic if and only if

nAb')

is bounded.

For

p - m - 1

this was

shown earlier and is much easier. 5) Other applications. The construction of Theta functions to prescribed (2m)-periodic divisor (Stoll [92]) was mentioned before. Later we wIll observe that Henson and Rubel [44) applied Nevanlinna theory. in particular the Lemma of the Logarithmic derivative to solve a problem in m"thematical logic. Of course both applications involve the FMTH.

·33·

A5.

Parabolic manifolds.

handle.

The exhaustion by condensors is difficult to

Hence exhaustions have been used in recent years.

Here the

parabolic exhaustions are opportune since they possess a rather easy harmonic condensor system. Let ., ~ 0

o :$;

r

M

be a connected complex manifold of dimension

m.

Let

COO

on

M.

be a non-negative, unbounded function of class E

~

and

A!;;; M

define

A(r)

(A.93)

(A.94)

A - Ix

(A.95)

If

E

A I .,(x) - r2}

w -

M[r]

is compact for each

exhaustion of

M.

A. - Ix

A I .,(x) > O}

E

dd c log ., r > 0,

The function

.,

the function

.,

is said to be an

is said to be parabolic if

(A.96)

on

M..

u

Then

~

0

on

If

M.

.,

(M,")

is said to be a parabolic manifold.

(A.97)

~., - Ir

Then

(A.98)

E

is a parabolic exhaustion, Define

~+Id"(x} ~ 0

v

X

E

M} .

....

~+ -~.,

aM(r} - M class

For

has measure zero. is

11

If

r

E

~T '

the boundary

compact, real, (2m-l}-dimensional submanifold of

cO. of M, oriented to the exterior of M, and c,-

I M

a>O

·34·

does not depend on

'" r e: -i1

If

~

r

0,

then

(A.99)

The cartesian product of parabolic manifolds is parabolic.

A

non-compact Riemann surface is parabolic if and only if each subharmonic function bounded above is constant. 1 0 (!) - 1I!1I2. map.

Then

Let

1 - 10

If: 0

If

(((m,10)

a: m be a surjective, proper, holomorphic

M -

is a parabolic exhaustion of

is called a parabolic covering manifold

c..

number is

(Stoll [105], [106].)

there is a biholomorphic map Thus

7

(M,7)

(a: m,1 0), Its generic sheet

of

(((m ,1 0)'

For more details see Stoll

A parabolic manifold is said to be strict if

THEOREM.

M and

Every m-dimensional connected, aff ne algebraic manifold is

a parabolic covering manifold of [103].

is parabolic where

If

(M,1)

h: M -

u >

o.

is a strict parabolic manifold,

((m

with

1 - 10

0

h.

is an isometry

Other proofs were given by D. Burns [9] and P. Wong [104]. A6.

The First Main Theorem

QIl

parabolic manifolds.

For

0 < s
define

(A.100)

(A.101)

(A.I02)

R(r,s) -

""2-m~--"-2

R(r,s) - log -;.

'" s

[

if

s

2m-2

m - 1

- Max(R(s"n'),O).

if

m

> 1

·35·

Then, except for the smoothness condition,

(M(r),M(s),o,b s)

is a condensor

with

(A.I03)

on

Mlr].

Thus the theory of section A4 applies and can be translated.

Strictly speaking, the smoothness condition is assured only if

...

~T'

belong to

b t

0 <

he Dirichlet problem is solved for all

Thus we have to requ're

rand

s

in

...

~T

rand S

s

< r.

only if boundary integrals

are involved. Now, we rewrite the theory in the new terms. nv

Nv

and the valence f n t'on

S - supp v

t > 0

are given for

and

r

n))(t) - t 2- 2m

(A.I04)

of a divisor

The counting function

)): M -

0 < s < r

by

J

vv m- 1 -

sit)

with

Z

vwm- 1 + nv(O)

s*[t)

(A.I05)

where

nv(O) -

11m t --.0

nv(t).

Since

one point, we cannot expect

MIO) 51!

ra

nv(O) - v(O)

may consist of more than

as in (A.38), but for strictly

If f

parabolic manifolds these are Kneser's identities. E

!P l ,

function of the a-divisor

~f

function on

Nf(r,s;a)

M,

if

a

8

E ~1"

(A.I06)

of

f

are denoted by

and these functions for the divisor

f 51! 0 are denoted by

....

8

nf(t)

is a meromorphic

the counting function and the valence

and

Nr(r,s).

~f

If

nc(t;a) 0

= ~f -

o<

00

~f

and of

,. s < r

E

~T

then (A.89) translates to the Jensen Formula

Nr(r,s) -

J M

log I flu -

J M<s>

log I flu

and

-36-

which implies Liouville's Theorem on Let

N.

Let

o<

Let

t

a

£

x

L be a hermitian line bundle with metric

complex manifold dim V· <....

is distinguished.

(x ,t)

M.

1P(V*)

and

V*

f: M -

f(M)!; ELla],

t > 0

with

Assume that

N be a meromorphic map.

If

denote the counting function and valence

u:

function of the intersection divisor respectively where

r(N,L)

va.

be a hermitian metric on

Let

on a connected

be a linear subspace of

by

nf(t;a,L)

0 < s < r.

and

and

Nfh·,s;a,L)

The compensation function of

;.

f

for

a

is given for

r

by

f.T

£

I

(A.I07)

log 0

1

f

0

,8

",t

u ~

o.

M

The spherical image and the characteristic of t > 0

and

0

<s < r

f

for

L

are defined for

by t 2- 2m

(A. lOS)

J

f*(c(L,"))

A

u m- l

M[t)

Tf(r,s,L,,,)

(A.109)

I

r

A

For

0 < s < r

£ ~1'

dt

s Ar(t,L,x) t

=

.

A

and

s

E:

-iT we obtain the First Main Theorem

(A.110)

which extends

mf

(A.11o) holds for all

If

elL, K) ~ 0,

r > 0

to all 0 <

S

then

as a continuous function such that

< r.

Af

and

Tf

are non-negative and increa

·37·

(A.Ul)

AC(O,L,x) - 1 im AC(t,L,x) t .... O

Af(oo,L,x) - 11

Af(t,L,x)

t .... O

exist and we have

(A.U2)

I

AC(t,L,x) -

C-(c(L,x)) '" wm- 1 + AC(O,L,K) .

M.lt)

If M - N and if C is the identity, we drop the index formulas (A.107) - (A.l 2). Let N - 1P(V)

be a her ltian vector space oC dimension

V

and

L - H - 0(1)

C in the

n + 1 > 1.

Take

as the hyperplane section bundle, then we

obtain

(A.U3)

r

Af(t) - t 2- 2m

Mit]

(A.U4)

J

C*(o) '" v m- 1 -

f-(o) '" w m- 1 + Af(O)

M.[t]

T fir ,s) -

J r

dt

Af(t) t

if

O<s
s

IA.U5)

mf(r,a) -

J

log 0 f. a 0 a

if

r

...

E

~7

M (A.U6)

Ac(O) -

1 i m Aflt) t -+0

where

Af ~

a

and

continuous function on o < s < r: (A.U7)

T f ;?; 0 IR+.

increase and

mf ~ 0

extends to a

Then we have the Flrst Main Theorem for

-38-

A

If

A

0 < s < r t:.f..,

s t:.f..,

and

and if

M -

«)

V

is a reduced

representation, we have

(A.UB)

J

Tf(r,s) -

log II

«)

J

lIa -

f

maY not exist.

uses the representation section hyperplane section bundle of fibers of

f

and

)" - 1 8

f. If.

«)

lIa .

M<s>

M A global representation of

log II

Ff

in

There is a substitute, which

VM X

Let

Lr

Lr

where

is the

be a hermitian metric along the

If.

is a hermitian metric along the fibers of

By Proposition 3.2 we have the sub it ute

(A.U9)

Tf(r,s) - Tf(r,s,Lf,lf.) -

J

J

log IIFfll)"a -

M The exhaustion of the limit

r'"

only.

00

M

M<s>

is easy to handle now.

We have to consider

For instance

(A.120)

where

Ar(oo)

>0

if and only if

strictly parabolic, that is, if and only if

Ar(oo)

<

00

f

(M, T) - (a: m ,TO)'

(Stoll [93)).

transcendental growth if and only if

The meromorphic map rum-dqenerate if

is not constant.

f: M -

f(M) ~ E[a]

Hence

f

Ar(oo) -

1P(V)

for all

a

then

If f

I a£FCV*)

mf(r,a)Cln(a) -

-t

is

is rational if

is said to have 00.

is said to be linearly E

F(V*).

If f

non-degenerate, we have

(A.IZl)

(M, T)

In 1 ] [ U-l U

C,

is linearly

·39·

IA.122)

I

T fIr,s) -

Nf(r,s;a)On(al.

ae:lP(V*)

For each

r > 0,

the incidence set

Br{r) - (a

E:

lP{V*) I f(M(r)) f"I E[dl ~ 121:

is measurable with

o~

br{d

I

=

on ~

1.

Bf ( r )

Since Bflr)

Nr{r,s;a) - 0 in (A.22).

if

a

£

lP(V*) - Br{d,

we can replace

If we integrate the FMTH over

Br{r),

lP(V*)

by

we obt ain

which implies (A.123)

Thus if

n - 6 and

c. -],

then

Tf(r,s)

of the hyperplanes are intersected by r

we obtain

"'00,

bf{d -

1

for

~

123

f(M(r)).

r'"

00.

assures us, that

Since

hence

T fIr,s) -

f(M)

00

99% for

intersects

almost all hYpe planes, which is a Theorem of Casorati-Weierstrass. A7.

The Ricci function.

a)

Notations.

example of the characteristic function.

We will give a most useful First some notations have to be

introduced which will be needed later also for other purposes. o ~ q E: l define

(A.124)

For

-40-

Let

M

be a connected complex manifold of dimension

m.

A

biholomorphic map

(A.125)

~

of an open subset

U of

called a chart of

M

(at

- (z1... , .zm)

M

onto an open subset

x

if

x E: U).

On

U

U'

of

a: m

is

of

M

define

(A.126)

(A.127)

t

where over

is a holomorphic frame of the canonic 1 bundle

IC

U with

(A.128)

Dependence on

~

is expressed by

U~.

U;. t~. x~ .

be any open covering of

M.

For

define

U~

(A.129)

I\.

- U~ '\ - U,\ "U~ " ... " U~ /\'O"'''P "0 /\'1 Ap

(A.130)

Alp) - Alp.1l1 - (>.. E: AP+1 1Ux. 'F- ell •

A chart atlas

(i}>")>"E:A

that

Il - (U>..I>"E:A

Abbreviate

is a family of charts

such

U{

is an open. so called associated. covering of

t>.. - t ix.

and

x>.. - Xi>..

only one zero free holomorphic function (A.131)

~x.: U>.. -

If

A>..J.l

(>...J.l) E: A(1).

exists on

U>..J.l

M.

one and such that

-41-

(6 Au (A,u)EAIlI '

Then

b) and

The Ricci form.

degree

each

A

2m

• > 0

Let

(ltA' AEA

M.

'" I UA - "'A x~ _ If

UAU .

Ric.

on

Let

Since

of class

•.

It does n t

The form

and

class

COO

M

For

on

M

'" U - .A 16 AU 12

then

exists on

Ric '"

M

such that

is called the Ricci form

p nd on the selection of the chart altas.

of the canonical line bundle q>

of

COO

one and only one form

(1,1)

One and only one hermitian metric

all forms

.A

(A,U) E AIlI

nd bidegree

Ric • I UA - dd c I g "'A. of

be a chart atlas on

d c log I 6 AJ.l 12 :: 0, COO

e.

be a positive form of cl

A there is a positive function

E

such that on

is a basic co cycle of the canonical b

im q> '" f - IC",('ll,E)'"

K such that

E of bidegree

is defined along the fibers

IC '"

(m,O)

on

M.

Then

Ric '"

for is the

Chern form (A_132)

Ric '" - c(K,IC.) .

The Ricci function_

c) dimension

m.

(A_133)

For

Let

0 < s < r

(M,T)

the Rlcci function of

Riclr,s,.) - Tlr,s,K,lC.'

function of the canonical bundle

Define

M+ - (x

E

is defined by

is the characteristic

K for the metric

M I u(x) > 01.

Then

but in order to define

Riclr,s,u m) we need

(M,T)

(a:m,T O'

is isometric to

•

(Ric .) ,.. u m- 1 t l - 2m dt.

Ric(r,s,.) -

From (A.132) we see that

be a parabolic manifold of

and that

IC y

Ric urn

.

is defined on

M'

M+ - M which means that

Ric u m ::

o.

Therefore we

have to Use an indirect method to define the Ricci function of

u m.

-42-

Lett> 0 v ~ 0

function

be dny form of degree of class

is defined by

COO

~~ - {r

(A.134)

For

0 < s < r

E:

~~

and class

COO

um

The set

VY.

=

on

M.

A

~1' I (log v)a integrable over M
E:

does not depend on the choice of zero.

2m

IR+ _ ~o

'II

and

s

E: ~T

with

has meaSure

l'

o

the Ricci function of

T

is defined by

(A.135)

t J

IbcT(r,s) -

(log v)a -

+J

(lOg v)a + Ric(r ,s, 'II)

M<s>

M

and does not depend on the choice of the form

Is

"to

Ric T

non-negative?

If

(M,T)

is a parabolic covering manifold of

proper, surjective, holomorphic map that

l' -

TO

0

7C.

The divisor

branching divisor of

((tm, l' 0)'

there is a

7C - (7C 1 ' of

p

d7C1

A

...

A

d7C m is the

Then

l' .

(A.136)

RicT(r,s) - Np(r,s)

~

0 .

Proofs are given in Stoll (103). AS.

Associated maps.

a)

General remarks.

to prove a Second Main Theorem.

There are several methods

Cartan's method [12] was extended to

several variables by Vitter [115] and Stoll (107) but is restricted to meromorphic maps

f: (tm -

"(V).

The method of

Cl1rlson-Griffiths-King [11], [351 works for meromorphic maps where

M

is a parabolic and

N

f: M - . N

is projective algebraic but is restricted

to dim M ~ rank f - dIm N. Noguchi's method [71J, (721 works in the same situation Without the restriction, but the defect bound is not explicitly calculated. The method of Ahlfors-Weyl 11], [109] works for meromorphic 1P(V) where M is a parabolic mamfold (Stoll [lOS]). The maps f: M -

-43-

original integral averaging method has been replaced by a curvature method introduced by Cowen and Griffiths [21] if

a:

M -

and modifled by

Pit-Mann Wong [123] and Stoll [108] for parabolic manifolds

M

We will

outline this modified Ahlfors-Weyl method. since we can extend it to movin~ targets.

Consciously or not. all the methods use the jet bundle which

shows up in our case in the concept of associated maps. b)

The def n'tion of associated maps ~ Riemann surfaces_

[118] and Ahlfors 11] f rst considered holomorphic curves Let

10:

a: -

V.

be a reduced representation.

Weyl

a: -

f:

1P(V).

0 ~ p ~ n

For

holomorphic maps

(A.137)

are defined. where

10

~

p

linearly non degenerated. fp : II: -

Gp(V)

M.

10:

for

0' p

If so. the

is defined.

a Riemann surface representation

0

pth

~

if and only if f

n.

associated map

For a holomorphic map

U-

V on a chart

zero free holomorphic factor. Gp(V)

c)

U ~.

z: U -

Hence the

pth

1P(V)

If we switch to

gains only a

10.1!.

0

f

if

1O.1!.

is

See Weyl [119).

The definition of associated maps

~

manifolds.

A distinguished

differential operator is missing on a connected. complex manifold dimension of bidegree

m > 1.

M

of

Therefore. Stoll [93] assume that a holomorphic form

(m - 1.0)

is given on

M.

holomorphic vector function on a chart

Let

10:

1t: U -

U ---- V

U'

of

by (A.127). One and only one holomorphlc vector function is defined by (A.l3S)

on

associated map

fp I U ~ 11'

is well defined by

linearly non-degenerate.

f: M -

the definition (A.137) still works for a reduced

another reduced representation on another chart.

fp : M -

is

10 ~

r -

d

10

A

B.

M.

B

be a Define

10 ~ :

U-

r V

-44-

The operation can be iterated:

10

(p) - ( 10 (p-ll) '.

Put

10

(0) -

10.

Abbreviate

(A.l39)

10

.2

10

-

~

Dependence on the chart

Let

f: M -

1P(V)

A

10 '

A

: U -

U',

then

10

fp : M -

Gp(V)

fp I U - 1P

0

10

J!.

10

(p)

10' ~'

be a meromorphic map.

J!. _ 0

to be leneral of order 2

A

is indicated by

one choice of a reduced representation ~

•••

10:

U-

10 (p) ~

If

and

10.2 ~

1O~.2 •

for

0,

V on a chart

for all possible choices and

f

is said

~. If so, the pth associated map

for

is well defined as a meromorphic mdp by for all possible choices of

A global definition can be given.

Let

10

and

~.

VM - M X V be the trivial

bundle

(A.l40)

admits a holomorphic section section of Let ~ : U -

f, 10:

U '.

Fp - Fpf'

which is uniquely defined by the property: U -

Define

V

be a reduced representation on the chart 10.2 €

r(Y. A VM) p+l

Define

(A.141)

t

called the 2th representatior

by (A.127).

Then

by

~ pIx) - (x,

10

nIx)). -

·45·

Usually we write

«> .l!..

general of order

p for

may not be reduced.

B.

p

f

for

> t f · The map f

Fpf

if and only If f

The representation

is of

«>.2.

f

cannot be identified

and

if - tf(B)

B such that

F p • 0,

Beware:

Thus

There is a number of

Then

called the I'enerali~ index

Z[O,nJ,

E:

Fp. 0 for

0, p

,t

is said to be general for

Fp:: 0 for

but

f

!! if if - n - dim

1P(V)

If so, f is linearly n n degenerate. If M admits m analytically independent holomorp 'c functions, and if f is linearly non-degenerate, then there exists a ho omorphic form B of bidegree (m - 1,0) on M such that f is general for B (Stoll (100)). Moreover if (M,"') is a parabolic covering mantfo d of

(a:m,.,.O)'

and if

f: M --. 1P(V)

is

linearly non-degenerate, a holomorphic form B of bidegree (m - 1,0) exists on M such that f is general for B and such that on

(A.142)

A9.

The Plucker Difference Formula.

and a divisor

UF p

If

If

p

- uF

p-1

- 2UF

is non-negative if 0 ' p < if • Here

o

- UF ;: 0

1P( A V)

and

then

Fp ill 0

~ 0 is defined. The p th stationary index divisor

(A.143)

UF

0, p ' i f '

M.

by nature. Gp(V).

Let

Cl p

UF

p

+ uF

-1

p+1

== 0

~ 0

by definition and

be the Fubini-Study form on

Then

p+l

(A. 144)

If

p -

Rf ,

then

Hif == O.

If 0 Ei p < 'f'

then

Hp > 0

outside a

thin analytic subset of M where we can compute the Ricci form of

Hp

·46·

(A.145)

(A. 146)

h

define

hP

-

M - M+.

0 on

If

(A.47)

If

exists.

0 ' p < Af

~

p

0

O:!:i

p

Sf

(r)

h 2lJm

on

p

< Af

and

r E

and put

the integral

p

0 < s < r

and if

E.f.~ with s E.f.~ we have

the Pliicker Difference Formula (A. 148) N

~f

-

(r,s) + T f P sf (r) p

p-l

(r,s) + T f

(r,s) p+l

Sf (s) + R~cT(r,s) p

H. Weyl and J. Weyl (118). (119) estdblished this formuld first in the

case

m - 1.

They liked the formula so much. that they called it the

Second Main Theorem.

We will reserve the name for the analogon to

Nevanlinna's Second Main Theorem. Hopefully.

Sf (r)

and

Ricr(r.s)

are "small" such that all the

p

characteristics

Tf

have dbout the same growth.

Then by addmg up the

p

second difference the SMTH follows. but there is still much to do.

If we

try to copy Ahlfors orignal proof. we are forced to require the identity mim_1B

A

B-

t./ m-

reveaI s th at on ly

1

which is impossible if

. m1m_l B" B- ....'"

1)

m-l

m > 1.

is required.

A closer inspection

With this condition.

·47·

Stoll (93) obtained the SMTH.

Later (Stoll (100)) it became c ear that a

weaker codition suffices, which we will discuss now. AIO.

Majorization.

every

>0

I'

The exhaustion

there is a constant

~

R,

is said to majorize

T

c

lf for

1 such that

(A.149)

on

The infimum of dll these constants is called

YO(r).

M(r).

Then

YO

increases.

Define (A. ISO)

Then

VIr) -

c = VIr). and

~

VIr)

Yo(r)

~

1 and

The funchon

Y

Y

llm YO(t). r
increases.

Also (A.149) holds with

is called the majorant associated to

T

B.

Let

(M, T)

((r m, T 0).

be a parabolic covering manifold of

be a complex vector space of dimension

n + 1 > 1.

Let

Let

V

f: M - !P(V)

be a linearly non-degenerate holomorphic map. Then there exists a holomorphic form B of bidegree (m - 1,0) on M such that f is general for B and such that T majorize B with d majorant Y satisfying VIr) ~ (1 + rl 2n 2

(A.151)

for all

r >

a.

See Stoll [1001. Let

(M, T)

be d parabolic manifold of dimension

complex vector space of dimension linear

y

(A.152)

Let

Let

f: M -

V

be a

!P(V)

be

non-degenerate map, which is general for the holomorphic form

of bidegree majordnt

n + 1 > 1.

m.

Y.

(m - 1,01 Take

on

£ >

o.

M.

Assume that

T

majorizes

B

Abbreviate

Q£(r,sl - Co log VIr) + RicT(r,s) + rCo log r

with a

~

B

-4Sfor

0 < s < r.

:e;

The inequality sign

means that the inequality holds r > O.

except for a set of finite measure in the variable the estimate (Stoll [lOS), Proposition 10.9) T f (r,s)

(A.153)

p

A.ll

pth

be the

a - V'(oU)

Take

ProJection.

~ 3PT f (r,s) E

t

+

1P(V*)

a J.lf

non-negative divisor

p

Then

J.lp

- J.lPpLU -

p

(3 P - l)Q£(r,s)

and

repres ntdtion section of f. p

~

0

Z[O,n).

E

respectively.

For

r

(A. 154)

p

.

Dc

fpla - IP(F p LoU) : M -

A meromorphic map

s

E

(A.155)

~

a

and

0

1P(AV)

is defined.

p

0 <s
The First Main Theorem for projections holds for with

(t,a) p

C fpla C

M exists.

A

tion function

log _ _...:.1_ _

J

mf (r;a)

Fp

is defined, whose

~T the compe

E

p

Let

Fplu • O.

counting function and valence function are denoted by Nf (r,s;a)

Then we have

~~

E

"" .f.T T f (r,s) - Nf (r,s;a) + mf (r;a) - mf (s;a) + Tf LaIr,s) p

p

p

p

Usually, (A.155) is derived from the FMTH's for

fp

p

and

fpLa;

compare

Stoll (108) pages 152-153. In Section 3 of this paper, we wlll prove a general First Main Theorem which contains (A.155) as a very special case. If

p - 0,

then

f - fO '

Also

We obtam the usual FMTH (A.1l7). constant,

:: 0

and

.

If

fLa

p - n,

C fnla 0 - 1.

A.12 The Ahlfors Estimates. Abbreviate

Take

is constant and

P E

then

fn

and

T fla :: O. fnLa

Thus (A.155) is trivial.

Z[O,n]

and

a

E

1P(V*).

arE

-49-

(A.15?)

Then (A.158)

(A.159)

(A.160)

For

0 < 8 '1, 82

we ob ain the fundamental estimate

.p+ 1 (a)

T.

p

IH

p

(8)1-.8

(A.161)

For

0 < s < r,

the inequality Yields the Ahlfors Estimate (Stoll [108])

(A.162)

Substituting (A.153),

Tf

p

can be replace by

Tf .

estimate is the cornerstone of the proof of the SMTH. the first version by integral average methods.

The Ahlfors Ahlfors [1] obtained

Cowen and Griffiths [21]

introduced the curvature method for holomorphic maps

f: It -

1P(V).

Pit-Mann Wong [123] modified the method for use on parabolic manifolds. This \Vas necessitated by the more complicated singularity sets and intersection divisors of meromorphic maps.

-50-

Define

S(r,a)

by

1/ S(r,a) - 1 + T f (r,s) + p

slr,a) ~ Sls,a) r > 0

If

0 ~ r < s.

Then

(s;a)

if

r;l;

5

and

P

0 < SIr,s) < 1.

For almost all

the Integral 2 41 +1 Ca)h p

I

Flr,a) -

Take

I: E fR 0,1).

P

•

M
mr

Ca)I-SCr,a)

a

p

By a method of Nevanlinna, the Ahlfors

Estimate gives log+ Flr,a) (A.164)

~ 3(1 + I:)(log T fIr,s) + log VIr) + log+ Ric'Tlr,s)) + I: log r . A.13 The Second Main Theorem. A finite subset A of 1PIV·) is said to be in general position if any subset S of A wIth IfS = p + 1 ~ n + 1 spans a projective plane of dimension p, in which case the family

(E[allaEA

of hyperplanes is said to be in general position.

Let A be a fimte subset of 1PIV·) in general position with k - If A ;l; n + 1. Take P E Z[O,nJ. Then there is a constant cp{A) that we have the Product to Sum Estimate:

Take any

each havE"

PIa) E fR[O,II.

(A.165)

a E A

IT

sla) E fRIO, 1)

PCa) aEA 0 xla 02-2.8Ca)

which implies

(A.166)

take any

~

and

cplA) [

x E Gp(V).

such

For

Then we

I PCa) ]n- p aEA 0 xla 02-2SCa)

·51·

Put

c - log Cp(A).

With

(A.155), integration over

M

in

(A. ISS)

implies (A.167' (n -

P)Sf (r) +

I

p

aEA

J

E: (n - p)

log

(mf (r,a) - mf (s.a)) P

[I

I SEA

.p+~ (~~:~a)]a (a)

aEA •

M

+

P

I

P

8(r,a)mr (r,a) + C P

E: (n - pIc. log

E: (n - pIC.

[-&

I

I

F(r,a))] +

1+

C

aEA

g + F(r,a) + const.

aEA

~ 3(1 + ~)(n - p)kc.(log Tf(r,s) + log V(r) + log+ RicT(r,s)) + ~ log r

for every

E

> O.

Let

9 ~ UF

n

be the Wronsky divisor.

If we sum the

Pliicker Difference formula (A.148), we obtain

(A.lSB)

n-l

I

P&Q

1

(n - P)(Sf (r) - Sf (s)) + 2" n(n + I)Ric T (r,s) p

P

Abbreviate (A.1S9) D(r,s) -

t n(n + l)kC.(1 + r)(log TfIr,s) + log V(r) + log+ RicT(r,s)) + t log r .

Then we have the Second Main Theorem (Stoll [l08))

·52·

(A.170) Ne(r.s) +

~

mf(r;a)

aEA

< (n + l)Tf(r.s) +

t n(n +

l)Ric.,.(r,s) + D(r.s)

The Ricci defect R the mpjorization defect Yf • the ramification ----f'defect Elf and the Nevanlinna defect ISf(a) are defined by

Ric.,.(r.s) T f
(A 171)

Rf

-

(A.I72)

Yf

- limr-+oosup

log Y~r2 T t
~

0

(A. 173)

Sf

- 11mr-+ooinf

N6 (r,s) T f
~

0

lI f (a) -

lim 1nf r-+oo

mf(r,s) Tf

lI f (a) ,

1.

(A. 174)

o,

where

and we assume

Rr

lin sup r-+oo

<

,..

The defects 00

and

Yf <

Rr 00

Nf(r,s) Tf

- 1 - lim sup r-+oo and

Yf are obstruction defects

at least.

The Second Main

Theorem yields the Defect Relation (A.175)

Sf

Hpre of

and

Yf

depend on

B.

but

me

and

Rr

are independent

B. If

(M •.,-)

is a parabolic covering space of

(g:m •.,-O)'

if

f

has

transcendental growth and if. as before, f is linearly non-degenerate. we can choose a holomorphic form B of bidegree (m - 1,0) on M such

·53·

that

f

is general for

B

and such that

majorizes

T

B

with

Yf - 0

Thus we obtain the Defect Relation

IA.176)

which is independent of

B.

If

1M, T)

Rr -

is affine algebraic, then

0

and

IA.177)

Rr -

> 0

If

V

O.

In the situation IA.176),

on

M

Then

is isometric to

(M,T)

Rr

<

00

if

f

((Cm,TO)

and

separates the fibers of

7C by a theorem of Noguchi (70). If the meromorphic function h separates the fibers of 7C and if Th(r,s)/Tf(r,s) 0 for r -

then

Rr -

0

N (}

Sr

was dropped since it depends on

can be disected to obtain better estimates.

be the intersection divisor Nrln)(r,s,a)

truncated at level

(/1~)(n).

be the valence function of

n

Let

(A.63).

B.

Let

The disection yields the

(A.178)

See Smiley [87], [88] and Stoll [108], (13.21) with truncated Nevanlinna defect

11m r

sup -+00

Nf

Yet

(/1~)(n)

Second Main Theorem

(A.179)

M

00,

by the same theorem of Noguchi.

In (A.176), the d"fect the term

on

p - 1.

(n)

Define the

(r,s,a)

Tf ( r , s )

-54-

with

0 E; 6£(a) E; 6£(n)(a) E; 1.

6 f (n)(a)

Now

6f(a)

can be replaced by

in (A.I75) - (A.l77).

A14. Co ments to the defect relation. The behavior of the defects of a meromorphlC map has been investigated in a number of papers. They cannol be reviewed here. Good introductions to the subject matter are: Griffiths [35]. Stoll [107]. Stoll [l01]. Stoll [108]. Shiffman [821. Stoll [103]. Wu [1251. Weyl [119). Now. we will list a few open problems: a) The Griffiths conjecture. Let (M. 'T) be a parabolic manifold of dimension m. Let N be a compact. connected. complex manifold of dimension n. Let L be a positive line bundle on N which is spanned N be a meromorphic map of rank f - n. by its sections. Let f: M Then n E; m. Then Griffiths and King [361 proved a defect relation for thE target family

(EL(a))at:I'(f(N.L))'

Does the same defect relation still hold

if we assume only that f(M) is not contained in any thin analytic subset of N. This is the hlI Griffiths conjecture. The problem is not easy. The conjecture becomes wrong if we only assume that f(M) g; EL(a) for all a

£

1P{f(N.L)) If

(Biancofiore [4]).

V is a complex vector space of dimension

n + 1.

if

N = 1P(V)

and if L - HP is the p > 1 power of the hyperplane section bundle the conjectured defect sum is (n + 1)/p and we speak of the small Griffiths conjecture. h) Higher codimension. Various types of Casorati-Weierstrass theorems have been proved for target families of higher codlmension. but no defect relation has been found. Perhaps there is none.

with

o<

c) Behavior of the defects. The set of deficient value a £ 1P(V·) Ii C(a) > 0 can be a continuum. How does this set look like? If 11 E; 1

what can we say about the set

(a

£

1P(V·) I IiC(a)

~

II}.

d) The Umkehr P.roblem. Given a countable subset A of 1P(V·) such that every finite subset is in general position. given a function Ii : A !R(O.I] with I lI(a) ~ n + I is there a linearly 8EA

-55-

non-degenerate meromorphic map lif(a) - Ii (a)

for all

a

E:

defects on the total space Al5.

Applications.

A.

f: (;m -

PI V) such that

In view of c) we do not perscrlbe the

!P(V*).

There are applications of value distribution theory some

of which were already mentioned:

The construction of Theta fucntions to

periodic divisors Stoll [921. normal families of non-degenerate divisors Stoll [951. the characterization of affine algebraic varieties Stoll [941. analytic

cycles of affine algpbraic varieties Corn alba-Griffiths [181. the transcendental Bezout theorem. Cornalba-Shiffman [191. Stoll [1001 and others. Tar'lki's High School Algebra Problem in Mathematical Logic Hpnson and Rubel [ 41. The last applicdli n is the most unexpected.

Rubel asked Stoll to

prove the Lemma of the Logarithmic Derivative for meromorphic functions or polydiscs.

In the paper. Stoll [l091. so commissioned, value distribution on

2

polydisc was recast and the Lemma of the Logarithmic Derivative proved. The value d stribution functions depend on a vector

1(' -

(r1' ...• rm)

exceptional sets are taken on a p-dimensional subvariety of vectors where

0 < p 'm.

and 1('

Latcr Spellecy 1891 proved the defect relation for

polydiscs. Are there more applications?

Perhaps the well-developed theory of

value distribution of ordinary differential equations in the complex domain can be extended to partial differential equations. In the next chapter. B. we will outline the theory of value distribution of movable targets developed in this monograph.

·56· Value Distribution Theory fur Moving Targets

B. Bl.

Initial remarks.

Nevanlinna [66) conjectured that his defect relation

remains valid. if the target points

a1' .,. ,a q

distinct, meromorphic target functions

g1' .. · ,gq

slower than the given meromorphic function

f

- 1, ... ,q.

a:,

on

a:,

on for

(B.1'

for

are replaced by mutually which grow

that is

r -

00

Nevanlinna proved the conjecture if

q

=

3.

He defined

(B.2)

Then

lif(gj' - lih(aj'

where

a1 - 0, a2 - 1,

and

a3 -...

Thus the

defect relation for fixed targets yields the defect relation for moving targets. Dufresnoy [251 proved a defect relation for polynomials of degree :!!; d, with a defect bound

d + 2.

span a vector space of dimension

If the meromorphlc functions p

obtained a defect relation with bound proved the conjecture of Nevanlinna if still open.

If

f

over

a:,

then C.T. Chuang (l5)

p(1 - lif(oo" + 1. f

gl' ... ,gq

is entire.

Hence he

The general case is

has finite lower order. Lo Yang [127] shows that there

are at most countable many deficient target functions satisfying (B. 1). In several complex variables, B. Shiffman [831. [84] proved the Nevanlinna conjecture under the assumptions (B.1) and

(B.3,

which reduces in one variable to the case of constant

gj'

In Section 9.

we will obtain Shiffman's defect relation as a special case of our general theory. However. we have to make a slight modification in the assumptions

·51·

S. Mori [63] extends the original Nevanlinna method (B 2) to meromorphic maps

f: a;m -

!P(V)

gj : a;m _

IP(V.)

where

is a complex vector space of d enSlOn

n + 1 > 1

and where

V

and to meromorphic target maps

gl' ... ,gn+2

are in general position sa Ify ng (B.1)

With a non degeneracy condition he obtains

(8.4)

In Section 11, we will extend Mori's method to meromorphic maps f : M(M, T)

IP(V)

and

gj: M _

for

IP(V·)

is a parabolic covering manifold of

j - I, ... ,n + 2,

where

«(tm, TO)'

In this monograph, we will establish a value distribution theory for meromorph'c maps

f: M -

for

We will encounter new and surprising concepts, methods

j - I, ... ,q.

and target maps

IP(V)

gj: M -

IP(V·)

Perhaps the journey is more valuable than its destination.

and results.

The small Griffiths conjecture was the prime motive for these investigations.

Clearly, they would not solve the conjecture, but would shed

new light on the problematic. This is the connection: Let H be the hyperplane section bundle on 'P(V). Take 1 < p E: 11-1. A section in

HP

can be identified with a homogeneous polynomial

degree

p.

We assume that the zero divisor of

a

~

V.

smooth, that is, that Since t.

£

da(t.)

da(t.,t.) - pa(t.)

V.,

which means

for da(t.)

0 for t. E:

£

V,

V:

all

t.

we have if

da g: M -

AID

'#-

~

Let 0

a;

of

is simple and

with

da(t.)

t..: V..

be a reduced representation of f. Then is a representation of a meromorphic map intersection divisor the divisor of < «> intersection divisor T g grows as (p -

£

a: V 0

jI!

a(t.) -

O.

0 for all «>: U -

U IP(V·). The «>

V

V·

for the incidence f(z) £ Elg(z)) is given on U by ,"<10> - da( «> , «» - pa 0 ~, Which is the of f with the hypersurface defined by a. Since l)T f we cannot hope to resolve the Griffiths

Conjecture, but \Ve may gain new insights.

-58-

82.

Hermitian geometry.

Section A4i1)_

Basic notations and definitions were given in

Here we introduce additional concepts which will be helpful

in our inve t gations. Let

V l' ... ,V q

and

W

be hermitian vector spaces.

A p- fold

operation (B.5)

is said to be projective, if there are maps

a:. -

p-

J

a:.

such that

(B.6)

for

)"

and if

a:.

Vj

with

t.1 c:l _.. c:l ~p ~ 0,

then

E

and

is well-defined_ (q1' ... ,qp)

E

Zp

t.j

E

j

If

I, ___ ,p.

=

Xj -

1P(t.}

1P(V)

E

The projective operation is said to be unitary of degree if

Ip

p . )I

q-

-

I >.. I J

>..

for all

a:.

E

If so, the

distance

(B.8)

is well-defined, but not a function of

xl c:l ___ c:l xp

by the dot over the operation symbol-

The projective operation is said to

be unitary if it is unitary of degree said to be homogeneous of deKl'ee for all

>..

E

a:.

and all

holomorphic function of

j

E

(q1' ... ,qp)

"'ll,pl

't.l' --- ,'t. p

operation is unitary of degree

(1, _.. ,I).

and if on

which is indicated

The projective operation i! E

Zp

if

Pj()")

t.l 0 _.. c:l t.p

=

)"

qJ

is a

VI X ." X V p _ Such an

(q l' --. ,qp)-

A projective operation

said to be homogeneous, if it is homogeneous of degree

(1, ___ .It

(;)

is

-59-

A homogeneous, projective operation is said to be p-linear If it is additive in each variable

~j.

A p-linear operation extends to a linear m· p

(B.9)

V l l8I···I8IV p - W

I:)

An example of a p-linear operation is the exterior product. which defines

0 xl ;.., ... ;.., xp

o.

Examples of bilinear operations are the

interior product which defines

0 xLy 0,

defines

with

0 x·y (J

0 <x;y> 0

and the inner product, which

~ 0

0

~

x;y 0

1.

An example

of an unitary operation is the hermitian product which defines

o

xly 0

o

(x Iy) 0

with

o~

-

0 xly 0 ~ 1.

Important for our Investigation is the contraction product (8.10)

III

( A

V) X (

which is defined as follows: 1:I

(B.11)

o ",···"'1:I p

1:Ij

Put

V·) -

(AV) X (AV·)

J.t -

p

q

Min(p ~ l,q+ 1).

Take

and

in

(-1)j

A

q+l

p+l

1:1

0 '" ... '"

1:1

j -1

A

1:1

j+ 1 " ... "

1:1

P

(B.12)

(B.13)

Then (8.14)

III

extends uniquely to a well-defined bilinear map. &I

(A V) lSI (A V·) p+l q+l

A linear map

(AV) lSI (AV*) p q

·60·

where

~

mP

results, which can be iterated mO is the identity and

mil

m =

mP - 1

0

for

P

=

W( 1:»

Gp(V)

E:

and

w

E(v) ~ E[wl.

if and only if

Also

dim E(v) " E[w) > p + 1 - p.

mT

(B.15)

B3.

0 v

mP

The First Main Theorem.

are explained in A4c). defined in A4d).

Gq(V"'),

W( '\lO) (:

=

1:>

mP

"It)

then -

0

m

If 1:>

m

'\lO

=

0

if and only if

Moreover

w 0

~

WDvmwD~1.

1

Hermitian line bundles and their Chern forms

Meromorphic mdps and their representations are

Divisors are discussed in A4e)

introduced in A5.

1 •. " .11,

Thus the iterations of

L.

reach from the tensor product to the interior product. v

~

Parabolic manifolds are

The value distribution functions on parabolic manifolds

are expounded in A6.

Here we outline the First Main Theorem for a

homogeneous operation of degree

(ql ..... qp)

and specialize to specific

operations. V l' ... ,Vk

Let

and

W

be hermitian vector spaces.

homogeneous projective operation (q1' ... ,qp)

For

j =

E:

k

Z.

I, ...• k.

Let

(M.T)

let

fj: M

hyperplane section bundle

Lf .

0: V 1 X ... Vk -

W

Consider a

of degree

be a parabolic manifold of dimension W(V}

m.

be a meromorphic map with

and representation section

f f.

J

(See A4d)).

J

The operation extends to a fiber preserving, holomorphic map

(B.16)

which defines a globdl, holomorphic section

(B.17)

If

F ~ 0,

F

then

If l' .. , .f k)

the operation divisor

is said to be free for

0,

in which case

·61·

(B.18)

J.tf

exists.

.

1

e

Its countinr function and valence function are abbrev'at d to

(B.19)

e, a meromorphic map

If (f l , ,.. .fk) is free for fl

e ... e fk

: M -

is defined by

I'(W)

(B.20)

for

If

'0

j : U -

j - I, '" ,k,

then

of

fl

e ... e

fk

Vj

is a reduced representation of

'0

'0

Ie... e

'0

k : U -

x

fj

E:

M - Z(F)

for

W is a representation

with

J.tf;" ;"f I U - J.t ~ . I':''''':' k ~

(B.21)

The compensation function

is defined for

I

(B.22)

r

...

E: ~T

by

log - - - - . . : . . - - - - " .

M In general the sign is undetermined, but this can easily be rectified. Since the distance function is continuous on the compact space .(V 1) X ... X I'(Vk) a constant c ~ 1 exists such that [J

xl

a '" e

xk D :Iii c

for all

Xj

E:

1P(V j)

and

j -

I, ... ,k.

The

____ "

~

1Il0dified compensation function (B.23)

m~ ::. ~ (r) 1 "'· .. \!I~k

I

M is non-dell:enerate with

log _ _ _ _

~c=__

0

·62·

me.

(8.24)

•

1 e ... Of k

f

(r)

In most applications. our maps

m.. O· ';f (r) + --~1 ...... k

fj

distance function is bounded by

log c ~

C,

o.

map into subspaces on which the

1

such that we can use (8.22).

The General First Main Theorem (Theorem 3.4) holds (8.25)

+ mf ';

';f

(r)

1... ••• ... k

0 <s < r

where

'"

~T

E:

and

s

'"

~T

E

The identity extends the definition r > 0

of the compensation function continuously to all holds for all

0 < s < r.

such that (8.25)

Since the constant cancels in the difference, the

modified compensation function can be used.

The First Main Theorem

contains only "essentially" non-negative terms, and it is a balancing statement. the term

1.

If

W

11:, then

=

T f e ... ef k 1

SPECIAL CASE.

n + 1. Vj -

Take AV

and

nj

E

vanishes.

Let IN

fl

V

1Po

is constant and

Now we can discuss special cases:

be a hermitian vector space of dimension

Define

W = AV.

:s: n + 1. Define

The exterior product

P

nj

" : VI X ... X Vk

fk : M -

p - n 1 + ... + nk

with

qj - 1.

e ... e

--+

W is a homogeneous projective operation.

obtain the First Main Theorem for the exterior product (8.26)

+

mf

•

•f

1'" ....... k

( r) -

m.

•

f l ",·· ."f k

( s)

We

-63-

If

p - n + I,

i - I, ... ,k, f1' ...• f k

then

f 1A

••• "

f (r,s) k

mr1~···;Jk

~

o.

A.

O.

If

If

n1

f·J : M - G

..,

=

- 1.

- nk

(V) 1

n·J

for

h n

.fk )

(f 1

This case will be discussed later.

k - 2.

if

a

£:

Gq(V)

Equidistribution.

f 1 :: a.

is constant and if

already proved by Ahlfors [II for

2.

-

are said to be in ,enerat position if and only if

is free for If

T

then

M -

(C.

the case was

He called it the First Law of

He used an integral averaging method.

SPECIAL CASE.

k = 2, p

T dke

Z[O,nl. q

£:

be a hermlt an vector space of dimension

£:

n + 1.

Z[O.nJ.

Let

VI

Define

V

A

=

V

p+l

and

V = 2

A

q+l

V*.

Let

be meromorphic maps.

f: M -

Define

!P( A V) p+l

U - Min(p+1.q+1).

As operalion take the iterated contraction and

W - (

and

A V) @ ( A V·). p+l-p q+l-p

mP .

g: M -

Take Then

P

11'(

A

q+l £:

V*)

IN[l.uJ.

q1 - q2 - 1

We obtain the First Main Theorem

for contra tion

Tf(r,s) + Tg(r,s) -

T

(8.27)

fmPg

N. (r,s) + m. (r) fmPg fmPg

and

3. that

SPECIAL CASE. q

product

~ p

and

(r,s)

m. (s) fmPg

then

g

Take the situation of the 2. Special Case, but assumE

P -

U -

q + 1.

Our operation becomes the interior and

W

=

A

V.

p-q

becomes the First Main Theorem for the interior product.

Now (8.27)

·64· t8.28)

then

~

m.

O.

If

g

is

flg

constant.

T g :: 0

and we obtain Theorem 4.9 of Wu (1261 page 112. which

is proved by taking integral averages. 4.

SPECIAL CASE.

of dimension and

n + 1.

g: M -

product.

Then

lP(V·)

~

2.

Se also (A.155).

Take

k

Let

V be a hermitian vector space

Take

VI - V and

V2 - V·.

be meromorphic maps.

qi - q2 - 1

and

W -

a:.

Let

f: M -

IP(V)

As operation take the inner We obtain the First Main

Theorem for the inner product (B.29)

If

g

is constant. then

Tg _ 0

and we obtain the classical

FMTH (A.117). Our proof of Mori·s. Theorem (631. Section 11. uses a special operation which will be helpful. but as an operation is not interesting.

B4.

Notations.

Associated maps were defined in A.8.

The Plucker

Difference Formula was stated in A.9. Here we introduce some additional notations concerning the interaction of two meromorphic maps. Let (B.30)

tM;,)

be a parabolic manifold of dimension

M+ - (x

£

m.

Define

M I vex) > 0) .

Let 8 be a holomorphic form of bidegree tm - 1. 0) on M. Let V be a hermitian vector space of dimenSion n + 1. The Fubini Study form ( GqtV) is denoted by Oq and on Gp(Va ) is denoted by Let

0;.

-65-

f : M ll f

index the

W(V)

qth

and

for

f

g: M -

t

and

W(V*) for

g

g.

For

f q

associated maps

fq : M -

(B.31)

IH q - mi m- 1f*(0) q q

(B.32)

II< p - mi m- 19 *(O*p) p IH f - 0 :: II< t g .

Then I-lq > 0

define

II< p > 0

and

hq ~ 0

I< p - k 2p u m'

On

II
by

Let

F

and

G p - F pg

the

pth

Z[O.t f )

E:

A

B

B

(If

q

M+ -(Ir

On

V Ig ) p

q

Gp(V*)

qf

Define

M.

gp

+ M - If

On

define

then

kp ~ 0

II
~: U -

U'

g by

Then

is a chart and if

is defined by (A.126). we have

(B.33)

Define

U - Min(p+l.q+l).

Fubini Study Form on

(B.34)

11'«" q+J

Take V) @ ( "

P

p+l

be

I the

exi t.

0 ~ p < t g•

and

M+ - I

If

Z[O •.e

- F

B ~ 0 .

A

define

V Iq)' p

E:

q

B ~ 0

A

0 ~ q < tf

h q2 u m.

p

gp: M -

and

A

and

outside a thin analytic on

q

M on

Gq(V)

If

B.

in respect to

g

representation section of

representation section of

be meromorphic maps with generality

E:

Let

Ol[O.U).

V·)).

Oqp

Define

•

_.

pq

(1)

pq

be the

-66-

Then

~ ~

0

~

fq EBP gp

(p) ~

pq

pq

(p)

exists.

"

[~]

>

o

o~

and

~ pq ~

J-t. M

outside a thin analytic subset of

and

Define

(8.35)

1

Abbreviate

IH pq '

IHpq

B5. Frenet frames and stress. By the method of Frenet frames. we will obtain curvature formulas and estimates which will lead to the Ahlfors estimates and the defect relation. We have reached a crucial point in the

If

proof.

a

ddc~p(a)

is not constant.

and

d"'p(a)

A

dC~p(a)

are

difficult to calculate and to interpret and the identities (A.159) and (A.160) become invalid. The Frenet frames and the operator EB will save us. A large amount of Frenet identities is proved in Section 5. As far as f alone is involved, they are known, Stoll [100J and (l08J. In this introduction, we shall avoid this horde of formulas as much as possible.

if

From the start. we will assume that f is general for B. that is. n. Denote ig = s. Define ff = ZIF n ) V Z(G s )' Our calculations

=

M - ff

occur on

{U}".'~A' '0 A'~>..}>..e:A

Let Here

{UA)>..e:A

AP+ 1

define

l' A '0

by (A.129).

V{

VA -

then

6

AI1

and (B.36)

V

of

f

Define

and

and

For

>.. - (>"0' .. , .Ap) in

AlpJ

by

(A.130).

For

Define

x>..

g

A (; A, and

We have reduced representations ~A :

is defined by (A.13l). WAJ-t

f

M.

is a chart, see (A.125).

by (A 126) and (A.127).

A : VA -

be a representation atlas of

is an open covering of VA

~>..:

the map

or subsets thereof.

are determined on

t\ --

V"

of

g.

If

(>".Si) (; AIIJ

Also zero free holomorphic functions UAl.(

by

-1i7-

For

p

E:

Z[O,s]

and

q

E:

Z[O,n]

we have 9 (9+ 1)

(8.37)

10,.,.9. - (v>..u

)q+1

2

(~u>..)

10 us

(8.38)

ID'~

(U,., - iJ-) -

(8.39)

for

q

E

Z[O,nj

(8.40)

for

0

~

q < j

for

q

E

Z[O,n]

(8.42)

for

p

E

Z[O,s]

(8.43)

for

0'

p < k ~ s

for

p

Z[O,s] .

Vector functions of class

COO

~~:

(U,., - J) -

and

" ...

,.,

(8.44)

A

Then

'1:.~ is called

>.. 1:.0'

At every point

x

E

U,., - 6,

an orthonormal base of

V.

which is also orthonormal. of class q

COO E:

V*

are uniquely determined by

(8.41)

and

V

on

Z(O,n]

and

,.,

E:

" 1:. p

Prenet frame of

the vectors

Let Then

U,., - 6

8.

•••

f

on

n

U>.. - 6,

t.~(x), ... ,to >..n (x) constitute

1:.~>"(x), ... '1:.~>"(x) be the dual base, 1:.0>.., ... ,1:.~ >..

are vector functions

caned the dual Frenet frame. A

E

~

define the stress coefficient

For

P

E:

Z[O.sl

-68-

(B.45)

where

I AX.PQ I ~ 1.

We have

for

(B.46)

p

£:

2[o,s] .

The stress coefficients describe the relative movement of one Frenct frame against another.

This ought to be a fundamental problem in

geometry, but I know no literature concerning these coefficients. can be calculated in terms of

A;q

and

(Lemma 5.8), which provides a definition of the stress coefficients without Frenet frames, but the method lacks geometric meaning.

The stress

coefficient matrix is unitary

(B.47)

Recall (B 34).

For

p

E:

2:[O,sJ

and

q

E:

it

p

=

Q

if

p

~

q

Z[O,nj

we obtain

(B.48)

Surprisingly. this unusual operation on a matrix, locally defined. gives the restriction of a global invariant.

(B.49)

(B.50)

Take

(>".J,I)

E:

A[lj.

On

U>"J,I

we have

·69·

(8.51)

(8.52)

If

a

t

Z[O.s]. p

p + q - a + b.

Z[O.n] and q pq ab then a function Spq = Sab E

Z[O.s]. b

E

primary stress invar nt. is uniquely defined on

Z[O.n)

with

of class

Coo.

M-J-

by

E

called a

ab

where

86.

-1 ~ Spq ~ +1.

The Stress Form I .

For

P

E

Z[O.s)

and

q

E

Ol[O.n]

we have

the CUNat re Stress Formula (8.54)

p

+ 2S p + 1

q+l q

/I(pq

The Curvature Stress formula surprises by its elegance. balance and simplicity. times.

The proof is difficult though it has been simplified several

In order to compute the gradient. higher order stress invariants have

to be introduced. For

dj

E

l[O.nl

j -

1.

with

.k.

take

aj

E

fRIO.s). b j t: Z[O.n]. c j

E

Z[O.sl,

-70-

(B.55)

c 1 d 1 "'C"k d k

A function

k,

level

S

a1b1···akb k

M-J

is uniquely defined on cldl···ckdk

(B.56)

Sa1b 1 · •.

8

k bk

2

J=!

k

j = I

s

called a stress invariant of

such that

k A>" -1 [ TI a

1"1

(B.58)

Coo,

of class

k c.d. + 1"1 j=! J J

;;,>..

J

b

J

A>"

c jd j

a .b . J J a b . ~1. ~

J

J

We introduce a repeatable summation convention

(B.59)

L

S .•.

j=O

For instance we have

11

2

• •• r •••

• • • J •••

r

j ...

0 fq II! gp 0

2

S

I

••• r . • •

pq

= SI I

For

P E:

pq

and

q

E:

Z[O,n]

we have the Stress Gradient Formula

(B.60) d~

=

A

pq

p

5

q p+l q I I II( p+l q P q P

p

+ 25

q I

p I

p+l q P

p

p q+l

+ 51 I P q+l p

q+l q

q

IK pq

-B IH q

q

Z[O,s)

;;,

ah

]

-71-

While (B.54) and (B.60) are difficult to proof,

\s easily

calculated

(B.61)

f*(O ) q q

(8.62)

B7.

II

The Ah fors E timates.

For

p E ZlO,s1

and

q

E

ZIO,n]

define

p ,q+l .p+1.qIKp + .p,q+l IHq + 2S p + 1 • q IKpq ~ 0 .

(8.63)

:::pq

Assume that

(fq gp)

is free for

analytic SE't.

Define

u

Then

lB.

Min(p+1,q+l).

.pq > 0

Take

outsidE' a thin

~ E iii? (0 , 1].

ThE'n we

have the estimate

82

[_1_. ] u pq

2

8- 1

='pq

(8_64)

If

T

majorizes

n

with majorant

Y

(see A.lO), we obtain the Ahlfors

Estimate (B.65)

~2

Jr J s

(~2

1

4>pq].B-l Zpq t ;-1

M[t)

~ 4(n + 1)3Y(r)(T f (r.s) + T (r,s) + ~) q gp

Which in particular implies the integrability of

(4) pq ) 8-1:: pq

over

MIt) for all t:> 0, which is remarkable. As seen in (A.162), the classical Ahlfors estimate involves the term • p,q +IlHq only (for p - 0). In view

-72-

of (B.63) it seems to be reasonable to drop take the term

2S

P

q+l

p+ 1.

q

measuring the movement of

+p+l,qll

in (B.65) and

to the otherside as a remainder term pq g in relation to f. This fine idea fails since

II<

each term

.8-1

(8.66)

p

q+l

2. pq Sp+ 1

may not be integrable over

M[t),

D
q

as an example in Section 10 shows.

So

it is not easy to proceed to the Second Main Theorem without running into a disaster.

After long considerations, the following procedure was chosen,

which avoids divergent integrals.

Define

(pq

~ 0

on

M+

by

'='pq

(B.67)

On

M+

define

(B.68)

Then the integrals

(B.69)

Ppq(r) - Ppq(r,g) -

t

I

(log ppq)O

M

J

(B.70)

k

log [ 1 +

~ J0 ~

0

M exist for almost all

r > O.

The terms

P pq

and

Rpq

are undesirablE

but measure the divergence and convergence of the jets of one particular case there is good news:

f

and

g.

-73-

(B_71) If p - J g • then kp - O. Ppq - 1 and Ppq

=0

After the difficulties of general position are cleared

UP.

Rpq

e proceed to

the Second Main Theorem as in the case of constant targets. which was outlined in Part A. 88.

General po't n.

easy to understand. operation

IIIP

constant

cp(A)

The classical expositions of general position are not

With the help of the exterior product. norms and the

we hope to present a simpler and clearer picture. in the product to sum estimates depends on

non-constru tive manner.

Also the

A in a

Here we construct a constant whose dependency

on the t get set A is explicitly known. which permits us to handle the moving target case . • ) The definit' of leneral pOsition. Let V be a hermitian vector space of d'mension n + 1 > 1. For (lJ 'I:- A ~ P(V) let span(A.1P(V)) be the intersection of all projective planes in span(A.1P(V)) equivalent.

is a projective plane.

a) If

L is a proper plane in

• ) If

(lJ

'I:-

B

S;;;

A with

v)

Every subset independent.

B 'I:-

Ii)

Take

S;;;

(lJ

'I:-

B

a hyperplane (B.72)

(lJ

1P(V)

E[a)

in

p(V).

then

liB' n + 1. of

with

A with

then

containing

A.

Then

IUA

n

L) , 1 + dim L.

liB - 1 + dim span(B.1P(V)).

liB' n + 1 is linearly

liB' n + 1.

F(V·).

dim

F(V)

The following four conditions are

then each

a

E

B defines

Then

"

aEB

E[a) - n - liB •

A subset A 'I:- (lJ of P(V) is said to be in general position if one and bence all of these four conditions are satisfied.

-74-

b)

The gauge_

k ~ n + 1,

Take

A!;;; W(V)

tdke an enumeration

(B.73)

A = (aI' .. , ,ak).

A.

Since

A

0

rIA) > 0

is finite,

c)

Incidence space_ ~

~

s

q

~

n.

Let

p,q,n

D(p,q,s)

Abbreviate

=

n + II . is in general position

be integers with

[

L:

[:

s

D(p,q).

=

q

+

+

j

Then

0

~

s

~

p

~

n

+

0~ D(p,q,s) ~ [n 1]. +

Then

p

Gq (v)

If

Define

J=O

D(p,q,O)

0 ~ rIA) ~ 1.

=

A

if and only if

p-s (8.75)

E:

If

The gauge

Then

rIA) = inf(r(B) I B !;;; A,liB

(B.74)

b

00.

r(a) = 0 HI ;...

does not depend on the enumeration of k > n + 1, the gauge is defined by

dnd

0 < flA = k <

with

D(p,q,s)

+

Take

1

is the dimension of the linear subspace

(B.76)

A v* I

5[h,p,s)

<~

A

p+l

A

of

V·

(Lemma 7.6).

~;~>

=0

Slb,p,s] ~ W(S[b,p,sJ).

Define

p+l

Then

(Lemma 7.7)

S[b,p,s]

(B.77)

Let

q);to 0

A

Gp (V'")

=

(

a

Gp (V'")IO b

E

be a finite subset of

Gp(V*).

ms +l

Then

a 0 - O} . q} ~ W(

A

V*)

p+l

Therefore it is defined when q) is in general position. Define k = #q). If a E Gp(V*), then E(a) ~ W(V*) and Gs(E(a» ~ Gs(V*). Take b

E

Gq(V).

Since

0

~

s

~

Min(p,q),

we can define

·75·

(B.78)

q}(b,s)

(B.79)

q}(b,s,r)

=

o

(a

E:

q} 10 blx 0

(a

E:

. +1 q} lob Ells a 0 ~ r) .

Then q}(b,s,O) = q}(b,s) (Lemma 7.8). Form these rather difficult results, we easily derive Theorem 7.8: "If q} is in general position, then 1Iq}(b,s) ~ D(p,q,s)." The theorem is known, Wu [126], but the proof here is easier to understand. Also we obtain a new result, Theorem 7.11. "If is in general position, then !lq}(b,s,r) ~ D(p,q,s) for all r E: IR with

o<

r ~ 2- 3n -3 r (q})."

The explicit domain for

of the Second Main Theorem. interval

0 < r

dependence of

d)

~

rO

rO

on

Classically, only the existence of some

Product to sum estimate. x+

Gp(V*).

Define

k =!lq}

=

If

q}

moves, the

has to be known.

QI.

deftne

E:

is crucial in the proof

is determined by Heine-Borel.

IR,

x

r

Max(x,O).

Take

Let

p,q

q} 'I: 0

Assume that

q}

and

s

as in c).

For

be a finite subset of is in general position

Abbreviate v

(B.80)

2(k

=

-

D(P.9

D(p,q,s

SS »

+

(B.81)

Let

b

E:

IJ: q} --+ IR(O,l) and P: () Gq(V). Theorem 7.12 asserts

D ( ) Cpq (1''1111) [ p,q,s

(B.82)

--+

I

I

b ffis+l

P (a)

'" ,£.

aE:QI. 0

b ~s+l

a

be functions on

P( a )

.,---r

aE:() 0

~-

IR[O,l]

a

q}

q}.

Take

] 0 ( p , q •s )

02-2/J(a)

+ (D( p,q,s) - k) +

02-21J(a)

which generalizes the theorem that the arithemetic mean majorizes the geometric mean.

-76-

e)

General position for moving targets-

manifold of dimension g: M _

maps

m.

1P(V*).

Let

QI

'#

121

Define

k

=

#QI.

Let

(M, T)

be a parabolic

be a finite set of meromorphic The indeterminacy of

QI

is

defined by I

(B.83)

z £ M - IQI

For

define

QI

-

V I g£QI g

QI(z) - (g(z) I g £ Q}},

Then

(B.84)

is a thin analytic subset of defined by

M.

r(QI)(z) - r(QI(z))

The gauge of

QI

z £ M - 6Q1'

for

is the function

r(QI)

0 ~ r(CJ) ~ 1.

Then

The degeneracy (B.85)

of

QI

is analytic in

and only if

CJ(zO)

M.

The set

QI

is said to be in general position il

1

is defined for all

f)

define

r

...

E: ~T

We shall estimate

0

~

Assume that

r CJ(r).

as the set of all subsets QI

~

Then

For

t- of QI with

is in general position, then

- {g 1, ... ,g hI.

0

It measures the decline of general positior

The First Main Theorem for general position. :Jlh(CJ)

Enumerate

M - 6CJ'

is in general position, the gauge measure

log rnJ)

00.

E:

6{lJ) ~ M.

(B.86)

r -

Zo

is in general position for some point

which is the case if and only if If

QI

~

h II~

E:

fN[l,k

- h.

is in general position

-77-

(B.87) Define

m.. - m 1.

• hand

N.. - N 1.

g A ••• Ag

Abbreviate

A" _ gl "

• h .

g A ••• Ag

"gh.

The First Main Theorem (B.2 ) for the

exterior product implies (B.88)

A

0 < s , r E:.f..,

where extends h -

n + 1.

s E: .f..,.

The formula continuously

r > O. such that (B.88) holds for all

m.. ) to all

I" -

A

and

then

T A.. (r.s) - 0

0 < s < r_

and we obtain

(B.89)

k < n + 1. have (8.89) with" If

we have (B.88) with .. - IJ. If k - n + 1. qJ In both cases we obtain the estimate

(B.90)

If

k > n + 1.

define

(B.91)

(B.92)

Addition implies the F rst Main Theorem for General Position: k > n + 1 and if 0 < s < r. then we have (B.93)

and the estimate

If

we

If

-78-

(B.94'

Surprisi gly, the First Main Theorem solves the problem of genera position.

tJ -

Up to here, only sel!.

(g)'\.E:A

finite if

of meromorphic mdpS

),. _

g).,

IJ - IJ

The Second Main Theorem. Let

(B2) (B3)

g),.: M -

is injective and if

general position in which case

(B1)

were considered.

I)

(g)., I)"

=

A}

E

We make these general assumptions

M

be a connected, complex manifold of dimension

Let

.,

be a parabolic exhaustion of

Let

V

be d hermitian vector space of dimension

f: M

Let

Cl}

be a finite set of meromorphic maps

[B6)

Let

B

be a holomorphic form of bidegree

[B7)

As!.ume Uldt

.,

majorizes

[B8)

Assume lhat

f

is general for

Then

-+

1P(V)

Define

p E: Z[O,i IJ1.

Define

0 < s < r.

n + 1 > 1.

be d meromorphic map.

If - n.

general position.

B

g : M -

(m

with majorant

IJp - (gp}gE:IJ' (f,gp)

and

Assume thdt

is free for

L

(B.97)

Rp(r,!J)

-

n-l

L

q=O

Ppq(r,g)

(k - D(p,q))+

L

BEqJ

k

=

R q(r,g) P

IIJ > O.

IJp

for all

l

n-l

L

M.

Y.

(B.95)

q=O gE:q}

on

1,0)

Abbreviate

~

!P(V·).

B.

ilJ - Minlig I g E: IJ}

Assume that

P p(r,q})

m.

M.

Let

(8.96)

is in

can be identified.

(B4)

Take

is said to be

1P(V·)

[B5)

Take

A family

A is finite, and is said to be in general position if the map

defined by

88.

Cl)

is in g

£

I).

·79·

(8.98)

+ 3

(8.99)

q ..

£ > 0

and r > 0

for all value

> O.

~

21] C.

++

n

"" ~ Iog + 8 E Q}

T g (r,s )

0-1 ~ (k - D(p,q))+ .

Y(p,k)

Take

(p

O

Recall that

~

means that the inquality holds

outside some set of finite measure.

Then we have the

Second Main Theorem (Theo~em 8.31). (8.100)

[[p : 1]

~

If

term

+

3(: :

p - O.

then

NF (r,s)

1]]T

+ [0 ;

~]kC.(1

+

f

(r.S) + [: :

£)log

Bp(r,s) - 0

and

Y(r) +

[

y(p.k)r~p(r)

1

P +

~]R1CT(r.S)

1.

]

The remaining

1

is the classical ramification term which can be used to

n

truncate the intersection divisors at the level [

1 p

+

]

_

0

and

Sp(r,s) > 0

n.

If

p > O.

then

is a new ramification term whose

1

significance is not yet known.

In any Cdse, the terms can be dropped.

The sum of the compensation functions. the characteristic. the Ricci function, the major ant are well-known classical terms.

The term

AQ} p

measures general position and can be estImdted by (B.90) respectively (B.93) and (A.153).

The terms

Pp

and

Rp

are new remainder terms which

account for the movement of the targets in respect to

f.

They are

-80-

difficult to estimate, but vanish if Sp(r,CJj)

p - tg

is a classical remainder term.

for all

g

qJ..

£

The terms

These terms exist under natural

assumptions.

89.

The Defect relation.

p

Z[O,tqJ."

£

free for

Assume that [B.11 - [B.8} hold.

Assume that

L for all

g

£

the Nevanlinna defect of

Q}p Q}. f

Take

is in general position and that

Then for

T f(r,s) -

gp

00

for

r -

(f,gp)

00.

is

Define

by

(B.101)

The First Main Theorem implies

(B.102)

0

~

N

Ii f(gp) - 1 -

1 i m sup r~oo

Lf(r,s) +

T

Lf(r,s)

-g-,p'i;:--:~--:-"'t--:--:..-&...JP~--~~Tf(r,s) + T (r,s) &p

We also assume

(B.103)

Ricr(r,s) T f
for

r -

00

(B.104)

for

r _

00

(B. 105)

for

r --

00

(B.106)

for

r _

00

(B.107)

for

r _

00

0

for all

•

g

£

Ci

1

·81·

Here (B.106) and (B.107) are trivially satisfied if g

£

I).

p - ig - iii for all

Assumptions (B.103) and (B.I04) are classical conditions on the

growth of f, Then

while (B.105) is the customary assumption.

l!§.

obtain the Defect relation (Theorem 8.12).

(B.108)

B.10

The ~ p - 0 - .II).

We assume that [B.11 - [B.81 are satisfied

We assume that I) is in general position and that f.g g

E:

Q}.

We assume that

the divisor

Uf,g

function of

).If,g(n).

(B.I09)

p - 0 - IIJ - Ig

truncated at level

I

Recall (B.91).

for all

n and let

g

Nf,g (n)

are free for each £

IJ. Let ).If,g (n) be the valence

Then we have (Theorem 8.7):

Nf g(r,s) ~ NF (r,s) + NnI(r,s) + I Nf g(n'(r,s) n '"/I gEQ}'

gEQ}'

Thus we obtain another version of the Second Main Theorem (B. 110)

------------------------------------~ (k - n - I)T f (r.s)

+ nk[k -n 1]

I

gEg)

T (r.s) + 2n(n + 8

l)k~ log Y(r)

+ 2n(n + l)k~(10g Tf(r.s) + 10g+ RicT(r.s»

The Nevanlinna defects are given by

+ t log r

·82·

(B. 111)

1 i m 1nf T r

'00

(r

s)

f'

T

+

g

(r,s)

Tf(r,s~ + Tg(r,S) ~ 1

SliP

1i m r ->00

(8.112)

If in addition we assume that (B.103), (B 104) and (B.105) hold. we obtain the Defect relation (Theorem 8.13)

(B.113)

The Second Main Theorem (B.ll0) and the Defect Relation (B 113) can be viewed as an extension of B. Shiffman's deff'ct relation [831. (84) from mC'romorphie functions f . M

-+

B11.

Analytic dependence. a)

IP(V)

on

M.

m > 1. If

B

IP 1

on parabolic manifolds

Functions.

dimension

f: ([m -

Let Let

M

fi

=

M.

be a connected. complex manifold of &M

be the field of mcromorphic functions

is a meromorphic form on &(B)

(B.114)

is a subfield of

&.

M.

{C{.l E:

=

& I dqJ

The meromorphic functions

are said to be analytically dependent if analytically independent if

dqJl

independent, then

k ~ m

if and only if

is constant.

qJ

to meromorphic maps

A

...

A

dqJl dCPk

A single function

to be analytically dependent on

Take 4>

are analytically independent functions

..

A

O.

on

dqJk == 0

M and

If they are analytically

rp

12I;I!: 4> !;

if either

B = O}

CP1' ...• qJk

A

~

A

is analytically dependent

tii..

cP

'+'1'''' ,qJk

Then

qJ

E:

tii.

is said

is constant or if there in



such that

-83-

C:P,C:P1' .. , ,C:Pk of all

c:P

40

~ ~(41)

c:P

E

If

V-

fR

E

E

a finite subset

&

• ~ fR

and

~(41 V (CPl).

E

(1'J ~

If

•

41 ~ Ijr ~~,

'"

of

and if

~(40) ~ ~(Ijr)

~(40 V (v-I)

E

Hence the assignment

~

is a subfiE'ld of then

exists such that

c:P c:P

tR(41) - ~M(41)

The set

analytically dependent on

~(~(41)).

~(41),

'"

are analytically dependent.

~("'l.

E

':P 1£ ~ 41),

but

~(.)

40 -

c:P

with If then

is a dependency

relation in the sense of Van Der Waerden (114) and Zariski-Samuel 11281. analytically indep ndent functions

the choicE' of th

40. of

The number

k

if and only if

B

said to define

~(.)

•.

is independent of Then

Analytically independent functions ~(.)

Here

constitute a base of

bac;e and is called the rank of

the same rank as constitute a b s

is defined.

CP1' --- ,C:Pk

~(40) - ~(C:P1' ... ,C:Pk)'

if and only if

a:

~(40) ~

Therefore the concept of a base of

~(40)

has

Cfl l' ... ,CPk

(8.115)

A meromo phic form

IS

is a ba'le of

Cfl k

Uj ~ 0 ~ Vj

holomorphic functions 1, ... ,k,

j -

~(.)

with

if and only if

tR(.) v{Pj

=

and if there are Uj

on

M

for

the hoI om orphic form

(B.116)

defines (R(40)

~(40).

b) dimenSion A

=

If

.!;

~(41) ~

then

Meromorphic vector functions. n + 1 > 1.

M - S

;a!:

a:

and the rcmk of

Let M

connected neighborhood function

~: U -

h(xl1O (xl

in which case

=

S

S

Let

V

40

Bnd

be a complex space of

be an analytic subset of

A holomorphic vector function

eJ.

to be meromorphic on

that

a:,

is defined to be zero.

V

~ (xl

if for each point

U

of

x

in

M

'0

XES

all

x

E

17 "A.

is called the pole ~

M A -

with V

is said

there is an open,

and a hoI om orphic vector

and a holomorphic function for

:

of

h ii! 0

We can take 10.

on A

U

such

maximal

·84·

c) Admissible bases. Let f: M !P(V) be a meromorphic map. -
Let

Define f c

bj ~ 1P(6j).

The base

f(M) e Elb o)'

if

: U-

V of

-
If so, then f.

Let

is said to be admissible for

< 0 ,6 0> ~ 0

-
f

f.

be an admissible base of

Then there are uniquely meromorphic functions coordinate functions of

for any representation

fl' ... ,fn'

called the

in respect to the admissible base

-
such that

for

(8.117)

for any representation

0

U -

V.

j -

1, ... ,n

Then

(8.118)

is a merom orphic vector function on M. Let S be its pole set. 'W) : (M S) V. is a reduced representation of f.

Then.

d) Maps. Let • 'I: IZl be a set of meromorphic functions on M. The meromorphic map f: M 1P(V) is said to be analytically dependent on ., if and only if there is an admissible base ~, ...•<-n of V for

f

such that the coordinate functions

fl' ...• f n

in respect to this

base belong to ~(.). admissible base. The independent of • if In -
If so. this is the case for any other choice of an meromorphic map f is said to be analytically and only if f is not analytically dependent on •. admissible base of V for f with coordinate

functions

then

only if

f l , ...• f n , fj E

~(.)

f

is analytically independent of

for at least one index

j

E:

11-111,n].

If

8

meromorphic form defining ~(.) and if c : U V is a representation of f. then f is analytically dependent on • only if (0 "" d 10) '" 8 :: 0 on U.

•

if and

is a if and

The meromorphic map f: M !P(V) is said to be free of • i1 and only if (f.g) is free for every Illeromorphic map g: M F(V.) which depends analytically on •. If f is free of ., then f is

-85-

analytically independent of V

for

41

if and only if

j

f

If


Let

fIg l + ... + fng n {~(4I) (gl' ... ,gn) ~

with

f: M -

use

B,

(m - 1,0) B.

Then

on B

if

f

f

is free of

is general for

holomorphic form

B

41

B

for all

M

m - 1.

is free of ~(~)

E

and

41 ~ I'2l

Let

Let

B

be a

be a meromorphic

~(4I).

Let

M - P.

P

41

if and only if

if and only if

M

Then

(Proposition

that is, if and onb

Thus if we can find a reasonablE ~(4I)

defining

If(B) = 0

If(B) - n,

be

Hence, we can

to define our differential operator.

(Theorem 9.4).

on

f

gj

which defines

is holomorphic on

M - p,

is analytically dependent on 9.1) and

then

be a meromorphic map.

W(V)

at least on

is an admissible base of

o.

set of meromorphic functions of rank form of bidegree the pole set of

.. • ,
f l' ... ,f n'

with coordinate functions

I, ... ,n

=

41.

we obtain a Second Main ~(4I).

Theorem and a Defect Relation over the function field

A

Poincar: Theorem with growth conditions proved by Skoda [R5] and (B.116) will provide a suitable form

B

in the covering case.

Of course the

general case can also be formulated, see Theorems 9.8 and 9.9. Unfortunately the theorem of Miles [55] has not been proved for several variables. B.12 [Ol) [D2)

A Second Main Theorem over function fields. We assume. Let

M

Let

41

be a connected, complex manifold of dimension

m > 1.

be a set of meromorphic functions on

of rank

~

I'2l

M

m - 1. [D3)

Let

J{

M _

:

a: m be a proper, surjective, holomorphic map with

generic sheet number

c.

and branching divisor

[D4) There are meromorphic functions "'j - CPj that

a

J{

belongs to

"'1 .... ''''m-1

~(4I)

CPj for

is a base of

[D5]

IJ~ I'2l

[06]

Each

[07]

t1 is in general position with Ill)

[081

f : M -

on

a: m

E

and such

~(4I).

tJ is analytically dependent on 1P(V)

such that

j = 1, ... ,m - 1

g

is a finite set of merom orphic maps g

p.

=

M- !'(V'").

~

k > n + 1

is a meromorphic map free of

4>.

-86-

These assumptions can be formulated and understood without any knowledge of value distribution theory.

7

=

70

0

7!" =

1171"112

distribution applies. Theorem

The function

is a parabolic exhaustion of

M

and value

Assuming [Dl) - [DB) we have the Second Main

(Theorem 9.15)

(8.119) n -

(k -

~.

z:

g E:qJ

l)T fIr,s)

N (n)(r,s) + nk [k f,g n

1]

z:

T (r,s) +

g E:CjJ

n (n

g

t

1) Np(r,s)

+ 2n(n + l)kC,k(Iog T fIr,s) + log + N p(r,s)) .. SIr)

where the remainder term (A) For each

~

> 0

SIr)

can be estimated as follows.

and

s > 0

there is a constant

cO(£'s) > 0

such that

(8.120)

SIr) ~ cO(£,s)(1 + r)2m-l

m-l

z:

j -1

(8) For each £ > 0

and

s > 0

T .~..Ir + £,s)

for all

r > s .

J

there is a constant

c 1(£,s) > 0

such that (8.121)

S(rl ~ cl(£,s)(1 + (log (1 + r2))2)

m-l

z:

j =1

(e) If

Ord tit j < >.. <

constant

(8.122)

Q¢

c 2(>..) > 0

for

T",.Ir + £r,s)

for all

r > s

J

j = 1, ... ,m -

1,

then there is a

such that

for all

r > 1 .

(D) If the functions

cpl'· .. ,CPm-l

c3 > 0

constant

are rational, there is a

such that

SIr) ~ c3 log r

(B.123)

for all

The Ricci (or branching) Defect of

(B.124)

If

f

Rf

=

11m

separates the fibers of

Noguchi (70).

f

r

~

is defined by

Np(r.s) sup T f ( r • s) ~ 0 .

r"'OO

K,

then

Rf <

00

0

for

by a theorem of

Now we assume in addition:

[D9)

Rf <

[010)

If

[011)

At least one of the following cases is satisfied.

g

2 .

00.

E:

then

Q}.

(A) A number

T g(r,s)/T fIr,s) -

£ > 0

r

--+

00.

exists such that

(1 + r)4m-1T~ (r + £.s) 1

_

Tfe r • s )

(B) A number

£ > 0

0

for

r

--+

< Ord f ~

00

is the lower order of (D) The functions

and

IN[l.m -1) .

E

exists such that

2 2 (I + (log(1 + r » )T~ (r + £r.s) -------......"T:-f-(~r-.-s...,,)-.....LJ------

(e) Ord ~j

00

cpl' ...

for

j -

--+

1. .... m - 1

0

for

where

r

--+

00

•

Qrd f

f.

,cr m- 1

transcendental growth, that is

are rational and T f(r,s)/log r

--+

f

has

00

for

r _

00

-88-

Assumtion [011] implies the transcendental growth of the (m-l)-dimensional component of a;m,

then [011) implies

Rr -

Therefore, if

is affine algebraic in

If there is a function

R f - O.

separates the f bers of

l'I'(supp p)

f.

g

II which

€

then (010) and Noguchi's Theorem [70) imply

1'1',

O.

Assume that [Dl) _ [DU) are satisfied.

Then we have the Defect

Relat on (Theorem 9.15)

(8.125)

If

n - 1

and

M - a;m

and

the identity, the Second Main

1'1'

Theorem (8 U9) and the Defect Relation (B.125) were obtained by B. Shiffman (83), (84) under a bit weaker assumptions. 813.

The Theorem of Mori.

functions

g: a;m -

I'(V)

A defect relation for

n + 2

moving target

in general position was proved by Mori [63].

[n Section 11, we improve Mori's result and extend them to meromorphic maps on parabolic manifolds.

Following Nevanlinna's method, Mori

transforms the given situation for moving targets into one for fixed targets without changing the value of the defects. (Bl] Let

(M,T)

[B2] Let

V

be a parabolic manifold of dimension

m. n + 1 > 1.

be a hermitian vector space of dimension

(B3) Let

f: M -

[B4) Let

IJ

~

1P(V) QJ

[B6) Let

(f.g)

[B7) Let

CAO, .. · ,CAn

CAn +1 - -

be a non-constant, meromorphic map.

'" be a set of meromorphic maps

[B5) Assume that

Let

We assume

is in general position with

be free for each

1\. - 1\.0 " ... " 1\.n·

Define

itO, .. ·• ~n+ 1

V·.

aj - II'(Uj)

be the dual base of If

ItQJ - n + 2.

Enumerate

be an orthonormal base of

uo - ... - un'

1\.0' .. · ,1\.n

gEl).

"(V·).

g: M I) -

Define

for

uQ.... ,CAn

are vectors in

(go' '" ,gn+l)'

0, '" ,n + 1.

j -

Put V·

define

·89·

(8.126)

for

j - 0,1, ••. ,n + 1.

A homogeneous projective operation V : V X V· X ... X V· -- V

(8.127)

is defined by n

1

(8.128)

to V too V ••. V "!n+1 - -;~=n:::::;:+:=;:

I

j=O

<~'"!J'><§J"~>~J'

Now, IE.4) and IE.5) imply that the Mori map (8.129)

h - f V go V ... V gn+ 1 . M -- 1P(V)

exists and is meromorph'c, (of course Mori (63) does not define this map by an operation).

Abbreviate

(8.130)

m(r)

(8.131)

If

0 < s < r,

the General First Main Theorem (B.25) implies n+l

Tf(r,s) +

(8.132)

I

j-O

T (r,s) - N,,(r,s) + m(r) - m(s) + Tb(r,s) gj

which yields the estimate

n+l

(8.133)

Th(r,s) ,

Tf(r,s) +

I

J-O

Define

tj -

(go' ••• ,gj-1,gj+1' ,., ,gn+1'

(B.89) reads (Lemma 11.8»

for

Tg.(r,s) + m(s) J

j -

0,1, ... ,n + 1.

Then

-90n+L

8.134)

L

k=O k;t:J

T gk(r,s) - N", .(r,s) + m~.Irl - m~.Is) . vJ

J

J

We estimate directly (Lemmata 11.9 and 11.11) n+L

L

NT/(r,s)::;

(8.135)

N", .Ir ,s)

J =0

YJ

n+l

L

m(rl ~

(8.136)

J=o

m", .Ir)

t

vJ

log(n + 1) .

Thus (B.1321. 18.134). (B.135) and (B.136) combine to (Corollary 11.12). (8.137) n+l

Tt(r,s)

~

Th(r,s) + n

n+l

L

T

J=O

For

E

Z[O,n + 1]

(r,s) +

L

me. (s) + log(n + 1)

j=O

gj

"J

we obtain similar estimates n+l

Nh a.lr,s) ~ Nf g.lr,s) +

(8.138)

, J

(8.139)

Nf,gir,s)

~

' J

L

Tg (r,s) + me. .(s)

k=O k;t:J

P

YJ

n+l

Nh aJr,s) .,. (n + 1) , J

n+l

L

Tgk(r,s) +

k-O

L

k=O

m"'k(s). v

Now we assume in addition IE8]

If

j E Z[O,n

+ 1],

then

T g.lr,s)/T fIr,s) -

0

for

r -.

J

Then we have (8.140)

(8.141)

for

r -

00

QO

•

-91The diagonal in

([n+2.

Let

D - (b, ... ,b)

£

([n+2 I b

£

([)

• IS

a complex line

E - ([n+2/D be the quotient vector space and let

p : ([n+2 _ E be the residual map. For each c £ 1Pm) one meromorphic map g : M 1P(V·1 is defined by c

"Take Let

6 £ [n+2

...0 j : U -

n+l

is a representation of c -

=

1P(p(6)).

Then

Then

11,,- ...

The map

c

6 - (b O' ••. , bn+1') be a reduced representation of gj for

V*

j - 0,1, ... ,n + 1.

with

one and only

gc gc

I

j-O

on

b.<~ .,,,,,>...0' J

J

U-

J

v·

U. n

is injective.

The meromorphic map f: M "f(V) is said to be linearly rum.-desene ate m!!I Ii if and only if (f,gcl is free for every c

£

(Mori's, non-degeneracy condition seems to be more rE'strictive.) The 1P(V) is linearly non-degenerate over CJI meromorphlc map f: M and only If the associated Mod map is not linearly degenerate (Theorem 11.15).

1P1E).

if

In addition to [El) - [EB) we assume: (E9)

A proper surjective holomorphic map

JC

M -

(E10)

that ., - II JC 112. Let p be the branching divisor of

JC.

Define

[Ell) [EIZ)

Let f be linearly non-degenerate over I). At least one map g £ Ii is not constant.

([m

Rr

exists such by (B.124).

Then [EB] and [E12] imply the transcendental growth of f. Then (A.176), (8.140) and (B.141) imply the Defect Relation of Nevanlinna-Mori (Theorem 11.16)

'i 1.

Let

"K:)

< t:)

and

The dual vector space

(OJ.

It.

V -

:

by

n + 1.

be a complex vector space of dimension

V

V. - V

Hermitian Geometry

,"K:»

For

(A V)* - AV* p

£

t:)

~(t:)).

-

V and

Identify

V*

V*

£

"K:)

V"

~

Define

consists of all linear functions

V

the inner product is defined by

< t:)

,~> ~ <~, 10

>

by

p

For

~

AV and

£

t:)

£

p

L~ ,~>

< t:)

,~>

t:)

10

Each

L"K:)

t:)

£

L~

£

A V

p-q

- < lO,~

-

1 E: q E: p E: n + I,

with

q

the interior product

< 10

AV·

£

Va

,,~>

for all

is defined by ~ £

A

p-q

va.

If

p

q,

=

then

It = AV.

o

spans a complex line

lP(

10) -

It

For

10.

A

S;;

V

define (1.2) Then

lP(A) - (lP( 1P(V)

lP : V. -

For (1.3)

10 )

I0

~

10

£

Al

is the complex projective space associated to lP(V)

p

£

is the natural projection.

Z[O,nJ,

If

n - 0,

the Grassmann £2Wl of order

V and

then

p

lP(lt) - too}.

is defined by

-93-

and

Gp(V) ~ 1P(G p(V))

into

!P( p~ 1 V).

is the Grassmann manifold of order

Observe that

G_I(V) ~ II:

and

Every

x -

Gn(V) = (oo)

Then Put

!P(~) E

E(x)

E(~)

E(x) - E(~I q - n

p

with

Gp(V)

-

II:~O

1P(E(x))

1.

For

~ ~ ~O

E(x)

of

Define

+ ... + lI:~p

definE's a

with base

(~ E

=

VI

~o'

'"

tJ

~

'"

- 01

is a p-dimensional projective plane in a

=

1P(
E

Gq(V*)

with

Ela)

of

- ('£I

Ela] - E[
A biholomorphic map defined by 15

0

E

!P(V).


by

E{x) - E[15(x)J.

V I
=

'0

0).

is a p-dimensional projective plane in

15: Gp(V) -

Gq(V*),

By symmetry

called the duality is uniquely 15: Gq(V·) -

These definitions extend to Grassmann manifolds.

Gp(V)

is defined

p + q < n.

For

x = !P(~)

Let E

p

and

Gp(V)

Eq(x) - Eq(t,) - 1P(E q (x)) EOM - E(x).

exterior product

For

is a thin dnalylic subset of v - 1'( '0)

E

Gq(V) - Eq(x)

Gq(V)_ the

q

define

11_6)

Obviously

!P(V).

15 - Identity.

non-negative integers with

Then

by

'~p

E[a) - E[
(1.5)

with

(iO(V) - !P(V).

" .. , " ~p

V

define a (p+ I)-dimensional linear subspace

Then

embedded

G I(V) - too).

(p+l) dimensional linear subspace

(1.4)

and

p

bE

-94-

(1.7)

v '" x - IP(

is defined.

Now, assume

O:r:::

10

:r::: p

q

t.) E Gp +q +1(V)

'"

"n.

For

w - 1P("'40)

E

Gq(V·)

define

E [wI - E ["'40] -

(1.8)

P

E p [w] ~ E p ["'40 I - IP(E p [w])

Then

EO[wl ~ E[w).

Obviously,

For

E

(10

P

Gp{V) I

L"'40 - 0).

10

is a thin analytic subset of

v - IP(

10) E:

Gp(V) - Ep[wl

Gp(V).

the

interior product is defined by (1.9)

( I )

A positive definite hermitian form

hermitian product or a hermitian metric on

by

V

J(t.1 t.)

1It.1I E:

For each

.

(~I t.)

t.

£

V,

= <~,t..>

V·

is defined by

t..

is an anti·linear isomorphism of

t. -

all

L8'

and

in

t.

V.

Then

t.... - t.

Also

("'40 118'*) - <18', "<10 >

If

IP(V),

then

x - IP(!',)

On

£

AV

x*

=

1P(t.·)

V

together with V

is defined

one and only one dual vector V

~ £

if

V.

V·.

onto

The map

V*

Here

(t.. I L8'.) - (L8' I t.)

and £

is called a

A norm on

for all

becomes a hermitian vector space by setting vector spaces.

V

and

is said to be a hermitian vector space.

( I )

t.*

on

for

V·· - V as hermitian 18'

IP(V·)

and

V

£

"<10

£

V·.

is defined.

a hermitian product is uniquely defined by the requirement

p

(1.10)

for all

and

t.j

Let

V

and

I8'j

W

in

V.

be hermitian vector spaces.

Then

V ® W become hermitian vector spaces such that for "'40 E: Wand 18' E: W we have

10

V Ell Wand £

V, t.

E

V,

-95-

(1.11) (1.12) Then

(1.13) (1.14)

1110 ®...oll

(V ® WI·

(V ED W)· - V· ED W* dnd hermitian vector spac s by

111011 11...011

=

V· ® W*

=

are ldentIfied as

(1.15)

(1.16' where

10

E

V,

V,

E

'lID

~ E

~ E

V*,

We shall define an operation p

Z[O,n)

E

and

q

E

Z[O,n).

III

V*. called contraction.

U - U(p,q) - Min(p+1,q+l)

(1.17)

E

"40 -

Take

Define

'lID 0

(1.18)

A

•••

10

A

'lID q

j

= (-l)j

E

Gq(V·)

10

0

A

and

•••

k

E

A

10

j-1

1;

if

Z[O,p),

Z[O,q)

A

10

if

abbreviate

j+1

A

•••

A

10 P

(1.19)

If

p -

.;ok -

0,

1.

this shall mean Define

10 • -

J

q - 0,

this shall mean

·96·

(1.20)

Then

(1.21)

which shows that

t:) /B...o

(1.22)

/B:

called contraction. /Br _ /B

0

/Br-1.

LEMMA 1.1.

If

"V 8

~

t:) /B q -

"V 8

p+l

1

'YO

p ~ q,

-

t:) L '\010

t:) Ffjq+ 1 ...0

p ~ q,

8

1 +

q

be the identity.

L

"V 8

p+l

Then

"V· -

q+l

"V

p-q

...0

i! the interior product, where

Lt:).

then J.t - q + 1. If q - 0, then by (1.21). Assume that the Lemma is proved for

Ffjq (t:)

Take

...0 -

£-

~~ ~~ G~q (V·) --"0 " ... " --q

II! "40) q

~ 1 tc.-O

-"';-'T

q

...0) -

P

then

q-p

0 ~ q ~ p.

-

FfjO

" V· - "V·

L( t:)

If

with

we let

q+l

"V 8 "V·

the operator can be iterated

~ q, then Ffjq+1 -

p

the convention

PROOF.

q+ 1

Z[1,J.t],

E:

i! the interior product. If FfjP+ 1 - L:

"V· -

P+ 1

r - 0,

If

If

r

is well defined and extends to a linear map

mq

(( t:) L...a k) 8

~ k)

~

·97·

+

q

EII q +1 - L.

By linearity, we see that q.e.d. Consequently if 'D

Ell 'YO

;t

O.

If

Ell

'YD

;t

'D

L'YO

v - 11'(

;t

'D)

O.

p ~ q,

If

respectively,

1P( /\ V)

£

we proceed likeWise,

and

'YO L 'D w

£

O.

;t

11'( 'YO)

£

p+l

with

'D

0,

(1.23)

q+l

define

v Ell w - 1P(

Ell 'YO)

'D

£

1P(/\V ® /\V·) p

If

'D

EII r 'YO

;t

0,

q

we define

v EII r w - 11'(

LEMMA 1.2.

IB r

'D

'YO)

£

and

If

is an orthonormal system such that

1P(

/\

p+l-r

'YO

£

'D

=

V ®

"V·,

"

q+l-r

Abbreviate

1;. -

'!-,o

_ II

'D

112

- II

'D

112

A

...

"

'!-,P'

V·)

and if

q+l

"

(1.24)

PROOF.

then 1P( " V*

Then

q

e.d.

.'0

A

'!-,p'

then

-98-

.If

LEMMA 1_3.

m

10

v - 1P(

- 0 if and

~

PROOF.

on~

t.O, ... ,t. p

Let

Gp(V)

10) £

if

be the orthonormal base of

which is the case iff

,p

m

10

= 0

"40

t.j £ F.[w)

E(v) ~ E[wl,

the case if and only if

for

£

Z[O,n).

~[n

Z[O,n)

~[q,nl

Let

2[O,n).

E:

~[q,nl

such that

is bijE'clive.

1.

q

1)

A pE'rm Itation £

Z(O,q)

if )(

E:

Z[q

If

q

~

n,

(>..,>.. 1.)

let

ll.

E: ~[q,n],

1:>.. = l' >"(0) Take

Min(p 1 1,q+l)

Ob'lerve that

If

~O'''' ,t-n

are verlors

define

(1.26)

LEMMA 1.4.

sign(>..,>..1.).

I

+

be the "empty" map

+ l,n]

1]

[" +

#~[q,n]

q - n,

=

which is

1m >.. " 1m A1. - 121

if )(

q

u

0, .. ,p

for

is defined by

(1.25)

>..

Lt.j - 0

mr and extE'nd the last

The signature of the permutation is denoted by

and if

=

be the inclu'Iion map.

A

-

If

j

"40

lhE're E'xists one and only one

for each

The map

iff

such that

be the set of all increasing, injective

LE't

q < n,

If

- q - l,n)

A1.1. = >...

Then of

E:

- {d.

~[n.n)

then >..1.

q

>... Z[O,q) -

E(v)

mr , which is much more difficult.

two LE'mmata to this operator Take

!heQ

q.e.d.

We shall provIde an explicit formula for

maps

Gp(V*),

E:

E(v) ~ ElwJ.

Then j ~ 0, '"

w - 1P(~)

and

p

E:

Take

Z[O,n]

and

A

q

...

E:

A

t.>"(q) .

Z[O,n]

D!'fine ~

E:

A V* q+l

Take

r

E:

Z[O,uJ.

Then

-99-

(1.27)

"40

If

PROOF.

r - I,

I

A £~ ( r -1

I

P1

sign(A,A 1.)1\,

this is true by (1.21).

r < p + 1.

proved for

-

1.

A

®

("AQ

L1\" ) I\.

Assume that the Lemma is

It shall be proved for

r + 1.

We have

_ u r

A map

if

A

£

manner:

+

a:

~[r

~[r

- l,p]

~[r,p]

- l,p] X Z[O,p - rl a d

j £ Z[O,p -

One and only one number

P

r]

is defined by

where

E

.,

a(A,j) = .,

is defined in this

Z[O,r - 1]

exists such that

A(O) < >"(1) < ... < >..(p) < A.l(j) < A(p + 1) < ... < A(r - 1)

Define

-rex) -

{

A.(x)

ifO~x~p+l

)•..i<j)

if x - P + 1

A(X -

1)

if

P +

1

<

x

~

r

•

r

-

1 •

Then ifO~x~j ifj~x~p

Therefore

-100-

(~L ...,,)Lt- 1.

,...

A (j)

1.

sign(A,A ) - (-1)

Also

[

lIa l(T) -- r

V.

+ 1

r

]t.

v.

=

1.

Take

Min(p-l-1,q-l-1).

p

l-tj

p

Therefore

~ -

mr-l-1

LEMMA 1.5.

-t

r

T £~

£

Take

I [

Z[O,n]

1.)

•

Sign(T,T1.)t- 1.

and

t. - ""0

(

obtain

WE'

I

r • p

.

sIgn T,T

A

A

q ...

A

t.p

£

q.e.d.

De'fi!J~

;l[O,n]

£

® ~Lt.T'

(' p(V)

anI!

~

E:

A V*. Q+)

Assume that

t-O, ... ,t-p

ll! an orthonormal system. Take' r

E

2:[0,1-1).

Then

(1.28)

PROOF.

Abbreviate

0(>") = sign(>..,A 1.)

for

>..

£

we have

where

A

and

T

run through

~[r

l,p I

q.e' d

~[r

l,p).

From (1.27)

-101-

Naturally. the operation

m is also defined when We regard V. as

the initially given vector space and we obtain the diagram

-

A v*

AV8

A V* 8

a

q+l

p+l

(1.29)

V" - V as its dual space.

lEe

is an isometry given by

a

Av* 8 Av

q

a(

p+l

1m I

p

Here

Av

q+l

a

Av 8 AV* q

p

8 ...a) - ...a 8

10

commutes by (1.21),

Hence the last two Lemmata yield:

LEMMA 1.6.

p

~

Take

€

- Min(p+l q+l).

Take

O.~].

Then

Take

r

E

Z

Z[O.n] 10

and

q

€

Z[O.n].

A V and

E

Hence

Define

18' - 18'0 '" '"

p+l

The diagram

10.

,.. I8'q

E

GqIV*)

(1.30)

LEMMA 1.7. ~

Take

- M nlp+l.q+l).

p

E

Z[O.n]

and

q

E

Z[O.n).

Define

T ke

Assume that 18'0' •• , .l8'q i! 3m orthonormal system.

Take

r

E

Z[O.Jl).

Then

(1.31)

LEMMA 1.B.

Take

U - Min(p+l.q+l).

10

Take

(1.32)

GpIV)

E

r

E

and

Z[O.U].

[ :] 112 II

10

mr

PROOF. W.1.o.g. we can assume that an orthonormal base

~O

'" .. , '" ~p'

....0

E

Gq(V*).

Def"ne

Then

'\010

p

II ~ II

~

q.

10

II 11....0 II .

Then

U - p + 1.

Lemma 1.5 implies

Take

-102-

q.e.c LEMMA 1.9.

If

u - Min(p+l,q+l) only

v - 11'(

«)

I

if r

and

E

Gp(VI

E

Z[O,u),

w - 1P(~)

and then

to

rIf

"tIC

Gq(V·),

E -

0

if

if and

if dim E(v) " E[w) > p + 1 - r .

PROOF.

Define

s + 1 - dim E(v) "E[w).

Abbreviate

A - E(v) " E[w) . Then we obtain the following exact diagram of vector spaces, where we have written the dimension under each vector space and where we abbreviate t=q+s+ 1. 0

1 0-

A

s+l

0

-

1 0 - E[w]

n-q

1 0_

D

n-t

...

0

1 E(v)

p+1

1 v n+1

1

..

-

- -

1 B

-0

p-s

1 c q+1

-0

1

E

F

n-p

t-p

1

1

1

0

0

0

-0

-103-

We can consider each of the exact sequences as an orthogonal splitting. Therefore we have E(v) - A ED B

v

E[w] ED C - E(v) ED E

E[w] - A ED D

c

B ED F

E-DEDF

V-AEDBEDDEDF. Hence we can take an orthonormal base

t.. s +1' .. ,t..p spans B t,~, ... ,t..~

E(w)

be t

Numbers

a

~

0

and

b

~

such that

The linear subspace

t..O' ... ,t..n

t..:+l' ... ,t..~, t..~-t+p+l' ... ,t..~

are linedr y independent and they cons titute a base of

V

t.. n -t+p+1' ... ,t..n spans F .

dual base of

conta·ns the vecto s

of

t..p+ I' ... ,t..n-t+p spans D

t..O' ... .." s spans A

Let

t..O' ... ,t..n

p - s + t - p - q + 1 E(w). 0

Also

t..O'

,t..p

which

in number. is a base of

Hence E(v).

exist such that

(1.33)

(1.34)

where

- b t..s+l'" * * ...<>n-t+p+l'" * ... ,.. ...<>p'" ... ,.. t..n*

'\10

n - t ~

Take any

A

o.

£ ~[r

Assume that - l,p].

s + 1 > P + 1 - r.

Thus

r > p - s.

Then

t..A - t..A(O) ,.. ... ,.. t.. A(r-1)

o Since

r > p - s,

:Ii: AlO) < A(1) < ... < AIr - 1) :E p .

we see that

'\10

Lt,~ -

O.

Now 11.27) and 11.33)

-104-

Assume that 'VD

Lt.>.. - 0

for all

Then

r ' p - s.

Then

>..

IB r

to

>..

o.

"40 -

Then (1.28) and (1.33) imply

~[r _ l,p].

£

>..: Z[O,r - 11 -

Define

i injective and increases.

which is a contradl t on.

Suppose that Z[O,p1

by

>..(x) - s + x + 1

We have

s + 1 > P + 1 - r.

Hence

s + 1 , p + 1 - r.

q.e.d.

(1.35)

If t.LIt

E

t.

£

Gp(V)

G p _ q _ 1(V)

and

It

£

Gq(V*)

with

0' q ,

p 'n,

then

and

(1.36) which follows from Lemma 1.1 and Lemma 1.8.

If For

t.

It £

E

A V,

q+l

A V

p+l

and

then Lt

£

It*

£

A V* was defined with

q+l

A V with

q+l

0,

q , p 'n,

metric contract·on. t. L*Lt - t.LIt*

(1.37'

with 0 EO q EO P 'n,

(1.38'

then

Lt·

£

Gq (V*,

£

A V.

p-q

lilt-II - IILtIl

define the

·105·

.Vp

and

Cl.39)

W be vector spaces.

A p-fold operation

e : VI

W

X •.• X Vp -

is said to be projective. if for each Pj : It. -

j £ 1N[1.p)

such that

It.

~j

for all

>..

for

1. ... .P. then the condition

j -

It..

£

there is a map

of the choices of

all

Vj

£

t.l' ... '~p

j - 1•...• p.

and

Xj - I'(~j)

If

~1 e... e ~p ~ 0

£

P(Vj )

is independent

and if the condition is satisfied.

(1.40)

is well defined independent of the choices of Xl e...

a

xp

~1' ... '~p.

We say that

exists.

The operation

e

is called unitary of degree

(ql' ... ,qp)

£

Zp

if

q.

1Pj(>") 1 - 1>"1 J for all VI' ... ,V p Vj for

and

>..

£

It.

and

j -

I, ... ,p.

Ware hermItian vector spaces.

j - I, ... ,p.

Take

in

Then

c:;

is well defined.

Observe that

a xp [] ~ 0 e

lI~p II

C ... C is not a function of

which we indicate by the dot over the operation sign.

operation

Xj - 1P(t.j)

•• _1I_~..:l~e_ _ _e_~.....p~.~1 cxle ... expc- ..... 1··ql qp

(1.41)

[]xl e...

Assume that

if and only if

Xl e... e xp

Xl e ... e xp

Here exists.

is called ~ if it is unitary of degree

The

Cl,l, ...• 1).

·106-

The operation if

pi>")

=

>..

q. J

t-l 0 ... 0 t-p

0

is called homogeneouS of degree

>..

for all

a:.

£

and all

is a holomorphic function of

V 1 X ... X V p'

11'(4/')

£

(t-I' ... ,t-p)

1P( "V),

If x

=

\P(t-)

£

(qI'

,qp)'

\P( A V)

and

then

q+l

Ox;" y 0 - 111:. " 4/'11 111:.11 114/'1I

U.43)

O~Ox;"yO~l

If

x - \P(t-) 0

with

£

1P( "V) p+l

~ q ~ P ~

n,

and

y

11'(4/')

=

£

11'( " V·) q+l

then

o

U.44)

xi.y

0 -

II1:.L4/'1I

lit-II 114/'1I

o~

(1.45) [f

x - 1P(1:.)

£

o~

(1.46)

if

o~

x

1P( "V) p+l

£

1P( 1:.)

£

q ~ p ~ n,

1P( "V) p+l

then

and

0 x;y 0 -

and

0 xLy 0 ~ 1

y - 1P( "V"'), p+l

1<1:..4/'>1 111:.11 114/'11

y - ]P(4/')

£

then

=

• 0 x LY 0 "" - 1.

1P( "V) q+l

zP

on

p+ 1

U.42)

E:

and if

Such an operdtion is also unitary of degree

A number of examples shall be given. y -

I, ... ,p

j -

(q1' '" ,qp)

with

·107·

•

(1.47)

o Ej

(1.48)

If

lit, L *11)'11

o xL·y 0 - 1It,1I 1111)'11

x - 1P(t,)

E

0 xL·y 0 :E 1

A V) and y - 1P(II)')

1P(

1PI

E

p+l

A V),

then

p+l

(1.49)

If x

1P(t,)

=

E

1P(

A V) and y - 11'(11)')

E

1P(

p+l J.l - Min(p+l,q+1)

A V·) and q+l

then

(1.50)

•

(1.51)

1

O:EDxlllyDEj-

.fit If

o Ej

x - 1P t,)

E

1P( A V) p+l

p :Ii: Min(p+1,q+1),

and

y - 1P(II)')

E

1P( A V·), q+l

if

then

(1.52)

• P yD:Ii: O:li:Dxlll

(1.53)

LEMMA 1.10.

II

x

E

"(V)

[pJ.l) -112 and

y E

"(V),

then

o x i y 0 2 + 0 x ;.. y 0 2 - 1 . PROOF. and y

Take E

1P(II)')'

0 ¢ ~ Then

E

V

and

0;11: II)'

E

V

such that

x - 1P(t,)

·108·

o xiy 0 2 + 0 x ~ y 0 2 _

I

111:.112 1I~1I2

[

1(1:.1 ~) 12 + 111:.112

1I~1I2 - 1(1:.1 ~) 12] - 1 . q.e.d.

Observe th t

Gn(V) - (00) - Gn(V*).

LEM A 111.

Take

(154)

• 2 1 DvlllooO -p + [

PROOF.

'lID

E

Let ""0.... '''''n

"" - ""0 ,.. ... ,.. ""n' that

v

"40 -

"40 0

Gp(V)

W E

Gq(V·).

o .. iii

W

Then

D2 -

be an orthonormal base of

Then

,.. ... '"

and

.. - 1'(",,)

"40 q ~

0

and

Gn(V).

E

Take

q

V.

+

1

Define

"40 j E

V·

such

w - 1P( "40 ) and such that

o' ... •"40 q is an orthonormal system. Then

1

q

q

(q

I I (""L'\IIljl""L'llDk)(~jl ~k) + 1)2 j-O k-O

(q

I II"" L '\Ill jll2 + 1)2 j-O

(q

I II I + 1)2 j-O k-O <""k''\AO l~kll2

q

q

n

·109·

q

n

CI

CI

(q + 1)2 j-O

q

1

q

+ 1 •

By symmetry, we ob am

LEMMA 1.12.

Take

v

m

0 v

00

02 -

P( ./\ V), w

E

p

q < p 'n

exist.

0, k

and

q.e.d.

1.

E

1P( /\ V·)

q+l

q - 1.

p -

:EO

+

1P( /\ V·), x

E

p+l

o,

II 2

lI...o j ll 2

I

+ 1)2 J-O

(q

Jt<1I.k''''O

k~O

Assume that

vLw

and

w

A

Then

.

.

O(vLw)Lx 0 0 vLw 0 - 0 vL(w '" x) 0 0 w ;.. x 0 .

(1.59)

PROOF.

0

Take

t;

«)

A V. 0

E

:f:.

...0

p+l

such that

v

=

1'(

E

V·

/\

and

1;.

A V·

E

q+l

k+l

«». w - 1P(...o) and x - 1Ph:'). Then we have

•

II(

o vLwLx

0 0 vLw 0

- II

«)

«) L-w» L~II II «) L...oII L-w> II 111;.11 1110 II 11...0 Ii

1110 L(...o ,...

~)l1

- II 10 II 11...0 ,...

11...0 '" 1;.11

~II 11"<10 II II~II

- 0 vL(w '" x) 0 D w ;.. x D •

On any complex manifold the exterior derivative to

where

k+l

dC

-

On

(il2ll'Ha -

A V

a)

define

and

Tp

dd c

by

Op

a+

a

twists

is the Levi operator.

T p(1;.)

p+l

only one position form

d -

q.e.d.

of bidegree

-

1I~1I2. (1,1)

on

Then there is one and 11'( A V) p+l

such that

x

·110·

(1.56)

1P*(o p ) = dd c log ., p

The form

Cl p

1P( "V*) p+l

is also denoted by

T ke map

E

(

"

n+l

V)~.

"V -

n-q

a:

0

P

Take

( " V)* p+l

The Fubini Study Form on

is called the Fubini Study Form.

u

Out.:

on

.

q

Z[O.n]

E

and

for all

Out.

E

"V.

E

A linear

q+l

is defined by

(1.57)

Therefore

t.

"V*. The map 0u:

La'

"V -

n-q

q+l

"V

E

n-q

"V* is a linear

n-q

isomorphism. If V is a hermitian vector space and if lIuli ~ 1. Du is an isometry. If "'" E ( "V). is dual to u. that is. if

then

n+l

<"'".u> - 1.

then

(1.58)

for

Let "'"0' --. ''''"n uo. -- .•u n dnd

t.

E

be a base of V with

be the dual base.

Then

define

= . ...

"V

q+l

(1.59)

t.

"'

t. -

(1.60)

[f

XII

Dut. -

E

" V q+l

and

a

E

A

__ •

A

£

Let

For

Then

I

xu"'"u

I

sign(u.u.l)xuu .1

I.tE~[q.nl

with

"V

q+l

"'"n - "'".

uo" __ . " un - u.

UE~[q.nl

p+l " V

"'"0

t.

U

0

~

p + q + 1 < n.

then

U

E

~[q.n

·111· ~ E

If

A

q+ 1

0-uI

The map

If

v

t).

and

maps

E (

A V*

0 , q < p 'n.

with

p+ 1

then

A

a:.

E

exists such that

At). -

-uI.

•

Therefore a biholomorphic map

(1.64)

II

W(

A V) -

II

D

1P - W

not depend on

II

D

Ii

The map

1P( A V*)

q+l

is uniquely defined such that -uI.

then

Grassmann cones into Grassmann cones:

A V)·.

n+l D-ul - ADt)..

Then

"! £

n-q

D

D-uI'

The map

Ii

does

is the identity; also

(1.65)

such that

E(x) - E[II(x))

for

x

previously defined dual ty map If

x

E:

w( A V) q+l

and

II. z

E

E

Gq(V).

Hence

Ii

coincides with the

We have the following identities. W( A V)

p+l

0 ' p + q + 1 , n.

with

then

(1.66)

o

Moreover if

0 ' p + q + 1 < n.

x

A

z

IixLz 0 = 0 x ;... z 0 - 0 lix ;... liz 0 - 0 IizLx 0 then

( Ii x) Lz

exists if and only if

exists and

(1.67) However if

Ii(x '" z) - IixLz . n ' p + q + 1.

then

-112-

o xLliz

11.68)

If

x

E:

1\ V) and

1'(

Z E:

0 - 0 lix ;... liz 0 - 0 zUix 0 . q ,

p 'n,

then

p+l

o

(1.69)

0"

Moreover if

1\ V·) with 0 '

11'(

q+l

q

hLliz 0 - 0 zLx 0 - 0 liz ;... x 0 .

< p "n,

then

(Ii z) ,.. x

exists if and only if

z Lx

exists and li(zLx) - (liz) ,.. x .

(1.70) If

q

E:

Z(O,n),

then

(1.71)

The projective space

V - I'(a:

$ V)

is called the proiective closur.

of V. We identify V - 1P(U} X V) by 1:. - 1P«(1,1:.)) for all 1:. E V and lP(V) - 1P({0) X V) by lP(t.) - 1'((0,1:.)} for all 1:. E V.. Then

v-

is the disjoint union, in which lP(V)

V V F(V)

is called the

infinite plane. Define

liJJ.p - 0

""JJ. - (IiJJ.O' ... ,liJJ.n)

such that

""0' ... '''''n

(a:n+1). - a:n+1

E

if

JJ. -

a:n+1_

and

liJJ.JJ. - 1.

Then

We choose the hermitian metric on

a: n+1

is an orthonormal base and we identify

suc h tha t

""0' •.. '''''n

obtains a conjugation by setting space is

is a self dual base.

Now,

a:n+1

i - 1:... The associated projective

11' - 1P(a:n +1 ) - 1[n - a: n V 11' n n-1

(1.72)

Here

P

!Po -

too}

and

1'1 -

a:

V (00),

which we realize as a sphere of

diameter 1 in (R3. For x E a: and a t: a: We have 1P(1,x) - x and 11'(1 ,a) - a. Let j(a) - 1'(-a,l) be the inversion. Here !P(0,1) _ 00 and 0 - j(oo) - 1'(1,0). The cordal distance is given by

·113· •

(1.73)

0

_

la -

Ox" a

I1 +

2

"

(1.74)

DXAooD

xl I

- 0

,,1 + Ixl2

lal

-~1

r:===~ + Ixl 2

-

x;j(a) 0

0 x;j(oo) 0

(1.75)

If

W is any holomorphic vector bundle over a complex space s is a holomorphlc section, the zero set

and if (1.76)

Z(s) - Ix

is analytic.

Let

1P(V} X V

as a subbundle, where

E: 1P(V) X V· -

"\10 I E(x)

E:

"\10

and

V·

...a

x

0(1)

E:

E(x))

of 0(1)

over

where

If

...a

1P(V)

E:

write

- "\10.

... "\10

where

Define 0(0) - F(V) X

w - 1P("\10)

(t.

For

t

if

p > 0

is defmed by

...a ~ O.

define

E:

"(V).

Sometimes we

O(p) - 8 0(1) p

and OJ-pI - 8 0(-1). p

Then

It is

is dual to

a global

V·,

vX Z(~) - E[w]

The

Ex( "\10) -

(1.78)

Then

F(V).

E:

The quotient map

F(V).

E:

"(V) X V.

0(-1) -

holomorphic secbon

F(V) X V I ~

is called the hyperplane section bundle.

a quotient bundle for

E:

E(x) - O(-l)x is the fiber over x

0(1) = O(-I}-

the inclusion map

n + I > 1.

has the tautological bundle

O( 1) - ((x,~)

(1.77)

dual bundle

M I sIx) - o}

be a hermitian vector space of dimension

V

Then trivial bundle

E:

M,

O(p)· - O(-p) for all

p

E:

Z.

The

-114hermitian product on 1P(Vl X V

which restricts to

metric

tp

p

Here

of

E

Z.

nO

c:Y(-1).

By tensoring and duality a hermitian

is consistently defined along the fibers of

1P(V) X V*.

where

V defines a hermitian metric along the fibers of

11

c:y(p)

for each

is the quotient metric of the metric along the fibers

The first Chern form computes as

is the Fubini Study Form on

F(V).

\2 Meromorphic Maps .2!! Parabolic Manifolds A)

Parabolic manu Ids. Let

T ~

and

M

be a connected, complex manifold of dimension

0 be a non·negative function of class A S;; M define

(2.1)

COO

- (x E A I T(X) <

A r)

on

M.

For

m.

Let

O:!$; r

IR

€

r2)

(2.2)

(2.3)

A - (x

(2.4)

E

A I -r(x) - r2}

A. - A - A[O] - (x

E

A I -rex) > O}

(2.5) (2.6) The function M[r]

is said to be an exhaustion if

is compact for all

pseudoball and radius

-r

r.

M(r)

(2.7) Then

~

In thls case,

M(r)

is called the closed

M

the pseudosphere of

-r

O.

is unbounded and if

the open pseudoball and

The function

r

-r

is said to be parabolic if

'" ~ 0 U ~

0

on

M and

u > 0

on some non·empty open subset of

M.

·116·

If

is a parabolic exhaustion.

T

manifold.

B

TO

E: a: m.

of

Up to biholomorphic isometry.

T

=

(a: m•T

0'

a: m

TOo tC -

m (Stoll [1061. Burns [91. P. Wong (124)). m.

be a sudective. proper holomorphic map.

Let

Then

IIrrll 2 is a parabolic exhaustion of M and

called a parabolic coverinJ[ manifold of algebraic manifold of dimension

a: m .

covering manifold of

A strictly

is the only strictly

M be a connected complex manifold of dimension

rr : M _

o.

v >

a: m is defined by Toht' - Ihl-1I2 for

parabolic manifold of dimension Let

is said to be a parabolic

A parabolic function is said to be strict if

parabolic exhaustion all

(M. T'

(M •.,.'

is

a: m. In particular any affine

m can be represented as parabolic

A Riemann surface is parabolic if every

subharmonic function which is bounded above is constant. product of parabolic manifolds is parabolic.

The cartesian

For more details see Stoll

(103). (105). (106). (107). [10B). Let

(M •.,.'

be a parabolic manifold of dimension

m.

if

(2.8'

Then

O
(2.9' (2.10,

do- Take

r >

o.

0

Then

aM(r,

is the boundary of

dM(r' - (x E: aM(r, I d.,.(x'

(2.11'

is open in the topology of

aM(r'.

smooth boundary manifold of

M(r'

Here

dM(r'

~

M(r,

and

0,

can be regarded as a

oriented to the exterior of

M(r'.

The

complement M - dM(r' is compact. Then M(r' is a Stokes domain in the sense of Tung (112), if for every point p E: M there is an open neighborhood U of p and a biholomorphic map 1t: U - . U' onto an open subset

U'

of

«:m

such that

B(\1

n

M'

has finite (2m-1I-

·117·

dimensional Hausdorff measure and such that zero (2m-l)·dimensional Hausdorff measure.

B(U" (M

dM(r)))

(2.12)

~T

-

Ir

E

IR+ I M(r) is a Stokes domain)

(2.13)

~T

-

Ir

E

IR+ I dT(x) ~ 0 for all x

M
E

....

~T C;;; ~T

Then

aMlr)

has measure zero.

and

~ dM(

M

=

then

M - ClM(r) - dM(r)

r)

Stokes doma n

M,

In

~T'

smooth if

r

a compll'x

pace.

E

M

If

is used.

r

~T'

E

In particular, this holds if r

E

~T'

(2.14)

If:

then

M

~T'

E

M(r).

M -

1f- 1(M(r))

while the boundary of

For

r

For most purposes

suffice. but in some proofs a proper modification thin analytic subset of

If

and Stokes Theorem holds for is smooth.

has

Let

If ~T

M

then

...

r

~T'

E

would along a

1f- 1(M(r))

is a

may not be

is not a manifold but

the number

J

c,-

a>O

M is constant.

If

r ~ 0,

then

(2.15)

In particular B)

Mlr] ~"

Divisors. I.et M

for all

r ~ O.

be a connectl'd, complex manifold of dimension

We can identify a divisor with its multiplicity function. holomorphic function on

M.

Take

germs of holomorphic functions at in

"x,

Then

r

defines a germ

x x.

E

M.

Let

0 ~ fx

Let - x E

"x

"x

Let

m.

On

f 'lObe

be the ring of

be the maximal ideal One and only one

M II

-118-

p+l Then f x E _ Px is said to be the ~ multiplicity of f at x. The function v: M is said to be a divisor on M if and only if for every point x E M

integer

p ;!: 0

-x

exists such that

there exists an open, connected, neighborhood functions

g iii 0

and

h iii 0

on

U

of

X

n

and holomorphic

such that

U

(2.16» The set

"M

of divisors on

v(x» ~ 0

divisor is non-necat ve if of

U

o

that

x

x E M, which is the case if

for all

and a holomorphic function

The support

vlU - JAg

S

~

supp v

is an analytic set of pure dimension

XES.

branches of

S.

each branch

BEt.,

Let "R(S»

for all

m - 1.

U

such

is Inversely, if

there exists one and

Let

t.

be the set of

be the set of simple points of

there exists an integer

x E B "!R(S).

on

for every simple point

v be a divisor with support S.

Let

vex) - kB

m - I,

Vs such that vS(x) - 1

only one divisor

g ii! 0

of the divisor

either empty or an analytic set of pure dimension S

A

x E M there is an open, connected

and only if for every point neighborhood

M is a module under function addition.

kB

1;:

0

S.

Then for

such that

Moreover, we have the locally finite

representation

(2.17»

v

Here

~

0

if and only if

kB > 0

for all

BEt..

Let Nand M be connected complex manifolds. Let f: M N be a holomorphic map. Let V be a divisor on N with f(M) II; supp v. Then there exists one and only one pullback divisor f'"(v) such that if

g ii! 0

subset

(2.18»

U

and of

h 51 0 M with

are holomorphic fUnctions on an open, connected II

IU -

JA~

-

JA~

then

·119·

~

C lit])

if

then

s i! 0

Let

E

M.

M,

Then

{}N -

f*

{}M

is a homomorphism.

be a holomorphic section of

Then

s

holomorphic Vl'ctor bundle

8

defines a (zero)-divisor

/.Is

For every point U

there is an open, connected neighborhood

holomorphic section h i! 0

U

on

t

to units and

W over

of

/.Is

m - 2.

'IJ:

M -

the trivial bundle

M X V.

defined.

E

Take

Here

b

V

f

of

holomorphlc functions 0 b /.If IU - /.Ig-bh

if

g b

~

h

U

on

and

pole divisor of

f.

divisor on

M

U

of 0

/.I h f.

Also

s.

~

hf

g.

f ii! 0,

If 00

/.If

be a parabolic manifold of dimension

with support

X

nv(r) - r

E

J

S (r

nv(r) -

(2.20)

~ ZES

Then

nv(r) -

nv(O)

for

[r

r -

vu m - l

Then then

m

v

Let

v

o.

If

m > 1,

then

be a

is definf'd

if

m > 1

if

m - I .

I

viz)

I

M

is called the

The counting function of

2-2m

If

M.

by

(2.19)

is

'IJ

and there are coprimf

x

such that

/.I7 1U

is called the divisor of

(M,T)

of

'IJ

For each point

is defined.

/.If - /.If - /.If

Let

/.I

be a meromorphic function on

f

and 00

t

are uniquely defined up A holomorphic

there is an open, connected neighborhood

00

t

can be viewed as a holomorphic section in

Let

the b divi or

0

and

Hence the (zero)-divisor

!Pl.

a

x,

and such that the zero set of

h

is well defined by

vector function

of

U and a holomorphic function

s IU - h • t

such that

has at most diml'n'lion

f ii! b,

v ~ 0,

If

o.

~

fa(v)

W over x

({).

·120·

(2.21)

J

n)l(r) -

r > O.

for

)lW m- 1 + n,,(O)

S(r)-S(O)

0 <

For

S

< r,

the valence function

If

and

)I

and

J,l

are divisors on

N_u(r,s) - - Nu(r,s)

N ,,(r,s) ~ 0

function on with

M,

s €.f. T

is defined by

dt

r

t

then

n_J,lIr) - - nJ,l(r)

)I ~ 0,

If

increase with define

)I

nJ,l+)I(r) - nJ,llr) + nll(r)

and

NJ,l+)I(r,s) - NJ,l(r,s) + N,,(r,s). and

M,

of

n (t)Js"

N)I(r,s) -

(2.22)

N)I

n)l(r) ~ 0

then

If f ~ 0

r.

Nf(r,s) - NUr(r,s).

and

is a meromorphic 0 < s < r

For

€

.f.T

we have the Jensen Formula

(2.23)

J

Nf(r,s) -

J

log I f I a -

log If I a.

M<s>

M (See Stoll [1031 IV 11.) C) Meromorphic maps. Let

M

and

N

be connected complex manifolds.

m - dim M

and

M

A - M - S.

Define

n!!Eh

closed graph 1C :

A(fl

Let

S

rf of

map into

M M N.

of

f.

be a proper, analytic subset of

Let f: A -

r f of the graph ((x,f(x» I x

closure closed

n - dim N.

Then

f

€

N A)

be a holomorphic map. in

is said to be meromorphic on

f

If

exists such that Define

maximal open subset of

A(f) M

The

M)( N is called the

r f is an analytic subset of M)( N and is proper.

Define

M

if the

if the projectio~

is meromorphic, a maximal open subset f

continues onto

1C- 1 CA(f))

such that

t:

r f'

1C:

A(fl

Then

Acn -

as a holomorphic A(f)

A(f)

is the is biholomorphi c

-121-

The complement

If

M - A(f)

is analytic with

f.

M is a manifold,

=

the indeterminacy of

Since

If

S

S;;;

and is called

dim If ~ m

2

and

(2.24)

dimy K- 1(x) > 0

in fact,

0/1: r f -

Let

fIB) - o/I(K -l(B». be a divisor on

y

If C t: N,

define

f (v)

Let

x

B

M,

t;

Assume that

is defined on

A(f).

If .

define

v

M.

Let

dim If

~

Hence m - 2,

M again denoted by

under the meromorphic

n + I > 1

V be a complex vector space of dimension

be a proper, analytic subset of

E

f(A(f)) II; C.

Since

contmues uniquely to a divisor on

f-(v) and c lled the pullback divisor of map f.

S

If

and

f-l(C) - K(o/I -1(C».

S.

N with support

f-(v)

K-l(X)

I:

N be the projection.

the pullback divisor the divisor

for all

Put

A - M - S.

Let

Let

f : AlP(V) be a holomorphic map. Let U be a connected, open, V is said to non-empty subset of M. A holomorphic map 10: U be a representation of f (at a point p E: M if p E: U) if i! 0

10 10

(x)

~

and if fIx) - 1'( 10 (x» O.

The map

f

for all

10:

E:

is meromorphic on

representation at every point of M. A representation

x

U -

UnA

with

M if and only if there is a

Assume that f

is meromorphic.

V is said to be reduced if

dim 10 -1(0) ~ m - 2 which is the case if and only if 10

-1(0) - U n If.

In other words, a representation

reduced if and only if

u 10

::

o.

representation at every point of M.

Let

10 j :

U1 n U2 ~ 0. U1

n U2 with

Uj -

V

10:

U-

A meromorphic map admits a reduced M and on every Cousin II open subset

be representations of

f

for

j -

Then there is an umque meromorphic function 10

1 -

V 10

2 on

V is

U1 n U2 ·

If

10

1,2 v

with on

2 is reduced, then

0

-122-

v

is holomorphic, if also

is reduced, the holomorphic function

~1

v

has no zeros. {U ~h, E A ~ II

Let

p t-1

. ,X.pl E A

be a family of su bsets of ,

M. If

define

(2.25)

(2.26)

Here

A[p] - A[p.ll]

~th

is called the

said to be open if

Ux.

to be a covering of

ru!!:Y!!. of

is open for each if and only if

M

ll.

X. E A.

M -

The family The family

V Ux.' X.EA

If

II

is

II

is said

U!; M,

define (2.271

M with general fiber

The trivial bundle over

VM = M XV.

If

a function, then

U

'I: (lJ

defines a section

~

;; (xl ~ (x, ~ (x))

for all

~

=

~

and identify

PROPOSITION 2.1.

If

x E U. ~

holomorphic if and only if (x.

Let

M

is a subset of

~

f: M -

~

~

: U -

is open. then

is holomorphic. (x)) -

is denoted by

: U - VM

~

U

and

V

V

by setting ~

is

Sometimes we write

(x).

lP(V)

be.!! meromorphic map.

there exists. uniquelY ~ to isomorphism • .!! holomorphic line bundle called the hYperplane section bundle of section

F

of

representatIOn Lf

QY!l..!:

frame

~

VM ® ~

Lr

over

: U- V

M

f.

over

M

-

such that

FlU

Il

!!!. unique.

Moreover

~

Le.

and.!! holomorphic

such that for each reduced ~Il

there ex'sts .!! holomorphic frame

U

Then

® ~ Il. If - Z(F)

Given

.:l

and

F.

the

!!!. the zero set of the

of

-123-

F.

f"'(O(l))

of the hyperplane section bundle

10

section frame of

of

~

:

Lf

0(1)

f

V of

v>..u

10

IT then the .!!

~"' ~ the dual ~ of

A -

A[l]

E

a: -

(0)

~,l~E:A

of reduced representations

II - IU~}~e:A

such that (>",u)

U>..u -

(10

exists such that

on v>..u • vu>" - 1

and

is an open covering

one and only one hal am orphic

(2.28)

Then

U ~ A(f),

pullback

Lf I A(f) - f"'(O(1)).

For e ch pair

function

!PIV).

Q!!

~

ill the subbundle f"'(0(-111 An!! i!

~.!! section

There ex ts a family

U~ -

M.

the bundle

~ U. Then

["'(0(-1))

~ identify

PROOF_ 10

in VM

10

if

A(f),

V ~ ~ reduced representation with

U -

:

10

Over

!! isomorphic to

section

v>..>.. - 1.

If

U>..u

(>",u.p)

E

A[2),

then

(2.29)

The cocycle

(v>..u)(>.. U)e:A[1)

M and a family U>..

(10

~1>"EA

determines a holomorphic line bundle of holomorphic frames

10

~ of Lf

Lf

or

over

such that

(2.30)

for all pairs

(>...u) E: A[l].

The transition formulas (2.28) and (2.30) show

that there exists a global holomorphic section

F

of

VM

e Lr

such that

(2.31)

Let

10: U -

V

be a reduced representation.

there is a hoiomorphic function 10

>.. - vA

10

on

u"

U>'"

vA: U " U>.. If

(>...U)

E

AU[lJ.

a: then

For edch CO}

>.. E AU

such that

vA - v>-.JJ.vU

on

-124-

U 1'\ U>..u· frame for

Hence of

-06.

A

E:

frame of

AU'

Lr

v>..

Lf

-0

~

over

Then over

V U -0

-

~

on

U 1'\ U>..u' -06. 1(U

U is defined by

FlU -

8

-0

U such that

Let

-0 6..

1'\ U>..) - VA

-0

~

be any holomorphic


C; 8
FlU -

U such that

function without zeros exists on

A holomorphic

c:P - h

-0

6.

.

Hence

-0 6. _

Thus

h:: 1

Let

-0

:

c:P -

Lf . F. ;; 6.

be a reduced a

and

-0 6..

be another solution.

representation of f.

Lf 1U

Lf 1U -

Let

U -

-0:

V

A bundle isomorphism

is uniquely defined by

a

-0

-0 6. _

0

;; 6..

Let ~: W V be another reduced representation of f with U 1'\ W ~ 1'lJ. A holomorphic function h without zeros exists on U 1'\ W such that ;; 6. _

Hence a :

'\010

h':;;;;l1.

h -0

-

On

a~ - a

Lr - 1;.

-0

on

U 1'\ W.

Then

-0

11 - h~11

U 1'\ W we have

over

U

is defined by

n W_ Therefore a clobal bundle isomorphism a

0

-0

11 -

;; 11 for all reduced

representations -0: U V. Obviously (Id 8 a) the construction is unique up to an isomorphism. The holomorphic map (2.32)

and

f-(O(-I)) - ((x.t,)

f E

A(£) -

1P(V)

M X VI

f;. E

0

"'" F - F.

pulls back

0(-1)

Hence

to

E(f(x)]} C M X V - VM

·125·

Let

U -

1:):

V..

~ (x) - (x,

Then

1:)

be a reduced representation of (x)) (; (x) X E(x)

holomorphic frame of of

f'"(('l(1))

f*(('l(-I))

U.

over

representation of

f

h'110

tJ

-

1:)

® ~ ..

=

F

section

of

f (('l(1))

reduced representation

4 I A(f)

-

Lr I A f)

W" U

and

on

U -

V..

a: Lf I A(f) -

Hence

The section

(Id ® a)

on

F - Ff

0.

'#;

Then there is

such that

U" W.

Therefore

~ ® ~..

FlU -

of

f"(('l(1I)

F I A(f) -

0

I frame

Consequently, a global holomorphic

exists such that 1:):

is a

1:)

be the d

W" U

h ~ .. -,;;..

U" W.

U!: A(f).

be another reduced

without zeros on Then

~..

Let

V..

on

Hence

f

with

F.

for each

U!; A(f).

F I A(f).

Trivially

Therefore a

exists such that

for each reduced representation

0

U !; A(f).

in

U.

with representation section

bundle isomorph' m

a

h

®,;;..

,;;

W ~ A(f)

with

W" U.

on

over

f*(('l(-l))x

'"4D: W -

Let

a holomorphic function

=

f

1:)

:

U-

V..

with

q.e.d.

is called the representation section of

f

Lr. Let

V l' •.• ,V k

and

e

W

be hermitian vector spaces.

VI X .,. X V k -

W

be an operation homogeneous of degree let

fj: M -

j -

1, '" ,k,

e

extends to a fber preserving, holomorphic map

A holomorphic section

1P(Vj )

Let

be meromorphic maps.

For The operation

·126·

is defined.

If

j - 1, ... ,k,

Ff

J

Then

-v.

J

is a reduced representation of

fj

for

then

e .. a Ff

I

U

10.

k

I U - ( 10 1

(f l , ... ,fk )

a .. , e

10

k

)

(

e

is said to be free for

which is the case if

10

Ie ... a

10

(and hence all) possible choices of

0,

if

0

k •

I' '"

10

6.)Ql

10 1

( 10 6. )Qk k

e ... e

Ffl

on ,10

e '" e

Ffk ill 0

U for at least one k'

If so, we write

for the divisor of this section and, on a parabolic manifold, denote the counting function and valence function of this divisor by

Now we will consider examples of such operations. Take

p

E

Z[O,n)

and

Q E

U - U(p,q) - Mm(p+l,q+l). Take line bundle over M. Abbreviate

Z[O,n\. P

E

Define

Z[O,U\.

Let

L

be a holomorphi

V(p,q,L) - ( " VM) 181 ( " Vti) 181 L p+l

q+l

A contraction

(2.33)

HIP -

is defined where if

a

E

Z[O,p).

mO If

IBP

X Id : V(p,q,L) -

is the identity,

p -

U - q + 1,

m1 then

-

m L -

V(p-p,q-p,L) and

mq +1

mll _ ma

0

mp-a

is the interior

·127·

product mapping into p

A VM ® L,

is the interior product mapping into

Let maps.

1P( A V) p+l

f: M -

F - Ff

Abbreviate

if

th n

p + 1,

I! -

P -

l

_ IIIP+1

q

and

A VM® L.

q-p

g: M

G ~ Fg

and

---0

1P( A V*) q+l

be lneromorphic

F ®G

Then

is a holomorphil

section of the holomorphic vector bundle.

(2.34'

with the proper commutation of terms.

Hence

F III P G - III P (F ® G)

is

a holomorphic section in

A V ®

(2.35'

If

p+l-p

~ 0,

p

F 1110 G - F ® G;

then

If

F 1111 G - F III G. P - I! - P + 1,

we write bundle

if

p

I! - q + 1,

p -

~ I,

abbreviate F III q +1 G - FlG.

we write

F III P+1 G - G IF.

we write

F III P +1 G - ,

4

M

If

p -

p + 1

=

If

q + 1,

which is a holomorphic section in the line

® Lg .

Let

U

10:

---0

A V be a reduced representation of f

and let

p+l ~

U

A V*

---0

be a reduced representation of

g.

Then

q+l

(2.36' The pair

(f,g,

is the case if

is said to be free of order 10

III P

if it is free of order order

I!.

If

P -

II! 0

'VIl

all) possible choices of

10

I,

and

on

p,

if

F III P G ill 0,

which

U for at ledst one (and consequently 'W).

The pair

f,g

is said to be free

and it is sald to be strictlY free if it is free of

J.t - P + 1,

we have

-128-

(G LF) I U - (...c L 10) ~

(2.37)

If

P - U - q + 1,

If

M. g

(f,g)

If on

10

U,

(2.39)

(2.40)

is free of order and

...c

~...c

A

we have (FLG) I U _

(2.38)

A 10

(10

P,

L...c)

~

10

the divisor

A

~ ...c A

UFIllPG

are reduced representations of

f

is defined on respectively

then IU - U U FIllPG IU U FIIlG

~

10

IIlP"'
10

1Il...c

U

if

p - 1

(2.41)

UGLF IU

U...cL1O

if

p-U-p+1

(2.42)

UFLG IU - U 10 L...c

if

p-u-q+l

if

p -u-p+1-q+l

(2.43)

If

(M,T)

=

U I U - U<

10

,'110>

is a parabolic manifold, we abbreviate the counting functions

(2.44)

(2.45)

if

p - 1

(2_46)

if

p-p+1-u

(2.47)

if

p-q+1-u

if

p-u-p+1-q+1

(2.48)

n

U

- nr

--x;g

and in full analogy for the valence functions.

·129·

LEMMA 2.2.

Let

maps.

x

Take

f: M -

M - (If V Ig)'

E:

E(f(x)) ~ Elg(x)].

n

dim E(f(x)) and only

Gp(V)

Also

E:

g: M -

Then

IT

GqCV·)

if

if

and only

q \!Ii p,

be meromorphi.

if and only if

(F III G)(x) - 0

(F III P G)(x) - 0

Blg(x)) > p + 1 - p.

if fIx)

and

(FLG)(x) - 0

then

if

Rplg(x)).

The proof follows immediately from (2.36), (2.37), Lemma 1.3, Lemma 1.8, and (1.8).

Then

J.tf,g

(2.49)

p - q,

If

permits another interpretation.

"'I : F(V) X 1P(V·) -

be the projections.

(2.50)

(c ,'YO)

E:

=

O.

Let

1P(V·)

They hft the line bundles and we obtain

cr(a,b)· - cr(-a, b). 'YO E: V=

p - q

"'2 : 1P(V) X 1P(V·) -

1P(V)

cr(a,b) - "'tcr(a)

where and

we can assume as well that

with

If

for

",~cr(b)

x - (v,w)

v - 1P( c)

cr(-l,-l)x'

e

and

E:

(a,b)

F(V) X 1P(V·)

w - 1PhlO),

Z2

E:

and

C

E:

V.

then

A global holomorphic section

"

of

cr(1,1)

is

defined by «c,"IO),,,(v,w»

(2.51)

If

a - F(
and

'lItO

E:

E:

F(V·)

V· - E[61,

b - 1P(6)

11"

E:

1P(V·)

and if

C

E:

V - Elal

then

,,(v,w) - < C

(2.52)

The divisor

and

- E: 11:.

,"10

>

~IE(V) .c.&IE(w) C ~~> '0' ~

<

has multiplicity one on its support


>

S,

i.e.

11" -

where

(2.53)

s-

supp " - ((v,w)

E

F(V) X 1P(V·) I v

E

Elw])

Us

·130·

is smooth, compact, complex submanifold of dimpnsion incidence manifold. LEMMA 2.3.

Let

With these notations wp have f: M _

maps.

A meromorphic

h(x)

(f(x),g(x))

if

=

and

Q!LY

if

1P(V)

and

h: M

map

for all

x EM·

h(M)!t S.

If~,

hIM) g; S,

PROOF.

callpd the

1

2n

g: M -

1P(V*)

1P(V) X 1P(V*) (If V Ig)'

1h!m

i§ defined

Then thE' I@ir

hY

E'xists such thilt

(If V Ig)

which is the case if and only if the pair f,g Assume that the pair f,g is free. Take K E M - Of V

is frep.

[(x) ( E[g(x)]

Il

a

1P(~)

E

E

1P(V*)

g(x) {Elb]. "0/0

:

U ~

and

b = 1P(1'Y)

1P(V)

E

such that

1£

of

R[6]

g

with

for all

x

E

1P( A V)

~

. U -

f.g A V

Jl

Define

=

1P( A V"'l

f

and

Min(p+ l,q+ 1).

A V*

: ll-

~

If

U

~

~

- <1J ~

Therefore ;; mP ~ - <1JI/; ~ one meromorphic map

A V and

(2.54)

f HIP g

M -

Assump

thpn

"010

on

V ~ P

and

(jJ

U.

with

g

~ I/; ~

U"

A p+ 1

A V*

'1+1

respectively

and

1P(

Z[O,~tl.

U

~

p+l

f

mP ~

E

g,

of

then there are holomorphic functions without zeros such that

P

'1+1

-

are other reduced representations of

Take

T dke reduel'd representations

p.

p+l

U.

be

'1+1

is free of order of

and

q.E'.d

p+l

that the pair

f

we hrlve

g: M -

and

and

E[~l

Jl I U

f: M -

meromorphic maps.

of

V,.

~ (z) {

such that By (2.52)

U.

Z E

h*(Jl n ) I U Again let

U

Take

[(x) (RIal

u-

~

Take reduced representations V:

(z)

and

i§ free

f.g

Jlf,g = h*(Jlnl.

x E M

if and only if

be merQ.morphic

on

I/;

U

n

on

U " l

U.

Hence one and only

A '1+ 1

V-) P

-131-

is defined such that U

if

"010

-

t:)

:

U -

U -

V*

III P

t:)

V·

is a representation of

"WJ

f

is a reduced representations of

is a reduced representation of

f III P

and if ~

However

g.

on

g

IIIP

"010

may not be reduced.

If

P

=

0,

q + 1 - I-l

we write

Let

P - P + 1 - U

If

f 1111 g - f III g. P -

f 111° g - f 181 g.

we write

f III P g

=



V I' ... ,V p

=

and

00

W

lPO

£

If

lP(VJ'

P -

if

P + 1 - q + 1

I-l -

hermitian vector spaces and let

be meromorphic maps.

(q I' .. , ,qpl.

(M - (If

is a non negative real valued function.

0

be a

Let

Then

.

G fp 0

(2.55)

we w Ite

which is the constant map.

p-fold unitary projective operation of degree fj : M -

P - I,

f III P g - glf

we write

f III P g - flg.

we write

If

1

V... V If I) -

IJ?+

p

The operation

0

can be specified

follow ng (1.421 - (1.55). Let

f: M -

lP( A V)

and

g: M -

p+l

maps with

p + q <

product (2.56)

is defined by (2.57)

n.

Abbreviate

lP( A VI

be merom orphic

q+l

F - Ff

and

G - Fg '

An pxterior

-132-

where

A V,

E:

C)

~

p+l

A holomorphic section

defined.

If

C)

:

A V, cP

E:

q+ 1

F ",G

u- A

representat·ons of

p+l

f

and

(

of

4 ' x' A

p+q+2

and

V

g

E:

ACJ

'"

E:

q+l

M.

over

M

x

VM) 8 Lf 8 Lg

:u- A

E:

and

Lg,x

are reduced

V

respectively, then

over

(2.58)

The pair

f,g

is said to be free for

the exterior divisor

is

uf;"g - UF",G

""

if

F '" G i1! 0,

U.

in which case

exists such that

On a parabolic manifold abbreviate the counting and valence functions by (2.60)

If the pair

f,g

is free for

(2.60)

f '" g

is defined such that 10

f

:

a meromorphic map

A,

A V and

U --

and

Gq(V),

g

C)

'"

If

respectively. f '" g

holomorphic at

x,

f

maps into then

A

p+q+2

VI

is a representation of

'\10: U -

p+l

then

oW)

M -- p(

f '" g

on

if

A V are reduced representations of

q+l

maps into

Gp(VI

Gp+q+ 1 (V)

and if both maps are

IF '" G)(x)

U

=

0

and if

if and only if

g

fIx)

maps into

E:

Ep(g(x)),

see U.6) and (2.36). LBMMA 2.4. with o ~ k

~ ~

P.q

and

p - q - 1

g: M-- P( A V·), q+l

and

Assume that the pair

(Lg)

k be integers with o ~ q < Let f:M-1P( A V). and p+l h: M - 11'( A V·) k+l

P ~

n

and

be merornorphic maps.

i! strictly free and that the

~

(g,h)

!! free

·133·

for "'. Then the pair (f,1 '" h) !! strictly free if and ~ if the ~ (fLg,h) i! strictlY free. !f so then (2.62)

fL(1 '" h) - (fLg)lh

M -

A

1P(

V)

P-q-k.-l

(2.63)

Take reduced representations

PROOF. ...0

U -

:

«>: U -

I

and

~: U -

A V of fU

and

q.: U -

A V·

of

q+l p-q

A V·

k+l

A

q+k+2

of

A V

p+l

of V·

1:1

and only if

l(~

'"

~)

(fLI,h)

-

(1:1

L~)l~.

is strictly free.

Hence Also

(f g '" h)

and

hand of

I

'"

Then (f,g '" h) is strictly free if and only if «> l(...o '" ~) and (flg,h) is strictly free if and only if (1:1 l ~ ) l ~ ~ 0 where

f

h.

~

0

is strictly free i

fl(g '" h) - (fll)lh

follows. Assume that (f,g '" hI is strictly free. Holomorphic functions Pli!O and Q II! 0 exists on U such that 1:1 l"tIC - Pg and ...0

~ -

"

Q~.

Then

PgL~

Hence

on

U.

q.e.d.

- (<<> L'W:)IL~ -

«> l(...o '" ~) - Q 1:1 lq. .

'i3 Let

The First Main Theorem

be a parabolic manifold of dimension

(M,"')

m.

Let N.

holomorphlc vector bundle over a connected, complex manifold be a holomorphlc section of

U

W

over If.

meromorphic map with indeterminancy holomorphic vector bundle

f-(W)

holomorphic bundle map

f:

uf

of

holomorphic section

i'

0

uf -

U

f.

0

over

f-(W)

--+

f-(WI

£1M - Ifl

divisor

the diVisor by

u

U~

Then

W

over

f: M

N

--+

be a Let

be a

W pulls back to a and there is a

f: M - If

M - If

--+

N

and a

such that

If we use the standard model, we have

~ Z(o),

U~ -

Let

M - If

over

Uf(x) - (x, a(f(x))) .

flx,w) - w

If

N.

W

then

a 51! 0

is defined on

of

and

a f 51! O.

M - If .

Since

continues uniquely to a divisor on

Hence a non-negative dim If M

~

m - 2,

again denotf'd

u~

THBORBM 3.1.

Green Residue Theorem.

along the fibers of

W.

v - u~

E:~.,

Take

s

Assume that and

r

Let

/(

be a hermitian metric

£1M - If) II; Z(a).

E:~.,

with

0 < s < r.

Abbreviate Then we have

·135·

rJ

(3.1)

s M [t

J

dd c log 11«

fll2

0

K

(6.43).

f

If

f

lJ

m-l

d t t 2m -

T + Nv(r,sl

I

log IIcr

I

fll,( ,,-

0

log IIcr

fll ,( "

0

.

M<s>

M PROOF.

A

is holomorphic, the theorem follows from Stoll (103), (108]

Assume that

with projections

f

is meromorphic.

rf -

7(:

Let

r f be the clo'led graph oj

"': r f -

M and

N.

Let

;..

J{

rf

0 : M -

be a resolution of singularities of

r f'

Thf'n

;..

~

fr

't -

T

frO : M -

0

~

surjective proper, holomorphic map and

8

parabolic exhaustion of the m-dimensional, connected, ,. ,. complex manifold M. The map f - '" 0 frO : M N is hal om orphic 0

is

is

M

8

;..

with

f(M) g; Z(cr).

1> = IJ,.cr

Hence

exists.

f

Therefore (31) holds with

M

=

0,)1

and

~·(o)

~-I(M
a

r

f. a,

E: ~'1

1>.

dim If :s: m - 1.

is biholomorphic.

M(t] - ~-l(M[t])

and

and

M, 0,

replaced by

is analytic with If

-

E: ~'t

s

;..

,. ~: M

The restriction

~*(o) - 0

M, 0, f, ,. Then If

Also

Hence

and

Hence we obtain (3.1) if we can show that

N1>(r,s) - N vir,s).

Define

S = supp

of all branches of

S

v and

,.

Let

So

be the union be the

which are contained in

union of all other branches of supp 1>j - Sj

,. S - supp 1>.

for j - 0,1

S.

There are divisors

such that

N1>(r,s) - Ni>o(r,s) + N1>l (r,s).

1> - 1>0 + 1>1'

1>.

J

~

Hence

Consider the standard models

0

with

-136-

f-(W) - {(y,w)

(M - If) X WI w ~ Wfly))

f:

where

f

0

S? - f

0

7C

0

7CO -

'"

7CO -

0

Ir(x,w) - (I?(x),w).

For

x

on

E

M-

if

i(niCx » - (f?(x),U(<<x))) - (f?(x),u(f(S?(x)'" -

Take

x

E

M - if.

Then

1)O(x) - 0

Since

dim(if

n

f

supp 1)1) < m - 1

- 4-) -

f-(W)

we have

Ur(S?(x)

and we have

1)1(x) - 1)(x) - Un,,(x) - U a (f?(x») -

f

M- ~ .

7r: r-(W) I (M

Hence a biholomorphic bundle isomorphism is defined by

r

and

v(f?(x» - f?-(v)(x) •

dim(Ir

n

supp v) < m - 1,

we

obtain

N1) (r,s) - NvIr,s) . 1

Let

j : If -

dim If ' m - 2,

M

and

we have

........

....

j:

M

If -

be the inclusion maps.

Since

·137· since

dim If

:$;

m - 2.

Therefore

NS) (r,s) - O.

We obtain

o

If W - L is a holomorphie line bundle, then

(3.2)

e(L,x) - - dd c log lIall;

is the Chern form of

L for

X

defined and of class COO on On M - supp v, w obtain

(3.3)

on

N - Z(a).

Of course,

t > 0,

(3.4)

t

2!-2

r

Hi

TfCr,s,L,x) -

t > 0,

For

0 <

5

For

r

.f. T ,

C

< r,

t

fa/e(L,x))" u m .

I

the ch raeteristic is defined by

C3.5)

For

fll;

0

the spherical image is defined by AfCt,L,x) -

For 0 < s < r,

the countin-: function the valence function

I

r

II

dt

AfCt) T

.

nrCt,u,L) - nvCt) NfCr,s, u,L) - NvCr,s)

is defined. is defmed.

the campen alion function is defined by

C3.6)

mfCr,a,L,K) -

J

log lIa

0

fllK a.

H

Then Theorem 3.1 becomes the First Ma'n Theorem for line bundles T C(r,s,L,K) - Nf(r,s,u,L) + mfCr,a,L,KI - mr's,a,L.K)

C3.7) for

0 <

S

< r

is

N and does not depend on the secbon

faCcCL,K» - - dd c log lIa For

c(L,K)

E.f.T

with

s

E .f.T •

II

·138· REMARK 1.

The compensation function

continuous function of

REMARK 2.

Nf(r,s,a,L) ~ 0

Af(l,L,K) ~ 0 compact,

K

r > 0

mf(r,a,L,K)

such that (3.7) holds for all

Ua ~

since

o.

Tf(r,s,L,K) ~ 0

increases and

extends to a

If

C(L,K) ~ 0,

increases in

II all.( ~ I,

can be taken such that

0 < s < r.

hence

then

If N is

r.

mf(r, a,L,.() ~ O.

For details see Stoll [103]. REMARK 3.

M - N and if

If

f

is the identity map, we omit

f

as

an index. REMARK 4. Let

Let

f: M -

section bundle of of

Lf .

V

lP(V)

f

be a hermitian vector space of dimension be a meromorphic map. on

Let

Lf

Take a hermitian metric

M.

Then the characteristic function

I.e

is trivial.

(3.8)

c 6

of

Lf

J

- T(r ,s,L f ,.( f) -

0 <

S

< r

~

1

along the fiberE

is defined. c: M -

over

M,

V

of f

in particular,

The Green Residue Theorem implies

E~.,

with

s

J

log II c 6 11 .( a -

M for

.(

T(r,s,Lf , K f)

Assume that there is a global reduced representation Then there is a holomorphic frame

n + 1

be the hyperplane

log II c 6 11 .( a

M<s> E ~., •

Now we will introduce the value distribution functions for a meromorphic map into projective space. vector space of dimension

n + 1 > 1.

Again let

V

be a hermitian

The hermitian metric on

V

induces a hermitian metric along the fibers of the hyperplane section bundle 0(1) whose Chern form is the Fubini·Study form nO' Let f : M lP(V) is defined by

be a lIIeromorphic map.

For

t > 0,

the spherical image

-139-

Ar(t) -

I

t 2m-2

f*(Oo)

A

~

tJm-1

0 .

M[t)

Ar

The function

increases.

(3.10)

Af(O) -

Observe that

Define

1 i m Af(O) O
M*lt) - MIt) - MIO).

I

Ar(t) -

Af(CO) - 1 i m Ar(t) . t ...00

Then

f*(Oo)

",m-1 + Ar(O) .

A

M.[t)

0 < s < r,

For

the characteristic of

f

is defined by

(3.11)

Then

(3.12)

If f

for

is constant, then

for

0 < s < r.

If f

and

Tf(r,s) > 0

for

Ar(oo) - Ar(t) - 0

for

Also

00

t > 0

and

Tf(r,s) - 0

Ar(oo) > Ar(t) > 0

is not constant, then 0 < s < r.

r -

Tf(r.s) -

for

00

r _

for

t > 0

00.

(Stoll (103) Theorem 12.8 and Stoll (108) Lemma 10.4.) PROPOSITION 3.2. M.

Let

IC

Let

Lf

be the hyperplane section bundle of

be a hermitian metric along the fibers of VM 8 Lf .

be the representation section of f

in

hermitian metric aloD8 the fibers of

VM - M X V,

hermitian metric on

>.. -

r

E:

£._

with

V.

0 < s < r.

Abbreviate

Then we have

fl

8

IC.

Lf .

Let

I

f

Let

on

F - Ff

be the

defined by the Take

s

E: ~T

and

·140·

(3.13)

J

Tf(r,s) - T(r,s,Lf,Kf) -

M -

10

(3.14)

V is a global reduced representation of

Tf(r,s) -

J

log II 10 1117 -

M

PROOF. have

Let

FlU -

10: U _

log IIFII}" 17 •

M<s>

M

If

J

log IIFII)..17 -

J

f,

we obtain

log II 10 1117

M<s>

V be a reduced representation of

f.

Then we

10 8 10 A and

dd c log IIFII~ IU- dd c log 1110 112 + dd c log 1110 AII~

(3.15)

Thus the Green Residue Formula, Theorem 3.1 implies (3.13). 10: M -

If

V is a global, reduced representation of f,

IIFII).. - 1110 II 1110 All K.

PROPOSITION 3.3. Let

N 10

Let

Hence (3.13) and (3.8) imply (3.14).

10: M -

J

Tf(r.s) -

M.

log II 10 II a -

M

for 0 <

S

< r

E:~.,

with

s

f

on

N

Assume that

Then

J

log 1110 1117 - N 10 (r.s)

E: ~.,

A reduced representation

function

h: M -

0:

10.

q.e.d.

M<s>

PROOF.

We have

be a representation of

be the valence function of the divisor of

the Theorem of Cousin II holds on

(3.16)

V

then

"C: M -

exists such that

V

exists.

A holomorphic

-141-

J

log II ~ II 0

f

-

M<s>

M

J

N tJ (r,s) -

log I h I 0

V I' '" ,V k

Let

-

J

log I h 10 .

M<s>

M Addition implies (3.16).

log II~ 110

q.e.d.

W be hermitian vector spaces.

and

be an o~ration homogeneou i - I , ... ,k (f 1, ... .fk ) divisor

let

fj: M -

Uf 0- ';f 1 ... '" k

(3.17)

For 1P(Vj )

8.

free for

IS

be a meromorphic map such that N

Let

f I 8- ... 8f k

be the valence of the

and define the compensation function by

J

mf 1·G.. ·Ofk (r) -

log

M for r

Let

----...!...---o . o fl 8

E ~., •

THEOREM 3.4. r E~.,

with

The General First Main Theorem.

0 < s < r,

If

s

E~.,

and

then

q

(3.18)

L

j -1

PROOF.

qJ.Tf.(r,s) - T f

Abbreviate

J

f:\

1"''''

Of (r,s) k

h - fl 8 ... 8 fk : M -

the hyperplane section bundle of

fj

1P(W).

and take a hermitian metrlc

be Xj

·142·

dlong the fibers of

Lj .

1l j

Let

induced metric along the fibers of >.. j - 1l j ~ K j'

Let

hermitian me ric along

Lt;

Lh

Kh

be the hermitian metric of VjM - M X V j.

Abbreviate

be the hyperplane section bundle of

along the fibers of

Lh ·

and thE

Vj

Let

IC

~

h.

Take a

be the metric

Then

dual to

(3.19)

IC

-

is a hermitian metric along the fibers of the line bundle

(3.20)

k

c(L,IC) -

(3.21)

j

L a

qjC(Lj,K j) - c(Lh,Kh) 1

which implies k

T(r,s,L,IC)

(3.22)

~

L

qjT(r,s,Lj,Kj) - T(r,s,Lh,Kh'

J=l Let

Fj - Ff .

be the representation section of

fj

for

j -

1, ... ,k

J

and let and

Fh

be the representation section of

>"h - llh

~

Kh

H2(U>..,Z) -

M

o.

For each

>..j : U>.. --. Vj

10

>"1

e ... e

function

L.

Let

(U>")>"EA

by Open, connected Stein subsets

10

1C

>..k

W>.. ~ 0

>.. j - 1l j ~ Kj

Let

be the respective hermitian metrics.

We shall construct a section in covering of

H.

of

fj

>..

E:

and

U>.. ~ 0

with

A we select reduced representations ~>..: U A --. W

is a representation of exists on

be an open

U).

such that

h

on

of

h

U>..,

on

U>...

Since

a holomorphic

-143-

on

u>.._

We have

(3.24)

(3.25)

...a ~*

Let to

be the holomorphic frame of

L~ over U). which is dual

Then

(3.26)

is a holomorphic frame of

Lover

U}".

For each pair

there are zero free holomorphic transition functions on

U},,/..l

=

U).

(I,

U/..l

such that

(3.27)

(3.28)

(3.29)

(3.30)

on

U},,/..l.

Therefore we obtain

v)./..d

().,jJ.) E A[l]

and

w)./..t

-144-

W>.. ~ >.. -

~ >..1

e... e

~ >..k

or

on over

U>..u.

Therefore one and only one holomorphic section WI U>.. - W>.. It>..

M exists such that

>..

for all

£

W

of

L

A.

Observe that

::.r 1"'···"'lk

Uf';

on

U>..'

hence

-

JAF

til"

-"1

t:\ OF 1"'··· k

::.r -

O· 1 ... "'lk

JA ~ ~

M,

on

JAW

NW(r,s)

t:\ t:\ ~ 1"'···'" ~ k

~

Nf ';

-

JAW

>..

-

JAW

which implies

::.r

1"'···"'lk

Therefore (3.7) and (3.22) imply

::.r

- Nf '; (r,s) + 1"'···"'lk

I M
Now. Proposition 3.2 lmphes

log

1 iiWir a IC

I

M<s)

log

1 iiWir a IC

-145-

I I

J

M

(3_33) Th(r.s) - T(r.s.Lh,xh' -

log IIF.II" a J

M<s>

log IIFhll"h a -

M

j

J M<s>

Define

(3.34)

Then (3.31), (3.32) and (3 33) combine to

- Nf '; ::.r Ir,s) + l""""'"k

For e ch

"E:

110).,

A.

I

we have

0

I

log ua -

M

II

0

J

log ua

M<s>

~II xh II~ ~·II x ~ _

1 on

U".

Hence

-146-

Hence (3.35) coincides with (3.18).

mf 10... 8r k

The compensation function continuous function on

IR+

q.e.d. ~T

extends from

such that (3.18) holds for al\

to a

0 < s < r.

In order to simplify the notation, we will also write T fIr,s) = T(r,s;f)

Nf(r,s)

etc. 1.

N(r,s;f)

Now, we will study the case of special operations. SPECIAL CASE:

The First Main Theorem for the exterior product.

Take positive integers V. -

A V and

nj

qJ' - 1

n.

J

"

~

with for

~

n1 + ... + nk 1, ... ,k.

j -

n + 1.

Define

Take the exterior product

J

=

a

fj : M -

as homogeneous operation of degree 1P( A V)

(1, ... ,1).

(f 1 , ... .fk )

be meromorphic maps such that

nj

for

A.

Put

p - n 1 + ... + nk

and

W - AV.

Let is fret

Then we have

p

k

(3.36)

I

j-l

T f .(r,s) - Tf J

+ mf

If

n1 + ... + nk - n + 1,

the Grassmannian cone.

1A

r ••• Al

k

(r,s) + Nf 1A 0

f (r) - mf 1""'" k 1A 0

0

then

fj: M -

0

Tf

0

••• "

f (s) k

f (r,s) - 0

1,,···A k

Gn . l(V),

then

J

o f 1 ;... ... ;... fk

0 ~ 1

Of (r,s) k

••• ,..

and the compensation function

If

fj

maps into

·147·

(3.37)

~

mf of (r) 1" ... " k 0

is non·negative.

0

The terms in (3.36) are invariant under permutation of the

f 1, .. · ,fk

maps

If

n1

n 2 - ... = nk - 1,

=

f1' ···.fk

then

are said to be in

(f l' ... .fk) is free for the op~ratlon

general position, if and only if

".

This situation will be explored in Section 7. Z,

SPECIAL CASE.

k - 2, p

;lID,n), q

E:

T e First Main Theorem for !! contraction, E:

;lID,n)

f - f1 : M -

1'( "V)

maps.

u = Min(p l,q+l)

and

p+l

Define

is free of order

w

III P

and take (f,g)

V2 =

11'( "V·) q+l

"V*.

q+l

Let

be meromorphic Then

P E 1N[1,u).

(f,g)

is free for the operation

III P ,

Define ~

( "

p+l-p

V) 18 (

"

q+l-p

V*)

(1,1).

We obtain

Tf(r,'!) + Tg(r,s) - T(r,s;f IIIP g)

=

M -

f

then

[J

N(r,s;f Hip g) + m(r.! Hip g) - m(s;f Hip g)

Gp(V)

eP

f

abbreviate

3.

and

is a homog n ous operation of degree

(3.38)

If

"V

p+l

g - fZ : M -

if and only if

p,

which we wlll assume now

Then

V1 -

and

Take

g [J

and

g: M -

Gq(V*)

~ 1 and m(r;f iB p

g)

map into Grassmann manifolds,

~ O. If

P

=

I,

we

III - 1Il1 .

SPECIAL CASH: The First Main Theorem for the interior product.

k - 2 p ,

E

Z[O,n), q

f - f1 : M -

E

Z[O,p)

and

and

g -

11'( "V)

p+l

V1 -

"V

p+l

f2 : M -

and

V2 - "V·. q+l

1P( A V·) q+l

Takf L('t

be meromorphic

·148·

maps.

Define

W -

A V. p

(f,g) Gl

is free for

of degree

L,

(1,1).

Then

(f,g)

is strictly free, if and only if

q

where we take the interior product

L as operation

We obtain

(3.39)

which is also a subcase of Special Case 2 with

o,

p - J.t - q + 1 ,

then

CI fLg CI ' 1

T g (r,s' - 0

p + 1. and

If

f: M -

mfL.g(rl ~ O.

If

Gp(V) g

and

g: M -

Gq(V*,

is constant, then

and (3.39) is well known (see Wu (126), Theorem 4.9).

We assumed 0 ' q 'p. The case 0 ' p ' q can be obtained by replacing V by V* and V* by V - V**. The details are left to the reader. 4.

SPECIAL CASE:

The First Main Theorem for the inner product.

k - 2, VI - V, V2 - V*, W - It.

Let

f: M -

1P(V)

Take

and

g : M1P(V*, be meromorphic maps such that (f,g, is free. The inner It is a homogeneous operation of degree product <, > : V X V* (1,1'.

We have

(3.40' Tf(r,s) + T (r,s) • Nf (r,s) + mf (r) - mf (s) 8 ,8 ,8 ,8

where

mf,g(r)

~

O.

If

g:: a

E:

1P(V·'

is constant, we write

(3.41' and obtain the classical First Main Theorem for HYperplanes (3.42'

-149-

[f we integrate over

we obtain

1P(V·)

(3.43)

J

Ti(r,s)·

(3.44)

Nt(r,s;a)Oo(a)

u:1P(V*)

We will Kive and Lemma (2.3). defined by

t

metric

§!l

alternative proof of (3.40).

A meromorphic map

h(x) - ( (x) g(x))

for all

Recall (2 49) - (2 53)

h: M x

E:

1P(V) X 1P(V·)

M - ([f V Ig)'

is

The hermihan

V mduces hermitian metrics along the fibers of the bundles

on

Vp(V)' VI'(V*)' ~(1)

and

H

~(1.1).

=

We have

hIM) ¢ Sand

Uf .g - h*(u 11 ) - u~ CLAIM: 01 E(x)

x ~ 1P~)

Take E: ~(l)x

1P(V)

E

~

PROOF OF THE CLAIM: with

~ - xO""O'

Xo

<~,""O> -

for map

(.II

11 __ 1E(x)11

(3.45)

V

and

j -

~(1)

Let

Let (~'''''l

and

1..... n.

V* -

E:

Then

V*.

and

Hence

x

.

0

for

.... 1......... n

Therefore

1101 E(x) II I -

_ I (~, 0> 1

- ...........,.,...II~II

'"'0' ... '''''n

.... 0......... n

-

t

I aOI -

be an orthonormal base of

be the dual base. j - 1..... n.

Hence

Then .... j 1 E(x) - 0

spans the kernel of the evaluation

It .... o is isometric to

I (""0'<.111> I -

I<~,<.III>I

II~II

O(l)x'

Wr have

q.e.d.

-150-

Take

<1:..«> ~ 0

x - !P(.~) and

&

E

!P(V,

V..

E

and

y - !Ph~)

£

<&.\11'> ~ O.

with

!P(V·,_

Pick

Now (2.52) implies

1I1 -, < I

1<6.\11'> I

= 1 <1:. .\11'> I - 0 X.Y 0

111:.11 11\11'11 If

0 < s < r

E

f.T

and

S

E

{' T'

then (3.7) implies

where mh(r.n.H.i)

f

log II n

f

log 0

Q

hll

0

o

0

M

f,g

~

mf,g(r)

M

Also Lemma 2.3 implies


Nh(r.s.n.HI - Nf.g(r,s).

We have

and we obtain

which proves (3 40). ThiS proof is already known. see Shiffman [84).

£

V:

with

'S4 Associated Maps

M

Let

be a ronnected, complex manifold of dimension

m

A biholomorphic map

(4.1)

U '*- 121

of an open subset called a chart

of

M

onto an open subset

U'

of

([m

=

IJ ~.

is

Defme

(4.2)

For

~ q E

0

Z

abbreViate

(4.3)

Then

(4.4)

Dependenc~

t

~

t ~' x

!, : V

"

)(>.,

=

~~

~

on ~

xit

-

x~~.

is indicated by an index etc.

V' ~~

r x.

11 = \Vx.1x.E:A

=

of

r ~>..

Let

(~~l~ E A

M.

Abbreviate

~

as nf'pd~d

be a family of chart., V, - V .... ,' V' = V'

"

0"

and u.,e the notation (2 25)

of open subsets of

M.

V

Then

x.

II>..'

(2.27) for the famib

\ilx.1x'EA

is call!'d a

-152-

chart atlas of (}",J.l)

on

M

if and only if

II

is a covering of

then there is a holomorphic function

A[l),

£

M.

A>..J.l

If

without zeros

such that

U}"J.l

(4.5) Then

A>..>.. - 1

on

U>..

and if

we have

(}",J.l,p) £ A(2),

(4 6)

r>.. is a holomorphic frame of the canonical bundle K of N

Each and

(A>"J.l}(}".U)E:A[1)

Let ~ : U -

is a basic cocycle of

K.

B

be a holomorphic form of bidegree

U'

is a chart. holomorphic functions

(m -- I,D) Bu

on

M.

exist uniquely on

such that

Let

V be a complex vector space.

holomorphic vector function. t::)'

:

U -

V.

Let

t::):

V

U -

A holomorphic vector function

called the B-derivative of

(4.8)

is uniquely defined by

t::).

t::)

The oprration can be iterated:

t::)

(p) ~

(t::)

(p-ll)'.

Put

t::)

(0) _

Abbreviate

(4.9)

t::)

Dependence on t::)

.a -

be a

t::)

ltR.

~

etc.

,.,

,

" ... "

t::)

is denoted by an index

~

.£

-

t::)

t::)

(p)

as

U- Gp(V).

t::) •

If U

-153-

Let f : M -

V 1P(V)

be a meromorphic map.

said to be a representation family of ~ ~ : U>.. -

chart and

n + 1 > 1.

be a complex vector space of dimension Then

f.

("&~. ~ >")~E:A

i~: U>.. -

if

U~

else is said. we assume that each

is

U~

V is a reduced representation of

Let

is a If nothing

f.

is connected, Stein and

H2(U~.Z) - O. Then U>.. is a Cousin II domain. each divisor on U>.. is U~.

principal and each meromorphic map has a reduced representation on The representation family is said to be a representation atlas if 11 - (U).)~EA

is a covering of

()..u) E A(1),

a holomo phic function

defined on

U).U

M.

which we will assume now. v).u

If

without zeros is uniquely

by

(4.10)

on

U).u _ Then

v).>.. - 1

on

U).

for all

).

E

If

A_

().,u,pl

E

A(2).

then

(4_11,

If

0' p

t:

Z

and

Let

Lc

(~.u) t:

A[l).

then

(4_12,

on

U).u'

be the hyperplane section bundle of f.

is a holomofphic frame of

4

over

U).'

If ()..u)

E

A(1).

Then

t:)

~

then

(4,13'

These transformation formulas imply the existence and uniqueness of a holomorphic section

Fp - F pf'

of the holomorphic vector bundle

called the pth representation section of f.

·154· p ( p+ 1 ) P+ 1 L f I] p - ( A VMI ® (LfI

(4.14)

.0.

K

2

'0'

p+l

such that

(4.15)

>-

for all

A.

E:

Here

then

Fptl :: O.

If

~

0

~

p

index of

f

o~

if

p ~

Hence

for

if

Band

ff

E:

M

f

Z[O,n]

Fp :: 0

if

P

We call

Fp :: 0 Fp i! 0

if the generality

is said to be general of order

admits

If

> n.

exists uniquely such that

p > if .

if

Fp = - 0

and

if

but

FO - F '1. 0

p

B if

for

m analytically independent holomorphic M,

functions, then for any finite sets of meromorphic maps defined on B

there eXists a holomorphic form

m - 1 on

of degree

the generality index of each of these maps

f

for

B

M

such that

equals the

dimen'lion of the smallest projective plane containing the image of (See Stoll [1001. Theorem 7.11.) if

The map

f

f.

is said to be leneral for

B

i f - n.

For each

p

Fp

E

M -

f p - 11'

0

dnd

AJ!. : U A -

~

Z[O,ifl. Gp(V) Gp(V)

pth

the of

f

associated map

is dpfined.

Here

is a representation of

fp

fp

is meromorphlc

for each

>-

E

A

and (4.151 implies (4.16)

In Pdrticular. ~ A.£. and let

F -1

u

o.

F 1

-

may not be reduced.

for all

>.. EA.

Define

Le[-11 - M X It - ItM

be the trivial section defined by

For

0 ~ p ~ If'

the

pth

F _l(z) - (z,ll.

s t a t'lonar~ divisor

Then

·155·

(4.17'

l

f

p

- ~F

p-1

- 2~F

p

+ ~F

~ 0

p+1

is non·negative (Stoll (93), (100)). Take

p

Z[O,n).

E

Let

non·negative form of class

CP: M -

COO

Gp(V)

is given on

be a meromorphlt mdp.

M - Icp

A

by

(4.18'

If

It: Ult -

on

Ult - Icp

U~

a chart, a non·negative function

Hlt (CP)

is defined

by

(4.19,

where

Hlt (cp)2

abbreviate f : M p

E:

is of cIa s

H" (CP) - H (CP). A ltA 1P(V)

on

UIt - If.

If

A E A

Returning to our given meromorphic map

and Its associated maps

Z[O,.I f ) we abbreviate

HpA - HltA (fp )'

COO

lip - H(fp)'

fp: M -

Gp(V)

for

Hplt - Hlt(fp )'

Then

(4.20'

(4.21'

(4.22'

where

Hp).. > 0

on the compLement of the thin analytic subset

-150-

(4_23)

Define

IHp - 0

if

For

p < 0

~

0

p > flf.

or if

p < flf

Also

we obtain on

(4.22) implies

M -

i~.

IHp - U

the identity

(4.24)

(4.25)

Let

bt' d parabolic manifold of dimension

(M,T)

M+ - (x

(4.26)

is not empty. on

M

Let

'"

be a positive form of degree of

...

Ric(r,s, y) - Jr

(4.27)

S

0 < s < r.

for

On

M,

The open set

M Iv(x) > 0) - Ix £ M I u(x)m > 0)

€

The RIcci function

m.

2m

and class

C"

is defined by

.f M[ t

(Ric y)

A

urn -1

J

a non-negative function

v

dt t 2m - 1

of class

C..

is

defmed by

(4.28)

Define

o

~., -

(429)

Then

€

~T I (log v)o is integrable over M
o does not depend on the choice of

~.,

measure zero. .,

(I'

Y

dnd

.

IR+ _ ~o

has

T

For

is defined by

0 <

5

< r

€

i~

with

s

€

.,

~o

the Ricci function Qj

·157·

(4.30)

Ric.,. (r,s) -

I

t

t

log va -

log vo + Ric(r,s, t)

H<s>

H does not depend on the choice of parabolic covering manifold of

I

In particular if

'if.

(tm,

71:: M -

if

(tm

(M, T)

,7I: m),

then

d7l: 1 " .. , " d7l: m ~ 0 p c: Z(O,n].

Take On

M+,

is the

is the divisor of the holomorphic form

IJ

on

a

is a p Oper,

surjective holomorphic map such that .,. - 1171: 112 and if .8 branching divisor of 71:, then Ric.,. (r,s) - N.s(r,s) ~ o. If

71: - (71:1' , ••

1

M.

(See Stoll [l03).)

Let. q>

M-

a non·negative function

Gp(V)

h(CP)

be a meromorphic map

is defined by

(4.31)

The functIon

h(q»2

is of class

COO

Sq>(r) -

(4.32)

on

I

M+,

For

r

E:

~T

define

(log h(CP))o

H provided the integral eXists. Returning to our given meromorphic map f: M 1P(V) and it, associated maps fp: M Gp(V) for p E: Z(O,tfl, we abbreviate h p-h (f) p •

r

E:

~o T

S E: ~o

.,.

(4.33)

Th en

on

M+.

exists.

For

2 m IHp - h ptJ

the integral

Sf (r)

For

0 ~ - p

0 < s < r c:

< Lf

and

~~

with

p

we have the Plucker Difference Formula Nf (r,s) + T f

p

p-1

(r,s) - 2T f (r,s) + T f (r,s) p p+l

- Sf (r) - Sf (s) + Ric.,.(r,s) p

p

-158-

where

T f (r.s) - 0

if

q

<0

q ~

or

flf.

(See Stoll (93) Satz 15.3

q

dnd Stoll (108) Theorem 7.6.)

f: M _

Let Then if

(~>..'

("&>...

t:)

t:)

P(V)

>...'Wl>..I>"EI\

>..1>..£1\

and

g: M -

g.

For each

6>..W v>..U

holomorphic functions

(f,.Id

is called a representation atlas of the pair

is a representation atlas of

a representation atlas of

be meromorphic maps.

F(V·)

and

(>...U) w>..U

f E

and 1\[1]

("&>.. •...0>..)>..£1\

is

there are

without zeros on

suc~

U>..U

that (45). (46). (4.10). (4.11). (4.12). (4.13). (4.14). (4.15) hold and such that

(4.34)

(4.35)

hold on

U>..u.

If

>..

E

1\.

then

w>..>.. - 1

(4.36)

The

on

on

p th

representation section

Fpa ..

of

g

U>...

If

U>..Up.

is abbreviated to

and is a section in

p(p+l)

(4.37)

L [pI - ( 1\ V*) ® (L )p+l g

P+l

(4.38)

Also we abbreviate (4.39)

M

g

(>".u.pI £ 1\[2

oQ

'CO

K

2

F

pg

=

G

p

159-

k 2pV m

Theil tl

For

on

non- negdlive form

IKpq

P

t:

of de~r(H;

Z[O.i g J dlld

2111

is defined on

M

by

(4.40)

for all

A.

A.

t:

Then

~ l.OH 2 q + IKp)

(4.41)

Take t

p

dnd

q

in

I

'2(p(p+ 1) + q(q 1)).

and

b = q

~

1

ZIO,n).

Tdke

p ~ O.

p

t:

Define

J..t -- Min(p+ l,q+ 1)

Z[O,J..t).

Then

and

a ~ p + 1 -

P ~ 0

Define the hoi om orphic vector bundle

(4.42)

Considermg (4.14) and (437) a contraction

(4.43)

is defined by

BlP - Bl P

® ld

and the proper commutation of terms_ F q BlP G p = IB P Fq

Hence we have a holomofphic section L~. g[q,p).

The pair

F q IB P G p ;E O. free of order

If

f,g

0 ~ q ~ if

(q,p,O)

if it is free of order of order

(q,p)

is said to be free of order

The pair (q,p,l).

0 ~ p ~ i g,

and f,g

® Gp of

(q,p,p) then

if

f,g

is sdid to be free of order

The pair

if it is free of order

F q IB O Gp = F q ® Gp

We write

F q 1111 G p - F q III Gp

If

p = q + 1

If

p -

f,g

p + 1

We have that is,

~ p + 1 ~

(q,p)

is said to be strictly free

(q,p,J..t1.

IBI - IB,

is

q + I,

we write we write

-160-

Fq mP+1 Gp - FqLG p . Fq

mP+1

Fq.G p and

Gp - .

and Ff

If

g

p _ q + 1 - p + 1.

which is a section in a line bundle.

Fq IIIP Gp

The sections

are not to be mistaken for the sections

111.0 G • The following lemma clarifies the situation. gp

LEMMA 4.1.

Take

inteR r with

0 E; p , Min(p+ l.q+ 1).

(P.q p)

we write

if and

q

E

Z[O.Jlfl

and

p

Z[O.Jlgl.

Let

P

Then the pair

(f.g)

i! free of order

E:

if the e.air (fq.gp) i! free of order

~

be !ill If 12. then

p-

(4.44)

PROOF.

Take

A

E:

A.

reduced representations

Since

UA is a Cousin II domain.

SA

of

Put

t - i'p(p+l) + q(q+l)).

fq

on

UA and c.t A of

there are Kp

on

such that and

o

UG I U A - UQ p

Hence

F q IIIP Gp ;& 0

have

on

.

UA •

q.e.d.

1

We have

>..

if and only if

Ff

IIIP G q

Kp

;&

O.

If so. we

UA

·161·

LEMMA 4.2. (O,p) all

Take

if

and only

£

Z[O,.ff J•

q

PROOF,

if

~:

Let

representations of (q,p)

if

Hence if

~

If

(f,g)

of order

£

U-

and

f

~

p ' q,

"40:

U -

V·

and

g

By Lemma 1.2 we have

q IB

respectively.

"4O p

(q,p)

(O,p), ., O.

Then

then Hence

for all

q

(f.g) £

is free of order for

(q.p)

be reduced (f,g)

is free of order

II "4O.E. L ~ II > 0

on an open

is free of order

Z[O,.ffl,

be the Fubmi Study Kabler form on on

(4.46)

If

i! free of order

II! O.

Define

p -

(f,g)

then

(f,g)

(q,p). is free

q.e.d

Oq - 0q,(_l)

For

V

is free of order

(O,p).

pq

Then the pair

(f,g)

is free of order

0

Z[O,.fgl.

the pair

"4O p

U and

Let Then

q IB

(f,g)

subset of

p

1,

then

write

1I\~1 V)

and

1'( A V 8

0; - O(_l),p

q+l

on

A V·).

p+l

'\~l V·).

-162-

(4.47)

If

mi'l' 1 (f q Lg I' )*([1 q p- "

q ~ P.

then

IH q + 1

(4.48)

pq

TakE' Take

'" H '" B

p £

q

£

ZIO.ifl

ZIO.ul

nnrl

E

il'IO.f gl

Dcfi II I'

U - Min(pl

Define

(4.49)

.(p)

(4.50)

•

[~]2

pq

_ .< 1 ) pq

pq

(4.51)

p

4>

q

(4.52)

4>

Oq

4>pO

o f"

ffi,o

- u2 0 f

- 0 f

o f

q

go

02

p

"

III gp 0 2

III g 0 2 ~ 0 f q Lg 0 2

•

III gp 0

2

Then Lemma 1.8 implies

(4.53)

(4.54)

(4.55)

[f

p ~ q.

thE'n

4o
)

If

q

~

p.

then

J,q1

Our theory could be carried out without Frenet frames.

HowevE'r

Frenet frames provide a deeper geometric understanding of the theory and lead to new global invariants which describe the relative movement of two Frenet frames.

Some of these invariants will appear in the Second Main

Theorem for moving targets.

Frenet frames Hnd Frenet formulas on

manifolds were introduced in (100).

For completeness sake the facts and

proofs shall be given here. Let B i1! 0

M

be a connected, complex manifold of dimension

be a holomorphic form of bidegree

Ip = Z(F p)

o~

~

p

Let

If - s.

Then

Let 0' s

and Ip

then

s,

(itA'

1:)

If

p

jI!:

~}~E:A

!;

Ip

M;

!;

if

for

Ip+1 p > s,

0

then

~

p

~

s.

-

1:).2.

LEMMA 5.1.

be a representation atlas of

(5.1'

etc.

If

P E: Z[l,s)

and

z

E

LE't

Let

f: M ~

n.

V P(V

Let

F p of

f

If

Ip - M.

no confusion is to be feared, we will omit the index 1:)~.2.

m.

M.

be the zero set of the pth representation section If

10

Then

Abbreviate

on

n + 1 > 1.

be a hermitian vector space of dimension be a meromorphic map.

(m - 1,0)

M - I p'

then

f.

On ol"cao;ion, if

A and write

·164· PROOF.

'D

Take

:x.,.E.::lJz)

A

with

~ P ) (z) -

'D

A

II

E:

l\.

Z E:

'D

A.l!. (z)

The identity

~ 0 implies

E(f p 1(z)) - (~

E:

VI~

A

'D:x..l!..::1. (z)

~ (~

E:

V I~

A

'D

For

z

I

M

E:

-

0)

E(fp(z))

A.l!. (z) - 0) -

q.e.d.

we have the associated flag

s

(5.2)

LEMMA 5.2.

Take

A

E:

II.

For

p = 0,1, '" ,s

vector functions

(5.3)

qf flass

COO

1i!:Q

uniquely determined .Qy

(5.4)

if

O~q
(5.5)

PROOF. 1"0'

Define

.. ,1"p-l

~O -

'D

/11

'D

II

on

U A - 10

are already constructed where

Assume that

p ~ s.

On

we have

Hence

...

'-p

=

~ (pI _ p- 1 ( I ~ ~ ('DPI~·'~·~O J =0

J

J

U:x. -

Ip

·165·

Then

t-p - ip/llipil

0 ~ q < p.

If

is of class

(t- p I t- q ) -

then

A

where

II

t:)

•..

IIt-O " '" " t-pll - lit-Oil ... lIt-pll

nil = II

.L

t:)

n--ltll II"'-pll.

We have

~.,

t:)

t-

"

=

on

o. p-l

1.

V - Ip

with

Also

"t:)

(p)

Hence

t:) .£ .II £ "11"'a

A

...

"'''''p'

Thus a solution is constructed.

~a'

Suppose that ~j -

Assume that Take

Z E

t-j

V - Ip'

which is spanned by true for

Hence

t-p(z).

a-I

'~p

.. ,

j - a,l, ...• p - 1 <

for

~p(z) £

Then

~p(z)

~p(z) -

The orthonormal frame f

for

A.

£

A.

For

E(fp(z))

S

By (5.5)

Z E

- at-p(z)

t-p(z).

A.

t-o'

VA. - Is'

~a - t-a

is already proved.

is perpendicular to

~a(z) - t-a(z), ... '~p-l (z) - t-p-l (z)

Therefore

and

is another solution.

with

Ia I

and the same is 1.

=

E(f p_ 1(z))

Also

q.e.d

is called the Frenet Frame of the system

t-~(z).

'"

.t-~(z)

is

an orthonormal flag base of the associated flag (5.2), but this condition dOe! not define the flag base uniquely.

We shall use the notations of Chapter 4

·166·

!f

LEMMA 5.3.

then

(A.U) E: All).

(5.6)

PROOF.

~ ~(z).

F reach

...

t

functIOns

~(z) Aq

U

Z E:

AU

_ I

p

~~(z) ••..• ~~(z)

both

and

are orthonormal flag bases of the same flag.

cO.

of class

exist on

UAU - Ip

Therefore

such that

N w (5.5) and (4.12) imply

_ [ ~ I ]P+1 IVAU

_ [ ~ ]P+l I vAUI

2 ~ 2+ II 2

[ 16 6 UA I ] uA

UA I ] [ 16 6 uA

~

UR.

II ~ J.lpll 2~2+1l 2

U '" ... '"

~o

~U p

or

If we replace

p

by

p - 1

and divide. we obtain (5.6).

q.e.d.

Our operations on vector spaces extend easily to vector valued forms. Where confUSIOn may occur. we indicate operations between forms by a dot Take of degree

(5.7.

A

1

E:

A

and take inteKers

and class

COO

P.q

is defmed on

in

Z[O.s).

A form

U A - (Ip V Ip.

by

-167-

If

p

£

Z - Z[O,s)

or

q

Z - Z[O,s)

£

a vector valued form of degree

(5_8)

11'

A p

-

define

1 and class

d~

A P

S

COO

is defined by

A

A

- q-O ! Bpq ~ q

A number of relations between these forms will be proved.

identities will be formulated and proved on may hold on larger sets. component of bidegree (0,1).

If

bidegree

Cl

If (1,0)

THEOREM 5.4.

such that

only, although they

w is a form of degree

i.i

and let

is a form of bidegree

(O,q)

UA - Is

Cl '"

Frenet Ident·ties.

Take

1,

let

w

be the

be the component of bidegree

(p,O)

8 - 0,

Our

and if 8

then A

Cl -

£

A.

is a form of

°

or

8 - 0.

Then we have

(5.9)

(5.10)

(5.11)

ifq>p+1

(5_12)

if q < p - 1

(5_13)

ifq:!i:p-1

(5.14)

if q

(5.15)

iflp-ql~2

(5.16)

ff 0

~

~

p + 1

p

~

sand 0

~

q

~

s

"168"

for p - 0, ... ,n if s - n

VA p - 0

(5.17'

(5.18'

A Vp

A

B

- 0

ifO'p's

(5.19'

'lC p A

B

- 0 - itA p

ifO,p's

.A

II

(5.20,

1

4'ir

A 9 pp

- dC log

- a log

··A

9 pp

(5.22'

A.£=l.1I

1110 A.£II

II

9"A pp - a log

(5.21)

10

10

if 0 < p , s

}...£ II

II

10

AR.=! II

II

10

AR.=! 11

ifO
ifO
Ii 10 }..,2.11 1

(5.23'

i A 4j( 9 00 - dC log 111O}..1I

- a log

9"A 00

(5.24'

9 00 - a log II

.)"

)..

B - 9 p ,p+l

(5.25'

9 p ,p+l

A

(5.26,

A 9 p + 1 ,p

AB-

(5.27'

IH p -

(5.28'

d~)..p -

(5.29'

··A

1

111O A"

a~A

p

B - HpA r)..

A

B - - "PArA

A

#) p+ 1 , p

.. ).. #}p+l,p

i m9 A p, p+ 1

-

_

A

s

A

B

I

9).. ~A + 1('A

s

eA ~).. + ~)..

q-O

I

q-p

pq q

pq

q

p

p

-

A

-

"B

10

A11

-169-

;;;. A

(5.30)

ut.

(5.31)

p

-

.. A

P

I

).

(}pqt.q

q-O

a.~p .A

,\'A A AB_(9· A ... A +() p.p+l~p+l ... )AB pp~p

dt.~

A

(5.32)

(5.33)

(9 A

B

B

A

A

P.p-lt. p-

REMARK.

1

+ (}A ... A + (}A A) pp~p p.p+lt.p+l

A

B

A

B

The equations (5.31) - (5.33) are called the FreDet Formulas.

The extensive list of identities is not yet complete (see Theorem 5.5).

The

list will be helpful and provides more information than given on previous occasions (Stoll (100), [l08)). If

PROOF.

p c Z[O,s)

and

q

E:

Z[O,s),

which proves (5.9) and implies (5.10).

then

Definitions (5.7) and (5.8) imply (5.16),

(5.17), (5.28) and (5.29) immediately.

For

Z E

t.O(z), ... ,t.n(z)

UA - Is

we select vectors

is an orthonormal base of

1:.s+1 (z), -.. ,t.n(z) V

if

s < n.

(5.34)

where

(5.35)

Rpq - ('If p I ~q)

are forms of degree

1.

We have

such that

Then

·170·

If 10

0 ' a < b , n, .l!. -

1110

put

t. b - t. a, a

J!.1It.O " '" " t.p'

A

...

A

t.b·

A

t.j+ 1 ,p

If

0' p 's,

then

We obtain

(5.36) 10

~

"t:),p+1I t -

t:)' t - dlO

.l!..l!.

A

B

f>,

B)

p

+ II t:)

n

"'"

II

~ t.0,j-1" 'dt.j J-O

If 0 ' r , p - 1.

then

t. r "

1O.l!..=.!. - O. Hence

+ 1

which implies

9 rk " B - 0

for

O'r'p-l

and

p

Also we have

Rrk " B - 0

for

o,

and

s + 1

Therefore

9 rk " B - 0

if

k > r + 1

r , p - 1

, k' ,

8.

k , n.

which proves (5.11) and implies

-171-

(5.12) and (5.15). 5 -

n.

then

Also we have

1(" p - 0

-.. ..!!.....-.-l "

for

-.. (s+l)

1("r

fA

B - 0

p - 0,1, ... ,n. and t:)

~

for

s < n.

Assume that

,- . .-. (s+l) -

If

O:$; r :$; s - 1.

-.. 1 .!.!..-

o.

Then

Hence

' +. .. + As t :(s) t:) (s+l) - AO t:) + A S t:) )·

Hence

t:) ~ - As t:) sand

(5.36) for

p - s

fl. s "

t:) ~ - fl.s " As t:) x ~ O.

Consequentl~

implies n

I

k-s+l

Consequently

Rsk

fA

(l)s(Rsk

B - 0

fA

for

B)fl.O " ... " fl.s " fl.k - 0

k - s + I, ... ,n.

Hence

1("s

fA

B

We proved (5.18) which implies (5.19). In (5 36) we take the hermitian product with

or

t:) J!..

and obtain

=

o.

·172· If

P

> 0,

we obtain

i 4ir

9 00 " B - d

C

I

og

n101 II

"

B

.

Degree considerations imply

9 00 " B -

Since B

jill

B

we obtain (5.22) and by conjugation (5.21) and by the same

0

method (5.24). If

1 a- log Ti151i "

Also (5.21) and (5.22) imply (5.20).

0 < P < s,

-

II

10

then (5.36) implies

~ II

II

10

.2±.!.II 10 l!."

10

(p+l)

r

·173·

We have

If

p

o<

p ~ s.

II

t:)

=

1 II

s,

t:)

then

If

9 s ,s+1 - 0 p - 0,

1t -

II

t:)

1 II

and

Hs -

o.

Hence (S.2S) holds for

then

t:)

At:)'

t - II

t:)

1 II II

t:)

II~O

At:)'

t

We have

which proves (S.2S) for

9 p ,p+1

reasons, we have have

epq

A

B - 0

conjugation (S.13).

IH p - H2pX m which is (S.27). trivially.

for

A

B

=

Conjugation proves (5.26). hence

0,

q ~ p + 1

9 p ,p+1 -

o.

For degree

By (S.l1), we

which implies (S.14) and by

We have 2

Hp~t

A

t

.

e)..

1m p. p+ 1

A

B

A

h 9 p+ 1. p

A

B

We have proved (S.9) - (S.29) which imply (5.30) - (5.33)

q.e.d.

THEOREM 5.5.

(5.37)

0 ~ p ~ s.

Frenet Curvature Formulas.

Take

)..

E: 1\.

Then

·174-

(5.38'

(5.39'

if

(5.40'

(5.41)

X.

(5.42'

d9 p • p 1 ~ B ~ B

(5.53)

d9~.P+l

(5.44'

d9 p • p _ 1

PROOF.

x.

~

B ~ B - a9 p • p _ 1 ~ B ~ B.

.. x.

We have

s

J!O which proves (5.37).

6 pj ~ 6I jq -

We have

(1Cp l1Cq )

B

~

B

A

B - a

9~.p+l ~

A

B

-

Ip-ql?;2

·175·

which proves (5.38) and implies (5.39).

If at least one term in I q - j I ~ 1.

and

d6 pq " B

I p - q I ~ 2,

Hence

d6 pq " B " B - 0

I p - q I ~ 3.

if

B

A

is not zero we hdve

Ip _ jI ~ ]

which implies

Here

d6 p,p+2 " B " B

since

6 p - 1,p+l" B "B

6 p ,p+l " 6 pi l,p+2 d6 p ,p_2

A

B "

O.

f

is general for

V.

is an orthonormal base of

V·.

>.. * 't- p : U>.. - In If

(5.45)

LEMMA 5.6.

(5.46)

V·

(>..,U) e: A[ll,

u 1I

~p

Take

and since

By conjugation we obtain Now (5.37) implies

Degree consideration give (5.43) and (5.44).

is an orthonormal base of

frame.

(2,0).

>.. e: A an d z

Take

6 p,p+2" B " B - 0

H nce we have proved (4.40).

(5.41) and (5.42) easily.

If - n - s.

and

has bidegree

B-

Assume that

0

Let

E

Hence we assume that

U>.. - In'

>..11 't-o (z), ...

Then

>..11

'~n

(z)

>..

~o(z),

>.. ... '~n(z)

be the dual base, which

The vector functions

are of class

C

00

and are called the dual Frenet

then Lemma 5.3 implies

v>..

[ 6 u >.. 16 u >..1

-~ p e: Z[O,nl

d~p>..*

B.

q.e.d.

and

r

>..

>..*

on

~p

E

n >.. A* ~ 9qp~q 0-0

A.

U>"u - In

Then

on

UA - In

·176·

PROOF.

=-=--

There are forms

A

on

pq

U

>..

- I

such that

n

Then q.e.d.

The Frenet Formulas for the dual frame follow immediately n

(5.47)

(6 >.. d~>"* qp A B)~>"* ~ q p AB-- q=p-l

(5.48)

d~r

(5.49)

d~>"* p A B

n

AB--

A

if

o<

p , n

(9 >.. qo A B)t.>"* q

~

q~O

B

>.. >..* >.. >..* (9)'' >..* p-l.pt.p-l + 9pp~p + 9 p+ 1 • pt. p +l) A BA B. Again, let

f: M -

1P(V)

be a meromorphic map general for

addition, we consider a meromorphic map Ig - s.

Then

atlas of the pair

0 Ei s Ei n. f,g.

Let

g: M -

Abbreviate

IP(V·).

(1->..,10 >..,"4C>")>..£A

be a representation

Adopt the notations of (3.34) to (3.44).

Define

J- - In(O V Is(g)· The J- {'\ UA is the union of the zero set of and the zero set of of

g.

(5.50)

.o..tO

A,!!.

Let

A 111'0'

A

·lII's

1(A p - dll'A p

10 >",!l

be the Frenet frame

Define A "pq - (dll'Ap III'A) q

B.

s ~

q~O

A

A

1! pq ll'q

Then Theorems 5.4 and 5.5 hold in analogy. For each A £ A and p £ Z[O.s) and q c Z[O.n) define the stress coefficient

In

-177-

15_51}

COO

which is a function of class

on

U>.. -

15.52)

if

Since

D:E;p:!!is

is an orthonormal system, we have

-

15.53)

If

8. Obviously we have

s - n,

the square matriX

{

o

if

p

-

if

p

'I:- q

q

IA~q) is unitary and we have also

15.54)

LEMMA 5.7. U>..U - 8"

Take

I>",u)

E

A[l]

and

p

£

:l[D,s]

and

q

E

Z[D,n].

On

we have

15.55)

PROOF.

If

a

E

a:

with

I a I - 1,

then

a - a- 1.

Hence 15.6) for

and (5.45) imply

q.e.d.

10'

·178· We will use this transition formula to introduce important invariants. but we will first derIve somp formulas which calculate the stress coefficients in terms of the representations LEMMA 5.8.

Take

).

E:

p

«>

Z{1.s)

and

1\

and

«

«> ).qL'" «> ),.9-1) • ...0),.>

E:

and

q

...0 .

E:

Z{1.n].

we have

(5.56)

(5.57)

(5.58)

(5.59)

II

«> ),..9. 11

11

«>

),.!L.l.1I II ...0 ),. II

( ( «> ).!l. L...o ).) I «>)..9.:.:.l) (-l)q -n---......--,..,.-....,.......,.,..-~:........ II «> ),.!l. II II ...0 ). II II «> ). s.:..l.1I

<( (5.60)

II

-

>,.

(5.61)

(5.62)

1\00 -

«> ),. • ( ...0

«> ).

II

)..E. L '" ...0 )..E..::.l ) >

11...0),..£ II

II ~ A.~ II

( ( ...0 ),..E. L «> A.) I ~ A.~ ) (-liP rr-~-..,.,,--.,.,,..--.......,,........,.-.."::;:;:=-...,..

II~)..E." II

< «>A·"'OA> II «> All lI"N:lAIl

«> >,.11

II~>".E..:::...!.II

Then

-179-

(5.63)

If

0 ~ p :!i: sand

O:!i: q ~ n,

then

(5.64)

(5.65)

(5.66)

(5.67)

PROOF.

on

17)"

We have

(5.66) 10.9, -

(5.67)

1110.9,11 ~O

~q

-

A

_.-

A

~q

lO n l*lO n _ 11 .;a...........

(5.68)

which proves (5.56).

(5.69)

Also we have

~.l!. ~ lI~pll ItO

~

.l!.

A

l*~

...

A

~

Itp

It p - -.::......--==~­

·180·

Taking the norm proves (5.62). «)

~

II., !L..!."t.O

-

A

•••

A

Taking the hermitian product with t.q_l

proves (5.57).

Also we have

(5.70)

Taking the norm proves (5.63). "40 ~ -

Taking the hermitian product with

II "40 L...l1I ID-O '" ... "ID-p-l

proves (5.58).

'" t.j-l "t.j+l

<., ,(...opL*"<1O p 1» 1110 II 1I"40'p' 1I II "40.2=1."

-

A

...

A

Also we have

t.q1t.O""·" ~q-l

<~O.ID-p> - ApO

I "4O'p'1I II., II II "40 .2=1. II

~ IID-O " .. , " ID-pL~O IID-O " ... " ID-p-l' p

.

- J!O ( l)J AjO IID-O,j_l" ID-j+l. p I ID-O""· - (-l)PA pO

'" ID-p-l)

-181-

<~!...o>

AOO - (lJI'O I !;.~) - - II ~ II 11...0 II

on which proves (5,59) (5.63) and (5 67). If 0 E; p ~ sand define u - Min(p+l,q+ 1). Then (1.20) implies

on

Ux.

0 ~ q ~ n,

Ux.

which proves (5.64) and implies (5.65) and (5.66).

q.e.d.

The identity (5.64) is remarkable. It is surprising that the rectangula norm sum can be E'xpressed as thE' contraction norm of the two associated meromorphic maps. TakE' integer'!

o,

Cj , s, 0 , d j

aj' hj' Cj' d j ~

n

for

a

with

j.w 1, ... ,k.

E; aj E; s, 0 ~ b j E; n.

Abbreviate

(5.71)

a - al + .. , + ak' b ~ b 1 + ... + b k , and assume

C

~

cl + ... + ck' d - d l ~ ... + d k

-182-

(5.72)

If

a + b - c + d

(A,Il) € A(lI.

we have

(5.73)

on

of class

Hence a function

a stress invariant of level

k

is uniquely defined on

M - 6

(5.74)

Obviously we hdve

(5.75)

(576)

We introduce the summation convention, which can be repeated • •• r •••

__ • j • • •

(5.77)

S •••

j •••

-

S

••• r •••

cO. ,

calle~

such that

·183·

Hence we have pq

u 2 0 fg III gp 0 2

(5.78)

(5.79)

o

(5.80)

~pq

fgLg 0 2

~q

0 gpLf 0 2

~

SI I pq

p

'itp

~

jzO

Oq

q

I Oq

L

- S

q

~

=

k=O

pO

p

SI pO

L

jzO

k~o

S

Ok SOk

jO SjO

(5.81)

p q p+l q I Sp+l I p q q

p j p+l k Sp+l j k P J=O k=O

(5.82)

q P P q+l SI q+l I q p p

q k q+l j SJ q+l k q .i =0 k=O

where the non negativity

In

q

q

L

L

p

p

L

L

(5.83)

S p+l

(5.84)

q P P q+l SI q+l I q I V). p

If

g

p

is constant, the function

distribution theory. new properties.

If

g

~

0

~

0

(5.81) and (5.82) follow from the local identities

q p+l q I p qIIV ).. q

p

Jk Jk

~q

q I L AA p+ I • j J-O

P

L

j-IJ

A Ajq+1

A).

Jq

12

are well known in value

is not constant, the functions are new.

The functions

AAP.l. 12

~q

acquire

Now we will establish a

curvature identity for the

Cl pq

which is remarkable for its symmetry and

If an index in

Aab

or

cd

simplicity.

Sa b

is outside its natural range,

cd

then define

Aab - 0

respectively

S 8 b - O.

-184-

THBORBM 5.9. q

E:

Z[O,n).

The Stress Curvature Formula.

Take

p

E:

Z[O.s)

and

Then we have

(5.85) +

(.p.q+l -

+

2S p + 1

2. pq + ·p.q_l)Hq

It<

q+l

p

q

pq

Also we state the formula for the gradient of

and we shall

.pq

prove both theorems together. THBORBM 5.10. q

£

ZIO,n).

The Stress Gradient Formula.

Take

mi m _ 1

deJ)

p

q

p+l

I q

-

S

.... pq "

p+l

q

P

I

+

IKp

q

P q

p

q+l

I p+ 1 q

I p

q

-

B "

5 I P

It<

B

q

p I

q+l

q+l

P

q

IHq

pq

We have dApq -

d(Lt p I t.~) s ~

j-O

"

~ )=0

(5.88)

de ...

pq"

p

+ 2S

(587)

Z(O,s)

E:

Then we have

(5.86)

PROOF.

p

OApq -

s ~

j-p

(dLt p I t.~) + (Lt p I dt.~)

11 pj(Ltj I t.~) -

n

~

j .. 0

(Lt p 1t."')9. + q Jq

n

Ajq ll p j +

.

Ajq ll p j +

l:

j =0

A.R. + (1t'plt.~) PJ qJ

n

l:

J-q

A pj 9 qj + (1i"p I t.~)

ht' P 11:."') q

and

-185-

(5.89)

OA pq

(5.90)

OApq

A ·9 . Pl ql

p

Ajq IIjp -

s ~

Ajqll jp -

j-O

(5.91)

q

-

~

J-P

~

Apj 9 jq

j-O

n

~

A pj 9 jq + It, * I if I q P

j=q

Theorem 5.4 implies

.

.

(5.92)

OA pq " B - IApqlllpp + Bqql + A p + 1,qll p ,p+-l

(5.94)

OA pq " B - -

(5.95)

OA pq " B

-

.

-1

Ap,q"ll &q,q+-l' " B

_.

-

(Apq(lIpp + Bqql + A p _l,qll p ,p_l

t-

Ap,q_l B q _ 1 q) "

Now (574) implies

- (ApqAp+-l,qllp,p+-l -

-

.

+ (A pq A p ,q+1 9 q q+-1 -

Summation yields

.

ApqAp_l,qllp_1,pl "

B

_. ApqAp,q_19q_l,q' " B .

F

-186(5.97)

(5.98)

a~

pq

A

B

(5.99) q

-

J~O "pt1,jl1 p,p+1 '" B

A

C>"pj

A

a"ptU

B

A

'1

t

l:

"pjl1 p ,p+1

q

-

J-O

B

A

+ J~O "p/ptU(-l)

A

B

B

A

m

allp,ptl

A

B

p

k~O

t

Since

(5.100)

im

. (l)m-l i mlrn_l'""2"";r'

mim_lddcoPpq

B

A

8

q

l:

L

x=l

where

"k,q+l e q,Q+l

J~O

A

A

B

A

aA kq

B

A

we obtain

B

=

mi m_ l

16

Aj(x) t

l:

x~9

i

2K

a60P pq

P

l:

k~O

Ak(x)

A

B

A

B

-187-

im Ap-Il,jAp-Il,jl1 p,p-t 1

A

B

A

I1 p-t1,p

imAp+l,jAp,j-t 1 I1 p ,p-l 1

A

B

A

e j + l,j

.

-

i m APJ·A P.J·11 p,p+l

A·(5) J

Aj(6)

=

A

B

B

..

A

A

B

A

ej,j_l

B

A

I1pp

A

B

A.

A

B

A

B

.

-

imApjAp+Ul1p,p+l

I1 p+l,p+l

A

B

.

Ak (l2) - imAkqAk,q+l eq,q+l

-

A

A

.

-

-

11 p+l,p

A

.

imApjApt},j_ll1p,p+l

AP) - imApjAp+Ul1p,p+l

Aj(8)

B

B

A

A

B

A

(l1kk + eq+l,q+l)

A

B

A

.

Ak (l3) - imAkqAk __ l,q+18q,q+l

I1k,k-l

A

B

A

B

-188-

A k (15)

=

-

-

-

irnAkqAk,qt18q,q+1 ,.., B

-

.

A k ( 6) - i rn A kq A k,q+ 1 B q,q-+ 1 '" 13

A

A

8 q +1 ,'1-+ 1

8 qq "

,.., 13

B

Here we have q

L (AJ(l) + A J"(4) ~ AP) + Aj(B)) J=o

q

L

J=u

A J"(21 -

p+l q SilK p+l q P

=

=

(4)p+ 1,'1 -

0

4>pqllKp

·189·

Consequently we have

-

m1 m_)

'" 8

(5.851

+ 25 P

q+l

q

p+l

IKpq

which proves Theorem 5.9. From (5.97) and (5.98) we obtain:

q

p

I

I

-

-

ApjAp+ l,jAkq Ak,q+ 1im II p,p+ 1 j-O k .. O q

p

I

L

j-O k-=O p

- S

A

B

-

p+l q

1

1

p

B '" 9q+1 ,q

AkqAk q+ 1ApjAp+ 1.im 9 q,q+ 1 ",BA II p+l,p '" 8

q

p+l q

A

q

p I

which proves Theorem 5.10.

q

p q+l 1

P q+l P

q.e.d.

q

Hq +

p 28

q p q+l 1 1

p+l q p

q

IKpq

·190·

If

p - 0 dud if

g

is constant, the formulas (5.85) and (5.86) are

well known.

See Cowen· Griffiths (123). Pit· Mann Wong (21) and Stoll (108);

but even if

p

=

0

and if

g

is not constant, the formulas are new and

were first derived without the use of Frenet frames. (5.101)

Spq - ~pq - ~pq-l .

The identity

~p-1,q + ~p-l,q-l

yields the Levi form of

+

p+1,q p,q] [ Sp+1,q - Sp,q IKp -

f

p,q+1 pq] [ Sp,q+1 - Spq IH q p , q+1

+ 2S p + 1 ,

q

[pq P-l,q] Spq - Sp_1,q IK p _ 1

[pq P,q-1] Spq - Sp,q_1 IHq _ 1

p-1,q+l IKpq - 2S

p

,

q

D

p , q p-1, q 2S p+1,q-1 IK p,q-1 + 2S p ,q-l IK p _ ,q_1 . 1

'!6 The Ahlfors Estimates If

g

is constant, the well known Ahlfors estimates for

ot q

iedd

to the defect relation.

We shall extend the Ahlfors estimates to the

functions

g

and

.pq

g: M -

where 1P(V*)

is not constant.

As before,

dre mcromorphic mdps and

f

f: M -

is general for

!P(V) B.

We use the curvature method of Cowen and Griffiths [211 as modified by Pit·Mann Wong [123] and Stoll [108].

We assume the notations and

constructions of the last chapter. LEMMA 6.1.

Take

Z[O,igl

E

and

q

E

Z[O,n].

Then

mi m_ 1 dd c log ot pq " B " B - IHpq - IKp - IH q .

(6.2)

Take reduced representations

PROOF. "'
p

U-

V*

of

g.

10

U -

V

of

f

and

Then we have

dd c log 4Ipq =

ddc log 1110

which proves (6.1).

.!l.

III

"'
~

112 - dd c log 1110

n

~

U2 - dd c log U ~

Now (4.20) and (4.46) imply (6 2).

q.e.d.

£

U'

-194-

+ 28 2 .8 1 S pq

8(8 +

PROOF.

1 +

p

q

II<

8

pq

pq

.8 ).8 pq IK p _ .8(.8 + 1 + • pq pq IH q

We have

LBMMA 6.4.

Take

o,

Then we have

£ £!R.

Abbreviate

~

q.e d.

p.: Z(O.lgl. q

v - (.pq

of

8

+ 8(1 - 8 + .pq}.pqlHpq

.8 ) • .8

Now (6.3) and (6 2) imply (6.5).

PROOF.

,q+l

p+ 1,

£

£)8.

v(1 + vI -1 ddc log v

Z[O.nl • .8

£

!RIO. 1)

and

Then Lemma 6.1 implies

·195·

~

LEMMA 6.5.

-

Take

8 g*(O ) + f*(Oq))

p

p

(6.7)

p

p

E

IO.lgl

PROOF.

tJ: U -

Let

representations.

For

and

,Q+ 1

21S p + 1 ,

Q

V

q.e.d.

g

q

E

ZIO,nl.

Then we have

p+l,q

IIKpq ~ S p + I

and

"to

,Q

U -

p,Q+l

IKp + S p , Q + 1 IH q

V·

be reduced

Then

-

p+l,q p,Q+l Sp+l,q I

q

E:

ZIO,lgl

and

q

E

Hq

ZIO,n].

q.e.d.

define

p+ 1,

(6.8)

:=:pq -

.p+l,q I

Q

,q+l IKpq'

·196·

LEMMA 6.6.

PROOF.

q

and

E

Z[O,n],

then

(5 78) implie!;

p,q+l 4>p,q+1 ~ ~pq + Sp, q+ 1

p+l ,q 4> p+ 1 q ~ 4> pq + S p + 1 , q

(6.10)

Hence

If

have

w('

p+l,q p,q+l p ,q+l =:pq ~ 4> pq OKp + IHq) + Sp+l ,q IKp + Sp,q+l IH q + 2S p + 1 , q IKpc

4>p+1.q ~ U ~ n + 1

Mort-uver.

and

4>p,q+l ~ J.l ~ n + 1

and

q+J q

I ~ 1

imply

q.e.d.

LEMMA 6.7. Definf'

lJ.

Take

p E Z[O.igl. q E Z[O,n)

M\n(p~1.q+ll.

Then we have

D2[.-L2 ..."'pq )~-1_'='pq

P

J.l

PROOF.

Trivially

1 :!;; U ~ n + 1

Lf'mma 6 3 and 6.4 Imply

and

and

~

E

!R(0.11.

-197-

~

U2 - 28 (1 +

~8pq )2(. m1m_1

~ 28 2 ... 8 - 1 S '...p q

-

dd c log(1 +

P ,q+1 IK pq p + 1, q

~8Pc) )

B '" B + IK P + IH q )

"

8 8 8 +. p q )~ p q IH pq

+ 8(1 -

(8(8 + 1) + ~8 ).8 UK + IH ) + (1 + ~8 )2UK + IHq )) pq pq p q pq p

>- U 2 - 28(8 2 .8-1 _ ,..

pq

-pq

+ ((1 +

~.8p q )2

_ 8(8 t

~.8p q )~8p q )(IK p

+ IHqll

q.e.d.

Let

(M,T)

be a parabolic manifold of dimension m - 1

be a holomorphic form of degree said to maiorize exists a constant

the holomorphic form c

~

1

M

on

B,

The infinum of all these constants is called YO

increases.

(6.12)

associated to

(6.13)

T

and

VOIr).

.,

is

there

if

M[r).

Then

VOIr) ~ 1.

lim Volt). r
The increasing function B.

B iI! 0

Define

VIr) -

VIr) ~ VOIr) ~ 1.

r > 0

if for every

on

The function

Let

such that

(6.11)

Then

m.

The exhaustion

r > 0, -

mi m_ 1 B " B

~

Y

is called the m!l.iorant

then

Y(r)U

m-1

on

M[r)

-198-

on

(6_14,

M.

um- 1 may not be positive definite. the existence of the constant c in (6.11' is not assured. If u m- 1 > O. then c exists. Since

urn 1 > 0 implies

however

(It m.1'O)

isometric to

u > 0 and (M.1') is biholomorphically

where

If

"'O(z) - IIz1l2.

proper. surjective and holomorphic and if

l' -

"'0

a

It m

M -

(I:

(I -

is

11_11 2•

then (M.,., is para 0 IC and a holomorphic form B if 0 of bidegree (m - 1.0) exists on M such that .,. majorizes B with VIr) E; 1 + r 2n - 2

(6.15'

If

m - 1.

that is, if

open parabolic Riemann surface. we take

B-1,

then

and such that

f

IS

B.

general for

M

is an

1 B '" 8 - 1 - u m- 1 and .,. majorizes B with V == 1. The idea mof an estimate (6.14' was first introduced by Stoll (93] and later refined

mi

in (100).

(Al) (A2' (A3, (A4, (AS, (A6, (A7'

Now the following general assumptions shall be made: Let M be a connected, complex manifold of dimension m. Let l' be a parabolic exhaustion of M. Let V be a hermitian vector space of dimension n + 1 > 1. Let f: M 1P(V) and g: M 1P(V*) be meromorphic maps. Let B be a holomorphic form of bidegree (m - 1,0) on M. Assume that .,. majorIzes B with majorant V. Assume that f is general for B.

Here (A7, implies that non-degenerate.

B ii! 0

and that

f

is linearly

LHMMA 6.S. Assume that (Al) and (A2) hold. Let N be a complex M be a meromorphic map with indeterminancy manifold. Let f): N If)' define

o<

Let

w

be a function of class C 2

u

by

u(x, - w(x,f)(x»

s < r

E~.,

with

s

£

for all

~.,..

on x

E

M X N. M -

Then we have

If)'

On

M - 1(1

Take

-199-

rI

(6.16'

s M [t

PROOF. '" : r on

r -

such that

Lel

r

I

I

7C

be the closed graph of

(1_'.

rltl -

I

'2

M

N be the projections. -1

1

uo

r.

On

Then

-. _

0

Let ~ -

a parabolic exhaustion

7C- 1(Mlt))

and

r -

"1<5>

7C: '"

A

T

7C- 1 (M
uo

r -

and -

T

M

w 0

and u

7C

Because

is 7C

~

0

g

ven

is a

modification we have

If

r

E ~1'

we obtain

I

uo -

M If

0 < s < r

E ~T

and

domains (Tung (112))

1 -'2

s

E ~.,.

I

wa.

r then

r(r)

and

Therefore we have

f

dC log 't .... dCw "" Clm - 1

r[r -rls)

- '21

I r(r -rls)

dw "" d C log 't '" "m-l !AI

r(s)

are Stokes

-200-

-t

J

w& -

J

ua -

J

t

r<s>

r

-t q

E

J

t

q.e.d.

ua.

M<S>

M LEMMA 6_9_

w&

Assume that (AI) - (A7) hold.

Z[O,nl,

8

E

IR(O,ll

and

r

E

IR

and

Take s

E

p

E

IR

with

t

2m - i

Z[O,igl, 0 < s < r.

Then we have

Jf

dd c log(l + .p8 q ) ,. u m- l

r

(6.17)

M

S

PROOF.

Define

meromorphic map

t

u - Min(p+l,q+l). tJ: M -

Define

> O.

A function

w

of class

for all x E M and (y,z) E N. Define x: E M - II f V Ig)' Then we have q

p

N - Gq(V) X Gp(V·).

A

N is defined such that for all

E

iii (n + 1)C. .

J

(618)

Take

dt

COO

u

x

E

M -

(Ir

V Ig ) q

is defined on

by

p

M X N

u(x) - w(x,tJ(x))

for

bl

·201·

Take

0 <

S

< r

£

~T

with

s

£

~T.

Then

- t M J

J

M<s>

The estimate (6.6) and Fatou's Lemma imply

I f r

~plJq)

dd c log(1 +

'" u m- 1

s M t J

:EO

Since

~ logIn + 2)

~T

:EO

is dense in

£

Z(O.lg1, q

free of order

£

Z(O,nl.

(p,q)

2m - l

c.(n + 1).

1Ii?+

we obtain (6.17) for all

THEOREM 6.10. Ahlfors Estimates. p

d t

t

IJ £

Then

IIi?(O,11

0 < s < r.

Assume that (AI) - (A7) hold. and

0 < s < r.

Assume that

q.e.d.

Take (f,g)

is

-202-

(6.19)

PROOF_

Lemma 6.7, (66) with

82[_1_ ..

pq

J,l2

on

82

MlrJ.

] 8-1

M [t

~

= 0

and (6.13) imply

=pq

Hence we have

Jr J S

I:

[J,l12

_ dt "pq] 8-1 '='pq t 2m-l

J

4(n + 1)2 Y (r)

[I:

J M[t)

+ Tg (r,s) + T f (r,s)] p

~

q

4(n + 1)2Y (r)((n + 1)c, + T g (r,s) + T f (r,s») p

q

~ 4(n + l)3v(r)(c, + Tg (r,s) + T f (r,s)). p

q

q.e.d.

·203·

We will establish an asymptotic Ahlfors Estimate which will involve the characteristic of f and g only. First we will recall some well known results. LEMMA 6.11. For

Let

0 < s < r

A

be a non-negative, increasing functIOn on

"'+ .

define

(6.20)

and

ID

T(r,s) -

A(oo) - 11 m A(t)

~

r s

dt

A(t) t

Then we have

00.

t-+oo

T ( r , s) _

(6.21)

J

log

A(oo)

for

r

r -

00.

The proof is left to the reader. LEMMA 6.12.

Assume that (AI) - (A6) are satisfied.

Take

p

£:

Z(O,l f ).

Then 1 o < ......,2'"'m~--.".2 r

(6.22)

J

IHp ~

Y(r)A f (r) p

~

Y(r)A f (00) p

M(r)

Tf (r,s) -

(6.23)

for

00

p

PROOF. subset.

Take

r > O.

Then

Hp > 0

on

r -

M(r)

00 .

except on a thin analytic

Therefore we have

which implies (6.22) and by Lemma 6.11 we also obtain (6.22). For

(6.25)

0 < s < r

we have

Jr J B

M (t

m

IHp t 2d t_l

I

~

Y(rlTfp (r,s)

q.e.d.

-204-

0 ~ s

Take lR(s,+oo).

E:

IR+

Let

III ~ '"

We write

measure in

IR.

if

~

dnd

there exists a subset

is self evident.

1 ~ min f r .....

(6.26)

be redl valued functions on

.p

lII(x) ~ ",(x)

such that

The calculus of

III

for all

x E lR(s,+oo) - E.

In particular

1II(r) ,

1

l.

of finite

E

cP ~ '"

implies

m sup .p(r). r ... OO

Nevanlinna [671 established the following well known result LEMMA 6.13.

Take

E

> 0

whlch is integrdble over F : lR[s,+oo) -

IR+

and

lR[s,rl

x

F(x) -

~ s.

Let

III j?; 0

r > s.

be a function

Define

by

(6.27)

for

s > O.

for each

J:

lII(t)dt

III ~ Fl+€:.

Then

PROPOSITION 6.14.

Assume that (Al) - (A6) are satisfied.

p E Z[O,.e f ) and

> O.

almost all

r

E

E.f. T

h!a

is integerable over

h!a

~

where

Sf (r) p

Sf (d p

for

r E(2m-1)(Y(r)T r (r,s))(l+E)2 p

M

(6.29)

M

and

J

(6.28)

Then

Take

~ (C,/2)(l + E)2UOg Tf (r,s) + log Vir)) + (C.I2)E log r p

is defined by (4.31) and (4.32).

·205·

PROOF.

Fubini's Theorem implies

J

IHp

M[rJ

~T.

hpu m

h p2 r7

J

m

=

M[rJ

which shows that

t £

J

-

M[rJ

is integrable over

M

for almost all

(6.25) and Lemma 6.13 imply

J

J

2

h p r7 ~ (1 (2m))r 1 - 2m [

M

f

IHp

r+E:

M[rJ ~

IHp

r2m l(Y(r)T (r,s)1+E: f p

M[ r ) Hence

J

2 h p r7 ~ (1/(2m))r £(2m-1)(Y(rlT f (r,s))O + £)

P

M which implies (6.28).

Sf (r) p

2

Also we have

f l o g hpo

=

(c.I2)log

[i:

M

I

h~O)]

M

~ (C./2)(1 + £)200g Tf (r,s) + log Y(r)) + (2m -

1)(C./2)£

P

+ (c.I2)lOg(C.I2)

~

(C.12)(1 +

£)2UOg T f (r,s) + log Y(r) + mc.£ log r. p

Replacing

c.

by

C./(2m)

implies (6.39).

q.e.d.

log

·206·

The following proposition was proved in Stoll [108\ Proposition 10.9. For completeness sdke, the proof shall be repeated here. PROPOSITION 6.15. P E

Z[O,ifl

and

> O.

E

(6.30)

for

Assume that

(AI) - (A6) hold.

Take

Abbreviate

Q£(r,s) - log VIr) + Ric 7 (r,s) + £c, log r

0 < s < r.

Th n

T f (r,s)

~ 3 P T fIr,s) + t(3 P

-

l)Q£ (r,s).

p

PROOF.

o~

The estimate (6.31) is trivial for

p < i

p + 1.

and that (6.31) holds for

p

p - O.

p.

0 < £ <

W.l o.g. we can assume that

Assume that

Then (6.31) shall be proved for

.f2 -

1.

The Pli.icker

Difference Formula (4.33) and (6.29) imply Tf

p+l

(r,s)

~

2T f (r,s) + Sf (r) - Sf (s) + Ric 7 (r,s) p P p

~ 2T f (r,s) +

(C,12)(1

p

~

+ £)2 log T f (r,s) +

C,

log V(r)

p

3T f (r,s) + QE(r,s) p

THEOREM 6.16. Ahlfors Estimates. 8

E IRW,l),

Assume that

E

s >

o.

> 0

and

(f,g)

is free of order

Assume that (A1) - (A7) hold.

Take

P

(p,q).

E ZIO,igl

and

Then we have

Take

q £ ZIO,i f ].

-G07-

(6_32)

The proof is easily obtained from Theorem 6.10 and Proposition 6 15. We need a refined version. THEOREM 6.17. p (; Z[O,.igl

and

(; > 0

Take

Ahlfors Estimates. q (; Z[0'-"f1.

and

s > O.

Assume that (Al) - (A7) hold

Assume that

(f.g)

IJ: IR+- -

Define

Take

is free of order

IR(O,1)

(p,q;

by

(6.33) l + Tf

1

(r,s) + T q

{

8(r) ,.

gp

~

1 + mf q

Define M - M+-

M+

~pq

q

Lf r

(s)

if

M+

On O.

Then

define

Epq

~

bt

F(r)

=

J M

exists.

s

O~r<s.

0

on

M.

~pqV

We have the estimate

m .

For almost all

the integral

(6.34)

~

p

p

by (4.26).

put

(r,s) + mf ~ (s)

_1_ 4> ] lJ(r)-l ~ [ ~2 pq

pq

0

On r > 0

-208-

log+ F(r)

(6_35)

~ 2(1 + E)(lOg Tf(r,s) + log+ Tg(r,s) + log Y(r) + log+ Ric.,(r,s) +

PROOF.

C,

log r.

11 ~ 8

Define

-IT : M

0

_

For

(0,1).

r > 0

the following

integral exists (Theorem 6.10)

FO(r) -

J

] 8(r)-1 [...L. J!2 pq

Spq

~

I H[ r

F 1(r) -

2m

Hence the integrdl

J! pq ] 11-1 ~ pq" m-1 d., ,... [l.2.

J:

F(t)

D

-

F1 (r )

I

rI S

- 2m

1 • ] 11-1_ [ J!2 pq '='pq

H[ r )

H[rJ

- m

J

[;r

.pq] 8(t)-1

~pqD)t2m-1dt

M

F(t)t 2m - 1 dt.

exists for almost all

t > O.

Lemma 6.13 implies

Define

for

r > s .

·209. Then

where

F 2(r)'

J r

8

FO(t)

Take a constant

dt

t 2

Co >

1 ~

4(n + .8(r)

1 + Co +

1)3 2 Y(r)(Tf (r,s) + T q

(r,s) + Co) •

lilt IB (s), Then q gp

F 2 (r) ~ 4(n + 1)3Y(r) T f (r,s) + Tg (r,s) + cO)3 q

gp

if

r > s ,

p

consequently we obta'o log+ F(r) ~ (1 + E)2 log + F 2(r) + c.f2m - l)1og+ r

~ (1 + E)2 log Y(r) + 3(1 + E)2 log +(T f (r,s) + T g (r,s)) q

p

+ E(2m - 1)log + r + log(4(o + 1)3) + 3(1 + E)2(lOg

Co

+ log 2)

~ (1 + E)2 log VCr) + 2mE log + r + 3(1 + E)2 log+(3 q T fer ,s) + 3 PT g(r ,s) + +c3 P + 3q - 2)Q E(r ,s))

~ (1 + E)2 log VCr) + 2mE log+ r + 3(1 + E)2(10g+ Tf(r,s) + log+ Tg(r,s) + 10g+ QE(r,s))

Here we have

-210-

log+ QE(r,s) - 10g+lC, log VIr) + RicT(r,s) + EC, log r) -'1+1 . '1t og og Vir) + log+RlcT(r,s) + 1+1 og og r + log+ c, + log+ EC, + log 3 E

~.

3(1 + £)2

log VIr) + log

+-

I)

Tg r,s +

mE

log

r

, 3(1 + E)"

We obtain log + F(r) ~ «(1 + E)2 + E) log VIr) + 4mE log r + 3(1 + E)2(10g Tflr,s) + log+ Tglr,s) + log+ RicT(r,s).

Without loss of generality we can assume that

~ 1 + 4E.

(1 + E)2 + E

Hence replacing

0 < E < 1. by

E

4Em

we obtain

log + F(r) ~ 3U + E)(log T r(r ,s) + log + T g(r ,s) + log VIr) + log + RicT(r,s)) + E log r .

On

M+

we have

IKpq - kphqu m.

on

M+.

(6.37)

Defme

II

and

q.e.d.

IHq - h q2 u m

Therefore we obtain

Ppq - 1

on

O~P

M

_

pq

~

2,9+ 1

t pq

h2 9

Then

and

·211· p+l. q • q + 1 ~ 0,

If

S p

attraction between Ppq ~

1,

then

f

and

P pq '

I

If

S

g.

which in some SE'n'le measures p+l. q p •q+1 h

q , - .p+l,q kp'

which in some sense measures distraction between

In general,

Ppq

measures the twisting of

f

and

g.

f

then

and

g.

Asympto cally we

define the obstruction terms 1

(6.38)

J

P pq(r) - "2

M

(6.39)

whenever the integrals ex t. and

g,

and

Rpq

Here

Ppq

the relative growth.

measures the twist between Both terms will appear in the

Second Main Theorem as unwelcome guests. Ppq == 0 == Rpq'

and

f

If

kp == 0,

then

Ppq == 1

Th s observation will lead to an important

applicallon (Chapter 9).

PROPOSITION 6.18.

Assume that (AI) - (A7) are satisfied.

p

E

Z[O,..egl

E

Then

and

P pq(r)

q

and

Z[O,n).

Rpq(r)

Assume that

(f,g)

exist for almost all

r > O.

We have the

estimates

(6.40)

(6,41)

- C. log(n + 1) ~ mf

~

q+l

If

s > 0

and

c > 0,

Sg (r) + Rpq(r) + P pq(r) p

mf Sg (r) + 2c. log 2. q

p

then we have

Take

is free of order

(p,q).

·212· (6.42)

Rpq(r) + Sf (r) q

~ (C,12)(1 + E:)2(log+ T f (r.s) + log+ Tg (r,s) + log Vir)) q p + (c,I2)E: log r.

PROOF.

on

M+.

The inequality (6.9) implies

Therefore

(6.43)

on

M+.

(6.44)

Also we have

o~

k

log [1 +

For almost all

r E: .f..,.,

integrable over

M.

consequently,

log

~p,q+1

+

the forms

hq + log

+

(log hq)o

The same is true for

log(1 + (k/hq))o

eXlsts for almost all

log

~] ~

and (log kq)o (lOg+ hq)o

is integrable over

r E: .f..,..

- log

kp - log hq + log 2.

M

and and

We have

[1 + ~]2 - login + 1) ~ log Ppq

I g Ppq ~ log ~p,q+l - log ~pq .

Define

JJ. p,q) - Min(p+l,q+l).

t

J

M

log ~pq

Take

0

-

-

r

E:

.f..,..

Hence

mC EBg Ir) + Co log J1(P.q) q

p

are (lOg+ kq)o Rpq(r)

·213·

exists.

Therefore

(log ppq)o

exists for almost all

- mf

r

q+l

t:

is integrable over

fT.

M

and

Ppq (r)

We have the estimat es

u( P, 9 ) R (1'( P ( !Hg (r I + c. log n + 1 pq r '" pq rl p

Ppq(rl ~ - me

m (r) + me ma (r) + c.log U(f'9+~) q+ 1 gp q "p up. q

which implies (6.40) and the left hand side of (6.41).

Also we have

k2

log Ppq ~ log .p,q+1

log .pq - log[l +;r] q

k

log .pq - 2 log [1 +

~]

+ log 2

and

P pq(r) ~ meqm p(rl

meq+1m g p(r) - ~q(rl + (C./2)log 2

+ C. I og u(f,g+l) U p,q)

where

U(p,q+ 1) u(p,q)

(6.41)

q.e.d.

~

2.

Hence we obtain the right side estimate of

PROPOSITION 6.19.

Assume that (All - (A7) are satisfied.

p

t:

Z[O,AgJ

t:

Take

and

s > 0

and

q E:

Z[O,n). E ~(O,ll.

Assume that

(f,g)

Take

is free of order

(p,q).

Then we have

(6.45)

~ Ppq(r) + 2c.(1 + El(log Tf(r,sl + log+ Tglr,sl + log Vir) + log+ Ric.,Cr,sll +

E:

log r

-214PROOF.

=

Define

IJ

I

1 -2

by (6.33) and

I (( og

jJ. (

F(r)

1 • p •q ) 2 pq

by (6.34).

Then we have

jlJ(r)-l ~ ja + P (r) pq

pq

M

+ C. log

jJ. (

1 p • q+ 1)

+ f:

£:

log r + P

f

£:

log r + P pq(r) + 1

2

+

pq

(r) + ,8(r)(Tf (r,s) + T ( ) + - ()I q gp r,s mfqlBgp S

·215·

+ P pq(rl + C.E log r.

If we replace

c.

by

(E/(l +

c.)l < E,

we obtain (6.441.

qed

We will extend Proposition 6.19 to the Second Main Theorem by using a family of meromorphlc dssociated maps

gp

in general position.

\7

General position

The properties of general position have been discussed in many places, for in tance, Ahlfors (1), Weyl [119), Stoll [93) and Wu (126).

We hope to

avoid the index acrob t'cs by which the reader is often confused. the theory will rema'n difficult enough.

However

Here the product to sum estimates

contain constants which we have to know explicitly, since our target spaces move.

Thus we are confronted with a difficult task, which however can be

solved satisfactorily. Let A ~

121

V be a complex vector space of dimension

be a subset of V.

n + 1 > 1.

The span for linear hull) of

intersection of all linear subspaces containing span(A V,.

A is the

A and is denoted by

LBMMA 7.1.

The following conditions are equivalent:

a'

We have of V.

II(A

b,

We have liB - dim span(B,V, for every subset B ~ 121 of with liB' dim V. Every subset B ~ fZJ of A Wlth liB ~ dim V is linear independent. If flJ~B!;A With HB ~ dim V, then

c,

3'

n L, , dim L for every proper linear subspace L

dim

(7.1)

where

E[a) - ker a

a'

Let

n

aEB

A

E[a] - n + 1 - HB

is a linear subspace of dimension

n

in

V•.

PROOF. ~ b,: Take flJ ~ B !; A with liB ~ dim V. Define L - span(B, V,. If dim L < n + 1. then L is proper and

-217-

dim L , liB , II(A " L) , dim L Hence

If

dim span(B,V) - liB.

dim L - n + I,

then

dim L , liB , dim V - dim L and

dim span(B,V) - liB

b) ~ c). Trivially, independent. b) ~ d).

Define

follows again.

liB - dim span(B,V)

EIB) -

"

EltA).

if and only if

Let

L

B

is linearly

be a linear subspace of

V·,

tAEB

then

is a linear subspace of LD D

-

L.

V with

dim L D + dim L - dim V

Moreover EIB] -

(~ E

V· I

-

a

v

tA E

V 18'

Hence, if

0

~

B !: A

dim EIB) d) ~ II). A "L

and

Let

L

and

liB' dim V,

then

E

B)

span(B, V))

liB - dim span(B,V)

- dim V - dim span(B,V) - n + 1 - liB. be a proper, linear subspace of

exists such that

span(BO'V) - span(A "L)

V.

A subset

and

IIBO - dim span(BO'V) - dim span(A " Ll , dim L , n

Assume that B - BO V 1&1.

and

BO ~ A "L. Then

Take

liB, n + 1

&

E

and

A " L - BO· B!: A.

Define

Therefore

BO

of

-218-

n

dim

E[iA) - n + 1 - liB.

iAEB

Since

"

E[iA)

spanD(B,V)

we obtain that

liB.

dim span(B,V)

Also

iAE: B

we have span(BO'V) ~ span(B,V)

span(A "L)

!;;

span(A

n

L)

or 1

-t

lIBO - IfB - dim span(B,V)

which is a contradictIOn.

Hence

=

BO - A "L.

A;t"

of

V

L)

=

IIBO

Hence

IIA " L - dim span(A " L) , dim L A subset

n

dim span(A

q.e.d,

is said to be in general position if and ani:

if one and therefore all the conditions aI, bl, c) and dl of Lemma 7.1 are Iioatisfied. Assume that a hermitian metric is given on If

k ' n + 1 write

A - (01' ... ,Ok)'

V,

Abbreviate

k - IIA.

Then

(7_2,

does not depend on the enumeration. ~[n,AI

If

n + 1 < k - IIA , +

be the set of all injective maps from

Z[O.nj

(7 3)

riAl - inf {!l1'(O) " - -. " 1'(n)!I 11.,.(0)11 .,. !I.,.(n)!!

(7.4'

riA) - inf(r(1'(Z[O.n)) 11'

Then

0 ~ riA)

,1.

The number

into

00,

A.

let

Define

E: ~[n.A)).

rIA)

is called the laure of

A.

·219·

LEMMA 7.2.

A ~

Let

be a subset of

12)

If

in general position.

V.

If

rIA) > 0

then

A is

A is finite and in general position, then

rIA) > O. PROOF.

Define

if and only if

k > n + 1

lfB ~ n + 1.

Then

bijective map

T:

rCA) ~ I'(B).

T

E ~[n,AI

Lemma 7.1 c).

Lemma 7.1 all

rIA) > O.

Take

B

Hence

is linearly indendent.

B

exists.

I position

w ch is thE

A

~

B - T(Z[O,n)) the set

0 < rIB) - rIA);

Hence

B ~ A with

lfB - n + 1.

A

Therefore

B is

is in general position. n + 1 < k <

eneral position and

tfB - n + 1,

~

By (7.4) we have

By Lemma 7.1 c),

A is i

12)

B!; Band

Z 0 nl _

exists such that Since

and

B!; A exists with

linearly independent. Assume that

A is in gene

rIA) > O.

Assume that

(See (7.4).)

then

A is linearly independent (see

case if and only if

o<

k ~ n + 1,

If

k - lfA.

B

A

and

Then

00.

rIA) - rIB).

is linearly independent by

q.e.d. ~

LEMMA 7.3. that

If

12) ~

A C V

A!; A C V with

with

itA

itA - n + 1

~

n + 1,

then

and such that

....

A - (uO' ... ,uk)

and

.up, '" ,.un +1

p ~ k

+ 1. Then (Uj l.uq ) - 0 if 0

Define

.... - CUo. ... ,uk,uk+ ~ ") A l' ... ,un·

exists such

rIA) - rIA) .

If IfA - n + 1, take A-A. If k + 1 - IfA be the linear subspace of V perpendlcular to span(A.V).

PROOF.

A

~

n,

Th en

and

W

Then

is an orthonormal base of ~ j ~ k

let

p ~ q ~ n

W with

+ 1.

·220·

A subset

A

~ 0

exists a subset

A

~ 0

is said to be in ,eneral position if there

1P(V)

of

A

in general position such that

of V.

A

1P : A - A

bijective.

IS

A

A

~

If

such that

If

position. A

!;

~

A

V.

V*

A

and only if

11': A _

A

II

A.

....

is in

then

A is bijective, then neral position.

A is in general rIA) > O.

A

Let

span(A,I'(V»

F(V)

containing

spdn(A,IP(V»

A

If

~

V.

If

is in general position if

intersection of all projective planes in is a projective plane.

is

rCA) - rCA)

rIA) > 0,

If

and if

"(V)

subset of

is fiOlte and in general position, then A

dnd

is

0

is bUective,

I' : A - A

independent of the choice of

A

A

be the A.

Then

such that

1P(A) - A,

then

span(A,II'(V)) - 1P(span(A, V)).

(7.5)

Hence Lemma 7.1 translates into: ~

LEMMA 7.4.

The following conditions are equivalent if

a)

We have

II(A 1"1 L) :IS 1 + dim L for every proper plane

b)

If

e)

121 ~ B !; A

Every subset

B

with

IIB:IS dim V,

~ 121

of

with

IIB:IS dim V,

A with

then

A

F(V); L in

1P V).

liB - 1 + dim span(B,F(V).

IIB:IS dim V is linearly

independent. d)

If

121 ~ B !; A

dim

(7 4'

e'

1"1

then

Ela] - n -

liB.

Gp(V*,.

Then

8E:B

A

IS

Take

in general position. p

E:

Z[O,n)

and

121 ~

Ii

!;

Ii!;

11'( A V·) p+l

and as such it is defmed when Ii is in general position. Then A V. Here 7.1 and 7.4 remam true if V is replaced by p+l

Lemmata

·221·

Ela] - Ela, A V]

is a hyperplane in

p+l

for

Ela,V)

REMARK.

and is not to be mistaken

p+l

which is a plane of dimension

n - p - 1

in

p V).

As the following example shows, the statement d) of Wu (126)

page 181 is wrong: 1\.0' 1\.1' 1\.2' 1\.3 for

1P( A V)

V - a;4, n - 3, h - 1, i - 3.

Take

a;4.

be the standard base of

j - 0, 1, 2.

h (aO,a1,a2) - (A)

Then

Define

Let

aj - 1P('"'J " '"'3)

is in general position.

We have

while statement d) of Wu 126) would imply that the intersection is empty h - i + 1 - - 1 < 0

since

and

i - 3

h + 2.

We assume that a hermitIan metric on q E: Z[O n)

If

a

E:

and

Gp(V*),

b

Gq(V).

E:

then

Take

E(a)!:; V*

m and

V is given.

Z

E:

with

Take

O:$; m :$; Min(p,q).

Gm(E(a))!; Gm(V·).

Hence we

can define

IJIb,m) - {a

(7.7) If

r

~

0,

E:

Q} I [] b L x [] - 0

we define

Q}(b,m,r) - {a

(7.8)

E:

• +1 Q} I [] b Ea m a [] :$; rl.

We shall estimate the number of elements of sufficiently small LEMMA 7.5.

r

Take

q

£

Z[O,n)

Min(p+l,q+l).

Take

b

G (V).

[] b mm+1 a [] - 0

Then

q

for all

x

E:

Gm(E(a));

m

Ilnd

J.t -

£

l1(b.m,r)

for

and we shall give an estimate for the bound of

i.e.

£

Z[O,J.t).

P

£

Z[O,n).

Take

a

£

Define Gp(V*)

if and only if

and

[] bLx [] - 0

r.

·222· !J(b.m.O) - lI(b.m).

(7.9)

PROOF.

Take

such that

& E Gq(V)

b - F(6)

and

and

a E GpIV·)

a - F(a).

be orthonormdl systems such that 6

~

aO' ...•a p

and


and

p ' q.

Th n

J.I -

E

0 b mm+l a 0 = 0

~[m.pJ.

frl
P +

1.

By Lemma 1.7 we have

if lind only if

((JI),.}),.E~[m.p)

Smce for all

E ~[m.p)

),.

6l
is a base of if and only if

6l'f:, - 0

which is the case if and only if

x

E

Gm(E(a)).

This proves the lemma in the first case.

J.I -

°m+1 [) b III a 0 - 0

E ~[m.q].

q + 1.

for all

By Lemma 1.5 we have

[q+l]2 116 IIIm+ 1 all 2 I lI"E~ [ m • q I

q+l]2 0 b mm+l a 0 2 m+ 1

Hence >.,

Then

for all

°

Gm(E(a))

p > q.

we see that

[) blx [) - 0

E

CASB 2:

for all

m~l E(a).

'f:,

[

6 0 , .•.• 6 q

P+l]2 [) b mm+1 a []2 _ [P+l]2 116 mm+1 all 2 I 116la},,11 2 m+l m+l }"E~[m.pl

Hence ),.

11611 - 1 - lIall

6 0 ",,, A6 q .

CASB 1.

[

Let

with

if and only if

al6>., - 0

for all

We have

(7.10)

Hence

al6),. - 0

Hence

[) b mm+l a 0 2 - 0

'"

E

~[m.p]

and all

if and only if

>..

E

<.u",.6>.,> - 0

if and only if

~(m.ql.

We have

for all

<.u",.6>.? - 0

'"

E

~[m.p).

for all

·223·

(7.11)

).

~[m,q].

E

IH~y,

o

- 0 for all

.

bLx 0 - 0

Take

and

G (E(a))

Z[O,n]

in

Z[O,m] -

Z[O,p]

2

- 0

if and only

(see Case 1).

m

and

1] _~~m J

m

[

0

m

E

I

q

+

+

j

E ~Ip.n] ITo

] [ + 1

l

If

q.e.d.

Z[O,p] "Z[O,q).

be the inclusion map.

tr - IT

Define

n p

-

q m _

]

j

.

Define E

it-{m

QUo

SIb p rn]

LEMMA 7,6.

E

Slb,p,m]

A V· I
p+l

A

}t,f:.> - 0

is a linear subspace of

A V·

Moreover, if

&0' ." '&n

JPI&} - b,

is a base of

V

with

p+l

dim Slb,p,rn] - D(p,q,m} < [: :

17.15}

17.16}

E

a 0

which is the case if and only

D(p,q,m) _ [n+ p+ 1

- If:.

and

x

q

(7.13)

(7.14)

E ~[m,p)

y,

for all

p

(7.12)

Let

. +1

0 b 111 m

Consequently

such that

:]

& - &0 " ... " &~

then S[b,p,m] - (t.

E

A V* I - 0 p+l

'"

.,

E

tr).

-224-

PROOF-

Take a base

such that

6 0 , _.. ,6 n

1P(6) - b_

V with

6 - 6 0 " --- " 6 q

Define

S[b,p,rn) - (~E

(7_17)

of

v .,

A V* I <6.,,~> - 0

tl").

E

p+l

Ob ous IY

~ E Slb,p,rn).

Take Hence Thus

6 11

E

T ke

Grn(E(b)

<6 .,;~> - <6 v

Therefore

Take

~

.,

dnd

Then

tf'.

E

11 -

11- - 6.,(rn+1) '" ...

11-;~> - 0

A

A1 V·. p+

"'(b ,p,rn ) are linear subspaces of S

SIb .p,rn ) an d

., A

~ E 'I-[rn,q).

0

6.,(p)

by the definition of

E

Gp _ m_ 1(V).

Slb,p,rn).

Slb,p mI.

E

~ E: S[b,p,rn).

Take

It E: Grn(E(b))

and

11- E: Gp _ rn _ 1(V).

Then

11- -

I

)"E:'I-( p-m-l ,n)

z)"6),, .

We obtain (7.18)

CLAIM 1:

If

11 E

'I-[m,q)

and

)"

E

'I-[p - m - l,n),

then

(7_19)

PROOF OF CLAIM 1. can assume that

6)1

If A

6 11

6)"

;f:

6)" - 0,

A

O.

then (7.19) is trivial.

An injective map

y: Z[O,p) _

Hence we Z[O,n)

is defined by Y(x) _ {V ( X ) },,(x

-

m

-

1)

if

x

E

Z(O.m)

x

E

Z(m + l,p).

·225·

There is a bijective map increasing.

Naturally,

'" - 11'-1.

Then

.,

0

71::

., '"

.,

is increasing. l:

Z[O,m) -

°, .,( Hence

.,

l

0

by (7.17).

viz) -

E:

.,

viz) ,

l (x))

=

In particular, :l[O,p)

v(y) -

.,(x) ,

E ~[m,q).

71:

",(y).

Thus

Take any

x

Ol[O,m)

E:

Then vim) ,

., E: tf'.

q.

Therefore

Claim 1 is proved.


S[b,p,m).

We have shown that

!-,~>

- 0.

Therefore

Slb,p,m] - Slb,p,m)

which proves

(7.16). Since

(&.,lTE:~[P n]

is a base of

the family

"V, p+l

is linearly independent and spans a linear subspace Then

S[b,p,m] - L 0

dim Slb,p,m) -

[: :

and

dim Slb,p,m) - dim

:] - litf'.

For each

s E: Z[m,p]

T s - (1' c: iI-(p,n] I {

Then

L

(&.,l1'E:tf'

of dimension

"V - dim L. p+l

litf'.

Hence

In order to prove (7.15) it remains to

calculate the number of elements of

tf'.

define

1'(X) , .,(x)

>

tf' - T m V T m+l V ... V T p

is

v(y) - .,(",(y)

m' "'(m).

By (7.13) we have

0

Define

then

.,(",(m)) - vim) -

T(m) ,

., _ V

~[p,n).

E:

",(z) ,

be the inclusion.

Claim 1 and (7.17) how that

~

such that

is injective and increasing we have

'" I Z[O,m]

and let

Ol[O,p) Hence

If 0 , z , y 'm,

v.

-

.,(",(z)) -

Since

Ol[O,p) -

is injective.

q

if

XEZ[O.S)

q

if

x

E Z[s

is a disjoint union.

+

}.

l,p)

Therefore

-226-

Take

s

~[m pI

E:

and define a map

ps : "tls,q) X "tIp -

If

v

E:

"t ,q]

{

VeX) /..I(x

., (; T

Then

U E: "tIp -

s -

l,n - q - 1) -

l,n - q - 11

Ts

define

by

ps(V,UI - . ,

.,(xl

lind

s -

s

x (; ZID,s)

If

s-I)+q+1

ObviouslY

X

is bijective.

Ill's - (1I"t[c;,q)HII"t[p - s - 1, n - q -

(;

Zls +

I,p)

Therefore

1)) -

[: :

:]C -:]

We obtam

a

p

< IIIf

L S=m

dim Slb.p,rn]

- IIIf

.J < J

[n I]. p

+ +

1

q.e.d,

Define

(7.20)

Slb.p,rn) - P(Slb p,mll

-227-

LEMMA 7_7. Take p and q in Take b E Gq(V). Then we have

(7.21)

S[b,p,m[ " Gp(V*) - (a

PROOF. if

1P(

for all


~

II' 0

=

E


Gm(E(b)).

jJ>.

A

jJ,

A

THEOREM 7.8.

o,

a

E

Take

m 'Min(p,q).

that

IJ

SIb p,m]; p

Z[O,p] " Z[O,q].

01.


Slb,p,m].

if and only i1 A V, p-m

for all

G(E(b))

E

such

generates

for all

0

=

Gp (V*)

Gm(E(b))

E

"


if and only

and

Hence

jJ

Gp _ m _I(V)

E

II'

E

" _ _ (V), G p m 1

jJ

E

Gp _ m 1 (V),

0 b mm+ 1 a 0 = (

q.e.d.

b

E

is in general position.

(7.22)

PROOF.

y

< - 0


'l[O,n)

E

Take

E

Gp_ m_ 1 (V)

for all II'

which is the case If a d only if if and only if

for all

Hence

~ 0

0","

Take

Since

if and only if

«ALII',jJ> - «A;II'

if and only if

Gm(E(b)).

E

m

0 b mm+l a 0 - 0

0 aLy 0 - 0

Then

for all

y

Take

Gp(V*) lob IiIm+I a 0

E

By Lemma 7.5 and symmetry

DaLy 0 - 0

that

'l[O,n].

and

Gq(V)

q

and

E

Z[O,n].

tlJ"'"

II

Take

!: Gp(V*).

m

E

Z

with

Assume

Then #Q}(b,ml , D(p,q mI.

Since

IJ

projective plane in

is in general position and since 1P( A V),

Slb,p,m]

is a proper

we have

p+l

IIIJ(b,m) ~ 1 + dim Slb,p,m) - dim Slb,p,m] - D(p,q.m) Basically, our proof is the same as Wu's (126).

q.e.d.

However, Wu does not use

the product 113 m+l and the norm 0 b 113Om+! a D. These new concepts simphfy the argument. The number D(p,q,m) shall be calculated in certain cases.

-228-

!&MMA 7.9. Define

Take

D(p,q,m)

p

and

in

q

by (7.12).

Z{O,n1.

Abbreviate

Take

m

E:

ZIO,p] (\ Z[O,q].

D(p,q) - D(p,q,O).

Then we have

< 2n+1

(7.23) 0 ~ D(p,q,m) -

(7.24)

D(p,q) _

p

If

0

~ m ~ q ~

p,

+

1

then we have

1

n

-

+ m +

-

m -

n

+

1] _[" - q]

p

+ 1

q

(7.25)

q]

[n -

+

D(p,q,m)

(7.26)

~

D{p,q,q) -

(7.27)

(7.Z8)

D(p,p,p) -

[

[

[

q

> 0

P

Q

1] _[n - Q)

" + P + 1

n

+

1]

p

q

- 1

P + 1

D(p,O,O) -

(7.29)

[f

0

~

m

~ P ~

q,

- D(p,O).

then we have

(7.3 )

(7.31)

D(p,q,m)

~

[

n

+

1] _[q 1] +

p+L

(732)

D{p,q.p) -

[

1>+1

n

+

1] [q 1]

p

+

1

+

p

+

1

·229· D(O,q,OI - n - q - D(O,ql

(7.33)

PROOF.

We have

[: : :]

p+l

!

j-O

n

-

p-m

q

]

+ 1 -

Hence

D(p,q,m) -

j

~ 0 [Q

j

1] [

+ j

in the case

m - 0

al

0

p

(7.24).

If

[

j~O

+

j

n - Q + 1 - j 0

~

m

~

q

+

+

1 +

]

which proves (7.23) and

q

~

p,

take

If

in the sum of (7.25) and the estimate (7.251 follows.

j - q - m

m - q,

this is

the only non·vanishing t rm in the sum and we have equality (7.271 which implies (7.28) and (7.2 I.

If

0

~

m

~

p

~

q,

take

j -

p - m in the

If m - p, this is the onb non·vanishing term in the sum and we have equality (7.321 which implies

sum of (7.30) and we have the estimate (7.301. (7.331 and (7.281. LEMMA 7.10. <JIl' ... ,<JIh

Let

q.e d. W be a hermitian vector space of dimension

be vector'! in

W.

Take

It

E

W*.

h.

Let

Then we have

(7.34)

PROOF.

"""

... '''''h

be an orthonormal base of

be the dual base.

Define

y U - <""U,It>

Define

Lf't

for

U

E

W.

IWI1,hl.

Let

""!, ...

Then

'''''~

-230-

Then

~I' ... '~h

is an orthonormal base of

A V.

A number

A

£

a:

h-l

exists such that

Therefore we have

If

oUI' _. ,oUh

are linearly independent, then

is the one and only element of

-

Gh l(W, - 1P(AW,. h

o ooLy 0 - 1

(7_35)

00

for all

Let

QJ;c"

pO"lItion

Take

(7_36)

Take

p

and

be a subset of

b

E

Gq(V,.

q

in Z[O,n].

Gp(V·'.

Take

Assume that

IR

Take

r

0 < r

~ 2- 3n - 3 r([J' .

£

with

Then we have (7.37)

Lemma 7.10 implies

y £ 1P(W·' _

We are ready to prove the required estimate. the ga ge (7.2' - (7.4,. THEOREM 7.11.

1P(oU1 " -•• " oUh'

MIJ(b,m,r' , O(p,q,m' .

Recall the defimbon of

m £ Z[O,p) " Z[O,q].

QJ

is in general

-231-

II is enlarged. 1!J(b.m.r) increases. By Lemma 7.3 we can

If

PROOF.

assume that

[n + 1]

~

I!J

p +

.

lVe abbreviate

1

(7.38)

d - D(p,q,m)

Define

11" by (7.13).

0 < h - d - 111" E; h.

Then

be an orthonormal base of

/.l - Min(p+l q 1)

V

such that

Let

&0' ... '&n

& - &0 " ... "&q

and

b - F(&).

CLAIM 1.

Take

E: Gp(V·)

U

a - 1P(u) E: Gp(V·).

T

(7.39)

I <&T'u> I E; [

PROOF OF Then

T

0

~ l

-

)I

e

1.

Let

E:

1I-(m,q].

T

E: 11".

l:

[I

~E:1I-[m.qJ

which proves Claim 1.

Define

Then

/.l ] [J b mm+! a [J . m + 1

Z(O.m] -

Define

pIx) - T(x + m -

E;

lIuli - 1.

with

1)

P

Z(O.p] E:

for

be the inclusion map.

1I-(p - m - l,n]

x

IIUL&~ 112] 1/2

E:

-

Z(O,m

[m

/.l+

by

-11

1] JI& IBm+1 ull

-232-

Assume that

tllJ(b,m,r) > d

where

r

is a real number

satisfying (7.36).

tllJ

Since aj

E

QI(b,m r)

such that

~

for

h,

there exists

j-

I,

...

E

IJ for

""j

Take

,d + 1.

linearly independent and constitute a base of

h

L -

(\

E[a J.,,, VJ - (t.

J-d+2

"V I

E

p+l

p+l

B-

(fJ.'T

=

I 'T

d ~ 2, ... ,h), E

ttl.

Then

subspace of dimension


d + 1

is a linear subspace of dimension A - (u J I j

I, ... ,h

j ~

with

Gp(V*)

E

of

"V*.

"""j" - ]

Therefore

p+l

= 0

V j

"V.

If we put

p+l

L ~ span 0 (A, "V*). p+l

then

p+l

h - d

~

of

"V.

p+l

=

d + 2, ... ,h:

Define

E - span(B, A V) - Slb,p,m)o

dim E (\ L

such that

IJ is in general position ""I' ... ,uh are

Since

a J - 1P(""j).

aj

is a linear

Therefore

dim E + dim L - dim

"V

p+l

- h - d + d + 1 - h - 1 There is a vector

III' E

E "L

with

for

11111'11 - 1.

j -

Then

d + 2, ... ,h

The exterIor product and the interior product over

"V

are denoted by

p+l A and L respectively to distinguish it from the operation over V. Abbreviate

"

and

L

-233-

for

£

Zl1,hl_

lI~jll ~ 1.

Then

We obtain

d+l <~,u.>~·11 - II L <~,u·>~·11 J J J J J~l j-l h

- II L

d+l

d+l

L

~

I <~,u·> I J

J-l d+l

L

~

[T

J-l

d+l

L

jal

[

d + 1

~

h < 2 n+1

T~O'"

~

Since

Y"<&T'u/ I

[m :

d

1] 0 b mm+ 1 aj 0 .

~

Jh <

2n + 1

and

h

-

d

aj

£

C)(b,m,r)

for

for

I, ... ,d + 1.

j

=

I, ... ,d + I,

have o b mm+1 a· 0 ~ r J

j =

Consequently we obtain

o< which is impossible. 23n+3r ~ r(QI).

r(C)) < 2 3n+3r ~ r(C))

Therefore

q.e.d.

2f/2

1<& T' u.> J 12r12

-

and

T£O'"

1<& T' u·> I Iy 12]1I2[ L J T T£O'"

d+l

Here

L

T£lf

~O'"

~ j ~1 ~ h

L

=

1IC)(b,m,r)

~

d - D(p,q,m)

if

-234·

The interval (7.36) is much too small, but it is amazinc that such a simple estimate holds for the difficult inequality (7.37) to be true. p - 0 - m,

2-3n - 3

then

If

l/(n + 1).

can be replaced by

Now, the product to sum estimates can be established. THEOREM 7 12. Let

~ 0

II

position.

Take

p

and

q

in

be a finite subset of

Define

be funct ons.

k - Ifl).

For

x £!R

Let

Z[O,n).

Gp (V·).

B:

II -

Take

!R(O,l)

and

x+ - Max(O,x).

define

m

E:

Assume that

Z[O,p) " Z[O,II) ~

is in general

P: ~ -

!R[O,l)

Also define

(7.40'

Take

b

E:

GII(V).

Then we have

(7.41) :f:.

~

_

,

_ _ _ _-=P-.;(~a=""-)_ _ _ _

a£~ 0

PROOF.

b mm+l

((

+

- 1.

We can cons der

+ D p,II,m) - k)

.

a 02-28(a)

Ca e 1: 0 < k ,D(p,q,m).

Then

cPII(~)

the product in (7.41) extended over D(p,II,m' factors by counting D(p,q,m) - k additional factors 1. Then (7.41) is true, since the arithmetic mean majorizes the geometric mean.

Case 2:

k > D(p,q,m'.

1 ] 3n+3r(~) 9 _ [-2

and

Then A _

(D(p,II,m, - k,+ - O. 2( k

-

D ( p •m) 9.

D ( p ,q,

m»

Abbreviate .

Th en

c pq (nI' ... - 9)., .

Because 1f!J(b,m, 8) , D(p,II,m) 'I!J, we can find a subset .. such that .. ~ IJ(b,m,9, and such that I" - D(p,II,m'. If a then

of IJ E:

Ii - ..,

·235·

a

Since

~

~

p

~

1,

I

we obtain

I

~

pea)

a£~ 0

Pea)

a£~ 0

b mm+l a 02-2A(a)

b mm+l

a 02-2A(a)

q.e d.

Let V be a hermit n vector space of dimension n + 1 > 1. M be a connected, complex manifold of dimension m. Let ~ ~ 0 be a finite set of meromo hlc maps g: M P(V·). Define k The indeterminacy of ~ IS defined by (7.42)

The set then

I~!;;;

IIJ' For each z

0

E:

M -

lJ(z) -

(7.43)

If

*-

~

IIJ is analytic with dim ICJ

~ ~

IJ, then

m - 2.

If

0

*-

~ ~

IJ,

IIJ' define the orbit (g(z) I g £ ~) •

~(z) ~ lJ(z) ~ 1P(V·).

The fusion of

IJ is

defined by (7.44)

Then

81J *- M.

then

811 - Ie) .

If

The Kauge of defined by

0

*-

~ ~

IJ, then 8 ~

Ii is the function

r(lJ)

~

81J' If k - I,

M - 81J -

!RIO,I)

Let II~.

-236-

(7.45)

fll)Hz)

If

rzJ 'f:. ~ !;;; I),

of

I)

then

r(~)

r(l)) ,

~(I)) - 81) V (z

rzJ 'f:. ~ ~ I),

k - 111)(,,,

Z E

M - 81) .

M - 81)'

on

The degeneracy

z I)

E

A I))

and let

Fg

II~

Gj - Fg.

If

z

if and only if

E

M - 81)'

I)(z)

and

J

Lj - L,

which

is in general position for some point

L,

be the hyperplane section bundle of

be the representation section of

- p , n + 1.

A(I)) 'f:. M

I)(zO)

let

gEl),

then

is not in general position.

is said to be in general position if

is the case if and only if

For each

M - 81) I r(I)Hz) - 01 .

E

~(~I!;;; ~(l)I.

then

and

The set

with

for

is defmed by

(7.46)

[f

r (I)(z))

=

~ ~

Enumerate for

j -

j

g.

Take

('I' ... "pi

I, ... ,p.

g

rzJ 'f:. ~ !;;; I)

and abbreviete

An exterior product

(747)

" - " S Id

(V~ S L 1) CD '" CD (V~ S Lp) -

("V~) S L1 S ... S Lp p

is defmed, which gives a holomorphic section (7.48)

The zeroset

Zit) - ZIG l " ... "G p)

analytic subset of

o,

s

(7 49)

M

For each set A E Z define

of the section

G l " .. _" G p

which does not depend on the enumeration of let

~(A)

~sIA) -

be the set of subsets of

IB

E

~(A) I liB - sl .

A.

For

is an ~.

·237·

LEMMA 7.13.

k - 1,

If

6q - I q . If k > 1, then

then

(7.50)

6q is a thin analytic subset of M.

In both cases, PROOF.

The case

Take

~

- (g,h)

If

Z E

E

Take

Z E

h

6q - I q , then

Z E

I q , then

q

E

k ~ 2.

Assume that

with

Ig

Z E

h

g.

'I;.

Z)

< k.

Define

for some

g

q.

E

Then

:Jl 2(q) and F g(z) " Fh(z) - O. Therefore

with g 'I;. h 10 g : U that

6q. If

Z E

F g(z) = O.

Hence

k - 1 is trivial.

Therefore

g

E

Z E Z(~)

q and h

!: W. E

q

exist such that g(z) - h(z). Take reduced representations va and 10 h : U - va of g and h respectively sud

U.

Since

6lJ - Iq we have

Z E

10

g(z)

'f;

0

'f;

10

h(z)

and (7.51)

1P(

which implies

We have

10

g(z) "

,,- (g,h)

E

10

10

,,- (g,h)

representations with Z E 10

Z E

II)'

g(z)

U. then

'I;.

0

'I;.

10

E

h(z) - O.

:JlZ(q) and z

We have shown that for some

g(z)) - g(z) - h(z) - 1P(

Then (7.52) holds. Z E t:)

h(z)

E Z(~)

of

g

Hence

6lJ' Assume that and

!: W.

aq!: W. Take z

va

h(z))

Therefore

:Jl2(q) where g

g : U -

10

'I;.

h. and 10

Z E

E

W.

Then

z

E

Z(,,)

Take reduced 10

v·

of

h(z) = O.

If

h : U -

g(z) "

10

Z(~) - IIJ'

Then

h

·238· glz) - 1PI ~ glz)) - 1PI ~ hlz)) - hlz) IIl)lz) < k.

which implies

6'1

;2

61). We have shown that

Z IE

W. Together we obtain 61) - W.

61J is analytic.

set

Hence

Hence (7.50) is proved and the

61J;t M,

Since

6IJ is thin analytic.

the set

q.e.d. LEMMA 7.14. then

k ' n ~ I,

If

then

6(1J) ~ Z(IJ)· If k > n + 1,

(7.53)

6(1J) is analytic.

In both cases, PROOF.

Case 1:

k ' n + 1.

Assume that

enumeration of I).

Abbreviate

Gj - Fg.

Let for

IJ - (gl' ... ,gk)

j - 1. ... ,k.

be an

Then

J

Z(IJ) - ZIG}

A

...

neighborhood of gj

for

j - I,

(7.54)

z

and ,k.

Take

z

~ j :

U -

Ig..

Z IE

IE

M. V·

Let

U

be an open. connected

be a reduced representation of

Then

Z(IJ) I U - (x

Subcase a: that

"G k).

Assume that ~

Hence

IE

U I ~ }(x) " ... "

z

£

j(z) - O.

Iq.

~

k(x) - O}

Then

j IE

Z[I.k]

By (7.54)

Z IE

Z(II).

Then

j

exists such

J

and

Subca e b:

Assume that

p

with

Z[l.k]

IE

~

j(z)

~

j(z) "

;t

0 t:)

~

j

< p

k(z)

and

p(z) - 0

with

;t

Z IE

6'1 -

1cJ.

exist such that

IE

zl1.kJ

g.(z) - g (z). J

p

Here

1P( ~ j(z)) - gj(z) - gp(z) - 1P( ~ p(z)). j

< p.

By (7.54)

z

IE

Z(IJ).

Hence

-239-

Subcase c: for all

j

E

Assume that

Z[l,k]

1 :Ii j < p :Ii k. lJ(z) -

,«)

Hence

and

Also

1 (z), ...

,«)

gj(z) - :PI

k - IIIJ(z) k(z)).

«)

1(z) " ... "

Subcase d: subcase

c.

Then

«)

and

and k(z)

j(z)) 'I- :PI

«)

The map

r([J(z)) - r(lJ(z)) - 0

Therefore

4(11) - 611_

Z E

r(lJ(z))

Assume that

Z E

lJ(z) C V:

and

«)

p(z))

=

o.

p(Z

0,

r

Defl e

1P: lJ(z) - . lJ(z)

[J(z) =

Th n

is bl cllv

is not in general pos tion. which implies

Z(IJ) - 61J. IIIJ(z)

=

k.

E Z(IJ).

Z

Define

lJ(z)

as in

The map

~

11' : lJ(z) - . lJ(z)

is bijectIve.

r(lJ(z)) - r(ij(z)) - O.

Subcase e: z E 61J

S;;

Hence

Since

Assume that

Z E

Take

t

Z

E Ig

Z E

.

E

t

E

Z E

Then

then

which proves Case 1.

Define

Zit) - 4(t)

4(1J).

S;;

Hence

~

W

4(1J).

E 4(1J).

Then

Assume that t

E :Pn+ 1(IJ)

Ig ~ It ~ Zit) ~ Subcase b:

g

n 611.

4(1J) - Z(IJ)

k > n + 1.

E :Pn+! (IJ),

Subcase a: Z

Z(I))

4(1)).

Assume that

If

we see that

z E 4(1J) - 61J .

These subcases show, that Case 2:

z E Z(IJ),

Z E

!:

!;;;

g E IJ

exists such that

gEt·

Then

w.

Assume that

ZIt)

then

exists such that

Z E

IJ and h E IJ exist such that :Pn+ 1(Ii) exists such that (g,h)

6 t - It

Iq.

W.

611 - IIJ. g 'I- h ~

It).

Then

k;!: 2

but g(z) - h(z). Then

and Also

-240-

Subcase c:

As ume that

z

(J(l)

is not in general position.

that

~(z)

Tog

Hence

Z E

~

Hence

is not in general position.

~(z) ~ !J(z).

Iq.

A(I) _

£

:Jln+1(!J)

E

A«(J) - W.

11I~(z) - k,

but

exists such

1I~(z) - n + 1

Here

A(~) - 6~ !; Z(~) !;;:

er we obtain

Then

and

w.

In particular,

A(!J)

is analytic.

q.e.d.

k' n + 1.

Assume that

Enumerate

!J - {gl' ... ,gk}·

Then

(7.53)

is defined and of class enumeration.

If

on

COO

M - I!J

6!J - I!J'

Z I:

and does not depend on the

then

o !J 0 (z) - 0 'l(z) ;.. ... ;.. 'k(z) 0 - o. If (7.54)

r«(J z)) - r«(J)(z) - 0 !J 0 (z)

Hence

r(!J)

setting

=

[]

Z E

for

z

I:

COO

j -

I, ... ,k

and

G1

A

on

M - I!J

by

6Q1 - I!J .

Assume that lJ is in ceneral position. for

then

'l(z) ;.. ... ;.. 'k(z) 0 , 1.

extends to a function of class

r(!J)(z) - 0

M - 6!J

•••

"

Gk _ O.

Then abbreviate

G. - F l

gj

The zero divisor

(7.55)

does not depend on the enumeration of I). Its support is the union of the (m-l)-dimensional branches of Z(I)) - A(QI). Hence supp

1t!J

t: Z«(J) - A !J).

If

in a line bundle and we have

k - n + I,

then

G1 " ... "G k is a section

supp Jill - Z(q) - A(lJ).

-241-

Also the meromorphic map (7.56)

does not depend on the enumeratl'on constant.

"'.

'11

If

k - n + 1,

Now, assume that (M,T) is a parabolic manifold. assumptions that k - IIIJ ~ n + 1 and that IJ is in we define the counting and valence function by

th en

'" QJ. is

Still u der the II' neral position,

(7.57)

t ~ 0

for

and

0 < s < r.

r

If

~T

E

we define the comp nsaUon

function and gauge mea re function by 1

(7.58)

log FnJ) a

provided the integ als exist, which will be shown below. mlJ(r) - r lJ(r),

which however fails if

Trivially

k > n + 1.

The First Main Theorem for the exterior product proved in Section 3. 1. Special case gives us immediately: THEOREM 7.15. First Main Theorem for leneral position (k ~ n + 1). Let (M,.,) be a parabolic manifold of dlmension m. Let V be a hermitian vector space of dimension finite set of meromorphic maps IIIJ ~ k :I:; n + 1. Assume that ~(r)

- r q(r)

S E ~T'

exists for all

r

n + 1 > 1.

Let

q

be a

11': M -

QJ.

1P(V*) with is in general position.

':I; 121

E ~T'

If

0 <

S

< r

Thpn

E: ~T

with

then

(7.59)

If

k - n + I,

then

"IJ is constant and T "II - O.

Hence

·242·

(7.60)

z

E:

M - 811'

f(q})(z) - f(lI(z)) - inf(f(~(z)) I ~

£

:Jln + 1 (q})) ~ 1

k - 1Iq} > n + 1.

Now, assume that

(7.61)

If

by the def nition of the gauge.

z

£

If

811 - Iq}' then

is also defined and the infinimum is zero. for

z

£

8 I) - II)'

For

z

M - IqJ

E:

Assume that M - 6(q})

we have

E

:fJn+1 (q})}

Also we defme the divisor (764)

sUPP UqJ - 6(q}).

We define the distance

(7.65)

Since

~ 1

tI i! in general pOsition. Then f(q}) > 0 on Also (7.53) implies

(7.63)

with

f(~)(z)

Hence we define

f(l))(z) - inf(r (~)(z) I ~

(7.62)

0 ~ 0 - f~)

(7.66' from (7.62) and (7.65).

for

~

E

:fJ n +1(q}),

o q} 0

we obtain

~

f(q})

then

f(lI)(z) - (

-243-

Now. assume that assumption that

(M. T)

is a parabolic manifold.

k - IIlJ > n + 1 and that

lJ

Still under the

is in general position, we

define the countins: function (7.67)

for

t > 0

and the valence function (7.68)

for

and the compensation fu

0 <

S

< r

t'on

J

(7.69)

log 0 tj 0 a

H for

E: ~T'

r

r c: ~T

Hence the (aure measure function

r lJ is defined for all

by

o ~ r lI(r)

(7.70)

-

J H

The estimate (7.70) is crude but helpful. Observe that

(7.71)

If

g E:

(7.72)

lI:Pn+l(lJ) -

[

k

]

n

+ 1

'"~

-

II is given, then 11("

E:

lP n+1(1J)lg

Therefore (7.60). (7.68) and (7.69) imply:

E:

[k L] -n

-244-

THEOREM 7_16_ (M, T)

Let

be a parabolic manifold of dimension n + 1 > 1.

hermitian vector space of dimension

g: M _

finite set of merom orphic maps Assume that S E .f. T

.

(k > n + 1).

First Main Theorem of General position

is in general position.

I)

1P(V·)

m.

Let

IJ"I:-"

Let

be a

IIIJ ~ k > n +- 1.

with

0 < s < r E.f. T

Take

V be a

with

Then

[k 1] g IEIJ T g (r,s)

(7.73)

n

(7.74'

The identity (7.73) extends

mlJ

such that (7.73) holds for all 0 < meromorphic map such that gEl).

Then

S

to a continuous function on

< r.

Let

Tg(r,s)/Tf(r,s) -

r I)lrl/T fir,s) -

0

for

r -

f: M 0 00.

for

1P(V) r -

IR~

be a for till

00

Although the estimllte

(7.74) is crude, it is surprising that such an explicit and still useful estimatl

r I) which measures the decline of general pO'lltion

exists for the function for

r _

00.

A family

II - Ig>.h.. EA of meromorphic maps

with fmite index set for all

},. E A, II

general position.

E

A

g},.: M -

is said to be in general position if

A with

},.

~ II

and if

Ig},.1 },.EAI

1PIV·) g},."1:- gil

is in

\S.

The Second Main Theroem

Our theory was guided bv the symmetry between

f

and

For the

Second Main Theorem we have to leave this convincing arrangemp t assnme that there is a family of meromorphic maps

g: M -

nd

W(V*).

This assumption is due to the very nature of the Second Main Thporem as exemplified in the case whert' the maps the following general

g

are constant.

(81)

Let

M

be a conne

(82)

(83)

Let

.,

be a parab hc exhaustion of

Let

V

be a herm t n vector space of dimension

(84)

Let

f: M -

(85)

Let

!J be a flO te set of meromorphic maps

(86)

Let

8

d, complex manifold of dimension

(87)

Assume that

.,

mdjorizes

(88) Assume that

f

is general for

E

B

If = n.

IJ).

n + 1 > 1.

be a meromorphic map.

WV

By (B8) we have

m

M.

be a holom rphic form of bidegree

lIJ - min (lg I g

We will makE>

u ptions:

8

For

with majorant

E

- 1,0)

!P(V*)

on

M.

Y.

B.

Define p

g : M -

(m

k - l!!J > O.

Z[O,lq}1

Define

define

(S.1)

as the family of associated maps of degree x + ~ Max(O,x) by affixing

g.

for a1l

x

E

JR.

For instance,

p

of

The dependcnct' on :E:pq(g),

4>q(g), "'p(g), Ppq(r,g), Rpq(r,g) etc.

~

q}. g

AgaIn put E!J

is indicated

pq(g), ppq(g), IKp(g), kp(g), .pq(g),

-246-

PROPOSITION 8.1. and (f,g)

q £ Z[O,n).

Assume that (B1) - (B8) hold. Assume that

is free of order

(p,q)

t)

Take

is in general position.

p

for all

g

E:

t).

For

£

p £ Z[O,it)1 Assume that

> 0 we have the

estimate

(8.2)

+ 3D(p,q)kC,(1 + £)(log Tf(r,s) + log VIr) + log+ Ricr(r,s))

l:: log+ T g(r,s) + £ log r .

+ 2D(p,q)C,(1 + £)

g£(J

PROOF.

Define

u - Min(p+1,q+1).

Define

(8.3)

By (6 9) we have (8.4)

o~

~ _____ t .s:.P.:a.9_(_&_)_ _ __

t P9 ( g ) ( n

CASE 1. and def ne

+ l)(kp(nI) + hq) ...

Assume that 8 r g)

2-

(n

llC)} - k :?: D(p,q).

by (6.33).

Put

8 5) Then Theorem 7.12 gives us the estimate

+ l)(k p (

Abbreviate

g )

+ hq)

2~1.

d - k - D(p,q) :?: 0

·247·

(~.

r(Q} )2d I T P gEQ}

~

c

o

[L [_1_. gEQ}

/..l2

(g)] .8(r.g)-l pq

/..l

(k

P

(P9 (g) (Q}) + h

~pq(g)

(g)].8(r.g)-l pq

(k

p

(Q})

+ h

q

)2

JDP.q q

)2

which implies

TI

0 fq

gEQ}

mg

02.8(r.g)-2~ P

(g) pq

The definition (6.37) co v r s this estimate to

-2d

- hq

[I T •

p! q+ 1

gEIJ

. [IT ~

mg

0 f

gEIJ

q

(g)h2][. IT f 0

q

tpq(g)

gEQ}

gEt)

hq

mg

0 f q

(g)

]

pq

0-2.8(r. g )]

p

. [L

02.8(r.g)-2( p

P

c r(t) )-2d[l + k p (IJ)]2d[r-r p o

III g

q

gEt)

02 .8(r,g)-2 ( p

pq

(g)]

[IT g EIJ

(g)] D(p,q) . pq

0 f

q

iii

g

P

0-2.8(r,g)]

-248-

Take the logarithm, i.ntegrate over

M

and divide by two.

This

gives us

(8.6) D(p,q)Sr (r)

~

+

I

gdJ

[mr

m (r)

g gp

-

meq+1 mgp (r)]

I Ppq(r,g) + ! .B(r,g)mr mg (r) gdl &£\1 q p

+ 1 D

2'

pq

J

[ log!

gEt)

[

1

.. 2 .pq(g)

] .B(r,g)-l

]

~pq(g) f1

...

H where

(8.7)

I

10C[1 + "Ph~I1)]f1 ~ g~1I

M

I M

-

I

gEt)

log

+ "p(g) -h

f1 + c, log(k + 1)

q

Rpq(r,g) + C. loc(k + 1)

Also we have

(8.8)

S(r,g)mr

q

mg p(r)

~ .s(r,c)(T f (r,s) + T (r,s) + m.. IB" (s) q gp ---x q gp

~ 1

-249-

The Ahlfors estimate 16.35) implies

(8.9)

I

log

M :$:

c,

(-1z. pq

I

&£11...

log!

I

(g)]

8Ir,g)-1~

[_1_.

I

gEII ... 2

pq

pq

(g)a

Ig)] 8Ir,g)-1~

pq

a

M

:$:

c,

I

gEQ}

log +

[~. (g)] 8Ir,g)-1 ~ a] ... pq pq

[I

+ C, log

_c,k

M ~ 3C,(l + E)k (l g Tf(r,s) + log Vir) + log+ RicT(r,s))

+

3C,(1 + E) I log + T (r,s) + CokE log r 1 C, log ~ g EQ} g

Now (8.6) - (8.9) yie d

18.10) D(p,q'Sf (r) + q

+

I

(mf!Hg (r' - mf

gEl)

q

P

q+l

!Hg (r)) p

2C,(1 + E)k(log T f(r,s' + log Vir) + log + RicT(r,s"

+ 2C,(1 + E)

I

gEqj

log+ T (r,s) + C,(k + 1)E log r g

where we abso pt the constants into the c 1 > 1 + Co(k + 1) > 1 impl es (8.2' in the case

and replace k

~

D(p,q,.

E

log r by

term.

EtC} <

E.

Take a constan' Then (8.10)

·250·

CASE 2. Assume that 0 < k < D(p,q). Define .B(t,g) by (6.33). Put

Abbreviate

d - Dlp,q) - k.

(8.11)

Then Theorem 7.12 gives us the estimate

which implies

11 gEII

+

E: Cl

f

0

q

IB g

p

D2.B(r,g)-2~

pq

(g)

[.lz. Ig)] .B(r,g)-Ie Ig)] D(p,q) u pq pq

I

gEIJ

[rT gE:1J

ppqlg)]

[11 [..1-.pq Ig)] -.Blr,g)] gEIJ u

• [d(k(II)+h)2+ p

q

2

I

gEQ!

[·pg(g)].Blr,g)-le

u2

Ig)]D(p,q) pq

-251-

The operator

f

t

log 1)0

is applied:

M (8.12) D(p,q) + Sf Ir) + ~ Imf IBg (r) - mf IBg Ir)) q g E:IJ q P q+ 1 p

J

Dpq

+ 2' 1

M

~

+

Dlp,q)log

[t

I

(d(kp(lJ) + h q )2

M +

[-.!z ~ (g)] .8(r,g)-I E (g)) JJ. pq pq

~

g IJ

~ C2 +

~

P pq(r,g) +

~ D(p,q) log +

gE:1J

+

~

0

J

M

~

D(p,q)

log+

g£1i

J

_1_ ~ (g)].8(r,g)-lE 0 [ JJ. 2 pq pq

M

We have (k (Ii) + h )2 p

q

~

(k + I)(h q2 +

~

g£~

kp(g)2)

·252· Prom \6.28) we obtrl.ln

J

ikpill) + h q )20

M

~

(k + 1)

J M

Consequently we hdve

J

log I

(kpill)

I

h q )2o

M

+ E:(2m - 1)1og r + 2 iog(k + 1)

Now (6.31) implies log T t (r,s) ~ log+ 1'r(r,8) + log+ log Yir) I log+ Ric.,(r,s) q

+ log+ log r + c 3

log T

gp

~ log+ T£(r,s) + log V(r) + log+ Ric.,(r,s) +

E:

log r

(r,s) ~ log + T g(r,s) + log Vir) + log + Ric.,(r,s) +

E:

log r .

·253· Therefore we obtain

log+

J

(kp(IJ) + h q )2o

M

which implies

(8.13) log +

J

(kp(IJ) + h q )20

M ~ (1 + £)(2 + k)(log Tf(r,s) + log VIr) + log+ Ric1'(r,s))

2:

+ (1 + £)

log

Tg(r,s) + £ log r .

g£IJ Now (6.35) gives us

(8.14) log+

J

[ _1_. (g)] .8(r,g)-l~ (g)o u 2 pq pq

M

+

£

log r .

Now (8.12), (8.13) and (8.14) imply

D(p,q) + Sfq(r) +

2:

g£IJ

(m r EBg (r) - mf IBg (r) q p q+l p

~ 3D(p,q)C.(1 + £)k(log Tf(r,s) + log VIr) + log+ Ric1'(r s»

·254·

+ 2D(p.q)C.(1 + E)

L log+ T (r.s) + L P pq(r.g)

gEl)

g

gECI

+ D(p.q)(C./2)(1 + k)E log r . By a change in the value of Let

Fq

fq : M -

be the

Gq(V).

L t

qth

E

we obtain (8.2).

q.e.d.

representation section of the map

NF (r.s)

be the valence function of the zero

q

divisor of

Fq .

LEMMA B.2.

Assume that (B1) - (B4). (B6) and (B8) are satisfied.

Assume that

0 <s < r

q.: Z[O.n).

and

E.e~

and

S

E.e~

and that

P E Z[O.lClJ

Then we have

(B.15)

1]

n-l L D(p.q)(Sf (r) - Sf (s)) + [ n + Ric1'(r,s) q q p + 2

q-O

PROOF. Recall (4.17). written as (N F

q-1

(r.s) + T f

q-1

Hence the Pliicker Difference Formula (4.33) can be

(r,s)) - 2(N F (r,s) + T f (r.s)) + (N F q

- Sf (r) - Sf (5) + Ric1'(r,s) q

Here Observe that

q

q

q+l

(r.s) + T f

(r.s)) q+l

·255·

n~1

(8.16)

If for

XO'X1'''''~

q-O

[n +- q] [n + 1] +

=

P

1

2

p

are indeterminates over the ring

Z - ZIO,nJ.

q E:

n~1

D(p,q) _

q=O

Z

and if

x

q

- 0

then we have the algebraic identity

(8.17)

n-l[ I

q

-1

n

p

q _

1]

-

.Xq +

[

P +

Therefore we obtain

1

+

n-l I

D(p,q)(Sf (r) - Sf (s)) +

q-O

By Lemma 1.11,

q

[

np +

q

n

constant and the constants

n

q.e.d.

+

mf mg (r) - (C,12)l1og(p + 1)

] Nf (r,s)

is a constant.

This

p

Sf (s)

can be absorpt by the remainder term

q

log r. Recall that (f,g) is free of order (q,p) for all q E: ZIO,n] if and only if (f,g) is free of order (O,p) (Lemma 4.2). Therefore if we sum (8.2) over q = 0,1, ... ,n - I, we obtain: E:

THBORBM 8.3.

satisfied.

Second Main Theroem. Assume that (81) - (88) are Take p E: ZIO,i ll). Assume that CJp is in general position.

Assume that (f,g) is free of order (O,p) and s > O. Then we have the estimate

for all

g

E

QI.

Take

E:

> 0

-25618.18)

+

n-l

+

I Ik

q-O

+ [n + l]RiC Ir,s' + 2[n + 1]c.(1 + p + 2 T p + 2

+

E

E,

I

log+ Tg(r,s'

gdl

log r .

Formula (8.18' is extremely long and needs interpretation. doubt

Ppq

and

~q

are the worst terms.

for all

(8.19)

Then q

E

kp(g' Z(O,n'

o. and

Hence g

E

Without

However if

P pq(r,g) - Rpqlr,g) - 0

g Ell. for all

r > 0,

all

II. We obtain:

THEOREM 8.4. §econd Main Theorem (Maximal version). Assume that IBl, - (B8, and (8.19, are satisfied. Assume that IJp is in general position. Assume that (f,g, is free of order (O,p, for all Take E > 0 and s > O. Then we have the estimate

g

E

IJ.

·257· (8.20)

nil[n q~1

q

l](Np

-

P -

+ Tf

(r,s) q

Ir,s)) q

+ [

1

p

]

1

+

F (r,s) n

n-l

r

(k - D(p q))+r .... (r) ~p

q-O

+ (; log r There is p operly.

8

geometrl

ROPOSITION 8 5. m.

B

Let

M

differential ;!:

121

CP: M -

Xo (;

Wand a point

dCP(Xo): if. Xo 1M) -

W

of

cP

at

Xo

M

Wand a

such that the

is injective.

Let

be a finite set of linearly non-degenerate merom orphic maps

hi : M -

1P(Vh)

"h + 1.

Define

cfifferential form

where

Vh is a complex vector space of dimension

n - Max(nh I h (; 6)"

B

of bidegree

I!olynomials of at most degree f»r

is chosrn

be a connected, complex manifold of d"mension

Assume that there is a finite dimensional vector space

holomorphic map

6

condition which implies (8.19) if

Pirst we need the following result.

B - CPS(S)

that is

1m - 1.0)

n - 1

J h - Dh

Then there exists a holomorphic whose coeffIcients are

such that each

for all

h

£

h

E:

6 is general

6. ~

tROOP. ~idegree

By Theorem 7.11 in (27] there exists a holomorphic form (m - 1,0)

polynomials of degree

OD

W such that

nh - 1

....

Bh

Bh

has coefficients which arr

at most and such that

h: M -

1P(V h)

if

-258-

is general for

cp(i\)

Bh -

h E ~_

for each

k

a complex vector space of dimension of

Y.

where

71 - h~~ 71hEh

m - 1

on

degree

n

W w

a base of function

h E ~

Vh

1\.~

-

h

...

Then

a hn - Dh (D,1l)-n.h h

~

...

B - 8{ 8)

O.

B{E h ' - Bh ,

Becouse 8 E Y

=

H( 8).

such that

-

1P(V h)

h: M -

is connected, ~

Dh(xO,8)

0

is a holomorphic form of bidegree

a hnh (X o)

Since

there is a

U X Y

WIth polvnomial coefficients of at most degree Dh(x O' S);i; 0

n - 1.

Define

in respect to

is general for

8,

B for each

q.e.d.

COROLLARY 8.6.

Assume that

(Bl) - (85) are satisfied.

is a finite dimensional complex vector space CJl . M -

be A holomorphic

is general for

Dh(zh' Eh)

Vh

Let

M.

"

Xo E U and a vector

the meromo phic map E~.

a h : U-

exists uniquely such that

It

Since

B(n)

h E H

8 - CP*(S)

h

1\. h

zh E U such that

1.0)

be a base

is a holomorphic form of bidegree

take a reduced representation

and define

there is a POint

(m

bE

coefficients are polynomials of at most

0'1

Dh . U )( Y -

for each

B(n}

Then

is an open, connected subset of

in respect to point

{Eh I h E~}

Y

1.

For each U

and let

Let

71 E Y define

For

where

k - 1I~_

Define

Wand a point

dCJl(xO) : ~ x (M) -

o

W

linearly non-degenerate.

of

Xo

€:

rp

at

Take

M

Assume there

Wand and holomorphic mal

such that the differential

xo

is injective_

p E Z[O,n).

Assume that

For each g

£

f

II assume

is

·259·

that

g(M)

is contained in a p-dimensional, projective plane in

but that

g(M)

plane in

!P(V*).

bidegree

(m - 1,0)

degree

Then there is a holomorphic differential form

n - 1,

and such that

!P(V*)

is not contained in any (p-1)-dimensional projective

on

=

of

W whose coefficients are polynomials of at most

such that .if

8

f: M -

nand

!P(V)

is general for

.i g - p for all

g

E

B

q:> (8)

in respect t

Q}

B.

Hence (B8) and (8 19) are satisfied. PROOF.

Let

V*g

be the smallest linear subspace of

g(M) (; !P(V;).

Then

map

!PIV;)

g: M -

dim V; - p + 1

p < n.

where

is hnearly non· degenerate.

8

there is a holomorphic fa m

of bidegree

V*

such that

The meromorphic

By Proposition 8.5

(m - 1,0)

on

whose

W

coefficients are polynoml f at most degree n - 1, such that f : M !P(V) and g M !PIV*) are general for B. Hence 8

If we conside

If - n.

gIM)!; ..(V~)

Since g

E

as a map into

g

we have

.i g ' p.

1PIV·),

then

p' 1 g

Hence

Ag - p

Then

[p!

for all

IJ. q.e.d. We want to study the case

term

NF Ir,s)

p - O.

appears in the Second Main Theorem.

1] - 1

and the

This term can be

n

used to modIfy the Second Main Theorem. Let

A;/:

0

be an analytic subset of pure dimension

Then there exists one and only one divisor r each simple point

x

E ~IA)

v be an divisor on M. e set of branches of ch that inite.

~.211

S.

Assume that

x

E

v A such that

A and such that

For each

vlx) - P A for all

We have

of

A

E

m - 1

S - supp

of

v A(x) - 1

supp v A-A. V ;/: 0.

Let

cr be

cr there is an integer PA

A "RIS).

The family

cr is locally

·260· ~ 0

Then

v

Take

n £ Z

PA > 0

if and only if with

~

n

O.

for all

A E

The truncated divisor

ex.

v

Assume that

)I(n)

~

is defined by

(8.22)

Obviously,

0 ~ v(n) ~)I

If

is a parabo c manifold,

(M 1')

and

)1(0) -

o.

If

)I _ 0,

put

v(n) _

o.

write

(8.23)

By a comblhation of the methods of L. Smiley [251 Lemma 3.1 and B. Shiffman [221, [231 (1.14), we obtain: THEOREM 8.7. Assume that (B1), (83), (B4), (B5), (B6) and (B8) hold. Let II be in general position with III; - k ~ n + 1. Assume that 19 - 0 for all let

UF

g

n

E IJ.

Fn

Let

be the

be its divisor.

Let

representation section of

nth

f

and

be the general position divisor by (7.56).

UIJ

Tht'n ~

(8.24) g

~

REMARK 1.

In (8.24) only

REMARK 2.

If

n

•g

>

B.

depends on

UF

(n>

u( f

is a parabolic manifuld. then (824) implies

(M,-,.)

I Nf g(r,s) ~ NFn(r,s) + Nm(r,s) + I N f(r,s) g£'1 ' .. gE'1'

(8.25)

f.RQQJ!.

We have

:jln+l(IJ) - (~ ~ lilli' - n + 1).

defined by (7.4 ) and sUPP UI; - 6(IJ) 6('1)

~

"EIJ Uf ,g '" UF n + Um .. + g "£IJ

IIJ

since

has pure dimension

by (7.42).

k ~ n + 1. m - 1.

Then

See also Lemma 7.13. Hence either

Let

If

U

-

6(1;)

D - supp

I

ge:QI

f,g

is

We have

is empty or

be the indeterminacy of f.

Define (8.26)

6('1)

v

8€QI

sUPP Uf,g .

(

-261-

If D - /ZJ, nothing is to be proved. Assume that has pure dimension m - 1. The set S - I (D) V I(~(QJ.)) V

(8_27'

is analytic with dim S :s; m - 2. of D - S. Take

D"#. /ZJ.

IIJ

Then

D

V If _

It suffices to verify (8.24) at every point

Xo E D - S. Take 'E 1l n+1(Q}). We claIm that

(8.28)

(8.29)

and

(830)

The maps go' _.. ,gn

and

fare holomorphic at

xo.

Take

n

a

(8.30'

E

1P(V) -

V E[gj(xO)) j-O

T1ere is an open, connected neighborhood

U

of

Xo such that the

f410wing conditions are satisfied.

(~ There is a chart ~ - (zl, ... ,zm) : U where

Xo E U

(i (

is a ball centered at

U' C;;;

0

U'

with

lJ-(xO) - 0,

and where

M - S.

The set U n D = Y n D - S ~ Ix E U I zl(x) go, ... ,gn and fare holomorphic on U.

() a E E[gj(x)]

for all

x

E

U

and

j -

0, ... ,n.

=

0)

is connected.

-262-

~

Take

representations

.:l

<~, ~ j(x»

that

P(~) - a.

with

V.

E:

:

0

'I:.

U-

By (3) and (4) there are reduced

V.

for all

of x

E

f U.

vt

;;. : U -

and

J

of

gj

such

Then

IV

(8.32'

'\10 j

is a reduced repre

rJ

<~. AID

(8.33)

<~.;; j >

-

ntation of

gj

rw

> AID

with

i-

"",,'

J

V~

U _

AID j

<~.

~~

,-v

<~ • AID 1 >"'40 2

IV

Since

1g

=

0,

for

j - 0, ... ,no

for

°

on

U" D.

~

11

E:

Z.

we have

AID j

"

'\10.1 -

O.

"

:;"

1

L~

Yj

Pj - Max(J.l j - n,O).

If

'\10.1 -

on

U

vanishes of order

v·J

Hence

The holomorphic function

Define

AID. 1

'\10 j> IV

Pj > 0,

then

0

v j > n.

Hence (8.34)

There are holomorphic functions

(8.35'

[f

Pj - 0,

(836)

~ - 0, ... ,n

for

X~

on

U

for

define

and

x

E

U "

such that

). - 0, ... ,n .

X~ - Yk(A) and (8.35) remains correct. Define

D

·263·

Then

Y ~

(8.37)

Let base.

1\.0'· .. ,1\.n

Define

be a base of

V.

zfx . 1\.~, '"

Let

,1\.~

VJ' - < 10 ,1\.*> J

(8.38)

be the dual

Define

v

w = det(w J,ll)

.

Then we have (8.39)

10

"

10 '

"

... "

10

n - V1\.O " '" " 1\.n

(8.40)

(8.39)

(8.42)

UF IU n

and we have

(8.43) j

V·w

< 10

,,10

,

"

... "

10

(n)

,'110 0

" " , , , 'lIOn

del(Y ]I (U)) - Y - zfX

o,fhich implies

~ence

we obtain (8.28).

>

·264·

Abbreviate

18.45)

A

n

(8.46)

'()

~

-

j~O

V~*j

~ j ~ (-1 J ~ 0

(8.47)

•..

U -

A

•••

on

U.

n

A

~ j-l "

Hj t 0

...0 j+ 1

"

'"

~n .

"

Hence we assume that

v n - O.

Obviously, (8 29) is correct if Holomorphic functIOns

(AV")*

U

exist on

We have

(8.48)

Since

t:)

is a reduced representation,

U.....

- O.

We obtain

t:)

(8.49)

which proves (8.29). Take an enumeration and

vI

~

~

)12

...

~

vk

Q} ~

O.

(gl' --. ,gk)

For

j

=

such that

n + 1. '" ,k

Vj -

Uf,g/XO)

define

(8.50)

where

~. ~~.

J

if

I

n+l

(8.51)

j

~ K

I

Uf,gJ'(x

By (8.28) we obtain

o) ~ u.."0+ 1(x O)

n+l

+ v F (x O) + 0

:L

j _ 1

U

(n) (

f • g j Xo

)

·265· If

n + 2 , j 'k,

we have

(8.52)

Hence (8.51) and (8.52) imply

q.e.d.

THEOREM 8.8. Seco d M Theorem (Maximal version for Assume that (81) - (B8) hId. Assume that Ig - 0 for all

0).

p -

g

E

CJ

Assume that (f,g) is free for all g E II. Assume that II IS in general position. Assume that ICJ - k ~ n + 1. For 0 < s < r abbreviate 8.53) Take

Q(r,s) - 2n(n + l)ke.(log Tf(r,s' + log V(r' + log+ Ric.,(r,s))

E

> 0 and

s > O.

Then we have the estimate

(8.54) Np (r,s) + ~ mf gIrl n BEC»'

~ (n + l)Tf (r,s) + + 2n(n + 1)e.

and

n(n

t

1)

Ric.,(r,s) + [nk -

~ log + T sIr,s) + Q(r,s, + BEC»

E

n(n

log r

t

1

)]rll(rl

·266·

(8.55)

l)T fIr,s)

(k - n -

I g £q}

Nf g(n)(r,s) +

n (n

+

1)

RicT(r,s)

2

,

L

T g(r,s) + Q(r,s) +

£

log r .

g Eq}

PROOF.

W.lo g.

0 <

<

E

t

Then Theorem 8.4 implies (8.54).

Now the

First Main Theorem (Corollary 3.3). Theorem 7.16 and Theorem 8.7 imply

(k -

'<:'

~

n -

l)Tr (r,s)

Nf g(r,s) - NF (r,s ) +

L

[nk -

+

L

gEq}

N

f,g

+

2

RiC (r,s) T

1)

+

n(n

2

+ 2n(n + 1)e.

~.

n(n

n

gEq}'

L

gEq}

(n)(r,s) +

L

+ 2n(n + l)e.

log + T (r,s) + Q(r,s) +

E

log r +

E

log r +

g

n (n

+ 1) R'

(

ICT r,s

2

)

log + T g(r,s) + Q(r ,s) +

g£q}

~ SEq} L Nf,g (n)(r,s) +

+ [[ nk -

n(n

+ 2

n (n

2+

I

mf g(s)

gEq}

,

1) R'ICT (r,s ) + Q() r,s + 2E 1og r

1)] [k 1] + 1] L -n

SEq}

Tg(r,s)

-267-

~

Replacing

I

gE:tJ

£

Nf ..(n)(r,s) + 'D

by

E/3

n (n

l

1)

RicT(r,s) + Q(r,s) + 3E: 1017 r ..

we obtain (8.55).

q.e.d.

The assumptions of Theorem 8.8 are quite satisfying and possible while this fact.

g

is not constant.

is

In Section 9, we will make use of

Thus in certain case , the undesirable terms eliminated.

I. g - 0

P pq ,R pq

are

However we may ask if it would be possible to fmd better

remainder terms.

For instance, we could split the integral

(8.56)

'nto the three integrals

(8.57)

J

M[ t)

(8.58)

J

M [t )

and try to incorporate (8.58) as a remainder term in the Second Main Theorem measuring the twist between f and g. Unfortunately. this idea does not work. From the Ahlfors estimates we know that the integral (8.56) exists. In Section 10, we will provide an example where all three integrals in (8.57) and (8.58) diverge while (8.56) exists as it must.

·268·

For the f urlher discussion we will assume (89) Take

p

E

ZIO,i Ci1.

Assume that

(8.59)

Pe9(r,g) Tt(r,s) - 0

(8.60)

R (r,g) EI Tf
(Observe that

Tf(r s) -

-

0

for

r -

00

for

r-

00.

by (B8).)

00

We d('fine the customary defects (Stoll 1281, [32]) by

R1c 7 (r,s)

(8.61)

Rf -

ll.m sup r->OO

Tf
(8.62)

Yf

11m sup

log Y~r2 Tf
Rf

r->oo

is called the Ricci Defect and

It is unknown, if

Ric 7 (r,s)

o.

~

If

Yf

is called the Maiorization Defect.

Rf <

00,

(8.63)

Since

for

y(r) ~

1,

we have

Yf

~

pardbolic space of

(8.64)

(a: m, 7 0)'

r -

00

•

O.

In the covering case, more can be said.

map

then

Let

(M,-r)

be a covering

Then there is a proper surjective holomorphi

-269-

such that

., - "0

0

cP

=

IICPIl2.

The divisor of

is called the branching divisor of

dq>l

'"

and denoted by

cP

...

Pq> •

(Stoll [103]) (8.65)

Hence if

f

separates the fibers of

separates the fibers of

Rr - a Define

So

Let

such that

11'

for

d,

S - supp Pcp'

Then

CP(S)

«;m.

The map

f

Thus if If

f

r _

is an analytic subset of

If

So

CP(S).

00

a:;m.

Observe that

Co

is affine algebraic of

cP

is biholomorphic,

B

11'*(8) - B

n - I,

00

for

r -

has transcendental growth and if

Assume that

degree

for

which is equivalent to

_

that

0

which

is said to have transcendental growth if

(8.67)

«;m

M

then we hdve

0 < s < r.

O.

on

Th(r,s)/Tf(r,s) -

be the 1m-I) d m nsional component of

Ar(OO) - 00

Rr -

h

by a theorem of

by the Sdme theorem of Noguchi.

is the sheet number of cp; M degree

Rf < 00

then

If there is a meromorphic function

Noguchi (18).

then

CP,

Ric.,(r,s)::

a

00 •

So is affine algebraic and

Rr - o.

is a holomorphic form of bidegree and that

8

(m - 1,0)

on

has polynomial coefficients of at most

then there is a constant

c > 0

such that

·270·

(8.68)

VIr) lEi 1 + cr 2n - 2

Hence

Vf

f

if

0

=

PROPOSITION 8.9. f

Assume that (81) - (B5) are satisfied.

Assume that

IJ is in general position. Assume that

(8.69)

g

~ 1 .

r

has transcendental growth.

is not constant and that

for every

for

for

r -

00

for

r -

00

IJ. Then

E

(8.70)

The proposition follows from (7.77).

If

> O.

p

•

the situation is not

so satisfactory. PROPOSITION 8.10. Assume that

IJp

1.

Assume that (B1) - (88) hold.

is in general position.

define

c

s > O.

Then we have

=

Ifk>[:J.

define

Define

c =

Take

k - IIIJ·

P

If

Take

Z(O •..eIJI.

E

k lEi [:

E

>0

J.

ant

(8.71)

I"tJ (r) p

I Tg(r.s) + ¥i3 P

gE

If

Yf - 0

~

Rf

-

1)(e. log VIr) + Ric.,(r.s) + Ee. log r) .

and if (8.69) holds for all

I"IJ (8.72)

11m

inf

r--too

g

(r)

P

Tf(r,s)

-

0 .

E

IJ. then

-271-

Theorem 7.14 and 7.15 imply

PROOF_

r CJ

(8.73)

~

c

(r) , p

0 < s < r

for

E

.f.T

8

T g (r,s) + mCJ (s)

ECJ

P

P

Proposition 6.15 implies

(8.74)

T g (r,s) ,

3 P T g(r,s) + t(3 P - 1)(C, log Vir) + RicT(r,s) +

EC,

log rl .

p

s > 0,

Now (8.73) and (8.74) imply (8.71) with any fixed absorbs the constants. Yf

0

=

o<

=

Rf

Af(oo) <

00

and we

EC,

log r

has transcendental growth, (8.67), (8.69)

If· f If

imply (8.72).

where

f

and

does not have rational growth, then

tam

0

(8.75)

for all

E

>

o.

Hence

0

IE: -

implies (8.72).

Assume that (81) - (88) hold. (f,g)

is free of order

(O,p).

Take

p

E

q.e.d. Z[O,1l(Jl.

Assume that

Define the Nevanlinna defects

(8.76)

0'

IIf[g p l - lim 1nf r-too

(8.77)

o ...~

Ii f (g p ) -

m

1i m

1

r-too

nf

Lf(r) 8p Tf(r.s) + T (r.s)

8p

Then Theorem 3.2 implies

N

(8.78)

o,

II fig pI - 1 -

8 p

Lf(r)

+

T

8 p

If(r.s)

11 m sup -=T~f"'(-r-.-s-)---:"+--=T=-g"&:""7"(-r-.-s....)~ , r~~

p

1 .

·272· Trivially we have (8.79)

If

T g (r,s)/Tr(r,s) -

0

p

for

r _

lif(i p ) - lif[gpl.

then

00,

Also

we define

(8.80)

(8.81)

If

Af(gp) < .. ,

the definition of the Nevanlinna defects implies

(8.82)

If

Yf - 0 - R f ,

Yf - 0 - Rf K f(gp) =

O.

then

and if If

A' flip)

:E: 3 P Ap(g)

T g(r,s)/Tf(r,s) -

>"f(g) <

00,

0

by (8.74).

Hence if

for

00,

r -

then

then

(8.83)

for

r -

00

Now, Theorem 8 3 implies immediately: THEOREM 8.11 D fect relation. with p E Z[O,IIi]' Assume that for all

g

E

Assume that (Bl) - (B8) and (B9) hold Rf - 0 - Yf' Assume that Af(g) < 0

Ii. As ume that lip is in general position. Assume that

rlJ (r,s)/Tf(r,s) -

(8.84)

p

Assume that have

(f,g)

is free of order

0

(O,p),

for

r _

for all

g

...

€

IJ. Then we

·273·

(8.85)

If T

RBMARK 1.

gp

(r,s)/Tf(r,s) -

then (8.84) is satisfied.

RBMARK 3.

If

0

for

Moreover we have

p = 0,

r -

00

lif[gpl

=

for all

g

E

QI,

lifhrp)'

then (8 85) reads

(8.86)

A different version of the defect relation can be obtained from Proposition 8.10.

We combine (8.71) and Theorem 8.3 to obtain

(8.87)

g~1i

mCL.g(r)

~

[[p :

1] + [0

;

1]]Tf(r,S) +

Co

RicT(r,s)

n-I

+

c~,

where

c!

and

I

I

q -0 gc:1J

c~

(Ppq(r,s) + kRpq(r,g)) +

are constants.

E

log r

This form of the Second Main

Theorem implies immediately: THBORBM 8.12. with

(8.88)

p

E

Defect relation.

Z[O,11J1.

Assume that

Assume that (Bl) .- (RB) and (Bq) hole

He for

0 - Yf · r -

00

Assume that for all

g

E

IJ .

-274-

Qp

Assume that order

(O,p)

is in general position.

for all

g E: ", 'It.

Assume that

(f.g)

is free of

Then we have

(S.89)

Also Theorem 8

THEOREM S.13. that

ig -

Assume that

0

!Li.....!

for all Q

leads us to another version of the defect relation.

i

g

relation. E:

Q.

Assume that (B1)

Assume that

m general position.

(f,g)

(B8) hold.

Assume that

Rf

=

0

Assume that (8 88) holds for aU

g

(S.90)

1 i m sup -~..... ' ~g---.-r ->00 Tf ( r , s )

E:

t).

Define

Nt

(8.911

Assume

is free for all

(n)

(r,s)

=

g E:

Yf .

q

Value Distribution ~ !! Function FI Id

\9.

Our theory can be used to prove a defect relation for hyperplanes m - 1.

defined over a field of meromorphic functions of rank

Our

application will extend a defect relation to meromorph c maps established b~ Shiffman for meromorphic functions only Let

6tM

M

form on

M,

m > 1.

be a connected, complex manifold of dimension

be the field of meromorphic functions on

If

M.

B

Let

is a meromorphil

then

(9.1) is a subfield of Then

cpl' ... ,CPk

6t M ·

Let

CPl' ... ,CPk

be meromorphic functions on

are said to be analYtically independent if

dCPl " ... " d'l'k i1! 0

and

analYtically dependent if

Let

•

cP

6t M is said to be analYtically dependent on

E

be a set of meromorphic functions on

M.

d'l'l""." dCl'k ==

•

such that

CP,CPl' •.. ,CPk

(CP

(9.2)

is a field containing

•.

E

if either

is

cP

CP1'

are analytically dependent.

,CPk

Then

6t M I cP analytically dependent on .)

The assignment

• -

6tM(.)

is a

dependency relation in the sense of Van Der Waerden [34J p. 204 or Zariski-Samuel [41J p. 97 (See Andreotti-Stoll[2J Section 4). CP1' ••• ,CPk

are analytically independent, then

o.

A meromorphic function •

constant or if there are analytically independent functions in

M.

k::eo m.

If

-276-

• ~ ~M

Therefore if independent

• g;

but

a:,

there are analytically

CPI'

(9.3)

Then

Ill l , ... ,CPk

independent of the c ~M(.)'

also of

tJl 1, ... ,CJl k

Th

a:

tR M(.)

then

tR M (.)·

is called a base of

The number

k

is

. e of the base and is called the rank of I" IS

and

•

a base of and

~M(.)

tRM(.),

is a basE' of

tR M (.)

containl"d in

•.

are said to have rank

•

or

If

• ~

O.

If

a:,

then

(9.4)

In general, a meromorphic form

B

~M(.)

is said to define

if

(9.5) If

M

is Stein, every meromorphic function on

holomorphic functions on

M.

If cpl' ... ,CJl k

then there are holomorphic fucntions XjCPj

=

"'j'

Vj ii!!

0

M

is the quotient of

is a base of

~M(.)'

and

such that

Xj'" 0

By (9.3) the holomorphic form

k

(9.6)

dept"'" '" dCJl k -

defines

~M(.)'

j

"(x·d",. - "'J.dxJ·) .. 1 J J

We have

(9.7) A subset

Let f : M -

• ~ 0

V

of

tRM

is said to be complete if I'RM(4o) - ••

be a complex vector space of dimension

IP(V)

be admissible for

be a meromorphic map. f

if

Q~ . . . . ,Q~

A base

n + 1 > 1.

uQ,... ,un

is the dual base and if

Let

is said to

-'1.77-

f(M) It E[1P(
~: U -

If

open, connected subset U only if

< ~ ,00> iii! O.

V

is any representation of

M then
of

Let 00' .. · ,on

fl' ... ,fn

for

admissible base for

f

is admissIble, if and

on

f,

ailed the

M

in respect to the admissible base such t at

(9.8)

~. U -

for any representation

on an

be an admissible b se for

then there are unique meromorphic functions coordinate function of f

f

of

V,

V.

I, '" ,n

j -

If .uO'

,.un

is any other

then there are complex numbers

cj.l".

I:

II:

such that

(9.9)

.(i* U

Then we have n

!

f.l

(9.10)

cj".f".

".-0 n

!

cO".f".

". .. 0

Let

•

~ Ii!I

be a set of meromorphic functions on

is said to be analYtically dependent on

•

there is an admissible base

of

coordinate functions


f l , ... ,fn

analytically dependent on

.,

V for

fj

I:

for any choice of an admissible base
f

tRM(.).

all belong to then

fRMC.)

and

fR M(.) for ...,~

M.

Then

f

if and only if such that the If f

is

j - 1, ... ,n

by the transition

formula (9.10).

•

The meromorphic map f is said to be analYticallY independent of and tRM(., if and only if f is not analytically dependent on •. If

QO' ... ,
is a base of

V admissible for

f

and if f I' ... ,fn

are the

·278·

coordinate functions, then if there is an index PROPOSITION 9.1. ~

•

j2J

j L t

for this base.

Ta e

0 ~ &-

df j " B :: 0 for

(3)

d< «)~6>

(4)

(<<)

~

bv

E

(3).

a:

A

A

1, ... ,n

j -

on

B :: 0

B

be a

be a base of

V

f(M) II; E(1P(&-)).

Let

Then the following statements

(1) <= (2).

u~,

Since fj

... ,u;:

on

M.

U.

on

d «») " B :: 0

Let

Let

be the coordinate functions of f

such that

V*

E

M.


Let

d pends analytically on &M(.)·

(2)

(2)

f I' ... ,f n

V be a representation of f.

are equivalent: (1) The map f

PROOF.

Let

if anly on

be a meromorphic map.

morphic functions on g Iit M(.).

and I t

f

1P(V)

•

fj E IitM(.)·

such that

f: M _

be a set of

admissible for

is analytically independent of

INU,n)

E:

meromorphic form de

«) : U -

f

U.

IitM(.)

E

if and only if

be the dual base to

df j

A

""'0' ... ,
n

exist such that

&- -

};

v-O

bvuZ.

n «)-

};

j-O

< «) ,a.>

g

-

< «) .u*>
n

};

j-O

b·< J

«)

'



On

B :: O.

U

we have

.",:;,. Hence

dg '" B = O.

We have

d[

< t:) P• (,)I~ >

i) '"

B

f

.:i g

oUI.] J '" B

gdf

(3) ~ (4).

f

g"

1

B

We have

(t:)

(4) ~ (2).

B -

"

]

'"

d'£l) " B- < '£l ,f,> '£l (,)I~,..

Let

(,)In

< '£l~ p; > '"

[d

"

be the dual base to

B] - 0 .

~''''

,(,)In'

Then

< '£l .oUI~>

- d

----=- '"

B

.oUI~>
.

< '£l

oUI*] > -

<'£l .oUIj>

'" B

* >2 <'£l .oUI O

< ( '£l

'"

d'£l)

'"

(,)1*

B•

A

0

(,)1* >

J

- 0

*>2 <'£l .oUI O

U.

on

Hence

df j '" B - 0

The meromorphic map respectively g :

M -

1P(V*)

LEMMA 9.2.

then

f

fit(.)

on

M.

q.e.d.

f: M - 1P(V)

if and only if

(f.g)

is said to be free of

which is analytically dependent on

If the meromorphic map

is analytically independent of

•

is free for every meromorphic map

f. M ~M(.)·

fit(.).

1P(V)

is free of

~M(.l.

·280· PROOF.

Assume that

CAD' ... ,CAn

f

be a base of

the coordinate functi ns. CA O'

••• ,CA~

for

j - 0, ... ,no

f

be the d

on the open, con e

Therefore

~

CAD' .. , ,CAn

: U _

V

'W:J

Hence

CAt, CA O'

IS

gl"2"3' ... "n

(.IIi, ...

,CA~

M.

of

1P(V·'

a representation of

g

~

with

<"oIO, CA 1> - < ~ ,CA O> iI! O.

LEMMA 9.3.

Let

f

g

for

g.

Let

for this base.

j - 2, ... ,n.

> - «(.1110 " (.IIIt)L

is not free.

~,~

~ "

Contradiction!

be a meromorphic function on

be a set of meromorphic functions on

M.

~

> :: 0 .

q.e.d. M.

Let

• ~ 0

Then the followmg statements

~M(.)'

The map

f : M - Fl

is free of

(2) The map

f: M - 11'1

is analytically independent of

(3) The meromorphic function

f

Hence

>

are equivalent: (1)

Then

We have

- «.1110 " CAt, (f,g)

Also

is defined as a meromorphic mal

gj:: 0

Also

~M(."

< ~ ,"010> - <'W:J,

Therefore the pair

< ~ ,CA O> iI! 0.

Then

be the coordinate functions of

is analytically dependent on

Let

a·J - 1P(CA~)

and define

is an admissible base of

g 1 - - fIE: ~M(.)'

be

fl' .. , ,cn

be an irreducible representation of

U

ed subset

Let

1, •.. ,no

j -

for

~M(.)

E:

1 Lf - g : M -

where

Therefore

fj

Then

and let

f

admissible for

V

base of

L t

~M(·)·

is analytically dependent on

does not belong to

~M(."

~M(."

II

·281·

(1) ~ (2).

PROOF.

(2) -

-UO.

Let

(3).

a:

1P(a: 2) - 1Pl -

by Lemma 8.2.

(.111

V (00)

be the standard base of 1P(uO + Z(AI) - z

by

for

coordinate function of the DleroDlorphic map independent of

f: M -!PI

there is a meromorphic map

and

such that the pair

"tIC: U -

on

U.

Let

If VI :: O. Therefore

where

g

base

ut.

P

a:.

&:

Then

u

f.

Since

v·

e*.

then

g: M (f.g)

IS the

is analytIcally

tRM(.).

is not free of

I't

is not free.

Let

10:

Then

be the dual base of uo. ul'

ut

f - 0

&:

tRM(.)

which is wrong.

V -

U -

< 10

• "WJ >

a: 2

:: 0

We have

Hence

VI - O.

Hence

is the meromorphic coordinate function for Hence

Then

which depends analytically on

be reduced representations.

Wo lI! 0 - WI'

u~.

f

f

tRM (.) we have f E tRM(.' as a function.

Assume that

(1).

tRM(.,

z

and identify

In respect to this base the meromorphic function

lP(u l , - 00.

(3) -

0;2

f - - if

&:

tRM(.).

g

in respect to the

Contradiction!

q.e.d.

Let B be a meromorphic form of bidegree (m - 1,0) on M. Let be the set of poles of B such that M - P - A in the largest open

sUbset of

M where

B

is holomorphic.

Let

f: M -

I'(V)

be a

·282·

meromorphic map. p

Then the concept of associated maps

0,1, ... .R f = .R fIB)

=

is defined on

to meromorphic maps

fp: M _

representatIOn, then

'D..2,: U " A -

U"

on

A

We say that

and f

'D..2,

is

THEOREM 9.4.

Let

B

f: M -

is a reduced

is a representation of

fp

meromorphic vector function on

U.

.Rf(B) ~ n.

if

W(V)

V

'D: U -

Gp(V)

II

for

and the associated maps extend

If

Gp(V).

ex t en d s t 0

'---..!tl for

A

fp

be a meromorphic map.

Let

• t.

ra

be a set of meromorphic functions of rank m - 1. Let B be a meromorphlc form of bidegree (m - 1,0) on M Assume that B defines ti M(.). Then f is free of ti M(.) if and only if f is general for B. PROOF.

a)

Assume that

poles of

B

and define

(~A.' 'D A.1A.EA

Let

~ A. : U A. -

U{

f

A - M - P.

and each

U A.

is a chart of

p

=

f.

M

Here

and

'D A.

(UA.)A.EA

Let

Abbreviate

P

be the set of

is holomorphic on f.

V

is a

is an open covering of

p - .RfIB).

The trdnsition We have to show

n.

Assume that

p < n.

For

(A.,/J.) E A[l)

define

on

UA./J.'

(9.11)

Then (9.12)

Here

'D A...2,

""0' ... '''"n

is hoiomorphlc on

of

V

such that

UA." A t.

A.

Then

U A. -

is Stein and a Cousin II domain.

rules (4.10) and (4.11) hold. that

B

Then

be a representation atlas of

reduced representation of M

i! free of IiM (4I).

ra. Take

A.O

E

A

and

-283-

(9_13)

For each u~

-

~

Then

E

A,

18'~

18'~

define

IJ~.£" 'ft.p+ 1 " ... " 11. n - 1

=

U~" A

is holomorphic on

on

and meromorphic on

U).

Also (9.14)

Since U~

u~u

on I8'~O ~ 0,

for all

~

E

the c nne tlvity of A.

11.~.... ,11.~

Let

be the dual base to ~~

meromorphic vector functIOn

: U~

(~,$A) E A(l) .

implies, that

M

18'~" 'ft.n ~ 0

S mllar

if

--0

for all

18'>... 0

>..

E

11.0'... ,11. n . V·

on

A.

A

is defined by

(9.15)

Here

~~

u~"

is holomorphic on

~~

9.16)

A.

u>"U...o /..l

=

If

(~.u) E A[l).

on

then

U~/..l'

We have

on Hence g : M of

g

~ --0

>..

~

l'(V*)

for each

0

on

U>.,'

Hence one and only one meromorphic map

exists such that

>..

E

A.

u~.

For

~ E

'\10

>.. I U~ n A is a representation

A we have

-284-

Hence the pair Take

-.. >..

'I)

(f,g)

>..

E

A.

A

tJ

>..,

is not free. On

A

...

U>..

S>..

such that

tJ

;It

functions

A>")I

>".2,(z)

0

10

of

(p) A

U>..

tJ,

"t:)

Z E:

U)" - S>..

( p+ 1 )

tJ )"

(p+l) A ....

exists such that

for all

exist on

A.2, ~ 0

-. . >.. (p) -

"

, A "" ~ A " •.• "

An analytic subset

we have

such that

~

A

-'&'),,)1

10

v-o

on

)"J!.)' -

U)" - S)"

10

>..

A

10

~

"

which implies

.. , "

tJ

~ p-l)

A

()I) >..

A.2.:!:.!. :: o .

S>.. ~ U>..

U)" - S)" 'I: 0.

Then we have

( tJ

1J

nP

and

Unique holomorphic

-285-

therefore on

UA." A,

which means

By Proposition 9.1, the meromorphic map is analytically dependent on ~(.).

Since

f

6lM(.),

is free of

contradicts an earlier ob e

atlOn.

b) Assume that f a meromorphic map Whl h

the pair

Hence

p

(f,g)

is free which

n.

general for B. Let G: M 1P(V·) be ends analytically on 6l M(.). Take reduced

representations t::I: U V of f and "tIC: U V· of g and assume that 1J: U U' is a chart. Take veclors 'I 0 and <...0 ,&> 'lOon U. In facl by taking U smaller we can assume that < t::I , and <...0 ,&> are nowhere zero on U. Then t::I are reduced representations of f Proposition 9.1 we have

t::I /< t::I , and ...0 - '110/<...0.&> respectively g on U. By

.;; '1' - d <"tIC...0.&> ABO ~':: O.

Hence

<;; ,~> all

:: 0

on

1, ... ,n.

j -

Assume that the pair U.

q.e.d.

is not free.

By induction, we see that

'" ,

f

U.

<;; (j),~>

Then :: 0

for

Here

t::IAt::I

since

(f,g)

on

is general for

B.

A

'" (n) _

... At::I

Therefore

t::I,!!

'\40

_

0,

'I 0

which is impossible.

·286· We may ask if the assumption in Theorem 9.4 that B has bidegree (m - 1,0) is a consequence of the other assumptions on •. In this repect we prove: LEMMA 9.5. Let • I! " be a set of meromorphic functions of rank m - 1 on M. Assume that tiM(.);I: tiM' Let B be a meromorphic form on

M defining

ti M(.).

Then

B

has bidegree

(m - 1,0).

PROOF. Let (p,O) be the bidegree of B. If p - m, then dCP A B :: 0 for all cP E tiM' Hence tiM(.) - tiM which is

impossible.

Hence

gl' ... ,gm-l

0 ~ p ~ m - 1.

be a base of tiM(.)·

An open, connected neighborhood on

Xo

Take

E:

M

Let

such that

gj

Xo and a holomo phic function

U of

U exist such that

is a chart. Let S be the set of all injective increasing maps u : bII[l,p] bII[l,m]. There are meromorphic functions Bu on for

U E: S

For

E:

Hence any

1W[1,m - 1) define

S.

U

such that

Bu:: 0

U E:

is

Xo and such that

holomorphic at

gm

p < m .- 1.

Assume that

on

Then

Sj - (u

U for all

U E:

E:

SIj

Sj

11m U = p < m - 1.

if

E

1m u).

Then we have

j E: bII[l.m - 1].

Hence

j

E

Take

IWU,m - 1]

·287·

exists such that

U

B == O.

implies

Hence

p - m - 1.

Hence

Sj.

E

Therefore

Bu == O.

Thus

tiMt.) - tiM[B] - tiM

B I U == 0

which

which is impossible.

q.e.d.

Let M and N be connected complex manifolds of dim n on Let f: M N be a meromorphic map of rank m and with Then dim If ~ m - 2. Then there exists a indeterminancy If non-negative divisor f

IJ

IJf

M - If'

~: U -

Let

an open neighborhood of ~

Then W.

Then IJ

M

called the ramification divisor of

tor also the branching d'V1 or) such that we have the following.

Xo E

on

on

Since

f

with

Xo

.fm)

flW - tf 1,

0

U'

has ra k

m,

M - If

Since

uniquely to a divisor on LEMMA 9.6.

Let

W ~ M - If

d'm If

~

ftxO)

f(W)!; U.

df 1 " .•. " df m ~ O. on

the divisor

W.

IJ

This defines

extends

M

iI' ... ,1m

be analytically independent meromorphic

functions on the connected complex manifold M of dimension A meromorphic map g of rank m is defined by I

(9.17)

Let

)

g - ,gl' ... ,gm IJ g

be the ramification divisor of

holomorphic functions j - I, ... ,m. gj for

j

=

Let

Vj

I, ". ,m.

W be

are holomorphic functiOn!

df 1 " ... "dfm m - 2,

Take

and let

such that

f1' ... ,fm

we know that

IJ I W is the zero d Visor of

on

be a chart of

where

Vj. 0

~

Wj

g.

M _

m.

I'm 1

Assume that there are

such that

Wjgj - Vj

on

M for

be the zero divisor of the representation

(Vj,Wj)

Define the holomorphic form

(9.18)

Then

y

~

O.

let

uy

m.

be the zero divisor

of

y.

Then we have

of

-288-

(9.19)

It suffices to prove (9.19) at every point

PROOF.

Then either

gj

or I/g j

Xo

or both are holomorphic at

E

M - Ig .

xO'

By a change of

enumeration, we can assume without less of generality, that holomorph c at 1 ~ j ~ n,

n + 1

~

j

xo

then

~

m,

and we have

and that (gJ 1

t

(vj,Wj)

n =

gn+ l' .. , ,gm

have poles at

is a reduced reprensentation of

g1' ... ,gn xO'

If

at

Xo

gj

g.

at

Also

Pg

is a reduced representation of

(l,l/gj)

(l/l/gj)vj .

0

Hence

)/j(x) - Ilv (x O)· j

is the divisor of

at

Xo

where

Therefore

ll y (xO)

= 2

n

I

j=l

Il~ (x O) + 2 J

m

I

j=n+l

0

Ilv . (xO) + Pg(x O) J

m

- 2

I

j-l

)/j(xO) + Pg(x O' .

q.e.d.

J

are

and

Xo

·289·

LEMMA 9.7.

Lt't

f,gl' ... ,gm-l

be analytically independent meromorphic M.

functions on the connected, m-dimensional, complex manifold Vj iiS 0 ~ Wj

be holomorphic functions on

j -

1, ... ,m - 1.

gj

for

j - I,

Let

))

,m -

1.

such that

M

be the divisor of the representation

j

Let

wjg j - Vj

A holomorphic form

B of bidegree

a meromorphic map.

Let

for

(Vj,Wj) (m -

01

1,0)

is defined by

(9.20)

Consider of the Let

p

f: M 1st

lP1

(and last) a

a

ted section

OCI

F1

of

f

JJF

be the divisor

1

in respect to

B.

be the ramificat on divisor of the meromorphic map

9.21)

Then we have

9.22)

ttF

PROOF.

Let

}}: U -

U'

reduced representation of t:l '

-

(vo,w o )'

Define

f

1

be a chart. on

m-1 ~ ))j'

- P + 2

U.

""0 - (1,0)

J-1

Let

Then and

t:l

wOf -

-

Vo

""1 - (0,1).

(vO,wO)

be a

Also Then

m-l (t:l

"

t:l

')t

-

(t:l

"d

t:l) ,...

B -

A

j-O

(vjdw j - wjdvj)""O ,... ""1 .

-290-

Since

Wo

v O'

are coprime at every point of

V,

Lemma 9.6 implies

m-l

UF If

f

2

P IV + 2

IU - JJ.1Q,,1Q'

is h om orphic on

w. - 1

and if

U

J

on

~

q.e.d.

Vj IV.

j" 1

U,

then

(9.23)

We will make the following Jeneral assumptions: (CI)

Let

M

be a connected, complex manifold of dimension

(C2)

LE't

l'

be a parabolic exhaustion of

(C3)

Let

• ~ 121

on

M.

be a set of meromorphic functions of rank

Let

V

(C5)

Let

f: M -

(C6)

Assume that

(C7)

Let

Ii

with

be a hermitian vector space of dimension

~

1P(V) f

is free of

fR M(.).

be a finite set of meromorphic maps

121

k = llii

~

(C9)

Assume that

QI

is in general position.

(ClO)

Assume that

B

is a holomorphic form of bidegree

which defines l'

g

E

QI

is analytically dependent on

if - n

M -

1P(V"')

majorizcs

in respert to

B

B.

(m - I,D)

on

with majorant

Y. f

is general for

By Proposition 9.1 (4) we have

o

(9.23)

fR M(.).

l'itM (.).

According to Theorem 9.4, the meromorphic map thid is

g

n + 1.

Assume that every

Aso;ume that

n + 1 > 1.

be a meromorphic map.

(C8)

(Cll)

m - 1

M.

(C4)

M

m.

for all

g

Eli.

Assumptions (ClI to (Cll) and Theorem 9.4 imply (81) - (B8), wherE' iff.

=

0

for all

g

E:

q).

Therefore Theorem 8.8 implies

B,

-Z91-

THBORBM 9.B. Second Main Theorem for functions fi Ids. Assume that (CI) - (Cll) hold. Take € > 0 and s > O. Then we have (9.24~

Np (r,s) + n

:L

mf gIrl

gEl)'

+ 2nln + l)C.lk log T fIr,s) +

I

log+ r glr,s)

I-

k log Rlc;Ir,s))

gEIJ

+ 2n(n + l)C.k log Y( ) +

log r .

€

Also we have (9.25~

(k -

n -

l)T f (r s)

+ 2n(n + l)c.k(log Tf(r,s) + log VIr) + log+ Ricr(r,s)) +

E

log r .

In addition, we assume (C12)

Tglr,s) Tf(r,s) -

0

for

r -

00

Ricr(r,s) (CI3)

11m

sup

r"OO

~

Tf(r,s)

for each

g E IJ

0

(C14)

Suppose that (CI) - (CI4) are satisfied. Nevanlinna defects mE

(926)

6f(g) -

6 f lgJ -

1 i min r ... oo f

Tf

Then we define the

t g(r) r,s )

-292-

(9_27)

1 -

11m r

(9_28)

"'00

11m sup r ... 00

Since

N( n )(r s) ~ N f • g'

(r,s),

-

we have

(9_29)

The Second Main Theorem 9.8 implies the defect relation. THEOREM

l i Defect relation. Assume that (Cl) - (C14) hold. Then

(9.30)

Our statement (9.53) does not contain any term dependmg on However, the assumptions (C10), (Cll) and (C14) depend on a situatlon deplored by Royden.

B.

B.

We are in

An assumption which has seemingly nothing

to do when the statement is made for the sake of the proof only.

In the

case

B

M = ([m

we will be able to eliminate these assumptions on

constructing a form

B

satisfying (CIO), (Cll) and (CI4) and to some

extent the rpsult can be established even if parabolic space of

T(Z) -

IIzII 2 -

a: m

where the exhaustion is

1zll2 + ... + 1~ 12.

majorizes any holomorphic form of bidegree

For any holomorphic vector function (9.31)

is a covering

([m.

First we consider the situation on given by

(M, T)

Then

u > 0

and

(m - 1,0).

g:

a: m

-

V

defme

T

by

·293·

LEMMA 9.10. on

a: m.

For

Let U

B

be a holomorphic form of bidegrpe

1N[1,mj

E

(m

1,01

define

(9.32)

Then

(9.33)

Define the holomorphic ve tor function Let

Y

be the majorant of

Y(r)

(9.34)

=

=

(B 1, ... ,Hm) ; a: m _

in respect to

T.

a: m.

Then we have

Max(l,mM(r,6)2)

log Y(r) ~ 2 log + M(r,6) + log m

(9.35)

PROOF.

We have

Take

r > D.

a:m[rl

if and only if

~

constants

E

c.

c > D,

Take

m

(9.36)

for all

B

fr

then

mim_1B"

B~

m

m

~

~

U~l

U,lJ",l

a:m[r) Then

and

~

E

a: m.

Let

YO(d

cum-Ion

I x 12 = cUt-liZ jJ.

be the infimum of all those

-294-

which i

mM(r.6 2 ~ YO(r).

lies

Max(1.mM(r.6)2)

(9.37)

If

mM(r.6)2

y(r) - 1 YO(r) ~ 1

Therefore

~

1.

then

~

m 1(6(ll-)I tJ I

and we have equality in (9.37). and YO(r) - VIr). Also

which implies

YO(r) ~ mM(r ,6)2.

In particular.

Y(r)

~ Ylr) .

IIt-II 2

If

and

YO(r)

mM(r.6)2 > 1

~

1.

Hence

then

Hence

is continuous in

r

which proves (9.34) and (9.35).

q.e.d. LEMMA 9.11. Let f be a holomorphic function on j £ INIl,ml. r > 0 and £ > O. Then

a:m. Take

(9.38)

PROOF.

Take

ll-

€

a:mlr]

such that

I fz.(ll-) I - Mlr.fz )' J

"'-1' ... ,"'-m h : a: -

be the standard base of

a: is defined by h( n

-

Let

J

a:m. A holomorphic function

f(ll- + l""'-j)'

Then

h '(0) - f (ll-). Zj

If

-295-

t E a:[El,

then

II~

I h(t) I li!: M(r + E,f)

M[r,

aClZfj ]

-

+ t1\}1 li!: 11'&11 + I t I :iii: v +

t E a:[El.

for all I fz.(~) I l

=

E_

Hence

The Cauchy estimates give

1 Ih '(0) I li!: E M(r + E,f) .

q e.d.

We need the following result of H. Skoda [851 Theorem 3 on THBORBM 9.12. Take E > O.

Let

f

a: m.

be a non-constant meromorphic function on

a: m.

a) There are entire functions g iii! 0 and h Ii! 0 on a: m such that hf - g and such that for each s > 0 there is a constant c( E,s) > 0 such that (9.39) for each

Max(1og+ M(r g) log+ M(r,h)) li!: c(E,s)(l + r)4m-1 Tf (r + E,S) r > s.

b) There are entire functions g iii! 0 and h iii! 0 on a: m such that hf - g and such that for each s > 0 there is a constant c( E,s) > 0 such that (9.40) Max(log+ M(r,g),log+ M(r,h)) li!: c(E,s)(l + (1og(1 + r 2))2)T f (r + Er,s) for each

r > s.

RBMARK. The pair (g,h) may have a common divisor, may depend on E but may not depend on s, and may be different in b) from the pair chosen in a). The case a) is good for rapid growth, the case b) is good for slow growth. PROOF. Skoda proved the theorem if f(O) - 1 and I - O. If and 0 'I: a E a:. Then there is a constant COla) such that

s > 0

·296·

Also we have TfIr + £,s) + T£,s,O) - T fIr + £,0) . s > 0

Hence the theorem extends to the case at

0

and

flO) 'I:-

A surne that eith r f

f

is holomorphic at

is not holomorphlc at zero.

that f function

is holomo h c at (.II

f

on

where

f

is holomorphic

o.

M by

0 with

flO) - 0

11(.1111 <

or tha1

Take (.II £ ([m

with

and f((.II) 'I:- O.

Define a merom orphic

tilt) -

f(~ + (.II)

for

It

IE:

such

£ ([m.

The function f is holomorphic at 0 with fiO) 'I:- O. Let i, h be the solut·on for a' respectively b,. Define entire function g and h

by

g('!' -

Then we have

ib - (.II,

hf - , • 0

h(lt' - h(lt - (.II, .

and

M(r", , M(r + £,i1

for all we have

r > O.

Define

([mlr,(.IIJ - ('!

([m[r,(.IIJ ~ ([mlr + E:J.

M(r,h) , M(r +

IE:

([m I lilt - (.1111 'r).

Hence f-(O) " v m- 1

I and

IE: ,h)

f-(O) " v m- 1

For

r > 0

-297-

,

[1 +

f) 2m-1

Tf(r + £,8) .

In case a) we obtain Max(log+ M(r,g) log+ M(r,h)) , Max(log + M(r + £,g1.l0g+ M(r + £,h)) , c(£,s)(l + r + £ 4m-1 Ti(r + 2£,s)

Thus if we replace obtain (9.39).

£

by

£/3

and redefine the constant

Now consider the case b): Max(log + M(r,g),log + M(r,h)) , Max(log+ M(r + £,i),log+ M(r + E,h))

Assume that

r > 1 + s,

then we contmue

c(£,s)

WI

-298-

Tf(r + 2E:r,s)

Since

there is a constant

s:E: r :E: s + I,

is bounded above and below for c 2(E:,s)

such that

Max(log+ M r,g),iog+ M(r,h))

Replace

E:

by

E: 2 and redefine the constant.

Then we have (9.40).

q.e.d. LBMMA 9_13. functions on

Let

vI'''' ,vm-l

and

wI' ...• wm-l

be holomorphic

a: m. Define

(9.41)

Let

Y be the majorant of

r > O.

B in respect to

.,.

Take

E: > 0

and

Then we have

(9.42)

log Ylr) :E: Z

m

L

+ log M(r + E:,Wi + 2

J-l

m

I

j-l

log

+

Mlr + l:.vJ,)

+ 21m - l)1og + 2E:m + log m

PROOF.

Let

19.44)

Let ""'1"" ''''''m

zl""'Zul

be the standard base of

be the coordinates of

a:

~j

a: m. Defme

mand define :vZu 1

v·

JU

b3

an<

·299·

Then we have

(9.45)

Define

Yju - WjVjIJ - VjWju

and

Itj - Yjl''''l + ... + Yjm""m'

m

& -

I

U-l

BU""IJ'

Then we have m

I B/''~IJ U-l ...

- Itl '" ... '" Itm-l

or

Hence we have

The estimate (9.58) implies log Y(r)

, 2 10g+ M(r,&) + log m

, 2

2m + 1og m . + 2(m - 1)1og + "7"

q.e.d.

Define

-300-

THEOREM 9.14.

Let

cpl' ... ,CPm-1

meromorphic functions on a)

he analytically independent £ > O.

Take

a;m.

B

There there is a holomorphic form

which d

mes

such that for each

tRa;m(cpl' .. · ,CPm-1)

c(£,s) > 0

is a constdnt

of hidegree

such that the majorant

of

Y

(m - 1,0) s > 0

B

on

there

for

T

can

on

a;m

he estimdted by

log VIr) ~ c(£,s)(l + r)

(9.46)

for all

4m-l m-l ~ j -1

Tcp,lr + E,S) J

r > s.

h)

There is a holomorphic form

which defines

&

a;m

(cpl' ...

c(£,s) > 0

constdnt

,Cp

m-

B

of hidegree

(m - 1,0)

such that for each

1)

Y of

such that the majorant

s > 0

B for

there is a T

can he

estimated by

log VIr) ~ c(£,s)(l + (log(1 + r2))2)

(9.47)

m-l

~

Tcp.lr + Er,s)

j -1

J

for r > s. c)

If

cpl' .. · ,CPm-l

holomorphic form & a;m (CP 1' .. · 'CP) m-1

B

for

of hidegree

(m - 1,0)

and a constant

c > 0

on

I[m

which defines

such that the majorant

Y

of

can be estimated hy

T

log VIr)

(9.48)

REMARK.

B

are rational functions, then there is a

In all there cases

Vj iii! 0 iii! Wj

~

B

clog r

r

~

2 .

can he obtained in the form (9.44) where

are holomorphic functions on

j - 1. .. , ,m - 1. on s

for

In a) and h), the form

a;m

B

with

WjCPj - Vj

depends on

£

for

but not

-301-

PROOF_

B is defined as indicated in the Remark, then

If

Hence it remains to choose

I, ___ ,m - 1.

j -

If

Cl'I'" - ,CI'm-1

are rational, then

taken as polynomials and (9.48) follows from (9.42). case c). a' Vj' Wj that

Each

is not constant.

Cl'j

does not depend on

j.

w{Pj - Vj

m-l

~

j

COlE)

T
is a constant. for

00

r -

Here

00.

for

J

v j' w j

can be

This settles the

;I

0_

We Cdn assume

The estimates (9.39) and (9.42) imply

log VCr) ~ 4C(E,S'(1 + r + E)4m-1

where

l'

By Theorem 9.12, we can choose

to satisfy Thereom 9.12 a) with C(E,S)

B defines

v- w-

=1

Tcp.(r + 2E,8) + COlE) J

1 + r + E

(l + r)(l + E).

~

Hence increasing

C(E,S)

Also

to a sufficiently

J

large constant

c 1(E ,s)

we obtain

m 1

~ 4cl(E s)(1 + r)4m-l j~l Trpt + 2E,8) .

log VCr)

If we replace b)

E

by

E/2

and redefine the constant, we obtain (9.46).

By (9.42) and Theorem 9.12 b) we obtain

log VIr) ~ 4c(E,s'(1 ~ (log(l + (r + E)2))2)

m-l

~

T g.((r + E)(l + E))

j-l

where the constant

COlE)

can be absorpt again by increasing

Without loss of generality we can a&sume that can replace

E:

by

£2/3.

J

Then

0 <

E < s < rand

E:

C(E,S).

< Min(l,s)

and we

·302·

2 2

, 1 + (log(1 + 4r ))

, 1 + (2 log 2 + 10g(1 + r2))2

Hence redefining the constant Let Jr : M -

M

c:m

yields (9.47).

C(E,S)

q.e.d.

be a connected, complex manifold of dimension be a proper, surjective holomorphic map.

is a parabolic exhaustion of coverinc space of as defined by (2.14).

c:m. Let

M and

(M,")

Ric.,.(r,s) - Nper,s)

(Stoll (103)). A divisor V ~ 0 n )1(00) < 00, that is, if N vIr,s)

=

(9.50)

f

Jr(suPP)I)

is given by Jr.

Then

M -

is affine algebraic.

P(V)

has rational Irowth if and only if

1i m

., - IIJrll 2

on (M,") has rational crowth if OUog r) which is the case if and only

if the (m-l)-dimensional component of A meromorphic map

r .... 00

Jr

be the branching divisor of

(9.49)

(9.51)

Let

is called a par bolic

The generic sheet number of p

Then

m > 1.

Tf(r,s) 10& r - At(oo) <

00

C.

·303·

and transcendental growth if order of

f

•A.(oo) "1

00.

The ---!ll: ord !Lf f

and the lower

are given by

(9.5Z) sup log Tf{r,s> ____ log r r -+00

lim

Ord f

--;--::"~

If there is a meromorphic map then

lim inf r-+oo

a: m -

h

1P(V)

log Tf
such that

Ord f

f - h

D

7t,

(9.53)

Hence f has rational growth if and only if h has rational growth which is the case if and only if h is rationaL Also f has finite order if and only if h has finite ord with Ord f - Ord h. THEOREM 9.15. The S cond Main Theorem for function fields.!rl.m: em. Let V be a hermitian vector space of dimension n + 1 > 1. Let M be a connected, complex manifold of dimension m > 1. Let be a surjective, proper, holomorphic map of sheet number

7C: M _ g:m c. and with

branching divisor p. Define T - 117C1I2. Then T is a pardbolic exhaustion of M. Let 'PI' ... ,<1'm-1 analytically independent meromorphic functions on

g:m.

Define

¥rj = q>j

D

7C.

Then

analytically independent meromorphlc functions on I; - ~M(¥rl' .. , ,¥rm-l)·

Let

¥rl' ... ,¥rm-l M.

are

Define

be a finite set of meromorphic maps

I)

1P(V*) with k - lit) > n + 1. Assume that I) is in general Assume that each gEl) is analytically dependent on 1;. Let 1P(V) be a meromorphic map which is free of 1;. Then we

g : M position. f : M have (9.54)

(k - n - l)T fIr,s)

E:.

~

gEt)

k[k

N f
-n

1]

"t' L.

gEl)

T (r,s) + g

+ 2n(n + l)c.k(log T fIr,s) + log + N p(r sll + SIr)

n


1)

N P (r, s)

-304-

where the following estimates of the remainder terms SIr) (A) Take E > 0 and s > O. Then there is a constant

are available. >0

COlE,S)

such that

SIr) :E: cO(s,E:)(l + r)4m-1

(9.55)

m-l

:r

j -1

(B)

T ke

E

> 0

s > O.

and

r > s .

for all

(r + £,s)

T "'j

c 1(E ,s) > 0

Then there is a constant

such that (9.56)

for

(C)

Assume that each

j - 1, ... ,m - 1.

'" j

II

constant

Assume that

constant

c3 > 0

CP1'''' ,CI'm_1

such that

are rational functions.

Then there is

such that SIr) , c3 log r

(9.58)

PROOF.

> 0

for

r > 1 .

for

(9.57)

(D)

C2(A)

r > s .

Ord '" j < A

has finite order and that

Then there is

all

for

Clearly (C1) - (C9) are satisfied with

holomorphic functions

V J"

wJ' on

«;m

r

~

2 .

• - ,.

such that

There are

w·CI'. - v. • 0 J

J J

and

such that

(9.59)

(9.60)

is a holomorphic form of bidegree liO - tR m(CPl' ... ,CI'm 1)

.:

-

(m - 1,0)

which defines

and which is majorized by

"0

on

.:m

with

II

·305· majorant Y satisfying the estimates (9.46) or (9.47) or (9.48) in Theorem 9.14. Then ., - "0 0 7C majorizes the holomorphic form B of bidegree (m - 1,0)

on

M

with the same majorant

is a holomorphic function on

M

Y.

Also

w - w

o

0

7C

iii 0

with

(9.61)

B defines { - liM("'I' ... ,!/1m-I)'

Hence

The assumptions (C10) and

(C11) are satisfied and (9.25) holds. If we assume without loss of generality, that 0 < E ' 1 and if we define SIr) ~ 2kn(n + 1)C. log Y(r) + log + r

(9.62)

then (9.25) implies (9.54)

Observe that

T!/I ,1r,s) -

00

for

s -

00

J

since

!/I j

is not constant, and that

T'P. - T",. . J

J

In the case of Theorem 9.14 a), the constant mcreased to a constant

CO(E,S)

as to absorb

C(E,S)

can be

2kn(n + 1)C.

and

and we obtain (A). In the case of Theorem 9.1 b), the constant can be increased to a constant c 1(E,s) as to imply (B). If

'PI'''' ,q>m-1

Assume that Ord 'P j < I. <

log+ r C(E,S)

are rational, Theorem 9.14 c) and (962) imply (C).

"'1' ... ''''m-1

have finite order and that

> 0 such that

00

for

j - I, ... ,m .- 1.

Take

Ord 'P j < >.. - TI

for

j - I, ... ,m - 1.

Then there are constants

and

Cs > 0

11

such that

T.,. ,1r,s) " c rA Tlfor .... J

4

r ~ 2

c4 > C

·306·

Applying 19.56) with

s -

E: -

1.

Then we have

q.e.d.

THEOREM 9.16.

Defect relation for function fields over

be a he m1t1an vector

n + 1 > 1.

pace of dimension

m > 1.

a 1fold of dimension

be a surjective, pro

r, holomorphic map of sheet number Let

p.

'l'l' ...• 'l'm-1

meromorphic functions on

([m.

Define

Let

7C:

'" j

=

'l' j

D

~ - I'RMI"'l' ...

''''m-1).

exhaustion of g : M on

~.

Let

Let

M.

!P(V).

Qj

!P(V)

and with

Then

.,

M.

Define

is a parabolic

be a finite set of meromorph1c maps

Assume that each

f: M -

([m

Then

7C.

are analytically independent functions on ., - 111'1"112.

be a

M _

c.

V

be analytically independent

"'I' ... ''''m-1

Let

Let

M

Let

connected, complex branching d1visor

«:m.

g

E:

Qj

is analytically dependent

be a meromorphic map which 1S free of

~.

Assume that

(9.63)

for

r ---

00

(9.64)

for

r ---

00

for all

g

E:

(J

•

Assume that at least one of the following assumptions (A) or (B) or (e) or (D) is satisfied. (A)

There is a number

(1

+ r)

(9.65)

for

E:

4m-l

> 0 such that

T",

Tf

- 1, ... ,m - 1.

(r.s) j

-0

for

r -

00

·307·

(B)

There is a number

E

> 0

such that

(9.66)

(1 + (108(1 + r2»2T~ (r + Er.s) ______________~~----~~1-----------

~

0

for

r ~

for

T f
00

1, ... ,m - 1.

j -

(C)

The functions

~1' ... '~m-1

Ord ~ l < Ord f ~

(9.67)

(D)

The functions

have finite order with

for

00

cp1' ... ,rfl m_ 1

j -

1, ... ,m - 1 .

are rational and

f

has

transcendental growth. Then we have

(9.68)

PROOF.

Main Theorem 9.15. Then

In Case (e) take

Tf(r,s)r->" _ . .

for

r -

00

A

by (9.62).

If

growth, then

cp1' ... ,CPm-1 Tf(r,s)/log r -

with

Ord

for

r -

are rational and if 00

~j

< A < Ord f.

and

(9.69)

(9.63).

Secon~

The defect relation (9.68) immediately follows from the

for

r -

0

f

00

has trancendental

and (9.69) follows from

q.e.d. Each of the condItions (A), (B), (C), and (D) implies, that

transcendental growth. If(SUPp p)

has

Thus if the (m-l)-dimensional component of

is affine algebraic, (9.64) is automatically satisfied.

separates the f bers of

f

11',

then

1i m

sup Np(r,s)/Tf(r,s) -

r ... oo

If

Rr

f

<

00

·308·

by a theorem of Noguchi [701 and the Ricci def('ct

If g

E

IJ

exists such that

g

Rf

appears in (9.68).

separates the fibers of

rr,

then

N (r,s) 0 ~ 2(C. - 1) T g (r,s)

1 i m sup r~oo

by the theorem of Noguchi [70].

Then (9 63) implies (9,64) and (9.68) hold!

Naturally. the Se ond Main Theorem and the Defed Relation simplif~ M - o:m

if we take Np(r,s) :: O.

Also

and

rr

1.

C, -

as the identity.

Then

oj!. = cpo

J

J

and

We leave it to the reader to formulatt> this

special case explicitly.

If M - ([;m

n - I, with

the assumptions again become simplicr. m >1

and

n - 1.

B. Shiffman already obtained the

two theorems under some what weaker results. (A) or (B) or (e) or (0) he needs only

(9.70)

Instead of the assumptions

for

for j - 1, ... ,m - 1. results [831. [841. THEOREM 9.17.

Tn this case,

r -

...

For comparison we st8t(' Shiffman's

ShIffman 1831. 1841.

meromorphic functions on

o:m

with

Let

f,gl' ... ,gq

m > 1.

be distinct

Assume that

(9.7U Then there is a constant

such that

(q - 2)T fIr,s)

(9,72)

If

cI > 0

T g .Ir,s)/TfIr,s) J

0

for

r _..

and

j -

I, ... ,q,

then we have

·309·

(9.73)

B. Shiffman shows that Theorem 9.17 follows from: THEOREM 9.18.

Shiffman [831. [841.

meromorphic functions on

([m

with

Let

f.g 1 •.... gq

q ~ m - 1 ~

o.

be d tmct

V

Assume that

(9.74)

Let

p

be the ramification divisor of the meromorphic map

Then there is a conc;tant

c > 0

such that

(9.76)

~

f

Nf g.lr.s) + c

J"'l'J

[.f J-l

T g.lr.s) + log Tf(r.s) + log+ J

r]

Our general theory forced us to make maximum modulus estimates for 8.

that is for the functions

Vj. Wj

such that

w{Pj - Vj.

Thus

applying Skoda's results we have to make assumptions (A). (B). (e). or (D) while Shiffman needs (9.70) only. will yiE'ld the same results.

Perhaps a modification of the B-method

Except for this small deviation Theorems 9.15

and 9.16 can be considered as an extension of Shiffman's Theorems 9.17 an( 9.18.

Hopefully, the assumptions that

"'I' ... ''''m -1

are lifted from

I[m

can be eliminated by further research. The B-method was invented almost 30 years ago. and has been used almost exclu'lively by the author (Stoll [93], [100], [l08\). The 8-method is justified again by the results of this section. Mori (63] obtains a defect relation for meromorphlc maps f : a;m _

1P(V)

and for moving targets

and extend his results in Section 11.

g.

a: m

-

1PIV).

We will discuss

';.10.

An Example

We WIll g ve an example where the integral in (6.19) cannot be split into three convergent Integrals according to (6.8).

We take Let

m - 1, M -

""0' ""1' ""2

the dual base.

a:,

B - 1. n - 2, V -

be the standard base of Define a holomorphic map

reduced representation

(10.1)

Then (10.2)

10

(10.3)

10 u

'It) - (O,1,2t) - ""1 + 2h2

(10.4)

(10.5)

We abbreviate

(t) -

(0,0,2) - 2""2

a: 3 f:

a: 3 ,

1P(V) - 1P2

and let

a: -

1P2

",,~, ""t, ",,~, be the

be

-311-

(10_6)

B = 1 + 41tl2 + Itl4

(10_7)

B- A+

C - 2 + Itl2

31tl 2

C2

3 _

Then we have

(10.8)

II

(10.9)

(10

10

112 - A

_1 L·

10 ) -

A meromorphic map

g

II

-

10

1112 - B II

-

"40

1P~

II: -

(10.13)

~

'It) - 1-1,O,2t)

II

=

-

It) - 10,0,2) - 21\.~

Then we have

2112

-

t(1 + 21 t 12)1\.0 + (1 -

4

1t 14)1\.1 + t(2 + 1t 12)-n. 2

is defined by the reduced

representation

(10_12)

10

1\.~ + 2t1\.~

-312-

(10.17)

(10.18)

('W)

.!. L

'W))

These maps are re

__

(1 _

-

2

I tl4)1\.~ - t(1 + 2 I tI )1\.

ed by the identities

(10.20) (10.21)

(10.22)

(10.23)

(10.24)

(10.25)

(10.26)

< 10 1. L·

(10.27)

<10

(10.28)

Recall

.!.' 'W) .!. L• 'W) > -

2(i2B + te)

-lL·lO'~2L·~1> -- -

2(te - t2B)

10

t

2 + t(2 + I t I )1\.~

-313-

>


(10.29)

Apq

=

.... "'""--=l!. ~ --:--:------=:.--::::=-_

II 0,9.11 II 0!l.=J..1I 1I...a.l!,1I II 0 ~ II

Hence t

(10.30)

AOI -

3 C + A 2 A _ t Ct • 02

A-f8

.fA

rs

(10.31)

(10.32)

(A jk ) is unitary.

where the matrix

lIo}1I

(10.33)

(10.34)

We have the volume elements

1I'"t10111

- - nA - - - k hO - 110112 II '\10 112 0 hI -

11011 110 1," 2 II 0 1."

W_

II~

R

II II "40 2 II II ~ 1."

Therefore we have (10.35)

- (21 "00 I

2

+ 1"10 + "Oll

2

B

)-:2 A

2

~

kl .

·314·

The

+

ma rix computes as

(10.37)

(10.38)

(10.39) Also we have

(10.40)

(10.41)

Hence

(10.42)

S~~ {

We have +11 > 0

(10.43)

<

0

1f

It I

<

1

-

0

if

-

> 0

1f

It I It I

1 1

+OO(t) > 0

everywhere.

J

(t(r)

Take

if

t ~ 0 0 < 8 <

I

(t(r)

>

SO 1 - _ 1 10

and

+01 > 0, +10 > 0

of

The the integrals

0,

but the integral

t - 0

and

I

a:( r )

do not exist since the integrands become infinite of order at

if

8 - 88 > 2

-315-

(10.44)

.

J [ .Olh~

u-

1.8+ 00

a:(r)

exists since the integrand becomes infinite of order and is of class

COO

hOlhi - 2.(A/.(B

(10.45)

elsewhere.

Since

hO - kO

I - 8B < 2

at

and since

is bounded we have

J

log [I +

:~] a

- log 2

J

log [1 +

:~] a

, c <

ROO(r) -

a:
(10.46)

R01 (r) -

00

a:
for some constant

10.47)

c > O.

Since

Poo(r) -

S 1. 0 ~ 0 O. 1

for

I t I > 1.

we have

J a:
for some constant

Co >

O.

In order to compute

(10.48) AB,fAB

(10.49)

P 01 (r)

we calculate

0

·316·

(10.50)

-, h (01

·02

c3~t3 + 2A 3

8 3 + 8 t- Itl 4C2 A3 4A 3

1 t-

l

"t 3

Hence

(01

2

·02 h l

A constant

cl > 0

--0

5

4'

for

t _

00

•

exists such that

(l0.51)

Thus the remainder terms

ROO' R 01 ' P OO ' POI

remain unbounded.

~ 11.

The Theorem of Nevanlinna-Mori

Let f be a non-constant, meromorphic function on a:. Let lJ be a finite set of meromorphic functions on a:. Assume th t f it g for all g £ lJ and that for

(11.1)

r _

00

•

Then Nevanlinna [661 wondered if his defect relation (11.2)

remains true.

He proved thls conjecture if

IJ consists of three elements

He defines

(11.3)

- 1,2.3

where

a 1 - 0, a2 - 1

Mori [63] extends this result to a meromorphic map and a set

IJ of meromorphic maps

g :

a: m

_

f

a: m

_

1P

1P* n

where IfIJ ~ n + 2 and where (11.1) holds for all g £ IJ. He assumes that IJ is in general posltion and that f satisfies a certain non degeneracy condition. Then (11.4)

anc

n

-318Mori enumerates lJ - (gl , ... ,g ) n+2 map

h :

a: m

-

an d hyperp Ianes

1P n

that

for

and he constructs a meromorphic

aj

E

in general position such

1'* n

j - I, ... ,n + 2.

Mori's theorem will be extended to maps defined on a parabolic manifold.

A hough the basic idea is the same, the modifications are

considerable, b t provi e better results and deeper insights. The following General Assumption g will be made (El, Let

(M,7,

(E2) Let (E3) Let

V

be a parabolic manifold of dimension

m.

be a hermitian vector space of dimension n + 1 > 1. f: M 1PIV) be a non-constant, meromorphic map.

(E4, Let Ii ~ 121

be a set of meromorphic maps

g: M -

1P(V*).

Ii is in general position with illJ - n + 2. (E6) Let (f,g) be free for each g E lJ· Enumerate Ii - (go' ... ,gn+l)· (E5) Assume that

(E7, Let

-<-'0.... .-<-'n

Define

be an orthonormal base of

-<-'n+1 - - -<-'0 - ... - -<-'n·

Define

V*.

aj - 1P(-<-'j)

for

j - 0,1, ... ,n + 1.

(E8, If j

E

Z[O,n + II,

then

a

Tg.lr,s,./Tf(r.s) -

for

r -

00.

J

REMARK 1.

These assumptions suffice for the construction of the

Nevanlinna-Mori map and the comparison of the defects. relation requires additional assumptions. REMARK 2.

The defect

It would be suffice to assume that

are in

general position, SLRce the theory does not depend on the choice of the hermitian metric on THEOREM 11.1.

V.

(Mori [631.)

exists a meromorphic map j ~ 0,1 .... ,n + 1

cp(s)

for

Assume that lEI) - (E7) hold.

h: M -

1P(V)

and such that for each

p - I, ... ,4

such that for all

such that s >

a

r > s

(h,aj)

Then there are free for

there are constants we have the estimates

·319·

n+l

Th(r,s) ~ Tf(r,s) +

(11.5)

Tg.(r,s) + cl(s)

I J"'O

J

(11.6)

Nh aJr,s) ~ Nf,g.(r,s) +

(11.7)

, J

J

n+l

I

k"O k;tj

Tg (r,s) + c3(s) k

n+l

(11.8)

Nh a (r,s) + (n + 1) , J

I

k"O

Tgk(r,s) + c 4 (s)

If in addition we assume that also (E8) holds, then we have (11.9)

for

r -

(11.10)

for

j -

00

0,1, ... ,n + 1 .

Basically the theorem is due to Mori, but does not give such explicit estimates. a)

The proof is obtained in several steps.

A special

(AO' ... ,~.

operat on.

The base

Let

""0' .. · '''''n

""0' ... '''''n

is orthonormal.

(11.11)

If "&0' ... '"&n+l

fot

(11.13)

E:

are vectors in

ZIO,n + 1].

If

j

E:

V*,

ZIO,n],

be the dual base of

define

define

Abbreviate

·320·

Then

§j

-

~ j " '& n + 1.

A homogeneous, projective operation

(11.14)

V : V X (V.,n+2 -

is defined for

'to

V, 'll-j

E

E

v·

with

V

0, ... ,n + 1

j -

by

n

't. V '&0 V ... V '&n+l - ,fn + 1 j~O <'t.,'II-l<§j"",>1\.j .

(11.15)

In fact, the operation is (n+3)-linear. LEMMA 11.2.

If 1:. t: V

and

'II- j

V·

E

for

j - 0, •. ' ,n + I,

then

(11.6)

PROOF.

If

j E: Z[O nJ,

we have

I <'t.,,&/I

I <§j'1\.> I ,

111:.11 II'II-jll

n

D

II'II-kll - 1It.1I 11'11-011 '"

lI'&n+ll1

k~j

Therefore we obtain

COROLLARY 11.3.

If

X E:

1P(V)

and

Zj E:

"(V·)

for

j -

0, ... ,n + 2,

then

(11.17)

LEMMA 11.4.

then

If

x

E: ..(V)

and

Zj £

F(V·)

for

j - 0,1, .. , ,n + 1,

·321·

n+l

TT

(11.18)

Zo ;.. .. , ;..

0

Zj_l ;.. Zj+ 1 ;.. ... ;.. zn+ 1 0

j-O

~ (n

PROOF.

Take

t.

. Zo V...

+ 1) 0 x V

E:

II!-J.II - 1 and

with

V

with

IIt.II - 1

z· - lP(it.) J

V ~+ 1 0 •

for

J

and

x - F(t.).

Take

j - 0,1, ... ,n + 1.

E:

Then

(11.19)

(11.20)

o

. Zo .

x V

V ••. V zn+ 1 0 -

!-o

lit. V

V

We have

(11.21)

n

(11.22)

Since

~

<1:.,o/aj .

AV·,

we obtain

oLt. - (00 " ... " 0n)Lt. -

a O,...

'~n

is an orthonormal base of

n

IIoLt.II -

(11.23)

jocO

[I

1<1:.,0/1 2 ] 112 - 1 .

j-O

If

have

j

I:

Z(O,n + II,

we have

lIij ll ~ 1.

lIiJ II - lI~j " itn+111 ~ lI~jll ~ 1.

If j

We obtain

E:

Z[O,nl,

we

V.

-322-

- (n + 1)1I~ v ~o v ~l V ... v ~b+lll

COROLLARY 11.5.

Take

x

j - 0.1. ...• n + 1.

Assume that

£

1P(V)

and

Zj £ 1P(V·)

ZO0 ...• ~+ 1

q.e.d

for

are in general position.

Then

v zn+l 0 > 0

(11.24)

and

x V

PROOF.

for all b) x

£

M

Zo

V ~+ 1

V

Since

z00 ...• zn+l

j £ ~[O.n + 1).

The Mori map. such that

gO(x). ... .gn+1 (x) exists.

Hence

exists.

are in general position.

Thus (11.18) implies (11.24).

q.e.d.

We assume that (El) - (E7) hold.

f.gO' ...• gn+l

are holomorphic at

is in general position

f.g O' ...• gn+ 1

is free for

Then V.

x

Take a poinl and such that

fIx) V go V... V gn+ 1 (x) A meromorphic map

(11.25)

h ~ f V go V ... V gn+ 1 : M -

called the Mori map.

is defined.

P(V)

We obtain the operation divisor

-323-

(11.26)

with valence function

and the distance

N"

(11.27)

with the compensation function

(11.28)

I

m(r) -

1

log y a > 0

for

r

E

.f.T

M r > 0

which extends to all THEOREM 11.6

as a continuous function.

Theorem 3.4 implies:

(The First Main Theorem for the Mori map). 0 < s < r,

If

that (El) - (E7) hold

We assume

then

n+1

(11.29)

~

Tr(r,s) + j

COROLLARY 11.7.

If

~

Tg.lr,s)

=

J

0

0 < s < r,

N 71 (r,s) + m(r) - m(s) + Th(r,s).

then n+1

(11.30)

Th(r,s)

~

T r(r,s) +

~ j =0

T g.lr,s) + m(s) . J

Thus (11.5) is proved. c) funcfon

Estimation of mer) and N ,,(r,s). go' ... ,gj-},gj+l' ... ,lfn+l

For

E

are free for

Z[O.n + II, ,...

Hence the divisor

(11.311

exists with valence funtion

Np. ~ o. J

(11.32)

'Y j -

D

go ;...

the

Also the distance

-324

does not vanish identically.

The compensation function

(11.33)

is defined for all function.

r

E:

~1'

r > 0

and extends to a\1

as a continuous

Abbreviate n+l

fJ.(r,s) -

(11.34)

L

k-O k;ej

Tk(r,s).

The First Main Theorem for the exterior product (3.36) respective!) (7.60) imphes:

LEMMA 11.8_

0 < s < r

If

and if

.... TJ.(r,s) - N

(11.35)

LEMMA 11.9.

r > 0

For

Pj

£

Z[D,n + 11,

then we have

(r,s) + mj(r) - mJ.(s) .

we have n+l

m(r) ~

(11_36)

PROOF.

L:

j .. o

mj(r) + logIn + 1)

Lemma 11.4, (11.7) and (11.32) imply

Hence (11.28) and (11.33) imply (11.36). LEMMA 11.10.

Let

U;e 0

of

h.

V·

of

gj

for

j -

Z[O,n + 11,

«>

0,1, ... ,n + 1

(11_37)

E:

(n + 1)y .

be an open, connected subset of

Then there is a holomorphic function

Moreover, if

~

q.e.d.

Assume that there arc reduced representdtions ~j : U -

Yn+1

YO'"

then

Y

:

U-

V of

and

I : U-

~

on

0

M.

U

f, V

such that

·325·

(11.38)

PROOF.

Since

111': U -

holomorphic function

V is a representation of

Y 1: 0

on

U

h

exists such that

on

U, YI.

It

have

(11.39)

If

E:

It=

Z\O,nl,

we obtain

< 10.'\10 .><.;) .,'ft,> - <~,.u.>

(11.40)

J

which proves (11.38) for

j E:

J

J

Xj 1: a

n

For

"40 n+ 1

E:

Z[O.nJ.

Y J

;lla.nJ.

There are meromo phic functions

(11.41)

=

-

I

J-O

X

on

U such that

r'40 j

.

we obtain v

v

(11.42)

n

n

I

j-O

< 10

• "40, j><.;) j"1\'>

which proves (11.38) for

j -

n + 1.

L

J"'o

q.e.d

< 10

• "40

·>X .<.;) + I'1\.>

J

J

n

a W

·326·

LEMMA 11.11.

If

j E Z[O,n +

11. then we have

(11.43)

n+l

(11.44)

~ Up. UT/ IE; j=O J

n+l

(11.45)

PROOF. U 'I: 0

NT/(r,s)

IE;

~

JED

N (r,s) Pj

for

O<s
It suffice'! to prove (1l.43) and (11.44) on open, connected subsets

of

M

where we have representations

described in Lemma 11.10.

10,

'Wl j'

It

and

We have

(11.46)

(11.47)

Thus (11.38) implies (11.43).

Also we have

(_l)n+l

(11.48)

n ~

j"'O

Since

UuL

U

10

Pn +l

-

IU

0,

we obtain

v

< 10

,'Wl l'Wl j

I

as

·327·

- Min(1l 1l 1U + Ilh , a·1 J U - J.t Pj I U I j

E

~

Un I U -

which proves (11.44).

n

~

IJ.

j=O

0 < s < r,

If

Z(O,n)}

Z[O,n))

Pj

IU

T Ivia ly (11.44) implies (11.45).

COROLLARY 11.12.

E

q.e.d.

then

(11.49)

ROOF.

According to (11.29), (11.45), (11.36), (11.35) and (11.34) we have

, Th(r,s) +

n+l ~

(Np.(r,s) + mj(r)) + log (n + 1)

J-O

J

n+1

- Th(r,s) +

I

j-O

n+1

i\(r.s) +

- Th(r,s) + (n + 1)

mj(s) + log (n +

n+l

I

j-O

which implies (11.49).

I

j-O

n+l

Tg.(r,s) + J

I

1)

ml(s) + log (n + 1)

j-O

q.e.d.

Thus Corollary 11.12 proves (U.6). d)

The estimation of the valence functions.

(11.43) and (U.35) imply

·328· which is 0+1

(11.50)

~

Nh a (r,s) , Nf ,.Ir,s) + ' J

, l

Thus (11 7)

1S

proved.

(11.43), (11.45) and (11.35) imply

E: N

Nf g (r,!» , J

T 'k(r,s) + mj(s) .

k=D k;t:J

,

Nh

,

Nh

(r,s) + NT/lr,!»

,dj

0+1

aj

(r,s) +

~

k=D

N Pk(r,s)

n+l

0+1

,aj

(r,s) +

~

Tk(r,s) +

k-O

~

k-O

mk(s)

which is 0+1

(11.51)

Nf ,.(r,s) , Nh a.lr,s) + (n + 1) , J ' J

Thus (11.8) is proved

e)

k~O

n+1

T'k(r,s) +

~ mkls). k-O

If we assume that (E8) holds in addition, then (11.5)

and 111.6) prove (11.9) trivially. immediately.

~

Theorem 11.1

i!

Also (11.7) and (11.8) proved (11.10)

proved.

Linear rum-degeneracy.

must find conditions on

f

In order to obtain a defect relation we

which assure that

h

is linearly

non-degenerate.

Take

fs. - (b O' ... ,bn +2)

open subset of

M.

Let

gj

j

ZIO,n ~ 1).

on

(11.52)

U for

E

'lAO

E

«;n+2.

j : 1J -

Let V*

U;t: 0

be a connected,

be a reduced representation of

A holomorphic vector function

·329·

is defined.

The map

6 -

(11.53)

96

D - Ib, ... ,b)

LEMMA 11.13.

96 _ 0

go' ... .£n+1

are in general position,

n

(11.54)

96 - / 0

<";; j'1Io> ~ 0 and

96 :: 0

if and only if

Since 6 - .".

6

([) •

D.

E

Xj

by (11.41).

Xj ~ O.

With

Since

(11.42) we have

(~{';;fn.> + bn+1Xj <.;)n+1,1Io»...oj

...0 k'

...

,...0 n

are linearly independent,

b j - bn +1 for all

is linear

96

E

in 6,

j

we have

E

Z[O,nl.

q.e.d.

if and only if

96 - 9.".

D.

E

Let

E - ([n+2/D

p : ([n+1 _

then

([n+21 b

E

if and only if

Define the diagonal

Define the meromorphic functions

PROOF.

Since

is linear.

E

be the quotient vector space.

be the residual map.

If

.".

is well-defined and the map

9 .... - 96

E

E

Let

with

.... -

9....

p(6) = .... ,

is linear and

injective. UO 'F- 0

If

u " uO 'Fg.

1

on

0,

is an open, connected subset of M with o . UO _ V* is a reduced representation and if ...0 j .

UO for

functions

Wj

If we define

on

j

E:

Z[O,n + 1],

then there are zero free hoi om orphic

U" UO such that

w - Wo '" w n+1

0

and

~~

= Wj'\1O j

on

U" UO.

-330-

(11.55)

thE:D

9g - W9ft

on

n U o.

U

Hence, take

one and only one meromorphic map

c

gc: M -

E:

I'(E).

1P(V*)

Then there is

defined by the

property:

"Take

ft

£

([;n+2

with

ft - (bO' ... ,b n +1) ;t 0.

of

gj

let

'\10

j - 0,1 ... ,n + 1.

for

representation of

Ic

on

LEMMA 11.14.

The map

j : U -

Then

V* 9ft

j : U -

J: - (gc I c

lI'(E) -

1:.

defined by (11.61) is a

£

~ft1 - G9ft2

on

U.

is defined.

lI'(E))

defined by

be reduced representations of

V·

Then there is a meromorphic function

G ~ 0

c -

Assume that

gj on

for U

for all

ft

*

D,

gc

is bijective.

gc

1

- Ic

j - 0,1,

2

,n+1.

such that

Hence (11.54) implies

(11.56)

Hence

be a reduced representation

By definition, the map is surjective.

PROOF.

Since

Then

U. n

A set of meromorphic maps

'\10

c - 1P(p(ft)).

we have

b~ - b~+ 1

G is a constant and

;t

°

for some

E:

Z[O.n] .

j £ Z[O,n].

p(b 1) - Gp(ft Z) which implies

cl - c2

q.e.d. Thus

J:

can be regarded as a projective space of dimension

n.

fo

·331·

The meromorphic map f: M 1P(V) is said to be linearly rum-degenerate ~ II (or go,· .. ,In +1) if and only if (f,Ic) is free for every

c

E:

F(H).

THBORBM 11.15. The meromorphic map f: M F(V) is hnearly non-delenerate over II if and only if the Mori map h: M P(V) linearly non-degenerate. PROOF.

a)

h i! linearly non-degenerate.

Assume that

is

Take any

c E: F(B). Take '" E a: n +1 - D with c - F(p(&-I). Let U ~ IZJ be an open connected subset of M such that there are reduced representations I ) : U V of f and '110 j : U V· of I j

for

and I : U V of the Mori map h. Define It Y as in Lemma 11.10 and g&- by (11.52). Then (11.54) implies

j - 0,1, ... ,n + 1

and

n

-

Hence

(f,Ic)

I

j-O

is free.

(b. - b +1)<';'·,11.><1),'110'> J

n

J

The map

f

J

is linearly non-degenerate over

Assume that f i! linearly !!!!!l-degenerate over I). 1P(V·). Then b - 1'("'1 with 0 ~ '" E V·. Then b)

b

E

'" - bO<110 + ... + bn~' Then as

lD

have

&-

t

D and

al and define

Put

bn+1 - 0

c - 1P(p(&-n g&-

E

1P(E)

by (11.52).

and exists.

Take

&- - (b O' ... ,b n+1) Take

By assumption

U,

II.

C,

E

I[n+1,

'\U)j' Y, 10',

< c ,g&-> ~

o.

I

As in al we

-332-



Therefore

(h,b)

is free.

f and only if

[J,

We shall restrict since the cond1tion "h

f.

condition on

(E9)

is linearly non-degenerate.

f

f

is linearly non-degenerate

is linearly non-degenerate.

r elf to the case of a covering parabolic space,

0

1S general for

B"

does not easily translate into

There is a proper, surjective, holomorphic map

Let

p

q.e.d.

Thus we assume in addition

such that (E10)

h

are constant, then

If over

The map

7f

M -

a;m

., - 117f1l2.

be the branching divisor of

7f.

Assume that

ND(r,s)

Rf-limsu p

(11.57)

r -.00

(Ell)

Assume that

(E12)

At least one map

REMARK 1.

f

is linearly non-generate over gj

Since the case of constant targets is already solved, it is

T (11.58)

g 1

gj

is not constant.

Then

(r,s)

-"'lL.. o -g - r -

-

A

gj

(00)

> 0

for

r -

..

Hence we obtain

j.

T (11.59)

Therefore

[J.

is not constant.

reasonable to assume that at least one

for this

-<00.

Tf ( r I s )

-

f

has transcendental growth.

has trancendental growth.

(r,s)

g 1 ... l .... o-g-r- -

..

for

r -

...

By (11.9) also the Mori map

h

i

-333-

REMARK 2.

By (Ell) and Theorem 11.11 the Mori map

h

ic; linearly

non-degenerate REMARK 3.

By (11.9)

REMARK 4.

If

1'1'

Rh -

Rr

<

00

is biholomorphic,

p - 0

and

Rf

~

O.

If

not biholomorphic, but if the (m-I)-dimensional component of is affine algebraic, then Theorem of Noguchi (70). j E Z[O,n + 1]

R

gj

<

00

Np(r,s) - O(log r). Rf <

if

00

exists such that

gj

by Noguchi's theorem.

f

By (11 58).

lI'(supp p)

Rf - O.

separates the fibers of

separates the fibers of

Then (E8) implies

is

1'1'

11',

By a 1'1'.

If

then

Rf - O.

The First Main Thporem for fixed targets (A.176) implies THEOREM 11.16.

Assume that (E1) - (EI2) hold.

(11.60)

n+1

~)

L.

I)f(gj

j~O

If (63).

M _!t m and

1'(ll) -

'n

118112,

+

1 +

Then

n(n + 2

1)

Rf

the theorem was proved by Mori

(He remarks that his original proof is incorrect and he is publishing a

correction.)

Our proof shows the power of the theory developed here,

provides better estimates and applies to a more general situation.

'H2.

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REMARK. When this manuscript was being completed for publication, Professor Shiffman sent me a preprint of the paper: (130)

Charles F. Osgood, Sometimes effect Thue·Siegel·Roth·Schmidt·Nevanlinna Bounds, or better, to appear in Journal of Number Theory, pp 51.

Charles Osgood asserts that he proved the Nevanlinna Conjecture (11.2) under the assumption (11 1) by means of number theory. The paper is difficult to understand and is still under investigation. The result was announced in: (131)

Charles F. Osgood, A fully general Nevanlinna N·small function theorem and a sometimes effective Thue·Siegel·Roth·Schmidt Theorem for solutions to linear differential equation, Contemp. Math., 25(1983) 129·130.

Although the announcement seems to indicate, that the Nevanlinna conjecture was proved, it lacks a clear cut formulation of the result and it IS not easy to understand.

-344Index

a-divisor 22. 119 admissible base 84. 276 Ahlfors estimat s 49. 71. 201. 206. 207 a-multiplicity 3 analytically dep ndent 82. 84. 275. 277 analyticdlly dependent on a set 82. 84. 275 analytically independent 82. 84. 275. 277 analytically independent of a set 84. 275 argument prmciple 4. 24 associated covering 40. 154 associated flag 164 associated map 43. 154 base 83. 276 - • admissible 84. 276 B-derivative 43. 152 big Griffiths conjecture 54 boundary integral representdtion 5 branching defect 87 brdnching divisor 42. 157. 269. 287 canonical bundle 40 Casorati-Weierslrass lheorem 39 characteristic function 3. 10. 12. 26. 137. 139 chart 40. 151 chart atlas 40. 152 Chern form 17. 137 chordal distance 2 closed graph 18. 120 closed pseudobaU 115 compensation function 3. 11. 26. 36. 48. 61. 137. 141. 241. 243. 323 - • modified 61 complete 276 complex projectIve space 92 condensor 26 contraction. metric 104 contraction product 59. 95. 96. 126 coordinate functions 84. 277 counting function 3. 4. 10. 26. 35. 36. 61. 119. 126. 137. 241. 243

covering 122 • associate 40. 154 _ • nerve 122 - • open 122 curvature stress formula 69. 184 defect 7 branching 87 - • majorization 52. 268 - • Nevanlinna 7. 52. 80. 81. 271. 291 - • ramification 52. 287 defect relation 7. 52. 53. 81. 82. 88. 91. 272. 273. 274. 292. 303. 306. 333 defect relation for function fields 303. 306. 309 defect relation of Nevanlinna-Mori 91. 333 defect. Ricci 52. 87. 268 - • truncated Nevanlinna 52. 29~ deficient 7 deficit 26. 27 define 83. 276 degeneracy 76. 236 derivative 2. 109 - • - B 43. 152 - • exterior 2. 109 distance 17. 58 - • chordal 2 - • projective 15. 105 distinguished 17 divisor 3. 20. 118 - • a 22. 119 - • branching 42. 157. 269. 287 intersection 22 non-negative 20. 118 of a function 22. 119 operation 60. 322 pole 119 prime 20 pullback 21. 118. 121 ramification 287 simple 20 stationary 45. 154 support of 20. 118

-345- , truncated 20, 260 - , zero 20, 21, 119 dual classification map 17 dual Frenet frame 67, 175 duality 93 , dual, Poincare 25 dual vector 94 dual vector space 92 exhaustion 33, 115 exterior derivative 2, 109 exterior product 5, 14, 59, 93 evaluation 17 finite family 78 first main theorem 4, 11, 27, 28, 36, 37, 48, 62, 63, 64, 77, 137, 241, 146, 147, 148, 141, 244, 323 for contraction 63, 147 for general position 77, 241. 244 - for hyperplanes 148 for line bundles 27, 137 for projections 48 for the exterior product 62, 146 - for the inner product 64, 148 - for the interior product 63, 147 - for the Mori map 323 , general 62. 141 - , unintegrated 25 free 15, 84, 60, 126. 127 - of a set 279 - of order p 127 - of order (q,p) 159 - of order (q,p,p) 159 - , strictly 127 , strictly of order (q.p) 159 Frenet curvature formula 173 Frenet formulas 169 Frenet frame 67, 165 - , dual 67, 175 Frenet identities 167 Fub ni-study form 15. 110 fusion 235 gauge 74, 76, 218, 235 gauge measure 76. 241. 243

general assumptions A 198 - B 78, 245 - (' 289290 - D 85 - E 88. 91. 318, 332 general first main theorem 62. 141 general for B 45. 154, 282 generality index 45. 154 general Jensen formula 29 general of order p for B 44, 94, 15 l general position 50. 63, 73, 74, 76 78, 218, 220, 236, 244 Grassmann cone 14. 92 - manifold 14, 93 Green residue theorem 23, 134 Griffiths conjecture, big 54 - - , small 54 growth, rational 302 - , transcendental 38, 269, 303 hermitian metric 16, 94 - product 5, 15. 59, 94 - vector space 15, 94 - vector bundle 16 hyperplane section bundle 18, 113 - - - of a map 19, 122 incidence manifold 130 indeterminacy 18. 76. 121. 235 index. generality 45. 154 - , stationary 45. 154 infinite plane 112 inner product 5. 14, 59. 92 interior product 14, 59. 92, 94. 126, 127 intersection divisor 22 Jensen formula 4, 24, 25, 35. 120 , general 29

, Lelong. theorem of Poincare 24 Levi operator 109 linear hull 216 linearly non-degenerate 38 linearly non-degenerate over IJ 91. 331 Liouville's theorem 36 lower order 302 maximum modulus 6

·346majorant 47, 197 majorization defect 52, 268 majorize 47, 197 mean value theorem ,29 meromorphic 83 meromorphic map 18, 120 - vector funct on 83 metric contraction 104 - hermiti n 16, 94 - , quot e t 16 modified comppnc;ation functIOn 61 Mori map 322 multiplicity, a 3 - , zero 19. 118 nerve of a covering 122 ~evanlinna conjecture 56 Nevanlinna defect 7. 52. 80. 81. 271. 291 truncated 53. 292 Npvanlinna Mori defpct relation 91. 333 non-negative divisor 20. 118 norm 5. 94 obstruction terms 211 open covering 122 open pseudo ball 115 operation divisor 60. 322

Poincar: dual 25 Pincar~-Lelong theorem 24 pole divisor 119 pole set 83 primarY stress coefficient 69 prime divisor 20 product. contraction 59, 95. 96, 126 , exterior 5. 14. 59. 94 hermitian 5, 15. 59. 94 inner 5, 14. 59. 92 interior 14. 59. 92. 126. 127 tensor 15 tensor symmetric 15 product to sum estimate 50, 75, 23l projective closure 112 - distance 15. 105 - operation 58. 105 - plane 14, 93 - , operation, homogeneous 58. 106 - operation. homogeneolls of degree (qt ... qp) 58. 106 - operation. unitary 58, 105 - operation. unitary of degree (Q1 .... ,qp) 58, 105 projective space. complex 92 pseudo ball. closed 115 - , open 115 pseudosphere 115 pth representation section 44, 153 pullback divisor 21, 118. 121

· projc('tive 58. 105 - , projective homogeneous 58, 106 - , projective homogeneous of degree (q1 •...• q p) 58. 106

quotient bundle 18 quotient metric 16

· projective unitary 58. 105 · projpctive unitary of degree (q 1" .qpl 58. 105

ramification defect 52. 287 rank 276

orbit 2 i:J oropr 302 · lowpf 30(' pdrabolic 33. 115 covpring manifold 34. 116. 302 manifold 33. 116 manifold. strictly 34. 116 plane. infinitp 112 · proipctlvc 14. 93 p-linedr 59

rational growth 302 reduced 16 - at a point 16 - representation 5. 19. 121 representation 18. 121 -- at a point 19 - atlas 153, 158 - family 153 - reduced 5. 19. 121 - section 19n 125 section pt 44. 153

I'liick('r diffrrcncp formula 46. 157

residue theorem 23

-347Ricci defect 52. 87. 268 - form 41 - function 41. 42. 156 Riemann sphere 2 second main theorem 7. 12. 42. 51. 53. 79. 81, 86. 255. 256. 265. 273. 291. 303. 308. 309 - over function fields 86. 291. 303. 308. 309 simple divisor 20 singular Stokes thoerem 23 small Griffiths conjecture 54 span 216 spans 17 spherical image function 3. 10. 26. 36. 137. 138 standard model 16 stationary index or divisor 45. 154 Stokes theorem 116 stress coefficient 67. 176 coefficient. primary 69 - curvature formula 69. 184 - gradient formula 70. 184 - invariant of level k 70. 182 strictly free 127 strictly free of order (q .p) 159 strictly parabolic manifold 34. 116 support of a divisor 20. 118 symmetric tensor product 15 tautological bundle 17. 113 tensor product 15 - . symmetric 15 theorem of Casorati - Weierstrass 39 - of Liouville, 36 - of Poincare-Lelong 24 transcendental growth 38. 269. 303 truncated divisor 20. 260 - Nevanlinna defect 53. 292 Umkehrproblem 8. 54 unitegrated first main theorem 25 unitary 105 - projective operation 58. 105 - projective operation of degree (q 1..... qp) 58. 105 valence function 3. 4. 10. 26. 35. 36. 61. 120. 126. 137. 241. 243. 323

zero divisor 20. 21. 119 zero multiplicity 19. 1U