The Equidistribution Theory of Holomorphic Curves. (AM-64) (Annals of Mathematics Studies)

THE EQUIDISTRIBUTION THEORY

OF HOLOMORPHIC CURVES BY

HUNG-HSI WU

PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1970

Copyright © 1970, by Princeton University Press ALL RIGHTS RESERVED

L.C. Card: 78-100997 S.B.N.: 691-08073-9 A.M.S. 1968: 3061

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

PREFACE These are the notes for a course on the Ahlfors-Weyl theory of holomorphic curves which I gave at Berkeley in the Winter quarter of 1969.

This is a subject of great beauty,

but its study has been neglected in recent years.

In part,

this could be due to the difficulty of Ahlfors' original paper [11; a subsequent poetic rendition of Ahlfors' work by Hermann Weyl [71 does not seem to be any easier.

The modest

goal I set for myself is to give an account of this theory which may make it more accessible to the mathematical public. My audience consisted of differential geometers, so these notes are uncompromisingly differential geometric throughout. I should like to think that differential geometry is the proper framework for the understanding of this subject so that I need make no apology for being partial to this point of view.

On

the other hand, I must add a word of explanation for the length of these notes which some readers would undoubtedly find excessive.

The reason is that great care has been taken

to prove all analytic assertions that are plausible but nonobvious, e.g. that certain constants in an inequality are independent of the parameters or that certain functions defined by improper integrals are continuous.

Although the experts

might think otherwise, I cannot help feeling that given a

v

TIlE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

vi

subject as intricate as this one, it is best to check through all the details rather than to let the correctness of the final

conclusions rest on wishful thinking. I assume that the reader knows a little bit about differential geometry, complex manifolds and complex functions of one variable, but not much of any of these is actually needed.

It

should be pointed out that Chapter II is essentially independent of the rest and gives a complete exposition of the Nevanlinna theory of meromorphic functions defined on open Riemann surfaces.

The pre-requisites for this chapter consist

merely of the most rudimentary knowledge of classical function theory and the differential geometry of surfaces.

Chapter I

is a disjointed collection of facts needed for the later chapters.

If the reader survives this chapter, he should

encounter no difficulty in reading the remainder of these notes. It remains for me to thank Ruth Suzuki for an impeccable job of typing. H. W.

INTRODUCTION By a holomorphic curve, we mean a holomorphic mapping x: V -

Pn~'

is the

where

V is an open Riemann surface and

n-dimensional complex projective space.

Pn~

The central

problem of the equidistribution theory of holomorphic curves, crudely stated, is the following: Pn~

in general position, does

them?

given

x(V)

m hyperplanes of

intersect anyone of

The motivation for this question comes from two different

sources. of an open

The first is algebraic geometric: V,

we let

holomorphically into

x

Suppose instead

map a compact Riemann surface

Pn~'

then

x(M)

is an algebraic curve

and it is a matter of pure algebra to check that intersect every hyperplane of a compact

M by an open

V

Pn~.

M

x(M)

must

Thus the replacement of

has the effect of transferring

the whole problem from algebra to the domain of analysis and geometry. Pl~

The second motivation comes from the case

is of course just the Riemann sphere and the above ques-

tion becomes: can

n = 1.

x(V)

given

m distinct points of the Riemann sphere,

omit them all?

Picard says that if

V

=~,

The celebrated theorem of Emile then

x(~)

cannot omit more

than two points or else it is a constant map. fore entirely natural to seek an

It seems there-

n-dimensional generalization

of this remarkable result. Yet the Picard theorem, like the above question. must be considered relatively crude in that it is only concerned with the extreme behavior of a point being omitted by the image of x.

Equidistribution theory, on the other hand, is much more

Vll

viii

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

refined and delicate as it seeks to yield information on how often each individual point is covered or how often each individual hyperplane is intersected by explicit:

x.

Let us be more

we will first explain this for the case of a

meromorphic function (i.e. a holomorphic

x: V -

Pl~)

and

then go on to do the same for holomorphic curves in general. On the outset, it is quite obvious that some restrictions must be placed on

V before meaningful statements can be made.

It has been determined elsewhere ([8], Part B) that the most suitable condition to impose on

V is that it carries a

harmonic exhaustion, i.e. that there exists a T: V - [0,00) set)

such that

(i)

T

= compact set) and (ii)

side some compact set of is a compact subset of

V.

COO

function

is proper (1.e.

T

T-l(compact

is a harmonic function out-

Then

V for each

V[r) r.

=

(p: p

€

(Example:

V, T(p) i r) If

V

= ~,

then such a harmonic exhaustion on «; can be chosen to be a COO function which equals log r outside the disk of radius three, say.

Then for all large values of r, C[r) is just the disc of radius e r In the same way, such a harmonic

.

exhaustion can be chosen on any

V obtained from a compact

Riemann surface by deleting a finite number of points.

Note

that what we have defined should properly be called an infinite harmonic exhaustion; see Definition 2.4 of Chapter II, §6. By a theorem of Nakai (Proc. Jap. Acad. 1962, 624-9), the Riemann surfaces carrying an infinite harmonic exhaustion are exactly the parabolic ones.)

One of the basic quantities in

this theory is the counting function

N(r,a),

defined as follows.

ix

INTRODUCTION

Let a

€

n(r,a)

the number of points in

If we fix an

Pl~.

v[roJ,

=

ro

so that

x-lea) nV[r], T

where

is harmonic outside

then by definition

Jrro n(t,a)dt.

N(r,a)

For the definition of the second basic function, we note that carries the classical spherical metric, which is a con-

Pl~

formal (hermitian) metric of constant Gaussian curvature. we denote its volume form by

f.m

= 1,

m,

If

and normalize it so that

then the order function

T(r)

is by definition

Pl~

T(r) As

r --

00,

N( r, a)

obviously measures how often the point

is covered by the points of

V.

On the other hand,

a

T(r) by

measures the average coverage of the pOints of

V·,

in mathematical terms, we have the following theorem:

(0.1)

T( r)

=

i N( r, a )m( a) .

Pl~

In other words,

T

is the arithmetic mean of

in mind, we introduce the defect function on 6*(a)

=

€

tha t

Pl~

- xCV),

6 * (a) = 1.

6 * (a) = 0

With this

Pl~:

lim inf(l _ N~(~»)). r-+oo

It will follow from a later result that

a

N.

then of course

N(r,a)

0 < 6* < 1.

o

for all

If r,

so

According to (0.1), the other extreme of

is to be interpreted as that the point

a

is covere

x

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

by the points of

V as often as the "average" point of

Pla;.

The main theorem of the whole theory can nm'1 be stated: suppose

V has finite Euler characteristic, then given any

m distinct points

al, ... ,am of ~. 5 * (a.)

(0.2)

l.

where a; -

Pla;,

<

l.-

+ X,

2

V = a;

X is a finite constant, which vanishes if

(oJ.

A first consequence is Picard's theorem.

consequence is that

5) 0

or

A second

only on a countable subset of

Thus except on a countable subset of are equally distributed over

Pla;,

Pla;

the values of

V

and hence the name

Pl~'

"equidistribution". There is also a beautiful by-product of the theory:

the

number "two" in Picard's theorem turns out to be accountable for by the Euler characteristic of the Riemann sphere.

To

put this in the proper context, let us go into the mechanics of the proof of (0.2) in some detail.

First note that we may

regard

83 ~

Pla;

as the quotient space of

unit sphere, by defining (wo'wl ) some

9.

[ZO,Zl]. of

Pla;

E

8

3

(zo,zl)

if and only if

E

a = [zo,zlJ,

orthogonal to

inner product


a,

then

Pl~

being the

wo,e

i9

wl )

for

we denote by

= [-zl,zoJ

is the point

orthogonal in the sense that the

in terms of representatives is zero.

One checks that function in

i9

(zo,zl) aL

83

to be equivalent to

(zo,zl) = (e

The equivalence class of If

83

a;2,

is a well-defined - (a1

and that

xi

INTRODUCTION

.

(0.3)

d C = .r:J.(d" - d l

where

An application of Stokes l theorem

).

to (0.3) yields:

(0.4) We shall always assume that on

oV[r].

where

*dT

>

r

r o'

so that

T

is harmonic

In that case,

denotes the conjugate differential of

this fact, we integrate (0.4) with respect to

r

dT.

Using

to arrive

at the first main theorem (FMT):

+

N(r,a)

(0.5)

1 ~

~

dV tl

where we have used the notation

~

can be shown that a.

x* U *dT a

=

h(r) -h(r o )·

It

is positive and is continuous

This together with (0.5) leads to an important inequality: N(r,a) < T(r) + c,

(0.6)

h(r)l~

T( r),

0

cV t] in

Ir

x * U *dT a ro

c

independent of

rand

a.

From (0.6), one deduces that the defect function satisfies

o

< 6*

~

1,

a fact mentioned above.

(0.5) relates

N

the topological data of second main theorem we can state it.

to V

(S~IT).

T.

We need to relate

T

also to

and this is the content of the We need some definitions before

The holomorphic mapping

x: V -

Pl~

is

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

xii

ramified at a finite number of points the compact set

V[t].

neighborhood of

Pj ,

of

x

at

If

x

we call

(Pl, ••• ,PmJ

is locally (mj-l)

rn jo to

inside 1

in a

the stationary index

Let m

nl(t) = ~j=l (mj - 1), and we define X(t)

Nl(r) =

I ror nl(t)dt.

the Euler characteristic of

E(r) = Jr X(t)dt. ro

Finally, on

the nonnegative function

h

x•(J)

Next, if we denote by

v[tl,

then we define

V - v[r o ]'

we introduce

by: h dT /\ • dT •

Then the SMT states that E(r) + Nl(r) - 2T(r) =

~

r (log h).dTlrro

•

~ oV[tJ

The proof of (0.7) makes use of the Gauss-Bonnet theorem. coefficient

2

in front of

T(r)

The

is the Euler characteristic

of the Riemann sphere and. eventually, this

2

makes its way

into (0.2).

(0.5) and (0.7) contain

The situation is now quite clear:

all of the information we need to derive the defect relations with the exception of the two line integrals and

r

oV'rtJ

(log h).dT.

r

ov'[ t]

x·u .dT

What we should do now is to obtain an

upper bound for their sum in terms of

T(r),

which is roughly

of the form

(0.8)

r x·ua.dT + ovtrJ r (log h).dT

oVlrJ

a

< Clog T(r).

xiii

INTRODUCTION

A substitution into the left side the values of the line integrals as given by (0.5) and (0.7) would then yield immediately (0.2).

(O.B) is to

The basic idea to obtain the inequality

integrate the inequality (0.6) over appropriately chosen function

p

on

Pl~ Pl~.

with respect to an It is a virtue

of this geometric approach that the choice of to be entirely natural:

log p

p

should be roughly

turns out ua •

The above theory was created by R. Nevanlinna in 1925. In the words of Hermann Weyl, this contribution of Nevanlinna constitutes one of the great mathematical achievements of the century.

The simplicity of the modern proof of (0.2) we owe

largely to Ahlfors; the beautiful idea of invoking the GaussBonnet theorem in Nevanlinna theory is also due to Ahlfors. The exposition of Nevanlinna theory given in Chapter II follows closely that of [B], which in t urn rests on the efforts of Ahlfors and Chern.

It should be pointed out that instead

of mapping into the Riemann sphere, one could equally well have chosen any compact Riemann surface as the image space (see

[B1), but there is no doubt that the Riemann sphere occu-

pies the central position of this theory.

In addition to this,

there are two reasons why I have exposed this special case with such deliberate care.

The first is that it gives us

same insight into the structure of the general equidistribution theory of holomorphic curves, which would otherwise have been lost sight of in a maze of technical details. I feel that there are a few that ere worth looking into.

obviou~

More importantly,

open problems in this direct

For one thing, the Nevanlinna

xiv

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

theory on the parabolic surfaces with infinite Euler characteristic has hardly gotten started.

Also, the relationship

between the defect function and the choice of the harmonic exhaustion should be clarified.

Finally, of the many remarkable

results established for meromorphic functions on

C in the

last decade (see [4) and a survey article in Bulletin Amer. Math. Soc. 1967, 275-293 by W. Fuchs), it seems that most would survive on those

V obtained from a compact Riemann

surface by deleting a finite number of points.

I

hope that

this exposition of Nevanlinna theory will stimulate someone into performing the task of transplanting these theorems from C to this class of Riemann surfaces. The equidistribution theory of holomorphic curves was first attempted by H. and J. Weyl in 1938, and was brought to essential completion by Ahlfors in 1941.

A short history of

the subject has been written by Hermann Weyl in his usual inimitable style in the preface to [71 and, despairing of bettering Weyl's lyricism, I will simply refer the reader to that monograph.

The exposition of this theory in these notes

follows the general guideline of [81.

Contrary to the prac-

tice of Weyl and Ahlfors, we deal directly with compact manifold rather than with greater conceptual clarity.

~n+l

,

Pn~

as a

and this results in

The outlook of Chapter IV has been

greatly influenced by two observations due separately to Chern and Weyl, and I take this opportunity to record them here. The first is Chern1s treatment in [21 of the first main theorem using the polar divisor

~B

as its core, and the second is

INTRODUCTION

xv

the remark by Weyl in [71 that the second main theorem is nothing but a glorified version of the classical Plftcker formulas.

Finally, the whole of Chapter V owes such a great

debt to Ahlfors' fundamental paper [11 that it would have been obvious even without my mentioning it. Let us describe the outline of this theory in some detail. So fix a holomorphic mapping

x: V -

Pn~'

where

V will be

assumed to carry a harmonic exhaustion as before. new phenomenon that appears when

n > 1

is that

The first x

no",

induces a series of holomorphic mappings into various Grassmannians.

To describe these, it is best to recall a well-known

fact from the differential geometry of curves in mn. 'Y: m _mn

be a smooth curve, then the one-parameter family

of tangent lines 'Y

'Y'

(the prime denotes differentiation) along

forms a surface, called the tangential developable of

Next,

'Y

which at

'Y.

induces a one-parameter family of planes along 'Y(t)

is spanned by the tangent

principal normal 'Y,

Let

'Y" (t) .

'Y'(t)

'Y

and the

These are the osculating planes of

and this set of osculating planes forms a three-dimensional

sub-variety of mn. osculating spaces of 'Y"(t)

By considering the three-dimensional 'Y

and the binormal

which at 'Y'''(t),

'Y(t)

is spanned by

y,(t),

we obtain a one-parameter

family of three-spaces, which forms a four-dimensional subvariety of mn , morphic curve

and so on. x: V -

Pn~.

Now let us return to our holoIn a way that will be made precise

in Chapter III, we will also be able to attach a

k-dimensional

osculating (projective) space to

p

x(p)

for each

€

V and

xvi

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

for each

k,

1 < k < n-1.

Grassmannian of

Thus if

G(n,k)

k-dimensiona1 projective subspaces of

then we can define a ho10morphic map kX(P)

is just the

The set of such p

where

k-dimensiona1 osculating space at

x(p).

v[tl

forms a

subvariety with boundary in

OX

= X.

which will be denoted by

This finite series of mappings

gonal to

Ak

Pn~'

k

A

G(n,k).

€

is an

oX' ••• 'n_1x,

A~.

The set of points of

and

Xk(t);

Pnt

ortho-

(n-k-1)-dimensiona1 projective subspace

~(t,Ak)

Let

are

x.

which is called the polar space of

noted by Ai

as

Finally, we shall agree on the notational convention

Now let

of

kX(P)

(k+1)-dimensiona1 analytic

Pnt,

called the associated curves of

of

Pnt,

kX: V - G(n,k),

k-dimensiona1 osculating spaces

wanders through

Xk(t).

denotes the

Ak

and is de-

denote the intersection number

note that these are subvarieties of com-

plementary dimensions, so their intersection number has an obvious meaning.

Define

Jror nk(t,A The next observation is that

G(n,k)

k

)dt.

carries a classical

Kahler metric called the Fubini-Study metric, whose Kahler form we denote by

Then the counterpart of the order

~.

function in the case

n

= 1 is the series of order functions: -I -rr

Jrrodt

V

J tl

k X*rok •

Now call a ho10morphic curve nondegenerate if its image lies

INTRODUCTION

in no proper projective subspace of can now be stated:

if

x

xvii

PniC.

The n1T of rank k

is nondegenerate, then

(0.9)

where the integrand of the line integral should be interpreted in this manner:

each

kX(P)

is a

sentable by a unit decomposable is true of on

Ak.

k-space, so it is repre-

(k+l)-vector, and the same

Since there is a well-defined inner product

(k+l)-vectors,

and

k

11

make sense.

The

proof of (0.9) is conceptually identical with that of (0.5). However, the counterpart of

(0.7) has a new element in it

because it now relates three consecutive order functions Tk _l , Tk , Tk+ l let

Sk(r)

index of

J:

to the topological data of

be as in

curve in

To state it,

sk(t)dt, where sk(t) denotes the stationary o kX: V - G(n,k) in V[t), i.e. the sum of the =

orders of zeroes of the differential of E(r)

V.

PniC,

(0.7).

kX

in

V[t).

Let

Then for any nondegenerate holomorphic

the second main theorem states that:

(0.10)

where

~k(t)

(E(t) + Sk(t) + Tk_l(t) - 2Tk (t) + Tk+l(t)) K, C are positive constants. =

and

xviii

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

(0.9) and (0.10) together will lead to the defect relations. The basic idea here is still to integrate the inequality No(r,a) < To(r) + Co function

B

over

defined on

Pnt

Pnt,

with respect to a well-chosen but the fact that

n > 1

now

makes this integration technically much more complicated than the case of

n

l."In a very ingenious mCinner, that often defies

=

belief, Ahlfors succeeded in choosing cuch a

B

to arrive at the

(o.n)

where

0 < a < 1,

b

is an arbitrary unit vector and

are positive constants which are independent of (Here

b.J 1 x

stands for that vector such that if

other vector,

=
1\ v>.

inequality lies in the fact that xCV)

a

)



intersects the polar space of

b

C, C'

and v

b. is any

The delicacy of this vanishes whenever (which is a hyperplane),

so that the integrand has singularities in a hyperplane. is the factor

Ib..J lxl

It

in the numerator that compensates for

these singularities and prevents the integral from being divergent. Although (0.11) is already difficult to come by, the road from (0.11) to the defect relations is still rougher. Ahlfors had to reach even greater heights in bringing this line of development to completion. be thus stated.

For each

The defect relations can

k_dimensional projective subspace

xix

INTRODUCTION

Again it is true that Ak

never meets any

0 ~ ~k < 1.

If the polar space

AL

k-dimensional osculating space of

x,

for all

then clearly 5 k (Ak )

=

1.

Now if

x: V - Pn~

finite Euler characteristic and

r

and consequently

is nondegenerate, (Ak}

of

V has

is a system of

k-spaces

in general position, then (0.12)

where each or

~ -

Xk

(oJ.

is a finite constant and vanishes if In particular, if

{An - l }

hyperplanes in general position and then

x(~)

must intersect one of

x:

V

= ~

is a system of ~

-

Pn~

(n+2)

is nondegenerate,

{An - l }.

It is impossible to adequately describe the difficulty that must be surmounted in order to pass from (0.11) to (0.12). I can only refer the reader to §4 of Chapter V to fully savour this virtuoso performance of Ahlfors. As has been remarked above, the case of

n

=

1

(i.e. mero-

morphic functions) suggests a lot of open problems and apparently will remain an active field for some time to come. However, the future of the general case certain.

n > 1

is far less

While these are a few obvious questions that remain

unanswered (e.g. can one obtain defect relations for holomorphic curves in Grassmannians?

can one replace hyperplanes by

xx

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

hypersurfaces of a fixed degree?*), the subject remains too narrow and too isolated and as such, it runs the risk of meeting an early and uneventful death.

The most pressing problem is

therefore to find applications for this theory.

One such was

given by Chern and Osserrnan in their study of minimal surfaces

(J. D'Analyse Math. 1967, 15-34). more are attempted.

In §5 of Chapter V, three

Of these, the most interesting should be

the two possible generalizations of Picard's theorem to n-dimensions.

These are problems intimately connected with

Kobayashi's theory of hyperbolic manifolds and ultimately with the intrinsic characterization of bounded domains.

I can

only hope that these notes will stimulate some interest in this subject, and that further work in this direction is forthcoming. In conclusion, I should point out certain notational conventions employed throughout these notes:

(1)

An asterisk

*

in front of a proof indicates that

the proof can be skipped without loss of continuity. (2) V.

There are three distinguished functions defined on

These are:

(3)

T

The sign

(p. 32), "

(J

(p. 35)

and

'Y

(p. 102).

in front of an inequality is defined

on p. 55 and p. 60.

* In

a private communication, Professor Wilhelm Stoll informed

me that he had solved the problem of obtaining defect relations when hyperplanes are replaced by hypersurfaces of a fixed degree.

This work is unpublished.

xxi

INTRODUcnON

e = IJ.(
( 4)

The symbol

is defined on p. 13l.

(5 )

The special usage of constants

C and

C'

is defined

on p. 168. (6)

The subscript in the symbol

Xk z

is defined on p. 69

and p. 170. (7)

The special usage of

a, b, E, F,

and

throughout these notes are defined on pp. 9-10.

A, B

CONTENTS Preface Introduction Chapter I.

§l

§2

§3

Generalities on projective spaces and Grassmannians The Fubini-Study metric Basic facts about Grassmannians Elementary inequalities

Chapter II.

§l

§2

§3

§4

§5 §6

§l §2

§3 §4

§5 §6

§l §2

§3 §4

§5 §6 §7

14

16 20

32 43 49

Elementary properties of holomorphic curves

Holomorphic curves induced by a system of holomorphic functions Nondegeneracy Associated curves Projection curves Contracted curves of the first kind Contracted curve of the second kind

Chapter IV.

4

9

Nevanlinna theory of meromorphic functions

The proximity function Holomorphic mappings into the Riemann sphere The nonintegrated main theorems Harmonic exhaustion and the integrated main theorems Two useful facts The defect relation

Chapter III.

1

62 67 71 74 75 79

The two main theorems for holomorphic curves

The The The The The

nonintegrated first main theorem integrated first main theorem FMT of rank k generalized FMT and two inequalities nonintegrated second main theorem and Plficker's formulas The integrated second main theorem The SMT of rank k

xxiii

81 87

99

109

115

124 131

xxiv

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Chapter V. §l §2 §3 §4

§5

The defect relations A basic theorem on integration over Pn~ The fundamental inequality The fundamental inequality of arbitrary rank Proof of the defect relations Applications

References Index of principal definitions

147 152 170 180 208

THE EQUIDISTRIBljTION THEORY OF HOLOMORPHIC CURVES

CHAPTER I

Generalities on Projective Spaces and Grassmannians We denote by

~n+l

euclidean space, i.e.,

~n+l

§l.

z=

(Zo, ••• ,Zn)

and

inner product on

Let

=

s2n+l

(Z:

the

(n+l)-dimensional complex

f(zo'Zl' ••• 'Zn): zA E ~J.

=

W = (wo' ••• 'wn ),

~n+l

then the canonical

is

denote the unit sphere in

Izi = lJ,

If

IZI

where

=+

mean the quotient space of

~n+l,

«z,z»1/2.

s2n+l

i.e.,

S2n+l

Pn~'

By

we

under the action of the

circle:

The equivalent class of such

Z we denote by

[zo, ••• ,zn'.

These are then the canonical principal fibre spaces: a: n+l

~

(1.1)

Here,

_ (oj

v(zo, ••• ,zn)

Z

Z

(~' ••. 'rZT)'

and

V 2 (Zo'···'Zn)

the diagram is commutative, i.e., T

is

that of

C*

a: - (oj, v2

the covariant

that of

vl

is the unit circle. 2-tensor J'

(l:AzAdzA)Q9(l:AzAdzA)J.

[zo,.··,zn'·

Therefore

v2° v l = v. The fibre of is the positive reals, and On

Cn+l - roJ,

consider

n 1 _ 2( (l:AZAzA)(l:AdZA ® dzA) (l:A=ozAzA)

It is hermitian, of type

(1,1)

and

llfE EQUIDISTRIBUTION llfEORY OF HOLOMORPHIC CURVES

2

in fact will be seen to be positive semi-definite. obvious reasons, we abbreviate

to this form:

1 2 r - }

F=

(1.2)

,

For



F

The associated two-form of

-

Again, we abbreviate

CD

will be denoted by

m,

i.e.,

to

Usually the context makes it clear whether tensor product or exterior product is intended in

(1.2) or (1.3), so that there

is no fear of confusion. T: Cn+l - (O} -PnC.

Consider the fibering principal fibre bundle with now demonstrate that it will follow that

Pn~

T

UA

=

as structure group.

We will

is a complex manifold, from which

is in fact a holomorphic mapping.

will exhibit a covering of Let

C*

This is a

([Zo, ••• ,Zn1: zA

P

n

~

We

by complex coordinate systems.

f oJ. Then the mapping [zo"",zn1

Zo zA_l zA+l zn n ~. A A A A sets UO""'Un define a complex structure on Pn~' The following even shows that P ~ is a Kahler manifold. n

- (z-, .•. '-z-,-z-"" 'z-) maps UA bijectively onto

Lemma 1.1.

= -F. then T *CD = CD. that

T* F

There is a Kahler metric If

(I)

F on

Pnt

is the associated Kahler form of

such

The ope

3

GENERALITIES ON PROJECTIVE SPACES AND GRASSMANNIANS

Remark.

F

Proof.

is called the Fubini-Study metric of

The second statement is quite clear. so we will

only prove the first.

(i)

with:

F (11)

is zero.

Pn¢.

invariant and

We will show three things to begin

restricted to the fibre of the action of (iii)

G: *

'IT:

G: n+ l - (oJ - Pn G:

on

leaves

the restriction of

complement of the vertical space (of

'IT:

~n+l)

with respect to the flat metric of

F to the orthogonal ¢n+l - (0] -PnG: is positive definite.

The first two will imply the existence of such an Pnt

and (iii) will imply that

F

F

on

is positive definite.

Since (ii) is immediate from definition (1.2). we will only prove (i) and (iii). Fix a point p = ( Po.Pl ••••• Pn ) E ¢ n+l - ( OJ. The vertical space at l:AP Ao¥:- == v.

p

is the complex line generated by

(As the reader readily percei.ves. rather than

A

working in the real tangent space with complex structure induced by space).

J.

we work directly with the holomorphic tangent

Note that

Ivl

=

Ipl.

Let an orthonormal basis

in the tangent space to 'p

be chosen so that

Furthermore. let

The orthonormality of

eo ••••• en {

l:A

j

l ••••• n.

then implies that

PA~PAj

l:A PAPAi tor i.

i = l •...• n.

6ij

= =

0

It follows that

3a

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

for

A

for

i. j = l, ... ,n.

=

O, ••• ,n

(1.4)

F relative to

Consequently, the matrix of

1 [011 J ~ "'1' F fact that

matrix

is proves (i).

The fact that

restricted to the span of

~ [1. • ] Ip I '1

(el ••••• e n )

The

has

proves (11).

It remains to show that clearly suffices to prove

F

dm =

is Kahler, i.e.,

°

because

*

11"

dw

=

0.

It

is injective.

To this end, let dC

=

>FT( d"

- d')

Then a simple computation gives

(1.6) Then of course

M

=

Q.E.D.

0.

We would like to draw a consequence from the above proof. From

(1.4). it follows that if F is restricted to the unit

sphere (i.e •• to those

p

such that

relative to an orthonormal basis

Ipi

= 1). its matrix with

4

TIlE EQUIDISTRlBurlON THEORY OF HOLOMORPHIC CURVES

[°1... 1] .

pointing in the radial direction is unit sphere

S2n+l

as a real manifold and let V denote the

vertical space of the fibration V = s2n+l nvI where a a a' space of the fibration over in

G:.

Consider the

a 'Ir:

'lrl

S2n+l

£

: S21'1+l and

G: n+l - (0) -

VI PnG:,

Pn Q:. Obviously is the vertical

V~ = span fe,

Le.,

Thus restricted to the orthogonal complement of

F

s2n+l a '

has matrix

~

Va

] relative to an orthonormal

L··.1

basis and therefore coincides with the usual inner product on S2n+l.

This proves the following:

Corollary.

S2n+l

If

is given the usual metriC and l : S2n+l -

is given the Fubini-Study metriC, then

'lT

Pn!C

Pn G: is

an isometry when restricted to the orthogonal complement of the vertical space of this fibration.

§2.

We shall need some elementary facts about Grassmannians,

so we will briefly mention them. dimension -

P IC n

p

in

pIC,

By a projective subspace of

we mean the image under

n

of a linear subspace of

(p+l)

'IT:

!C n+1 - f 0 1 G: n +l •

dimensions in

Hence, by definition, there is a one-one correspondence between the set of linear spaces of dimenSion the set of projective

p-spaces in

(p+l)

P!C n

in

!C n+1

and as a rule we

identify these two sets and denote this common object by Note that

G(n,O) = Pn !C.

€A = (0, ••• ,0,1,0, ..• ,0) all.

A=O, ••• ,n.

and

G(n,p).

Let us fix the canonical basis (A-th spot) of a:n +l once and for

Then we can coordinatize

G(n,p)

as follows.

5

GENERALITIES ON PROJECTIVE SPACES AND GRASSMANNIANS

Call a e

(p+l)-vector decomposable iff it can be written as Then each (p+l)-dimensional subspace fl of p defines a decomposable (p+l)-vector up to constant

1\ . . . I\e.

o tn+l

multiples. of

A,

More precisely, let

be any basis

then

L

(1.7)

i o ("'(ip If

e

(eo, ••• ,e p )

{e~, ••• ,ep)

"i ••• i Ei 1\ ••• I\E i • 0 pop

is a different basis of

then clearly

= pe I 1\ ••• 1\ e I pop So if we write:

for some nonzero complex number

then

for all

o

1\ ••• 1\ e

fl,

Ph iI ···i = " i · · · i o pop

the canonical basis identified with

( Eo, ••• ,E } , n

tl(p),

exterior power of

~n+l

and

Now via fl

p+l ~ n+l

fl P+ l

where

£(p)

tification, (1.7) implies that

p.

is naturally

denotes the

=

(n+l) .p+l •

e 1\···l\e

o and consequently

p

(p+l)-th

Under this idencorresponds to

fl itself corresponds o P to a line in ~£(p) - (0): (p( ••• ,"i ... i , .•. ): pEt * J. o p Under the projection '11": C£(p) - (1 0 .... P l (p ) -1 C, fl therefore

( ..• , "i .•. i , ... )

corresponds to a unique point in G(n,p)

into

Pt(p)_lC'

is injective (Jenner

map, we again identify G(n,p)

This map from

[51).

with its image in

Using this P£(p)_lC,

Before proceeding further, let us elaborate on (1.7). SUPPose a

=

O, ... ,p

then the coefficients of (1.7) satisfy

6

THE EQUIDISTRIBtrrJON THEORY OF HOLOMORPHIC CURVES

(1.8)

We shall also need the fact that submanifold of

P£(p)-l~.

Its dimension is

G(n.p)

is a compact

(It is in fact a quadric.

(n-p)(p+l).

As we saw above.

Jenner [51) ~

P£(p)-l

is

AP+l~n+l,

just the projective space formed from the vector space so this is merely the statement that the condition for a

AP+l~n+l

member of

to be decomposable is an algebraic condition.

We should now mention that the natural inner product on ~n+l

can be extended to

way.

If

p

hP+l~n+l

Ao~n+1 =~.

= -1.

=

fo

1\ •••

l\f

p

in a natural

and the inner produ~t is just If

B

(-1 ~ p <~)

are decomposable

1\ •• • 1\ e and o p (p+1)-vectors. then by

fI. =

e

definition (1.9 )

a.b

= O••••• p.

This inner product is then extended to all of linearity.

{e o ••••• e n l

if

~n+1

• •• 1\

then {e 1\ • io basis of AP+1~n+1. have identified

in ~

~.e(p).

and

~

by

An equivalent way of describing this inner product

is the following:

members of

AP+1~n+1

e i : i <. .• < i 1 is an orthonormal pop Yet another interpretation is this. We

AP+1~n+1 with ~£(p).

AP+1~n+1

Then

is an orthonormal basis of

So let

A. ~

and let them correspond to



~

be two and

e

is nothing but the inner product of

in the natural inner product of

~.e(p).

As usual.

7

GENERALITIES ON PROJECTIVE SPACES AND GRASSMANN1ANS

We now relate this inner product to the differ&ntial geo.etry of

G(n.p}.

covariant

AP+len+l -

{oJ.

there is the usual

2-tensor corresponding to (1.2):

F_ where

On

1 2{ _ J.

A .. (···'),i ••• i •••• ) o p

of AP+1en+l - (01.

As before

denotes the coordinate function w: AP+len+l - {O) - P£( p) -1 e

is a principal fibration. and so there is the Fubini-Study ..tric

F on

P£(p)-l e

a Kahler structure on structure on geometry.

G(n.p)

such that

G(n.p).

y

•F - F.

F then induces

We now show that this Kahler

is homogeneous in the sense of Riemannian

The unitary group

U(n+l)

acts on projective

n-space

as a group of isometries with respect to the Fubini-Study metric, traasitively but not effectively.

This is clear from (1.2).

U(n+l)

G(n.p};

also acts transitively on

because any

(p+l)

dimensional linear space can certainly be

moved to any other by the unitary group. transfo~ation

AP+len+l thus

of

a: n+l

induces a unitary transformation of

also acts on

Combining these two facts, Lemma 1.2 U(n+l)

P"(P)-l G:.

But each unitary

(this is immediate from the definition (1.9»

U(n+l)

on G(n.p)

this is also clear

P£(p)-l e we

and

as a group of isometries.

see that the following holds.

is a transitive group of isometries

with respect to the induced Kahler metric from

8

TilE EQVIDISTRIBlTTION TIIEORY OF 1I0LOMORPIIIC CURVES

We now corne to the notion of P IF".

projective subspace of [wo""'wn '

pna:,

€

the polar space of a

If

n

then the relation



0

=

is clearly

independent of the particular representatives chosen. fore one can speak of two points of If

A

Pn a;

orthogonal to

subspace of dimension

point of

Pn

a:

o

1\ ••• 1\

o

1\ ••• 1\



=0

b

of

p

B

where

(b O ,

=

B,

0

for

G(n,p).

€

~B

A

... , €

bp )

o 1\

~

~B'

••• 1\

The

a

of A and p A € G(n,p) satisfying B

in

G(n,p)

A trivial but important remark

is the intersection in

~B

B.

is again independent of

G(n,p) a

(n-l)-space,

is a basis of

and the set of all

and the hyperplane defined by Consequently,

space of a

Up to scalar factor,

will be called the polar divisor of

and will be denoted is this:

is clearly a projective

called the polar

the particular representatives b

the set of

In particular, the polar

Now let

b

p'

condition

A

p,

is a hyperplane, Le., a projective

and vice versa. b

n-p-l,

A~.

to be denoted by

=

being orthogonal.

is a projective subspace of dimension

points in

B

pna:

There-

<M,B>

is a subvariety of

P£(p)-l a;

=0

(M

G(n,p)

€

of

G(n,p)

AP+la: n + l ). of (complex)

codimension one. There is a second interpretation of

~B

useful below:

~B This follow from

(A

€

G(n,p): A

II BJ.. ~ ¢}.

that will be

9

GENERALITIES ON PROJECTIVE SPACES AND GRASSMANNIANS

Lemma 1.3 ~n+l

of

E and

Let

and

Then

respectively.

E

iff the intersection of

F

F

be

(p+l)-dimensional subspaces

( eo' •.• , e p 1 and

and let

F

E

be bases of

<e

1 \ . . . 1\ e , f 1\ ••• 1\ f ) = 0 o pop and the orthogonal complement of

is at least one dimensional. To prove this, we may as well assume

orthonormal basis of normal basis in terms of

E.

Then extend

(eo, .••• e n } eo, ••• ,e n •

~n+l

of

(e o , ..• ,ep l

(e o , •.• ,ep l and express

is an

to an orthofo, ••• ,fp

The lemma immediately follows.

Using this. we now give a third interpretation of ~B

is in fact the generator of the

homology group of Schubert variety

G(n.p).

~B'

2[(n-p)(p+l)-1)-dimensional

In Chern's notation [2). it is the

(N-p-l,N-p •••• ,N-p) ~

constructed from

p

LN_p _l C LN_p + l C LN_P+ 2 c ...

B.l. =

C

LN

LN_p +p '

This fact

will be useful in Chapter IV.

§3.

At this point, we wish to set up some notational

conventions that will be in force throughout the rest of these notes.

a, b

points of p

n

~

will denote unit vectors of P~.

n

A. B

~n+l

as well as

will denote projective subspaces of

as well as any of the unit decomposable vectors representing

them.

Similarly,

E. F

will denote linear subspaces of

~n+l

as well as any of the unit decomposable vectors representing them.

A superscript will indicate dimension.

for both an

h-dimensional projective subspace of

a unit decomposable Ek

Thus

(h+l)-vector representing it.

will stand for both a

Ah Pn

«:

stands and

Similarly,

k-dimensional linear subspace of

10

TIlE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

a: n+l

and a unit decomposable

k-vector representing it.

G. H. K. A. M will be symbols for general multi-vectors without any special conventions. Our goal in this section is to prove some elementary but useful inequalities about multi-vectors.

First and easiest

to prove is of course the Schwarz inequality:

M. K E AP+la: n+l

if

then (1.10 )

I<M.K>I .s.. IMI IKI·

There is a more subtle inequality: H E A~n+l.

K E APa: n+l

and

then. IKI\HI .s.. IKI IHI.

(1.11)

First recall that the tensor power Q9Pa: n+l

* Proof.

the following inner product: normal basis of il' •••• i p )

a: n+l •

if

(eo ••••• e n )

then the set of

ei

1

n E ®~n+l,

has

is an ortho-

(2) ••• (2) e i

is by definition an orthonormal basis for

Same for ®~n+l. and

let

(all p

Gfa: n+l •

Then it is routine to check that i f I Ie ® n I = IIC I I nl.

Let

Ie

e Qfa: n+1

'Ir: (2)sa: n+l - Asa: n +l

be the natural projection for any s. Clearly 1'Ir(IC) I < llel and also if G E ASa: n+l is given. there is a unique ~ E(2)sa: n+1 such that G ~

I'Y~

IGI

'Y i ••• i e i 1\ 1 s 1 'Yi 1 e i /VI • • • 1· •• s 1 'CI = ~

that since

'Ir(.~)

'Ir

= K.

and

'Ir(~) =

G.

(In fact. if

e i • just take 'Y to be s e • ) Now. let us pick /C and .n is

••• 1\

tOI

'CI

'Ir(n) = Hand

llel:; IKI,

is an algebra homomorphism,

Inl

'Ir( /C ® Tl)

IHI.

so

Also,

11

GENERALITIES ON PROJECTIVE SPACES AND GRASSMANNIANS

Thus I'IT(.~)I\'IT(TI)I =

IKI\HI

1'IT(Ie(8)TI)1

Next we turn to the interior product. H

€

h~n+l,

product of that, if

and assume K and

G

H,

0 ~ p ~ q. K

~

H,

P

=

0,

K

€

APC n+ l ,

We define the interior

to be that

=

(q-p)-vector such

.

K -1 H = irH

of course

by definition. K...JH

Let

Aq-PC n+l ,

€

If

Q.E.D.

= llel ITII = IKI IHI·

< 11e® TIl

and i f

On the other hand, if

to be that

p = q,

p > q

(p-q)-vector such that if <M, K ~ H>

=

~

0,

M

€

K...l H = we define

hP-~n+l,

.

There is a simple 'lemma which we shall need in Chapter III. Lemma 1.4. vectors, then

*~. assume Cn+l

=

If

Let

K € hPC n+ l ,

Write

H ffi H.i..

cn+l

H € A~n+l

and for definiteness,

as an orthogonal direct sum:

This leads to an orthogonal projection If

p(K)

is of dimension smaller than

K must contain an element of to see that

H are both decomposable multi-

K...J H is also decomposable.

p < q.

p: Cn+ l -H.

K and

H.i..

p,

then

In this case, it is easy

K ~ H is zero and there is nothing to prove.

let dim p(K) = p. Choose orthonormal basis feo ••••• en } eD+l , so that feo, ••• ,e _ } is an orthonormal basis of p 1

So of p(K)

12

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

(e o ••••• eq _1 J is an orthonormal basis of

and so that

H.

Then clearly. K

ae o 1\

=

I\ep-1 + (terms involving e q •...• e n ).

•••

It follows that

K..J H

=

aep 1\

•••

l\e q _1 •

Q.E.D.

Our next inequality is the analogue of (1.11): (1.12) This generalizes Schwarz's inequality (1.10).

* Proof. Let K If

E

E

Aq - PCn+1

E

APcn+1.

is a unit

H

E

A~n+1

and assume

p ~ q.

(q-p)-vector. then (1.10) and

(1.11) imply that II

= II < IHI IKI\EI < IHI IKI lEI IHI IKI.

Taking the case

E

to be p > q

K..J BIlK ..J HI.

we are done.

is similar.

The proof of Q.E.D.

We close this chapter with another inequality that will be needed in Chapter V. and

Let

E be a unit decomposable

F be a unit decomposable

subspace of

F.

Let

K be a

(q+1)-vector so that p-vector.

q-vector

E is a

Then

(1.13)

* Proof. decomposable

We first observe that if q-vectors and

a

A and

G are two

is a unit vector orthogonal

to

A,

wr1 te

E.

= .

then

F = E 1\ a,

Assume

p

~

where q

a

and let



By our assumption, we may

is a unit vec tor in EI

be a unit

F

normal to

(q-p)-vector, then

<E,KI\EI> = <EI\ a,KI\EI

1\

a>

= • If we let

EI

K .J E/IK ..1 EI,

then we obtain

(K .J E) I K ..J E I =
r 1\ a> •

Now

I~K

IK ~ FI

E? I\a\ -< 1 K ..1 :J E =

max

IBI~l

because of (loll) and since

,

The proof of the case

p > q

we have

IK.J EI < IK ~ Fl.

is similar.

Q.E.D.

CHAPTER II Nevanlinna theory of meromorphic functions §l.

OUr ultimate concern is with holomorphic mappings

of Riemann surfaces into

Before taking up the general

P C. n

case, however, we want first to deal with the case i.e., the Riemann sphere

S

n

=

1,

hoping thereby to gain

= Pl~'

some insight into the structure of equidistribution theory. The exposition of this chapter follows closely that of [8), and the reader may consult that paper for a more general account of Nevanlinna theory on Riemann surfaces. In Chapter I, we defined the for "Fubini-Study") on

<1: 2

F

(2.1 )

Where

Z

form of (2.1) ,

Let

F

=

~2

'11":

- (0)

_ (o) _ S

S

=

this

Pl <1:,

is the usual

is the hermitian covariant tensor

1 2«Z,Z> _ l

(zo,zl)

F

metric (abbreviation

In the case of

Pn~·

simplifies to the following. fibration, and on

F-S

and

dZ

=

(dzo,dz l ).

The associated two-

according to (1.6) is

as = ~ be the

ciated two-form.

dd c log Izi =

F-S

metric on

S

Then of course

is also the volume element of

r-r

F

d'd" log IZI

and let 'II"*ro = as, on

terms of a local coordinate function

S. z

ro

'II"*F =

F

and

ro

(In general, if in =

x +

hermi tian metric assumes the form g dz Q9 dZ,

14

be its asso-

~

y,

an

15

NEVALlNNNA THEORY OF MEROMORPHIC FUNCTIONS

g dxl\dy,

then its volume element is

.r-r g

two form i s us fix an

dz 1\ d-z.

~

a

So they are trivially equal).

We wish to construct a function

S.

€

while its associated

as

so that

ua(p) -

00

S - (a)

equals

ill.

p - a

1

on

ua c

S

on

2' dd u a

and such that

Let

Such a function was first found by Ahlfors

and Shimizu in 1929, and can be described in the following way. (ao'~)

Let

be a representative of

e~e.

mined up to a multiple of so

quotient space of a

in

S

(-81 '&0)'

a;

(Recall that

unchanged if


( a o ' ~)

= O.

is deterS

is the

The polar

1) .

aL

is of course just the point i.e.,

(ao,al )

space of

represented by I
The function

is

( e Ae a o ' e (.:Teal) ,

is replaced by

we may legitimately consider it as a function of

a,

a

so S.

€

Denote by na(z) the following function defined on (:2 _ (oj _ (,r- l ( a»):

Since

Ua(Z)

= na(~Z)

is seen to come from a function in a unique function ".* u a

= tl a •

ua

on

S,

COO

S - (a), in

Now the condition that

be eqUivalently described as ua

~,

for any nonzero complex number

u a (p) Z,

so

00

as

00

is easily seen to have this property.

is a holomorphic function of

i.e., there is

S - (a),

u a (p) -

such that as


p - a

may

0,

and

_

Furthermore,

dId" log 1
Therefore according to (2.1)1, the following holds in e2 _ (OJ _ (v-l(a»): ~ ddcn = .J=T dId" log Izl 2

=

~ dId" log IZI =~.

1a, a.!...) 1

a

Consequently,

0a

v*(l ddcu ~

a

_ ill)


O.

16

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

1 '2 dd c 11"* u a

* = '2 1 dd c u a

11" (1)

c

1

in

'2 dd u a below.

(1)

=

S - (a).

Theorem 2.1. defined on

* is injective.

Since

11"

We summarize this in the theorem

then there is a function

a e: S.

~ ddcua = (1) in S - (a l • form of the

(i11)

= O.

S with the following properties: is COO in S - (a).

(i) (ii)

If

~

F-S

metric on then

If

is the point in 11":

where

S

(:2 _ (0) _ S

is the volume

(1)

S.

= 10g..J!L

11 (Z) a

I rZ I

orthogonal to

a,

where

,

and

is the usual fibration. la~ZI

In (iii) above, we have written

for

ll, a

convention we shall adhere to from now on.

----

into

--------

§2.

To put this to use, we turn to holomorphic mappings

S.

We first clarify a close relationship between holo-

morphic mappings into

S and those into

open Riemann surface and mapping. x

(:2.

Let

V be an

x: V - S a nonconstant holomorphic

As is well-knOwn, this is equivalent to saying that

is a meramorphic function on

x: V - C U (00)

V.

More precisely. if

is a meramorphic function. let

be the stereographic projection.

Then

to be a holomorphic mapping into

S.

sox

s: C U (00) - S is easily seen

It is the converse of

this fact that

interes~us.

Let

be a holomorphic mapping. and we will show how

x: V -S

so we present it in greater detail.

16a

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

it leads to a meromorphlc function on be the usual fibration. if

Do

= ([zo,zl': Zo

neighborhood on such that V*

S

V.

Let

~: ~2 - (0) - S

As we have seen in Chapter I, §l,

~ 01,

then

Do

is a valid coordinate

withzcomplex coordinate function

~([zo,zl]) = Xo'

denotes the complement in

Now V of

S = Do U [[0,1]).

~:

Do - ~

If

16

(I)

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

* = 2" I dd c u a

I 2" dd c 11"* u a

11"

I c 2" dd u a

in

=

(I)

S - (a).

~

= O.

* is injective.

Since

11"

We summarize this in the theorem

below. Theorem 2.1. defined on

then there is a function

S,

€

~ ddcua = (I) in S - (a l , form of the

(iii)

a

S with the following properties: is COO in S - (a).

(i) (ii)

If

F-S

metric on then

I:t"

is the point in 11":

where

S

C2 _ (0) - S

(I)

is the volume

S.

(Z) = log~ , where a Irzi orthogonal to a, and t1

is the usual fibration. la~ZI

In (iii) above, we have written

for

lI, a

convention we shall adhere to from now on.

- ---

into

--------

§2.

To put this to use, we turn to holomorphic mappings

S.

We first clarify a close relationship between holo-

morphic mappings into

S

open Riemann surface and mapping. x

and those into

Let

V be an

x: V - S a nonconstant holomorphic

As is well-knOwn, this is equivalent to saying that

is a meromorphic function on

x: V - II: U (00)

V.

More precisely, if

is a meromorphic function. let

be the stereographic projection.

Then

to be a holomorphic mapping into

S.

this fact that Let

C2 •

interes~us.

sox

s: II: U (00) -

s

is easily seen

It is the converse of

so we present it in greater detail.

x: V - S be a holomorphic mapping. and we will show how

16a

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

it leads to a meromorphic function on be the usual fibration. if

Uo

= ([zo,zl': Zo

neighborhood on such that V*

S

~

V.

Let

~: t 2 - (0) - S

As we have seen in Chapter I, §l, 01,

then

Uo

is a valid coordinate

Withzcomplex coordinate function

~([zo,zl]) = Xo.

denotes the complement in

Now V of

S = Uo U ([O,lJ)'

~:

uo-t

If

17

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

eO

then Let

P

~ox

E

* V - V.

x: V* - t

Since

then

b E Uo '

x

such that

b

x(A)

S C.

f

b

C of

- (p}) =

~

property of

fl.

Let

A - (p}

n x(A)

B

Now

~(B)

is an open set in

Uo

onto

a:,

implies that

E(B)

n (E

Casorati-Weirstrass, This means

E

0

fl.

then

E(B)

b.

A of V,

B be a neighborhood

C,

omits the open set

p.

and

is discrete in

=

In particular,

a:

because

and the one-one x)(A - (p)) =

0

~

This means that the holomorphic function

0

so that

not containing

V - V*

Because

is a homeomorphism of

E x.

(p}

(= [0,1])

x(p) x(p)

* A - (p} ~ V.

disjoint from

n x(A

~

over

is holomorphic, there exists a neighborhood

we may assl.DIIe that

B

x

0

becomes a meromorphic function in a neighborhood of

there exists a neighborhood

of

~

We want to extend

To this end, let

P

is a holomorphic function.

t.

in

0

x

fl.

defined on

By the theorem of

p

is not an essential singularity of

x

can be extended to a meromorphic

a: u (m}. Do this for every p E V - V* and we have extended E x to a meromorphic function ~ Since V is open, ~ .. x: V - a: U (CD}. /by the well-known theorem of Behnke-Stein,

function

~ E x: 0

A-

0

there exists a pair of holomorphic functions

. x ~ = ....!. • Then the map Xo

~: V -

x o ' xl

a: 2

on

V

Such that _ x(p) = (xo(p),~(p»

becomes a holomorphic mapping.

Let

VI

be the complement in

V of the common zeroes of

and

xl

such that

Xo

(which is also a discrete set), then we have the diagram:

>-

x./a: 2

(2.2) VI

X

-

!T

) S

(oJ

18

TIiE EQUlDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

We claim that the diagram is commutative. 'IT

0

x =x

on

V',

set, say on V*

n

it suffices to show V'.

Let

p

V*

€

n

In order to show

'ITO i = x

on an open

V';

we will prove

xl'

so we are done.

(~o'IToi)(p)

(~o'IToi)(p)

--

~(p)

x:n;r' o

while and

We may therefore regard ('IT o~)

IVI

to all of

x: V- S

V.

as the extension of

In general, whenever diagram (2.2)

is commutative, we will say that the holomorphic mapping

~: V - c 2 induces the holomorphic mapping x: V - S. Conversely, suppose mapping such that

~: V_C 2

is a given holomorphic

~(p) = (xo(p),~(p)),

holomorphic functions, we will show that holomorphic mapping

x: V - S

where

in the above sense.

so V - VIis a discrete set in

gives a map

~: V' - C2 - (O).

V.

x

= 'IT

the diagram (2.2) is commutative by definition. extend

x: V' - S

p

€

xo(p)

= xl(p) = O. Let

p-centered coordinate function in a neighborhood and let

Xl

~

in

xo '

0

x.

V'

Then

If we can V,

(The extension, if possible, is unique).

V such that

Xo

V'

to be a holomorphic mapping on all of

we will be finished. So let

Let

restricted to

~

Define now

are

induces a unique

~

again be the complement of the common zeroes of V;

x o ' Xl

am~ + bnC

n

am+l

+ bn+1C

rn+l n+l

+ + ...

~

be a

U of

p

19

NEVANUNNA THEORY OF MEROMORPlDC FUNCTIONS

where

~

m, n

and

1

p

so small that

am

F 0,

bn

F O.

We may assume

is

U

is the only zero of

x o ' ~ in U. For Consider the new functions

definiteness, let us say m < n. * and xl* in U such that Xo

* Xo

am + am+l~ +

x*l = b ~n-m + b ~n-m+l + n'" n+l'" Xo

Since in U.

only vanishes at x*: U - 0: 2 - (0)

Let

'If" " x*

We claim:

extends

* Xo

p,

is clearly free of zeroes

from

U -

this, let again Uo = ([zo,zll: Zo

F 01

~([zo,zll) = z?'o.

and

q

F p,

then

* * x* (q) = (xo(q),xl(q)).

be such that

x

(p 1 to

~: Uo -G:

and

We now show two things: (~o

'If"

0

I

U.

(ii)

is one-one, (i)

'II" 0 X* :

is a holomorphic map, and we have the desired

result. So.

S

U -

x

~

q e U

('If"

U -

(p)

Since

and

if

be

implies that

0

x*)

(i)

X* )(q) = (;" x)(q)

is a holomorphic function on

To prove

U.

To prove (i), note that

(~o

'II'

0

and (ii) shows that

implies that

qFP

* * X* )(q) = ~('II"«xo(q).~(q))))

a

"(

~(q)

F O.

* * (xo(q) xl(q) 1) w • ---iwr.-Ix (q)1 I,x (q)1

* (q) • ~m(q) xl (q) xl --.--() = *( m =x:rcrr"~('If"(xo(q),xl(q))) = (~o'lf"ox)(q) Xo q Xo q) • ~ (q) 0 * (q) ~

- (~o x)(q)

by definition of

to note that never zero in on U.

~

eo'll"

u,

ox·

V-

To prove (ii), it stlffices

X~: by the above. and since x*o

~"'II"ox·

S.

is

is evidently a holomorphic function

Since we can do this for each

8UCceeded in extending It:

=

x.

*

x: V' - S

p E V _ V',

we have

to a holomorphic mapping

20

TIlE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

Following Herman Weyl, we call a holomorphic mapping x* :

such that

of

x

in

U.

'Ir

X

0

*

=

a reduced representation

X

The above procedure of extension proves inci-

dentally that if

x: V -+ 8

given any sufficiently small duced representation in

-x:

is induced by

U.

US

V,

x

c: 2 ,

V _

then

always admits a re-

We want to reprove this fact

without the assumption that

induces

x

x.

80 let

be an

V

arbitrary Riemann surface (in particular it may be compact) and let

x: V -+ S be a holomorphic mapping.

open set of

Let

V and for definiteness, let us assume that ([z o ,zlJ: z0

I oJ c- S. As usual, let

be the coordinate function on ~

0

x becomes a holomorphic function on x*: U_c: 2 - (oJ

map

U be an

such that x*

Uo.

Then

U and the holomorphic

x*(p) = (l,(~o x)(p))

'Ir"

representation of

U and we have proved that if

x: V -+ S then

x

~!:.

in

holomorphic mapping, where

phic of

is a reduced

V is arbitrary,

always admits !:. reduced representation in any suffi-

ciently small open set of

§3.

Thus

x*

enjoys the property that x

x.

clearly

V.

From this point on, let a fixed nonconstant holomor-

x: V -+ S be given and let V with

COO

boundary.

D be a compact subdomain

In this section, ~ allow

arbitrary, so in particular, it can be compact. we are interested in the counting multiplicity.

n~ber

Now

of pre-images of

x(p)

= a iff

V to be

If

a

a

in

<x(p),a~>

€

S, D,

= o.

NEVANLINNA TIlEORY OF MEROMORPmC FUNCTIONS

21

One can even show that the number of times p covers a is multi-valued equal to the order of zero of the/holomorphic function <x.a~ at

p.

Instead of proving this simp1e statement. we simply

take as our definition:

(2.3) n(D.a) = sum of the orders of zeros of <x.a-L> We also define of the

LS

=

ill

relating

}~ X*In.

=

metric on

F-S

is that

v(D)

in

D.

where

ill

1 if The following theorem

(The reason for the factor of

S.

c.f. (2.10) below).

'IT;

to

n(D.a)

is the volume form

is often called the non-integrated

v(D)

First Main Theorem. Theorem 2.2. such that

f(oD)

Assumption as above. suppose

n (a)

n(D.a) +

(2.4)

where

"a

1 c =~ d ua '

Proof.

Let

(a1' •••• a n ) cD U£

of

where

51.

=

a

is

then

ioDx*" a

=

v(D)

(See Theorem 2.1)

= (al ••••• a n ).

x-l(a)

- oD. and

By hypothesis.

so that we can choose a neighborhood f-l(U£)

with the property that

a j £ Uj '

a £ S

n Uk

Uj

=

51

if

Ul U ••. U Un' Then we have j ,i k.

by Stokes' Theorem and Theorem 2.1 that v(D)

=

!. 'IT

=

LD x*

!. lim 'IT

(J)

=

!. lim 'IT

f

x* (J)

E+O D-Ul-"'-Un

f

£+0 D-U1-"'-Un

x· (~ddCu ) a

=

f

11m dx·" a E+O D-Ul-"'-Un

22

THE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

It remains to show that the last sum equals

n(D,a).

For

this purpose, it is sufficient to prove that: lim

(2.5)

£+0

r -x

=

*),

orl j

a

the order of zero of

To this end, we may clearly assume

Y be a holomorphic mapping 1/"

0

y

=

x,

where

1/": 11: 2 -

x

a~

represented by

eo.

also write

y

yoe o + ylel'

functions.

Now

=

* ua)(z)

(1/"

Let

eo' e l

in

11: 2 -

-

S.

(0) -

viation for "orthonormal") Then

y: U j

) , aL>

is so small that we

Uj

can take a reduced representation of

<xC a j

(0)

O.N.

Uj ,

i.e., let

such that basis (abbre-

be chosen so that

is represented by where

Yo' Yl

log~.

I a..LZ I

el •

a

is

We may

are holomorphic

So relative to

e 0

Hence,

Because

IYI

is never zero in

so clearly lim £+0

Jd

C

OU j

equivalent to proving:

log Iy I

Uj =

,

O.

log Iyl

is

em

To prove (2.5), it is then

23

NEVANLINNA THEORY OF MEROMORPmC FUNCflONS

(2.6 )

11m £+0

k ClUjJd

C

But this is now immediate because since x(a j ),

we have

<x( a j ), e:->

log Iyll = the order of zero of

y(a j )

represents

Ix(aj),a.J.1 = ly(aj),a.J.1 =IY1(aj)l.

So i f

aj-centered coordinate function S is used, and if m m+l Yl = bm~ + bm+l~ + ..• , then the right side of (2.6) is an

of course

On the other hand, if

Yl = t~.

then

small that lim

m.

Jd

C

h

Since

h(O) ~ 0,

is zero-free in

log Ihl = O.

h = bm + bm+l t + .•• ,

we may assume

Uj ,

Uj

is so

so that again

Hence,

£+0 ClUj

lim £+0

k ClUjJd

C

log I yll =

i;r lim ~

lim

11.

J

+ =

But if

~ = ICle~e,

the real functions

f mdc log

£+0 ClUj

then

u, v

~

£+0

fd

C

£+0 OU j

I~ I

log Ihl

c

dC log I I •

j

dC log lei = de.

(In general, if

on the plane are such that

u +

4=I

is holomorphic then a simple computation gives

So since

log I ~ I +

result.)

Then,

..r-r e

is holOllOrphic, we have the above

This proves (2.6), and hence the theorea.

Q.E.D.

v

TIlE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

24

The nonintegrated v(D)

FMT

relates the important quantity

to the analytic quantity

n(D.a).

We now head towards

the nonintegrated Second Main Theorem. which relates the topological data on subdomain on

D. C~

V with

To state it. let boundary

no extra work in mapping

~D

aD.

M.

with respect to

of

~

f: D - M Let

df

K

is nonsingular

f * G becomes an hermitian metric in a

aD.

aD

So let

with Gaussian curvature

We will assume that

so that

neighborhood of

S.

M a compact Riemann surface.

G be an hermitian metric on

along

There is

D into an arbitrary Riemann surface

be a holomorph1c mapping.

O.

to

D be a compact

as above.

this time instead of the Riemann sphere

and volume form

v(D)

Thus the geodesic curvature form f * G is defined.

is given below).

Inside

of

~

(A precise definition

D however.

df

will in

general have critical pOints. i.e •• there will be a finite number of pOints j

= l, ... ,n.

(a l ••••• a n } CD - oD

These

their images

at which

df(a j )

aj's are exactly the points Which cover with multiplicity greater than one.

covers

m(j)-times. it is reasonable to call

the stationary index of

f

at

aj •

the stationary index of

f

in

D and will be denoted

Then the nonintegrated

SMT

Theorem 2.3.

Lj(m(j)-l)

will be called ~

Assumption and notation as above. the

21rX(D) -

(IC +

ern

~ (D)

=

fr* em).

D

If (m(j)-l)

states:

follOwing is Valid: (2.8)

= O.

n l (n).

25

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

Furthermore, if (2.9 )

G has constant Gaussian curvature then

~J Ie

xeD) -

The

X

oD

M = S,

=

X(M)v(D).

in (2.8) and (2.9) of course denotes the Euler

f

v(D) ;: -1 f *11, c D

characteristic, and if

~ (D)

+

this

v(D)

where

c

J

m

coincides with our previous

Before proving the theorem, let us note that the on

S

S = Uo

has-constant Gaussian curvature

U ([0,11) where

such that on

Uo'

+4.

v(D) .

F-S

metric

To see this, let

{[zo,zl]: Zo ~

~: Uo - C

OJ.

is the usual coordinate function

C([zo,zlJ) Using (2.1), we find

(2.10 )

CD

Now in general, if

G is an hermitian metric,

local coordinate function if

(2.11)

and __

z

=

x + ~ y,

G = g( dx 2 + dy2)

then 11

uA

11

its volume

K its Gaussian curvature, then with respect to a

form and

Where

Thus

11.

=

=

g dx" dy

=

=

they are related thus:

g dz ® dz,

~

g dz" dz

~ 2g 6. log g

K

~ ~_~ + ~ ,~ OX Oy

=

2 4 _0 is the usual Laplacian.

ozoz

Combining

(2.10) and (2.11), one readily sees that the Gaussian curvature of

F

is equal to

4

in

Uo'

Since Gaussian curvature is a

continuous function, it has to be equal to

4

at

[0,1]

also,

26

TIlE EQUIDISTRIBlJTJON THEORY OF HOLOMORPHIC CUKVES

and this proves our assertion above. Corollary.

Let

f: D - S

Consequently,

be a holomorphic mapping into the

Riemann sphere equipped with the

F-S

metric then

(2.12) Theorem 2.3 will be proved via the Gauss-Bonnet Theorem. Before stating the theorem, we fir3t give a precise definition

of the geodesic curvature form of an oriented (or parametrized) curve

in an oriented surface with respect to some Riemannian

y

metric

<,

so that

>.

ds

We assume

y

parametrized by arc-length

s,

t = d/ds

be

is the volume element of

the unit tangent vector field along field

n

so that

t

1\

n

y.

y.

Let

Choose a unit normal

defines the same orientation as the

surface i.e., if m is the volume form of m( t

1\ n)

> O.

= , Then = KgdS Kg

~

<,

>,

then

By definition, the geodesic curvature of where

y

is

D denotes covariant differentiation.

becomes a one-form on the submanifold

called the geodesic curvature form of

y,

y.

Now we quote a weak form of the Gauss-Bonnet Theorem. Let

~

be a compact Riemann surface with

equipped with an hermitian metric

G.

Let

the Gaussian curvature and volume form of

C~ K

boundary and

n

G as usual.

~

denote Then

the theorem states that 21rX(D) -

J I ro. K

=

Two comments about this theorem are in order.

First, with

27

NEVANl.INNA THEORY OF MEROMORPHIC FUNCTIONS

r-r

z = x +

respect to a local coordinate function

y,

(2.11)

implies that: (2.13)

~ g dz /\ dZ,

n

if

KSl = -~ da c log g.

then

~

Second, the following situation often presents itself.

is

a compact subdomain of a Riemann surface on which is defined

an hermitian metric on which

z - x

G.

U be a coordinate neighborhood

Let

+ "/":1: y

= (x = E, E a constant}

and that

nu

Ot>

is valid and suppose that

~ () u = (x < El.

Then in

the notation of (2.11), we claim that the geodesic curvature ~

form

06

of

~

(2.14)

U has the following expression:

in =

~ 0 10°!

= ~dc

g dy

log g)

restricted to

o~.

The equality of the last two quantities is immediate because dC log g while

06 () u

Since

~ (Iu o~

fora of

=

=

= _

0 log g dx + 0 log g dy

Oy

ox'

(x = a constant}. (x < El

and

We now prove

G = g(dx 2

is simply Jg dy.

As

to prove that the geodesic curvature Now, if

t

~

+ dy2),

= ~gdS, ~g

n

=

-1

..rg

0

Ox'

Then using

ox

~

1

06

o log ox

-fg

coherent

is the unit normal field of

Let us write for simplicity we get:

<,

t

.

the volume

is

of

coherent with its orientation, then clearly n

-

it suffices

is the unit tangent vector field of

with its orientation and

_ 1 0 logg dy If'

1

0

and

= ..rg Oy'

> for

G(

,

).

g

28

THE EQUIDISTRIBlJflON THEORY OF HOLOMORPHIC CURVES

thereby proving (2.14). Proof of Theorem 2.3. from (2.8). = Kcv(D), c = {n. gives

We first show how to derive (2.9)

K is constant, then { f* (Iffi) = K~f*n

Suppose

where it lIay be recalled:

v(D) =

~

£

f*n

But the Gauss-Bonnet Theorem applied to

21rX(M)

=

~

Iffi = Kc.

Thus

..£

and

M itself

f* (KG) = 21rX(M)v(D)

and

(2.9) follows. Now to the proof of (2.8). critical points of function around the set

aj

(!Zj! < E1

and let

f

such that and let

(al, ••• ,am)

Let Zj

be a local coordinate Denote by

zj(a j ) = O. W = U~=l Wj •

f * G is an hermitian metric, and so

be the

f

Wj

On

D - (al,···,am), becomes an isometry

D - (al ,··· ,am) equipped with f * G into M equipped with G. Hence f * (KG) is exactly the product of the Gaussian curvature of f * G and the volume element of f * G in from

D - (al, .•• ,am). curvature form of

~

So retaining the symbol

aw,

with respect to

f

* G,

for the geodesic we apply the

29

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

Gauss-Bonnet Theorem to get:

rlr. of,

2Ir(X(D-W)) -

to =

X(D-W)

if*(m).

=

D-W

is due to the fact that we

with respect to

OWj

Clearly,

D-W).

rIC oW

JIC

The plus sign in front of orient each

+

(and not with respect

Wj

f f* (m)

= XeD) - m and lim

1f* (m). Hence, 1f* (lm) = 2IrX(D) - onr

£+0 D-W

D

It

+ 11m

D

£+0

~=l (

r

O~j

2Ir)

/C -

To prove (2.8), and hence the theorem, it suffices to prove: (2.15)

lim(~ £+0

rIC - 1)

O~j

=

stationary index of

To prove this, let us first agree to write Next, let ~

of

f(a j )

=

p,

and let a

z

f

in place of

Zj.

p-centered coordinate function

C,

be chosen so that relative to

the volume element

n

G assumes the form:

..r:r a n ="2

T del\d.,.

Consequent ly,

The stationary index of

f

write

where

phic in

r*c

=

Wj •

Zm(j)h(Z),

at

aj h(O)

is

(m(j)-l),

f 0,

and

Then,

* d~zC

=

zm(j)-l(h(Z) + z~z».

h

so we may is holomor-

30

TIlE EQUIDISTRlBUfION TIlEORY OF HOLOMORPHIC CURVES

Let

b

assume

and

= (h + z~). Wj

b(O)

is so small that

(f* a • Ib 12)

is

simplify this, let

" + {.:r 6

z

1z 12d" 1\ de.

F 0,

we may as well

is nowhere zero in

curvature form

log 1z I.

Wj - (slit),

is the volume form of

~

of

OWj

f * G,

-

~z/\dz

with respect to

the geodesic f*G

therefore

=

~ dC loge f* a' 1b 12 • 1z 12m( j) 1

=

1 c m(j) dc log Izi + 2 d log(f* a· Ibl 2 )

=

m(j) dC" + ~ d C log(r*a. Ib1 2 )

=

m(j)d6 + ~ dC 109(f*a' Ib1 2 ),

where in the last step, we made use of (2.7). is nowhere zero in

Wj ,

the last item is a

Hence, l1m(k e+o

and

when

OW j ,

restricted to

Wj'

Then

...r-r

Now a simple computation gives

assumes the following form in accordance with (2.14):

~

Thus,

To further

= Izle{.:r e and let function in

Wj •

Hence,

f *n

But since

b

= h(O)

and zero-free in

COO

is a coordinate

dWj = (n = log e). =

Since

r~ _1)

O~ j

*r

=

lim e+o

=

m(j) + 0 - 1

O~ j

de + 11m e+o

Since COO

one-form in

rii:

O~ j

r*a· Ibl2

dC

log(f*a. Ibl)_l

stationary index of at a j .

f

31

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

This proves (2.15), and hence the theorem.

Q.E.D.

Before closing this section, we should point out that Theorems 2.2 and 2.3 are quite interesting when we restrict D to be a compact Riemann surface without boundary.

So let

be a compact Riemann surface without boundary and

M'

f: M' - S

a holomorphic mapping (which, by the preceding section, is just a meromorphic function on for all

a

E

S.

covered exactly

Theorem 2.2 gives

M').

This is the statement that every

n(M',a) a

E

S

=

v(M')

is

v(M')-times, thus giving the simplest instance

of equidistribution.

Actually, a more general statement can

be proved without appeal to Theorem 2.2, namely, let

f: M' - M

be a holomorphic mapping between compact Riemann surfaces, then the excision axiom in homology theory readily yields that each m

E

M is covered exactly the same number of times (which is

just the degree of

f).

some positive integer

J

So let us say n.

IJ

because

n

to one for

c =

Jn.

Hence (2.9)

M

~(M')

c

nx(M),

a relation known as Hurwitz's Formula. this on the assumption that

is

Then in the notation of (2.9),

V(M') = -1 f * n = n· n =n c M c M' becomes:

X(M') +

f

We have only proved

M admits an hermitian metric of

constant Gaussian curvature, but the uniformization theorem shows that this is true for all compact holds in general. in M'

Now ~(M')

so Hurwitz's Formula

is the stationary index of

and is by definition nonnegative.

and X(M') > X(M),

M,

f

Consequently, if 0 ~

there exists no nonconstant holomorphic

X(M)

THE EQUIDlSTRIBlJflON THEORY OF HOLOMORPHIC CURVES

32

mappings

f: M' - M.

We shall generalize Hurwitz's Formula

in Chapter IV; these are the so-called Pldcker formulas.

§4.

So far the open Riemann surface

V has been arbitrary.

To get deeper results, we must impose a restriction on V. Taking

as a model, which is exhaustible by the level lines

~

of the harmonic function Definition 2.1.

log r,

A CfIO

we introduce:

function

T: V - [0, s)

(s.s.. flO)

is called a harmonic exhaustion on the open Riemann surface

v

iff

(1)

T is onto

(2)

T is proper, i.e., inverse image of compact set is compact.

(3)

T is eventually harmonic, i.e., there exists a number

0.s.. reT) < s,

reT),

harmonic on

such that

T is

T-l([r(T),s)).

As usual, a fUnction

T is harmonic iff

respect to a local coordinate fUnction (l1T)dxl\dy,

where

l1

z

ddcT

=x +

=

~

O.

y,

is the Laplacian.

cOincides with the classical definition of harmonicity.

With we

So this It

should be remarked that at the moment,le do not as yet distinguish between the cases done later. and

ai

E

M,

If

V

such a

s

= flO and s < "",

M - (~I" •• ,aqJ, T with

s

CfIO

function which equals

M is compact

= "" can be easily constructed:

choose holomorphic coordinate functions then a

where

but this will be

zi

- log IZil

about each in

ai'

(I zi I .s..~)

33

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

will do.

A similar procedure shows that if

V is obtained

from a compact one by deleting a finite number of discs, such a

s <

with

~

can be constructed.

~

Finally, if

is

V

the Riemann surface of a multivalued function such as it also has a harmonic exhaustion with ~,

be the canonical projection, and let on

which equals

~

Then

~ =

on V.

~'o

log r

=

00:

v:

Let

be the

C~

V-~

function

outside of the disc of radius two.

is obviously a harmonic exhaustion function

v

Note that this particular

the pOints whose images under v

s

(sin z)l/n,

has critical points at

~

are integral multiples of

v

(= 3.14159···).

The last example shows that we must in general make allowance for which

d~

points

~

let

z

to have many critical points, i.e., pOints at

~

O.

=

However, where

~

is harmonic, the critical

all isolated because of the following:

be a coordinate function near

The function

f

o~ +

= Oy

.rT

'V-.!.

p,

z

Let

d~(p)

= x + {.;r y.

o~ is holomorphic because the Ox

02~ + 02~ = 0, which is ox 2 Oy2 is harmonic. Since the critical points

Cauchy-Riemann equations are just satisfied because of

~

coincide with the zeroes of Convention. ~

Which

is harmonic.

V and we work only in that part of

r EE

Contains a critical point. T

in

which

r(~).

is called a critical value of

critical values of

V on

Thus all parameter values of

arise are always greater than An

we are done.

From now on, we fix a harmonic exhaustion

function

on

f,

~

iff

~-l(r)

The above implies that all the are discrete.

Hence

O·,

34

THE EQUIDISTRIBlTflON THEORY OF HOLOMORPHIC CURVES

except for this discrete set of numbers, T(p) ~ll

r) r

~

oV[r)

(= (p: p E V,

is an imbedded one-dimensional submanifold for reT). (Since

oV[r)

disjoint union of circles).

is compact, it has to be a Thus

V[r)

(= (p: p

V, T(p)

E

~

r))

is a compact surface with

C~

boundary except for the discrete

set of critical values of

T.

Furthermore, if

this exceptional discrete set, submanifold of

oV[r)

V except for a finite set of pOints, and these T on

Before discussing the passage from holomorphic function

cr

such that

its conjugate one-form

Re cr

T,

we digress a

If

~

is a one-form

is defined as follows:

*~

take any local coordinate function

= p dx + q dy;

oV[rJ.

T to its associated

bit to define the conjugate differential.

~

belongs to

is a one-dimensional

are precisely the critical points of

on V,

r

z

=

then by definition

*~

x +

~

y

and let

= p dy - q dx.

A

simple computation will show that this definition is independent of the particular

z

chosen.

will show that a real function Thus returning to

T,

~

Another Simple computation is harmonic iff

we see that

*dT

d*d~

= o.

is a closed one-form

on V - V[r(T»). Now we want to generalize the situation that in any annulus of function

t,

one can construct frOll

log r +

~

log r

e (multi-valued) which gives a coordinate

function on the annulus-minus-a-radial-slit. (rl ,r 2 )

the holomorphic

So suppose

does not contain any critical values of

W is one component of

Int(V[ r 2) - V[ r l ) .

be one of the level curves of

T in

T and suppose

Let 'Y

W for some

W n oV[ r J

=

r

E

(r1 ,r 2 ).

35

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

J*d~

We call

'Y

=f

harmonic function a fixed point of

p

~

a = (p

Let Thus

a

~

of

'Y.

by

f

and define the conjugate

pep)

d(*d~) =

Since

integral multiples of

a =~ +

*d~

a period of

and

=

J: *d~,.

0,

P

= *d~.

dp

where

Po

is

o

is defined up to

Consequently

P is a (multi-valued) holomorphic function on W. E

W: pcp)

= 0

(mod f)J

is a slit containing

Lemma 2.4. X (O,f) ~ ~

For

a

= ~

pcp) = nf,

(1.e.,

E

,E).

Po'

+ .(.:'I p as above,

is biholomorphic.

n

a: W - a - (rpr2)

(i.e., one-one, onto and holo-

morphic) • *Proof.

Clearly

a

is holomorphic, single-valued, onto

and everywhere non-singular.

We have to prove one-one-ness.

Suppose

~(p) = ~(p,) =

p

a(p)

and

p'

=

a(p').

lie on the same level curve 'Y*

If we can check that one-one on and 'Y*,

Then

W-a. then

p

Let

is one-one on 'Y*,

say.

Thus

=ov[r* 1 n (W-a). then

a

will be

be the point of intersection of

p*

=JPoP *d~ = JPo p*

pcp)

r*,

*d~

+

JPp

**d~.

a

The path of

integration of the second integral may be chosen to be along

'Y*. Since d~ is nowhere zero in W, *d~ is nowhere zero on 'Y*. But it is easy to see that *d~ is actually nowhere zero when restricted to the submanifold 'Y*

so that the second

integral must be either strictly increasing or strictly decreasing as

p

traverses

Corollary.

'Y*.

J

o

r)

This proves that

p

is one-one on 'Y*. Q.E.D.

*d~

-

L

positive is a/constant for all

36

TIlE EQUIDISTRIBlTflON THEORY OF HOLOMORPHIC CURVES

Proof.

It is a continuous function of

r

and the lemma

says that it is constant in between critical values of T. SO L is constant. Since *dT is coherent with the orientation of oV[rJ, L is positive. Q.E.D. Let

Definition 2.2. of

T.

contain no critical value

(rl'r 2 )

The holomorphic coordinate function of Lemma 2.4 in a T- l ((rl'r 2 ))

component of

is called a s:eecial coordinate

function. We shall always denote this special coordinate function by

(1 =

T+

.r-r

p.

Two

remarks should be made here:

is defined only up to a translation in

( i)

(1

but in all appli-

p

cations. this ambiguity is irrelevant and will therefore be ignored.

(ii)

(1

can be aonstructed locally wherever

T is

free of critical pOints. and in the sequel we will have to invoke this local existence of Now we return to

(1.

x: V- S.

Our immediate task is to

integrate the formulas in Theorems 2.2 and 2.3.

The first

step in this direction is to transform the line integrals that appear in these formulas.

So given

compact surface with boundary V[rJ

a

€

S.

consider the

satisfying the following

two conditions: (61..)

x(oV[rl)

(£ )

r

(£ )

n (al

=¢

is not a critical value of implies that oV[rJ is a COO

T. submanifold of

so according to Theorem 2.2:

(2.16)

v(r)

V.

37

NEVANLINNA THEORY OF MEROMORPIDC FUNCTIONS

1 2i dc u a '

and

vCr)

for

Since

(rl ,r 2 )

x

is holomorphic,

(mod r).

= 0

,. =

X*}. a

r. =

T- l «rl ,r 2 )),

W of

dinate function p

rl , r 2

X*~a

n(V[r],a)

(J

=

..r-r

T+

In W,

p

oV[rJ

such that

1

x*"

T.

r

€

(rl ,r 2 )

Working in

we employ special coor-

valid on W minus the strip becomes the coordinate curve

Since dCX*U a ox*u ~ ~ dp when restricted to

W() oV[rJ

~ dCX*ua •

=

contains no critical values of

one component

for

Under (ot) and (ct),

Since (£) holds, there exist and

n(r,a)

v(V[r]).

Lemma 2.5.

Proof.

and we have written

we see that oV[r).

Hence,

1 J(r,r) ox*u a dp a - 2i (r,O)--o;ro 1 J(r,r) = ~2i (r,o)x*ua dp) = =

o 1 J(r,r) * ~21r (r,O)x u a *dT)

rrco 2i Wn1oV[r) x*u. a *dT). 1

Since this is true for each component of

oV[rl,

we are done. Q.E.D.

Thus

if

(01..) and (-£) hold,

n(r,a)

+...2..rJ=, r x*ua or' ovlr)

*dT)

vCr)

C"lI

We can now carry out the afore-mentioned integration:

we integrate

38

TIlE EQUIDISTRIBUTION TIlEORY OF HOLOMORPHIC CURVES

both sides with respect to

r.

So let

in which (~) and (~) hold for all

r

[rl ,r 2 )

be an interval

[rl .r 2 J.

€

Then the

above leads to:

Jrl2 n(t.a)dt + k <3V[t) I X*Ua *dTl:21 r

(2.7) where

J:2 v(t)dt, 1

h(t)l:~

have used the standard notation:

we

=

=

h(r 2 ) - h(r l ).

To extend (2.17) to arbitrary intervals. we need a technical lemma: Lemma 2.6.

r

<3virJ function of

r

x*u a *dT

for all

r

~

for a fixed

a

is a continuous

r(T).

We do not prove this lemma here for two reasons. more general lemma will be proved in Chapter IV.

(2)

(1)

A

We are

trying to gain an understanding of holomorphic curves by looking at this special case of mapping into the Riemann sphere, so we should not be distracted by such technical details. Now let X-l(a)

(ro.rn )

~

and the critical pOints of

relatively compact set for which say,

(r(T),s)

(~)

-r

-r-l«ro,rn )),

The points

are both finite in the so the points in

[ro,rn '

and (£ ) do not both hold are finite in number,

(rO·rl····,rn_l'rnJ·

vi"tJ x*ua *dTlri+l ri

<3

Then for each

(Cf. Def. 2.1).

(ri,ri +l ),

We define

vi"tJ x*ua *d-rl:

lim ctri+l,d,l.r i <3

(2.17) holds and thereby giving

equations of the type (2.17).

We add these

n

n

equations and

39

NEV ANLINNA THEORY OF MEROMORPIllC FUNCTIONS

using Lemma 2.6 we see that the sum of the middle terms of these equations telescopes and (2.17) becomes true for itself.

[ro,rnJ

In other words:

This forces the definitions: T(r) =

Definition 2.3.

x and

function of

N(r,a)

Jrro v(t)dt

=

is called the order

Jrro n(t,a)dt

is called the

counting function. The choice of

ro

(ro

~ reT»~

it is fixed once and for all. to it in the sequel. First Main Theorem.

is immaterial so long as

We will suppress any reference

(2.18) is essentially the so-called Let us note that we can actually simplify

the middle term a bit morel namely, according to diagram (2.2), and

'lr

0

X=

X

holds in V' ,

the set of the common zeroes of let

oV' [tJ

=

oV[t) () V'.

and

oV[t] - oV' [tJ

where ~.

For any

t

~

T(r),

is then a finite

set of points and so in particular is of measure zero in Therefore, by (iii) of Theorem 2.1:

is

V - VI

oV[t).

40

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Combining this with (2.18), we have Theorem 2.7 (FMT)

For every

r

~

r(T):

Observe that the terms on the left depend on T(r)

does not.

deficiency of

a

whereas

Hence the middle integral compensates for the N(r,a),

(e.g.,

N(r,a)

=

0

f(V[rJ) () fa)

if

=

¢),

and for this reason is sometimes referred to as the compensating term. Now it is easy to see that, since line of

T,

*dT

ov[tl

is the level

induces a positive measure on

ov[tl.

is by definition of the way one orients the boundary of

oV[t)

Furthermore, Schwarz's inequality (1.10) implies

V[t)).

that

~ > 1 (one must never forget that

that

109.J..!.L

la~il

-

Ia..Lil

(2.19 )

(This

> O.

-

r

oV''rt]

log

la~1

=

1),

so

Hence,

~ *dT > 0 Ia~i I

for all

t.

Another technical lemma we need at this point is this: Lemma 2.8.

r 109.J..!.L * dT I a~il

for a fixed

r

is a

ovtrJ

continuous function of

a.

We will not prove this lemma here for the same reasons as those given after Lemma 2.6.

In any case, combining these two

facts, we have arrived at the following basic inequality:

41

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

N(r,a) < T(r) + const.,

(2.20)

independent of Proof.

r

and

where the constant is a.

By Theorem 2.7,

(2.19 ) ~

T(r) + const.

where the constant is chosen to be the maximum of

r log ~ *d,. la1.il

1

~ ovr'ro] S.

as

a

varies over the compact surface

(Lemma 2.8).

Q.E.D.

This inequality will dominate what is to follow.

For

the moment, however, we turn to the integration of (2.12). Let us first introduce some potation: write

x(r)

for

x: V-S

= x(V[r]), nl(r) = nl(V[r]).

as above,

Then under assump-

tion (-j,) above and ([ )

df

is nowhere zero along

oV[rl,

we know from (2.12) that (2.21)

Where metric

x(r) + nl(r) - 2V(r) ~

=~

r

o~[r]

is the geodesic curvature form of x*F.

(F

is the

F-S

metric on

~ oV[r]

S.)

tbe last line integral, we introduce a function

in the

To transform h

on V - V[r(,.)J:

42

TIlE EQUIDISTRIBlTfION THEORY OF HOLOMORPHIC CURVES

(2.22) where m is of course the volume form of the on

S.

h

x

h

is

~~

T in

negative function, for the following

is holomorphic, so

orientation on V,

log h

F

(which is discrete), but the important thing to

note is that

dTA*dT

metric

is not defined at the critical points of

V - V[r(T)]

reason:

F-S

X*ill

is coherent with the

while a simple computation also gives that

is coherent with the orientation on V.

Consequently,

is a well-defined function on V - V[r(T)].

Lemma 2.9.

Under assumption (o£) and

~

[ ), for every

r ?. r( T):

Proof. is a

COO

for every

Because of the presence of ('£) and ([ ),

log h

function, and because of (2.14) and (2.22),

r?. reT).

So the lemma can evidently be proved in

the same way as Lemma 2.5.

Q.E.D.

The situation now parallels that of the FMT:

we have by

virtue of Lemma 2.9 and (2.21) that x(r) + nl(r) - 2v(r)

= ~(J= r v

't'lT"

r (log o¥tr]

h)*dTJ.

The analogue of Lemma 2.6 states that Lemma 2.10.

r

o1ft r]

(log h)*dT

is a continuous function

43

NEVANUNNA THEORY OF MEROMORPIDC FUNCTIONS

of

r

for all

r

~

reT).

(This lemma will not be proved here, for the same reasons as above).

So using this lemma, an integration leads to:

Introducing the following notation: E(r)

~ Jr

ro

x(t)dt,

we have arrived at the Second Main Theorem. Theorem 2.11 (SMT)

For

E(r) + N1 (r) - 2T(r)

§5.

x: V - S

~~ ~~

and

( (log ovttJ

r

~

r( T),

h)*dTlr • ro

The program now is to integrate the inequality (2.20)

with respect to a well-chosen function on the sphere is the basic idea of Nevanlinna and Ah1fors.

S.

This

Before doing

that, however, we need to know something about integration of differential forms. Lemma 2.12.

Suppose

f: D -V

is a

compact oriented manifold with boundary manifold

V

of the same dimension.

grable form of top degree in V, n(a)

C~

map from a

D into another oriented

Suppose

~

is an inte-

and suppose for each

denotes the algebraic number of preimages of

Then

~f*~ ~

I

n(a)t

a

a

E

V,

in D.

44

TIlE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Remark.

The meaning of algebraic number of preimages is

as follows. n(a)

n(a)

0

=

is defined only when

(ii)

df(a j )

df

f-l(a)

then

is a finite number

(al, .•.• a p+q )

say,

j.

and

In the event that

preserve the orientations of

D

and

M

q

of

j

Then by definition,

seen from the proof below that almost all points of

V.

n(D.a)

I t will be

p - q.

n(a) n(a)

is thus defined for

For a holomorphic

surfaces, this definition of the above

a € Im f,

a IS and reverse them at the remaining

of the

the

If

is nonsingular for each

both hold, let p

(i)

aD.

of points disjoint from

at

a ~ Im f.

if

n(a)

f

between Riemann

clearly coincides with

except at the critical points of

f

(which

is only finite in number and therefore has nil effect on the integration) .

* Proof. image under

df

V -

e - f ( aD ),

e

a compact set.

and because

D,

So

VI

Hence

V - VI

D

in

D

(Le.,

is compact it is thus

e

f(aD)

f(aD)

Of course

is an open submanifold of

by SardIs theorem, well-known that

f

is the

being the image of this compact set is

itself compact and hence closed. closed.

e

where

The set of critical points

is singular) .

is closed in

f

=

of the critical points of

f

points at which of

VI

Let

V.

is also

Furthermore,

is a set of measure zero and it is is also a set of measure zero in

has zero measure.

If

v € VI.

is closed, and discrete (because

df

is nonsingular on

and hence finite (because v

€

VI

•

n(v)

of points in in

VI

is defined. rl(v),

onto which

f

D

is compact).

Let

then each maps

n(v)

n(v) v

€

then

V.

f-l(V) f-l(v))

Thus for every

be the total number VI

has a neighborhood

open sets diffeomorphically.

45

NEV ANLiNNA THEORY OF MEROMORPHIC FUNCTIONS

Let

(V j )

be a locally finite covering of

sets and let (VjJ.

Then

(~j}

{f*~j}

VI

by such open

be a partition of unity subordinated to is a partition of unity subordinated to

the locally finite covering

(f-l(V j)}

of

D_r-l ('~ ) -oD.

So

by definition of the integral,

J

f*41

D_f-l(t:. )-oD =

Jf*41,

D

where the last step is because

= ~41 = O.

Jf*41

Now, the definition of

=

n(a)

0

and gives:

Q.E.D.

Lemma 2.12 has very interesting implications, which we now give.

Recall that

N(r,a)

= Jr n(t,a)dt,

to integrate both sides relative to

and we propose

ro

ID

over

S.

In order to

46

THE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

invert the order of integration of the right side, we shall need the following plausible fact: Lemma 2.13. where

[rl ,r 2 ]

n(t,a)

[rl ,r 2 ] x S

is a measurable function on

is any finite subinterval of

[a,s).

A more general statement will be proved in Chapter V. Now since

n(t,a)

~

0,

Fubini's theorem gives:

1N(r,a)CD Jrro (r~ n(t,a)CD)dt Jrro(vr{JrX*CD)dt =

S

(Lemma 2.12)

=

=.r:ro'JTV(t)dt ='7I'T(r). In other words, (2.23)

~

T(r) = }

N(r,a)CD.

Account being taken of the fact that

I CD

= 11",

we see that

S

this says the order function is the arithmetic mean of the counting function.

As a corollary, the arithmetic mean of

the compensating term over

S

is equal to zero, a fact which

a180 follows directly from the homogeneity of the Riemann sphere. Armed with (2.23), we are gOing to show that if V carries an infinite harmonic exhaustion (i.e.,

8 =

~

in

Def. 2.1), a very strong form of the Casorati-Weirstrass theorem holds on V. " * (a)

For

a

E

S,

define

;;: inf (1 -

NH~;».

47

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

~*(a)

is called the defect of

*

meaning of

a.

Before explaining the

we first show that:

~,

o < ~* < 1 It is obvious that

on

S.

* <1

because both N and T are posiTherefore, to show < 1 + const. tive. By (2.20), T(r) * li const. = 0, and that o ~ ~ , it suffices to show that T( r) ~

.

NHrj)

r!

this amounts to showing that

T(r) -

is obvious in view of the fact that

as

00

r -

vet)

00.

But this

is strictly increasing

and positive, so that

When is N(r,a)

* equal to l? First of all, if a , 1m x, then 0 for all r and clearly ~ * (a) = 1 in this case. ~

=

In general, if

a

is very sparsely covered by

that the growth of 1. e., if

N(r,a)

11m sup N( r" a)

=

lags far behind that of 0,

'I"\rT"" other extreme, when ~ * (a) = 0, r+co

x very often.

is Covered by

x: V - S,

~

then again

* (a) = 1.

T(r), The

then obviously means that The main observation now is

that: ~

This implies

*

0

xCV)

Bays much more.

almost everywhere on is open dense in

S,

S.

but of course it

To prove this, observe that (2.23) implies

that

~(l

-

NHfj))(\)

O.

so

48

THE EQUIDlSTRIBUTION THEORY OF HOLOMORPHIC CURVES

On the other hand, the integrand considered as a family of functions on

S

depending on the parameter

r

is uniformly

bounded because (2.20) implies that 1

> (1 - NH~j)) > cT(~)' - 0

as

r -"'.

Hence by Fatou's lemma,

O. Since

5*

~

[5*(a)CJ)

0,

=

5*

The statement that

which is equivalent to

0,

5*

vanishes almost everywhere on

o S

is the essential content of the FMT; it says that the points of

V are evenly distributed on all of

of measure zero.

S

except for a set

This is the first equidistribution statement

we have proved thus far.

Later on, we shall bring the proof

of a much sharper statement:

La€S 5* (a)

~

2,

provided certain

reasonable conditions are met. We conclude this section with a very useful fact: Lemma 2.14 (Concavity of the logarithm). positive measure on a topological space Suppose

~-measurab1e.

function defined on

k

~~~I

* Proof.

E,

f

Let

R and let

is a nonnegative and

~

be a E be

~-integrab1e

then

J(log f)d~ < log(~ If d~).

E

-

~~~I

The following is a trivial rephrasing of the

classical proof as given in Nevan1inna [6, p. 251].

So let

a.e,

49

NEVANLINNA THEORY OF MEROMORPmC FUNCTIONS

c

=~

(i)

l

[cp dlJ.

dlJ.,

and let

= 0

and

log(l + t) i t.

~

IJ.\""I

§6.

I

cP =

(11)

qJ

f-c.

Ie >

Hence by (11),

(log f)dlJ.

Note two things: -1.

But for

t

~

-1,

log(l + cp/c) i cp/c,

and so,

i

Jlog c dlJ. + EJlog(l + !)dlJ.) c log c + ~ It dlJ. log c

'=

10g(~

=

~(

IJ.\"'I

E

We will now integrate

arrive at the defect relations.

I

f dlJ.}.

Q.E.D.

N(r,a) < T(r) + const. to Let

p: S -JR

be a function

satisfying two conditions: (i) (11)

it is integrable and nonnegative.

!

pro = 1.

We shall later give

p

explicitly.

Since the constant in

N(r,a) < T(r) + const. is independent of

r

and

a,

JN(r,a)p(a)m < T(r) + const.

S

because of (ii).

But the left side equals

because of (i), Lemma 2.13 and Fubini's theorem.

Hence it is

equal to

(Lemma 2.12) (x* p

is positive)

50

TIlE EQUIDISTRlBUI'ION TIlEORY OF HOLOMORPHIC CURVES

(2.22)

where the last step may be proved by invoking the special

a

coordinate function

(2.24)

J:

o

dtJ! dS( 0

=

r

avlsJ

T+

.r-r p.

Hence,

(x·p)h .dT) < T(r) + const.

We now apply Lemma 2.14 and the Corollary to Lemma 2.4:

or, log L +

f( r

r log h .dT} ~log r (X·p)h.dT avts)

log x·p .dT +

avls)

So

av'rsl

by (2. 24) :

(2.25 )

Jr dt Jt exp[log L + ro ro

f avlsJ r log(x· p).dT + ~ av1s) r log h .d·dd!

< T(r) + const. We wish to bring

r

log(x· P).dT to the form of the av'r s J line integral in the FMT. By the argument preceding Theorem 2.7.

x•p

Should be

~. More correctly, let [~, •••• aq) be

laJ..il

a finite set of distinct pOints of function

p

on

c 2 _ CO}:

S.

Consider the following

51

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

where

c

o <~

< 1.

e

€

C*

is some positive constant to be determined l and p(eZ)

p(Z)

=

=

'Ir

-PI

where

s.

(;2 - (O) _

'Ir:

P is an integrable function on S.

and so

(~I"'laq)'

p

-1!L

We first check that

The only places that may

cause trouble are exactly the points and inspect

Thus we

is continuous near

!aiZ !

Choose C.N. basis

e ol e l

by

is represented by

then

aj

in

«:2

the coordinate function given by Clearly

is an

~

so that

aj

el •

~([zoeo

Let

..J!L

Z + z1Zl)~ 0

_

=

~: U 0 -

+ zl e l J)

(1 +

zlzl

le.jZI

II:

be

z~ • o Since

=

,-2 )\

,z

/

0

we see that with respect to the coordinate function

{~)2~ assumes the form:

!e:jz ! •

is represented

aj-centered coordinate system.

(..J!L)2~ = ( z0

!ajz!

Let us

i ~ jl

Now i f

need only concentrate our attention on the factor

e ol

for all

P on S such

implies the existence of a function

*P

that

As usual l the fact that

(1 +

!~!-2)~.

~I

It is consequently

equal to

Since that

0 < ~ < 11

!~1-2~

polar coordinates in the plane clearly says

is integrable near the originl i.e' l

(~)2~

Ie:jZI

52

THE EQUIDISTRIBtrrlON THEORY OF HOLOMORPHIC CURVES

is integrable near integrable on

S.

aj •

Since this holds for each

j,

p

is

As

p

is clearly nonnegative, condition (i)

of the requirement of

p

is met.

make an appropriate choice of Thus we have chosen our

To cope with (ii), we simply

c. p.

The usual arguments then

give:

Combining this with (2.25) and letting Irr dt o

J:

ro

exp(const. +

+

~

t 1,

we obtain:

t Lf-l- avtrs ) log Iatlilx I.dT ~

r log h *dT}ds

avis]

< T(r) + const. Substituting into this inequality the expression of the line integrals in the FMT and SMT, we have clearly proven the following:

if we define

+ const., where the constant is independent of (2.26)

f~,

... ,aq}

and

s,

then

J:rodt Itroe~(S)ds -< T(r) + const.

We now bring to (2.26) a minor refinement.

Define

n(r,a)

to

53

NEVANLINNA TIlEORY OF MEROMORPIDC FUNCTIONS

<x,a~>

be the number of zeroes of

in V[rJ

without counting

multiplicity, and let

We clearly have:

n(r,a) - n(r,a) = ~

0,

so that

~

1 (stationary PEX- (a)nV[r) nl(r) = ~pEV[r) (stationary

index of

x

at

p)

index of

x

at

p) ~ ~=l(n(r,~) - n(r,~)l ~ O.

Nl(r) ~ ~=l(N(r,~) - W(r,~»,

Hence,

which implies that

Combining this with (2.26), we have the final result: Theorem 2.15.

If

~(r)

denotes the quantity:

then

where the constants are independent of

rand

(al, ••• ,aql.

From Theorem 2.15, we are going to derive bounds of directly in terms of

T(r).

~(r)

We seek inequality of the type:

e~(r) ~ (T(r) + const.). Naturally,

we must make allowance

for the fact that such an inequality may not hold for all

r.

To this end, we distinguish two cases. Definition 2.4. exhaustion if

s =

~

V is said to admit an infinite harmonic in Definition 2.1.

If

s

<~,

we say

54

THE EQUIDISTRlBUfION THEORY OF HOLOMORPHIC CURVES

V admits a finite harmonic exhaustion. CASE 1.

The infinite case.

increasing function on real number

k

continuously is a once/differentiable, positive,

If t

Lemma 2.16.

>

with

[ro '''')

ro ?. 1,

then for any

t' < t k holds on [ro '''') - I,

1,

is an open set on which -[ d log x < "'.

where

depends on

(I

I

k. )

J.

Proof.

Let

t' > vt k where

I C [r ,"') -

is any positive function and

v

arbitrary positive number. v

be the open set on which

0

is an

Then,

tK

1 (""l-k) , < 1 "" ~ = (1 - k) ~

=>

k > 1

~ v <~ ~

on

I,

(tl - k ), =

< '"

where we have written is because

t' ~ wk.

t

I =Ui(ai,b i )

and the last inequality

is positive and increasing.

If we let

v =

1

x'

ro?. 1

So outside of

implies

v <1

I,

on

and the lemma is now obvious. Successive applications of Lemma 2.16 to and

t(r) = fr dt ro

J: e~(s)ds

e~(t) ~ (J:oe~(S)dS)k

J: e~(s)ds ~

yield:

if

k > 1,

then

0

for

t . I'

(T(t) + const.)k

for

such that t . I"

tfd log x < "';

such that

o

tilrd

log x < "'.

The latter made use of Theorem 2.15.

Let

55

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

I

I' U I",

then

e~(r) ~ for

r f I

such that

(T(r) + const.)k 2

Jd log x < co.

In general, we shall

I

let

"II" stand in front of an inequality if the latter only

holds in

[ro'co) - I.

Thus,

1I~(r) < k 2 log(T(r) + const.), 2

1I~~=l(T(r) - N(r,~)) ~ 2T(r) - E(r) + \~ 10g(T(r) + const. + const. by virtue of the expression of

~(r)

in Theorem 2.15.

Equi-

valently,

"~k(l -

N(r,~) -E(r) 1 Lk2 T(r) )~ 2 + ~ + T[rT("'""4ir log(T(r) + const.)

+ const.). The important thing to observe is that since I

does not comprise a neighborhood of

inequality holds in an unbounded set.

co,

~d

so that the above

We are therefore prompted

to introduce:

~(a) = lim inf(l _ 1'!(r,a)) r+co """"T'[FJ X =

log x < co,

lim~uP(-~f~')·

The above inequality therefore reads: ~=l ~(~) ~ 2 + X + 11m sup ( J. r+co

56

THE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

Since

T( r) - '"

as

r - '"

are all independent of

r

and since the constants in (~,

and

••• ,aq}'

implies that the last term vanishes.

l'HOpital's rule

Consequently,

Theorem 2.17 (Defect Relations). morphic and if

(}

If

x: V - S is holo-

V admits an infinite harmonic exhaustion, (~,

then for any finite set of distinct pOints

••• ,aq}

of

S,

(2.27) 5,

like

5*

defect function. 5* < 5.

Hence

introduced in §5 above, is also called the Since

~

R(r,a)

0 <5 <1

N(r,a),

again holds.

it is clear that 5(a) > 0

lightly covered compared with the other points. we say

a

is a Picard point.

If

a

iff it is If

S - xCV),

€

5(a)

1,

surely

a

is a Picard point, but the converse is by no means true (e.g.

ze z

assumes the value

0,

but

5(0)

=

1).

The reader

will easily convince himself that from (2.27) follows the relation:

Rrwhen

X is finite,

now elaborate on

X.

5

=0

except on a countable set.

It evidently measures the relative growth

of the Euler characteristic of the order function.

V[rJ

against the growth of

If the number of components of

does not stay constant for large r - "'.

We

r,

then

oV[rJ

x(V[rJ) - '"

as

Hence in this case, the usual direct-limit argument

with homology groups gives

xCV) = -"'.

Thus if

xCV) > -"',

57

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

there is an

ro

X(V)).

constant (equal to i.e., if

r > ro ==> x(V[r))

so that

V is

C or

Therefore, when

C - (oJ,

L~=l ~(~) < 2.

reduces to

zation of Picard's Theorem.

X~ 0

is a fixed

X(V)

~

0,

so that (2.27)

This is a far-reaching generaliX>0

In general,

is to be

expected because we may delete as many points as we wish from

s

to obtain an open Riemann surface

S'

which admits an

infini te harmonic exhaustion; the natural injection of

S' - S

certainly cannot obey any defect relation of the type ~ ~(~) ~

2.

So we should seek a condition on

insure the vanishing of

X

= O.

all

x

~

roo

o.

is transcendental and

In fact let

r

Here is one.

;!: T'[rT =

transcendental iff that if

X. r

ro

x

itself to

We call

x: V - S

Then one can easily prove

X(V)

is finite, then

be so large that

x(V[rl)

=

X(V)

for

Then,

X = 11m sup lim sup

-~t~~

= 11m sup

T(;)

T(;' . x(v)(r-ro )

f:o X(t)dt

= 0

In a special case, the notion of transcendency coincides with the classical notion of essential singularity.

For there

1s this result: Lemma 2.18. Surface then

If

V is obtained from a compact Riemann

M by deleting a finite number of pOints

x: V - S

is transcendental iff

to a ho10morphic mapping

Xl:

M - S.

x

(~,

••• ,am)'

is not extendable

58

TIlE EQUIDISTRlBUfION TIlEORY OF HOLOMORPIUC CURVES

Proof.

If

x: V - S

is transcendental, we will first

show that it is not extendable to prove a more general statement: harmonic exhaustion and for every real number

x': M - S. if

x: V -S r

€

V admits an infinite is transcendental, then

xCV - V[r])

is dense in

Suppose false, then there exists an borhood r

€

m.

of

U

such that

~

t

S

€

r

= roo

=

¢,

and a neigh-

¢

=

S.

for some

So

n(t,a)

= n(ro,a)

Hence,

t ~ ro => x(ov[tl) () u

Next, since ov[tl

r o'

a

unx(V - V[rl)

There is no harm in letting

for all

to

a

In fact, we

for all such

(See Theorem 2.1)

t

x* u a

restricted

is bounded above by a constant

K.

By (2.18),

T(r)

by the corollary to Lemma 2.4. ~

1

n(ro,a) > O.

So clearly,

if

x

then it is extendable to an

x': M - S.

= ~ > O.

=

for all

a

E

S

r TtrT

This contradicts transcendency.

We now prove the converse:

Then

lim sup

11m inf T(:) «2.20»,

nCr,s)

~ < "".

is not transcendental, So let Since

11m inf N(r/a) < d Or N(r,a)

d

Orr

r

11m sup T[rT

N(r,a) < T(r) + const.

~ < "".

But

59

NEVANLINNA THEORY OF MEROMORPllC FUNCTIONS

r_oo

has a 11mi t as

n( r,a)

and

because it is monotone

11m N(rla} exists and equals 11m n(r,a) by r 1 < 00. But the number l'H&pita1's rule. Hence lim n(r,a) < 13" ~ is independent of a, so the number of pre-images of a increasing, so

!E!. all a

E

Now let z.0 /: 0)

is bounded by a universal constant.

Uo

and let

~(( zo' zl J)

of §2,

S

zlL

/Zo'

x: V - S

be the open set in ~:

Uo -C

S

such that

Uo

=

r[Zo'Zl]:

be the usual coordinate function

U0 U ((0,,1 J) • After the reasoning r--is ho1omorphic iff x: V - G: U roo) is a Then

S

=

eo

meromorphic function.

According to the preceding paragraph,

this meromorphic function

~

~

0

x

has the property that its

preimage of any member of the extended complex plane is a ~inite ~

~

0

x

number of pOints.

But

V

=

M - (~, ••• ,am)'

has an essential singularity at any

aj ,

so if

the Casorati-

Weirstrass theorem coupled with the Baire category theorem would imply that there is at least one is an infinite set.

a

E G:

This not being the case,

dab1e to a meromorphic function on

M,

Whose preimage ~

~

0

x

is exten-

and consequently

itself is extendable to a ho1omorphic mapping

x

x': M-S. Q.E.D.

For further applications and examples concerning defect values, points of ramification and uniqueness theorems, the reader is referred to Nevan1inna [61, Hayman [41, [81, as well as a forthcoming dissertation by Edwardine Schmid (Berkeley 1969). CASE 2.

The finite case.

Lemma 2.19.

Suppose t

continuous isa once/differentiable positive

60

THE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

increasing function on k > 1,

number where

[O,s),

<~.

s

t'(r) i s:r tk(r)

is an open set on which

I

Then for any real

holds for all

r

~d

-~.

10g(s-r) >

I

[O,s) - I,

E

The proof of this theorem is analogous to that of Lemma 2.16. As before, the main thing we should note is that comprise a full neighborhood of t'(r) i s:r tk(r)

s

in

does not

so that

is true for numbers arbitrarily close to

By applying this lemma successively to

Jrrodt Jtroe cp( s) ds '

[O,s)

I

JTro

e~(s)ds

and

we obtain:

IIcp(r) i k 2 log(T(r) + const.) + (k+l)10g(s:r) where

"II"

now means

"outside of

I".

This together with

Theorem 2.15 imply that

2

i 2 + -~f~~ + ~(L4~ 10g(T(r) + const.)

== 2

+

-H ~ ~ +

+ (k+1)10g s:r + const.} E(

r) •

So we introduce again:

~(a) X

=

11m inf(l _ N(r,a)) r+s 'I"(r')

lim sup r+s

E =

-~frr~

11m sup ~(c1 log(T(r) + c 2 ) + 2 log s:r + c 3 }, r+s -L\ J -L

where

are same positive constants.

The above

s.

61

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

inequality leads to Theorem 2.20 (Defect relations). morphic and

If

£ =

0

(~,

••• ,aqJ

If (2.28) holds and So in this case,

S:

xCV) >

O.

-~,

~ ~(~) ~

it is obvious that 2

again holds.

clear what the exact meaning of (2.28) is. one can show that if

of

if 1 log s-r lim sup T(r) I*s

(2.28)

also.

is holo-

V admits a finite harmonic exhaustion, then for

every finite set of distinct points

Furthermore

x: V -S

x

For

V

X

=0

It is not

= unit disc,

has a very bad essential singularity

on the unit circle, then (2.28) is satisfied.

Yet. there are

meromorphic functions with an essential singularity on the unit circle for which (2.28) does not hold. c~

show that if (2.28) does hold. then

in

S

in a very strong sense.

In general. one

xCV - V(rJ)

See [81 for details.

extreme of (2.28) is the case where

T(r)

is bounded.

is dense The other For

Such meromorphic functions defined on the unit disc. there is a vast literature.

One class1cal theorem due to Nevanlinna

1s that such a meromorphic function is the quotient of two bounded holomorphic funct1ons.

The reader should consult

Hayman (4J for this and related matters.

CHAPl'ER III

Elementary properties of holomorphic curves ~l.

an

OUr object of study is a holomorphic curve, i.e.,

x: V -

Pnt,

where

x

an open Riemann surface.

is a holomorphic map and

(xo' ••• 'xn )

function this.

x

Let

fxo' ••• 'xn }

(n+l)

x=

(xo' •.• 'xn )

x:

t n+ l

V-

is

and the vector valued

is not identically zero. V'

holomorphic func-

not all of them identically zero; in

other words, a holomorphic map given such that

is

The most natural way to generate

such a map is given by a system of tions

V

We now elaborate on

be the complement of the common zeroes of

in

V,

then we have the following commutative

diagram:

(oJ (3.1)

x =

where by definition, x

x.

In a moment, we will extend

to be a holomorphic map on all of

holomorphic

Pn~

x: V -

x

commutative, we say V

'Ir.

V.

Whenever we have a

x: V - t n + l

and

induces

x.

The extension of

is achieved in the following manner.

= xn(p) = O.

Since

we may choose a coordinate function in a neighborhood

U

of

p

(3.1)

such that

Let

p

€

x

is to

V - V',

V - V ' i s discrete, z

such that

centered at

p

U - fpl CV'.

defined In

U,

O.

We

we have the following factorizations: xA(z) where

5 A > 1,

YA

=

z

5A

A=O, ... ,n,

YA(z),

is holomorphic in 62

U

and

YA(O)

I

63

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

may assume

U is so small that

U for every A.

YA

is zero-free in all of

Among the integers

there is a

~o""'~n'

smallest one; there is no harm in assuming that it is This gives rise to a map

*

x (z)

(Yo(z),z

=

We now claim that

11" 0 X

x*: U - (:n+l - (0) ~l-~

*:

U.

and

(11)

11" 0

Let

Uo

x*

is holomorphic.

([zo, ... ,zn): Zo

I

0)

E

U - (pJ.

(i)

1I" o

-

Pn a:

x*1

U

-'(pJ

We will do both simultaneously. ~: Uo

and let

_a: n be the zlL

~([zo,···,znJ) = ( iZ o"'" (~o 11"0 X * )(z) = (~o

To prove (i), i t suffices to prove z

°yn(z)). x: U - (p J

extends

coordinate function such that

for all

such that

We need only prove two things:

to all of

=

~ -~

°Yl(z), ••• ,z n

~o'

znL /Zo)'

x)(z),

But

(~ox)(z).

To prove (11), we must show that in U.

~ 0 11"

Q

X

*

is holomorphic

But this is obvious because in the above expression,

YO""'Yn are all holomorphic functions and Yo is never zero in U. This proves that we have extended x over p. Since

p

extended unique.

is an arbitrary point of x: VI - pna:

to all of

V - VI,

V.

we have in fact

The extension is obviously

Hence,

Lemma 3.1.

A holomorphic mapping

x:

V - a:: n+ l

not identically zero induces a unique holomorphic map

which is x: V - Pnt.

x,

64

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

x: V -+ Pnil: be holomorphic. and let U * U -+ il: n+l (0) such that V. A map x:

In general. let

-

be an open set of * = X on U is called a reduced reEresentation of 'lroX U.

In the course of proving

x:

that if

U -+ il: n+l

small.

in

Lemma 3.1. we have clearly proved

induces

reduced representation in

x

x: U -+ Pnil:.

U. provided

then

x

admits a

U is sufficiently

For a later purpose. we want to prove the existence

of reduced representations in greater generality and in a simpler way. and that

x:

So let us drop the assumptions that U -+ IC n+l

induces

be a holomorphic mapping. open set in that Uo

s:

x(U)

V.

where

([zo ••••• znJ: Zo U0 -~ ",n '" such that

s.

x

V arbitrary.

Uo

J oJ

~

PnlC

x*: U -+lC n+l - (0)

~o

x

where

U decomposes into

= (~ ••••• xn)'

such that

Let

x*(p) 'Ir

(l.xl(p) ••••• xn(p))

= 0

n

Then the mapping

X

*

x

on x

in

U.

sufficiently small open set in

In other

U.

V.

then

If

Thus.

x

U is a

admits a reduced

U.

We now prove the converse of Lemma 3.1. Lemma 3.3.

is

x: V -+ Pna: . be a holomorphic mapping.

V is an arbitrary Riemann surface.

representation in

U is so small

( zlL znl ) fZ o ... • • .no •

is a reduced representation of

Lemma 3.2.

U be a small

is the open set

s( ( zo' .... zn J)

clearly holomorphic and satisfies x*

Let

x: V -+ Pnil:

on which is the coordinate map

restricted to

holomorphic functions

words.

and simply let

For definiteness. let us say

~Uo'

Relative to

x

V is open

Given a holomorphic curve

x: V -+ PniC.

65

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

,mere

x:

V is open, there exists a map

induces

V _ G: n+l

x.

* Proof. Let Uo = ([Zo, .•• ,ZnJ: Zo f 0) C:

UO -

which

(Cn

and let

~([zo,···.znJ)

be the usual bijection:

= (

zlL znL fZo'···' IZo).

It is quite clear that

xCV)

cannot be contained

in every hyperplane of

PnG:,

so we may as well assUme that

xCV)

is not contained in

Zo

O.

In that case, the multi-

valued holomorphic function on

V: p - <[l,O, •••• O],x(p»

only a discrete set of zeroes.

Let V*

x·• V* - U0 C- Pn G:

these zeroes.

Then

to

is then a collection of

x: V* - (Cn

* on V.

~o

say.

= (fl, ••• ,fn ).

x

can be extended to be prove this, let ta.ining

p

p

so that

€

n

and let

U - (pl <,;; V* •

and there is at least one

be the complement of is well-defined and n

holomorphic functions

We claim that

fl, ••• ,f n

meromorphic functions on V.

V - V*

aj

has

To

U be an open set conNow x(p)

=

[O,al, ••• ,an ],

f O. Consider Uj = ([zo, ••• ,zn]:

Zj

f 0). Then by continuity. x maps some neighborhood of

p

into

x: U - Uj be welldefined. On Uj is the coordinate function ~ : Uj _ G:n such Zo Zj_l Zj+l zn that ~([zo.·.·,znl) = ( - , ••• ., then .. .. •••• - ) , Zj Zj Zj Zj e• x: U - G:n gives rise to n holomorphic functions on U:

to x

Uj •

Shrink U if necessary, let

(ho •••• ,hj_l,hj+l' •••• hn). If we denote by +1 * X : U - a: n _ (0) the map x (q) (ho(q), ••• ,hj_l(q),l,hj+l(q), * ••• ,hn{q» for q € U, then 'lroX = x; for clearly =

.•

e. lie

x* = ~ox and ~ is a bijection. Since ho(p) = 0. may assume that p is the only zero of ho in U (again

T

0

Ibrink U i f necessary).

It follows that for

q

€

U - (p):

66

TIfE EQUlDISTRIBUfION TIfEORY OF HOLOMORPHIC CURVES

•

But

(~O

X)( q)

(~O

X)(q) = (fl(q), ••• ,fn(q»

fl' ••• ,fn ,

by definition of

so the two ordered sequences of holomorphic functions on ~ hi.-l 1 hi.+l hn U - (p), (fl' ••• ,fn ) and (il" .. ' 'lL' ' ... 'lL)' are o 000 0 in fact equal. As the latter are in fact meromorphic functions on all of

U,

they are therefore the unique extension to Do this for every

and we see that n

~

0

x

0

-

-

~

~

x = (fl" •• ,fn )

~

~

0

€

V - V*

= (fl , ••• ,fn ) is extended uniquely to

meromorphic functions

1.e.,

p

U

x: V - 0: U (co} x ••• x G: U rcol

on V,

is meromorphic.

By the theorem of Behnke-Stein, there exist holomorphic (~'~l, ••• ,an'~n}

functions

such that

-

fi

=

aU

(?Pi)

on

V.

Now define:

i

and let

x:

V -

o:n+l

The claim now is that

= 1, ••. ,n,

x

induces

complement of the common zeroes of V*

=

Let

xo' ••• 'Xrt

'!T.

-

x: V*

need only prove that

n VI

-

'!TaX

the latter is open in V.

-U

and

o x

Now for

and

x: V*

agree on p

€

V*

be the

VI

in V and recall

is the complement of the discrete set

Then of course

(xo , ••• , xn ) •

x.

It is perfectly simple to prove this.

that

i

be the holomorphic map

x-l(PnG: - Uo)'

n VI V*

n VI,

-

n VI

Uo.

We

because

67

ELEMENTARY PROPERTIES OF HOWMORPHlC CURVES

(~o 1T o X)(p)

=

(~"1T)(Xo(p), ••• ,xn(p))

a l (p)

a (p)

(~'···'e:(p)) (fl(p), ••• ,fn(P)) (~

by definition of 1TOX(P)

x(p)

=

(fl(p),···,fn(p))

=

because

p

E

V*

ox)(p)

fl, ••• ,fn • for all

p

Since

is a homeomorphism,

~

V*nV'.

E

Q.E.D.

Because of Lemma 3.3, we will always assume that our given

x: V - PnG:

will eventually need both

x

and

we should keep (3.1) in mind. uniquely determined by x. see that -x: V - Cn+l and V-PnG: and

x:

is induced by a certain x

V-

cn +1.

We

in most situations, so

Note that

x

is far from being

In fact, it is not difficult to

-y: V -",..n+l

induce the same map

iff there exist nonconstant holomorphic functions

g

on

V

such that

hi

= gy.

h

To avoid non-essential

complications, however, we will agree that once one choice of

i

is made, it will be kept fixed throughout the discussion.

§2. those

For the most part, we shall be interested only in

x: V - PnC

such that

does not lie in lower

xCV)

dimensional projective subspaces of

P G:. n

This is a natural

Condition to impose since we are interested in the question (tt

how many hyperplanes

IUbspace of

PnC

we say x: V-PnG: Clearly

x

xCV)

can fail to meet.

that contains

xCV)

is

PnC

If the only itself, then

is nondegeneratej otherwise, degenerate.

is nondegenerate iff

x:

V _ IC n+l

is not contained

68

THE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

~n+l.

in ! proper subspace of for nondegeneracy.

We need an analytic criterion

To this end and for other purposes, we

introduce the following quantities: borhood

U in

fix a coordinate neigh-

V and a coordinate function

z

on

U,

we

let

(3.2)

k=l, ••• ,n

-xC j)

where by definition

X~

We record here the effect on coordinate function on w is given on zero on

U.

U.

of a different choice of

Assume a second coordinate function is nowhere

then it is clear that

U,

If we let

y(k)

(4, ... ,4), dk

dk

=

dw

k

l, ••• ,n,

dw

then obviously, -(1) Y

dz -(1) Ow x ,

-( 2) y

Ow Y

d -(1)

=

and in general,

If in analogy with above, we introduce the notation

- -(1) 1\ Xk =yl\y w

then

•••

-(k) , l\y

69

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

(3.3) We formalize here the convention that the subscript symbol

z

in the

xk

z will always indicate the local coordinate function

with respect to which

xkz is formed, as in (3.2); when the particular coordinate function used is immaterial, we will omit the subscript and simply write

Xk.

Now fix a coordinate neighborhood function

z

on

U.

value in

~£(k) ~ Ak+l~n+l

U and a coordinate

Then the function on

takes 1

(where as usual

see §2 of Chapter I).

If a second coordinate function

used on

is related to

(~)

U,

then

x!

is nowhere zero in Lemma 3.4.

p ~Xk(p)

U,

x~ by (3.3).

1)' w is

Since

we have proved

The holomorphic mapping

U - ~£(k)

given by

is defined up to multiplication by a zero-free

holomorphic function on

U.

As a consequence of this lemma, it makes perfect sense Xk

to speak of the zeroes of say:

"Xn

on

is identically zero on

In particular, we can

V. V".

(Notation:

Xn == 0) •

The criterion of nondegeneracy which we are after is then the following: Lemma 3.5. Proof.

x: V - Pn~

Suppose

x

subspace of dimension borhood

U of

is degenerate, then n

in

~n+l.

i(k)

x

== O.

lies in a

Over any coordinate neigh-

V with coordinate function

defined as in (3.2) and

xn

is degenerate iff

z,

-(k) x

can be

lies in the same subspace of

70

THE EQUlDlSTRIBUTION THEORY OF HOLOMORPHIC CURVES

~n+l in

as

for

= l, ••• ,n.

k

~

x,i(l), ••• ,i(n),

over

V.

i

U and hence

x~

xl\i(l)I\ ••• l\i(n)

=0

a coordinate neighborhood (k

in

U

=

(e , •••• E

vectors

has to be zero

Consequently.

=0

Xn

in

As above. let us operate in

with coordinate function

x ••.•••. x o n x(l) ..... x(l) o n

. x(n) •..•• x(n) o

di~ ~. dz such that

n + 1

l, ..•• n ) are defined as in (3.2).

det

where

U.

Xn - O.

Conversely, suppose

that -(k) x

Since there are

e

o

1\ ••• l\e

z

so

By (1.8).

n

n

1 is the canonical basis of ~n+l and

n

Since

x~

= 0,

there is an integer

£,

0 < £ < n,

while

Shrink zero in

U if necessary, we will assume that U.

Expand

A£+l

A£

is nowhere

by the last column; this gives a

differential equation, homogeneous of order

(£+1),

which is

71

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

satisfied by

x£+l'

Because of the alternating property of determinants. this same differential equation is also satisfied by Hence.

xo ••••• x£+l

equation of order

are solutions of a homogeneous differential (£+1)

whose Wronskian

By a basic theorem of such equations. dependent in

U.

xo' •••• x£.

Since

(= A£+l)

vanishes.

xo ••••• x£+l

U is an open set.

are linearly

xo ••••• x£+l

are

linearly dependent in V (because these are holomorphic functions.) Consequently xCV) has to lie ina proper subspace of «:n+l , and x is degenerate. Q.E.D.

§3. curve

From now

x: V- PnC

~ ~

will assume that

is nondegenerate.

~

holomorphic

To every nondegenerate

holomorphic curve is associated a whole series of holomorphic mappings into various affiliated projective spaces of

Pn~'

and we now describe some of these. The first such is the so-called associated holomorphic ~ of rank

(£(k)

= (~

k.

! i».

coordinate function

kX: V - G(n,k) ~ P£(k)_lC, Let

z.

be defined as in (3.2).

k

= 0 .... ,n.

U be a coordinate neighborhood with then

Xk z

=

i.
Identifying

\k+lq:n+l

with

can

~£(k)

as usual. we have defined a map Lemma 3.5. a map

kX:

x~ U -

Xk. z' U _ (!£(k) • By is not identically zero. x~ therefore induces

P£(k)_lC,

by Lemma 3.1.

Suppose

U'

is a

second coordinate neighborhood with coordinate function

z' •

then the above procedure leads to a map

k xt :

U' -

P.t(k)-l(!'

The content of Lemma 3.4 is clearly that

kX

and

It:x'

agree

72

TIlE EQUIDlSTRlBUTION THEORY OF HOLOMORPHIC CURVES

Hence the collection of local maps in fact defines a unique holomorphic mapping We can say more: kX(V)

x~ is a decomposable

since each

in fact lies in

G(n.k).

that we identify G(n.k) holomorphic mapping

kX: V -P£(k)_lC,

(See §2 of Chapter I; recall

with its image in

kX: V -G(n.k)

S

P£(k)-l~')

P£(k)-l~'

kX'

if

k.

So by

is any coordinate neighborhood

U

with any coordinate function UI

The

k = O••••• n.

is called the associated holomorphic curve of rank the definition of

(k+l)-vector.

z.

Xkz

defined as in (3.2) and

is the complement of the zeroes of

Xk z

in

U (which

makes sense by Lemma 3.4). then

(3.4)

is a commutative diagram. Note that in the above. if is trivial. x

So we only get

= ox.lx•••• 'n_lx,

From

n

~

k

=

n,

the whole situation

nontrivial curves: on. we exclude the

~

k

=n

from our consideration. The associated curve of rank curve

x: V_

Pn~

k

of the nondegenerate

may well be degenerate. i.e ••

lie in a lower dimensional projective subspace of However. we now show that divisor

~

of

G(n.k)

kX(V) if

from Chapter I §2 that for

x B

may

P£(k)_lC,

can never lie in a polar is nondegenerate.

€

kX(V)

G(n.k).

Recall

73

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

(3.5)

(A

G(n.k): AnB.L

€

intersection in

J

¢)

Pi(k)-l~

of

the hyperplane defined by (M

€

=

O.

<M.B>

and

=0

Ak+la:n+l).

So suppose for one

G(n.k)

B

Let

G(n.k).

€

S

kX(V)

~B'

Equivalently.

U be a coordinate neighborhood on which

is a coordinate function

z is defined as in Shrink U if necessary. we may assume x~ has no

(3.2). zero in

U.

By

property that of

kX(P)

on

U.

(eo .... • e k )

such that

xk

(3.4). the map x~: U _tiCk) - (01 has the

k1T

0

x~

for each

Let

z

kX'

=

p

€

Thus

x~(p)

is a representative

eo ••••• e n

be an O.N. basis of

spans

Since

B.

<X~.B>

U and hence by the above.

<X~.B>

=

a: n+l

such that

<x "x(l)" .•. "x(k).

eo" .•• "e k >. (1.9) implies that

on

U.

Let

n

~A=O

x

YAeA'

then this is equivalent to

det

- 0

i

where

y(i) j

yo •.••• yk

=

~. dz

By Lemma 3.5 (or rather. by its proof).

are linearly dependent in

U.

A fortiori.

=0

74

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Yo""'Yn V.

Thus

are linearly dependent in x(V)

U,

and hence in all of Cn+l ,

lies in a proper subspace of

dicting the nondegeneracy of Lemma 3.6.

If

Hence we have proved

x.

x: V -Pnt

is nondegenerate, the asso-

ciated holomorphic curve of rank

La

not lie in a polar divisor

contra-

k

(k

O, ••• ,n-l)

=

for any

does

B E G(n,k).

§4. A second type of related curve of x: V- Pnt is obtained by projecting

x

subspace

More precisely, let

Ah

of

pnt.

into an

h-dimensional projective

the fixed holomorphic map which induces be an O.N. basis of E

t n+l

such that

stand for the subspace of

x i

n

~A=O

YAeA'

t n+l

if

x

where

Let

coordinate function

eo""

0

(eo, ••• ,e n }

I\eho

Let

~

eo,ooo,eho If x into Ah is,

of

AX: V-A

induced

= ~~=O Ya(p)e a • Obviously, AX.

z,

so that

(3.2).

x~

AX~

and

Then for

k

~ h,

=

AX"Ax(l)" 000 "AY/

IAX~I

§3). x(j) = ~A yij)e A,

=

IX~ ~ Ahlo

is the interior product of Chapter 1, *proof.

yi j )

and let

be

U be a coordinate neighborhood with

are both defined as in (~

AX(P)

is nondegenerate, then so is

Lemma 3.70

=

a: n+l

V_

spanned by

then the projection

AX: V -E,

x,

Ah

by definition, the holomorphic mapping by the map

x:

djYA =

--;::r-' dz

Since

x

Similarly

=

~~=O YAeA' AX

h

~a=O yaea ,

and

where

-( j) AX - ~a yij)eao

xk in terms of eo,···,e , i t xk n A z and z are equal is clear that their coefficients of e " •• "e io 1k If we expand both

0

75

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

io < ••• < i k ~ h.

provided only terms

e

therefore involves

where at least' one of jk But since Ah = eo 1\ ••• 1\ e h •

1\ • •• 1\

jo is bigger than h.

(X~ - AX~)

e

(X~ - AX~)

interior product of

with Ah

jo ••••• jk taking the

obviously has the

effect of annihilating every single term of the latter type. Thus

(X~ - AX~) ...J Ah

clearly

O.

=

or

X= -.I Ah

AX= .J Ah.

=

But

IAX~ ~ Ahl = IAX! I because AX lies in Ah. Q.E.D.

§5. We now come to contracted curve of the first kind. Let

Ah

be an

h-dimensional projective subspace of

let an O.N. basis

(eo ••••• en J of

h A =e o l\···l\e h •

Let

h < k ~ n-l.

as in (3.2).

For

is a mapping

U - Ak-hr,n+l.

decomposable

(k-h)

=

function

xkz

is defined

p ~ (X~...J Ah)(p)

the function

P£(k_h_l)_lr,.

Xk J Ah z

is a

where

£(k-h-l)

If a different representative

w were chosen in

and a different coordinate U.

th e· res ulti ng

Xkw .J Ah

would

X~...J Ah of above by a nowhere zero holomorphic

function (Lemma 3.4). P£(k_h_l)_lr,

so that

By Lemma 1. 4.

were used for

differ from the

S

S

(Lemma 3.1).

e~ 1\ . . . 1\ ~

z.

vector and so this induces a holomorphic

U -G(n.k-h-l)

(~ ~ ~).

be chosen so that

U be a coordinate neighborhood on

which is a coordinate function

mapping

r,n+l

and

pnr,

Hence the resulting mapping

would have been the same.

U -G(n.k-h-l)

This implies that

we actually have a holomorphic map uniquely defined on all of V:

for

h < k

~

n-l.

76

TIlE EQUIDISTRlBUTION THEORY OF HOLOMORPHIC CURVES

and we cul this a contracted Lemma 3.8.

Xk

~

of the first kind.

J Ah does not lie in a polar divisor of

G( n. k-h-l) • Proof. By (3.5).

Suppose it lies in <Xk -.J Ah.B>

means that

Xk

=

O.

LB'

Le..

where

B

<xk.AhI\B>

lies in the polar divisor of

by the decomposable

k-vector

E

Ahl\ B.

G(n.k-h-l). O.

But this

G(n.k)

defined

contradicting Lemma 3.6. Q.E.D.

The most interesting special case is when this case. space

of

Xh+l.J Ah: V _ Pnf-. Ah

by

A~

k

h + 1.

In

We denote as usual the polar

and claim that in fact.

(3.6) be an O.N.

This is quite clear because if we let (eo ••••• e n ) basis of '"~n+l such that Ah = eo 1\ ••• 1\ e h • then By definition of the interior product. if <ea , Xh+l -.J Ah>

=

<Xh+l • e 0 1\

••• 1\

a

=

Ai.

O••••• h.

e h 1\ e a >

=

then

O.

Return now to the coordinate neighborhood U with coordinate function z. then Xh xh+l xh+2 are defined as in z' z • z Ah and y(l) _ dy We have (3.2). Let us write y xh+1.J z - Oz· the following important lemma.

Proof.

We have

hence

77

ELEMENTARY PROPERTIES OF HOLOMORPHlC CURVES

Consequently, y

=

/1)

(Xh 1\ 3{(h+l)).J Ah z

(X~l\x(h+2)).J Ah

=

To prove the lemma, it suffices to show that for any two vectors

e

and

f

of

Cn +l :

i.e., _ =

•

z

z

Using the preceding observations, this is equivalent to:

(3.7)

We will prove let let

(3.8)

9

(3.7) by proving the following general statement:

and

and

be two decomposable vectors in and

Vl

<91\

, v2

be any four vectors.

~ll\ ~2,3

1\

vll\ v2>

<91\ ~1'3 1\ vI>

-

Then

<91\~1'3

• <91\

~2,3

I\v 2> •

I\v 2>


1\ vI>

Because of definition (1.9) of the inner product for decomposable

78

THE EQUIDISTRIBlTTION THEORY OF HOLOMORPHIC CURVES

vectors, (3.8) is equivalent to an identity on determinants, which we formulate as follows. matrix,

A

=

(aijJ,

and let

in the upper-left corner of

i.e., A'

Furthermore, let of

Aa~

~

Let A'

A be a

(p+2) x (p+2)

denote the

p x p

submatrix

A,

.l,

all ••• alPl

[

';p_'

denote the

A obtained by deleting the

(p+l) x (p+l) a-th row and

submatrix

~-th

column.

Then (3.9)p+?!! (,det A· det A'

= det

~det ~+l,p+l det Ap+ 2,p+l

det

Ap+l,P+~.

det Ap+ 2 , P+2J

This is known as Sylvester's theorem on compound determinants. We will prove it by induction on

p.

For

p

=

1,2,

both

(3.9)3 and (3.9)4 can be proved by a Simple computation. we assume

p > 2,

and suppose that (3.9)3' (3.9)4' ••• , (3.9)p+l

have already been proved. in the

(p+l)-th and

Both sides of (3.9)p+2 are linear

(p+2)-th rows and columns, therefore

we need only consider those

A's with only one

1

the rest of the entries equal to zero in any of the and (p+2)-th rows and columns.

l's lie outside of the

right corner. the first

p

and with (p+l)-th

After disposing of the trivial

cases, we find that we only have to consider those which the

Now

A's in

2 x 2 box in the lower

Since permutations of the first

p

rows and

columns leave (3.9)P+2 invariant, we are reduced

79

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

to proving (3.9)P+2 only for an

A of the following form:

o 0 1

A a pl

a pp

0···0 1 0 o· .... 0 1

0

0 1

0 0

0 0

The matrix in the upper-left corner of AI.

A is of course just

Using the notation above. let us denote by

matrix of column.

AI

obtained by deleting the

Furthermore. let

A"

be the AI

in the upper-left corner of

l

Q-th row and (p-2) x (p-2)

· · ••• a· p-2,1

All

Then for this particular

A.

the sub~-th

submatrix

•

al l

i. e ••

A~~

1!P-2

J

ap~2,p_2 •

a

Laplacels expansion of each term

of (3.9)P+2 reduces (3.9)P+2 to

det AI • det All

det Ap_l,p_l

det Ap _l • P]

det AIp, p - 1

det AI p P.

det [

But we recognize this as (3.9)p' which is true by the induction assumption.

§6.

Hence (3.9)p+2 is true.

Q.E.D.

Finally, we define contracted curve of the second

80

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

kind.

The details here are almost identical with those of the

preceding section and so will be left out. h-dimensional projective subspace of

Let

and let

Pn~

coordinate neighborhood with coordinate function k < h ~ n,

the function

U _ Ah-k(:n+l •

p

k

Ah

(X~ .J Ah)(p)

be an U be a

z.

For

leads to a mapping

Ah .J Xk is a decomposable z (h-k)-vector and so this induces a holomorphic mapping

U -

By Lemma 1.4,

G(n,h-k-l) <;. P£(h-k-l)-l(:.

By Lemma 3.4, this gives rise

to a well-defined holomorphic curve:

J

k

X for

k < h

~

n,

h

A: V - G(n,h-k-l) <;; P£(h-k-l)-l~ and we call this a contracted curve of the

second kind. Lemma 3.10.

xk.J Ah

does not lie in a polar divisor of

G(n,h-k-l) • We consider the special case V

-Pn~.

h

k + 1.

=

Actually the image lies in

Ak+l,

Then

xk J Ak+l:

so that

(3.11) Staying in

U,

let

y

x~ .J Ak+l

and

y(l) =~.

Then

Sylvester's theorem on compound determinants (3.9) implies Lemma 3.11.

k+l Ak+l>(X k - l J Ak+l ) y"y (1) -_ <X z ' z •

Lemmas 3.9 and 3.11 will be of critical importance in Chapter V.

CHAPI'ER IV The two main theorems for holomorphic curves §l.

The first goal of this chapter is to establish the

First Main Theorem for a holomorphic mapping

f: V -+ G(n.k)

~ P£(k)-l~' k

=

is the set of

k-dimensional projective subspaces of

and

O••••• n-l.

where

(Definition 2.1).

that each holomorphic curve

(~!

i).

G(n.k) Pn~'

From Chapter III. §3. we learn

x: V -+ Pn+1C

series of associated curves of =

=

V is an open Riemann surface admitting a harmonic

exhaustion.

k

£(k)

O•.••• n-l.

G(n,k)

ox.lx •...• n_lx.

x

Recall from Chapter I,

has a naturally given Kahler metric. which is

the restriction of the F-S metric on

po

( 4.1)

=

fibration and function on

P£(k)-l~.

If on

C£(k):

1 2 ( <1\.1\ > - <1\ • dll. >) , <11..11.>

F,

k'lr* F

then

k X : V-+G(n.k) ~ P£(k)-l~'

Thus we are considering all at once the holo-

morphic curves that

rank k.

gives rise to a

is the usual

where 11.

= ( ...• \i ..• i ••.• ) is the coordinate

~£(k).

o k By (1.6). the associated two-form

m

of

F has the property that

m = !.2 dd c

(4.2) If B

ill

£

is the Kahler form of

G(n,k).

by (3.5). defined by

logl1\l.

F.

then

we have the polar divisor

k'lr* ill ~B

In particular, if we denote by (1\: <1\,B>

= oJ,

then 81

~

=

on TIB

m.

Given

G(n,k)

given

tne hyperplane

= TIB nG(n,k).

Denote

82

THE EQUlDlSTRlBlITlON THEORY OF HOLOMORPHIC CURVES

by

n B the function defined on

~£(k) - (0) - (kw-l(ITB)J

such that:

(4.3) lA, B I

where we now write

for

that although the multi-vector multiple of Because

eJ:Ie,

(A,B>

IA,BI

I (A, B> I

for convenience.

(Note

B is only defined up to a

suffers from no suCh ambiguity.)

is a holomorphic function of

A,

ddclA,BI

~ d'd"logIA,BI = J:I(d'd"log(A,B>~ = J:I{d'd"log(A,B>

-

d"d'log~ =

1 ddcn ~

B

O.

Hence

(4.2) and (4.3) imply that

lOg~

=~

dd c

=~

dd c log IA I _ ~ dd c log IA,BI

=~

dd c log IA I

=~ Moreover, for every clearly

~(~A) = ~(A),

defined on

C~

and so there is a

~

- ITB such that kW* u B = nB• * lC) _ lc kW (ru - ~d u B = ru - ~d nB = O. dkw

is surjective),

P£(k)_lG: - IT B• Now restrict all this information to agree to denote the restriction of

F, w,

ID

*,

=

uB

Thus on

*

Since kW 1 2 dd c u B on

G(n,k). ~,

~

function

P£(k)-l~

P£(k)_lC - IT B, is injective (because

e

If we

etc. to

G(n,k)

still by the same letters , then we have clearly proved: Theorem 4.1. function

uB

If

B e G(n,k),

such that:

k

=

O, ••• ,n-l,

there is a

THE two MAIN THEOREMS FOR HOLOMORPHIC CURVES

u B is COD on G(n,k) - l:B· 1 c '2 dd u B =~ in G(n,k) - l:B'

(i) (ii)

83

where

"(I)

is the

Kahler form of the restriction of the.F-S metric to (iii)

G(n,k).

If we denote by

k"lf*u B'

the function

~

k"lf: q:.£(k) - (oJ - P.£(k)_lG:,

then

where

ilB(II.) = log

~.

Now we return to the consideration of a holomorphic mapping

f: V -G(n.k)

not assume COO

P.£(k)-lq:.

V to be open.

For the moment. we need

D be a compact surface with that boundary in V and assume/ f(V) does not lie in ~ for

a fixed on

S

B

G(n.k).

€

D such that

Let

Thus the multi-valued holomorphic function

p .....

is not identically zero and

consequently its zeroes are isolated.

(4.4)

n(D,B)

We define

sum of the orders of zeroes of

=

We now give a motivation for this definition.

If

in

D.

x: V- PnG:

is our original holomorphic curve. the prime question of interest in equidistribution theory is: of pnq:

pnq:,

does

xCV)

TI?

which is the polar subspace of

intersects

TI

generally, let curve of

if and only if

PnG:.

kX(V).

<x, a>

kX: V -G(n,k)

rank k.

For each

projective subspace of of

intersect

pnq:

p

given a hyperplane

Let TI.

a

€

V.

be the point of

Then clearly.

has a zero in V.

~P.£(k)_lG:

kX(P)

xCV) More

be the associated is a

k-dimensional

U kX(P) is a subset peV By abuse of notation, we also denote this union by and so

(One can in fact show that

kX(V)

TI

is locally a

84

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Pn~)'

(k + 1)-dimensiona1 subvariety of B~

dimensional projective subspace if

kX(V)

let

B~

intersects

B be the polar

or not.

of

So given an PnC,

Bl.,

then

(3.5), the above question is equivalent to: p

1-+

we want to know

Rephrasing this a little,

space of

valued ho10morphic function

(n-k-1)-



B

€

G(n,k).

By

does the mu1tihave a zero in V?

One can in fact prove that the intersection number of the Bl-

singular chains

and

in

kX(D)

the sum of the orders of zeroes of are led to the consideration of Now define

v(D)

D

'IT

of the Fubini-Study metric of due to the fact that if

P1~

then



n(D,B)

= -1f f *co,

is exactly equal to

Pn~

where

in

D.

Thus we

as given in (4.4). co

P£(k)-l~'

is the Kahler form The factor

~ is

'IT

is anyone-dimensional projective

i

p 1 a:

(I)

= 'IT.

This easily proven

fact will also follow from §1 of Chapter V.

The following

theorem is then the non-integrated First Main Theorem. Theorem 4.2.

Let

f: V - G(n,k) ~ P£(k)_la:, k = 0, •.• ,n-1, V is arbitrary. Assume that for some

be ho10morphic and B

does not lie in the polar divisor }':B' I f D is a compact subdomain with Cf7) boundary in V such that €

G(n,k), f(V)

f(aD)

n }':B

= ¢,

then

(4.5) where

n(D,B) AB Proof.

1

c

~ dUB'

Let

+I

aD

f * AB = v(D),

(See Theorem 4.1).

g: D -E

be the real-valued function

85

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

g(p) of

If(p),BI.

=

g

Since

are isolated in

(a l , ... ,am),

Since

feD)

and

Uj

n Uk

=

disjoint from

¢

f(oD)

n~

~.

v(D)

g-l(U E ) if

the zeroes

D and hence form a finite set =

¢,

Hence we may choose a neighborhood the property that

LB,

does not lie in

j

I

=

(al , .•• ,am) S; D - oD. UE

of zero in ill

Ul U .•• UUm,

k.

where

with

a j E Uj

feD - Ul U •.. U Um)

Thus

is

By Stokes' theorem and Theorem 4.1,

=

It remains to prove that the last sum is equal to

n(D,B).

For this purpose, it is sufficient to prove that

(4.6)

the order of zero of lim E-+O

We may clearly assume that



r f*}..B'

~j Uj

is very small so that by

Lemma 3.2, there is a reduced representation of i.e., there is a holomorphic map y: Uj _ Ci(k) that

k 1T

so that

0

Y = f.

B

=

Now choose O. N. bases

eo" ••• "ek •

Write

y

=

f

-

(0)

(eo,···,e n )

yle o "'"

Uj ,

in

"e k +

such in

Cn +l

86

TIlE EQUIDISTRIBlITION TIlEORY OF HOLOMORPHIC CURVES

then for every

p e Uj ,

By (i11) of Theorem 4.1 I

Y*k~*~ = y* log ~ = log ~

f*U B

log

II~II = log

Iyl - log IY11,

so that,

Since

Iyl

so that

dC10g Iyl

is never zero,

11m e-+O

r d 10g

ob j

C

Iyl

presentative of the projective subspace If(aj),BI = ly(aj),BI = IY1(aj)l.

y(a j )

Furthermore,

O.

=

is clearly

f(a j )

of

in

COO

Uj ,

is a rePnt,

hence

So the order of zero of

is equal to the order of zero of

Y1

at

aj ,

To

prove (4.6), it suffices to prove: the order of zero of =

lim e-+O

~

Y1(a j )

r d 10g C

"iv j

IY11.

This is essentially the argument principle. let

z

be a local coordinate function centered at

there is an integer where

h

m(j)

such that

is ho10morphic and

so small that zero of

In greater detail,

Y1(a j )

h

h( 0)

J

is nowhere zero in is just m(j).

Now

+ dC log Ihl = m(j)d9 + dC log Ihl,

Y1(z) O.

aj •

= Zm(j)h(Z),

We may assume

Uj •

Uj

is

So the order of

dC10g IY11 where

Then

z

=

= m(j)d c log Izl I z I e..r:ie •

87

TIiE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

Remembering that

(See (2.7)). because

h

COO

is

log Ihl

is nowhere zero, we have =

k lim oOjr m( j)de + lim k otjr dClogi hi e+O

~ m( j) • 2'IT +

= Th~s

Now we assume

T(p) ~ reT)). ~

T which is harmonic on

We recall this notation:

oV[rJ = (p: T(p) = r).

r),

in

V - V[r(T)J,

T,

so that all parameter values

reT).

In

lated.

V[r]

(p: p e V,

We shall work exclusively

rare assumed greater than

the critical pOints of

Also recall that if

p e V - V[r(T))

holomorphic function

cr

T are iso-

and if

then in a sufficiently small neighborhood of

p,

dT(p)

10,

there is a

= T + J:Ip which serves as a coordi-

(Lemma 2.4 and the remarks after Definition 2.2)

Now return to our previous situation. and fixed

=

(p: p e V,

i.e., only in the domain of harmonicity of

V - V[r(T)J,

nate function.

Q.E.D.

V is open and has a harmonic exhaustion

function (Definition 2.1)

T(p)

0 = m( j).

(4.7) and there with the theorem.

proves

§2.

e+O

f

B e G(n,k),

is holomorphic. f(V)

We have

If we assume that for a

does not lie in

~B'

that (et)

f(oV[rl)

(~)

r

n~

= ¢,

is not a critical value of

thenTheorem 4.2 implies that

f: V - G(n,k)

T,

and furthermore

88

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

I

n(r,B) + where we have written v(V[r])

and

function

a

n(r,B) c

1

AB

oV[r]

dUB.

= ~

f*~B

=

for

n(V[r],B),

v(r) v(r)

for

The use of the special coordinate

= T + J:Ip leads to:

Lemma 4.3.

Under assumptions (Gi.) and (£),

r f*A B -;&<2:.... r f*UB*d T). - r ~ ovir]

ovir]

The proof is identical with that of Lemma 2.5. have that if

(~)

(~)

and

hold for

d 1 n(r,B) + ~~

r

Suppose now

[rl ,r 2 ]

hold for all (4.8)

r

E

2 n(t,B)dt

then

f * uB*dT)

v(r).

is an interval in which

[rl ,r 2 ],

Jrr l

i

oV r]

r,

+

Thus we

WO and

('£.)

then an integration leads to:

~ oV[t] J f*U B*dTlrr l2 = Jrr l2

v(t)dt,

C/I

where we have used the standard notation: To extend (4.8) to all subintervals of

h(t)I:~

(r(T),s)

= h(r 2 ) - h(r1 )· without the

restrictions of (0{) and (3b) we need the following Lemma 4.4.

I

f*UB*dT

for a fixed

B is a continuous

cV[r]

function of

r

for all

r.

* Proof. We first show that the lemma follows from the following two facts: (a)

f

V[r]

d(f*UB*dT)

is finite for all

r.

89

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

(5)

If

f(p) e

~B

disc flzl < eJ then

and about

r f*UB*dT

lim

We

e-.o O~e

is a holomorphic p

such that

flzil < eJ, On

V[r] - Ue ,

form.

Jj

So let

f-l(~B) () V[r]

disc U~ .£ i zi(P i ) = 0, and let Ue = Ui=l Ue· is COO and so f * uB*dT is a COO one-

Enclose each where f * u~

0,

= O.

As always, we invoke Stokes' theorem. f PI' •.. 'P.e J •

z(p)

Pi

by a holomorphic

Thus, (by (a))

(Where oV[rJ

cve[r]

denotes the subset of

outside of those

intersect

cV[rJ;

which

consequently we have

oV[rJ) . (by (EI)).

This shows that the last integral is finite for all prove continuity in (4.9 )

I

oV [ r i) If

r,

it suffices to show that

f*UB*dT -

r. ri

t r

To implies

(f*UB*dT. ovt r J

denotes the characteristic function of

Lebesgue's bounded convergence theorem implies that

then

90

TIlE EQUlDISTRlBUflON TIfEORY OF HOLOMORPHIC CURVES

J d(f*UB*dT)

=

V[ri ]

JXid(f*UB*dT) V[r]

=

- V[r]J d(f*UB*dT) This proves (4.9) and hence the lemma. (n) and

(~).

The singularities of

f-l(~).

at the discrete set

two-form at apE V[r]

p

a reduced representative of k~. y = f.

so that Thus


But

are located

rep)

at

f

U.

E~.

Let

y: U _~£(k) - {oJ (Lemma 3.2).

(eo ••••• enJ

in

~n

be

So be chosen

y = yle o "··· "e k + •.•.

and write

k~*)UB

y*

log

~ = log 11;;.11

d(f* ~*dT) = d(log IYI*dT) - d(log lyll*dT) Iyl

U be a

and

f*UB = (y*!

Thus

f*uB*dT.

It remains to prove

d(r* uB*dT)

and let

Choose O.N. basis

B = eo"··· "ek

I

oV[r]

and we need only examine this

such that

coordinate neighborhood at

=

is never zero. so

d(log IYI*dT)

Therefore i t suffices to show that

L d(log

is

... C

in

U•

in

I yll*dT)

U.

is

U

finite. that

Let

Yl(z)

z =

be a coordinate function centered at

z~(z).

where

We may as well assume that d(log lyll*dT)

h h

is holomorphic and is never zero in

U.

=

d{(m log Izi + log Ihl)*dT)

=

md(log Izl*dT) + C... form

=

1:1

p h(O)

such

F O.

Then

dlzl"*dT + m log Izld*dT + Coo form.

91

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHlC CURVES

In terms of polar coordinates, both forms are obviously integrable in

U.

(a). For

This proves

notation and have that in

(~),

we use the same

U:

...

= C form - m log lzl*dT. If

OWE = (p:

I z(p) I

£l,

then

r c... form

~£

There is no question that as

£

- m log

f

£

*dT.

OWE

-0,

the first integral on

the right approaches zero, so it remains to prove that lim log £..0 in

£

~

*dT = O.

We use polar coordinates

£

U minus a radical slit, where

z = reJ:Ie

log r + J:Ie Then

-dT = *(OT OT dO) OT dOu-01T OT d log r. Or·rd 1 ogr+01T u =r Or lim log £..0

because

fr.

£

£ •

[rl ,r 2 ]

and lim e log e = O. Q.E.D. e..o is an interval such that all U,

obey (~) and (~) while

r e (rl,r,) (e<,) or (£).

1

r *dT = lim log !w r fr. de e..o e 11m e log e J fr. de = 0 e..o oWe r

3W£

is bounded in

Now suppose

r

Consequently,

rl

and

r 2 may violate

We define: f*

~ ov1tJ ~

r

*dTlr 2 = lim 1 f*u *dTld rl df r 2,c./. rl 2i' cvttJ B c

With this definition, (4.8) is obviously extended to such

92

THE EQUIDISTRlBUI'lON THEORY OF HOLOMOIlPHlC CURVES

[rl ,r 2 J If

where the end-points

[r 2 ,r,J

r l , r 2 may violate (e{) or (~).

is another interval such that each interior

satisfies (dK) and

(~)

while

r 2, r,

r

may not, then Lemma 4.4

implies that

Now let

[ro,rnJ

The pOints in in number, say,

be an arbitrary sub-interval of

[ro,rnJ

(r(T),s).

at which (e{) or (~) fails is finite

(ro,rl, ••• ,rn ).

By the preceding discussion,

we have

for

i = O, ••• ,n-l.

Add these

n

terms on the left telescope into

equations and the second

~

r

alit t]

f·~.dTlrn.

Thus

r0

we have:

We are therefore led to the definition of the order function:

and the counting function: N(r,B)

=

fr n(t,B}dt. ro

93

TIlE TWO MAIN TIlEOREMS FOR HOLOMORl'HIC CURVES

The number

ro

is always assumed to be above

once chosen. it will be fixed. Theorem 4.5 (FMT). holomorphic. such that in the polar divisor tion. then for any

~.

r

N(r.B) +

~

but

We have thus proved f: V - G(n.k) ~ P£(k)_lIC

Let k

reT).

= O, •••• n-l. Suppose

and

f(V)

be

does not lie

V admits a harmonic exhaus-

reT):

~r f*~*dTlr = T(r). (lv-h] ro

As in the case of meromorphic functions. we call the second term on the left the compensating term. compensate for? is an

Let

BL be the polar

What does it

space of

(n-k-l)-dlmensional projective subspace of

the remarks after (4.4). if each B.L

meets

•

then

i f the subset

then

N(r.B)

N(r.B)

O.

f(p)

so PnC.

(p € V[r])

By

never

The same reasoning shows that

U f(p) of PnC p€V[rJ is small. But T(r)

meets

BL very rarely.

is independent of

B,

so

the above identity implies that the compensating has to be relatively large in this case.

Thus the compensating term compen-

sates for the deficiency in the intersection of

U f(p). p€V[rl Now observe that for any (IBI

=

1

because

A € tiCk).

B € G(n.k) ~ P£(k)_lIC.

B.L with

IA.BI ~ IAI IBI

~ ~ O. By Theorem 4.1(iii). ~ > O.

becaUse

*dT

induces a positive measure on

is coherent with the orientation of

IAI

and the latter is

by definition the quotient space of the unit sphere). log

=

(lV[rJ).

So

Consequently. (lV[r]

(since it

we have

94

TIlE EQUlDISTRIBlITlON TIIEORY OF HOLOMORPHIC CURVES

(4.10) There is one more fact we need before we can derive the bas1c inequa11ty.

This fact is

r

LeDDDa 4.6.

ov'[rJ function of

f*~*dT

for a fixed

r

is a continuous

B.

Let us assume th1s for a moment and prove the sought for inequality: (4.11)

N(r.B) < T(r) + const.. independent of

rand

where the constant is B.

For, N(r.B)

= T(r) (4.10)

< T(r)

So we may choose the constant to be the maximum of the cont1nuous funct10n man1fold

B 1-+

G(n,k).

* Proof

.J,r

f

oV[roJ

f* ~*dT

defined on the compact

(LeDDDa 4.6).

of Lemma 4.6.

By virtue of facts (a) and (e) of

the proof of Lemma 4.4, it is equivalent to proving the cantinu1ty in Let

B of the integral

r-l(~)

n V[rJ

=

r

d(f*uB*dT). V(rJ (pp •••• P.e1 and let each

Pj

be

surrounded by a coordinate neighborhood on Which is def1ned a fixed coordinate function

Zj

such that

Zj(P j )

=

o.

Let

95

TIlE TWO MAIN llIEOllEMS FOR HOLOMORPHIC CURVES

and define

Now

is disjoint from

f(VrrJ-W)

function of

'!hen,

B on Vrrl-W

~B'

so

*

is a

f~

COO

(see (i11) of Theorem 4.1) , so

there is no question of the continuous dependence of the first integral on

We only have to examine each summand of the

B.

last sum carefully.

sentation of f in Wj • _IYI f * u B = log~. So

Jd(f*~*dT)

and let

be a reduced repre-

By now, it is familiar that

j

Iyl > 0,

y

wfd(lOg IYI*dT) - wfd(lOg ly,BI*dT).

=

j

Since

j

Fix a

j

log Iyl

is

and independent of

COO

B.

So

the first integral of the right side may be left out of consideration.

Therefore what we must prove is the following:

Bj

be a sequence of projective

to

B (in the sense that we can pick representatives of

and

B in

C£(k)

tatives of

Bj

k

spaces in

Pn~

let

converging Bj

so that the coefficients of the represen-

converge individually to those of

B),

then

Jd(lOg ly,Bjl*dT) - wfd(lOg ly,BI*dT). j

j

Now recall that

y(a j )

has a zero at assume Wj in

Wj •

€~,

so the holomorphic function

For convenience, we shall also

is so small that

aj

is the only zero of

Furthermore, it is obvious that

uniformly to



on Wj •



converges

To prove the above (and hence

96

TIlE EQUIDISTRlBUTlON TIlEORY OF HOLOMORPHIC CURVES

the lemma), it therefore suffices to prove the following: Let

(gj}

be a sequence of ho1omorphic functions defined

on the closed unit disc g,

and let

~

be a

6

C=

and converge uniformly on one-form on

and vanishes nowhere else. of radius

Then if

6

Assume that

6.

to g(O)

is the closed disc

6'

~ about the origin,

J6' d(log

Igj!~)

-f6' d(log

Now the left side equals .[. I ~j! d! gj!

Igl~) A

~

+ {lOg ! gj I dcp.

So

it is equivalent to proving:

( 4.12)

J log

(4.13)

J log

Igjl dcp -

6'

!g!dcp.

6'

By assumption, there is a positive integer

= zmh(z),

g(z)

zero in pOints that

6.

where

h

is ho10morphic in

By Hurwitz's theorem, for each

a j1 ,· •• ,a jm

6.

little simpler, let us assume that aj •

and has no

j,

there are

and

is free of zeroes in

simply call it

6

(possibly not all distinct) such

of

a jl -o, ••• ,a jm - 0

where

m so that

To make the notation a a jl

= ••• =

a jm,

and we

The reader will perceive that this sim-

plification by no means restricts the generality of the subsequent discussion. and

hj

So we have

never zero in

6.

gj(Z)

=

(z-aj)~j(z),

We now claim that

hj

aj-o

converges

0

97

THE TWO MAIN THEOREMS FOR HOWMORPHIC CURVES

uniformly to aj

h

in

6'.

To begin with, we may assume that all

are in the interior of

(~)

h

= _1_

h(C)

(hj(Z)dZ

~ ~6 z -

j

= _1_

6'.

e

Jh(Z~dZ

~ 06 z-

For every

= _1_

r

~

6'.

€

g/z)

~ ~6 (z_aj)m(z_t)

= _1_

r

g(z)

2'rPoI-l 0'6 zm(z_C)

Since the integrand of the integral of

dz.

dz.

hj(C)

converges

uniformly to the integrand of the integral of

h(C)

on

06,

we have proved our claim. (4.12) now reads:

Since

hj

and

h

for

are zero-free and furthermore.

of its derivatives converge uniformly to

h

on

h j and all 6.

it is

obvious that the second integral on the left converges to the second integral on the right. to prove:

for

So to prove (4.12). it suffices

a j - 0, I

Let

aj

= a j + J:I~j.

The above simplifies to:

J(X-aj)dXI\CfI + (Y-~j)dYI\CfI _ JXdXI\CfI + ydyI\CfI.

1z -a j 12

6'

6'

1z 12

To prove this. it is clearly sufficient to prove the following: let

f

implies

be a

COO

function on

6'.

then

a j + J:l~j

E

aj

_ 0

98

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHlC CURVES

(4.14)

1. J, I

_(_X_-a-'jl/....).... 2 fdx , lz-ajl

{

(y-e ) ----'j........ 2 fdx z -a j I

A

1. ~ J6

dy -

A

dy -

' Iz I

Let us prove the first one, say.

21 about

radius

function of and let

~

t.j

aj

and let

n t.,.

fdx

A

dy,

fdx

A

dy.

' lzl

Define

Let

Xj

be the disc of

be the characteristic

fj(~)

=

f(~

+ a j ),

denote the complex plane as usual.

where obviously

Ej

- 0

as

Xj(~)

Then,

In view of Lebesgue's

bounded convergence theorem,

LXjfj ~ dxdy - j ~ fdxdy I I ' z

C

and (4.14) is proved. g

IZI

It remains to prove (4.13).

= z~ and gj = (z-aj)~j' jlOg

Since

hj

free in

lhjld~

+

(4.13) becomes

m~lOg lz-ajld~

converges uniformly to t.,

h

~lOg lhld~ on

t.

+ mJ.IOg

lzld~.

and both are zero-

the first integral on the left clearly converges

to the first integral on the right. &j - 0

USing

So it suffices to prove:

implies

J log ~,

lz-ajld~ -Jt.' log lzld~.

But the method of proof of (4.14) applies equally well to this Situation, so the lemma is completely proved.

Q.E.D.

99

TIlE TWO MAIN TIlEORl!MS FOR HOLOMORPHIC CURVES

§3.

In this section, we specialize Theorem 4.5 to the

associated curve of

(Chapter III, §3) to obtain

rank k

various refinements.

Recall that we assumed our original

holomorphic curve

x: V -

Lemma 3.6,

does not lie in any polar divisor

to be nondegenerate.

~

of

G(n,k),

€

where we have attached a subscript function rank. that

By

Theorem 4.5 therefore implies that for every

G(n,k). Ak

kX(V)

Pn~

k

to both the counting

T to distinguish their

N and the order function

We propose to simplify the compensating term. ro

and

r

are both above

reT),

Recall

so the line integral

of the compensating term is taking place in the domain of harmonicityof function

T.

a

=

But there,

T + J:Ip

we

can use the special coordinate

(Lemma 2.4 and the remarks after

Definition 2.2) except at the critical pOints of therefore define pOints of

T.

a

As noted before,

is defined only up to a p,

but (3.3) shows that

is well-defined despite this ambiguity.

Pl-+X~(p)

into

and by (3.4),

c£(k);; flk+lCn+l k"" 0 X~

set of pOints in

We can

x~ as in (3.2) outside of the critical

translation in the imaginary part

Xak

T.

=

kX

we still denote by

X~

makes sense except on a discrete

V - V[r(T»),

Union of the zeroes of

The mapping

(this discrete set being the

and the critical points of

Since integration always ignores finite point sets, the following 1s therefore valid:

100

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

by virtue of Theorem 4.1 (iii). mk(r,Ak )

(4.15)

=

1 2rr

Ak

€

So if we define:

{. log

oV

t]

then we obtain:

(4.16) which holds for all

G(n,k)

if

x

is nondegenerate.

Our next task is to derive two other expressions for Tk(r).

Let

U be a coordinate neighborhood in V on which

is a coordinate function

z.

Let

x~ be defined as in (3.2),

then (4.2) and (3.4) imply immediately that outside the zeroes of

xk. z·

(4.17) Since

we may write in rather

suggestive notation (but not-too-correctlY) that =

..!.. 2rr

1:ro dtvr'tlr dd c log

Tk(r)

Ixkl. z

Still keeping the same notation as above, we obtain from the following:

Now by (3.2),

outside the zeroes of

and so

101

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

for

= l, ••• ,n-l.

k

To extend the validity to

k

0,

we

simply define: X-I:: l.

z

Thus on

U minus the zeroes of

Taking into account of the fact that

we

may apply Sylvester's theorem on compound determinants to conclude:

outside the zeroes of

.r::i

( 4.18)

IX~14

where we have written as usual, etc.

Xkz

IXk-112Ixk+112 z z

"2

(3.8)

dz Adz

IX~-112,

Again, it is tempting to write that

dz Adz.

The trouble with this, as is the trouble with

= ~fr ell

ro

dt

f dd c log

V[t)

Ixkl, z

is that

z

Tk(r)

is not a globally

defined coordinate function, so that the integrand does not make sense on all of

V.

This suggests that we should look for a function on

V

102

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

which can serve as a coordinate function at every point of

V.

Such a function is provided for by a theorem of Gunning and Narasimhan: Theorem 4.7 [31.

On every open Riemann surface, there

is a ho10morphic function whose differential vanishes nowhere. Let us seize such a function notation once and for all. open subset

U of

coordinate function.

V,

~

on

V and fix the

Thus in every sufficiently small

the restriction of Then

to

~

U is a

x~ makes global sense on V and

(3.4) implies that the following diagram is commutative:

( 4.19)

where

V'

is the complement of the zeroes of

in

V.

Furthermore. by virtue of (4.17) and (4.18). we now have (4.20)

Consider the first expression of

Tk(r)

will apply Stokes' theorem to the integrand

in (4.20).

r ddc log

vr'tl

in exactly the same way as we did in Theorem 4.2. detail,

xk ~

In greater

will vanish in a finite number of pOints

(P1 •••.• Pm} ~V[t).

So we enclose each

Pj

by

a small

103

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

W =Uj=l Wj • On v[tl-W, we can apply Stokes' tneorem to the COO form dd c log IX~I. Then take the €-ball

Wj •

Let

limit of the integral as of

IXkl 'Y

(4.21)

do not fall on

r

1 2ir vr't] dd

€ - O.

The result is:

ov[t 1,

then:

~

Jd

c 1 og IX'Ykl =

ClI

C

log IXkyl

if the zeroes

-

ov[tl

nk (t)

where by definition: (4.22)

nk(t) = the sum of the orders of zeroes of in

V[t].

Note that (4.22) makes sense because each component of is a holomorphic function on

V.

Now repeating the proof of

Lemma 2.5 almost word for word, we can show that once and

ov[t)

IX~I,

contains no zero of

.J:rr

r

ovtt)

dC log IX'Ykl =

c&< ~

t

~

reT)

then

I

ov[tl

log IXykl ).

Substitute this into (4.21) and integrate, we get:

(4.23)

~Jr

ro

dt

r ddc log IX'Ykl

vr'tl

where we have written if every

t € [ro,rl

none of the zeroes of

=

~

I

oV[t]

Nk(r) = J:onk(t)dt.

log IX'Ykl*dTI:

- Nk(r) 0

(4.23) is only true

has the property that

oV[t]

contains

IX~I. But now the analogue of Lemma 4.4

is valid; again the proof can be transferred to this case almost verbatim. function of

t.

Therefore

r

ovl t]

log Ixkl*dT

is a continuous

'Y

The standard arguments that led to Theorem 2.7

and Theorem 4.5 now show that (4.23) is valid for any subinterval

104

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

[ro.r]

of

( 4. 24)

(r(T).s). Tk(r)

=

Consequently. (4.20) implies that

~ ovitJ r log

ixki*dTI r - Nk(r). ~ ro

We now summarize (4.16)-(4.18) and (4.24).

Recall first

the various definitions. k

Nk(r.A ) where

k nk(t.A )

where

~(t) =

l~l

v[tl.

in

=

Jrronk(t.A )dt k

= sum of the orders of zeroes of in

v[ tJ.

sum of the orders of zeroes of the function where

~

is a fixed function enjoying the

property of Theorem 4.7.

where

vk(t)

=

r

vr'tJ

~(r.Ak) where

cr

=

-k ov[t] flOg

is a holomorphic function having

Theorem 4.8 (FMT of rank k).

Let

degenerate holomorphic curve and let be its associated holomorphic curve of Let

V admit a harmonic exhaustion.

and for

r

~

reT):

T as its real part.

x: V -+ PnlC

be a non-

kX: V -+ G(n.k) rank k.

k

Then for each

=

~

P£(k)_lIC

O••••• n-l. Ak

€

G(n.k)

105

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

If

Y is the fixed function on

V having the property of

Theorem 4.7. then

Furthermore. let

U be a coordinate neighborhood in

coordinate function

z.

V with

then except on a discrete point set:

dz I\dz

=1

X- l

where

z

Remarks.

by definition.

(1)

In the same setting. we may restate (4.11)

as follows: where

(4.25)

independent of (2)

k =0

The case

ck

rand

is a constant Ak

is notable for its simplicity. so

we state the above for this case separately. =

~ crl

~ x* CD

1

= ~ =

r log rx;ar . I xl. *d-r! r

cvi t]

I

ClV[t)

r

0

log IX1*dT!r - N (r) roo

dd c log I-xl

..r:r I x1\ x( 1) 12 2

For any

Ixl4

dz 1\ dz

a

€

pnt.

106

THE EQUIDISTRIBlTTION THEORY OF HOLOMORPIDC CURVES

(3)

We have introduced the holomorphic function

x~: V _C£(k)

V and with it, the mapping kX: V - P£(k)-l~'

nk(t,Ak )

we now give an equivalent

sum of the orders of zeroes of the function

=

in

Take a

p

zero at

€

p

1~.Akl.

X~(p) I 0,

if

V;

V[ t] .

then obviously the order of

of the quotient equals the order of zeroes of equals the order of zero at

in a small neighborhood of each

which induces

~(t,Ak):

definition of

(4.25a )

x~,

Vsing

on

y

X~(q)

p,

kiT

0

p

X~

is a representative of

of kX

<X~,A~.

by (3.4), so for all

kx(q)

But

q

in

this neighborhood, which in turn simply means the order of zero of p.

<Xk Ak) at y' Next, let ~(p)

V

neighborhood

of

p

is the order of zero of 0,

=

at

then in a sufficiently small

p,

we can wrtte

x~

=

zIDy.

where

z

p-centered coordinate function on V and Y is such that i t is never zero on V. Then Y: V _ a::£(k) - (0) is a

is a

reduced representation of that

kiT

each

kX( q),

0

Y

=

k X in V (this is obvious) so Thus each y(q) is a representative of

k X' q € V,

and so the order of zero at

is just the order of zero at

p

of


p

of

k
The latter is

equal to the order of zero at to the order of zero of never zero in 1

TYT

k

IY,A I

U).

Since

I?T IXk Akl 1

'V'

'V

p of IY,Akl , and hence equal I TYT IY,Ak I at p (because Y is

IX~I = Izml Iyl,

it follows that

and we have again proved that the

107

TIlE TWO MAIN TIlEOREMS FOR HOWMORPHIC CURVES

order of zero at of zero at

~(t,Ak) (4)

p

of

p

1 IXKI 'Y

.

of

IXk,Akl

is equal to the order

'Y

According to the definition of

given before Theorem 4.8, we have proved (4.25). Although only incidental to our purpose, we neverthe_

less give an extension of the integral formula (2.23). noted in Chapter I §2, the Fubini-Study metric of induces a Kahler

structure on

form of this Kahler metric on by

n.

If now

g

G(n,k)

'rTf.

G(n,k)

g

g

- (G(n,k) f n)

we shall denote generically G(n,k),

we

to be: -1 (

J gn )

G(n,k)

Sometimes we write also 17( g(Ak) Ak of integration.

P£(k)_lt

The volume

is an integrable function on

define the arithmetic mean of (4.26)

G(n,k).

As

to emphasize the variable

The main observation here is:

"l Nk(r,Ak )

(4.27)

=

Tk(r)

A

According to (4.16), this is equivalent to: (4.28)

To prove this, first note that if of

G(n k) , ,

the function

and measurable on of

g

G(n,k).

g(Ak)

B =

log

is an arbitrary member IBI IB,AKI

is nonnegative

We have noted the nonnegativeness

on previous occasions; it is measurable because it is

c0ntinuous in

G(n,k}

-~.

Hence for some

~,

0 < ~ ~ 00,

108

THE EQtnmSTRlBlTI10N THEORY OF HOLOMORPHIC CURVES

~.

Our first claim is that to two facts:

is independent of

~

U(n + 1)

(i)

B.

This is due

log~

acting on the function

IB,A I

as a transformation group of U(n + 1)

G(n,k)

leaves the function

invariant.

(ii)

of

(Lemma 1.2) and in particular, leaves

G(n,k)

Next we claim that

J log oV[t)

Ixkl cr

IX~,Akl

is a transitive group of isometries

< "'.

~

* d-r

n invariant.

For Lemma 4.6 says that

is a continuous function of

Ak

and

therefore

But

IXkl log ---,~crT.:-

IX~,Akl

is nonnegative and measurable on

ov[t] x G(n,k)

(again because it is continuous except on a set of measure zero), so Fubini' s theorem gives

r - r

rfl:

mUsing AK o~[t) - o~(tJAK •

the first claim and the corollary to Lemma 2.4. we deduce that the above inequalities may be rewritten as

o Since

o

< L < "',

<

~L

< '"

we conclude that

mAk ~(r.Ak)

=

lit Ak

~

~

<

00.

Therefore, by (4.15).

IXkl r cr flog *d-rl ro ov[t] IX~,Akl

109

TIIE TWO MAIN TIIEOREMS FOR HOLOMORPHIC CURVES

0,

thereby proving (4.28).

~4.

The FMT of rank k introduces the quantity

as given by (4.15).

Since by definition

mk(r,A k )

<x~,Ak>.= Ak J x~,

we may rewrite (4.15) as 1 mk(r,Ak ) = ~ 1r

Now let

h

I

oV(t]

log

be an integer obeying

a unit decomposable

(h+l)-vector.

-1 < h < n

and let

Ah

In Chapter V, we shall

encounter expressions of the type: ( 4. 29)

and the purpose of this section is to give an interpretation of this.

Invoking the distinguished function

may write as a consequence of

Recalling (4.24), we have,

(3.3) that

~

on

V,

we

be

110

TIlE EQUIDISTRlBUfION TIlEORY OF HOLOMORPHIC CURVES

Now introduce sum of the orders of zeroes of the function

h

and write

nk(t,A )dt.

h

= k,

(4.25)

coincides with the usual counting

shows that function.

When

We further define:

Then upon adding the various terms, we find

(4.30)

h

h

h

mk(r,A ) + Tk(r,A ) + Nk(r,A )

Since the FMT says that for

h

=

k,

=

Tk(r)

k k mk(r,A ) + Nk(r,A )

=

Tk(r)

(4.30) leads to:

(4.31) We now wish to give a geometric interpretation of the formal object -~(t,A h )

=

h Tk(r,A).

Let us define

sum ot' the orders of zeroes of the function

I Xk .J 'Y

Ahl

in

V[t],

h then as a matter of comparing definitions, we see that ~(t,A) h h + Nk(t) = -Nk(t,A), h + nk(t) = ~(t,A). Hence Nk(t,A) where

111

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

J

r

-Nk(t,Ah ) =

r

o

-nk(t,Ah )dt.

So we may rewrite

as:

(4.32) Keeping. (4.32) in mind, we separate our considerations into two cases: Case 1:

k < h.

curve of

x

E be the to

A,

In Chapter III §4, we introduced the projection

into

Ah,

which was denoted by

Cn+l

(h+l)-dimensional vector subspace of

then the mapping

AX~: V _hk+1E

induces the associated curve of

IAX~I

AX: V - Ah.

I~ ~ Ahl,

rank k

Let

corresponding

defined in Lemma 3.7 of

AX.

By that lemma,

consequently, the interpretation of the

order function as given in (4.24) shows that

h Tk(r,A)

in this

case is exactly the order function of the associated curve of rank k

of

AX.

We can also look at this from a second point of view.

In

§6 of Chapter III, we introduced the contracted curve of the second kind,

k

h

X ..J A : V - G(n,h-k-l) S P.e(h_k_l)_lC.

X~ .J Ah: V - h h-kC n+l

clearly induces

Xk.J Ah,

pretation in (4.24) of the order function (for shows that curve

h

Tk(r,A)

Since

the interk

= 0) again

is the order function of the holomorphic

xk.J Ah.

Case 2:

h < k.

We introduced in §s of Chapter III the con-

tracted curve of the first kind,

S P.e(k_h_l)_la:·

Obviously,

xk ..J Ah.

lAh ..J ~I

Since

Xk ~Ah: V -G(n,k-h-l)

xk.J Ah: V _ hk-hCn+l 'Y

=

lX~.J Ahl,

induces

the interpretation

112

TIlE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

in (4.24) of the order function for

k

=

0

is the order function of the holomorphic curve In any case, we have shown that if

h

Tk(r,Ah )

shows that

f k,

Xk ~ Ah. then

is actually the order function of some holomorphic curve. is a holomorphic curve, the order

Recall that if

f: V

function of

was originally defined to be

f

-Pn~

Jrro dt vr'tlr

f* (I),

so the order function is always positive and strictly increasing. Hence h

Tk(r,A ) > 0

(4.33)

if

k

f h.

We now summarize the foregoing in the next theorem. Theorem 4.9. a unit decomposable define for

nk(t,Ah )

k

=

Assumptions as in Theorem 4.8, let (h+l)-vector,

h

be

= -l,O, ••• ,n, and

O, ••• ,n-l,

= sum of the orders Of. zeroes of

Similarly, let

Ah

Nk(r,A h )

be defined with the zeroes of

Then the following identity holds:

in

V[t].

113

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

Furthermore,

h > k,

and i f

Tk(r,Ah ) = the order function of the associated holo-

(4.34)

morphic curve of curve of

x

rank k

into

of the projection

Ah.

the order function of the contracted curve of the second kind and if

Xk ~ Ah.

h < k, the order function of the contracted curve

( 4.35)

xk ~ Ah.

of the first kind

In connection with the above theorem, there are two basic inequalities which we shall need. Lemma 4.10.

We now prove ihem here.

There exist constants

k=O, ..• ,n-1,

such that

~

IXkl r log -Ix"""k-=--.J';;""cr-A'--h-I * dT av'[r]

.s..

Tk ( r) + ak

cr

holds for all

rand

Lemma 4.11.

Ah.

There exist constants

bk ,

k

o, ... ,n-l,

such that

holds for all

Ah

and

r.

Proof of Lemma 4.10. Also,

by

By (4.31) and (4.33)

its very definition,

Nk(r,Ah ) > O.

We therefore

114

TIlE EQUlDISTRIBUTION TIlEORY OF HOLOMORPHIC CURVES

deduce from (4.30) that

which by (4.29) is equivalent to

I

1 2'Jr

oV[r)

J

IX~ I 1 IX~ I h *dT < Tk(r) + 2'Jr log -Ix.....kr--J=--A"'"h-I *dT IXcr A I oV rol cr

log

k J

The right-side integral can be proved to be a continuous function of

in exactly the same manner that Lemma 4.6 was proved.

Ah

So we may let over

ak

be the maximum of this function as

G(n,h).

Ah

varies

Q.E.D.

Proof of Lemma 4.11.

We begin by recalling (4.24) and

(4.32): Tk(r)

r log Ixkl*dTlr - Nk(r) ~ oVI[ loglxk J Ahl*dTl rr -

~

=

011'[ t 1

ell

Tk(r,Ah ) By (1.12),

=

'Y

t )

ell

'Y

~

0

-'Y

o~r)lOg IX~ J

every zero of Nk(r)

'Y

IXk J Ahl < IXkliAhl

our notation convention that Hence

r0

I~I

-Nk(r,A). h

or equivalently,

Ahl*dT

Ah

=

Ixkl 'Y

(one must remember

is a unit decomposable vector!)

~ ovtrJlOg IX~I*dT.

is of course a zero of

Furthermore,

IX~ ~ Ahl,

Combining these two facts, we have:

so

115

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

The situation now is identical with that of the preceding lemma: the last integral is a continuous function of bk

to be the maximum of this function as

so we take

Ah

wanders over

G(n,h).

Q.E.D.

§5.

We now head towards the Second Main Theorem. Tk _l , Tk , Tk+l

relates the order functions characteristic of

V[r).

to the Euler

In the first part of the derivation

of this formula, however, we need not assume that

V is open

G(n,k) ~ P£(k)-l~'

and that the receiving space is

In fact.

the exposition is smoother without these assumptions. let

M be an

metric

F

whose associated two form we denote by ID. F and

Let

Pn~'

arbitrary) and let

C~

boundary.

D be a compact subdomain on V with

We consider the restriction

f: D -M.

in general vanish at a finite number of pOints

f(a j )

We call these the critical pOints of

df

will

(al •••• ,am) f

in

D.

If

= p, let z be an aj-centered coordinate function and

let

U be a coordinate neighborhood of

map

~: U -

is a

is inten-

f: V - M be a nonconstant holomorphic mapping

(V

D.

(This

which have been employed to

ID,

denote the F-S metric and its Kahler form on

of

So we

n-dimensional hermitian manifold with hermitian

redundant usage of

tional).

It

0

cen

such that

C(p)

en_valued function such that

there will be a positive integer

=

p

with coordinate * (0 ..... 0). Then f C

f* C(O) s

=

(0, .•.• 0)

such that

and

( z - s df* C) ( 0 )

THE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

116

~ (0 ••••• 0)

(z-(S-l)df*~)(O) = (0 ••••• 0).

while

clearly independent of the choice of fore an intrinsic invariant of ~of

s

is

and is there-

We call it the stationary

We define:

D - ral ••••• a m).

Let

~

and

sum of the stationary indices of

seD)

(4.36) In

at

f

f.

z

This

f *F

=G

f

in

D.

becomes a hermitian metric.

n its volume form.

K be its Gaussian curvature and

n is also the Kahler form of G and so

Of course

(4.37) The nonintegrated form of the Second Main Theorem may now be stated as follows.

(The reader may consult at this point the

discussion of the Gauss-Bonnet theorem in Chapter II. §3). Theorem 4.12. mapping. where

Let

f: D -M

be a nonconstant holomorphic

D is a compact subdomain with

in an arbitrary Riemann surface

V and

complex manifold with hermitian metric nowhere zero on

oD

with respect to

G

M is an F.

boundary

COO

n-dimensional

Suppose

so that the geodesic curvature =

f *F

df ~

is of

oD

Notation as in (4.36)

is defined.

and (4.37). the following holds: 21rX(D) + 2'rrs(D) -

fie

oD

XeD)

where

Proof. and let

Zj

=

L

Kn D

denotes the Euler characteristic of Let

ral ••••• a m}

D.

be the points where

be a local coordinate function near

df aj

vanishes with

117

TIlE TWO MAIN TIlEOREMS FOR HOWMORPHIC CURVES

Zj(a j ) = O.

Denote by Wj

m

W =LJj=l Wj •

On

D - W,

the set

(IZjl.s. €)

and let

G is a hermitian metric and we can

apply the Gauss-Bonnet theorem:

(I!: + (I!: =

21TX(D-W) -

c!w

a'D

f

Kn

D-W

I!: the geodesic

where, for convenience, we also denote by curvature form of

oW

due to the fact that we orient each (and not with respect to and

f

lim €~

D-W).

fDKQ.

fD

21TX(D) -D(IC + lim a'D €~

Kn

OWj

lim(~ €~

Hence,

L~=l (

J

0 j

IC - 2'Jr) ~

(IC - 1) = stationary index of

oiv j

For this, we explicitly compute we denote Let

by Z =

Wj

X(D-W) = XeD) - m

To prove the theorem, it suffices to show that for each

(4.38)

is

oW with respect to

Clearly

D-W

Kn =

fl!:

and the plus sign in front of

SO

Z.

..r:'i(}

re

IC

of

OW j •

at

For simplicity,

Z is a coordinate function centered and let

n

=

log r,

is a multi-valued holomorphic function in OW j = (p: n(p) = log E).

f

j,

In

Wj

then

Wj - (0)

e

n + ..r:'ie

and

minus a radial slit,

~

is

a coordinate function and, we can write (see (2.11)): G

g(dn 2 + d( 2 ),

n

g dn

I\de.

By (2.14), the geodesic curvature form of

OW j

has the expression:

118

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

(4.39)

_ 1 -"2

It

It remains to find out what ~l'

j

•.• '~n

m

Around

g

de

is.

Let p

n

~j(p) = 0,

so that

is the associated two form of

decomposes into

= P and let

f(Oj)

we may write

p,

hermitian positive definite. f

3i1

be local coordinates at

= l, ••• ,n.

where

0 log g

and

F

Relative to

z

(aij )

(Sl' .•• '~n)'

and

holomorphic functions f

is

= (fl, ••• ,f n )

so that

Because

f(Oj)

=

p,

vanish at

all

smallest order of zero among

such that

h

f

fi(Z)

hi

z5 hi (Z),

=

fl, ••. ,f n ,

is holomorphic and

the stationary index of where

O.

is

fl

have the

say, and let

~

h(O)

Let

O.

It is clear that Similarly let

(5-1).

is holomorphic in Wj ,

i

= 2, ••• ,n.

Then

for bl

k

=

l, ••• ,n.

is never zero in a neighborhood of

assume that the holomorphic function Let

Since

b

bl

O.

bl(O)

= ~(O) ~ 0,

We may clearly is never zero in

denote the row vector of functions:

b

=

[b1···bnl.

Wj •

119

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

and let

A denote the hermitian positive definite matrix

(f* a ij }.

Then

Clearly bAFt because

b

Furt~rmore

dz /\ dz

is a

COO

function which never vanishes in

is never zero there and

A is positive definite.

a simple computation shows that in

=

I z 1 2dTl/\ de

n

=

(recall that

Wj

z

=

re.r.:ie

(oJ,

Wj and

,,= log r),

so

Since g,

f *en

by (4.37) and

we see that

g

_ 1 '2

II:

-

5

n

g dTl/\ de

=

= IzI25(bASt).

by definition of'

Consequently, by (4.39),

0 109(\z1 25 (bASt )) de II

o logall \z\

de + 1 0 log bAst de '2 all

5 de + COO form because

"

log \z\

and because

bASt

is

COO

and zero-free.

Hence:

l1m(~ e-+O

r ~ - 1)

~j

=

5 - 1 •

As remarked above, the stationary index of so this proves

(4.38).

f

at

aj

is

(5-1),

Q.E.D.

The rest of this section constitutes a digression and may be omitted without loss of continuity.

We would like to elaborate

120

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

on Theorems 4.2 and 4.12 for the case of a compact Riemann surface

D without boundary.

mapping

x: M -+ P a: n

(i)

x

So let us fix a holomorphic

with the explicit assumptions that

is nondegenerate (i.e., the only subspace of

containing

x(M)

is

pna:

itself} and

Riemann surface without boundary. the associated curve of

rank k

(ii)

Pnt

M is a compact

We would like to define of

x,

i.e.,

kX: M-+ G(n,k}

The definition given in §3 of Chapter III does not apply because it made use of a map induces

x.

Since

x

x~

which defines i

-x

is compact, no such

must modify the definition somewhat.

which exists, so we

In the formula (3.2)

in a coordinate neighborhood

U,

throughout by an arbitrary reduced representation

t n+l _ (0) by

~.

of

x

in

U.

To show that this collection of

~

the help of a coordinate function

Xv*

does indeed piece

kX: M -+ G(n,k}.

z

we must

~ is defined with

prove the analogue of (3.3), i.e., suppose

and

* U-+ x:

The resulting function we denote

together to define a global mapping

x*

we replace

and a reduced representation

is defined with the help of a coordinate function

wand a reduced representation

* y,

then on

prove the existence of holomorphic functions that

This is not difficult:

U n V, hl' h2

we must such

by the remark at

the end of Chapter III §l, there exist holomorphic functions such that tation shows that

~~

=

glxit = g2 Y* ' k(k;l)

g~(~;)

associated holomorphic curve of

~.

rank k,

so a simple compuThis shows that each kX: M - G(n,k)

c:= P£(k)_lr ,

121

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

The analogue of Lemma 3.6 can be proved in

is well-defined.

a similar manner, so ~

of

kX(M)

never lies in any polar divisor

G(n,k).

NOw, according to Theorem 4.2,

(4.40) for every

B

€

G(n,k),

where

= the number of zeroes of



A, B

In particular, for any

and in

M,

counting multiplicity.

G(n,k),

€

We proceed to give several interpretations of nk(M,B)

on the basis of

(4.40).

then

homology class

1 -(I) 'Ir

as

f

P Q: 1

and

First of all, we have remarked

previously (above Theorem 4.2) that i f line in

vk(M)

ko=l.

Pl~

is any complex

This shows that the co-

'Ir

H2 ( P £( k) -1 ~ ;2: ) •

is in fact the generator of

Jr ) (7F

1 (I))

kx~M

'

we see that

Vk(M)

is

nothing but that integer which, when multiplied with the fundamental two-cycle of now

TIB

P£(k)-l~'

be the hyperplane

generates the homology group 2(£(k)-1) - 2,

gives the cycle

(h:

= oJ

in

H*(P£(k)_l~;~)

and is hence the

Poincar~

kx(M). P£(k)-l~;

and

By

dual of

vk(M),

is therefore the intersection number of the cycles TIB•

Since

B is arbitrary, we see that:

TIB

in dimension

POincare duality and the preceding interpretation of Vk(M)

Let

kx(M)

122

TIlE EQUlDISTRIBUTION THEORY OF HOLOMORPtuC CURVES

Vk(M)

(4.41)

=

the order of the algebraic curve P£(k)-l~

kX(M) S G(n,k)

Now

equate kX(M)

vk(M)

out that

in the sense of algebraic geometry.

and

TIB (') G(n,k)

~B

in

~B.

=

So we can also

In §2 of Chapter I, we pOinted

G(n,k).

is the generator of

integral homology group of dual of

in

with the intersection number of the cycles

~

and

kx(M)

~B'

cl

2(n-k)(k+l)-11-dimensional

G(nJk)J

so if

is the generator of

H2(G(n,k);~).

be the dual element of

cI

H2 ® H2 -~,

is the generator of

then

Y2

is the Poincar~

cl

Let

Y2

under the natural pairing

which we call the fundamental two-cycle of

H2(G(n,k);~), G( n, k) •

Thus

Poincar~ duality easily leads to:

kX(M)

(4.42)

Now each

=

vk(M) • Y2

kx(m)

(m

as homology classes.

M)

€

is a

k-plane in

PnC,

so the

U kx(m) is a subset of Pnt which is in fact an m€M algebraic variety (called the variety of osculating k-spaces union

of the algebraic curve of

PnC

by

kX(M) •

x).

We also denote this subvariety

As remarked after (4.4),

~(M,B)

fact the intersection number of kX(M) and the polar B..L of B. BJ... is a projective subspace of dimension

is in space (n-k-l),

but is otherwise arbitrary, so (4.40) leads to: (4.43)

Vk(M)

=

the order of the variety

kx(M)

geometry.

(

(k+l)-dimensional algebraic in

Pnt

in the

sen~of

algebraic

123

TIlE lWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

NOw, what does Theorem 4.12 have to say about

nk

=

Kk

kX*m and let

kX?

Let

be the corresponding Gaussian curvature.

Then,

~(X(M)

( 4.44)

+ sCM))

Take a coordinate neighborhood

=

J Kkn M

U of

k •

M with coordinate

z'• by (4.18) we have:

function

Consequently, (2.13) implies that

Kkn k

=

~ dd c loge

Ixkz-112Ixkz+l12 k 4 ) IXzl

(dd c log IX~-ll _ 2 dd c log IX~I + dd c log Ix~+lIJ

where the last step made use of (4.17).

Hence by definition:

This coupled with (4.44) give us the final result: (4.45) for Vn(M)

k = O, ••• ,n-l, =

0

where

v_l(M) = 0

for obvious reasons.

of the interpretations of

Vk(M)

known as the Plncker formulas.

by definition and

(4.45) together with anyone given in (4.41)-(4.43) are Their beauty lies in the fact

124

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

that the left-side of (4.45) consists of analytic invariants of

x,

of

M.

while the right-side is a purely topological invariant

§6.

We now resume the assumption that

Riemann surface with a harmonic exhaustion integrate Theorem 4.12.

T and proceed to

Again recall that we always work in

the domain of harmonicity of

T,

i.e., in V - V[r(T)],

all parameter values are greater than

so

r(T).

f: V -M be a nonconstant holomorphic mapping as

Let

in Theorem 4.12. and if

V is an open

r

If

oV[r]

contains no critical point of

is not a critical value of

T,

f

then in the notation

of that theorem,

2'rrX ( r) + 2'lrs ( r) -

(4.46)

r

Ie =

ovtrJ

r

KQ

vrrJ

where for convenience we have written x(r)

x(V[rl),

s(r)

s(V[rJ).

Let us define a function (4.47) Clearly

f

h

*

h

in

V - V[r(T)J

(1)

is not defined at the isolated point set which

consists of the critical points of thing to note is that and of

dT !\*dT V.

by

h

f

and

T.

The important

is nonnegative because both

n

= f

*(1)

are coherent with the naturally given orientation

As in the FMT, we convert the line integral of (4.45)

125

TIiE 1WO MAIN TIiEOREMS FOR HOLOMORPHIC CURVES

into a derivative: Lemma 4.13. points of

f

Proof.

If

and

oV[r]

T,

does not contain any critical

then

We use special coordinate function

on a component

T- l ((rl ,r 2 ))

W of

where

r

€

= T + J:Ip

a

(rl ,r 2 ).

Then

n = h dT A dp so that G = f * F = h( dT 2 + dp 2 ). By (2.14), ~ = ~ 0 lo~g h dp. The argument given in Lemma 2.5 now serves Q.E.D.

to conclude the proof of this lemma also. The analogue of Lemma 4.4 is

r

Lemma 4.14. of

r

for all

* Proof. T,

r If

~

(log h)*dT

p

oV[r]

such that

contains no critical point of

f

and

So assume that it contains both.

dT(p) = 0,

and

df(p) = O.

Let

= T + J:Ip be the usual holomorphic function in a neigh-

borhood of

p.

a

is no longer a coordinate function at

because

dT(p) = O.

Let

v

=T

so that

pcp) = O.

Then

~

v

tion near m,

is a continuous function

reT).

the lemma is trivial.

Take a a

01[ r]

p

z = (~)l/m

W of

p.

such that

~(p)

T(p)

and let

p

p be chosen

+ ..r:i p is a holomorphic func-

= O.

For some positive integer

will be a coordinate function in a neighborhood

From the proof of Theorem 4.12, we know that

n = f*ro = !z!2(5-1)(bAot )dZ Adz, tion which vanishes nowhere in

where W and

bAot (5-1)

is a

C~

func-

is the stationary

126

THE EQUIDlSTRIBUfION THEORY OF HOLOMORPHIC CURVES

index of =

q

f

d~ Ad~.

h

~

p.

By definition of Since d~ = mz m-1 dz,

IzI 2 (5-1)(bA'Ot)dZAdZ

h, we see that

=

n =~ h d~Ad~

=

~ h. m2 • IzI 2 (m-1)dZAdZ.

h = ~ IzI 2 (5-m)(bAOt ), and so log h m2 2(5-m)10g Izl + COO function. Thus on aV[r], log h

Hence in =

at

W,

at worst only a simple logarithmic singularity.

has

Therefore

this lemma may be proved in a manner similar to Lemma 4.4.

Q.E.D. The situation here parallels that of the FMT.

We have by

(4.46) and Lemma 4.13 that

~(~ r (log ar 't'lT" av'[ r]

x(r) + s(r) provided

aV[r]

contains

r:.

o

X(t)dt

r

+.r:

ro

=

~

~

s(t)dt -

Jrro dt Vit] r

S( r)

Im

rIm

vr'r] f

and

T.

and use Lemma 4.14 to

k a1'[rt] (log 'IT"

E( r)

and we have:

r

r(T):

Introduce the notation:

1 2'Ir

no critical point of

We integrate this with respect to conclude that for

h)*dTl

127

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

(4.48)

Now recall that for

r

(Corollary to Lemma 2.4).

~

r

*dT = L is a constant. ov1r] By the concavity of the logarithm r(T).

(Lemma 2.14).

>.Ji: ovirl r log h *dT

-

'II"

i,,-Jrro dt v[t)r + .Ji: r (log h)*dT. ovr'ro 1 ~(E(r) + S(r) _ ~Jr dt r ro v(t] E(r) + S(r) -

=

Kn

'II"

So if we let

~(r)

=

Kn + const.}.

then

e~(r)

<;.. -

<J

.l.J

r h *dT.

ovtrl

r

r dt LJ: e~(t)dt h *dT. Now h dTA*dT = f*ro. ro - ro ovit] which is certainly integrable on V[r) - V[roJ. So Fubini's

implying.

theorem gives:

Combined with the above. this leads to

128

THE EQUIDlSTRIBUTION THEORY OF HOLOMORPHIC CURVES

For this holomorphic map

f: V -M,

let us define

vCr)

1'Ir

T(r) If

M is taken to be

vCr) and

=

Jr v(t)dt. ro

G(n,k)

S P£(k)-ll[!,

this

T(r)

of course cOincide with the previously defined vCr).

and T(r)

Now integrate (4.49) once more, and we obtain:

LJr dSJs e!P(t)dt < T(r) ro r0 -

(4.50)

where

!pet)

=

~(E(t) + Set)

- i:rr

J:.

ro

ds

r Kn + const. J.

vr'sl

(4.48) and (4.50) constitute the Second Main Theorem. We summarize these in the following. Theorem 4.15 (SMT).

Let

holomorphic mapping, where exhaustion and

on

m.

h

is defined on

Furthermore,

be a nonconstant

V is open and admits a harmonic

Let

G and

F and

G denote the hermitian metric

V minus the critical pOints of

Gaussian curvature of

where

f: V-M

M is complex hermitian with metric

associated two form f *F

£rJ

f

V

n

=

f *m.

V - V[r(T)l

f.

Let

K be the

Then

by n

h dT"*dT.

*

U)

'

129

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

where

~(E(t)


~Jt

+ Set) -

r0

ds

J Im + const.).

V[ s J

We now specialize this theorem to the associated holomorphic curves of

rank k

of a fixed nondegenerate holomorphic curve kX: V - G(n,k) ~ P£(k)_lG:

x: V - PnG:.

So consider

replace the

M of Theorem 4.15

is

h

of

T,

in this case?

~

G(n,k),

f

and

~

What

Operating outside the critical pOints

we always have

function in V - V[ rJ.

cr

+ J:Ip

= T

as a local coordinate

We can write equiva1 ently

'"' = J:I ,. ~ h dcr A dcr.

On the other hand, we have by (4.18) that outside a discrete point set,

n Thus for

dcr Ada •

kX'

IX~-112Ix~+112 (4.51)

h

IX~14

except on a discrete point set (which is negligible in all matters concerning integration). (4.52) where

Next, we claim that for

kX:

JIm

1

~ V[tJ vk(t)

=

~ 'IT

almost everywhere

J kX*OO

Kn

prove the latter locally. with coordinate function discrete set,

etc.

V[tJ

It suffices to prove that

2(k_1X*00 Let z.

2kX* 00 + k+lx*00), and we U be a coordinate neighborhood -

Then by (4.18), except on a

130

THE EQUIDISTRIBurJON THEORY OF HOLOMORPHIC CURVES

By (2.13) we have, Kn =

~ dd c log(

= _(dd c

IXkZ-ll 2 IX kZ+l 12 k 4 )• IXzl

log IX~-ll _ 2dd c log IX~I + dd c log IX~+ll).

By (4.17), except on a discrete set, etc. so we have the desired result.

*

kX m

=

k 21 dd c log IXzl,

Combining (4.48), (4.51)

and (4.52), we have (4.53)

where we have written

Sk(r)

for the function

f:

o

sk(t)dt,

where sk(t)

=

sum of the stationary indices of

kX

in

V[t).

Combining (4.50)-(4.52), we obtain the following: same positive constant (4.5 4 )

f:odS f: oexp where

for

C,

(!£!. k(t»)dt.s. C Tk(r) k(t)

=

(E(t) + Sk(t) + (Tk_l(t) - 2Tk (t) + Tk+l(t»)

(4.53) and (4.54) essentially constitute the SMT of rank k, but further refinements Ofl(4.54) are possible and we devote the next section to these.

131

TIfE 1WO MAIN TIfEOREMS FOR HOLOMORl'HIC CURVES

§7. We begin by introducing a notation first suggested by H. and J. Weyl.

Let

defined on

s

[O,s),

8

and

be continuous functions

~

as in Definition 2.1.

We say

8 =

~(~)

if and only i f

for some positive constants Weylls wrote

8

= w(~),

K,

C

and

(Actually the

Ct.

w

but we have already used

denote the Kahler form of the F-S metric on

Pn~)'

to

Here are

the basic properties: LeDDlla 4.16. (i)

If

81 < 8

compact set, then

off a compact set and

8 = ~(~) ~(~),

implies

off a

81 = ~(~.l)'

8 + 0(1)

then

~ ~ ~l

= ~(~).

(ii)

If

8

(iii)

If

C is a positive constant and

If

8 =

8

=

~(~),

then

C8 = ~(~).

(iv) then

~(~)

and

81

is positive off a compact set,

e - 81 = ~(~). (v)

Suppose

Proof.

8

=

~(~)

and

(i)-(iv) are obvious.

81

=

~(~),

then

9 + 91

= ~(~).

For (v), use Schwarz's

inequality twice.

Q.E.D.

e = ~(~), we have iDDllediately a comparison between the growths of e and log ~ as r - s. See LeDDI!as 2.16 and When

2.19.

This will be made preCise later.

us rewrite (4.54) as:

For the moment, let

132

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

(4.55) It may be recalled that T_ 1 (r) a consequence of the definition X- 1 1, and Tn(r) for

k

0, ••• , n-1.

=

dd c log Ixnl

because

-

0

(for

Xn

=0 =0

as

is nothing but

o A··· A en' where t is a ho10morphic function, so 'I~I Now Sk is given by the stationary index of kX' so by

te

definition

Sk

~

It I).

We may therefore use Lemma 4.16 (iv) to

0.

simplify (4.55) to: (4.56) valid for

k

=

0 •••• ,n-1.

We will now show that this inequality

leads to the remarkable fact that the various order functions To , •••• Tn _1

are essentially of the same order of magnitude.

To this end. we define a new function for each

T: [0, s) -lR by:

r e [O,s);

By Lemma 4.16(i). (4.55) and (4.56) imply: (4.57) ( 4.58)

for

k

=

O, •••• n-1.

Lemma 4.17.

For

(k+1)Tk+1

=

k

=

0 •••• ,n-2,

(k+2)Tk - (k+2)~k+1) E + ~(T),

133

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

and for

k

l, ••• ,n-l,

=

(n-k)Tk _l = (n-k+l)T k - (n-k)~-k+l) E + ~(T).

(4.60)k

Proof.

We first prove (4.59).

because in this case it reads: is (4.58) for

k > O.

for

above as (k+l)

= 2To

Tl

k

=

O.

kTk

=

(k+l)Tk _l - k(k;l) E +

- E +

k

~(T),

0

=

which

Now assume (4.59)k_l' i.e., assume ~(T )

We will prove (4.59)k by induction.

k(k;l) E - (k+l)Tk _l + kTk = ~(T).

times (4.58) gives

= ~(T).

It is true for

(k+l)E +

Rewrite the

By Lemma 4.16(iii),

(k+l)T~_l

- 2(k+l)Tk + (k+l)Tk+l Adding these two and using Lemma 4.16(v) we obtain

exactly (4.59)k. To prove (4.60), we use backward induction. k

=

n-l

reads:

Tn _2

=

2Tn _l - E +

restatement of (4.58) for

~(T).

k = n - 1.

The case of

But this is

(Recall

Tn

j~st

a

=0).

Now assume (4.60)k+l' i.e., assume (n-k-l)Tk

=

(n-k)Tk+l - (n-k-ld(n-k) E + ~(T)

(n-k-ld(n-k) E + (n-k-l)T k - (n-k)Tk+l

Rewrite this as

Multiply (4.58) by

(n-k)

=

~(T).

and use Lemma 4.16(iii) to obtain:

(n-k)E + (n-k)Tk _l - 2(n-k)Tk + (n-k)Tk+l = ~(T). Add these two and use Lemma 4.16(v), we get exactly (4.60)k. Q.E.D. First Corollary. (k+l)T£

=

If

£

~

k,

then

(£+l)Tk - ~k+l)(£+l)E + ~(T)

134

THE EQUIDISTRIBtrrION THEORY OF HOLOMORPHIC CURVES

(n-k)T£ = (n-£)Tk - (k £)(n-£)(n-k)E + ~(T).

2

Proof.

Straightforward induction from the lemma.

Second Corollary. or an annulus, then for

£

(n-k)T£ Proof.

E(r)

=

~

~

x(r) > 0

r, 0

for

r

or

C - (01,

(n-£)Tk +

~(T).

xCV)

~

0,

Sk

* Proof.

=

so for suffi-

(see discussion after Theorem 2.17).

large.

The desired result now follows

from the First Corollary and Lemma 4.16(iv).

(4.61)

or a disc,

k,

In any of these cases,

ciently large Thus

V = C,

If

Q.E.D.

(n+l) (n+l)". () (k+i)(n-k) Tk - ~ + ~ T

So = 2T0 - Tl - E + while (4.60)1 gives by virtue of Lemma 4.16(iii) that For

k = 0,

(4.57) gives

- Tl = - (n~l) To - ~ E + ~(T). So

=

(~~l) To - (n;l) E + ~(T),

process.

gives

These two together give

using Lemma 4.16(v) in the

So (4.61) is proved for

is proved in a similar manner.

~(T),

k = O.

Now let

The case of 1 < k < n-2.

k = n-l (4.59)k_l

135

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

- Tk _l

k = - \K+IT

kE+ ~ () Tk - ~ T ,

while (4.60)k+l gives _ T - _ (n-k-l) T (n-k-l) E + ~(T) k+l (n-k) k 2 (We have used Lemma 4.16(i11) twice).

Now substitute these

into (4.57) and use Lemma 4.16(v) to simplify, we get (4.61). Q.E.D. Corollary. then for

k

If

V is

~

C,

- (0),

a disc or an annulus,

= O, ••• ,n-l, _ (n+l) ( ) Sk - (k+l)(n-k) Tk + ~ T

Proof.

Same reasoning as the Second Corollary of the

preceding lemma. At this point, we must distinguish between the cases of infinite and finite harmonic exhaustions. §6 of Chapter II).

(Definition 2.4 of

Leaving the finite case to the reader as

an exercise (cf. the end of Chapter II), we concentrate on the infinite case. 9

= ~(~)

Thus let

V-

[0,(0).

Recall that

if and only if

r:.

ro

ds

fSro exp(K9(t) }dt

for some positive constants

K,

Lemma 2.16 twice we see that ( 4.62)

T:

II 9(r) <

/C

< e»(r) + C' C and

= ~(~)

9

Ct.

By applying

implies

log(c~(r) + C')

where as in §6 of Chapter II,

/C

> 1,

and the Sign

"II" in

TIlE EQUIDISTRIBlITION TIlEORY OF HOLOMORPHIC CURVES

136

front of an inequality means that the inequality is only valid

[O,~)

in

- I

with [d log x

<~.

Applying (4.62) to the

Second Corollary of Lemma 4.17, we get:

«: - (oJ,

if

«: or

is

V

then

(k+l)T.£(r) < ('£+l)Tk(r) +

/C

10g(CT(r) + CI)

for

.£

II (n-k)T.£(r) < (n-'£)Tk(r) +

/C

10g(CT(r) + CI)

for

.£ < k.

"

~

k,

Obviously, they together imply that for some positive constants cl

and

c 2'

II T(r) < clTk(r) + c 2 10g(CT(r) + CI) Now by definition, as

r-+~

because

Tk(r)

f

increasing.

v[tl A fortiori,

lim log T)r)

= O.

Te r

r-o

= ;:

ro

kX*ill

~

dt

V tJ

as a function of

T( r) -+

~

as

r -+

=

O, ••• ,n-l.

t

~

is strictly

Consequently, r,

r,

"II" indicates inequality

Note that since

only valid outside an exception set we

Tk(r) -+

Combined with the inequality

above, we have that for sufficiently large

k

~•

Thus for sufficiently large

c 2 log( CT( r) + CI) < 12 T(r).

for

so

kX*ill,

I

on which

I

may as well incorporate any compact. interval into

d log x <

~,

I.

Therefore, the preceding inequality is valid without the redundant phrase:

"for sufficiently large

there exists a positive constant

e

r".

So we can say that

such that

137

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

II T(r) < eTk(r)

(4.63) for all

k

=

O••••• n-l.

Next. we apply (4.62) to the Corollary of Lemma 4.18: if

V

= ~

or

~

- (01.

',I Sk() ( n+l) r < (k+l)(n-k) Tk ( r ) +

log « CT r ) +) C'

~

By (4.63), the last term may be replaced by log Tk(r) But since lim T (r) = 0• given any €. k

r~

~

log(CeTk(r) + C') <

€

Tk(r)

ciently large compact set.

~

log(CeTk(r) + C'). the inequality

would hold outside a suffi-

This implies

(4.64) We may summarize (4.63) and (4.64) into Lemma 4.19.

If

exists a positive constant

Moreover. given any

€

In this sense all magnitude. and of

Tk •

Sk

~

V is e

or

~

- (0).

then there

such that for all

k

=

O••••• n-l.

> O.

To ••••• Tn _l

are of the same order of

does not grow faster than a fixed multiple

As an application of this. we prove

Lemma 4.20.

Let

is impossible to have

V be either Tk + CE

=

C or

~(T2)

C - (01.

Then it

for any positive constant

138

TIlE EQUIDISTRIBUTION TIlEORY OF HOLOMORPHIC CURVES

C and for any Proof.

k

=

0 •.••• n-1.

We noted previously that

for sufficiently large

r

in case

E(r) ~

V=

is nonnegative or

by Lemma 4.16(iv). the hypothesis implies that

t - (01. Tk

=

so

~(T2).

By (4.62) this entails

Lemma 4.19 implies that, " /C 10g(CT2(r) + C') < /C 10g(ce2T~(r) + C')

< 2/C 10g(T~(r» 4/C log Tk ( r). Thus

II Tk ( r) < 4/C log Tk ( r), and so 1 ~ l1~UP

4/C log Tk(r) Tk(r)

= O.

a contradiction. Now suppose that

Q.E.D. V

~

C or

~

-

(oJ.

(but we still assume

V has an infinite harmonic exhaustion.)

We cannot

expect Lemmas 4.19 and 4.20 to hold without further restrictions because in this case,

E < O.

Motivated by the definition of

transcendency in Chapter II, we are led to imposing the same condition on ho1omorphic curves. Definition 4.1.

A ho10morphic curve

transcendental if and only i f

r lim T,JrT r-- 0

x: V - Pnr=

O.

is called

139

TIfE TWO MAIN TIfEOREMS FOR HOLOMORPHIC CURVES

xCV) is finite and x is transcen-

We claim that if

lim !H!'l = 0'. The proof is the same as in r-+co ~ let ro be such that r ~ ro ~lies x(r)

dental, then Chapter II: then 11m .

.r-+co

0

X(t)dt x(r )(r-r ) r o o 0 r = lim T (r) = 11m T (r)

Hence assuming VI

~

f:

AA ~

or

r-+co

0

r-+co

=

=

xCV),

o•

0

xCV) finite and x transcendental, if

- (0),

we have for all sufficiently large

r

that

2

- (.e k)(k+l)(£+1)E(r) .s. €To(r) .s. €T(r) for

0 < k < .e < n-l

~n-.e)(n-k)E(r) .s. €To(r) i €T(r)

for

and

O.s..e.s. k .s. n-l,

number.

where

€ is any preassigned positive

Therefore according to the First Corollary of Lemma 4.17

and (4.62),

II (k+l)T.e(r) .s. (£+l)Tk(r) + (€T(r) + if

k <.e

.£ < k.

constants

log(CT(r) + C'»

Ie

log(CT(r) + C'»

and

II (n-k)T.e(r) .s. (n-'£)Tk(r) + (€T(r) + if

Ie

Together these two imply the existence of positive c1

and

c2

(independent of

€)

such that

140

THE EQUIDISTRIBUTlON THEORY OF HOLOMORPIDC CURVES

T( r) -

Then again. ~

implies large

r.

GO

10g(CT(r) + CI)

r _ ~

lim log T,r)

implies

GO

ET(r)

r--

O.

=

T(r

for all sufficiently

Hence.

II T(r) If we choose because

as

c2

~ clTk(r) + 2EC~?(r)

E so small that is independent of

(we may do this 2EC 2 < !. 2 E) we have:

( 4.65) for some positive constant

of

cl '

for all

k

Still assuming the transcendency of

x

= O, ••• ,n-l. and the finiteness

xCV), we next apply (4.62) to Lemma 4.18.

In a similar

way to the above, we see that for any preassigned

E > 0,

Since

E,

E is arbitrary and

cl

is independent of

we see

that ( 4.66) for any prescribed Lemma 4.21. and

E > O. Suppose

We summarize (4.65) and (4.66) in

V F C or C - (0),

V has an infinite harmonic exhaustion.

xCV) If

is finite

x: V - PnC

is transcendental, then there exists a positive constant such that

/

e

141

TIlE TWO MAIN TIlEOREMS FOR HOLOMORPHIC CURVES

for all

k

= O, ••• ,n-l.

Corollary. for all

k

Proof.

=

If assumptions as above, then

11m in!' r--

r

~ = 0 "k\~'

We have 1

1

1

TiJTT .s.. e • T"{'r) .s.. e • TJTT '

lim sup ~ r-"o\r,

=

by assumption of transcendency of

0

So the conclusion is immediate. Lemma 4.22. and

> O.

E

l ••••• n-l.

II while

Furthermore. given any

Suppose

V

=

x.

Q.E.D. ~

~

or

xCV)

- [oj.

V has an in!'inite harmonic exhaustion.

If

is finite

x: V -

Pn~

is transcendental. then it is impossible to have Tk + clE

=

Proof.

~(T2)

for any constant

cl

and for any

k

O, •••• n-l.

By (4.62). we have

By Lemma 4.21, there is a positive constant

e

such that

II } T(r) + ~E(r) .s.. /C 10g(CT2(r) + C') For sufficiently large

.s.. 2/C log T2(r)

=

r.

we know that

4/C log T(r).

~ 10g(CT2(r) + C')

Hence,

II } T(r) + ~E(r) .s.. 4/C log T(r)

142

or,

TIlE EQUlDISTRlBUTlON llIEORY OF HOLOMORPHIC CURVES

!. +

lim inf c ~

e

1

~) <

11m sup 4/1: r~

TfTT -

and we have seen above that

11m

~

E(r)

log T)r). T(r

T'J'FT

=0

if

x

is trans-

cendental, so

The

sup

11m

left with

on the right is of course zero.

1

e i o.

We are therefore

a contradiction.

Q.E.D.

Finally. we explain the meaning of transcendency by proving the analogue of Lemma 2.18 in Chapter II. Lemma 4.23.

Let

V

=

M - (~ •••• ,amJ, a i E M,

compact Riemann surface, and x: V - PnC

Proof.

then

•

Xl.

x

a

E

of such a to

U

V - V[r]

cannot be

is transcendental.

IT

of

PnC

there exists a neighborhood

Unx(V - V[r]) PnC

x

Then

lim xCV - V[r]) = (x'(fi-) ••••• x'(am)J. r~ .~

following property:

Let

l, •••• m.

Suppose it is extendable

there will be at least one hyperplane

such that

M is a

x I: M - PnC.

We first prove that if

then it is not extendable to Xl

=

is transcendental if and only if

extended to a holomorphic mapping

to

i

where

=

¢

and so

with the U of

IT

for all sufficiently large

be the point orthogonal to

IT.

The existence

clearly implies that the restriction of is bounded below for all large

r.

r.

l<x.a>1

and hence

log_l xl ~. to V - V[ r] is bounded above TT,"iT by K for all large r. Let us assume that ro is sufficiently the restriction of

large.

for all

t

~

ro'

143

TI-IE TWO MAIN TI-IEOREMS FOR HOLOMORPHIC CURVES

and so

Now we use

(4.16) and (4.10) to obtain:

where

ovtr)

L

=

r *d't'

as usual.

So

This contradicts transcendency. To prove the converse l let us assume that

x

tions

(fol ••• lf n )

m 1-+ (fo(m)I ••• Ifn(m)) Minto

MI

(fi)1

meromorphic func-

induces in a natural way a holomorphic pnr..

For if we let

we have a mapping

where

'IT":

It suffices to extend

(f i )

-f: M' -

r. n+l _ {oJ - P

n

f

to all of

over the common zeroes of the

(f i ).

M'

be the complement

and the common zeroes

r. n+l - (0).

c:

f

Define

is the usual fibration. M.

We first extend

f

This can be done in

exactly the same way as in the proof of LeJlDDa 3.1. extend

We begin

then the function

of the finite set of the poles of of

(n+l)

= ~ > 01

defined on a compact (or for that matter l

arbitrary) Riemann surface

mapping of

x': M -+ pnr..

to

with the observation that if we have

is not

lim sup ~ !"'i'OO "o\r,

transcendental l or more precisely that and we will show how to ext end

x

over any of the poles of the

(f i )

Then we

by treating the

144

THE EQUIDISTRIBUTION THEORY OF HOLOMORPIDC CURVES

poles as we did the zeroes.

There is no need to write down

the details. Back to the proof. all

a

€

lim inf

Pn~'

No(r,a) < To(r) + Co

By (4.25),

for

so by our assumption above, we have

N (r,a) 0 r <~ <

But l' H8pi tal's rule implies that

co.

r-+oo

in fact:

lim no(r,a) < 1~ < co. (See the proof of Lemma 2.18). By r-the very definition of no(r,a), we see that xCV) intersects so

~

hyperplane only!. finite number of times, and this

intersection number is bound above by intersects the hyperplane

x=

So if x,

Xo Let

(xo' ••• 'xn )

(zo

O}

=

l' ~.

only finitely many times.

is the mapping into

has only a finite number of zeroes Uo =

([zo' ••• ,znJ: Zo ~

oJ,

=

-

t n+1

z

which induces

(P1' ••• 'P£}

and let

the usual coordinate mapping such that

z

xCV)

In particular,

~([z.o!"

~:

Uo

in V.

- tn

be

.,znJ)

~o"'" %0)' Consider x~o x: ~ - (ap ••• ,am} - [pp""P£} 4: n • Then clearly ~ 0 i = (.....!., ••• ,....£). Of course, ~ 0 i (

Xo

Xo

has poles at (P1, ••• ,P£1. We will show presently that none x of the (~ol can have an essential singularity at ~ x so ~o"'" ~o are meromorphic functions on M. By the remarks above, the

(n+1)

meromorphic functions

induce a ho10morphic mapping of mapping obviously extends

M into

x: V - pnt.

Pn~'

~

x

(lrx-" .. , xn) o 0 and this

145

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

~, ••• ,xn have no essential Xo Xo xl has an essential singularity singularities on M. Suppose Xo at ~. By the Casorati-Weirstrass theorem and the Baire It remains to show that

category theorem, there is a dense set in t each of which x has an infinite pre-image. So let ~ assume a certain value Xo ~ an infinite number of times. But this would mean that xCV) ~Z 1

an infinite o number of times, contradicting a conclusion reached above. intersects the hyperplane defined by

(Zl

x

~

So --, ••• ,~ have no essential singularities on Xo Xo

M.

Q.E.D.

We now summarize the main conclusions of §6 and §7 in the following. Theorem 4.24 (SMT of rank k).

Let

nondegenerate hoI om orphic curve and let be its associated curve of Suppose

and i f

rank k,

k

=

O, ...• n-l.

V admits a harmonic exhaustion. then for each

k.

T(r)

=

T - 0) n

a: -

(0)

or

vI

t

(T 1 -

x: V -+ P t be a n kX: V -+ G(n,k)

or

(b)

Furthermore, assume that either

(a)

V

= t

or

V has an infin1 te harmonic exhaustion,

t - (0).

xCV)

is finite and

x

is transcendental.

146

TIlE EQUlDISTRIBtrrlON TIlEORY OF HOLOMORPHIC CURVES

Then there exists a positive constant k

=

If

e

such that for all

.

0, ••• ,n-l

E

is any prescribed positive number, the following inequality

also holds:

for

k = O, ••• ,n-l.

Finally, under assumptions (a) or (b),

it is impossible to have an inequality: where

0 < k < n-l

and

Tk + CE

= ~(T 2 ),

C is any positive constant.

assumption (b), this is impossible even for nonpositive

Under CIS.

CHAPTER V The defect relations We now have both the FMT and the SMT.

§l.

Following the

pattern established in Chapter II, the next move will be to integrate (4.25) for respect to a function

= 0 (No(r,a) < To(r) + co)

k

S over

to integrate directly over

pnt.

Pnt,

with

While it is possible

for technical reasons, it Cn+l •

will be simpler to lift the domain of integration to

To this end, we will apply the following lemma to the principal fibration 'IT 2'. S2n+l _ Pn t of Chapter I. Lemma 5.1. with

G

Let

'IT: P -

M

as structure group such that

have riemannian metrics and the map

(11)

be a principal fibre bundle

g ...... p • g

(i)

G,

P

and

M

G acts as a group of isometries,

of

G into

P

is an isometric imbed-

ding for each pEP and (iii) if H denotes the distribuin tion/ P which is the orthogonal complement of the tangent space to the fibre, then of

P.

Let

0p'

manifolds and let

Proof.

~

f

and

d'ITIH 0G

is an isometry at each point denote the volume forms of these

be an integrable function on

M.

The proof is trivial because if all these condi-

tions are met, then for every trivializing neighborhood M,

'IT-l(U)

metric.

Then

is isometric to

U of

G x U equipped with the product

So Fubini's theorem clearly says that

147

148

TIlE EQUIDISTRIBunON TIlEORY OF HOLOMORPHIC CURVES

A partition of unity then does the rest.

Q.E.D.

Now the Corollary to Lemma 1.1 says that the principal fibration

~2: s2n+l - Pn~

with the usual metrics and

(s2n+l Pn~

with the F-S metric) satisfies

all three conditions of Lemma 5.1.

where nS

n

and the circle equipped

Hence if

B is an integrablp

denotes the volume form of the F-S metric of

denotes the volume form of

circle is obviously

s2n+l

Pn~'

and the volume of the

Again from Chapter I, we have a second principal fibering ~l: ~n+l _ (O) _ s2n+l. Let Sr ~.

be the sphere of radius and let (thus form of

nCr) ns

r

imbedded in

be the volume form of

= n(l)).

Let

dL

Sr

~n+l

(thus

S2n+l)

denote the usual Lebesgue volume

(5.2)

=

=

in the induced metric

i.e.,

dL

Sl

drAn(r),

so that in terms of the usual fibration (see (1.1)), we have:

u

149

mE DEFECT RELATIONS

=

J""0 e

-r 2

dr

j 'lr*l

* ('lr2~)n(r).

r

Now

'lrl!Sr

is a diffeomorphism of

~~ns

is quite clear that

Sr

~ nCr),

=

onto i.e.,

S2n+l

and it

nCr)

r

r2n+l'lr~ns.

Thus, 2 e- r dr

=J~

J'Ir*~e-dL cn

j r2n+l'lr*1 ('Ir*~' n ) 2 S

r 2 r 2n+l e- r dr

=J~

2

j* * r

'lrl('lr2~·nS)

=J~ r 2n+l e- r dr J2n+l('Ir;~)ns S 2n+l _r2 cfo r e dr)' Pncf~n 2'Ir'

n!'Ir

(by (5.1))

f ~n.

Pnc

f~n

i. e.,

P C n

=

Jn+l ('Ir*~)e-dL.

b n.'Ir

We summarize this into

C

the following theorem. Theorem 5.2. and let

nand

Let

be an integrable function on

PnC

denote respectively the volume form of the Fubini-Study metric on PnC and the flat metric on c n+l respectively.

dL

~

Then

150

TIlE EQUIDISTRIBUTJON TIlEORY OF HOLOMORPHIC CURVES

Corollary.

Pn~

The volume of

Before we can integrate

~n/n!

in the F-S metric is

No(r,a) < To(r) + co'

we need

one more bit of preparatory material. Lemma 5.3.

* Proof.

no(t.a)

is a measurable function on

It is obvious that a semi-continuous function is

measurable, so we need only prove that nuous. x: V -

no(t,a)

is semi-conti-

Recall that we have a holomorphic and nondegenerate Pn~.

For

a e

Pn~'

of the orders of zeroes of isolated because

x

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

~

no(t,a) <x,a>

is nondegenerate. no(t,a)

and

ri.?. t,

is by definition the sum in V[t].

for

The zeroes are

We will prove

(tn,a n ) - (t,a),

is equivalent to semi-continuity. where

[O,s) x Pn[.

Let

(tn }

=

and this

(r i } U (Sj}'

It suffices to prove:

(5.3) (5.4) We will first prove (5.4).

In the following, we can localize

our considerations to a neighborhood of each of the zeroes of <x,a>

and in each neighborhood, we may choose a reduced repre-

sentation of

a

y

of

x

(Lemma 3.1) and fix a representative

so that the zeroes of

<x,a>

~

in this neighborhood become

exactly the zeroes of the single-valued holomorphic function

/

151

TIlE DEFECT RELATIONS

.

However, in the interest of notational simplicity, we

prefer to abuse the language a bit and refer to itself as a holomorphic function on have in mind such

's.

-

when

fa

a,

j

converges to

==

<x,a>

V even though we actually

Now it is self-evident that

for all

aj

fa

f

j

since

Sj i t.

Because

uniformly on compact sets.

So

is sufficiently large, Hurwitz's theorem implies that

j

n (t,a j ) o

=

1:

P€V[t)

(order of zero of

fa (p)) j

(order of zero of 1: < -P€V[t) =

fa(p))

no(t,a).

n (sj,a j ) < n (t,a) for j sufficiently large. o - 0 This clearly disposes of (5.4). As for (5.3), we have to prove

Altogether,

it by a contradiction.

Assume it 1s false, then

11m sup no(ri,a i ) > no(t,a). i-+co we may assume that we have rl

~

r2

~

••• > t

Whenever we have

where

and furthermore, that

same number of zeroes of = no(t,a).

Passing to a subsequence if necessary,

fa

as does

VTt),

Again by Hurwitz's theorem, i

is sufficiently large.

no(rl,a i ) i no(t,a).

nO(ri,a i ) i no(rl,a i ),

Since

we now have

we have arrived at a subsequence

member is smaller than or equal to ~!! nO(ri,ai ) > no(t,a)

v[r l )

encloses the

Le.,

nO(rl,a)

no(rl,a i ) i no(rl,a)

Combined with the above, rl

~

ri

obviously implies

no(ri,a i ) i no(t,a).

Thus

(nO(ri,ai ))

such that each

no(t,a).

This contradicts

and proves (5.3).

Q.E.D.

152

TIlE EQUIDlSTRlBUfION TIlEORY OF HOLOMORPHIC CURVES

§2.

Let n

We are now ready for the integration.

the volume form on function

on

~

Pn~

Pnt

as in §l.

be

We choose an integrable

satisfying two conditions: 1

(5.5) ~

(5.6)

We will specify

later, but for the moment, we simply use

~

these two properties of respect to

~n

>0

over

~

and integrate the following with

pnt:

NO(r,a) < To (r) + c 0 , where

a

€

Pnll:

and

is independent of

r

~No(r,a)~(a)n(a) < To(r) +

c0

c0

Pnl(: because of (5.5). n(t,a)

Now

N (r,a) o

=

fr n (t,a)dt ro 0

is nonnegative and measurable on

more by (5.5),

~

and

a.

Hence

and by Lemma 5.3,

[ro,r) x pnt.

Further-

is integrable and nonnegative, so Fubini's

theorem applies and we have:

By Theorem 5.2, this is equivalent to

where

C and

C'

are positive constants.

We want to evaluate

153

TIlE DEFECT RELATIONS

the inside integral on the left. of

and let

Choose an arbitrary O.N. basis W = span(e o }'

We emphasize the fact that we start off with an arbitrary O.N. basis rather than with the canonical basis (Eo ••••• En) of ~n+l. Let ~: V _ t n+l be the map which induces x: V - Pnt as usual. and let

~

relative to

(yo •••• yn)

be the coordinate functions of

(eo ••••• e n }.

i.e •• for

be the complement in Then

V[r] - vo[r)

zeroes of

V[r]

~(p) = ~~=OyA(p)eA.

p E V.

of the zeroes of

is a finite set which includes the common

(Yo' •••• YnL

Zl = (~~=lziei) E W...L - (O}.

For

define a holomorphic function

gZ: Vo[t] - t

by

1

<x(p) ,Zl> yo(P)

ZlYl(P)+···+znYn(p) yo(P) Let

gz.

z

denote the number of pre-images of

n(t.z o )

counting multiplicitlBs.

We claim:

under

o

if we denote by

Z

1

the vector

Let

then

ql ••••• q£

and let

aj

be the pOints in Vo[t]

such that

be the unique positive integer such that if

is. a local coordinate function centered at -a j

~j

<x(qj)'Z>

But clearly. = O.

<x(qj)'Z>

I

O.~.

By definition.

<x(qj)'Z> = 0

if and only if

the pre1mages in

Vo[r]

of

!ja j

Zo

So

under

~j

then is

if and only if

SZl(qj) = zoo

0

no(r.~(z)).

zoyo(qj) +

ql ••••• q£

gz. 1

+ znYn(p)

are exactly

Furthermore. the

154

11IE EQUlDlSTRlBUTION 11IEORY OF HOLOMORPHIC CURVES

~

multiplicity with which that positive integer

covers

~.6l

~J

such th~~

and this is true if and only if ~ O.~.

at •

qJ

~j'

is exactly

-~

~j j(&zl(qj) - Zo) ~ O.~.

Cj j(-~yo(qj)

aj =

So it is clear that

Z0

- ••• - ZnYn(qj))

and we have proved

our claim. In view of (5.2), we have: no ( t,'IT ()) Z dL

=

. o Adz- o ) (..r:'i "2 -( n t,zo ) az /\ (.r:I) n( dz1 /\ dZl /\ ••• /\ dz /\ dz ) n

2

where dL 1

of

Z

n

= ~A=ozAeA

as above.

n

Let us further agree to let

stand for the volume form

wL.

So the integrand of (5.7) becomes

and by Fubini's theorem:

(5. 8 )

By Lemma 2.12. the last integral over

1

9:~ (~~('IT(Z 0 eo +

V [t]-.6 1 o

where

~

-.61

t:

: V [tl - a:. 0

and

is equal to

£

Zl))exp(-Iz 12)dZ o /\dZo )

Vo[t)

0

1s the complement 1n

v[tl

155

TIlE DEFECI' RELATIONS

of the zeroes of apE Vo[t] around

p.

Yo.

Let us evaluate this

in~egrand.

~

and let a coordinate function

So pick

be introduced

Then in this coordinate neighborhood,

.I

n

-,

n-

YO~i=lZiYi - y'~. o J.=lZiYil2de I\dY

2 Yo

.

"

dYA where we have written Yl for CT(". If following (3.2), we dxo dXn ( 1) write i for (cn;"."CT(")' the above then becomes:

Now substitute this into (5.8) and apply Fubini's theorem once more, we obtain:

Where

v

is a

1 oca11 Y v =

C~

..r:i ~ h

this neighborhood,

two form on vo[t] d~

Y I\d",

such that if we write

then for each

p

of

V0 [t]

in

156

TIlE EQUIDISTRIBUTION TIlEORY OF HOLOMORPfUC CURVES

Now recall that we had started off with an arbitrary O.N. basis [eo, ••• ,enJ

of

cn+l

last integral formula for

W~ of

sional subspace

W~ = span[el, ••• ,enJ.

and

h(p) Cn+l .

Thus the

is valid for any

Our task now is to make a

each choice being dependent on

judicious choice of

so that the integrand assumes its simplest form. so by definition

yo(p)! 0;

orthogonal complement of n-dimensional subspace W~

choose

0

<x(P),Zl> =

Ix(p) I

ponent of

in particular

x(p) -x(p)~

x(p)~.

to be

for every

cn+l

in of

Cn+l •

p,

Now p

x(p)! 0

€

vort],

and the

is therefore an For this

p,

we

With this choice, it follows that Zl

€

if one recalls that x(p)

n-dimen-

W.L. Yo(p)

What is more,

is nothing but the com-

in the direction orthogonal to

the above integral formula for

h(p)

Iyo(p) I W.L.

Hence

simplifies to:

h(p)

At this point, we can change notation a bit without fear of confusion.

We let

dL

be the generic symbol of the Lebesgue

measure in all complex euclidean spaces and let

Z be the

generiC symbol of the coordinate function in the same spaces. We can then rewrite the above in a more civilized fashion:

(5.10)

h(p)

157

TIlE DEFECT RELATIONS

or more simply:

Return now to (5.9).

Using the notation there, we have

by virtue of (5.7) that

C and

where

C'

are positive constants.

Since

VEt)

I

differ by a finite point set, we may replace by

r v.

vrt)

and v

vo[t] Furthermore (5.6) and (5.11) imply that

and since locally

v =

.r:;;- h

J

d~ Ad~,

v[tl-V[r o '

v <

h ~ 0,

we see that

r v,

- v(t]

so that

r:.

ro

But in

J

dt v < CT (r) + C'. V[tJ-V[r o ] 0

V - v(r o ]'

T

is harmonic and we can locally introduce

the holomorphic function

a = T +

J:I

p

as a coordinate func-

tion, except on the discrete set of critical points of Therefore, except on a finite point set of

T.

V[t] - v(roJ,

can write v = ~ h do Ada = h dT A.dT

With

h

given by (5.11), with the understanding that

i(l)

we

158

TIlE EQUIDISTRIBUTION TIlEORY OF HOLOMORPIUC CURVES

dx

is now given by

dx

(a:' ... ' a:).

As usual, we ignore such

exceptional finite point sets in integration, so that

where the last step is due to Fub1n1's theorem and the existence of special coordinate functions (Definition 2.2).

Putting

these together, we have the following:

where

C,

C'

are positive constants and if we write

then

Thus far, we have not specified

~

but rather, we have

proceeded solely on the assumption that (5.5) and (5.6) are valid for

~.

We now make an initial step toward the explicit

determination of a suitable function

cp:

[0,1) -lR

~

by introducing an integrable

and fixing a unit vector

b

E

a:n+l •

Then we require that

for all

a

E

PnC.

The right side inner product

!a,b!

can be

159

TIlE DEFECT RELATIONS

taken with any unit vector representing

a,

and is clearly

independent of the particular choice of the representative. Then for

Z

E

~n+l,

• ~(T(Z))

Z = ~(~) =

IZ.bI 2

~(~).

IZ I

I L> I

This choice of

~,

that it be dependent only on a single

coordinate, is motivated by the choice of the density function p made in Chapter II, §6 in the case

n

= 1.

Now

~

so

(5.5) and (5.6) if (5.12) is to remain valid. We claim that if ~ satisfies (5.14) and (5.15) below, then ~ would indeed satisfy (5.5) and (5.6). defined should still obey

(5.14)

~

J~ ~(t)(l

>a

- t)n-ldt

(5.14) implies (5.6) is obvious. We will show that (5.15) is equivalent to (5.5) by an explicit computation. By Theorem 5.2, (5.5) is equivalent to:

That

J ~(T(Z))e-dL

=

nh

cn+l By

(5.13)', this equals

.ow the Lebesgue integral on unitary group. = (1,0, ••• ,0)

~n+l

is invariant under the

So if we move the unit vector

b

to

EO

by a unitary transformation, the above may be

160

THE EQUIDISTRIBlTTION THEORY OF HOLOMORPHIC CURVES

rewri tten as:

f .. n+l

Z Z

11>(

0

)e

0

z z + ••• +z z

...

0

n n

0

/\

(~

-(z Z +···+Z z)..r::il 0 0 n n (_-_ dz /\ dz ) /\ ••• 2 0 0

dZ n /\dz n )

Introduce polar coordinates:

Let

zA

n!.". •

=

.r:'ie A

=

rAe

,

then

then the above becomes: n! .".n

clearly

m~ps

(0,1) x (O,m)

(O,m) X ••• X (O,m)

X ••• X

((n+l)-times).

(O,m)

one-one and onto

By a simple induction

argument:

Observe further that (to+···+tn ) = (sl+···+sn)' so that to T = t + ••• +t Hence we may transform the above integral into: o n

161

THE DEFECT RELATIONS

-n! , '!Tn

raJ

Because Joe equivalent to

raJ

-s _

as = 1

n·

J~

_co

and) 0 se -ds = r(2) = 1,

q>Cr)( l-T )n-ldS

n!,

=

this is

which is precisely

'!Tn

We now return to the computation of of (5.13)', we can write:

for any

p

€

h

in (5.12).

V[t) - v[r o )

not among the finite set of critical pOints of

where it is understood that Let us choose O.N. basis span(e o } = span(x(p)l,

q>

T or the zeroes

in

~n+l

so that

span(eo,e1l = span(i(p),b},

span(e o ,el .e 2} = span(x(p),b,i(l)(p)l. ~~=O zAeA

Then as

b

=

=

(~,O,

and

x(p)~ = span(el, ••• ,enl. (zo, ••• ,zn).

Thus in this notation, we may write: i(p)

which is

satisfies (5.14) and (5.15).

eo, ••• ,e n

For simplicity, we will write

Because

... ,O)

(&2'&3,0, ••• ,0)

x(l)(p) = (&4'a5 .&6'0 •••• ,0).

162

TIlE EQUIDISTRlBtmON TIIEOIlY OF HOLOMOIlPHIC CUIlVES

Z

With this simplification, and noting that

Z ; (O,zl' ••• 'Zn)' h(p)

E

i(p)~ implies

we have:

zlzl·a3i'3 la5z1+a6z212 J exp(-zlz l-···-znz n) • cp( _ _ ). -( )~ z z +. ··+z z la112

~-"---?r-~

11

xp

because of antisymmetry considerations.

nn

Hence

h(p)

Writing

f(a3 )

for the last integral, we have: la 12

~ f(a 3 ) < h(p)

(5.16)

I~I

-

We claim: ( 5.17)

f(a 3 )

{""["J[")

if

n d

n(n_1)~n J~ tcp(I~I2t)(1_t)n-2dt

if

n > 2.

163

TIlE DEFECT RELATIONS

Let us prove the claim for n > 2. To this end, employ polar J:Ie as before, and let ti = r 2 . Then coor di na t es zi = rie i tle

-tl-···-tn tll~12 ~(tl+···+tn)dtl···dtn

Then use the same transformation such that

t2 = (1-T)s2' •.•• 'tn

T(s2+,··+ s n)'

=

(0,1) x (O,m) X ••• X (O,m)

to map

(O,m)

tl

(T, s2' ••• , sn) - (tp ••• , t n )

X ••• X

(O,m)

dt A ••• Adt 1

n

=

(n

times).

(l-T)sn'

=

diffeomorphicallyonto

Observing as before that

(1_T)n-2(s +.··+s )dTAds A···Ads, 2

2

n

n

we see that

+ 2 =

Jm L Jm ••• o 0 i<j J~

s2e- s ds .

+ 2 (n-l)Jn-2)(

80

J~

e-sds

we get:

i

j

-s _ .. ·-s

2

nds "'ds J 2

n

~n J~ T(1_T)n-2~(Tla312)dT • ((n-l)

But

sse

(J~ J~

e- s dS)n-2

se- s ds)2(

J~

e- s dS)n- 4 J.

164

TIfE EQUIDISTRIBUTION TIfEORY OF HOLOMORPHIC CURVES

which is exactly (5.17).

(5.17) prompts us to introduce a function t: [O,l]-lR such that wm(S)

(5. 18 )

t(s)

'lrn

=

{

'lrn

(n-2)!

if

I1

n

1

=

0 t~(st)(l-t)

n-2

dt

if

n > 2

is a normalizing factor to insure (5.22) below).

(n-2)!

Then combining (5.16)-(5.18) and (5.12), we have: Jr dt ro

(5. 1 9)

where

C,

C'

Jtro ds ov[f s] h *dT <

(r) + C'

CT 0

are positive constanmindependent of

b,

and

2

at

p,

2 l a 51 h(p) = ~ t( la3 1 ), la1 1

provided

O.N.

basis

eo" •• ,en

is so chosen that x(p)

(a1 ,0, ••• ,0)

=

b = (a2'~'0, ... ,0) x(l)(p)

= (a 4,a5,a6 ,0, ••• ,0).

We now rewrite this expression of Relative to the same

O.N.

basis,

h

in intrinsic notation.

165

TIlE DEFECT RELATIONS

So, we may rewrite (5.19) as:

where

are positive constants independent of -(1) dx o dXn is given by (5.18) and x = (crcr""'(fcJ)' C,

C'

t

b,

Recall that we began the integration with an integrable

we determined

-E

~

in terms of a real-valued function

(cf. (5.13)).

(5.14) and (5.15), wise

~

satisfying (5.5) and (5.6) and, then

Pn~

B on

function

~

(5.21)

~.

uniquely, and if

two conditions, then

~

Now we make this claim:

t

t

satisfies the following

would satisfy (5.14) and (5.15):

The derivatives of positive there.

(5.22)

other-

In (5.18), we defined a new function

W: [0,11 -E in terms of ~

satisfies

will satisfy (5.5) and (5.6).

is arbitrary.

determines

~

We proved that so long as

[0,11

~:

f~

t

exist on

t (s) ds

=

(0,1)

and are

1.

Granting this claim for the moment, we bring our search for a Suitable

B satisfying (5.5) and (5.6) to an end by locating

a Suitable t such that (5.21) and (5.22) are fulfilled. SuCh a t turns out to be exceedingly Simple:

t(s) = (l-a)(l_s)-a,

a
1.

166

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

It is easy to see that this (5.22).

The constant

a

satisfies both (5.21) and

t

is arbitrary for the moment and will

be allowed to vary later on.

On the basis of this choice of

t, we may rewrite (5.20) as (5.23)

< C'""'0'(r'J + where b.

C,

C'

are positive constants independent of

Now note that since

b

is a unit vector,

for every choosing O.N. basis =

span(b,Z}.

"t v

a

and

!Z!2 _ !bAZ!2

this is most easily proven by eo, ••• ,e n

so that

b

=

eo

span(e o ,e1 )

and

This being the case,

lil2 a

(~! -!) b,x

.

Furthermore, we have the identity:

where

~

denotes the interior product of Chapter I §3.

this identity is easily proven pointwise at each V[t] - v[r o )

by choosing O.N. basis as in (5.19).

p

Again

of It should

be pOinted out that t he above does not make sense at those (finite number of)

piS

where

x

vanishes, but since one

need only worry about the behavior of the integrand almost everywhere, such exception may be ignored. identities, the integrand of (5.23) is:

Using these two

167

THE DEFECT RELATIONS

l
lxl

13C\2 Ib ..J(xAx(1),,2 _ Ib~xI2IxAx(1)12. (.lxl.)2o Ixl 4 l bAx i l"D,"Xf > liI 2 Ib.J(xAi(1»1 2 - bb,x I2I xAX(1)1 2 ( IxL)2o Ixl 4 lxl ro;ff Ib A xl2 ~ Ixl 2 lbl 2 = Ixl2 by (1.11» Ib.J(XAx(1»)I 2 I Ixl)2o Ix Ax(1),2 > Ixl4 • 'TD,XT lil4 (because

and the last step is because ex < 1

(~) 20 ~ (~) 2

and Schwarz's inequality (1.10) says

since

~ ~ 1.

(5.23)

therefore simplifies to: (l-ex)

1 J (-xAx -(1) ) 12 ( ~ . x )2o*dT frrodt JtrodoVsisb sJ Ixl4 , Jt ds [1-XAX-(4 I *d-r + CTo(r) + C' < (l-ex) J dt ro ro oV sJ Ixl I 1-XAX-(1) I2 dTA*dT + CTo(r) + C' (l-ex) J dt ro v[tJ-v[roJ Ixl4 1) 2

r

=

r

where it may be recalled

-(1) dxo dXn x = (crcr' ••• 'crcr).

Now according

to Remark (2) following Theorem 4.8, we have:

except on a discrete set (which is the critical points of and the zeroes of

x).

T

As usual, such discrete sets are ignored

in the integration so that:

168

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

(1-<1)

;: dt r

o

V[t]

(1-<1)

-(1) ,2 J ,-_x I\x,-,4 _ dT l\*dT -v ro x [

]

JrrodtV[t]-V[r J ) x*(J) o

< (1-<1)

JrrodtVr't]r x*(J)

(l-<1)To( r)

< To(r). Using this, we may rewrite the previous inequality in its final form, which is inequality (32) in Ahlfors' paper r1]: (5.24)

where

0 < a < 1.

of

and a ~

cr

b

=T

~

C'

are positive constants independent

and in terms of the special coordinate function -x(l) dxo dXn (crcr••••• crcr). We note that C and C'

+~-lp.

will be

C.

generic symbols of positive constants.

in the sequel, we will denote by the same letters

Although C and

C'

the two constants that appear in the right side of the many variants of (5.24), these constants actually change their values several times. ~round

determines

off the proof of (5.24). we must now show that ~

uniquely and that if

(5.22). then indeed not difficult.

~

t

satisfies (5.21) and

satisfies (5.14) and (5.15).

For by (5.18).

t(s)

n

=

~ (n-2)!

Jl0 t ... (st)(1_t)n-2dt • m

t

This is

169

TIlE DEFECT RELATIONS

Letting

T

=

st,

we get, 'll"n

t(s) = (n-2)!

fS0 n1

Tep(T)(S-T)

n-2

dT,

S

or equivalently,

n.._ s .,.(s)

n (S n2 (n~2)! Jo T(S-T) - ep(T)dT •

=

Recall this formula from calculus: d JX ax 0

f(x,Y)dY

=

JX rx<x,y)dy df + f(x,y) 0

•

We therefore have that

or, ep(s)

n 2

=

1 d n --n ~s -;(s)) •

ds

'II" S

t

Thus

completely determines

that (5.21) implies (5.14). valent to (5.15).

ep

and this formula also shows

We now show that (5.22) is equi-

By (5.25) and Fubini's theorem:

n JS fo t(s)ds J10 dsf(n~2)! sn 0 T(S_T)n- ep(T)dTJ n J1 Tep(T)d-r J1 (S_T)n-2 (n-2)! sn n J1 J1 ::.)n-2 s2 (n-2)! 0 Tep T dT T 1 - s i

1

=

'II"

'II"

2

ds

0

T

(

)

(

ds

170

mE EQUIDISTRIBUTION mEORY OF HOLOMOIlPHIC CURVES

Our assertion that (5.22) is equivalent to (5.15) is now obvious.

§3.

Q.E.D.

The goal of this section is to generalize (5.24) to

the associated holomorphic curve of rank k this by employing a trick due to Ahlfors. sable Ak - l

k-vector

k = 0, •••• n-l.

will also stand for a

space of

Pnt

Ak - l )

by

(X~..J Ak

x,

and we do

Fix a unit decompo-

Of course, if

k > 0,

(k-l)-dimensional projective sub-

by our notational convention.

curve of the first kind xk J Ak of

of

-1

Then the contracted

(r

V - A.J...

=

polar space

may be defined as in (3.6). We now apply (5.24) k-l, then -x should be replaced to the holomorphic curve Xk.J A

-1.

be replaced by of Xk..J Ak-~

and Lemma 3.9 implies that

(X~+l.JAk-l). by (4.35), is simply

(x Ai(l»

should

The order function

Tk(r,Ak - l ).

But so far

as the inequality (5.24) is concerned, Lemma 4.11 says that we may replace

Tk(r,Ak - l )

constant independent of and agreeing to write

Xk

by

Ak-l • for

Tk(r) + b k ,

where

bk

is a

Thus putting all these together.

X~ from ~ on, we obtain:

171

TIfEDEFECT RELATIONS

0 < a < 1,

where

dent of

a,

band

C,

to all

A£,

C'

Ak-l.

our purpose.

general~r

Ak - l

and

are positive constants indepen-

This is still not sufficiently

We wish to extend the left side from

-1 < £ < k-l.

a general averaging process.

For this, we must discuss

~(Ah)

Let

h-dimensional projective subspaces of -lR)

4>

Ah - 1.

and let us fix an

over all

Let

AJ..

Ah::> Ah-l.

be a function of

Pn~

~

(i.e.

: G(n,h)

We want to take the average of

This is done in the following manner.

be the polar

Then a moment's reflection will show that for each there is a unique

a

E

AJ..

such that

Ah

=

Ah - l A a.

when we restrict the domain of definition of containing the fixed

Ah-l,

4>

thus giving rise to a function: denote by

4>(Ah - 1 A a).

Therefore,

to such

becomes a function of AJ.. -lR,

The average

Ah,s a E AJ.. ,

which we simply

rrrt

4>(Ah)

of

4>

Ah ::::>Ah - l ~

all

h-dimensional projective spaces contal ning the fixed

Ah - l

is by definition the arithmetic mean of

over

AJ..

where n

4>(Ah - 1 Aa)

i. e.,

denotes the volume form of the F-S metric on the

projective space

AJ..

(The notation agrees with that of (4.26)).

In practice we have to lift the domain of integration from to

E,

of

~n+l

where

E is the

corresponding to

equivalently define:

AJ..

(n-h+l)-dimensional vector subspace AJ...

Thus by Theorem 5.2, we may

172

TIlE EQUIDlSTRlBtmON TIIEORY OF HOLOMORPHIC CURVES

where

dL

denotes the Lebesgue measure of

A~ represented by ~.

[~]

E.

canonical coordinate function of

E and

Z the

is the point of

(See Chapter I, §l).

The following lemma is basic. Lemma 5.4. let

h

Let

be such that

Ai be a decomposable £ > h > O.

Then for a fixed

~£ is a constant independent of Ah - l

where

£. Write ~n+l ~ A£ @ A~

depending only on *Proof.

h

be the

A~.

Since

If

dim p(F) < h,

F ~ Ah - l

p:

then

F

~n+l

Let

F

corresponding

as multivectors, in this case, both

So let

O.N. basis

Cn+l _Ai.

contains an element of

sides of the above identity clearly equal nothing to prove.

A£ and

(orthogonal decomposition).

h-dimensional vector subspace of

Ah-l.

and

and

This leads to an orthogonal projection

to

(.i-n)-vector and

~

dim p(F)

of

h.

_00

and there is

We may then choose

so that

p(F) and so that (eo, ••• ,ei ) is a basis of Obviously, Ah-l = ae o /\ ••• /\ e h _l + (terms involving e£+l, ••• ,en ) and Ai = eo /\ ••• /\ e £ so that Ah-l...J A£

basis of

=

ae h /\ ••• /\ e £.

let

a

Hence

IAh - l -.J Ail

=

lal.

be a unit vector orthogonal to

= Ah and let a

=

a0 e0 +

•••

+ an en'

F

On the other hand, so that Ah - l /\ a

Then since

Ah.J Ai

173

TIlE DEFECT RELATIONS

Let

E

be the orthogonal complement of Cn+l

vector subspace of A.L

of

where

Ah-l.

Z

Since

a

is the coordinate function of

E.

and we write

Ai.

and

Since each of

eh ••••• e£ each of

F

itself and consequently

part of an O.N. basis of of

that we wrote such that

E

and

We first show its indepen-

gonal to each of

{fh, .••• fnJ

Ah - l

This amounts to showing that the last

integral has these two properties.

is orthogonal to

space

The last summand is by definition

and that it is finite.

Ah - l

is the

by (5.27):

E.

It remains to prove that it is independent of

dence of

E

corresponding to the polar €

••• +

then

F.

E.

is orthoeh·····e£

(eh ••••• e£J

So we may pick O.N. basis fh = ~, ••• ,f£ = e£.

so that

Recall

Z

h < B < £,

zB

=



rewrite the last integral as:

=

.

Hence if we

is

174

lHE EQUIDISTRIBUfION lHEORY OF HOLOMORPHIC CURVES

which in turn may be written as

where

Z = (zo, ••• ,zn_h)

Cn+l - h ,

is the usual coordinate function on

the independence of the integral from

is obvious.

r

A£

It is equal to

e-log(lzoI2+ ••• +lz£_hI2)dL -

f

e-logIZI 2dL

+

f.

Cn+l-h

Break up each of these integrals into first summand

and

As to finiteness, let us prove that this last

integral is convergent.

cn~l-h

Ah - l

r

The Izi <1 in the case of either integral is con-

r

I zi.?l

Izf~l

vergent, as the use of spherical coordinates readily shows.

r

in both cases has a negative inte<1 e-logl z 12dL. It grand and is bounded below by Izf <1. 0 suffices to show that this is finite. In turn, it is equivaThe second summand

rZ

lent to showing that P

r

J" e-loglz

p

is the unit polycylinder

0

12dL

is finite, where

(Izol < 1, ••• ,lz£_h l < l}.

But

now the integrand is measurable and negative, so we may apply Fubini' s theorem and transform the integral into an

(n-h+l)-

fold interated integral each over the unit disc in the complex plane. [ e lzo <1

Therefore, it suffices to show that in -Iz 12 0 10glzol2.

.r:r

-4 dz o Adzo

is finite.

t,

By use of

175

TIlE DEFECT RELATIONS

4~ f~

polar coordinates, this equals

e- r2 (r log r)dr,

which

is certainly finite because the integrand is continuous

Q.E.D. Armed with Lemma 5.4, we are going to apply the averaging process to (5.26).

First, we rewrite it as:

where

Now fix an We apply

Ak - 2

and choose

111

b

to be orthogonal to

Ak-2.

to both sides of this inequality:

the

Ak - l :::)Ak - 2

right side remains unchanged while the left side changes to the following:

(This is just Fubin1's theorem and this application is permisSible because the integrand is visibly measurable and positive). By Lemma 2.14:

>

IJlf

log <1>( Ak - l )

- Ak - l :::)Ak - 2 =

log (Ak - 2 ) + d,

where the last step made use of Lemma 5.4 and d is some constant Ak-2 and dependent only on k. Thus,

1 n d ependent of

176

TIlE EQUIDISTRJBUTION TIlEORY OF HOLOMORPHIC CURVES

Combining this with (5.28), we obtain:

1:rodt ftrodsavrs)f-4>(Ak- 2 )*dT -< CTk(r) + C',

(l-a) C,

where and

C'

Ak-2.

are positive constants independent of

We may repeat this process and bring

4>(Ak - 3 )

down to

for every integer

o

C,

£ = 0,

£

J: f:

(l-a) where

etc.

C'

dt

ds

~

k,

r 4>(A£-l)*dT ~ CTk( r) + C' A- l

a, as a

b

and

•

A£-l.

(If

O-vector of unit

I n eac h s t ep. we keep b t 0 be a un i t

vector orthogonal to the fixed 4>(A£-l)

0 <£

remain independent of eJ:i8) •

4>(Ak-2)

airs)

we of course interpret

1 eng th , i .e.

b

The end result is clearly the following: such that

0

a,

A£-l.

But since we may rewrite

as

k l A b).J Xk+l, 2( , A£-l) -1 xk, 2a 4>( A£-1 ) __ IA£-l .-I x - ,2, fA£-l _ , IA~- .-I

each factor subspace

A£

A£-lAb of

Pn~'

I (A~-~ A

xltl tj.

becomes an

b)..J

xkl

i-dimensional projective

Taking this into account. we may sum-

marize the foregoing into Theorem 5.5. such that of Then

Pn~

Let

0 < £ < k. of dimensions

k = O••••• n-l Let £-1

A£-l ~ A£ and

£

and

£

be an integer

be projective subspaces respectively

(A- l == e.r:J.. e ).

177

TIlE DEFECT RELATIONS

where

C,

C'

Ai,

and

are positive constants independent of

while

There is a similar inequality for

i > k.

is dual to the above, so we will only sketch it. the contracted curve of the second kind given by (3.11) of Chapter III, §6. holomorphic curve Xk ~ Ak+l , so x (Xk....J Ak+l)

and

xt\x(l)

The order function of (4.34).

Its derivation We consider

xk~Ak+l: V _Ak+l

We apply (5.24) to the will be replaced by (Lemma 3. 11 ).

<Xk+l,Ak+l> (Xk-l....J Ak+l)

Xk~Ak+l

is simply

Tk(r,Ak+l ),

by

However, so far as the inequality (5.24) is concerned,

Lemma 4.11 says that where

by

A£-l

x~ etc.)

(Xk etc.

0 < a < 1.

a,

bk

Tk(r,Ak+l )

may be replaced by

is a constant independent of

Ak+l.

Tk(r) + b k ,

Putting all

these together, we obtain (5.29)

~e~

0 < a < I,

pendent of

a,

and

band

C,

C'

are positive constants inde-

Ak+l.

We are going to subject (5.29) to a similar kind of averaging process as (5.27).

Let

~(Ah)

be a function of

178

THE EQUIDISTRlBUTION THEORY OF HOLOMORPHIC CURVES

projective subspaces of dimension and consider the set of Ah

such an

versely each Ah

Ah+1

in a

€

Ah+1 ,

Fix an

a

Ah+ l

€

space of

a ¢

Ah+l.

in

Ah+l

space of

,

and con-

also uniquely determines an

Ah C Ah+1:

If we restrict

to only such

Ah,s contained

therefore becomes a function of

in the fixed €

PnC.

The polar

is just a point

the domain of definition of

a

in

Ah CAh+l.

Ah+ l

is the polar

h

which we simply denote by

1TI

the average

¢( Ah)

of

¢

¢(a).

By definition,

~ all

Ah

contained

Ah C Ah+1 itl the fixed Ah +l ,

1.e.,

where

n

E

Ah+ l

is the arithmetic mean of

¢(a)

is the volume form of the F-S metric on

be the vector subspace of

Cn+ l

over

Ah+l.

Let

corresponding to

then Theorem 5.2 says that we may equivalently define:

where

dL

denotes the Lebesgue measure of

coordinate function and by

z

[z 1

rzr

the point of

E,

Z

Ah+l

its usual represented

The following analogue of Lemma 5.4 may be proved

rzr·

in a similar manner. Lemma 5.6. and let

h

Let

Ai

be a fixed decomposable

be an integer such that

i < h < n-l.

(i+l)-vector Then for a

179

mE DEFECT RELATIONS

fixed

Ah+l, £ h+l loglA .J A I + a£h'

where

a£h

is a constant depending only on

independent of

A£

and

£

and

h,

and

Ah+l.

Rewrite (5.29) as: (l-a)

.r: r:. ro

dt

ro

ds

r

avis]

~(Ak+l)*dT

< CTk(r) + C',

-

where ~(A)

Now hold

b

fixed and take the average of both sides of this

inequality over all use of

Ak+ l

contained in a fixed

log(rr-n~) ~ m(log~)

Ak+2.

Making

and Lemma 5.6 as above, we

arrive at:

where

C,

C'

remain independent of

a,

band

Ak+2.

application of this process leads us to the following: every integer

£

such that

(l-a) Jr dt ro where rewrite

Jtrods avts) r ~(A£+l)*dT ~ CTk ( r) + C'

are i n dependen t

~(A£)

in this form:

~(A£+l)

=

for

k < £ < n-l,

C'

C,

Repeated

0f

a,

b

and

A£+l.

Let us

IXk-l..J (b...JA£+l)1!lx k+l..J A£+112( Jxk.JA£+~ )2a 1xk..J A 14 1X ...J (b .J A 1) I

180

TIlE EQUIDISTRIBUfION TIlEORY OF HOLOMORPHIC CURVES

What remains to be done is clearly to choose that the factor

b.J A£+l

to be a point of

A£+l.

b

in

A£+l.

becanes meaningful. then

projective subspace of

b

b....J A£+l

A£+l.

is an

cleverly so We choose

b

i-dimensional

in fact. the polar

space of

With this choice. we have arrived at the follow-

ing counterpart of Theorem 5.5. Theorem 5.7.

Let

= O, •.. ,n-l and P be an integer

k

such that

k < £ < n-l.

spaces in

Pn~

Let

of dimensions

A£ C A£+l £

and

be projective sub£+1

respectively.

Then (l-a)fr dtft ds r0 r0

r

ov1 s 1

IXk-l.JA£12Ixk+l.JA£+112 Ixk .J A£+l14

I k ..J A£+1 I ) 20.* d-r • (X Ixk..J All .s.. CTk(r) + C' 0
where dent of

§4.

a,

A£

and and

C,

C'

are positive constants indepen-

A£+l.

We now head towards the defect relations.

Through-

out this entire section, we will follow closely Ahlfors [11, p. 2Dff.

This part of Ahlfors' paper constitutes, in my opi-

nion, one of the most brilliant chapters in the annals of mathematics. We must first discuss the notion of a system of sional projective spaces fAh) <; G(n,h)

~ P£(h)-l~'

fAh) where

in general position. £(h)

=

(~~i).

The

h-dimenNow,

181

mE DEFECT RELATIONS

correspond to points in

P£(h)_lC

one-dimensional subspaces of

j

always a

e£(h)

via the projection map

hV: C£(h) - CO} - P£(h)_lC.

of the fibration given any

and hence to a system of

one-dimensional subspaces of

j-dimensiona1 subspace of

In general,

C£(h),

e£(h)

there is

which contains

them all, e.g. any

j-dimensiona1 subspace that contains

their common span.

We now adopt as formal definition that a

system of

h-dimensiona1 projective subspaces

[AhJ

of

PnC

is said to be in general position if and only if, when considered as one-dimensional subspaces of subspace of for all

j

C£(h)

C£(h),

each

contains no more than

such that

1

~

j

~

£(h)-l.

j

j-dimensiona1 of the

(Ah}

The following lemma

somewhat clarifies this definition. Lemma 5.8. a system of

The following are equivalent conditions on

h-dimensiona1 projective subspaces

[Ah}

( a)

The

(b)

Considered as one-dimensional subspaces of j

(c)

for

of the

[Ah}

of the 1

~

of

P C: n

are in general position.

(Ah} j

~

e£(h) ,

are linearly independent, for

Considered as one-dimensional subspaces of j

(Ah}

span a

any

1 ~ j ~ £(h).

C£(h),

j-dimensiona1 subspace of

any e£(h),

£(h).

Any of the above implies that (d)

The intersection of any subspace of

pne

1 < i < h+2. dimension

-1

i

of the

fAh}

of dimension at most

is a projective h-i+1,

for

(By definition, a projective subspace of is the empty set.)

182

TIlE EQUIDISTRIBunON TIlEORY OF HOLOMOIlPlDC CURVES

In general, (d) does not imply any of (a)-(c), but it does in a special case.

For a later reference, we single this

out. Lemma 5.9.

A system of hyperplanes

CAn-l)

Pn~

of

is

in general position if and only if the intersection of any of them is a projective subspace of dimension

n-i

in

i

Pn~

(1 ~ i ~ n+l).

The proofs of all these statements are simple exercises in linear algebra and so will be left to the reader.

The

following is a bit deeper. Lemma 5.10. tive subspace of

Let Pn~.

jective subspaces

Bk

be a given

k-dimensional projec-

For a system of

h-dimensional pro-

CAh)

in general position, the number of

Ah with the property that for a fixed every Al CAh satisfies Al~Bk = 0,

where the binomial coefficients

B

=

Al.JBk of

Choose O.N. basis

eo /\ ••• /\ eke

Ah

= 0

Let

for all

v

(l ~ h,

l ~ k),

does not exceed

are defined for all

(l+x)~ = ~~::= (~)xv.

integers by the expansion Proof.

(~)

l

eo, ••• ,en

of

en

so that

~ ai i e i /\ ••• /\ e i • Then 0··· h 0 h Al C Ah i f and only i f the coefficients Ah

=

satisfy the condition: whenever at least (£+1) of the subscripts are smaller than or equal to k.

183

THE DEFECT RELATIONS

This is obvious.

Now for

which have exactly

j

~

(£+l)+j

0, the number of components

subscripts smaller than or equal

k+l)( n-k ) (k+l)( n-k ) to k is ( (£+l)+j (h+l)-((£+l)+j} = £+j+l h-£-j. So the tota~number of coefficients satisfying the condition stipulated in (5.30) is

(~

'Let

. io < ••• < i h } io···i h ·

tions on

C£(k).

be the usual coordinate func-

Consider the subspace

S

of

C£(h)

defined

by: ••• ih = 0 for all io,· •• ,ih { ~i ofOthem are smaller than or equal

to

k.

We have just seen that there are exactly independent) linear equations, so have also seen that an A£ C Ah S.

satisfies

Since

fAh}

Ah

dim S

Ph(k,£) =

such (obviously

£(h) - Ph(k,£).

0

if

and only i f

Ah

lies

in

is a system in general position, it follows Ah

satisfying

cannot exceed the dimension of

A£.JB k S,

=

0

for every

which is

- Ph(k,£).

£(h)

Q.E.D.

Now let there be given a projective subspace of dimension

We

has the property that every

A£.J Bk

that the number of such A£ CAh

( £+1)

where at least

h.

Suppose

h

A

= eot\···

t\e h ,

Ah

where

in

Pn~

eo, ••• ,eh

are O.N. and will be fixed throughout the remaining discussion. For a fixed lIhere

£

such that

0 ~ /, ~ h,

0 ~ io < ••• < i /, ~ h,

and let

let

A£

=

A/'-l C A£

ei

t\ ••• t\ e i

o be such

£

184

that A- 1

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

=e

!\···!\e , jo < ••• < j£-l' (Recall that jo j£-l is just a scalar of absolute value one) , We apply A£-l

'Iheorem 5.5 to this pair obtaining: (l-a)

~

0

dent of

~

0

< £ < k,

thereby

IA£-1.Jxk-112IA£.Jxk+112 I ror dt Itro ds aJsl . . . . IA.t-1.J xkl ~ • (

where

A£-l C A£ ,

IA£-l..Jx k l 2a :e k ) * d-r IA .J X I

CTk(r) + C',

< a < 1,

and

C,

A£-l

and

At.

a,

C'

are positive constants indepen-

We may rewrite the integrand as:

IA£-1 .J Xk - 1 1 2 • IA£ .J xk+1dx2 IA:e .J xk I IA:e-1 .J Xk 14- 2ii Now for each choice of

io <

...

< i£

subject only to the conditions that A£-l ~ A£, io, ••• ,i£

0

and

jo <

~

io, ••• ,i£

we have an inequality as above. and

jo, ••• ,j£-l

' ,

. < j £-1' hand

~

If we now let

take on all possible integral

values and sum all of the resulting inequalities, we get (1-a)

I rr odt t:.r dsavts] r 0

~

a

J:

IA£-l.J Xk - 1 12 • IA£..JXk+1 IA:e-1.J~KI4-2Ci IAE.JXKI

*d-r

CTk(r) + C'

with new values of dent of

L

and

Ah.

C and

C'

which, however, remain indepen-

The finite sum in the integrand is greater

than every single term, and clearly there is a term which is at least equal to

185

THE DEFECT RELATIONS

where the maximum and minimum refer to the finite number of A£-l

and

A£

over which we have summed.

The expression can

be written in the equivalent form:

1Xkl2

£1

IA -

...J X

k 2 because

I

( by (1. 12) ) •

4-2a > 2

and

Having done this, we

obtain:

For simplicity. let us denote the quotient of the two minima by

¢(Ah ).

We now let

Ah

range over a finite system in

general position and form the sum of the corresponding inequalities:

(5.31)

186

TIlE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

with new constants

C,

C'

which are still independent of

lx Ik

a.

k

By Lemma 5.10, it is impossible to have for more than exists an of the

Ph(k,£)

of the given

l

such

Ah,s.

=

00

IA .Jx I Therefore, there

k

x I > M for at most Ph(k,£) IA -1 xkl independently of Xk. The remaining Ah,s satisfy

M such that

Ah,

Min

Ah.

Min

Let us say there are

P

Because

Furthermore,

in view of (1.13). one.

Therefore its minimum will still exceed

Consequently, referring back to the expression of

~(Ah),

we get

for these

P

such that

Min

if we use

1:'

we have:

of the

Ah,s.

Let there be

q

of the

Ah,s

fk l > M. By the above, q ~ Ph(k,£). SO, IA ~xkl to denote summation over these q of the Ah, s,

187

lHE DEFECT RELATIONS

> log l:' ¢(Ah )

log l: ¢(Ah) Ah

=

log(~ l:' ¢(Ah)l + log q

> ! l:' log ¢(Ah) + log q

-q

(Lemma 2.14)

> !( l: log ¢(Ah ) - 2ap log Ml + log q

- q

h A 1

h

> (k £)( l: log ¢(A ) - 2ap log Ml + log q, - Ph' h A l.e.

h

1

log ~ ¢(A ) ~ Ph(k,£)

A

h

l:h log cl>(A )

a constant depending only on the system :.,U8ual the constant

+ cl

where

cl

is

A

r

ov'r s)

* dT

for

of the Logarithm again, we have:

s

~

(Ahl. r( T ) •

Let

L

be as

By the Concavity

188

THE EQUlDlSTRIBUTION THEORY OF HOLOMORPHIC CURVES

where

c2

is a constant depending only on the system

(Ah ).

Keeping this inequality in mind, we inspect the integral

r

log

a/r s 1

~(Ah).dT.

It is equal to

def

~

I

aVr sJ

I .dT -

r

avi s1

II .dT

Obviously,

Now by Lemma 4.10,

where

ak

is independent

where

c3

and

~

Ai and

s.

Therefore,

are constants depending only on the system

189

TIlE DEFECT RELATIONS

£Ah)

alone and not on a.

satisfy

(l-a)Tk(s)

depending only on

r- *

ovt s)

I

d-r > -

=

1.

(Ah)

We now choose

a:

a

should

Hence there is a new constant such that

c4

I:

r -log Min( IAr..Jk X I 2

ov'r s)

and therefore

so that taking into account of a previous inequality, we obtain:

Where

c5

Row define:

is a constant depending only on the system

(Ah ).

190

TIlE EQUlDISTRIBlTfION THEORY OF HOLOMORPHIC CURVES

Then the above may be rewritten as:

By (5.31), this implies

(1-0) J:OdtJ:

O

exp(tpJlLE) 9(S) +

Recall that we have already chosen

a

C

so that

5}dS

~ CTk(r)

(l-o)Tk (r)

+ C'. 1.

Hence:

where because

C"

is some new constant and the last inequality is

Tk(r)

is monotone increasing. In the notation

of Chapter IV, §7, this may be written as:

or in view of Lemma 4.16(ii),

Recall that we have defined a function

T(r)

=

maX(To(r), ••• ,Tn_1(t))·

TIlE DEFECT RELATIONS

191

So by Lemma 4.16(i), we obtain

In greater detail, we have the following:

We wish to point out explicitly that (5.32) is only valid for

o<

i < k.

Introduce the notation:

'(k,i)

Then (5.32) may be written as:

(5.32)ik

Y(k,£) - Y(k-1,i-1) Ph(k,i)(-E - Sk - Tk _1 + 2Tk - Tk+1 ) + ~(T2)

=

In a similar fashion, we obtain:

(5.32)~:i Y(k-l,i-l) - Y(k-2,i-2) =

pp(k-l,i-l)(-E - S~ - Tk _2 + 2Tk _l - Tk ) + ~(T2)

(5.32)~-i-l Y(k-£+l,l) - Y(k-i,O)

= Ph(k-£+l,l)(-E - Sk_£+l - Tk_i + 2Tk _£+l - Tk_£_2) + ~(T2)

192

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

(5.32)~-£ Y(k-£,O) Ph(k-£,O)(-E - Sk_£ - Tk_£_l + 2Tk _£ - Tk_£+l)

+ IJ.( T2) The last is because

Y(k-£-l,-l)

=

0

(This explains why we

have been so careful all along about the case

A-I).

Adding

all these inequalities (5.32)~, .•• ,(5.32)~-£, we obviously get:

where use has been made of Lemma 4.16(v). £

=

h.

In this case, there is only one

mum need be taken. h

mk(r,A)

Finally, A£ C Ah

~

let

and no mini-

Recall at this point the definition of

in (4.29): mk(r,Ah )

=

~

r

ovtt]

log

IA

hxk'k *dT\r .J X I ro

So the left hand side of the above becomes:

By (the analogue of) Lemma 4.6, the last two terms are continuous functions of in Definition 2.1). conclude:

Ah

and so both are

0(1)

as

r - s

(s

We may therefore apply Lemma 4.16(ii) to

as

193

TIlE DEFECT RELATIONS

L (mk(r,Ah ) - mk+l(r,Ah)J Ah h

= Li=OPh(k-i,h-i)(-E - Sk_i - Tk _i _l + 2Tk _i - Tk _i +l ) + IJ.(T 2 ) This is valid for

o

upon summing over

k

for

~(r,Ah),

< h < k. from

k

Note that to

n-l

=0,

mn (r,Ah ) and writing

so

mk(Ah )

we finally arrive at:

(5.33) + L~:~ L~=O Ph(m-i,h-i)(-Tm_i _l + 2Tm_i - Tm-l"+1) + IJ.(T 2 ) Once more, we emphasize that (5.33) is valid only for

< h < k.

0

This is essentially one-half of the sought for defect relations. In all applications, we are only interested in the case

h = k,

but there is as yet no direct method of proving (5.33) only for this case. OUr next objective is to apply some combinatorics to simplify the last double sum of (5.33). last double sum to all i <0

and for

i > n-l.

i,

_00

< i < 00,

We will extend this taking

Ti = 0

for

Recall that

n+l ( k+l) n-k) Ph ( k,£ ) = ( h+l) - Lj~O £+j+l (h-£-j , Where

(IJ.) is defined for all integers by the binomial series v if v < 0.) So (l+x)1J. = L~::: (~)xv. (Note that

this yields

194

TIlE EQUlDlSTRIBUTlON TIIEORY OF HOLOMORPHIC CURVES

n+l) ( m-i+l ) n-m+i (h+l - ~j~O h-i+j+l ( i-j )

(5.34 ) We consider the case

i > h+l.

In the last sum,

j

~

0

as

it stands. But if j < 0, then h-i+j+l < 0 and hence m-i+l ) = O. Thus the last sum may as well be extended to ( h-i+j+l all integral values of

j.

It therefore equals

where the identity is obtained by comparing the coefficients of

xh+l

in the expansions of

• (l+x)n-m+i.

(l+x)n+l

and

(l+x)m-i+l

Hence, Ph(m-i,h-i)

= 0 if i > h+l.

Therefore, extending the summation to all

i

in the last

double sum of (5.33) means we must add to the left side of

(5.33) the following quantity:

But if

i < 0, O.

Ph(m-i,h-i)

= (~!i) because in (5.34),

So the above equals

n+l) ~n-l ~-l (T + 2T T ) ( h+l m=k i=_m - m-i-l m-i - m-i+l n+l n-l ( ) (h+l) ~=k -Tm + Tm+l

(~!i)( -Tk )· Hence (5.33) is equivalent to:

195

THE DEFECT RELATIONS

+ ~~:~ ~r:-~ Ph(m-i,h-i) (-Tm_i _l + 2Tm-i -Tm_i +l ) +

~(T

2

).

We proceed to simplify the last sum.

By choosing the

TIS as the running subscript we may rewrite

subscript of the the double sum as

~:i ~r:-~ (-Ph(i+l,h-m+i+l) + 2ph(i,h-m+i) - Ph(i-l,h-m+i-l))Ti • Now we apply the recursive relation among the binomial

(~) + (v~l) = (~:l).

coefficients:

inspecting the coefficient of

(l+x)~+l

(l+x)(l+x)~.

and

XV

This can be proved by in the expansions of

This implies:

Ph(i,h-m+i) - Ph(i+l,h-m+i+l) =

~J~O

(

1+2 ) ( n-i-l ) 1+1 ) n-i) J h-m+i+J+2 m-i-J-l - (h-m+i+J+l (m-l-J

(

i+1

) ( n-i-l)

(

i+l

) (n-i-l) J

= ~J~O h-m+l+J+2 m-l-J-l - h-m+l+J+l m-l-J = -

(

i+l ) (n-i-l) h-m+l+l m-l

In a similar fashion:

196

TIlE EQUIDISTRIBUfION THEORY OF HOLOMORPHIC CURVES

Hence the above double sum equals +ex> n-1 f( i )( n-i) ( i+1 )(n-i-1)J Li=_m ~=k h-m+i m-i+1 - h-m+i+1 m-i Ti

_ L+ex> Ln - 1 f( i )[(n-i-1) + (n-i-1)) - i=_m m=k h-m+i m-i+1 m-i

[(h_m!i+1)(h_~i))(n;:i1)JTi

-

+ex> n-1 f( i )(n-i-1) ( i )(n-i-1)J = Li=_m Lm=k h-m+i m-i+1 - h-m+i+1 m-i Ti

i )(n-i-1) ( i )(n-i-1)J f - ( h-k+i+1 k-i + h-n+1+i n-i Ti

Now observe that

(

~

) -

~+1

-

~

= -1

1

if

0

otherwise.

{

So the second sum has every coefficient equal to zero except for

i = n,

but then

identically.

(~) and

= 0

if

Tn = O.

Thus the second sum vanishes h
As to the first sum, remember that v

< O.

(nk:i1) = 0

So

unless

(h-k!i+1) = 0 k > i.

unless

i

~

and

(k-h)-l

So the above equals

Lk (i )(n-i-1) i=k-h-l - h-k+i+1· k-i Ti We may now rewrite (5.35) in its final form:

k (i )(n-i-1) ( 2) - Li=k_h_1 h-k+i+1 k-i Ti + ~ T .

197

mE DEFECT RELATIONS

When

h = k.

we claim that this reduces to _ ~n-kl ~ki n Pk(m-1.k-1)(S 1 + E) ( n+l)T k+l k m= =v m-

+~(T2). Th1s 1s because ~k ( i )(n-1-l)T i=-l 1+1 k-i 1

has every coefficient equal to zero due to the presence of (1!1)

except when

i = -1.

But then

T_l = O.

So the whole

sum vanishes. Mak1ng use of the F1rst Corollary of Lemma 4.17. it is poss1ble to derive a variant of on the right side.

(5.36) where only Tk appears

It goes as follows.

_ ~k (1 )(n-i-l) 1=k-h-l h-k+i+l k-1 Ti r

k

-~i=k-h-l -

(1 )(n-i-l)(i+l)J h-k+1+l k-i K:+r Tk

r~ ~~=k-h-l

k-h

k

(h_k!i+l)(n

k:i l)(k-i)(1+l)JE +

( i + l )(n-i-l)J h-k+i+l k-i Tk

r

- K:+r ~i=k-h-l

r

k-h k (i+l)(n-i-l)J - K+I ~i=k-h-l k-h n-k-l Tk _ r(n-k)Jk-h) ~k (1+l)(n-1-l) JE + ~(T) i=k-h-l k-h n-k

In order to simplify the coefficients of invoke the following identity: integers, then

if

Tk

P. q. r

and

E.

we

are positive

~(T)

198

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

To see this, recall that if ( -m) p

definition,

m is a .nonnegative integer, by

(-l)P m' (m+l) ..... p! (m+p-l).

every nonnegative integer

Hence for

m,

Now, (l_x)-(p+l) = (1 + (_x))-(p+l) = ~v (-(~l))(_l)VxV = ~v (p;V)XV • Similarly,

(l_x)-(q+l) = ~

~

(q~)x~ q

and

(1_x)-(P+q+2)

-- ~n ~ (p+q+l+n) p+q+l x n •

S0 compar i ng th e coef fi cents i of

(1_x)-(P+q+2)

(l_x)-(p+l). (l_x)-(q+l),

and

xr

we get the

above identity. Applying this identity, we obtain: _ ~k (i )(n-i-l) i=k-h-l h-k+i+l k-i Ti

(- (~~¥) ~~~~-h (n~k:l)(k:h)}Tk _ ((n-k)jk-h)

k+l

~~2~--L- ~j=k-h

(n- j )( j )) () n-k k-h E + ~ T

_ (k-h) (n+l)T K+T h+l k _ (n-k)Jk-h) ((n+l) + (k+l)(n-k-l pE + ~(T) h k-h n-k _ (~-r) (n+l)T _ (n-k)Jk-h) (n+l)E + (T) + I h+l k h ~,

in

199

mE DEFECT RELATIONS

where the last step is due to the fact that n-l,

(n~~kl)

so that

O.

=

k

never exceeds

Therefore from (5.36) we deduce

where use has been made of Lemma 4.16(i} and (v) to get

~(T) + ~(T2)

=

~(T2).

(5.36)-(5.38) constitute the defect

0 < h < k.

relations for

To complete the picture, let us deduce also the defect k < h < n-l.

relations for the case

Because the details are

somewhat similar to the preceding, we will be brief.

We first

need the counterpart of Lemma 5.10. Lemma 5.11. subspace of subspaces

Let

h i £)

k-dimensional projective

h-dimensional projective

in general position. the number of

the property that every k < i.

be a fixed

For a system of

Pn~.

fAh)

Bk

Ai;? Ah

satisfies

Bk.J Ai

with

a fixed integer and

(i

O.

=

Ah

does not exceed

This lemma reduces to Lemma 5.10 if we replace each by its polar

space

polar

B.!.

space

Ai.

(dim B.!.

(dim A.!. =

=

Let

are part of an O.N. basis

A£+l =>A£ ~Ah

and

Bk

by its

n-k-l).

We now suppose that we are given eo ••••• ~

n-h-l)

Ah

Ah

=

e o /\ ••• /\e h •

eo ••••• e n

in

where

(n+l.

be projective subspaces corresponding to

200

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

the subspaces of (£+2)

and

(£+1)

with

k < i < n-l:

where

0 < a < 1,

pendent of

a,

cn+l

spanned by an arbitrary choice of

members of

and

Ai and

C,

reo, ••• ,en ).

We apply Theorem 5.7

C'

are positive constants inde-

A£+l.

Proceeding in similar manner

as above, we first obtain:

C,

where of and

C'

are new positive constants still independent

and the minimum refers to the finite number of Ai h Letting Ah range over a finite A£+1 containing A.

a,

system in general position, we obtain the analogue of (5.32):

201

THE DEFECT RELATIONS

'(k.,e) = L

Let

~

Ah

J log Min( I kXkl:e .J I

oV[ sJ

X

A

then the above may be written

(5.39)~

'(k.,e) - '(k+l.£+l)

= p n-h-l (n-k-l.n-,e-l)(-E-Sk-Tk - 1+2Tk -Tk+1) + ~(T2). k+l )k+n-£+l k Adding (5.39),e.(5.39),e+l.···.(5.39 n-l • '(k+n-,e.n)

=

O.

we get

1 ~ log Min( kxk I : Ah 2'Ir oV [ s J IX ~ AEI =

,e

=

h.

kxk-11

I X -1.J AEI

).dT

L~:~-l Pn_h_l(n-k-i-1.n-h-i-1)(-E-Sk+i-Tk+i+1 + 2Tk+i-Tk+i_1) +

If

and noting that

~(T2)

no minimum need be t.aken. so this reduces to

Lh (~(Ah) - mk _1'(A h )} A

= L~:g-l Pn_h_l(n-k-i-1.n-h-i-1)(-E-Sk+i-Tk+i_1 + 2Tk+i- Tk+i+l ) + II(T2). ... mk(r.Ah )

where we have written

mk(Ah )

Summing over

from

0

-L:=o

L~:~-l Pn_h_1(n-m-i-1.n-h-i-1)(E+Srn+i)

k

to

in place of k

and noting that

as usual.

m_1(Ah)

= O.

we have:

k n-h-l Pn-h-1 ( n-m-i- 1 .n-h-i-1) + Lm=o Li=O • (-Trn+i_i+ 2Trn+i- Trn+i+1) + ~(T2) This is valid when

k < h.

We now extend the last double sum

202

THE EQUlDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

over all

i.

setting as before

i > n-l.

When

Ti = 0

When

The coefficient vanishes When

i <0

and

i > n-h

and

(n+l) = (n+l) when i < O. So after we have extended n-h h+l the sum to all i, we should add to the left the following

equals

quantity:

~:=O l:1:~

Pn_h_l(n-m-i-l,n-h-i-l)(-Tm+i_l+ 2Tm+i- Tm+i+l)

n+l) k ~-l ( ) ( h+l ~=O ~i=-~ -Tm+i_l+ 2Tm+i- Tm+i+l n+l) - ( h+1 Tk • We have therefore obtained the following: (5. 40 )

h) = (n+l) k n-h-l Pn-h-l ( n-m-i-l,n-h-i-l ) h+l Tk - ~m=O ~i=O

~h ~ ( A A

• (E+Sm+i) +

~=O ~r:-~ Pn_h_l(n-m-i-l,n-h-i-l)

It remains to simplify the last double sum. and let (5.41)

~

= n-l-m.

Let

a = n-h-l

Then this double sum may be written as:

~~:~-l-k ~r:-~ Pa(~-i,a-i)(-Tn_l_(~_i_l)+2Tn_l_(~_i) -Tn_l_(~_1+1) )

It may be recalled that previously we have computed the last infinite double sum of (5.35) and found that (5. 4 2)

~~:i ~r:-~ ~(m-i,h-i)(-Tm_i_l+2Tm_i-Tm_i+l) k (i )(n-i-l) = ~i=k-h-l - h-k+i+l k-i Ti •

203

'DIE DEFECT RELATIONS

The left side of (5. 4 2) would be identical with (5.41) if only we replace

h

by a l

k

by

and

(n-l-k)

Tj

by

Tn _l _ j •

So (5.41) equals Ln-k-l i ) ( n-i-l )T i=n-k-l-a_l -(a_(n_k_l)+i+l (n-k-l)-i n-l-i = n-k-l ( i )( n-i-l )T Li=h_k_l - k-h+i+l n-i-k-l n-l-i _ Ln - h+k (n-l- j )( j )T - j=k - k-h+n-j j-k j _ Ln - h - k _(n-l-j)(j)T - j=k h-k-l k j Substituting this into (5.40)1 we obtain the counterpart of (5.36) : ( 5. 4) 3

k n-k-l Pn-h-l ( n-m-i- 1 , n-h-i-l ) L ~ ( Ah) = (n+l) h+l Tk - Lm=o l:i=O Ah

This is valid for

k < h.

When

~

= hI

(5.43) again reduces

to (5.37). Again we can transform the last sum using the First Corollary of Lemma 4.17 so that only -

Tk

appears.

l:n-h+k (n-l-i)(i)T i=k h-k-l k i (_Ln - h+k (n-i)(n-l-i)(i)JT i=k n-k h-k-l k k -

(~L~:~+k (~:i:i)(~)(i-k)(n-i)JE + ~(T)

(_ h-k Ln-h+k (n-i)(i)JT n-k i=k h-k k k (T) _ r(k+l)Jh-k) Ln - h +k (n-i)( 1 )JE i=k h-k k+l + ~

204

TIlE EQUIDISTRIBUfiON TllEORY OF HOLOMORPHIC CURVES

ih-k)(n+l)T _ (!ik+l)(h_k)(n+l)JE + (T) k 2' h+2 IJ.

~h+l

where we have used the previously proved identity: ~q+r

v=q

(p+q+r-v)(v) = (p+q+r+l) p q r·

Substituting into (5.43) the above, we have arrived at the counterpart of (5.38): (n-h) (n+l) = ~ h+l Tk

(5. 44)

-

~:=O ~~~-l Pn_h_l(n-m-i-l,n-h-i-l)Sm+i

_ (k+1)Jh-k)(~~) + lIl=O ~k ~n-k-1 Pn_h_1(n-m-i-l,n-h-i-1»)E i=O

(5.43) and (5.44) are the defect relations for the case (n-l)

~

h

~

k.

We now summarize the above into a comprehensive

theorem, which is the main result of the whole development. Theorem 5.12 (Defect Relations).

Let

nondegenerate ho1omorphic curve and let exhaustion.

Let further

(Ah)

siona1 projective subspaces of

x: V - Pnll:

be a

V admit a harmonic

be a finite system of

h-dimen-

Pn C in general position.

the generalized compensating terms satisfy the following inequalities:

~(Ah) If

0

= ~(r,Ah) ~

h

~

k,

k (i )(n-i-l) - ~i=k-h-l h-k+i+l k-i Ti +

of (4.29) then

IJ.

Then

(T2)

205

mE DEFECT RELATIONS

(~)(n+1)T ~+~

h+1

k

_ ~n-1 ~hi n Ph(m-i,h-i)S i m=k =v m-

_ (n-k)Jk-h) (n~l) + ~:~ ~~::O Ph(m-i,h-i»)E + ~(T2) If

k.s. h .s. n-1,

then n+1) k n-k-1 ( ) ( h+1 Tk - ~m=O ~i=O Pn-h-1 n-m-i-1,n-h-i-1 (E+Sm+i) _ ~n-h+k (n-1-i)(i)T + U(T2) i=k h-k-1 k i ( n-h)(n+1)T n-k h+1 k -

~km::O ~n-k-1 p h 1(n-m-i-1,n-h -i-1)S i i::O n- m+

1, )( )(n+1) k n-k-1 - ( 2'k+1 h-k h+2 + ~m=O ~i=O Pn-h-1 • (n-m-i-1,n-h-i-1)}E Finally, if

h = k,

+ ~(T2).

then

As we mentioned before, the last conclusion of the preceding theorem is the most important in application.

We can

rephrase it in an essentially equivalent way, as follows.

Let

V admit an infinite harmonic exhaustion (Definition 2.4). For each

k-dimensiona1 projective subspace

define the defect of

Then

~k

Ak

Ak

of

PnC,

to be:

is a measurable real-valued fUnction on

G(n,k).

we

206

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Clearly

15 k

~

o

follows that 8 k : G(n,k) -

(Nk(r,Ak ) < Tk(r) + c k ) ,

From (4.25)

1. ~

8k•

[O,ll.

(See §5 of Chapter II).

Thus

AJ.. be the polar

space of

Let

From the definition of

k Nk(r,A)

k

I5 k (A

)

=

p € V,

then

1.

k

I5 k (A) = 0

k Nk(r,A) =0

for all

r

kX(P) () AL = ~ and so

is then to be interpreted as

kX(P) () AJ..

f ¢ for "many" p €

cases,

=0

15 k

V.

We now show that in many

except on a countable subset of

Theorem 5.13 (Defect Relations).

Let

G(n,k).

x: V- PnC

nondegenerate holomorphic curve and suppose either or

(oJ, or else (ii) V

C -

(a i € M)

and

system of

€ > O.

(i)

be a V

=~

compact M - (al , ••• ,aml is transcendental. If (Ak} is a finite =

k-spaces in general position

Proof.

where

x

Ak.

(see the paragraph preceding

Theorem 4.8 and the discussion after (4.4», if for all

it

~

(0

k

~

n-l),

then

Suppose not, then

Thus outside a compact set, k

Nk(r,A ) 1 l: (1 _ ) > (n+ ) + Ak Tk(r) - k+l or

€

l: (Tk(r) - Nk(r,Akp ~ (~!i) + €}Tk(r), Ak

(4.16),

l:k ~(r,Ak) ~ ((~i) + €}Tk(r). A

,

or in view of

By Lemma 4.l6(i)

207

TIlE DEFECT Il.ELATIONS

and the last Conclusion of the preceding theorem:

(~!i)

+ E)Tk(r)

= (~i)Tk - ~,i Pk(m-i,k-i)(Sm_i+E ) + IJ.(T 2 )

ETk Now

~ ~,i

sm-i

_>

0

Pk(m-i,k-i)(Sm_i+E) = IJ.(T 2 ). by its definition, so Lemma 4.l6(iv) implies

E > 0,

Since

by Lemma 4.16(iii).

Note that each

constant, so the coefficient of

E

Pk(m-i,k-i)

is a positive

is a positive constant.

This contradicts the last conclusion of Theorem 4.24. We mention in passing that if we only know that

Q.E.D. V has

an infinite harmonic exhaustion, we can still obtain defect relations.

In fact, define

Then one can prove that for a finite system of

k-spaces

(Ak)

in general pOSition, the following holds:

(5.45)

and if one of

(Xo, ••• ,Xn_l)

is finite, so are the others.

Because this seems to be too complicated to be of much use,

we leave it as an exercise to the reader.

208

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

§5.

Ahlfors mentioned at the beginning of his paper [11

that the equidistribution theory of holomorphic curves suffers from a lack of applications.

The situation has not much

improved in this respect in the intervening thirty years. this concluding section of the notes:

In

we attempt to give a

few simple consequences of the defect relations, some of which are classical. We have as usual a nondegenerate holomorphic curve Pn~.

x: V -

For each

p

€

V,

kX(P)

sional projective subspace of

Pn~'

sional osculating space of

at

Proposition 5.14. holomorphic curve. (ii)

x

Let

x

is of course a and we call it a

(i)

is transcendental and

k-dimen-

p.

x: V - PnC

If either

k-dimen-

be a nondegenerate ~

V

or

C - CO),

or

V is a compact Riemann surface C(n+l) + 1) k+l in general position,

with a finite number of points deleted, then given k-dimensional projective subspaces of (0

~

k

~

n-l),

PnC

at least one of them meets an

sional osculating spaces of

x.

hyperplanes in general position,

(n-k-l)-dimen-

In particular, given

(n+2)

xCV) must intersect one of

them. Proof.

Let

CAk}

be the original finite system of

k-spaces and let

CAL)

be the corresponding system of

spaces formed from the polar

spaces of

CAkJ.

Each

(n-k-l)-

A~

has the same Grassmannian coordinates as the corresponding so if the

CAk) CAL)

are in general position, Lemma 5.8(b) shows that are also in general position.

If

(n_k_l)x(P)

does

209

THE DEFECT RELATIONS

not meet each

Ai

each

~.

(Ak)

then

Nn_k_l(r,Ai

)

and therefore

5n_k_l(A~)

=

for any

and each

r,

p

((~i) + 1)

Since there are

~'s, we obtain

V,

€

1: 5 n _k _l (Ai)

0 1

for for

(= ((~~~) + 1))

= (~~~) + 1.

such

This contradicts

A~

Theorem 5.13.

Q.E.D.

From this proposition follows the classical theorem of Borel.

We formulate it in this fashion.

function

xo, ••• ,xn +l

on V,

Given holomorphic

we call a linear relation among

them an identity of the form:

ao, ••• ,an+l

where

not equal to zero. and only if

~

are complex numbers with at least one We call it a special linear relation if a o , ••• ,an+1

of the

Proposition 5.15 (E. Borel). and let

(n+2)

be given on

(n

xo , ••• ,xn+1

among

is equal to zero.

V

Let

= ~

~

or

nowhere zero ho1omorphic functions

V,

ai

~

1).

- (0) xo, ••• ,xn+l

Then a special linear relation

implies a linear relation among any

(n+1)

of them. Proof. + a n+1xn+1

Let the special linear relation be

= O. After renumbering if necessary, let us say

that we want a linear relation among ho1omorphic mapping If

1T:

q:n+l

xo""'xn •

x: V - ~n+1 - (0)

{oJ - PnC

a holomorphic curve in (n+2)

aoxo +

as usual, then Pn~'

Obviously,

where x:: x

Consider the

x 1T'

x

(xo "" ,xn ). is of course

never meets the

hyperplanes in ~neral position given by:

zo=O'zl=o"",zn=o,

210

TIlE EQUJDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

aoz o + ••• + anz n

and x

= o.

Proposition 5.14 implies that

mast be degenerate and so the vector-valued function

takes value in an

n-dimensional subspace of

Cn+l

x

defined

o

Proposition 5.16.

Let

V be a compact Riemann surface

M with a finite number of points deleted and let xo, ••• ,xn+l

nowhere zero holomorphic functions

V (n

~

1).

(n+2) be given on xo, ••• ,xn+l

Then a special linear relation among

implies either that a linear relatlon exists among (n+l) them, or that every quotient XiLj ;X (0 ~ i,j ~ n+l) can be extended to a meromorphic function on Proof. V

C or

C - (oJ.

Let V

= M - (al, ••• ,amJ, ai

is that if the holomorphic curve

V- Cn+l

or both.

Of course we may exclude on the outset the cases

a compact Riemann surface and

x:

M,

of

xx

(where

then every quotient

=

E

M.

~

M is

Our first observation

x: V-

(xo""'xn ))

where

Pn~

induced by

is not transcendental,

(0 ~ a,~ ~ n)

is extendable to a

meromorphic function on M. To prove this, note that actually so that

x

= 'II"

0

x,

for instance that

where

'11":

Xl~o""'~o

to be meromorphic functions.

~ Zo

~n+l - (O)

x: V _ ~n+l - (oJ, - PnC.

are all extendable over

So take a neighborhood

and take the usual coordinate neighborhood

that

Uo

=

a1

U of ([zo"",Zn):

with coordinate function ~: Uo - ~n such zlL zn/ ~([zo"",znJ) = ( /Zo"'" -lZo). Because Xo never

I O)

on

Pn~

Let us show

211

TIlE DEFECT RELATIONS

vanishes, on

~ Uo '

x(U - (al )

so that

C. x

is well-defined

U - (~).

Quite obviously, C. x: U - (al ) - Cn is xl xn Cox = (-, ••• ,-). Since by assumption x is

such that

Xo

Xo

not transcendental, Lemma 4.23 implies that to a holomorphic mapping defined on all of

x

is extendable

U into

PnC.

(The proof of that part of the lemma remains valid regardless of whether

x

Lemma 3. 3,

Cox

tions on U,

is nondegenerate or not). is then extendable to

By the proof of n

meromorphic func-

which proves our claim.

Now to the proof of our lemma. relation be

Let the special linear

aoxo + ••• + an+lxn+l = O.

It suffices to prove

that if there is no linear relation among any (n+l) of xi / xo, ••• ,xn+l' then every quotient 7X j can be extended to a meromorphic function on all of M (0 < i,j < n+l). Let ~ xn us show that - , ••• , - can be so extended, the proof of Xo

Xo

the others being similar. x: V - PnC

induced by

Since actually

'Ir

X

0

not meet any of the and

Consider therefore the curve

x: V _ Cn+l

(0),

x,

it is easy to see that

(n+2)

hyperplanes given by:

aoz o + ••• + anz n = O.

x=(xo,· .. ,~)·

where

x

zo=O, ••• ,zn=O,

These hype:rplanes are in general

position, and so Proposition 5.14 says that either degenerate or

x

is not transcendental, or both.

there is no linear relation among take value in a hyperplane of that

x

does

is not transcendental.

xo, ••• ,xn '

tn+l;

x

x

is

Since does not

we have to conclude then

By our initial observation,

~

xn o Xo functions.

X-, .•. ,--, are all extendable to all of M as meramorphic Q.E.D.

212

TIlE EQUlDlSTRIBlTflON TIlEORY OF HOLOMORPHIC CURVES

One can find an application of this theorem to the uniqueness problem of meromorphic functions in the thesis of Edwardine Schmid (Berkeley 1969).

Our next proposition depends

on a lemma which has not been completely proved. Conjectural Lemma 5.17.

H1 ••••• Hn+ 2

Let

hyperplanes in general position in the set of of

PnC

contains fewer than pOSition.

through

(k+2)

n > 2.

Ak

(Ak ~ Hi'

n Hi:

(Ak

hyperplanes of

(A hyperplane of

Pn~).

in

PnC.

k-dimensional subspaces

which have the property:

Ak

i

Ak,s as

i

1 ••••• n+2)

=

1 ••••• n+2}

=

Ak

(n+2) Consider

in general

is of course a

Then the union of all such 1 ••••• n-1,

be

(k-1)-space k

runs

is the union of a finite number of distinct

proper projective subspaces of

PnC.

The proof of this lemma for

n

2.3,4

is not difficult;

it is also relatively easy to prove that the number of hyperplanes than

H such that (n-1)+2

(H

n Hi:

i

hyperplanes of

finite in number.

1, ••• ,n+2}

=

contains fewer

H in general position is

However, the general proof for

n

=

5

starts to get very long and I have not carried it through.

In

any case, the following proposition follows from this conjecture. Suppose

Proposition 5.18. mapping such that position, then of

x(~)

x(C m)

avoids

x: ~m _ P ~ n

(n+2)

is a ho10morphic

hyperplanes in general

lies in a proper projective subspace

PnC. Proof.

Let us say that

x(~m)

the latter are in general position.

avoids

H1 ••••• Hn+ 2

and

We may assume on the outset

213

11IE DEFECT RELATIONS

that

x

is not a constant map.

Let

p

be any point of

Consider the set of complex lines passing through ~m.

union is all of

Hence the restriction of

one of them should be nonconstant. assuming that

~p

x:

used the notation:

z

€

~J.

-

Pn~

if

p

which is parallel to

~p = (( z, P 2' ••• , Pm) :

Cp

is the complex line through

(1,0, ••• ,0).

Now

x:

~p

- PnC

(n+2)

1 < i < n-1. for which

Say i t lies in an i

If we choose

x(C p ) ~ Ai.

ho10morphic curve.

= 1, ••• ,n+2J.

hyperplanes of

hyperplanes

x: ~p - Ai

then

x(~p)

Since

i-space

Ai

of

still avoids

n Hi:

(Ai

the latter must contain fewer than Ai

(£+2)

in general position by Proposition 5.14. Ai

lies in the union of

guaranteed by Conjectural Lemma 5.17, say, p

over an open se t

U'

of

~m. ~

cient1y small, we can clearly assume that nonconstant for each is an open set implies that

If

U

x(U)

to show that s

=

U AS _ l

1,

PnC,

is a nondegenerate

the finite number of proper projective subspaces of

s

is a

to be the smallest integer

We are thus forced to conclude that

us vary

p

in general position and is therefore degenerate, by

Pn~

Proposition 5.14.

i

to at least

is nonconstant, where we have

nonconstant ho10morphic curve that avoids of

x

their

So there is no harm in

(P1' ••• , Pm) ,

In other words,

p;

em.

p

€

U'.

C A_

x(U)

x:

a:P -

The union of all

--1

is

Pn It

G: p '

P

€

U',

U ... U As • We now use induction on actually lies in one of the

there is nothing to prove.

implies that

A1 , ••• ,A s ' Let If U' is suffi-

and the preceding argument obviously

of -

Pn~

x(u)

So suppose

lies in one of the

Al •••• ,A s • x(U)

~

Al U

Al , ••• ,A s _1 '

214

TIlE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

We now show that i f for some x(q)

j

between

1

and

s.

U ... U As _1 l.

As - (A 1

€

X(U) ~ A1 U ••. U As'

A1 U •.. U As _1 •

set into

by the ho10morphy of

- (A1

U ... U

U AS_1

As _1 1

q

into

But if

x

As

q

x

The ho1omorphy of

x

carries

and we

Then

x(q)

I

As

x(u) SAl

and our induction hypothesis implies that

1 < i < s-l.

U.

that is

x.

U.

€

€

carries an open

So we may assume that

for any

q

then by continuity.

disjoint from

are done in this case.

x(U) S Aj •

Suppose for some

a sufficiently small neighborhood of

As'

then

U ...

x(U) S Ai'

now implies that

Q.E.D. Corollary. of

Pnt

Let

H1 ••..• Hn + 2

in general position and

be

(n+2)

x: t n -

hyperplanes

Pnt

is a holomorphic

mapping whose differential is nonsingu1ar somewhere. x(<<:n)

must intersect one of

Then

H1 •...• Hn + 2 •

This generalizes the classical Picard theorem. We wish to generalize the Picard theorem in yet another way.

For each positive integer

n.

let

p(n)

tive integer greater than or equal to (2n+1)

be the posisuch that the

following lemma is valid: Lemma 5.19. Pnt

Let

H1 ••.•• Hp (n)

in general position and let

Ak

k-dimensiona1 projective subspace of any (k+2)

Hi.

Then

{Akn Hi: i

hyperplanes of

Ak

=

be

p(n)

hyperplanes of

(1 < k < n-1) pne

1 •..•• p(n))

be a

not contained in contains at least

in general position.

215

THE DEFECT RELATIONS

For

n

~

4.

it can be shown by a straightforward case

by case examination that

p(n)

= 2n

mination of the smallest possible remains unsettled.

+1

would do.

p(n)

The deter-

for a general

n

In any case. on the basis of this lemma.

we can prove the following. Proposition 5.20. mapping such that of

Pnt

Let

x(tm)

x: t m - PnlC

be a holomorphic

fails to intersect

in general position.

Then

x

pen)

hyperplanes

must reduce to a

constant. m = n = 1.

When

this says that a meromorphic function

on the Gaussian plane must be a constant if it omits three distinct points of the Riemann sphere (because we have remarked above that

p(l)

=

3

would do). which is Picard's theorem

again. Proof. morphic

x: IC - PnlC.

says that then

x

x(lC)

Choose x(t).

It suffices to prove the proposition for a holo-

Ak then

Since

p(n)

must be degenerate.

(n+2).

If

x

Proposition 5.14 is not a constant.

lies in a projective subspace

k

1 < k < n-l.

A.

so that it is the smallest subspace containing x: t - Ak

is a nondegenerate holomorphic curve

which avoids all the hyperplanes But at least

~

(k+2)

(Ak

n Hi:

i

=

1 ..... p(n)J.

of the latter are in general position by

Lemma 5.19. so this contradicts Proposition 5.14.

Q.E.D.

Proposition 5.20 is interesting in view of Kobayashi's theory of hyperbolic (complex) manifolds.

(cf. a forthcoming

monograph of his; see also J. Math. Soc. Japan 1967. 460-480).

216

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

A hyperbolic manifold has most of the essential properties of a bounded domain, including the fact that every holomorphic mapping of

~

has proved that

into it reduces to a constant. PnC

minus

2n

P.J. Kiernan

hyperplanes in general position

is not hyperbolic, (see his paper "Hyperbolic submanifolds of complex projective space", to appear in Proc. Amer. Math. Soc.). This is why we required above.

p(n)

to be at least (2n+l)

in the

Motivated by Proposition 5.20, we conclude these notes

with the following conjecture:

Pn~

in general position is hyperbolic.

minus

p(n)

hyperplanes

REFERENCES [1]

L.V. Ah1fors, The theory of meromorphic curves, Acta Soc. Sci. Fenn. Ser. A, vol. 3, No.4.

[2]

S.S. Chern, Complex manifolds without potential theory, Princeton, 1967.

[3]

R.C. Gunning and R. Narasimhan, Immersions of open Riemann Surfaces, Math. Ann. 174(1967), 103-108.

[4]

W.K. Hayman, Meromorphic functions, Oxford, 1964.

[5]

W.E. Jenner, Rudiments of algebraic geometry, New York, 1963.

[61

R. Nevanlinna, Eindeutige analytische Funktionen, Berlin, 1953.

[7]

H. Weyl and J. Wey1, Meromorphic functions and analytic curves, Princeton, 1943.

[81

H. WU, Mappings of Riemann Surfaces (Nevan1inna theory), Proc. Sympos. Pure Math. vol. XI, "Entire Functions and Related Parts of Analysis." Amer. Math. Soc. 1968, 480-532.

217

INDEX OF PRINCIPAL DEFINITIONS algebraic number of pre-images 43 associated curve of rank k 71 average of a function over all A~ containing a fixed Ak -; 1171 average of a function over all contained in a fixed A + l7B binomial coefficients

lB2 n canonical basis of t +l 4 compensating term 40, 93 concavity of the logarithm 48 conjugate one-form 34 contracted curve of the first kind 75 contracted curve of the second kind 79 counting function 39, 92 critical point 24, 33, 115 crl tical value 33 degenerate curve 67 decomposable vector 5 defect 47, 56, 205

finite harmonic exhaustion Fubini-Study metric 3

54

Gauss-Bonnet theorem 26 general position lBl geodesic curvature form 26 harmonic exhaustion 32 Hurwi tz' s formula 31 hyperpl.ane B induced mappings lB, 62 infinite harmonic exhaustion interior product 11 ~(~)

(the symbol)

nondegenerate curve

53

131 67

order function 39, 92 osculating space 122, 20B Pldcker's formulas 123 polar divisor B polar space B projection curve 74 • prOjective subspace 4 reduced representation

20, 64

special coordinate function 36 special linear relation 209 stationary index 24, 115 Sylvester's theorem on compound determinants transcendental curve 138 transcendental mapping 57 219

78

Annals of Mathematics Studies Number 64