The Equidistribution Theory of Holomorphic Curves. (AM-64), Volume 64 9781400881901

Table of contents :
CONTENTS
Preface
Introduction
Chapter I. Generalities on projective spaces and Grassmannians
§1 The Fubini-Study metric
§2 Basic facts about Grassmannians
§3 Elementary inequalities
Chapter II. Nevanlinna theory of meromorphic functions
§1 The proximity function
§2 Holomorphic mappings into the Riemann sphere
§3 The nonintegrated main theorems
§4 Harmonic exhaustion and the integrated main theorems
§5 Two useful facts
§6 The defect relation
Chapter III. Elementary properties of holomorphic curves
§1 Holomorphic curves induced by a system of holomorphic functions
§2 Nondegeneracy
§3 Associated curves
§4 Projection curves
§5 Contracted curves of the first kind
§6 Contracted curve of the second kind
Chapter IV. The two main theorems for holomorphic curves
§1 The nonintegrated first main theorem
§2 The integrated first main theorem
§3 The FMT of rank k
§4 The generalized FMT and two inequalities
§5 The nonintegrated second main theorem and PIücker's formulas
§6 The integrated second main theorem
§7 The SMT of rank k
Chapter V. The defect relations
§1 A basic theorem on integration over PnC
§2 The fundamental inequality
§3 The fundamental inequality of arbitrary rank
§4 Proof of the defect relations
§5 Applications
References
Index of principal definitions

Citation preview

Annals of Mathematics Studies Number 64

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 A L L R IG H T S R E S E R V E D

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 Ahlfors1 original paper [ll; a subsequent poetic rendition of Ahlfors1 work by Hermann Weyl [7 1 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 inde­ pendent 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

yi

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

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 varia­ ble, but not much of any of these is actually needed.

It

should be pointed out that Chapter II is essentially Inde­ pendent 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 C,9 where V is an open Riemann surface and Pn€ * is the n-dimensional complex projective space. The central problem of the equidistribution theory of holomorphic curves, crudely stated, is the following: in general position, does them?

givenm

x(V)

hyperplanes of

intersect any one 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(D,

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

P^C.

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. P^(C

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

The celebrated theorem of Emile

V = (C,

then

x( (D)

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

vii

v iii

t h e e q u id is t r ib u t io n t h e o r y o f h o lo m o r p h ic c u r v e s

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 -+ P^C)

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,°°)

such that

(I)

set) = compact set) and side some compact set of is a compact subset of

V

V.

is proper (i.e.

t

(ii)

r

C

function which equals

Then

t ~1(compact

V[r] = (p: p e V, r(p) £ r}

for each

log r

r.

(Example:

If

V = £,

Pm 1

is locally

m . to J

neighborhood of of

x

at

m ..

p.j.*

we call

(m.,-1) j

inside 1

in a

*

the stationary index '

Let

J

nl(t) = z“=1 (“j - 1), and we define

n^(t)dt. Next, if we denote by o the Euler characteristic of V[tl, then we define

X(t)

N^(r) = J"

E(r) =

f x(t)dt. Finally, on J ro the nonnegative function h by:

V - V[r ], 0

we introduce

*

x oo = h dr A * dr, Then the SMT states that (0.7)

E(r) + Nx(r) - 2T(r) = ^

(log h)*dx o

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 ). The situation is now quite clear:

(0.5 ) and (0 ,7 ) contain

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

J (log h)*dx. dv[t]

/ x*u *dr avtt] a 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)

/ x*u *dx + f (log h)*dx < C log T(r). dV[r] a dVTr]

x iii

INTRODUCTION

A substitution into the left side the values of the line inte­ grals as given by (0 .5 ) and (0 *7 ) would then yield immediately (0.2).

The basic idea to obtain the inequality (0.8) is to

integrate the inequality (0 .6 ) over appropriately chosen function

p

P^C

on

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 u .

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 [8 ], 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 [8 ]), 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

some 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.

More importantly,

I feel that there are a few obvious open problems in this direct that are worth looking into.

For one thing, the Nevanlinna

X iv

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

theory on the parabolic surfaces with infinite Euler charac­ teristic 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

€

in the

last decade (see [b] 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 €

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 Weyl1s 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.

(Cn+\

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

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 Pldcker formulas. Finally, the whole of Chapter V owes such a great debt to Ahlfors’ fundamental paper [ll 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 —► P^C,

where

assumed to carry a harmonic exhaustion as before. n > 1

new phenomenon that appears when

is that

V

will be The first

x

now

induces a series of holomorphic mappings into various Grassmannlans.

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

fact from the differential geometry of curves in IRn . Let 7: IR —+3Rn

be a smooth curve, then the one-parameter family

of tangent lines 7

(the prime denotes differentiation) along

forms a surface, called the tangential developable of

Next,

7

which at

7.

induces a one-parameter family of planes along 7(t)

is spanned by the tangent

principal normal 7,

y1

y"(t).

7 1(t)

7

and the

These are the osculating planes of

and this set of osculating planes forms a three-dimensional

sub-variety of JRn . By considering the three-dimensional osculating spaces of 7 ”(t)

7

and the binormal

which at 7tn(t),

7(t)

is spanned by

7 ’(t),

we obtain a one-parameter

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

and so on.

x: V —*•P^C.

Now let us return to our holo­ In 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 e V

x(p)

for each

and

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

xvi

for each

k,

1 £ k £ n~l.

Grassmannian of

Thus if

G(n,k)

k~dimensional projective sub spaces of

then we can define a holomorphic map ^.x(p)

is just the

The set of such p

^x: v

PnC_,

&(n,k),

where

k-dimensional osculating space at

x(p).

k-dimensional osculating spaces

wanders through

V[tl

forms a

subvariety with boundary in Xk (t).

denotes the

^.x(p)

as

(k+1)-dimensional analytic which will be denoted by

Finally, we shall agree on the notational convention

Qx = x.

This finite series of mappings

called the associated curves of

0x*****n

are

x.

Now let A^ € G(n,k). The set of points of Pn ' A A A'

Again, we abbreviate (1.3)

to this form:

■'"—■"Tyf — } ) holds, there exist and

n(r,a)

v(V[r]).

Lemma 2 .5 .

- . 1 tt .n:

and we have written

= r.

we see that Hence,

f x*> _ ± f(r>r ) Z x *s d p V WOdVfr] a 2*Ur*°) -(r,r) = r,0 )x*ua dp> = ^ A r / ( r , 0 )x*ua *dT) =

^ (W

w n 4 [ r ] X*U a *dT)'

Since this is true for each component of

9V[r],

we are done. Q.E.D.

Thus if (fri ) and (^>) hold,

n(r*a) + P w

dVfr]X % a *dT) = V(r)

We can now carry out the afore-mentioned integration:

we integrate

38

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

both sides with respect to in which (firt.) and

r.

So let

) hold for all

[r^r^]

be an interval

r e [r^r ^ ] .

Then the

above leads to:

(2 .7 )

/ 2 n(t,a)dt + i JT1 9v[t]

f

2 = / 2 v(t)dt,

I lrl

rl

X^ where we have used the standard notation:

h(t) r-, = h(r d.) - h(rn -L)

To extend (2,17) to arbitrary intervals, we need a technical lemma: for a fixed function of

r

for all

a

is a continuous

r >_ r( x).

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~1 (a)

(rQ,rn ) c: (r(x),s)

and the critical points of

relatively compact set for which (Dt) and say,

a Then for each

t

t*1 ((rQ,rn )),

The points

are both finite in the so the points in

[r0 ,rn 1

) do not both hold are finite in number,

f V rl'- 9**rn~l*rn ^ x*u

(Cf. Def. 2.1).

We define

*dT T,i+ 1 = lim J I ri cT'r1+i'dJ'ri 5v[t]

(r±>ri+i)>

(2.17) holds and thereby giving

equations of the type (2.17).

We add these

n

n

equations and

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

39

using Lemma 2.6 we see that the sum of the middle terms of these equations telescopes and (2 .I7 ) becomes true for itself.

[rQ,rn ]

In other words:

f n n(t,a)dt + L f >Jro 2* dVIt]

f no v(t)dt

(2 .18 )

This forces the definitions: Definition 2.?.

T(r) =

v(t)dt

is called the order

o function of x

and

N(r,a) = /

n(t,a)dt

is called

the

ro counting function. The choice of

rQ

(rQ >_ t (t ))

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 more, namely, according to diagram (2.2), x = (xQ,x1 ) and

7r° x = x

holds in

the set of the common zeroes of let

dVf[t] - dV[t] Pi V 1.

xQ

V f, and

where x^.

dV[t] - dV»[t]

V - V ! is

For any

t >_ x(r),

is then a finite

set of points and so in particular is of measure zero in

dv[t].

Therefore, by (iii) of Theorem 2.1: I

dVTt]

x*u

a

*dx =

/

x*u

f

X*

dv'[t] dV'[t]

/

dVTt]

log

*dx =

/

dvqt]

x*(ir*u )*dx

lo g -1Z I »dT = |a Z !

lpl

|a x|

*dr.

a

f

dvqt]

lo g - l.p l - *dT |a x|

40

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

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

For every

r

r(t ):

r = T(r).

r

o

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

if

f(V[r]) H (a) = 0)»

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

t,

*dT

dV[t]

is the level

induces a positive measure on

$V[t].

is by definition of the way one orients the boundary of

V[t]).

(This

dV[t]

Furthermore, Schwarz1s inequality (1.10) implies

that

— Ijj--- >_ 1 |a^x]

that

log

(one must never forget that

>_ 0.

|a-1-] =1),

so

Hence,

for all

(2.19)

t

Another technical lemma we need at this point is this: for a fixed continuous function of

r

is a

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

(2.20)

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

Proof.

r

and

where the constant is a.

By Theorem 2.7,

N(r,a) = T(r) + ^

*dT o (2.19) o

) above and (£ )

df

is nowhere zero along

dV [r 1 ,

we know from (2 .1 2 ) that

(2 .21) where metric

k

is the geodesic curvature form of x*F.

(F

is the

F-S

metric on

dv[r]

S.)

the last line integral, we introduce a function

in the

To transform h

on

V - V[r( t )!

42

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

(2 .2 2 ) where on

x*o> = h dT A #dr is of course the volume form of the

cd

S.

h

x

h

in

is. a non negative function, for the following

is holomorphic, so

orientation on dr A * dx

%

F

(which is discrete), but the important thing to

note is that

log h

metric

is not defined at the critical points of

V - V[r(*r) ]

reason:

F-S

V,

x*cd

is coherent with the

while a simple computation also gives that

is coherent with the orientation on is a well-defined function on

Lemma 2.9*

V.

Consequently,

V - V[ t (t )].

Under assumption (^g ) and ( tl ), for every

r >_ t (t ):

Proof. is a

C°°

for every

Because of the presence of ( &) and ( C ),

log h

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

r

r( t ).

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 ) + nx (r) - 2v(r) =

(log h)*dT).

The analogue of Lemma 2.6 states that is a continuous function

43

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

of

r

for all

r >. r(t).

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

So using this lemma, an integration leads to:

Jl ox(t)dt +JIon]_(t)dt

- 2T(r) = L

J

0 v[tJ

r (log h)»dT r

o

Introducing the following notation:

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

For

x: V -► S

E(r) + tijTr) - 2T(r) =

§5 .

'

and

r >_ r(r), r

nnak

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 Ahlfors.

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 n(a) Then

V,

C°°

D

Suppose

map from a

into another oriented $

is an inte-

and suppose for each

denotes the algebraic number of preimages of

a

a e V, in

D.

44

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Remark. The meaning of algebraic number of preimages is as follows. n(a)

n(a) = 0

if

is defined only when

of points disjoint from (ii)

df(a .)

the

p

df

of the a.1s.

dD,

f“1(a)

say,

then

is a finite number

(a^,...,ap+q3 j.

and

In the event that D

and

ajfs and reverse them at the remaining definition,

seen from the proof below that almost all points of

V.

n(D,a)

n(a) = p -

n(a)

n(a)

M

q

of

q.It will be

is thus defined for

For a holomorphic

surfaces, this definition of the above

a e Im f,

preserve the orientations of

Then by

J

(i)

If

is nonsingular for each

J

both hold, let at

a | Ini f.

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. Let image under

f

f

df

is singular).

is closed in

a compact set.

D,

So

Vf

well-known that

in

D

(i.e.,

ofcritical points

D is compact it is thus

is

Of course

f(dD)

an open submanifold of V. ^

is also

Furthermore,

is a set of measure zero and It is

f(dD)

is also a set of measure zero in V. -1 V - V ’ has zero measure. If v £ V f, then f~ (v)

is closed, and discrete (because df and hence finite (because v € V f,

n(v)

of points in in

f

being the image of this compact set is

by Sard’s theorem,

Hence

is the

Q

The set

and because

itself compact and hence closed. closed.

where

of the critical points of

points at which of

V f = V - )£ - f(dD),

is defined. f~1(v),

V 1 onto which

f

D

is compact).

Let

then each maps

is nonsingular on

n(v)

n(v)

f~^(v))

Thus for every

be the total number

v e V*

has a neighborhood

open sets diffeomorphically.

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

Let

CVj}

be a locally finite covering of

sets and let (Vj).

Then

C J

r

dt ro

(Lemma 2 .1 2 )

J

(x p)(x m)

V[t]-V[rQ]

(x p

is positive)

50

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

J

J

dt

o

(x*p)h dTA *d T

(2.22)

V[tl-V[r0] rt

J

f dt f ds (x*p)h *dx Jro Jro dVfs] where the last step may he proved by invoking the special coordinate function

f_

(2.24)

dt J

ro

ro

a = t + •PT p.

Hence,

ds( f (x*p)h #dr) dvts]


0

and so

'i 3

p —

near

a^.

The only places that may {a^,...,a^3. Now if

Let us

i ^ j,

is continuous near

a ..

|a£z |

Thus we

3 17;I 1 1

need only concentrate our attention on the factor Choose O.N, basis by

eQ,

then

a^

p

e , en in C so that a. o jj is represented by e^. Let

the coordinate function given by Clearly

£

is an

£: U q -*■ (D be

X

£( ^z0eo + zlel^) ~

a^-centered coordinate system.

(_[Zj_)2A = (zozo ^ Z1Z1 )X = (1 + IajZ I

r is represented

fi zo

zlzl

' c

Since

-2

>\

we see that with respect to the coordinate function (--1-? I— )2*

assumes the form:

(1 + |C|~2) \

£,

It is consequently

i * jz i equal to

( | £ ! 2 + 1 )^|£|~2"^ = continuous function • |C|~2\ Since that

0 < \ < 1, |e|“2A

polar coordinates in the plane clearly says

is integrable near the origin, i.e.,

(JiLL)2* Ia jZ 1

52

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

Is integrable near integrable on

S»

ay

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:

log .1-*X L *dr 4* const.

= 2A S? 1

1=1 dVTs] Combining this with (2.25) and letting

| ^ f 1,

we obtain:

av[sj £ T(r) + const. Substituting into this Inequality the expression of the line integrals in the FMT and SMT, we have clearly proven the fol­ lowing:

if we define

_ 0 ,

index of

x

at

p) >_ Sk=l^n ^r,ak^ ”

so that

N-^r) >_ S ^ 1 (N(r,a^) - F(r,a^.)),

n^(r) = 2 pcV[r] >. 0.

(stationary (stationary Hence,

which implies that

Zk (T(r)-N(r,ak )) + N ^r) > Zfc(T( r) -1T( r, a ^ ). Combining this with (2 *26 ), we have the final result: Theorem 2.15.

If

cp(r)

denotes the quantity:

1 (T(r) - TT(r,ak )) - 2T(r) + E(r)} + const. then f o

dt f r e ^ s)ds < T(r) + const., o

where the constants are independent of

r and

{a^,..,,a^).

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

T(r).

1,

real number

holds on

_ 1 ,

with

^ d log x < °°.

then for any

[r0,°°) - I, (I

whereI

depends on

k. )

Proof. Let I c [rQ, 1 ft > arbitrary positive number. v
rQ ==> x(V[r])

(equal to x(v ))•

if V

reduces to

Is €

or

Therefore,

€ - {0},

^L=l ® (a&) £ 2*

zation of Picard1s Theorem.

is a fixed

when x(v ) >. 0,

X 0

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

infinite harmonic exhaustion; the natural injection of

S f -► S

certainly cannot obey any defect relation of the type 2^. 5(a^,) £ 2.

So we should seek a condition on

insure the vanishing of transcendental iff that if X = 0. all

x

lim

x*

Here is one. = 0.

r >. rQ *

rQ

We call

itself to x: V

S

Then one can easily prove x(v)

is transcendental and

In fact let

x

finite, then

be so large that

x(V[rl) ~ X(V)

Tor

Then,

x = lim sup

fT

= lim sup

= lim sup t^TX* X(V)(r-rQ )

x(t)dt

=0

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

For there

Is this result: Lemma 2.18. surface then

M

If

V

is obtained from a compact Riemann

by deleting a finite number of points

x: V —► S

is transcendental Iff

to a holomorphic mapping

x f: M

S.

x

Ca^,.*.,am },

is not extendable

58

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC 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 f: M if

x: V —► S

S.

V admits an infinite is transcendental, then

r ex(V - V[r])

is dense in

Suppose false, then there exists an borhood r € 1.

U

of

a

such that

Uf1x(V

t >_ rQ,

a € S

- V[rl) =0

There is no harm in letting

for all

In fact, we

r = r.

So

S.

and a neigh­ for some n(t,a) = n(rQ,a)

Hence,

N(r,a) = J r n(t,a)dt = n(ro,a)(r-rQ). o Next, since to

9v[t1

t — >~ rO = > x( dV[t1) Pi U = 0 9 for all such

(See Theorem 2.1)

t

x*u 81 restricted

is bounded above by a constant

K.

By (2.18),

T(r) = N(r,a) + i2w 9vj |tl x*ua*dx

1 1,(r' a) + s 7 a X / ' 1*’ " 1 £ n (rL ,a)(r-r + J2t*"ro^) + t .L by the corollary to Lemma 2.4. > — 7— 0 — ~-n(r ,a)r > 0.

So clearly,

lim sup

This contradicts transcendency.

We now prove the converse:

if

x

is not transcendental,

then it is extendable to an

x f: M -+ S.

= P > 0.

Then

lim inf —

= ^ < °°.

for all

a € S

((2.20)),

So let

Since

lim inf ^ (,,*>ft) < ^

/ s W N(r»a ) n(r,a) = -g-------- , 3r r

lim sup

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

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

and

n( r,a)

has a limit as

increasing, so

lim

l fH&pltalfs rule.

3

ft?

Hence

is independent of

for

all a

e S

r -*■

1 The proof of this theorem is analogous to that of Lemma 2.16. As before, the main thing we should note is that

I

doesnot

comprise a full neighborhood of s in [0 ,s) so that 1 k f 1 (r) < — — f (r) is true for numbers arbitrarily close to s-r ^ By applying this lemma successively to J r e^^s^ds and r t 0 f dt f r e ^ s^ds, we obtain: 1!t(r) £ k 2 log(T(r) + const.) + (k+l)log(~r) where

”||n

now means

"outside of

I".

This together with

Theorem 2.15 imply that

I’V

F(r,ak )

------

2 < 2 +

+ T'fr)

^°S(T(r) + const.)

+ (k-fl)log

5 2 +

+ const. 1

+ €(r)^

So we introduce again: 5(a) = lim inf(1 - M ) r+s -nr X = lim sup T+S € = lin^sup

where

c^, c^, c^

E( r)

log(T(r) + cg) + 2 log JL- + c^l,

are some positive constants.

The above

s.

NEVANLINNA THEORY OF MEROMORPHIC FUNCTIONS

61

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

V

If

x: V

S

is holo-

admits a finite harmonic exhaustion, then for (a^,...#0^}

every finite set of distinct points

of

S:

Sk=l C (ak ) £ 2 + X + GFurthermore

€ = 0

if log -iLlim sup — —- = 0. r*s

(2 •28 )

If (2 .28) holds and also.

So in this case,

x(v ) > -°°> 5(a^) _< 2

again holds.

clear what the exact meaning of (2 .28 ) is. one can show that if

x

X - 0

it is obvious that

For

It is not

V = 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. can^ show that If (2 .28 ) does hold, then in

S

in a very strong sense.

In general, one

x(V - V[r])

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 classical theorem due to Nevanlinna

is that such a meromorphic function is the quotient of two bounded holomorphic functions.

The reader should consult

Hayman [^] for this and related matters.

CHAPTER III Elementary properties of holoinorphic curves 51. an

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

x: V -► P^C,

where

x

an open Riemann surface.

is a holomorphic map and

(XQ,...,xn )

function this.

x

Let

{x ,...,xn J

(n+1)

x: V *-+ C

x = (xQ,...,xn )

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:

where by definition, x

In a moment, we will extend

to be a holomorphic map on all of

holomorphic

x: V -+ P^C

commutative, we say V

x = ir® x.

x

and

V.

Whenever we have a

x: V —►

induces

x.

such that (3.1) is The extension of

is achieved in the following manner.

then

xQ(p) = ... = *n (p) = 0.

Since

we may choose a coordinate function in a neighborhood

U

of

p

Let V - V*

z

such that

x

to

p e V - V T, is discrete,

centered at U - (pi C V L

p

defined In

U,

yA (0) / 0.

We

we have the following factorizations: 5A

XA(Z) = z yA(z)’ where

5A >_ 1,

yA

A =

is holomorphic in 62

U

and

ELEMENTARY PROPERTIES OF HOLOMORPHIC CURVES

may assume U

U

is so small that

for every

A.

y^

63

is zero-free in all of

Among the integers

5 ,...,

there is a

smallest one; there is no harm in assuming that it is 8 . * n+1 , * This gives rise to a map x : U -* € - [0J such that

x*(z)

We now claim that to all

of U.

/

\

, •. . , Z

= ( y D( z ) > z

*

ir * x : U -► extends

We need only prove

tt ®

*

and

(ii)

Let

UQ = {[zQ, .. .,zn l: z Q £ 0}

z € U - {p).

(i) rr ©

We willdo both and let

P^€ x# | U

fp]

simultaneously.

UQ -► (Dn

be the

Z7 /

Z

.

£( [z Q, ..., zR ]) = ( /zQ, ..., !%0 ).

To prove (i), it suffices to prove

for all

[p]

two things:

xis holomorphic.

coordinate function such that

f %

x: U -

iro x )(z) = (£ o x) (z),

But

= ( C 0 x ) ( z )* £ © tt 0 x

To prove (ii), we must show that in

U.

But this is obvious because in the above expression,

yo,..., y zero in Since

p

extended unique.

is holomorphic

are all holomorphic functions and U.

yQ

This proves that we have extended

is an arbitrary point of x: V* -+ Pn

so the two ordered sequences of holomorphic functions on h, h - , h. , h U - fp}# (f •••sfn ) ^h"~* — 'E~'~E— ~9 ****Tp~)9 are o o o o o in fact equal. As the latter are in fact meromorphic functions on all of of

U,

they are therefore the unique extension to

(fqj•9**fn )

and we see that n

on

U - (p}. Do this for every

£ o x = (f^,...,fn )

meromorphic functions

i.e.,

U

p € V - V*

is extended uniquely to

(f^Tx = (f^, ... *? n )

v~*(C U(°°} x ••• x €U{°°]

on

V,

is meromorphic.

By the theorem of Behnke-Stein, there exist holomorphic a,/ functions ...*an*Pn 5 such that f^ = ( /p ^) on V. Now define:

’*^i-l^i^i+l ***^n# and let

x: V

-►

(Cn+1

^ ~ 1,.• •, n,

he the holomorphic

The claim now is that x

induces

map

x

=

(x q ,

...,xn ).

x.

It is perfectly simple to prove this.

Let

Vf

he the

complement of the common zeroes of xQ,..,,xn in V and recall * „1, , that V is the complement of the discrete set x (Pnfn (p))

= (p^Tp)’" ' ^ C p T ^ = (f1 (p)^..-,fn (p))

because

p e V*

= (? * x )(p ) by definition of ir o x(p) = x(p)

f^,.. • for all

Since

t

is a homeomorphism,

p€ V* f| V 1.

Q.E.D.

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

x: V

Pn€

is induced by a certain

will eventually need both

x

and

we should keep (3.1) in mind. uniquely determined by x: V -►

O

A ... A €

X^ = 0,

z

so

By (1.8),

n

n —

€n+1

is the canonical basis of

Since

in

As above, let us operate in

a coordinate neighborhood x^)

vectors

there is an integer

and £,

0 '

= 0

det eo>' on

€n+1

U.

Let

x =

^kek 9

*(k 3^

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

are linearly dependent in

U.

A fortiori,

74

THE EQUIDISTRIBUTION TH EORY OF HOLOMORPHIC CURVES

y , • «*•>yn V*

Thus

are linearly dependent in x(V)

and hence in all of

lies in a proper subspace of

dicting the nondegeneracy of

Lemma 3 . 6 .

If

x: V

x.

Pn ix>—

that

G(n,k)

*n-lx *

Reca11 from Chapter I,

has a naturally given Kahler metric, which is

the restriction of the F-S metric on (4.1)

then

F = -- —

feTr*F = F,

fibration and function on F

If on

(D^^).

- ),

where

^tt: C ^ k ) -►

is the usual

A = (...,X. . ,...) is the coordinate o *’,;Lk 0 giij B We may clearly assume that

is very small so that by

Lemma 5.2, there is a reduced representation of i.e., there

is

that

vir°y = f. k so that B = e A O

aholomorphic map

y: Uj-*■

Now choose O.N. bases (e •*• A e, . KL

Write y = yn e x

O

f

in

- {o]

Uj, such

,.. .,e ) in Cn+'1' o' * n A ••• A e + •*•, K,

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

86

then for every

p e U ., J 3

= y, (p ).

f*uB = y V * uB = y* l0® = log

By (iii) of Theorem 4.1

-L

= log

= log |y| - log Iy1 1,

so that,

’f%B ’ * ^

4

“ Since

W 1 dClog *u3

Iyl l -

w «

d° log ly '

1 ^

|y|

is never zero, dclog |y| is clearly C°° in U ., f c lim J d log |y| = 0 . Furthermore, y(a.) is a ree*>0 dUj J

so that

presentative of the projective sub space |f(a^),B| = |y(aj),B| = |yi(a^)|.

f(aj)

hence

So the order of zero of

is equal to the order of zero of

y^

at

a^.

To

prove (4.6), it suffices to prove: (4.7)

the order of zero of = llm e-O W^

This is let there where

y^a^.)

J d°log IylI 9V. x * 3

essentially the argument principle. Ingreater detail,

zbe a local coordinate function centered

1. .n h

ihteger

such th„t

is holomorphic and

so small that zero of

»(J)

h

y1(aj)

h(0)

m(j).

0. Uj.

Now

+ d° log |h| = m(j)d0 + dc log |h|,

a..

Then

yl(s) =

is nowhere zero in Is just

at

We may assume

Uj is

So the order of

dclog|y^( = m(j)dc log |z|

where

z = |z|e^“^0.

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

(See (2.7)). because

Remembering that

h

log |h|

is

87

C°°

in

U. J

is nowhere zero, we have f dClog|h|

= -gp m (3) 1 2ir + 0 = m( j). This proves (4.7) and there with the theorem.

§2.

Now we assume

V

is open and has a harmonic exhaustion

function (Definition 2.1) >_ r(r)}.

T ( p )

T(p) £ r}, in t

,

t

which is harmonicon [p:

We recall this notation:

dV[r] = Cp: r(p)

V - V[r(T)],

= r),

In

lated.

V - V[r(x) ],

Vfr] ={p: p e V,

We shallwork exclusively

r

are assumed greater than

the critical points of t

Also recall that if

p € V - V[t(t)] and

then in a sufficiently small neighborhood of p, holomorphic function nate function.

cx = t + J~-ip

fixed

f

B € G(n,k),

dr(p) ^ 0,

there is a

is holomorphic. f(V)

We have

does not lie in

G(n,k)

and furthermore

(eo f(dvtri) n sB = 0, is not a critical value of

then Theorem 4.2 implies that

f: V

If we assume that for a

that

r

if

iso­

(Lemma 2.4 and the remarks after Definition 2.2)

and

(~£ )

are

which serves as a coordi­

Now return to our previous situation. cz

p € V,

i.e., only in the domain ofharmonicity of

so that all parameter values

r(ar).

Q.E.D.

t

,

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

88

J

n(r,B) +

dV[r] where we have written v(V[r])

and

function

dcu^.

~

a = x +

Lemma 4.3*

n(r,B)

Ip

f**R = v(r) B for

n(V[r],B),

v(r)

The use of the special coordinate

leads to:

Under assumptions (61) and (£), f

= iL{i-

f*\

cm r ]

B

< ^ 2lr

/

f*ur)*dT),

e v fr]

B

The proof is identical with that of Lemma 2.5. have that if (©() and (^ ) hold for

n0 E =

f f*uB*dT avTr] B

(by (0)).

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

If

Xj[

r,

it suffices to show that

r.

To

r^f r

implies

J f*Ur**dT -+ / f*u *dx. aVTr± 3 B dvTr] B denotes the characteristic function of

V[r^],

Lebesgue* s bounded convergence theorem implies that

then

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

90

f U-ofcdT =

J

B

/ d(f U-o*dx)

vr^]

B

‘ vr/]*ld(f‘UB * « > —

J d(f*u„*dT) =

V[r]

B

This proves (4,9) and hence the lemma. (cx) and (f3).

The singularities of

at the discrete set two-form at a

f~^(2B ),

p € V[r]

suchthat

coordinate neighborhood

at

a reduced representative of tt* k.

y = f.

so that Thus

and

f*u *dr.

B

It remains to prove

d(f u0*dT)

are located

we need onlyexamine this f(p) € 2^.

p and

let

f

U.

at

J

dV[r]

y: U

Let

U he

-►

„ (o)

(Lemma 3.2).

So

be

Choose O.N. basis

{e .... ,e 1 in (Dn be chosen o n andwrite y «= y1eQ A •• • A efe+

B = eQ A ... A

= y1

and

f■*„ uB = (y • = log

)uB = v* y w log _|AJ___=-log ly Ty-| Iy| - log |y1 |.

Thus

d(f*uB*dr) =

d(log |y |* dT) - d(log |y^ |* dT)

But

|y | is never

zero, so

d(log |y |* dT) is

in

C°°in

U. U.

Therefore it suffices to show that / d(log |y. |* dT) is JU finite. Let z be a coordinate function centered at p that

a

y ^ z ) = zmh(z),

where

We may as well assume that

h h

is holomorphic and is never zero in

U.

such

h(0) / 0. Then

d(log |y1 |*dT) = d((m log |z| + log |h|)*dT)

= md(log |z|*dT) + C°° form =

dI zI A*dT + m log |z|d*dT + C00 form.

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

In

termsof polar

integrable in

U.

91

coordinates, both forms are obviously This proves (a).

notation and have that in

For (p), we

use the same

U:

f*u0*dT = C°° form - (log jy^l^dr) = C°° form - m log |z|*dT. If

5W€ = [p: |z(p)| = e},

then

I f * u * dT = I C°° form - m log c J *dx# 9¥€ B 5¥€ 3w € There is no question that as

€ -+ 0,

the first integral on

the right approaches zero, so it remains to prove that lim log € j *dx = 0. €-K) aw

We use polar coordinates

log r +

€

in

U

minus a radical slit, where

*dT = * ( 5 7 a rd log r +

z = re"^~^.

d9) = r

d0 -

Then

d log r.

Consequently,

lim log € J *dT = lim log € • J r ^ d0 e-o gw€ €-K> aw€ ° r = lim € log € J dT d9 ~ 0 €~*o awe5F because

is bounded in 0T

Now suppose r € (r^>rg) (0 O or (^>).

[r^r^l

U,

and lim € log 6 = 0 . €->0

Q.E.D.

is an interval such that all

obey (&( ) and (^>) while

r^

and

r^

may violate

We define:

/ f*u *dT = lim -r '’ * rl dtr0,cj.r,

/ f*u. dV[t]

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

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

92

[r-pTg] If

where the end-points

[r^,r^]

r^,

may violate ($•{) or

is another interval such that each interior

satisfies (£K) and

) while

r^, r^

)« r

may not, then Lemma 4,4

implies that [

i -

f*ir*d T

27r a v it]

^

j

2

+

' ri

i

/

^

^ $ v [t]

Now let

[rQ,rn ]

The points in

J "

f*u *dr

5

dvrt]

B

r2

f’ug^dxl^.

^

l rl

he an arbitrary sub-interval of

[rQ,rR ]

in number, say,

i

at which (0{ ) or

( . . . ,rn3 .

(r(t),s)e

) fails is finite

By the preceding discussion,

we have + ^ f f-V d / “ dlr dv[t] * |ri

ri for

i = 0,,*,,n-l,

Add these

n

terms on the left telescope into

- f ™ i

v(t)dt

equations and the second i

f

^ dVTt]

f*u *drI J 1.

B

lro

Thus

we have: III

»(».*> « + £

- I I I

Tdt-

We are therefore led to the definition of the order function: T(r) = f v(t)dt = ~ f dt J f*05 V ' Jro K 1 r J r 0 V[t] and the counting function: N(r,B) = f

n(t,B)dt.

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

The number

rQ

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

Let

r(x)9

but

We have thus proved f : V -+ G(n,k) (Z

k = 0,...,n-l, and 2^.

93

Suppose

V

be

f(V)

does not lie

admits a harmonic exhaus

r >_ r(r):

"(r'B) + 5F

■ T(r>'

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

Let

BJ' be the polar

space of

(n-k-1)-dimensional projective subspace of

the remarks after (4.4), if each meets

B-1",

if

subset

the

then

What does it

then

N(r,B)

N(r,B) = 0. U

f(p)

pcvfr] is small.

of

But

f(p)

(p c V[r])

B,

so

P^C.

Bx

By

never

The same reasoning shows that P € n T(r)

meets

B"1"

very rarely,

is independent of

B,

so

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

Thus the compensating term compen­

sates for the deficiency in the intersection of U r

B1' with

f(p).

p€V[rl Now observe that for any (|B| = 1

because

A €

|A,B| _< ]A ||B| = |A |

B e G(n,k) (Z P ^^^ ^C,

and the latter is

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

>_ 0. *dr

By Theorem 4.1(ili),

uB

0.

induces a positive measure on

is coherent with the orientation of

dvfr]),

So

Consequently, dV[r]

(since it

we have

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

94

J

(4.10)

dVfr]

f*uT,*dT > 0 B

for all

r >

t

(

t

) .

There Is one more fact we need before we can derive the basic inequality.

This fact is

f

Lemma 4.6.

dV[r] function of

f*up*dx B

for a fixed

r

is a continuous

B.

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

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

r

and

where the constant is B.

For, N(r,B) = T(r) + i ^

/ f*V 0,

be a reduced repre­

^/d(log |y|*dr) - ^jd(log |y,B|*d-r).

j Since

y

So

f*uB - log Ty?ir

J

j

log |y|

j is

C°°

and independent of

B.

So

the first integral of the right side may be left out of consi­ deration.

Therefore what we must prove is the following:

Bj

be a sequence of projective

to

B

and

in

tatives of

Pn€

converging Bj

so that the coefficients of the represen­ Bj

converge individually to those of

^/d(log (y,B.j|*dT) — t)

J

B),

then

d(log |y,B|*dx). J

Now recall that

y(a^) € 2^,

so the holomorphic function

has a zero at

assume in

spaces in

(in the sense that we can pick representatives of

B

k

let

Wj.

Wj

a.. For convenience, we shall also u is so small that a^ is the only zero of

Furthermore, it is obvious that

uniformly to

on

Wj.

converges

To prove the above (and hence

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

96

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

fg^}

be a sequence of holomorphic functions defined

on the closed unit disc g,

and let

cp be a

A

C°°

and converge uniformly on one-form on

and vanishes nowhere else. of radius

~

Then if

A.

A*

Assume that

A

to g(0) = 0

is the closed disc

about the origin, / d(log (gJcp) -*/ d(log |g|-

We now summarize (4.l6)-(4.l8) and (4.24).

Recall first

the various definitions.

rr

J Tonk (t,Ak )dt

Nk(r,AK ) = where

n, (t,Ak ) = sum of the orders of zeroes of = * v, (t) = J .x a>s K Vft]

where

w fl o and

X■hi r m. (r,Ak ) = ~ f log — L-2R_*dT k 2lr dV[t] IXk,A |

where

a

is a holomorphic function having

Theorem 4.8 (FMT of rank k).

Let

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

V

and for

admit a harmonic exhaustion. r >_ r(r):

x: V

as its real part.

r

Pn\(r>Ak) = Tk(r)

is the fixed function on

V

having the property of

Theorem 4.7, then

T* (r) = Furthermore, let

*

U

coordinate function

I

a l ] 108 ,X^ * aT

- \(r). o

be a coordinate neighborhood in z,

V

with

then except on a discrete point set:

kx*a> = i- ddc log |X^| k-1 |2|xk+1 ,2 jzi ix“ - r i x jp dz Adz

ixzl

X "”1 = 1 z

where

Remarks.

by definition, (1)

In the same setting, we may restate (4.11)

as follows: (4.25)

Jr N^(r,A ) < T^(r) + c^,

where

independent of (2)

The case

k = 0

cfc is a constant k and A .

r

is notable for its simplicity, so

we state the above for this case separately.

- Br

w ( t ) lo e T 0 r

T»(r) ■Wsv-L10g x*co =

75 -

*dT r r

-

o

ddc log |x]

J ~ T Ix A x h ) I2

^

lil’a'

I

*

F

_

For any

dzAdz .

N

(r ) ov 1

a £ P aL

- v r >-

Now introduce n^(t, Ak ) = sum of the orders of zeroes of the function |Xk J Ah | X— ---

—

and write

V[t]

N. (r,Ab) = f n. (t,Ab )dt. k J T0 K

shows that this function.

in

1^1

N^(r, A )

When

h = k,

(4.25)

coincides with the usual counting

We further define:

Tk(r"4h) - 4

5T( t]l0S |x* J Ah|*di

0 - v

p.Ah) - V

1-)-

Then upon adding the various terms, we find (4.30)

mk(r,Ah ) + Tk (r,Ah ) + Nk(r,Ah ) = Tk(r)

Since the FMT says that for

h = k,

m^.(r,Ak ) + N^(r,Ak ) = T^.( r ),

(4.3 0 ) leads to: (4.31)

\ ( r , A k ) = 0.

We now wish to give a geometric interpretation of the formal object

T, (r,Ak ). 'k(

Let us define

n^(t,A^) = sum of the orders of zeroes of the function |Xk j A h |

in

V[t],

then as a matter of comparing definitions, we see that 4- n, (t) = n, (t,Ah ).

Hence

N^( t,Ah ) + N, (t) = N,( t,Ah ),

n^(t,A*1) where

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

111

rT

N, (t,Ah ) = J

n, (t, Ak )dt.

So we may rewrite

o

T (r, A*1) K

as:

T (r,Ah ) = i f log |X* J Ah |*dx rr " Nk(r'A s »h >• K dV[t] T o

(4.32)

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

k < h.

curve of

x

E to

In Chapter III §1, we introduced the projection

into

A*1, which was denoted by

^x: V

A*1.

Let

be the A,

(h+1)-dimensional vector subspace of €n+^ correspondir k k+l then the mapping : V -+ A E defined in Lemma 3 .7

induces the associated curve of -J A*11,

IaX*y I = I

rank

k

of

pX. By that lemma,

consequently, the interpretation of the

order function as given in (4.24) shows that

T^,(r,Ak )

in this

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

of

^x.

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,

Xk J

A*1: V

Xk J A*1; V —► A k “k€n+1

1 ) i®-

G(n,h-k-l) c :

clearly induces

Xk J

A*1,

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

T^(r,Ak )

Xk J

Case 2:

the inter­

k = 0)

again

is the order function of the holomorphic

A11.

h < k.

We introduced in §5 of Chapter III the con­

tracted curve of the first kind, Obviously,

C= Xk j A h .

Since

Since

Xk _J Ah : V

Xk J Ah : V

|Ah J X k | = |Xk j A h |,

G(n,k-h-l)

Ak~hCn+1

induces

the interpretation

112

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

in (4.24) of the order function for

k = 0

shows that T (r,A^) K. k h is the order function of the holomorphic curve X J A . In any case, we have shown that if

h £ k,

then

T^(r , A*1)

is actually the order function of some holomorphic curve. Recall that if

f : V -*> P C

function of

was originally defined to he

f

so the order function

is a holomorphic curve, the order

r

r

*

I dt J f a>, ro vrt] always positive and strictly increasing.

is

Hence (4.33)

Tk(r,Ah ) > 0

if

k / h.

We now summarize the foregoing in the next theorem. Theorem 4.9.

Assumptions as in Theorem 4.8, let

a unit decomposable define for

K -

(h+1)-vector,

h = -1,0,...,n,

Ah

be

and

0,. .. ,n-. 1 Sr

f

•

9VTt]

nr-hirr*dT r lX a'

r o

X_ f Tk(r^ h) = h 2ir ,.i. ,loe !xy j AV dT - \(r»Ah) avtt] where

=i

f N^(r, AlA) ^ J ^

nv (t,A“)dt

and by definition,

n^.( t, A^1) = sum of the orders of, zeroes of

|

J A^|

h Similarly, let

W,( r, A )

be defined with the zeroes of

in

V[t].

1 J Ah | — 2-— — — |x*|

Then the following identity holds: mk (r,Ah ) + Tk(r,Ah ) + Nk(r,Ah ) = Tfe(r).

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

k Furthermore,T^( r,A ) = 0, (4.34) T^( r, A*1)

and if

h

113

> k,

= the order function of the associated holo­ morphic curve of curve of

x

into

rank k

of the projection

A*\

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

X^1_J A*1.

h < k,

(4.35)

T^.(r,A^) = the order function of the contracted curve of the first kind

X^ _l A^1.

In connection with the above theorem, there are two basic inequalities which we shall need. Lemma A. 10.

We now prove th em here.

There exist constants

a^.,

k = 0,.. ., n-1,

b^.,

k = 0,.. .,n-l,

such that

holds for all

r

Lemma 4.11.

and

A*1.

There exist constants

such that

\ ( r>kh) < Tk(r) + \ holds for all

Ah

and

r.

Proof of Lemma 4.10.

By (4.31) and (4.33)

Also, by its very definition,

N^.(r,A^) > 0 .

T^(r,A^) >_ 0.

We therefore

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

114

deduce from (4.30) that mk O , A h ) < Tk(r) which by (4.29) is equivalent to i

f

lx k l

r

i

lx k l

_±_ J log — Xra X, *dr < T, (r) + -±/ log — - -«• - *dx 2lr 9V[r] ]Xk J A | ^ dVrT0] |Xk J A | The right-side integral can be proved to be a continuous function of

A^1 in exactly the same manner that Lemma 4.6 was proved.

So we may let over

a^. be the maximum of this function as

G(n,h).

A*1 varies

Q.E.D.

Proof of Lemma 4.11.

We begin by recalling (4.24) and

(4.32):

r r

I

|X^ J A | < | | jA | = |X^ | A*1

log IX^ J A ^ Udr
cd

, , | a logd.pMSb) .. R $ log 1z|

j. 1 9 log t|AFt

= 5 d0 + 0°° form because

T) = log |z|

and because

bAF^

is

C°°

and zero-free.

Hence:

* O'TT

€“K) dlT

* ~ 1) = 6 - 1

I j

As remarked above, the stationary index of so this proves (4.38).

f

at

aj

is

(6-1),

Q.E.D.

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

We would like to elaborate

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

120

on Theorems 4.2 and 4*12 for the case of a compact Riemann surface

D

mapping

x: M

(i)

x

without boundary* P^€

So let us fix a holomorphic

with the explicit assumptions that

is nondegenerate (i.e.,

containing

x(M)

is

P^C

the only subspace of

itself) and

Riemann surface without boundary. the associated curve of c: P^^)

.

induces

x.

rank k

(ii)

M

Pn €

is a compact

We would like to define of

x,

i.e.,

^.x: M

G(n,k)

The definition given in §3 of Chapter III does * * M n+1 which not apply because it made use of a map x: € Since

x

is compact, no such

must modify the definition somewhat. k

which defines

x

exists, so we

In the formula (3.2)

in a coordinate neighborhood

U,

throughout by an arbitrary reduced representation « {o} by

k

X^.

of

x

in

we replace *

x : U

U.

The resulting function we denote k To show that this collection of X^ does indeed piece

together to define a global mapping prove the analogue of (3.3)*

^x: M -► G(n,k), we must k suppose X^ is defined with

the help of a coordinate function

z

and a reduced representation

#-

x w

and

Xy.

is defined with the help of a coordinate function

and a reduced representation

*

y ,

then on

prove the existence of holomorphic functions that

h^Xy = h^X^.

This is not difficult:

U f }V, h^, h^

we must such

by the remark at

the end of Chapter III §1, there exist holomorphic functions sl# g2

on

u ^ v

tation shows that

such that

= g^y*, so a simple compuk(k+1) g^xjj = g^(^~) ^ X^. This shows that each

associated holomorphic curve of

rank k,

^.x: M -^G(n,k) CP ^i(k) -1^*

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

is well-defined.

The analogue of Lemma 3.6 can he proved in

a similar manner, so 2b

of

121

^x(M)

never lies in any polar divisor

G(n,k).

Now, according to Theorem 4.2, (b.hO)

nk(M,B) = vk(M)

for every

B e G(n,k),

------

= the number of zeroes of In particular, for any

1 f * 7T K

v.(M) = — J ,x o>,

where

K

in

M,

and

n. (M,B) K

counting multiplicity.

A, B e G(n,k),

nk(M,A) = nk (M,B). We proceed n, (M,B)

to give several interpretations of vk(M)

on the

kl

basis of (4.40).

First of all, wehave remarked

previously (above Theorem 4.2) that if line in

Pi(k)-l />/,„% ' 1

homology class Rewriting

— on

v^.(M)

then

J k

nothing mental now

but

that

two-cycle

nB

be

the

generates

the

2( i ( k ) - 1 )

-

Poincare v^(M) and

is Eg.

duality

(~

^

integer

cjd) ,

which,

and

when

anycomplex

This shows that the co~

is

group hence

the

see

that

multiplied the

2

H (

^G;£).

v^.(M)

H *( P i ( k )

the

preceding

the

Since

is arbitrary,

with

cycle

fA : = 0}

therefore B

we

gives

hyperplane

and

is

is

^

of

homology

2,

/ TT = 1. p-/^

P^G

is in fact the generator of

as

and

^.x(M).

dual

number see

Let

in

nB

of

interpretation

we

funda­

1 ® * ^ ^^ ^ m e n s ^ o n

Poincar6

intersection

the

of the

that:

~

o>.

of

By

v^(M),

cycles^x(M)

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

122

(4.41)

v^(M) =

the order of the algebraic curve P$(k) 1®

Now

^.x(M) c: G(n,k)

equate ^x(M)

v^,(M) and

out that

and

in

sense of algebraic geometry.

nB H G(n,k) =

So we can also

with the intersection number of the cycles

2^ SB

in

G(n,k).

In §2 of Chapter I, we pointed

is the generator of

integral homology group of dual of

^ x ( i n

Zg,

c^

2f(n-k)(k+1)~1)-dimensional

G(n,k),

so if

is the generator of

c^

H (G(n,k);Z).

be the dual element of

c^

H2 _ r(r).

Proof. If r,the lemma Take a

p

$v[r]

such that

borhood of

contains no critical point of

is trivial.

a - t + J~^±p p.

is a continuous function

f

and

So assume that it contains both.

dt(p) = 0 ,

and

df(p) = 0 .

Let

be the usual holomorphic function in a neigh­ a

is no longer a coordinate function at

because

dt(p) = 0.

Let

v = t - r(p)

and let

so that

p(p) = 0.

Then

£ = v + sT^lp

is a holomorphic func­

tion near m, ¥

p

such that

C(p) = 0.

p

p

be chosen

For some positive integer

z = ( £ ) will be a coordinate function in a neighborhood of p.

From the proof

of Theorem 4.12, we know that

Q = f*co = |z |2(5~1 )(bAF^)dz tion which vanishes

Adz", where

nowhere in ¥

and

bAF^ (5-1)

is a

C°°

func­

is the stationary

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

126

index of -

f

at

p.

h d^ Ad^. L X

By definition of

Since

h,

d^ = mzm ~^dzs

Q ~ h dx A *dx

we see that

Iz| 2(5_1)(bAFt )dz Adz = a = 'fB- h d £ A d t =

Hence in

W,

h = — m

h • m 2 • | z | 2 (m - 1 ) d z A d z \

|z |2(5“m )('bAFt),

= 2(S~m)log |z | + C°° function.

and so

Thus on

log h

dV [r ],

log h

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. (4.46)

and Lemma 4.13 that x(r) + B(r) - jjLf^ dV[r](log h)*drl = ^

provided

$V[r]

contains

no critical point of

We integrate this with respect to conclude that for

r

^

m

f

ro "

'

+

r >_ r( t ):

s(t)dt - -jb f (log ' ro ' ' w dV[tlv ~ m 2rJvo

.

vlt [t]

Introduce the notation: E(r) = I r

X(t)dt o

s(r) = / r s(t)dt o we have:

and

x#

and use Lemma 4.14 to

Ir x(t)dt

and

We have by

h ) * d x L

'

lro

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

(4.48)

E(r) + S(r) -

127

i f (log h)*dx 4ir dV[t]

=

/

dt

v rt t l

^ J ro

Now recall that for (Corollary to Lemma 2.4).

r

>

r( t

),

I *dT dV[r]

=

L

is a constant,

By the concavity of the logarithm

(Lemma 2.14),

^

i o e ( ^ s X i ’” '’" 1 1 ^

d X i ioi! h ,dT

J

= E(r) + S(r) - L dt X ' V* * J t q y[/t] + w So if w© let

YU

J (log h )* dT . svrro r

cp(r) = -y [E (t ) + S(r) — — ^ ^

/" dt / KX2 4* const.}, ro V[t]

then e ^ 1') < J. / h *dT, “ L dVlrl implying,

L/

e ^ ^ d t < f dt / h *dx. Now h dxA*dx = f*a>, ro _ ^ ro dvtt] which is certainly integrable on V[r] - V[rQ]. So Fubini* s theorem gives: f dt [ h*dT = f h dr A *dT Jro dvftl V[r]-V[r0] < J h dT A * dx ~ Vfr] # f CD, Vfr] I Combined with the above, this leads to

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

128

For this holomorphic map

f: V

M,

2.

let us define

v(r) = —

j

*

^ ^f

T(r) = J r v(t)dt. o If

M

is taken to he

v(r) and

G(n,k) _ k, then Iiz^l(k+l)(i+l)E

+

[i(T)

Add these Q.E.D.

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

134

If

i

£

k,

then

(n-kjTg = (n-i)Tk - i*z!)(n -i)(n-k)E + u(T). Proof.

Straightforward induction from the lemma.

Second Corollary.

If

or an annulus, then for

V = C,

or

(D - {0},

or a disc,

i £ k,

(k+l)^ = (f+l)Tk + n(T), and for

I . 0 for

x(V) £ 0,

(see discussion after Theorem 2.17).

r large.

The desired result now follows

from the FirstCorollary and Lemma 4.l6(iv). Lemma 4.18.

For

For

Q.E.D.

k = 0,...,n-l

St ■ -(tAKn-i;) Tlt Proof.

so for suffi­

k = 0,

+

(4.57) gives

S^ = 2Tq - T^ - E 4- jJt(T),

while (4.60)^ gives by virtue of Lemma 4.16(iii) that _

= - iTzll T0 - (.P-. 1 ).E + n(T).

SQ = process.

Tq So (4.61) is

is proved in a gives

E + pi(T),

using Lemma 4.l6(v) in the

proved for

similar manner.

These two together give

k = 0.

Now let

The 1 £ k_< n-2.

case ofk = n-1 (4.59)^, ^

THE TWO MAIN THEOREMS FOR HOLOMORPHIC CURVES

- V i

- - irar \

135

- 1 E - *(T >*

while (^*60)k+1 gives

(We have used Lemma 4.16(iii) 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

If

V

is

C,

(D - {0},

a disc or an annulus,

k = 0,..♦,n-l,

Proof.

Same reasoning as the Second Corollary of the

preceding lemma. At this point, we must distinguish between the cases of infinite and finite harmonicexhaustions, §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. Thus let 0 = p,($)

Recall that

if and only if

o

o

for some positive constants Lemma 2.16 twice we see that (4.62)

T: V *-► [0,«>).

K,

C

and

0 = p,($)

C!,

By applying

implies

II 0(r) < k log(C(r) 4- C f)

where as in §6 of Chapter II,

k

> 1,

and the sign

”11"

in

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

136

front of an inequality means that the inequality is only valid in

[0,°°) - I

with

d log x
^n )

*

such that

t0 = t(si + ••• + sn ) tx = (1 - x)Sl

tn m ^ clearly maps

- T ^sn

(0,1) x (0,°o) x ••• x (0,*>)

(0,oo) x ••• x (0,«>)

((n+1)-times).

one-one and onto

By a simple induction

argument: dt A ••• A dt = (1-T )n“^( Sn+ •*•*f*S )dT Adsn A ..• Ads • o n v ' v 1 n' 1 n Observe further that (t +•* *+t ) = (sn+ *•*+s ), so that ^ v o n' v 1 n y9 = . Hence we may transform the above integral into t o n

THE DEFECT RELATIONS

f° °

Because J q e equivalent to

—S

f° °

161

-S

ds = 1 and J Q se ds = r( 2 ) = 1, this is rl t n *J0 t(t )(1 -T )n~ ds = ~ , which is precisely

(5*15)• We now return to the computation of of (5.13)?, we can write:

for any

h

in (5.12).

p e V[t] - V[rQ]

not among the finite set of critical points of of

t

Because

which is

or the zeroes

x,

where it is understood that Let us choose 0 #N. basis span{e0) = spanfx(p)},

en

in

£n+1

so that

span{eQ,e1 } = span(x(p),b],

span(e0,e1,e2) = span{x(p)^x^1 )(p)}. For simplicity, we will write

Then

z^eA

as

x(p)x = s panCe^.#.,en). (zQ,...,zn ).

Thus in this notation, we may write: x(p) - (a2.,0, •••,0) b = (a2 ,a3 ,0 ,...,0 ) ^

^(P ) = (

9

S

9® *

• • •

and

fQ) •

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

162

With this simplification, and noting that Z = (0,z^,#,.,zn )#

h(P ) =

We have

implies

we have: z., z•» * a~7a~7 znzn ) •

r exp(-z / -z, I x(p)x

zi I 21

I

2

^— =r) zlzl+ ---+znzn

dL

la5 i2 , z -z n) • ( T | a :5 |2 ) d T

roo ‘

=

^

/ o

•

- s

r oo

J 0 ’‘

’ J

-•••-S

e

0

' r ( l - T ) n “ 2 =

b,

f

Recall that we began the integration with an integrable function

P

on

we determined —►HR

P

in terms of a real-valued function

(cf. (5.15)).

wise

We proved that so long as

and (5 .I5 ),

(5.14)

satisfying (5.5) and (5 .6 ) and, then

Pn€

in terms of

determine

uniquely, and if

two conditions, then (5.21)

Other­

In (5 .1 8 ), we defined a new function

f: [0,1] -~*3R cp

satisfies

will satisfy (5 .5 ) and (5*6).

p

is arbitrary.

cp

cp

cp: [0,1]

cp

cp.

Now we make this claim: f

f

satisfies the following

would satisfy (5.14) and (5 .I5 ):

The derivatives of positive there.

f

exist on

(0,1)

and are

(5 .2 2 ) Granting this claim for the moment, we bring our search for a suitable

P

satisfying (5.5) and (5 .6 ) to an end by locating

a suitable

f

such that (5 .2 1 ) and (5 .2 2 ) are fulfilled.

Such a

f

turns out to be exceedingly simple:

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

166

It is easy to see that this (5*22)*

The constant

a

be allowed to vary later f,

if

satisfies both (5*21) and

isarbitrary for the moment and will on.

On the basis of this choice of

we may rewrite (5.20) as

(5.23)

(1-a)

fl odtP ods dv[fs] l

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:

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

168

dT A*dT

(1-a)

f r dtvrti J J r o

= (l-a)T0(r) < To(r)e Using this, we may rewrite the previous inequality in its final form, which is inequality (32) in Ahlfors1 paper f11:

< CTo(r) + C \ where

0 < a < 1,

of

and

b

a

a = t + sTUp,

C,

C*

are positive constants independent

and in terms of the special coordinate function /1 \ ax ax x ' = (-g-jj-j ••• e W® no^e that C and C 1

will be our generic symbols of positive constants. in the sequel, we will denote by the same letters

Although C

and

Cf

the two constants that appear in the right side of the many variants of (5 .24), these constants actually change their values several times. T> round off the proof of (5.24), we must now show that determines

cp uniquely and that if

(5 .22), then indeed not difficult.

cp

f

satisfies (5.21) and

satisfies (5.14) and (5 .I5 ).

For by (5 .1 8 ),

t

This is

THE DEFECT RELATIONS

Letting

t

= st,

169

we get,

f (s) =

ir11

"IT Tlog| z 12dL. It

|zf_ ed: 77^(log 0)

and Lemma 5.6 as above, we

arrive at: (l-ct) r

dt f* ds

Jro

where

C, C f

f

|A

because

J X | j

—

IA

I

L

IA J x k | ^

Mln(

Now

IYk ,2a

|A

> l

J X I “

|A

Ix^

J X X| 2

|Yk+1,2 71YI 1 .5 )

:- 4

’ |A J X ,

IA

|yk| 2 J~r ■' g--c. because

J X |

(by (1.12)).

£■

i f l4-

J X ^ l ^ 2®

A-2a > 2

)

and

Having done this, we

obtain:

[l-a) f

Ixk_112 1xk+112

dt / ds / s] |Xk |4 o o dVL£ jYkt 2a jYk+112 Min(— I 1y : — jr— drr-p) |A J X | A J X 1 lxk-r g 1 rpri *dT Min( — n1j- — "a : -— gly ^ "Tr a ) U r " 1J X 1 12 Ia .J X I

< CT, (r) + C 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 )

(l-a) / ' « II ro ro

/ I X ^ V LX^ l lf dvts] |XK P Ah

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

186

with new constants

C,

C?

which are still independent of

a.

xk | Min — *-|A^JX | Therefore, there

By Lemma 5.10, it is impossible to have for more than exists an of the

M

A51,

p^( k, Jl)

of the given

Ah .

>_ M for at most p, (k, i) A£ J Xn independently of Xk . The remaining Ah| s satisfy such that

Min

Min

Let us say there are

p

such

A 1s.

(by (1.12)), we have for these ,k. 2a M in ( - 4 A U X 1^

p

Iy-k4*l I -— y— — IA J X

Because

>_ 1

Ah,s:

xk+l|2

^

i± r r ? ) < M ' lA^JX1

Furthermore,

[a

in view of (1.13). one.

X"

Ixk _ 1 | 2 J X 15 ’ lA^

J xk| 2

■)2 > l Ai-ij

X ¥rn

Therefore its minimum will still exceed

Consequently, referring back to the expression of

O(A^),

we get M.By the above,q< p,(k,i).So, |K J l X | ~~

if we use we have:

Z ! to denote summation over these

- h

q of

theA ^ !s,

THE DEFECT RELATIONS

187

log 2 log S' (Ah ) A

= log(| S' 4>(Ah )} + log q > i S' log (Ah ) + log q

(Lemma 2.14)

If Slog 4>(Ah ) - 2ap log M) + log q Q Ah — p~ (k,TX '^ log °(Ah) "

i.e.

log S $(Ah ) > Ah —

n

. ,1 >S log $(Ah ) + Ph' K> Ah

loS

c, where 1

a constant depending only on the system(A*1}. usual the constant

/ *dT for dVTs] of the Logarithm again, we have:

E SVts] >1

|5*p

J

Let

s > r(r).

c. 1 L

is

be as

By the Concavity

ah

| ^ | » jg

■ hurts]

+ log

iT v

V

. t

„

An

i / iogh ! ± i ^ ! h £ . « + 1 / i„gS ,{#v Vts] |xV Wfs] Ah

a,

= ^LfE(s) + Sk (s) + (Tk_1(s) - 2Tk(s) + Tk+1(s)) (yk-1,2 1vk4-l,2 log L --L --- 1— *dx) + ilog S $(Ah )#dT d V[ x Qj ^Ah Tr-] ||A F r| V QVlOJ [s]

J

f

(by Theorem 4.24) > ^ f E ( s ) + Sk ( s ) + (T k _1 (s ) - 2Tk ( s ) + Tk+ 1 ( s ) ) }

+ Lph(k,i) rs~ A- ~rr ^s y J[s] los ‘Ka*1) * ^ + c 02

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

188

where

is a constant depending only on the system

{A51}.

Keeping this inequality in mind, we inspect the integral

f/ ^log $(Ah )*dT. ] avis]

It is equal to

f |Yk,2a |„k+li2 J log Min(— i • L ■ net : ~ V ~k+l )*dT dV[s] |A J X | |aLj X | -

f log Min(— dV[s] |A def

~ _l a “

j

=

g : — |A

I *dr -

dv[s]

I

II

1 k g)*dT J X |2 *dx

dV[s]

Obviously, IYk|2 |Yk+1|2 I > log M l n ( - 4 * -I.-7 : - fe '-viT ?) |A J X |

~

|A

IX

|

- (l-a) log Max — |A J X | Iyk|2 Iyk+1|2

> 1o«

; j l ' f T x A 2’ -(l-a)

2

log

A i G Ah

g• i F T F p

Now by Lemma 4.10, Ter W

where

a^

^ log -- 1-- ^r=— * dT < T, (S) 4* a.. d v [s ] |AJ j F | ~ k k

is independent

f J I dV[s]

f * dr> J ~ dVls] -

where

c^

and

a^.

A

0

and

s.

Therefore,

lYk l2 !yk+l|2 log Min(— i— irjr-vy :— p ^ lA^JX^p |A J XK+±| (l-«) =3 (Tk (s) + a^.)

are constants depending only on the system

THE DEFECT RELATIONS

[Ah ]

alone and not on

satisfy

(l-a)T^.( s) = 1.

depending only on f

a*

-I *dT >

dvts]

[Ah ]

189

We now choose

a:

a

Hence there is a new constant

c^

such that

log Min(— p- Ifr g

J

should

|A J X |

— dVTs]

^

-g)»dT +

|A J X

|

4

and therefore / log (Ah )*dt dvtsi > f log Mln(— : - 4 X^ .g)»dT “ dVIs] |A J X |2 ]A^JX + |2

.

/

dVTs]

log Hint t ' f ' T x , : l A ^ J X 1""1 !2

j 'P

jA^"1 J X K |2

* c ,, ^

so that taking into account of a previous inequality, we obtain:

r dv[s]

[F j2

> ^CE(s) + Sk (s) + +

Ah ( T ^ b ) - 2Tk (s)

S ( [ log Lph^ ’ ) Ah dVTs] .

f -log dVIs]

|A

+ Tk+1(s)l

|PrZlX*| 2

JX

|

|P? J X*+ |2

| Ar J X |

+ C5 where

cr is a constant depending only on the system b Now define:

(A*1}.

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

190

e(s) = ph (k,^){E(s) + Sk (s) + ( \ _ 3_Cs) - 2Tk(s) + \ + 1 (s))}

+ I i

/

Ah ^

dV[s]

l o g Mint .

, :

|A J X |

l^ v li

|A _J X

- 2 i. / log Min(— l;^"1 1f- . -■ : — Ah ^ $V[s] |A J X | |A

8

|

).J t

f. ^)>dr J X |

Then the above may be rewritten as:

n

W

C

0 (s )

+ c

T

1
1 (r)} ,

THE DEFECT RELATIONS

191

So by Lemma 4.16(i), we obtain

e =

h (t 2).

In greater detail, we have the following:

(5.32)

Z — ■ jh

2

log Min(—

J

dVIs]

g : —

|A J X |

- 2 ~

Ah ^

J

i [-r-g)»dT

"|A J X

|

log Mln(—

dVIs]

|A

; -

J X |

J-jk lgR g)»d

|A

_J X |

= P h (k,i)(-B - Sk - (Tk _x - 2Tk + Tk + 1 )) + n ( T 2 ).

We wish to point out explicitly that (5.32) is only valid for Introduce the notation:

0 < £ _ h+1.

Therefore, extending the summation to all

i

in the last

double sum of (5*353) means we must add to the left side of (5.35) the following quantity:

C i But if

sl L . ph("-1-h-1>(-T. - i - i + s v i i < 0,

(n^Tn^i) = 0,

p^(m-i,h-i) =

- T„.i+i>-

because in (5.54),

So the above equals

C i = Cl>

(“V i - 1 + 2Tm-i " Tm-i+l) (-Tm + V l >

= 0 ( - v Hence (5*55) is equivalent to:

THE DEFECT RELATIONS

(5.35)

2 V

195

Ah)

Ah

= C X

~ * £ ± Et o

+ E)

+ Sm=k Z£ L « ph^m “i,h“i^ ~ Tm-i-l + 2Tm-i - V i +1> + ^ t2)We proceed to simplify the last Siam. subscript of the

By choosing the

T* s as the running subscript we may rewrite

the double sum as XiLcc (-Ph(i+l,h-m+i+l) + 2ph(i,h-m+i) - ph (i-l,h-m+i-l))T1. Now we apply the recursive relation among the binomial coefficients:

(*J) + (v^i) -

•

inspecting the coefficient of (l+x)^1

and

(l+x)(l+x)^.

xv

This can be proved by

in the expansions of

This implies:

ph (i,h-m+i) - ph (i+l,h-m+i+l) _ _

f/ i+2 \/ n-i-1 \ _ j>0 X-m+i+j+2' ^m-i-j-l'

/ i+l \/ n-i \n X-m+i+j+l^m-i- y

_ _ rr/ i+l \ , / i+l \i/ n-i-1 \ ^ j>0 'h-m+i+j+2' 'h-m+i+j+1' 'm-i-j-1' / i+l \r/n-i-1 \ / n-i-1 \i^ *h-m+i+j+1' 'm-i-j' + ^m-i-j-l'n _

f/ i+l w n-i-1 \ ^ / i+l wn-i-1 \-j j>0 X-m+i+j+2' 'm-i-j-1^ 'h-m+i+j+1' 'm-i-j' / i+l w n-i-1 v ''h-m+i+l^ m-i '

In a similar fashion:

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

196

Hence the above double sum equals

4!L » C k

= y+0°

ff

i=-°°

m=k

^

\/n-i-1\ m /

i

\/ n-i-L im

^h-m+i''m-i+1'^h-m+i+l'' m-i '

i

— X+CO ^ \yn-i-1% / 1 wn-i-l^-Jrp ~ zi=:«co 1 ^h-k+i+lM k~i } ^ lh~n+l+iM n-i ';ii -f

— ” *4=-°°

y+°°

/

i

\/n-i-1 *_

ih»k+l+l^ k~i ' 4 + 4=-°° \h«n+i+l'' n-i ' 4

Now observe that [l

-

0

if

li= »1

otherwise .

So the second sum has every coefficient equal to zero except for

i = n. 3 identically.

but then

(!J) = 0

V < 0.

and

if

(n^ ^ )

T = 0. Thus the second sum vanishes n As to the first sum, remember that h _ i.

unless

i >_ (k-h)-l

So the above equals

\ / n - i - 1 \ rp ^ h -k + i+ lM k - i ' 4 9

f

We may now rewrite (5*35) in its final form: (5-36)

Z mk(A ) = (h+l)Tk ~ Zm=k Si=0 ph^m"i,h"i^ Sin-i + E ) A ~ Si=k-h-l^h-k+i+l^nk-i )Ti + >1('1'2)-

THE DEFECT RELATIONS

When

h = k,

(5.37)

197

we claim that this reduces to

2 *>k (Ak ) = ( k S ) Tk Ak + H(T2).

PkC^n-i^-DCS^, + E)

This is because k , i )fn-i-l)T i=-l 4+1'^ k-i ;ii has every coefficient equal to zero due to the presence of (i ^ )

except when

i = -1.

But then

T ^ = 0,

So the whole

sum vanishes. Making use of the First Corollary of Lemma 4.17, it is possible to derive a variant of (5*56) where only on the right side.

T^

appears

It goes as follows.

~ Si=k-h-l ^h-k+i+1^ k-i )Ti = ^"2i=k-h-l ^h-k+i+1^ k-i ^ k + T ^ Tk

f k-h k / i+1 wn-i-1., “ l" k+T zi=k-h-l 'h-k+i+1'^ k-i >’Ak -

)

*tk-h-l (h-kS+l)(kIi:l)lE + ^ T >

r k-h „k /i+h/n-i-lvi. ~ l" k+T i=k-h-l 'k-h''■n-k-1' k

In order to simplify the coefficients of invoke the following identity: integers, then

if

p, q, r

and

E,

we

are positive

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

198

To see this, recall that if definition,

m

is a -nonnegative integer, by

("“ ) = (-l)p m V (m+A)'‘j Jl

B-l U .

Hence for

E' •

every nonnegative integer

m,

(? ) - (-i)p (m;p-:1 ) Now,

- *v ( T V • Similarly,

(l-x)”(q+1^ = S

M'

(q^M')xM' and

Q

(1-x) "(P+q+2)

= Xfi (^p+q+i^) *

So comparing the coefficients of

(l-x)"(P+q+2)

(l-x)-(P+ 1 ) • (l-x)-(q+1),

and

xr

we get the

above identity. Applying this identity, we obtain: / i \ /n-i-lx,p i=k-h-l ^h-k+i+lM k-i ;ii -

^

sjS-h

/k-hwn+l^T - " (k + r ^ h + i ^ k

-

- - ^ r 1C H

+

-

+ „(T)

i " ;1)* + »(t>.

in

THE DEFECT RELATIONS

199

where the last step is due to the fact that n-1,

so that

(5.38)

( “ ^ ) = 0.

s m^A*) =

k

never exceeds

Therefore from (5.36) we deduce

zj =0 ph (m-i,h-i)Sni_i

(^)Tk -

A - flB —

^ =0 Ph (n-i,h-i))E + p(T:

where use has been made of Lemma 4.16(1) and (v) to get p.(T) + ji(T^) - \l (T2).

(5.36)-(5.38) constitute the defect

0 £ h £ k.

relations for

To complete the picture, let us deduce also the defect relations for the case

k 0 Lemma

*

5 J r o J r o dvtsl

IX _1 A I

|3CJA^| £ CTk(r) + C»

0 < a < 1,

where

and

C,

i pendent of

a,

A

and

C * are positive constants inde0. ]_ A * Proceeding in similar manner

as above, we first obtain:

(i-ofdtfds

1

/

dV[s]

, Jr o Jro

fxV

,Yk ,2a Min(- |X 1 iP J F l® ¥FTr5

IYk-1 1 2

1

• |X

IXCT1J

1

'

J A^l --------

12 ' |Xk J

*dT

|2

a

£ CTk(r) + C' where of and

C,

a,

are new positive constants still independent o and the minimum refers to the finite number of A

A^*1

C1

containing

A*\ Letting

system ingeneral position,

Z i

we obtain the analogue

f log » l n ( _ 4 j H r

Ah 2ir 9 v r s i

p r j

- Z Ah

= Pn-h-l(

1

Ah range over a

a

|

:

of (5.32):

j ^ l. ).« |x

/ log Mln( ■

W dV[s]

finite

|X

j a

|

1L r _l A

n-^-1)(-E-sk-Tk_l+2V

|

|X J A

|

Tk+l) + ^ t2) *

THE DEFECT RELATIONS

201

k k~l log Minr- i u .-.. ; - 1 C — L )»dT, avis] jX J A | |X 1J A i!|

/

Ah

then the above may be written (5.39)5

- ^(k+l^+l) =

+ ^ ( T 2 ),

Adding (5.^9)^,(5.39)^11,... , ( 5 . 3 9 ) ^ ' £+1, ¥(k+n-i,n)

=0,

and noting that

we get

Z 1 log M n ( . J . * X Ah ^ av[s] |x 4 j a 4 = Si=0

: - K ~ 1lr ) . «

pn-h-l^n"k “i~1,n“h"1“1 ^ " E_Sk+i“Tk+i+l

+ 2Tk+i-Tk+i-l) + p (T ) If

I = h,

no minimum need be taken, so this reduces to

ahv Z (m (An) - m Ah

*/Ah x 1(A )}

+ a w

W

l

where we have written

m^C A*1)

Summing over

0

k

from

to

+ in place of k

m^.(r,A^)

and noting that

as usual,

m ^(A^1) = 0,

we have: Z ^ ( A 1*) . -Z*,0 A + Zm=0 Si=0

Pn_h _1 (n-m-i-l,n-h-i-l)

• (-Tm+i-i+2Tm+ i-Tm+i+l) + ^ This is valid when

k < h.

We now extend the last double sum

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

202

over all when

setting as before

±$

i > n~l.

equals

when

The coefficient vanishes when = (fo+i)

the sum to all

= 0

i,

when

i < 0.

i < 0 i >_ n-h

and and

So after we have extended

we should add to the left the following

quantity:

Sm=0

pn-h-l^n"m "1_1,n-h"i"1 ^ “Tm+i-l+2Tm+i"Tm+i+l) = (h+l) 2m=0

- - O

^“Tm+i-l+2Tm+i ~Tm+i+l^

v

We have therefore obtained the following: (5.40)

z mk(Ah ) = ( ^ ) T k - ^ =0 A

Pn_]1_1 (n-m-i-1,n-h-i-1)

• (E+fW + 2m=0 2ilcc Pn_h_1(n-n.-i-l,n-h-i-l)

' (-Tm+i-l+2TnH-i-Tm+i+l) + ^ T2) It remains to simplify the last double sum. and let

(5-41)

p = n-l-m.

Let

a = n-h-l

Then this double sum may be written as:

2p=n_i_k

Pa(p“i'a “i )(-V.l-(3-i-l)+2Tn-l-(e-.i)

-Tn-1-(P-1+1)) It may be recalled that previously we have computed the last infinite double sum of (p.35) and found that

(5. *2)

C k

Z£ L « Ph (m - ^ h-i) (-Tffi-l-l+2V l - Tm-l+l) = Zi= k -h -l

“ ^ h -k + i+ l^ H k -l1 ^ ! '

THE DEFECT RELATIONS

The left side of

(5.42)

only we replace

h

203

would be identicalwith(5.41) if

by a,

k

by

(n-l-k)

and

Tj

by

So (5.41) equals yn-k-l / i w n-i-1 ^ i-n-k-l-a-1 “^a-(n-k-l)+i+l''(n-k-l)-i' n-l~i ^ _n-k-l i=h-k-l "

/ i w n-i-1 vT k-h+i+1''n-i-k-1' n-l-i

yn-h+k / n-1-j w J \m j=k "^k-h-f-n-j^j-k^j

n-h-k /n-l-JwJxT = zj=k -vh« k - l ^ k ;ij Substituting this into (5.40), we obtain the counterpart of (5.36): (5.43)

Z ^(A*1) = ( H l ) \ - I*=0 A •

This is valid for

pn_h_1(n-«-i-l,n-h-i-l)

- 2? S +k (h'i:i)(k)Ti + ^ t2) • k _< h.

When

% = h,

(5.^3) again reduces

to (5-37). Again we can transform the last sum using the First Corollary of Lemma 4.17 so that only

"

T^

appears.

_n-h+k ,n-l-iwiiT i=k ^h-k-ln k ;ii r vn-h+k ,n-iwn-l-iwi>lT = 1 i=k ^nlk^h-k-l^kj u k ~

2i=k+k (hIkIl^k)(i_k)(n"i)lE +

r h-k _n-h+k , n - i w i n . = l“ n^Sc Si=k ^h-kM k ;,1k _ (.(.^ H K jl-kX sn-h+k (^"J)(ki;L))E + n(T)

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

204

- - { f e £ j O Tk - ( I d M - l H h - l O O l E * It(T) where we have used the previously proved identity: q-kr ,p+q+r-VwVv _ ,p+q4-r+lv v-q * p ''q' ' r '* Substituting into (5.43) the above, we have arrived at the counterpart of (5.38): (5.W)

2 V A

aH) " 2m=0 Si=0

pn-h-l(n“m-1“1,n~h“1"1 )Sm+i “ h+2'

■2 + 4=0

Pn_h_1(n-»-i-l,n-h-i-l))E

+ |X(T2) (5.43)

and (5.44) are the defect relations for the case

(n-1) >_ h ^ k.

We now summarize the above into a comprehensive

theorem, which is the main re suit of the whole development. Theorem 5.12 (Defect Relations).

Let

nondegenerate holomorphic curve and let exhaustion.

Let further

{A*1}

sional projective sub spaces of

admit a harmonic

be a finite system of Pn-i,h-i)(Sm_.+E) " ^ i = k - h - l ^h - k + i + 1 ^ ^ " k - i 1 )

+ M-C1 ,2 )

Then

THE DEFECT RELATIONS

- 4:i 4 m

-

i>h(”-i.h-i)s„-i

- (lE=i^f*=E). ("«) +

If

k ■< h £ n-1, ^

205

^

ph(„.l,h.l))E + b (t s

then

>\(Ah ) = (^+i)\ - S*=0 S±=q 1 Pn _h_1(n-ln-i-ljn-h-i-l) • (E+£W „n-h+k /n-l-i\/iv_ n(T*h ^ k '■h-k-1 ''■k' i + ; _ ,n-hu n+lv - 'n^k'^h+l' k " Sm=0 Zi=0

pn-h-l^n_m-i"1,i:i"h "i“1 ^Sin+i

- t^k+i)(h-k)(£|) + s*=0

pn_h_1

• (n-m-i-l,n-h-i-l)]E + p,(T2). Finally, if

h = k,

then

As we mentioned before, the last conclusion of the pre­ ceding theorem is the most important in application.

We can

rephrase it in an essentially equivalent way, as follows. V

Let

admit an infinite harmonic exhaustion (Definition 2.4). k For each k-dimensional projective subspace A of P^C, we define the defect of k* 6 k (AK)

Then

5^.

A^

to be:

= limJLnf

, (1

Nk (r'A\ - ~ T fr) )

is a measurable real-valued function on

G(n,k).

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

206

Clearly

6^

1.

follows that

From (4.25)

0 ^

( \ ( r,Ak ) < Tfc(r) 4- cfe),

(See §5 of Chapter II).

Thus

i

5^: G(n,k)

[0,1] .

Let

A

it

if-

he the polar

space of

A .

\r

From the definition of

N^(r,A )

(see the paragraph preceding

Theorem 4.8 and the discussion after (4.4)), if for all

p e V,

5^.(Ak ) = 1.

It

then

N^( r,A ) = 0

5^(A^) = 0

for all

r

and so

is then to he interpreted as

^x(p) C\kx 0 0

for "many"

cases,

except on a countable subset of

5^. = 0

^x(p) H A1 - 0

p € V.

We now show that in many

Theorem 5.15 (Defect Relations).

Let

G(n,k).

x: V —► PnC

nondegenerate holomorphic curve and suppose either or

ak

Proof.

Suppose not, then

N.(r,Ak ) lim inf 2 (1 - — — r-*~ Ak Tfe(r) where

e > 0.

Thus outside a compact set, v (1 ' ^ Ak

or

M

’ - ' \ . ,«i, + e IC'fr) '- 'k+1' ’ k

2 (Tk (r) - Nk (r,Ak )} >_ Ak

(4.16),

1 (HI) + € ,

+ e)Tk (r )>

2 mk (r,Ak ) >_ C(k^ ) + €)Tk (r). Ak

or i-n view of

By Lemma 4.l6(i)

THE DEFECT RELATIONS

207

and the last conclusion of the preceding theorem: + e)V

r) = C l > Tk - \ , i Pfc C n - l ^ - D f S ^ + E ) + H(T2)

€Tk 4 Zk>i pk (m-i,k-i)(Sm_1+E) = n(T2). Now

i >_0

Since

by its definition, so Lemma 4.16(iv) implies

eTk + ^ , 1

= p.(T2).

Tk +

= n(T2),

€ > 0,

by Lemma 4.16(iii).

Note that each

constant, so the coefficient of

E

p^(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 =

Si=0 Pk (»-i,k-i))linj*up

Then one can prove that for a finite system of

k-spaces

fA^}

in general position, the following holds:

(5.45)

and if one of

2 ek(A*) < (Si) + *k' A* (xo#•

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.

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES

208

§5 .

Ahlfors mentioned at the beginning of his paper [l]

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 x: V

P^(C.

For each

^.x(p)

is of course a

sional projective sub space of

Pn