The equidistribution theory of holomorphic curves

THE EQUIDISTRIBUTION THEORY OF HOLOMORPHIC CURVES BY HUNG-HSI WU PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF TO...

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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


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;,

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,

= .

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


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.( 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


(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

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


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


~*(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 - 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


where

c

o 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 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


increasing function on k > 1,

number where

[O,s),

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


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

~ = 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


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


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 -;& 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


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


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


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


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


[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


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

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


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 )


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


(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


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


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 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


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


(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)

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 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


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

~)
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

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


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


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
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-(

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


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

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


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-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 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 (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 (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


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 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


£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))·


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
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


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(

The equidistribution theory of holomorphic curves

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

J-holomorphic curves and quantum cohomology

Model Theory of Holomorphic Functions

J-Holomorphic Curves And Symplectic Topology

J-holomorphic curves and quantum cohomology

Arithmetic Theory of Elliptic Curves

Elements of the Theory of Algebraic Curves

Arithmetic Theory of Elliptic Curves

Capacity Theory on Algebraic Curves

Arithmetic theory of elliptic curves: Lectures

Geometric Theory of Algebraic Space Curves

Extension of holomorphic functions

The Curves of the Needle

Holomorphic function theory in several variables

Holomorphic Dynamics

The arithmetic of elliptic curves

The Arithmetic of Elliptic Curves

Holomorphic functions of several variables

Moduli of curves

A Book of Curves

Atlas of Fatigue Curves

Singularities of plane curves

Arithmetic of elliptic curves

Book of Curves

Elliptic Curves: Function Theory, Geometry, Arithmetic

Geometry of Algebraic Curves

Geometry of algebraic curves

Moduli of Curves

Moduli of Curves

Moduli of Curves

The equidistribution theory of holomorphic curves