TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 169, July 1972
HOLOMORPHICMAPSINTO COMPLEX PROJECTIVE SPACE
OMITTINGHYPERPLANESt1)
BY
MARK L. GREEN
ABSTRACT. Using
methods
akin to those
of E. Borel and R. Nevanlinna,
generalization
of Picard's
theorem
to several
variables
is
a lemma on linear relations
among exponentials
of entire
cally,
it' is shown that a holomorphic
map from Cm to P
hyperplanes
has image
lying in a hyperplane.
If the
more hyperplanes
in general
position,
the image
will
space
of low dimension,
being
forced
to be constant
2n + 1 hyperplanes
in general
position.
The limits
sion of the image
are shown
to be sharp.
Introduction.
Theorem
planes
linear
We will establish
1. Let
in general
position,
subspace
Furthermore,
hence
in a proper
Corollary
linear
image
Theorem
resolve
Received
in a linear
map f: Cm—» P
conjectures
in a projective
mean greatest
that
misses
subspace
that
integer.
72 + 2 hyperplanes
of dimension
misses
image
lying
of S. S. Chern
1 improves
of the linear
2. A holomorphic
has
n + k hyper-
\n/2\,
subspace.
when the hypothesis
hyperplanes
omits
of f is contained
the brackets
map f: Cm—► P
contained
[72/2] in Corollary
on the dimension
obtained
map that
theorem:
2n + 1 hyperplanes
is constant.
corollaries
The dimension
of Picard's
is sharp.
2. A holomorphic
position
These
bound
has
the image
< [72//e], where
1. A holomorphic
postion
in general
bound
be a holomorphic
k > 1. Then
of dimension
this
Corollary
in general
f: Cm —» P
two generalizations
a
proved by reduction
to
functions.
More specifiomitting
n + 2 distinct
map omits
n + 2 or
lie in a linear
subif the map omits
found for the dimen-
Chern's
subspace
of general
map f: Cm —> P
in a proper
by the editors October
AMS 1970 subject classifications.
original
generated
position
projective
[3] and> H. Wu ([7], [8]).
is dropped,
that
conjecture.
by the image
omits
linear
A weaker
will be
as follows:
any distinct
n + 2
subspace.
18, 1971.
Primary 30A70, 32H25; Secondary
32H20.
Key words and phrases.
Picard
theorem,
value distribution
theory,
exponential
Nevanlinna
theory, hyperplanes
in general position,
holomorphic
curve, Nevanlinna
teristic
function.
(l)This
research
was done while the author held an NSF Graduate
Fellowship
Procter
Fellowship
at Princeton
University.
Copyright
© 1972, American
89
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
function,
characand a
Mathematical
Society
[July
M. L. GREEN
90
The
proofs
exemplified
fonctions
are entirely
analytical,
by R. Nevanlinna's
méromorphes,
inspired
Le théorème
by the approach
de Picard-Borel
and will be parallel
to Emile
to the subject
et la théorie
Borel's
des
proof of Picard's
theorem in 1897 [l].
Before
embarking
by a brief
Using
on the full proofs,
discussion
of the case
homogeneous
coordinates
we will give
of n + 2 hyperplanes
z.,
they
coordinates
as
may be expressed
(/„,
Cm —►C are holomorphic.
Further,
\z. U + , ---+z
no poles,
r
isfying
= OS and has
n
position
in P .
may be assumed
to
= 0}. Writing the map / in global
• • ■, / ), since
as exponentials
key to what follows
in general
• • • , z , the hyperplanes
be \z. = O}, i = 0, • • • , n, and \zQ + ■• • + z
homogeneous
a skeleton
the
exp(gQ),
because
/. are without
poles
• • • , exp (g ), where
their
it likewise
sum is never
zero,
is an exponential
r
or zeros,
the
as
g .:
/ omits
expron+l
(p
,) sat-
the equation
exP (g0) + • • * + exp (gn) + exp (z77+ gn + l) = 0.
What must
be done
that
g. — g. is constant,
into
a proper
deed
a reduction
defect
tion
linear
relations
(independent
tunately,
fact
by the last
we differentiate
of shorter
holomorphic
is a method
themselves.
Since
tives
of the
g ., they
will
not serve
designed
Once
follow
"rates
in later
such
the theorem
from some
linear
(see,
in order
to show
in a crude
Proposiby
the impossibility
some
equation
way,
of the
be proved
of holomorphic
is to find
and in-
result
and will
such
/ goes
line,
for example,
directly
the differential
expign
the domain
functions
way to utilize
it satisfies.
namely
For-
we divide
through
track
small
that
will
the Nevanlinna
and indeed
works
is established,
and combinatorial
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possible
the coefficients
functions,
line)
providing
of the equation
to the exponentials
involve
so that
quotients
of deriva-
the maximum
modulus
characteristic
in this
obtain-
of the exponentials
is still
with respect
the coefficients
on exponentials
algebra
the coefficients
of the fact
However,
purposes
g. is the complex
An induction
of growth"
steps
- gn + 1) = 1.
of the
but where
functions.
will be meromorphic
our purpose.
for just
is a classical
curves
- é„+l)>•••+
length,
of keeping
have
this
Vz 3/
and hence
the complex
exponentials
to do this
(assuming
ing an equation
of exponentials
is constant
is just
by a constant),
unless
term getting
are nonconstant
there
/.//.
idea,
an exponential—
exp(g0
Then
essential
it is only necessary
the equation
is absurd
is accessible
"independent"
not differing
about
case
of holomorphic
The
among
equation
will be possible,
the result
means.
means
the crucial
case
However,
relation
this
When the domain
in the theory
elementary
of a linear
that
i?¿ j, in which
subspace.
to this
5.15 of [7]).
fairly
is to show
function
the two main theorems
fiddling.
was
case.
§ 1 will
do this
will
for
1972]
HOLOMORPHIC MAPS INTO COMPLEX PROJECTIVE SPACE
Theorem
tions,
2, where
it is simplest.
§ 2 will indicate
position
theorem
tion of Theorem
which
(Theorem
[n/k]
linear
contain
characteristic
relations
function
among
functions,
Let
§ 3 gives
§ 4 delineates
through
the reduc-
the manner
a construction
§ 5 lists
we will need.
§ 6 establishes
in
more or less
of holomorphic
the properties
functions
of the
the theorem
as a corollary
on
of the
theorem:
g .: C —> C be holomorphic
functions,
h.: C—» P
meromorphic
i - 1, • • • , N, satisfying
(l)hlexpigl)
+ •••+
¿Nexp(gN)
s 1,
(2) Tib., r) is o(SN=1 T(exp(g.),
Nevanlinna
characteristic
we exclude
as values
Then
some
function
of r a set
proper
linear
Let
Tj-linear
algebraically
total
its usual
measure
except
that
in R .
of the terms
a result
meaning
h. exp (g .) is constant.
on algebraic
independence
of
functions:
/.,-•-,
fN be holomorphic
combination
functions
of them is constant.
independent
§ 7 incorporates
of finite
o has
of § 6 comes
of meromorphic
Theorem.
r)) for all i = 1, • • • , Ai, where T is the
and little
combination
Out of the considerations
exponentials
trivial
case.
by equa-
in the general
the analysis.
exponentials
one variable
Theorem.
at a specific
1 is sharp
be camouflaged
so surprisingly
proof.
The next two sections
following
the geometry
drops
on exponentials.
oí Theorem
that theorem's
Nevanlinna
lest
1) by looking
1 to the theorem
the bound
reversing
Then,
why the dimension
91
functions
a few loose
over
ends
on Cm such
Then
exp(/.),
that
no non-
• • • , exp ifN)
are
C.
and suggests
some improvements
in the
theorems.
During
tence
the last
of a paper
the case
stages
of preparation
of J. Dufresnoy
when the domain
particular,
Corollary
Dufresnoy's
normal
of the results
1. Proof
of holomorphic
Theorem.
2 of Theorem
news
1 should
methods,
arrived
although
has some overlap
certainly
H. Fujimoto
be attributed
has
of the exis-
restricted
to
with mine.
In
to him.
Using
independently
found
some
paper.
of Theorem
functions,
Let
paper,
from 1944 [4]. His work,
is one dimensional,
families
in this
of this
2. We will
whose
proof
assume
the following
is found
in a later
g.: Cm —> C be holomorphic
lemma
on exponentials
section:
functions,
i = 0, • • • , n, satisfying
identically
exp(g0(z,,
Then for some
'
distinct
' • *. zm)) + • • • + exp(gn(zi;
i, j, g {z .,-■•,
'
°2
1
z
m
• • •, zj)
) - g.iz ,,•■-,
ö;
1
z
m
constant.
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= 0.
) is identically
J
92
M. L. GREEN
Recall
the statement
Theorem
of Theorem
2. A holomorphic
hyperplanes
has
Proof.
image
map
lying
2:
f: Cm —» P
in a proper
Let the hyperplanes
[jul5
that
projective
linear
be, in homogeneous
Z aitzi = ° '
omits
any distinct
n + 2
subspace.
coordinates,
' - 1.•••»«+ 2.
i=0
Writing
never
/as
ifQ, •••,/),
zero,
the exponential
each
if / is to miss
nor can it have
a pole
since
these
the
exp (g ) of a holomorphic
t = 1, • • • , n + 2 yields
hyperplanes,
/. have
function
none;
then
2"=Qa.
/.
we can conclude
g : Cm —» C. Doing
this
is
it is
for
a set of equations:
n
Z aitf, = exp(ëtî>
t=l,---,
n+2.
£=0
Considering
the
be at least
n + 2 vectors
one linear
a = iaQ , • • • , a
dependence
among
them,
) in C"+
say
, we note there
-£?*, c a
must
= 0. Then
n+2
Z ct exP(Sz^- °t=l
By choosing
logarithms
for the nonzero
c , we are in possession
of a relation,
if
S = \t\ct=éO\,
¿j exp(g( + l°gcP K 0
tes
that
falls
under
the umbrella
We are free to conclude,
exp(c)exp
of the theorem
for some
sjé
on exponentials
t, that
-g
that
began this section.
is a constant
c. So exp(g
)=
ig().
To return
to the system
of equations
by which
n
the
g's
were
defined,
we have
n
Z ais U = exP(s^-
Z ait fi = exP(si)-
i= 0
Subtracting
g
i= 0
exp (c) times
the second
equation
from the first
produces
n
Z {ais- ^P^KM
-°
z'=0
which is the linear
relation,
relation
for otherwise
same hyperplane,
2. Geometric
make the geometric
among
the
as = exp ic)a
contrary
/. the theorem
, which
to hypothesis.
interlude—Theorem
content
of Theorem
would
requires.
imply
a
It is a nontrivial
and
fl
determine
Q.E.D.
1 for P
1 clearer
This
section,
and render
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it is hoped,
its conclusion
will
more
the
1972]
HOLOMORPHIC MAPS INTO COMPLEX PROJECTIV£
plausible.
We will take
hyperplanes
pared
as our starting
in question
down still
section—here
point
are in general
further.
we will
The general
spotlight
Theorem
2 and see
position,
the dimension
case
be treated
the first
will
interesting
SPACE
how,
when
the
of the image
formally
case,
93
may be
in the next
P,.
Thus
that
misses
Theorem
1 be-
comes
Theorem
in general
general
position
Proof
H
H
a further
no three
planes
involved
and
thus
Cl H
and
that
noting
when
the domain
which
planes
joining
have
in
the image
point
number
is not particular
to P,,
and becomes
(~)
H2 and
is no 2-plane
in P,
passing
engineered
in at least
of
four
special
class
of three
of the other
if we knew
Theorem
on the lines
image
through
lines
the reasons
when
natural
attention
when
/(0).
that
on any
in Cm,
This
argu-
the transition
is confined
a less
2 only
for maps
the origin
through
It is
hyperplanes,
Theorem
1 for P,
of
hyper-
two.
on one of the 5 omitted
illuminates
rather
H
with
the best
of intersection
Hence,
conclude
Cm is difficult
were
But the lines
so there
of intersection
of permissible
the four hyper-
in common
in one of a very
not lying
theorem
Further,
even
goes
(by general
of / to lie in a line.
/ lies
lines.
still
the 5 hyper-
for if the lines
position
But in
intersect
by the image.
one of the points
of such
line).
that
position
in P,.
if the two-plane
two lines
in general
C , we could
to an arbitrary
position
a common
be distinct,
on the line
a fixed
number
must
only
in common,
image
in general
of the 5 hyperplanes
is omitted
no point
the one-dimensional
of the finite
in general
H
2 compels
through
is
2-plane
We are led to conclude
lines
are only a finite
C
would
as
/ is constant,
there
domain
share
the 2-plane
Theorem
worth
because
5 hyperplanes
1 hyperplanes
in a 2-plane
happen
of pairs
must
the five to a point
Cm by using
This
five hyperplanes
those
among
can
we exclude,
unless
namely
planes
2, this
H, D H . contain
lines.
In fact,
misses
be 5 hyperplanes
of / lies
lines.
of the hyperplanes
intersects
distinct
a map that
of Theorem
of intersection
both to be found.
2-planes
line;
H , • ■■ , H
in the intersections
H2, which
through
ment,
dose
distinct
H2 O W,, then
be
lines,
2. Let
two of the lines
position,
f: Cm—> P,
2, we know the image
in only three
through
into a complex
from Theorem
to avoid
map
is constant.
By Theorem
planes
A holomorphic
goes
position
in P,.
order
1 for P..
restricted
from
to hyperplanes
class
of hyper-
is considered.
If six hyperplanes
in general
dimension
of the image
say
H
H. n
planes
in just
CI H
the line
and
H
two distinct
tion of a seventh
to intersect
position
is not further
hyperplane
O H
points.
O H,,
There
in general
the system
in P,
reduced,
are omitted,
as a line
intersects
the collection
is no slack
position
of hyperplanes
the permissible
through
in this
case
with respect
in three
distinct
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two triple
points,
of six hyperso that
the addi-
to the others
points.
forces
By the
94
M. L. GREEN
classical
Picard
3. Proof
theorem
theorem,
/ is then
of Theorem
on linear
constant.
1. The only analytic
relations
among
[july
fact needed
exponentials;
the rest
in this
section
is linear
is the
algebra.
We are
to prove
Theorem
planes
1. Let
in general
linear
subspace
Proof.
Let
position,
k > 1. Then
< [n/k],
(/0,
If / is to miss
•••,/)
in general
these
map that omits
the image
the brackets
be homogeneous
coordinates
position
be
then
\"Ln_.a.
integer.
of /: Cm —►P . Let
z . = 0} where
exp (g ) where
in a projective
mean greatest
"%."_. a. f. is never
an exponential
n + k hyper-
of f is contained
where
hyperplanes,
is therefore
This yields
be a holomorphic
of dimension
n + k hyperplanes
expression
f: Cm—» P
the
t = I, • • • , n + k.
zero
or infinite;
the
g : Cm —►C are holomorphic.
n + k equations:
n
Z aU U = exP^z^'
t = 1, • • • , n + k.
i =0
We group
they
the
differ
g
into collections
by a constant.
so that
g
there
are
Assume
and
g
are in the same
s such
clusters,
cluster
iff
each one as large
as
possible.
The equations
from each
side
cluster.
is always
altered
are now relabeled
so that
Equations
so that
are multiplied
g.,
•••, g
ate representatives,
by constants
so that
one of the functions
exp (g ), r = 1, • • • , s.
/.,•••,/
the set
partition
il,
the right-hand
The notation
• • • , n + k\ with
one
can be
the property
that
n
Z
^/z=eXP(^
iff teir-
i =0
The following
properties
of hyperplanes
(1) Any n + 1 of the vectors
a = ia
(2) Any
fl
cients
n + 2 of the vectors
is zero
(or else
there
Now, if the complement
or more elements,
it would
of n + 2 indices
relation
among
and after
linear
the coefficient
/
equation
possible
consisted
coming
among
relation
between
\l,
to pick
a subset
of precisely
induce
a linear
from the same
which
Such an equation
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none
are recalled:
independent.
of whose
relation
cluster,
contradict
coeffi-
n + 1 of the vectors).
S oí \l,
one element.
is a consequence
would
in P
• • • , n + k\ contained
the exp (g ). The relation
) is nonzero,
element.
in the set
position
) ate linearly
a linear
be a dependence
of S would
the terms
of exp(g
a unique
become
, ••• , a
satisfy
of any
S C\ I
the vectors
we regroup
a nontrivial
contains
so that
would
in general
• ■■ , n + k]
The linear
among
there
n + 1
the exp (g ),
would
is nontrivial
of the fact
the theorem
still
be
because
S P\ I
on linear
1972]
HOLOMORPHIC MAPS INTO COMPLEX PROJECTIVE SPACE
relations
of exponentials,
g
by a constant
differ
for after
most
for all
[in + k)/k]
in
volving
in the same
Tr
/.
cluster
S"z=0xAa. i,t . - a.i,t+\
yields
#(T) -s
the equations
we chose
of general
position.)
of codimension
pick
to form
4. Sharpness
T were
significant
be realized
it is indeed
linear
To see
let
dimension
of P
s = [n/k\
by vectors
be a partition
that
the
/..
(The
the original
which
72+ 1
was a consequence
subspace
of P
So the image
of / is con-
Q.E.D.
the dimension
[n/k]
than a geometrically
possible
bound
position.
of the image
of Theorem
1 may
and is always
In fact,
attain-
ta/Zs] can always
of a nondegenerate
and not merely
containing
among
s clusters
of proof rather
in general
the hyperplanes,
subspace
this,
be represented
I
omitting
in-
equations
s - 1.
the sharpest
72 + k hyperplanes
of the
in a linear
< [n/k].
1. Although
for each
equations
S — Ï < [n/k].
of the method
as the actual
to Pn
so
equations
adjacent
with end product
independent,
of dimension
s is at
T of tz + 1
linear
subtract
from the fact
linearly
of dimension
of Theorem
number,
independent
of / is contained
s < [(72+ k)/k],
to be a consequence
ed for any given
follows
So the image
subspace
any subset
t, t + 1 £ T r . Doing6 this
independent
in the
k
T = T D I . Then the #(T )
out the exponentials
know
in a linear
IS*
To do this,
to cancel
linearly
/
at least
of clusters
and
integer.
#(T ) — 1 linearly
are independent
no g
of any
contains
the number
greatest
of the exp (g ), for we can
n + 1 - s, hence
We already
1 ,,•••,
yield
= 72+l-s
equations
minimal
s.
/
and none
)/.'z = 0 where
reason
CLn/
is to interpret
each
only the
appear
mean
of \l, • • • , n + k\. Write T = Ur^r> where
equations
tained
Therefore,
In consequence,
the brackets
in clusters,
so the complement
at most 72 elements.
r = 1, • • • , s.
where
All that remains
equations
are collected
for t ¿é u. Contradiction,
set Í1, • • • , 72+ k\ contains
equations
the terms
95
map from
as the dimension
of the
the image.
+ 1. Let the
in homogeneous
n + k hyperplanes
coordinates
of 11, •• • , 72 + k\ so that
in general
a = iaQ , • ■ • , a
each
/
m
contains
position
). Let
k or more
elements.
Select
holomorphic
functions
g
: CL"'
—►C,
m = 1, • • - , s, so the map
/ 1
(expgj,
example
rWfe]^
• • ■ , exog):
gx = zv
CLn
p^ so that Víw0ai¿i
t = 1, • ■ • , 72 + k. If we can
to only
72+1
the properties
Let
—> PrAj
■■ ■ , gs_ , = zs_v
equations,
= dtexpigm)
find nonzero
independent
d
(such
maps
a map /=
where I £ ¡m and d^O
so that
and compatible,
these
abound—for
ifQ, ■■■ , fj:
fot all
n + k equations
then
reduce
/ will be a map with
we seek.
r. = (r.
hyperplanes,
is nondegenerate
gs = 1). We seek
, ..-, r
- , ) be a basis
for the linear
relations
among
the n + k
b = \, • • • , k - \. We want to pick d , t = 1, ■■• , ?2+ k, to satisfy
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
96
M. L. GREEN
the system
[July
of equations:
2* rtbdt = °
iot ail m = 1, • • • , s and b= !,•••,
k - 1.
teim
This
is a system
get by keeping
of ik - l)s
m fixed.
k unknowns.
Finding
d.jt
0. It would
tion
of the equations
hypothesis
a nontrivial
position
n + 2 of them.
I . Therefore
the situation
d =¿ 0, tel
share
a solution
were
(n + l) x (n + l)
tion,
then
Aif)
/.
its
image
ih).
we started
position
Then
I..
picked
a solution
that
lie in
with
position.
Because
the
we can put the solutions
together
to
n + 1 of them.
vector
with
no d
£"
a./.
Letting
this
'—*P
subspace
zero.
by general
posi-
t = 1, ■•• , n + 1.
to be the definition
goes
of P
being
= d exp ig ),
A be the
id t exp (gm))>
gm, we take
of maps
constructed,
of the complex
line to P
existence
was pointed
of n elements
in which
of
nondegenerately
and omits
the
map from
lies
indexed
C to this
we remark
minus
to
n + k
that
• • ■ , 2n}, L
the
[5].
those
We take,
through
and the corresponding
line omitting
in
n remaining
is the one passing
by /,
for k = n we
2n hyperplanes
out by P. J. Kiernan
of il,
the image
of the hyperplanes
Any nonconstant
at least
must
of the subsystem
of general
/: C'"
linear
The
involves
have
entries
all
combina-
any relation
1 < t < n + 1, nonsingular
the
one with
linear
rb must
) >
with.
maps
the line
of intersection
of the
n + k equations
h is the column
of the class
whose
the
(a. ), 0 < z < n,
s = 2, /. = any subset
ments.
prevent
the first
dimensional
get the nonconstant
as
that
2 of the nonzero
in common,
to (say)
Having
[n/k]
As an example
general
which would
from the construction
in an
hyperplanes
combination
—
set up so that
matrix
A~
It follows
m
if some
we
in #(/
one of the variables.
implies
out by the assumption
= h, where
/=
only
) > k, at least
subsystems
but we want
only
d , t = 1, • • • , n + k, of all the equations
+ k, reduce
Therefore
a solution
involved
s
of k - 1 equations
is no problem,
such
any linear
no variables
The equations
t = I, •••,«
#(/
, is ruled
s subsystems
obtain
As
up into the
of the hyperplanes
Hence
entries.
breaks
consists
solution
to find
of the subsystem
n + 2 nonzero
all
that
subsystem
be impossible
of general
at least
equations
Each
ele-
the point
point
two points
gives
for
an
example.
5. The Nevanlinna
niques
that
enliven
and simplicity
properties
Let
follows
complex
An ideal
carries
over
reference
we have
function.
in mind,
tech-
the elegance,
Even
its most
so it is these
power,
elementary
we will
briefly
is [6].
be a meromorphic
to several
Among the many beautiful
only a few can rival
characteristic
for the purposes
/: C —> P
function.
analysis,
of the Nevanlinna
suffice
recapitulate.
characteristic
function
variables).
of one variable
Consider,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
for a £ P.,
(much
of what
1972]
HOLOMORPHIC MAPS INTO COMPLEX PROJECTIVE SPACE
72(7, a)=
#jz
£ f
rr
1ia)\ \z\ <r\
nit,
a) - niO,
Ai^r' ^ = Jo-7-+
ju
(
counting
a)
r
multiplicities,
,
n '
1
dd
+ fiz)-
772(7, a) =
ii a/oa,
/|,|=7l0g+l/U)l^
log x = max (log x, 0).
ing function
function
a) is called
it measures
how closely
the integrated
mir, a) is called
/ approaches
the value
count-
the proximity
a on the circle
r. If we let
it is a consequence
of a in the sense
First
Nir,
in r. The function
Tir) = Nir,
then
If**-,
The function
and is continuous
because
of radius
gr*
,1
./Ul=7l0g
where
97
that
Main Theorem
characteristic
We can define
Tir)
Tir)
so that
theorem
the difference
of Nevanlinna
function
normalized
of Jensen's
a) + mir,
that
this
of any two such
theory
way.
fp a) = 1. If B(r)
expression
is independent
is bounded.
and it allows
by the above
a different
a),
This
us to define
is the
the Nevanlinna
equation.
Let
a> be the Fubini-Study
is the disc
of radius
metric
on P.,
t, then
Tir)= JoT Jf BU)/*U*.t
Thus
Tir)
is the logarithmically
of radius
< r, where
time
are hit.
they
(1) Tir)
in the image
representation
is a continuous,
(2) lim ^^Tir)
Some other
that
points
This
later
volume
of the images
repeatedly
assumed
of the characteristic
increasing
= 00 unless
fairly easy
will be used
averaged
function
/ is constant,
properties
under
/ of discs
by / are counted
function
shows
each
that
of r,
in which
case
of the Nevanlinna
it remains
bounded.
characteristic
function
are
(3) Tif + g, r) < Tif, r) + Tig, r) + log 2,
(A)Tifg,r)<Tif,r)+
Tig,r),
(5) T(l//,
r) = T(f, r) + constant.
Two less
elementary
functions
of derivatives
two conventions
after
of R
Air) = OiBir))
excluding
(6) Tif',
be needed,
of finite
will
in this
mean
total
and
to the characteristic
/. In order
lim
a constant
00A(r)/ß(r)
as values
c so that
Then
is the derivative
to state
and the next will
are excluded
exists
as above.
r) = OiTif, r)) where /'
section
B(r)^0
measure
will mean that there
a set of intervals
relating
of a function
to be used
Air) = o(ß(r))
a set of intervals
will
and exponentials
of notation
Convention.
wise,
properties
of /,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
these,
assist
us:
= 0 after
of r. Like-
Air) < cBir)
98
M- L. GREEN
(7) for / a nonconstant
The exceptional
in Nevanlinna
appearing
tently
intervals
theory,
lemma
is likewise
the congenial
tervals
Because
used
theory
to prove
of entire
duce
have
linna.
Since
mathematics.
Theorem.
orthodox
1897,
approach
reason
I will give
In the course
classical
Let
of finite
symbols
exceptional
The theorem
that
theorem
goes
back
proof,
will re-
to Nevan-
of dimly remembered
albeit
somewhat
in one variable
of Picard's
sinews
in the
of proof
a complete
in-
on exponen-
The results
to which
of it, a result
total
retain
The analytic
section.
we are aiming
0
and indeed
into the penumbra
improvement
on exponentials
of sets
as the
of proof.
be proved.
and the method
has fallen
intervals
o and
functions.
variable
from
is due to a persis-
number
counterparts,
of one complex
many of the proofs
be quarantined
the modified
1 and 2 will itself
since
For this
to E. Borel's
of a finite
measure,
r)).
exceptional
in this
their
The theorem
the union
plague
presence
such
will be our concern
ent from the originals.
similar
Their
exacts
of entire
known
then,
here
of exponentials
functions
been
intruded
can most often
of the method
Theorems
r) = oiTie',
they
no modification
relations
of the main theorems
tials
Tif,
have
that
total
of their
necessitate
6. Linear
Borel
of finite
properties
will
function,
of the theorems.
of Emile
of its conclusion.
measure
which
but fortunately
in the statements
useful
price
holomorphic
[July
differ-
will surface
theorem.
for is
g. : Cm —> C be holomorphic
functions,
i = 1, • • • , n, satisfying
identically
expig)
Then
for some
Proof.
The first
the functions
restriction
distinct
of exponentials
tinct
g¿\£
line.
— g'\f
of the
B..
contains
an open
of lines
set on which
and the function
set.
in this
consider
g\c
still
on all of Cm.
for some
to hold
for every
takes
out a region
completes
dis-
_,;
B .. ate closed,
g .(0) - g (0).
This
the
B.. = \^eP
theorem
sweeps
g- - g- is the constant
hence
Let
as the
of
On £, the equation
holds;
is assumed
theorem,
the origin
case,
the origin.
But now the identity
through
are identical
the domain
gXo) - g (0).
the theorem
j. By the category
constant.
m = 1, where
the result
functions
Because
i
¿; through
is the constant
IS constant!.
0.
is identically
to the case
Assuming
for the restricted
set in the set
ing an open
1
g . : Cm —» C to a line
is all of P
exp(gn)=
g. — g.
is a reduction
i and /, g\f-g\g
\Jii.-Bi.
open
step
is a complex
of the
+•••+
i, j, the function
line,
one
over,
for an
in Cm contain-
Thus
the constant
the reduction
to one
through
by one
variable.
The inductive
term,
differentiating,
eventually
process
dividing
a contradiction—may
sketched
through
in the Introduction—dividing
again
getting
now be applied.
an ever
Before
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
shorter
tackling
equation
the real
and
thing,
it
1972]
HOLOMORPHIC MAPS INTO COMPLEX PROJECTIVE SPACE
is instructive
of finite
to work out the case
order,
i.e.
exponentials
where
the exponentials
of polynomials.
exp(P1U))+.-.+
Dividing
through
by the last
99
in question
are entire
So assume
expiPNiz))=
term and differentiating
0.
gives
N-l
£
iP'tiz) - P'Niz)) expiPfz)
- PNiz)) ^ 0.
z'=l
The coefficients
equation
can vanish
is nontrivial.
only when
Continuing
P {z) — PNiz)
the process,
is constant,
at the
so the new
(zV + 1 — ze)th stage
we get
k-1
£
where
the
tion,
if we lose
This
is impossible
nonconstant
any terms
it means
unless
polynomial
P -iz) - P Az)
duction,
RI.U)exp(P.U)-PjkU))=
¿=1
R .iz) ate nonzero rational
leads
P iz) — P Az)
is constant,
function.
we may as well
inexorably
For at the (¿V- l)th stage,
assume
can only happen
line,
a P .iz) - P .iz) is constant,
instead
if P Az) - P Az)
the exponential
As the proof
we have
just
of a
is done
tried
if any
the in-
to reject.
P2iz))=
is constant.
1.
Thus
somewhere
along
the
i'w* j.
proof will work to show
of our equation
of constants,
equa-
we get
This
The same
since
this
is constant.
it is not and continue
to the conclusion
R1(z)exp(PI(z)-
coefficients
When we differentiate
R .iz) exp ÍP .iz) - P Az))
is a transcendental
is constant,
which
functions.
some
1,
that the conclusion
of exponentials
holds
of polynomials
if we assume
to be rational
the
functions
so
N
Y
Riz)expiPAz))
=0
z=0
will imply
that
P. - P.
is constant
will be generalized.
products,
derivatives,
and the fact that
and quotients
the exponential
only if it is constant.
have
Nevanlinna
functions
function
of the rational
in the preceding
of rational
of a rational
functions
functions
function
remain
of exponentials
then
the role
functions.
To be precise,
what
can be shown
whose
is
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
result
that
sums,
rational
functions,
function
coefficients
o" of the characteristic
the properties
will perform
stronger
the fact
can be a rational
that are "little
themselves,
section
z, /'. It is this
is how to generalize
If we take equations
characteristic
of the exponentials
listed
for some distinct
The difficulty
of the characteristic
of the needed
features
100
[July
M. L. GREEN
Theorem.
tions,
Let
g.:C
—►C be entire
functions,
h.:C
—» P
meromorphic
func-
i = 1, ■• • , N, satisfying
(1) hlexpigl)
+ ••• + hNexpigN)
= 1,
(2) Tih ., r) is o(S _ T(exp(g .), r)) for all i = 1, • • • , N, where T denotes
Nevanlinna
characteristic
we may exclude
function
as values
and little
of r a set
o has its usual
of intervals
of R
meaning
of finite
except
total
the
that
measure
ias in § 5).
Then
some
proper
nontrivial
linear
combination
of the terms
h. exp(g .)
is constant.
Remark
further,
1. The relation
of the conclusion
for the hypotheses
cannot
are not hereditary.
gx = z4, h2 = - 1, g2 = z4 + z3, ¿3 =(ez + l)/ez,
satisfy
the theorem's
while
hypotheses.
Tih A) is o(2¿ = 1 T(exp(g¿),
T(exp(g4)>
Remark
2. There
relation
mills
theorem
and thus
now applies,
o(T(exp
(g. - gN)))
be done.
constant,
as is the sum over
to - 1, they
z different
from
z, there
N, exp(g¿
exists
a jpti
The
Assume
h^xpig.)
thus
until
for
again
This
to prove
In fact,
strong
again
N = 1, as
, r) is assumed
„ exp (g . — g
) is
add up
that,
gives
of
for some
a conclusion
it says,
a simple
for
induction
The
is
"clusters"
form.
on the length
h exp (g.)
to be
case
on whichever
in that
from the weak.
to this
ate
the two constants
and nonzero.
constant.
Tih.)
in which
the conclusion
we are trying
is clear
is easy.
X. ~ , exp (g . — gN) =
so the
may be used
we reap
was grist
oiTih
of the relation,
s 1 implies
, r)), which
Tih.,
cannot
r) =
be,
notwithstanding.
true for lengths
as above
x.2 = h.Z + h 2°.g..Î
is constant.
the requirements
theorem
induction,
Tih
X
of S. Since
Another
as
constants,
combination
g. - g.
that
g. - gN is constant,
form of the theorem
1 are equivalent
the theorem
1, with hypotheses
with
theorem
intervals
linear
the hypotheses
The verification
h. being
unless
the theorem
the strong
of Theorem
the
The
for which
may be rephrased
- gN) is constant
than
x), r), but then
exceptional
0 where
r) +
exponentials
few sections.
is nonzero
ol the theorem.
be used.
Tiexpig
),
.) m 1, but
o of T(exp(g,),
among
the complement
be zero.
stronger
separates
of the proof
both
proper
combinations
superficially
Proof
So some
cannot
the two linear
all that
+ h ,exp(g
subcase
relations
of the first
we would
every
è,exp(g,)
an important
on linear
2. _. exp (g .) = 0. This
- 1. The present
constant
h. = expCz
g3 = z, h^ = - 1, g4 = z, which
in fact,
r)), it is not little
is, however,
theorem
for the combinatorial
Assume
be shortened
for example
r)-
are hereditary—the
will
Then
necessarily
Consider
and
We note,
But this
shorter
than
N > 1. Differentiating,
N, and let
we get
X
S._
as we are working ö in one variable,
would
of the conclusion
be a constant
would
be met,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
proper
linear
¿exp
x.exp(g)
'
that
=
x.I = 0 iff
combination,
so we may assume
(g.) =
and
this route
1972]
HOLOMORPHIC MAPS INTO COMPLEX PROJECTIVE SPACE
of escape
is closed
by the last
amenable
o(S
off and
term and get
function
o(X.
listed
of the inductive
for all
but this
section
rewrite
equation
relation
that
would prove
Tix./xN)
is
of the character-
T(x./x„)
is
enough.
is what makes
we have
the proof work—that
on T(x./x„)
is good enough
ig.)) is 0Í1N~ l Tiexp (g . - gN))), which is
the latter sum contains
the original
we may divide
provided
to give
this
the estimate
after all. It turns out that S/^Tiexp
since
good
surprise—and
a brief bit of legerdemain,
surprising
hypothesis
combine
is not quite
We are in for a pleasant
after
Therefore
z = 1, • • • , N - 1. The properties
in the last
T(exp(g.))),
zero.
— %.~ Ax ./xN) exp ig ■- gN) = 1. This
to an application
~ ¡T(exp ig. - gN)))
istic
x. is not identically
101
fewer
functions.
To see
this
estimate,
we
as
N-l
exp(-£N)==
Z
hiexP(-gi- 8N) + hN-
i=l
Therefore,
T(exp(gN)).
Tiexpi-gN))
N
N-l
< ¿j Tib)*
i'=l
Using
the hypothesis
Tiexpig.-
on the
£
£=1
Tih),
Tiexpig.-
and noting
that
gN)) + constant.
T(exp (g .)) < TiexpigN))
+
g N)), we get
/N-l
TUxpigN))
< oiTiexpigN)))
\
-t o I £
T(exp(g.
- gN))\
N-l
2j
But the little
o's
Tiexpig;-
can be incorporated
gN)) + constant.
into the dominant
terms
to give
N-l
T(exp(g/V^
< 2 Z
T^exP^(
- gN)) + constant
z'= l
(except
possibly
Since
for exceptional
T(exp(g¿))
intervals).
< T(exp(gN))
+ Tiexp ig{ - gN)), we conclude
l,N_xTiexpig))
is 0(^:{T(exp(g¿-gN))).
As we now know that
shorter
linear
relation
all of the necessary
we constructed,
combination
of the shorter
hypotheses
the inductive
equation's
hypothesis
terms
Z a/V*N)exP^; ~ 8n^= c'
are verified
ensures
is constant,
that a proper
i.e.
SSll, ..., N- lj.
i £S
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
for the
102
M. L. GREEN
This may be rearranged
as
2j fl¿x¿exp(g¿)
- cxNexpigN)
i£S
that ih . exp (g .))' = x.exp(g.),
Recalling
[July
2^ a.h{expig^
= 0,
fl¿'s and
c constants.
we may integrate
- chNexpigN)
= d,
and get
da
constant.
i£S
But this
tion,
is a constant
proper
so we are done.
Although
it is not really
ing theorem
drops
Theorem
Let
linear
germane
be holomorphic
independent
functions
jfák,
over
A polynomial
of the original
of this
paper,
7=1
\l=l
"
on linear
that
This
among
= °>
eral
position
that
can happen
the exponentials
may be rewritten
aij e Z' cj € C*' M some
of exponentials,
integer-
this
implies
that,
This
elements
of a linear
since
the right
bound
- %kMt =
hyperplanes
2. An instance
F:Cm—>
be proved
James
in other
than gen-
of the strange
Picard
spaces
in P
to the
's theorem
of view
number
things
omits
three
hyperplanes
image
lying
in a hyperplane.
has
shown
and in fact
that
must have
image
lying
P . parameterizing
in this
paper
the images
to have
it should
reduces
a map omitting
to
three
in one element
the pencil
clearly
for any set of hyperplanes
containing
be nice
that
2 has
and miss-
applies.
developed
of steps
on the intersection
is conjectured
P
by my methods,
Carlson
of divisors
It would
based
formula
omitting
of codimension
a map is induced
of linear
hyperplanes.
for this
pencil
and point
in a finite
the dimension
those
theorem.
so that
The method
determine
map
subspace
may easily
Picard
points,
of maps
by Theorem
A holomorphic
proposition
of the pencil,
ing three
settled
for some
the hypothesis.
The case
in a linear
the classical
as
is this:
Proposition.
intersecting
contradicts
is hardly
1-linear
are algebraically
j
relations
remarks.
functions).
no nontrivial
exp if ), • • • , exp (/N)
loge. + ^Ni=laijfi = logck + ZNi=la.kf., and hence S^tyj
7. Further
the follow-
of holomorphic
on Cm such
Then
relation
/ N
\
j V aijfi)
l°g ch ~ loge.
equa-
C.
M
V c.exp
'
of exponentials
functions
of them is constant.
By the theorem
to the main topic
independence
combination
Proof.
of the terms
out for free:
(algebraic
/,»•■">/«
combination
Q.E.D.
something
properties
follow
of maps
better—a
allows
a sharp
into
P
one to
bound
omitting
general
formula
of the set of hyperplanes.
by the methods
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
for
developed
Once
here.
1972]
HOLOMORPHIC MAPS INTO COMPLEX PROJECTIVE SPACE
Theorems
that
1 and 2 might
the hyperplanes
each
have Nevanlinna
curves.
This
ear relations
defect
should
follow
on P
of degree
Another
or polydisc,
class
we used
functions.
to pursue
Considerable
is to consider
maps
omitting
on this
work with
defined
generalizations
of the Schottky-Landau
It is a pleasure
to acknowledge
of holomorphic
the theorem
of maps
progress
joint
the hypersurfaces
theory
to prove
is the case
[2] and his recent
of problems
and obtain
the hypothesis
that
of the classical
of holomorphic
direction
thesis
by replacing
assumption
from the theorem
n + 2 or greater.
Carlson's
improved
by the weaker
1 in the sense
of exponentials
A more interesting
in James
be slightly
are omitted
103
on lin-
divisors
problem
Phillip
is made
Griffiths.
only locally,
theorem
on a ball
and results
on hyperbolicity.
Acknowledgments.
of Phillip
Carlson
Griffiths.
Of great
help,
too,
have
been
the generous
many conversations
assistance
with
James
and H. Wu.
BIBLIOGRAPHY
1. E. Borel, Sur les zéros des fonctions
entières,
Acta Math. 20 (1887), 357—396.
2. James Carlson,
Some degeneracy
theorems
for entire functions
with values
in an
algebraic
variety,
Thesis,
Princeton
University,
Princeton,
N. J., 1971.
3. S. S. Chern, Proceedings
International
Congress
of Mathematicians
(Nice,
1970).
4. J. Dufresnoy,
Theorie nouvelle
des famillies
complexes
normales.
Applications
à l'étude des fonctions algébroides,
Ann. Sei. Ecole Norm. Sup. (3) 61 (1944), 1-44-
MR
7.289.
5.
Peter
Kiernan,
Hyperbolic
submanifolds
of complex
projective
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Proc.
Amer.
Math. Soc. 22 (1969), 603-606. MR 39 #7134.
6.
R. Nevanlinna,
Gauthier-Villars,
7.
Le théorème
-,
de Picard-Borel
et la théorie
des fonctions
méromorphes,
1929.
H. Wu, The equidistribution
Princeton,
8.
Paris,
theory
of holomorphic
curves,
Princeton
Univ.
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N. J., 1970.
An n-dimensional
75 (1969), 1357-1361.
extension
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Bull.
Amer.
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MR 40 #7482.
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