Jun 18, 1983 - (n+ l)-dimensional, oriented, real analytic manifold such that 8r is an orientation ..... PROOF. Define z x y. Then. T where the j coordinate of the vectors z, x or y is positive if ..... A meromorphic function f on D(b) is the quotient of two holomorphic functions g ... These properties will be ...... Consequently log(b x.) ...
Internat. J. Math. & Math. Sci. Vol. 6 No. 4 (1983) 617-669
617
VALUE DISTRIBUTION AND THE LEMMA OF THE LOGARITHMIC DERIVATIVE ON POLYDISCS WILHELM STOLL Department of Mathematics, University of Notre Dame Notre Dame, Indiana 46556 U.S.A.
(Received June 18, 1983)
ABSTRACT.
Value distribution is developed on polydiscs with the special emphasis
that the value distribution function depend on a vector variable.
A Lemma of the
logarithmic derivative for meromorphic functions on polydiscs is derived.
Here the
Bergman boundary of the polyd+/-scs is approached along cones of any dimension and exceptional sets for such an approach are defined.
KEY WORDS AND PHRASES.
Value distribution theory, valence function, Jensen formula,
characteristic function, counting, function, spherical image, compensation function, Lemma
of the logarithmic derivative, and approach
cone.
1980 AS MATHEMATICS SUBJECT CLASSIFICATION CODES. 32H30, 32A22. i.
INTRODUCTION. Value distribution for polydisc exhaustions has been studied by Ronkin [i],
Stoll [2
and others.
They emphasized the growth of entire holomorphic and
meromorphic functions and the representation of canonical functions to a given divisor in
Cn.
C. Ward Henson
For applications to mathematical logic, Lee A. Rubel and
[3], [4] inquired if the Lemma of
the logarithmic derivative could
be established for a meromorphic function on a fixed given polydisc.
In the classical one variable theory, the Lemma of the logarithmic derivative has been one of the basic tools for a long time.
analogous Lemma was proved only recently.
In several variables, the
For ball exhaustions of
En,
Vitter
5
proved the Lemma for differential operators with constant coefficients and derived the defect relation for meromorphic maps f
n
/
m
See also Stoll
6].
618
W. STOLL
Vitter’s Lemma extends easily to differential operators with polynomial or, with
proper modifications, to differential operators with entire coefficients. Earlier, a weak version of the Lemma was proved by Gauthler and Hengartner [7] for the special operator
D’
/z
z
I +
+
Zn /Zn,
extension to a general differential operator.
which does not permit the
D’
The differential operator
suffices
for meromorphlc functions, but imposes unnecessary restrictions for meromorphlc maps
f
cn
/p
m
Recently, Shlffman has shown how to derive Vitter’s Lemma from the
[8],
result of Griffiths and King
which also can be interpreted as a Lemma of the
Biancoflore and Stoll [9] gave an elementary proof of
logarithmic derivative.
The same method will be used to obtain the Lemma of the
Vitter’s result.
logarithmic derivative for polydiscs.
A theory should reflect the intrinsic algebraic, geometric and analytic Our development of
structures of the mathematical landscape under consideration.
value distribution theory on polydiscs will adhere to this principle.
This is
an important feature of this paper, which is mostly self contained and requires only a minimal knowledge of several complex variables.
Let us outline the main result. by its coordinates define
II
(Iz
Denote by
II
z
nl).
llrll
The euclidean space the length of r in
/
Rn
partially ordered
For z
(z
z
I
n)
e
n define
For 0 < h e
RnO,)-- (r
n is
n
o 0, then
Fn.
(r,F)
If 0 < r
#
0, then o
e
T(p,n) with p
e
(2.17)
El n]
Take 0 e
Cn-p
Abbreviate
Then
(r,F)
(F
m< If r
() >
(2.18)
0, then
(0,F) Take p e lq[ l,n] and T e
Hence F
o0)flP
TZ
is defined on
exists for almost all
z
T(p n).
).
D
F(0).
(2.19)
Take z e m
then
By Fubini’s theorem
Tz0D)
M(r(r)
F
and we have
M(r,F)
M((r),H).
(2.20)
More explicitly, this is written as
M(r F)
M(n(r)
M( (r)
F
Tz ))
(2.21)
If r > 0, then
m
(2.22)
zm m
Sometimes we shall write (r,F) as in (2.18) even if r has some zero coordinates. Then
(2.21) writes as
in
(2.22) which
is more instructive.
VALUE DISTRIBUTION ON POLYDISCS
3.
625
PLURISUBHARMONI C FUNCTIONS.
n.
Let B be a subset oflR
B +R u {_oo} is said to be increasing
A function g
+ if g(x) _
0
Naturally, if we consider divisors on complex manifolds and spaces the definition of divisors has to be localized.
+
Let V be a complex vector space of dimension m Then
,
operates on
V,
by multiplication.
> i.
Define
V,/,
The quotient space (V)
{0}.
V
V,
is a
connected, compact complex manifold of dimension m called the compl.e.x projective
(A)
(A n
{IP(z)
V,)
0
vector functions V
w(b)
vector functions v
O,(b,v)
V
^
W- 0.
V,
The residual map IP
s_pace of V.
+z
V.
/
A}.
Define
and w e
Let O(b,V) be the set of all holomorphic
O,(b,v)
..representation of f.
Two holomorphic
are called equivalent v
O,(b,v)
from)(b)
{0}.
O(b,v)
O,(b,V).
This defines an equivalence relation on
is said to be a meromorphic map
If A c_ V, define
/(V) is holomorphic.
into
w,
if
An equivalence class f
(V) and each
V in f is said to be a
The representation v is said to be reduced if for each
representation w of f there exists a holomorphic function g such that w
meromorphic map has a reduced representation and if v and
gv.
are reduced
representations, there are holomorphic functions h and h on )(b) such that v
v
hr.
g
and h e O (b) has no zeros.
Hence hh
Therefore the non-negative divisor
the choice of the reduced representation
Let v and
.
Moreover w
w g
Each
gv
ghv.
hv and
Hence
is well defined independent of
Moreover w is reduced if md only if
be reduced representations of the meromorphic map f, then the
in det e rmin a cy
If is well defined
[z
ID(b)
v(z)
analytic, with dim
f()
If
(v())
O}
{z
n- 2
(z)
]D(b)
If z
((z))
)(h)
IP(V)
O}
If
(3.16)
then
(3.17)
631
VALUE DISTRIBUTION ON POLYD%SCS
)(b)
is well defined and the map f
If
(V) is holomorphic.
+ O,
()
e )(h) with
representation of f and if
/
where
(V).
e
V,
The projective space (W)
O.
(V*) and if f()(h)
I, then (W)
Take a e (V*).
Then E[a]
If)
is said to be
ov depends on a and f only and
g
linearly
V,
then
is called the
The valence function of f for a is defined by
-
N
a(r’q)" f
(3.18)
If w is a representation of f, there is a holomorphic function g
w
P()
(ker ) is a hyperplane in
E[a], then f
Nf(r,q;a)
Hence
is called a
Then a
If v is a reduced representation of f and if
The divisor
a-divisor of f.
m-
E[a] is a bijective parameterization of all hyperplanes in (V).
non-degenerate for a. o v
V
is a linear map
The map a
If a
.
If p
Let V* be the dual vector space of V.
hyperplane.
IP(()).
then f()
Let W be a (p+ l)-dimensional linear subspace of V. is a project.ive, plane of dimension p in IP(V).
If m is any
v.
By definition
w
g
a
Take a positive hermitian form (i)
V
V
/zlz ).
llzll
gv.
Hen ce
m + f"
(om
The associated norm is defined by
0 with w
(3.19)
,
called a hermitian product on V.
If x
V and z
V, we have the
Schwarz inequality
(3.20) If
e V* and
B
V*, vectors
(z)
=
(zla)
e V and
B(z)
h
V exist uniquely such that
(zlh)
for all
e
V.
(3.21)
A dual hermitian product on V* is defined by
(IB)
(alb)
(3.22)
W. STOLL
6 32
I,
If
If z
E
then
I1=11.
1111
(V) and a
e
Therefore (3.20) and (3.21) imply the Schwarz inequality
V,
() with
(z) and a
(V*), then z
and
V,.
The
projective distance from z to E[a] is defined by
(3.24)
and e.
independent of the choice of the representatives
+
The compensation function of f for a is defined for r e
0 -
0
or0(3( r (r))) O
If
and
then
(3.29)
-Tf(p,r)
Tf(r,p)
Tf(q,p)
Tf(r,q) The function
with
+
If r, p, q belong to
If.
q
n(b)
if 0 < q _< r < b.
(3.30)
r and decreasing in q
is logarithmic convex and increasing in
and continuous where 0 < q _< r < b.
Let m be a representation of f. holomorphic function g
$
Take a reduced representation v of f.
A
By definition
w
0 exists uniquely such that m
gv.
Dg"
Then
logllwll
)
]D
C
n
ii II lg"V’n
lglgiSn-
+
]D
]D 0 of
+
and
Fubini-Stud Kaehler
metric
We have
(3.33)
Denote the corresponding form on (V*) also by
.
Then
ram(a)
log
(3.34)
An exchange of integration implies
mf(r,a)m(a)
(3.35)
=i
Integration of the First Main Theorem yields
Nf(r,,a)n(a)
Tf(r,q)
(3.36)
a(V*) For V
m+l
ZlW + setting
write 1P
+
z w m m
m
1P(V) and 1P
m
IP(V*).
On
m+i
Identify the compactified plane
define (l:
{}
with IP
by
VALUE DISTRIBUTION ON POLYDISCS z
m(z,w)
W
if z
and 0
+
w e
identify
?I with
u
,
setting
and a
,
(Zl,Z2)
(I 0)
If e e
-
{oo} by
2
IP()
If z e
.
0 and
z
(z,l)
z
(2)*,
define
re(w,0)
(I,0)
re(l,0)
(3.37)
and (0,I)
el
2
and
setting
i#
if
2
az
Then (e)
2. i
IP()
and
0
then (z i)
(0 i)
635
with
(z,l) a and
if
0
I
(2)*
Also
z.
e(z,l)
+
z
a.
so_-
is defined by
Define
oo
by
Then
Ic(z,1) IIII ll
g
637
(=) there is a homogeneous polynomial P. of J
degree j such that
P. (z
g(z)
(4.1)
a)
J
j= (a)
g
for all z in a neighborhood of a in )(b) and such that
g
0 if and only if g(a)
(a)
0.
Ng+h(a) Therefore the map
Moreover Let on
)(b).
+
@,(b)
gh(a)
g(a)
(4.2)
Min(g(a),h(a)).
(4.3)
g(a)
if and only if h(a)
Then v
is a valuation of the ring
@(h).
0.
f
0 is a meromorphic function
where f
Let g and h be holomorphic functions on )(h) which are coprime at every
point of )(b) such that hf
meromorphic map f. u
Obviously
g(a) + Dh(a)
defined by g
be a divisor on ID(b).
0.
0 is a holomorphic function on B3(b) then
If h
gh(a)
P g ()
g.
Then (g,h) is a reduced representation of the
Let f, g and h be another choice.
@*(b) on )(b) without zeros exists such that f
reduced representation of f.
A holomorphic function uf.
Then
is another
O*(b) exists such that (g,h)
Hence v
Cons equent ly
9(a)
(ug,h)
(a) -D(a)= g(a) -h(a)
(vug,vh).
.
(4.4)
is well defined independent of the choice of f, g, and h and is called the
.
multiplicity of v a__t a. If
e
D(h) and p
(PVI
The map D(b)
defined by v
+
2
(=)
P)l(a)
+
2 (a)
The divisor v is non-negative if and only if (a)
v
(a) is a homomorphism:
7z, then
is the zero divisor if and only if
(a)
0(=)
_>
0 for all a
0 for all a e )(h).
identify a divisor v with its maltiplicity function a way of defining divisors on complex manifolds.
(a) and
o. )(h).
(4.5)
The divisor
Therefore we can in fact this is one
W. STOLL
638
Let f be a meromorphic function on ]D(h). a
f
is defined.
divisor
a
f()
Then
f
The function f
f(=)
f()
0 is holomorphic if and only if
-f().
0 and
fl
If
f2
a__t
a, the a-divisor
.
0, the
If f
a__t =.
is said to be the _multiplicity of f
holomorphic and without zeros if and only if
I/f()
If f
is called the a-multiplicity of f
is defined and
f
?I"
Take a e
> 0 for
(), and f
all
0 for all t e ID(I).
llf(i)
Also
0 are meromorphic functions on ]D() then
flf2 () flCa) + f2Ca)" The map
K,(b)
defined by f
/
/
Let V be a divisor on )(b).
9(z)
+
f(a)
(4.6)
is a homomorphism.
0 is called the support of 9 and denoted by supp 9.
0, then the support of
)(b).
Let R(supp ) be the set of regular points of supp
.
a.
a
and c 6R
p.
divisor v on D(b) such that
)(b) such that
f.
If
f
supp
f +
lq
of is
If 9 > 0, then
If.
Take a e
I"
Then
O, then supp
Take p
9
Then the function
{z D(b) () > 0}. Let f be a meromorphic function on )(b) with indeterminacy
supp
If a
R(supp ).
0 then supp
If
pure dimension n
is an analytic subset of
constant on the connectivity components of
Assume that f
.
The closure in )(b) of the set of all z e D(b) with
If
supp
is
Of
u {Z e ID(b)
u supp
Let
(D(c))
Then f
f(z)
If
a}.
(4.7)
f.
)(c) /)(b) be a holomorphic map. supp
9.
Take a
Let f be a meromorphic function on
0 is a meromorphic function on D(c).
The
pullback diviso r
q*())
(4.8)
foq
is well defined and independent of the choice of f.
pj
and
j
are divisors onD(b) with
(ID(c))
{
If
supp
9
V.3
>- 0, then *(v) >- 0. for j
i, 2, then
If
639
VALUE DISTRIBUTION ON POLYDISCS
(m(=))
i supp(pl91
+
and
P22
+
*(Plgl
.
all u that
_
En
For 0
+
If 0
()(t0(r)))
)(h).
said to be restrictable
,
Z
direction
For z
If u e
is defined.
and 0 -< t
exists such
supp w.
.
us for
R (r), the pullback divisor
variation on each compact subinterval of [0
If
>
(u)
as the finite sum
The function t
If 0
t0(r)
Take
(D(t0(r)))
)
n[z,t]
for all 0
is defined by
be a divisor onD(h).
9
if and only if
real analytic subset of ).
%()
En
and 0 < r < h, a largest number
Let
__t
+
) such that w is not restrictable to
the set of all
V[]
z
an injective linear map
ID
(4.9)
pl#*(l) + p2*(92).
P292
and if p. e
for
W. STOLL
640
n
Let f
z
pI vl+p22
[z t]
p n
i
0 be a meromorphic function on )(b) such that
- Rv(r).
Then [z]
Df.
t0(r)
For 0 < q < r
0, then
B and t
+ t(h
and
0 such that for each
log(I/6)
e B we have
< oo.
dB
(6.15)
B
The closure B is compact and contained in
n[h].
If
e
h + (
B, then b
N)
B
+ which contradicts (c).
Hence
Define the associated approach cone by
M
If
{ + t(h
B, then x < h and EXAMPLE I. Take
)
+ t(h
I, with p
with k
_>
B()
< m.
B(B)
and with
ran((1
B
For each %
(0,i] define
)b).
+
Assume there are constants c > 0 and q > 0 such that %
657
VALUE DISTRIBbION ON POLYDISCS
(0, I].
B(B(%))
_
< I.
_< t
>
q0 b > q
and
q.e.d.
c > 0
T(s,q0h
T(sb,q)
T(r,q)
Now, the Lemma of the logarithmic derivative follows easily, which constitutes the main result of this paper.
THEOREM 6.18. function on 3(b).
Take b
+
+n
U
un(b)
and q
Let M be an approach cone.
log+Idzf/ fl D
n
Let f be a non-constant meromorphic
log+Tf(r,q)
-< 17
lq[l,n].
Take
+
Then
+
(6.42)
19 log
for r e M.
PROOF.
Then c > 0 and s e
Take q > 0 with qb < q.
r > q and T(r,q)
>_ c
for all r
6
(0,I)
n
M +IR
For T(r,q) define p
M[s,l).
exist such that
by (6.32) and
@(r) as the largest number such that r < O(r)r
_
