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pq].B-l Zpq t ;-1
M[t)
~ 4(n + 1)3Y(r)(T f (r.s) + T (r,s) + ~) q gp
Which in particular implies the integrability of
(4) pq ) 8-1:: pq
over
MIt) for all t:> 0, which is remarkable. As seen in (A.162), the classical Ahlfors estimate involves the term • p,q +IlHq only (for p - 0). In view
-72-
of (B.63) it seems to be reasonable to drop take the term
2S
P
q+l
p+ 1.
q
measuring the movement of
+p+l,qll
1.
be d meromorphic map.
If - n.
general position.
B
g : M -
(m
with majorant
IJp - (gp}gE:IJ' (f,gp)
and
Assume thdt
is free for
L
(B.97)
Rp(r,!J)
-
n-l
L
q=O
Ppq(r,g)
(k - D(p,q))+
L
BEqJ
k
=
R q(r,g) P
IIJ > O.
IJp
for all
l
n-l
L
M.
Y.
(B.95)
q=O gE:q}
on
1,0)
Abbreviate
~
!P(V·).
B.
ilJ - Minlig I g E: IJ}
Assume that
P p(r,q})
m.
M.
Let
(8.96)
is in
can be identified.
(B4)
Take
is said to be
1P(V·)
[B5)
Take
A family
A is finite, and is said to be in general position if the map
defined by
88.
Cl)
is in g
£
I).
·79·
(8.98)
+ 3
(8.99)
q ..
£ > 0
and r > 0
for all value
> O.
~
21] C.
++
n
"" ~ Iog + 8 E Q}
T g (r,s )
0-1 ~ (k - D(p,q))+ .
Y(p,k)
Take
(p
O
Recall that
~
means that the inquality holds
outside some set of finite measure.
Then we have the
Second Main Theorem (Theo~em 8.31). (8.100)
[[p : 1]
~
If
term
+
3(: :
p - O.
then
NF (r,s)
1]]T
+ [0 ;
~]kC.(1
+
f
(r.S) + [: :
£)log
Bp(r,s) - 0
and
Y(r) +
[
y(p.k)r~p(r)
1
P +
~]R1CT(r.S)
1.
]
The remaining
1
is the classical ramification term which can be used to
n
truncate the intersection divisors at the level [
1 p
+
]
_
0
and
Sp(r,s) > 0
n.
If
p > O.
then
is a new ramification term whose
1
significance is not yet known.
In any Cdse, the terms can be dropped.
The sum of the compensation functions. the characteristic. the Ricci function, the major ant are well-known classical terms.
The term
AQ} p
measures general position and can be estImdted by (B.90) respectively (B.93) and (A.153).
The terms
Pp
and
Rp
are new remainder terms which
account for the movement of the targets in respect to
f.
They are
-80-
difficult to estimate, but vanish if Sp(r,CJj)
p - tg
is a classical remainder term.
for all
g
qJ..
£
The terms
These terms exist under natural
assumptions.
89.
The Defect relation.
p
Z[O,tqJ."
£
free for
Assume that [B.11 - [B.8} hold.
Assume that
L for all
g
£
the Nevanlinna defect of
Q}p Q}. f
Take
is in general position and that
Then for
T f(r,s) -
gp
00
for
r -
(f,gp)
00.
is
Define
by
(B.101)
The First Main Theorem implies
(B.102)
0
~
N
Ii f(gp) - 1 -
1 i m sup r~oo
Lf(r,s) +
T
Lf(r,s)
-g-,p'i;:--:~--:-"'t--:--:..-&...JP~--~~Tf(r,s) + T (r,s) &p
We also assume
(B.103)
Ricr(r,s) T f 1.
M - S
;a!:
a:
and the rcmk of
Let M
connected neighborhood function
~: U -
h(xl1O (xl
in which case
=
S
S
Let
V
40
Bnd
be a complex space of
be an analytic subset of
A holomorphic vector function
eJ.
to be meromorphic on
that
a:,
is defined to be zero.
V
~ (xl
if for each point
U
of
x
in
M
'0
XES
all
x
E
17 "A.
is called the pole ~
M A -
with V
is said
there is an open,
and a hoI om orphic vector
and a holomorphic function for
:
of
h ii! 0
We can take 10.
on A
U
such
maximal
·84·
c) Admissible bases. Let f: M !P(V) be a meromorphic map. - 0
such that
(8.120)
SIr) ~ cO(£,s)(1 + r)2m-l
m-l
z:
j -1
(8) For each £ > 0
and
s > 0
T .~..Ir + £,s)
for all
r > s .
J
there is a constant
c 1(£,s) > 0
such that (8.121)
S(rl ~ cl(£,s)(1 + (log (1 + r2))2)
m-l
z:
j =1
(e) If
Ord tit j < >..
..) > 0
for
T",.Ir + £r,s)
for all
r > s
J
j = 1, ... ,m -
1,
then there is a
such that
for all
r > 1 .
(D) If the functions
cpl'· .. ,CPm-l
c3 > 0
constant
are rational, there is a
such that
SIr) ~ c3 log r
(B.123)
for all
The Ricci (or branching) Defect of
(B.124)
If
f
Rf
=
11m
separates the fibers of
Noguchi (70).
f
r
~
is defined by
Np(r.s) sup T f ( r • s) ~ 0 .
r"'OO
K,
then
Rf
0
0
for
r
--+
< Ord f ~
00
is the lower order of (D) The functions
and
IN[l.m -1) .
E
exists such that
2 2 (I + (log(1 + r » )T~ (r + £r.s) -------......"T:-f-(~r-.-s...,,)-.....LJ------
(e) Ord ~j
00
cpl' ...
for
j -
--+
1. .... m - 1
0
for
where
r
--+
00
•
Qrd f
f.
,cr m- 1
transcendental growth, that is
are rational and T f(r,s)/log r
--+
f
has
00
for
r _
00
-88-
Assumtion [011] implies the transcendental growth of the (m-l)-dimensional component of a;m,
then [011) implies
Rr -
Therefore, if
is affine algebraic in
If there is a function
R f - O.
separates the f bers of
l'I'(supp p)
f.
g
II which
€
then (010) and Noguchi's Theorem [70) imply
1'1',
O.
Assume that [Dl) _ [DU) are satisfied.
Then we have the Defect
Relat on (Theorem 9.15)
(8.125)
If
n - 1
and
M - a;m
and
the identity, the Second Main
1'1'
Theorem (8 U9) and the Defect Relation (B.125) were obtained by B. Shiffman (83), (84) under a bit weaker assumptions. 813.
The Theorem of Mori.
functions
g: a;m -
I'(V)
A defect relation for
n + 2
moving target
in general position was proved by Mori [63].
[n Section 11, we improve Mori's result and extend them to meromorphic maps on parabolic manifolds.
Following Nevanlinna's method, Mori
transforms the given situation for moving targets into one for fixed targets without changing the value of the defects. (Bl] Let
(M,T)
[B2] Let
V
be a parabolic manifold of dimension
m. n + 1 > 1.
be a hermitian vector space of dimension
(B3) Let
f: M -
[B4) Let
IJ
~
1P(V) QJ
[B6) Let
(f.g)
[B7) Let
CAO, .. · ,CAn
CAn +1 - -
be a non-constant, meromorphic map.
'" be a set of meromorphic maps
[B5) Assume that
Let
We assume
is in general position with
be free for each
1\. - 1\.0 " ... " 1\.n·
Define
itO, .. ·• ~n+ 1
V·.
aj - II'(Uj)
be the dual base of If
ItQJ - n + 2.
Enumerate
be an orthonormal base of
uo - ... - un'
1\.0' .. · ,1\.n
gEl).
"(V·).
g: M I) -
Define
for
uQ.... ,CAn
are vectors in
(go' '" ,gn+l)'
0, '" ,n + 1.
j -
Put V·
define
·89·
(8.126)
for
j - 0,1, ••. ,n + 1.
A homogeneous projective operation V : V X V· X ... X V· -- V
(8.127)
is defined by n
1
(8.128)
to V too V ••. V "!n+1 - -;~=n:::::;:+:=;:
I
j=O
for all
is defined by ~ £
A
p-q
va.
If
p
q,
=
then
It = AV.
o
spans a complex line
lP(
10) -
It
For
10.
A
S;;
V
define (1.2) Then
lP(A) - (lP( 1P(V)
lP : V. -
For (1.3)
10 )
I0
~
10
£
Al
is the complex projective space associated to lP(V)
p
£
is the natural projection.
Z[O,nJ,
If
n - 0,
the Grassmann £2Wl of order
V and
then
p
lP(lt) - too}.
is defined by
-93-
and
Gp(V) ~ 1P(G p(V))
into
!P( p~ 1 V).
is the Grassmann manifold of order
Observe that
G_I(V) ~ II:
and
Every
x -
Gn(V) = (oo)
Then Put
!P(~) E
E(x)
E(~)
E(x) - E(~I q - n
p
with
Gp(V)
-
II:~O
1P(E(x))
1.
For
~ ~ ~O
E(x)
of
Define
+ ... + lI:~p
definE's a
with base
(~ E
=
VI
~o'
'"
tJ
~
'"
- 01
is a p-dimensional projective plane in a
=
1P("(0) Take
Min(p 1 1,q+l)
Ob'lerve that
If
~O'''' ,t-n
are verlors
define
(1.26)
LEMMA 1.4.
sign(>..,>..1.).
I
+
be the "empty" map
+ l,n]
1]
[" +
#~[q,n]
q - n,
=
which is
1m >.. " 1m A1. - 121
if )(
q
u
0, .. ,p
for
is defined by
(1.25)
>..
Lt.j - 0
mr and extE'nd the last
The signature of the permutation is denoted by
and if
=
be the inclu'Iion map.
A
-
If
j
"40
lhE're E'xists one and only one
for each
The map
iff
such that
be the set of all increasing, injective
LE't
q < n,
If
- q - l,n)
A1.1. = >...
Then of
E:
- {d.
~[n.n)
then >..1.
q
>... Z[O,q) -
E(v)
mr , which is much more difficult.
two LE'mmata to this operator Take
!heQ
q.e.d.
We shall provIde an explicit formula for
maps
Gp(V*),
E:
E(v) ~ ElwJ.
Then j ~ 0, '"
w - 1P(~)
and
p
E:
Take
Z[O,n]
and
A
q
...
E:
A
t.>"(q) .
Z[O,n]
D!'fine ~
E:
A V* q+l
Take
r
E:
Z[O,uJ.
Then
-99-
(1.27)
"40
If
PROOF.
r - I,
I
A £~ ( r -1
I
P1
sign(A,A 1.)1\,
this is true by (1.21).
r < p + 1.
proved for
-
1.
A
®
("AQ
L1\" ) I\.
Assume that the Lemma is
It shall be proved for
r + 1.
We have
_ u r
A map
if
A
£
manner:
+
a:
~[r
~[r
- l,p]
~[r,p]
- l,p] X Z[O,p - rl a d
j £ Z[O,p -
One and only one number
P
r]
is defined by
where
E
.,
a(A,j) = .,
is defined in this
Z[O,r - 1]
exists such that
A(O) < >"(1) < ... < >..(p) < A.l(j) < A(p + 1) < ... < A(r - 1)
Define
-rex) -
{
A.(x)
ifO~x~p+l
)•..i<j)
if x - P + 1
A(X -
1)
if
P +
1
..: Z[O,r - 11 -
Define
i injective and increases.
which is a contradl t on.
Suppose that Z[O,p1
by
>..(x) - s + x + 1
We have
s + 1 > P + 1 - r.
Hence
s + 1 , p + 1 - r.
q.e.d.
(1.35)
If t.LIt
E
t.
£
Gp(V)
G p _ q _ 1(V)
and
It
£
Gq(V*)
with
0' q ,
p 'n,
then
and
(1.36) which follows from Lemma 1.1 and Lemma 1.8.
If For
t.
It £
E
A V,
q+l
A V
p+l
and
then Lt
£
It*
£
A V* was defined with
q+l
A V with
q+l
0,
q , p 'n,
metric contract·on. t. L*Lt - t.LIt*
(1.37'
with 0 EO q EO P 'n,
(1.38'
then
Lt·
£
Gq (V*,
£
A V.
p-q
lilt-II - IILtIl
define the
·105·
.Vp
and
Cl.39)
W be vector spaces.
A p-fold operation
e : VI
W
X •.• X Vp -
is said to be projective. if for each Pj : It. -
j £ 1N[1.p)
such that
It.
~j
for all
>..
for
1. ... .P. then the condition
j -
It..
£
there is a map
of the choices of
all
Vj
£
t.l' ... '~p
j - 1•...• p.
and
Xj - I'(~j)
If
~1 e... e ~p ~ 0
£
P(Vj )
is independent
and if the condition is satisfied.
(1.40)
is well defined independent of the choices of Xl e...
a
xp
~1' ... '~p.
We say that
exists.
The operation
e
is called unitary of degree
(ql' ... ,qp)
£
Zp
if
q.
1Pj(>") 1 - 1>"1 J for all VI' ... ,V p Vj for
and
>..
£
It.
and
j -
I, ... ,p.
Ware hermItian vector spaces.
j - I, ... ,p.
Take
in
Then
c:;
is well defined.
Observe that
a xp [] ~ 0 e
lI~p II
C ... C is not a function of
which we indicate by the dot over the operation sign.
operation
Xj - 1P(t.j)
•• _1I_~..:l~e_ _ _e_~.....p~.~1 cxle ... expc- ..... 1··ql qp
(1.41)
[]xl e...
Assume that
if and only if
Xl e... e xp
Xl e ... e xp
Here exists.
is called ~ if it is unitary of degree
The
Cl,l, ...• 1).
·106-
The operation if
pi>")
=
>..
q. J
t-l 0 ... 0 t-p
0
is called homogeneouS of degree
>..
for all
a:.
£
and all
is a holomorphic function of
V 1 X ... X V p'
11'(4/')
£
(t-I' ... ,t-p)
1P( "V),
If x
=
\P(t-)
£
(qI'
,qp)'
\P( A V)
and
then
q+l
Ox;" y 0 - 111:. " 4/'11 111:.11 114/'1I
U.43)
O~Ox;"yO~l
If
x - \P(t-) 0
with
£
1P( "V) p+l
~ q ~ P ~
n,
and
y
11'(4/')
=
£
11'( " V·) q+l
then
o
U.44)
xi.y
0 -
II1:.L4/'1I
lit-II 114/'1I
o~
(1.45) [f
x - 1P(1:.)
£
o~
(1.46)
if
o~
x
1P( "V) p+l
£
1P( 1:.)
£
q ~ p ~ n,
1P( "V) p+l
then
and
0 x;y 0 -
and
0 xLy 0 ~ 1
y - 1P( "V"'), p+l
1O
M is constant.
If
r ~ 0,
then
(2.15)
In particular B)
Mlr] ~"
Divisors. I.et M
for all
r ~ O.
be a connectl'd, complex manifold of dimension
We can identify a divisor with its multiplicity function. holomorphic function on
M.
Take
germs of holomorphic functions at in
"x,
Then
r
defines a germ
x x.
E
M.
Let
0 ~ fx
Let - x E
"x
"x
Let
m.
On
f 'lObe
be the ring of
be the maximal ideal One and only one
M II
-118-
p+l Then f x E _ Px is said to be the ~ multiplicity of f at x. The function v: M is said to be a divisor on M if and only if for every point x E M
integer
p ;!: 0
-x
exists such that
there exists an open, connected, neighborhood functions
g iii 0
and
h iii 0
on
U
of
X
n
and holomorphic
such that
U
(2.16» The set
"M
of divisors on
v(x» ~ 0
divisor is non-necat ve if of
U
o
that
x
x E M, which is the case if
for all
and a holomorphic function
The support
vlU - JAg
S
~
supp v
is an analytic set of pure dimension
XES.
branches of
S.
each branch
BEt.,
Let "R(S»
for all
m - 1.
U
such
is Inversely, if
there exists one and
Let
t.
be the set of
be the set of simple points of
there exists an integer
x E B "!R(S).
on
for every simple point
v be a divisor with support S.
Let
vex) - kB
m - I,
Vs such that vS(x) - 1
only one divisor
g ii! 0
of the divisor
either empty or an analytic set of pure dimension S
A
x E M there is an open, connected
and only if for every point neighborhood
M is a module under function addition.
kB
1;:
0
S.
Then for
such that
Moreover, we have the locally finite
representation
(2.17»
v
Here
~
0
if and only if
kB > 0
for all
BEt..
Let Nand M be connected complex manifolds. Let f: M N be a holomorphic map. Let V be a divisor on N with f(M) II; supp v. Then there exists one and only one pullback divisor f'"(v) such that if
g ii! 0
subset
(2.18»
U
and of
h 51 0 M with
are holomorphic fUnctions on an open, connected II
IU -
JA~
-
JA~
then
·119·
~
C lit])
if
then
s i! 0
Let
E
M.
M,
Then
{}N -
f*
{}M
is a homomorphism.
be a holomorphic section of
Then
s
holomorphic Vl'ctor bundle
8
defines a (zero)-divisor
/.Is
For every point U
there is an open, connected neighborhood
holomorphic section h i! 0
U
on
t
to units and
W over
of
/.Is
m - 2.
'IJ:
M -
the trivial bundle
M X V.
defined.
E
Take
Here
b
V
f
of
holomorphlc functions 0 b /.If IU - /.Ig-bh
if
g b
~
h
U
on
and
pole divisor of
f.
divisor on
M
U
of 0
/.I h f.
Also
s.
~
hf
g.
f ii! 0,
If 00
/.If
be a parabolic manifold of dimension
with support
X
nv(r) - r
E
J
S (r
nv(r) -
(2.20)
~ ZES
Then
nv(r) -
nv(O)
for
[r
r -
vu m - l
Then then
m
v
Let
v
o.
If
m > 1,
then
be a
is definf'd
if
m > 1
if
m - I .
I
viz)
I
M
is called the
The counting function of
2-2m
If
M.
by
(2.19)
is
'IJ
and there are coprimf
x
such that
/.I7 1U
is called the divisor of
(M,T)
of
'IJ
For each point
is defined.
/.If - /.If - /.If
Let
/.I
be a meromorphic function on
f
and 00
t
are uniquely defined up A holomorphic
there is an open, connected neighborhood
00
t
can be viewed as a holomorphic section in
Let
the b divi or
0
and
Hence the (zero)-divisor
!Pl.
a
x,
and such that the zero set of
h
is well defined by
vector function
of
U and a holomorphic function
s IU - h • t
such that
has at most diml'n'lion
f ii! b,
v ~ 0,
If
o.
~
fa(v)
W over x
({).
·120·
(2.21)
J
n)l(r) -
r > O.
for
)lW m- 1 + n,,(O)
S(r)-S(O)
0
",u)
U>..u -
(10
exists such that
on v>..u • vu>" - 1
and
is an open covering
one and only one hal am orphic
(2.28)
Then
U ~ A(f),
pullback
Lf I A(f) - f"'(O(1)).
For e ch pair
function
!PIV).
Q!!
~
ill the subbundle f"'(0(-111 An!! i!
~.!! section
There ex ts a family
U~ -
M.
the bundle
~ U. Then
["'(0(-1))
~ identify
PROOF_ 10
in VM
10
if
A(f),
V ~ ~ reduced representation with
U -
:
10
Over
!! isomorphic to
section
v>..>.. - 1.
If
U>..u
(>",u.p)
E
A[2),
then
(2.29)
The cocycle
(v>..u)(>.. U)e:A[1)
M and a family U>..
(10
~1>"EA
determines a holomorphic line bundle of holomorphic frames
10
~ of Lf
Lf
or
over
such that
(2.30)
for all pairs
(>...u) E: A[l].
The transition formulas (2.28) and (2.30) show
that there exists a global holomorphic section
F
of
VM
e Lr
such that
(2.31)
Let
10: U -
V
be a reduced representation.
there is a hoiomorphic function 10
>.. - vA
10
on
u"
U>'"
vA: U " U>.. If
(>...U)
E
AU[lJ.
a: then
For edch CO}
>.. E AU
such that
vA - v>-.JJ.vU
on
-124-
U 1'\ U>..u· frame for
Hence of
-06.
A
E:
frame of
AU'
Lr
v>..
Lf
-0
~
over
Then over
V U -0
-
~
on
U 1'\ U>..u' -06. 1(U
U is defined by
FlU -
8
-0
U such that
Let
-0 6..
1'\ U>..) - VA
-0
~
be any holomorphic
E: 11:.
,"10
>
~IE(V) .c.&IE(w) C ~~> '0' ~
l(...o '" ~) and (flg,h) is strictly free if and only if (1:1 l ~ ) l ~ ~ 0 where
f
h.
~
0
is strictly free i
fl(g '" h) - (fll)lh
follows. Assume that (f,g '" hI is strictly free. Holomorphic functions Pli!O and Q II! 0 exists on U such that 1:1 l"tIC - Pg and ...0
~ -
"
Q~.
Then
PgL~
Hence
on
U.
q.e.d.
- ( = IJ,.cr
Hence
exists.
f
Therefore (31) holds with
M
=
0,)1
and
~·(o)
~-I(M(r,s) - N vir,s).
Define
S = supp
of all branches of
S
v and
,.
Let
So
be the union be the
which are contained in
union of all other branches of supp 1>j - Sj
,. S - supp 1>.
for j - 0,1
S.
There are divisors
such that
N1>(r,s) - Ni>o(r,s) + N1>l (r,s).
1> - 1>0 + 1>1'
1>.
J
~
Hence
Consider the standard models
0
with
-136-
f-(W) - {(y,w)
(M - If) X WI w ~ Wfly))
f:
where
f
0
S? - f
0
7C
0
7CO -
'"
7CO -
0
Ir(x,w) - (I?(x),w).
For
x
on
E
M-
if
i(niCx » - (f?(x),U( 0,
(3.4)
t
2!-2
r
Hi
TfCr,s,L,x) -
t > 0,
For
0
0
for
Ar(oo) - Ar(t) - 0
for
Also
00
t > 0
and
Tf(r,s) - 0
Ar(oo) > Ar(t) > 0
is not constant, then 0 < s < r.
r -
Tf(r.s) -
for
00
r _
for
t > 0
00.
(Stoll (103) Theorem 12.8 and Stoll (108) Lemma 10.4.) PROPOSITION 3.2. M.
Let
IC
Let
Lf
be the hyperplane section bundle of
be a hermitian metric along the fibers of VM 8 Lf .
be the representation section of f
in
hermitian metric aloD8 the fibers of
VM - M X V,
hermitian metric on
>.. -
r
E:
£._
with
V.
0 < s < r.
Abbreviate
Then we have
fl
8
IC.
Lf .
Let
I
f
Let
on
F - Ff
be the
defined by the Take
s
E: ~T
and
·140·
(3.13)
J
Tf(r,s) - T(r,s,Lf,Kf) -
M -
10
(3.14)
V is a global reduced representation of
Tf(r,s) -
J
log II 10 1117 -
M
PROOF. have
Let
FlU -
10: U _
log IIFII}" 17 •
M<s>
M
If
J
log IIFII)..17 -
J
f,
we obtain
log II 10 1117
M<s>
V be a reduced representation of
f.
Then we
10 8 10 A and
dd c log IIFII~ IU- dd c log 1110 112 + dd c log 1110 AII~
(3.15)
Thus the Green Residue Formula, Theorem 3.1 implies (3.13). 10: M -
If
V is a global, reduced representation of f,
IIFII).. - 1110 II 1110 All K.
PROPOSITION 3.3. Let
N 10
Let
Hence (3.13) and (3.8) imply (3.14).
10: M -
J
Tf(r.s) -
M.
log II 10 II a -
M
for 0
PROOF.
We have
be a representation of
be the valence function of the divisor of
the Theorem of Cousin II holds on
(3.16)
V
then
"C: M -
exists such that
V
exists.
A holomorphic
-141-
J
log II ~ II 0
f
-
M<s>
M
J
N tJ (r,s) -
log I h I 0
V I' '" ,V k
Let
-
J
log I h 10 .
M<s>
M Addition implies (3.16).
log II~ 110
q.e.d.
W be hermitian vector spaces.
and
be an o~ration homogeneou i - I , ... ,k (f 1, ... .fk ) divisor
let
fj: M -
Uf 0- ';f 1 ... '" k
(3.17)
For 1P(Vj )
8.
free for
IS
be a meromorphic map such that N
Let
f I 8- ... 8f k
be the valence of the
and define the compensation function by
J
mf 1·G.. ·Ofk (r) -
log
M for r
Let
----...!...---o . o fl 8
E ~., •
THEOREM 3.4. r E~.,
with
The General First Main Theorem.
0 < s < r,
If
s
E~.,
and
then
q
(3.18)
L
j -1
PROOF.
qJ.Tf.(r,s) - T f
Abbreviate
J
f:\
1"''''
Of (r,s) k
h - fl 8 ... 8 fk : M -
the hyperplane section bundle of
fj
1P(W).
and take a hermitian metrlc
be Xj
·142·
dlong the fibers of
Lj .
1l j
Let
induced metric along the fibers of >.. j - 1l j ~ K j'
Let
hermitian me ric along
Lt;
Lh
Kh
be the hermitian metric of VjM - M X V j.
Abbreviate
be the hyperplane section bundle of
along the fibers of
Lh ·
and thE
Vj
Let
IC
~
h.
Take a
be the metric
Then
dual to
(3.19)
IC
-
is a hermitian metric along the fibers of the line bundle
(3.20)
k
c(L,IC) -
(3.21)
j
L a
qjC(Lj,K j) - c(Lh,Kh) 1
which implies k
T(r,s,L,IC)
(3.22)
~
L
qjT(r,s,Lj,Kj) - T(r,s,Lh,Kh'
J=l Let
Fj - Ff .
be the representation section of
fj
for
j -
1, ... ,k
J
and let and
Fh
be the representation section of
>"h - llh
~
Kh
H2(U>..,Z) -
M
o.
For each
>..j : U>.. --. Vj
10
>"1
e ... e
function
L.
Let
(U>")>"EA
by Open, connected Stein subsets
10
1C
>..k
W>.. ~ 0
>.. j - 1l j ~ Kj
Let
be the respective hermitian metrics.
We shall construct a section in covering of
H.
of
fj
>..
E:
and
U>.. ~ 0
with
A we select reduced representations ~>..: U A --. W
is a representation of exists on
be an open
U).
such that
h
on
of
h
U>..,
on
U>...
Since
a holomorphic
-143-
on
u>.._
We have
(3.24)
(3.25)
...a ~*
Let to
be the holomorphic frame of
L~ over U). which is dual
Then
(3.26)
is a holomorphic frame of
Lover
U}".
For each pair
there are zero free holomorphic transition functions on
U},,/..l
=
U).
(I,
U/..l
such that
(3.27)
(3.28)
(3.29)
(3.30)
on
U},,/..l.
Therefore we obtain
v)./..d
().,jJ.) E A[l]
and
w)./..t
-144-
W>.. ~ >.. -
~ >..1
e... e
~ >..k
or
on over
U>..u.
Therefore one and only one holomorphic section WI U>.. - W>.. It>..
M exists such that
>..
for all
£
W
of
L
A.
Observe that
::.r 1"'···"'lk
Uf';
on
U>..'
hence
-
JAF
til"
-"1
t:\ OF 1"'··· k
::.r -
O· 1 ... "'lk
JA ~ ~
M,
on
JAW
NW(r,s)
t:\ t:\ ~ 1"'···'" ~ k
~
Nf ';
-
JAW
>..
-
JAW
which implies
::.r
1"'···"'lk
Therefore (3.7) and (3.22) imply
::.r
- Nf '; (r,s) + 1"'···"'lk
I M ~ O.
with
!P(V·,_
Pick
Now (2.52) implies
1I I
= 1 ..J.l
If
without zeros
such that
U}"J.l
(4.5) Then
A>..>.. - 1
on
U>..
and if
we have
(}",J.l,p) £ A(2),
(4 6)
r>.. is a holomorphic frame of the canonical bundle K of N
Each and
(A>"J.l}(}".U)E:A[1)
Let ~ : U -
is a basic cocycle of
K.
B
be a holomorphic form of bidegree
U'
is a chart. holomorphic functions
(m -- I,D) Bu
on
M.
exist uniquely on
such that
Let
V be a complex vector space.
holomorphic vector function. t::)'
:
U -
V.
Let
t::):
V
U -
A holomorphic vector function
called the B-derivative of
(4.8)
is uniquely defined by
t::).
t::)
The oprration can be iterated:
t::)
(p) ~
(t::)
(p-ll)'.
Put
t::)
(0) _
Abbreviate
(4.9)
t::)
Dependence on t::)
.a -
be a
t::)
ltR.
~
etc.
,.,
,
" ... "
t::)
is denoted by an index
~
.£
-
t::)
t::)
(p)
as
U- Gp(V).
t::) •
If U
-153-
Let f : M -
V 1P(V)
be a meromorphic map.
said to be a representation family of ~ ~ : U>.. -
chart and
n + 1 > 1.
be a complex vector space of dimension Then
f.
("&~. ~ >")~E:A
i~: U>.. -
if
U~
else is said. we assume that each
is
U~
V is a reduced representation of
Let
is a If nothing
f.
is connected, Stein and
H2(U~.Z) - O. Then U>.. is a Cousin II domain. each divisor on U>.. is U~.
principal and each meromorphic map has a reduced representation on The representation family is said to be a representation atlas if 11 - (U).)~EA
is a covering of
()..u) E A(1),
a holomo phic function
defined on
U).U
M.
which we will assume now. v).u
If
without zeros is uniquely
by
(4.10)
on
U).u _ Then
v).>.. - 1
on
U).
for all
).
E
If
A_
().,u,pl
E
A(2).
then
(4_11,
If
0' p
t:
Z
and
Let
Lc
(~.u) t:
A[l).
then
(4_12,
on
U).u'
be the hyperplane section bundle of f.
is a holomofphic frame of
4
over
U).'
If ()..u)
E
A(1).
Then
t:)
~
then
(4,13'
These transformation formulas imply the existence and uniqueness of a holomorphic section
Fp - F pf'
of the holomorphic vector bundle
called the pth representation section of f.
·154· p ( p+ 1 ) P+ 1 L f I] p - ( A VMI ® (LfI
(4.14)
.0.
K
2
'0'
p+l
such that
(4.15)
>-
for all
A.
E:
Here
then
Fptl :: O.
If
~
0
~
p
index of
f
o~
if
p ~
Hence
for
if
Band
ff
E:
M
f
Z[O,n]
Fp :: 0
if
P
We call
Fp :: 0 Fp i! 0
if the generality
is said to be general of order
admits
If
> n.
exists uniquely such that
p > if .
if
Fp = - 0
and
if
but
FO - F '1. 0
p
B if
for
m analytically independent holomorphic M,
functions, then for any finite sets of meromorphic maps defined on B
there eXists a holomorphic form
m - 1 on
of degree
the generality index of each of these maps
f
for
B
M
such that
equals the
dimen'lion of the smallest projective plane containing the image of (See Stoll [1001. Theorem 7.11.) if
The map
f
f.
is said to be leneral for
B
i f - n.
For each
p
Fp
E
M -
f p - 11'
0
dnd
AJ!. : U A -
~
Z[O,ifl. Gp(V) Gp(V)
pth
the of
f
associated map
is dpfined.
Here
is a representation of
fp
fp
is meromorphlc
for each
>-
E
A
and (4.151 implies (4.16)
In Pdrticular. ~ A.£. and let
F -1
u
o.
F 1
-
may not be reduced.
for all
>.. EA.
Define
Le[-11 - M X It - ItM
be the trivial section defined by
For
0 ~ p ~ If'
the
pth
F _l(z) - (z,ll.
s t a t'lonar~ divisor
Then
·155·
(4.17'
l
f
p
- ~F
p-1
- 2~F
p
+ ~F
~ 0
p+1
is non·negative (Stoll (93), (100)). Take
p
Z[O,n).
E
Let
non·negative form of class
CP: M -
COO
Gp(V)
is given on
be a meromorphlt mdp.
M - Icp
A
by
(4.18'
If
It: Ult -
on
Ult - Icp
U~
a chart, a non·negative function
Hlt (CP)
is defined
by
(4.19,
where
Hlt (cp)2
abbreviate f : M p
E:
is of cIa s
H" (CP) - H (CP). A ltA 1P(V)
on
UIt - If.
If
A E A
Returning to our given meromorphic map
and Its associated maps
Z[O,.I f ) we abbreviate
HpA - HltA (fp )'
COO
lip - H(fp)'
fp: M -
Gp(V)
for
Hplt - Hlt(fp )'
Then
(4.20'
(4.21'
(4.22'
where
Hp).. > 0
on the compLement of the thin analytic subset
-150-
(4_23)
Define
IHp - 0
if
For
p < 0
~
0
p > flf.
or if
p < flf
Also
we obtain on
(4.22) implies
M -
i~.
IHp - U
the identity
(4.24)
(4.25)
Let
bt' d parabolic manifold of dimension
(M,T)
M+ - (x
(4.26)
is not empty. on
M
Let
'"
be a positive form of degree of
...
Ric(r,s, y) - Jr
(4.27)
S
0 < s < r.
for
On
M,
The open set
M Iv(x) > 0) - Ix £ M I u(x)m > 0)
€
The RIcci function
m.
2m
and class
C"
is defined by
.f M[ t
(Ric y)
A
urn -1
J
a non-negative function
v
dt t 2m - 1
of class
C..
is
defmed by
(4.28)
Define
o
~., -
(429)
Then
€
~T I (log v)o is integrable over M..1>..£1\
and
g: M -
g.
For each
6>..W v>..U
holomorphic functions
(f,.Id
is called a representation atlas of the pair
is a representation atlas of
a representation atlas of
be meromorphic maps.
F(V·)
and
(>...U) w>..U
f E
and 1\[1]
("&>.. •...0>..)>..£1\
is
there are
without zeros on
suc~
U>..U
that (45). (46). (4.10). (4.11). (4.12). (4.13). (4.14). (4.15) hold and such that
(4.34)
(4.35)
hold on
U>..u.
If
>..
E
1\.
then
w>..>.. - 1
(4.36)
The
on
on
p th
representation section
Fpa ..
of
g
U>...
If
U>..Up.
is abbreviated to
and is a section in
p(p+l)
(4.37)
L [pI - ( 1\ V*) ® (L )p+l g
P+l
(4.38)
Also we abbreviate (4.39)
M
g
(>".u.pI £ 1\[2
oQ
'CO
K
2
F
pg
=
G
p
159-
k 2pV m
Theil tl
For
on
non- negdlive form
IKpq
P
t:
of de~r(H;
Z[O.i g J dlld
2111
is defined on
M
by
(4.40)
for all
A.
A.
t:
Then
~ l.OH 2 q + IKp)
(4.41)
Take t
p
dnd
q
in
I
'2(p(p+ 1) + q(q 1)).
and
b = q
~
1
ZIO,n).
Tdke
p ~ O.
p
t:
Define
J..t -- Min(p+ l,q+ 1)
Z[O,J..t).
Then
and
a ~ p + 1 -
P ~ 0
Define the hoi om orphic vector bundle
(4.42)
Considermg (4.14) and (437) a contraction
(4.43)
is defined by
BlP - Bl P
® ld
and the proper commutation of terms_ F q BlP G p = IB P Fq
Hence we have a holomofphic section L~. g[q,p).
The pair
F q IB P G p ;E O. free of order
If
f,g
0 ~ q ~ if
(q,p,O)
if it is free of order of order
(q,p)
is said to be free of order
The pair (q,p,l).
0 ~ p ~ i g,
and f,g
® Gp of
(q,p,p) then
if
f,g
is sdid to be free of order
The pair
if it is free of order
F q IB O Gp = F q ® Gp
We write
F q 1111 G p - F q III Gp
If
p = q + 1
If
p -
f,g
p + 1
We have that is,
~ p + 1 ~
(q,p)
is said to be strictly free
(q,p,J..t1.
IBI - IB,
is
q + I,
we write we write
-160-
Fq mP+1 Gp - FqLG p . Fq
mP+1
Fq.G p and
Gp - .
and Ff
If
g
p _ q + 1 - p + 1.
which is a section in a line bundle.
Fq IIIP Gp
The sections
are not to be mistaken for the sections
111.0 G • The following lemma clarifies the situation. gp
LEMMA 4.1.
Take
inteR r with
0 E; p , Min(p+ l.q+ 1).
(P.q p)
we write
if and
q
E
Z[O.Jlfl
and
p
Z[O.Jlgl.
Let
P
Then the pair
(f.g)
i! free of order
E:
if the e.air (fq.gp) i! free of order
~
be !ill If 12. then
p-
(4.44)
PROOF.
Take
A
E:
A.
reduced representations
Since
UA is a Cousin II domain.
SA
of
Put
t - i'p(p+l) + q(q+l)).
fq
on
UA and c.t A of
there are Kp
on
such that and
o
UG I U A - UQ p
Hence
F q IIIP Gp ;& 0
have
on
.
UA •
q.e.d.
1
We have
>..
if and only if
Ff
IIIP G q
Kp
;&
O.
If so. we
UA
·161·
LEMMA 4.2. (O,p) all
Take
if
and only
£
Z[O,.ff J•
q
PROOF,
if
~:
Let
representations of (q,p)
if
Hence if
~
If
(f,g)
of order
£
U-
and
f
~
p ' q,
"40:
U -
V·
and
g
By Lemma 1.2 we have
q IB
respectively.
"4O p
(q,p)
(O,p), ., O.
Then
then Hence
for all
q
(f.g) £
is free of order for
(q.p)
be reduced (f,g)
is free of order
II "4O.E. L ~ II > 0
on an open
is free of order
Z[O,.ffl,
be the Fubmi Study Kabler form on on
(4.46)
If
i! free of order
II! O.
Define
p -
(f,g)
then
(f,g)
(q,p). is free
q.e.d
Oq - 0q,(_l)
For
V
is free of order
(O,p).
pq
Then the pair
(f,g)
is free of order
0
Z[O,.fgl.
the pair
"4O p
U and
Let Then
q IB
(f,g)
subset of
p
1,
then
write
1I\~1 V)
and
1'( A V 8
0; - O(_l),p
q+l
on
A V·).
p+l
'\~l V·).
-162-
(4.47)
If
mi'l' 1 (f q Lg I' )*([1 q p- "
q ~ P.
then
IH q + 1
(4.48)
pq
TakE' Take
'" H '" B
p £
q
£
ZIO.ifl
ZIO.ul
nnrl
E
il'IO.f gl
Dcfi II I'
U - Min(pl
Define
(4.49)
.(p)
(4.50)
•
[~]2
pq
_ .< 1 ) pq
pq
(4.51)
p
4>
q
(4.52)
4>
Oq
4>pO
o f"
ffi,o
- u2 0 f
- 0 f
o f
q
go
02
p
"
III gp 0 2
III g 0 2 ~ 0 f q Lg 0 2
•
III gp 0
2
Then Lemma 1.8 implies
(4.53)
(4.54)
(4.55)
[f
p ~ q.
thE'n
4o,
B)
p
+ II t:)
n
"'"
II
~ t.0,j-1" 'dt.j J-O
If 0 ' r , p - 1.
then
t. r "
1O.l!..=.!. - O. Hence
+ 1
which implies
9 rk " B - 0
for
O'r'p-l
and
p
Also we have
Rrk " B - 0
for
o,
and
s + 1
Therefore
9 rk " B - 0
if
k > r + 1
r , p - 1
, k' ,
8.
k , n.
which proves (5.11) and implies
-171-
(5.12) and (5.15). 5 -
n.
then
Also we have
1(" p - 0
-.. ..!!.....-.-l "
for
-.. (s+l)
1("r
fA
B - 0
p - 0,1, ... ,n. and t:)
~
for
s < n.
Assume that
,- . .-. (s+l) -
If
O:$; r :$; s - 1.
-.. 1 .!.!..-
o.
Then
Hence
' +. .. + As t :(s) t:) (s+l) - AO t:) + A S t:) )·
Hence
t:) ~ - As t:) sand
(5.36) for
p - s
fl. s "
t:) ~ - fl.s " As t:) x ~ O.
Consequentl~
implies n
I
k-s+l
Consequently
Rsk
fA
(l)s(Rsk
B - 0
fA
for
B)fl.O " ... " fl.s " fl.k - 0
k - s + I, ... ,n.
Hence
1("s
fA
B
We proved (5.18) which implies (5.19). In (5 36) we take the hermitian product with
or
t:) J!..
and obtain
=
o.
·172· If
P
> 0,
we obtain
i 4ir
9 00 " B - d
C
I
og
n101 II
"
B
.
Degree considerations imply
9 00 " B -
Since B
jill
B
we obtain (5.22) and by conjugation (5.21) and by the same
0
method (5.24). If
1 a- log Ti151i "
Also (5.21) and (5.22) imply (5.20).
0 < P < s,
-
II
10
then (5.36) implies
~ II
II
10
.2±.!.II 10 l!."
10
(p+l)
r
·173·
We have
If
p
o
.. - In If
(5.45)
LEMMA 5.6.
(5.46)
V·
(>..,U) e: A[ll,
u 1I
~p
Take
and since
By conjugation we obtain Now (5.37) implies
Degree consideration give (5.43) and (5.44).
is an orthonormal base of
frame.
(2,0).
>.. e: A an d z
Take
6 p,p+2" B " B - 0
H nce we have proved (4.40).
(5.41) and (5.42) easily.
If - n - s.
and
has bidegree
B-
Assume that
0
Let
E
Hence we assume that
U>.. - In'
>..11 't-o (z), ...
Then
>..11
'~n
(z)
>..
~o(z),
>.. ... '~n(z)
be the dual base, which
The vector functions
are of class
C
00
and are called the dual Frenet
then Lemma 5.3 implies
v>..
[ 6 u >.. 16 u >..1
-~ p e: Z[O,nl
d~p>..*
B.
q.e.d.
and
r
>..
>..*
on
~p
E
n >.. A* ~ 9qp~q 0-0
A.
U>"u - In
Then
on
UA - In
·176·
PROOF.
=-=--
There are forms
A
on
pq
U
>..
- I
such that
n
Then q.e.d.
The Frenet Formulas for the dual frame follow immediately n
(5.47)
(6 >.. d~>"* qp A B)~>"* ~ q p AB-- q=p-l
(5.48)
d~r
(5.49)
d~>"* p A B
n
AB--
A
if
o
.. qo A B)t.>"* q
~
q~O
B
>.. >..* >.. >..* (9)'' >..* p-l.pt.p-l + 9pp~p + 9 p+ 1 • pt. p +l) A BA B. Again, let
f: M -
1P(V)
be a meromorphic map general for
addition, we consider a meromorphic map Ig - s.
Then
atlas of the pair
0 Ei s Ei n. f,g.
Let
g: M -
Abbreviate
IP(V·).
(1->..,10 >..,"4C>")>..£A
be a representation
Adopt the notations of (3.34) to (3.44).
Define
J- - In(O V Is(g)· The J- {'\ UA is the union of the zero set of and the zero set of of
g.
(5.50)
.o..tO
A,!!.
Let
A 111'0'
A
·lII's
1(A p - dll'A p
10 >",!l
be the Frenet frame
Define A "pq - (dll'Ap III'A) q
B.
s ~
q~O
A
A
1! pq ll'q
Then Theorems 5.4 and 5.5 hold in analogy. For each A £ A and p £ Z[O.s) and q c Z[O.n) define the stress coefficient
In
-177-
15_51}
COO
which is a function of class
on
U>.. -
15.52)
if
Since
D:E;p:!!is
is an orthonormal system, we have
-
15.53)
If
8. Obviously we have
s - n,
the square matriX
{
o
if
p
-
if
p
'I:- q
q
IA~q) is unitary and we have also
15.54)
LEMMA 5.7. U>..U - 8"
Take
I>",u)
E
A[l]
and
p
£
:l[D,s]
and
q
E
Z[D,n].
On
we have
15.55)
PROOF.
If
a
E
a:
with
I a I - 1,
then
a - a- 1.
Hence 15.6) for
and (5.45) imply
q.e.d.
10'
·178· We will use this transition formula to introduce important invariants. but we will first derIve somp formulas which calculate the stress coefficients in terms of the representations LEMMA 5.8.
Take
).
E:
p
«>
Z{1.s)
and
1\
and
«
«> ).qL'" «> ),.9-1) • ...0),.>
E:
and
q
...0 .
E:
Z{1.n].
we have
(5.56)
(5.57)
(5.58)
(5.59)
II
«> ),..9. 11
11
«>
),.!L.l.1I II ...0 ),. II
( ( «> ).!l. L...o ).) I «>)..9.:.:.l) (-l)q -n---......--,..,.-....,.......,.,..-~:........ II «> ),.!l. II II ...0 ). II II «> ). s.:..l.1I
,.
(5.61)
(5.62)
1\00 -
«> ),. • ( ...0
«> ).
II
)..E. L '" ...0 )..E..::.l ) >
11...0),..£ II
II ~ A.~ II
( ( ...0 ),..E. L «> A.) I ~ A.~ ) (-liP rr-~-..,.,,--.,.,,..--.......,,........,.-.."::;:;:=-...,..
II~)..E." II
< «>A·"'OA> II «> All lI"N:lAIl
«> >,.11
II~>".E..:::...!.II
Then
-179-
(5.63)
If
0 ~ p :!i: sand
O:!i: q ~ n,
then
(5.64)
(5.65)
(5.66)
(5.67)
PROOF.
on
17)"
We have
(5.66) 10.9, -
(5.67)
1110.9,11 ~O
~q
-
A
_.-
A
~q
lO n l*lO n _ 11 .;a...........
(5.68)
which proves (5.56).
(5.69)
Also we have
~.l!. ~ lI~pll ItO
~
.l!.
A
l*~
...
A
~
Itp
It p - -.::......--==~
·180·
Taking the norm proves (5.62). «)
~
II., !L..!."t.O
-
A
•••
A
Taking the hermitian product with t.q_l
proves (5.57).
Also we have
(5.70)
Taking the norm proves (5.63). "40 ~ -
Taking the hermitian product with
II "40 L...l1I ID-O '" ... "ID-p-l
proves (5.58).
'" t.j-l "t.j+l
pqllKp
·189·
Consequently we have
-
m1 m_)
'" 8
(5.851
+ 25 P
q+l
q
p+l
IKpq
which proves Theorem 5.9. From (5.97) and (5.98) we obtain:
q
p
I
I
-
-
ApjAp+ l,jAkq Ak,q+ 1im II p,p+ 1 j-O k .. O q
p
I
L
j-O k-=O p
- S
A
B
-
p+l q
1
1
p
B '" 9q+1 ,q
AkqAk q+ 1ApjAp+ 1.im 9 q,q+ 1 ",BA II p+l,p '" 8
q
p+l q
A
q
p I
pq OKp + IHq) + Sp+l ,q IKp + Sp,q+l IH q + 2S p + 1 , q IKpc
4>p+1.q ~ U ~ n + 1
Mort-uver.
and
4>p,q+l ~ J.l ~ n + 1
and
q+J q
I ~ 1
imply
q.e.d.
LEMMA 6.7. Definf'
lJ.
Take
p E Z[O.igl. q E Z[O,n)
M\n(p~1.q+ll.
Then we have
D2[.-L2 ..."'pq )~-1_'='pq
P
J.l
PROOF.
Trivially
1 :!;; U ~ n + 1
Lf'mma 6 3 and 6.4 Imply
and
and
~
E
!R(0.11.
-197-
~
U2 - 28 (1 +
~8pq )2(. m1m_1
~ 28 2 ... 8 - 1 S '...p q
-
dd c log(1 +
P ,q+1 IK pq p + 1, q
~8Pc) )
B '" B + IK P + IH q )
"
8 8 8 +. p q )~ p q IH pq
+ 8(1 -
(8(8 + 1) + ~8 ).8 UK + IH ) + (1 + ~8 )2UK + IHq )) pq pq p q pq p
>- U 2 - 28(8 2 .8-1 _ ,..
pq
-pq
+ ((1 +
~.8p q )2
_ 8(8 t
~.8p q )~8p q )(IK p
+ IHqll
q.e.d.
Let
(M,T)
be a parabolic manifold of dimension m - 1
be a holomorphic form of degree said to maiorize exists a constant
the holomorphic form c
~
1
M
on
B,
The infinum of all these constants is called YO
increases.
(6.12)
associated to
(6.13)
T
and
VOIr).
.,
is
there
if
M[r).
Then
VOIr) ~ 1.
lim Volt). r 0
if for every
on
The function
Let
such that
(6.11)
Then
m.
The exhaustion
r > 0, -
mi m_ 1 B " B
~
Y
is called the m!l.iorant
then
Y(r)U
m-1
on
M[r)
-198-
on
(6_14,
M.
um- 1 may not be positive definite. the existence of the constant c in (6.11' is not assured. If u m- 1 > O. then c exists. Since
urn 1 > 0 implies
however
(It m.1'O)
isometric to
u > 0 and (M.1') is biholomorphically
where
If
"'O(z) - IIz1l2.
proper. surjective and holomorphic and if
l' -
"'0
a
It m
M -
(I:
(I -
is
11_11 2•
then (M.,., is para 0 IC and a holomorphic form B if 0 of bidegree (m - 1.0) exists on M such that .,. majorizes B with VIr) E; 1 + r 2n - 2
(6.15'
If
m - 1.
that is, if
open parabolic Riemann surface. we take
B-1,
then
and such that
f
IS
B.
general for
M
is an
1 B '" 8 - 1 - u m- 1 and .,. majorizes B with V == 1. The idea mof an estimate (6.14' was first introduced by Stoll (93] and later refined
mi
in (100).
(Al) (A2' (A3, (A4, (AS, (A6, (A7'
Now the following general assumptions shall be made: Let M be a connected, complex manifold of dimension m. Let l' be a parabolic exhaustion of M. Let V be a hermitian vector space of dimension n + 1 > 1. Let f: M 1P(V) and g: M 1P(V*) be meromorphic maps. Let B be a holomorphic form of bidegree (m - 1,0) on M. Assume that .,. majorIzes B with majorant V. Assume that f is general for B.
Here (A7, implies that non-degenerate.
B ii! 0
and that
f
is linearly
LHMMA 6.S. Assume that (Al) and (A2) hold. Let N be a complex M be a meromorphic map with indeterminancy manifold. Let f): N If)' define
o
O.
Then
Hp > 0
on
r -
M(r)
00 .
except on a thin analytic
Therefore we have
which implies (6.22) and by Lemma 6.11 we also obtain (6.22). For
(6.25)
0 < s < r
we have
Jr J B
M (t
m
IHp t 2d t_l
I
~
Y(rlTfp (r,s)
q.e.d.
-204-
0 ~ s
Take lR(s,+oo).
E:
IR+
Let
III ~ '"
We write
measure in
IR.
if
~
dnd
there exists a subset
is self evident.
1 ~ min f r .....
(6.26)
be redl valued functions on
.p
lII(x) ~ ",(x)
such that
The calculus of
III
for all
x E lR(s,+oo) - E.
In particular
1II(r) ,
1
l.
of finite
E
cP ~ '"
implies
m sup .p(r). r ... OO
Nevanlinna [671 established the following well known result LEMMA 6.13.
Take
E
> 0
whlch is integrdble over F : lR[s,+oo) -
IR+
and
lR[s,rl
x
F(x) -
~ s.
Let
III j?; 0
r > s.
be a function
Define
by
(6.27)
for
s > O.
for each
J:
lII(t)dt
III ~ Fl+€:.
Then
PROPOSITION 6.14.
Assume that (Al) - (A6) are satisfied.
p E Z[O,.e f ) and
> O.
almost all
r
E
E.f. T
h!a
is integerable over
h!a
~
where
Sf (r) p
Sf (d p
for
r E(2m-1)(Y(r)T r (r,s))(l+E)2 p
M
(6.29)
M
and
J
(6.28)
Then
Take
~ (C,/2)(l + E)2UOg Tf (r,s) + log Vir)) + (C.I2)E log r p
is defined by (4.31) and (4.32).
·205·
PROOF.
Fubini's Theorem implies
J
IHp
M[rJ
~T.
hpu m
h p2 r7
J
m
=
M[rJ
which shows that
t £
J
-
M[rJ
is integrable over
M
for almost all
(6.25) and Lemma 6.13 imply
J
J
2
h p r7 ~ (1 (2m))r 1 - 2m [
M
f
IHp
r+E:
M[rJ ~
IHp
r2m l(Y(r)T (r,s)1+E: f p
M[ r ) Hence
J
2 h p r7 ~ (1/(2m))r £(2m-1)(Y(rlT f (r,s))O + £)
P
M which implies (6.28).
Sf (r) p
2
Also we have
f l o g hpo
=
(c.I2)log
[i:
M
I
h~O)]
M
~ (C./2)(1 + £)200g Tf (r,s) + log Y(r)) + (2m -
1)(C./2)£
P
+ (c.I2)lOg(C.I2)
~
(C.12)(1 +
£)2UOg T f (r,s) + log Y(r) + mc.£ log r. p
Replacing
c.
by
C./(2m)
implies (6.39).
q.e.d.
log
·206·
The following proposition was proved in Stoll [108\ Proposition 10.9. For completeness sdke, the proof shall be repeated here. PROPOSITION 6.15. P E
Z[O,ifl
and
> O.
E
(6.30)
for
Assume that
(AI) - (A6) hold.
Take
Abbreviate
Q£(r,s) - log VIr) + Ric 7 (r,s) + £c, log r
0 < s < r.
Th n
T f (r,s)
~ 3 P T fIr,s) + t(3 P
-
l)Q£ (r,s).
p
PROOF.
o~
The estimate (6.31) is trivial for
p < i
p + 1.
and that (6.31) holds for
p
p - O.
p.
0 < £
o.
> 0
and
(f,g)
is free of order
Assume that (A1) - (A7) hold.
Take
P
(p,q).
E ZIO,igl
and
Then we have
Take
q £ ZIO,i f ].
-G07-
(6_32)
The proof is easily obtained from Theorem 6.10 and Proposition 6 15. We need a refined version. THEOREM 6.17. p (; Z[O,.igl
and
(; > 0
Take
Ahlfors Estimates. q (; Z[0'-"f1.
and
s > O.
Assume that (Al) - (A7) hold
Assume that
(f.g)
IJ: IR+- -
Define
Take
is free of order
IR(O,1)
(p,q;
by
(6.33) l + Tf
1
(r,s) + T q
{
8(r) ,.
gp
~
1 + mf q
Define M - M+-
M+
~pq
q
Lf r
(s)
if
M+
On O.
Then
define
Epq
~
bt
F(r)
=
J M
exists.
s
O~r<s.
0
on
M.
~pqV
We have the estimate
m .
For almost all
the integral
(6.34)
~
p
p
by (4.26).
put
(r,s) + mf ~ (s)
_1_ 4> ] lJ(r)-l ~ [ ~2 pq
pq
0
On r > 0
-208-
log+ F(r)
(6_35)
~ 2(1 + E)(lOg Tf(r,s) + log+ Tg(r,s) + log Y(r) + log+ Ric.,(r,s) +
PROOF.
C,
log r.
11 ~ 8
Define
-IT : M
0
_
For
(0,1).
r > 0
the following
integral exists (Theorem 6.10)
FO(r) -
J
] 8(r)-1 [...L. J!2 pq
Spq
~
I H[ r
F 1(r) -
2m
Hence the integrdl
J! pq ] 11-1 ~ pq" m-1 d., ,... [l.2.
J:
F(t)
D
-
F1 (r )
I
rI S
- 2m
1 • ] 11-1_ [ J!2 pq '='pq
H[ r )
H[rJ
- m
J
[;r
.pq] 8(t)-1
~pqD)t2m-1dt
M
F(t)t 2m - 1 dt.
exists for almost all
t > O.
Lemma 6.13 implies
Define
for
r > s .
·209. Then
where
F 2(r)'
J r
8
FO(t)
Take a constant
dt
t 2
Co >
1 ~
4(n + .8(r)
1 + Co +
1)3 2 Y(r)(Tf (r,s) + T q
(r,s) + Co) •
lilt IB (s), Then q gp
F 2 (r) ~ 4(n + 1)3Y(r) T f (r,s) + Tg (r,s) + cO)3 q
gp
if
r > s ,
p
consequently we obta'o log+ F(r) ~ (1 + E)2 log + F 2(r) + c.f2m - l)1og+ r
~ (1 + E)2 log Y(r) + 3(1 + E)2 log +(T f (r,s) + T g (r,s)) q
p
+ E(2m - 1)log + r + log(4(o + 1)3) + 3(1 + E)2(lOg
Co
+ log 2)
~ (1 + E)2 log VCr) + 2mE log + r + 3(1 + E)2 log+(3 q T fer ,s) + 3 PT g(r ,s) + +c3 P + 3q - 2)Q E(r ,s))
~ (1 + E)2 log VCr) + 2mE log+ r + 3(1 + E)2(10g+ Tf(r,s) + log+ Tg(r,s) + 10g+ QE(r,s))
Here we have
-210-
log+ QE(r,s) - 10g+lC, log VIr) + RicT(r,s) + EC, log r) -'1+1 . '1t og og Vir) + log+RlcT(r,s) + 1+1 og og r + log+ c, + log+ EC, + log 3 E
~.
3(1 + £)2
log VIr) + log
+-
I)
Tg r,s +
mE
log
r
, 3(1 + E)"
We obtain log + F(r) ~ «(1 + E)2 + E) log VIr) + 4mE log r + 3(1 + E)2(10g Tflr,s) + log+ Tglr,s) + log+ RicT(r,s).
Without loss of generality we can assume that
~ 1 + 4E.
(1 + E)2 + E
Hence replacing
0 < E < 1. by
E
4Em
we obtain
log + F(r) ~ 3U + E)(log T r(r ,s) + log + T g(r ,s) + log VIr) + log + RicT(r,s)) + E log r .
On
M+
we have
IKpq - kphqu m.
on
M+.
(6.37)
Defme
II
., - 0
for all
We have
(7.10)
Hence
al6),. - 0
Hence
[) b mm+l a 0 2 - 0
'"
E
~[m.p]
and all
if and only if
>..
E
.? - 0
'"
E
~[m.p).
for all
·223·
(7.11)
).
~[m,q].
E
IH~y,
o
- 0 for all
.
bLx 0 - 0
Take
and
G (E(a))
Z[O,n]
in
Z[O,m] -
Z[O,p]
2
- 0
if and only
(see Case 1).
m
and
1] _~~m J
m
[
0
m
E
I
q
+
+
j
E ~Ip.n] ITo
] [ + 1
l
If
q.e.d.
Z[O,p] "Z[O,q).
be the inclusion map.
tr - IT
Define
n p
-
q m _
]
j
.
Define E
it-{m
QUo
SIb p rn]
LEMMA 7,6.
E
Slb,p,m]
A V· I - 0
is a linear subspace of
A V·
Moreover, if
&0' ." '&n
JPI&} - b,
is a base of
V
with
p+l
dim Slb,p,rn] - D(p,q,m} < [: :
17.15}
17.16}
E
a 0
which is the case if and only
D(p,q,m) _ [n+ p+ 1
- If:.
and
x
q
(7.13)
(7.14)
E ~[m,p)
y,
for all
p
(7.12)
Let
. +1
0 b 111 m
Consequently
such that
:]
& - &0 " ... " &~
then S[b,p,m] - (t.
E
A V* I - 0
A
A1 V·. p+
"'(b ,p,rn ) are linear subspaces of S
SIb .p,rn ) an d
., A
~ E 'I-[rn,q).
0
6.,(p)
by the definition of
E
Gp _ m_ 1(V).
Slb,p,rn).
Slb,p mI.
E
~ E: S[b,p,rn).
Take
It E: Grn(E(b))
and
11- E: Gp _ rn _ 1(V).
Then
11- -
I
)"E:'I-( p-m-l ,n)
z)"6),, .
We obtain (7.18)
CLAIM 1:
If
11 E
'I-[m,q)
and
)"
E
'I-[p - m - l,n),
then
(7_19)
PROOF OF CLAIM 1. can assume that
6)1
If A
6 11
6)"
;f:
6)" - 0,
A
O.
then (7.19) is trivial.
An injective map
y: Z[O,p) _
Hence we Z[O,n)
is defined by Y(x) _ {V ( X ) },,(x
-
m
-
1)
if
x
E
Z(O.m)
x
E
Z(m + l,p).
·225·
There is a bijective map increasing.
Naturally,
'" - 11'-1.
Then
.,
0
71::
., '"
.,
is increasing. l:
Z[O,m) -
°, .,( Hence
.,
l
0
by (7.17).
viz) -
E:
.,
viz) ,
l (x))
=
In particular, :l[O,p)
v(y) -
.,(x) ,
E ~[m,q).
71:
",(y).
Thus
Take any
x
Ol[O,m)
E:
Then vim) ,
., E: tf'.
q.
Therefore
Claim 1 is proved.
- 0.
Therefore
Slb,p,m] - Slb,p,m)
which proves
(7.16). Since
(&.,lTE:~[P n]
is a base of
the family
"V, p+l
is linearly independent and spans a linear subspace Then
S[b,p,m] - L 0
dim Slb,p,m) -
[: :
and
dim Slb,p,m) - dim
:] - litf'.
For each
s E: Z[m,p]
T s - (1' c: iI-(p,n] I {
Then
L
(&.,l1'E:tf'
of dimension
"V - dim L. p+l
litf'.
Hence
In order to prove (7.15) it remains to
calculate the number of elements of
tf'.
define
1'(X) , .,(x)
>
tf' - T m V T m+l V ... V T p
is
v(y) - .,(",(y)
m' "'(m).
By (7.13) we have
0
Define
then
.,(",(m)) - vim) -
T(m) ,
., _ V
~[p,n).
E:
",(z) ,
be the inclusion.
Claim 1 and (7.17) how that
~
such that
is injective and increasing we have
'" I Z[O,m]
and let
Ol[O,p) Hence
If 0 , z , y 'm,
v.
-
.,(",(z)) -
Since
Ol[O,p) -
is injective.
q
if
XEZ[O.S)
q
if
x
E Z[s
is a disjoint union.
+
}.
l,p)
Therefore
-226-
Take
s
~[m pI
E:
and define a map
ps : "tls,q) X "tIp -
If
v
E:
"t ,q]
{
VeX) /..I(x
., (; T
Then
U E: "tIp -
s -
l,n - q - 1) -
l,n - q - 11
Ts
define
by
ps(V,UI - . ,
.,(xl
lind
s -
s
x (; ZID,s)
If
s-I)+q+1
ObviouslY
X
is bijective.
Ill's - (1I"t[c;,q)HII"t[p - s - 1, n - q -
(;
Zls +
I,p)
Therefore
1)) -
[: :
:]C -:]
We obtam
a
p
< IIIf
L S=m
dim Slb.p,rn]
- IIIf
.J < J
[n I]. p
+ +
1
q.e.d,
Define
(7.20)
Slb.p,rn) - P(Slb p,mll
-227-
LEMMA 7_7. Take p and q in Take b E Gq(V). Then we have
(7.21)
S[b,p,m[ " Gp(V*) - (a
PROOF. if
1P(.
A
jJ,
A
THEOREM 7.8.
o,
a
E
Take
m 'Min(p,q).
that
IJ
SIb p,m]; p
Z[O,p] " Z[O,q].
01.
- «A;II'
if and only if
Gm(E(b)).
E
m
0 b mm+l a 0 - 0
0 aLy 0 - 0
Then
for all
y
Take
Gp(V*) lob IiIm+I a 0
E
By Lemma 7.5 and symmetry
DaLy 0 - 0
that
'l[O,n].
and
Gq(V)
q
and
E
Z[O,n].
tlJ"'"
II
Take
!: Gp(V*).
m
E
Z
with
Assume
Then #Q}(b,ml , D(p,q mI.
Since
IJ
projective plane in
is in general position and since 1P( A V),
Slb,p,m]
is a proper
we have
p+l
IIIJ(b,m) ~ 1 + dim Slb,p,m) - dim Slb,p,m] - D(p,q.m) Basically, our proof is the same as Wu's (126).
q.e.d.
However, Wu does not use
the product 113 m+l and the norm 0 b 113Om+! a D. These new concepts simphfy the argument. The number D(p,q,m) shall be calculated in certain cases.
-228-
!&MMA 7.9. Define
Take
D(p,q,m)
p
and
in
q
by (7.12).
Z{O,n1.
Abbreviate
Take
m
E:
ZIO,p] (\ Z[O,q].
D(p,q) - D(p,q,O).
Then we have
< 2n+1
(7.23) 0 ~ D(p,q,m) -
(7.24)
D(p,q) _
p
If
0
~ m ~ q ~
p,
+
1
then we have
1
n
-
+ m +
-
m -
n
+
1] _[" - q]
p
+ 1
q
(7.25)
q]
[n -
+
D(p,q,m)
(7.26)
~
D{p,q,q) -
(7.27)
(7.Z8)
D(p,p,p) -
[
[
[
q
> 0
P
Q
1] _[n - Q)
" + P + 1
n
+
1]
p
q
- 1
P + 1
D(p,O,O) -
(7.29)
[f
0
~
m
~ P ~
q,
- D(p,O).
then we have
(7.3 )
(7.31)
D(p,q,m)
~
[
n
+
1] _[q 1] +
p+L
(732)
D{p,q.p) -
[
1>+1
n
+
1] [q 1]
p
+
1
+
p
+
1
·229· D(O,q,OI - n - q - D(O,ql
(7.33)
PROOF.
We have
[: : :]
p+l
!
j-O
n
-
p-m
q
]
+ 1 -
Hence
D(p,q,m) -
j
~ 0 [Q
j
1] [
+ j
in the case
m - 0
al
0
p
(7.24).
If
[
j~O
+
j
n - Q + 1 - j 0
~
m
~
q
+
+
1 +
]
which proves (7.23) and
q
~
p,
take
If
in the sum of (7.25) and the estimate (7.251 follows.
j - q - m
m - q,
this is
the only non·vanishing t rm in the sum and we have equality (7.271 which implies (7.28) and (7.2 I.
If
0
~
m
~
p
~
q,
take
j -
p - m in the
If m - p, this is the onb non·vanishing term in the sum and we have equality (7.321 which implies
sum of (7.30) and we have the estimate (7.301. (7.331 and (7.281. LEMMA 7.10. <JIl' ... ,<JIh
Let
q.e d. W be a hermitian vector space of dimension
be vector'! in
W.
Take
It
E
W*.
h.
Let
Then we have
(7.34)
PROOF.
"""
... '''''h
be an orthonormal base of
be the dual base.
Define
y U -
Define
Lf't
for
U
E
W.
IWI1,hl.
Let
""!, ...
Then
'''''~
-230-
Then
~I' ... '~h
is an orthonormal base of
A V.
A number
A
£
a:
h-l
exists such that
Therefore we have
If
oUI' _. ,oUh
are linearly independent, then
is the one and only element of
-
Gh l(W, - 1P(AW,. h
o ooLy 0 - 1
(7_35)
00
for all
Let
QJ;c"
pO"lItion
Take
(7_36)
Take
p
and
be a subset of
b
E
Gq(V,.
q
in Z[O,n].
Gp(V·'.
Take
Assume that
IR
Take
r
0 < r
~ 2- 3n - 3 r([J' .
£
with
Then we have (7.37)
Lemma 7.10 implies
y £ 1P(W·' _
We are ready to prove the required estimate. the ga ge (7.2' - (7.4,. THEOREM 7.11.
1P(oU1 " -•• " oUh'
MIJ(b,m,r' , O(p,q,m' .
Recall the defimbon of
m £ Z[O,p) " Z[O,q].
QJ
is in general
-231-
II is enlarged. 1!J(b.m.r) increases. By Lemma 7.3 we can
If
PROOF.
assume that
[n + 1]
~
I!J
p +
.
lVe abbreviate
1
(7.38)
d - D(p,q,m)
Define
11" by (7.13).
0 < h - d - 111" E; h.
Then
be an orthonormal base of
/.l - Min(p+l q 1)
V
such that
Let
&0' ... '&n
& - &0 " ... "&q
and
b - F(&).
CLAIM 1.
Take
E: Gp(V·)
U
a - 1P(u) E: Gp(V·).
T
(7.39)
I n + 1,
(7.53)
6(1J) is analytic.
In both cases, PROOF.
Case 1:
k ' n + 1.
Assume that
enumeration of I).
Abbreviate
Gj - Fg.
Let for
IJ - (gl' ... ,gk)
j - 1. ... ,k.
be an
Then
J
Z(IJ) - ZIG}
A
...
neighborhood of gj
for
j - I,
(7.54)
z
and ,k.
Take
z
~ j :
U -
Ig..
Z IE
IE
M. V·
Let
U
be an open. connected
be a reduced representation of
Then
Z(IJ) I U - (x
Subcase a: that
"G k).
Assume that ~
Hence
IE
U I ~ }(x) " ... "
z
£
j(z) - O.
Iq.
~
k(x) - O}
Then
j IE
Z[I.k]
By (7.54)
Z IE
Z(II).
Then
j
exists such
J
and
Subca e b:
Assume that
p
with
Z[l.k]
IE
~
j(z)
~
j(z) "
;t
0 t:)
~
j
< p
k(z)
and
p(z) - 0
with
;t
Z IE
6'1 -
1cJ.
exist such that
IE
zl1.kJ
g.(z) - g (z). J
p
Here
1P( ~ j(z)) - gj(z) - gp(z) - 1P( ~ p(z)). j
< p.
By (7.54)
z
IE
Z(IJ).
Hence
-239-
Subcase c: for all
j
E
Assume that
Z[l,k]
1 :Ii j < p :Ii k. lJ(z) -
,«)
Hence
and
Also
1 (z), ...
,«)
gj(z) - :PI
k - IIIJ(z) k(z)).
«)
1(z) " ... "
Subcase d: subcase
c.
Then
«)
and
and k(z)
j(z)) 'I- :PI
«)
The map
r([J(z)) - r(lJ(z)) - 0
Therefore
4(11) - 611_
Z E
r(lJ(z))
Assume that
Z E
lJ(z) C V:
and
«)
p(z))
=
o.
p(Z
0,
r
Defl e
1P: lJ(z) - . lJ(z)
[J(z) =
Th n
is bl cllv
is not in general pos tion. which implies
Z(IJ) - 61J. IIIJ(z)
=
k.
E Z(IJ).
Z
Define
lJ(z)
as in
The map
~
11' : lJ(z) - . lJ(z)
is bijectIve.
r(lJ(z)) - r(ij(z)) - O.
Subcase e: z E 61J
S;;
Hence
Since
Assume that
Z E
Take
t
Z
E Ig
Z E
.
E
t
E
Z E
Then
then
which proves Case 1.
Define
Zit) - 4(t)
4(1J).
S;;
Hence
~
W
4(1J).
E 4(1J).
Then
Assume that t
E :Pn+ 1(IJ)
Ig ~ It ~ Zit) ~ Subcase b:
g
n 611.
4(1J) - Z(IJ)
k > n + 1.
E :Pn+! (IJ),
Subcase a: Z
Z(I))
4(1)).
Assume that
If
we see that
z E 4(1J) - 61J .
These subcases show, that Case 2:
z E Z(IJ),
Z E
!:
!;;;
g E IJ
exists such that
gEt·
Then
w.
Assume that
ZIt)
then
exists such that
Z E
IJ and h E IJ exist such that :Pn+ 1(Ii) exists such that (g,h)
6 t - It
Iq.
W.
611 - IIJ. g 'I- h ~
It).
Then
k;!: 2
but g(z) - h(z). Then
and Also
-240-
Subcase c:
As ume that
z
(J(l)
is not in general position.
that
~(z)
Tog
Hence
Z E
~
Hence
is not in general position.
~(z) ~ !J(z).
Iq.
A(I) _
£
:Jln+1(!J)
E
A«(J) - W.
11I~(z) - k,
but
exists such
1I~(z) - n + 1
Here
A(~) - 6~ !; Z(~) !;;:
er we obtain
Then
and
w.
In particular,
A(!J)
is analytic.
q.e.d.
k' n + 1.
Assume that
Enumerate
!J - {gl' ... ,gk}·
Then
(7.53)
is defined and of class enumeration.
If
on
COO
M - I!J
6!J - I!J'
Z I:
and does not depend on the
then
o !J 0 (z) - 0 'l(z) ;.. ... ;.. 'k(z) 0 - o. If (7.54)
r«(J z)) - r«(J)(z) - 0 !J 0 (z)
Hence
r(!J)
setting
=
[]
Z E
for
z
I:
COO
j -
I, ... ,k
and
G1
A
on
M - I!J
by
6Q1 - I!J .
Assume that lJ is in ceneral position. for
then
'l(z) ;.. ... ;.. 'k(z) 0 , 1.
extends to a function of class
r(!J)(z) - 0
M - 6!J
•••
"
Gk _ O.
Then abbreviate
G. - F l
gj
The zero divisor
(7.55)
does not depend on the enumeration of I). Its support is the union of the (m-l)-dimensional branches of Z(I)) - A(QI). Hence supp
1t!J
t: Z«(J) - A !J).
If
in a line bundle and we have
k - n + I,
then
G1 " ... "G k is a section
supp Jill - Z(q) - A(lJ).
-241-
Also the meromorphic map (7.56)
does not depend on the enumeratl'on constant.
"'.
'11
If
k - n + 1,
Now, assume that (M,T) is a parabolic manifold. assumptions that k - IIIJ ~ n + 1 and that IJ is in we define the counting and valence function by
th en
'" QJ. is
Still u der the II' neral position,
(7.57)
t ~ 0
for
and
0 < s < r.
r
If
~T
E
we define the comp nsaUon
function and gauge mea re function by 1
(7.58)
log FnJ) a
provided the integ als exist, which will be shown below. mlJ(r) - r lJ(r),
which however fails if
Trivially
k > n + 1.
The First Main Theorem for the exterior product proved in Section 3. 1. Special case gives us immediately: THEOREM 7.15. First Main Theorem for leneral position (k ~ n + 1). Let (M,.,) be a parabolic manifold of dlmension m. Let V be a hermitian vector space of dimension finite set of meromorphic maps IIIJ ~ k :I:; n + 1. Assume that ~(r)
- r q(r)
S E ~T'
exists for all
r
n + 1 > 1.
Let
q
be a
11': M -
QJ.
1P(V*) with is in general position.
':I; 121
E ~T'
If
0
n + 1 and that
lJ
Still under the
is in general position, we
define the countins: function (7.67)
for
t > 0
and the valence function (7.68)
for
and the compensation fu
0
1.
be a meromorphic map.
WV
By (B8) we have
m
M.
be a holom rphic form of bidegree
lIJ - min (lg I g
We will makE>
u ptions:
8
For
with majorant
E
- 1,0)
!P(V*)
on
M.
Y.
B.
Define p
g : M -
(m
k - l!!J > O.
Z[O,lq}1
Define
define
(S.1)
as the family of associated maps of degree x + ~ Max(O,x) by affixing
g.
for a1l
x
E
JR.
For instance,
p
of
The dependcnct' on :E:pq(g),
4>q(g), "'p(g), Ppq(r,g), Rpq(r,g) etc.
~
q}. g
AgaIn put E!J
is indicated
pq(g), ppq(g), IKp(g), kp(g), .pq(g),
-246-
PROPOSITION 8.1. and (f,g)
q £ Z[O,n).
Assume that (B1) - (B8) hold. Assume that
is free of order
(p,q)
t)
Take
is in general position.
p
for all
g
E:
t).
For
£
p £ Z[O,it)1 Assume that
> 0 we have the
estimate
(8.2)
+ 3D(p,q)kC,(1 + £)(log Tf(r,s) + log VIr) + log+ Ricr(r,s))
l:: log+ T g(r,s) + £ log r .
+ 2D(p,q)C,(1 + £)
g£(J
PROOF.
Define
u - Min(p+1,q+1).
Define
(8.3)
By (6 9) we have (8.4)
o~
~ _____ t .s:.P.:a.9_(_&_)_ _ __
t P9 ( g ) ( n
CASE 1. and def ne
+ l)(kp(nI) + hq) ...
Assume that 8 r g)
2-
(n
llC)} - k :?: D(p,q).
by (6.33).
Put
8 5) Then Theorem 7.12 gives us the estimate
+ l)(k p (
Abbreviate
g )
+ hq)
2~1.
d - k - D(p,q) :?: 0
·247·
(~.
r(Q} )2d I T P gEQ}
~
c
o
[L [_1_. gEQ}
/..l2
(g)] .8(r.g)-l pq
/..l
(k
P
(P9 (g) (Q}) + h
~pq(g)
(g)].8(r.g)-l pq
(k
p
(Q})
+ h
q
)2
JDP.q q
)2
which implies
TI
0 fq
gEQ}
mg
02.8(r.g)-2~ P
(g) pq
The definition (6.37) co v r s this estimate to
-2d
- hq
[I T •
p! q+ 1
gEIJ
. [IT ~
mg
0 f
gEIJ
q
(g)h2][. IT f 0
q
tpq(g)
gEQ}
gEt)
hq
mg
0 f q
(g)
]
pq
0-2.8(r. g )]
p
. [L
02.8(r.g)-2( p
P
c r(t) )-2d[l + k p (IJ)]2d[r-r p o
III g
q
gEt)
02 .8(r,g)-2 ( p
pq
(g)]
[IT g EIJ
(g)] D(p,q) . pq
0 f
q
iii
g
P
0-2.8(r,g)]
-248-
Take the logarithm, i.ntegrate over
M
and divide by two.
This
gives us
(8.6) D(p,q)Sr (r)
~
+
I
gdJ
[mr
m (r)
g gp
-
meq+1 mgp (r)]
I Ppq(r,g) + ! .B(r,g)mr mg (r) gdl &£\1 q p
+ 1 D
2'
pq
J
[ log!
gEt)
[
1
.. 2 .pq(g)
] .B(r,g)-l
]
~pq(g) f1
...
H where
(8.7)
I
10C[1 + "Ph~I1)]f1 ~ g~1I
M
I M
-
I
gEt)
log
+ "p(g) -h
f1 + c, log(k + 1)
q
Rpq(r,g) + C. loc(k + 1)
Also we have
(8.8)
S(r,g)mr
q
mg p(r)
~ .s(r,c)(T f (r,s) + T (r,s) + m.. IB" (s) q gp ---x q gp
~ 1
-249-
The Ahlfors estimate 16.35) implies
(8.9)
I
log
M :$:
c,
(-1z. pq
I
&£11...
log!
I
(g)]
8Ir,g)-1~
[_1_.
I
gEII ... 2
pq
pq
(g)a
Ig)] 8Ir,g)-1~
pq
a
M
:$:
c,
I
gEQ}
log +
[~. (g)] 8Ir,g)-1 ~ a] ... pq pq
[I
+ C, log
_c,k
M ~ 3C,(l + E)k (l g Tf(r,s) + log Vir) + log+ RicT(r,s))
+
3C,(1 + E) I log + T (r,s) + CokE log r 1 C, log ~ g EQ} g
Now (8.6) - (8.9) yie d
18.10) D(p,q'Sf (r) + q
+
I
(mf!Hg (r' - mf
gEl)
q
P
q+l
!Hg (r)) p
2C,(1 + E)k(log T f(r,s' + log Vir) + log + RicT(r,s"
+ 2C,(1 + E)
I
gEqj
log+ T (r,s) + C,(k + 1)E log r g
where we abso pt the constants into the c 1 > 1 + Co(k + 1) > 1 impl es (8.2' in the case
and replace k
~
D(p,q,.
E
log r by
term.
EtC}
0,
all
II. We obtain:
THEOREM 8.4. §econd Main Theorem (Maximal version). Assume that IBl, - (B8, and (8.19, are satisfied. Assume that IJp is in general position. Assume that (f,g, is free of order (O,p, for all Take E > 0 and s > O. Then we have the estimate
g
E
IJ.
·257· (8.20)
nil[n q~1
q
l](Np
-
P -
+ Tf
(r,s) q
Ir,s)) q
+ [
1
p
]
1
+
F (r,s) n
n-l
r
(k - D(p q))+r .... (r) ~p
q-O
+ (; log r There is p operly.
8
geometrl
ROPOSITION 8 5. m.
B
Let
M
differential ;!:
121
CP: M -
Xo (;
Wand a point
dCP(Xo): if. Xo 1M) -
W
of
cP
at
Xo
M
Wand a
such that the
is injective.
Let
be a finite set of linearly non-degenerate merom orphic maps
hi : M -
1P(Vh)
"h + 1.
Define
cfifferential form
where
Vh is a complex vector space of dimension
n - Max(nh I h (; 6)"
B
of bidegree
I!olynomials of at most degree f»r
is chosrn
be a connected, complex manifold of d"mension
Assume that there is a finite dimensional vector space
holomorphic map
6
condition which implies (8.19) if
Pirst we need the following result.
B - CPS(S)
that is
1m - 1.0)
n - 1
J h - Dh
Then there exists a holomorphic whose coeffIcients are
such that each
for all
h
£
h
E:
6 is general
6. ~
tROOP. ~idegree
By Theorem 7.11 in (27] there exists a holomorphic form (m - 1,0)
polynomials of degree
OD
W such that
nh - 1
....
Bh
Bh
has coefficients which arr
at most and such that
h: M -
1P(V h)
if
-258-
is general for
cp(i\)
Bh -
h E ~_
for each
k
a complex vector space of dimension of
Y.
where
71 - h~~ 71hEh
m - 1
on
degree
n
W w
a base of function
h E ~
Vh
1\.~
-
h
...
Then
a hn - Dh (D,1l)-n.h h
~
...
B - 8{ 8)
O.
B{E h ' - Bh ,
Becouse 8 E Y
=
H( 8).
such that
-
1P(V h)
h: M -
is connected, ~
Dh(xO,8)
0
is a holomorphic form of bidegree
a hnh (X o)
Since
there is a
U X Y
WIth polvnomial coefficients of at most degree Dh(x O' S);i; 0
n - 1.
Define
in respect to
is general for
8,
B for each
q.e.d.
COROLLARY 8.6.
Assume that
(Bl) - (85) are satisfied.
is a finite dimensional complex vector space CJl . M -
be A holomorphic
is general for
Dh(zh' Eh)
Vh
Let
M.
"
Xo E U and a vector
the meromo phic map E~.
a h : U-
exists uniquely such that
It
Since
B(n)
h E H
8 - CP*(S)
h
1\. h
zh E U such that
1.0)
be a base
is a holomorphic form of bidegree
take a reduced representation
and define
there is a POint
(m
bE
coefficients are polynomials of at most
0'1
Dh . U )( Y -
for each
B(n}
Then
is an open, connected subset of
in respect to point
{Eh I h E~}
Y
1.
For each U
and let
Let
71 E Y define
For
where
k - 1I~_
Define
Wand a point
dCJl(xO) : ~ x (M) -
o
W
linearly non-degenerate.
of
Xo
€:
rp
at
Take
M
Assume there
Wand and holomorphic mal
such that the differential
xo
is injective_
p E Z[O,n).
Assume that
For each g
£
f
II assume
is
·259·
that
g(M)
is contained in a p-dimensional, projective plane in
but that
g(M)
plane in
!P(V*).
bidegree
(m - 1,0)
degree
Then there is a holomorphic differential form
n - 1,
and such that
!P(V*)
is not contained in any (p-1)-dimensional projective
on
=
of
W whose coefficients are polynomials of at most
such that .if
8
f: M -
nand
!P(V)
is general for
.i g - p for all
g
E
B
q:> (8)
in respect t
Q}
B.
Hence (B8) and (8 19) are satisfied. PROOF.
Let
V*g
be the smallest linear subspace of
g(M) (; !P(V;).
Then
map
!PIV;)
g: M -
dim V; - p + 1
p < n.
where
is hnearly non· degenerate.
8
there is a holomorphic fa m
of bidegree
V*
such that
The meromorphic
By Proposition 8.5
(m - 1,0)
on
whose
W
coefficients are polynoml f at most degree n - 1, such that f : M !P(V) and g M !PIV*) are general for B. Hence 8
If we conside
If - n.
gIM)!; ..(V~)
Since g
E
as a map into
g
we have
.i g ' p.
1PIV·),
then
p' 1 g
Hence
Ag - p
Then
[p!
for all
IJ. q.e.d. We want to study the case
term
NF Ir,s)
p - O.
appears in the Second Main Theorem.
1] - 1
and the
This term can be
n
used to modIfy the Second Main Theorem. Let
A;/:
0
be an analytic subset of pure dimension
Then there exists one and only one divisor r each simple point
x
E ~IA)
v be an divisor on M. e set of branches of ch that inite.
~.211
S.
Assume that
x
E
v A such that
A and such that
For each
vlx) - P A for all
We have
of
A
E
m - 1
S - supp
of
v A(x) - 1
supp v A-A. V ;/: 0.
Let
cr be
cr there is an integer PA
A "RIS).
The family
cr is locally
·260· ~ 0
Then
v
Take
n £ Z
PA > 0
if and only if with
~
n
O.
for all
A E
The truncated divisor
ex.
v
Assume that
)I(n)
~
is defined by
(8.22)
Obviously,
0 ~ v(n) ~)I
If
is a parabo c manifold,
(M 1')
and
)1(0) -
o.
If
)I _ 0,
put
v(n) _
o.
write
(8.23)
By a comblhation of the methods of L. Smiley [251 Lemma 3.1 and B. Shiffman [221, [231 (1.14), we obtain: THEOREM 8.7. Assume that (B1), (83), (B4), (B5), (B6) and (B8) hold. Let II be in general position with III; - k ~ n + 1. Assume that 19 - 0 for all let
UF
g
n
E IJ.
Fn
Let
be the
be its divisor.
Let
representation section of
nth
f
and
be the general position divisor by (7.56).
UIJ
Tht'n ~
(8.24) g
~
REMARK 1.
In (8.24) only
REMARK 2.
If
n
•g
>
B.
depends on
UF
(n>
u( f
is a parabolic manifuld. then (824) implies
(M,-,.)
I Nf g(r,s) ~ NFn(r,s) + Nm(r,s) + I N f(r,s) g£'1 ' .. gE'1'
(8.25)
f.RQQJ!.
We have
:jln+l(IJ) - (~ ~ lilli' - n + 1).
defined by (7.4 ) and sUPP UI; - 6(IJ) 6('1)
~
"EIJ Uf ,g '" UF n + Um .. + g "£IJ
IIJ
since
has pure dimension
by (7.42).
k ~ n + 1. m - 1.
Then
See also Lemma 7.13. Hence either
Let
If
U
-
6(1;)
D - supp
I
ge:QI
f,g
is
We have
is empty or
be the indeterminacy of f.
Define (8.26)
6('1)
v
8€QI
sUPP Uf,g .
(
-261-
If D - /ZJ, nothing is to be proved. Assume that has pure dimension m - 1. The set S - I (D) V I(~(QJ.)) V
(8_27'
is analytic with dim S :s; m - 2. of D - S. Take
D"#. /ZJ.
IIJ
Then
D
V If _
It suffices to verify (8.24) at every point
Xo E D - S. Take 'E 1l n+1(Q}). We claIm that
(8.28)
(8.29)
and
(830)
The maps go' _.. ,gn
and
fare holomorphic at
xo.
Take
n
a
(8.30'
E
1P(V) -
V E[gj(xO)) j-O
T1ere is an open, connected neighborhood
U
of
Xo such that the
f410wing conditions are satisfied.
(~ There is a chart ~ - (zl, ... ,zm) : U where
Xo E U
(i (
is a ball centered at
U' C;;;
0
U'
with
lJ-(xO) - 0,
and where
M - S.
The set U n D = Y n D - S ~ Ix E U I zl(x) go, ... ,gn and fare holomorphic on U.
() a E E[gj(x)]
for all
x
E
U
and
j -
0, ... ,n.
=
0)
is connected.
-262-
~
Take
representations
.:l
·264·
Abbreviate
18.45)
A
n
(8.46)
'()
~
-
j~O
V~*j
~ j ~ (-1 J ~ 0
(8.47)
•..
U -
A
•••
on
U.
n
A
~ j-l "
Hj t 0
...0 j+ 1
"
'"
~n .
"
Hence we assume that
v n - O.
Obviously, (8 29) is correct if Holomorphic functIOns
(AV")*
U
exist on
We have
(8.48)
Since
t:)
is a reduced representation,
U.....
- O.
We obtain
t:)
(8.49)
which proves (8.29). Take an enumeration and
vI
~
~
)12
...
~
vk
Q} ~
O.
(gl' --. ,gk)
For
j
=
such that
n + 1. '" ,k
Vj -
Uf,g/XO)
define
(8.50)
where
~. ~~.
J
if
I
n+l
(8.51)
j
~ K
I
Uf,gJ'(x
By (8.28) we obtain
o) ~ u.."0+ 1(x O)
n+l
+ v F (x O) + 0
:L
j _ 1
U
(n) (
f • g j Xo
)
·265· If
n + 2 , j 'k,
we have
(8.52)
Hence (8.51) and (8.52) imply
q.e.d.
THEOREM 8.8. Seco d M Theorem (Maximal version for Assume that (81) - (B8) hId. Assume that Ig - 0 for all
0).
p -
g
E
CJ
Assume that (f,g) is free for all g E II. Assume that II IS in general position. Assume that ICJ - k ~ n + 1. For 0 < s < r abbreviate 8.53) Take
Q(r,s) - 2n(n + l)ke.(log Tf(r,s' + log V(r' + log+ Ric.,(r,s))
E
> 0 and
s > O.
Then we have the estimate
(8.54) Np (r,s) + ~ mf gIrl n BEC»'
~ (n + l)Tf (r,s) + + 2n(n + 1)e.
and
n(n
t
1)
Ric.,(r,s) + [nk -
~ log + T sIr,s) + Q(r,s, + BEC»
E
n(n
log r
t
1
)]rll(rl
·266·
(8.55)
l)T fIr,s)
(k - n -
I g £q}
Nf g(n)(r,s) +
n (n
+
1)
RicT(r,s)
2
,
L
T g(r,s) + Q(r,s) +
£
log r .
g Eq}
PROOF.
W.lo g.
0
OO
Tf"f(g)
1.
uQ,... ,un
is the dual base and if
Let
is said to
-'1.77-
f(M) It E[1P(
(4)
( -
'" B
* >2 - ... 0
>..
E
11.0'... ,11. n . V·
on
A.
A
is defined by
(9.15)
Here
~~
u~"
is holomorphic on
~~
9.16)
A.
u>"U...o /..l
=
If
(~.u) E A[l).
on
then
U~/..l'
We have
on Hence g : M of
g
~ --0
>..
~
l'(V*)
for each
0
on
U>.,'
Hence one and only one meromorphic map
exists such that
>..
E
A.
u~.
For
~ E
'\10
>.. I U~ n A is a representation
A we have
-284-
Hence the pair Take
-.. >..
'I)
(f,g)
>..
E
A.
A
tJ
>..,
is not free. On
A
...
U>..
S>..
such that
tJ
;It
functions
A>")I
>".2,(z)
0
10
of
(p) A
U>..
tJ,
"t:)
Z E:
U)" - S>..
( p+ 1 )
tJ )"
(p+l) A ....
exists such that
for all
exist on
A.2, ~ 0
-. . >.. (p) -
"
, A "" ~ A " •.• "
An analytic subset
we have
such that
~
A
-'&'),,)1
10
v-o
on
)"J!.)' -
U)" - S)"
10
>..
A
10
~
"
which implies
.. , "
tJ
~ p-l)
A
()I) >..
A.2.:!:.!. :: o .
S>.. ~ U>..
U)" - S)" 'I: 0.
Then we have
( tJ
1J
nP
and
Unique holomorphic
-285-
therefore on
UA." A,
which means
By Proposition 9.1, the meromorphic map is analytically dependent on ~(.).
Since
f
6lM(.),
is free of
contradicts an earlier ob e
atlOn.
b) Assume that f a meromorphic map Whl h
the pair
Hence
p
(f,g)
is free which
n.
general for B. Let G: M 1P(V·) be ends analytically on 6l M(.). Take reduced
representations t::I: U V of f and "tIC: U V· of g and assume that 1J: U U' is a chart. Take veclors 1.
be a meromorphic map.
(C8)
(Cll)
m - 1
M.
(C4)
M
m.
for all
g
Eli.
Assumptions (ClI to (Cll) and Theorem 9.4 imply (81) - (B8), wherE' iff.
=
0
for all
g
E:
q).
Therefore Theorem 8.8 implies
B,
-Z91-
THBORBM 9.B. Second Main Theorem for functions fi Ids. Assume that (CI) - (Cll) hold. Take € > 0 and s > O. Then we have (9.24~
Np (r,s) + n
:L
mf gIrl
gEl)'
+ 2nln + l)C.lk log T fIr,s) +
I
log+ r glr,s)
I-
k log Rlc;Ir,s))
gEIJ
+ 2n(n + l)C.k log Y( ) +
log r .
€
Also we have (9.25~
(k -
n -
l)T f (r s)
+ 2n(n + l)c.k(log Tf(r,s) + log VIr) + log+ Ricr(r,s)) +
E
log r .
In addition, we assume (C12)
Tglr,s) Tf(r,s) -
0
for
r -
00
Ricr(r,s) (CI3)
11m
sup
r"OO
~
Tf(r,s)
for each
g E IJ
0
(C14)
Suppose that (CI) - (CI4) are satisfied. Nevanlinna defects mE
(926)
6f(g) -
6 f lgJ -
1 i min r ... oo f
Tf
Then we define the
t g(r) r,s )
-292-
(9_27)
1 -
11m r
(9_28)
"'00
11m sup r ... 00
Since
N( n )(r s) ~ N f • g'
(r,s),
-
we have
(9_29)
The Second Main Theorem 9.8 implies the defect relation. THEOREM
l i Defect relation. Assume that (Cl) - (C14) hold. Then
(9.30)
Our statement (9.53) does not contain any term dependmg on However, the assumptions (C10), (Cll) and (C14) depend on a situatlon deplored by Royden.
B.
B.
We are in
An assumption which has seemingly nothing
to do when the statement is made for the sake of the proof only.
In the
case
B
M = ([m
we will be able to eliminate these assumptions on
constructing a form
B
satisfying (CIO), (Cll) and (CI4) and to some
extent the rpsult can be established even if parabolic space of
T(Z) -
IIzII 2 -
a: m
where the exhaustion is
1zll2 + ... + 1~ 12.
majorizes any holomorphic form of bidegree
For any holomorphic vector function (9.31)
is a covering
([m.
First we consider the situation on given by
(M, T)
Then
u > 0
and
(m - 1,0).
g:
a: m
-
V
defme
T
by
·293·
LEMMA 9.10. on
a: m.
For
Let U
B
be a holomorphic form of bidegrpe
1N[1,mj
E
(m
1,01
define
(9.32)
Then
(9.33)
Define the holomorphic ve tor function Let
Y
be the majorant of
Y(r)
(9.34)
=
=
(B 1, ... ,Hm) ; a: m _
in respect to
T.
a: m.
Then we have
Max(l,mM(r,6)2)
log Y(r) ~ 2 log + M(r,6) + log m
(9.35)
PROOF.
We have
Take
r > D.
a:m[rl
if and only if
~
constants
E
c.
c > D,
Take
m
(9.36)
for all
B
fr
then
mim_1B"
B~
m
m
~
~
U~l
U,lJ",l
a:m[r) Then
and
~
E
a: m.
Let
YO(d
cum-Ion
I x 12 = cUt-liZ jJ.
be the infimum of all those
-294-
which i
mM(r.6 2 ~ YO(r).
lies
Max(1.mM(r.6)2)
(9.37)
If
mM(r.6)2
y(r) - 1 YO(r) ~ 1
Therefore
~
1.
then
~
m 1(6(ll-)I tJ I
and we have equality in (9.37). and YO(r) - VIr). Also
which implies
YO(r) ~ mM(r ,6)2.
In particular.
Y(r)
~ Ylr) .
IIt-II 2
If
and
YO(r)
mM(r.6)2 > 1
~
1.
Hence
then
Hence
is continuous in
r
which proves (9.34) and (9.35).
q.e.d. LEMMA 9.11. Let f be a holomorphic function on j £ INIl,ml. r > 0 and £ > O. Then
a:m. Take
(9.38)
PROOF.
Take
ll-
€
a:mlr]
such that
I fz.(ll-) I - Mlr.fz )' J
"'-1' ... ,"'-m h : a: -
be the standard base of
a: is defined by h( n
-
Let
J
a:m. A holomorphic function
f(ll- + l""'-j)'
Then
h '(0) - f (ll-). Zj
If
-295-
t E a:[El,
then
II~
I h(t) I li!: M(r + E,f)
M[r,
aClZfj ]
-
+ t1\}1 li!: 11'&11 + I t I :iii: v +
t E a:[El.
for all I fz.(~) I l
=
E_
Hence
The Cauchy estimates give
1 Ih '(0) I li!: E M(r + E,f) .
q e.d.
We need the following result of H. Skoda [851 Theorem 3 on THBORBM 9.12. Take E > O.
Let
f
a: m.
be a non-constant meromorphic function on
a: m.
a) There are entire functions g iii! 0 and h Ii! 0 on a: m such that hf - g and such that for each s > 0 there is a constant c( E,s) > 0 such that (9.39) for each
Max(1og+ M(r g) log+ M(r,h)) li!: c(E,s)(l + r)4m-1 Tf (r + E,S) r > s.
b) There are entire functions g iii! 0 and h iii! 0 on a: m such that hf - g and such that for each s > 0 there is a constant c( E,s) > 0 such that (9.40) Max(log+ M(r,g),log+ M(r,h)) li!: c(E,s)(l + (1og(1 + r 2))2)T f (r + Er,s) for each
r > s.
RBMARK. The pair (g,h) may have a common divisor, may depend on E but may not depend on s, and may be different in b) from the pair chosen in a). The case a) is good for rapid growth, the case b) is good for slow growth. PROOF. Skoda proved the theorem if f(O) - 1 and I - O. If and 0 'I: a E a:. Then there is a constant COla) such that
s > 0
·296·
Also we have TfIr + £,s) + T£,s,O) - T fIr + £,0) . s > 0
Hence the theorem extends to the case at
0
and
flO) 'I:-
A surne that eith r f
f
is holomorphic at
is not holomorphlc at zero.
that f function
is holomo h c at (.II
f
on
where
f
is holomorphic
o.
M by
0 with
flO) - 0
11(.1111
O.
Take
a;m.
B
There there is a holomorphic form
which d
mes
such that for each
tRa;m(cpl' .. · ,CPm-1)
c(£,s) > 0
is a constdnt
of hidegree
such that the majorant
of
Y
(m - 1,0) s > 0
B
on
there
for
T
can
on
a;m
he estimdted by
log VIr) ~ c(£,s)(l + r)
(9.46)
for all
4m-l m-l ~ j -1
Tcp,lr + E,S) J
r > s.
h)
There is a holomorphic form
which defines
&
a;m
(cpl' ...
c(£,s) > 0
constdnt
,Cp
m-
B
of hidegree
(m - 1,0)
such that for each
1)
Y of
such that the majorant
s > 0
B for
there is a T
can he
estimated by
log VIr) ~ c(£,s)(l + (log(1 + r2))2)
(9.47)
m-l
~
Tcp.lr + Er,s)
j -1
J
for r > s. c)
If
cpl' .. · ,CPm-l
holomorphic form & a;m (CP 1' .. · 'CP) m-1
B
for
of hidegree
(m - 1,0)
and a constant
c > 0
on
I[m
which defines
such that the majorant
Y
of
can be estimated hy
T
log VIr)
(9.48)
REMARK.
B
are rational functions, then there is a
In all there cases
Vj iii! 0 iii! Wj
~
B
clog r
r
~
2 .
can he obtained in the form (9.44) where
are holomorphic functions on
j - 1. .. , ,m - 1. on s
for
In a) and h), the form
a;m
B
with
WjCPj - Vj
depends on
£
for
but not
-301-
PROOF_
B is defined as indicated in the Remark, then
If
Hence it remains to choose
I, ___ ,m - 1.
j -
If
Cl'I'" - ,CI'm-1
are rational, then
taken as polynomials and (9.48) follows from (9.42). case c). a' Vj' Wj that
Each
is not constant.
Cl'j
does not depend on
j.
w{Pj - Vj
m-l
~
j
COlE)
T 1.
Tf(r,s) 10& r - At(oo)
1. Let be a surjective, proper, holomorphic map of sheet number
7C: M _ g:m c. and with
branching divisor p. Define T - 117C1I2. Then T is a pardbolic exhaustion of M. Let 'PI' ... , O. Then there is a constant
are available. >0
COlE,S)
such that
SIr) :E: cO(s,E:)(l + r)4m-1
(9.55)
m-l
:r
j -1
(B)
T ke
E
> 0
s > O.
and
r > s .
for all
(r + £,s)
T "'j
c 1(E ,s) > 0
Then there is a constant
such that (9.56)
for
(C)
Assume that each
j - 1, ... ,m - 1.
'" j
II
constant
Assume that
constant
c3 > 0
CP1'''' ,CI'm_1
such that
are rational functions.
Then there is
such that SIr) , c3 log r
(9.58)
PROOF.
> 0
for
r > 1 .
for
(9.57)
(D)
C2(A)
r > s .
Ord '" j < A
has finite order and that
Then there is
all
for
Clearly (C1) - (C9) are satisfied with
holomorphic functions
V J"
wJ' on
«;m
r
~
2 .
• - ,.
such that
There are
w·CI'. - v. • 0 J
J J
and
such that
(9.59)
(9.60)
is a holomorphic form of bidegree liO - tR m(CPl' ... ,CI'm 1)
.:
-
(m - 1,0)
which defines
and which is majorized by
"0
on
.:m
with
II
·305· majorant Y satisfying the estimates (9.46) or (9.47) or (9.48) in Theorem 9.14. Then ., - "0 0 7C majorizes the holomorphic form B of bidegree (m - 1,0)
on
M
with the same majorant
is a holomorphic function on
M
Y.
Also
w - w
o
0
7C
iii 0
with
(9.61)
B defines { - liM("'I' ... ,!/1m-I)'
Hence
The assumptions (C10) and
(C11) are satisfied and (9.25) holds. If we assume without loss of generality, that 0 < E ' 1 and if we define SIr) ~ 2kn(n + 1)C. log Y(r) + log + r
(9.62)
then (9.25) implies (9.54)
Observe that
T!/I ,1r,s) -
00
for
s -
00
J
since
!/I j
is not constant, and that
T'P. - T",. . J
J
In the case of Theorem 9.14 a), the constant mcreased to a constant
CO(E,S)
as to absorb
C(E,S)
can be
2kn(n + 1)C.
and
and we obtain (A). In the case of Theorem 9.1 b), the constant can be increased to a constant c 1(E,s) as to imply (B). If
'PI'''' ,q>m-1
Assume that Ord 'P j < I.
C
·306·
Applying 19.56) with
s -
E: -
1.
Then we have
q.e.d.
THEOREM 9.16.
Defect relation for function fields over
be a he m1t1an vector
n + 1 > 1.
pace of dimension
m > 1.
a 1fold of dimension
be a surjective, pro
r, holomorphic map of sheet number Let
p.
'l'l' ...• 'l'm-1
meromorphic functions on
([m.
Define
Let
7C:
'" j
=
'l' j
D
~ - I'RMI"'l' ...
''''m-1).
exhaustion of g : M on
~.
Let
Let
M.
!P(V).
Qj
!P(V)
and with
Then
.,
M.
Define
is a parabolic
be a finite set of meromorph1c maps
Assume that each
f: M -
([m
Then
7C.
are analytically independent functions on ., - 111'1"112.
be a
M _
c.
V
be analytically independent
"'I' ... ''''m-1
Let
Let
M
Let
connected, complex branching d1visor
«:m.
g
E:
Qj
is analytically dependent
be a meromorphic map which 1S free of
~.
Assume that
(9.63)
for
r ---
00
(9.64)
for
r ---
00
for all
g
E:
(J
•
Assume that at least one of the following assumptions (A) or (B) or (e) or (D) is satisfied. (A)
There is a number
(1
+ r)
(9.65)
for
E:
4m-l
> 0 such that
T",
Tf
- 1, ... ,m - 1.
(r.s) j
-0
for
r -
00
·307·
(B)
There is a number
E
> 0
such that
(9.66)
(1 + (108(1 + r2»2T~ (r + Er.s) ______________~~----~~1-----------
~
0
for
r ~
for
T f this
special case explicitly.
If M - ([;m
n - I, with
the assumptions again become simplicr. m >1
and
n - 1.
B. Shiffman already obtained the
two theorems under some what weaker results. (A) or (B) or (e) or (0) he needs only
(9.70)
Instead of the assumptions
for
for j - 1, ... ,m - 1. results [831. [841. THEOREM 9.17.
Tn this case,
r -
...
For comparison we st8t(' Shiffman's
ShIffman 1831. 1841.
meromorphic functions on
o:m
with
Let
f,gl' ... ,gq
m > 1.
be distinct
Assume that
(9.7U Then there is a constant
such that
(q - 2)T fIr,s)
(9,72)
If
cI > 0
T g .Ir,s)/TfIr,s) J
0
for
r _..
and
j -
I, ... ,q,
then we have
·309·
(9.73)
B. Shiffman shows that Theorem 9.17 follows from: THEOREM 9.18.
Shiffman [831. [841.
meromorphic functions on
([m
with
Let
f.g 1 •.... gq
q ~ m - 1 ~
o.
be d tmct
V
Assume that
(9.74)
Let
p
be the ramification divisor of the meromorphic map
Then there is a conc;tant
c > 0
such that
(9.76)
~
f
Nf g.lr.s) + c
J"'l'J
[.f J-l
T g.lr.s) + log Tf(r.s) + log+ J
r]
Our general theory forced us to make maximum modulus estimates for 8.
that is for the functions
Vj. Wj
such that
w{Pj - Vj.
Thus
applying Skoda's results we have to make assumptions (A). (B). (e). or (D) while Shiffman needs (9.70) only. will yiE'ld the same results.
Perhaps a modification of the B-method
Except for this small deviation Theorems 9.15
and 9.16 can be considered as an extension of Shiffman's Theorems 9.17 an( 9.18.
Hopefully, the assumptions that
"'I' ... ''''m -1
are lifted from
I[m
can be eliminated by further research. The B-method was invented almost 30 years ago. and has been used almost exclu'lively by the author (Stoll [93], [100], [l08\). The 8-method is justified again by the results of this section. Mori (63] obtains a defect relation for meromorphlc maps f : a;m _
1P(V)
and for moving targets
and extend his results in Section 11.
g.
a: m
-
1PIV).
We will discuss
';.10.
An Example
We WIll g ve an example where the integral in (6.19) cannot be split into three convergent Integrals according to (6.8).
We take Let
m - 1, M -
""0' ""1' ""2
the dual base.
a:,
B - 1. n - 2, V -
be the standard base of Define a holomorphic map
reduced representation
(10.1)
Then (10.2)
10
(10.3)
10 u
'It) - (O,1,2t) - ""1 + 2h2
(10.4)
(10.5)
We abbreviate
(t) -
(0,0,2) - 2""2
a: 3 f:
a: 3 ,
1P(V) - 1P2
and let
a: -
1P2
",,~, ""t, ",,~, be the
be
-311-
(10_6)
B = 1 + 41tl2 + Itl4
(10_7)
B- A+
C - 2 + Itl2
31tl 2
C2
3 _
Then we have
(10.8)
II
(10.9)
(10
10
112 - A
_1 L·
10 ) -
A meromorphic map
g
II
-
10
1112 - B II
-
"40
1P~
II: -
(10.13)
~
'It) - 1-1,O,2t)
II
=
-
It) - 10,0,2) - 21\.~
Then we have
2112
-
t(1 + 21 t 12)1\.0 + (1 -
4
1t 14)1\.1 + t(2 + 1t 12)-n. 2
is defined by the reduced
representation
(10_12)
10
1\.~ + 2t1\.~
-312-
(10.17)
(10.18)
('W)
.!. L
'W))
These maps are re
__
(1 _
-
2
I tl4)1\.~ - t(1 + 2 I tI )1\.
ed by the identities
(10.20) (10.21)
(10.22)
(10.23)
(10.24)
(10.25)
(10.26)
< 10 1. L·
(10.27)
-- -
2(te - t2B)
10
t
2 + t(2 + I t I )1\.~
-313-
>
a
r > s
(h,aj)
Then there are free for
there are constants we have the estimates
·319·
n+l
Th(r,s) ~ Tf(r,s) +
(11.5)
Tg.(r,s) + cl(s)
I J"'O
J
(11.6)
Nh aJr,s) ~ Nf,g.(r,s) +
(11.7)
, J
J
n+l
I
k"O k;tj
Tg (r,s) + c3(s) k
n+l
(11.8)
Nh a (r,s) + (n + 1) , J
I
k"O
Tgk(r,s) + c 4 (s)
If in addition we assume that also (E8) holds, then we have (11.9)
for
r -
(11.10)
for
j -
00
0,1, ... ,n + 1 .
Basically the theorem is due to Mori, but does not give such explicit estimates. a)
The proof is obtained in several steps.
A special
(AO' ... ,~.
operat on.
The base
Let
""0' .. · '''''n
""0' ... '''''n
is orthonormal.
(11.11)
If "&0' ... '"&n+l
fot
(11.13)
E:
are vectors in
ZIO,n + 1].
If
j
E:
V*,
ZIO,n],
be the dual base of
define
define
Abbreviate
·320·
Then
§j
-
~ j " '& n + 1.
A homogeneous, projective operation
(11.14)
V : V X (V.,n+2 -
is defined for
'to
V, 'll-j
E
E
v·
with
V
0, ... ,n + 1
j -
by
n
't. V '&0 V ... V '&n+l - ,fn + 1 j~O I ,
111:.11 II'II-jll
n
D
II'II-kll - 1It.1I 11'11-011 '"
lI'&n+ll1
k~j
Therefore we obtain
COROLLARY 11.3.
If
X E:
1P(V)
and
Zj E:
"(V·)
for
j -
0, ... ,n + 2,
then
(11.17)
LEMMA 11.4.
then
If
x
E: ..(V)
and
Zj £
F(V·)
for
j - 0,1, .. , ,n + 1,
·321·
n+l
TT
(11.18)
Zo ;.. .. , ;..
0
Zj_l ;.. Zj+ 1 ;.. ... ;.. zn+ 1 0
j-O
~ (n
PROOF.
Take
t.
. Zo V...
+ 1) 0 x V
E:
II!-J.II - 1 and
with
V
with
IIt.II - 1
z· - lP(it.) J
V ~+ 1 0 •
for
J
and
x - F(t.).
Take
j - 0,1, ... ,n + 1.
E:
Then
(11.19)
(11.20)
o
. Zo .
x V
V ••. V zn+ 1 0 -
!-o
lit. V
V
We have
(11.21)
n
(11.22)
Since
~
which proves (11.38) for
j -
n + 1.
L
J"'o
q.e.d
< 10
• "40
·>X . ~ 0 and
96 :: 0
if and only if
Since 6 - .".
6
([) •
D.
E
Xj
by (11.41).
Xj ~ O.
With
Since
(11.42) we have
(~{';;fn.> + bn+1Xj
Therefore
(h,b)
is free.
f and only if
[J,
We shall restrict since the cond1tion "h
f.
condition on
(E9)
is linearly non-degenerate.
f
f
is linearly non-degenerate
is linearly non-degenerate.
r elf to the case of a covering parabolic space,
0
1S general for
B"
does not easily translate into
There is a proper, surjective, holomorphic map
Let
p
q.e.d.
Thus we assume in addition
such that (E10)
h
are constant, then
If over
The map
7f
M -
a;m
., - 117f1l2.
be the branching divisor of
7f.
Assume that
ND(r,s)
Rf-limsu p
(11.57)
r -.00
(Ell)
Assume that
(E12)
At least one map
REMARK 1.
f
is linearly non-generate over gj
Since the case of constant targets is already solved, it is
T (11.58)
g 1
gj
is not constant.
Then
(r,s)
-"'lL.. o -g - r -
-
A
gj
(00)
> 0
for
r -
..
Hence we obtain
j.
T (11.59)
Therefore
[J.
is not constant.
reasonable to assume that at least one
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REMARK. When this manuscript was being completed for publication, Professor Shiffman sent me a preprint of the paper: (130)
Charles F. Osgood, Sometimes effect Thue·Siegel·Roth·Schmidt·Nevanlinna Bounds, or better, to appear in Journal of Number Theory, pp 51.
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Although the announcement seems to indicate, that the Nevanlinna conjecture was proved, it lacks a clear cut formulation of the result and it IS not easy to understand.
-344Index
a-divisor 22. 119 admissible base 84. 276 Ahlfors estimat s 49. 71. 201. 206. 207 a-multiplicity 3 analytically dep ndent 82. 84. 275. 277 analyticdlly dependent on a set 82. 84. 275 analytically independent 82. 84. 275. 277 analytically independent of a set 84. 275 argument prmciple 4. 24 associated covering 40. 154 associated flag 164 associated map 43. 154 base 83. 276 - • admissible 84. 276 B-derivative 43. 152 big Griffiths conjecture 54 boundary integral representdtion 5 branching defect 87 brdnching divisor 42. 157. 269. 287 canonical bundle 40 Casorati-Weierslrass lheorem 39 characteristic function 3. 10. 12. 26. 137. 139 chart 40. 151 chart atlas 40. 152 Chern form 17. 137 chordal distance 2 closed graph 18. 120 closed pseudobaU 115 compensation function 3. 11. 26. 36. 48. 61. 137. 141. 241. 243. 323 - • modified 61 complete 276 complex projectIve space 92 condensor 26 contraction. metric 104 contraction product 59. 95. 96. 126 coordinate functions 84. 277 counting function 3. 4. 10. 26. 35. 36. 61. 119. 126. 137. 241. 243
covering 122 • associate 40. 154 _ • nerve 122 - • open 122 curvature stress formula 69. 184 defect 7 branching 87 - • majorization 52. 268 - • Nevanlinna 7. 52. 80. 81. 271. 291 - • ramification 52. 287 defect relation 7. 52. 53. 81. 82. 88. 91. 272. 273. 274. 292. 303. 306. 333 defect relation for function fields 303. 306. 309 defect relation of Nevanlinna-Mori 91. 333 defect. Ricci 52. 87. 268 - • truncated Nevanlinna 52. 29~ deficient 7 deficit 26. 27 define 83. 276 degeneracy 76. 236 derivative 2. 109 - • - B 43. 152 - • exterior 2. 109 distance 17. 58 - • chordal 2 - • projective 15. 105 distinguished 17 divisor 3. 20. 118 - • a 22. 119 - • branching 42. 157. 269. 287 intersection 22 non-negative 20. 118 of a function 22. 119 operation 60. 322 pole 119 prime 20 pullback 21. 118. 121 ramification 287 simple 20 stationary 45. 154 support of 20. 118
-345- , truncated 20, 260 - , zero 20, 21, 119 dual classification map 17 dual Frenet frame 67, 175 duality 93 , dual, Poincare 25 dual vector 94 dual vector space 92 exhaustion 33, 115 exterior derivative 2, 109 exterior product 5, 14, 59, 93 evaluation 17 finite family 78 first main theorem 4, 11, 27, 28, 36, 37, 48, 62, 63, 64, 77, 137, 241, 146, 147, 148, 141, 244, 323 for contraction 63, 147 for general position 77, 241. 244 - for hyperplanes 148 for line bundles 27, 137 for projections 48 for the exterior product 62, 146 - for the inner product 64, 148 - for the interior product 63, 147 - for the Mori map 323 , general 62. 141 - , unintegrated 25 free 15, 84, 60, 126. 127 - of a set 279 - of order p 127 - of order (q,p) 159 - of order (q,p,p) 159 - , strictly 127 , strictly of order (q.p) 159 Frenet curvature formula 173 Frenet formulas 169 Frenet frame 67, 165 - , dual 67, 175 Frenet identities 167 Fub ni-study form 15. 110 fusion 235 gauge 74, 76, 218, 235 gauge measure 76. 241. 243
general assumptions A 198 - B 78, 245 - (' 289290 - D 85 - E 88. 91. 318, 332 general first main theorem 62. 141 general for B 45. 154, 282 generality index 45. 154 general Jensen formula 29 general of order p for B 44, 94, 15 l general position 50. 63, 73, 74, 76 78, 218, 220, 236, 244 Grassmann cone 14. 92 - manifold 14, 93 Green residue theorem 23, 134 Griffiths conjecture, big 54 - - , small 54 growth, rational 302 - , transcendental 38, 269, 303 hermitian metric 16, 94 - product 5, 15. 59, 94 - vector space 15, 94 - vector bundle 16 hyperplane section bundle 18, 113 - - - of a map 19, 122 incidence manifold 130 indeterminacy 18. 76. 121. 235 index. generality 45. 154 - , stationary 45. 154 infinite plane 112 inner product 5. 14, 59. 92 interior product 14, 59. 92, 94. 126, 127 intersection divisor 22 Jensen formula 4, 24, 25, 35. 120 , general 29
, Lelong. theorem of Poincare 24 Levi operator 109 linear hull 216 linearly non-degenerate 38 linearly non-degenerate over IJ 91. 331 Liouville's theorem 36 lower order 302 maximum modulus 6
·346majorant 47, 197 majorization defect 52, 268 majorize 47, 197 mean value theorem ,29 meromorphic 83 meromorphic map 18, 120 - vector funct on 83 metric contraction 104 - hermiti n 16, 94 - , quot e t 16 modified comppnc;ation functIOn 61 Mori map 322 multiplicity, a 3 - , zero 19. 118 nerve of a covering 122 ~evanlinna conjecture 56 Nevanlinna defect 7. 52. 80. 81. 271. 291 truncated 53. 292 Npvanlinna Mori defpct relation 91. 333 non-negative divisor 20. 118 norm 5. 94 obstruction terms 211 open covering 122 open pseudo ball 115 operation divisor 60. 322
Poincar: dual 25 Pincar~-Lelong theorem 24 pole divisor 119 pole set 83 primarY stress coefficient 69 prime divisor 20 product. contraction 59, 95. 96, 126 , exterior 5. 14. 59. 94 hermitian 5, 15. 59. 94 inner 5, 14. 59. 92 interior 14. 59. 92. 126. 127 tensor 15 tensor symmetric 15 product to sum estimate 50, 75, 23l projective closure 112 - distance 15. 105 - operation 58. 105 - plane 14, 93 - , operation, homogeneous 58. 106 - operation. homogeneolls of degree (qt ... qp) 58. 106 - operation. unitary 58, 105 - operation. unitary of degree (Q1 .... ,qp) 58, 105 projective space. complex 92 pseudo ball. closed 115 - , open 115 pseudosphere 115 pth representation section 44, 153 pullback divisor 21, 118. 121
· projc('tive 58. 105 - , projective homogeneous 58, 106 - , projective homogeneous of degree (q1 •...• q p) 58. 106
quotient bundle 18 quotient metric 16
· projective unitary 58. 105 · projpctive unitary of degree (q 1" .qpl 58. 105
ramification defect 52. 287 rank 276
orbit 2 i:J oropr 302 · lowpf 30(' pdrabolic 33. 115 covpring manifold 34. 116. 302 manifold 33. 116 manifold. strictly 34. 116 plane. infinitp 112 · proipctlvc 14. 93 p-linedr 59
rational growth 302 reduced 16 - at a point 16 - representation 5. 19. 121 representation 18. 121 -- at a point 19 - atlas 153, 158 - family 153 - reduced 5. 19. 121 - section 19n 125 section pt 44. 153
I'liick('r diffrrcncp formula 46. 157
residue theorem 23
-347Ricci defect 52. 87. 268 - form 41 - function 41. 42. 156 Riemann sphere 2 second main theorem 7. 12. 42. 51. 53. 79. 81, 86. 255. 256. 265. 273. 291. 303. 308. 309 - over function fields 86. 291. 303. 308. 309 simple divisor 20 singular Stokes thoerem 23 small Griffiths conjecture 54 span 216 spans 17 spherical image function 3. 10. 26. 36. 137. 138 standard model 16 stationary index or divisor 45. 154 Stokes theorem 116 stress coefficient 67. 176 coefficient. primary 69 - curvature formula 69. 184 - gradient formula 70. 184 - invariant of level k 70. 182 strictly free 127 strictly free of order (q .p) 159 strictly parabolic manifold 34. 116 support of a divisor 20. 118 symmetric tensor product 15 tautological bundle 17. 113 tensor product 15 - . symmetric 15 theorem of Casorati - Weierstrass 39 - of Liouville, 36 - of Poincare-Lelong 24 transcendental growth 38. 269. 303 truncated divisor 20. 260 - Nevanlinna defect 53. 292 Umkehrproblem 8. 54 unitegrated first main theorem 25 unitary 105 - projective operation 58. 105 - projective operation of degree (q 1..... qp) 58. 105 valence function 3. 4. 10. 26. 35. 36. 61. 120. 126. 137. 241. 243. 323
zero divisor 20. 21. 119 zero multiplicity 19. 1U