Concept of Toroidal Groups
1. The
The
the
of toroidal groups
general concept irrationality
Irrationality
The fund...
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Concept of Toroidal Groups
1. The
The
the
of toroidal groups
general concept irrationality
Irrationality
The fundamental too]
complex
KoPFERMANN
by
Lie groups which
over a
pseudoconvexity
and
cohomology
groups.
and toroidal coordinates
are
by irrationality
fibre bundles
by
holomorphic functions and contributed basic properties of them.
KAZAMA continued the work with
of others
introduced in 1964
condition. MORIMOTO considered in 1965
lack non-constant
1.1
was
toroidal coordinates which allow to select toroidal groups out conditions. Toroidal groups
complex
torus group with
a
can
be
Stein fibre
represented
isomorphic
to
as
principal C'.
a
Toroidal groups The concept of complex
torus groups leads to
1.1.1 Definition
A toroidal group is
an
Abelian
complex
Lie group
on
which every
holomorphic
function is constant. Toroidal groups have several means
all
holomorphic
quasi-torus
simply
group of
a
as
(11,C)-proups,
quasi-torus'.
that
Sometimes
a
functions
are
constant is Abelian
unique connected and Abelian real Lie
complex
(Remark
Lie group
1.2.3
on
group of dimension
n
p
on
18).
which
connected and the Cn the unique connected and Abelian complex Lie
complex dimension
Also called Cousin p
constant,
or
theorem of MORIMOTO is that every
holomorphic
R' is the is
in literature such
are
is any connected Abelian Lie group.
A consequence of which all
names
functions
n
quasi-torus,
which is
simply
connected.
because COUSIN had
1)
Y.Abe, K. Kopfermann: LNM 1759, pp. 3 - 24, 2001 © Springer-Verlag Berlin Heidelberg 2001
an
example
of such
a
group
(see
Concept of Toroidal Groups
The
1.
4
Proposition Every connected Abelian complex 1.1.2
to Cn /A where A is
Lie group is
subgroup of
discrete
a
isomorphic
as
Lie group
complex
Cn.
Proof If such
a
covering
Lie group X has the
group with
Therefore A:= ker
projection -
7r
7r,
(X)
complex dimension Cn
7r :
is
discrete
a
n, then Cn is its universal
X which is
---
subgroup
complex homomorphism.
a
of Cn such that X
Cn/A.
-
Q.E.D. A lattice A C R' is
a
the Abelian
Lie group X
ordered set For
a
complex
matrix P
or
be the
the
(A,,
=
fx,Al
:=
-
complex rank of rank of
the coordinates
subspace of
a
Cn/A. A basis of a lattice A C Cn is A,) of R-independent Z-generators of A.
(A,,
+
-
-
-,
A,)
xrAr
+
an
let
xi E
:
basis P is said to be the lattice A C Cn is
a
of A
that the
R,
7
Xr E
RI
change
< n, then after
m
complex
of the coordinates
that Cn /A
so
Cn/Zn
=
rank of
complex
C-span CA
coordinates. If the
m
linear
a
Z-generators
-
,
we can assume
the first
n, then after
-
:=
of A.
R-span
complex
of R7n. A lattice A c Cn represents
subgroup
lattice A C Cn with basis
RA
The
discrete
:=
RA
lattice A. If
a
linear
+
iRA of A is the
change
and real rank of A C Cn
we can assume
-
a
(C/Z)n
that el,
C*n
-
-
-
-,
en
of
are are
by exponential
map
e(z) where C* is the If the
subgroup 7r :
CA
(exp(27rizi),
multiplicative
complex rank of A
of coordinates
Let
:=
el,
-
-
-
,
exp(27riZn))
group of the
C Cn is
get
-
-,
n
complex
and the real rank
en E
A and then A
r C Cn of real rank q. We say that A
Cn
=
we can
-
--,
RA
Cn/A iRA
be the natural
+
=
(z
E
Cn),
numbers. + q, then after
n =
Zn E) F with
n
+ q.
projection.
The maximal compact real
subgroup
of
Cn /A
is
=
the
RA/A
maximal real torus K:=
RA
7r(RA)
I MCA
change
discrete
Zn (D r has the rank
K
I
a
a
=
RA
n
iRA
RA. Then MCA n
RA/A,
that is the
projection of the real
RA of A. Moreover let MCA
MCA/(MCA
:=
:=
RA
n
Ko
becomes
complex subgroup
a
MCA/(MCA
n
A)
iRA be the maximal C-linear subspace of
n A is discrete in
A)
==
span
MCA so that the projection KO complex subgroup of Cn /A. KO is
of the maximal real torus K.
:=
7r(MCA)
=
the maximal
Irrationality
1.1
Proposition
1.1.3
Let A C C' be
If the
1.
discrete
a
complex
rank
subgroup.
m :=
rankCA
C'/A where A is considered Let
2.
5
and toroidal coordinates
rankCA
=
as a
RA/A,
maximal
RA
subgroup
n
of C'.
iRA be the maximal complex subspace
discrete
a
C*m ED
-_
is not dense in
A)
n
(Cn-m/r) of
subgroup
complex subgroup MCr/(MCF
Rr1r
torus
(C'/A)
then
Cn/A -when .1' -C Cn-?n is
than
complex subgroup MCA/(MCA
of RA- If the maximal
maximal real torus
=
n
Cn-,
-_
discrete
and MCA
n
<
n
rank
complex
r)
n
-
m
and the
is dense in the maximal real
Cn-m/r.
of
Proof 1.
If
m
then A spans
rankcA,
=
plex dimension m so that subspace U. We get Cn/A 2.
The closure Ko
RA
=
n
K" of
a
C
complex subspace V
U (D V with
V/A. RA/A
=
RA/A.
certain real dimension
an n
of the
iRA of RA is
of the maximal real torus K torus
=
U E)
7r(MCA)
=
plex subspace MCA
Cn =
a
C Cn of
CA
:=
projection
of the maximal
connected and closed real
a
So K
splits
COM_
m-dimensional C-linear
-
into K'
=
com-
subgroup
Ko and another
m.
A' ED A" such that decomposition A m. Since MCA C RAI) K' RAI' /A" where rankRA" RAI /A' and K" 0. Then rankCA" m. MCA` MCA and MCA" n. So we get Cn CA' E) CA", where the C-spans We assumed rankCA m or m, respectively. CA') CA, have the complex dimensions n Because K
=
K' E) K"
we
get the
=
=
=
=
=
=
=
=
-
The real and the the other
K' r
hand,
RAI /A'
complex rank of A" coincide.
the
projection Of MCA'
which is maximal in
=:
CA' /A'.
Then
CA"/A"
-_
C*-. On
MCA is dense in the real We get
CA' /A'
-
Cn-m/.p
torus
with
Q.E.D.
A'. This proves the proposition.
A consequence of this proposition is that for every toroidal group Cn /A the lattice A has maximal complex rank n. KOPFERMANN introduced in 1964 the concept of n-dimensional toroidal groups
with the
irrationality
condition
[64].
1.1.4 Theorem
Let. A mal
c Cn be a discrete
complex subspace
equivalent:
subgroup
of
complex
rank
of the real span RA. Then the
n
and MCA the maxi-
following
statements
are
1. The
6
CI/A
1.
of Toroidal
Concept
Groups
is toroidal.
2. There exists
C'\ f 01
E
no a
so
that the scalar
product (9, A)
for all A E A. 3. The maximal
real torus
E Z is
integral
(Irrationality condition) complex subgroup MCA/(MCA
RA/A
n
A)
is dense in the maximal
C'/A.
of
(Density condition)
Proof If
I >- 2.
complex
a
exponential
vector
function
:6
a
e((a, z))
=
periodic holomorphic function. If the projection of the 2>-3. dense in
RA/A,
one
a
that
(a, A) E Z (A E A), then the exp(27r o-, z)) (z E Cn) is a non-constant Aso
maximal
complex subspace MCA C RA is not one splits (Proposition 1.1.3). After change R-independent set of generators of A so that the
then at least
of coordinates there exists unit vector el is
0 exists
C*
and the others
orthogonal.
are
Then for
a :=
el all scalar
products (01, /\) (A E A). Let f be holomorphic on Cn/A. Then f is bounded on the compact real 3>-1. torus RAIA and therefore f o 7r constant on the maximal complex subspace MCA C RA. Now f must be constant an RA/A by the density condition. Then the pullback f o -7r is constant on RA and because the complex rank of A is n the holomorphic function f must be constant on Cn. Q.E.D. E Z
A
complex
Lie group is
Stein manifold. The
a
products C
Every connected Abelian second factor is
a
Stein group, if the x
C`
real group is
are
underlying complex
isomorphic
to
R1
x
real torus. For connected and Abelian
get the
of Abelian Lie groups
1.1.5
Decomposition
Every
connected Abelian
complex C
with
a
toroidal group Xo. The
Lie group is X
C*M
X
a
Abelian Stein groups.
(R/Z)m
where the
complex Lie
following decomposition proved by REMMERT [64, and by MORIMOTO in 1965 [74]. 1964]
we
manifold is
groups
cf KoPFERMANN
(REMMERT-MORIMOTO)
holornorphically isomorphic
to
a
XO
decomposition
is
unique.
Proof Existence.
Propositions 1.1.2 and 1.1.3 together with Theorem 1.1.4. Uniqueness. Let Xi := Sj x Tj (j 1, 2) where Sj are connected Abelian Stein groups and Tj toroidal. If 0 : X, -+ X2 is an isomorphism, then obviously O(T1) C T2 and therefore O(Tl) T2, T, and T2 are isomorphic and thus S, Q.E.D. XUT, and S2 X2/T2 are isomorphic. =
=
=
=
A consequence of the toroidal groups is
decomposition theorem and
the
density condition for
Irrationality and toroidal coordinates
1.1
7
1.1.6 Lemma
For any connected Abelian
complex
Lie group X the
following
statements
are
equivalent: Stein group.
1. X is
a
2. X is
isomorphic
3. there exists
no
to
0
x
C*m.
complex subgroup of positive dimension
connected
maximal compact real
subgroup
(Stein MATSUSHIMA and MORIMOTO
in the
of X.
group criterion for Abelian Lie
proved
in 1960 the
groups)
following generalization
of this
[70]
lemma
(MATSUSHIMA-MO RIM OTO)
1.1.7 Theorem Let X be
a
connected
Lie group. Then the
complex
following
statements
are
equivalent: 1. X is
a
Stein group.
2. The connected component of the center of X is 3. X has
connected
no
compact real For the
complex subgroup
subgroup
(Stein
of X.
of this theorem refer to the
proof
Stein group.
a
positive dimension
of
original
With the previous Lemma the Stein groups Lie groups where the connected center is
are
in any maximal
group
criterion)
paper.
exactly the
isomorphic
to
a
0
connected
complex
C*m.
x
Complex homomorphisms Complex homomorphisms be described
universal 1.1.8
by
of connected and Abelian
covering
spaces in the Abelian
Proposition complex homomorphism
For any
A C C' and A' C the commutative
C"
T
there exists
:
a
7r :
Cn
--,
Cn /A and 7r'
will be
Lie groups
--
C"/A'
17r
17r, Cn' /A'
Cn'
with discrete map
C'
the
subgroups
--+
C"
with
C n'
Cn /A :
can
description by
prefered. Consider first
unique C-linear
diagram
, is called the lift of
case
C/A
Cn
where
complex
Hurwitz relations. Instead of tangent spaces the
--,
Cn' /A'
are
the natural
projections.
r.
Conversly, a C-linear map f : Cn --+ Cn' with f (A) C A' induces a complex homomorphism -r: Cn /A -* Cn' /A' such that the diagram becomes commutative.
8
The
1.
of Toroidal
Concept
Groups
Proof By the path lifting theorem there
(O)
with
0 such that 7r' o,
=
phism and that Let X
:=
is
=
exists
a
T o 7r.
Hence -
unique continuous becomes
CI
map
complex
a
homomor-
C-linear map.
a
Cn /A and X'
Q.E.D.
Cn' /A'
:=
C"
-4
and
:
-r
X
X' be
--+
complex homomor-
a
phism. Then:
covering map, iff its lift is bijective. Then X is A. isomorphism, iff i is bijective and (A) a complex Lie subgroup of X',
-r
is
a
-r
is
an
X is
iff there exists ,
:
X is
Cn a
Cn'
--.>
closed
Now let P
X
Lie
-r:
-
-
is
C)
homomorphism
X, iff there
of
X
-r :
-+
X,
iff
n (Cn). is
embedding
an
Cn'
can
be described
is defined
:=
by
P'M"
=
and ho-
(A',,
-
-,A,,)
-
be
a
the matrix relation
(Hurwitzrelations) (z)
-
by
=
E Cn
and M' E
map, iff C
GL(n, C)
Cz
(Z
M(r, r; Z)
matrix. Then:
complex homomorphism bijective,
iff C E
T
is
The group X
GL(n, C)
=
Cn /A is
a
C=n and CA=Aln maps
T
:
covering
a
and M' E
P'= CPM
Holomorphic
A,
=
basis of A CCn and P'
--->
.
integral
The
(A)
subgroup
(A,, -, A,) be a Cn' Then : Cn
where C E M (n', n;
T
group of X'.
covering
X'.
--+
CP
an
immersion and
injective and
complex
:=
basis of A' C
(injective)
an
is
momorphism
is
a
--
M
with
Lie
complex
GL(r, Z) =:
so
M'-'
subgroup
G
is
regular.
that
GL(r, Z).
of X'
=
Cn' /A',
iff the rank
(Cn).
Cn /A
Cn' /A'
--+
of toroidal groups
essentially
are
com-
plex homomorphisms. 1.1.9
Let
T:
Proposition C" /A' Cn /A --+
toroidal, Cn' /A' Cn /A,
any
be a holomorphic map with -r(I) 1, complex Abelian Lie group and where 1, 1' =
Cn' /A', respectively.
Then
T
is
a
where Cn /A is are
the units of
complex homomorphism.
Proof By path lifting theorem there
, (O)
=
0
so
that
T o 7r
=
any A E A the difference
Let A'
=
T(A).
Then
ir'
-?(z
o
exists
-?,
+
a
holomorphic
where 7r, 7r'
A)
-
-?(z)
are
Cn
map
the canonical
must be constant,
---
Cn'
with
projections.
namely i (A)
E
For
A.
, j (z for the components
A-periodic
A)
(j
of
j (z)
=
1,
=
-
-
-
,
(Z
Aj
+
T a
a
partial derivatives ak- j toroidal group. Then
describes the Stein
are
is
a
Q.E.D.
complex homomorphism.
following proposition
9
Cn)
E
Now the
n).
and therefore constant since Cn /A is
C-linear map and The
-j
+
toroidal coordinates
Irrationality and
1.1
factorization for toroidal
groups.
Proposition Cn' /A' any complex Abelian Lie group and Cn /A be toroidal, X' ---> X' a complex homomorphism. Then the image T(X) is a toroidal group.
1.1.10 Let X -r :
X
=
=
The connected component
(ker T),,
of the kernel of
induces
T
a
factorization
X
X1 (ker -r),,
X'
-
Proof '
: Cn
Let
Cn
_+
C' ,a C-linear
be the lift of
subspace and
discrete. The map -
:
X
The
Cn --+
___>
-r.
Then - is
C-linear, the image V :=, (Cn)
V n A' discrete in V. Therefore
Cn'
V/(V
induces
n -
(A))
a
-,
- (A)
C
c V n A' is
homomorphism
v/(v
n
A')
--+
X'
V/(V n A') c X' must be toroidal image -r(X) holomorphic functions. Moreover the map =
because X has
non-
constant
V/ (V is
a
covering
map and
n
X1 (ker -r),,
(A)) -_
--+
V/ (V
V/ (V n -
n
A) Q.E.D.
(A)).
Toroidal coordinates and C*n-q -fibre bundles
Standard coordinates
are
used in torus
theory whereas toroidal
coordinates
re-
spect the maximal complex subspace MCA of the R-span RA of the lattice A c Cn. 1.1.11
Standard coordinates
Let A C Cn be
a
discrete
subgroup of complex
rank
n
and real rank
change of the coordinates we obtain A R-independent Z-generators -/j, -lq E _V of F. Then
After
a
linear
=
P
=
(In, G)
Iq
0
T
0
In-q
T
n
Zn (D F with
+ q. a
set of
1. The
10
with unit In are
an
we
can
invertible
Groups
GL(n,C)
E
:=
R-independent, iff the
coordinates has
of Toroidal
Concept
assume
they
course
An immediate consequence of the
A basis P
(1)
(In, G)
:=
there exists
defines E
no a
('Y1i-)^1q)
:=
Thus, after matrix i of
that the square
imaginary part Imi .
coordinates of A. Of
and G
rank of ImG is q.
are
These coordinates
Zn\ f 01
M(n,q;C)
permutation of the
the first q
rows
of G
called standard
are
irrationality condition 1.1.4(2)
so
E
uniquely determined.
not
toroidal group, iff the
a
a
following
is:
condition holds:
that 'o-G E Zn.
(Irrationality
condition in standard
coordinates)
1.1.12 Toroidal coordinates
Toroidal coordinates where introduced GHERARDELLi and ANDREOTTI in KAZAMA refined them in in 1984
KOPFERMANN in 1964 and then
by
1971/74.
by
VOGT used them since 1981 and
slightly by transforming MCA
with
[64, 33, 115, 116, 53] Let P
be
G) i
=
of the last n
-
q
n
ones v
-
q
by
a
standard basis of A
of the first q
square matrix
After
:=
:=
(u, v
-Jmt)(Imi )-'
changing
:=
change
the first q coordinates
+
E
Ri u)
M(n
-
(u q, q;
Cq, V
E
R).
C
u
and the last
(
(1q, t)
Iq t
0
In-q R, R2
In-
0
M(q, 2q; C) is the basis R := (Ri, R2) E M(n
E
T. The real matrix
-
of q,
Cn-q)
We get toroidal coordinates.
the order of the vectors the basis of the
P
where B
imaginary part of of the invertible, and let i be the matrix
the shear transformation
l(u, v) where R,
rows
of G. Then
rows
that the
so
of G is
a
q
given lattice becomes
B) R
q-dimensional complex
2q; R)
is the so-called
torus
glueing
matrix. The lattice becomes 0
A
=
(Zn-q)
Toroidal coordinates have the 1-
MCA
=
dinates,
JZ
E Cn
:
Zq+1
(D rwithbasis
(B)
of r.
R
following properties: Zn
=
01
is the
subspace of
the first q
coor-
1.1
2.
RA real
3. Cn
E Cn
fZ
:IMZq+l subspace generated by
-::::
z:::
ED V ED iV
MCA
Of course toroidal coordinates groups have many
We
if
same as
Of
::--:
q units eq+1
MCA
:=
not in the least
the order of the basis 0
a
basis of the
The
toroidal
That is
(u
periods
by
Cq, v
E
Cn-q).
E
obtain
we
0
B,
=
and B2
torus T
same
advantage
-
( In-q B2) ( In-q B)
(Imt) -'
B,
A,
uniquely determined,
R
R, R2
now
is the
V, where V
complex subspace MCA with
((Imt)-lu, v + Rju)
P=
where
ED
en E
I
transform the standard coordinates
we
changing
-:::
symmetries.
l(u, v) After
n
-
11
iV-
are
transform the maximal
can
the
the
RA
=
In1Zn
:--
*
and toroidal coordinates
Irrationality
as
:=
(Imfl 'Ret + i1q.
before and R
the
(RI, R2)
:=
(BI, B2)
Then B same
glueing
is
matrix.
of these toroidal coordinates with refined transformation from
standard coordinates is
I(Im-yj)
(t)
for the basis 71,
=
(j
ej
=
q)andl(ej)
1,
(1)
on
the
there exists
glueing
no a
E
the torus T
1.1.13 Real
=
toroidal group, iff the
a
condition in toroidal
depends only
on
the
glueing
,
I
,
*
parametrizations a simple real parametrization of
7
An be the first and
complete the
basis
following
by -yj
71,
'
coordinates) matrix R and
B.
generated by
Toroidal coordinates allow
A,
n)
q +
such that 'o-R E Z2q.
Zn-q\ 101
It is to remark that this condition on
(j
matrix R holds:
(Irrationality
not
ej
*,'lq Of F-
''
In toroidal coordinates the lattice A defines
condition
=
*
iej (j
*
Cn /A. For this let
the last q elements of P so that we can i 'Yq 2n Then = + 1, n) to a R-basis of the R q .
the R-linear map n
(L)
z
L (t)
(Aj tj
+ 7j tn+j)
(t
E R
2n)
j=1
induces
a
(R/Z )n+q.
real Lie group
isomorphism
L
:
T
x
Rn-q
__ ,
Cn /A, where T
12
If
1.
The
Concept of Toroidal Groups
denote with
we
u
the first q toroidal coordinates and with
ones, then the real toroidal coordinates
change
of the real parameters t1 i
(- R)
*
*
*
)
LR(t)
Reu, Imu, Rev, Imv
are
n
-
given after
q a
t2,, by
At
--::::
the last
v
(t
E R
2n)
with
Iq ReS ImT
0
0
0
0
0
(Imi )-' Jmt)-'Ret
0
0
Iq
0
0
R,
R2
In-q
0
0
0
0
In-q
0
0
In-q
A
or
R, R2 In-q 0
0
In the second
0
case we
get real toroidal -coordinates- as- given- by -reftned-transfor-
mation from standard coordinates. In both
first
+ q real t-variables become
n
1.1.14 C*n-q -fibre bundles
Toroidal coordinates define rank
n
+ q
subspace
as
a
over a
torus
representation of
C*n-q -fibre bundle. The
that the
:-=
MCA/BZ2q
-4
any toroidal group Cn /A with
projection 13
MCA of the first q variables induces
onto the torus T
functions Zn+q -periodic in the
A-periodic.
P:X=Cn /A
so
cases
T
a
:
Cn
-4
MCA
onto the
complex homomorphism
MCA/BZ2q
=
with kernel
Cn-q/Zn-q
-
C*n-q closed in X
diagram Cn
MCA
IX/
lir
X
T
becomes commutative. It is well known that every closed
defines
a
[105, 7.4] Thus,
principal fibre bundle or
as an
HIRZEBRUCH
Cn /A with A
a
with base space
XIN
N of
and fibre N
a
Lie group X
(see
STEENROD
[45, 3.4]).
immediate consequence of toroidal coordinates every Lie group X =
Zn ED F of rank
complex q-dimensional Such
complex Lie subgroup
bundle is defined
cocycle condition
a,+,,
n
torus T
by
(z)
an
=
+ q is
as
a
automorphic factor
a,,
(z
C*n-q -fibre bundle
principal
over
the
BZ2q), fulfilling 7, 7-' E BZ2q).
the
base space.
+
T) a, (z)
(z
a,
E
(,r
E
MCA)
Irrationality and toroidal
1.1
coordinates
13
1.1.15 Lemma
Let X
Cn /A be of rank
=-
+ q and let A in toroidal coordinates be
n
Z-generated
by
with the basis B
MCA/BZ2q Then the
morphic
and the
e
=
-
group T
torus
-
C*n-q -fibre bundle X __P+ T is
principal
e(rk)
a,
the
M(q, 2q; C) of the (rl, -, r2q) E M(n
E
matrix R
glueing
R
-
2q; R).
q,
given by the
constant auto-
factor
H where
T2q)
(TI,
=
B
In%
P
C*n-q is the
Rn-q
:
The bundle is
I-sphere.
C C*n-q
(Sl)n-q
E
(k
map and
exponential
2q),
1,
=
Jz
S,
JzJ
E C:
=
11
trivial.
topologically
Proof Let
u
Cn
=
projection
Then
L:
we
get
The
=
v,
of the
MCA/BZ2q
T
=
7r*X
=
J(Ui X)
E
Cn-q the last
E
v
pullback
--+
MCA
n
q variables of the
-
bundle X 4 T
principal
along
the
is
MCA
7r(U)
X:
X
=
P(X)J
trivialization
a
MCA
--+
.
X
C*n-q
E Cn-q is any
v
Indeed,
ED Cn-q
ir:
7r*X
where
q and
MCA be the first
E
MCA
with
v
(u)
c
V
mod Zn-q iff
V2
L(u, x)
by
(u, e(v))
:=
(U
MCA,
E
x
E
X),
7r'- I (x) by
(U) (U)
mod A
=
so
that
t
becomes
bundle
a
V2
V1
isomorphism. Now let
(u)
E
V
T
=
i'_1(x)t 0
where aBor
Let
phic
Ba
period with
a
Instead of
(u)
a a
E
we can
Z2q Define T(u, x) -
(v+Ra) u+Ba
take
a
V
T(U, X)
=
(u
Ba, e(v
+
+
Ro-))
=
e(Ra) acts componentwise. Therefore e(Ra) (0- EE Zn) defines X _P_+ T.
(u
+
c
:=
i`
1
(u
(x)
Ba, e(Ro-)
the constant
o
+ T, so
x)
for any
that
e(v))
automorphic
factor
:=
Lj
be the
topologically
trivial line bundle
Lj
T defined
by the
automor-
factor
aj(Tk) where R
logically
=
(rik)
trivial
E
M(n
sum
L
=
-
q,
L,
=
e(rjk)
2q; R) (D
...
(D
(k
=
1,
2q)
is the
given glueing
Ln-q
is
a
matrix. Then the topo-
vector bundle L
TZ' T,
defined
by
the to
Concept of Toroidal Groups
1. The
14
factor
automorphic
a
diag(al,
=
-
-
-,
a,,-q)
E
GL(n
-
q;
C)
and associated
the given bundle X -P-+ T which is then topologically trivial.
If the line bundles
product L,,, defined
L'
on
an
automorphic factor
If
T :
Cn /A
Cn'/A'
Cn/A
by aX,,3,x, respectively, defined by the product a,\,3), (A E A). If a
is
Lp
defined
are
by
A).
(A
E
0
L, Lp
Cn'/A'
--+
is
then the tensor line bundle L is
aX, then the dual bundle L* is defined
defined
by
a-'
line bundle
complex homomorphism and the
a
by the autmorphic factor ax,, then defined by the automorphic factor ax := - *a.' (A)
is
Q.E.D.
pullback r*L'
the
on
T
1. 1. 16 Remark
Let X=Cn /A bundle
over a
defined
by
as
in the
previous lemma
torus T and
the
Lj
automorphic
Then X is
=
M(n
E
-
q,
represented
previous proof
as
natural C*n-q-fibre
be the line bundle
Lj P4
T
e(rjk)
=
2q; R)
(k
is the
=
2q)
1,
matrix.
given glueing
get by the irrationality condition (I) for toroidal groups:
we
a
(rik)
as
be
factor
aj(Tk) where R
in the
toroidal group, iff for all
o-
\ 101
G Zn-q
the
trivial line
topologically
bundle n-q
n-q
L'
11c)
L Oaj
a'(Tk)
givenby
=
e(E ujrjk)
is not
analytically
(Irrationality
trivial.
Maximal Stein We have
seen
C*n-q
closed
as
2q)
(k
j=1
j=1
subgroups
that every group X
subgroup.
=:
We shall
condition for line
of toroidal groups
Cn /Awith A= Zler of
see
bundles)
that the dimension
n
-
rankn+q
has
q is maximal for
toroidal groups.
SERRE fibre
conjectured
are
proved
in 1953 that
Stein manifolds is
in 1960 that
a
complex
a
complex analytic fibre
Stein manifold
[100].
are
X is
a
Stein
manifold, if
Stein manifolds and its structure group G is
Lie group. A
space B and structure group G
space whose base and
MATSUSHIMA and MORIMOTO
complex analytic fibre bundle
base space B and fibre F
connected
a
principal
are
bundle is
connected
a
Stein
complex Stein
a
manifold, if base groups.
[70]
1.1
1.1.17
Let X
Irrationality and toroidal coordinates
15
Proposition
C'/A
=
be
a
toroidal group of rank
1. For every closed Stein 2. If N C X is
a
subgroup N
=-
n
C
+ q. Then: x
C`
have 2f +
we
maximal closed Stein subgroup, then
XIN
is
a
<
m
n
-
q.
torus group.
Proof Let V
1.
=
C+' be the universal covering of N
induces the inclusion of the lift V
--+
so
that the inclusion N
Cn. Then V n A has rank
m
VI(V n A) has the same rank. So (V + A)IV has the rank n + q M. the quotient XIN must be a toroidal group because X is toroidal. Because XIN (Cn/V)/((V + A)IV) the rank of XIN must be n + q f On the other hand the dimension of XIN is n m and therefore its < 2(n-f -m). So n+q-m < 2(n- -m) what was to be proved. N
=
-
=
-
If
2.
S
-
XIN
is
C*r with
induces
a
a r
-
non-compact toroidal group it contains > 0
as we
have
seen
N-fibre bundle
principal
a
closed Stein
in the section before. Then
7r-
1
(S)
with Stein manifold S
With the mentioned result of MATSUSHIMA and MORIMOTO submanifold
bigger
than N
compact
torus group.
We shall
see
Stein
later
(p 70)
so
that N cannot be maximal.
7r : as
Now
-
m.
rank
subgroup
X
-+
XIN
base space.
7r-'(S) XIN
X
---
because
is
a
Stein
must be
a
Q.E.D. that
subgroups isomorphic
to
quasi-Abelian C
x
varieties
C` with 2f +
m
can =
n
have maximal closed -
q and
> 0.
1.2 Toroidal
Every complex
Lie group has
q of the group. Such a
maximal torus
and
subgroups
a
maximal toroidal
a
group is
exactly (n
an a
n-dimensional connected
period
group
and the rank of
Pf
=
is the closed
of
tion of the
tively,
complex Lie
group
group. Then p E X is
a
left
forallx E X.
complex subgroup
of all left
periods of f
is
f
group of all
:= n
-
meromorphic
or
groups of all functions
period
contained in
are
X if
dim
The function f is non-degenerate, if rankf totally degenerate, if f is constant.
period
a
complex Lie
on
f (X)
rankf
The
Some toroidal groups have
sets of toroidal groups
subgroup
meromorphic function f
f (px) The
which determines the type
subgroups.
The maximal toroidal
Let X be
subgroup
q)-complete.
-
subgroup. Compact analytic
translations of these torus
period of
pseudoconvexity
Pf. =
n
otherwise
degenerate
and
holomorphic functions is the intersecor holomorphic, respec-
meromorphic
X.
on
The type is
an
important invariant of complex Lie
groups.
First,
we
give the
definition of the type for toroidal groups, which also holds for any connected Abelian complex Lie group. 1.2.1 Definition
A toroidal group X
MCA of the real For
a
span
=
Cn /A is of
type
q, if the maximal
RA has the complex dimension
toroidal group X
==
Cn /A with A
=
complex subspace
q.
Zn ED.V the type is the real rank q of
the lattice F. It is well known that
a
complex
Lie group X is
a
Stein group under
conditions:
a) b)
X is
holomorphically separable.
X has at every
MORIMOTO
[74] proved
1.2.2 Theorem
Let X be
a
point local coordinates given by global functions. in 1965:
(Holomorphic reduction) (MORIMOTO)
connected complex Lie group with unit 1, and let
one
of the
1.2 Toroidal
X0
:=
Ja
G X
f (a)
:
be the constant set of all 1. 2.
for all
f (1)
pseudoconvexity
holomorphic functions
f in
17
XI
holomorphic functions. Then:
X0
is the
X0
is the smallest closed normal
group of all
period
and
subgroups
holomorphic functions.
complex subgroup
X1X0
such that
is
a
Stein group. 3. 4.
toroidal
subgroup contained in the center of X. Every complex homomorphism 0 : X -+ Y into a Stein group Y can be split as 0 0 o 7r with the natural projection 7r : X --+ X1X0 and a complex homomorphism 0 : X1X0 -- Y of Stein groups. X0 is
a
=
X0 is called the maximal toroidal subgroup of X and
X1X0
the
holomorphic reduction of
X.
X is said to be of type q, if its maximal toroidal
is of
subgroup X0
type q.
Proof
a)
X0 is closed in morphic functions. Xo is
a
subgroup
X0 is the intersection of the period groups of all holo-
X:
of X: For
a
C X
and
holomorphic f
the functions
f-(x) := f(x-1) (x E X) are holomorphic. f(ab) fa(b) fa(l) f(a) f(l) and f(a-1) and
=
such that
=
ab, a-'
=
E
=
For
=
a,b
f_(a)
fa(x)
:=
X0
E
f-(I)
=
f (ax) get
we =
f(I)
Xo.
a complex subgroup of X: Let X0 and X be the Lie algebras of X0, X, respectively. We want to prove that X0 is a complex subalgebra of X. Let U E Xo
Xo is
and V
:=
W E X. Moreover let exp sU be the map which maps
1-parameter subgroup
0(s in
certain connected
a
For
holomorphic fo (s) fo (0) and a
=
for all t E R
X0 is
so
E R
s
on
a
of X with tangent U at 1. Then define
it)
+
:=
of 0 E C.
neighborhood
function then
f
fo (z)
that V E
:
X
=
exp(sU) exp(tV)
C let
fo := f o 0. Now exp sU fo (0) locally around 0 E C. Then -4
E
X0
so
that
exp tV E
X0
Xo.
characteristic
subgroup of X: For a E AutX and a holomorphic function f holomorphic function f (x) : f (a (x)) (x E X). For a E X0 we f '(a) f '(1) get f (o-(a)) f (1) so that a(a) C- Xo. X0 is the period group of all holomorphic functions: Suppose f (px) f (x) (x E X) for a fixed p E X and all holomorphic functions f Then f (p) f (1) so that the other hand On let all for E Then X0. f (p) f (1) f. p f (px) f (xx-'px) f,, (x-lpx) fx (1) f (1) (x E X), hence p is a period for all holomorphic we
a
define the =
'
=
=
=
=
=
.
=
=
=
=
=
functions.
b) X1X0 is a Stein group: As we mentioned before it X1X0 is holomorphically separable. By the definition
is sufficient to show that
of X0
we can
define the
1.
18
natural
Concept of Toroidal Groups
The
homomorphism
W
W(f ) 7r 7r(a) and L
R (XlXo) with
'H (X)
:
for
f
o
f
E
R (X),
7r(b) be distinct X1X0 is the projection. Let -d such that there exists Then E f (a-'b) :A f (1) for f H(X) X1X0. The a-'b Y separates -a and L. Xo. We set 1:= p(fi) E H(XlXo). X0 is the smallest closed and normal complex subgroup of X, such that X1X0 is a Stein group: Let N be a closed and normal subgroup of X such that XIN is a Stein group. Take a G X \ N. Moreover let f E H (XIN) such that f (ir (a)) =A 7r separates a XIN. Then f := f f (ir(l)) with natural projection 7r : X where
7r :
X
=
--4
elements of
*
*
*
*
o
-+
and 1. Hence
X0,
a
XOO
X0 is connected: Let 1. Then
X00
so
C N.
Xo
be the connected component of X0
XIXOO
is normal in X and
a
group of
covering
containing the
X1X0. By
a
unit
result of
XOO is connected. Stein, XIXOO is a Stein group. Then Xo C XOO, therefore Xo On X0 all holomorphic functions are constant: Let X00 C X0 be the constant set of 1 of all functions holomorphic in X0. Then X00 is closed and normal in X0. So =
X1X0
and
X01X00
be considered
can
Stein groups. So
X0
is
XIXOO
p 14
functions
on
Stein groups and
are
by are
(X/Xoo)/(Xo/Xoo). XlXoo
-
hence
X00,
X00
adjoint representation Ad
:
X
--+
GL(X)
algebra of
of X is
a
Lie
the kernel is the center Z of X. It is well known that
X1Z
subgroup
c)
Let
is Stein.
such that
0:
X
group Y and is
X1X0
Y be
Xo
C
kero.
X0. All holomorphic
a
C X
is
a
But then
0
a
complex
be the maximal toroidal
0
image of the
GL(X)
a
GL(Y)
subgroup
of
is
and
Stein group.
Stein group. Then X0 C Z.
homomorphism of
=
X. The
is the smallest closed and normal
By the previous result X0
connected and Abelian complex
a
X0
--+
=
on
constant.
is in the center Z of X: Let X be the Lie
Thus
are
the mentioned result of MATSUSHIMA and MORIMOTO
Stein group. Then X0 C
a
X0
XlXo
fibre bundle whose base space and fibre
principle
a
as
o 7r
subgroup
XlXo
with
0
only
constant
:
Lie group X into
subgroup
of X. Then
of Y and then toroidal --+
a
Stein
O(Xo) so
that
Q.E.D.
Y.
1. 2.3 Remark
Every complex
Lie group with
and connected and therefore
a
Every compact and connected complex
For
a
connected
complex Lie group
compact subgroup
subalgebra
with Lie
algebra
of 1C. Then there exists
a
Lie groups
X with Lie
is Abelian
Cn/A.
Lie group is
holomorphic function rings of complex
The
holomorphic functions
toroidal group
torus.
are
Algebra X
those of Stein groups.
let K be
a
maximal real
1C. Moreover let ICO be the maximal a
uniquely
defined
complex complex subgroup Ko of
K associated with
/Co which is independent of the choice of K [75]. We get the
following diagram
in which
case.
The lattice A in the
general and behind diagram gives always a toroidal group. we
note first the
the Abelian
1.2 Toroidal
subgroups
Lie group
group of dim
max.
C
X
n
'`
'
Abelian
X
Ct+-+no
I
I
XO
19
algebra
general
I CnO
X0
of dim no
a
X
pseudoconvexity
toroidal
subgroup
re
C-
X
I
I
Lie
Abelian
general
and
su
'
bg
v _c
/A
CA /A
/C + ir,
X0
Cno
RA + MA
=
t
roup
K
RA/A
Ko
MCA/(MCAnA)
of real dim no+ q
complex subgroup
RA
max.
of dim q
The maximal toroidal
ICO=Knir,
subgroup X0 and
the
MCA=RAniRA
type
q of
a
Lie group X
Another immediate consequence of the previous theorem is the 1.2.4 Lemma
Let X be
a
connected
Lie group with the maximal toroidal
complex
X0. Then X is holomorphically
convex, iff
subgroup
X0 is compact.
Proof If X0 is not compact let
>-.
fal
is
X0 is
-<.
a
X0. Then the holomorphically
E
convex
hull of
Xo. So X is not a
closed
principal bundle
holomorphically convex. subgroup of X so that XIXO
with base space
XIXO
is
a
Stein group. Now X is
and fibre X0. Then X is
convex.
a
holomorphically Q.E.D.
Pseudoconvexity Let X be
a
complex manifold of complex
A real C2 -function
X
R is
dimension
n.
plurisubharmonic,
iff the
complex Hessian
form n
a20 (Xxjd-(OZji94
E
H(O)(x)
j,k=1
is
positive semi-definite for all
x
E
addition at least p positive
subharmonic,
iff it is
The C2 -function
0:
Ix
X
X, p-plurisubharmonic for all
eigenvalues n-plurisubharmonic.
--+
E X
R is
:
an
O(x)
(Levi form)
-
x
E
X and
cl
C3-- X
for all real
c.
0
has in
strictly pluri-
exhausting function for X, iff
<
iff
The
1.
20
Concept of Toroidal Groups
p-plurisubharmonic functions
These
which have at least p
special
are
positive eigenvalues,
of p-convex functions
cases
but that
necessairely posi-
not
are
tive semi-definite. It is to remark that in literature the fact that
a
function is
p-convex is counted in different ways. So GRAUERT and ANDREOTTI counted n
-
p+ I
eigenvalues
for the benefit of the formulation of
We follow HENKIN and LEITERER with p
Plurisubharmonic functions
set K C X has
compact
principle for connected analytic
the maximum
obey
so
[44, 4.3].
property for plurisubharmonic functions implies that every
sets. The exhaustion
a
compact plurisubharmonic hull.
Definition
1.2.5
A
eigenvalues
in the first p of suitable coordinates
convex
cohomology theorem. that 0 becomes linearly a
complex manifold
p-complete, iff it has an exhausting p-plurisubpurely p-complete, iff it is p-complete but not (p+ 1)-
X is
harmonic function. X is
complete. special completely p-convex manifolds. Those are defined by exhausting p-convex functions only. GRAUERT showed in 1958 with a supplement of NARASIMHAN in 1961 that Stein manifolds are n-complete. Compact These manifolds
manifolds functions It is to
never
that here also the
means
Therefore
5.1].
but
because all
1-complete
plurisubharmonic
constant.
remark,
p-convex
an
0-complete
are are
are
also, a
that X
complex
can
in literature. So
counting differs
be exhausted
by
manifold is said to be
(p
a
+
1)-convex
completely
function
weakly 1-complete,
[44,
if it has
exhausting plurisubharmonic C'-function only.
For toroidal groups of type q, the real dimension
n
of real dimension
n
maximal real subtorus in 1973
proved 1.2.6
RA/A
-
q of the space outside the
+ q is
important.
KAZAMA
[521:
Proposition
Every Abelian Lie group X
=
purely (n
Cn /A of type q is
-
q)-complete.
Proof In toroidal coordinates the
RA The
:__
R-span of A
fZ=X+iy
G
Cn
is
:
=
Yq+1
''
*
:Yn
=
01-
A-periodic plurisubharmonic C'-function n
2
kz)
(z
Yk
=
x
+
iy
E
Cn)
k=q+l
induces
a
group X
C'-function =
>
Cn 1A, because
0
fZ
on
X
G Cn
=
:
Cn /A, which is
(Z)
<
C2
I
C
RA
exhausting x
on
Dc with D,
the :=
1. The
22
Concept of Toroidal Groups
The maximal
Every
complex
toroidal group has
subgroup.
complex T
complex subtorus
imal
MCA/(MCA
A),
n
group
point
as
MORIMOTO
as
in
proved
Lie
[751.
(MORIMOTO)
connected
a
a
complex Lie
a
maximal closed subtorus
uniquely defined
a
But this torus may reduce to
1.2.9 Theorem
Let X be
subtorus of
Lie group. Then there exists the
max-
subgroup, situated in 7r(MCA) compact and connected complex analytic
complex
as
unique
which contains all
Lie
::::::
sets A C X with 1 E A.
Proof part of the maximal toroidal subgroup X0 of X, but
every A must be
Of
course
we
want to prove the theorem for all Abelian
If
n
1, then X
=
=
C*
C,
or a
Lie groups X
complex
C'/A.
=
torus T and the theorem is true. So let the
theorem be true for all connected Abelian
complex
Lie groups of dimension <
n
C'/A group, defined by a discrete subgroup A of rank r. The cases 1. and 2n are trivial so that we can assume 1 < r < 2n r 0, 1 1.1.12 we get By the decomposition theorem 1.1.5 and toroidal coordinates and X
a
=
-
=
Cn
coordinates. V
V'E)iV'
MCA
W)
(VI
E)
is the linear
=
V' the R-span of its unit subspace of the last m coordinates. Now it is easy to
see
E) W
subspace MCA of RA is the subspace of the first q subspace of the next p : r q coordinates, vectors so that RA MCA ( V' and W the linear
where the maximal linear =
=
that
set A c X with 1 E A
analytic
connected compact
a
-
=
RA/A:
is in
p+2m
Let
z
=
u
+
w
where
u
E
MCA
(D V1
RA and
=
xjwj E WE) W with
E
w
j=1 a
Wp+2,rn. The functions
R-basis W1,
fj are
:
R, defined by fj(7r(z))
X
plurisubharmonic on X
Now
we
=
(j
xj
and therefore constant
want to prove A C
ir(MCA).
Because
on
1,
=
.,p +
RA /A. (RA) n MCA/(MCA A) is
A. So A c
7r(MCA)
Abelian Lie group of dimension < n we are ready. So let Ao C Cn be the connected component of the lift
=
2m)
7r
=
an
A
=
7r-1 (A) with 0
E
Ao.
P
Now
we
consider
z
=
u
+
v
with
u
E
MCA and
v
E j=1
vi,
vp of V
=
V' + W. The functions
gj(z)
=
Vj
U
=
1,
-
-
-,P)
zjvj with
a
C-basis
1. The
22
Concept of Toroidal Groups
The maximal
Every
complex
toroidal group has
subgroup.
complex T
complex subtorus
imal
MCA/(MCA
A),
n
group
point
as
MORIMOTO
as
in
proved
Lie
[751.
(MORIMOTO)
connected
a
a
complex Lie
a
maximal closed subtorus
uniquely defined
a
But this torus may reduce to
1.2.9 Theorem
Let X be
subtorus of
Lie group. Then there exists the
max-
subgroup, situated in 7r(MCA) compact and connected complex analytic
complex
as
unique
which contains all
Lie
::::::
sets A C X with 1 E A.
Proof part of the maximal toroidal subgroup X0 of X, but
every A must be
Of
course
we
want to prove the theorem for all Abelian
If
n
1, then X
=
=
C*
C,
or a
Lie groups X
complex
C'/A.
=
torus T and the theorem is true. So let the
theorem be true for all connected Abelian
complex
Lie groups of dimension <
n
C'/A group, defined by a discrete subgroup A of rank r. The cases 1. and 2n are trivial so that we can assume 1 < r < 2n r 0, 1 1.1.12 we get By the decomposition theorem 1.1.5 and toroidal coordinates and X
a
=
-
=
Cn
coordinates. V
V'E)iV'
MCA
W)
(VI
E)
is the linear
=
V' the R-span of its unit subspace of the last m coordinates. Now it is easy to
see
E) W
subspace MCA of RA is the subspace of the first q subspace of the next p : r q coordinates, vectors so that RA MCA ( V' and W the linear
where the maximal linear =
=
that
set A c X with 1 E A
analytic
connected compact
a
-
=
RA/A:
is in
p+2m
Let
z
=
u
+
w
where
u
E
MCA
(D V1
RA and
=
xjwj E WE) W with
E
w
j=1 a
Wp+2,rn. The functions
R-basis W1,
fj are
:
R, defined by fj(7r(z))
X
plurisubharmonic on X
Now
we
=
(j
xj
and therefore constant
want to prove A C
ir(MCA).
Because
on
1,
=
.,p +
RA /A. (RA) n MCA/(MCA A) is
A. So A c
7r(MCA)
Abelian Lie group of dimension < n we are ready. So let Ao C Cn be the connected component of the lift
=
2m)
7r
=
an
A
=
7r-1 (A) with 0
E
Ao.
P
Now
we
consider
z
=
u
+
v
with
u
E
MCA and
v
E j=1
vi,
vp of V
=
V' + W. The functions
gj(z)
=
Vj
U
=
1,
-
-
-,P)
zjvj with
a
C-basis
1.2 Toroidal
real
are
A
=
7r(Ao)
Every
AO
on
and
therefore
subgroups
constant
so
that
pseudoconvexity
AO
23
MCA
C
7r(MCA)-
C
connected
and
Q.E.D.
complex Lie
group X has the
T which is contained in the maximal
subgroup
and
unique maximal complex
torus
complex subgroup of the maximal
compact real subgroup of the maximal toroidal subgroup of X. This maximal torus contains all
Furthermore
we
compact and connected analytic get the following proposition
as
sets A C X with 1 E A. a
consequence of the above
argument. 1.2.10
Let X X
Corollary
Cn/A
=
has
the
'7r(MCA)
be
toroidal group.
a
positive
dimensional
MCA/(MCA
=
A)
n
maximal
has
a
complex torus subgroup, iff positive dimensional complex torus
subgroup. 1.2.11 Definition
A
complex Lie
group without
torus
a
subgroup
of positive dimension is called
torusless.
The maximal
RAIA
of
a
complex subgroup 7F(MCA)
dimension q
subgroup
of the maximal real compact
non-compact toroidal group X <
Because of the
n.
cannot be
a
density
=
C'/A
condition
Therefore the maximal
torus.
subgroup complex the maximal complex torus subgroup T has
of type q has
dimension < q, if X is not compact.
As
an
immediate consequence
Example: Every 1.2.12
get the
toroidal group of dimension
n
>
2 and
type 1 is torusless.
Proposition
For every toroidal group X
XIT
we
=
C'/A
with maximal subtorus T the quotient
is torusless.
Proof Let S C
XIT
complex
submanifold of
be
a
positive dimensional subtorus. Then
contradiction for dim
X,
where
a-'(S)
>
a
:
dim T.
X
-4
XIT
is the
a-'(S)
is
a
compact
projection. This
is
a
Q.E.D.
2.
Bundles
Line
Line
torus
automorphic eral.
VOGTstudied
factors.
between
calculated
cohomology
the
the
cohomology
their
maximal
2.1
Line
bundle
can
have
wild
bundles
of theta
groups
Let A
into
bundle
set
be
a
autornorphic of
trivial
and
line
a
with
groups
in gen-
VOGT and KAZAMA
groups.
by studying
Lie group
a
:
A
x
(z)
multiplication
cycles
Z1 (A, -H* (Cn)). a
function
&,\(z)
of the
Toroidal
groups
line
not
bundle.
homogeneous.
factors
C'
:--*
C*
or
(/\
E
1-cocycle
of A is
given
by
a
A)
condition
=
the
Under
trivial axe
get the decomposition
functions
non-vanishing
a .x+),,
we
subgroup.
discrete
cocycle
the
the automor-
system which deflnes
which
which
=,H*(Cn)
exists
of
toroidal
on
complex
any
topologically
bundles
ax E
if there
the
KAZAMAand UMENodetermined
exponential
tool
the
is
factor
holomorphic
which fulfils
way from
decomposition
significant
groups
of the
bundle
theta
a
topologically
C C'
and wild
even
Automorphic
An
a
characteristic
bundles
line
groups.
and
toroidal
on
decomposition
of the line
line
theta
toroidal
in
the
complex subgroups.
The characteristic
phic factor
1981182
in
groups
of toroidal
differ
groups
1964 used
in KOPFERMANN
case.
distinguished
He
of toroidal
cohomology
and
bundles
compact
Cohomology
and
ax,
(z
+
A) a., X(Z)
(z
E
Cn, A, A'
E
automorphic factors form the Abelian automorphic factors a.X, &,\ (A E A) 0 E H (Cn) with
The
*
=
O(z
+
A)a,\(z)iP-1(z)
Y.Abe, K. Kopfermann: LNM 1759, pp. 25 - 56, 2001 © Springer-Verlag Berlin Heidelberg 2001
(z
E
Cn).
A). Lie group are
of
co-
cobordant,
26
Line
2.
The
Bundles
factors
automorphic
B'(A,,H*(Cn))
Cohomology
and
cobordant
H'(A,,H*(Cn))
Let X be The
:=
cohomology
becomes the first
sheaf
exponential
e(z)
-
The Picard
Z
--*
induces
--+
the
isomorphism
of all
Chern
integral
long
:
X
class
(see
that
(LI
cl
f
H2 (X,
Y is
--+
is
=
(Cn))
H*(Cn).
in
holomorphic
--->
functions.
0
cohomology
exact
c--', H2 (X,
sequence
Z)
--+
....
H1 (X, 0*)
:=
classes
a
:
holomorphic
of
H1 (X, 0*)
&
of the line
Cn
Z)
we
0
=
L2)
cl
=
(LI)
holomorphic
and
bundle
bundles
line
on
X, the
H2 (X, Z)
cl(L) is Pic(X).
(f *L)
the
first
[39,
p
class
or
139])
(L*)
and cl
=
complex manifolds, =
Chern
L E
(L2)
+ cl
map of
-cl
(L).
then for
the
pullback
f *cj (L).
0 because of Theorem B for Stein manifolds get H1 (X, 0) that H1 (X, 0*) 0. Therefore every line bundle on X
and
=
so
analytically
--4
GRIFFITHS and HARRIS
cl
For X
H*
values
0*
+
H1 (X, 0*)
hornornorphism
class
Chern
Weremark
If
(Cn ))IB'(A,
homomorphism cl
the
coboundaries
group
group
combining
is
0-
--+
Pic(X) is
H*
of A with
the
H1 (X, 0)
--+
-
of
sequence
exp(2,7riz)
:=
subgroup
the
and 0 the sheaf of germs of
0
with
Z'(A,
group
complex manifold
a
form
I
to
that
so
=
=
C'
trivial.
Proposition
2.1.1
Let A c Cnbe
a
subgroup
discrete
and X =Cn/A.
Then there
exists
a
canonical
isomorphism H1 (A, H*(Cn)) which
maps
phism Proof
class
Let
a :=
bundle
of line
lax
Cn
class
a
X
:
A C
C
by
of cobordant bundles
Al
be
on
an
__.
piC(X),
automorphic
factors
to
a
holomorphic
isomor-
X.
automorphic
factor
for
A. Now A acts
on
the trivial
w)
A (z, Then
L,,
7r*L,,
-
(Cn
:=
Cn
C)/A
X
A,
+
defines the
given by
C
x
(z
(z)w)
ax
a
fibre
bundles
Line
2.1
(z bundle
27
groups
C).
WE
Cn/A
over
pullback
trivial
with
factor.
automorphic
same
Cn,
E
toroidal
on
The map az
by az(a)
defined
L,,
:=
is
Z' (A, H* (Cn))
:
piC(X)
_+
homomorphism which induces
a
homo-
injective
the
morphism aH
Now let
Cn
Pic(X).
L E
C be such
x
H'(A,H* (Cn))
:
pullback
Then the trivialization
a
7r* L is
t(z, w)
with
Pic(X).
---*
analytically
trivial.
(z, h, (z)w),
=
Let
t
7r* L
:
where h, E H*
(C"')
Then
(z)
a,x
defines A
h, (z factor
automorphic
an
holomorphic
holomorphic automorphic
(differentiable, (differentiable, factor
A,,
(a)
a,
(z
(Z) =
L,,
E
so
Cn,
that
E
z
A)
function s continuous) automorphic continuous)
Cn
:
Q.E.D.
surjective.
aH is
--+
C is
called
a
to
an
by
an
belonging
form
+
A)
a,\
=
(z
(z)s(z)
E
Cn,
forms
automorphic
of all
A E
A).
belonging
to
a.
Proposition
2.1.2
A c Cn be discrete
Let
with
C-vectorspace
be the
1
A) h',
if
a,\
s(z Let
+
factor
automorphic
and L,,,
(A
a,\
E
A).
be
line
a
bundle
Then there
Ho (X, L,,)
exists
X
on a
=
Cn/A
defined
isomorphism
canonical
A,,.
--+
Proof Let
a
the
pullback
t
X
:
be
L,,
--+
7r*u
:
Cn
(z, h, (z) w)
(z, w)
a
7r*L
--+
define
by a,\(z)
s(z for
A)
+
every
L,,.
s
E
h,(z
+
hjz
+
a
trivialization S :
Cn
h,,(z)u(7r(z))
:=
A)(h,(z))-'
A)a(-7r(z))
=
continuous) t --+
(z as
in
C E
:
--+
Cn
we
get
X
C with
by
Cn).
Proposition
a,\(z)hjz)o-(7r(z))
ir*L
Then
section.
=
2.1.1
a,&)s(z)
(z
E
Cn)
A E A.
Obviously let
=
=
and
the function
s(z) Indeed,
(differentiable,
holomorphic
(a) a(7r(z)) H'(X, L,,)
the map h,,
A,,.
Then h,
Then :
s
is
an
injective
s(z)(h,(z))-1 A,,
is
an
homomorphism. is well
isomorphism.
defined,
On the other
and
a
is
a
hand
section
of
Q.E.D.
28
Bundles
Line
2.
decomposition
The characteristic Let A
c C'
be
a
factor
system
e(z)
exp(27riz)
=
Automatically
H exponential
The
E
with
(z
(z)
a,\+,,
(A
a,\
system
A) gives
rise
to
uniquely
not
a
determined
a,\
condition
=-
(z
a,,
system is
an
(z)
=
e(ax)
:=
(A
A)
E
C).
E
cocycle
the
exponential
an
of functions
a,\
where
of
subgroup.
discrete
Every automorphic
exponential
Cohomology
and
+
for
A)
induces
a,\
(z)
+ a,\
mod Z(z
automorphic
E
Cn)
if the additive
summand,
cocy-
condition
cle
a,\+,,
(z
a,,
+
A)
(z ECn)
(z)
+ a,\
holds.
exponential systems a,\,dX (A E A) iff there factors are cobordant, automorphic
Two
d.x(z) They
are
map
=
e:
Zn (D F be Cn
-+
+
A)
+ a,\
(z) a
=
h(z
(z)
h(z)
-
A)
+
a
h(z)
-
C*n induces
E
Cn/A.
=
E
associated
W(Cn)
with
A).
if
(z
subgroup and let X
discrete
(A
mod Z
their h E
function
systems,
(z)
+ ax
Cn). Then the
exponential
isomorphism
an
H'(A,,H*(Cn))
H'(F,,H*(C*n))
-+
that
so
H'(F,,H*(C*n)) becomes
working siderations 'Y'
exists
exponential
cobordant
strictly a.x
Let A
h(z
-=
if
cobordant,
are
^//
Let
c- r
A
:=
an
isomorphism
with
the whole
to the
the
same
way
automorphic factor Zn-periodic a., E W*(C*n)
in
as
(/\
a,\
(^/
E
Proposition Cz
r)
A)
we can
with
Instead
2.1.1.
reduce
cocycle
of
our
con-
condition
for
only. Zn (D.V be
A r-reduced
periodic
in
pic(X)
-
a
automorphic
functions
a,,
subgroup
discrete
E
factor
W* (Cn)
(-y
is E
F)
of Cn. a
set
with
of
holomorphic
cocycle
property
non-vanishing for
all
-y,
Zn-
-y'
E F.
A I-reduced functions
automorphic
of
in the
case
2.1.3
The
summand is
I)
E
theorem
wild
missing
Z"-periodic
for
property
29.
groups
all
-Y'
-Y,
E r.
in by KOPFERMANN
1964
[64].
decomposition
characteristic
toroidal
on
holomorphic
of
set
introduced
was
summands
a
cocycle
additive
with
decomposition
following
The
-H(Cn) (-y
ay E
bundles
Line
2.1
of
exponential
an
system
(KOPFERMANN) A
Let
belonging position
to
(z)
following
the
1.
(X-,, z)
=
X
I,
A
r
r
x
We say that
Cn)
the
A the r-reduced
Sy E H(Cn) O-coefficients
2.
(1y
r)
of its
'Y+-Y'
and d: r
(-f
dy
+ i
+ c.,
(Xy, -y')
:=
-
E
E
r)
decom-
I)
characteristic
r-reduced
characteristic
Fourier
where Xy
form
bilinear
X(-Y) (7
:=
E
homomorphism, linear
alternating automorphic
r-reduced
a
alternating
the
(X..y,, X)
r-reduced
is
and
form
bilinear
form.
summand with
vanishing
expansion,
such that
ay,
Zn-periodic
function,
A
:=
=
depend only
e(a,)
(,y
a
on
a
summands
(,y
line
bundle
r)
are
a
same
strictly
a
r-reduced is
constant
characteristic
cobordant
unique and
with
as
E
homomorphism the
and
have the
change
mod Z
E
(Xy, z),
form
linear
0-coefficient
systems
c1Y
(-y, 7'
mod Z
homomorphism.
vanishing
The wild
(m, -y)
a
-y')
of ay into
and the real
cy +
is
exponential
The characteristic
Remark.
form
d(-y)
:=
homomorphism.
a,y
+1A(-y, 2
+ ay,
d-,
Cobordant
R
--->
c.,
summands'Y with d-, are unique.
Cobordism. r
-
decomposition
The
automorphic
=
R with
--+
Uniqueness.
a,y
(z)
+ s,,
characteristic
E
C
:
(xy, -/)
(7
system a'Y
characteristic
wild summand of a,,. s,y is called the r-reduced with R constants E are E r) (7 Cy
3.
d
2
by A(7, -y)
X is the
E
-
exponential has the
homomorphism which defines
a
Z
,
(X,y, z) (z
+
factor
properties:
Zn is
__.
Then every
lattice.
a
automorphic
F-reduced
a
a,, with
C Cn be
Zn E) r
=
r)
with
X and the
L defined
mE
Zn-
characteristic
bilinear
automorphic
by
the
r),
which is not
factor
E
Proof Let ay
(,y
Zn-periodic.
E
r)
be
an
exponential
Because the a,
are
system for a.,
Zn-periodic
(,y we
c
get
necessarily
a
Bundles
Line
2.
30
X,y,j with
unit
vectors
(z
ay
(j
ej
Cohomology
and
1,
-
-
,
fy (z)
Obviously
(z)
ay
:
(xy, z)
-
(z)
a-, is the
of
sum
linear
a
form and
unique.
f-y (z)
Now let
(')
E f
-
e
I
*
*
'
i
7
X-j,n)
(X-y, z)
such that
fy (z)
+
Such
function.
Zn-periodic
((a, z))
Cn)
E
EZn.
Zn-periodic
is
=
a
(z
E Z
that
so
(X-y,
=
(Z)
a.,
-
n)
-
tX-y
ej)
+
be the Fourier
a
of
expansion
decomposition
ft.
Set
is
f(o)
ky
OIEZn
and s-,
f'Y
:=
ky (-y
-
E
F)
such that
fy (z) cocycle
The
for
condition
+
k7.
implies
a,,
(z)
a-,+-,,
(z)
sy
=
-=
(z
a,,,
-y)
+
+ ay
(z)
mod Z
and therefore
z)
(X-y+-y,,
(z)
+ s-,+-y,
+
ky+,y,
=-
(X,,z)+(X,,,z)+s,(z)+s-y,(z+-y)+(X,y,,-y)+k,y+k,y, Now as before
the linear
(X-Y+-Y" Z) Thus,
X
without
:
V
Zn becomes
--+
0-coefficients
(Z)
is fulfilled.
The set
(,y
s,
E
k,y+-y, Let we
c., + i
d-t
:=
ky
-
1 2
=
r) -=
(xy, -y)
=
so
that
By changing
Finally
dy+.y,
Cobordism.
-y to =
Let
-y'
dy
we see
+
a-y,
=
the
S7 (Z) + S'Y' is
an
ky
+
with
(z
ky,
+
real
c,,,
cy + ay,
A(-y, 7')
+
(X-Y" Z)
The part
+
+1 A(-y, 2
(Z
7)
of the Fourier
E
Finally
by the previous
mod Z.
E Z.
dy,. h-,
be cobordant
exponential
we
get
mod Z.
Then
-y')
expansion
Cn)
summand.
(Xy,, -y dy.
-
condition
cocycle
automorphic
get c-,+.,,
(X-Y' Z)
homomorphism.
unique
is
8-Y+-Y'
a
such that
unique
is
part
modZ.
systems.
Wehave
congruence
Line
2.1
&.(z) with
-H(Cn)
h E
_=
h(z
e(h)
such that
-y)
+
mE
Zn and
d. (z) Because of
=_
I
of a^,,
j_Y
(z)
equation
coefficients.
The rest
E
with
an
=
0
a
of the
A)
E
A
x
the
A in the
X
=
way
Cly
=
Cn/A
For the extended
The a,\
integral
(A
E
A)
f (Z)
-
0vanishing Q.E.D.
with
1(X'Y'
in
E
bilinear
A)
we
defect
cx + c,\,
of the
+
Zn'
X
2
A')
:
=
a,\ +,\,
(z)
-
a,\,
(z
+
A)
re-
r
x
I'
to
by
A). on
the line
bundle
L
same.
(A, A'
for the extended
-
x-y
automorphic R, but
E
A).
exponential
is
D(A,
=
relation
mod Z
relation
a.,
an, --->
theorem
E
depends only
A does the
:=
systems
Xm+7
A from
form
(A, A'
a,,,+.y
r)
decomposition
same
-A(A, A')
y E
bilinear the
form
get the
cocycle
E
(XA,, A)
-
that
remains
defined
_V is
exponential
to
s,,
(D
system
we see =
Zn
=
exponential
(m
M)
(Xx, A')
A
with
homomorphism d: A
a
is defined
=
A)
Zn and sm+,y
dy
The wild
same.
-
considerations
1
=_:
c,x+,x,
the
are
as
systems
E
its
homomorphism
(A
c,\
(A
a,\
2
alternating
the
decomposition
a
mod Z.
Fourier-series
alternating
Because the characteristic X
f (z)
-
decomposition
A
-
it
as
A(A, A')
on
:
characteristic
same
(Z)
only
our
dm+,y
and
cM+-Y We extend
+ S-Y
similar
characteristic
homomorphism
(A
7)
exponential
factor
(m E Zn' -y E F), We can restrict E r). E (m Zn).
summand sx
get
we
is clear.
automorphic
By uniqueness mains
7)
+
ay
Zn'
+
transform
we use
of F-reduced
Zn-periodic
a.+-,
(m
because
strict
Extension
2.1.4
The
by
is
f (z
=
mod Z
homomorphisms
,
31
groups
such that
f (z
+
&-y respectively, 9-Y (Z)
This
(m, -y)
+
toroidal
f (z)
+
f
the characteristic
uniqueness
summands s-Y
a.
h(z)
-
on
As above
(m, z)
=
function
Zn-periodic
a
(z)
Zn-periodic.
is
h(z) with
+ a,
bundles
a,\
(z)
E
Z(A,
A'
E
A).
system
32
So
Bundles
Line
2.
easily
we can
by direct
prove
A(A, A')= D(A, A') integral
remains
A C C'
(z)
+a,\,
(z +A)
-a.\,
a,\(z)
-
A.
x
be
subgroup
discrete
a
of C'
2
H
which
maps
defined
by
the integral especially factor automorphic
Cn/A.
=
Then there
exists
a
proof
Chern class
Cn,
(z
+
A')
(z)
+ a,\,
a,\,
-
only
not
with
rank
In
contrast
general. =
to the
of the torus
case
toroidal series
S,Y(z)
wild
lattice
a
Fourier
Zn E9 F be
summand with
of
bundle
line
a
system
(A
a,\
E
L
A)
X
on
to
the
(z
+
pp
A)
a,\
-
25/26.
(A, A'
(z)
E
works for
It
A)
any discrete
2n.
Automorphic
in
cl(L)
exponential
with
LANGE-BIRKENHAKE[66]
see
A E
subgroup
a,\
=
Alt2 (A, Z),
---+
form
alternating
A(A, A') For the
(X, Z)
an
characteristic
A
and X
isomorphism
canonical
Let
the extended
that
D(A', A)= a,\(z +A')
-
A
calculation
Lemma
2.1.5
Let
on
Cohomology
and
summands summands need not
and sy
(-y
E
F)
a
be cobordant
r-reduced
to
0
automorphic
expansion
E s,(,O')e((u,
=
z))
(Z
E
Cn).
O1EZn\jOj The a
E
irrationality
Zn\ 101
Z. The
cocycle
for
exists
a
+
-yj.,)
-yj,,
S-Y(Z)
-
with
e((a, -yj,))
Wewant to construct
standard a
fixed
-
a
(z
Sfj.,
=
coefficients
the Fourier
s(') [e((#-T, Y holds
for
of
garanties
coordinates
basis
yq of I'
-yj ......
that
such that
implies
condition
8-f(z such that
(I)
condition there
1
:A
-yj,,))
of
+
an
-s(')
7)
SY.?,,,
-
(Z)
automorphic
[e((oF, -y))
0.
Fourier
h(z)
series
=
E h(')e((a, ,EZn\jOj
z))
(Z ECn), summand
for
(.7,
every
bundles
Line
2.1
toroidal
on
33
groups
with
h(z +,y) MCAor
On
If it
tively.
C'
even
is true
so
that
in
shall
classical
the
We remark
first
on
(a
case
h does not
Y.7.1
MCAor C',
defined
respec-
by
Zn\ jo}).
E
converge
in
general.
general are cobordant to 0 only RA, as VOGTproved [115, 116].
MCAof
complex subspace
maximal
s(')
=
torus
to 0
uniquely
are
summands in
wild
the
that
1]
-
to
contrary
see
h(')
the
then
h(') But
s.,(z)
=
becomes cobordant
s.,
Cn,
the
on
h(z)
-
that
a
Fourier
E h(0')e((a,z))
series
We
on
the
everywhere
converges
ffEZn
Cn, iff
in
associated
the
real
lh() I k'
E
series
Laurent
for
converges
every
DrEZ'
k E
R 0* (VOGT)
Proposition
2.1.6
Let
X=Cn/A be
toroidal
a
constant
b)
G :=
(-Yj
a
real
r
Let
r-l'rl
an
summand
automorphic
where A =Zn ED r.
of I'
summand is cobordant
to
a
constant,
iff
there
exists
<
dist('Go-,
J'Ga =inf,' -TEZ
Zq)
-
-r
I(u
EZn\ 101).
coordinates. glueing matrix of A in toroidal automorphic summand is cobordant to a constant,
R be the
Then every a
to
such that
> 0
(TS) c)
Yq)
......
be the basis
automorphic
Then every
Then
subspace MCAof RA-
C-linear
the maximal
on
q.
cobordant
summand is
a) Every automorphic Let
of type
group
real
r
(TT)
iff
there
exists
> 0 such that
r-I'l
<
Z2q)
dist(tRa,
'infEZ2 ItRa
=
-
'
-rj
(a EZn-q\ 101).
Proof As shown in
1.1.12
has the basis
(In, G)
1(u, v) into
toroidal
change
we can
=
with
a
((Imt)
coordinates
z
refined -
1
u,
the
+
R, u)
(u
where A has the basis
P
( In-q 0
=
w
transformation
linear
v
coordinates
standard
B,
B2)
R, R2
E
C",
v
ECn-q)
where A V
34
t
Bundles
Line
2.
is the square
with
B2
Im
of the first
matrix
Iq
:--
The maximal
Cohomology
and
the basis
C-linear
of
q
a
of G with
rows
Imt,
invertible
(RI, R2)
and R :=
torus
the real
subspace MCAof RA becomes the
B
:=
(B1, B2)
glueing
matrix.
of the first
space
q
variables. In
coordinates
want
to
get
(-y
E
_V).
s-y
after
z
refined
MCAor
on
(9h
,9zj for j to
h(z
=
(z
function
+
1(-y))
+
prove
There
(9h
=
q
or
for j
=
1,
for
=
n, if the
p,
If the statement
1(Zn )-periodic,
an
standard
to
F)
-V)
For this
it
purpose
is sufficient
hj(tzl,
hj (j
(z)
h (z + I
(-Y
general
(TS)
or
E
(TT),
After
=
1,
-
-
p)
-
,
(-y))
+ s.,
so
(z)
-
E
F)
h (z)
summand constant
change 1-1
linear
get the Fourier
I(W)
1]
h'(,')
1(w) OrEZn
(**)
is
equivalent
b)
and
hold.
that
w we
0
zj
s,(Y')
in the first
q variables.
toroidal
coordinates
from refined series
e
((a, w))
and o
cases
respectively,
Zn)dt
OIEZn\101
h
I)
and in the
tzp, zp+,,
automorphic
coordinates
(z)
with
0
-Lh.9zj
statement.
in
hj
I
E
:=
87
such that
we
cobordant
the function
is true,
1(Zn )-periodic
Proof of the
s
=
j=1
z
coordinates
F)
means
functions
entire
hj (z)
+ 1
P
to
E
-9zj
respectively.
n,
(,y
-'Os'y (Z)
namely in case a) for p q conditions given irrationality
h(z)
defines
=
C-
h such that
h(z)
This
(Z)
Zj
I(Z')-periodic
are
hj (z
is
s,,(z)
9,y (-y
the
Statement.
Indeed.
holomorphic
47))
p, where p
p
a
-
standard
summand
Cn, respectively.
on
from
automorphic
We want to find
,(z) is constant
transformations
I(Zn )-periodic
an
to
Vol
e
((u, w))
c
c)
M') [e((o-, for
j
1,
=
Now for
-
-
-y))
11
-
p where p
-
,
every
=
or
q
Zn\ 101
E
a
-27ris(')
=
(o-, 1-1 (ej))
(-Y
toroidal
on
F,
E
a
35
groups
Z,\ 10})
E
respectively.
n,
p
choose
we
bundles
Line
2.1
a
of
-yj,,
-y
a
fixed
basis
ly
q)
Of
F such that
(j this
Besides
coordinates
we
to
1-1(ej)
(t)
a)
Case
coordinates
standard
=
(j
Im-yj
1,
=
-
-
M')
q)
-27ris(')
:=
(j
everywhere
-,
1-1(ej)
q).
ju
ej
=
with
series
(j
For that
=
first
E Zn
n).
1,
11-1
-
we
q +
coefficients
(o-jm-yj)[e((a,-yj,,))
1,
J>
and
the Fourier
We have to show that
converges
toroidal
refined
back from
changing
after
that
remember 1.13
real
fix
a
>
E}.
E
> 0
and define
Then
E I sy, I
I (a, Imyj) I I e ((a,
))
-yj,,
-
11
-
1
V
<
E I syj,, I
-
E
aEJ>,s
obviously
is
1,
0,
-
-
-
,
q)
for
convergent
k E
every
Rn>0
because
I (a,
the
Now define
j(j)<E The map
x
:=
--+
jor
exp
Zn
Ej
x
(x
E
0 <
R)
is
je((u,,yj)) a
I (a, u) I with
a
c
> 0.
127r (o-,
Im
constant
Therefore
- j) I I e ((o-,
-
11
<
:!
c
Im
so
I exp ((o-, u))
with
(*
-yj,,))
*
-
(j
e}
diffeomorphism
locally
that
-
*) 11
-
1
<
(a
cj
E
P)),
such that
a
EJ
These series
converge
Cj 27r
because
Is(') I V ^(ja
the series
q).
(j of s..y,
do
(j
-yj) I
o9s,,1i9zj
series
everywhere.
converge
Irn
V
EJ>,
1,
q).
(j
36
Case
b)
the Fourier
h(')k
^fill
-2ris(')
=
the
the series
The map
with
z
(k
coefficients
o-
J>,
E
converge
have to
we
study
e(z)
-*
ja
:=
as
E Zn
map
je((or,-yj,))
0 <
C/Z
11
-
C* and its
Ej.
<
inverse
diffeomorphisms
are
so
locally
dj je((u,
-yj,))
suitable
dj,
If the
11 :5 dist((a,
-
> 0
and
((or, -yj),
Z)
cj
irrationality
J'Ga
TEZq
G=
Remember that
N-I'l
(-yj
......
<
vfn-inf
yq)
a
suitable
> 0 such
c
7-1
=
irrationality
there
exists
a
J<,)
c
natural E
N > 0 with
Zq).
of F. Then
le((o-,
-yj,,))
11
-
Y,.'j (o-,ek)jN1'1k'
08^1j /09Zk is not
SW
(k
converge
fulfilled,
then
q +
=
for
every
n).
1,
natural
N> 0
constant
d > 0
EZq with inf
7-E zq
We can choose
Of
(TT)
condition UN
a
(a
because the series
convergent
E
that
OrEJ<.
If the
exists
Zq)
1 :5
18(-)
E
(a
q).
basis
Tj,,,
11
-
I
rj
dist('Ga,
-
-yj))
there
fixed
is the
l(o,, -yj,,)
7*j,,,
-
fulfilled,
is
-
je((a,
:5 cj
I (o-, -yj)
inf 7-jEZ
(TS)
inf
<
^1j), Z)
=
condition
N-I'l
is
that
so
n).
1,
q +
=
coefficients
dist
with
coefficients
with
series
(o-, ek) [e((a,
with
series
with
j<E
that
Cohomology
and
We consider
a)
As-in
Bundles
Line
2.
the
JtGUN
UN
fl
-
pairwise
=
dist(tG'TN,
different.
Zq)
< N-
Then there
JINJ.
exists
a
such that d We want
r-reduced
le((UN, to
^fj)
-
show that
11 :5 dist((UN)'Yj); there
automorphic
a-,(z)
exists
a
Z)
< N- JINJ
divergent
(07N
Fourier
series
summand
:=
h(z +,y)
-
h(z)
(,y
E
I)
E
J<E)-
h such that
the
is
Because h is
convergent.
be cobordant
t6
a
uniquely
determined
h(z)
E e((O'N
toroidal
on
37
groups
the summand ay cannot
by ay
constant.
obviously
Indeed,
bundles
Line
2.1
:=
Z))
,
N
is
divergent,
but
(z)
a-,.,
h(z
:=
-yj)
+
h(z)
-
E[e((0*N,'Yj))
=
Z))
1]e((6N,
-
N
everywhere
converges
because
le((07N,'Yj))
-
11 kl"vl
for
convergent
c)
Case
It
proved
was
q variables
-
in
From
n
that
every
on
So
MCAand
b) holds,
then
Zn-q-periodic
b)
apply
we can
if
Q.E.D. toroidal
for
groups
decomposition
characteristic
an
automorphic G by matrix
all
to
replace
we
to
in the last
R.
matrix now on
But
properties.
summand is cobordant
holomorphic
coordinates.
toroidal
these
summands with
glueing
a)
in
)
R ,O*
summand which is constant
automorphic n
k E
every
16IN
N
N
N
is
k
< d
we can
are
assume
constant
on
that
the
summands in the
the wild
maximal
C-linear
subspace
MCAof RA-
Theta
and
bundles
topologically
bundles
line
trivial
factor defining a line system of an automorphic exponential In the gento an exponential bundle is cobordant system of linear polynomials. wild summands can be reduced to constants only on the eral toroidal case the
On
a
torus
maximal
every
complex subspace MCAof RA-
Definition
2.1.7
C'/A
Let
X
An
automorphic
=
The
or
factor
=
Z'
(D F.
is
a
theta
factor
or
theta
factor
and theta
bundle
are
if it
linearizable,
A line bundle polynomials. can be given by a theta factor.
if it
linearizable,
names
L
on
X is
used in classical
is a
torus
given
by
theta
bun-
an
theory.
Lemma
2.1.8
For
A
system of linear
exponential dle
with
a
line
equivalent:
bundle
L
on
X
=
Cn/A
with
A
=
Zn
r the
follwing
conditions
are
38
Bundles
Line
2.
theta
1.
L is
2.
The wild
a
Cohomology
and
bundle.
summand of the
automorphic
factor
of L is cobordant
to constants.
Proof Consider
the characteristic
On the other
hand
decomposition
2.1.3.
Q.E.D.
get:
we
Lemma
2.1.9
For
a
bundle
line
L
X
on
CI/A
=
A
with
=
Zn .p
following
the
statements
equivalent:
are
trivial. topologically Chern class cl(L) integral
1.
L is
2.
The
3.
The characteristic
4.
The
0.
and
=
0.
=
The
5.
=
form A alternating F-reduced characteristic homomorphism X 0. factor automorphic defining L can be given by bilinear
automorphic
an
sum-
mand.
Proof 2x3.
By
1 -3-
If the line
without
tion
f
Lemma2.1-5.
C'
:
--+
bundle
zeros
C* exists
topologically by Proposition
L is
that
so
be
a,\
exponential
an
h(z Wecalculate =
h(z
=
a,\,
=
=
+ A +
(z
+
A)
+
h(z A')
A)
+ n,\,
h(z + A + A') a,\ (z + A') +
=
-
nx
A)
+
(z)f (z)
a,\
=
system for
ax(z)
h(z)
+
A') h(z + A) + a,\ (z) h(z + A') + ax, (z)
+ A + -
then a
it
has
a
continuous
automorphic
continuous
sec-
form
with
f (z Let
trivial, 2.1.2
-
+
h(A) h(z
(A
(A
ax
A)
and
with
+ nx
+
A)
E
A).
E
f
=
e(h). (A
nx E Z
Then E
A).
h(z)
-
+ nx
h(z
+
+
A')
h(z)
-
+ nx,.
Then a,\
The characteristic
3>-4.
+
A')
bilinear
The restricted
characteristic
(z
+ ax,
and
Alrxr
=
,
Then
=
ax,
alternating 0, iff the
(z
A
+
=
A)
+ ax
we
=
(X'Y" -Y)
get for the extended
bilinear
(z).
0.
commutation
property
homomorphism Xjr
(X,Y ly') holds.
(z)
(-j' -Y' form
G
F)
for
the
restricted
A(A, A') If
we
X,y
take
-y
=
(-y
E
F).
0
=
If X
4>-5. a.y
(z)
-/'
and m'
0, then L
=
sy
=
(XI, m')
=
(z)
dy (7
E F be
G
Let
given by E
F)
homomorphism
a
:F
A
unit
=
on
m+
vector
exponential
-y,
toroidal
groups
A'
+ -Y
39
1 .
n),
ej
we see
system
y'
mod Z
c,, + cy,
c,,+,y,
E
-P)
+
+
mlc-yl
mqc-y ,
R with
F)
mod Z
c,,
=
bundles
Then
:=
--+
a,y
all
with
6ml,yl+***+Mq'Yq defines
for
the
basis.
a
Line
with
ej
m=
-
is
+ cy + i
(Xy,, m)
-
2.1
changing the F-reduced auby .y without c., can be substituted i L. Then factor + + defining tomorphic d-, is a F-reduced automorphic 6,y s., summand aX (A E A) definsummand which can be extended to an automorphic so
that
ing
L.
the
5 -L
As
there
exists
have
we
seen
a
h(z
(Uj)jEj
let
For that
projection
7r
C'
:
be
h
:
+
A)
a
h(z)
-
we
7r
enough
is
it
show that
to
C with
--+
=
finite
locally
C'/A
--+
C'
proof
of the
beginning
the
at
continuous
(A
ax(z)
E
A).
of X
covering
C'1A
=
such that
with
have
-1
U (Vj
(Uj)
A),
+
AEA
Vj
where
Moreover of units
A)
+A
(A
E
let
-rj
:=
are
belonging
we
define
that
(A
E
by cocycle
A)
is
Uj
--+
and
Vj (j
: 0, supppj
the continuous
a
h3-
C
C'
pj(u)ax(-rj(u))
hj(z) Now a,\
:
7r
J)
E
:
Vj
and
+A
-4
Jpj
:
Uj j
E
biholomorphic.
is
J1
be
a
partition
(Uj)jEJ:
to
pj
Then
disjoint
pairwise
(7rJVj)-1
0
summand of
condition
Uj
-+
and
C
pj
=
1.
by
for
z
=
-rj(u)
+ A
elsewhere.
automorphy
and
z
-
rj
(u)
E A
a
period
so
40
Line
2.
hj (z
covering
The
h(z
it)
+
+
tt)
and
Bundles
hj (z)
-
Cohomology
hj [Tj (u)
=
+
=
pj
(u)az-,j
=
pj
=
pj
(u) (u)
(z
(u)+,,
a,,
(,rj (u)
a,,
(z).
+
(Uj)jEJ is locally finite. h(z) a,, (z) (ti E A).
Picard
the
hj [-rj (u)
-
pj
-
7j
-
With
(u)az-,j
(z
+
(u))]
-rj
-
(Tj (u))
(u)
(u))
hj
h
continuous
get finally
we
Q.E.D.
of cobordant
The characteristic
automorphic
system of
bundle
line
bundle
theta
of
L
on
L,9 and
a
A line
bundle
T,, (x)
x
+a
L
on
X
(x
E
X)
=
trivial
is
=
implies by
defined
C'1A
line
La
=
to
that
an
every
exponential
by
an
expo-
To
bundle
this
see
homogeneous
the translated
is the
bundle
product
tensor
of
a
Lo
Lo.
(9
unique.
is not
CI/A
X
group
topologically
a
decomposition
this
factor
factor,
isomorphic
is
summands.
116])
toroidal
L
general
automorphic
theta
a
bundles,
automorphic
an
of
(VOGT [115,
Theorem
2.1.10
Every
line
and an automorphic defined polynomials, factor, a wild automorphic summand. So we get the
system of linear nential
trivial
product
is the
Pic(X),
F-reduced
decomposition
factor
C
topologically
of
group
of classes
the group
Ta* L
on
is
X,
we
define:
if for
translation
every
holomorphically
isomorphic
L. Remark
2.1.11 Let A
=
z, (Dr and L
factor
ay conditions
2.
The ay The ay
3.
The
1.
(-y
E
are
P)
a:
r
--+
a
line
bundle
E C
P) r) (,y
the line
are
constant.
can
be substituted
E
X
C'
=
/A.
For a.P-reduced
system
a.,
(-y
E
automorphic
V)
following
the
by
a
homomorphism
a: r
is
a
homomorphism
a:
C without
r). factor
bundle
(,y
a., L
can
E
r)
be defined
by
the
r
representation
C*. a:
r
--+
C
C*.
Proof Indeed, we have to see only 1 >proof of Lemma2.1.9. Because and
on
exponential
of L with
equivalent:
changing a., automorphic
Wesay that or
z
it]
=
-
Pico(X)
to
+
subgroup
The
In
(u)) (Tj (u)) rj
-
2. But this
a-,
=
can
cy + i
d-,
be done in the
with
same
way
homomorphism d
as :
r
in the --+
R
-yl,
the desired
the
bundle
[116].
iff
given by
is
F)
by c.,
E
r)
bundle
is
homogeneous,
every
line
d-t (,y
+ i
a
line that
assumption
becomes
ABE proved
1989 that
in
a
in any
case
iff
a
basis
topologically
is
it
Cn/A
=
bundle
line
a
on
homomorphism with Q.E.D.
X
on
cy,,
......
is
a
theta
homogeneous
is
[3].
representation
a
bundle
41
groups
Theorem
2.1.12
For
,y
:=
1982 that
in
under
it
6,.y
that
so
E
property.
VOGTproved
trivial
homomorphism Z-generated
the
as
7q of F
-,
-
a.,
take
we can
toroidal
on
(-y, -y'
mod Z
cy + c,,,
=-
c.y+..y,
bundles
Line
2.1
bundle
line
a
L
on a
1.
L is
homogeneous.
2.
L is
a
3.
L
topologically
following
the
group
trivial
by
be defined
can
toroidal
statements
equivalent:
are
bundle.
theta
representation.
a
Proof Let
A= Zner.
a,*y (z) T:,*: L phic
=
(z
a.,
factors
(z)
This
+
and this
is
s.,
(z
Now as in
means
(xy, x)
+ s.,
equivalent +
x)
proof
x)
(z
+
x)
+
(Xy, x)
s.y
Such
a
(z
+
x)
(z)
-
s.,
decomposition
Z'-periodic
s,,
(z)
+
x)
defining
the
a
=-
(z)
+
is
(z
-= -r.
there
(z)
e(,rx)
-
s.,
(z).
F-reduced
where
-rx,
automor-
C'
:
---+
(z
7-,
+
-y)
+ a.,
(z)
-
-r,,
(z)
mod Z,
=
+
-y)
exists
(m., z)
-
a
-r,,
(z)
E
F).
decomposition
unique
+ q.
(,y
mod Z
(z)
qx such that
Z'-periodic
(Xy, x)
unique
(Xy, x) and
-
of the theorem
and
mx E Z'
iff
L,
to
(z
+ s.,
that
,r.
with
(Xj, x)
+
system
to
sy
-
(z)
a-,
=
with
cobordant
are
holomorphic.
+
isomorphic
holomorphically
is
a.y
:
by the exponential
T*L is defined
bundle
The translated
1>-2.
=-
so
=-
(mx, ^j)
that
(m, -y)
we
+ q,
(z
+
y)
-
get
mod Z
(y
E
F)
q,
(z)
mod Z.
C is
42
Bundles
Line
2.
sy
sy(z
Then Let
+
x)
On toroidal
x)
+
(z)
sy
-
strictly
are
=
(z
qx
+
-y)
(z).
q,,
-
cobordant.
(*)
(x,y, x)
=-
(m, 7)
(m, -y)
-=
(n, -y)
groups
because of the
m= n
(z
and sy
consider
us
Cohomology
and
irrationality
(-Y
mod Z
(-y
mod Z
F).
G
r)
E
for
m,
implies
E Zn
n
condition.
So the map Cn 3
by (*) becomes Let
a
with
0 <
this
But
E
homomorphism.
a
X-11N
:=
with
N
big
so
be because
cannot
(Xy, a) one
to
Then X
a
-a
the real
one on
Zn is countable.
exists
Then
< 1.
-y E I` with
:A
m,a
F-+
interval
-a
=
0 and L is
q,
(z)
XY
0 for
(0
<
-
0
0.
all <
E
1).
topologically
(Lemma 2.1.9).
trivial
(**)
consider
Next
(z
s,, which
to
+
that
demonstrates
cobordant
x)
all
=
the
qx
(z
+
-y)
automorphic
every
E
o-
-
summands
s(')
Zn\ fol with a suitable -yj, =A 0. Then we can define
73.1
[e((tT,
a
fixed
of
basis
s-,(z
+
x)
cocycle, groups
-yj ......
strictly
are
condition
for
is
yq of F
so
that
1
q(x)
Especially
(z)
of the a consequence s'Y* Remember that summands sy(z) = E s Y toroidal on z))
s(') [e((u, Y
e((a,
+ sy
)e((u,
automorphic
for
Zn
E
7n.
Suppose that there
that
Our map becomes
< 1.
X _4
p(-)
:=
:=
qx(O)
s(')
Y-1.1
(z)
q.
-yj,,.))
[e((u,
-
has to converge
1:
=
1]
-1
(a
absolutely
p(') [e((a, x))
Z'\ 10}).
C-
for
all
x
E Cn , and
by
1] e((o-, z)),
-
o,EZ'\101 in
particular
q(x)
E p(')
=
[e((u,
x))
-
1].
OIEZ'\101
Indeed,
for
R>1
any k E
n
k' Then there a
:A
0. This
exists
a
<
we
I21'1k'
suitable
shows that
have
the
-
x
part
11
E Cn with
(o-
>
0, but
210'1 k'
=
a
=A 0).
e((u, x))
for
all
a
>
0, but
lp(') I V
or !0'0'00
because
be convergent,
must
In the
convergent.
way
same
cEZ'\101
of the
part
same
we can
jp(') I
1:
of
see
43
groups
V
for
series
the other
that
toroidal
on
parts
absolutely
q__ is
are
for
convergent
So
< 0.
o-j
some
the
bundles
Line
2.1
p(')
P(X)
e
((a, x))
OIEZ'\101
defines
Fourier
convergent
a
everywhere. Finally condition
(**)
series
implies
(x)
s,, With
(-I)
q.,
& + -y)
=
q(-I)
-
s,,(x) 2>-3. then
L,
X
=
If
In
qx
q(x
only
toroidal
a
of a
factor
is
get
q(x)
-
dy (,y
E
r)
is
bundle.
theta
a
theta
constant.
of
a
bundle
against
line
and
bundle a
the
representation,
a
bundle the
By
exponential implies Q.E.D.
decomposition
characteristic
the
a^, is invariant
homogeneous
X
group
we
have to
difference
a
we
representation.
translations. L
on
topologically
a
toroidal
trivial
group
line
into
bundle
is
bundle.
line
CI/A
=
Pic(X)/Pico(X)
:=
A
with
(z)
=
identify
Cn/A
defines with
A
=
(x-y, z) those
=
a
+
subgroup.
-
2
(X'Y
who differ
(X-Y, Z)
1 +
-
2
=
Z'
,
Y) by
(X'Y' -/)
ED r
+ C'y a
can
be
represented
+
'Y
Zn E) r C
(,y
E
by
expo-
-P), But for
representation.
and vice
representation
NS(X) is
(0).
q(-I)
-
+ i
c.,
:=
But then
theta
a
&Y (Z)
=
sy(O)
qx
-
q(x)
=
+
a.,
a
holomorphic
systems
where
X
must be
group
a-,
the
7)
decomposition
the
up to
by
NS(X) nential
(0)
+ sy
qx(0)
that
that
so
The N6ron-Severi
of
(y)
+
p(x) -p(O)
0
=
automorphic
r)
E
product
tensor
unique
L is defined
(,y
0
=
consequence,
the
=
and
so
must be constant.
0, sy
=
0
=
defining
the
system a,, X
0 and s-,
remark
previous 3>-1.
z
=
to zero. The line bundle must be s,, (y E r) is cobordant trivial If the exponential system a., defines a topologically
that
so
=
for
q(x)
that
so
Hom(F, Zn)
(,y versa.
E
any other
F)
So for
toroidal
groups
Cohomology
2.2
cohomology
The Dolbeault toroidal
theta
topology.
with
those
comparable
is
A toroidal
line
every
the
X
group
X is
on
In
cohomology groups
have differential
groups
of torus
of toroidal
only
groups
wild
for
have
groups
cohomologous
forms not
and wild
theta
is lineatizable.
bundle
groups
general
In the
case
we
define:
Definition
2.2.1
L
the
coefficients.
constant
torus
a
with
cohomology
wild
Toroidal
Toroidal
Over
groups
The Dolbeault
groups.
non-Hausdorff to
of toroidal
a
about
section
proved that automorphy
Cn /A is
=
bundle
theta
Lemma2.1.9
to
know that
we
group,
wild
iff
line
every
bundle
group.
line bundles trivial we topologically theta bundle, iff every summand of 2.1.6 and constants. By the results of Proposition iff some special conditions hold. So we this is true,
bundle
cobordant
is
theta
toroidal
a
bundles
theta
line
every
toroidal
a
otherwise
L
and
X is
on
a
get the
(VOGT)
Theorem
2.2.2
For any toroidal
X
group
=
Cn /A of type
following
q the
statements
are
equiv-
alent: theta
1.
X is
2.
Every summand of automorphy
3.
For any basis
a
group.
real
positive
G
=
yq)
......
r
<
dist(tGa,
Zq)
(Condition 4.
For any basis a
positive
in toroidal
real
r-Il
(TT)
for
this
wild
that
remember
we
principal
constants.
there
exists
a
<
groups
C*n-q -fibre
7'EZ"
theta
in
groups
with
(a
101),
Z'\
coordinates)
standard
glueing
real
E
R there
matrix
exists
such that
r
theorem
toroidal
JtGa -Tj
inf
=
coordinates
Z2q)
dist(tRa,
(Condition With
to
coordinates
such that
r-Il
(TS)
(-I,
of A is cobordant of F in standard
it
is
for
=
for
possible
theta
to
bundle
that with
n
every a
torus
-
-rl in
groups
construct
any dimension
[see 1.1.12]
'inf,EZ2 J'Ror
> 1
toroidal
(a
group
Zn-q\ 101). coordinates)
toroidal theta
and type
toroidal
E
group
groups
q with
of type
of dimension
as
0 < q <
q
q is as
a
well n.
as
For
natural
base space.
Cohomology
2.2
VOGTconstructed C*-fibre
as
the
over
KAZAMAand UMENOconstructed
wild
toroidal
which
groups
bundles
groups
generated
is
by the
wild
iIn-1)
-
toroidal
l)-dimensional [56]
for every
in 1991 that
of dimension
groups
-
and q with
n
and type
n
theta torus
and
groups
0 < q <
n
[20]
q.
Proposition
2.2.3 For
toroidal
exist
(n
over
(In-l,
by the periods
and CATANESEproved CAPOCASA there
wild
and toroidal
which
torus
1984 n-dimensional
in
C*-fibre
as
generated
are
theta
onedimensional
45
groups
[115]
(1, i).
periods
toroidal
in 1981 2-dimensional
bundles
of toroidal
every
and q with
n
dimension
q there
toroidal
are
and for
n
theta
C*n-q-fibre
principal
natural
0 < q <
wild
complex
T of
group
and toroidal
groups
bundles
torus
every
which
groups
are
T.
over
Proof
Obviously
the
(a, 0,
R=
such that
and other
ELSNER[29] In the
2.2.2(4)
case
numbers p,
A theorem
for
holds
glueing
an-q)
and it
some one
of
=
does not
a
matrices n-q E R
hold
for
other
with
E
Zn-q\ 101
E
Zn-q\ 101 3TN
Q-1inearly
constructed of toroidal :=
'(a,,
independent
give examples
N>o 317N
with
cases
Q-1inearly
the construction
N-Il
Vr E Z:
<
1(a,a)
+,rl
with
ones
VN E
both
with
equivalent
is
N>o Vor
3N E
(W)
with
have to
we
(T)
in
0)
-
the property
So
ones.
proposition
2,P2
independent
the
theta
3,-
:=
following groups
-
such that
M
V(a, -r)
E
(T)
we
Ce :
(log
Zm+'\ 10 1 : 1,r
pi,
-
-
log
1 + al
with
start
-
,
Theorem
-
TNI
< N-
IINI,
On-q-
*)
*
+
examples:
-,pm
of ALAN BAKER[15,
cm > 0
01,
and define
=
1(6N, a)
E Z
the first
n
-
q
prime
log P,,,).
3-1] guarantees
pi
m :=
+
-
-
-
the existence
of
a
real
+ am log pmI > S-'-
with S:= 2max 11,T1, If If
17J-r
+ +
(a, a) I (a, a)j
>
<
1, then there
1,
then
Jrj
-.5
exists 1 +
jam,11
lull,
nothing
I (a, a)l
to
:5 1 +
> 2.
prove.
11all 11all
with
maximum
norm.
46
If
2
2(1
J,r
Jrj + +
Bundles
Line
2.
S,
<
(a, a)l
Finally
>
we can
especially
S
then
m) ljo-11 11all
for
and
Cohomology
2
=
because 2
Ilo-11 11all
ljo-11'
k,,, find
otherwise
sufficiently
a
In the
=
case
pl,
am
of toroidal
wild si
with
m:= n
1
2 I-rj so (m > 1).
that
k,,,
[2(1
N c N
so
:=
that
in
N"
any
(*)
With
+
we
(x -> 1),
:=
=
(o-, a) I
+
log
> N-
pm must be
groups
(W)
1, sj,+i
:=
we
107 1.
independent.
Q-1inearly with
start
strictly
(A
jLm2`11
>
monotone sequence
1)
q and define
-
00
a,
aj
2s,,
aji
:=
U
M))
am).
(a,,
a :=
By N
(2 SN)-?,
07NJ
U
2N
TN'j
=
1......
M)
TN:=ETN,ji
0`N:=(0`N,1C'*)0`N,m)7
j=1 we
get "0
0 < a,UN,l
-
E
TNJ
2 SN-Sp <
2SN-SN+1+1
<
1(N
tt=N+l
Because
jai I
<
1, 0
0 < ajUNJ
< TNJ < UNJ,
-
TNJ
=
aja,1i
N
1
have
we
TNj,l
-
<
jo
jNj-'(aO'N,l
and m
E jo-
j-1 Nj
<
M2,m-1
-
Nj
j=1
Then
(a7 UN) Now x2N'
<
S <
get
m) jjajj]-11'-.
km-lxc-
>-
case
jo-1
x :=
log
=
'.
where
big
J,r Then a,
jo-j-'-
! km
S
2logp,.,,
=
2Nx (x
>
9)
so
-
that
M22MSN-SN+I+l
7-N
for <
I< M22MSN-SN+I+l x
:=
m2MSN(N >
M222MSN-SN+l
<
3)
N-m2-N
TNJ)
>
1).
of toroidal
Cohomology
2.2
47
groups
and then
(a; UN)
(**)
because of
Finally
TN
-
we
I
<
fixed
every
Q-1inearly
are
sufficiently
I and all
big
191; HARDY-WRIGHT] a,
Theorem
a
(N
3).
>
N. So after
<
-I
a
TNJ
theorem
[42,
of LIOUVILLE
be transcendental.
must
Then a,,
-
independent.
-
am
-,
Q.E.D.
cohomology
Dolbeault Let X be
SNm2-'N
28N+1-1
O'Nj for
N- JINJ
<
get
TNJ
a,
< N- 'IN,-
complex manifold
of toroidal
of dimension
n,
theta
f2m the
groups
sheaf
of germs of holo-
m-forms
morphic
(m
hidzi
w
<
n),
III=m where 0
=
S?O is the sheaf of germs of holomorphic
of germs of C"O
Em,P be the sheaf
(m,p)-forms
complex valued
1:
W=
functions,
fi,jdzi
A
(0
d-zj
<
n)
m,p:!
III=-,Ijl=p where 9
:=
EO,O is the sheaf of germs of C'-functions m-forms
holomorphic beault's
V)
0
0
of
called
E-,P(X)
induces
,
global
5-Poincar6's
om
__4
_"_>
+
Em'O
the
space
sequence
Z(X,.EmP)
91,P)
=
Following an isomorphism
exists
a
i -4
em'l
em,2
em,O(X) _-5> SM,1(X) This
sections.
space B (X,
X.
and
f?M(X)
finally
on (m,p)-forms exists there lemma)
C"O
X. an
Then exact
be the
by
Dol-
sequence
Em,n
-4
o,
DOLBEAULTsequence
the
S?,rn(X)
I -
H',P(X) with
the
sheaves
of fine
which
(also
lemma
and
=
defines
:=
Z(X, Sm,P)/B(X, "5 -
-a Em,P(X))
of DOLBEAULT[36,
Sm,n(X)
DOLBEAULT-&-cohomology
the
ker(S',P(X)
im(Em,P-1 (X)
theorem
Sm,2(X)
p
>
__
0
groups
Sm,P)
-Em,P+'(X)) of"&-exact
of
-5-closed
C"O
and the
(m,p)-forms
on
204; GRAUERT-FRITZSCHE] there
48
Bundles
Line
2.
Cohomology
and
H',P(X) Nch cohomology.
to
For
X
groups
forms
w
<
lifts
the
A
d-zK
of
(m,p)-forms
X= Cn /A the
groups
let
w
Eljl=,,.,
=
Then the coefficients
Jdzi
set
To calculate
must
cohomology
all
C*n-q-fibre
principal Lie
that
=
over
All
is
Hm,O(X)
of
basis
a
used the
holomorphic
are
n
torus
fibre
<
Cn.
on
toroidal
m<
groups in
seen
n). are
that
1.1.12
isomorphic
subgroup
closed
a
toroidal
that
Wehave
group.
+ q has a
(0
fact
the
to
group.
the
idea,
principal
of toroidal
groups
1982-1984
in
same
the
in
n
M
cohomology
the
in 1981-1983
authors
torus
a
becomes
KAZAMAand UMENOcalculated
which
fJK-
( )
=
we use
Cn /A of rank
X/C*n-q
T:=
VOGTdetermined
groups
bundles
X
group
so
groups.
ml
=
(m,p)-
the
be a D-closed A-periodic fidzi (0, m)-form So for and therefore be holomorphic constant.
dimHO(X, Rm)
C*n-q
exactly
are
Coo-coefficients
I JI
:
groups
every
X
on
A-periodic
with
C'
on
n).
m<
For that
C'/A
=
E fJKdzi
=
For toroidal
(0
HP(X, Q')
-
of the
coordinates
theta
cohomology of namely to work
groups.
toroidal
all
forms
with
C*n-q -bundle.
natural
[117, 56] In this
space X
fibres
considered KAZAMAand UMENO
sense
a
over
which
biholomorphic which
.F of C'-functions germs of Coo
of S21 and In details
0
A
in
nZn_q
the
)
as
a
Stein
T of
along in
F
a
locally
complex
manifold
coefficients
with
S. With
the fibres
trivial
dimension the
sheaf
fibre q with
of germs
and the sheaf T',P
they got
an
analogous
of
resolution
HP(X, f?').
of
with
ED F
the basis
(z, w)
P
( In-q
(Z1,
=
Ig
0
=
Zq)
S
R, R2
W1
i
)
where det ImS
Wn-q)
so
that
with
0
0
B
projection 7r:
induces
a
[58]
take
coordinates
toroidal
(Iq, S)
(m,p)-forms
to
holomorphic
are
representation
we
complex manifold
paracompact
are
1992
in
principal
base space.
=
MCA/BZ2q
C'1A bundle
C"7-fibre
with
(z, w) C holomorphic m-forms
Moreover
sheaf of germs of all
X
let
be all
=
w
=
E JIJ=M
hid(i
the n
=
T
q-dimensional
variables
and
torus as
before
group
T
nm the
(m
<
n).
Now we define
respect u
of germs of C'-functions
sheaf
the
as
along the fibres and F',P to Idzi, -,dzn)d Zli -,d-Zql
morphic
Let
Y
-
be the first
q and
the last
v
5-operator
We decompose the
n
Lemmaapplied
0
Now we beault
can
on
5u
f2m-t+jrm,O
__.
holo-
are
(m,p)-forms
with
coordinates.
in toroidal
au
=
+
av.
the base space
5u
17m,l
7u
Fm,2
the
(60)
as
-5u
with
groups
get
we
Fm,q
__+
0.
lemma of Dol-
following
type.
Z(X,.F',P)
let
For that
B(X,.F',P)
"5u
ker(.F-,P(X)
:=
im(.Fm,P-'(X)
and
2.2.4
Lemma (KAZAMA-DOLBEAULT)
X=Cn1A be
:=
a
toroidal
*.F',P(X))
Z(X,.Fm,P)/B(X,
-
be the
of
-Ou-
space of all
all'au-exact
forms.
Then
group.
HP(X, fl')
.Fm,P+1(X))
-5u
closed
Let
X which
F.
in
complex variables
q
-
cohomology
the
calculate
and coefficients
49
groups
into
a
By Dolbeault's
on
of germs of
sheaf
the
as
-
*
-
of toroidal
Cohomology
2.2
YM,P
Proof Let U
=
JU,, I
be
a
finite
ir
of the torus
covering
open
-1(Uc,)
Uot
X
trivialization
T with
C*n-q,
to be identified.
of
Sm',P
let
Moreover
m"-forms
holomorphic
sheaf
be the
of Coo
C*n-q
on
.
(m',p)-forms
U,, and
on
fl*m"
sheaf
the
Then J?
.FM'PjU,,,XC*n-q
'MI/
MI+M11=M
where
(
denotes
the
topological
By KfJNNETH'S formula k
H
(Uc,
X
product.
tensor
[49, KAUP]
we
get
Hk(U a (&
C*n-q,.Fm,p)
C*n-q,
SMIP
(3)
D
S? M
M,+MII=M
H'(U,,,Em',P)
)
Ht (C*n-q,
S?m")
+t=k
Then V
=
JUo,
X
C*n-ql
is
a
LERAY covering
for
Fm,P
on
X.
=
0
(k
>
1).
50
Line
2.
Let
I
For
a
I
g,,
be
Bundles
a
Cohomology
and
to the covering unity subordinate Z' (V, we put
of
partition
I
cocycle
U"' I of T.
U
G
E 011*r(XW-C. ',
W:=
9010011***Clk-1
C, "
...
W.
-
Ci
E Ck-1(V,.F',P) fgct0cj1,**cjk_jj MOCZ1**'ak } because I f,,0 c,,...o,, } is This and (Y) proves the theorem.
Then
For the next
step
(m, p)-forms
Two
We want to show that
5-cohomologous
to
5-closed
consider
a
bfg,oal
and
...
=
0
ak-l} (k > 1). Q.E.D.
5-cohomologous,
are
cochain
a
Hence Hk (V,.FM,P)
VOGT[117].
follow
we
becomes
cocycle.
a
with
difference
their
(m,p)-form
0-closed
every
form
iff
on
coefficients.
constant
a
is
-6-exact.
toroidal
For that
theta it
is
group
is sufficient
to
(0, p)-forms.
with
Indeed,
E Fjjdzr
A
d-zj
dzr
Fjjdz-j)
A
ij
Ej Fjjdz-j
all Let
u
are
be the first
5-closed. q und
the last
v
from standard
formation
n
-
coordinates
where B of
=
LR(t)
=
1.1.13
(LR),
(t
At
E
B,
with
T and R
torus
a
(BI, B2)
=
R2n)
(Imt)
=
(RI, R2) of the
lint -R,ImT
is
Cn/A
be
a
toroidal
5-cohomologous
and ReB2
=
(Imt)
Ret
We get
matrix.
a
is the
basis
parametrization
(Reu, Imu, Rev, lmv) according
0
0
Iq
0
0
RjReT
-
R2 In-q
0
0
In-q
0
(VOGT)
Proposition
2.2.5
R
-Ret
0
Let
trans-
is
0
A-'
M
q
coordinates
matrix
refined
B
glueing
the
real
where the inverse
1
-
after
A has the basis
that
0 In-
P,
coordinates
q toroidal so
to
a
theta
group.
form
with
Then every constant
5-closed
A-periodic
(0, p)-form
coefficients.
Proof
Eljl=p
Fjd-zj
Let
w
a)
The coefficients
=
be
Fj
o
A-periodic.
LR1
are
Zn+q-periodic
in the first
n
+ q variables
t'
of
Cohomology of toroidal
2.2
the real
t
parameters
=
(t', t")
E
R'n.
So
develop
we can
Fj
51
groups
into
a
real
Fourier
series
Fj
o
*(o,)
1:
LR1 M
(t ) e((u, 11
t
1
0,EZn+,l
of
(*),
v)
Fj(u,
We get
=
fj*(')
E,Ez,,+,
(Imv) e[E(O') (u, v)
(0-3) iIMV)l
-
with
the
help
where
E(') (u, v)
[(tUl
:
=
Rl)lmt]
-t93
Re u
[( tUl
-
and 0711 CT2 EZqand 93 E Zn-q The
previous
last
n
-
q
Lemmashows that
complex variables
must be constant
v.
tU3 RI)Ret
-
(t
-
U2
of
the components
are
the Fj But
93
E
a
R2)]
j(o')
the
fj*(')
I111
U
+
(U3) V)
i
Zn+q.
be assumed to be
can
then
t -
holomorphic
in the
(Imv) e(- (073, iIMV))
that
so
f J,
Fj
e(E('))
0'
O'EZn+q
with
coefficients.
constant
Now define
w(-)
f
jlo )e(E('))d-zj,
ljl=p
E
fjl
(a)
azi,
ljl=p
C(-)
((tf l
2
t93R,)Imi
i
[(to-,
tU3R,)Ret
-
-
(t92
tU3R2)
and q
C(-)
d-z
j=1
Then
U') Since
w
=
EIEZ"+q W()
is
5w
=
27rie(EW) (O')
&-closed 27ri
A
(O)
and
=
E e(E('))&)
79(a).
0, A
79(')
=
0.
Croo
It
would contradict
b)
It
C-linear
is well
C(')
that
is to remark
the
known
map D
:
V
0 0 for irrationality
[77, --+
p
o-
:A
0. Otherwise
condition
1.1.12(1)
7; MUMFORD]that
C there
exists
a
tal
map
for
any
=
for
t93R, toroidal
and
tO'2
=
tU3R2
groups.
C-vectorspace
V and any
52
Line
2.
Cohomology
and
Bundles
M-1
M
:AV--,
D]
A
V
the properties:
with
M
Dj (XI
(*)
A
A
...
Xm)
E(_I)m-k
=
D(Xk)XI
A
A
...
k
A
,
*
*
A
Xm
k=1
and If
D(Xo)
1, then for
=
D] (a D]
is the so-called
a
E
A'
XO)
+
(Dja)
every A
V
multiplication
interior
A
by
X0
a.
=
D. Now define
the
C-linear
space
q
Eajd-zj:
V:=
(j=l,..-,q)
ajEC
j=1
the
C-linear
map
D(')
V
:
C
--*
by
q
&)
q
D
d-zj)
aj
aj j=1
j=1
Zn+q\ f0j)
2
I C(,)
and 77
.-(_1
(a)
e(E('))D(')]
27ri
(79('))
(a
E
Zn+q\ jo}).
Then
Because
D(O')(&))
D(')j so
1
=:
(,d(')
get with A
&))
(D(')] (d(-)))
+
A
&') =,d(o')
that
e(E('))D(')j and with If
we
we
w(O)
=
(t)
define
is
A
formally
q
fj("d-zj,
a
convergent
Remember that
every
&))
+
E,00
E,,,o
Eljl=p
E,00 n(')
(?9(')
677(')
W(,)
=
E,9,0 77(l), form for
with
toroidal
coefficient
of
W
theta is
(or
w
becomes
coefficients. groups a
Ej
Zn+q\ 101)
W(O).
-
then
constant
n(')
(,)(')
=
finite
5-cohomologous
But
we can
only. sum
of summands
to
show that
Cj(U I
k E
every
R' ,o
(TT)
r- 10'31
independent
of
Fj
of Theorem 2.2.2
(93
-
k ff3
holomorphic for
that
so
I f J10'
and
a
are
'infEZ2 I tT t93RI
<
e(E(')).
j
because the coefficients
remember
Moreover
53
groups
,
The number of the summands is for
( )
C-(')Tf)
21ri
of toroidal
Cohomology
2.2
\ fo})
convergent
v.
real
suitable
a
Zn-q
E
in
> 0
r
-
,
With
(172
T:=
E
Z2q
t
so
we
have
1
C(0')
2
with
a
real
for
We have
that
seen
addition there
exists
Indeed, so
that
a
for all
iIq
0
On toroidal tion
A
=
j
harmonic
the
case
I kO'3
with
fjd-zj
0)
d
E,7:0 I f(
< -
Of 0'3 on
)IrIO'3lkO'3
10' i
a
is
Q.E.D.
0-
=
toroidal
theta
coordinates
group
is
E C.
In
fj
coefficients
constant
toroidal
in
0`3
fj
every
=
0, if
> q.
fj
function are
form
which
is
H',P(X)
of
A-periodic
and
theta
represented
is
uniquely
not
toroidal
about
is
zj
2)fj
=
dfz-j
D-exact.
element
every
result
If
FC(-T)I
(0,p)-form
that
D(fjd'Tj)
groups
get the final
Eljl=p
assume
> q the
j dz-j
theta
invariant
Now we
form
E J with
every
in
even
Zn+q with
E
E,,,o
A-periodic
we can
j
dTj
a
(o-
Then
R1>0,
every
to
that
to
k E
every
2)-cohomologous
by for
determined
a
q <
translan.
groups
Theorem
2.2.6
X=Cn/A be If
a
toroidal
X 24 T represents
induces
an
theta
X
as
a)
Every differential
termined
group
of type
C*n-q -fibre
q.
bundle
Then:
over
the
torus
T, then
7r
isomorphism 7r*
2.
Im -iRet
0`3R)
I C(O')
d > 0.
constant
convergent
1.
t -
that
1 U31 < d
Let
(t'r
-
form
:
HP(T, 0) in
-+
Z(X,,E',P)
HP(X, 0). is
represented
by
a
uniquely
de-
54
Line
2.
E
P!
M!
P
M
1
1
Cohomology
and
Bundles
CJK
dzj
dZK
A
-
/\
E
Cf dzi,
dZn}
A
/\
Cf d-zj,
-
dzql.
-
IJI=IKI=p
b) P
M
--ACf
HP(X, flm)
dzi,
-
-
-
7
AACf dz-1,
dznj
then
f2m)
dim HP(X,
(q)
(n)
=
d-Zq}
-
P
M
Proof
for
all
and
surjective
T
on
coordinates
that
so
we can
Eljl=p
coefficients
constant
forms
are
HI,P(X)
-_
represented
is
previous
and the
2.2.5
tion
with
the
from
lift
restrict
our
and j :5 q
cjdz-j
T to X is
also
injective.
HP(X, S?m)
2.
forms
Their
E J.
j
(0,p)-forms
to
and in toroidal
isomorphism
With Dolbeault
1.
considerations
in toroidal
coordinates
Proposi-
after
E ci,jdzj
by all (m,p)-forms
remarks
A
dz-j
with
III=IJI=p
coefficients
constant
where all
dzi
w
w
)-exact,
is
iff
the wj
MALGRANGE took manifold
with
mology
group
in
only
A
(1:
E J in
j
of
1975
the
sum.
A
KAZAMAdetermined
1984 the
a
complex
cohomology groups
group
of type
by 1).
Lie
Q.E.D.
groups
pseudoconvex complex proved, that its cohonon-Hausdorff topology [67, 84].
quasi-Fr6chet
wild
wi).
get the result
functions
holomorphic
has
toroidal
we
j:(dzi
example of GRAUERTof
an
constant
in
about
=
cohomology
all
0)
result
cjjdz-j) So
for
so
H'(X,
a
and
groups
of all
toroidal
goups
X be
infinite
a
toroidal
dimensional
Proof According
to
(TT)
wild vector
spaces
of Theorem
with
2.2.2(4)
Then for
q.
1 <
non-Hausdorff
a
toroidal
q the
p:!
HP(X, 0)
VN E
N>o 31Y3,N
E
Zn-q\ 101
,rN
E
Z2q
group
:
are
topology. is
iff
wild,
in
toroidal
coordinates
(W)
[53].
is the
Proposition
2.2.7 Let
all
I.
are
Dolbeault
Their
q for
j :5
decomposition
Wehave the
ItRU3,N
-
77N
I
<
N-10'3,N
Cohomology
2.2
glueing
with
t0'
(t
:::--:
had
01
i
that
seen
R. Denote
matrix
it0 2
tG`3) E 577(o)
(-)
:
27ri
in the
dI &7N) so
for
that
30
I
IC(aN) 12 Take
1 f J,(0' N)
Assertion.
Co
( )
f J10,
and
we
wild
is easy to
explain
that
groups.
get by (W) 10'3,Nl
< N-
(-7N)
0 for
=
but
k E R >0
cD is in the
the
I
ka3,N
d
>
the
other
I fj
7q
-
Then
a.
I NI
k(OrN)
0'3,N
Ik
173,N.
ENf J(O'N )kO'3,N
cD
proof of
Eq(O*N) divergent.
so
closure
that
D -exact.
is not
=
D7 with
HO,P(X) -y
the
proposition
is
(O,p)-forms
proved, but not
be Hausdorff.
cannot
Eljl=p-l
=
Gjd-zj
of these
By uniqueness
2.2.5.
EN77 (6N )
divergent
Wefollow
proved
This
of the space of the a-exact
O'q (OrN)
the
proof
of
but
=
Proposition
and
g(-)e(E('))
E,
=
we
as
if
we
q(O'N)
take
as
above.
and get
=
J)P-ld(ON
27ri(-
)
9
A (-YN)
with
,0(-YN)
f J,( N)d-Zj
f
fd(ON)
=
9
Eg(ON)d-ZJ. 1
ljl=p Wewant to compare the coefficients
proof
of
Proposition
of d-zj
=
P
k=1
d-zj
A
...
A d-zp
only and get by
2.2.5
)D (ON)j (d-zk) E f J,( 'N) C(ON I
P
=
in
get
2),y(6N) EN'Y (0-N),
2.2.5
Gi
decompositions
convergent
)) A&N) D(6N)j (d(ON f
of the
is
assertion.
Assume that
with
It
convergent.
I C(ON)
V/ q
EN7)77 (ON)
space itself
Proof of the
>
n-q
every
:=
because then
this
is
is the
Decisive
in
We
2.2.5.
D(')j (?9('))
toroidal
fj
ko'3,lv
N-JaN1 for
convergent
I
N)
fJ
Proposition
jo
certain
a
&7N)
(E('))
e
=
that
so
of TN and
with
be
can
proof
of
55
groups
071,N, 172,N E Zq the components remember the
especially E Z)(ON) in the case of divergent same way as in that proof
is convergent
Indeed,
us
.
77
n(ffN)
with
Zn+q Let
E
of toroidal
27ri
E(_j)k-1g ON)C(6N) Jk k=1
k
56
Bundles
Line
2.
where the
side
right
of the
sum
Cohomology
and
is taken
jk
all
over
k,
(1,
=
p).
Then
P
fj(")
j:(_l)k-1g ON)C(UN)
21ri
--
k
J"
k=1
g AN )k 0'3,N
Because ery
k E
seen
R -01.
But this
in Theorem 2.2.6
must be
for
convergent
sequence
(o-N)
ev
have
as we
wild
of toroidal
for the characterization groups
toroidal
groups
finite of cohomologically groups of cohomologi-
complex
Lie
groups
define:
we
Definition X be
theta
X is
a
Lie
wild
theta
For Lie
group,
and wild
X is
a
a
Stein
following
X is
toroidal
3.
X is
Lie
complex iff
theta
the maximal
but
For the
proof
subgroup X0.
toroidal
group,
group.
subgroup
[571:
of type
Then:
1990 the
in
following
wild
toroidal
group,
iff
a
toroidal
(p
>
p:! ,
0).
q:
have theta
topology. subgroup X0 :A
iff
--
all
HO(XlXo, 0) HP(X,
topology. we
0
HP(X, 0)
all
not
=
1 <
q.
a
positive
group,
iff
finite all
dimension.
HP(X, 0)
are
Hausdorff
HP(X, 0) Lie
group
p be with
with
With
a
wild
toroidal
Lie
group,
group
dimensional
X is
theta
toroidal
a
HP(X, 0)
all let
cases
theta
infinite
Hausdorff
toroidal
KAZAMAand UMENO got
maximal
group,
2.
a
X0 is
maximal
with
group a
(KAZAMA-UMENO)
connected
In the a
iff
Lie
Xo is
groups
their
Theorem X be
iff
group,
by studying
2.2.9
complex
connected
a
Lie
4.
N
special
the
them toroidal
toroidal
the
general
of
case
a
1.
ko'3
-
type.
X is
Let
..........
also sufficient
are
So KAZAMAcalls
infinite
2.2.8
result
IC(N)l
Q.E.D.
groups.
For the
Let
('N)
f -1
for
possible
is not
type and corresponding
cally
E
convergent,
above.
The results theta
is
quote the original
paper.
0)
are
X of X 0
HP(Xo, 0).
infinite
dimensional
with
non-
Varieties
Quasi-Abelian
3.
Quasi-Abelian used in
varieties
the
1964
relations
period
quasi-Abelian
CATANEsEadded in
1991 the
factors
KOPFERMANN
establish
to
the
of toroidal
groups.
in 1987-
theory
is
CAPOCASAand
meromorphic
non-degenerate
a
in
of his
The consequence
reduction of
existence
ABE characterized
forms.
standpoints.
different
meromorphic
the Main Theorem and the
function.
Ample Riemann forms
3.1
phic
determine
factor
Ample Riemann
relations.
period
characterized
in
a
lirst
The Hermitian
of A. WEIL Theorem
3.1.1
(The
of
factors
of theta
an
by
the
can
be
by KoPFERMANN
introduced
was
factor
automorphic
of
decomposition
Hermitian
automor-
which
varieties
theta of Z'-periodic decomposition used for this generalization. [64, 119]
were
an
be described
theorem.
the fibration
decomposition
properties
can
quasi-Abelian
deflne
forms
decomposition
the characteristic
on
erations
by
step
The Hermitian
basing
Hermitian
a
whose
form
of
decomposition
Appel]-Humbert
and the
decomposition
The Hermitian
an
factors.
Consid-
fac-
automorphic
(KoPFERMANN)
tor) Let
A
(D F C C'
Z'
=
automorphic a,\
using ample Riemann
from
bundles.
line
positive
automorphic
of
GHERARDELLI and ANDREOTTI contributed
groups.
theorem varietes
with
groups
decomposition
toroidal
for
the fibration
1971-73 89
toroidal
are
Hermitian
(Z)
=
with
the
1.
H is
a,\
2i
[(H
P(A)e following a
bilinear S is
factor
a
be
a
E
A)
(A +
rank
of
lattice
n
has the Hermitian
S) (z, A)+
1 2
(H
+
S) (A,
Then every
+ q.
A)]
Zn-periodic
decomposition +
sA(z)
+
h(A))
(A
E
properties:
Hermitian
form
and alternating
symmetric
on
Cn such that
A: A
C-bilinear
x
form
A
-+
on
C, Cn.
Y.Abe, K. Kopfermann: LNM 1759, pp. 57 - 92, 2001 © Springer-Verlag Berlin Heidelberg 2001
Im HI A x A
=
A is the characteristic
A)
58
Quasi-Abelian
3.
2.
(A
s,\
A)
E
Varieties
the
is
wild
Z'-periodic
automorphic
summand with
vanishing
0-coefficients. The map
3.
A
o:
,q(A and h
C'
:
L(z, A) uniquely
only
The wild
factor
are
The
3.
factors
form
characteristic
by
s,\(z)
1
semi-character
A))
A(A,
2
the
A, hence
of
(A, A'
A),
E
on
(Xx, z)
(z
E
C)
homomorphism (X.\, z) which depends given automorphic factor a,\ (A E A).
RA-
uniquely
is
of h is
the
with
same
cobordant
with
y(A)e(h(A)) a,\ (A E A).
is
strictly
product
decomposition linear
(
e
a
by the given automorphic
determined
A).
E
For cobordant mands
is
S) (z, A)
+
determined
summand
(A
a,\
(H
defined
bundle
uniquely
The ImH is 2.
1}
form.
2i
determined
the line
on
o(A)p(Al)
=
C-linear
a
=
form
The C-linear
is the
IzI
and cobordism.
Uniqueness 1.
jz:
:=
A')
+
C
--*
S1
--+
unique only
cobordism
uniquely
determined
decomposition
The
the theorem
in
as
Zn -periodic
the
on
properties
into
the maximal
C-linear
form
Im
the wild
sum-
functions. constant
part
semi-character
a
subspace
in
the
Q and
MCAOf
a
RA-
Proof
i)
Definition
Let
L (z,
A)
of the Hermitian :
=
(Xx, z) (Z
L defined
bundle
composition
by
Cn )
(=-
L(z, A) (A
Then every
2.1.3.
L(z, v)
HIAXA
factor
E
(v
A) E
linear as
has
A.
:--
characteristic
automorphic
given
the
H with
be the
a
the
in
unique
form
of the
characteristic R-linear
line de-
extension
RA)
and then
A(u, v) is the istic
unique bilinear
A: A
nating
R-bilinear
for
x a
A
L(v, u)
--->
iff
R-bilinear
v
E
RA)
extension
of the character-
form R and
R-bilinear
a
real-valued
and alter-
A the condition
R(u, v) holds,
(u,
RA x RA.
Z to
real-valued
form
L(u, v)
-
alternating
and
real-valued
Remember that
=
=
A(iu, v)
=
-A(u,
iv)
H := R + iA is Hermitian.
ImL(u, v) is symmetric because L(u, v) is C-linear
But
MCA. Hence, if
we
define
because in
u.
R(u, v)
A(u, v)
ImL(iu, v) =ReL(u, v) 0 for all u, v E A(iu, v) + A(u, iv) A(iu, v) (u E MCA) V E RA))
Therefore :=
is
real
and
=
H(u, v) is Hermitian
To extend
valued and
(RA
RA
MCA(D V with
=
MCA) and by Rv
X
H(u, v) fixing
has after
any
of the
S(z, v) C-linear
in
symmetric
and
z
extension
v)
on
of S to Cn
X
A
subspace.
this
on
any real
V and take
the R-valued
given
(MCA X RA) U
R on
(u,
extension
C'
to
X
Cn.
factor.
(z
H(z, v) RA
RA)
E
v
Cn,
E
RA)
there
RA Therefore
x
E
v
-
exists
a
unique
Cn Then .
(Xx, z)
1
(H
=
2i
S) (z, A)
+
decom-
form of the characeristic
linear
2.1.3.
position
2) 3)
RA)
V. Then define
x
iA(u, v)
characteristic
determined
uniquely
-
symmetric
L(z, A) is the
+
automorphic
2iL(z,
:=
G
59
V. Then
x
R(u, v)
:=
=
R-vectorspace
an
RV a unique Hermitian
The decomposition ii) 1) Obviously
is
V
on
ImH
with
one
R on RA x RA by
extension
R-bilinear
MCA, V
E
RV on V
form
R-bilinear
symmetric
symmetric
(U
iA(u, v)
+
MCAX MCA, the unique
on
H put
and
R(u, v)
:=
Riemann forms
Ample
3.1
(A
sx
E
A)
The real
is the
summand of the characeristic
wild
d
homomorphism
:
A
--+
R has
decomposition
2.1.3.
R-linear
extension
unique
a
d: RA --> R. Now
h(u) is the
unique
C-linear
be any R-linear
form
fixing
r
a
d(iu)
on
+
(u
id(u)
MCAwith
Imh
=
RA)
E
d. Moreover
let
r
:
RA
R
---
of Reh from MCAto RA. Then
extension
h(u) has after
:=
unique
:=
r(u)
C-linear
+
id(u)
extension
(u to
RA)
E
Cn.
Finally
p(A) with
iii) Of
c,\
as
in the
The rest course
used in
for
characteristic
:=
e(c,\
-
r(A))
decomposition
(A
E
2.1.3
is
A) a
semi-character
lattices
A.
Q.E.D.
is clear.
P-reduced
for
A
form.
=
Zn ED F C Cn the
Hermitian
decomposition
can
be
60
Quasi-Abelian
3.
Now it
is
classical
torus
Let
X
Then every
(Appell-Humbert be
line
Abelian
an
bundle
L
decomposition
Lie X
well
known
in
decomposition) be
can
given by 1
A)
A
with
group
[H(z,
2i
H, wild
Hermitian
on
I
:=,Q(A)e( with
decomposition
theory:
C'/A
=
following
the
get
to
easy
very
Corollary
3.1.2
Varieties
+
Z'
an
A)]
H(A,
2
=
+
ED F.
factor
automorphic
sx(z))
(A
summand sx and semi-character
in
o as
G
A)
the Hermitian
3.1.1.
Proof Define
the
polynomial
quadratic
1
q(z) q(z
Then
dant
to
+
a,\
q(z) E A).
A) (A
-
4i
:=
[S(z, A)
S(z, z)
4i +
1
S(A, A)]
2
factor automorphic characteristic homomorphism the
The lost
of the
part
h(z).
+
h(A)
Ox becomes cobor-
that
so
Q.E.D.
The APPELL-HUMBERTdecomposition cause
+
is
be used in
cannot
Z"-periodic.
nomore
form,
be-
about
the
F-reduced
Informations
go lost.
factor
given automorphic
in the
Appell-Humbert
decompo-
sition e
obviously
is
2i
cobordant
called
is therefore
and
a
wild
X
with the
+
theta the
is
A)]
S(A,
defines
1. It
factor
2
+
h(A))
(A
analytically
the
A)
c
bundle
trivial
X
x
C and
its
type,
factor. product
of
a
theta
factor,
defined
by
factor:
Definition
3.1.3
Let
to
A)
trivial
a
Every automorphic
[S(z,
=
Cn /A be
a
Chern class
A theta
cl
factor
,O,x (z)
=
9(A)e
with
group
alternating
characteristic
the
A of rank
1.
H is o a
a
form
A
:
A
x
A
---+
bundle
on
Z defined
(L).
of
type
(H,
1
( [(H 2i
+
p,
S, h)
S) (z, A)
for
+
A is
-
2
(H
where
2.
+ q and L be theta
n
and bilinear
Hermitian
semi-character
with for
ImHjAxA A,
=
A,
an
+
automorphic
S) (A, A)
+
factor
h(A))
(A
E
A)
X
by
I
S is
4.
h
symmetric
a
a
A reduced
C'.
on
factor
theta
is
(H,
(H,
of
1
of of
exponential
Together we
get
Theorem line
2.1.6
on
by
reduced
(H, p)
type factor
The wild
toroidal
a
Lq and
bundle
theta
(A
Werestrict
t,\
Lo by
(H,
line
q,
factor
wild
an
=
trivial
of type
factor
and can
X
topologically
a
theta
a
Cn be
factor
given by morphism
the
Of VOGT
t,\
=
X
:
by
--+
2.1.3, :
=
M(r, Z)
characteristic of
values
(XA,) Ak) (PRC)
A
P
A E
-
a
the
(XAk Aj) i
Cn
if
bundles
r
can we
(A,,
-
-
,
restrict
A,)
of
of the
alternating
A
on
=:
tXP
A(Aj, Ak) -
YX
=
relations
so
A
0
Lo
by
(A
a
E
factor
theta
of
A).
-
by
represented
be
can
=
A
on
values
to
a
theta
this
according in
A)
E
factors.
be
reduced
a
Chern class
=
-
-
-
,
by P
=
X
:=
(A,,
cl(L) homo-
characeristic
Now we
situation.
1,
A(Aj, Ak) (j,
and on
the
to
special
Aj (j
entries
basis
(A
i9,\
theta
A. The characteristic
lattices
integral
on
by reduced
Cn /A with
X
homomorphism
characteristic
Lq
=
Lo.
MCA
defined
R-independent
the matrix
(Period
L
alternating have complex
don't
-
e(sx) On
L
product
well
as
+ q and
n
:=
bundle
line
form
characteristic
the
Z-basis
the
defines
the
relations
to line
of rank
lattice
a
which
decomposition
of
A).
E
bundle
S, h)
be assumed to be constant
considerations
our
A c
note
instead
Theorem 2.1.10
bundles) C/A is
of line
group
Period
a
(A
summand s,\
of the N6ron-Severi group NS(X) which is defined by a reduced theta factor.
bundle
fix
(H,,Q)
A)
E
Every element
theta
A)
E
write
we
and the Decomposition
(Decomposition
bundle
LV is defined
Let
(A
simplicity
For
e(s),(z))
=
automorphic
an
Proposition
with
3.1.4
a
before.
A)])
H(A,
the
Every of
2
factor
tx,(z) is the
+
0, 0).
q,
A wild
as
p
1
A)
2i
0, 0) with H and
p,
factor
automorphic
an
( [H(z,
t9,\ (z) =,Q(A)e of type
61
form and
C-bilinear
form
C-linear
Riemann forms
Ample
3.1
r).
Let k
(XXl
=
1, *,
7
A,).
us
de-
-, r) X,\,)
Then
that for for
a
the
X E
M(n,
r;
characteristic
Q.
homomorphism)
62
Quasi-Abelian
3.
The solutions
fixed
a
on
as we
have
the
S) (z, A) fore
analytically proof of Corollary
the
A
factor
theta
with denote
we
respectively.
form
Hermitian
a
their
entries
Then X
2i
ImtPHP
(PRH)
SP]
+
A
=
basis
the
on
[H75
-L
=
without
course
analytically
trivial
for
we
q <
n as
the r-reduced
take
a
bundle have
the
A
we
symmetric X
M(n, (X.\, z) G
X.1-
bilinear
Cn
by
on
X
for
a
the
form
same
q;
Z). -L
=
(H
2i
+
As be-
S.
symbols H, S,
by (PRC) matrix
H
for
the
relations
As before
Hermitian
x
in the chararac-
seen
proved
we
C. The Hermitian
x
in the
seen
by SP C
homornorphism
with
Hermitian
part.
X
the
part
Hermitian
H)
-LSP defines 2i
the
uniquely theorem,
H is not
decomposition
determined in
not
even
case.
We can write
If
the
(0, Xv)
of the
a
X and
defined
bundles
have
symmetric
(Period of
as we
3. 1.1
that
so
with
=
a
solution
3.1.2.
determined
H and
line
trivial
characteristic
Zn E) p the
=
uniquely
is
special
a
homomorphisms
the
decomposition theorem, namely X theorem Hermitian decomposition
teristic
In
after
coordinates
r-reduced
The characteristic
P of A define
seen
X + SP with
exactly
are
M(n, C).
S E
basis
standard
In
equality
of this
symmetric
any
Varieties
('A2
A,
=
alternating
characteristic
-A2 A3
standard
basis
) P
A,
with
(In, G)
=
A G
c
M(n
+ q,
M(n; Z),
of A =Zn
A3
ED.V,
Z) E
in
block
form
M(q; Z).
then
we
get readily
from
(PRH)
'GAjG +'GA2 -A2G
(PRS)
(Period From here
we can
F-reduced
theta
(XI,,
i
XG)
So most of the 3.1.5
with
following
Theorem
existence
of
to
Then
For
that
we can
take
Xj-n
(Period relations
period a
line
theorem
bundle
the Chern class
cl
L
A3
relations
(PRC).
get (PRC) by using
we
Each of the
go back
factor.
+
=
standard
in
we =
0-
remark
01
XG
:=
that
A2
coordinates) A,
=
a
X
(PRS) only. is
proved:
relations) (PRC), on
0 for
and with
(PRH) and (PRS) is equivalent X =Cn /A defined by a reduced theta
(L) given by
the
integral
and
alternating
with
factor
matrix
A.
the and
Ample Riemann forms
3.1
Proof Subsequently, Then obviously
by (PRC) z) :=,q(A)e((X.\,
i9,\(z)
defining
factors
theta
are
To construct
a
only
have to construct
we
or
+
(PRH) (X,\, A))
such
semi-character
a
line
it
is
or
enough
I
(A
e(c,\)
Q(A)
clearly
Then
[H(z, A)
2i
A)
real
to construct
A')
2
E
p(A)e(
:==
A for
on
0
+
given A.
a
H(A, A)])
bundle.
-A(,\,
C'N + Cv +
C,\+,\,
semi-character
a
63
(A, A'
mod Z
becomes
a
(A
c,\
E
A)
with
A).
E
desired
of the
semi-character
type. let
For the construction, co
0 and any collection
=
A, be
A,,
(j
E R
c,\,
a =
r).
1,
with
of A. Start
Z-basis
R-independent
Then define
cM1A1+--+M'A' + mc,\,
+
mic,\1
+
mlm2 2
+MlMrA(A,,,\,) 2
A(Al, A2)+
2""'
+M' and
-2' A(Aj, Ak)
use
2
A(Ak) Aj)
j, k :5 r)
(1
mod Z
A (Arto
Ar)
1,
required Q.E.D.
the
see
congruence.
Now
calculate
we can
toroidal
bundle
on a
by
P-reduced
P
the =
X E
(In, G) M(n, q; Z)
X
group
A
with
=
with
coordinates
of A in standard
a
of the r-reduced
the values
toroidal
a
X. The theta
homomorphism
characteristic
as
Cn/A
=
of
group
homomorphism
characteristic
the
using
N6ron-Severi
the
Zn ED _V is X
r
:
group
which
factor
So
G of r.
basis
characteristic
a
q
line
determined
uniquely
Z'.
-+
of type
defines
we
take
a
basis
define
Finally homomorphism we
on
G. Weget
NS(X) for
toroidal
groups
For
Example: identify
q
-
of type =
NS(X) for
Non-compact dimensional
GL(q, R)
E
M(n,
condition
_
Zn
for
groups
T.
We write
X
=
:
txG
(XT) 0
in
groups
are
-
tGX
E
M(q, Z)j
is
empty
bracket
NS(X)
XT E
in
with
C*n-q-fibre standard
M(q; Z)
so
X E
that
we can
M(n, 1; Z):
1,
Z.
-
so
all
X of type
groups
natural
G
with
the
toroidal
all
torus
toroidal
and take
Z)
homomorphisms
characteristic
1-dimensional
torus
q;
q.
1 the
the r-reduced
especially
IX
that
bundles
coordinates
over
with
a
Imi
qE
64
Quasi-Abelian
3.
NS(T)
IXT
s-
further
of all
calculations
line
the
compact
those
case
IMHIAXA
2.
H is
form
are
exactly
Abelian
later
of
the
Hermitian
is well
the
a
general
of
concept
p 101.
on
subgroup
discrete
toroidal
variety
A C C'
of complex
rank
subspace MCAOf PI-A-
C-linear X
group
does not
represented
are
8].
The basis
H to the
Cn /A with
=
depend
on
by lattices
an
ample
Rie-
the representation
CA with
P of A transforms
Hermitian
Hc
matrix
ample Riemann forms of toroidal
elementary
form
=
GL(n; C)
C E
PC
to
CP and
=
1C_1HV_1 belonging
substitute
whole
Cn
the
groups.
imaginary
So, if the Hermitian nonsingular. complex subspace MCAof a toroidal
maximal
easily
geometry
that
is
Im H has the
rank any
at
least
ample
part form
of
a
H is
group
positive positive
of type
q,
2q.
Riemann form
by
one
that
is
positive
definite
[33, 87]:
Lemma
Let A C Cn be
for
a
imaginary Proof
A. Then there
As in the so
part
RA
exists
H + H is
on
proof of that
subgroup
discrete
RA, such that
nates
The
varieties.
4.1
lattice.
about
the
on =
Wecan
form
In
A C Cn:
p
on
matrix
Hermitian
A
a
the maximal is
groups
known from
definite
on
lattice
relations
speak
definite
on
a
the transformed
We can
H for
quasi-Abelian
a
toroidal
[see
3.1.7
an
H on Cn such that
variety
Cn /A with
=
Hurwitz
on
Section
in
with
A.
for
Isomorphic
then
ample Riemann form.
Riemann forms
groups
definite
positive
The definition of X
It
the
Z-valued,
is
quasi-Abelian
mann
to
form
Hermitian
a
1-
A
see
groups
Definition
3.1.6
is
of toroidal
group
toroidal
An ample Riemann form n
C NS(X)
bundles.
are
bring
we
M(q, Z) I
G
98].
varieties
Riemann forms
ttXT
-
Ngron-Severi
Ample Quasi-Abelian
tXTf
:
lifted
of the
Of SELDER[97,
papers
M(q; Z)
E
subgroup
becomes the For
Varieties
RA
as
a
complex
Hermitian
positive
definite
rank
form on
fl
and H an
n
on
Cn
-
ample Riemann Cn, which is symmetric
of
course
with
the
same
H.
the Hermitian =
of
3.1.1 let us take toroidal coordidecomposition form RV on V x V, MCA(D V. Take a symmetric R-bilinear
by
defined
and
R-valued
an
and
diagonal symmetric
real
a
R has
:=
The determinants
of the
principal
a
unique
Hermitian
extension
minors
of the first
j
a
torus
group.
V
on
V
x
C'.
to
H+
Of
hq+2,q+21
hq+l,q+l,
given toroidal
to
group
a
of
lattice
the
prove
we
ft
to
Lemma
3.1.8
A C C'
Let
degenerate a
For this
be extended
can
Q.E.D.
of the
the lattice
aim is to extend
65
of the matrix
rows
if the entries
for j := q + 1, j := q + 2, positive ft successively are chosen sufficiently big. definite. At least H + ft must be positive
are
The first
RV
entries.
form R on RA x RA by R:= RV
bilinear
This
0 elsewhere.
positive
with
matrix
Riemann forms
Ample
3.1
be
form
A'
so
that
exists
a
C'
on
Then there
subset.
on
complex rank
of
lattice
Hermitian
countable
valued
a
rank
and real
n
ImH is Z-valued
on
U F}
\ IRA
E C'
A0
<
r
2n, H
non-
a
A x A and F C Cn so
ImH is Z-
that
A e ZAO.
Proof Let to
P
basis
a
A,)
(A,,
=
P
=
be
(Al)
Consider
/\2n)
7
the R-linear
used the
we
Obviously
it
Assertion.
there
is
once
to
show the
an
x0 E R
any
a) -ro
=
E Zr
*
'
*
i
so
Im'POHP0
then
because the
1
b)
reason
the
Abelian
r.
this
In
take
we can
L(x)
any
=
Rr is
xo
dim ker
integral
Ax
:=
(x
E
R7)
because
surjective
nonsingular. unique
the
To has
2n such E R
because
an
of the Hermi-
part
L(x)
Let --+
U Fl.
becomes
imaginary
H is
Hermitian
case
RA U F. This is possible
that
L
=
2n
solution
for
ro
with
L(xo) -
r
=
> 0 and F is
set.
A <
rank take as
nonsingular
of the =
Then
E Z".
-Pxo V
we can
A
\ IRA
PXO E Cn
:=
proved,
Ar Ao).
matrix.
that
Ao
and
has been
(Al
:=
part
rank
countable
Case
In
2n
Axo
=
the coefficient
H for
exists
A:= ImtPHP^.
where
E R
symbol
the assertion
imaginary
Case
a
2n),
(:
Proof of the assertion. alternating, 2n A :=Im'PHP E M(r, Z). Wenote that L: R
with
A0
D A.
H is
form
tian
the
A
lattice
torus
a
enough
where Po
matrix,
A
same
.L(xo) Indeed,
of
E Cn
A2n
Ar+1
We add
< 2n.
r
map
.Q: ):= Here
of A with
basis
a
a
solution
r.
Let xo
ro
of
E Z'
be any
L(xo)
To
=
integer
and A0
above.
next
step
varieties.
we
extend
the
lattice
of
a
image of L. Then Pxo V RA U F by the same Q.E.D.
not =
quasi-Abelian
in
the
variety
to
lattices
of
66
Quasi-Abelian
3.
3.1.9
Proposition
Let X
=
C'1A
form H for
A,
for
quasi-Abelian
a
A, which
Al, A2 with
groups
well
be
A,
there
For that
take
By
we can
A, and induction
period periods
basis
is
exist
theorems
nearly
A toroidal
Abelian
results
an
lattices
of torus
ample Riemann form
as
=
R-span
Ar2
A,,,
of its
lattice
we
a
take
and not
in
care
the
in the
1971-73
years
a
proofs.
The first
quasi-Abelian
theorem
iff
variety,
about
seminar
the
"Variet6
published
and
of their
it
that
new
Q-span of the Q.E.D.
one.
(CHERARDELLi-ANDREOTTI [32, is
group
=
of ANDREOTTI and himself
without
Theorem
3.1.10
H becomes
of the lattices
r
in the
not
CHERARDELLi arranged with
that
=
of the the other
Abeliene"
so
ample Riemann
an
are
lattices
the rank
over
chosen
A2
A
=
with
n
Cn. Then there
on
Al, A2 of rank n+q+ I such that A, nA2 A. A ED ZA, with A, A any A, RA and A2 ZA2 with the Q-span of A,.
in
not
n
of rank q <
variety definite
positive
A2-
for
as
is
Proof By Lemma3.1.8 A2
Varieties
results
in
Quasi1974 two
is the
33]) the
is
covering
of
group
an
variety.
Proof of an Abelian quasi-Abelian variety and A be a lattice A (Proposition that the so 3.1.9) imaginary part of the ample Riemann form H for A, which is positive definite remains on C', Then the identity on A x A. E : Cn --+ Cn with E(A) c A induces integral a covering homomorphism t: Cn/A -- C'/A by Hurwitz relations (p 8). the lift E of the covering --<. By Hurwitz relations map t is a linear map with E(A) c A and an ample Riemann form for A is an ample Riemarm form for A. Let X
-.
variety
T
C'/A
be
C'1A
with
=
=
a
A c
Q.E.D. Quasi-Abelian Abelian
varieties
Proposition
of
is the theorem
3.1.9
between
[43],
of HEFEZ in 1978
POTHERING [87]
obtained
of
quasi-Abelian
ANDREOTTI. Also
by
conjectured
the covering hierarchy holomorphic Fourier series.
in
and the space C*n of
varieties
A consequence was
situated
are
in
the
which
1977 this
theorem. 3.1.11
The
Theorem
meromorphic
X has infinite
(HEFEZ) function
transcendental
M(X)
field
degree
a
non-compact
variety
C.
over
Proof Let
X
tice
of
=
an
C'/A
be
Abelian
Riemann form
a
quasi-Abelian
non-compact
variety
H. Moreover
according let
P
=
Proposition
(A,,
-
-
-,
A,)
variety 3.1.9 be
a
with
and a
standard
A
D
A
common
basis
a
lat-
ample
of A and
Ample
3.1
a
standard
period
new
A. Transforming
of
basis
h B
Cn
:
=:
(bl,
-
-
b2n)
-
,
1, do, di, d2
h, ho, hi
7
we
get the
*
'
construction
i
h(z
+
with
Aj)
=
with
known.
sequences
by
equations
the
z))h(z)
e((bj,
functions
2n)
1,
=
complex numbers and
defined
functions
well
Jacobian
(j
h(z)
cj)
+
C2n E C is
independent
of Jacobian
h2
7
start
of
existence
z)
e((bj,
C1 we
Q-1inearly
of
*
Aj)
+
-HP_.
B:=
the
functions
any constants
following
For the
by h(z
C defined
--
A with
=
of theta
theory
classical
the
-77
with
simultaneously
both
67
relations
ImtRH-'B In
Riemann forms
(j
=
1,
2n)
(j
=
1,
2n
and
=e((bj,z))hk(Z) hk(Z+Aj) hk (z + A2n) e((b2n) Z) + dk) hk (Z) =
additional
have the
The
Proof of
the
P(x) P(fO,fll..-,f.)
Let
P(fo(z
+
(k
fk
E
+
1A2n)
E
N)
are
(k
e(ldk)fk(Z)
=
E
Ejjzj<degaAx/' =0.
Put
C[x] be d:=(do,dj,.--,dn). E
=
polynomial
a
with
P
the t-th
P(fo,
--fm"'
f,,,)
fl,
fixed
N)
(1,
fit
--->
E
Nm+1 a,,:A
01,
z
=
a,
E Cn
(')'
f0
we
(z)
have
a
... 1
f
M
(Z)
(t
=
1,
-
-
-,
N).
system of equations
N
(-rt, d)) St (z)
=-
0
(1
=
0, 1,
-
-
-,
N
-
1)
t=1
constant
V:=
0
coefficient
e((,ri, e(2(,rl, e((N
-
matrix
d)) d))
e((-r2, d)) (2 (r2, d))
e
1) (rj, d)) e((N
-
1) (T2, d))
e((-rN, d)) e(2(-rN, d))
...
...
...
e
((N
-
(z)
nonvanishing
with
summand becomes
E e(I a
:A
and
Then
Falptl
N summands of
the
St (z)
with
which
independent.
algebraically
1A2n)i-Jm(Z+1,X2n))
-r:
For every
N),
N).
by
efficients
that
hk1h (k
:=
assertion =
We can enumerate
so
fk
property
fk(Z Assertion.
1)
-
functions
meromorphic
A-periodic
the
Then define
-
1) (TN, d))
=_
0.
co-
Quasi-Abelian
68
3.
But all
the St cannot
dermonde's
Varieties
det V
everywhere
in the
C'
d)))
=
N(N-1)
(e((Tj,
11
2
d))
e((Tk,
-
1<j
At least
jo
factor
one
has to vanish
ko. The 1, do, di,
<
Maximal In the last form
-
-
section
d,,
-
,
Stein
we
become
that
seen
(7j,),
that
so
d)
Q-1inearly
subgroups
have
rank
has at least
so
that
Van-
has to vanish
(_J)
=
simultaneously
vanish
determinant
imaginary
2q. The following
mod Z for
dependent.
varieties
of
part
definition
certain
Q.E.D.
quasi-Abelian
of the
(7k, d)
=_
0.
ample Riemann
an
is due to GHERARDELLiand
ANDREOTTI [33]. Definition
3.1.12
C'/A be a quasi-Abelian variety ample Riemann form H for A is said
Let
X
An
2q
+ 2f
For
a
lattice
fixed
A C Cn
Frobenius
to
the Frobenius
integers
For the
ImH has the
that
A
rank
basis
so
ImH becomes
=
dk
-,
-
_O
of this
basis
A C-linear
map -
di I d2l
that
so
a
unimodular
matrix
A becomes unimodular-
D=diag(dl,---,dk)-
with
0 0
the
classical
(p 8)
Idk
exist
form
D 0 0
are
2k there ...
DO
0
alternating
special
Her(Cn)
-
relations
Hurwitz
3.1.14
dj,
proof
characteristic a
special
a
of rank
normal
f
WAM
on
f, iff
form:
A C M(r, Z) alternating and GL(r, Z) positive integers
equivalent
By
take
we can
normal
any
The
q.
be of kind
Lemma (FROBENIUS)
3.1.13
ME
to
RA-
on
the wellknown
For
of type
=
elementary lemma
we can
form
A
of A.
GR6BNER[40]
see
that
assume =
divisors
ImH is
or
A:= Irn'PH P
the matrix
given
in
LANG [651.
Frobenius
its
of the
normal
form
P of A.
:
Cn
_-,
of the Hermitian
Cn' forms
induces
a
contravariant
by T* (H')(x,
y)
:=
map -r*
H(T(x),
T(y))
:
Her(Cn') (x,
y E
Cn)
Definition
Let X be A surjective
a
toroidal
group
and X'
a
complex homomorphism
quasi-Abelian r
:
X
--*
variety. X'
is
a
homomorphism
pre-
serving
form H' of
ample Riemann
an
preimage
its
(H)
-r*
is
69
ample
an
of X.
Riemann form
of
The kernel
X, if
forms
AmPle lUemann
3.1
bilinear
a
Iv
KerH:=
form
Hermitian
or
H(v,w)
E V:
0 for
=
Proposition complex homomorphism -r X' the following statements Abelian variety
space V is defined
vector
on a
all
E
w
as
V1.
3.1.15
For
a
1.
-r
is
r
is
2.
on
homomorphism preserving
a
quasi-
a
equivalent:
are
form.
ample Riemann forms,
all
X onto
group
ample Riemann
one
homomorphism preserving
a
toroidal
a
which
positive
are
Cn.
of
The kernel
3.
of
surjective
is
r
connected
a
subgroup.
Stein
Proof X
Let
Cn/A
=
Let the
1 >- 3.
preimage
KerH
(E
(H')
-r*
of
Let
Cn ,
H its
H'
preimage group
=KerH
covering
(E
x
Lie
of ker-r
space
RA).
n
course
Stein
on
MCAso that
(Lemma 1.1.6)
ker-F
kernel
by Stein
dimension that
so
H is
Stein
a
criterion
group
positive
which
Example:
according
previous
the
generates
=
EI(E
-_
0.
By
n
definite
RA has
En
1.1.6 =
proposition,
definite
of X with
a
groups
toroidal
group
generated
A)
no
0. E is the kernel
on
as an
complex
of the lift
Q.E.D.
MCA-
example
on
universal
Riemann form
variety is
a
under
connected
of ABE [61 shows.
The basis
P=
torus
an
C-linear
F
positive
is
subgroup
MCA
En
that
A',
for
form
ker-r
the
group
A' be
Then E C
C X.
an ample preimage of a homomorphism preserving a quasi-Abelian However, the image of a quasi-Abelian variety. even if the kernel homomorphism need not be quasi-Abelian,
Of
variety.
H' for
maximal
F be the
Let
H> 0
groups
ample Riemann
so
quasi-Abelian
a
Riemann form
group.
and the
E. Then
subspace of positive therefore
Stein
be any
Cn'/A'
=:
ample
an
universal
RA)
n
Abelian
for
connected
a
3>-2.
a
on
criterion
group
covering
=
of E n RA. Then F C MCA. But
subspace must be
:
A.
for
H= 0
that
so
and X'
group
H
E C Cn be the
let
Moreover
is
toroidal
a
Riemann form
ample
Stein
be
by
(
100
0
010
i
0 0 1
v/'2-
X, which allows
5i
0
+
v/3-i
two
V7_ projections
to the
2-dimensional
70
1
P1
Quasi-Abelian
Varieties
(1005i)
and
=
0 1 i
The torus
generated
Therefore
period
the
isomorphic
One main result
[33].
together
with
called
other
Fibration
3.1.16
1.
X is
2.
X has
a
toroidal
quasi-Abelian a
is
ABE
of type
with
variety Stein
Abelian
an
not
of two C*.
to
algebraic
of the
second
a
proof
in
1980
[6].
(GHERARDELLi-ANDREOTTI)
group
closed
maximal
XIN.
and
a
is
kernel
DODSON brought
Theorem. Wefollow
Theorem
X=Cn/A be
if the
even
-
isomorphic
by P2
generated
torus
product
of GHERARDELM and ANDREOTTIwas Theorem
[27].
results
projection
is
C*.
a
Fibration
it
respectively.
because it is the
fulfilled,
not
to
of the seminar
They
the
5i
v17-
0-i
+
of the first
But are
v/'2-
algebraic,
proective
relations
is also
projection
is
0
0 1
The kernel
one.
quasi-Abelian.
X is
because
Let
by P,
of dimension
groups
2 of
0
10
P2
q. Then
equivalent:
are
ample Riemann form for A of kind t. N -_ 0 x C*I with 2f + m n q
an
subgroup
=
of dimension
variety
-
q + t.
Proof 2 an
1. 3.1.15 Proposition ample Riemann form.
1 >- 2.
H be
Let
1"step.
Using
A such that
because
definite
positive
Frobenius'
with
m
n
q
-
If
m,
> 0
we can
period
An+q+l
lattice
A,
with
0 Im
tp, HP,
0
-D,
nonsingular
remains a
basis
P,,
=
(Al
and *
*
*
7
becomes
2nd step.
}mmws
same
the
OD1 0
0
0
0
)
of
-D,
as
basis
},rn-1
choose
basis
a
P
=(All***
7An+q)
with
D=diag(d1,---,dq+t).
in the
proof of Lemma3.1.8
PI
a
lattice
D,,, 0
=
(A,;
-
7
where DI
rw.,
Continuing
integral.
A2n-V)
way
new
0
lm'P.HP.
(Lemma 3.1.7).
can
Of
0
in the
that
so
homorphism preserving
a
2f
-
00
0
select
is
OD
-DO
tPH75
projection
C'
on
Lemmawe
0
Im
the
A.,,, with
this so
D..
=
=
further
of the extended
diag(di,
dq+ +1)
procedure that
a
An+q+l)
integral
diag(d,
till
m=
0,
we
get
valued
dn-1))
nonsingular. If f > 0,
we
get in the
same
way
as
in the
proof of
Lemma3.1.8
a new
period
/,tl
so
with
that
of the extended
the
P,,+,
basis
(A,,
=
Riemann forms
Ample
3.1
An-1)
p,
I
1mtP",+JH-P",+
=
1
(_O Ej) tE
with
Then
valued.
the basis
Pm+2
=::
get by the
we
(A,)'*
*
0
i
An-i)
same
lemma
a
dn-1
...
0
...
new
)
period
A2n-21571)
An-1+1)
pi)
0
...
di E2
0
IMtP.,,,+2H!5,,+2
-tE2
0
0
dn*-f
0
...
E2
with
0
0 0
0
...
dn-j+1
of a torus exists there a basis and integral. Finally nonsingular -P := Pm+2t jA2n-2,6'Y1C**i'Yi) -iAt An-i+W** (All* -)An-i7P1)' E:= E,,,+21 and nonsingular extended latticeA= Am+2X- With integral is
=
di 0
Im'PHTP
-tp
BE
where
0
Cn/Ais V step.
As
a
Riemann form
A:=
In this
situation
consequences 1.
U is
2.
CP is
regular =
a
variety consequence P
corresponding
is
a
the
basis
we
get
dn-1
dn
same
of the
Riemann form
Abelian
as
X.
k
variety
CnIA,
=
H
forAand Im
'P- HP
we
define
P
of the classical and with
(tP, W) basis
with
Abelian
an
group
of the
0
0
C-P
that
so
-Y,
of the extended
Am+2
lattice
a
A2n-2t)
E,
with
,
0
I
0
integral
i
A,,,+,
lattice
di
is
An-i+li
71
with
C:=t
(-tk 0) 0
=:
B
(U, V)
ofCAwhich defines part in CP. We get
with
relations
period BU-1
symmetric
integral
and
nonsingular.
U, quadratic PA-1 tP-
=
V. Then there 0 and
are
iPA-1 1P
two
> 0:
Wand Im W> 0. the
same
Abelian
variety
asA.
CP is the
72
3.
Quasi-Abelian
Varieties
di
0
W1'1
dq+l
Wl,q+f
Wq+l,l
CP
dn-f
Wn-1,1
0
Wn-f,q+t
WnJ and ImW positive
Because Wis symmetric
Wn,q+t
...
definite
principal
the
W1'1
Wl,q+,e
Wq+tj
Wq+t,q+t
minor
W*
has the If
we
define
T
the first
according
(D, W*)
:=
of
lattice
the
generates onto
properties.
same
q + I variables
CP with
matrix
shows the
The theorem
D
diag(di,
=
induces kernel
_-
of fibre
importance
above,
as
The
variety.
homomorphism C' (D C*m.
a
N
dq+1)
Abelian
X
Y
-->
then
T
projection
Cn/TZ2q+2f
:=
Q.E.D.
bundles
Stein
with
fibres
and
a
base space.
algebraic
projective
with
q + f-dimensional
an
Definition
3.1.17
A
X complex manifold projective algebraic variety
A consequence
X such that
of the Fibration
X is
iff
it
an
analytic
is
submanifold
a
in
set
of
a
X.
Theorem is the
(ABE [4, 51)
Theorem
3.1.18
quasi-projective,
is
Every quasi-Abelian
variety
is
quasi-projective.
Proof The fibration
fibre
compactify bundle
over
these
X with
an
A consequence with
a
general of
shows that
theorem
bundle
KODAIRA].
Abelian
fibres
to
Abelian
Pj+m variety
algebraic
group
as
quasi-Abelian
the
variety
of KODAIRA's
projective linear
an
so as
with
X variety C'1A is a C' (9 C*m. We can -=
fibres
X becomes
that
a
submanifold
base space and fibres
embedding
base space,
structure
Stein
group
theorem
projective is
projective
is
that
space
in
a
fibre
Pt+m. every
fibres
algebraic
fibre
and
[60,
bundle
projective Theorem 8
Q.E.D.
The
In the
general
the
ample
to
of de Rham cohomology
Hodge decomposition
pings
case
to
Let X
=
C'/A
be
Abelian
an
coordinates
in toroidal
B
(Iq, t)
=:
So the
matrix.
is
of
q.
forms of A
The basis
LR(t)=
parametrization
T and R
torus
a
At
(t
Ret Imt
A:=
0
Ri
E
R2n)
functions
A-periodic
transforms
t'
+ q variables
A-periodic
r-form
W(t)
of t
E
r!
0 0
In-q
:=
can
(Ri, R2)
of the real
which
functions
into
(t', t")
0
0
0
0
be
given
glueing
the
written
toroidal
are
Zn+q -periodic
in
the
ER2n.
be written
can
1 =
=
0
R2 In-q
0
with
of differential
R, R2
In-q
basis
the
(LR)
A
Main
of the
Iq
0
I.
n
map-
proof
with
coordinates
first
Holomorphic
as
P
where
forward
groups.
corresponds
varieties.
of type
group
bundle.
line
theta
bundle
line
positive
straight
average
Lie
a
same a
quasi-Abelian
and the
Cohomology
the
allow
spaces
the
characterizes
Theorem which
by
determined
Riemann forms
complex projective
the
of
only for
works
groups
of the Chern forms
the average
varieties
quasi-Abelian
of
Characterization
3.2
the real
in
(t)dtil
fi'...i,
A
t-parameter ...
A
as
(t
dti,
E
R2n)
I
Zn+q -periodic
A
...
ir
have Fourier
which
coefficients,
M
')'ir
fil
series
expansions
e((.Y, t')).
,,EZn+q
3.2.1
Lemma (ABE)
The Fourier
series
function
R2',
on
EOIEZn+q f(') (t") e((a, t')) iff
for
all
s
E R and all
(1 o,EZn+,i
+
1,T12y
to
converges
I
multiindices
2
aZn+q -periodic
C'-
74
Quasi-Abelian
3.
uniformly
converges
Varieties
compact subsets
on
of R-q
Proof f
Let
be
Zn+q-periodic
a
C'-function
and
f (') (t")
f (t)
e
((a, t')).
O'EZn+q
Then for
E
any it
Zn+q,
we
have
IT f (t) (R/Z)n+q
where T:= any multiindex
(/)
same
f (") (t"),
holds
equality
for
all
01II/Ot"If
DIf
with
fT (Djf)(t)e(-(It,t'))dt'
(t")
Dj e(- (jL, t')) dt'
(Dj2f)(")(t") we
=
=
f
27ri/-tj
way
same
equality
By Parseval's
dt'
integration
f and in the
t'))
(-(A,
By partial
I > 0.
(Dj f)
The
.
e
applied
-4
on
7r2
lLj2
f
EDj2)'f
(47r2-
F
for
k
integer
any
0
get
(47r 2)2
j,
+
(til)2 1
1
IIZ12)k
1(4,7r2
right-hand
The
the left-hand course
this
side side
is true
By Sobolev's
is
continuous
even
for
in
uniformly
converges
any
-
2+q )kf(t)12
D2
dt
D
n
'1 T
0'
t".
So
on
Theorem the
by Dini's
every
compact
subset
series
C C Rn-q.
on
of
DIf.
embedding theorem
1: D,f(')
[72,
p
78/79]
(t")e((o-,
t'))
the series
0'
converges
of R
n-q
Now let
tangent in
absolutely so that f K
:=
bundle
C'-category,
and is
a
RA/A
uniformly
x
C, where C
is
any
compact subset
Q.E.D.
Coo-function.
be the
maximal
of K. The tangent
where
T
on
TR.-,,
real
bundle
compact subgroup TX of X is identified
is the tangent
bundle
of
Rn-q
of X and TK the with
TK
x
TR,
quasi-Abelian
of
Characterization
3.2
varieties
75
Definition
3.2.2
Cn/A
Let
X
real
subtorus
=
be
K
Abelian
an
RA/A.
=
1
Av(w)(t")
-
r!
)ir<2n
1
any C'
r-form
n
the
w
maximal
+ q and the
of
average
w on
K
by
(fK
E
-
A of rank
with
group
defined
coordinates
is in toroidal
Lie
Then for
fi1-j,(t,t")dt')dti,
A
A
...
(til
dti,
E Rn-q
Remark of
1. The average
toroidal
Coo-form
a
coordinates
is
the last
on
n
which
depends
in
only.
q variables
-
coefficients
with
C"o-form
a
(m,p)-form
2. For any C'
W(t)
m!
fijdzj
P!
A
dz-j
ljl=p we can
For
C'
a
r-form
1
so
:=
1:
rl
K.
w on
a,
t))dtil
A
A
...
dtir
that
In this
case
we
EZn+q
obviously
obtain
Av(w)(t")
(Av)
=
w(')(t)
g, Sr be the sheaf of
Let
X.
By Poincar6
which
germs of
-
the
B(X, Er)
space =
Z(X,E'r)
iM(S,_l
Two r-forms
exists
+
an
-
(X)
Wl, W2 are
d -
+
ker(S'(X) Er(X))
__4
+
global
A
...
or
dti,.
COO r-forms
sequence d
...
__4
S2n
sections
__+
so
0,
that
Z(X, Sr) IB(X, Sr)
(X, C)
=
exact
dE2d of
the de Rham. sequence
induces
A
complex valued C"o-functions
cSO del
Hr
with
AV-ir (t") dti,
r!
lemma there 0__ C
d -+
Sr+l (X))
of d-exact
d-cohomologous
(w,
-
of d-closed
C"O r-forms
W2)
iff
on
space
X.
difference
their
and the
is d-exact.
Lemma
3.2.3 Let
(t ) e((
A
.7
on
of
the r-form
define
w we
W(,)(t)
Av(w)(t")
the average
similarly
define
X=Cn/A be
r-form
w on
X
we
an
Abelian
have:
Lie group
with
A of rank
n
+ q. Then for
any C'
76
Quasi-Abelian
3.
1.
The average
2.
If
w
of
w
is
=
1
(t")
Av (w)
w(O). Av(w) Av(w)(t") =
then
d-exact,
is
Varieties
n+q+l
Av(w)
w
is d-closed
and
4.
If
w
is
d-closed,
then
w
=
(0)
fil
-
(t")dti
A
...
A
dti,.
some i
0, then
is d-exact.
w
d-cohomologous
is
i.e.
tr<2n
1
If
(TK)',
on
E
-
3.
0
Av(w).
to
Proof 1. As 2. Let
before.
seen w
be d-exact
M
77
such that
w
=
dq
with
(r-l)-form
C"O
a
77 and
(t") e((a, t))dti,
(r
A
A
...
dtir,
Then
(tit) E'k= 1(-l)k+ where I
maxfk
:=
if
Especially,
ik
:
1 <
i,
121ri
<
<
n
fil
3. Now let
w
follows
o-
be d-closed
E
Zn+q
(1)
from
so
\ 101 for
that
...Zr
.
Ir
...
(tit)
dw(l)
that
=
127ri
fil
(0-) 1k-?,r+1
...
where I a
multiindex
(3)
:=
fixed
21ri
maxf
k
:
ik :5
multiindex
(i(U), S(o-)
il,
-
-
',
n
qj
+
(ij, ir)-
-
-
as -
,
i,)
,
(tit)
+
af
E (_I)k+l
S(a)
1 < i
o-i(,).
:=
ir+
1,
1
(t
I
atik
k=1+1
(tit),
above. with
1
<
il,
ir
E( _1)k+l
O[f
(-) i(17)il
k=1
With
.
(tit)
It < 2n
(-) Zl**"k"*Zr+l
<
2n
we
consider
Weget
(t") e((u, t'))
g a)
and
-
k=1
For
ir
...
atik
0.
(or E Zn+q). maxji : ai 01 (i 1, i,+ 1) with
any multiindex
ok
il
then
r+1
1:(_l)k+
ag(-)
Er=,+,.(-J)k+lk
0
i(a)
set
we
(0)
I
0
+
(2)
For every
il
(tit)
qj.
+
...
(0)
0-k 9
(tit)
f i(O-) 2-xi
S (a)
...
Zk
...
r
(t"
e((o-, t'))]
the
3.2
we
of
Characterization
quasi-Abelian
By (3)
we
(t") e((u, t'))dti,
j,
(r
a[g(-)i1-ik-ir
E(_l)k+l
t'))] ati,
by (1)
that
have
w
Av(w
have
we
general
for
If
d77
E,CZ,,+,\j0jw(0').
=
to
converges
w(l)
Since
element
an
Av(w)
=
of
=
0,
dq. =
Av(w)
-
Av(w)
=
by 3)
0. So
w
Av(Lo)
-
following
the
2.2.5
Proposition
in
seen
5-closed
Q.E.D.
is d-exact. is
proposition
not
true
in
(O,p)-forms.
X=Cn/A with
d-closed,
is
w
A of rank
then
X
which -
is -
there
+ q and
n
exists
a
be
w
a
C"o r-form
determined
uniquely
on
X.
r-form
X
on
X with
coefficients
constant
dtl,
'q(')
:Zn+,\Iol
Proposition
3.2.4
Let
77
and satisfies =
Av(w))
-
(4)
and
Lemma 6.1
H0(X,Sr-1)
As
dti,-,.
get
k=1
4.
A
...
i,-1<2n
1
r
By
A
-
(t") e((tT, t'))
we
77
define
(4)
so
varieties
-
,
E
-
r!
d-cohomologous
cil
...
j,
dti,
A
A
...
dti,,
1
w.
X is
the
w(o')
all
sum
of
of terms
Av(w) containing
only
dtn+q
Proof Let
w
be d-closed. the
Existence.
In
Obviously
dt,w(')
sum w
0. Then
dt,,w(l)
=
w(7)
are
closed.
dw(')
0 for
=
We set
d
=:
0.
We decompose
W(0)
fi(,.).,,(t")dtil 0
-
r! +
+
A
...
A
dti,
I
(r
-
1)!
1: 1
j(.,
W
0)
iA
Z1-*?,r-1
1
wio)
A
dti
+
wo(
0
0) 1
dti,
A
...
A
dti,-,
dt,
+
dt,,.
78
JO)
where with
Zl
*
4-1
...
the
to
we
by
w(')
have
we
that
by (2)
The
cohomology
X1
-
dt2n
w(O)
=
0
Cil
...
ir
0
=
respectively,
(0) fill..." (0)
dti,
A
jwj
dt, AI... is constant
,
andwO aredt,exact.
A
...
dtir
ir
dn(O)
with
q(O).
(r-l)-form
Coo
a
d?7(o)
(0)
d77 2
X2 +
,
then
X1
X2
-
d(77(o)
=
(0))
77 1
-
2
of any differential
class
s0
Q.E.D.
0-
X2
with
(0)
that
so
(0)
*
degree
the
0. Therefore ,
E I
X, +
Considering
-
(0)
Lemmathe
r-form,
by dt,w(O)
obtain
0 and dt,
=
2
!
x +
=
w(O)
If
we
1-
:=
(r-l)-form,
1-form,
a
dt.,,+q+l,
w o)
Poincar6's
set
Uniqueness.
are
differentials, dt,
x
and
0
z
differentials
0
and
ci,...i,
Hence
1
same
w o)
0, dt,
Varieties
2), w(O)
,
*
1
to the
respect
respect
=:
Quasi-Abelian
3.
form
Z(X, E')
in
is
represented
by
a
determined
uniquely
r
x
so
=
1:
r!
EACIdti,
dti
ci
dtn+q I
III=r
that r
/\
Hr(X, C) Let
(u)
zj
=
=
A-It,
V
t2n
Uj
(j
+iUq+j
1,-,q), dtn+qj
Zq+k
=
Cjdtj,
=
dtn+q}-
Cf dti,
(k
U2q+k +iVk
=
dUn+ql,
Cf dul,
1,.-.,n-q).
Then
tn+q+l)
VI
Vn-q
=
-
Since Vk
are
C'-functions
Therefore
dZq+k.
on
X, dVk
dtn+q
dtl,
are
iaZq
and vice
dtn+ql
3 X
dzj,---,dZnid'ZI)*
-
d-cohomologous Wehave
versa.
ClZq+k
dZq+k
0 such that
a
linear
to
and
dU2q+k
compositions
-
of
1-1-mapping
r
r
ACIdti,
Xc
EACJdzj,
dZn) CtTl
&Zq I
i
i
where
XC M+P=r
Thus in
3.2.5
Let
we
1993
get the following
Theorem X=Cn/A be
The
...
3-ki
...
kp
Theorem about
dz 31 A
...
A
dzj_
A
d_Zk,
de Rahm cohomology
A
...
A
dZkp
(UMENO) a
toroidal
group
statements following cohomology class of 0
hold G
of type
q.
in toroidal
Z(X, Sr)
is
coordinates:
represented
-
due to UMENO
[113].
Then the
a)
c'71
M!P!
by uniquely
determined
quasi-Abelian
of
Characterization
3.2
varieties
79
r
ACIdti,
x E
I
-dt,,+q
and r
ACfdzi,
XC E
dz,,,
iLrZql-
d ,,
'r
C)
Hr(X,
b)
ACIdti,
_
dtn+q},
r
ACI&I, H'(X,
C)
=
0
H'(X,
C)
=
(
(r
dZqj (1
dZn, dZ1,
>
n
+
< r
<
n+q).
q),
then dim
For toroidal
theta
n
+ q
X =Cn /A
groups
r).
<
r
got after
we
P
M
_-ACjdzj,
HP(X, Qm) The consequence
wrong for
obviously 3.2.6
Theorem
For toroidal
theta
dznj
following
the
is
toroidal
Theorem 2.2.6
wild
AACjd_Z1 of
Theorem
dZq}-
i
VOGT in
1983
[1171 (which
is
groups).
(VOGT) groups
X
=
C)
H'(X,
Cn/A
of type
q
(Hodge decomposition)
HP(X, Qm).
(D -+P=,
-:5-,p
Proof As
we
have
seen
any r-form
before
xm,P
xC
XC has
X',P
with
a
---
decomposition 1
1
M!
p!
M+P=r
-:5n,p
so
that
the
a
CJK dzi
A
dZK
IJl=rn IKI=p
mappings 4im,P
induce
E
H'(X,
:
C)
D
[X]
--+
[Xm,P]
E
HP(X, S?')
homomorphism -P
:
H'(X,
(D
Q
HP(X, S?m).
m+p=r
-:5n,p
Considering
the
dimensions,
we see
that
!P is
an
isomorphism.
Q.E.D.
80
Quasi-Abelian
3.
Varieties
Chern forms Let X be
a
complex manifold
A Hermitian
metric
of dimension
X is
on
of fibre
n.
coordinates
in local
given
metrics
zj,
by
zn
n
ds
2
hiT(x) dzjd-Zk i,k=l
where the C'-coefficient
Every complex manifold A Hermitian with w
Let
form
is
a
L
--+
X be
belonging
to
hv
Uv
:
be fibre
(,,
then
Every
100].
:
=
bundle
line
LI, L2
on
bundle
and
definite
positive
are
line
metric
hjh2.
the fibre
metric
11h.
f
then
h is
o
A Hermitian
(0)
Oh
:=
a
fibre
metric
-W log
h,
on
form is
a
1-cocycle
/-t
E
N). product
Hermitian
transformations,
coordinate
because
fibre
on
h
the
on
L defines
on
A
metric
Hermitian
hl, h2 h,
then
X the
-
on
a
resp.,
the
map of
a
09jak log h, dzj"
defined
is
metrics metric
fibre
Hermitian
a
holomorphic pullback f *L of
Y is
--
has
line
then
dual
L,
[76,
p
bundle. &
bundle
L2 has L*
has
complex manifolds,
L. Hermitian
Ej,k=l n
dz-vk
form
92 1,g _8z1d-z1 k
41 dz ' A dz-' k
X and d-closed.
Definition
3.2.7 Let
globally
(v,
The induced
metric
fibre
X with
X
:
X:nj,k=l This
N).
E
by
of C'-functions
set
a
IgvJ112
hv
=
fibre
Hermitian
metric
fibre
C" (X)
C-
Of X-
given by
against
L has the
f
If
w
is defined
X which
on
complex manifold
a
If
C"o-form
(v,IiEN).
bundles
the fibre
Uv (v
a
form.
fUv}vEN
hL
X is
on
83].
p
(h
L is
on
invariant
any
with
d_Zk
A
=
bundle
with
on
hv I (v
A line
h
[76,
metric
real
a
covering
coordinates
12 remains hv(vZ,=hvgvt,(Ogv,,Z,,=hl,(,,Zl, (C., Cv)
line
R>0
--+
H
iw is
metric
fibre
/ .T dzj
matrix
open
an
Hermitian
a
En i,k=l
=
holomorphic
a
(Hermitian)
Let
w
form,
Hermitian
(h)
X has
coefficient
Hermitian
a
Ig,,,l A
If
is Hermitian
X.
on
If
(hjk)j,k=l,---,n
matrix
L be
Then
-495log
X -'T
a
line :=
h the
bundle
on
-aj'5klogh curvature
X with
(j,k form
a =
fibre
1,-,n)
and
(9h
h.
metric is ::--
called
--l-oh 2-7ri
the the
curvature, Chern
form
Oh of
h.
Characterization
3.2
The inclusion 2
H
(X, C)
Z
so
that
so
C)
(L)
cl
of
line
a
H'(X, Z) H2(X, Q. By de
Xj / Id-exact (1, 1)-forms obviously
--*
C' 2-forms
on
part
as
on
of H2 (X,
Xj Q.
Oh:-' any fibre
with
confer
and KODAIRA [76, We use the
L
on a
X in H2 (X,
complex manifold
C)
I
I
27ri
2,7ri
--Oh:::--
0,9
log
h
h of L.
metric
proof
For the
bundle
Chern form
every
(0)
GRIFFITHS and HARRIS
127].
p
following
In the first
of
definition
[39,
line
positive
a
148f]
p 141 and
book -Oh is used
MORROW
or
form.
curvature
as
bundle.
Definition
3.2.9
A line
bundle
on
complex
a
whose curvature
metric
Every line bundle
L
manifold
on a
H. A line
bundle
is
iff
has
it
fibre
Hermitian
a
definite.
group is defined
toroidal
3.1-1. decomposition form H on C', however uniquely
L determines
positive,
is
form is positive
Hermitian
tian
81
inclusion
be assumed to be in
can
C' 2-forms
the d-closed
(L)
cl
by
represented
the
the
varieties
Proposition
3.2.8
The Chern class is
Id-closed
_-
consider
we can
Oech cohomology
in
the Chern class
that
Rahm H2 (X,
C induces
--+
quasi-Abelian
of
by
an
automorphic
decomposition
This
only
on
ample,
iff
factor
determines
RA- Wesay that the Hermitian
with
the Hermi-
the line
form
H is
bundle
positive
definite. We want to establish a
line
bundle
same
a
factor
line a,\
Then the
functions
lemma is due to ABE (Remark
bundle
(A
E
L
A)
on a
with
A
on
C'
toroidal an
fibre
Hermitian
forms
of the
fibre
1 in
be defined
by
determined metrics
by
of the
[2]).
group X
exponential metrics
h
=
C'/A
system on
L
of functions
correspond
ax
to
the
with
A(z which
the Hermitian
Chern forms
Lemma
3.2.10
Let
averages
of the
bundle.
following
The
between
connection
a
and the
+
A)
-
A(z)
=
41r Im aA (z)
(A
satisfies
aZj19"T_k
)j,k=l,. .,n
,92A
(9Zj0Tk_)j,k=1,---,n*
E
A),
an
(A real
automorphic
E
A). valued
C'-
82
Quasi-Abelian
3.
Varieties
Proof >-
First
.
that
note
we
systems is (see
(z)
of the
disjoint
pairwise
to
U, by
U,,
n
then
(h)
Wedefine
U U,*
V
on
A*(z
and
Uv*
E
z
A,,v)
+
For any
z
z
+
A*
-
Alv (z)
E Cn there
Finally --<
.
with
the additional ABE [6]
proved
X
=
Hermitian
0 v) and set g, U,,*, n (U* + A,,,)
be described
can
(p 80)
L
we
If
0.
X
as
have
(x))
log(h,
-
on
(7rJu*)-1.
:=
(x EUv n U,,).
on
(x))
one U,*, biholomorphic
is
(x
E
uv
u,,).
n
(z
A*
Uv).
E
then
log Ig,,,(7r(z))l
2
=
log(h,(7r(z))
:=
a
A E A with
(z
+
A)
(z)
-47rIm
=
Uv*
+ A E
z
+ 4-7rIm
(z)
a,\
for
(z
suitable
a
E
(z).
47rIm a,\,,,,,
=
v
so
that
Cn)
once
hv
:=
more
exp(A
to
see
the
o,,)
(v
o
Cn/A form
desired
N)
E
of A.
property and
just
this
fibre
in
1989 that
be
fulfils
metric
Q.E.D.
condition.
Theorem
3.2.11
L
metric
every
form H on Cn which is
Hermitian
Let
(*)
need
we
0 (/.t
=
finite
a
by the help of (*).
defined
Directly
U,,*,
exists
:
UAEAIUv* Al components of 7r-'(U,)
exists
where
+
the C"0 -function
E
=
A(z) is well
exponential
A).
c:
map. there
=
(p,, (x))
aA,,,
log(h,,
A*(z) If
A,,,
(A,) A2
covering
a
E A with
defining =
=
u,-,
n
a
of the fibre
log Ig,,, (x) I
many
u,*
functions
is
7r-'(U.)
all
exists
gv,L(x)
2
of
property
(z)
+ Im ax,
C'/A
--+
that
assume
there
transition
By definition
C'
:
so
A,)
+
countable
We can
-7r.
U,, 0 0,
Now the
7r
of X
(z
Im a),
=
projection
(UV)VEN
covering
cocycle
28)
p
Im ax, +,x,
The natural
of the
consequence
a
bundle
theta
given by
a
on a
certain
toroidal fibre
group metric
determines
on
this
a
bundle.
(ABE) a
toroidal
group
H. Then there
7r
and L
exists
(H -T) j,k=l,-.-,n=(_
a
a
fibre
theta metric
OZjO-Zk
bundle h
on
)j,k=l,-. ,n*
on
X
determining
L such that
the
quasi-Abelian
of
Characterization
3.2
varieties
83
Proof Appell-Humbert
The
bundle
?9,x(z) with
e(ax(z))
=
Let
A(z)
and
-i9'Alazia- 7k
(z)
7rH
=
for
the fibre
By
Lemma3.2.3(4)
[H(z,
2i
-
defining
factor
the theta
Then
Using
3j.
of
+
S'
-+
7r
A(z
foregoing
lemma
+
A(z)
-
A)
=
47r Imax
(z)
get
we
aZjaZk A(z).
to
Q.E.D. the
Chern form represents
a
This
E
that
so
A)
calculate
(_
(A
H(A, A)).
we
corresponding
the average
A)
A
the
=
L
on
o :
A)])
H(A,
2
by definition
has
average
same
Chern class form
associated
an
theorem
we
need
a
Lemmaof ABE
[101.
Lemma (ABE)
3.2.12
Let
L be
1.
If L is
a
bundle
line
on a
toroidal
trivial,
topologically
for
any Hermitian
TK
on
x
fibre
Av(19h)
The average
X=Cn/A.
group
Then:
then
Av(19h)
is
+
is Hermitian.
For the last
2.
A)
Re H(z,
(21r
(H -T)
h
metric
=
the Chern form itself.
as
which
z).
Re H(z,
-7r
:=
7r
(L)
automorphic
an
H and semi-character
form
Hermitian
a
p(A)e(
=
4,7r Im ax
cl
of
decomposition
L is
h
metric
of the
TK independent
0
=
TK
on
x
TK
L.
on
Chern form
of
a
Hermitian
fibre
metric
2.1.9
on
h
on
L
of h.
Proof 1. If L is
the
topologically
Chern form
then cl
trivial,
19h of
any fibre
Av(19h) 2.
Let
a
fibre
obtain Let
hl, h2 metric
the
(9h
Then its
=
be two fibre on
the
metrics,
trivial
line
(L)
=
on
=
0
by Proposition
is d-exact.
metric 0
on
the
bundle.
TK
given
x
By
bundle.
1)
to
Then h
the
;'
949
average
log
:=
have
hj/h2
Chern form
19h
is we
Q.E.D.
conclusion.
2 7ri
we
TK.
line
Applying
p 38. Then
Lemma3.2.3(2)
h be the
Chern form of
a
fibre
metric
h
on
a
line
bundle
L.
84
Quasi-Abelian
3.
Varieties n
19h :=Av(Oh)
is
by
Lemma3.2.12
We state
this
Zq+k The relation
especially
(k
iVk
vi
(9h 6h
n
-,
generated
has the
h-invariant
in terms
dul,
=
(u, v)
t2n,
-
JK
(u)
is
and
=
V
A-It,
19h,
Of
representation
W,
(9Zja k
t1l)
the
dt'.
and also by a/19U1, by 49/194) *) 09/19tn+q only of terms dui A duj, part consisting
-
7
form
(t', t")
=
the
in
1,...,q)
=
t"
on v
hjT(t")
and t
Then,
.
dun+q, dvi, -i dVn-q. fl h (Z' W) on Cn defined by -
(j
zj= Uj + iUq+j in toroidal coordinates.
q)
-
=
of h.
Let
coordinates 5
TKis
Hermitian
-
...
hjT(v) Since
-
t?i+q+li Vn-q are dependent only
=:::
k.T
coefficients
1,
=
real
between
precisely.
more
d-Zk
A
j,k=l
TKx TKindependent
on
meaning
U2q+k +
=
2
E k-jdzj
.
average6hgives
The
i
if
19/(9Un+q)
we
rewrite
the associated
'V
n
-h
(Z7 W)
HV`
3T(V)ZjTk-
7r
=:
i,k=l
The above fact
means
fthl
that
V
fth I RAxRA= 0
particular,
In
V )
forms
Two Hermitian
RA xRA is
HI and H2
Cn
on
H11RAXRA=--H21RAXRA
following
The
toroidal
Theorem
3.2.13
A-equivalent,
abbreviated
as
the
characterizes
iff
Hi-AH2-
line
positive
bundles
on
the
Hermitian
a
associated
(ABE)
holomorphic
L be
on
[10]
of ABE
are
trivial.
groups.
Let
h
theorem
of h.
independent
topologically
if L is
form
with
H(z, w)
A-equivalent
L is
then
to
H is
=
on a
toroidal
group
Enk=l H ` ZjTk. j,
of the
the average
If L is positive,
bundle
line
Chern form
X
Then the
19h of
=
Cn /A
determining form
Hermitian
fibre
any Hermitian
ftvh
metric
H. an
Riemann form
ample
for
A.
Proof Let L
ically a
fibre
=
line
metric
Then the
of L into
L,9 OLO be the decomposition
trivial
h,3
bundle on
average6h,,
L,0
a
theta
Lo (see Theorem 2.1.10). so :=
that
19h,,
Av(19h,,)
2-7ri
is
By Wlog hv
obviously
the
bundle
L'
and
Theorem 3.2.11 2i same:
En i,k=l
a
topolog-
there
exists
H Tdzj A dZk3
Characterization
3.2
quasi-Abelian
of
varieties
85
n
19h,9
Therefore For
==
the Hermitian fibre
any
of the Chern form
Now h,
:=
hvho
is
fibre
a
Lo,
on
average
E HiTdzj
2
H,,` h,9
form
ho
metric
19h,9
19h, is just H. -h form H,,",o associated
Hermitian
vanishes
metric
with
the
RA x RA-
on
The associated
L.
on
dZ_k-
with
associated
the
(9h,,
A
3
j,k=l
H," hi
form
ko
is
+
H
ho
Then H,"hi
I RA
H,"
X RA
h,9
I RA
x
-ho
H,`
RA +
I RA
X RA
H. For any fibre
h
metric
L,
on
we
rh
-
rh
Hv Finally 19h
let
L be
only
need is
to
form.
fl
form
h
H> 0
Then its
V,
holomorphic
projective
L be
for
a
HO(X, L)
any
lemma
1993
[8].
For that
let
L be
sition
2.1.2
forms
belonging
on
MCAC RA,
of line
L be
on
to
bundle
line
on
projective
to
bundles
was
a
which
associated
the
Q.E.D.
MCA-
spaces
define
complex
maps to
X
=
section
C'1A.
sections
V E
of L
an
Then
automorphic
Ho (X, L) such that
is
factor
isomorphic
to
the
V)
Ho (X,
obviously
As
we
space
A,,
a.
is
L
generates
p(x)
proved by CAPOCASAand CATANESEin
H'(X, L) a.
on
holomorphic
exists
given by
p 27
a
holomorphic
H'(X, L) =A 0,
have
isomorphic
then
line
bundle
H'(X, V)
on
X
generates
=
0
1991
seen
of all to
in
on
on
X for
=
2.
(The general
proof
works in the
same
way.)
any
integer
X,
0.
[20]
and
Propo-
automorphic
A,,i.
C'/A.
L'
Proof For I
Since
H> 0
Lemma
3.2.15
If
19h is also positive.
mappings
of all
there
E X
x
ABE in
Let
H is an ample Riemann form for A, we MCA. Let h be a fibre metric on L such that
average
sections
holomorphic
following
The
H.
A
spaces.
The space iff
Hv
Definition
3.2.14
Let
OR
and H coincide
Holomorphic Westudy
-A
To know which
know that
positive
a
Hermitian
positive.
by Lemma3.2.12
obtain
f > 2.
Quasi-Abelian
3.
86
H'(X, L) 0 F(a, z) := f (z
0 there
Because
sider
z(')
for any fixed
ax
=
Vx
topologically
fo
(a('))
t),
(A
trivial
F(a(o),
a(0))
-
AC,2
F(a(l),
and
=_
0
0
_=
z(o))
on
Cn. So there
exists
defining
L'q
on
an
6,2 )
(z
t),
-
:A
in the
proof
a(0))
(z
tx
0 tx
Lo the
a(0)).
+
(H, 9)
of type
F(a(O), z)
is
is the
two wild
last
of the
product
MCA So
E
fo(do))
-
factors
theta
and the
factor
form for
automorphic
a(0))
+
=
in Theorem 3.1.4.
described an
L
Lq and the wild
bundle
the theta as
factor
automorphic
be the
two reduced
=
us con-
:=
RA, and then
on
0. Then let
fo (a) F(a, z(1)) (a E Cn) because MCA+A MCA. Otherwise,
C')
and
0.
Vx (z
t9' ,\ of type (2H, t2 (Z) for a(0) factor
wild
the
be
Lo
factor
theta
E
even
:A A)
E
of the first
product
z
E Cn ) becomes
79,\ (z The
a) (a,
cannot
bundle
z) (z
+
f $
form
automorphic
an
?9,x defines
factor
where the theta
Then
fo
E Cn.
E MCAwith
Now let
exists
a)f (z
-
RAi f would be
is dense in
a(')
Varieties
reduced factors
form
automorphic
an
is
of
Q.E.D.
0.
Remark
a(0)'s
Taking suitable V and
sections
N
HO(X, L)
Indeed,
is
complete
a
JVjJ
let
as
IVIC define
semi-norms a
We say
that (P
P
(N)
pings
to
we
set
(N
that +
get relatively
E
J)
prime
-
-
n
X
locally
be
finite
Then
compact)
cnvi
by which HO(X,L)
-
,
V
(N))
E
Ho (X,
becomes
a
Fr6chet
space,
HO(X, L) N.
also
L)N
has
the
iff
zeros
sections
common zeros.
projective the
zero
a
map-
a
map F
n
[46,
p
L
on
X
=
of
set
compact
a
fN)
(fl,
manifold
complex :
X
CN is
--+
a
1141.
(ABE)
H'(X, L) generates 1)-tuples
image of
if N >
holomorphic
spaces.
holomorphic
set,
of
the existence
of ABE guarantees
under
Proposition
Suppose of
(V(1),
Wj.
(CC
JVJ
(j
CE X
every
space and then
metric
remember that
compact Lebesgue 3.2.16
i
Wj
0:-: on
sup
:=max
dimensional
X of dimension
Vj
L is trivial
proposition
higher
For that
=
have
following
The
we
space.
with
HO(X,L),
on
complete
therefore
V
metric
JWjJ
and
of X, such that
coverings
of the above lemma,
0 of L'.
of sections
(P E
=
CI/A
H'(X, L)N+1
and let
with
N >
zeros
on
n.
Then the
X is of first
category.
Proof It
is
sufficient
to
prove
the
statement
for
any
compact
subset
C C X,
because
X is the countable
f
M:=
Statement
M is
a.
For each
j take xj
Because
Oj
of compact
union
E
closed
in
E
C,
a zero
87
HO(X, L)
-P in
--+
that
assume
Oj(xj)
=
C1.
on
4ij
M E)
get 0
we
varieties
consider
0. Wecan
=
C
on
Let
:
Oj (xj)
such that
we
!P has
:
HI(X,L)N+l
uniformly
converges
L)N+1
HO(X,
4
So
sets.
quasi-Abelian
of
Characterization
3.2
xj
O(xo)
--+
=
N+1.
xo c
--+
0
so
C.
that
0 E M.
Statement
0(0),
-
b.
without
Statement N>
((p(O),
Let
c.
no
zeros
loss
without
considered
UO n C with
=
0
according :
=
Now
Op (UO we
M has T1
=
O.O(Cj)
finish
(00),
-
the
aNO)
-
aN)
the
,
compact
proof
of
Let
E
the
0
E
L)'+'
Theorem
N+r+2
HO(X, L)
-
Ix
we
set
(N)
0
W
-,
coordican
be
O(x) 0 01.
G U:
If
UO n C =: UjOO=1 Cj CN+1 is a holo-
U, 0
:
-
__+
(0)
The
zeros
we can
show that
to
b) Applying
there
C.
on
is
rest
statement
exists
:...
a
statement
that
conclude
1: aNiOj))
-
HO(X,
E
L)N+1
j=0
and
suitable
arbitrarily
small
coefficients
So 4i is
aij.
no
Q.E.D.
(ABE)
Suppose that HO(X, L) generates holomorphic mapping
[e]
:=
-
of M.
3.2.17
:
(P(N))
:
L
on
X and N >
V(j)
PN with
of the N-dimensional
is the
hyperplane
bundle
proposition
there
exist
sections
E
n.
Then there
HO(X, L)
X
Proof By the previous
where
assume
some
Lebesgue measure zero in CN+1 Then we can take proposition.
After
without
%OW, ..' W(N)
C for
on
point
(O
we can
in
r
-I:
interior
=
Uo
proposition. M.
j=0
'P
of this
r
zeros
W(O),
O(N) /0)
...'
sets
formulating
HO(X,
E
(0, TI)
to
(W(0) no
U. Let
0. If UO n C 0 0,
(OM/0,
before
point.
V'))
-
c) successivly
has
C is contained
The sections on
C and
HO(X, L)N+1
E
set
in
zeros
CN+1 \ Z: For that
E
functions
are
without
CN+1 such that
Z C
set
zero
L is trivial.
Z
U:
in
HO(X, L)N+2
compact
0,0
Because
interior -
-,
common zero
number of sections
finite
a
C).
n
can no
that
the remark
to
-
-
ready with
Cj.
map, all
morphic
(a,,
any
exists
compact U by assumption. E
(P(N)
-..'
U on which
we are
compact
Lebesgue
holomorphic
be
to
a
generality
of
neighborhood
nate
Z
C for
on
in
(p(N), 0)
aoO,
-
any
even
-,
-
exists
( p(o) has
-
X there
without
common zero
Then there
n.
U Cl--
open
any
OME HO(X, L)
-,
-
Indeed,
For
W(O),
-
and L
projective
-
-
,
V(N)
E
=
exists
-!P*
space
a
[e], PN-
HO(X, L) hav-
Quasi-Abelian
3.
88
ing
no common zero
The fact
L
that
Varieties
on
[e]
P*
=
They define
X.
the theta
ABE and
has
factor
generalized that
proved
The additive
1960 for
in
a
of X defines
the exact
which
cohomology
the
-
-
-
-+
Cn/A
0
Let
M* (U,,)
fmlf,
:=
[D']
that
so
by
a
be
a
determines
Suppose
0-
a
0
open
an
covering
f Um
[D].
D, D'
bundle
line
-5* [D]
the map D
If
becomes
sequence.
-4
0
5 H1 (X, 0*)
Div(X)
-4
we
get the
same
...
homomorphism
Pic(X).
and H a Hermitian
group we
group
-4
form
on
Cn with
X
=
imagi-
write
:=
AIRA XRA
(ABE)
the Hermitian that
functions
[10].
0.
cohomology
-4 M*/O*
of the Picard
toroidal
holomorphic
a
on
of
HO(X, M*) ?4 HO(X, M*/O*)
Theorem
L be
0-
functions
HA := -UIMCA XMCA and AA 3.2.18
of the
sequence
A:= Im. H. Then
part
nary
E
g,,,
[D]
J:
Let X
fm
0* -"* M*
-4
and the definition
Using
groups
fibres
sequence
0
induces
bundle
the
meromorphic
of germs of
be described
can
toroidal
on on
by
holomorphic
D + D' defines
then
homomorphism
Indeed,
sheaf
functions
if
HO(X,M*10*),
functions
local
the transition
divisors,
=
of germs of
by
D given
A divisor
a
be identified
can
groups,
semi-definite,
[63, 641.
constant
are
toroidal
positiv
bundles
line
by the given line
is determined
multiplicative
the
and 0* the subsheaf
is
form
all
functions
Div(X)
where M* is
factor
automorphic for
groups
and in 1964 for
theta
a
property
of divisors
group
toroidal
on
groups
by
meromorphic
the
M
are
torus
non-trivial
form which
Hermitian
functions
1995 this
in
PN+l.
-+
Q.E.D.
form determined
the Hermitian
that
X
is trivial.
Meromorphic KOPFERMANN proved
map 0:
a
H'(X, L)
line
bundle
form
generates
over
a
toroidal
group
suppose
H'(X, L) =7
C'1A
which
H on Cn.
L
on
X
or
0
.
Then:
there
1.
exists
A-equivalent For
2.
H'(X,L)
E
constant'
01 /02
is
C'/A
is the
quasi-Abelian
ft
form
Hermitian
varieties
which
C'
the
on
89
is
H.
to
01,02
any
semi-definite
positive
a
of
Characterization
3.2
xO +
on
(KerHA)
7r
all
for
xo E
f
function
'02 0- 0 the meromorphic
with
X, where
7r
:
C'
X
--
projection.
Proof
a)
H'(X, L)
If
L, then by Theorem
generates
there
3.2.17
exists
holomorphic
a
mapping 4
(01
Since
The
P,
:
02
o(o)
assume
(P(N))
(0)
(,p
=
positive
is 2
02
exists
h
metric
h
o
E
1
H. Since any
(
t
positive
is
0
a
Lemma3.2.15
L2determines
The
the
Note that
2The
set
in
e
positive
a
=
P,,-,
C
curvature
-P)
azj C9Zk
%,
Jacobian
of !P and
tiT(t', Cn, fl,
on
*
k
=
1,---,n
by
Theorem 3.2.13
V)dt')j,k=l,...,n
is
is also
semi-definite.
t
a)
of
positive
A-equivalent
Hermitian as
such that
f2
form
a
meromorphic indeterminacy.
Chern form
of the
of the
xO
form
2H.
a),
in
we
For
such that
f
and then the last
are
function
Hermitian
Fubini-Study
E (,9fl
to
For
(dt.
0 for E (afl( f is constant on xo + 7r(KerHA). of a) if HO(X, L) :A the assumption
point
a
L 2 satisfies
two theorems
of its
(1, 1)-form
is the
A, given by (fK
ft
Hermitian
following
the
7r
!P and then
b) By
same as
h with
matrix
a2 log(h
(
-
semi-definite
neighborhood
Therefore
the
1 :=
H((,
=
definite.
definite
we can
HA C MCAwe get
G Ker
Then in
divisor
hyperplane
the
metric
the Hermitian
(E -T(z))
form
the Hermitian
for
by
fibre
a
'i9!P E &P, where a(fi
We have
property
same
[e].
(_49Xii9Tk)j,k=1,---,N
-
(P induces
=
-!P*
=
define
we
7r
pullback
HO(X, L) and L
has the
PN+2
X
P, defined
on
E:=
and the
E
-
there
that
so
the fibre
With
[e]
bundle
hyperplane
form.
P)
V),
=
(p(N))
(0)
(P
O(j)
P N with
X
is
ft
--A
even
metric
semi-
2H. The rest
of the
proof is Q.E.D.
Pn.
exists
on
to
constant
on
0. Besides
positive
steps
h
positive
is
a
2H there
decisive
on
=
is constant
called
fibre
metric
have
[e]
xo +
the Main Theorem. on
is
ir(KerHA).
every
given
set
by
contained
the
associated
in
90
Quasi-Abelian
3.
(ABE)
Theorem
3.2.19
Varieties
Any meromorphically Proof Let X
C'/A
=
set
of all
are
positive
be
theta
separable
toroidal
meromorphically
a
factors
79 with
semi-definite
quasi-Abelian
a
toroidal
determining
type
a
is
separable
MCA,
on
group
variety. Define
group.
Hermitian
T
H' ,
forms
the
as
which
and let
nKer(H o)A-
Ker T:=
'OET
It
suffices
Indeed, there
dimc Ker T
if
Cn
:
7r
Ker T
show that
to
X
-4
exists
Cn/A
==:
0.
0, then
>
is the
line
pole
the t9 is
theta
a
non-zero
we
HIA
choice
> 0
of
KerHA
f.
toroidal
Abelian
a
by
the
assume
meromorphically separable, Let f (,7r(x)) -7 f (7r(y)).
divisor
zero
of
f,
and then
on
By
the
same
the projection
simultaneously L O (9 LO,
and
Theorem 2. 1. 10 that
L
=
by where
form H. Now there
By
=
d E T.
where
[(f),, ]
=
by
7r(y),
:?
X such that
on
[(f)o]
X is
type which determines the Hermitian plo. p, 0 E HO(X, L) such that f
such that
Theorem 3.2.18
Theorem f is constant of KerT.
But this
on
the
contradicts
Q.E.D.
(CAPOCASA-CATANESE)
Theorem
3.2.20
Any
of
sections
of
projection the
of f. Wemay
factor
exist
get
determined
bundle
divisor
f
7r(x)
Ker T with
E
x, y
Since
function L:=
be the
take
projection.
meromorphic
a
=
with
group
a
non-degenerate
meromorphic
function
is
quasi-
a
variety.
Proof f be
Let X L
f
=
is
(W10)
o 7r
by
determined If
KerHA :
ABE in
proved
1987
[1]
is not
the
in
existence
bundle
meromorphic
Then there
holomorphic projection
an
all
ample
results toroidal
7r
z
is
exist
:
groups
Cn
function
+
KerHA for
1989
[4]
all
of
[33].
meromorphic
functions
form
z
by
E C'
the
A.
H
same
Q.E.D.
in
toroidal
theta
supplements
with
their
CAPOCASAand CATANESEadded the
non-degenerate
bundle
line
0 of L such that
Hence HA is positive
GHERARDELLIand ANDREOTTI contributed
1974
where
by Theorem 3.2.18.
Main Theorem for in
C',
X. The Hermitian
-4
non-degenerate. for
on
holomorphic W and
semi-definite
Riemann form of the
a
sections
L is positive
f is constant on possible because f
main
and for
predecessor rem
the line
and therefore
definite
group.
prime
the natural
with
0, then
This
theorem.
toroidal
a
relatively
X and
on
A-periodic
non-degenerate
a
C'1A
=
classical
1991
[20].
in
Fibration
aspect
groups
[10].
As
Theo-
of the
toroidal
a
1.) 2.) 3.) 4.)
(Characterization Cn /A the following
Main Theorem
3.2.21
For
X is
a
X has
X
group
=
quasi-Abelian line a positive
X is the
covering
X has
closed
a
quasi-Abelian
of
Characterization
3.2
of
varieties
91
quasi-Abelian varieties) are equivalent:
statements
variety. bundle. of
group
Abelian
an
subgroup
Stein
N
variety. C' x C*m
-
so
XIN
that
is
an
Abelian
variety.
5.) 6.) 7.)
X is
quasi-projective.
X is
meromorphically
X has
a
separable. meromorphic non-degenerate
function.
Proof Theorem 3.2.11.
1
2.
2
1.
Theorem 3.2.13.
3.
Theorem 3.1.10.
1
Theorem 3.1.16.
1 >- 4.
Fibration
4 >- 5.
Theorem 3.1.18.
5 >- 6.
trivial.
3 >- 7.
trivial.
6 >- 1.
Theorem 3.2.19.
7 -
Theorem 3.2.20.
In
1.
showed in which
-
section,
iff
and
rank
n
it
is
special
the
in
a
-
trivial
0.
bundle
line
has
a
prove
for
proposition
the
3,
non-trivial
and MARGULIS [47] HUCKLEBERRY
trivial.
COUSIN
of rank
2 and lattices
of dimension
case
topologically
1983 to
HO(X, L)
assumption
used the
used his
any dimension
n
and
+ 1.
in
L is
that
assumption
3.2.22
in
language analytically
a new one
ABE proved
Let
[261
1910
we
in modern
idea
The
propositions
previous
some
Q.E.D.
[10]
1995
topologically
trivial
following
more
general
on
toroidal
the
Proposition (ABE) line a holomorphic
bundle
L be
determines
Hermitian
a
Then Ho (X,
L) :A 0,
iff
form L is
H on C'
analytically
is not
a
and suppose
necessary.
result:
X
group
HA
Cn /A
=
which
0.
trivial.
Proof Ho (X, L) 0 0, then take any o E HO(X, L) with p HO(X, L2) generates L2 on X and let 21-1 be the Hermitian
If
L2. By Theorem 3.2.17 !P
:
(,p
(0)
:
...
we
:
have
O(N))
a
:
holomorphic
X
---+
PN
0.
By
form
Lemma3.2.15
determined
mapping with
W(j)
E
Ho (X,
L
2
by
92
3.
for
which
Assume
Quasi-Abelian we can
now
P(xo
that
and then The
+
y)
there
=
X. This
converse
is trivial.
The
following theta
exists
4i(xo)
for
2
W
=
zero
a
!P is constant
on
trivial
o(O)
suppose
that
Theorem 3.2.18
Varieties
the
on
all
y E
example of ABE in 1989 [6] shows the with LIA 0, then HO(X, L) =
The basis
toroidal
in
1
0
O v'2-
1
0
1
a
toroidal
generated
.
A
form
C3
on
--
A(el,
e-3) ie3)
A(x, y) get
a
Hermitian
form
H(x, y) Consider theta In
the theta
bundle
general
discuss
this
it
L. is
=
factor
not
MCASO
7r(MCA)
easy
in
=
to
the
=
0
0
=
of
non
topologically
0.
0
=
-A(e3,
el)
A(ei,
e3),
=
0
otherwise.
H on C3 defined
A(x, iy)
+
0, but L is discuss
next
existence
iV2 i-v/-3 iV3- 0
iA(x,
V whose Hermitian
Then HA
problem
XO +
R' by
A(iel,
we
XO +
=
on
of
C3/A. Let Jel, e2, e3l be the natural complex by f el, e2, e3) iel, ie2 I. Wedefine an alternat-
X
group
of C3 Then RA is
ing R-bilinear
0
-=
p
proof
coordinates 0
By A
Therefore
the
Q.E.D.
P
basis
of xO + KerHA
projection
7F(MCA).
W. As shown in
possible.
is not
bundles
Example:
generates
E X of
xo
the
chapter.
not
by
y), form
x, y E
is
topologically
conditions
for
C3.
H, and trivial
the
corresponding AA 0 0-
for
H(X, L) :A
0.
We shall
and Extension
Reduction
4.
can
be reduced
non-trivial
sections. it
questions
defined
form
Hermitian
proved
useful
is
by
Automorphic
4.1
For the
morphic
live
forms
an
a
have
t > 3. For
toroidal
if and
possible,
group
if the
only
ABE condersidered
of kind
Riemann form
ample
conditions
problem of
the
them,
satisfies
factor
with
from
group.
always
L
integer
any
condition.
certain
a
is
toroidal
a
he
t.
forms
automorphic
of
existence
fulfils
is associated
where the fibration
case
bundle
for
bundles this
a
bundles
line
bundles
Riemann form
of
reduction
line
M. STEIN showed that
the
concept
ample
very
holomorphic
extend
to
is
of
line
in
sections
positive
that
V
that
also
general
a
meromorphic
conjecture
the
compactifications.
to standard
the
He
With
of the
existence
TAKAYAMAproved
Recently
some
of the
proof
a new
gave
of non-trivial
existence
bundles.
line
positive
to
of the
question
the
ABE showed that
If
necessary.
are
is reduced
existence
to
the
a
given
case
auto-
of
ample
Memann forms.
Reduction Let X
=:
bundle
over
First
we
Wehave
(CO) is
Cn /A be
consider
the
already
seen
HA
necessary
We shall
a
X which
=
to
toroidal
group
determines necessary
that
the
HIMCAXMCAis
for
prove
positive
the
of type
definite
form
a
holomorphic
line
H.
HO(X, L) :A
for
conditions
L be
and let
q,
the Hermitian
case
0.
condition
positive
semi-definite
and not
zero
HO(X, L) 0 0 (Theorem 3.2.18). four
that
other
(Cl)
conditions
-
(C4)
are
also
necessary
for
HO(X, L) :A 0.
By
Theorem 3.1.4
given by bundle
a
we
reduced
given by
an
may
factor
assume
that
L
=
L,9 (& LO, where LV is
0,\ of type (H, p) and LO
automorphic
factor
t,\
=
Y.Abe, K. Kopfermann: LNM 1759, pp. 93 - 124, 2001 © Springer-Verlag Berlin Heidelberg 2001
e(s,\).
is
a
topologically
Consider
a
theta
trivial
the condition
bundle line
Reduction
4.
94
(Cl)
Ker(AA)
and Extension
D
Ker(HA),
It
is obvious
iff
MCAn Ker(AA)
(X, L) :A 0,
Proof Suppose Then
that
L')
Ho (X,
we
L
then
=
consider
Let
prime
L,
7r
:
C'
L
=
V) f
be the canonical
is
of
group
Lemma3.2.15
contradiction.
Moreover
there
Wedefine
projection.
rela-
exist
Lemma3.2.15).
after
a
non-constant
by
C'
on
(C 1). By (C 1).
a
(C 1) is not satisfied. H'(X, L) (see Remark
E
C'/A
function
period
the
(Cl)
satisfy
not
and obtain
situation
0
f We set
ImH.
The condition
the condition
does also
X and
on
p,
satisfy
not
X and L'
on
following
the
X
)
=
(C 1).
the condition
Lo does
(9
L'
sections
meromorphic
Ker(HA).
C
A
xRA with
Ker(HA).
=
L satisfies
generates
Ho (X, L) generates
tively
that
AIRA
(ABE [8])
Theorem
4.1.1
=
MCAn Ker(AA)
Remark.
satisfied, If HI
AA
where
0
Ir
:=
O
0 7r
f
Pf :=JaEC':f(x+a)=f(x)f6rallxEC'JObviously
Pf a
projection
T :Cn
period
with
group
epimorphism
-r
toroidal.
7r'
Let
Pf.
)
Cn IF
by
gives
a
f
=
)
exists g
the
a
T. Let
o
X'
be the
relatively L'
bundle
of
Cn,
write Pf FEDF, subgroup. Consider the meromorphic function g on Cn IF a
we can
A'
:=
T
T(A). :
Since
canonical
Then A' is
Cn
)
X' and
Cn IF
a
an
must
be
The function
projection.
functions.
relatively
subgroup
induces
X'
toroidal,
X is
prime holomorphic over
=
discrete
projection
(Cn IF)IF.
:=
two
line
and F is
There
X1
subgroup
closed
Therefore
X
represented
representation of L'
Cn IF.
)
a
subspace
F such that
A C
locally
is
complex linear
of F because
is
Pf
A. Since
::)
where F is
prime
This
g
local
sections
0,
0'
such that
071 g 0
From
f
=
g
o
T it
0'7r'
0 7r
0'0
0 7r
Since L
--
Let
(1)
the
pairs
7r/
follows
(W, 0)
and
are
0
T
7r'O T*
both
relatively
prime
pairs,
we
obtain
T* L'.
H' be the Hermitian
form
AA
=
determined
AA,
by T*L'.
where A'=
By
Im H'.
L
_-
T*L,
we
have
Ker(HA) F. A'
f
3.2.18
Theorem
By
pull-back
is the
is
f
Since
C F.
constant
is
of
a
hand,
On the other
Ker(HA) and (2)
but
Ker(AA)
xo
(Cl)
is
(CI)
Ker(AA)
E:=
Then E is
Cn IF.
on
contained Then
we
complex
a
in
have
.
yo
exists
A(xo, yo) 0
RA with
A'(xo, yo)
=
Then, there
satisfied. E
x0
E
By (1)
0.
0.
=
satisfied,
is
set
we
Ker(AA)-
Ui
subspace of
linear
Ker(HA)
the property
Cn with
c E.
Proposition
4.1.2
H be
ft
take
exists
Ker(ft)
with
=
satisfying
C'
on
E, if HA is
Furthermore,
in
this
on
Cn with
case
we can
definite.
positive
not
ft
form
Hermitian
(Cl).
condition
the
(CO).
the condition
semi-definite
positive
a
H satisfies
iff
H,
-A
form
Hermitian
a
there
Then
ft,
form
Then
C'.
E
x
not
Q.E.D.
When the condition
Let
any
MCA is
contradiction.
a
(E)
is not
We can take
A(xo, yo) This
for
function,
95
f6rallxEKer(HA)andyERA-
condition
the
Ker(HA)
+
x
alternating
R-bilinear
AA(x,y)=O,
(2)
on
non-constant
a
forms
Automorphic
4.1
Proof proof
The If
is
definite,
HA is positive
(Cl)
the condition
the
Weconsider inite
nor
Suppose
case
there
0,
automatically.
(Cl)
the condition
that
RA
=
=
exists
a
On the other
> 0
hand,
=
and neither
semi-definite
> 0
such
is satisfied.
Wenote that
.
f
q
=
-
k, k
real
There exist ED V2 i Cn
MCA( V1
=
linear
MCAED V1
(D
positive
def-
we
=
A(w', iw')
!
=
=
allw'112
for
all
w'
E
have b > 0 such that b
jjxjj
Ilyll
for
all
x, yE Cn.
as
c
-
2b2
b -
2
2
> 0
and
subspaces V, (D iV1 ( iV2
V2
Ker(HA) ED V, in this Ker(AA) Since HjCi,,C4 dimc Ker(HA).
such that
IA(x, iy) I :! c
=
zero.
H(w', w')
Take
Ker(HA)
Since
itself.
Lemma3.1.7
is
HA is positive
that
Ker(HA) E) V, iV, MCA Ker(HA) E) C,
and E
this
then
is satisfied
and V2 of Cn such that Let
of Lemma3.1.7.
generalization
a
c
-
2
b
2b 2 -
b
> 0.
C.
case.
>
0,
96
Reduction
4.
Let
V2
T:
x
and Extension
V2
R be
T(V21V2) Now,
define
we
>
AI(v,x):=O
f6rvEVj(DV2andxERA,
A, (V2, iV1)
A, (vj,
A, (V2, 'V21)
T(V2)
-A(vi
=
V21)
-Aj(x,ivj)
Aj(ivj,iv):=
Aj(vj,v)=0
Aj(iV2,X):=
-Aj(x,iV2)
for
0
=
7
exists
H + Hi.
that
=
ft
Then
Ker(A)
=
show
Ker(A)
E C
Ker(A).
-A
E.
=
iV2
V Ker(A). Ker(A). Therefore
Next
we
C'
V2
ED
(D
iV2
Noting
ft
is written
E is
=
(3)
(4)
following
V2,
V,
V2.
Eq
x, y EE Cn.
=
A,,
positive
uniquely
iAj (x, y).
for
(A,)A
A + A,.
=
0-
Ker(H-)
Since
=
Ker(A),
=
it
=
\ 101. (A,)A
Since
=
=
=
i
definite
on
0
(D
V2
ED
W2. Every element
as
wherewECandu,vEV2.
inequalities
ft(w, w)
i.e.
Ker(HA) E) V, 0 iVj, we can easily see Ker(AA)nV2 f 01, there exists 0 it follows that A(V2, Y) A(21 Y) 0 0we have A(iV2) iY)= A(V2 Y) =A 0. Then,
E
hand
is
part
Ker(A).
w+u+iv Wehave the
ft
Im
C:
all
+
H is obvious
A:=
E. Let
On the other
show that
V2 E
H, whose imaginary
A, (x, iy)
=
v
for
form
Take any V2 E V2
Then V2
V, and
E
vi
V2 and
iy)
A, (ix,
RA with A(V2, Y) 0 0. From
y E
for
V2,
V2 E
Hermitian
a
to
We show
suffices
iV2)
f6rV2EV2andXERA,
H, (x, y)
fl:=
R by
Cn
X
f6rVjEVjandvEVj0)V2,
A, (x, y)
Let
Cn
forviEVIandxERA,
A, (iV2, iV) : A, (V2 V) Then A, has the property
there
,
V2, v' 2 E
for
Aj(ivj,x):=
Therefore
V2.
f6rVjEVjandVEVj V2,
iV2)
=
:
form with
symmetric
V2 E
A,
form
f6rXEMCAandyE Cn,
-A(vj,iv)
all
for
AI(x,y):=O Aj(vj,iv):=
R-bilinear
CJJV2 112
alternating
R-bilinear
an
definite
positive
a
H(w, w)
Ifl(w,u)+fl(u,w)1=12A(w,iu)1<2blIwIlIjulI
!
al1w
112
for
w
c
0,
f6rw(=-C'anduEV2,
of
(5)
Jft(w,iv)+ft(iv,w)
(6)
Ift(u, ft(u, u)
(7)
iv)
ft(iv, we
fl(w
+
+
al1w 112 +(c
for +
w
(c
U
+
iv)
IIUI12
+
(c
b)
-
C' and
u,
v
b)
IIUI12
for
v
-
E
E
E
V2,
V2,
for
u
E
V2,
V2.
11ull JJvJJ
11wJ[ JJvJJ 11-2blI
IIVI12
(11WI, IJUJI) +(c-b-2b2) IIUI12 2b2IIVI12 a (11wl, JIV 11) +(c-bc bHull ( -2b--b 11V112+ (c-b -2 2
2b
a
V
97
0,
+ iv
u
for
IIVI12
b)
-
JJwJJ 11ull 11-2blI
2b
-
b)
-
iv,
:
+
we
T(u, u) ! (c
+
w
for
11ull JJvJJ
:, 2b
A(iv, i2V)
obtain
U
I
u)
A(u, iu)
iv)
(8),
Using (3)
2bJJwJJ JJvJJ
:5
ft(iv,
+
A(u, iu)
=
(8)
1
forms
Automorphic
4.1
_
2
2
a
a
2
2b
_
2
2
a
a
2b2
-
2
C
IIVI12
b
-
> 0.
Conversely we assume that there exists ft on C' with ft -A H. Suppose that there
exists
0. Let
y
xo E
iyo
:=
Ker(HA)
ft(y,
Y)
A(Y, iY) A(yo, iyo) A(yo, iyo)
=
=
=
If
we
take
Suppose tion
we
In this
t > 0
may
case
sufficiently
the
that
Ker(A)
large,
(CO)
conditions
assume
n
(Cl)
condition
Ker(AA).
xo
semi-definite
positive
Take yo E
is
Hermitian
form
fulfilled.
Then
not
RA such
as
A(yo, xO)
<
t E R. Then
for
+ txo
with
a
the
that
H is
RA
=
+
2tA(yo, xo)
+
2tA(yo,
then
fl(y,
y)
and
(Cl)
are
<
xo).
0,
a
By the
satisfied.
semi-definite
positive
Q.E.D.
contradition.
on
C'
above
and
proposi-
Ker(H)
=
E.
Ker(AA).
Proposition
4.1.3
following
The
(C2) the
A*
:=
statement
&(A)
projection
is
a
with
holds. discrete E
:=
subgroup
Ker(AA)
Ui
of Cn 1E, where
Ker(AA)
-
Cn
Cn IE is
98
Proof Suppose For
discrete.
dimensional
any
&(Aj)
A* is not
that
positive
a
and Extension
Reduction
4.
E
x
such that
&(x) Since Ker(H) Ho o (& x &).
H=
A* be the closure
Let
subspace
linear
RA with
&(x).
--*
real
S there
E
E,
=
exists
A0
Then there
(&
o
&),
x
exists
Take any A C- A.
JAj}
Hermitian
a
Im H =
=
of A*.
S C A*.
sequence
a
take
we can
Then A
S with
A such
c
form
Ho
where
Ao
=
that
C'IE
on
Im
Ho.
Wehave
A(A, x)
=
&(x))
Ao (&(A),
Ao (&(A),
lim
=
&(Aj))
A(A, Aj)
Since
A(A, x)
k.
=
E
x
Ker(AA).
From the Cn /A It
)
V, by
=
&
,0,\
X0 is
and t,\
a
toroidal
e(,s,\)
=
automorphic
=
there
k'
=
0. There-
Q.E.D.
obtain
an
toroidal
a
that exists
subgroup
epimorphism
a
for
so.
group
of
Then
E.
X is
by
X
:
REMMERT-
'dE,,x
*
C*12
and tE,A
E
e(SE,A) be the restrictions Ker(H) Ker(A), we
=
=
A C= A n E and
all
tEA
X0,
X
=
Since
for
'dE,A
factor
X
respectively.
o(A)
=
C"
=
of A n E
on
E
have
E E.
x
we
of
get the
Lemma
4.1.4
If
E)
Let
An E
'dE,A (X) For the
n
group.
on
we
discrete
a
Then k
argument.
such
x
hand,
1.1.5
EI(A where
C'IE
)
other
C E.
Here Y is also
A n E is
that
MORIMOTO's theorem
same
A and
on
On the
rx.
=
Ker(AA)
C'
:
depending
Z
E
the
contradicts
(Cn IE)IA*.
y:=
k
exists
J-00
0, A(A, rx)
>
r
This
projection
obvious
is
any
A(A, rx)
V E Z such that fore
there
Z,
c-
For
A(A, Aj).
lim
=
J-00
Ho (X, L) :A 0, then the automorphic
'dE,A
factor
*
tE,A
is cobordant
to
1.
Proof Consider there
that
the above
exists F
-
LIF
Then
a
EI(A is
Changing
the
'dE,,\
=
a
'
tE,A
fibre
projection F
=
a :
o--'(y)
X
for
y
=
(Cn 1E) /A*.
E), LIF is given by'dE,A'tE,), trivial analytically by Proposition
and
automorphic 1 when
homomorphism
go
'dE,A
factor
H'(X, L) =A 0, by :
A
)
go
(A)
S'
=
fz
*
1,E,A)
:
JzJ
LIF
is
if
=
necessary,
L) 0 0,
0. We know
topologically
trivial.
Q.E.D.
3.2.22.
the virtue
EE C
Ho (X,
HO(F, LIF) :A
y G Y such that
n
Since
we
may
assume
that
of the above lemma. Wedefine
11 by
9(A)
ifAEAnE
1
otherwise.
Now
we
may
we
Ho (X,
consider
tA
assume
from
L) :A
and 0,\
1 for
=
as
respectively.
0,\,
and
t,\
following
the
satisfies
o
'dE,A
'tE,A
Theorem 3.2.18
improved
is
the facts
to
E
=
tE,,\
by
follows,
as
that
and
1
=
Ker(H)
Ker(AA)
Ker(H)
=
HO(X,L) generates L on X or H'(X,L) Ho (X, L) with W2 :A 0 the meromorphic function
Suppose that E
P2
on x
+ 7r
Let
f
(E)
=
for
all
X
:
L2 A*
Cn /E,
on
and such
(,0,*,
0'
as
(,X) H*
,
where V*. A
&)
-
(t;(,)
is
&
a
&)
o
02 and 79*
m :
A
e(s,*\*
Then
A E AnE
we
are
exists
=
and it
for
by
o o-
projection.
=
.,0*
By
a
:
(Cn IF)IF X
)
we -
in
Y and
have
L*-.'
Ho
=
o
for
Since 02 t2 (H*,,o*). take H* x we can &), (&
of type
factor
homomorphism
tt(A)
any
is constant
.
ti
A
:
)
S' such that
&(A)
factors
with
the
Hermitian
same
We can take
cobordant.
a
form.
homomorphism
s,\(x)
=
=
all
have
(X))
8
=
*(&(X))
=
0
L
hand
we can
is the
X1
-r'
-r
=
take
factor
pull-back
e
m:
+
is* 2
&(A)
automorphic
=
p(A)
AEAnE. Thus A E A n E.
all
R such
A
an
on
for
1
(_21M(A) of
1
.,0* &(,\)(&(O))
,(A)
=
p(A)=
IL(A)
automorphic
/W2
and t,*\*(V) e(s,\(x)) e(m(A)). Let t,\(x) It(x) and -(1/2)m(A) + (112)s,*(,\) (&(x)) are cobordant. For
that
AEAnE.By (0)
Therefore
a
are
o(A )2 =,O,\(0)2 forall
Then,
(P1
of Theorem 4.1.1.
X
-r
and H
theta
o,67
sta(,\)(& On the other
proof
the
into
theta
cobordant
reduced
[t(A)-1
R such
)
(y)).
&
o
Hence t92A and
in
0.
f
Cn /A is the natural
=:
projection
reduced
are
,d2A (X) for
RA-
n
of the line bundle over X' by -r', pull-back factor L' is given by an automorphic 0,*\.
M. Then there
=
function
decomposed
is
L' the
Letting
X
C'
The
D E.
We assume that
u*L'.
=
Pf
we see
X.
)
:
7r
meromorphic
the
of Theorem 4.1.1
proof
the
X, where
E
x
W10 be
Theorem 4.1.5
TI
proof
of the
modification
trivial
the
and
1-
=
Theorem
4.1.5
7
if
1 becomes
=
19E,A
(9)
(P1
Then
(C3),
condition
A E A n E.
all
Then the condition
according
go(A)-l
-
that
first
the
99
0.
p(A)
(C3)
go(A)
-
forms
Automorphic
4.1
as 0
m(A)
&)
factor
for
=
A for
A E A n E.
0 for on
A*
C' on
is
Then,
cobordant
C'IE.
the
to
Furthermore
tx,
100
Reduction
4.
(--lm(A)
e
Therefore
(C4)
There
topologically
t*
o o-
L
pose that
y
a
A*
with
for
Y
over
Cn/E
on
(Cn/E)/A*
=
where
E
Hermitian
over
toroidal
a
(CO)-(C4).
defines
which
such that
onto
toroidal
a
Ker(AA)
and
t,\
:
that
)
go is
H
projection
There
=
g(A)
:=
Ho
for
o
exists
(&
some
semi-character
a
reduced
Sup-
epimorphism
an
Ker(H).
=
Cn/A.
=
by the
Y
group
U
form go
have
we
iKer(AA) Ho on Cn 1E such S' by Qo(A*) A*
=
X
group
Then
Then By (C3) go is well-defined. Im Therefore there exists := Ao Ho.
&(A). with
A*
conditions.
the
bundle
line
a
homomorphism
a =
associated
of
(Cn/E)/A*
=
Cn/E,
positive
t,*\*
factor
the conditions
definite
We define
condition:
bundle
sufficiency
the
L satisfies
X
line
trivial
forallAcAnE.
I
o
cobordant.
are
Cn
s,*(,,)
automorphic
an
Lq 0 Lo be
=
2
following
the
exists
a
1 +
2
satisfies
tx
Now we discuss
Let
and Extension
x
&).
A E A of A*
factor a t9o,,\* &. Let L,'O. be the theta bundle (Ho, go) such that 0.X 790,&(),) Let t,*,* be the automorphic over Y given factor for a*L,' bydo,.X.. Then Lq A* in (C4), and let Lo be the topologically trivial line bundle over Y given by
A* of type
for
o
=
=
Obviously
t**. A
Lo
Thus
o-*L. 0
--
L
Theorem
4.1.6
Let X.
X
=
C'/A
Suppose
=
.
we
0
have
(Lt .
Lo
toroidal
a
and let
group,
L satisfies
79
as
L'vo
& L'o
LO).
=
LV
0
Lo be
790
bundle
over
&L')00, 0
and the line
group
line
a
Then
ifHO(YLl
0
the toroidal
are
L
(CO)-(C4).
conditions
Ho(X,L)--Ho(Y,L'.0L') where Y and
0
(ABE [8])
be
that
Lq
theta
bundle
over
Y defined
above.
Proof The
epimorphism
Ho(X, L). If there
a
:
X
For any WE
exists
meromorphic Theorem 4.1.5.
a
the
injection
Ho(Y, Li OOLo), u*W is 0
section
function
Y induces
f
Hence o-*
E
HI (X, L) which is
:=
01o-* p
:
Ho (Y, L'
is not 90
(2)
o-*
constant
not
constant
Ho (Y,
:
on
fibres
constant on
on
a-1(y).
Ho (X, L) is
Lo)
L,'30 0 Lo) a-'(V) (y E Y). a-' (Y), then the This
an
contradicts
isomorphism.
Q.E.D. By the above theorem, to
the positive
definite
the existence case.
Weshall
problem discuss
of
holomorphic problem
this
sections
later.
is reduced
properties
Further The following
Riemann forms
for
definition
meromorphic
of
forms
Automorphic
4.1
is natural
101
functions
as we
showed before.
Definition
4.1.7
X
Let
CI/A
=
be
A:= Im. H is Z-valued
(1),
If HA > 0 in
By Proposition definite
on
H is
then 4.1.2
we
A
x
H
on
C'
is
called
a
A,
(CO)
the conditions
H satisfies
form
A Hermitian
group.
X, if
Riemann form for
(1) (2)
toroidal
a
(Cl).
Riemann form
ample
an
may
and
that
assume
for
X.
Riemann form
a
semi-
positive
H is
C'.
on
Lemma
4.1.8
H1, H2 be Riemann forms for H, + H2 is also a Riemann form for
Let
Ker(AA)
a
toroidal
=
C'/A.
Then H
X with
(Ker((AI)A))
=
X
group
(Ker((A2)A))
n
.
Proof
(1)
The conditions
before
positive
semi-definite
on
we
lemma,
the
Then,
C".
Ker(H) Using the facts
it
was
to
been
we
proof
the
by
the
proved
0
X
:
=
C/A )
Lie
X,
the
manifolds.
Ker(H2).
n
MCAn Ker(AA),
=
we
by
a
over as
to
for
theorem
reduction
extended
was
non-compact
ho-
due to
for
reduction
ABE
[4]
shows
more
another
that
We mention
Lie groups.
How-
precisely
proof
of the
and CATANESE[201. CAPOCASA
toroidal a
meromorphic It
meromorphic
of the
reduction)
(Meromorphic
be
group
(Ker((A2)A)),
and SNOW[48]. HUCKLEBERRY
theorem following meromorphic reduction.
Theorem
Abelian
are
that
n
of the
is obtained
theorem
X
and H
Q.E.D.
know the existence
how to get
Let
fulfilled
are
proved.
mogeneous manifolds
4.1.9
Ker(HI)
(Ker((AI)A))
=
compact homogeneous complex
ever
is trivial
=
GRAUERTand REMMERT [371
Then
may
(CO)
H1, H2
that
assume
Ker(H) n RA and Ker(HA) Ker(AA) for H and (Cl) is satisfied
Ker(AA) as
=
it
we
the condition
that
see
and
of the Riemann form
the definition
in
note
H. As
for
group.
quasi-Abelian
fibres,
which
Then there
variety has the
exists
X,
following
with
a
holomorphic
the
properties:
connected
fibration
complex
102
Reduction
4.
toroidal a homomorphism between gives the isomorphism p* : M(Xi)
1.
q is
2.
p
3.
If
T
X
:
there
the
into
homomorphism quasi-Abelian variety
a
M(X),
)
homomorphism
a
meromorphic
the
groups.
unique
such
that
is called
Y is
)
exists
means
X,
and Extension
f --+ f 0 quasi-Abelian
a
a
: X, X, exists
0.
Y with
)
Y, then
variety a
T
This
op.
uniquely.
of X.
reduction
Proof If there
exists
lytically In this
may
all
H'(X, L)
have
we
X, is the trivial
case
Suppose
Riemann form for
no
trivial
that
there that
assume
Riemann forms
for
0
by
any line
the previous
bundle
L
X not
ana-
Hence M(X)
results.
C.
=
group.
exists
H is
X, then for =
Riemann form
a
H for
semi-definite
positive X which
are
By Proposition
X.
C1. Wedenote
on
semi-definite
positive
on
by
4.1.8 R the
C'.
Let
that
E
we
of
set
E:=nKer(H). HER
By A*
there
Lemma4.1.8
Consider
the
(A)
=
is
an
mann
form
f
f
--+
Lie
o
p is
group
X. Then
R,
x
E C
EI(E X,
X
C'IE.
of Cn 1E. Let
X1. Since is of
course
Then
we
=
X,
closed
a
=
4.1.3,
Then
Lie
any Rie)
M(X),
subgroup
El (E
fibre
with
H,
complex Abelian
complex
bundle
in-
form
an
A connected
Theorem 4.1.5.
using
By Proposition
(Cn IE)IA*.
:=
have
Ker(fl).
=
ample Riemann For a quasi-Abelian variety. can see that p* : M(Xi) we
A) (E + A)/A is X1 (Ker q) is a fibre
n
E R such
C'
:
)
), X, Ker(H).
isomorphism
an
Ker p o:
X
o:
=
HE
Riemann form
a
projection subgroup
discrete
a
epimorphism for X, with ft H, o ( duces
exists
canonical
of
A) (cf.
n
HIRZEBRUCH[45]). Next
we
show the
quasi-Abelian above.
(3).
property
Y, and let
variety
For any x,
E
Let
Xi,
p
is constant
T
-r :
we can
for
x
a/
some :
X,
Since
)
4.1.10 Let
define
&)
with
a
Since
)
X,
It
xi.
is
E
a
=
fibration
in
If it
given is not
so,
a
the
there
meromorphically separable f (T(x)) 54 f (T(x')).
the fibre a
that
homomorphism with
is onto,
into
such that
mapping obvious
homomorphism
o-1 (xi).
Y is
M(Y) on
a
be the
the fibre
on
holomorphic
=
Y be another
X
o:
X,
E
Therefore
Y be
X
p-1 (xi) with T(x) 0 T(x'). Theorem (Main 3.2-21), we can take f However f o T E M(X) must be constant x, x'
exist
X
:
:
p-
=
(xi),
X, a'
a
contradiction.
Y by a(xi) := T(x) homomorphism. Let
)
is
o-
T
1
a o
p.
Then
o-
o
o
o7'
=
o
p.
Q.E.D.
u.
Corollary
X be
a
toroidal
group.
Then
M(X)
=
C iff
X has
no
Riemann form.
Proof If
there
exists
a
Riemann
form
for
X,
then'
the
quasi-Abelian
variety
X,
Automorphic
4.1
M(X)
M(Xi)
--
Conversely, f
0 by L.
there
that
f
exists
take
can
we
with
Thus
dimensional.
positive
is
L
bundle
(CO)-(C4),
conditions
function
X and two
-4
H must be
sections
determined
form
Hermitian
H be the
Let
meromorphic
non-constant
a
line
a
W10.
=:
the
L satisfies
Since
X,
---+
C.
Then
Ho (X, L)
E
X
:
o
103
Riemarm form
a
Q.E.D.
X.
for
meromorphic
The
but
varieties, It
54
suppose
M(X).
E
W,
reduction
meromorphic
the
in
forms
can
variety quasi-Abelian the meromorphic
this
well-known,
is
degree
of CHOW, if
n
reduce
the
t in
limits
quasi-Abelian
of
point. a complex
a
of
0 < t <
meromorphic
> 1, hence the
to
field
function
those
are
groups
can
that
have any transcendental
orem
of toroidal
fields
function
n
reduction
torus
group
by
given
a
The-
have any such
can
dimension.
only
has
which
Example:
[64]
1964
KOPFERMANN gave in constants
as
generates
a
toroidal
,F2 tv 5 i.\,F3 i 001iV7
C'1A
=
i
i
0 10
=
X
group
group
coordinates 100
P
toroidal
non-compact
a
functions.
meromorphic
in standard
The basis
example of
an
on
which
meromorphic
all
functions
are
constant.
automorphic
of
Existence
forms
and Lefschetz In this
section
morphic
and Lefschetz
sections
Werefer X
C'/A
=
over
X. As
form
H is
there
exists
compact
have
a
toroidal
seen
Khhler
form
manifolds,
Unfortunately,
on
in
the
the
Then
Consider
group.
in Theorem
definite
more
Our purpose
manifolds.
groups.
need
general give the
only results
state
we
is to
here.
proofs.
for
papers a
positive
Khhler
&9-Lemma. groups.
we
be
toroidal
about
original
the
to
type
of holo-
existence
proofs
His
theorems.
weakly 1-complete
for
knowledge
systematical
of TAKAYAMAon the
results
recent
state
and technic
results
Let
we
theorems
type
a
if L is
3.2.13,
holomorphic positive,
MCA, equivalently the
Chern
'first
converse
is
shown
89-Lemma does not
KAZAMAand TAKAYAMA[54]
proved
on
that
by
in a
general
toroidal
words,
other
H'(X,R). of
virtue
hold for
E
Hermitian
its
In
C'.
cl(L)
class
L
bundle
line
then
the
For
so-called
for group
toroidal X the
104
Reduction
4.
00-Lemma holds
following
the
X, iff
on
X is
[1101)
in
L be
3.1
[110])
in
TAKAYAMA[110]
group.
proved
weak 09(9-Lemma
the
using
of the A9-Lemma.
instead
holomorphic
a
determines
line
Hermitian
a
the
above
H
on
any
proposition
(Theorem
manifolds
conditions 1. L is
toroidal
a
Suppose
relatively
X
group
that
H is
TAKAYAMAproved
which
definite
positive
for
theorem
Cn/A
=
compact open subset
ampleness
an
[109]),
6.6 in
H be the
L and
on
on
Cn.
on
on
of X.
weakly
1-complete
the
(TAKAYAMA [110])
Theorem
4.1.12
bundle
form
MCA. Then, L is positive
Let
theta
Proposition
4.1.11
By
toroidal
a
(Theorem
proposition
(Lemma 3.14 Let
and Extension
same
as
Proposition
in
following
Then the
4.1.11
two
equivalent:
are
positive,
2. H is
positive
About
the
definite
MCA
on
of
existence
non-trivial
the
sections
following
is
conjecture
well-
known:
Conjecture.
Let
determines
a
non-trivial
section.
Partial
results
conjecture
form
non-trivial
a
a
toroidal on
X which
group
MCA,
by COUSIN [26] and ABE [8]. Recently general case by TAKAYAMA.
then
L has
a
the mentioned
in the
line
Suppose
bundle that
on
H is
a
toroidal
definite
positive
X which
group
on
determines
a
MCA. Then L has
a
complex
linear
HO(X, L) has the infinite
space
dimension,
if
X
compact. bundle
line
with
connection
4.1.14
type
any
a
the
3.10).
to
one-to-one
be very
ample,
if
HO(X, L)
immersion)
holomorphic
into
existence
of
sections,
TAKAYAMAproved
the
following
(TAKAYAMA [110]) line
bundle
on
a
toroidal
group
X.
Then
L'
is very
ample
t > 3.
TAKAYAMAimproved
Theorem
a
said
space.
positive
integer
X is
theorem.
Theorem
L be
L
embedding (i.e.
a
Lefschetz
Let
on
definite
(TAKAYAMA [110])
holomorphic complex projective
In
for
the
holomorphic
gives
bundle
line
section.
Moreover,
A
H.
holomorphic
known
holomorphic
a
Hermitian
a
form H. If H is positive
proved
was
L be
is not
were
Theorem
4.1.13
Let
L be
Hermitian
the
above theorem
in
the
next
paper
(Theorem
3.4
and
Automorphic
4.1
Let 1. 2.
X be
a
toroidal
L is very L
2
is very
that
The second Theorem
[85)
with
group
ample, if
X is
ample,
if there
(A, LIA)
is
statement
for
105
(TAKAYAMA [111])
Theorem
4.1.15
forms
a
bundle
line
L. Then
torusless,
principally in
positive
the
the compact
does not exist
polarized
a
Abelian
above theorem case.
non-trivial
is
subtorus
A of X such
variety.
known
as
OHBUCHI's Lefschetz
Extendable
4.2
A toroidal
dles
group
bundles,
the
and next
X
1.1.14
=
Cn/A
that
bundle
on
of fibre
has structures
a
q-dimensional
(Iq i )
phism
Cn
p:
gives Cq
P(Z1 Then
p(A)
subgroup Lie
groups.
7
)
...
Zn)
is the
properly
In-q
of X
as
:=
Cq
We define
T.
(Zl)
e(Zq+l),
Zq,
...
follows
a
ical
We define
projection.
X
homomor-
group
e(zn)).
....
pn-q 1
X
automorphisms. point
)lp(A).
an
Cq
pn-q 1
complex projective
and fix
pn-q 1
(Cq
commutative
X
one-dimensional
discontinuous
in
C*n-q-principal as
X
,
manifold
seen
)
i
R, R2 torus
We have
q.
by
for the group of these extended is
of type
P of A is written
Iq
0
a
n:
where P,
(
bun-
f.
on Cq subgroup of Cq X C*n-q and acts naturally X Thus C*n-q have -_ we automorphisms. (Cq X )lp(A) Any 77 E p(A) can be extended to an automorphism
is
of
C*n-q
X
=
line
of C"'-fibre
case
bundles
representation
a
q-dimensional
a
of kind
group
The basis
torus.
p
The basis
toroidal
define
coordinates
extendable
the
discuss
of C*n-q-fibre
case
non-compact
a
We can consider
We first
Riemann forms
ample
with
case
be
toroidal
bundles.
of fibrations.
The Let
bundles
compactification
each
in
line
embedding
-k t
X
p(A)
of
gives
a
also
Cq
on
be the
X
through
X
P(A) pn-q 1
complex
compact
pn-q 1
X
:
complex
as
We write
space.
Then it Cq
:
as a
7
The action
free.
Let
C*n-q
canon-
following
the
diagram Cn
P
Cq
X
pn-q 7r
X
where
ir
:
Cn
Now, consider on
t(X).
The
X
=
Cn /A is the
X
projection.
holomorphic line bundle L, on problem was studied by following a
We see X. Then
t(X)
(t-')*Ll
=,k(Cq is
X
a
C*n-q).
line
M. STEIN 1994 in his thesis
bundle
[108].
When is there
Problem.
(t-')*Ll in this
The results Take
a
subset
p(l)
:=
Cq
Ll,.(x)
-_
section
n
(Xq+I(I)
X
(I) is c X(I)
Xp
qJ. Letting F := where x X,,(I)),
-
...
have
t(X)
The
projection
&
Consider
X(r)
line
holomorphic
a
Cq
=
p(J)
X
OL of L
pull-back
The
C*, d,\(z)
(I)), we =fr(Cn).
Xp
the
t(X)
space
of the
q variables
first
bundle
Cn-q -fibre
T and the
given
p(A)
X(I)
by
(Cq
X
I C
p(p)
n
-
q}.
Since
factor
automorphic
an
X
with
C*.
on
factor
given by the automorphic
X is
d
:
A
x
Cn
ap(,x)(p(z)).
:=
L, A
)
x
X be
Cn
)
(t-')*Ll
with and
:
_k (Cq
X (I)
Letting
Lemma
4.2.2
:
X
Lemma
4.2.1
0
L
Cq
set
lemma is obvious.
following
Let
bundle
we
i(CqxP(J1,...,n-qJ))
X:=
a
Cn , L is
-
a :
The
.
Cq onto
bundle
q} \ I,
-
if i E Ic.
OfCqXpn-q
pn-q 1
-principal
\ 101
P,
Wealso define X
n
T.
)
:
Cq
:
C*n-q
the
induces
.
c
&
with
on
if i E I
C :=
open subset
an
L
?
Xq+i(I) Then Cq
bundle
line
107
due to M. STEIN.
are
I C
X
holomorphic
a
bundles
line
Extendable
4.2
an
a
C*.
holomorphic
line
Then there
exists
Ll,(x),
c:--
automorphic
iff
factor
there a :
a
exist
p(A)
x
factor
given by an automorphic line bundle L holomorphic
bundle
function
holomorphic
a
(Cq
p(l))
X
W(z+A),3,\(z) o(z)-'=ap(,\)(p(z))
)
V
:
X (I)
Cn
C*
C* such that
f6rall(A,z)C:Ax
Cn.
Proof If
(t-')*Ll
_-
LJ,(X),
then
automorphic
be the
factor
L,
2!
which
t*(Ll,(X)). defines
Let L.
a :
Then
p(A)
a o
(Cq
X
p and
0
X are
p(J))
C*
by
cobordant
Lemma4.2.1.
The
is
converse
Let the line
:
A
x
proved by the
Cn
bundle
)
on
of Cn. Weconsider
(C)
C* be
a
same
reduced
X determined
the
following
Q.E.D.
lemma.
by )3.
theta
factor
of type
Take the canonical
condition:
ImH(A,ej)=OforallAEAandq+l<j:5n.
(H, p), unit
Lp
and let
vectors
el,
.
.
.
be ,
en
108
Reduction
4.
and Extension
Lemma
4.2.3
Let
0: A
the
condition
line
bundle
C* be
C'
X
(C),
then
for
X(I)
L
reduced
a
any I
f 1,
C
(t-1)*Lp
with
factor
theta .
.
.
n
,
-
of type
q}
there
(H, p)
which
exists
a
satisfies
holomorphic
Lj,(X).
_-
Proof decomposition
Wehave the
Cn
=
(el,...'eq)c
where MCA =
have Im HI RA X V
of the choice
MCA(D V (D W RA (D W)
-::-
=
and V
0. We may
Hlivxv.
of Im
that
means
the
assumption
we
0 because of the freedom
f6r1
independent
H is
en)R. By
....
Im HICnXV=-
Then
H(ej,eq+i)=O This
(eq+l,
=
assume
Zq+l,
on
7
...
Zn-
We can write
,Q(ej)=e(k(ej)),
k(ei)ER
f6ri=q+1,...,n.
Letting
g(z) we
:=
-k(eq+l)Zq+l
define
01,,\(z)
e(g(z
+
A)),3,x(z)e(g(z))-l
( 1H(z, 1
e(g(A)),o(A),e Then,31 is also From
(A+
reduced
a
91(eq+l)
,ol
Of
=
=
...
+
a,eq+l
0 and 01
course
the above property. is well-defined
(en)
Since,%(z)
is
see
obtain
=
cobordant.
Then
an
(H, pi),
of type
2
H(A, A)
(A)
where '01
e(g(A)),y(A).
follows
1 it
..-an-qen)
are
2i
p,
(A)
for
A E A and a,,...,
Therefore
we
form
automorphic
may
a :
assume
p(A)
X
(Cq
an-q that X
E Z.
'o itself
C*n-q
has C*
by
Thus the line
Wecan
=
1
A)+
-
factor
theta
101
ap(,\)(p(z))
we
k(e,,)Zn)
-
independent bundle
that
the
a
L
on
:=,3,\(z) 7 Zq+l, X(I) defined
on
is extendable
...
to
A G A and
for
p(A)
Zn I
a
by X
z
is extendable
a
(Cq
has the X
pn-q)
E Cn.
to
required
p(A)
X
(Cq
X
property.
in the above
proof
p (I)).
Q.E.D. Then
Extendable
4.2
exists
a
0
A
:
subset
a
109
L
satisfies
(t-')*Lp
X with
on
the condition
LI,
--
(C),
then
there
If
there
exist
X(I)
with
qX).
Lemma
4.2.5
Let
bundle
line
holomorphic
(H, p)
of type
factor
theta
reduced
a
bundles
Corollary
4.2.4
If
line
(t-')*L,3
C* be
C'
x
I
C
n
Ll,(X),
--
reduced
a
qj
-
and
0 satisfies
then
factor
theta
(H,,q).
of type
bundle
holomorphic
line
the condition
(C).
a
L
Proof 0
Suppose that
(q
ej
automorphic Cn
0:
n)
such that
factor
b
:
A
x
p(z
H(Ao, ej) (Cq X p(j))
0.
+
ej)
+
hand,
On the other
b.\0 (z
a
we
+
ap(,\)(w) e(b,\ (z)).
ej)
b,, (z
-
b,, (z)
+
have
an
function
holomorphic
L
ej)
+
+
Ao)
+
be, (z)
have
a
6
define
(z)
ej)
b,\. (z) Lp
--
H(Ao, ej) :A
Im
+
we
we
:
:
a
A
(Cq
X
X
Cn
x
6,,
and
mapping
p(A) (z)
C
Z,
E
we
0.
-
b,j (z
-
If
E Cn.
z
b.\0 (z)
-
be (z
-
,.,
+
Ao)
-
b,\. (z)
0.
Q.E.D.
contradiction.
3: A
x
Ljqx), L, 2--
iff
exists
a
0 satisfies
be
L,,,
C* be
Cn
Then there
L,
we
the
in
Theorem
4.2.6
Let
Ao)
+
by
obtain
Also
e(ap(;k)(w)). Since 6,\ (z
=
We sum up the above results
Let
a
E A and
We can take
op.
a
=
e(b,\(z)).
=
=
=
is
By
A0
exist
Lemma4.2.2
any A E A and
for Then
0,\(z)
b.\0 (z
This
Then there
C* and
A)O,\(z) o(z)-l
+
A)0,\(z) p(z)-1-
(z) bei (Z)
p, then
b.\0 (z
have
+
C such that
a o
X
C such that
Cn
P(I))
p(z
=
,\(z) )
by
(C).
Im
C* such that
)
ap(,\)(p(z)) We set
p(A)
a:
the condition
satisfy
does not
j :!
+ 1 <
a
0
where
and
automorphic ,3 of type (H, p). We say
topologically
line
in the
trivial
a
L,,
Lp
proof of factor
L,.
L
on
X
(or X)
with
(t-')*Lp
(C).
a
is
factor.
bundle
bundle
is
that
theta
line
the condition
factor
The argument
reduced
holomorphic
holomorphic
Lp,
a
on
a
toroidal
topologically a
theta
the condition
Lemma4.2.5 we
is also
obtain
line
valid
the
a
we
bundle
given
reduced
theta
(C) for
Then
X.
group
given by
bundle
L, satisfies Then
trivial
if a
0 does
line
have
by
an
factor
it.
bundle
L, with
and Extension
Reduction
4.
110
Lemma
4.2.7
L,, 0 Lp be L, does not satisfy
Let L, If
bundle 4.2.8
Let
L
exists
a
line
n
bundle
then
qj
-
L
toroidal
line
no
toroidal
on a
Ljqx),
!--
above.
as
holomorphic
line
Ll,(x).
--
bundle
X
group
exists
(t-')*Ll
(t-')*Ll
with
on
on a
there with
holomorphic
trivial
homomorphism A
bundle
(C),
condition
any I C
Proposition L, be a topologically
If there a
X(I)
on
the
for
line
holomorphic
a
then
group
X.
L, is given
by
C*.
Proof that
We suppose
Proposition
3.8,
the
By
01
=
=0 for
for
(Cq
X
=
Zn);
1
all
any I C
p(j))
M1i***iMn-qEZ;
+Mn-qen)
+'**
=
x
factor automorphic By ax(z) e(ax(z)). summand a: A x Cn the automorphic
properties: ...
assumption,
an
that
assume
A =Mleq+l for j is Zej-periodic
p(A)
:
a,\(Zq+l,
by
given
is may
following
C have the
1) ax(z) 2) a,\(z) 3) ax (z)
L, we
q +
.
11,
n
C* and
n.
,
.
q I there
-
exist
an
automorphic
W,
function
holomorphic
a
factor
Cn
:
C*
such that
0,'(A)(p(z))
H
(Z)
ae,
j
=
=
q +
fj(Z) Cq
Eni=q+
_
X
Cn-q
.
1
f I(z
Then
n.
kizi
1 for +
ej)
j -
(j
=
=
1,...'n,
kj
E Z
q +
of
part
W'(z) WI(z)
Cn with
on
f I(z)
periodic
the
A)ox(z)W,(z)
q +
=
Zej-periodic
is
Expanding
+
f,
function
a holomorphic 0P1(,,)(p(z))
We can take
W'(z
=
=
e(fT(z)).
Since
for Zej-periodic Therefore (j n). q + n). We write z (z', z") is
=
=
I
f (z),
obtain
we
n
f,,(z')e((o-,
f,(z)
+1:
Z"))
We have also
the
1:
(z)
a,\
zi.
(z)
of a,\
expansion
Fourier
ki
i=q+l
,EZ'-q
z")).
a,\,,e((a,
CEZn-q
Take
V
p(A)
:
(j
periodic
=
X
q +
(Cq
X
1,
p(j)) n), we
b,(,,)(p(z))
e(b,).
C with have the
similar
Since
b,(,,)(p(z)) P
is
Zej-
expansion
E bI,,(z')e((a,z")).
=
A
P
o,EZ'-q
If i E o-i
1, then Xq+i(I)
< 0.
It
follows
from
=
(*)
C. Therefore that
b,,\,, (z)
=
0 for
a=
t
(al
...
7
O'n-q)
with
Extendable
4.2
b,
ill
z"))
(z')e((u,
(a,\,, UEZn
bundles
line
fi(zl
+
A')e((o-,
+
A"))
f,, (z)) e((u, z"))
-
"
-
n
E
+
(mod Z).
ki Ai .
i=q+l
Thus
obtain
we
for
a
=
variable
Un-q)
t
(al Zk (1
....
I
<
k <
fli(Z'
+
a,\,,
with
q),
on
T
(t
we
t2q)
trivial
+
fi(e((u,
equality
above
by the
=
aZk
(rl,.
-
r2q).
-,
homomorphism tj of L,. section holomorphic
a
by
a
(see
Remark 1.
_=
f,,
fo
A"))
a
r
1)
-
e((a, A"))
A with
not
depend
j
F,
E
the
16).
Hence
be
Let L,
a
line
e(-(u,rj)).
--+
For X is
I
Of. /OZk
toroidal,
0 for
_=
bundle
Then
k
=
L,
1,
we
is q.
put
ax,,
some
0
=
Of
A"))
(Ri R2)
and
=
f1i for
A')e((a,
+
defined as
f,I ( z') Since
1
ji)Z2q
analytically
Then
17
(9f.IlaZk
consider
can
not
(Iq Cql(jq
=
f,,(z 1)
-
Differentiating
< 0.
o-i
A"))
have
we
Og (z Zk Weset
A')e((o-,
+
f
the
on
we can
:A
same
e((a, A"))
-
1
( a 11
Un-q)
....
we can
with
o-i
< 0.
write
ax,,
-
e((o-, All))
1. The
ax,,
I
0,
of such
choice
get by the
t
=
a
A for
a,\
of the
side
right-hand
(z)
is
an
above
equality
does
summand. If
automorphic
argument for
a=
t(Oli
some
i E I
...
Un-q)
)
with
ui
> 0.
We set
Z(J) Then
:=
jor
E
Zn-q;
F-aEZ(I) fie((u,
or,
z"))
< 0
is
for
convergent
on
or
Cn-q
o-j
.
> 0 for
For
any
some
a
j
E Zn-q
define ax,,
e((u, All))
for
some
A E A with
E
e((u, A")) :A
1.
Icl. \ 101
we
that
We note
\ 101
Z,-q
\ 101
EZn-q
or
and Extension
Reduction
4.
112
U,
=
such that
f (Z)
and there
Z(1),
C
a
Z (I)
f,
fi.
=
E
:=
11,
I C
exists
n
q}
-
for
any
Then
f,
e
((o-, z11))
0rEZn-q\f0J converges
Cn-q
on
f (z
+
A)
+
and satis-fies
,
a,\(z)
f (z)
-
[a,\,, a.\,
a,\(z)
Thus
1)] e((u, z"))
-
+ a.\,o
o.
is cobordant
to
homomorphism A
a
E) A
Q.E.D.
E C.
a.\,o
F-+
Theorem
4.2.9
L, be
Let
A"))
f,(e((tT,
+
holomorphic
a
holomorphic
satisfying In
this
L,
!--
line the
case,
L',
o,*
L
where
o-
take :
toroidal
on a
(t-')*Ll
with
on
X. Then there
group
Ll,,(X),
-
iff
L, is
exists
theta
a
a
bundle
(C).
condition we can
bundle
line
bundle
T is
X
L'
bundle
theta
a
C*n-q
a
T
on
-principal
flZ2q
Cql(jq
=
that
such
bundle.
Proof Assume that
Ll,,(X). the
By
there
exists
Lemma 4.2.7
representation
holomorphic
a
satisfies
L,
We have
above.
as
by
line
Theorem 4.2.5.
is
a
theta
In this
type the
a
(H, p).
we
may
assume
Furthermore
A E A and a,,...,
)
C*
first
by
has the desired The
converse
that
we
of Lemma4.2.3,
space of the
Cq
L,
Let
L
bundle
line
L'
=
0
c-_
Lo
-k
on
be
with
Then
is cobordant
,o(A for
(C).
(t-')*Ll
with
on
to
0 L-
A
C*.
reduced
theta
homomorphism
a
:
Hence L,
bundle.
case,
proof
4.2.8
L
holomorphic
a
(L By Proposition
bundle
condition
the
i.e.
may
=
a,,-q
Let
Then :=
3,\(z).
where 3 is that
+
On
an-qen)
Cn
we can
define
The theta
a
H and Q have the
independent
H is
E Z.
Lp,
assume
+ a, eq+1 +
q variables.
a',(,\)(&(z))
L,
)
Zq+l, =
-
-
properties
of in
and
Q(A)
Cq be the a
Zn
-,
factor
theta
bundle
projection
factor L'
a/:(Iq
onto
flZ2q
T defined
the X
by a'
property. is obvious.
Q.E.D.
The
A toroidal
principal
bundle
Cn/A
=
as seen
of type
before.
of kind
q has the
It has of
113
f of the natural
structure
other
course
bundles
fibrations.
In this
C*n-qsection
we
associated problem of line bundle concerning the fibration f. form of kind We Riemann must some modify parts of the ample
the extension
consider with
X
group
case
line
Extendable
4.2
an
in the
argument
[12]
used in
previous
study
to
The results
section.
in this
section
admitting
functions
meromorphic
are
due to ABE and
algebraic
an
addition
theorem. of type q (1 < q < n), and let L, quasi-Abelian variety Theorem 3.1.4 we have the decomposition it. on a positive By L, L, 0 LO, where L, is defined by a reduced theta factor ?9,x of type (H, g) trivial. and LO is topologically Then H is an ample Riemann form (Theorem definite that H is positive We assume on Cn (Lemma 3.1.7). By 4.1.12). may of theta factor, there exist extension Lemma 3.1.8 and the natural a discrete subgroup A of rank 2n and a theta factor 0_ on A x Cn such that ?9,\ is the and A of restriction on A x Cn, A c A as subgroup Cn/A is an Abelian Let
X
Cn/A
=
be
be
a
bundle
line
=
=
variety. Now we
by Hn the Siegel
Wedenote 4.2.10
Let
Normal
P and
1
(0 half
upper
< V <
n
q).
-
degree
space of
n.
form
of
P be basis
using invertible Theorem
H is of kind
that
assume
we
and A
obtain P
the =
After
respectively.
and unimodular
matrices
3.1.16),
A
normal
(W D),
forms W
a
(see
matrices
change of basis proof of the Fibration
suitable
the
of P and P
(Wij)
G
as
follows
'Hn
di D
dn where di
(i
1,
.
.
.
,
n)
are
positive
integers
with
d, I d2l
...
Idn,
and
di W/
dq+t
(P/ Pit),
dq+,+,
P
wit
dn0
114
Reduction
4.
where
and Extension
put
we
Wl,q+t
W11
W/
Hq+j,
E
Wq+ J
Wq+i,q+t
Wq+i+l,l
Wq+f+l,q+
WI/ Wnl
Then the
Ci
X
projection C*n-q-21 -fibre
T
Cn
:
pn-q-'-fibre 1
associated
Cq+f
bundler
bundle
(q
first
the
onto
A.
X
:
Wnq+t
...
Compactifying A. We shall
T
f)-variables
+
fibres,
the
study
gives obtain
we
a
the
extendability
the
-
of L,
X.
to
We first
define
P(ZI) The
...
Zn)
i
(Zl)
=
;
...
precisely
more
homomorphism
group
a
problem
the
formulate
Cn
p:
as
Cq+f
)
X
e(Zq+i+1),...,
Zq+i,
previous section. C*n-q-21 x C' by
e(zn-q)7
Zn-q+l
on Cq+t p(A) acts properly discontinuous X Then X pn-q-t) we have an isomorphism lp(A). 1
subgroup
(Cq+ CTp(A)
which
Cq+f
and fr:
Weconsider
as
x
considered
gives
=
as
P,
(z, w) a
real
embedding
natural
pn-q-t 1
be
the Problem
A, ,
of Ox. Let of Cn
X
the
on
P
as
=
(zl,.
basis x
=
We assign x
W1i
=
-
-
( x,, )
t
we
2n
for
E R
.
Let
C*n-q-V
X 7r
Zn)
:Cn
X
X
give the
form
normal -
E Cn
x
i
o 7r.
Wn-q-f).
-
x', x"
Wx' + Dx",
(Cq+
p
o
...
complex coordinates (zl, Zn) The basis P (W D) is
the
We write
.
fr
7
pn-qI Let
First
situation.
, Zq+i, of Cn. Then any .
X
X
:
with
projections
p 106 in this
above.
t
We
the
in
-
-
1
=
be
can
C- R
represented
as
n .
x
E Cn.
theta
factor
X
It
is well-known
that
is cobordant
790,&) for is
E
A
to
the
?90,.\
restriction
have
A' where
e
I-
WA'+ DA" and
with
cobordant
we
=
=
to
'A'x
x
0
,
-
2
E Cn
do'
of
(a)
the
(cf.
on
A,
=
'A'WA'
do'
I
Theorem 3.1 in
A xCn.
(b) 0
We note
[631). that
Thendx for
A E A
Extendable
4.2
Zq+f
C
a
b=
,
E
explicitly
writedo,.\
A
Pa + P"b and
=
dX be
then
a
we can
theta
factor
take
a
(z, w)
with
basis
P of A
-
2
Hence
we
obtain
the
We say that a
(N)
subset
is the
normal
I C
n
q
-
1 2
E C'.
form
of
2f}.
an
ample theta
:=
represented
f,
as
factord),
Wedefine
C
Xq+t+i(I)
ample and of kind
taW'a
-
(z,w)
-
H is
above and Vx is
taz
e
H. If
form
in the
as
=
A=P'a+P"bEAandx=
Take
E Cn.
Hermitian
dx (x)
(N) for
=
taWal
1
taz
e
=
Proposition
4.2.11
Let
x
zn-i.
follows
as
,do,.x(x) for
115
bn-i
aq+t we can
bundles
bi
a,
Then
line
if i E I
Pi \ 101
if i E IC.
We set P (I)
Xq+t+l (I)
:=
X
*
,
,
X(J):=,k(Cq+i (Cq+f =
Then
we
have
t(X)
C
X(I)
C
-
A* where T
Cq+f
X
pn-q-i
X
Xn-t (1)
X
x
X
(P1 \ f 01)f)
p(J))
lp(A).
P (I))
Let
:=
-f
(p(A))
=
T-
(A),
Cq+i is the projection.
Then the basis
d,
dq+f and A
=
Cq+f /A*.
Wedefine
,Ooo*,,\*
a
=
theta
1-
factor
eta*z
0*,),,, 0 2
on
ta*Wa*
A*
X
Cq+t by
P* of A* is
for
A*
W'a* + D'b*
=
by 00*0'A*
and Extension
Reduction
4.
116
Let
*
I C
n
L
:=
q
-
-
and
T*Lv.* VI
Cq+i
E
z
,di
:
p(A)
x
(Cq+
,dj,,7(y) hand, d,\ (x)
On the other
the
obtain
we
bundle
defined
A. For any
T
LIX(j)
Cn. Then
-
is
given by
an
=
C*'
=
(-T op(x)).
i90*',j:.P(,\)
following
P(I))
x
Hence
which
proposition,
we
(t- 1) L'9
have
1,(X)
*
Lv is extendable
says
the
to
.
compactification
Proposition
4.2.12
L, there A* X Cq+t
For
exists
the
line
T*L, *,
where
C* such that
L,
investigation
of topologically
-
A given
LO*0
bundle
T :
0
by
Cq+t
X
factor
theta
a
Cn-q-i
0*0
Cq+i is
projection.
the
The rest
X
Let
is the
trivial
line
bundles.
above,
and let
Proposition
4.2.13
Cn /A
=
be
trivial
topologically holomorphic
variety quasi-Abelian line bundle holomorphic
bundle
line
Then there the
exists
a
as
a
Suppose
that
C* such that
Lo
X.
on
there
X be
a
exists
a
p
o -T
:
Ll,(X).
-
homomorphism 0
homomorphism
Lo
such that
L
(L-TLO
by
p(l)
X
L,00. by
of
pull-back
be the
have Cq+t
we
A be the theta 0
factor
automorphic
Thus
Let L,9*
.
A*
:
A
X is
given
C*.
Proof Let
di Dn-t dn-t We assume that
summand a,\(x)
automorphic 1. a,\ 2. a,\
(x) (x)
=
0 for
is
Dn-I
Takeasubset
factor,3I
:
Lo is defined A E Dn-f Zn-t; Zn-t-periodic
I C
p(A)
X
J1,...'n-q-2tJ. (Cq+t X p(J))
by
has the
with
factor
automorphic
an
(see
properties
respect
By
to
xi,
Lemma4.2.2
C* and
a
...
a,\(x)
Section
=
e(a,\(x)).
The
2.1):
)Xn-k-
there
holomorphic
anautomorphic
exists
function
p
I
:
Cn
C* such that
0P1(A) (P(X))
pI(x+A)a,\(x)WI(x)-1
forall
(A,x)E(Ax
(Cn
n
(Cq+
x
P(I))).
f,
Let
be
holomorphic
a
automorphic
summand
e(bI(',)(y)). the line
E
bundle
E(p(Al),p(A
=
(Cq+l x P(I)) 2(X(j), Z) represent given by,3I. Then
b,(A,)(y)
+
bundles
e(f,(x)). first
the
(Y)
P
+
P
Lo
is
P
is
=
Then
=
7
of
bI( A2)(Y).
0 for Al A2 E A by (1). trivial, topologically E(p(Al) P(A2)) that PI Therefore trivial. assume we may topologically Lp, Ad, ej) (P(X)) n f, where ej is the j-th unit vector of C'. Hence we have 1, j
Since
an
Chern class
b,(Al)(p(A2)Y)
-
P
117
We have
C such that
x
bp'I(,\2)(p(Al)y)
=
p,(x)
with
E H
=
2))
C'
on
p(A)
:
cl(Loi) Lp, on X(I)
Let
P
function
V
line
Extendable
4.2
=
1 for
-
.
.
,
.
fI(x+djej)-fI(x) Wedefine
a
=kj
E
Z, i
1 ....
=
n
f.
-
function
Dn-,eZ7--periodic
n-f
k,(x)
:=
kj
E dj
I(X)
f
Xi.
j=1
Let
x
=
function
(x', x", x",
of
x
) ECq+t
...
we
obtain
Cn-q-21
X
the Fourier
C'.
x
k,(x)
Considering
as
a
periodic
expansion n-t
fliw,
fi(X)
x"Id"))
x"')e((a,
IEZn-q-2t
where
we
3
j=q+t+l
set
x"Id" Similarly
d'xj'
E
+
we
t
Xn-,e/dn-i)
(Xq+i+l ldq+t+l,
=
have a,\
(x)
E
=
(x',.T
ax,,
..
x"Id")),
)e((o,,
uE Zn-q-21
bi(A) P
b,,, (x', x111)
(p(x))
A
e((or, x" Id")).
o,EZn-q-21
By the right-hand with
respect
On the other
Since the is on
Thus
of
we
P
1.... is
element
on
bI(A)
(1),
Xn-t+j
hand, V P(A) (Y)
(n-f+j)-th
holomorphic x"'.
to
side
P 1 with
(P(X))
is
extendable
(p(x))n-t+j to
on
of
P, \ 101 with
p(x)
Xn-t+j
is
equal 1,
bA,,(x')e((or,
(p(x)) 0rEZn-q-2f
respect
to xn-t+j,
t),
obtain
bPI(A)
on
I
holomorphic
respect
holornorphically
x"Id")).
hence
to
Yn-i+j-
bI(A)(p(x)) P
independent
C
For i E
1, Xq+i+i(I)
Yq+i+i-
If
on
j
E
\ 101
P,
and Extension
Reduction
4.
118
=
Xq+t+j(l)
then
Ic,
with
to
respect
Yq+i+j.
j
a
E Ic.
=
t
(Ul,
-
from
follows
It
Un-q-2t)
,
-
-
E
(3)
(1)
bI(,,)(y)
with
=
(0
o-i
for
to
holomorphic
is
P
respect
have
we
(x)
C with
on
P, \ 101. Therefore Hence
b, all
holomorphic
is
P
=
(2) for
b,(,)(y)
C. Then
0
some
i E I
or
0 for
with
0-j
)e((o-,
A"/d"))
>
some
that
bA,, (x')e((u,
Id"))
x"
UEZ-q-V
I:
[a,x,, (x', X11/)
+
I,
(x'
A, x"'
+
..
+ A
O-EZ-9-2f n-i
f,, (x,
-
x
)] e((u, x"Id"))
...
k.7
-7
+
dj
j=q+f+l
By (2) and (3)
(4) for
ax, all
some
(5)
o-
j
,
t
+
f,,(x'
A', x"'
+
Un-q-2f)
(Ul
+ A
with
ai
fo, (x'
+
..
A"/d"))
)e((u,
< 0 for
fi(xl,X111)
-
some
E I
i
=
(x', x1l')
a.\,o
+
go
9,(X)
A', x"'
+ A
(x)
:=
fo' (x',
21 ),
Y-
:=
fo, (x',
..
x
for
o,,
f,,(xl,
...
x"Id"))
xl")e((u,
Zn-q-21
,C:
some
i
I
fli(X,x
+
I
I//
),e((o,,x"1d")).
aE Zn-q-2f
(x)
disjoint
and
g, (x)
subsets
.
some
j
holomorphic
are
I,,
for
o
or.,>
.
.
,
IN Of 11,
Ic
functions n
-
on
q
-
U a=1
we
set
11,
n
-
q
-
Cn.
2fJ
N
And
=
with
or
Wedefine
Take
E
Z).
o-j
0 > 0
1c, and
b,.X,0 (x)
Then go
(n,\
+ n,\
obtain
(x', X111)
=
E
we
Aj
2fl.
with
)
+ constant.
for
line
Extendable
4.2
bundles
119
N
g(X)
:
go
=
E g'-
W+
W -
U=1
p(x)
Then
e(g(x))
:=
is
Zn- -periodic
D,,_
a
d,\(x) Then & is that
ao
automorphic
an
(x)
d,\
factor
A*
:
Cq+t
X
factor
depend
does not
W(x
:=
holomorphic
The
trivial,
)
(x",
on
x
)
C* such that
Combining L, =
=
L,,,
Abelian
Therefore
follows
It
a.
there
(4)
from
exists
an
and
(5)
automorphic
on
ao,j:(.\)(i (x)). A
0
ao and
Cq+'IA*
=
by ao is topologically homomorphism 0
given
Then there
variety.
exists
a
cobordant.
are
the above proposition
=
L,9
C'/A
Lo be
(9
of type form
Hermitian
of
A and
X
bundle
a
with
Q.E.D.
Proposition
L
kind t:
holomorphic
q, where
H and
form
Riemann
Lo f
4.2.12,
there
bundle
Lo is determined is
(0
topologically <
X be
X
line
V < as
by
on a
we
q).
-
above.
a
obtain
the
quasi-Abelian
reduced
trivial.
n
theta
We assume H is
Let
A
T
Then there
exists
a
variety
factor
=
19 with
ample
an
Cq+1 /A*,
holomorphic
line
such that
(t-')*Ll iff
to
Theorem
4.2.14
Let
an
).
...
C* such that
bundle
line
and A is
A*
T :
function
A)a,\(x)W(x)-'.
+
cobordant
d,\(x)
X
holomorphic
C*-valued
Cn. We define
on
exists
a
theta
bundle
Ll,,(X)
-_
L'
A with
T*L'.
Proof By Proposition T*Lv.. 0
4.2.12
Then L,9
morphic
line
there
is extendable
bundleto
exists to
a
.
theta
bundle
Since
LO
that
we
obtain
Lo is given
a
by
homomorphism V) the
L,
(9
L-1, 79
A such that there
exists
L,9 a
holo-
such that
Lojqx).
(t-')*Lo Thus
=
Lg*0
:
A*
homomorphism 0
)
o -r
:
by Proposition
C*
A
)
C*.
4.2.13
such
Q.E.D.
:
and Extension
Reduction
4.
120
Explicit In the previous
representation section
obtained
we
formula give the explicit of Proposition application
of
Weshall as
an
Let a
X
=
Cn/A
Then
be
we can
quasi-Abelian an ample
a
take
a
P of A
basis
normal
the
of
form
ample
an
forms for
automorphic
forms
an
theta
ample
theta
factor.
factor
4.2.11.
H of
form
Hermitian
automorphic
of
of type
variety
dx
factor
theta
(I
q
< q <
is of kind
f
n). Suppose (0 < 2f < n
that -
q).
as
di W/
dq+t
(PI Pit),
dq+i+l
P
wit
dn-f 0
and dx is
represented
as
,d,x (X) A= P'a+P"bE
for
The C-vector
holomorphic
A and
A, ,
space
f (x
(1) Let
f
c-
A,9,.
Since
on
+
Ox(x)
A) =
taz
e,
(z,w)
x=
of
f
functions
=
1
2
E Cn
4.2.11).
(Proposition belonging
forms
automorphic
1
taW'a
-
consists
of all
dq+l+j Z-periodic
with
to
0,\
Cn with
=
VA(x)f (x),
E
A,
x
ECn.
1 for
di
zn-t,
AE
dn-t 0
f
is
di Z-periodic
respect
to wj
with
(j
=
respect
to
zi
(i
1,...,n-q-2t).
1,
q +
We put
di
dq+f
f)
and
Extendable
4.2
Then
f
following
has the
(2)
bundles
line
121
expansion
Y
f(x)=
f,(w)e((o-,z)),
O'Ed-lZ,]+'6
f,(w) is a holomorphic function of w and dq+t+jZ-periodic towjforj=l,...,n-q-21.ForanyA=P'a+P"bEAandx=(z,w)E side of (1) is the left-hand where
E
(3)
f,(w
W"a)e( tWa)e((u,
+
with
respect
Cn,
z)).
9Ed-1Z"+'6
And the
right-hand
(4)
taz
e
(1)
of
side
taW'a]
-
e
Comparing (3)
Therefore
we
+
we
E
d-1 Zq+' and
Consider
a
taWa
e
((a, z))
(w)e((o-
z))
-a,
obtain
W"a)e( W'a)
fo-+a(W) a
(w) orEd-1Zq+f
=
[_
e
2
taW'a
fo-+a(W)-
have
(5) for
(4),
with
f,(w
2
1Z'1+1
O'Ed
is
a
=
1 t(o-
+
Ia)
W'a
2
fo. (w
+
W"a)
Zq+t.
C-
holomorphic
e
function
g(w)
Cn-q-i
on
satisfying
the
following
dition
(*)
g is
dq+l+j Z-periodic
For any R > 0
we
with
respect
wj for
to
j
=
1,
.
set
HR:= 1W E Cn-q-1
;
jImwjj:5R(i=1,...,n-q-2f)
jwjj:5R(j=n-q-2f+1,...,n-q-f)j. Wedefine
fa(w):=g(w+W"a) If
we
foraE
put
Ma(R)
:=
sup
Ifa(w)j,
wERR
Zq+t.
n
-
q
-
2f.
con-
Reduction
122
4.
then
M,,(R)
and Extension
< +oo
by (*).
by C.F the
set
lal
We set
Ejq:j
:=
Jail
for
a
'(a,,...,
=
aq+i)
E
Zq+i. Wedenote
(6)
I
lim sup jal-+oo
functions
holomorphic
of all
(*)
satisfying
g
a /exp [-7r ta(Im WI)a] Ma(R)
=
and
0
aE zq+e
for
R > 0.
all
CY is
Obviously
f,
functions we
complex linear
a
(2) belong
in
C.F.
to
It
space.
is
to
easy
Then CY 54 0 if
that
see
Av, 0 0.
the
coefficient
On the other
hand,
have the Lemma
4.2.15
CY :A
0,'
in
fact,
CY has
uncountable
an
basis.
Proof For any
complex numbers
at,
a,,...,
we
define
n-q-2f
g(w)
:=
t
E
e
Wi +
j=1
i=1
(*).
Then g satisfies
E ajWn-q-21+j
Since
Eq+t i=1
aiWq+f+l,i
W11a
Eq+' we
aiw,,,i
have
Jfa(w)J
W"a)l
=
Jg(w
+
=
exp
[-27r
(Im
n-q-211
E
-
dq+l+a
a=1
q+ w.,
+
E ajIm q+f
i
E
-2-7r
Wq+i+,,i
i=1
Im
ap
Wn-q-21+,O
+
E aiWn-t+)3,i
Weset
A:=
maxflapl;,3 andi=1,...'q+fj.
B:=MaXfJWq+t+,,iJ;a=1,...'n-q-f Let
R > 0. Then
n-q-2t
E a=1
I
dq+,e+a
we
(
obtain
the
following
estimates
on
HR
(R
+ B
n-q-V
q+,e
Im Wa +
ajIm Wq+f+ci,i
lal)
1: a=1
1
dq+,+a'
Extendable
4.2
Im
+
Wn-q-2t+)3
ao
M(R +
ajWn-i+)3,i
B
jal).
have
we
Ma(R) :5
(7)
Im W' is
Since
123
q+t
i
Therefore
bundles
line
exp
I
(R
21r
(n-q-V
Ia
follows
it
_L-0-4-
(7)
from
la /exp F[-_i7ta _((_Im
lim sup jal-+oo aEZ'+'
W-+fA q+1+.,
a=1
definite,
positive
+ B
that
Ma(W) WI)al Ma(
0-
=
Hence g E C.F.
We can take
arbitrary
uncountable
basis.
U(d)
Let
Rpv(d-1 Zq+ /Zq+l
:=
=
di
dq+1-
...
C.F has
show that
is easy to
)
be
a
complete
The number of elements
.
U(d)
We set
an
=
6
system
of
representative
OU(d)
of
U(d)
:=
is
of
given
by
(CT)6
as
S61.
1
(ABE [11])
Theorem
4.2.16
Then it
E C.
Q.E.D.
d-1 Zq+1 mod Zq+,f 6
ak
a,,
Then the series
E C.F.
Take any
6
f(X)
(8)
E Y_
:=
fs,+a(w)e((sj
+ a,
z)),
x
=
(z, w)
aEZ'I+l
converges
on
fsi+a(W)
(9) Conversely, complex Proof
any
linear
right-hand satisfied by (9). the
side
is
an
to find
the
ample
an
general
of
2
A, ,,
where
1 t(si 1a)Wa] +
e
is
f, (w
2
represented
is
a)Wa
(8) f
already
as
(8).
in
E
+
W"a). A,9,
Therefore
on
t(si
by (6).
Cn
+
1a)Im
Wa
2
(5)
The condition
is
for
is still
trivially
A,9,. Q.E.D.
seen.
formula. it
27r
exp
converges
an
ample theta
Riemann form
explicit case
1 +
Then
The above formula with
A,&,
E
1 t(si
e
converse
f
:=
to
spaces.
Since
The
belongs
Cn and
For open.
H has some
a
factor.
wild
special
If
an
summand,
factor
automorphic then
type ABE [11]
it
is
gave it.
very
aX
difficult
However, for