Toroidal Groups Line Bundles Cohomology and Quasi abelian Varieties

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