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




.

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


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

-

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


,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
0.

127r (o-,

Im

constant

Therefore

- j) I I e ((o-,

-

11


0

constant

d > 0

EZq with inf

7-E zq

We can choose

Of

(TT)

condition UN

a

(a

-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


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


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
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

>


2.

prove.

11all 11all

with

maximum

norm.

46

If

2

2(1

J,r

Jrj + +

Bundles

Line

2.

S,




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
-

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


M222MSN-SN+l




1).

of toroidal

Cohomology

2.2

47

groups

and then

(a; UN)

(**)

because of

Finally

TN

-

we

I




N. So after




__

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




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


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
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
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
0 and F is

set.

A
- 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 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

n

+

< r




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 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)


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
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 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


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,,