t 1 9 l J く I

H O M O MO R PHtS M S O F T ORO I D A L G RO U PS

Yuk itaka A B E

A t o r oi dal g ^{r o u}p is a c o n n e ct ed c o mp l ^{e x c o m m utat}iv e

L i^e g ^{r o u}p ^w ith o utJn O n C O n Sta nt h ^ol o T n O rP h i ^c fu n ctio n s . 工n th is p ^ap ^{e r} 事 W e Sh al l sho w s o m e

l^{p r OP e rt ie s of h} o m o m o rp h i^{s m s}

of 亡o r oi d al g ^{r o ll}p ^{S . We s}hal l al s o sh o w a fu rth e r fe s ult c o n c e r ni ng ^{a n e w c}la s s a n alog^{o u s t o t}h e r ef in ed C b e r n cla s s d ef i^{n e}d in t³1 ^{. And w e sh al l} g i^{v e a n ot}h e r p^{r o o}f 9^f G b e r a rd el l i ^‑Andr e ot ti f i b r a亡io n th e o r e m .

C ON T E N T S I N T RO D UCT tON

1 . TO R O工DAL GRO口PS

2 . L t N E BUN DLE S AN D F A C T O R S O F AUT O MOR P H Y 3 ^. STA ND A R D R E P R E SEN TAT IO N O F THE T A FA CTORS 4 . E X T E N SIO N OF T H E TA FA C TO RS

5 . G tl E R AR D E L L 工^‑AN D R EO T T 工 F 工B R AT工O N THEOR E M

6 6 6 7 7 1 7 6 8 4 8 9 6 . A N EW CLA S S ANAL OG O U S T O T H E R E F t NED Cti E RN C LA SS 1 0 0

7 . HOMOMO RP HIS MS O F TO RO IDAL G RO U P S 1 0 7

REFE RE NC ES

tNTROD UC T 工O N A q ^{u ot}i e nt C ⁿl^{r of C n} b y ^a lat ti c e

r i s c al led a

. ^{亡o r oi d al g r o up i f al}l h olo m o rp b i^c fu n ctio n s o n i_t a r e c o n st^{a n}t く^{s e e D E F IN I T工O N} 1 ^.1J ^{. Th e fa ct that}

e v e ry p ^{e r}i^od ic fu n cti o n o n C

n Wi亡h 2 n ^‑p ュ^e p ^{e r}i od is a q ^{u ot}i e nt

of _tw o tb eta fu n cti _{o n s}

, h ad al_{r e a}d y b ^{e e n} kn o w n in th e e a rl y day^{s o}f th is c e n tu r y く^{cf .} t^{1 9}コI ^{. P . Co u sin} くt⁴コ,r⁵コ1 ^{stud i ed}

p ^{e r}i od ic fu n cti o n s with p ^{e r}iod r

, ^{s m al le r th}ム^{n 2n} , a nd d i s ^‑ c o v e r ed a g ^{r o u}p ^wh i ch d o e s n ot c o n tain a a nd C ^央 a s d ir e ct

s u m m a nd s . Su ch a g ^{r o u}p is e x a ctl y ^{a to r o}i dal g ^{r o u}p ^{. L ate r J} the stud y ^of t o r oi dal g^{r o u} p^{s w e r e} take n up b y ^{K . Ko}p f e r m a n n

1^{1 4}コ ^{in 19 64 . A . M o riT n OtO} F^{1 6}コ ^{sho w ed that a t o r oi dal} g ^{r o u}p

ap p ^{e a r s a s t}h e S teiniz e r of ^a c o n r l e Cted c o mp l^{e x} L ie g ^{r o u}p ⁹

a nd c al l ed it a n く^{H .C}J ^‑g^{r o u}p ^. F . G he r a rd el l i a nd A . Andr e o t ti

f⁸コ ^{d ef i n ed a q u a si ‑ abel ia n v a rie}t y ^{a n}d p ^{r o v e}d s o m e p ^{r o}p ^{e rt}ie s of it . tL K a z a m a r^{l l}コ ^{sho w ed that a to r oi d al} g ^{r o u}p is a

w e ak l y 1 ^{‑c o m}p l ^{et e} m a ni fol d く^{s e e} f^{1 8}コ ^{fo r th e d ef ini tio n}l ^.

G . P oth e ri ng r^{2 0}コ ^{a nd A . 甘efe z} r⁹コ ^{s亡ud ied} m e r o m o rp h i^c fu n cti _{o n} f i el d s o n to r oi dal g^{r o u}p^{s .} T hey ^sh o w ed th at

th e m e r o m o rp h i ^c fu n ctio n f iel d o n a n o n c o mp ^{a ct} q ^{u a s}i ^‑ab el ia n v a ri ety h a s th e inf inite 亡r a n s c e n de n亡al deg ^{r e e o v e r} a ^. C h . V og^t r^{2 1}コp ^{r o v e}d that e v e ry holo m o rp h i^c l i n e b u nd l e o v e r a t o r oi d al g ^{r o u}p ^{X 三} Cⁿl^{r i s def in ed b} y ^a th eta

f a ct o r i f a nd _{o n}l _y i f a c oh o m olog y g ^{r o u}p ^H

lく^X ,OJ ^is

f inite d i _{m e n s}io n a l . T h e c oh o m o log y g^{r o u}p ^s Hp

く^X ,OJ くp 之1J

a r e cl a s si f i ed b y ^{H . Ka z a m a} r^{1 2} ト ^{T he a utho r} r³コ ^{ch a r a c ‑}

te ri z ed to r oi d al g ^{r o u}p ^{s w}ith p ^{o s}itiv e l in e bu nd l e s a nd

p ^{r o v e}d _the m e r o m o rp h i^{c r e}d u ctio n th e o r e m fo r t o r oi d al g ^{r o u}p ^{s .} The s e r e s ult s s ug g ^{e st 亡}h at 亡h e n otio n of q^{u a s} ト ab el i a n

v a riety i s a n at u r al e xte n sio n of _th e n otio n of ab el i a n v a riet y ^. Th e a uth o r r²コ^{d ef in ed a n e w cla s s}

c

i^J

rく^Ll ^{a n a}l ^‑

og^{o u s to t}he r ef i n ed C h e r n cla s s fo r a l in e bu nd

.^{1 e L ov e r}

a to r oi dal g ^{r o u}p ^{X ‑‑} C ⁿl^r f^{s e e s e ctio n 6}1 , a nd st ud.^{l ed}

h olo m o rp h i^{c s e ct}i o n s of L . 工f L i s a t op ^ol og ^{l C a}l l_y trivial h olo m o rph ic li n e b u nd le o v e r a to r oi dal g ^{r o u}p ^,

へノ

the n c

rt^{Ll ‑ 0 くr31 I , But th e} c o n v e r s

l

e do e s n ot h ol d in g^{e n e r a}l .

In th is p^ap ^{e r w e s}h al l c o n si de r h o m o T n O rP h i^{s m s o}f to r oi dal g^{r o u}p ^{s .} T h e g ^{r o u}p ^of h o m o m o rp b i s m s i s stud i ed

け^{h e o r e m 7 ‑2}J ^{. Le}t 入 ニ A ^{. ‑ +} T b e a h o m o m o rp h i^s_Jⁿ f r o m a n ab el ia n v a riet y A o nt o a c o m pl e x to r u s T ^. Th e n T is als o a n abel ia n v a ri et y ^. In g ^{e n e ra}l i t do e s n ot h ol d fo r to r oi dal g^{r o u}p^s く^{s e e} Ex a mp l ^e 7 .3J ^{. We sh al l} g i^{v e a n}

e x a mp l^{e o}f th e亡a b u nd le L o v e r a to r oi dal g ^{r o u} p ^{X ニ} C ⁿl^r

s u ch th a亡 i亡 is n ot top^ol_lo

g i^{c a}l l _{y t}ri vial b ut 苫

rくい ^{‑ 0 .}

W e sh al l al s o g ^{i v e a n ot}he r p ^{r o o}f of 亡h e G h e r a rdel l i ^‑ And r e o亡t i f ib r atio n T h e o r e m く^{Th e o r e m 5 .4}1 ^.

1 , TO _{R O 工}D A 工J GRO ロPS Let r b e a d is c r ete s ub g^{r o u}p

of C ⁿ g ^{e n e r ate}d b _y v e cto r s pl ^{, . . . ,P}r 6 C ⁿ wh ich a r e

l i n e a rl_y i nd ep^{e n}d e n t ov e r 3i. T h e n w e c al l i亡 a lat tic e of r a nk r i n C ⁿ . B y g^{e n e r at o r s} p

l ^{, . .. ,}P

r

W e g ^{et a n} く^{n , r} ト^{m at rix}

p ^‑ く^tp

l ^{, . . . ,} t

P r1 ,

wh ich i ^s c al l ed a p ^{e r}iod m atrix of I^l

, o r al s o of X ^ニ Cⁿl^{r .}

D EF I N ITI ON I .1 . Let r be a l at t ic e in Cⁿ . T he n x ^三 C

n

l^{r i s c al l ed a} to r oi dal g ^{r o u}p i f e v e ry h olo m o rp h i^c fu n cti o n o n X is c o n sta nt .

Fo r g ^{e n e r ato r s} p

l ^{, , . . ,p} r

Of r , w e d e n ote b_y ^く_p

l ^,

‑ . . ,p

r ,

a th e c oLmP le x l in e a r s ub sp^{a c e o}f J

e ⁿ sp ^{a n n e}d b y

t_p

l ^{, . . . ,P} r

l . If X ^ニ C

nl^{r i s a} t o r oi d al g ^{r o u}p ^{, t}h e n

d im

c

くp

l ^{, . . . ,}P _r ^, c

コ n . Sup p ^{o s e t}hat d im

c

くp

l ^{, . . . ,}P

r 5

c 5 n ^‑1 .

T h e r e e x ists a c o mp l ^{e x} l in e a r s ub sp^{a c e} V of C

n

wi th d i^m

c

V 2 1 s u ch th at

c^{n ニ く}p

l ^{, . . . ,}p

r ン

c e^{i V .} T h ^{e n} X ^‑‑ C

nlr ^{ニ く}_p

l ^{, . . . ,}P _r^ン

cl^r O ^V , it c o n tr ad icts th at x i s a to r oi dal g^{r o u}p ^. T h e r ef o r e r 2 n . tn fa ct r 5 n .

工f r ^ニ n

, then C^{n 王} C p

l 曲 . . . 由 C p

n

. H e n c e

cⁿI^{I. ニ C}l³ O . . . 0 CI^D ,

it al s o c o nt r ad i ct声 ^th at X i s a to r oi d^al g^{r o u}p ^. F r o m abo v e w e c a n w ri te r ^ニ n+m , 1 S T n 5^{‑ n} . W he n m ^ニ n _, x ^‑ cⁿl^{r i s a c o m}p l ^{e x to r u s .}

L et r ^{a n}d r ^T p ^{e r}i od m at ^ri c e s a r e P cⁿl^{r a nd X}

‑

7 ‑ c

nl^r‑

e xi s亡 M 色G Lく^r, Zl ^{a nd}

Tw o p ^{e r}iod m a亡rlc e s 王^, i f th e ab o v e c o n d itio n

f o m s of p ^{e r}iod m atri^x

E^{1 4}l ^, t^{1 7}コ ^{a nd} r^{2 1}コJ ^.

P RO P O SIT I O N i .2 .

く^{1 S m く n}J ^{in C n .}

b e lat 亡1c e s of r a nk r ln C

n

wh o s e a n d P ^I r e sp ^{e ct}iv el y ^. T h e n X ^ニ

a r e is o m o rp b i^c i f a nd o nl y i f ^th e r e

A,^{e G L}く^{n ,C}I ^{s u ch} th at P ^{I 三} APM .

a nd P ^‑ a r e s ai d t o b e eq ^uiv a le nt i^{s s a}tisf i ed . W e c a n t ak e n o r m al

a s iⁿ th e f ol l o wing p ^{r o}p ^{o s}i tio n く

Let T b e a l at ti c^{ie o}f r a nk n+m The n X ^ニ Cⁿl^r

.

i s a t ^{o r o}i dal

生 空吐 出 生 旦

th e fol l o wing p ^{e r}iod

m atrix of r i s eq ^uiv al eh t m atrix

al ^{P ‑} く^工

n

Sl 生 吐

く^{1 .1}l ^oS車^{d n} 王竺 室 主 ^{o 良} れt^ol,

wh e r e 工 is ^.the u nit

‑ ‑‑‑. ‑ ^{n ‑} . くⁿ ,n ト^{m a} trix

g ^{r O}TtP

工旦

聖吐 ^{S is a n くn ,mJ ‑}

m at^ri^{x w}h o s e im ag i ^{n a r}y 且 聖土 ^{工m S h a s r a nk m .} 些坦 ^aJ 坐 ^{eq uiv al e nt} j里 坐 ^{fol lo w ing p} e^{ri od}

bl

く^{1 .2}, oR i

三

に ^j

whe r e T i s a p ^{e r}i od m atrix 坐

m a trix

孤^‑d im e n sio n al c o mp l ^{e x to r u s}

a n d R i s a r e al く^{n ‑m} ,2m ト^{m a}trix .

Let T be a lat ti c e of r a n k n 佃 く^{1 皇 m く n}J ^{i n} Cⁿ

wh ich c o ntain s n ^‑v e ct o r s l in ea rl y iⁿdep^{e n} d e nt o v e r 4I.

T h e n w e c a n tak e a p ^{e r}i ^od 占^at^ri^x P of r a s bJ ⁱⁿ

P RO PO S 工TI O N 1 .2 witho ut c o nd i tio n く^{1 .2}l ^{. L 6t} 吋^{m b e}

th e r e al l i^{n e a r} s ub sp^{a c e o}f C

n

sp ^{a r m e}d^. b y r . T ak e th e c o o rd in ate s z ^ニ く^z_l , ^{. . .} , Z

nl ^{of C n a nd s et}

z .

ニ X _. + I^‑1 y

j

コ コ

f o r j ^{こ 1 , . . . ,n .} B y ^{a m a}p p iⁿg く^z

l ^,

x n ^,y

l ^{, . . . ,}y_nl ^{w e i d e nt}i f y C ⁿ with

工 S

m

RI R

う

whe r e T ^ニ く^工

m

SI , R ^‑ く^R

I R

2J ^{a nd}

.

去^主こ7^n,wH 三^{u xl ,. .‑.,}

‑ く誕盲₁ ,

Let U ^‑ R ^e 音 ^{a nd} V ^‑ 工m 音^. I f w e r eg^{a r}d C ⁿ a s 択²ⁿ b y ^the abo v e i d e nti f i c ati o n 9 th e p^{e r}i^od m at ^rix P b e c o m e s the fol lo wing ^{r e a}l く^{2 n} ,n+m ト^{m a}trix