(4.8) F{U) := {/ | / : C°° in U and |^ = 0, for i = q + 1,... ,n}.

COHOMOLOGY OF THE FUNDAMENTAL GROUPS OF

TOROIDAL GROUPS

Masanori MUTA

Abstract

We construct isomorphism betweem d-cohomology and the coho mology of the fundamental groups of toroidal groups, and get standard forms of p-cocycles, which was given by Vogt [6] in case of 1-cocycle.

Using differential forms via the above isomorphism enable us to obtain the new results in higher dimensional cases. An isomorphism between Cech cohomology and the cohomology of the fundamental groups of complex tori are given in [5]. Our results give a generalization of the isomorphism to toroidal groups.

1. Introduction

Kazama and Umeno [3] calculated the 9-cohomology of toroidal groups using the Fourier expansions of d-closed forms and characterized the toroidal

2000 Mathematics Subject Classification : 32M05

groups of cohomologically finite type, namely those of having finite-dimensional cohomology groups. For getting the results, they proved that a toroidal group is of cohomologically finite type if and only if every d-closed (0, p)-forms on it are d-cohomologous to constant (0, p)-forms. Further, they proved that toroidal groups not of cohomologically finite type have non-Hausdorff topol

ogy.

On the other hand, Vogt [6] obtained the standard forms(in Theorem 2.2)of the 1-cocycles of the cohomology of the fundamental groups of toroidal groups, solving difference equations.

In this paper we construct isomorphisms between <9-cohomology and the cohomology of the fundamental groups of toroidal groups, and get the stan dard forms of the p-cocycles. This gives an another proof of the above result of Vogt, in case of 1-cocycles and a generalization to the cases of p-cocycles.

Using differential forms via the isomorphism enable us to odtain the new results in higher dimensional cases. By the isomorphisms, every d-closed (0, p)-forms on toroidal groups correspond to p-cocycles which are cohomol- ogous to the standard j9-cocycles. We hope this correspondence could shed some light on study of cocycles in toroidal groups which have infinite dimen sional cohomology groups. The isomorphisms between Cech cohomology and the cohomology of the fundamental groups of complex tori are given in [5], and these are valid for toroidal groups. Our results are regarded as another approach for the cohomlogy of complex tori.

2. Definition and statement of results

At first, for a toroidal group X of complex dimension n, we recall the defi nition of the cohomology groups Hp(iri(X), H°(Cn, O)) of tti(X) with values in the additive group H°(Cn, O) of the holomorphic functions on Cn, accord ing to [5] (p. 22). We simply call them the cohomology of the fundamental group of a toroidal group X.

Definition 1. A connected complex Lie group X is called a toroidal group if every holomorphic function on X is constant.

Let n be the complex dimension of X. Since any toroidal group of complex dimension n is a complex abelian Lie group , there exists a discrete subgroup F of Cn such that X is isomorphic onto the quotient group Cn/F and F is the fundamental group of X. Let r = rank F, then n + 1 <r <2n and there exists a basis Ai,..., Ar for F. We write F = Z{Ai,..., Ar}. The matrix P = [Al5..., Ar] is called a period matrix for Cn/F.

Definition 2. A toroidal group Cn/F is called of type q (1 < q < n) if rank F = n + q.

Let H = #°(C\ 0), C°(F, H) := H, and

Cp(T, H):={f\f :Tp —> H}, where Tp = F x ... x F (p-times).

The coboundary 5 : CP(T, H) —> CP+1(T, H) is defined as follows:

(2.1) Sf(X)(z) = f(z + A) - /(*), for p = 0,

p-i

I](-l)i+1/(Ao,.. •, Ai + Ai+1,..., Xp)(z)

i=0

...,\P-i)(z\ forp>0,

where A, Ao,..., Ap G T. Put Z?(r,H):=Ker(6), for p > 0,

B*(I\ if) := Im (5), HP(T, H) := Zp(r, H)/&>(r, H), for p > 0, and H°(r,^) = Z°(r,if) = H°(Cn/T,O).

Definition 3. We call HP(T, H) the cohomology of T with values in H.

Let Cr> s be the sheaf of germs of C°° (r, s)-forms on a given complex manifold. Put Dr>s = H°(Cn,Cr's)- Then CP(T, Dr>s), for p > 0, and the coboundary <5 are similary defined. Then d : CP(T, Dr's) —> Cp(r, Dr's+1) and — : CP(T, Dr<s) —> CP(T, Dr's) are defined as follows:

C/Zi

(5/)(Ai,...,Ap)(2):=a(/(A1,...,Ap)(z)), and

(

for / € Cp(T, Dr>s) and Ai,..., Xp G T.

Then

(2.2) 55 = 59 and — 5 = 5-^-.

By a suitable hnear change of Cn, Cn/F has a period matrix

(2.3) P=[In,V],

where In = [ei, • • •, en] is the n x n unit matrix and (2.4) V = [Vij; l<i<n,l<j<q} = [vu ■ • -,

is a n x g matrix. We can assume

(2.5) det Im [vy; 1 < z, j < g] ^ 0.

Put v* = ^T-lei (q+1 <i<n),[vu...,vn} = [%; 1 < i < n, 1 < j < n] and /?t = Imvi (1 < z < n), then A,... ,/?n are C-linearly independent. Define two coordinates zi, • • •, zn and tu • • •, t2n in Cn, where ti G R (1 < i < 2n) as follows,

(2.6) z = ziA + • • • + znpn

H h tnen + tn+ii;i H h

Hereafter, we write (zi,..., zn) instead of zi^i H h 2n/3n and t(tu ..., t2n) instead of tiei H \- tnen + tn+iVi + • • • + t^n-

We denote by Z^(M,Cr's) the space of the enclosed C°° (r, s)-forms on a complex manifold M and by S^(M,Cr>s) the space of the 9-exact C°° (r, s)- forms on a complex manifold M. Then by the Dolbeault theorem, we have

(2.7) HP(M, O) =

Let

{Pa\...av&Zai\ A ... A dzap G Z^iC™/T,C 'P).

<n

Looking at <p as an element of C°(T, D°'p), 8(p(X)(z) — (p(z + A) — (p(z) = 0, for any A € I\ Put

We define y><*) G Zk(T, D°'?-k), satisfying <pik\\u ..., Xk) G %(C\ C°'"-fc), for each Ai,..., Xk G F, inductively on A; (1 < k < p).

Since y>(°> is a d-closed form on Cn, there exists $(°> G C°(I\ Z)0^-1) satis fying

(2.8) ¥j(°)

Put

(2.9) ^:

Then clearly <5(/?(^ = 0, and

(2.10) d<pw(X) = 6(8&0))(X) = <V(0)(A) - 0. for A e T.

Hence, for each A G T, (/j(1)(A) G ^(C",^0^-1). Assume we get

<pW G ^*(r, D°'p-k), (l<k<p) satisfying

(p{k)(X1,..., Afe) G %(Cn, C0-p-fc), for each Ai,..., Afc G F. Then there exists

satisfying (2.11) Put (2.12)

then

(2.13) ^(fc+1)(Al5..., Afc+i) = Vfc)(Ai, • • •, A,+1) = 0.

Thus we get <^fc> G Zk(T, D0'p~k) satisfying <^(fc)(A,..., Afc) G Zg(Cn, C°'p-k),

1 < k < p. Since (p{p)(Xu ..., Ap) G %(C\ C°'°), </?(p) 6 ZP(F, #).

We denote by 77 the mapping Zg(Cn/T,C°'p) —> ZP(T, 77) such that

(2.14) II(<p) = (p, forp = 0,

= (p{p\ forp>0.

Note that the definition of 77 depends on the choice of $^ (0 < k < p).

But we have the following

Theorem 2.1. Zei C"/F fee a toridal group. Then the mapping U defined by (2.14) induces an isomorphism for each p,

H°'p(Cn/T) ^ Hp(T, H)

Then we get the following

Theorem 2.2. Let Cn/F be a toroidal group of type q. Then every f^ 6 ZP(T, H) is 6-cohomologous to a(p) € ZP(T, H) satisfying for all Ai,..., Xp € r andz = (zu...,zn) eCn,

a(Ax,..., Ap)(z) = a(Ai,..., Ap)(0,..., 0, zq+u ■ • •»^i*)-

3. Proof of Theorem 2.1

Before proving Thorem 2.1, we note some facts about cohomology of F with values in Dr's = H°(Cn, Cr's). Let tt : C" —>• C^/F be the projection, be an open coverings of Cn/F and {Wi} an open subsets of C" such that

(3.1) ttj = ?r| Wi : Wi —>• Vi is a homeomorphism and

7T 1(Vi) = y^(Wi + A) is a disjoint union

Aer

At first, we show the following

Proposition 3.1. Let Cn/T be a toroidal group. Then HP(T, Dr^s) = 0, p > 0 Proof. We take coverings {V;} and {Wi} as in (3.1).

Let / G ZP(T, Dr's). For any z G tt""1^), there exists a unique Ao G Y such that z G Wi + Ao and tt"1^^)) = z - Ao. Put for z G tt"1^)

ft(Ai,.. •, Ap_i)(z) := /(Ao, Ai,... .Ap-iXTrr1^*))).

Putting w^^z)) in place of z in (2.1), since Sf — 0, we get for any

(3.2) /(Al7..., Xp)(z) = ^(-l)V(Ao, • • •, A,- + Ai+lj..., Ap)(7rr1(7r(2)))

j=o

Let {7^} be a partition of unity subordinate to the covering {Vi}. Put for any z G Cn,

i,..., Ap_i)(z) := ^2 7t W

Then

Thus the proof is completed. □

Then we begin to prove Theorem 2.1. For p > 1, we need the following

Lemma 3.2. Let Cn/T be a toroidal group and </?, iff G Zg(Cn/T, C°'p), p > 1. Suppose two forms ip^ and ip'^ in Zk(T, D°'p~k) are constructed from ip and iff respectively as in (2.11) and (2.12) for 1 < k < p.

If ip and if! are d-cohomologous, then there exists p(fc-D e c*"1^, JD°'"-fe-1) satisfying

(3.3) v,(*)-^*)=gffp(*-D.

Proof. For ip^ and ^k\ there exist <S>^ and &^ in Ck{T, D°'P-k~1), respectively which satisfy (2.11) and (2.12), 0 < k < p. By the assumption, there exists V € H°(Cn/T, C0^'1) such that <p - iff = dip. We shall prove the lemma by induction on k. For k = 0, there exists p(0) € C°(r, D0>p-2)

satisfying (3.4) Then

Thus the lemma is proved for k = 1.

Suppose there exists p^~^ e Ck-2(T, D°'p~k) satisfying (3.5) <^-1)-<//(fc-1

for k > 1. Then there exists p^~^ € C*"1^, D0-^^1) satisfying (3.6) S**-1) - $/(fe"1) = 8p(k-V + dp^-'l

Then we get

y,(fc)_¥/(*)

Hence the lemma is proved for k and the proof is completed. □ Then we show the following

Proposition 3.3. Let Cn/F be a toroidal group. Then the mapping IJ defined by (2.14) induces a homomorphism

(3.7) 77 : H%p(Cn/T) —> HP(T, H).

Proof. We may assume p > 0. Let ip and (pf are d-cohomologous in Z5(Cn/r,C0'^). Suppose <pW and ^W are constructed as in (2.11) and (2.12) by $(fc) and <&'(k) respectively, 0 < k < p. For p = 1, there exists i/j G i7°(Cn/r,C0'0) satisfying

if — (p' — dip.

Since

{) and <// = (0)

Hence

Accordingly 77 defines a homomorphism for p = 1.

Next we consider the case p > 1. By Lemma 3.2, there exists Cp"2(I\ L>°'°) satisfying

Then there exists h G C"1^, 77) such that

Hence

This means 77 induces a homomorphism (3.7) and the proof is completed. □ Next we show the following

Proposition 3.4. Let Cn/F be a toroidal group. Then the homomorphism II is injective.

Proof. We may assume p > 1. Let ip, </?' € %(Cn/r,C°'p). Suppose <^(fc) and yp'W are constructed as in (2.11) and (2.12) by $^ and $'^ respectively, 0 < k < p. Assume (p^ := Ilip and (p1^ := II<p' are 5-cohomologous. Then there exists h € Cp-1(r, H) such that

p(p) - (f/W = 5h

Then there exists a^~^ G Zp~l(Y, D°<°) such that

Then

In case p = 1, cr(°) G Z°(r, D0-0) = i7°(Cn/r, C°<°). Note that y? = pW and yj' = <^'(°). Hence </? and <^' are d-cohomologous.

If p > 1, by Proposition 3.1, there exist 7)^-^ e Cp~2(r, D°-°) such that o-fr-1) = <57/(p-2). Then there exists a^~^ g Zp~2(r, .D0-1) such that

Continuing this way, we have

Since ff(°) e Z°(T, D0'?-1) = H°(Cn/T, C0*'1), <p and tf are

d-cohomologous, and the proof of the proposition is completed. □ To complete the proof of Theorem 2.1, we need to show the following

Proposition 3.5. Let Cn/F he a toroidal group. Then the homomorphism II is surjective.

Proof. Let / € ZP(T, H). Then by Proposition 3.1, there exists Cp-\r, L>°'°) such that

Put

^(p-D := d&v-V G Cp-\T, D0'1).

Then Vp~1} = 0, hence there exists &p-V e Cp~2(r, D0'1) satifying

^(p-i) = (j$(p-2)> if p>i.

Continuing this way, we get <pW e C°(I\ L>0'P) and $(°) e C°(T, D0'?'1) satisfying

(o) (<>) and ^(o) = o.

Then y)(°) G %(Cn/r, C°-p). By the construction we see 77^(0) = /. Hence

II is surjective, and the proof is completed. □

Prom Propositions 3.3, 3.4 and 3.5, the proof of Theorem 2.1 is completed.

4. <9-cohomology of toroidal groups

In this section, we summarize some facts about d-cohomology of toroidal groups([3] and [4]) to prove Theorem 2.2.

Let Cn/F be a toroidal group of type q with the period matrix P = [In,V].

From (2.6), for i = 1, • • •, n

%JT{^lii S«^)' and

Hence

and

(4-3)

Let

be a C°°-function on Cn/r, where t = *(*i,..., f2n), * = *(<i, • • •, tn+q), and t2// = *(tn+g+i,..., t2n). For each m G Zn+<?, we put

(4.4) Kmti : = *m«i - mn+i (l<i<q), and Km : = Max.{\Km>i\; 1 < i < q}.

Ifm^O

(4.5) * "' = -irKm^it2") exp(27r\/^l < m, i >) (1 < i < q)

1 dcm(t2")

= v-l(7rmiCm(r//) -^—-)exp(27rv-l <m,t>) (q + 1 < i < n),

and

(4.6) J v ^ = TrKm,iCm(t2") exp(27rv/zl < m, t >) (1 < i < 9)

2ttv —1 < 771, £ >) (9 + 1 < 2 < n).

In case m = 0,

U 7)

^^ = 1 ac^^ and

We have the following([3],p.95))

Proposition 4.1. A complex Abelian Lie group Cn/T with a period matrix P = [In, V] is a toroidal group if and only if

Km>0 for any m G Zn+9\{0}.

For each open subset U C Cn/T

(4.8) F{U) := {/ | / : C°° in U and |^ = 0, for i = q + 1,... ,n}.

We denote by J7 the sheaf defined by the presheaves

Suppose f(t) e ^(C^/r,^7), then from (4.6) and (4.8) we can write

n

(4.9) /(*)= ^ cmexp(-2n ^2mitn+i)exp(2TrV::l<m,t'>),

where cm are constants satisfying for any real number R > 0 and any positive integer k ,

(4.10) C(k,R) :=sup{|cm| ||m'||fc/2l|m//|l;m e Zn+q} < oo,

where, ||77i'|| = max{|mi|, |mn+i|; 1 < i < q}, and

||m"|| = max{|mj|;g+ 1 < j < n}.

(4.10) is obtained similarly to Lemma 7 in [2] (cf. Proposition 2.1 in [4]).

Conversely, a function satisfying (4.9) and (4.10) is in H°(Cn/r,F). Let J**8 the sheaf of (r, s)-forms with coefficients in T, Z^(Cn/r, ^s) the space of the ^-closed ^r's-forms on Cn/T and %(Cn/r, ^s) the space of the d- exact ^^-forms. Then by [4] (Lemma 2.2), we have

(4.11)

Suppose if = ^ (/?m be a <9-closed ^^-form. Then there exist a unique

constant (0,p)-form

X = — ^2 coti...apdzai A ... A dzap, and ^"P-^-forms ^m satisfying

where for each m G Zn+9\{0}, (pm — d^f1 and for m — 0, <f>° = x +

We note that ^ ^m does not converge generally. Then we have the

following(Theorem 4.3 in [1])

Theorem 4.2. Let Cn/T be a toroidal group of type q with a period matrix P = [7n, V]. Then the following conditions are equivalent.

(1) There exists a > 0 satisfying

sup exp(—a||7n*||)/i;Cm < oo,

Zn+<?\{0}

where \\m*\\ = max{|ra;|; 1 < z < n).

v

(2) HP(Cn/T,O)^ /\C{dzu...,dzq}, (l<p<q)

= 0, (p>q).

5. Proof of Theorem 2.2

Suppose 1 < p < q < n are positive integers.

Let

K...ivdzn A • • • A P' l<ht...tip<q

be a holomorphic p-form on Cn. Put

1 Yl Yl ^dA dzn A • • •A

p\

Then we have the following

Lemma 5.1. Suppose dq£lp = 0. Then there exists a holomorphic (p — 1)- form f^"1 on Cn satisfying

Proof. Let I := max{ik | 1 < ik < q, 1 < k < p, 9filAp ^ 0}. We shall prove this lemma by induction on L If I = p, we have

Then 9 f)Qp

—-—-—aZi A dz\ A ... aZr, = 0 i=P+i dz-l

QQP

Hence, for p + 1 < i < g, we have — '"p = 0. Then

Put

^12...p(^? ^2, . . . , Zp, 0, . . . , 0, 2:g_f_i,

0

and Of'1 := 9f2^p(z)dz2 A ... dzp. Then

Assume that the lemma holds for I — 1, I > p and

^idzh A...Adzi

is a holomorphic form on Cn such that dqW = 0. Since for

^ = 0, we have

^...ipl^) = ^...ip^l' ■ • • ' *<> 0' • • • ' 0'

Put

and

w := / _ 1v| E ^...i^dziz A ... A dzit

Then

= h E («U - P^f^W A ... A dzip.

If we put Q,'p := £lp - dqu, we have dqQ,'p = d9Qp = 0. Then by the induction hypothesis, we have Sl'p-1 satisfying Q!p = dqQ'p-\ Hence Qp = dq(QfP-1+u)

and the lemma is proved. □

Next, we show 77 in (2.14) defines a mapping %(Cn/T, ^°'p) —>■ Zp(r, 77) which induces an isomorphism for each p > 1,

77g(Cn/r, ^°>p) s 77p(r, 77).

As in (2.7) and (4.11), we have two isomorphisms

1: 77p(Cn/r, O) ^ 77§>p(Cn/r) and J : 77p(Cn/r, O) ^ 7%(Cn/r, ^°-p).

Let {K} be a Leray covering of Cn/r for O. Then 77p(Cp/r, 0) is isomorphic onto the Cech cohomology

HP{{V} o) = zp{{v^ o)

U lh ] 73p({^},0)-

We have the following (cf. Proposition 4 in [7] and Lemma 2.4 in [4])

Lemma 5.2. Suppose fp e Zp({Vi}, O), <p € %(Cn/r, C°'p) and

<p e Zg(Cn/T, ?°'p) satisfy

(5.1) 7([/p]) = \<p) and J([fp]) = [0], for p > 1,

where we denote by [u] the cohomology class of a cocycle u. Then there exists V> G H°(Cn/T, C0'p~l) satisfying

(5.2) (p = ip

In particular, if 0 e Bg(Cn/T, C°'p), then 0 G Bg(Cn/r, JF°>p).

Proof. We also use the symbol 5 as the coboundary for the Cech cohomology Hp{{Vi}, S), where S = O, C°'r or J"°'s, r, s > 0. For f G Z*({Vi}, O), there exist (pp~1 G Cp~1({Vi}, Co>0) and in case p > 1, a sequence of cochains ipk G Ck{{Vi}, a0'*"1*-1), (k = p - 1, p - 2, ..., 2, 1, 0) such that

(5.3) fp = 5^, and d<pk = Sip1"1.

Since dip0 is a ^-closed C°° (0, p)-form on Cn/T, we get

(5.4) I([fP]) = V

Similary there exist <fP~l G Cp~l({Vi}, ^r0>0) and in case p > 1, a sequence of cochains <pfe G Cfe({^}, JF0^-*-1), (fc = p - 1, p - 2, ..., 2, 1, 0) such that (5.5) fp = 5<po-\ and d(p* = Sip1"1.

Then

(5.6) J([fp}) = [dip0].

From (5.3) and (5.5), in case p = 1,

(5.7) <p° - <p° G F°(Cn/r, C°'°).

In case p > 1, we get a sequence of cochains ipk G Cfe({Vi}, C°'p~fe~2),

(k = p — 2, p — 3, ..., 1,0) such that

(5.8) if"-1 - c^"1 = <5VP"2, and (pk - <pk -W = Sip1"1 (Jfe>0).

Then

(5.9) dip1 - dip1 = 5d^°, and

tpV-tpO- di)0 e H°(Cn/T, CQ'p-1).

Prom (5.7) and (5.9), there exists V' G #°(Cn/r, C0^"1) such that dip0 - dip0 = dtf.

Combining this with (5.4) and (5.6), for any (p G %(C"/r, C°'p) and ^ € Zd(Cn/T, F°>p) satisfying (5.1), there exists if> 6 H°(Cn/T, C°^~l) which satisfies (5.2). Next, suppose <p G %(Cn/r, C0)P). We take fp and </? satis fying (5.1), then by (5.2), tp € BgiW/T, C°'p). Hence /* G Bp({Fi}, 0) and

<p G Bg(Cn/r, ^°-p). Thus the lemma is proved. D

Then we have the following

Lemma 5.3. II in (2.14) defines a mapping %(Cn/I\ F°>p) —> ZP(T, H) which induces an isomorphism for each p > 1,

/%(C7r, F°>p) ^ Hp(T, H).

Proof. Suppose 0 G %(C"/r, ^°-p). Since <p G %(Cn/r, C°-p), 77^3 G Zp(r, if) is denned. Let / G ZP(T, H). Then from Proposition 3.5, there exists <p G %(C"/r, C°-p) such that i7</? = /. By Lemma 5.2, there exist

<p E ZB(Cn/T, F°'p) and ip G H°(Cn/T, C0^-1) such that <p = <p + dip. Then Flip and II (p are <5-cohomologous in ZP(F, H). Hence 77 defines a surjective homomorphism Hg(Cn/T, F°'p) —> HP(T, H).

Suppose II<p and II f/ are <5-cohomologous, where <p, iff G ZQ(Cn/r, TQ'P).

Then by Proposition 3.4, there exists ij> e H°(Cn/T, C0-^1) such that <p - {p' = dtp. Then by Lemma 5.2, there exists ^ € H°(Cn/T, T0^'1) such that

ip — ip' = dxp. Hence the above homomorphism is injective, and the lemma is

proved. □

Hereafter we write <p G Zy(Cn/r, T0'9) instead of <p. We put Fr's = H°(Cn, 7r*^r's). Then CP(T, Fr's), for p > 0, and the coboundary <S are defined similarly to §2. Let <p G Zg(Cn/r,^°'p). Since 7/p(Cn, n*^'*) = 0, for p > 1, we can construct ip^ e ^fc(F, F°'p~k) for 0 < fc < p, using

$(*) e £0^ Fo,P-fc-i) as in (2.H) and (2.12). Namely

(5.10) <pW = <p = d$l°\ (pW = 6&°\

(5.11) <p{k) = d${k\ and </?(fe+1) = <5$(fc), for 1 < k < p - 1.

Then

For

</> = -r_pi 52_{^i^ <} <Pai...apdzQl A... Adza G

put

Then we have the following

Lemma 5.4. Le* y> <E ^(C/r,^0-"), and yj^ G ZP(I\

Then

<Pi € ZB(Cn/T,T°'p), and [77^] = [-|—], for \<i<q.

Proof. Since

we have ipt £ Zg(Cn/T,F°'p).

Prom (5.10)

Continuing this for (5.11), we have

(5.13) ^jfor0<i:<p.

Hence [Ilipi] = [^ ] = [——] and the lemma is proved. □

C/Zi

In the proof of Lemma 5.4, (p^ and </?^ are defined by $^ (0 < k < p)

() ^

and (p\ ' = ——. Hereafter we identify <pf' with ——. Then we have the following

Proposition 5.5. Let Cn/T be a toroidal group of type q.

Suppose <p e Zg(Cn/T,T°'p). Then <pt e Bd(Cn/F, T°<p), forl<i<q.

Hence (pV eBp(T,H).

Proof. Put

V = ~. Y\ <Pai...apdzai A...Adzape %(Cn/r, f°'p),

^ 0), and

where \Km^m)\ = Km.

Then by (4.6)

ipm = dip™, for each m ^ 0,

Let

(5-14) «.«„_> == - -^(^-^-^ m ^ o, and

Then by (4.5)

and applying (5.14) to (4.10), we see ^ := ^ ^ converges. Hence

(5.15) <Pi = dil>i.

Then (^Jp) is 5-exact, and there exists h^~x) G Cv-l(T,H) such that ^ =

Shi1'. Hence the proposition is proved. D

Further we have the following

Proposition 5.6. Let Cn/F be a toroidal group of type q.

Suppose ip e Z^(Cn/F,^°'p).

Then, there exists h!f~l) e Cp~l{Y,H) such that

= Sht1] and Q\- = °\-, far l<i,j<q.

To prove Proposition 5.6, we need some facts. Let Hr>° be the space of holomorphic r-forms on Cn. Then CP(T, Hr>°) and the coboundary 5 are defined similarly to §2.

Lemma 5.7. Let Cn/T be a toroidal group of type q.

Suppose (f e Z-B(Cn/r,Jr0^), andp> 1. Then, there exist

q

Eh (p-l) , ^-l/p rrl,0\ _„ J ti\ dZi E O [V , n ) ana t=i

such that

pV> = Sh^, for l<i<q, and

Proof. Let 1 < i < q. Prom Proposition 5.6, we have ip{ e H°(Cn/T, T0*'1) satisfying (5.14) and (5.15). Further, we have $^ G C°(T, F°'p-k~1), (0 <

k<p-l) satisfying (5.10) and (5.11). Then

(5.16) && d

Since p > 1, there exists pf] € C°(T, Fo'P~2) such that

Hence

In case p > 2, there exist pf] e C1^, F0'p~3) such that

(5.19) ^1-^.5^).

Suppose we have pf~l) € Ck~\T, F0'^'1) for pf~2) G Ck~2(r,

p > 2 and 2 < k < p — 1 such that

Then

Hence if p > 3 and A; < p - 1 there exists p\k) G Ck{T, F°'p-k-2) for pf 1} e

Cfe-a(r, F0-?-^1) such that

(5.20) ^_Vf-»=5pf>

Prom (5.19) and (5.20) we see inductively that (5.20) holds for 1 < k < p-2.

Since

there exists /iSP-1) G C"1^, H) such that

(5.21) ^^ - 5pt2) = h!f-l\ and <^p) = ^f-1), for p > 3.

Prom (5.18) and (5.19), (5.21) is valid for p > 1. Then

,B221

Next, we shall describe -^- (1 <i,j< q)} using (5.14). Put

3

(5-23) «"■„,_, := ^P^ = ,->^->-^-, m ^ 0,

Then applying (5.23) to (4.10), we see ^ := ^ ^ converges and

From (5.23), we have -0™ = ij)™ for 1 < i, j < q. Hence Vji = Vty

(5.24) ^^

Hence

(,25) ^-?)-».

Then there exists hf e C°(T, H) and pf G C°(r, F0^"3) (p > 2) such

that

= ap^, for p > 2.

Further from (5.20), we have

In case p > 2, from (5.26) and (5.27), we have

Hence if p > 3, there exists p\)} <= C^F, F°'p-4) such that

Combining (5.29) with (5.27), inductively on k, we get p\f € Ck(T, F°'p-fc-3)

(1 < k < p - 2) such that

Q-(fc) p. (fc)

(5.30) %-%- = -^i"^ + ^ ^ P > 3-

Prom (5.27) and (5.30), there exists h{?~2) G CP~2(T, H) such that

~ ¥ = ~sp^3)"^a)> for p

Prom (5.28), (5.31) is valid for p = 3.

In case p > 2, from (5.22) and (5.31), we get

dZj 5hdhij

From (5.22) and (5.26), (5.32) is valid for p = 2.

Put

Then (5.32) means that

and the lemma is proved. □

Lemma 5.8. Let Cn/F be a toroidal group of type q and p > 1.

Suppose ip <E Zg(Cn/r,^°'P).

Then there exist

(1 < ix,..., ir < q, and r = 1,... ,p),

which are skew-symmetric in all indices, such that if we put

^.j /_^ i\...ir l\ ' ' ' lr \ '

then

(5.33) ^ = 8h{rl\ i = l,...,q (in case r = 1),

(5.34) dqn{p~r+1) = Sn{p-r\ r = 2,...,p and (5.35) dqQ,{0) = 0 (in case r=p).

Proof. By Lemma 5.7, we have h!f~X) e C^"1^, /f) satisfying (5.33). Sup pose /ijf:^ G Cp-r+1(T, H) is given (r > 2). We claim that for each r (r> 2), there exist pS1s)..ir_1 G CS(T, F°<p-s-r), for 1 < ilf... ,ir-i < Q and

s = 0,... ,p — r, which are skew-symmetric in all indices, satisfying

dh{p-:+1) * dP{p-:]

(5.36) ^(l)*1 Y"^ ^(l)*1 y

fc=l ozi* fc=l

(5.37) 0(£(_i)*-i_i-fc-L) = o, and

-

(5.38) a

fc=i

for 1 < s < p — r.

In case r = 2, in the proof of Lemma 5.7, we have p\s) G <7S(I\ F0^-5"2) for

1 < i < q, and s = 0,... ,p - 2 satisfying (5.22), (5.25), and (5.27). These conditions correspond to our claims.

Suppose (5.36), (5.37) and (5.38) hold for r (r > 2). Let p> r. By (5.37),

there exists pf^ ir e C°(T, F0^'7"1) such that

(5.39) X/"1) —li^lh^lr. = dpy' ir.

k=i Zik

Then by (5.38)

(5.40) I

Hence, if p > r + 1, there exists p}]'Ar e C^T, F°'p~r-2) such that

.i -t TA- j, A _ (r\\ — /1^

(5.41)

fe=l ^ tfe

Continuing this, using (5.38), we get p|^>ir G Cs(r, /ro.p-r-s-i^ for

5 = 1,... ,p — r — 1, such that

(5.42) _

From (5.38) and (5.42), there exists hf~2 e Cp-r(T, H), such that

/r >)o\ X "^/ -i \/c—1 ii ...iu...ir r (p—v—1) 7 (p—v) c / 1 -1 \

(5.43) > (-1) —— = -6p\r, - h?> for (p>r + 1).

From (5.40), (5.43) is valid for p > r. From (5.36) and (5.43), we have

(5.44) EC"1)*"1 V""tr = 5ht2 (P > r).

fc=i OZik

Namely if h!?'^ e Cp-r+1(r, H) is given with (5.36), (5.37) and (5.38), then we can construct h(p~2 e Cp~r{T, H) satisfying (5.44). Then (5.34)

holds for r (r > 2).

Next we show (5.36), (5.37) and (5.38) hold for r + 1. From (5.43)

-+1 dh(p-:] *•+! dP{p-:~l)

(5.45) ^(-l)*-1 »y--^+i = -^^(-i)fc-i M^-tr+i ^ for p>r

*=i ifc fc=i **

On the other hand, from (5.39)

dp{0), .

(5.46) d(} (-1) — ) = 0-

Further from (5.42), for s = 1,... ,p — r — 1

(5.47)

k=l "k k=l

Thus we have /ijf"^ with (5.36), (5.37) and (5.38) for r +1. Then similary to

getting (5.43), we can construct ^".X+i w^ich satisfies (5.44) in case r + 1.

Hence (5.34) holds for 2 < r < p.

Next we show (5.35). Since r = p, from (5.37), we have

ii-ik-ip _ 7 (0)

dz. -nh...

k=l

Hence

Thus (5.35) holds for p > 1, and the lemma is proved. □ Then we begin to prove Proposision 5.6.

In case p = 1, from (5.16), h\0) := ^^ - V; e C°(r, i/). Then ^ = 5h\0)

and dhT dhf

dz{ d

Hence the proposition holds for p = 1.

In case p > 1, by Lemma 5.8, we have holomorphic r-forms fi(p~r) (r —

l,...,p) satisfying (5.33), (5.34) and (5.35). Hence by Lemma 5.1, there

exists a holomorphic (p - l)-form e<°> € C°(I\ f/p-1)0) such that

Then

Hence, there exists G^ € C\T, Hp-2'0) such that

= o.

Continuing this, we get e^2) G CP~2{T, H1'0) such that dq(n{p-l) - se{p-2)) = o.

Put

i=\

Then

Hence

Hence the proof of the proposition is completed. □ Proof of Theorem 2.2. By Lemma 5.3, for every /<p) € ZP(T, H), there exists (p € Zg(Cn/T, T°>p) such that y?(p) and /(p) are <5-cohomologous. It

suffices to prove the theorem for (p(p\ Then for each 1 < i < g, by Proposition

5.6 there exist /i|p~1} G C^O^tf) such that

(pf> = 5hf~i] and

Put

2=1

then dqn(p-V = 0. Hence by Lemma 5.1, there exists h^~1^ G Cp~l(T, H)

such that

namely

V=a1p

Hence

^ dZi

Put

Then

and we complete the proof of Theorem 2.2. □

Acknowledgement. The author wishes to thank Professors Abe, Furushima and Kazama for fruitful discussions during the preparation of this paper.

References

[1] H. Kazama: d-Cohomology of (H,C)-groups, Publ. R.I.M.S. 20(1984), 297-317.

[2] H. Kazama and K. H. Shon: Characterization of the d-cohomology groups for a family of weakly pseudoconvex manifolds, J. Math. Soc.

Japan 39(1987), 686-700.

[3] H. Kazama and T. Umeno: Complex abelian Lie groups with finite- dimensional cohomology groups, J. Math. Soc. Japan 36(1984), 91-106.

[4] H. Kazama and T. Umeno: d Cohomology of Complex Lie groups, Publ.

RIMS, Kyoto Univ. 26(1990), 473-484.

[5] D. Mumford: Abelian varieties. Oxford Univ. Press, 1970.

[6] C. Vogt: Line bundles on toroidal groups, J. Reine Angew Math.

335(1982), 197-215.

[7] C. Vogt: Two remarks concerning toroidal groups, manuscripta math.

41(1983), 217-232.

Masanori MUTA

Department of Mathematics Kyushu Sangyo University Fukuoka 813-8503, Japan

e-mail: [email protected]