Concept of Toroidal Groups 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 - 24, 2001 © Springer-Verlag Berlin Heidelberg 2001 an example of such a group (see Concept of Toroidal Groups The 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, 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 discrete a complex rank subgroup m := rankCA C'/A where A is considered Let and toroidal coordinates rankCA = as a RA/A, maximal RA subgroup n of C' iRA be the maximal complex subspace discrete a C*m ED -_ is not dense in A) n (Cn-m/r) of subgroup complex subgroup MCr/(MCF Rr1r torus (C'/A) then Cn/A -when 1' -C Cn-?n is than complex subgroup MCA/(MCA of RA- If the maximal maximal real torus = n Cn-, -_ discrete and MCA n < n rank complex r) n - m and the is dense in the maximal real Cn-m/r of Proof If m then A spans rankcA, = plex dimension m so that subspace U We get Cn/A The closure Ko RA = n K" of a C complex subspace V U (D V with V/A RA/A = RA/A certain real dimension an n of the iRA of RA is of the maximal real torus K torus = U E) 7r(MCA) = plex subspace MCA Cn = a C Cn of CA := projection of the maximal connected and closed real a So K splits COM_ m-dimensional C-linear - into K' = com- subgroup Ko and another m A' ED A" such that decomposition A m Since MCA C RAI) K' RAI' /A" where rankRA" RAI /A' and K" Then rankCA" m MCA` MCA and MCA" n So we get Cn CA' E) CA", where the C-spans We assumed rankCA m or m, respectively CA') CA, have the complex dimensions n Because K = K' E) K" we get the = = = = = = = = - The real and the the other K' r hand, RAI /A' complex rank of A" coincide the projection Of MCA' which is maximal in =: CA' /A' Then CA"/A" -_ C*- On MCA is dense in the real We get CA' /A' - Cn-m/.p torus with Q.E.D A' This proves the proposition A consequence of this proposition is that for every toroidal group Cn /A the lattice A has maximal complex rank n KOPFERMANN introduced in 1964 the concept of n-dimensional toroidal groups with the irrationality condition [64] 1.1.4 Theorem Let A mal c Cn be a discrete complex subspace equivalent: subgroup of complex rank of the real span RA Then the n and MCA the maxi- following statements are The CI/A of Toroidal Concept Groups is toroidal There exists C'\ f 01 E no a so that the scalar product (9, A) for all A E A The maximal real torus E Z is integral (Irrationality condition) complex subgroup MCA/(MCA RA/A n A) is dense in the maximal C'/A of (Density condition) Proof If I >- complex a exponential vector function :6 a e((a, z)) = periodic holomorphic function If the projection of the 2>-3 dense in RA/A, one a that (a, A) E Z (A E A), then the exp(27r o-, z)) (z E Cn) is a non-constant Aso maximal complex subspace MCA C RA is not one splits (Proposition 1.1.3) After change R-independent set of generators of A so that the then at least of coordinates there exists unit vector el is exists C* and the others orthogonal are Then for a := el all scalar products (01, /\) (A E A) Let f be holomorphic on Cn/A Then f is bounded on the compact real 3>-1 torus RAIA and therefore f o 7r constant on the maximal complex subspace MCA C RA Now f must be constant an RA/A by the density condition Then the pullback f o -7r is constant on RA and because the complex rank of A is n the holomorphic function f must be constant on Cn Q.E.D E Z A complex Lie group is Stein manifold The a products C Every connected Abelian second factor is a Stein group, if the x C` real group is are underlying complex isomorphic to R1 x real torus For connected and Abelian get the of Abelian Lie groups 1.1.5 Decomposition Every connected Abelian complex C with a toroidal group Xo The Lie group is X C*M X a Abelian Stein groups (R/Z)m where the complex Lie following decomposition proved by REMMERT [64, and by MORIMOTO in 1965 [74] 1964] we manifold is groups cf KoPFERMANN (REMMERT-MORIMOTO) holornorphically isomorphic to a XO decomposition is unique Proof Existence Propositions 1.1.2 and 1.1.3 together with Theorem 1.1.4 Uniqueness Let Xi := Sj x Tj (j 1, 2) where Sj are connected Abelian Stein groups and Tj toroidal If : X, -+ X2 is an isomorphism, then obviously O(T1) C T2 and therefore O(Tl) T2, T, and T2 are isomorphic and thus S, Q.E.D XUT, and S2 X2/T2 are isomorphic = = = = A consequence of the toroidal groups is decomposition theorem and the density condition for Irrationality and toroidal coordinates 1.1 1.1.6 Lemma For any connected Abelian complex Lie group X the following statements are equivalent: Stein group X is a X is isomorphic there exists no to x C*m complex subgroup of positive dimension connected maximal compact real subgroup (Stein MATSUSHIMA and MORIMOTO in the of X group criterion for Abelian Lie proved in 1960 the groups) following generalization of this [70] lemma (MATSUSHIMA-MO RIM OTO) 1.1.7 Theorem Let X be a connected Lie group Then the complex following statements are equivalent: X is a Stein group The connected component of the center of X is X has connected no compact real For the complex subgroup subgroup (Stein of X of this theorem refer to the proof Stein group a positive dimension of original With the previous Lemma the Stein groups Lie groups where the connected center is are in any maximal group criterion) paper exactly the isomorphic to a connected complex C*m x Complex homomorphisms Complex homomorphisms be described universal 1.1.8 by of connected and Abelian covering spaces in the Abelian Proposition complex homomorphism For any A C C' and A' C the commutative C" T there exists : a 7r : Cn , Cn /A and 7r' will be Lie groups C"/A' 17r 17r, Cn' /A' Cn' with discrete map C' the subgroups + C" with C n' Cn /A : can description by prefered Consider first unique C-linear diagram , is called the lift of case C/A Cn where complex Hurwitz relations Instead of tangent spaces the , Cn' /A' are the natural projections r Conversly, a C-linear map f : Cn + Cn' with f (A) C A' induces a complex homomorphism -r: Cn /A -* Cn' /A' such that the diagram becomes commutative The of Toroidal Concept Groups Proof By the path lifting theorem there (O) with such that 7r' o, = phism and that Let X := is = exists a T o 7r Hence - unique continuous becomes CI map complex a homomor- C-linear map a Cn /A and X' Q.E.D Cn' /A' := C" -4 and : -r X X' be + complex homomor- a phism Then: covering map, iff its lift is bijective Then X is A isomorphism, iff i is bijective and (A) a complex Lie subgroup of X', -r is a -r is an X is iff there exists , : X is Cn a Cn' .> closed Now let P X Lie -r: - - is C) homomorphism X, iff there of X -r : -+ X, iff n (Cn) is embedding an Cn' can be described is defined := by P'M" = and ho- (A',, - -,A,,) - be a the matrix relation (Hurwitzrelations) (z) - by = E Cn and M' E map, iff C GL(n, C) Cz (Z M(r, r; Z) matrix Then: complex homomorphism bijective, iff C E T is The group X GL(n, C) = Cn /A is a C=n and CA=Aln maps T : covering a and M' E P'= CPM Holomorphic A, = basis of A CCn and P' -> integral The (A) subgroup (A,, -, A,) be a Cn' Then : Cn where C E M (n', n; T group of X' covering X' + CP an immersion and injective and complex := basis of A' C (injective) an is momorphism is a M with Lie complex GL(r, Z) =: so M'-' subgroup G is regular that GL(r, Z) of X' = Cn' /A', iff the rank (Cn) Cn /A Cn' /A' + of toroidal groups essentially are com- plex homomorphisms 1.1.9 Let T: Proposition C" /A' Cn /A + toroidal, Cn' /A' Cn /A, any be a holomorphic map with -r(I) 1, complex Abelian Lie group and where 1, 1' = Cn' /A', respectively Then T is a where Cn /A is are the units of complex homomorphism Proof By path lifting theorem there , (O) = 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 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 T In-q T n Zn (D F with + q a set of 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 In-q R, R2 In- 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 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 RA real 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 a basis of the The toroidal That is (u periods by Cq, v E Cn-q) E obtain we 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 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 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 (Imi )-' Jmt)-'Ret 0 Iq 0 R, R2 In-q 0 0 In-q 0 In-q A or R, R2 In-q 0 In the second 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) Extendable 4.2 exists a A : subset a 109 L satisfies (t-')*Lp X with on the condition LI, (C), then there If there exist X(I) with qX) Lemma 4.2.5 Let bundle line holomorphic (H, p) of type factor theta reduced a bundles Corollary 4.2.4 If line (t-')*L,3 C* be C' x I C n Ll,(X), reduced a qj - and satisfies then factor theta (H,,q) of type bundle holomorphic line the condition (C) a L Proof Suppose that (q ej automorphic Cn 0: n) such that factor b : A x p(z H(Ao, ej) (Cq X p(j)) + ej) + hand, On the other b.\0 (z a we + ap(,\)(w) e(b,\ (z)) ej) b,, (z - b,, (z) + have an function holomorphic L ej) + + Ao) + be, (z) have a define (z) ej) b,\ (z) Lp H(Ao, ej) :A Im + we we : : a A (Cq X X Cn x 6,, and mapping p(A) (z) C Z, E we - b,j (z - If E Cn z b.\0 (z) - be (z - ,., + Ao) - b,\ (z) Q.E.D contradiction 3: A x Ljqx), L, iff exists a satisfies be L,,, C* be Cn Then there L, we the in Theorem 4.2.6 Let Ao) + by obtain Also e(ap(;k)(w)) Since 6,\ (z = We sum up the above results Let a E A and We can take op a = e(b,\(z)) = = = is By A0 exist Lemma4.2.2 any A E A and for Then 0,\(z) b.\0 (z This Then there C* and A)O,\(z) o(z)-l + A)0,\(z) p(z)-1- (z) bei (Z) p, then b.\0 (z have + C such that a o X C such that Cn P(I)) p(z = ,\(z) ) by (C) Im C* such that ) ap(,\)(p(z)) We set p(A) a: the condition satisfy does not j :! + < a where and automorphic ,3 of type (H, p) We say topologically line in the trivial a L,, Lp proof of factor L, L on X (or X) with (t-')*Lp (C) a is factor bundle bundle is that theta line the condition factor The argument reduced holomorphic holomorphic Lp, a on a toroidal topologically a theta the condition Lemma4.2.5 we is also obtain line valid the a we bundle given reduced theta (C) for Then X group given by bundle L, satisfies Then trivial if a does line have by an factor it bundle L, with and Extension Reduction 110 Lemma 4.2.7 L,, Lp be L, does not satisfy Let L, If bundle 4.2.8 Let L exists a line n bundle then qj - L toroidal line no toroidal on a Ljqx), ! above as holomorphic line Ll,(x) bundle X group exists (t-')*Ll (t-')*Ll with on on a there with holomorphic trivial homomorphism A bundle (C), condition any I C Proposition L, be a topologically If there a X(I) on the for line holomorphic a then group X L, is given by C* Proof that We suppose Proposition 3.8, the By 01 = =0 for for (Cq X = Zn); all any I C p(j)) M1i***iMn-qEZ; +Mn-qen) +'** = x factor automorphic By ax(z) e(ax(z)) summand a: A x Cn the automorphic properties: assumption, an that assume A =Mleq+l for j is Zej-periodic p(A) : a,\(Zq+l, by given is may following C have the 1) ax(z) 2) a,\(z) 3) ax (z) L, we q + 11, n C* and n , q I there - exist an automorphic W, function holomorphic a factor Cn : C* such that 0,'(A)(p(z)) H (Z) ae, j = = q + fj(Z) Cq Eni=q+ _ X Cn-q f I(z Then n kizi for + ej) j - (j = = 1, 'n, kj E Z q + of part W'(z) WI(z) Cn with on f I(z) periodic the A)ox(z)W,(z) q + = Zej-periodic is Expanding + f, function a holomorphic 0P1(,,)(p(z)) We can take W'(z = = e(fT(z)) Since for Zej-periodic Therefore (j n) q + n) We write z (z', z") is = = I f (z), obtain we n f,,(z')e((o-, f,(z) +1: Z")) We have also the 1: (z) a,\ zi (z) of a,\ expansion Fourier ki i=q+l ,EZ'-q z")) a,\,,e((a, CEZn-q Take V p(A) : (j periodic = X q + (Cq X 1, p(j)) n), we b,(,,)(p(z)) e(b,) C with have the similar Since b,(,,)(p(z)) P is Zej- expansion E bI,,(z')e((a,z")) = A P o,EZ'-q If i E o-i 1, then Xq+i(I) < It follows from = (*) C Therefore that b,,\,, (z) = for a= t (al O'n-q) with Extendable 4.2 b, ill z")) (z')e((u, (a,\,, UEZn bundles line fi(zl + A')e((o-, + A")) f,, (z)) e((u, z")) - " - n E + (mod Z) ki Ai i=q+l Thus obtain we for a = variable Un-q) t (al Zk (1 I < k < fli(Z' + a,\,, with q), on T (t we t2q) trivial + fi(e((u, equality above by the = aZk (rl, - r2q) -, homomorphism tj of L, section holomorphic a by a (see Remark _= f,, fo A")) a r 1) - e((a, A")) A with not depend j F, E the 16) Hence be Let L, a line e(-(u,rj)) + For X is I Of /OZk toroidal, for _= bundle Then k = L, 1, we is q put ax,, some = Of A")) (Ri R2) and = f1i for A')e((a, + defined as f,I ( z') Since ji)Z2q analytically Then 17 (9f.IlaZk consider can not (Iq Cql(jq = f,,(z 1) - Differentiating < o-i A")) have we Og (z Zk Weset A')e((o-, + f the on we can :A same e((a, A")) - ( a 11 Un-q) we can with o-i < write ax,, - e((o-, All)) The ax,, I 0, of such choice get by the t = a A for a,\ of the side right-hand (z) is an above equality does summand If automorphic argument for a= t(Oli some i E I Un-q) ) with ui > We set Z(J) Then := jor E Zn-q; F-aEZ(I) fie((u, or, z")) < is for convergent on or Cn-q o-j > 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 Icl \ 101 we that We note \ 101 Z,-q \ 101 EZn-q or and Extension Reduction 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' = c-_ Lo -k on be with Then is cobordant ,o(A for (C) (t-')*Ll with on to 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,\(z) where is that + On an-qen) Cn we can define The theta a H and Q have the independent H is E Z Lp, assume + a, eq+1 + q variables a',(,\)(&(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, 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 (0 half upper < V < n q) - degree space of n form of P be basis using invertible Theorem H is of kind that assume we and A obtain P the = After respectively and unimodular matrices 3.1.16), A normal (W D), forms W a (see matrices change of basis proof of the Fibration suitable the of P and P (Wij) G as follows 'Hn di D dn where di (i 1, , n) are positive integers with d, I d2l Idn, and di W/ dq+t (P/ Pit), dq+,+, P wit dn0 114 Reduction where and Extension put we Wl,q+t W11 W/ Hq+j, E Wq+ J Wq+i,q+t Wq+i+l,l Wq+f+l,q+ WI/ Wnl Then the Ci X projection C*n-q-21 -fibre T Cn : pn-q-'-fibre associated Cq+f bundler bundle (q first the onto A X : Wnq+t Compactifying A We shall T f)-variables + fibres, the study gives obtain we a the extendability the - of L, X to We first define P(ZI) The Zn) i (Zl) = ; precisely more homomorphism group a problem the formulate Cn p: as Cq+f ) X e(Zq+i+1), , Zq+i, previous section C*n-q-21 x C' by e(zn-q)7 Zn-q+l on Cq+t p(A) acts properly discontinuous X Then X pn-q-t) we have an isomorphism lp(A) subgroup (Cq+ CTp(A) which Cq+f and fr: Weconsider as x considered gives = as P, (z, w) a real embedding natural pn-q-t be the Problem A, , of Ox Let of Cn X the on P as = (zl, basis x = We assign x W1i = - - ( x,, ) t we 2n for E R Let C*n-q-V X 7r Zn) :Cn X X give the form normal - E Cn x i o 7r Wn-q-f) - x', x" Wx' + Dx", (Cq+ p o complex coordinates (zl, Zn) The basis P (W D) is the We write fr pn-qI Let First situation , Zq+i, of Cn Then any X X : with projections p 106 in this above t We the in - - = be can C- R represented as n x E Cn theta factor X It is well-known that is cobordant 790,&) for is E A to the ?90,.\ restriction have A' where e I- WA'+ DA" and with cobordant we = = to 'A'x x , - E Cn do' of (a) the (cf on A, = 'A'WA' do' I Theorem 3.1 in A xCn (b) We note [631) that Thendx for A E A Extendable 4.2 Zq+f C a b= , E explicitly writedo,.\ A Pa + P"b and = dX be then a we can theta factor take a (z, w) with basis P of A - Hence we obtain the We say that a (N) subset is the normal I C n q - E C' form of 2f} an ample theta := represented f, as factord), Wedefine C Xq+t+i(I) ample and of kind taW'a - (z,w) - H is above and Vx is taz e H If form in the as = A=P'a+P"bEAandx= Take E Cn Hermitian dx (x) (N) for = taWal taz e = Proposition 4.2.11 Let x zn-i follows as ,do,.x(x) for 115 bn-i aq+t we can bundles bi a, Then line if i E I Pi \ 101 if i E IC We set P (I) Xq+t+l (I) := X * , , X(J):=,k(Cq+i (Cq+f = Then we have t(X) C X(I) C - A* where T Cq+f X pn-q-i X Xn-t (1) X x X (P1 \ f 01)f) p(J)) lp(A) P (I)) Let := -f (p(A)) = T- (A), Cq+i is the projection Then the basis d, dq+f and A = Cq+f /A* Wedefine ,Ooo*,,\* a = theta 1- factor eta*z 0*,),,, on ta*Wa* A* X Cq+t by P* of A* is for A* W'a* + D'b* = by 00*0'A* and Extension Reduction 116 Let * I C n L := q - - and T*Lv.* VI Cq+i E z ,di : p(A) x (Cq+ ,dj,,7(y) hand, d,\ (x) On the other the obtain we bundle defined A For any T LIX(j) Cn Then - is given by an = C*' = (-T op(x)) i90*',j:.P(,\) following P(I)) x Hence which proposition, we (t- 1) L'9 have 1,(X) * Lv is extendable says the to compactification Proposition 4.2.12 L, there A* X Cq+t For exists the line T*L, *, where C* such that L, investigation of topologically - A given LO*0 bundle T : by Cq+t X factor theta a Cn-q-i 0*0 Cq+i is projection the The rest X Let is the trivial line bundles above, and let Proposition 4.2.13 Cn /A = be trivial topologically holomorphic variety quasi-Abelian line bundle holomorphic bundle line Then there the exists a as a Suppose that C* such that Lo X on there X be a exists a p o -T : Ll,(X) - homomorphism homomorphism Lo such that L (L-TLO by p(l) X L,00 by of pull-back be the have Cq+t we A be the theta factor automorphic Thus Let L,9* A* : A X is given C* Proof Let di Dn-t dn-t We assume that summand a,\(x) automorphic a,\ a,\ (x) (x) = for is Dn-I Takeasubset factor,3I : Lo is defined A E Dn-f Zn-t; Zn-t-periodic I C p(A) X J1, 'n-q-2tJ (Cq+t X p(J)) by has the with factor automorphic an (see properties respect By to xi, Lemma4.2.2 C* and a a,\(x) Section = e(a,\(x)) The 2.1): )Xn-k- there holomorphic anautomorphic exists function p I : Cn C* such that 0P1(A) (P(X)) pI(x+A)a,\(x)WI(x)-1 forall (A,x)E(Ax (Cn n (Cq+ x P(I))) f, Let be holomorphic a automorphic summand e(bI(',)(y)) the line E bundle E(p(Al),p(A = (Cq+l x P(I)) 2(X(j), Z) represent given by,3I Then b,(A,)(y) + bundles e(f,(x)) first the (Y) P + P Lo is P is = Then = of bI( A2)(Y) for Al A2 E A by (1) trivial, topologically E(p(Al) P(A2)) that PI Therefore trivial assume we may topologically Lp, Ad, ej) (P(X)) n f, where ej is the j-th unit vector of C' Hence we have 1, j Since an Chern class b,(Al)(p(A2)Y) - P 117 We have C such that x bp'I(,\2)(p(Al)y) = p,(x) with E H = 2)) C' on p(A) : cl(Loi) Lp, on X(I) Let P function V line Extendable 4.2 = for - , fI(x+djej)-fI(x) Wedefine a =kj E Z, i = n f - function Dn-,eZ7 periodic n-f k,(x) := kj E dj I(X) f Xi j=1 Let x = function (x', x", x", of x ) ECq+t we obtain Cn-q-21 X the Fourier C' x k,(x) Considering as a periodic expansion n-t fliw, fi(X) x"Id")) x"')e((a, IEZn-q-2t where we j=q+t+l set x"Id" Similarly d'xj' E + we t Xn-,e/dn-i) (Xq+i+l ldq+t+l, = have a,\ (x) E = (x',.T ax,, x"Id")), )e((o,, uE Zn-q-21 bi(A) P b,,, (x', x111) (p(x)) A e((or, x" Id")) o,EZn-q-21 By the right-hand with respect On the other Since the is on Thus of we P is element on bI(A) (1), Xn-t+j hand, V P(A) (Y) (n-f+j)-th holomorphic x"' to side P with (P(X)) is extendable (p(x))n-t+j to on of P, \ 101 with p(x) Xn-t+j is equal 1, bA,,(x')e((or, (p(x)) 0rEZn-q-2f respect to xn-t+j, t), obtain bPI(A) on I holomorphic respect holornorphically x"Id")) hence to Yn-i+j- bI(A)(p(x)) P independent C For i E 1, Xq+i+i(I) Yq+i+i- If on j E \ 101 P, and Extension Reduction 118 = Xq+t+j(l) then Ic, with to respect Yq+i+j j a E Ic = t (Ul, - from follows It Un-q-2t) , - - E (3) (1) bI(,,)(y) with = (0 o-i for to holomorphic is P respect have we (x) C with on P, \ 101 Therefore Hence b, all holomorphic is P = (2) for b,(,)(y) C Then some i E I or for with 0-j )e((o-, A"/d")) > some that bA,, (x')e((u, Id")) x" UEZ-q-V I: [a,x,, (x', X11/) + I, (x' A, x"' + + A O-EZ-9-2f n-i f,, (x, - x )] e((u, x"Id")) k.7 -7 + dj j=q+f+l By (2) and (3) (4) for ax, all some (5) o- j , t + f,,(x' A', x"' + Un-q-2f) (Ul + A with fo, (x' + A"/d")) )e((u, < 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,,