§4.1 T he Long Cohomology Sequence of a Com pact Complex F la t Manifold.
In w hat follows, M denotes a connected, com pact, com plex, flat manifold. According to Theorem (1.3.9), there is a n atu ral exact sequence associated w ith M
(4.1.1) 0---►A ---► G ---► $ ►1 ,
in which $ , the holonomy group of M is finite and A ~ is the tran slatio n subgroup of G ~ 7r^(M) . W e m ain tain th e sam e n o tatio n as in § 2.3 and let V denote th e complex vector space of dim ension n and complex stru ctu re t whose underlying real vector space is A (g) g R . Let Z , O , O* be the sheaves of locally constant Z-valued, holom orphic, and non-vanishing holomorphic functions on M . Following sim ilar lines as in § 3.2, th e short exact sequence of sheaves
0 ---► Z O — ! - * O* ---►0 ,
where [j(f)](x) = 27rif(x) and [e(g)](x) = exp{g(x)} , w ith x G M and f , g G Z , O respectively, gives rise to a long exact sequence in cohomology
0 -► H®(M;Z)-» H“( M ; 0 ) --- ► H ® (M ;0*)--- ►
H *(M ;Z )----►h1(M ;0) H ^ (M ;0 * )---► H ^(M ;Z )---> .
R em ark (4.1.2): R em ark (3.1.5) is always tru e and because of Corollary
(1.3.8) R em ark (3.2.1) is true. Therefore, we can freely replace Z w ith Z and O [resp. O* ] w ith O [resp. O* ] , where O is th e C-algebra
O := { f I f : V ► C , f holomorphic } and O* := { f I f : V ► C* , f holomorphic }
its group of units. So, in the case where M is a flat manifold, G is obviously th e Bieberbach group 7t^(M) and H*(M;Z) H *(7T]^(M);Z) , H *(M ;0) - H*(7Ti(M);0 ) [resp. H *(M ;0*) - H *(7Ti(M);0 *).
Furtherm ore, Proposition (3.2.2) holds tru e for precisely th e sam e reasons as before.
Therefore th e long cohomology sequence above splits again into tw o p arts 0 ---► Z ► C ► C * ► 1
0 ►h1 (M ;0 )/h1(M;Z) — » ► H ^ (M ;Z ) ► • • • . By use of th e known identification Pic(M ) = H ^(M ;0*) and by lettin g
PicO(M) := h1 (M ;0 )/h1(M;Z) one gets
(
4
.1
.4
) 0 ---►PicO(M)---► Pic(M )---> h 2 (M ;Z )---►••• .W e now w ant to take a look a t H (M;Z) . As topological groups, they are all discrete because their topology comes from Z . For p > 2 , H^(M ;Z) are no longer torsion free. However, one can easily com pute th e rank^[H^(M ;Z)] by m eans of th e following lem m a. It also tu rn s out th a t this depends only on th e holonomy representation p : $ ► GLg(A) and not on th e cohomology class defining (4.1.1)
L em m a (4-L 5): If G is a Bieberbach group as above and p : 0 ---►
GL^(A) th e holonomy representation associated w ith (4.1.1), th en one has H ‘(G;Q) = [A <h1(A;Q)]® ,
th e ^-in v arian t elem ents of i-th exterior power of H (A;Q) .
w ith (4.1.1). The E2-page is given by = H-^($;H*(A;Q)). Because Q is divisible , everything collapses except th e i-axis. Therefore
H'(G;Q!) ~ H "($;H '(A ;Q )) - [H'{A;Q)]*
By Lem m a (3.2.5) and th e universal coefficient theorem H*(A;Q) A * H^(A;Q) , where H^(A;Q) Hom(A (g) gQ,Q) , because 0 acts trivially on Q .
A. T he Hodge Decomposition
Let, denote again th e sheaf of germs of holomorphic p-forms on M. The cohomology groups H*(M;f)*) rem ain one of th e m ost im p o rtan t invariants of any com pact complex manifold. The Hodge Theorem rem ains tru e and one gets th e following Hodge Decomposition Theorem for any K ahler m anifold. A proof of the Theorem can be found, for exam ple, in [G- H] page 116.
Theorem (4-1.6): If M is a com pact K ahler manifold
H'(M ;C) - © H^(M;H^)
p-\-q = i
w here = H^(M;n*’)
L em m a (4.1.7): If M is a n-dim ensional, complex, com pact, connected,
flat manifold, th en we have
dim ^ H^(M; O) = i rankg H^(M; Z) and PicO(M) = H^(M; O) / H^(M; Z)
is a complex torus.
h1(M; C) ~ H \M ; O) ® H “(M; w here H®(M; ÎÎ ) is th e space of holom orphic 1-forms and
H '(M ; O) = H®(M; f i ') . T hus,
dim g h1(M; C) = 2 d im ^ H '(M ; O) (♦)
B y th e Poincaré D uality H"(M ;Z) Hq(M;Z) and because M is connected Hq( M ; Z ) ~ Z . It follows th a t Hyj_j[(M;Z) is torsion free. For
H "(M ;Z) ~ Hom(H„(M ;Z),Z) 0 E xt(H „_i(M ;Z ),Z ) ,
and Ext(H^_;^(M;Z),Z) is isomorphic to th e torsion subgroup of H^_]^(M;Z) . By Poincaré D uality once more, one concludes th a t H^(M;Z) has no torsion. By th e universal coefficient theorem
H '(M ; Z) ~ H '(M ; C)
=J> r a n k j H^(M; Z) ~ d im ^ H*(M; C) (**) By (*) and (**) now
dim,Q H^(M; O) = ^ rankgH^(M ;Z)
Because H^(M:Z) inherits th e topology of Z , it is discrete and th e statem e n t of th e Lem m a follows.
One can actually find th e dimension of PicO(M) ,
L em m a (4-1.8): dim^Pic(^(M) = ^ rank^[H^(A;i
proof: V is th e universal covering of M and M = V /A is a complex torus which is a finite holomorphic covering of M , Com bining (3.2.3) and (4.1.3), we get th e following com m utative diagram
(4.1.9)
0 ---► H '( M ;Z )---►h1 (M ;0 ) ---- ► P ic (M ) ► H ^(M ;Z )---► ••• M and M are K (7rj^,l)-spaces and from th e Lyndon-H otschild-Serre spectral sequence, we have th a t
One easily sees th a t in this case
= h1($;H0(M;Z)) = H ^ $ ;Z )) = Tors(H o($;Z)) = 0 E ^ ^ = E^’^ = K er([H l(M ;Z )]^ ► H ^($;Z)) .
Because $ is a finite group H^($;Z) is a finite group and thus H^(M;i is a subgroup of finite index of the torsion free group [H^(M;Z)]^ . Thus
rankg H^(M;Z) = rank^ [H^(M;Z)]^ = rankg [H^(A; and the proof of th e Lem m a follows now from Lem m a (4.1.7).
If we let H^(M; Z) = A @ ^ , where A is th e free abelian p a rt and ^ torsion group, we have
Proposition (4-1.10): The Im{ S : H^(M; O*) —► H^(M; Z) } is a direct
sum m and of H^(M; Z) and contains all its torsion elem ents. In fact, m ore is tru e, Pic(M )/PicO(M ) is discrete and
M I S ) = Tors(H2(M; Z)) ^PicO(M)
proof: H^(M; Z )/Im {6} —► H^(M; O) is injective by exactness, and
H^(M; O) being a vector space over C is torsion free. T hus all torsion elem ents of H^(M; Z) are contained in Im {6} . H^(M; Z )/Im {6} is finitely generated abelian and thus we can w rite
h2(M ; Z) - H^(M; Z )/Im {6} © lm {6 } .
Since H^(M; Z )/Im {6} is torsion free, Tors(H^(M ; Z)) C Im {6} . For th e second p a rt of the statem en t, it suffices to notice th a t from (4.1.4) we get an exact sequence
Pic(M ) ^ . xj2/
0 — p i ^ — ^ H^(M;Z) — > H -(M ;0 ) , where H^(M; O) is a vector space.
F u rth erm o re, exploiting th e topological stru ctu re of the groups in (4.1.4), we have
Proposition (4-1.11): Ker{ 6 : H^(M; O*) —► H^(M; Z) } is precisely th e id e n tity com ponent (H ^(M ;0*))^ of H ^(M ;0*).
proof: Clearly, because H^(M;Z) is discrete ^^(H^(M;0*))^^ = 0 . W e
now need to show th e other direction. Since H ^(M ;0) is a complex vector space, it is connected and so is e*^H^(M;0)^ C H ^(M ;0*) . B ut Ker{ 6 : H^(M; O*) —► H^(M; Z) } = e*^H^(M;0)^ is connected and contains (H1(M ;0 *))q , thus
Ker{ S : H ^ M ; O*) —► H^(M; Z) } = (H ^(M ;0*))^ .
R em ark (f.1 .1 2 ): Propositions (4.1.10) and (4.1.11) are still tru e even if
6 is th e connecting hom om orphism H ” ( M ;0 * ) ---► H ” "*” ^(M;Z) W e thus have, keeping the above notation, th e following two
^ © Al ~
l m {6},
Al © A2 ~ A ,
where
Ai
,A2
are free abelian groups.Proposition (4.L I S ) : rank^ H^(M; Z) = rank^ Z)]^ .
proof: From th e Lyndon-Hochschild-Serre spectral sequence of (4.1.1), we
have
h2(M; Z) = © E ^ 2 . E ^ ” = E^’“ = E^’“ / Im (E ^’^ —► Eg'")
E ^ ^ = = K e r ( E y —, e| ’")
g0^2 = = Ker(E^’^ —» e| ’")
Now, th e groups Eg’" = H^(0;H"(M ;Z)) = H^($;Z) a n d Eg’^ =
H ^($;H ^(M ;Z)) are finite, thus
ra n k j H^(M;Z) = rankg E ^ ^ .
e| ’ " = e| ’ " / Im (E ^’ ^ » e| ’ ") w ith Eg’" = H®($;H®(M;Z)) = H^(#;Z) a finite group. W e thus have,
ra n k j H^(M;Z) = ra n k j E ^ ^ = rank^ e"’^ ,
where E ^ = K er(E^’^ —» e| ‘) w ith E^’' = a finite group. It follows th a t
rankg H^(M;Z) = rankg = rankg
§ 4.2 Line Bundles and A lgebraizability of F lat Manifolds,
Let M denote a connected, com pact, complex, flat R iem annian m anifold. Corollary (1,3,8) says th a t M is isom etric to one of th e form ^ \ ^ 2n / ^2n where 0 2 „ is the isotropy group of th e origin, J^2n th e group of rigid m otions of and G 7r^(M) a torsion-free, discrete, cocom pact subgroup of ^ 2n • According to Theorem (1,3.9), th ere is a n a tu ra l exact sequence associated w ith M
(4.2.1) 0 ►A ---► G ---► 0 ---► 1 ,
in which $ , the holonomy group of M is finite and A ~ is th e tran slatio n subgroup of G . In w hat follows we m ain tain th e sam e n o tatio n as in § 2,3 and let V ~ denote the complex vector space of dim ension n and complex stru cture t whose underlying real vector space is A ® ^
R .
If V is th e universal covering of M , M = V /A is a complex torus which is a finite holomorphic covering of M , If fu rth er M is a flat algebraic manifold, then so is M , The m ain result in this section is th a t we can un d erstan d Pic(M ) as the ^-in v arian t elem ents of Pic(M ),
Associated w ith (4,2,1) one has th e cohomology class c G H^(^;pA)
th a t classifies th e extension and which we assume to be non-trivial. If i : A —► A = V is the m ap defined by sending Ai - > A ®1 , A G A , then th e class i:^(c) G H ^($;V) is trivial for $ is finite and V is divisible. W e
choose
since $ is always finite, th e above definition makes sence. It is not difficult to show th a t c G H ^(#, V) is a canonical choice, am ongst several ones, of a 1-cocycle such th a t
( < 5 c ) ( g i , g g ) = c ( f i ( ( g ^ , g g ) )
(4.2.2)
= b ) - * , E c(g^g,, l ) + * ^ E ^ c ( g ^ , k)
1^ ^ E jg i C ( g2, h) - c(gig2, h) + c(gj, g^h)) e 0
®2)> for c e H ^($, A) and thus
(4.2.3) g^c(g^, h) 4- c(g^, ggh) = c(g^, g^) + c(g^gg, h)
for ail g p gg, h G $; ^ is th e second differential in th e b ar resolution and 6 th e corresponding differential from the 1-cochains to th e 2-cochains. T he elem ents of G are being represented as the elem ents of th e cartesian product A x $ , where the addition is being defined by
{K g) + (A', g') = (A + g • A'+ c(g,g') , gg') , for all A, A' G A , g, g' 6 $.
T he action of G on V , G x V —► V is given by (A,g) (z) = g(z) -f |$|(A -H c(g)),
which induces th e action of $ on M , $ ~ G / A x V / A —►V/A given by
g N = [g(z) + l^l(A + c(g))] = [g(z)] ,
since |$ |c(g ) G A (recall th a t V = A (g) gIR inherits its G -m odule stru c tu re form A) .
For any G-module A the action of $ on H^(A;A) is defined by m eans of th e action of G on H ” (A;A)
G X H"(A;A) ---► H^(A;A) given by
(7*c) (A]^, ...,Ayi) = 7 'C (—7+Aj^+7, 7 + A „+ 7) for 7 6 G and A^- G A . For n = 1 and A = O*, this gives us
0 X h1(A;0*) ---► H '(A ;0 * ) as follows
(g*e„)(z) = e^_i„(g“ lz)
where e G H^(A;0*) , g G $ , u G A , and z G V .
R em ark (4.2.4): One can easily see th a t th e skew, IR-bilinear form F
which is integral on A and invariant under th e complex stru c tu re is 0- in v arian t if and only if its associated herm itian form H , H (x,y) = F(tx,y)-f-iF(x,y) is ^-in v arian t, for
F (tx ,y )+ iF (x ,y ) = H (x,y) = (g*H)(x,y)
= H (g“ ^x,g“ V ) = F (t(g “ ^x),g“ V )+ iF (g “ ^x,g“ V ) ' T he im aginary p arts have to be equal, thus
F(x,y) = F(g~^x,g“ V ) = (g*F)(x,y) .
T he other direction is obvious.
L em m a (4.2.5): If F as in th e R em ark (4.2.4) above is 0 -in v arian t th en so is th e 1-cocycle
e^(z) = a(u) exp{7rH (z,u)-h|H (u,u)} ,
where u G A , z G A (g) gR , H(x,y) = F (tx ,y )+ iF (x ,y ) and a is a sem icharacter for H .
inv arian t) .
Proposition (4.2.6): Let H G [NS(M)]^ , F = ImH and a be a
sem icharacter for H . Define
• = ^ |0 |(A + êW )W where f„(z) = e_„(z) = e„^(z—u) and
e„(z) = a(u) exp{7rH (z ,u )+ |H (u ,u )} , u,A G A , z G V , g G
T hen h G H ^(G ;0*) . F urtherm ore, th e Chern class of th e line bundle defined by h is F .
proof: F irst from th e R em ark (4.2.4) and Lem m a (4.2.5) a G [Pic^(M )]^ and e G [Pic(M )]^ . One easily sees th a t
(4-2.7) fu + i,(z) = 4 ( z - u ) fu(z) .
T he cocycle condition for h is as follows
,) + (V, / ) ( " ) = (^ ’8) • h(V, g p ) ■ W e th en have “ + gX' -\- c{g,g'\gg')^'^^ ^ I $ I (A + + c{g, g ' ) + c ( y / ) ) ( ^ )
- l | $| ( A + a(») + sA' + ,c(,'))^^^
+ f | $ | ( A + cW )W V i ( » A ' +•f 0
f G [ P i c ( M) ]$
' ^ | $ | ( A + c(,))W
(where we recall th a t (A,g)“ ^ = ( - g “ U - c ( g “ ^,g) , g“ ^) ).
For th e second p art of th e proof, first notice th a t th e line bundle L(H ,a) corresponding to H and a as above is obtained as the quotient of C x V for th e action of 7rj^(M) = G given by
f I « I (A + c(ff))(^ ) ■ " ' g z + l ^ l ( A + e ( g ) ) )
where th e notation is being kept as above. To find th e C hern class of L(H ,a) we look at the com m utative diagram
P ic (M ) ► T he vertical m aps are inclusions. In particular,
j(ci(L (H ,a))) = c^(i(L(H ,a))) = c^(i(h)) = c^(f) = F .
Also, c^(L(H,a)) is a torsion free elem ent of H^(G;Z) , thus by th e proof of Proposition (4.1.13)
C i ( L ( H , a ) ) e E « i,2 < = [ H ^ ( A ; Z ) ] * .
com pletes th e proof.
One can have further
L em m a (4.2.8): Let L^ and L2 G Pic(M ) such th a t C j(Lj) and
Cj(L2) are torsion free. T hen Cj^(L^) = c^(L2) if and only if th ere exists a (A,g) G G such th a t th e one line bundle is th e tran slate of th e o th er, i.e.
•
proof: Since for any (A,g) G G the translation : M
hom otopic to th e id en tity
M is
To prove th e other direction, it suffices to show th a t any line bundle w ith chern class zero can be realized by constant m ultipliers. To this end, notice th a t th e diagram of short exact sequences of sheaves
0 -4 C 4 c
1 2
4 0
4 0
induces for any com pact K ahler manifold a com m utative diagram as follows
i î
T h e m ap i , represents th e projection of H^(M;C) = H^’®(M) © H®’^(M) on th e second factor and so is surjective. It follows th a t any cocycle h G H ^(M ;0*) in th e kernel of c^ is in th e image of i^ , i.e. is cohomologous
to a cocycle w ith constant coefficients and this proves th e Lem ma.
By Proposition (4.1.11) and the com m ents at th e end of §1.2 we deduce th a t PicO(M) is precisely th e iden tity com ponent of Pic(M ) . F u rth erm o re, Pic(M )/PicO(M ) is finitely generated abelian and therefore discrete as a Lie group. So, in conjunction w ith Proposition (4.1.10), we m ay w rite
^ Tors(H2(M ;Z)) © .
Now, let ij> : P ic (M ) ► P '‘ o^M) th e identification m ap.
Define P = ^ “ ^(T ors(H ^(M ;/)) , then P occurs in th e following tw o extensions:
(4 .2 J 0 )--- 0 ---► PicO (M )--- ► P ---► T o rs(H ^(M ;Z ))---► 0 , and
(4.2.11) 0 ►P---► Pic(M )---► 0 .
T he 2-cocycle in H^(Tors(H^(M;Z));PicO(M)) th a t classifies extension (4.2.10) is trivial. In th e opposite case Pic^(M) would be a subgroup of finite index in a connected Lie group and so contradicts Proposition
(4.1.11). Since Pic(M ) is an abelian topological group we can w rite
(4.2.12) Pic(M ) ~ PicO(M) © T ois(h2(M ;Z )) ® .
This, for a com pact abelian topological group, follows directly from a theorem of Pontrjagin, Theorem 55 on page 213 of [P]. T he case of a general abelian topological group A w ith id en tity com ponent a Lie group
A
q such th a tA /A
q be finitely generated, can be reduced to th a t of a com pact abelian topological group through the following reasoning:Let (j) : A ---► A /A ^ 2=: $ 0 Z” be the identification m ap. Define = . T hen G ^ < A , A / G | ~ Z ” and A G^ x Z ^ . Now, G j is a Lie group w ith finitely m any com ponents, so it has a m axim al com pact subgroup, say K , see relatively [M4]. K < Gj^ w ith G^ abelian, so G j / K ~ IR^ and finaly G^ ~ K x . The decomposition now follows from
P o n trjag in ’s theorem on K .
W e have now proved, Proposition (4.2.6) and extension (4.2.11), the following version of the A ppell-H um bert Theorem (3.4.5) for flat manifolds
Theorem (4.2.13): If M is a com pact, K ahler, flat manifold, and P th e
extension of Tors(H^(M ;Z)) by PicO(M) as above. Then th ere is a short exact sequence as follows
0 ►r ---► Pic(M ) ---» [NS(M)]* > 0 .
Corollary (4.2.14): If Pic(M ) is w ritten as in (4.2.9), then N is th e rank of [NS(M)]^.
Corollary (4.2.15): Let II be a ^-in v arian t herm itian form on V ,
integral on A and L(H,t/j) th e line bundle on M associated to (H ,^) , ip
a sem icharacter for H . Then H is positive definite if and only if th e m ap induced by a linear system of ^-in v arian t holomorphic sections of L ^ gives an im bedding of M as a closed subm anifold in a projective space for each n > 3.
proof: By m eans of G rothendieck’s spectral sequence, see page 2 0 2 of
[Or], we have
H °(M ;0(L )) = [H“(M ;0 (L ))]* . T he proof now follows from (3.6.8) and (4.2.10).
Chapter V
§ 5.1 Projective F lat Manifolds of Prim e Holonomy.
In this section, we classify th e projective flat manifolds whose holonom y group is either a cyclic group Cp , or dihedral Ü2p , w ith p prim e, give an estim ate for the size of the set their positive line bundles and constuct explicitly and directly in term s of th e representation space of th e holonom y group some interesting imbeddings.
W e first recall th e basic facts of th e integral representation theory of th e cyclic group of prim e order.
A. Integral R epresentations of Cyclic Groups of Prim e Order.
T he m ain result here is th a t the arb itrary Z[Cp]-module is essentially composed of three m uch sim pler types of Z[Cp]-modules.
Throughout this section , let denote a p th prim itive root of u n ity over Q, and let K = Q(Cp)- We also let R denote th e ring of all algebraic integers in K which we know to be a Dedekind dom ain w ith Z- basis {1, Cp, Cp^~^}r so th a t (K:Q) = (R:Z) = p —1.
Suppose A is any fractional ideal of K. T hen we tu rn A, th a t has a Z-rank p —1, into a Z[Cp]-module by defining th e scalar m ultip licatio n by
g ' a = Cpa , a € A
and g a generator of Cp. From th e above definition, it is obvious th a t th e m odule stru ctu re on A is well defined and also th a t two such ideals A and B in K are the sam e as Z[Cp]-modules if and only if A and B are isom orphic as R-modules; and this in tu rn is equivalent to A and B being in th e same ideal class.
T he second type of Z[Cp]-module is constructed as follows:
Let again A be an ideal in K as previously and a^ a fixed elem ent in A. Consider now th e external direct sum A 0 Z as Z-modules and tu rn this in to a Z[Cp]-module by defining th e scalar m ultiplication
g • (a, n) = (Cpa + na^, n) , a e A , n e N.
Because is a root of the cyclotomie polynom ial of order p one can easily check th a t the above definition is indeed ’’good” . T he construction