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, ~ (e n J e ) . Now we discuss the ad(JZ)-action on J e n e. From the integrability condition it follows t h a t [JZ, J C ] = J [ J Z , C] for all C E J e n e. Hence J e n e is ad(JZ)-stable. In generalizing the eigeuspace decomposition for not necessarily diagonalizable endomorphisms we define weight spaces gx. Let V be a real vector space, an endomorphism, and Vxc := {z E V c = V |
C I : I N : (~ - Aid)N(x) = 0}.
Then
v -
(v n ( v f + v f ) ) . l m ~._~0
Thus we obtain a complete decomposition of V. Now assume t h a t there is a Lie algebra structure on V and t h a t ~ is a derivation. T h e n
v. l c We use this to analyse the ad(JZ)-action on J e n e. These methods finally yield a complete description of the Lie algebra g and its J - s t r u c t u r e . This determines the complex structure on G . To realize this homogeneous manifold in C6 , we then give a concrete realization of this Lie algebra as an algebra of globally integrable holomorphic vector fields on C 3 . This yields a G - a c t i o n on C 3 . A given point in C 3 lies in an open orbit of G iff the g - v e c t o r fields span the whole tangent space at this point. T h u s it is possible to determine the open orbits by explicit calculations with determinants.
18 G r e a l s o l v a b l e , dims(G) > 6 In the solvable situation we still have to discuss the subcase that no real sixdimensional Lie group acts transitively. This is possible only under special circumstances. In particular this implies that there is a fibration
G/HX+XCG/J. A key point in the further analysis is the group A := ( N n j ) 0 . We prove that this group A is actually an abelian normal Lie subgroup of G. Furthermore g = < A 1 , A ~ , A s > s ~ a for appropriately chosen A~ E g. Now the ad(A~)actions must stabilize the totally real subspace a in ,~ = a + is. From this we finally deduce that dims(A) < 4. Since dims(G) > 6 this implies dims(A) = 4. Using explicit calculations with vector fields on C s we then classify the poesible manifolds, which are only the following two.
~+ = { ( z , ~, ~) I z m ~ - R~ ~ I m ~ + (Re z)~ > 0} n - = {(~, ~ , ~) I Zm z - Re ~ Zm 9 + (Re ~)~ < 0}
G mixed Next let us discuss the case where G is neither solvable nor semisimple, i.e. G = St< R with dims(R) > 0, dims(S) > O. The nilpotent normal Lie subgroup A := (G ~ N R) ~ plays a key role in this case. Since we now may assume that the above described "maximal holomorphic" fibration is injective, the center of G is discrete. This implies that R is not central in G , hence dims(A) > O. Furthermore G / H ~ G/[-I ~-* FN. Observe that .4~ can not act transitively on positive-dimensional A-orbits (This is in fact true for any nilpotent group A). From this we can deduce that either there exist fibrations
G/H
~
G/~I
~
GIi
1 GIz
1
or the group A is abelian and acts transitively on G//?/. In the latter case we obtain G --/7i K A with .4 = (C s , + ) . Consequently G C GLa(C) D4(7/.~, +) or
M-
1
m , n E Z,
zEC
Proof. Let ~r denote the natural projection M ---, M / M ~ = Z2 &nd choose elements a , 8 E M such t h a t ~r(a) = (1, 0) and 7(8) = (0, 1). T h e n there exists p, A E C* such t h a t ~-x-~-I
~_. ~ x
and 8.z-8
-1 = ~ z
for all z E M ~ - (C, + ) . Furthermore there exists p E M ~ -- C such that a " 8 = 8" or. p. Now the group structure of M is completely determined by the p a r a m e t e r s p , ~ and p. We will now replace a and fl by a - s and 8 " t where s , t E M ~ - ( C , + ) in order to obtain new parameters p ' , A' and/71 defining the same group as p , )~ and p. Clearly Ai = A and p ' = p , since M ~ is sbelian. Now let us calculate p ' :
=:,
a. s . 8.t
'
= /3.t .a .... =
5
p = ~ 41+ s +
=>
p =
p'
1
p, -- 1 -- t P 1) -
1 _ 1) = p'
Hence either p - ~ - 1 or we m a y achieve p - 0 by replacing a and 8 by a . s resp. 8" t for appropriately choosen s, t E M ~ --~ C. If p = 0, then or and 8 commute. Hence they generate a subgroup isomorphic to (Z ~, + ) . Therefore M --- M ~ ~ (Z 2, + ) for p = 0. Finally we want to construct an isomorphism ~b:M
~Go=
{(' + 1
m, nEZ,
zEC
}
33
(11)
for the case p ~ 0, X = p = 1. This m a y be done in the following way:
r
1
1 1
~:])~
r't In b o t h cases of the above l e m m a it is easy to show the following
L e m m ~ 4. Under the assumptions of the abo~e lemma there ezists a threedimensional complez Lie group G* and an injective Lie group homomorphism p: M -~ G" such that p ( M ) is normal in G" and G ' / ( p ( M ) ) ~- (C" • C ' ) . T h e preceding L e m m a t a now imply:
Under the basic assumptions there exists a three-dimensional solvLie group G* with the following properties: G* "~ C a ~_ G I N ~ • N ~ ~ as manifolds, N / H o c G" as a Lie subgroup with ( N / H ~ o normal in G ' , and N / N ~ -_ Z ~ C G ' / ( N ~ ~ ~- ( C a + l + ) , such that G ' / ( N / H ~ ~_ (C') h •
L e m m a 5.
able (i) (ii) (iii)
Construction
o f r : G* -% G / H ~
Thus there exists a N/N~
biholomorphic m a p
O: G * / ( N ~
~ -* G / N ~
We want to construct a N / H ~ biholomorphic m a p G* ---, G / H ~ . First we note t h a t b o t h G* ~ G * / ( N ~ ~ and G / H ~ --, G / N ~ are N~176 bundle over a base biholomorphic to C t x C r . Since the base is a contractible Stein manifold, these principal bundles are globally trivial. Let : G * / ( N ~ ~ :---~ G* and ~r: G / N o ~ G / H o be sections in these principal bundles. Together with the N~176 right actions they induce global trivializations. We will use this fact to define a biholomorphic m a p r : G* --, G / H ~ in the following way: For each point P0 E G* there exist uniquely determined elements a E G * / ( N ~ ~ and n E N ~ ~ such t h a t p0 = ~(a) . n . T h e n we define
r162
34 where R , denotes the right action of N ~ follows immediately t h a t r is N ~ ~ following c o m m u t a t i v e diagram
~ on G / H ~ . From the definition it invariant. Furthermore we have the
G*
~
G/H ~
G*/(NO/H o)
~
G/N o
1
1,
where d is an N / N ~ biholomorphic map. Our goal is to obtain a N/H~ map from G" to G / H ~ . Since @ is already N~176 problems m a y arise only if N is not connected. Hence let us assume t h a t N is not connected, i.e. N / N ~ ~_ Z k with k >_ 1. First let us discuss the case k -- 1. Then there exists a "generating ~ element o E N such t h a t N -- UkEZ ~ " NOL e n u n a 6. Assume that both ~ ( z . o ) = ~ ( z ) . o for all z E G * / ( N ~
~ (where a. b denotes group multiplication in G* reap. G * / ( N ~ 1 7 6 and cr(Ra(y)) = R a ( ~ ( y ) ) for all y E G I N ~ (where R . : z --, R . ( z ) denotes the natural right action o / N / H ~ on G / H ~ reap. G / N ~ Then the map ~ : G* --, G / H ~ defined by ~ ( ~ ( x ) . . ) = R . ( ~ o d(z)) commutes with the o-right action. Proof. Observe t h a t
~(r 9. - o ) =r -o..~) =~(~(~. o ) n ~) =R..(~(~(x). o)) =R.~ o ~(~)). o) = R . . o Ro(~ o 0(x)) =R~ o ~ ( ~ o 0(~)) =R~(~(~(~)- .)) for all z E G * / ( N ~ O). r'l
~ and n E N ~
~ . (Here n " denotes n conjugated by
T h u s it suffices to prove t h a t there exists a section ~ : G * / ( N ~ ~ ---, G* such t h a t ~ ( z . o) = ~ ( z ) . o for all z E A := G * / ( N ~ 1 7 6 To prove this we s t a r t with arbitrary section ~ : G * / ( N ~ ~ ---, G*. Observe t h a t we m a y modify this section in the following way: Let X : G * / ( N ~ 1 7 6 ~ N ~ 1 7 6 be an arbitrary holomorphic map. T h e n we obtain a modified section by
~(x) := ~(~)-x(~) N~
Next we introduce an "aberration measuring" function ~ : G * / ( N ~ ~ ---, ~ which for a given section ~ : G * / ( N ~ ~ ~ G* measures whether
35 commutes with the a-right action. To be precise we define ~b : G ' / ( N ~ G" such that
~ -+
C(x) 9 a = C(x. a ) . ~(~)
for all z E G * / ( N ~ 1 7 6 Then ( commutes with the a-right multiplication iff r is a constant map to eH ~ . Now let ~ and ~ denote the aberration measuring functions for the original section ( and the modified section ( . Then ( ( ~ . a ) . X(Z" a ) . ~(x)
= r
x ( ~ ) 9a
-- C(x). a . x ( z ) ~
-- c(~. a ) . r for all z E G * / ( N ~ 1 7 6
x(=) ~
where X(Z) a denotes X(z) conjugated by a . Hence x(=-a).
~(z) = ~(~). x(~) ~
for all z E G * / ( N ~ 1 7 6 Thus to achieve ~ - e we have to find a holomorphic map X : G * / ( N ~ 1 7 6 -"* N ~ ~ with X(Z" a) = ~ ( z ) - X ( z ) a. Recall that N ~ ~ is a complex solvable simply-connected Lie group. Hence there exists a sequence of Lie subgroups {e} --- A0 C A 2 . . . C Am = N ~
~
where A4-1 is normal in -4/ and Ai/Ai+l ~- (C, + ) . Thus we may first prove the existence of the desired function X for the special case N ~ ~ ~_ (C, +) and then apply induction to prove this in general. L e m m a 7. Let ~ E O(C). Then there exists a holomorphic function X E O(C) snch that X(= + 1) - X(X) = ~(z)
Proof. Let ao : C --, C denote a C ~176 such that a0(x) = 0 for all z E C with Re(z) _> 1 and a 0 ( t ) = - 1 for all z E C with Re(z) _< - 1 . Define an(z)--
{ao(z- n)+ 1 ifn>O a0(z--n) if n _ < 0
This implies in particular that an(z) = 0 for all n with In[ >_ l + ] R e ( z ) l . Hence the infinite sum nEZ
36 is everywhere a locally finite sum and therefore convergent. Now one can explicitly verify that X0(x + 1) - X(x) = $(=). Since # is holomorphic it follows that Ox0(z + 1) = OX0(z). Next observe that C / Z - C* is a Stein manifold, i.e. a O-problem on C* is solvable. Thus there exists a function g E C~176 such that 0g = 0X and 9(x + 1) = g(x) for all z E C. Now X := X0 - g is the desired holomorphic function with X(z + 1) - X ( = ) = # ( = ) . o C o r o l l a r y 8. Let 0 E O(C) and ~ E C*. Then there exists a holomorphic function X such that X(z + 1) - AX(=) = ~(x) for all z E C.
Proof. Let
"=
By the preceding L mma there e a t s a holomorphic
function X1 such that Xx(z + 1) - Xl(z) = ~ l ( z ) . Now X(x) := Xl(z))~ ~ is the desired holomorphic function, n This completes the proof for the following Lemma L e m m A 9. There exists a section ~ : G ~ 1 7 6 ~) = ~ ( z ) . a for all z E G * / ( N ~ 1 7 6
~
, G / N ~ such that ~ ( z .
C o r o l l a r y 10. Assume that N / N ~ ~_ (Z, +). Then there ezists a N / H ~ variant biholomorphic map ~ : G ~ --~ G / H ~ . T h e c o n s t r u c t i o n o f ~ : G* -~ G / H ~ f o r N / N ~ ~_ Z 2 Now let us assume N / N ~ ~_ Tfl. Let ~, ~ E N / H ~ such that ~r(~) -- (1, 0) and ~r(~) -- (0, 1). From the above considerations we can deduce that there exists a section ~ : G~176 ~ ~ G* such that ~ commutes with the ~-action. Now to controll the ~-action we want to modify the section ~ by a function X : G~176 ~ -'* N ~ ~ To preserve the ~-equivariantness X need to be -invariant. First let us assume that ~ commutes with ~. Then we introduce an "aberration measuring function" ~ : G ~ 1 7 6 ~ ---, N ~ ~ such that
for all z E G * / ( N ~ 1 7 6 Now we can proceed exactly as above. Then it finally turns out that we can modify ~ such that ~ commutes with both ~ and because G * / ( N ~ 1 7 6 1 7 6 ~_ C* x C* is a Stein manifold. In the case where ~ does not commute with ~ we introduce coordinates on G*/(N~ ~ "~ C2 such that (z, y ) . ~ = (z + 1, y) and (z, y ) . j8 = (z, y + 1). By the same methods as above we may achieve that
y).
_-
y).
37 and y).
=
r
#.
=
with z E C -~ N ~ ~ . This again yields a biholomorphic N/H~ map ~ : G* --* G~ / H ~ . The result Thus we proved the following P r o p o s i t i o n 11. Assume that X is a complez manifold with d i m e ( X ) c with the Lie algebra structure given by [A,B] = C and [A,C] = D . Let h = < C > c and X = G / H . T h e n no proper Lie subgroup of G acts transitively on X .
38
Chapter The case G semisimple, complex In this chapter we handle the case, where G is a complex semisimple Lie group. In this case we have the great advantage that there exists a detailed classification of all complex semisimple Lie groups. A key point in our considerations is the fact that only few complex semisimple Lie groups can act non-trivially on three-dimensional complex manifolds. 1. Basic A s s u m p t i o n s In this chapter we assume the following: G is a semisimple complex Lie group and H a closed complex Lie subgroup such that d i m c ( X ) _< 3 with X = G / H , G is connected and simply-connected, G acts almost effectively on X = G / H . 2. T h e c o m p a c t case In this paragraph we assume in addition to the basic assumptions that X is compact. Here we can use a result of J.Tits. He classified all compact complex homogeneous manifolds up to dimension three in [Ti2]. In particular he proved P r o p o s i t i o n 1. Let X = G / H be a compact complex homogeneous manifold. Assume that G is semisimple and dime(X) < 3. Then (i) X is bihoiomorphic to one of the following homogeneous-rational manifolds: ~1, ~2, Ps, Q3 or a direct product, or (ii) X is biholomorphic to a torus-principal bundle with one-dimensional fibre over P1, ~2 or ~x x P1, or (iii) X ~_ SL2(C)/F , where F is a discrete cocompact subgroup of SL2(C). Remarks. (i) Conversely every manifold listed in the above proposition is a complexhomogeneous manifold on which a semisimple complex Lie group acts transitively. (ii) Torus-principal bundles over homogeneous-rational manifolds with higherdimensional fibres are also complex-homogeneous manifolds, but not necessarily by a semisimple Lie group. (iii) Borel has proven that every semisimple complex Lie group (thus in particular SL2(C)) admits a discrete cocompact subgroup (cf. [B1]).
39
3. T h e n o n - c o m p a c t case Here we a~sume that X is non-compact. We start with the following result, which is true without any restriction on d i m e ( X ) : P r o p o s i t i o n 1. Let X -- G / H be a non-compact complez-homogeneons manifold. Assume that G is a simply-connected semisimple complez Lie group acting almost effectively on X . Then there ezists a homogeneous-rational manifold Q and a Lie groep homomorphism p : G ~ A u t o ( Q ) with a discrete kernel kerp such that dime(Q) < dime(X). Proof. (i) Let G / H --* G I N denote the G-anticanonical fibration of G / H . Let L denote the connected component of the ineffectivity of the G-action on G I N . Now L is a connected normal Lie subgroup of G. Since G is semisimple and simply-connected, this implies that there is a normal semisimp|e Lie subgroup Go of G such that G " L • Go. (ii) Observe that L C N implies that H ~ n L is a normal Lie subgroup of L. Since G - L • Go, the group H ~ n L is also normal in G. Hence H ~ O L acts trivially on X = G / H . Therefore H ~ n L is discrete, i.e. ~ : L ---* N / H ~ is almost injective. Since L is semisimple, it follows that o Ad : L ~ SLc(n/h)
is almost injective. (iii) Next consider the G0-orbit Y - G I N -- G o / ( N A G o ) "--, PN. The group Go is represented in A u t o ( ~ v ) as a linear-algebraic group, because Go is semisimpie (see [Hu, Prop.I.D.5]). Thus Y is the orbit of a linear-algebraic subgroup of P S L N + I ( C ) in PN. Therefore Y is a constructible set in the algebraic Zariskitopology of PJv (see [Hum, Th.IA.4]). This implies in particular that either Y is a projective-algebraic variety or the Zariski-closure Z of Y in P/v contains a compact G0-orbit E with d i m e ( E ) < d i m e ( Y ) . (iv) By a result of Hironaka (see [Hi]) there exists an equivariant desingularization Z of Z. Consider the G0-action on Z. Again Z contains a compact G0-orbit /~ with either dimr < d i m e ( Z ) or J~ = Z. (v) L e t L1 denote the connected component of the ineffectivity of the G0-action on E. Then there exist a normal semisimple Lie subgroup G1 of Go such that Go ~ L1 • Gx, i.e. G ~ L • L1 • G1. Let z0 E E. Then z0 is a fixed point for the Ll-action. Since L1 is semisimple, the natural representation qJ : L1 --* GLc(Tzo(Z))
is faithful (see [HO1, Prop. 1.1.5.1.]). Note that Lx stabifizes the vector subspace T~o(J~) of T,o(Z) pointwise, because L1 acts trivially on J~. Consequently the semisimple group Lx stabilizes a transversal vector subspace V of Tree(Z) such that the representation ~o : La --, G L c ( V )
40
is faithful. (vi) Recall that G - L x Lx x G1. Thus we have a faithful representation
To : G ---, S L c ( n / h ) x S L e ( V ) x Auto(E,). Let k = d i m e ( N / H ) and n = dime(V). Then dime(X) = dime(Y) + k and = d i m e ( 2 ) = n + dime(~:). Note that Y is compact, if n = 0. Since X is non-compact, it follows that n = 0 implies k > 0. Thus k + n > 0. Observe that there is an almost injeetive Lie group homomorphism dime(Y)
,~ : S L k ( C ) x S L . ( C ) .-, P S L . + k ( C )
= Auto(IP.+~-l)
This yields an almost injective Lie group homomorphism Tx : G ~
Auto(P.+k-1
x
~).
Note that d i m e ( X ) = n + k + dime(/77). Therefore
dime(•n+t-1 x E,) < dime(X). Thus the proof of the proposition is finished, if E is homogeneous-rational. (vii) Finally let us discuss the case where ~7 is not a homogeneous-rational manifold. Then let J~ -* Q0 denote the Gl-anticanonical fibration o f / ~ . The same arguments as above imply that there exists an almost injective Lie group homomorphism from G1 to SLv(C ) x Auto(Qo) with p = dime(/7~') - d i m c ( Q o ) . Thus there is an almost injective Lie group homomorphism
r2 : G ~ Auto(Pn+t+v-1 x Qo) with dirnc(l?.+k+v_l • Qo) < d i m e ( X ) , o C o r o l l a r y 2. Assume in addition to the basic assumptions that X = G / H is non-compact. Then either G ~- SL3(C), or G ~- SL2(C) or G = SL2(C) x SLy(C).
Proof. By the above proposition there exists an almost injective Lie group homomorphism p from G to Auto(Q), where Q is a homogeneous-rational manifold with dime(Q) < d i m e ( X ) . Since dime(X) _< 3, it follows that d i m e ( Q ) _< 2. Thus Q is biholomorphic to one of the following: 1~1, 172, I71 x 171. Hence there is an almost injective Lie group homomorphism from G to PSL3(C) or PSL2(C) x PSL2(C). Since G is simply-connected and semisimple, this implies that G is isomorphic to SL2(C), SL3(C) or SL2(C) x SL2(C). [] By an explicit check of Lie subgroups of SL2(C) it is easy to prove the following L e m m a 3. Let G "~ SL2(C) and let H be a closed Lie subgroup of G such that G / H is a non-compact manifold. Then either (i) H is discrete, or (ii) H is conjugate to
-}
41
and X is biholomorphic to P1 x P1 \ A with A = {(~, y) e ?1 • ?1 I z = y } ,
or
(iii) H is conjufate to
and X is biholomorphic to
?~ \ Q1 = {[zo: zl: z~] I zoz~ # z~}, or
(iv) H is conjugate to {/1
1)[z 6C} '
x _~ c ~ \ {(o, o ) } , or
(v) H is conj.gate to
and X ~- Cr \ {(0, 0))/Zn for some n 6 I~. R e m a r k . The class of discrete subgroups of SL~ (C) is (at least to the present time) too large for a fine classification. For instance, one can show that for each Riemann surface M the fundamental group lrx(M) can be embedded in PSL2(C) ~- SL2(C)/Z2 as a discrete subgroup. To prove this assertion note that there is a natural embedding of z i ( M ) in Auto(l~4), where ~/ is the universal covering of M . Recall that there are only three different simplyconnected Riemann surfaces ( ? I ( C ) , C, C* ), all of which are equivariantly embeddable in P , . Hence there is an embedding of z l ( M ) in A u t o ( ? : ) which is isomorphic to PSL2(C). 4. T h e case G ~_ SL2(C) x
SL2(C~
Here we assume G ~ G, x G2 with Gi ~- SL2(C). We furthermore assume that H is a closed Lie subgroup such that X = G / H is non-compact with d i m e ( X ) _< 3 and assume that G acts almost effectively on X . From the preceding paragraph we know that under these circumstances d i m e ( X ) = 3. Hence d i m e ( H ) = 3, since dime(G) = 6. Therefore either H ~ is solvable or H ~ is isomorphic to SL2(C) or PSL2(C). The subcase H ~ solvable
L e m m a 1. Under the above assumptions either H = Ht • H2 with Hi C Gi or B' C H ~ C B, where B is a Borel froep of G.
42
Proof. (i) Note t h a t
") (" ')1
}
is a Borel group of G ~- SL2(C) x SL2(C). Since H is solvable, we m a y assume t h a t the connected component H ~ of H is a Lie subgroup of B . Observe t h a t dime(B) -- 4, di,T~(H) = 3 and dimc(B') = 2. Therefore 1 < d i m c ( U n
B') < 2. (ii) First assume d i m c ( H n B ' ) = 2. Since dimc(B') - 2, it follows immediately t h a t B ~ C H ~ C B . This proves the proposition for the speci~l case dirnc(H n B ' ) - 2. Hence, from now on we assume that dimc(H n B') : 1. (iii) We can choose basis vectors X, Y , Z, W for the Lie algebra b of B such t h a t IX, Y] = Y and [Z, W ] = W are the only non-trivial c o m m u t a t o r relations between the basis vectors. Furthermore we can require t h a t b n g l = < X, Y > c and b N g2 = < Z, W > c . (iv) Recall t h a t we m a y assume dimc(H n B') = 1. T h e n d i m c ( B / B ~) = d i m c ( H / ( H n B ' ) ) and consequently h:
c
with A , B , C E b ' and C ~ 0. Now C = q Y + ~ ; V C 2 \ {(0, 0)}. Since h is a Lie algebra it follows t h a t
for some (~,/3) in
[X + A, ~ Y + / 3 W ] = IX, ~ Y +/TvV] = ~ Y E h and [Z + B , ~ Y + / 3 W ] : / T A r E h. Hence either ~ : 0 or 13 : 0. (v) Let us wlog assume that /3 = 0. Then h = < X + 7 W , Z + 6 W , Y >C for some 3', 6 E C. Next observe that IX + -yW, Z + 6W] = - - y W E h. Hence 7 = 0 and h = < X , Y , Z + 6 W > c . (vi) It follows t h a t after conjugation h = < X, Y , Z > c and therefore
This implies
,.,,,.,:,,ou{( :_,);(_._,
Hence N G ( H ~ ~ _ Z2 and therefore either H - H ~ or H - NG(H~ In b o t h cases we obtain H -- H1 • H2 with H~ C G~. []
43 Note t h a t X - G / H ~_ ( G 1 ] H I ) x (G2/H~), if H -- H1 • with Hi C G~. Hence the only interesting case is that where B ~ C H ~ C B . Here we continue our considerations in order to prove that B ~ C H ~ C B implies H C B . Since H C Nv,(H~ we start with the following Lemma. 2. Assume that B' C H ~ C B .
Then N a ( H ~ = B . Proof. T h e inclusion B C NG(H ~ is obvious, since B ~ C H ~ 9 On the other hand, B ~ C H ~ C B implies t h a t nbn -1 E B for all n 9 NG(H ~ and b 9 B ' . Now observe t h a t 1
~, -c2)~
It follows t h a t c = 0 for all (:~) 9 NG(H~
1 + ac)~/"
Hence NG(H ~ C B . o
C o r o l l a r y 3. The assumption B t C H ~ C B implies B I C H C B . L e m m A 4. Assume B I C H C B .
Then the basic assumptions imply that H / H ~ is finite and that G / H is a C* -principal bundle over Pl x P1. Proof. Note t h a t H / B I C B I B I ~- C* x C*. Consider the h o n m t o p y sequence ~2(B/H) --. , I ( H / B ' ) --. ,r~(l~/B') --. , ~ ( B / H ) --. ~ o ( H / B ' ) - - . ~ o ( B / B ' ) ={e}
=z~
=H/uo
={e}
Observe that H is normal in B . Therefore B / H is a Lie group and consequently ~r2(B/H) -- (e}. Since G / H was assumed to be non-compact, it follows f r o m G / B ~. II~1 • P l t h a t B / H is a non-compact homogeneous Riemann surface. This implies t h a t either ~r,(B/H) " Z or ~rx(B/H) = {e}. Furthermore H / B ' is a holomorphically separable Riemann surface, since H / B I C C" • C*. T h u s either l r l ( H / B ' ) ~_ 7/. or ~rl(H/B') - {e}. Now from the exactness of the h o m o t o p y sequence and ~rx(B/B') ~" Z 2 it follows t h a t ~cx(H/B') ~_ Z and ~rl(B/H) " ~.. Hence H ~ ' ~_ C" and B / H "~ C*. Furthermore it follows t h a t 7ro(g/B') ~_ H / H ~ is finite, n L e m m a 5. Let A denote a Lie subgroup of C* • C* which is isomorphic to C'. Then A = {(e p ' , e q~) [z e C}
for some p, q E 2 .
44
Proof. The group A is a one-parameter-subgroup of C* • C*. Hence
A = {(e ~', : ' )
I z r C}
for some A, p E C. Next consider the Lie group homomorphism ~b : (C, + ) ---, A given by r = (e ~s, e~Z). The kernel is exactly 21riz f3 2~riz ker ~b = ~ P . Since A _ C*, it follows that ker~b __ g . Thus [~ : p] E P I ( Q ) . Therefore
A = { ( e ' , e " ) I z ~ C} for some p, q E Z with [p: q] = [A: p]. ra From the two preceding Lemmata we deduce L e m x n a 6. From the basic assmmptions it follow ttiat
H/B' = {(e'*+2"'~ ', e,'+2"'e) I z E C, m , . E ~ } for some p, q, k, h E Z. Finally we prove a statement about the structure of G/H~ L e m n ~ a 7. In the notation of the preceding Lemma, the manifold G / H ~ is a
quasi-a~ne v . . e t y i~ pq < O, isomo.~hic to P, • C 2 \ {(0, 0)} i # pq = 0 aria a quasiprojective variety wdh O ( G / H ~ = C iff pq > O. Proof. Recall that G / B ' ~_ C2 \ {(0, 0)} • C2 \ {(0, 0)} and that B I B ' ~_ C* • C* acts on G / B e by scalar multiplication on both factors. First assume that H~
'
= {(~',
e")
I z E C}
with p, q E Z and pq > 0. Then we may assume that p, q E 1~1. Thus there is an injective map
~: (c ~ \ {(o, o)} • c ~ \ {(o, o)})/H ~
, ~5
defined by
,/,([(xo, ~,); (~, ~,~)]) = [~,: ~o~;-' : ~I: ~ : ~ : 4 -~: ~ ] . This morphism realizes G / H ~ as the quasiprojective variety
v = {[~o : . . . : z~] I~o~1-1 = ~ , .~z~"-~ = ~, (~o,~, ~) # (o,o, o) # (~, z~,~5)}. Let 17r denote the gariski-closure of V. Now dime(V) = 3 and d i m c ( V \ V ) = 1. Hence o ( v ) = o(::) = c. Next let us discuss the case pq = 0. Here we may assume p = 0. T h e n H~ with
Hi={( 1 1)IzEC},
and
H,=(( A
AV,)IAEC',vEC}.
45 Therefore G/Ho ~- G1/H1 x G~/H2 ~- Ca \ {(0, 0)} x ~1. Finally we consider the case pq < O, Let us assume that p > 0 and q < 0. Now with H ~ ' = {(epZ, e " ) [ z E C} we obtain an embedding of G / H ~ in C e by the morphism ~b: (Ca \ {(0, 0)} x Ca \ {(0, 0 ) } ) / H ~
' ]?5
defined by
r
=1); (-3, *a)l) = (=o'4, ={'4, *o'4, ={+4, =o~-'=x4, =o'=2p
-
lzs).
This realizes G / H ~ as the quasi-afline variety
W = {(zo : ... : zs) E ce lZozs = zlz2, z4' = Z o l - ' z l , ~ = Zgo-lz2, }
(z0,...,~5) # (o,...,0)
Note that W is not Stein, since it arises by removing one point from a threedimensional Stein manifold. 13 The subcase H ~ semi-imple
Lenuna 8. Assume that H ~ is a three-dimensionai semisimple connected complez Lie subgrotp of G = G1 • G~ = SL2(C) x SL2(C) and that G acts almost effectively on G / H . Then H ~ is conjugate to
GD ---- {(gl, 92) E G1 x G2 l gl - g2}. Proof. Since G = G1 xG2 is a direct product, it is clear that the projection maps xi : G ---, Gi are Lie group hotnomorphisms. Note that H is semisimple and G - SL2(C). Hence either x i ( S ~ = {e} or lri(H ~ = Gi. Clearly S ~ = Gi if x j ( H ~ = {e} for j ~ i. Hence the assumption that G acts almost effectively on G / H implies that both homomorphisms Xl and ~2 are surjective. Then dimc(H ~ = 3 implies that both 7i are Lie group isomorphisms. Hence
is a Lie group isomorphism from SL2(C) to SL2(C). Thus
H~ = fig, ~(g)) I g c SLy(C)} for some Lie group automorphism ~b E Aut(SL~(C)). Observe that every Lie group automorphism of SL2(C) is actually an inner automorphism (see e.g. [Ti3]). Hence there is an element go E SL2(C) such that ~b is just conjugation with go. It follows that (e, g0). H ~ (e, go z) = G D = {(gl,ga) I gl = g2}.
13
46 L e m m a 9. In the above notation
NG(GD) = {(gt,g2) l gX = +g2}. Proof. Let ( g t , g 2 ) E NG(GD). T h e n gl "h'gi -1 = g2" h . g ~ l for all h (E SL2(C). This implies that g~lg2 commutes with every h E SL2(C). Since the center of SL2(C) is just {(1, - 1 ) } , it follows that gl = -t-g2. o This immediately implies 10. Let G = SL2(C) x SL2(C) and H a closed three-codimensional semisimple Lie subgroup of G Assume that G acts almost effectively on G / H . Then G / H b biholomorphic to one of the manifolds SL2(C) and PSL2(C).
Lemma
R e m a r k . Actually there is a very natural action of SL2(C) x SL2(C) on the manifold SL2(C): Just let (gx,g2) E SL2(C) • SL2(C) act by group multiplication from the left with gl and multiplication from the right with g{ 1. 5. T h e case G ~ SL3(C) In this p a r a g r a p h we discuss the case closed Lie subgroup of G such t h a t G / H d i m c ( G / H ) < 3. By 4.3.1 we know that dime(G) = 8. Hence dime(H) = 5. First we prove a result about maximM
G ~ SL3(C). Let H denote a is a non-compact manifold with d i m c ( G / H ) = 3. Observe that connected Lie subgroups (see also
[Hum, p.186]). L e m m A 1. Let G be a semisimple linear-algebraic group. Let M be a mazimai
connected Lie subgroup of G. Then M is either reductive or parabolic. Proof. First let M denote the algebraic closure of M . Since M is connected, it follows t h a t J~/' = M ' (see [Hu, Prop. I.D.8.]). Hence M ~ G. Thus the m a x i m a l i t y assumption implies M = AT/, i.e. M is an algebraic subgroup. Since M is algebraic, it has a Levi-decomposition L g V, where V denotes the unipotent radical and L a m a x i m a l connected reductive Lie subgroup of M . Now let N denote the normalizer of V in G . Since G is semisimple and V is solvable, it follows t h a t N = G iff V = {e}. Furthermore maximality implies t h a t either N = G or N o = M . Thus either V = {e} or N o = M . If V = {e}, then M = L, i.e. in this case M is reductive and the L e m m a is proven. Finally we have to discuss the case where dimc(V) > 0 and N o = M . Since V is solvable, it is contained in a Borel group B of G . Since V is unipotent, it is furthermore contained in the unipotent radical of B , this is B ' . Now N O = M implies N Of3 B ' = V. But B ~ is nilpotent. Hence d i m c ( N w (V)) > dimc(V) if 0 < dime(V) < dimc(B'). T h u s N O = M implies V = B ~. Since B ~ is normal
47 in B , it follows from V = B' that B C N O = M . Therefore in this case M is a parabolic subgroup of G. o L e m m a 2. The group H ~ is contained in a maximai parabolic subgroup P of G.
Proof. (i) First we want to show that any connected Lie subgroup A of G with dime(A) = 6 is maximal parabolic. Let A C G with dime(A) = 6 and A = A ~ . Let SA K RA denote a Levi-Malcev-decoml)osition of A. Observe that any Borel subgroup of SL3(C) is complex five-din~ensional. Hence A is not solvable. Thus dimc(SA) = 3, because no complex six-dimensional semisimple Lie group can be embedded in SLs(C). Consequently dimc(BA) = 5, where BA is a Borel subgroup of A. This implies that BA is already a Bore] subgroup of G . Therefore A is parabolic. Now observe that a six-dimensional parabolic Lie subgroup of SL3(C) is already maximal parabolic. (ii) Thus either H ~ is a maximal connected Lie subgroup of G , or H ~ is contained in a maximal parabolic subgroup P of G. Assume that H ~ is a maximal connected Lie subgroup of G. Then from the preceding Lemma it follows that H ~ is either maximal parabolic or reductive. For dimension reasons it cannot be maximal parabolic. But if H ~ is a complex five-dimensional reductive group, then a maximal torus in H ~ is at least complex three-dimensional. Since a maximal torus of the whole group G ~- SL3(C) is only complex two-dimensional, this is also impossible. Hence H ~ cannot be a maximal connected Lie subgroup of G . Therefore H ~ is contained in a maximal parabolic subgroup P of G. [3 R e m a r k . Since G "~ SLs(C), all maximal parabolic Lie subgroups of G are conjugate. In fact every maximal parabolic subgroup of SL3(C) is conjugate to
P=
A
A E GL~(C),p, q E C
.
Now we are in a position to prove L e m m - 3. Let X -- G / H be a non-compact complex-homogeneous manifold with dime(X) 3 In this chapter we will first discuss in general line bundles over complexhomogeneous manifolds S / H where S denotes a semisimple complex Lie group. T h e n we will apply the results about line bundles in order to deal with the case d i m e ( S ) > 3. More precisely we will assume that X is a complex-homogeneous manifold with G a complex Lie group of minimal dimension acting holomorphically and transitively on X and that S is a maximal connected semisimple Lie subgroup of G . Furthermore a ~ u m e that dime(G) > d i m c ( S ) > 3. We will prove that under these assumptions X is biholomorphic to a line bundle over
P2orPlxP1. 1. L i n e B u n d l e s
Line bundles over homogeneous-rational manifolds Let L denote a line bundle over a homogeneous-rational manifold Q. For every line bundle there exists the associated C*-principal bundle, which is embedded in the line bundle as the complement of the zero section. From 4.6.1 we know that this embedded C* -principal bundle is a homogeneous manifold by a S-action, where S denotes the universal covering of the connected component of the automorphism group A u t o ( Q ) . L e m m a 1. Let L denote a line bundle over a homogeneous-rational manifold Q. Then the following assertions are equivalent: (i) There exists a connected Lie group G acting transitively on the total space of L such that the projection ~r : L ---, Q U G-equivariant. (ii) The line bundle has a non-zero section cr : Q --, L . Proof. (i)=~(ii) Let a0 : Q --" L denote the zero-section. Let z t E L \ ~r0(Q) and z0 E a0(Q) such that ~r(z0) = ~r(zl). Let g E G such that g(z0) = x t . T h e n O" := g o o'0 og -1 : Q - - * L is a non-zero-section. (ii) =~ (i) Since Q is homogeneous, the existence of a non-zero section implies that for every point q0 E Q there exist a section ~ E I'(Q, L) such that ~r(q0) ~ 0. Hence with the natural action the additive group ( r ( Q , L), + ) acts transitively on the fibres of the line bundle L --* Q. Let S = ( A u t o ( Q ) ) ~ . Recall that by 4.6.2 the S-action on Q can be lifted to a S-action on L. Therefore the group Go = S K (I'(Q, L), + ) acts transitively on the total space of L. Finally note that both S and (I'(Q, L), + ) are finite-dimensional complex Lie groups, because Q is assumed to be compact. Hence Go is a finite-dimensional complex Lie group. [3
52 R e m a r k . Let Do E Div(P2) given by Do = { [ z o : z l : z2] I zo = 0 )
with multiplicity one. Let D1,D~ E Div(P1 x P1) given by D1 -- {([zo : Z'I]~ [t/)O : t01D [ 2:0 -- O}
D~ = {([~o : "d, [,,,o : ,~]) I ,,,o = 0}, both with multiplicity one. Then every divisor in P~ resp. PI x Pt is generated by Do resp. D1 and S2.
C o r o l l a r y 2. Let L denote a line bundle over P2 or Pt X P1. Then L is a complez-homogeneons manifold iff L is generated by a divisor D with D E< Do >N, reap. D E< D1, D2 >N. Line b u n d l e s over t h e affme q u a d r i c Next we will discuss line bundles over the atfme quadric
Q2 -" { ( z l , z2, z3) E C ~ [ Z l Z 3 - z~}., which is biholomorphic to
P1 x P1 \ A = {([Z0 : Zl], [W0,'Wl]) [ [Z0 : Zl] :~: [W0 : U)I]}" This manifold is homogeneous under the natural SL~(C)-action given by
A : ([z], [w]) *-~ (A([z]), A([w])). Obviously this action is extendible to ]~1 X P1- Thus we have an equivariant embedding of Qa in P1 x P1. L e m m a 3. Let L denote a line bundle over P1 x P1 \ A , which is eztendible to a line bundle L o v e r ~1 X P1. Then L is a homogeneous manifold. Proof. Let D denote a divisor on PI x P1 generating L. Then D = riD1 + roD2. Observe that
A = {([zo : zx], [,,'o, ~d) I ZOWl = ZIU)0} is linear equivalent to D1 + D2 as a divisor. Choose k E H such that k > max(Im[, Inl):Then ]~ = D + k A is a positive divisor on P1 xP1. Therefore the line bundle L generated by /) has a non-zero section and is a complexhomogeneous manifold under the action of (SL2(C) x SL2(C))K F(PI x P1, L). Thus L l r l x r l \ a is a manifold homogeneous under the action of SLy(C) K r(P1
• P1, L).
53 But from b = D + kA it follows that
LIr,xrl\a --- LIr, xr~\a --- L. Hence L is a complex-homogeneous manifold, t3 Actually the extendnbility condition in the preceding Lemma is not necessary, because it is always fulfilled:
L e m m a 4. Every line bundle over 02 is eztendible to a line bundle over P I • Proof. To prove this statement it suffices to show that every line bundle over Q2 is generated by a multiple of the divisor D1 = { ( [ z 0 : zx], [w0: wl]) I z0 = 0}. Consider the projection wl : P1 x P1 --* P1 on the first factor. The restriction 0rl [Q2 : 02 ~ P1 realizes Q2 as an affme bundle over P1. This bundle has a C ~176 ~ : Px x C --* Q~ defined by
~: ([x0 : ~], ~) ~ ([~0: ~d, [~ + y~0:-~0 + yxl]). Therefore we obtain an isomorphism
x" : H2(~1, 25) -% H2(Q2,25). Now observe that D1 = lr*({[0 : 1]}). Hence H2(Q2,25) is generated by D1. Since Q2 is Stein, it follows that the group homomorphism
H i ( Q 2 , 0 " ) ---* H2(Q2,25) induced by the exponential sequence is in fact an isomorphism. Thus every line bundle over Q2 is generated by a multiple of the divisor D1. rl L i n e b u n d l e s o v e r P2 \ QI Now let us discuss line bundles over the homogeneous manifold
P2 \ Q1 = {[x0: x l : ~2] I ~1~ - 4~0~2 # 0), which is a quotient of Q2 by 252 realized by the projection
~ : ([z0: z~], [wo: w~]) ~ [zo~o : zow, + ~1~o: ~ 1 ] . The action of 252 = {id, ~} on Q2 is given by
~: ([~o: ~], [w0: ~ ] ) ~ ([w0: ~,], [z0: ~d). L e m m • 5. Let D G Div(P2\Q1). Then the pulled back divisor x*D G Div(Q2) is linearly equivalent to the zero divisor.
54
Proof. Let again D1 denote the divisor {z0 = 0} on Qa. It is easy to verify that D1 is linearly equivalent to - ~ * D 1 . Since D1 generates Pic(Qa), this implies that D + ~b*D N 0 for all D E Pic(Qa). Next choose a divisor D E P i e ( X ) with X = IPa \ Q1. Then x ' D E Pie(Qa) is ~-equivariant. Therefore x * D + ~6*x*D ~ 0 implies that 2~r'D ~ 0. Since Pie(Qa) ~ g is torsion-fzee, it follows that x*D ~ 0 for all D E P i e ( X ) . [3 Thus for any D E Div(X) there exists a meromorphic function m on Qa such that ~r*(D) = O(m). Clearly ~r'(D) is r Hence ~ E O'(Qa). Now we define D e f i n i t i o n . Let f E O*(Qa) and D E DIv(X). Then f is called an associated function to D , if there exists a meromorphic function m E A4 "(Qa) such that x*D ,,- a(m) and f = ~--m" '~ Next we want to show that a divisor is determined by its associated function. L e m m a e. Let f E O*(Q2) and D1,Da E D i v ( X ) . Assume that f is an associated function to both D1 and Da. Then Dx and Da are linearly equivalent.
Proof. Let m l , m 2 E .Ad (Qa) such that Ir*Di ,,- a(m/) and f = ~ for i ~- 1,2. Then r * = ~ and consequently ~ = ~b*~-~2. Hence ~ is a ~b-invariant meromorphic function, i.e. there exists a meromorphic function m E A4 (X) such that ~r*m - ~ Then D1 = Da + a ( m ) . [3 L e m m a 7. Let D E D i v ( X ) . Then either f+ - 1 or f_ -- - 1 is a function associated to D. It is not possible that both f+ and f_ are associated to the same divisor.
Proof. (i) Let D E Div(X) and f be a function associated to D . Now Qa is simplyconnected, hence H I ( Q a , Z) = {0}. Therefore exp : O(Qa) --" O*(Qa) is surjectire. From this fact it can be deduced that there exists a function h E O*(Qa) such that h a = ~. Now by definition ~b*f = ~. Hence ~b*ha = ~1~, and consequently ~ = 4-~. Since ~r*D = a(m) and h E O*(Qa), it follows that ~r*D = a(mh) and that y_
mh
r162
h - f~-~h = + 1
is a function associated to D. (ii) Now we show that 1 and - 1 can not he associated functions to the same divisor D E Div(X). Assume that D E Div(X) is such a divisor with both 1 and - 1 as associated functions. This implies that ~r*D = ~(m) = ~(mh) with h E O*(Qa). with ~ m = 1 and r247 _-- - 1 . Therefore there is a function h E O ' ( Q a ) with ~
- - 1 . Now let "f: [0, 1] -* Qa be a curve with
-r(t) = ~(-r(t - ~))
55 for all t E [89 1]. In particular 7(1) = r
=
o
= 7(0).
Hence 7 is a closed curve. Thus h o 7 : [0, 1] --* C* is a closed curve. Now h -- --I implies that 1 h o 7(0
= -h o 7(t -
for all t E [ ] , 1]. Hence h o 7 : [0, 1] --* C" is not homotopic to zero. This yields a contradiction to the fact that Q2 is simply-connected and h : Q2 " " C* a smooth map. Hence it is not possible that one divisor has both 1 and - 1 as associated functions, n There exists a divisor on X with - 1 as associated function: D = ([Zo: Zl:
z2llzo
-
0}Ix.
Hence D i v ( X ) "~" 7Z2 . R e m a r k . Alternatively, the non-triviM line bundle over Q2/Z2 can be defined as Q2 • C / ~ - with ([w0: wl], [z0 : zl],t) ~, ([z0 :zl], [tOo : w x ] , - t ) . Since X is a Stein manifold, every line bundle has a section and therefore is generated by a divisor. Thus due to the above considerations we are in a position to prove the following: P r o p o s i t i o n 8. There ezists an wnique non-trivial line bundle over the manifold ]P2\Q1. This line bundle is generated by the divisor D - {[z0: Zl: z2] [ z0 " - 0). Its total space is a complez-homogeneous manifold. Proof. We have already seen that the line bundle generated by D is the only non-trivial line bundle over X . Since D is the restriction of a positive divisor on ]?2, it is clear that L ( D ) extends to a line bundle L over P2 with F(]?2, L) ~ (0}. Now F(~2, L) is finite-dimensional, because ~2 is compact, hence the additive group (F(~2, L), + ) is a finite-dimensional Lie group, which acts transitively on the fibres of L. Consider the embedding ~b : SL2(C) ---, SLs(C) given by
c2
2ed
This yields an SL2(C)-action on ~2 with exactly two orbits: P2 \ Q1 and Qx. Thus SL2(C) acts transitively on P2 \ Q~. By 4.6.2 we know that the SL2(C)-action can be lifted to an action on the total space of the line bundle L. Hence there is an action of S L 2 ( C ) ~ (F(172, L), + ) on I, with the total space of L = L[r2\q~ as an open orbit. Thus L is a complex-homogeneons manifold. []
56 2. T h e c a s e dimc(S) > 3 In this paragraph we assume the following B a s i c A s s n m p t i o n s . G is a simply-connected complex Lie group, H a closed Lie subgroup, such that X = G / H is a manifold with dime(X) 3 and dime(R) > 0. Finally G is required to be of minimal dimension among all complex Lie groups which act transitively on X . L e m n m 1. Let G / H ---, G / N denote the Tits-fibration.
Then d i m c ( G / N ) ~_ 2 and S acts almost effectivel~ on
GIN.
Proof. The assumption of minimality implies that G acts almost effectively. Since the basic assumptions imply dimc(G) ~_ 7, it follows that H ~ is not normal in G, i.e. dimc(G/N) > O. Now d i m e ( N / H ) _< 2 implies that N ~ ~ is a solvable Lie group, hence (N~ k C H ~ for some k > 0, where (N~ k denotes the k t h group of the derived series, i.e. (N~ 1 = (N~ ~ and (N~ k+l = ((N~ ~. Let L denote the connected component of the ineffectivity of the G-action on G I N . Then L C N O implies that L k C H ~ Since L k is a n o r m a l Lie subgroup of G , the minimality assumption implies that L k = {e}. Therefore L is solvable and consequently S acts almost effectively on G / N . Since dime(S) > 3, it follows that d i m c ( G / N ) >_ 2. [] Under the basic assumptions X is biholomorphic to a line bundle over 171 x 171 or 172.
L e m m a 2.
Proof. First we want to prove that the R-orbits in G / H are closed and onedimensioned. Assume that dimc(G/N) = 2. Then G I N is one of the homogeneousrational manifolds 171 x 171 and 172- (Prop. 4.2.1). Consequently in this case the R-action on G I N is trivial. Thus R acts only along the fibres of the bundle G / H ---* G I N . These fibres are connected, because GH is connected and G I N is simply-connected. Since R is normal in G , all the R-orbits in G / H are biholomorphic to each other. It follows that R acts transitively on all fibres, i.e. R H = N in this case. Next assume dime(G/N) = 3. Then it follows from Lemma 2.2.15 that the R-orbits in G I N are closed. Now the minimality assumption implies that S acts almost effectively on G/ R N . Therefore either G/ R N ~_ 172 or G/ R N " 171x171. Consider the i b r a t i o n G / H --* G / R N . Note that the fibres are connected, because G / R N is simply-connected. Hence the fact that R is normal in G implies that R acts transitively on the fibres. Thus in any case we have a fibration
G/H
G/RH
57 with G / R H ~_ ~2 or G / R H - F1 x l?x. Note t h a t the assumption t h a t X = G / H is not biholomorphie to a direct product implies t h a t this bundle is not trivial. First assume t h a t R H / H ~- C" or R H / H ~_ 7"1. Since G / R H is simplyconnected, it follows t h a t HI(G/RH, r ) = {e} for all discrete groups r . Therefore the structure group of the bundle G/H --* G / R H m a y be reduced to (Auto (RH/H)) ~ . For R H / H ~ C* and R H / H ~_ 7"1 this implies t h a t G / H --* G / R H is a principal bundle. From the basic assumptions it follows t h a t the bundle is non-trivial. Furthermore from the minimality assumptions it follows t h a t S does not act transitively on G / H . T h u s there exists a lower-dimensional S-orbit. Now recall t h a t GIN is simply-connected and t h a t S acts transitively on GIN. Therefore a lower-dimensional S - o r b i t must be two-dimensional and induces a section in the principal bundle G / H --* GIN. This yields a contra~ diction, because a non-trivial principal bundle has no sections. T h u s we can conclude R H / H ~_ C. Since S does not act transitively on G / H , it has a lower-dimensional orbit in G / H . Note t h a t dimc(RH/H) = 1, t h a t S acts transitively on G / R H and t h a t G / R H is simply-connected. Hence a lower-dimensional S - o r b i t is onecodirnensional and yields a section in the bundle G/H --* G / R H . T h u s this C-bundle is a line bundle, o
58
The
mixed
Chapter 6 c a s e w i t h S _~ S L 2 ( C )
and
R abelian
1. B a s i c A s s u m p t i o n s In this chapter we assume that (i) X is a complex manifold with dirnc(X ) < 3, on which a complex Lie group G acts holomorphically and transitively (ii) The group G is connected and simply-connected. It has a Levi-Malcevdecomposition G = SK R with S ~_ SL2(C), dime(R) > 0 and R' - {e}. (iii) The group G is of minimal dimension among all complex Lie groups acting holomorphically and transitively on X . (iv) X is not biholomorphic to a direct product of lower-dimensional complexhomogeneous manifolds. We will obtain a classification for this case in the following way: First we study the representation of SL2(C) in Aut(R) given by conjugation. This is a linear representation since from our basic assumptions it follows that R - (C TM, + ) for some k E l~l. We will distinguish two cases: Either this linear representation is irreducible or R is the vector space direct sum of two or more S-stable vector subspaces. We will start with the latter case. Exploitating the minimality assumption we will show that there are at most two irreducible S-stable subspaces and finally deduce that X -~ H A ~ H i in this case, i.e. the manifold X is biholomorphic to the direct sum of two line bundles over P1. T h e n we t a k e a closer look to the case X _~ H A ( ~ H z with k = l. It turns out that there is a transitive SL2(C) Kp (C t+2, +)-action on H A ~ H k with an irreducible linear representation p : S --, GLk+~(C~. This is the first of three subcases of the case where the representation of S in Aut(r) is irreducible. The other two subcases yield quotients of C x C 2 \ {(0, 0)} by discrete groups and certain bundles over Q2 and P2 \ Q1. 2. R e p r e s e n t a t i o n s o f SLy(C) Let G = SK R denote a semidirect product with S ~- SLa(C). Then the group structure of G is determined by the group structure of R and the natural group homomorphism p : S ---, Aut(R) given by conjugation. The latter is determined by the linear representation Ad : S --, Aut(r) ~-* G L ( r ) where r denotes the Lie algebra of R. Hence it is useful to study linear representations in order to understand semidirect product structures. This linear representation is associated to a representation of the Lie algebra: ad : s --, gl(r). It is a standard result of the theory of semisimple Lie groups that for each such representation there exists a decomposition of r into irreducible ad(s)-submodules r i . Moreover, for every irreducible Ad(S)-module a there exists a basis Z 0 , . . . , Z , such that the ad(s)-action can be described in the following way:
59 Let H, X, Y denote the standard base of s with [H, X] = 2X, [H, Y] = - 2 Y and IX, Y] = - H . Then the ad(s)-action on a is given by [H, Zj] = (n - 2j)Zj IX, Zj] = (n + 1 - j ) Z j - x [Y, Zj] = (j + 1)Zj+l Reall.-stion with polynomials For explicit calculations it is often convenient to realize an irreducible SL~(C)module in the following way: Consider the usual action of SL2(C) on C~ . This induces an SL2(C)-action on the ring of polynomials C[X,Y]. Now the irreducible SL2(C)-submodules are exactly the vector subspaces C[X, Y]d of all homogeneous polynomials of a given degree d. Thus one obtains an irreducible SL2(C)-representation for each dimension, since dimc(C[X, Y]a) = d + 1. 3. T h e c a s e R a b e l i a n , r e d u c i b l e S - r e p r e s e n t a t i o n In this paragraph we assume in addition to the basic assumptions that the radical R of G is abelian and that the representation of S in G L ( r ) is reducible. Since R is abelian and simply-connected, we obtain R ~ (C t , + ) . Therefore S acts linearly on R. Let (~i~i R~ denote the decomposition of R in irreducible S-submodules. Observe that for every subset J C I the direct sum (~i~.r R~ is a normal Lie subgroup of G.
Under the assumption of minimality the Ad(S)-modwle r has at most two irreducible submodules.
L e m m ~ 1.
Proof. From the minimality assumption it follows that the R-orbits in G / H = X are at most two-dimensional. Let R~0 and P~t denote two non-trivial irreducible subspaces of R. Then minimality of G implies that their orbits are one-dimensional. Furthermore SK (Bi~oR~ can not act transitively on X . This implies that the orbits of (Bi#ioR~ are also one-dimensional and therefore coincide with the R~t-orbits. Thus S K (R~o ~ R~t) acts transitively on X . Now minimality implies that I = {i0, ix}. o Therefore G = S K (Rt + R2), where the R~ are irreducible S-submodules. Furthermore it follows from the above proof that the R~-orbits are onedimensional, while the R-orbits are two-dimensional. Hence ( r + h ) / r is a two-dimensional subalgebra of g / r _ s = sl2(C). Now we may assume that with the usual base H , X , Y we obtain r + h = < H , X > r + r . Hence H + A, X + B E h for some A, B E r . Since r is abelian, it follows that h f3 r is both a d ( H ) - and ad(X)-stable. Together with the fact that the R~-orbits are one-dimensional this sumces to prove the following statement.
60 L e m m a 2. Let X , H , Y denote the tJsual basis for sis(C). Let g 0 , . . . , Z n resp. W0,...,Wt denote the basis for rx resp. r 2 such that SLy(C) acts on gi and
W i as described above in parafraph 2. Then after conjt~gation h = < H + a Z , + / S W t , X, Z 0 , . . . , Z , - x , W0, 99 9 W t - 1 > c
for some o~,~ E C. Proof. It is clear t h a t h =< H+~Z,
+/SWt,X + 7Z, + 6Wt,Z0...Zn-l,W0,...,Wt-1
•c 9
Observe t h a t [ H + a Z , + ~ v V t , X + 7 Z , + 6 W t ] = 2X - n T Z , - k 6 W t ~Z,-1 -/~Vt-1. Since this c o m m u t a t o r m u s t be contained in h , it follows that 7 = 6 = 0 o C o r o l l a r y 3. Every S-orbit in G I N is biholomorphic either to P1 or to C 2 \ {(0, 0)}/X~ with p ~ ~ . Now we want to discuss the normalizer-fibration. For this purpose note t h a t
SL2(C0 acts trivially on an irreducible SL2(C)-module iff the module is onedimensional. Therefore the center it of g is the direct s u m of all one-dimensional irreducible S-submodules of r . It follows that g = g ' + 9 with 9 f3 g ' = {0}.
Let N denote the normalizer NG(H ~ of H ~ in G. Under the basic assumptions it follows that n -- h + z, where 9 denotes the center of g. Moreover G I N is a C ~-bundle over ]?1. (0 < ! < 2). In particular G I N is simply-connected.
LemLma 4.
Proof. From the above L e m m a it follows by explicit calculations t h a t n = h + z. Since g = g ' + s, it follows t h a t G ' acts transitively on G I N . As the base of the Tits-fibration, G I N is equivariantly embeddable in some Pro. Now A = (G~N R) ~ acts linearly as a unipotent group on Pm (see 2.2.12). But an orbit of a unipotent group acting linearly on a projective space is always biholomorphic to some C~ (see [Hu, Prop. 1.E.1]). Hence the R-orbits in G I N are biholomorphic to some C~ (p E l~l). By 2.1.1 these orbits are closed, since A is a linear-algebraic normal Lie subgroup of G . Therefore there exist s fibration G I N --, G / A N . From the preceding results it follows t h a t d i m e ( G / A N ) = 1. Since SLy(C) acts transitively on G / A N , this implies G / A N ~- P1. Now from a h o m o t o p y sequence for the bundle G / N c " G / A N ~_ P1 it follows t h a t G I N is simply-connected. D Lernrn~ ft. From the basic assumptions it follows that the center Z of G is
discrete.
61
Proof. Assume to the contrary that the center Z is positive-dimensional. Consider the normalizer fibration G / t t ---, G I N . Since G I N is simplyconnected, it follows from n = h + s that N = N O = Z - H = Z - H ~ . Hence N / H ~ is an abellan group. Consequently H is a normal subgroup of N and therefore G / H ---, G I N is a principal bundle. Recall that G' acts transitively on G I N = G / Z H and that G = G ~ 9 Z is the smallest Lie subgroup of G acting transitively on G / H . Form the minimality assumption it follows that dirnc(Z(z)f3G'(z)) = 0 for all z E G / H . Therefore the G'-orbits in G / H are coverings over G / Z H . Since G / Z H is simply-connected, the G~-orbits are sections in the holomorphic principal bundle G / H ---, G / Z H . Thus this principal bundle is trivial, i.e. G / H is biholomorphic to G / N • N / H . This contradicts the basis assumptions. Hence under the basic assumptions the center Z can not be positive-dimensional. 13 From now on in this paragraph we assume that the center of G is discrete. Note that the R~ are normal Lie subgroups of G. Since they are contained in the unipotent group R = R f3 G ~ they are themselves unipotent. Hence it follows that the R~ are linear-algebraic normal Lie subgroups of G. Thus their orbits in G I N are closed, and biholomorphic to C. Furthermore the radical R is also a linear-algebraic unipotent normal Lie subgroup of G . Therefore the R-orbits in G / N are closed and biholomorphic to C 2 . Moreover G / R N is an one-dimensional SL2 (C)-orbit, hence biholomorphic to P1. Thus we obtain: L e m m a 6. The base G / N of the Tits-Jlbration is a C2 -bundle over Pt and in particular simply-connected. Hence G I N = G / H , i.e. the Tits-]ibration is injective. Let us now consider the fibrations G / N ---, G/R~N: L e m m a T. Under the above assumptions G / R 1 N ~- H t and G / R 2 N "" H n, where H i denote the ith power of the hyperplane bundle over PI(C).
Proof. This follows immediately from the fact that S K R~ acts transitively on G / R 1 N with (sK r2) f3(rl + n ) = < H + ~ W t , X , W 0 , . . . , W t - 1
>c-
Now consider the following G-equivariant commutative diagram
X ~_ G / H
1
H ~-G/R1H
,
G / R 2 H ~- H '~
,
G/RH~-P1
1
From this splitting it follows that the bundle G / H ---, G / R H is just the direct sum of the line bundles H t and H " . In particular this implies that G / H --,
62
G / R H is a vector bundle of rank 2 over P1, where the structure group GL2(C) can be reduced to the group of diagonal matrices. Thus we obtain: P r o p o s i t i o n 8.
Under the above assumptions G / H is bihoiomorphic to the
total space of a direct sum of two positive line bundles over Pl. Conversely let E denote a vector bundle of rank two over P1, which is the direct sam of H rn and H t . Then E is a homogeneows complez manifold iff m>0 andl~_O.
Proof. To prove that E is not homogeneous, if i < 0 or m < 0, consider the holomorphic reduction. It has then fibres of different dimensions. (This follows from the fact that for k < 0 the holomorphic reduction of H t is actually blowing down the zero-section). [] R e m a r k . By a theorem of Grothendieck (see [GrR, p.237]) every vector bundle over P1 is a direct sum of line bundles. The special case k = ! The vector bundle H k @ H t is defined by the transition functions f:
9
,-,
In particular, for k = ! the structure may be reduced to
This is the center of GL~(C). In particular this implies that the natural SL2(C)action on the fibre C 2 is well-defined, i.e. invariant under the transition functions. Thus we may extend the group G -- SL2(C) K (C TM x C TM) acting transitively on H k ~ H k to the larger group
Go = (SL2(C)
SL2(C~ K (C h+1
x
•
Ck+1).
In the following we will denote the first SLs (C)-factor by Sz and the second SL2 (C) by $2. The Lie algebra is
< H I , X l , Y1, H~, X2, Y2, Zo,..., Zk, W o , . . . , W k >c with isotropy algebra
< H1, Xl, H2, X~, Y2, Z0, 9 9 Zk-1, W 0 , . . . , W k - 1 ~>C 9 Here the Sl-action is the usual one, i.e. [Hz, Zj] = (k - 2j)Zj
IX1, Zj] -- (k -[- ] -- j ) Z j _ 1 ['~rl, Zj] (j -4" 1)Zj+x -
-
63 and the S2-action is given by
[H2, Z~] = Z~ [H2, Wi] = - W i
IX2, W,] = Z~ [Y2, Z,] = W~ Now consider the Lie subalgebras
gl = < H1 + H 2 , X 1 + X 2 , Y 1 + Y 2 > c + r and g2 --< H i + H 2 , X 1 + X 2 , Y x + Y 2 >C +
< Zo, Zx + W 0 , . . . , i ! Z i + ( i - 1 ) ! W i _ l , . . . , k ! W k >C Clearly g~ 4914g0. Hence G24G14G0. Moreover dimc(g2/(g~ f3h)) = 3. Thus there is an open G2-orbit in the G-orbit G / H . Therefore G2 acts transitively on G / H . This yields a transitive SL2(C) K (C k+2' +)-action on H k (~ H t .
T h e m a n i f o l d H 1 (B H 1 - I?3 \ L Finally let us mention that HI(~H 1 is biholomorphic to P s \ L , where L denotes a complex line in P3. Let L = {[z0 : . . . : z3] I z2 = z3 = 0}. Then the projection on P1 is given by [z] ~-* [z2: z3] and the projections on H 1 _~ P~ \ {[1: 0 : 0]} are given by [z] ~ [z0 : z2 : z3] resp. [z] ~ [Zl : z~ : z3]. The action of G = SLy(C) t< (C2 x C2) is given by an embedding ~ : G ---* GL4(C) with
The action of G2 = SLy(C) K Ce is given by
4. T h e c a s e R a b e l l a n , i r r e d u c i b l e S - r e p r e s e n t a t i o n
Generalities In this paragraph, we assume in addition to the basic assumptions that the representation of S in G L ( r ) is irreducible. This implies in particular that R is ahelian since Ad(S) must stabilize r ~. Hence R -~ ( C t + i , + ) (k E 1~1). We use the usual basis < Y , H , X > for s and < Z 0 , . . . Z k > for r as described above. From the basic assumptions it follows that the R-orbits are either one-
64 or two-dimensional. Hence s n (r + h) is a subalgebra of dimension one or two in s _ sl2(C). Thus we obtain after conjugation:
sN(rq-h)=
< H, X >c or < H > r or < X > c
(i) (ii) (iii)
From the ad(X)-action ad(X) : Z~ --, (k + 1 - i)Z~-I it follows that h N r --
{c Z 0 , . . , Z~-I > c
incase(i) in case (iii)
In case (ii) h N r is not necessarily stable under the ad(X)-action, but ad(H)-invariant. Hence in this case hf'lr is the direct sum of one-dimensional ad(H)-eigenspaces < Z~ > c . Thus hNr =< Z0,...,Z~-I,Z~+I,...,Zk >c for some 0 < i < k in case (ii). These considerations yield L e m m a 1. Under the abore assumptions the isotropy algebra h equals one of
the following algebras after conjugation in s: (i) < H + otZk-1 ~" ~Zk, X -~-7Zk_l -~- 6Zk, Z0, 9 9 Zk-2 >C, (ii) c , ( i i i ) < X'~- O~Zk,Z 0 , . . . Z k _ 1 >C
(~,~,%6ec) The property of h to be a Lie subalgebra causes certain restrictions on the paranmters a, ~, 7, 5. We will investigate these restrictions now. For case (i) consider [ H + a Z k - t + / ~ Z k , X+TZ/~_l+6Zk ] = 2X-k6Zk+(27-kT-/~)Z~_l-2aZk_2 E h. It follows that 27 = 2 7 - t 7 - ~ and 25 = - k 6 . Hence k7 = -/~ and 6 = 0 since k = dime(R) - 1 can not equal - 2 . This implies in particular that any S-orbit in G / H = X is either open (Case (i) with a, ~ ~ 0, Case (ii) and (iii) with a ~ 0) or biholomorphic to Ca \ {(0, 0 ) } / T ~ for some m E 1~1 (Case (i) with a ~ 0 = fl = 7, Case (iii) with a = 0) or to the affme quadric Q2 or Q2/z~ (Case (ii) with a = 0) or to PI(C) (Case (i) with a = fl = 7 = 0). By the basic amumptions S does not act transitively on G / H . Hence after conjugation dime(S/H) < 3, i.e. dimc(H N S) >_1. Thus we may sharpen the above Lemma to the following statement: L e m m a 2. Under the above assumptions the isotropy algebra h eqwals one of
the following algebras after conjugation in g : (i) < H + a Z t - 1 , X, Z 0 , . . . , Zk-2 > c ( a E C), (ii) < H , Z 0 , . . . , Z~-I, Z ~ + l , . . . , Zk > c ,
(iii)< X, Z0,...Zk-~ >c Now the normalizer algebras can be calculated explicitly. This yields:
65 L e m m R 3.
Under the assumptions of the preceding Lemma the normalizer
algebra n = n s ( h ) of h equa/s
- =
h h ~ < Zl > c h h ~ < Z~ > c
h ~ < H, Zt >c
Cue
in in in in in
case (i) iH k # 2 case (i) iH k = 2 caseCii) i ~ t # 2i case (ii) iff k = 2i case (iii)
(i)
We will now discuss case (i) in detail. Our first goal is to construct a fibration G / H --* G / A with G / A ~_ H t . Let Ao := ( H ~. ( H N R)) ~ and A := NG(Ao). Since b o t h H ' and H N R are normal in H , it follows t h a t A0 is a normal subgroup of H . Hence H C A. Furthermore A is a closed Lie subgroup of G , because it is the normalizer of a connected Lie subgroup in G . Thus we obtain a fibration G / H --, G / A . From the preceding L e m m a we deduce t h a t a0 = < X, Z0, 9 9 Z t - 2 > c a=
c"
This implies t h a t G / A ~_ H t .
Next let us discuss the second case, i.e. < H , Z 0 , . . . , Z~-l, Z ~ + l , . . . , Zk > c . T h e n n = h iff 2i ~ k emd n = h + r iff 2i = k. In b o t h cases the R-orbits in G / N are closed, because R = G ~n R implies t h a t R acts on G / N ~-* Ply as a unipotent normal linear-algebraic subgroup of G . Thus G / N is a C t -bundle over G / R H . Furthermore either G / R H ~_ Q2 or G / R H ~_ Q2/Z2 "" P2 \ Q t L e m m a 4. Assume k ~ 2i. Then G / H is biholomorphic to the total space of a non-trivial line bundle over Q2.
Proof. F r o m R = G ' n R it follows t h a t R acts on G / N ~ P~v as a unipotent normal linear-algebraic subgroup of G . Hence the orbits of R in G / N are closed and biholomorphic to C. From the structure of h we know t h a t either G / R N ~_ Q2 or G / R N ~_ Qa/ga - Pa \ Qx. In b o t h cases we obtain G / ( R N ) ~ ~_ Q2. T h u s G / H ~ is a C-bundle over Q2. Note t h a t S acts transitively on G / ( R N ) ~ and t h a t G / ( R N ) ~ is simplyconnected. It follows t h a t every two-dimensional S - o r b i t in G / H ~ is a section in the C = b u n d l e G / H ~ --, G / ( R N ) ~ Since S does not act transitively on G / H ~ , it has at least one such two-dimensional orbit. Hence this C-bundle has a section and therefore is a line bundle.
66 Recall h = < H , Z 0 , . . . , Zi-1, Z i + l , . . . , Zk > c and consider the Ad(Zi)action on h . It follows that for some points z E G / H ~ the isotropy algebra h , equals < H +~Z~,Z0,...,Z~_I, Zi+I,...,Zk >c with c~ ~ 0. Hence there exist an open S-orbit in G / H ~ This implies that S acts transitively on the complement of the zero-section of the line bundle G / H ~ ~ G / ( R H ) ~ . Thus the universal covering of the complement of the serosection is biholomorphie to SL2(C). Since the complex manifolds SL2(C) and C x Q2 are not even homeomorphic, it follows that the line bundle G / H ~ --+ G / ( R H ) ~ can not be trivial. Finally we have to show that G / R N ~- Q2. For this purpose assume the contrary, i.e. G / R N ~_ Q2/Z2. Then we have the following commutative diagram G/H o ----,z2 G / N
1
G/(RN) o
z,
G / R N - " Q2/Z2
Now, since ~rl(Q2/Z2) = 77-2, there exists a two-dimensional S-orbit in G / N such that the restriction of the projection G / N ~ G / R N is either biholomorphie or a 2 : I-covering. Let us consider the case where this orbit is a 2 : 1-covering. Note that the mean value or(z, y) = ~ in C is invariant under biholomorphie transformations, because every biholomorphie transformation on C is automatically affine-linear. Thus by taking the mean value in each fibre of the C-bundle G / N ~ G / R N the 2 : 1-covering yields a section. Therefore G / N --~ G / R N is a line bundle even if this two-dimensional S-orbit is only a 2 : 1-section over Q2/T.2. Hence the line bundle G / H ~ ~ G / ( R H ) ~ ~_ Q2 is the p u l l b a c k of a line bundle over Q2/Z2. But by 5.1.5 every such pull-backed line bundle is trivial over Q2. Thus the a~sumption G / R N "" Q2/Z2 yields a contradiction, n R e m a r k . The line bundles over Q2 occuring in the preceding Lemma seem to be parametrized by two parameters: k and i. But in 5.1.4 above we proved that the group of all line bundles over Q2 is isomorphic to ( Z , + ) . The ez-
planation for this is that the homogeneous manifold for given parameters k, i is actnaily biholomorphic to the total apace of the line handle L (~-2i), ~ohere Lo is s generating element of Pie(Q2) -- (Z,
+).
It rerrmius to discuss the subease 2i = k. L e m n m 5. Assume that 2i = k. Then the base of the normalizer fibration is Q2/Z2. F , rthermore the group N / H ~ is a semi-direct prodsct Z2Kp (C, + ) The defining group homomorphism p : Z2 --', C* = Aut(C, +) is given by p(~b) = ( - 1 ) i, where ~ denotes the non-trivial element of Z2.
Proof. First observe that n = h + r . Recall that h = < H, Z 0 , . . . , Z ~ _ l , Z i + l , . . . , Z k
>c
67 and n - h ~ < Z~ > c . Hence either G / N ~- Q~ or G / N ~- Q2/T.~. Let
Then P • N o , but P normalizes ezp < H > c . Furthermore the Ad(P)-action on r is given by Ad(P) : Zj ---, ( - 1 ) J Z k _ j and in particular Ad(P)(ZI) = ( - 1 ) i Z i . Thus P also normalizes h . Hence P E N G ( H ~ i.e. N = NG(H ~ is not connected. Therefore G / N " Q2/77.2. To determine the group structure of N / H ~ , note that p2 E H ~ 9 Hence N / H ~ = { e , P } K p ( N ~ 1 7 6 Observe that U ~ _~ C* K ( C h , + ) and N O ~_ C* K (C TM, + ) . Thus N ~ ~ = (C, + ) . Finally the structure of p follows from
ad(P)(Z,) = (-1)~z~. Q L e m m a 6. i.e. N~
The fibre of the normalizer fibration G / H --+ G / N is connected,
Proof. Recall that N / H ~ = Z ~ ( N / H ~ ~ . Hence either H ~ C N or N ~ = N. We will show that the assumption H ~ C N leads to a contradiction to our assumptions. Assume H ~ C N . Then we obtain a fibration G / H --, G / N ~ ~Q2. Now H is normal in N o , because ( N / H ~ ~ is an abelian group. Hence G / H ---, G / N ~ is a principal bundle. Now S acts transitively on G / N ~ , but not on G / H . Thus there exists a two-dimensional S-orbit in G / H . Since G / N ~ is simply-connected, this orbit yields a section in the principal bundle G / H --+ G / N ~ This implies that this bundle is trivial, i.e. G / H ~- G / N ~ • N ~ in contradiction to the basic assumption. [3 Lemms~ 7. Assume that i is even and 2i = k. Then G / H ---, G / N ~- Q2/Z2 is a non-trivial C* - or Tx -principal bundle.
Proof. The group homomorphism p : Z2 --+ Aut(C, + ) is trivial iff i is even. Hence N / H ~ ( C , + ) is abellan. Thus H is normal in N . Consequently G / H --* G / N is a principal bundle. From the preceding Lemma we know that N / H is connected. Furthermore by the basic assumptions G / H is not biholomorphic to G / N • N / H , i.e. the principal bundle G / H --+ G / N is not trivial. Since Q2/Z3 is Stein, every C-principal bundle over Q2/Z2 is trivial. Hence either N / H ~_ C" or N / H "" Tl.n L e m m a 8. Assttme that i is odd and 2i = k. Then G / H is a quotient ( Q ~ x C ) / F , where r is a discrete subgroup of Z~K (C, + ) with the foilowinf Z2K C-action:
9 : (([,], [w]), y) ,-, (([~], [w]), y + ~)
for
=, y E
c, ([z], [w]) ~ P1
x P~ \ a -,, q 2 ,
and
~: (([z], [w]), ~) ~. (([~], [z]),-y)
68 for the non-trivial element ~ E Z2. Proof. Note that G / H ~ ~ G I N ~ is a principal bundle. Since G I N ~ ~Q2 is simply-connected, this principal bundle has a section induced by a twodimensional S-orbit. Thus G / H ~ ~_ Q2 x N / H ~ . Furthermore N ~ ~ ~_ C, because G / H ~ is simply-connected. Since the trivialization of G / H ~ ~ G I N ~ is induced by the section and the principal right action of N ~ ~ it follows that the N ~ 1 7 6 action on Q2 x C is just given by
~: (([w], [z]), y) ,.-, (([w], [z]), 9 +
~).
Now N / H ~ ~_ Z2 K C. Let ~0 denote the non-trivial element of Z2 and ~b = (~0, 0). Then
r (([w], [z]), y) ~ (([z], [w]), ~(y)) where ~ E A u t o ( C ) . Furthermore ~ o ~ = id. Hence ~2 = id]c. This implies that (after an appropriate change of coordinates ~ = y + a) ~(y) = - y . Now the Lemma follows from the fact that G / H ~- ( G / H ~ 1 7 6 where H / H ~ acts from the right as a discrete subgroup of N / H ~ . o
cue
(m)
Finally let us discuss the third case, i.e. < X, Z 0 , . . . Z k - 1 > c 9 Here we obtain n - < H, Zk > c 9 h . Thus d i m c ( G / N ) -- 1 and hence G / N ~P1. Furthermore N / H ~ ~ C* K pC. The group structure is induced by [H, Zk] = - k Z k . It follows that the group homomorphism p : C* ~ Aut(C, +) ~_ C* is given by p : ~ ~-* ~k. Hence N / H ~ is abelian i f f k - 0. Now ( N / H ~ ~ is a closed connected Lie subgroup of N / H ~ . Therefore N~H ~ is a closed connected Lie subgroup of G and induces a fibration sequence G / H ~ --* G / N ' H ~ --* G / N ~_ P1. Observe that n~+h - < X > c + r . It follows that S acts transitively on G / N ~ H ~ and furthermore G / N ' H ~ ~_ Ca \ {(0, 0)). By the minimality assumption S acts not transitively on G / H ~ Hence there exists a two-dimensional S-orbit in G / H ~ . Since G / N ' H ~ is simply-connected, this orbit yields a section in the bundle G / H ~ --* G / N ' H ~ . But this bundle is a principal bundle, because H ~ is normal in N ' H ~ . Thus the bundle is trivial and G / H ~ ~_ C x Ca \ ((0, 0)). Now we are in a position to give a concrete realization of G and G / H : Let X = C x Ca \ {(0, 0). Let R = (C[X, Y]k, + ) , where C[X,Y]k denote the additive group of homogeneous polynomials of degree k. Let R act on C x Ca \ {(0, 0)) by P : (z, v) ~ (z + P(v), v) for P E C[X, Y]k. Now define an SL2(C)-action by A : ( z , v ) ~ ( z , A . v). Then we obtained a SL2(C) K (Ca+l, +)-action of the desired kind. Moreover the N / H ~ action on G / H ~
~: (c" ~ c) • (c • Ca \ {(0, 0)}) -~ (c • c ~ \ ((0, 0)})
60 is given by
r (~, x): (z,,~) ,--, (.x~ § x, ~,~,~).
70
Chapter 7
T h e m i x e d case w i t h S -~ SL2(C) a n d R n o n - a b e l i a n 1. B a s i c A s s u m p t i o n s In this chapter we assume that (i) X is a complex manifold with dime(X) < 3, on which a complex Lie group G acts holomorphically and transitively (ii) The group G is connected and simply-connected. It has a Levi-Malcevdecomposition G = S K R with S ~_ SL2(C). The radical R is positivedimensional and non-abelian. (iii) The group G is of minimal dimension among all complex Lie groups acting holomorphically and transitively on X . (iv) X is not biholomorphic to a direct product of lower-dimensional complexhomogeneous manifolds. 2. G e n e r a l i t i e s
Lenuna 1. The R-orbits in G/H are two-dimensional and the R'-orbits are one-dimensional. Proof. By the minimality assumption neither R nor S t< R' acts transitively on G/H. It follows that dimc(R/(HnR)) < 3 and R'(zo) ~ R(zo) for z0 E G/H. Furthermore the minimality assumption implies that /~ acts non-trivially on G/H. Hence 0 < dimc(R'(zo)) < dimc(R(zo)) < 3. [3 C o r o l l a r y 2. The commutator 9roup R' of R is abelian.
Proof. Note that R' is nilpotent. Hence its commutator group /~' can not act transitively on the /~Y-orbits (see 10.2.2). Since the R'-orbits are onedimensional, this implies that the R ' - a c t i o n on G/H is trivial. Thus R " = {e}. [] Now S is a semisimple Lie group which acts linearly on r by the adjoint action. Since S is reductive and r' an Ad(S)-stable subspace of r , it follows that there exist an Ad(S)-stable subspace V0 in r such that r = 1" ~ V0. Furthermore it follows from the minimality assumption that the representation Ad : S --* GLc(V0) is irreducible. Thus there are two subcases: Either the S-action on 1/0 is trivial (i.e. dimc(Vo) = 1) or not (i.e. dirnc(Vo) > 1).
71 3. T h e c a s e d i m c ( R / R ' ) = 1 Let us first discuss the case dimc(Vo) = 1. Since V0 is one-dimensional, it is already a subalgebra of g . Hence G = (S x A)t< R ' with A - ezp(V0) __. (C, + ) . L e n n n a 1. The algebra av can not contain a proper Ad(S x A)-stable sub~ec-
torspace. Proof. Assume t h a t there exists such a subspace r0. Since R ~ is abelian, r0 is a subalgehra of r ' . Since r0 is stable under Ad(S • A), it furthermore follows t h a t r0 is an ideal in g . Hence (s x V0)t< r0 is an ideal in g . Recall t h a t the R ' - o r b i t s in G / H are one-dimensional. Thus it follows t h a t Re = ezp(r0) acts transitively on the R*-orbits. Therefore Go = ( e z p ( s + V0 + r 0 ) ) acts transitively on G / H contrary to the minimality assumption. D Now let Wt denote the sum of all k-dimensional irreducible S-submodules of r ' . T h e n Is, V] = {0} implies t h a t these spaces Wt are also Ad(A)-stable. Hence by the preceding L e m m a Wt = r ~ for some k. Furthermore [s, V] = {0} implies that r ' = V1 | V~ as a vector space where the representation Ad(S x A) on r ' is given by Ad(S x A) = p l ( S ) | P2 (A) with Pl : s --~ GL(V1) and p2 : A ---, GL(V2). The preceding L e m m a now implies t h a t b o t h representations Pi (i = 1,2) are irreducible. But A is abelian, hence solvable and therefore stabilizes a full flag in V2. Thus dimc(V2) = 1. Hence the representation Ad : ( S x A) --, G L( Vx) is actually an irreducible representation of GL2(C). Therefore there exists a basis < Z 0 , . . . , Z t > of r ~ such t h a t with the usual basis < H , X , Y > of s and < V >C = V we obtain [V, Zi] = Zi and the usual S-action. Now from the minimality assumption it follows t h a t G' does not act transitively on G / H . Hence the G ' - o r b i t s are two-dimensional and therefore h C g ' . It follows t h a t wlog h = < H , X, Z 0 , . . . , Z k - l > r Hence n=
+h +h
ifk#O ifk=O
Now we are in a poeition to prove 2. The assumption dimc( R / R ~) = 1 leads to a contradiction to the basic assumptions.
Lemma
Proof. We will show t h a t from the assumption d i m c ( R / R ~) = 1 it follows t h a t X = G / H is biholomorphic to a direct product of lower-dimensional homogeneous manifolds. First consider the case k = 0. Then the S-action on BY is trivial, hence G = RxS. Furthermore H ~ = B • {e} and NG(H ~ = B x R , where B denotes a Borel subgroup of S. Thus H = B x ( H f3 R ) . This implies t h a t G I n ~-- ( S / B ) x ( R / ( R f3 H)). Next let us discuss the case k # 0. Here we obtain a normaliser fibration G / H ---, G / N with G / N ~_ H t . Observe t h a t G / N is simply-connected. Thus N is connected. Consequently the one-dimensional Lie group N / H ~ is abelian. Hence H is normal in N, i.e. G / H ---, G / N is a principal bundle. Now the
72 G~-orbits in G / H are two-dimensional and G ~ acts transitively on G/N. It follows t h a t the projection • : G/H ---. GIN realizes every G~-orbit in G / H M a covering over GIN. Since GIN is simply-connected, this yields a section in the principal bundle G/H ~ GIN. Thus the bundle G/H ---, GIN is trivial, i.e. G / H "" GIN • N / H . o
73 4. T h e c a s e dimc(R/RY) > 1 The structure of G We will now consider the case dimc(R/RY) > 1. Let V0 denote the irreducible S - m o d u l e transversal to ta. Since the Ad(S)-action on V0 is non-trivial, it follows t h a t G - G ' . T h u s G ' f3 R = R . It follows t h a t R is nilpotent. Let R t , . . . , R ~ denote the central series of R (i.e. R / = [ R , R / - I ] ) . Since R is nilpotent, there exists a p G I~l such t h a t R p ~ ( e ) = / / ~ + x . T h e n Z = BY is central in R . L e m m a 1. Let Z = BY. Then the Z-orbits in G / H coincide with the t~-or-
bits. Proof. T h e group R* has one-dimensional orbits and Z is s normal Lie subgroup of G acting non-trivially on G / H . This implies t h a t Z acts transitively on the R~-orbits. n L e m m a 2. The isotropy algebra h n r of the R-action on G / H is stable under Ad(B), where B is an appropriately choosen Borei group of S ~-- SL2(C).
Proof. Since the G = St< R-orbit in G / H is bigger than any R-orbit, it is clear t h a t no S t< Z - o r b i t coincides with a Z-orbit. Therefore every S t< Z - o r b i t in G / H is at least two-dimensional. On the other hand the minimality assumption implies t h a t S t< Z does not act transitively on G / H . Thus there exists a twodimensional St< Z - o r b i t in G / H . Hence we m a y assume t h a t dimc(St< Z)/((St< z ) n H))) = 2. Observe t h a t the Z - o r b i t s are one-dimensional and hence of codimension one in this St< Z - o r b i t . Therefore ( z + h ) n s is a subalgebra of codimension one in s _ sl2(C), i.e. a Borel algebra b . Note t h a t Z centralizes R and t h a t h n r m u s t be stable under the Ad(H)-action. Hence h fl r must be stable under the Ad( Z H)-action. T h u s the statement follows from ( Z H fl S) ~ = B . v 3. Let c denote the center of r. module. In particular c = z -" r p .
Lemma
Then c is an irreducible Ad(S)-
Proof. Obviously c is Ad(S)-stable and c + r ' ~ r . Since r = r ' + V 0 , where V0 is an irreducible Ad(S)-module, it follows that c C r ~9 Thus the C-orbits in G / H are one-dimensional. Now assume that the Ad(S)-action on c is not irreducible. Recall t h a t h n r is stable under Ad(B), where B denotes a Borel group of S. Hence Ad(B)(h n c) C (h f3 c). Thus it would follow from dimc(e/(c f3 h)) = 1 t h a t h f 3 c contains an Ad(S)-stable subspace co. But since C is central in R , this subspace co would be an ideal in g , hence ineffective. Therefore the Ad(S)-representation in G L ( c ) must be irreducible. Let again p E 1~1 such t h a t dime(BY) > 0 but BY+l = [R, BY] = {e}. Since BY is Ad(S)-stable and BY C C , this implies C = BY. o Let W o , . . . , W k denote the usual basis for C. Now let us discuss the irreducible S - s u b m o d u l e Vo. Let Z 0 , . . . , Z , denote the usual basis. T h e n h A V0 = < Z 0 , . . . , Z , - 1 > c .
74 L e m m a 4.
Let r be a Lie algebra on which sl2(C) acts by derivations. Let [W0,W1] is s-stable.
Wo, W1 denote s-stable subspaces. Then
Proof. Recall that for every derivation ~b
~([x, Y]) = [~(x), Y] + IX, ~(Y)]. ra Recall that R is nilpotent with R" = {e} and r = V0 ~ r I. Furthermore ~ {e} = / ~ + 1 It follows that
r = ( ~ ld with IV0, l~] = V~+I. All these suhspaces V~ are Ad(S)-stable and Vp = •. L e m m a 5. Under the above assumptions it follows that
[ h N r , r I] C h. Proof. Recall that the Z-orbits coincide with the R~-orbits. Hence r ~ C (s-Fh). Now observe that [h N r, h] C h and [r N h, z] = {0}, since Z is central in R. v
A key step in the investigation of this case is the following: L e m m a 6. Let a o , . . . , % E { 0 , . . . , n } . Assume that [Za,, [..., [Z~, [Za~, Z,o]...] ~ h Then ai -" n for all i with 2 < i < p. Proof. From r " = {0} it follows that
[z, IV, w]] = IV, [z, w]] for all Z , V E r, W E r ~. Hence the value of X = [Zo,, [..., [Z,~, [Zo,, Zoo]...] is invariant under permutation of the a 2 , . . . , ap. Thus it suffices to prove that ap - n. But a~ - n follows immediately from Zi E h for i < n and [h f3r, r~ C h.[3 We m a y even sharpen this result: L e m m ~ 7. Under the assumptions of the preceding Lemma, one of the numbers a0, al must equal n. Proof. This follows from
[Z,, [..., [Z~, [Zoo, Z , , ] . . . ] = [Z,, [..., [Zo~, [Z,, Z , . ] . . . ]
-[zn, [..., [z~176 [z~, z~ [] L e m m , , 8. The above assumptions imply {a0, a l ) = {n, n - 1).
75
Proof. Assume to the contrary that [Z~,
with
i < n -
[..., [Z,, [Z~, Z,]...] r h,
1. Now Z~ = a[X, Z~+I] with a = n1-1-7_i.Thus
[z~, [..., [z,, [z~, IX, z,+~]...1 r h. Now observe that
[z~,[...,[z,, [z~, [x, z,+l]...1 = [z,, [...,[z,, [z,, x], z,+d 99.1 + [z., [..., [z~, [x, [z~, z,+l]...l. Here
[z., [..., [z~, [zn, x], Z,+ll...] e h, by the preceding LeInma, because [Zn, X] is a multiple of Zn-1 and we assumed n>i+l. Thus
[z~, [..., [z~, [x, [z,, z,+l]...1 ~ h. Induction by similar arguments yields IX, [Z,, [..., [Z~, [Z,, Z,+I]...1 r h. But this yields a contradiction, because [X, s] C h. Hence i < n - 1 is not possible, i.e. i -- n - 1 is necessary. [3 L e m m a 9. The smbspaces ~ of r are irreducible Ad(S)-modules. Moreover Vi C h for I < i < p - 1. Proof. Recall that K = [Vo, [Vo,..., [Vo, Vo]...]. Since Zj is an ad(H)-eigenvector for the eigenvalue n - 2j, it follows that the element
[Z,,, [..., [Z,I, Zool...] is an eigenvector for the eigenvalue (i + 1)n - 2 ~ aj. Hence 2 - (i + 1)n, the eigenvalue for
[z., [..., [z., z.-d...1 is the lowest occuring eigenvedue for ~ . Note that the eigeuspace for this eigenvalue in l~ is one-dimensional. Thus ~ = ViI $ Vi" , where V~~ is an irreducible ad(s)-submodule with 2- (i+ l)n as lowest eigenvalue (i.e.dimc(V~') = (i+ l)n1 ) and V~" is the direct sum of lower-dimensional irreducible ad(s)-submodules of V~. Recall that the commutator of two ad(s)-stable subspaces is again ad(s)stable (see 7.4.4). Note further that the lowest ad(H)=e'Igenvalue of [V0,VLq can not be smaller than the sum of - n and the lowest ad(H)-eigenvalue of V~". This implies that [Vo, Vi"] C V ~ I . Recall that Vp = 9 is an irreducible ad(s)-module. Hence V~' = {0}. Let
.:=(~v,.. i~_1
76 Since IVY,V/'] C Vi~t for all i, it follows that a is an ideal in g. Observe that r l n h is a one-codimensional ad(H)-stable subspace of r' and that [Zn, [Zn, [..., [Zn, Zn-1]-- .] r h. It follows that r ~n h is the direct sum of all ad(H)-eigenspaces of higher eigenvalues than 2 - (i + 1)n. In particular V~ C r' n h for all 1 < i < p - 1. Now a = HV/' is an ideal in g which is completely contained in the isotropy algebra h . Hence a is contained in the ineffectivity, i.e. a = (0} due to the basic assumptions, o L e n n n a 10. The Lie algebra strwctsre of g is completely determined by the parameters n and p .
Proof. We choose a base H , X ~ Y of s and Z 0 , . . . Z , as usual. Furthermore for each Vm we choese a base V ~ n , . . . , V ~ . such that we obtain the usual sl2(C)action on Vm. We m a y wlog assume that [Z., [.
[Z.,Z.-I]
.]= V "
W e want to prove that this determines completely the value of all commutators of the form [ Z ~ [..., [Z,,, Z~ For each m we will prove this by induction ~ a~. We assume that our assertion is true for all (ao,...,am) with ~ a i _> N . Now choose (ao,...,am) with a~=N-1. Let
w = [z.., [... [z~ Z~ Then W is an ad(H)-eigenvector and [Y, W] ~ 0 , since ~ ai < q,~ = (m + 1)n - 2. Thus the value of W is determined by the value of ['I/', W ] . Now an iterated application of the Jacobi-identity shows that [Y, W] is a well-determined linear combination of [ Z , . - 1 , [ . . . , [Z~1, Z , . ] . . . ]
[z.., [..., [z.,_l, Z.o]...] and
[z~ [..., [z~ Z.o-d...1. By induction these values are all uniquely determined. This determines the value o f W . r~ The Tits-fibration We have already proved that
g = h H < Y, Zn,Wk > c
77 where
w k = [ z , , [..., [ z , , z , - x ] . . . 1 . Here k = (p + 1 ) n - 2. Furthermore
Now h = rrlh~
< H + ctZn -I- j ~ W h , X -F 7 Z . -l- 6Wk > c 9
Observe that [H + a Z n + ~ W k , X + 7Zn + 6Wk] = 2X - nTZn - k6Wk -- a Z . _ I --/~Wk- I E h This implies 7 = 6 = 0. L e m m a 11. The normalizer algebra n of h in g is tlte folloutinF n={
h'<w'>c
if'~Oifk=O
Note that k = 0 if and only if (n,p) = (1, 1). equation k = (p + 1)n - 2 .)
Ce
m XZ A m m e
(This follows from the
( . , p ) # (1,1). TAe. H = H ~
Proof. For (n,p) = (1, 1) we deduced n = h . Thus GIN is three-dimeusional and G / H is a covering over GIN. Now consider the fibration GIN ~ G / R N ~P1.
The group R is unipotent, because R = R N G t. Hence the bundle C ~ -bundle over P1. In particular GIN is simply-connected. Therefore N = H = H ~ . n
GIN ~ G/ R N is a
L e m m m 13. After appropriate conj9 of H in G we oMaim cr = O. /j' (n,p) # (1,1), rye can/urtherrnore acAieve that ~ - O, i.e. h =< H,X >c ~(r Nh).
Proof. Since SK Z does not act transitively on G / H , there must exist an SK Z orbit of dimension less or equal two. Hence alter conjugation s + 9 + h ~ g . This implies a = 0. Recall that 9 is an irreducible Ad(SLa(C))-module of dimeusion k + 1 = (p + 1)n - 1 and that h N (s -[- 9 = < H -t- j~Vcrk,X, W 0 , . . . , W k _ l
>c
for a = 0. It follows that for k ~ 0 this S g Z-orbit is biholomorphic to H k , the k t h power of the hyperplane bundle over P1 with the usual action of SL2(C) K F ( P h H~). In particular for k ~ 0 there exists an one-dimeusional S-orbit. Hence wlog ~ = 0. v
78 Lemma /~--I.
14.
Let ( n , p ) = (I, I ) .
Then we may assume that either ~ = 0 or
Proof. For ( n , p ) = (1, 1) we obtain g = < Y , H , X , Z0, Z I , W > c and h - - < H + ~ W , X, Z0 > c 9 Assume t h a t ~ ~ 0 and let A E C* such t h a t ~2 = 8. W ' = ~ W . Finally note t h a t
Let Z~ - l z i
and
h = < H + W ~,X,Z~ > c 9 and t h a t the m a p ~b : g --, g defined by 0 ( H ) -- H , 0 ( Y ) -- Y , ~b(X) = X , ~b(Z0) -- Z~, ~b(Zl) - Z~ and ~b(W) - W ~ is an Lie algebra automorphism. This completes the proof. [] Thus we obtain P r o p o s i t i o n 15. Under the basic assumptions the structure of G is completely determined by the parameters n , p , where n = d i m c ( R / R ~) and p = length of central series, i.e. t~0 ~ {e} = R p+1 . Furthermore the structure of G / H ~ is completely determined by the parameters n, p, /3 , where/3 E {0, 1} for n = p = 1 and ~ = 0 in all other cases. Concrete Realization of G/H ~ for ~ = 0 We will see t h a t G / H ~ m a y be realized as a certain C2-bundle over ]71. To prove this we will describe the transition functions for this bundle and give a realisation of the Lie algebra g as an algebra of globally integrable vector fields on the total space of this bundle. Let /~ = (U0, U1) denote the usual open covering of P1, i.e. define Ui {[Z 0 : Z1] [ Zi ~ 0). Let Vi = Ui x C 2 . We introduce global coordinates z, w, z on ~ and z I, w I, z ~ on VI such t h a t the coordinate functions z, z I are actually the projections of ~ on U~. Hence z -- ~ and z I =- ~ Now we p a t c h VI and V2 together by the following transition functions 81
1 Z
1
tO t --- _ ~ t O Xn
z,
1 -- Znp+'-~_2Z
1 Znp.l.n_l top4"I
"
79 T h e transition functions imply d x o - - - -- 1z.~d2z
1
1 dw I =Zn d w + n ~ . .
z-+1 d z
d e = ,-,+l--,d" - (p+ 1 ) ~ d ~
-("p + " - 2)"" + ("p + " - 1)~'+~d, znP+n
Now ~o; ' ( d z ,) = 1, o '(d" '~ = 0 etc. Consequently
O'
O'
#
20
O---~ = -z =
_
-~z - nwx'ffjw
z"~ a-
- ((rip +"
-
2)zz + ~ + 1
oz
~ + 1 ) ~ . --~ Oz
0__' znp+n_,L =
8z
Oz
Thus we obtained a Ca - b u n d l e over P1. Next we will realize g as a Lie algebra of globally holomorphic vector fields on 1:1 and V2 Let H = -2z
- nw~--~w- (rip + n - 2)z
y-__ 0 0z
20 x = = ~+
0 .~=b-j + ((.p+.
- 2)~z + ,~+'
T h e n < H , X, Y > c is an Lie algebra of globally holomorphic vector fields isomorphic to sl2(C) Furthermore let Oz, v o = < z . O~--4w+x. - 1ta'bS
..
nz._l~+(n_l)z._,wvO,
...(n)- I)-" ~(nz'.0-~+, kzt'-lWV~z
,...
and
=< z"C++l)-2u~'-i~z,-.., zu~-i~z, u~-i~z >c
c ~ r . But for the elements in < H, X > c @r the global integrability follows from Lemma 2.3.1. T h e s p e c i a l e a s e n = p = 1, # = 0
In the subcase n = p = 1 with j8 = 0 there exists a very special realization of Here G / H ~ ~_ Q3 \ L where Qs denotes the projective quadric in P4 and L a complex line in Q3. In appropriately choosen complex-linear coordinates: G/H~
q ~ = {[xo : . . . : ,,,] I ~o~ + ~ ' ~
+ x~,,~ = o}
and L - {[zo :...:
x4] I ~ 1 = ~2 = ~o - 0 } .
Then G -- S L 2 ( C ) K R acts transitively on Qa \ L where R denotes the threedimensional Heisenberg group. The S-action is given by
[
ZO :
for A E S L 2 ( C ) . (a,b,c)
X2
:
X4
,-*
[
x0 : ( a - l ) '.
(.1) X2
:A-
X4
The R-action is given by
9 [ z 0 : Z 1 : Z2 : Z 3 " Z 4 ]
[z0 + a z l + bz2 : z l : z2 : z 3 + cz2 - 2azo - a 2 z l : z 4 - c z l - 2bz0 - b2z2]
for a, b, c E C. Now G / H ~ --* G / R ' H ~ is the C-principal bundle r : Q3 \ L --, P~ \ {[1 : 0 : 0]} where the projection is given by -:
[-o : ... : ~,] '-, [xo: x,:
~,~].
81
Furthermore the projection on G / R H ~_ P1 is given by ,~: [~o : . . . : ~,1 ~
[ ~ , : ~21.
T h e s u b e a s e /~ ~ 0 The group G = SL2 (C) K H3 may be described as follows G "- ((A, (~1, ~2), z) E SLy(C) • Ca • C} with the group law
(A, ~, z). (A', ~', z') = (AA', ~' + ~A', z + z ~+ vA~E(v') +) whe,e E = (-Vo). Using the fact that A E A t = E for all A E SLy(C) one can easily check explicitly that this is a group l&w. There is a very natural embedding of G/I~ in SLs(C):
: (A, v, z) ~
A
Now from the structure of the isotropy algebra it follows that after conjugation '
-~ (((+'
Observe that
It follows that ~b(H ~ is precisely the intersection of r with the isotropy of the SL3(C)-action on P2 at [0 : 1 : 0]. Hence G I N is biholomorphic to the ~(G)-orbit through I0 : 1 9 0] in P2- Thus G I N ~_ P2 \ {I1 : 0 : 0]}. T h e tra_n~ition f u n c t i o n s Our next goal is to calculate the transition functions for the C*-bundle w : G / H ~ G / N ~ ~2\ L. For this purpose we consider the following open covering of G / N ~- P2 \ {I1 : 0 : 0]): / / = {U1, [72) with Ui --
{[zo :Zl: z2] ~ ~2 I~, # O}
Note that U~ - C 2 . In order to find a trivialization of ~r : G / H ~ --+ G / N restricted to U1 we consider the following Lie subgroup of G:
82 Note ~.hat a([0: 1: 0]) =
1 8
for
9
=
1
((:
This implies that the A-orbit A([1 : 0 : 1]) equals UI. Furthermore g C A, hence A acts transitively on the preimage r - I ( U I ) in G i n ~ . Hence w-l(U1) _ A / ( A f'l H ~ Now observe that
,)
}
Note that A N H ~ C Z and A / ( A N H ~ = AI x Z / ( A n n ~ with
),(t,O),O)[s,tEC)
A1 = { ( ( s
Observe that A1 acts freely and transitively on U1 C I72. It follows that the natural projection
*r-l(U,) ~_ ( A / ( A n H~
r A / A t " (A N H ~ ~_ C*
given by
1) yields a trivialisation of the C' -bundle G / H ~ ~ G I N restricted to U1. Next let g0= ((_
1
1),(0,0',0)
9
Observe that go(U1) = U2. Hence we obtain a trivialization of the bundle G / H ---* G I N restricted to U2 by the map r ~ := r o go 1 : ~r-l(U2) --. C*. Now we are in a position to start calculating the transition functions. Let p0 he a point in ~r-l(U1 N U2). Then
Po = aoH ~ = g o a l H ~ for some a0, al in A and go defined as above. Hence aoh = goal for some h E H ~ 9 Now choose s, t, u, d, t', u ~, p, a, b E C such that a 0
((1)
,)
s
1
,(t,0,u
1
1),(t',O),u')
83 Then l"(po) = e", ~(po) = 9u' and goal
and aoh =
((,,
see
( ( s '-1
=
1 ) ( t' ' , O ) , u ' )
, ) ,,,,,,+o,
sb + e-e
'
Now we obtain the following equations: e" = s'
(1)
b= 1
(2)
sb + e-~ = 0 t' = t~,
(4)
(5) (6)
o = tb + a u ~ = u+p+
Thus
u ~ - u = p + ateP .
(7)
ate p
Hence r'(po) = e . , _ . = e~e~ ~(po)
Now from (6) and (2) it follows that a = - t . Together with the equations (1) and (5) this yields ~'(po) = eee~ ~(po)
= s'e-"'
Next recall that Ir(po) = [zo: Zl: z2] -- It: 1: 8] = It': s ' : - 1 ] It follows that r'(po) ~(po)
L e m m a 16.
~
Zl "~ I z2 x2
---,--ez
T h e C" - p r i n c i p a l bmndle G / H ~ --, G / N
_ ~
P2\{([1 : 0 : 0])} is an
e a e . s i o n of the C'-prineipat S.nate S L ~ ( C ) / Z -~ S L 2 ( C ) / U ~- C2 \ {(0, 0)) where
z~-r--{( Proof.
Observe that
1 1)InEZ l
and
U--{( 1 1)IzEC}
84
conjugated with
.0_((11) equals
Note t h a t
iff e~ = 1, a - b - 0 and p + era p E 2~riZ. Hence
-
0.
This is equivalent to b --
-a
-
p
and
'1)1, Furthermore one can show th&t
Thus we found an open S-orbit isomorphic to S L 2 ( C ) / F in G / H ~ over an open S-orbit isomorphic to S L 2 ( C ) / U in G / R ~ N . Finally we observe that ff a point zo G G / H ~ lies in the open S-orbit then the whole fibre of 7 : G / H ~ --~ G I R T H ~ through z0 is contained in the open S-orbit. This is clear, because R ~ commutes with S. D
P A R T III
The Classification of Three-Dimensional Homogeneous Complex Manifolds X = G / H Where G is a real Lie group
86
The structure of the classification
N S-dim.
V (p" 113) N 4-dim. = 6 - - l - - ( p . 114)
__I dimlt(G)
G solvable --(p. 104)
N 5-dim. (p. 124) direR(G) > 6
(p. 131)
/~ transitive --on G///
--
_ ~ 0///~ Oil / (p" 149) [_ 3 0///--. 0 / I (p. 153)
s ~_ SL2(~) (p. 165)
G mixed__/~-orbit
2-dim.
I
S~_SU~
(p. 1~8)
direR(S) > 3
(p. 181)
_/~-orbit 1-dim. (p. 190) 3 holo. --fibration (p. 190) G ~_ SL4(C) -- G semisimple--
__/B holo.
- ( p . 208) G = SLs(C) --(p. 211)
fibration --(p. 214) G ~_ SO5(C) --(p. 222)
87
Chapter 8
Preparations 1. C omplexifications
Let X = G / H be a homogeneous complex manifold, where G is a real Lie group. Our goal here is to study complexitications, i.e. G-equivariant open embeddings G / H --, G / f / , where 8 is a complex Lie group and g = g + ig in their representations in Uo(X, T o X ) . Later on we will use such complexilicatious to obtain fibratious G / H ---, G / I which are compatible with the complex structure on G / H . G - o r b i t s a r e Zariski-dense in G / H
L e n n n a 1. Let G/~I be a complez-homogeneons manifold and G a real Lie subgro,~p of 8 stlch that g + ig = g. Then in the complez-analytic Zariskitopology any G-orbit in G/I-[ is dense. Furthermore the ineffecti~ity of the G-action on any two G-orbits coincide. Proof. Let f2 denote a G-orbit and ~ its Zariski-closure. Consider the group of all g E 8 which stabilize ~. Since ~ is a closed complex-analytic subset of G/~[, this is a closed complex Lie subgroup of 8 . But since G acts as a group of holomorphic transformations and stabilizes f~, it follows that necesearily G also stabilizes ~. Hence the stabillzer-group of ~ which contains G and is a complex Lie subgroup of G, must be the whole group G. Therefore ~ = G / H and the first assertion is proved. Now denote the ineffectivity of the G-action on f~ by L. Since L fixes f~ pointwise, it must also fix ~ = 8//"/. Canonical subgroups
Definition 2. A Lie subaigebra a of a complez or real Lie algebra g is called canonical if it is in~ariant ender all automorphisms of g as a real Lie algebra. R e m a r k . For instance, the radical, the commutator and the center of any Lie algebra are canonical. L e m m ~ 3. Let g be a complez Lie algebra with a totally real snbalgebre g s.th. = g ~ ig, and it a canonical complez Lie sabaigebra of g. Denote g N A by a. Then i = a ( ~ i a .
88
Proof. Let us consider X E & with X = Y + i Z ,
Y , Z E g . Note t h a t conjugation in g with respect to g is an a u t o m o r p h i s m of g as a real Lie algebra. Since ,~ is canonical, it is stable under this conjugation Thus Y - iZ E / t and therefore Y E ~. The same argumentation applied to i X shows t h a t Z E ~. Hence X E a + i a . [3 C o r o l l a r y 4. If A is a canonical Lie subgroup (i.e. its Lie algebra is canonical) then the A-orbits are analytically Zariski-dense in the A-orbits. 2. M i n l m a l i t y C o n d i t i o n s During the classification of homogeneous manifolds X = G / H we will always assume t h a t among all Lie groups which act transitively on X the group G is of minimal dimension. In this p a r a g r a p h we develop some tools to exploit this assumption. L e m m A 1. Let G be a real Lie group acting transitively
a s a group of holomorphic transformations on a complex manifold X = G / H and let G / H -~ G / I be a fibration compatible with the complex structure. Assume that no positivedimensional complex Lie group acts nontrivially on I / H . Furthermore assume that the G-action on G / H is almost effective and denote by L the ineffectivity of G on G / I . Then L acts almost effectively on every fibre of G / H --* G / I .
Remark. For example the condition on the fibre I / H holds if I / H C~ \ ~ " or a bounded domain.
equals
Proof. For any z E G / I let F= denote the fibre of the bundle G / H --* G / I over z. Choose x0 E G / I . For each z E G / I there exists a Lie algebra h o m o m o r p h i s m r : l --, to(F=, TOE=) with kernel Ix. Now choose Zl,...,Z, E 1 such t h a t the ~bz0(Z~) are all real-linearly independent and l = l=o+ < Z 1 , . . . , Z , > = . Observe that the r are already complex linearly independent, because no complex Lie group can act nontrivially on I / H . Obviously all these l= have the same dimension and vary smoothly in their dependence on z . Hence we find an open neighborhood U of x0 such t h a t for all z E U the vector fields r are complex linear independent. Now we choose local coordinates z = ( z l , . . . , x , , ) on V and fibre coordinates y = ( Y l , . . . , Y~) (if necessary only locally around some point of the fibre over z0). In these coordinates we obtain 0
z, =
/
)oyi
Now take any X E l=o. Since l = l=(~ < Z I , . . . , Z , >It, for all = E U we find some well-defined real-valued functions f~ on U such t h a t X = ~"~ f~(x)Z, on U. Hence
x= .. s ,,7
~
Oyj
89 Now X is a holomorphic vector field. Thus
~
~O~k z)gi,(z,
y) = 0
for all j and k.(Observe that the gO are holomorphic). But now the complex linearly independence of the ~br(Zi) shows that ~ ( z ) = 0 for every z 9 U. Hence the fi are real-valued holomorphic functions vanishing at z0, i.e. fi - 0. Thus the vector field X vanishes identically on U, and hence X - 0 on the whole manifold G / H. c] Without imposing any conditions on the fibre we can obtain the following result, which is in particular useful in the non-solvable situation. L e m m a 2. Let G be a real Lie group acting transitively as a group of holomorphic transformations on a complez manifold X = G / H and let G / H -* G / I be a fibration compatible with the complez strmcture. Assume that the G-action on G / H is almost effective and denote by L the ineffectivity of G on G / I . Furthermore denote by L~ the ineffectivity of the L-action on I / H . Then L~ C R, where R denotes the radical of G.
Proof. Let Se denote a maximal semisimple subgroup of Le. Now S~ stabilizes the Lie subalgebra I of g, since 1 is an ideal in g. Hence there is a transversal Sestable subspace V C g such that g = V ~ 1. Observe that [V, s~] C V N 1 = {0}. Since 1, is an ideal in 1, it follows that [g, Se] C le. Hence Se acts ineffectively on the whole G / H . Thus L, is solvable. Finally observe that L, ,~ L and L 4 G imply that Le C RL C R, where RL denotes the radical of L.[3 L e m m a 3. Let X = G / H be a homogeneous complex manifold. Assume that A is a normal Lie subgroup of G with totally real orbits in G / H . Then A f3 H is discrete, i.e. A acts almost freely on its orbits.
Proof. (i) Let k = dimg(A(z)), z0 9 X and denote the isotropy algebra of g in z by h r . Choose X 1 , . . . , X t 9 a such that <Xl,...,Xt
>ltaNh=a-
Then < X x , . . . , X t >R $ aNhx = a holds for all z 9 U where U is a sufficiently small open neighbourhood of z0 in X . Furthermore for all z 9 U the tangent vectors X i ( z ) are C-linearly independent, since the A-orbits are totally real. (ii) Choose X 9 a. It follows that
Xlu = ])~'~f,(x)X~lu i
with real-valued functions fi on U. Choose local holomorphic coordinates z~. Then x,
= $
90 with gij E O(U). Consequently x
= $
Since X is a holomorphic vector field it follows that the functions
(:) i
are holomorphic. Thus =0. t
Note that the g~j are holomorphic. Hence
But this implies
for all z E U. Since the tangent vectors X~(=) are C-linearly independent, it follows that a--~kfi ~ 0 for all k, i. Since the functions fi are real-valued this implies that they are constant. Hence a :
C. d,,,(z, ~) V , , y e A". (viii) Assume that N n=l
This implies that
p,~ E r -1 (Bt(zo)) and a . E Z~" for all n. Hence N
E
n=l
N
>
r~=l
94 Observe that therefore ~ , ( A ' ) C T for all n. Hence dG/H(p, q) >_~ or dG/H(p, q) >_ C . dT(p, q). Both imply dG/H(p, q) > 0 if p ~ q. Thus G / H is hyperbolic. [] R e m a r k . In the proof given above we must require I / H is not only a bounded domain but even symmetric. For d i m c ( I / H ) _< 3 this is no restriction, since any bounded homogeneous domain of dimension less than four is already symmetric. However, for applications in higher dimensions it is interesting to know whether the condition of symmetricity can be eased to boundednese. In [W5] we were able to prove this.
Lamina 2. and G~ are G1 and G2 group which
Let ~ be a bownded homogeneons domain. Assume that both G1 connected Lie groups which act freely and transitively on t~. Then are conjegate in Auto(f~). Moreover there always ezists a solvable acts freely and transitively.
Proof. See [PS, P.51 and [Vi]. [] Hence to each bounded homogeneous domain there is uniquely associated a freely acting solvable group. L e m m a 3. Assume ~ ~_ G / H "--, G/IY[. Assume furthermore that G is solvable. Denote the nilradical of G by N . Then there ezists only one N* eq.iva.ant ~bration N/(N o H) - , N/i.
Proof. Observe that ~" acts transitively on 6//?/_~ C2 . Since G / H ~_ ~ it follows that N equals the three-dimensional Heisenberg-group. Hence N' has complex one-dimensional orbits in G//~/. But since the automorphism-groups of the homogeneous Riemann surfaces do not permit any N~-action on N / I , the fibres of this fibration must coincide with the ~"-orbits. Hence this fibration is unique. [3 4. A - a n t i c a n o n i c a l fibrations Lemana 1. Let G be a real Lie group, G / H a homogeneous manifold and A a normal subgroup which acts transitively on G / H . Denote the intersection of H and A by Hx. Then the fibration A / H I --* A / N A ( H ~ is G-eqeivariant. R e m a r k . This is equivalent to the assertion that NA(H~ 9H is a subgroup of G and not only a subset.
Proof. Since H1 = H N A, it follows that HI and hence H ~ is normal in H . Thus both H and NA(H ~ ~re contained in the group NG(H~ Furthermore NA(H ~ = A f l NG(H ~ is a normal subgroup of NG(H~ Hence NA(H~ . H is a Lie subgroup of NG(H~ []
95 5. L e t t - i n v a r l a n t c o m p l e x s t r u c t u r e s Let G / H be a real homogeneous manifold, g and h the corresponding Lie algebras. Now any G-left invariant complex structure corresponds to a linear automorphism J of the quotient vector space g / h such that J2 - -id IX, JY] - J[X, Y]
(I)
V X E h, Y E g
[JX, JY] -_ J[JX, Y] + J[X, JY] + [X, Y]
(2) (3)
Now let K x denote the map a d ( X ) o J - J o a d ( X ) . Then the last two conditions may be reformulated as Kx(Y)--0
VXEh, YEg
(K~x-JoKx)(Y)-0
VX, Y E g
(2') (3 ~)
Observe that K x o J equals - J o K x for all X by definition. Hence to verify condition (3) it suffices to check this for vectors X 1 , . . . , X , such that X 1 , J X 1 , . . . , X , , J X , is abase for g. L e m m a 1. Let a be any ideal in g. Then Ja + a is a subalgebra of g. Proof. It suffices to show that [Ja, Ja] C J a + a. This follows from the integrability conditions (3).n L e m m a 2. Assume that G / H has a lefl-invariant complex structure given by J :g--~g. Let a be any ideal in g. Let i denote the algebra Ja q - a - t - h . Then the condition (2) applied to i is fulfilled, i.e. G / I has a compatible lefl-invariant complex structure. Proof. For any X E h we have Kx(g) C h. Furthermore for any X E a it is true that K x ( g ) C a + J a . Since K x - J o K x -- 0 rood h it also follows that Kx(g) C a + J a + h for all X E J a . Hence Kx(g) C J a + a + h for all X E J a + a + h .
D
Finally we want to state two standard results about the relationship between these J-structures and complexifications. L e m m a 3. Let G I f t be a complex-homogeneous manifold, i.e. a quotient of complex Lie groups. Assume that G is a real form of G with open orbit in G/[-I. Then from the induced complex structure on G / H with H = G f3 [t we obtain ia={XEg=g~iglX+iJX E h~,]3}.
96 L e m m a 1. Under the assumptions of the above Zemma let G / H ---, G / I be a ]ibration compatible with the complez structure on G / H . Then ~={XEt=g+iglX+iJX
Ei@ii}.
6. S o m e r e s u l t s f r o m l i n e a r a l g e b r a For the classification we will need a generalization of the eigenspace decomposition to non-diagonalizable endomorphisme. Let V be a real vector space of finite dimension and ~ an endomorphism. Let If c denote the complexitication V ~ l t C. Then ~ extends to a complex-linear endomorphism of V c , which we again denote by ~. We define the weight space V;~ as follows : = {,, E V c I 3 , ,
:
_
= 0}
Then a standard result of linear algebra states that one can find a basis for V where every basis vector is contained in some V t3 (V~ + VX) with A E C. From the Leibnitz rule it follows that if ~ is a derivation of a real Lie algebra V then for any X E Va and Y E V~ the commutator [X, Y] is contained in Va+~. Especially the following holds L e m n m 1. Let g be a real Lie algebra with complezification g . Assume that is a derivation on g. Let ~ be a non-real complez number such that W = g f3 Va + lira is totally real and of dimension two. Let X E W \ {0}. Then W = < X, OX >It. Furthermore, X is the sum of two eifenvectors, and [X, OX] is an eigenvector with eigenvalue 2 R e ( a ) .
97
Chapter 9 Holomorphic fibre bundles Let X = G / H be a homogeneous complex manifold, where G is a real Lie group acting as a group of biholomorphic transformations. Let 7 : G / H --~ G / I be a fibration which is compatible with the complex structure, i.e. G / I is a homogeneous complex manifold and the projection w is a holomorphic map. In general such a fibration is not a holomorphic fibre bundle, i.e. there is no local holomorphic trivialisation. But there are important exceptions, especially the anticanonical fibration and fibrations induced by the complex ideal. We will study these fibrations in this chapter. D e f i n i t i o n 1. Let X = G / H be a homogeneous complez manifold, where G is a real Lie group. The manifold X is called trivial, if o n e o f t h e following holds - The manifold X is complez-homogeneous, i.e. there ezists s complez Lie group which acts transitively o n X . - The manifold X is a directproduct of two or more lower-dimensional homogeneous complez manifolds. 1. A b s t r a c t complexification Let G be a simply-connected real Lie group acting holomorphically and transitively on a complex manifold X = G / H . Now there exist a natural representation of the corresponding Lie algebra g in the Lie algebra r o ( X , T o X ) of global holomorphic vector fields on X which represents the elements of g as G-left-invariant vector fields on X . Let g denote the complexifcation of g in the Lie algebra F o ( X , T o X ) of global holomorphic vector fields on X. Let !, be the isotropy algebra of g, and (~, / / the corresponding simply-connected Lie groups (which exist by e.g. Theorem 3.15.1 [Va]). In general it is not clear w h e t h e r / f is a closed Lie subgroup of (~. But under the additional assumption that either G is solvable or S "" SLy(C), where S is a mammal semisimple subgroup of G, it follows from I.emma 2.2.9 that /2/ is closed in G. Hence in this case we can study the complex-homogeneous manifold G / / f . We will denote the intersection G N/2/ by H1. Then from the construction it follows that the corresponding Lie algebras hi and h of H1 and H coincide. Denote the connectivity component of e in H by H ~ . Then we have the following diagram of G-equivariant fibrations.
G/H ~
,/ GIH
\ GIH~
~
GIH
98
2. T h e a n t i c a n o n i c a l f i b r a t i o n Let G be a real simply-connected Lie group acting transitively and almost effectively on a three-dimensional complex manifold X 0"~ G / H . For z E X denote the isotropy-algebra by h~. Now the complexified Lie algebra g = g ~ ig is represented on X as an algebra of holomorphic vector fields. Denote by l~r its isotropy algebra, i.e. the algebra of those elements of g, whoee corresponding vector fields on X vanish at z. Next we introduce an equivalence relation on X . Two points z and y are equivalent iff h . = i ~ . Then the projection X --* X/..~ is a G-equivariant holomorphic fibre bundle (see [no2]). Moreover, X / . ~ = G / I with I = G A Nd(/~ ). Hence I normalizes H ~ = G A H . The anti-canonical fibration gives a G-equivariant embedding of G / I in a P , . This yields also a natural complexification G / I ~ G / i . 3. T h e c o m p l e x ideal Let G I H ---, G / I ~-, G / i ~-, P . denote the anticanonical fibration of G I H . Consider the representation of the Lie algebra g in ro(]Pn, ToI?.). The representation of g is not necessarily a totally real subalgebra of g in ro(l?,, T o P . ) . In this paragraph we will discuss the case that this representation is not totally real. We will see that under these circumstances there is a fibration G / H ---* G / J with I C J such that the representation of g on G / J is totally real. Moreover, this fibration G / H ---, G / J is a holomorphic fibre bundle. L e m m - 1. Let G / H be a homogeneous complez manifold. Let
GIH
1 G/I
Oil
denote the G-anticanonicai fibration of G / H . Assume that there are fibrations Gilt
1 al: 1
GN
01i l
GIi
such that J / ? = J / I . Then G / H ---, G / J is a holomorphic fbre bundle. Proof. (i) Let (~ be the simply-connected complex Lie group corresponding to the Lie algebra g = g + ig. Denote its isotropy algebra by i~. Let if/ he the connected Lie subgroup of 0 corresponding to li. Define Hi = H A G. Observe that (Hi) ~ = H ~ We will start with proving that x : G/H1 ---, G / J is a holomorphic fibre bundle.
99
(ii) The quotient topology on G / / t may be very bad, e.g. not hausdorff, but nevertheless G / H 1 is a subset of G//:/. For any gill E G / H 1 we define Te = {a E G [ a(gUl) ~. G / H 1 } . Note that TI = {a E G [ ag E G~I). Hence
T, - G[-I9 -1. (iii) Recall the assumption that J / I = J/i. Since G / H --} G / I is the Ganticanonical fibration, it follows that J / H - j / ~ I . Hence J / : / - J . Observe that G/r/g -1 = GJ~I9 -1 - G J g -1 . Moreover, for any j E J one has T r - Tij , because G[-I(gj) -1 - G f l j - l g - 1 -_ G J j - l g - 1 - G j 9 - 1 , since j E J . (iv) Next choose a vector subspace V C g, such that g - V ~ ~. Define - V N e z p - t ( T i ) . Observe that Ts contains an open neighbourhood of 9 in G. Thus VI contains an open neighbourhood of 0 in V. (v) Now consider the holomorphic map ~bj : V t -* G / J defined by ~,(X) = ezp(X)gJ E GJ/J = G/J. Since g = j ~ V, it follows that for a sufficiently small neighbourhood W of 0 in V with W C VI , the restriction of ~l to W is biholomorphic onto its image in G / J . Denote Ce(W) by U. Now we have a biholomorphic map Cs : W x J / H --~ ~r-l(U) defined by r
a l l ) = ezp(X)galYI.
(vi) Let 91 E G / J be another point neat g. Define r anMogonsly. To determine the transition function for the fibre coordinates we compare the following.
exv(x)
aH = exv(Xl)glalH
= g-l v(-X)e V(Xx)g axU g-le p(-X) v(Xl)gl e / Since X and X1 depend holomorphically on the base point, it is clear that the transition functions are holomorphic mappings into J . Hence ~r : G / H x --} G / J is a holomorphic bundle with structure group J . (vii) Since G / H 1 and G / H coincide modulo coverings, it is clear that via the same methods the bundle G / H --} G / J is also a holomorphic fibre bundle with structure group J . [3 L e m m A 2. Let X = G / I be a homogeneous complez manifold which is eqeivariantly embeddable in some Pn. Consider the representation of g in F o ( X , To X).
I00
With respect to this representation define m = g n ig. Assume that this intersection m is positive-dimensional. Then there are fibrations with positivedimensional fibres
l
GII
1
GIJ
such th,t J / I = J / i and moreover J / I is equivariantly embeddable in some Pro. Proof. (i) Since both [g, g] C g and [g, ig] C ig, it follows that m is an ideal in g . Moreover, m is an ideal in g. Let M denote the connected Lie subgroup of G corresponding to m . Furthermore define A = ( M A G~)~ . (ii) Note that the M-orbits in G / I are not necessarily closed. But by Lemma 2.2.11 A is a linear-algebraic group. Hence the A-orbits are closed due to L e m m a 2.1.1. Observe that g~ = g a g ~. Thus a = g ~ n g A i g = g ~ n m , i.e. a is a complex idea] in g. It follows that G / I --, G / A I is the restriction of the holomorphic fibre bundle ( ~ / i ---, ~/AI to the open subset G / A I C ~/AI. Hence G / I ---, G / A I is a holomorphic fibre bundle. Observe that by Lemma 2.1.1 G / A I has an equivariant embedding in some ~m. Thus we are finished, if dim(A) > O. (ill) Assume that A -- {e}. It follows that [G, M] C A = {e}, i.e. M is central in G . Now direR(M) > 0 implies that the G-anticanonica] fibration of G / I has pveitive-dimensioned fibres. This gives the desired fibration. [3 L e n u n a 3. Let G / H be a homogeneous complex manifold. Denote the antica-
nonical fibration by x : G / H ---, G / I . Assume that I / H is discrete. Then the G-anticanonicai fibration of G / I is injective. Proof. Recall that the anticanonica] fibration identifies two points iff their complex isotropy algebras coincide. Since ~r : G / H ---, G / I is a covering, it follows that I~= = i s ( s ) for any x E G / H . Hence for any x, y E G / I with x ~ y it follows that [= ~ i v . D Mammal holomorphic fibration Now we are in a position to prove the main result of this paragraph.
Let G / H be a homogeneous complex manifold. Then there exists a fibration G / H ---, G / J with the following properties. The bundle G / H --, G / J is a holomorphic fibre bundle. G / J is equivariantly embeddable in some P~. The representation of g in the algebra of global hoiomorphic vector fields on G / J is totally real. The G-anticanonicai fibration of G / J is injective.
P r o p o s i t i o n 4.
101
Proof. Apply the anticanonical fibration resp. the fibration granted by Lemma 9.3.2 on G / H until we obtain a fibration G/H --* G / J , such that both the G-anticanonical fibration is injective and the representation of g is totally real. The fact that G / H is finite-dimensional, and Lermna 9.3.3 imply that this is reached by finitely many steps. Now the other assertions follow immediately from Lemma 9.3.1 and Lemma 9.3.3. [] We will call this fibration max/ran/holomorphic fibrntion. 4. H o l o m o r p h i c f i b r a t l o n s in t h e case dima(S) 1 such that g(k) + h g(k-1) .~. h = g. B u t g(k-1) q_ h -----g implies g' C g(k) _{_h ' . Hence we obtain a contradiction: g = g ' + h C g(k) q_ h . n
C o r o l l a r y 2. On a nilmanifold the commutator group never acts transitively. C o r o l l a r y 3. Let N be a nilpoient Lie group. Then the commutator group N ' has at least codimension two. Proof. Otherwise n would be the sum of n' and a one-dimensional subalgebra of n in contradiction to the Lemma. rl
L e m m a 4. Let g be a nilpotent Lie algebra and a C g a positive-dimensional ideal. Then a has a positive-dimensional intersection with the center of g . Proof. There exist a k _> 0 such that
dirnzt(a N g(k)) > 0 = dirnI(a N g(t+D). For this k the positive-dimensional algebra a f3 g(t) is central in g. o L e m m a 5. Let n be a real nilpotent Lie algebra. three-dimensional Heisenberg algebra.
Then n' can not be the
Proof. Assume the contrary. Since n is nilpotent we can find a two-dimensional Lie subalgebra a such that n " C a C n' and [a, n] C n " . Now choose vectors X E a and Y E n' such that n' = < X , Y > z + n " . Observe that n " = < [X, Y] > z +[n', n '~] and [n', n ' ] = {0}, because n ' is nilpotent and d / m R ( n ' ) = 1. Hence < [X,Y] > a = n ' . Since Y E n' there are vectors Z, W E n such that [Z, W ] - Y . Now [[Z, W], X] ~ 0. But X E a implies that [ W , X ] , [Z,X] E n " . Since d/mR(n")--- 1 it follows that both [Z, [W, X]] and [W, [Z, X]] equal zero. Hence we have a contradiction to the Jacobi-identity. a
C o r o l l n r y 6. Let n be a nilpotent Lie algebra with n " ~ {0} . Then n " is at least three-codimensional in n ' .
106
Proof. Assume that dirna(n ~) = d i r n a ( n " ) + 2 . Since n is nilpotent, there exists a normal subgroup a of n such that a C n " and dim(a) = d i r n ( n ' ) - 1. Now n#/a is a non-abelian nilpotent three-dimensional Lie algebra, hence isomorphic to the three-dimensional Heisenberg algebra. Thus we obtained a contradiction to the above Lemma. [] C o r o l l a r y 7. Let N be a niipotent Lie group with dirne N _ 4. Hence we obtain a contradiction since there doesn't exists a four-dimensional nilpotent subgroup in Auto (B,~). o
111 6. R e d u c t i o n t o t h e c a s e dimlt(G) = 6 We would like to prove that the minimality assumption already implies that d i m l ( G ) = d i m m ( G / H ) . Unfortunately this is only true under an additional assumption. Suppose we have the following fibrations (all fibres one-dimensional) G/H
1 GIJ
'---, ( ~ / t l
1 1
GIJ
It turns out that for J / H ~_ C x H + we cannot assume dimlt(G) = 6. L e m m a 1. Under the basic assumptions it follows that dimlt(G) = 6 if there ezists a fibration
G/I
1
GIJ
G/H
l
0/]
with one-dimensional base and J / H ~ C • H + . Proof. (i) Since G ~ C N it suffices by Lemma 8.2.6 to show that N n H is discrete. (ii) First assume that N is abelian. From Lemma 10.5.2 it follows that ~r acts transitively on ( ~ / f / . Hence N n f / would be ineffective on G / / ~ . Thus (N n/~/)0 = {e} and consequently also (N n H ) ~ = {e}. (iii) Assume that N is not abelian. By Lemma 10.4.1, we know that J / H 7~ C 2 \JR 2. By the assumption of this Lemma, we have J / H ~ C • H + . Hence by nontriviality, J / H is biholomorphic to either the ball ] ~ , its complement or H + • H + . In any of these cases N n J must act almost effectively on J / H , since it is ineffective on the base. (iv) Checking the automorphism groups of the ball ~ , its complement and H + • H + now yields that N n H is discrete. [3 C o r o l l a r y 2. Under the assumptions of the Lemma, N ~ is abelian. Proof. Apply Corollary 10.2.7. []
112
Classification
Chapter II for G solvable
and
dimR(G) = 6
1. Gp.neralities In this chapter we will assume in addition to our basic assumptions that dima(G) = 6. For this case we will give a complete classification in the following way: First we will analyse the Lie algebra structure of the nilradical n of g. Then we choose Z E z(n) and study the eigenspace decomposition of g with respect to the ad(JZ)-action. Here we will exploit the integrahility conditions for J and the Jacobi identities. This leads to a complete description of the Lie algebra g and its J-structure. Finally, we give a concrete realization of g as a subMgebra of r o (TX, X) with )C = G / f / _ Ce . Then one can easily calculate where g spans the whole tangent space. This gives us a description of the open G-orbits in G / H . In one special case, that means if N is a Heisenberg group, we make these calculations easier by applying results of Tanaka [Tan] and Tolimieri [To]. First we need some auxiliary Lemmata.
L e m m ~ 1. Assnme that A is a complex canonical Lie swbgronp of N with one-dimensional orbits in G / t t . Then dimc(A) = 1. Proof. Consider
G/~
l
~
G/H
!
By Lemma 8.1.3 it follows that I / H 7k C*. Thus I / H ~- H + . Hence the real nilpotent group A := Gf3/] acts almost effectively on I / H . Thus dimx(A) = 1. But A is a canonical subgroup, hence dimc(A)= dirnx(A). D L e m m a 2. Let Z denote the center of 1~. Then Z f3 f I is discrete.
Proof. Since ~r acts transitively on G//~/ the group Z f3/7/ is ineffective on
G/h.v
L e n n n a 3. Asswme that g contains a full real flag of normal Lie subalgebras. Let 9 be any one-dimensional ideal in g. Choose Z E 9 Then all eigenvalwes of ad( J Z ) are real and any eigenspace transeersal to < JZ, Z >R is J-stable.
Proof. (i) Denote be c the centralizer of 9 in g. Obviously e is of real codimension one in g. Observe that according to the non-triviality of G / H the commutator [JZ, Z] cannot equal zero. Hence g = < J Z >x ~ e and therefore g=< JZ, Z >x~(cnJc). (ii) Since ad(JZ) must stabilize the full flag it is triagonalizable, i.e. it has only real eigenvalues. Now choose any vector X E c f3 J c . Consider the integrability condition [JX, JZ] = J [ J X , Z] + J[X, JZ] + IX, Z].
113 Since Z commutes with both X and J X it follows that
ad( JZ) o JlcnJc = J o ad( JZ)[cnjc. Hence the eigenspaces are J-stable. [] C o r o l l a r y 4. Assnme nnder the assnmptions of the above Lemma that ad( JZ) stabilizes a vector snbspace V of c. Then ad( JZ) also stabilizes the vector space JV+V.
L e n n n a 5. Let Z denote the center of N . Assnme that dimlt(Z) = 2. Then it follows that d i m l ( N ) = 4. Proof. Consider the fibration G/H
!
!
GlI
/2H =
Since Z is central in N any one-parameter subgroup of Z is normal in N . Hence there exist infinitely m a n y N-equivariant fibrations ~Q/I ---, .~r/j with d i m e ( N / J ) = 1. This proves that i is abelian and I / H ~ ~,~ (see Lemma 8.3.3 ). Neither is I / H biholomorphic to the complement of ~ . Hence we know that I / H is either biholomorphic to C 2 \ R 2 or to H + x H + . Now N is not abelian, because direR(N) >_ 3 > 2 = direR(Z). Thus the case I / H ~_ C2 \ R 2 is ruled out by Lemma 10.4.1. By Lemma 8.2.1 the ineffectivity L on G / I must act almost effectively on I / H ~_ H + • H + . But G / I is a homogeneous Kiemann surface. Note that for any homogeneous Riernann surface N 13 1 acts trivially on G / I . Hence L contains N 13 I . Thus direR(N) ffi is an ideal in g . Denote < Z, J Z >R by i and let I denote the corresponding Lie subgroup of G . Since < Z > s is an ideal in g , it follows from Lemma 8.5.2 that the fibration G / H ---, G / I is compatible with the complex structure. Hence we have a fibration
G/H
l
G/I
G/H
l
dli
114 with d i m c ( I / H ) = 1. Observe that Z is centralized by N , which acts transitively on ( ~ / H . Hence the vector field generated by Z vanishes identically on G / I . Thus the inetfectivity L of the G-action on G / I is at least real one-dimensional. Hence any /-orbit in I//~/ is at least real one-dimensional. Consequently either I / H __ C or I / H ~- H + . Hence by non-triviality wlog I / H ~ H + . I f L is two-dimensional, then it follows by Lemma 8.2.5 that the fibration G / H --, G / I is holomorphically trivial. Hence it remains to prove the Lemma for dimit(L) = 1, i.e. the case L = ezp(< Z >it). (iii) Assume that dimit(L) = 1. Now denote the centralizer of Z in g by c. We have dimit(c) = 5 because < Z >It is an ideal and J Z r e. Now choose any X E c 13 go. Consider the integrability condition [JX, J Z l = J[JX, Z 1 + J[X, J Z l + IX, Z]. Since g~ 13 Jg~ = {0} it follows that [JX, JZ] = IX, Z] -- 0. Hence < Z, J Z >it is an ideal in g . This contradicts the assumption that the ineffectivity of the G-action on G / I is one-dimensional. Thus L is always two-dimensional and therefore the fibration G / H --. G / I is holomorphically trivial, n 3. T h e case dimit(N) = 4 There are only two non-isomorphic real four-dimensional non-abelian nilpotent Lie algebras. (1) the direct product of the three-dimensional Heisenberg algebra and the onedimensional algebra (2) the algebra n = < A, B, C, D >it with [A, B] = C and [A, C] = D . Observe that in case (i) we have dirnit(N ~) = i. The
case
direR(N)
- 4
dimR(N')
-
1
Here we have dimit(N) -- 4, dimR(N ~) -- 1 and d i m l ( Z ) = 2 . Let e be the centralizer of n ' in g. Then e is five-dimensional. Observe that n 13 J n is twodimensional, because dimit(n) -- 4 and g -- n + J n . Note, that [JZ, Z] = 0 would imply that the orbit of I = ezp(< J Z , Z >it) is not isomorphic to H + and therefore equals I / H . Thus the assumed non-triviality of G / H implies Z ~ J n . Hence there are two possibilities. Either n = z ~ (n 13 J n ) or z 13 J n is one-dimensional. Under the above assumptions, the first case can't occur. L e m m - 1. Assume that n = z (~ (n 13 J n ) .
a homogeneous surface.
Then G / H ~_ fl x H + , where fl is
115
Proof. (i) Choose Z in n' such that [JZ, Z] = Z. Next choose W E z N J c and X E n n J n such that g = < X, W , Z, J X , J W , J Z > l . (ii) The algebra g decomposes g = < J Z > a 9 < Z > a (~ < W > 1 ~
< J W > a 9 < X, J X >~t,
where all these vector subspaces are ad(JZ)-stable. This follows from the fact that ad(JZ) must stabilize s, n I , c N J c and n N J n . (iii) Consider the integrability condition :~
[JZ, J W ] = J [ J Z , W] + J[Z, J W ] + [Z, W] [JZ, J W ] -- J [ J Z , W].
Now J[JZ, W] E J s = < J Z , J W >g, but [JZ, J W ] E g' C n = < Z , W , X, J X >a 9 Thus [JZ, JW] = [ j z , w ] = 0. (iv) Observe that, since both J W and W commute with J Z , the commutator [ J W , W] also commutes with J Z . Hence [ J W , W] E< W >1 and < J W , W >1 is a subalgebra of g. Moreover < J W , W >z commutes with the algebra < J Z , Z >n. (v) Since W is central in n, it follows from an integrability condition that [JW, JX]-
J[JW,X] E nNJn=(X,
JX)lt.
Now [ J W , JX] = J[JW, X] implies that the fibration induced by the Lie subgroup I corresponding to the Lie algebra < W , J W >z is compatible with the complex structure on G / H . Hence consider the fibration
G/H
1
G/I
,--. G/EI ,--.
l
G/i.
Note that < W >a is an ideal, because W is central in n, [JZ, W] = 0 and [ J W , W] E< W >ffi- Therefore any /-orbit in i//:/ is analytically Zariskidense in I / / t . Thus by non-triviality I / H ~- H + and consequently I//?/--~ C. Furthermore I / H ~_ H + implies [ J W , W] ~ 0. (vi) On the other hand [JW, JX] = J[JW, X] E < X, J X >a shows that J = (exp(< JZ, J X , Z, X >1)) is an ideal in g. (Observe that [JX, X] E n ~ = < Z >a-) Hence it also induces a fibration compatible with the complex structure. Now [ J W , W] ~ 0 implies that the base of this fibration is isomorphic to H + .Therefore the map
G/H--* G / I • G / J ~- H + x G / J
116 is biholomorphic, o W e will now show
Lemnaa 2. Asswme diml(N) = 4 and dirna(N') = 1. Assume farther that diml(s n J n ) = 1. Then G / H is biholomorphic to {(z, w, z) E Ca [ Ira(z) < 0 and Im(z - ~w) < 0}.
Proof. (i) Choose Z E n ~ such that [JZ, Z] = Z. Denote the centralizer of Z in g by c. Next choose W E 9 n J n and X E n N J e such that X is contained in an ad(JZ)-weight space. This is possible because ad(JZ) stabifizes c O J c . (ii) An integrability condition implies that [JZ, JX] = J[JZ, X] E g~ A J g ' C n A J n . Now n N Yn = < J W , W >R. Thus ad(JZ) stabilizes J n A n = < J W , W >a and maps < JX, X >a into < J W , W >a- Hence < J X , X >a is contained in the 0-weight space of ad(JZ). (iii) Observe that n' = < [JW, X] >a- Hence [s X] -- AZ with A E R \ { 0 } . We can substitute a real multiple of W for W . Thus we may wlog assume that A = 1, i.e. [ S W , X] = Z. (iv) Note that < W > z = s n (nA Jn) is stabilized by ad(JZ). Hence W is an ad(JZ)-eigenvectorfor some eigenvalue p E R. From the integzabilitycondition it follows that [JZ, J W ] = J[JZ, W] = S p W = p J W . Thus J W is also an eigenvector for the eigenvalue p. Since X is contained in the 0-weight space, it follows from [JW, X] = Z that Z is contained in the weight space for the eigenvalue p + 0 = p. Now [JZ, Z] = Z implies p = 1. Consequently Z, W and J W are all ad(JZ)-eigenvectors for the eigenvalue 1. (v) N o w g is the vector space direct sum of the 0-weight space J0 = < JZ, JX, X >a and the 1-weight space J1 = < J W , W , Z > a 9 Hence the 1-weight space is an abelian ideal in g. The 0-weight space J0 is a subalgebra with J0 n n = < X >a. It follows that
[SZ, JX], [JZ, X], [JX, X] E< X > a . Next consider the integrability condition [JZ, JX] = J[JZ, X l + J[Z, JX] + [Z, X]. Since X E c n J c implies [Z, JX] = 0 J[JZ, X]. Hence
IX, Z], it follows that [JZ, JX] =
[JZ, JX], J[JZ, X] E g~ n Jg~ n J0 C n n J n G J0 = {0}.
117 Therefore [JZ, JX] = 0 = [JZ, X] and [JX, X] = a X with a 9 R. (vi) Now J W and W are contained in the 1-weight space and [JZ, W] 9 W >R. Thus the integrabflity condition [JZ, J W ] = 3[JZ, W] + 3[Z, J W ] + [Z, W] implies [JZ, W] - W and [JZ, J W ] - J W . (vii) Another integrability condition yields [JX, J W ] = J [ J X , W] + J[X, J W ] . Hereby [JX, J W ] E< J W , W , Z >a, J [ J X , W] 9 J Z , J W >a and
J[X, J w ] = - J z .
Hence for some T 9 R we obtain [JX, J W ] = T J W and [JX, W] - 7 W + Z. (viii) Consider the Jacobi identity 0 = [JX, [X, J W l ] + [X, [3W, JXll + [dW, [JX, Xll = "fZ + a Z Hence T = - - a . (ix) Since n is the nilradical it follows that the ideal c = n ~ < J X >a is not nilpotent. This implies that 7 # 0. Hence replacing X by a real multiple of X shows that we can wlog assume that T = 1. Thus the Lie algebra structure and the J-structure on g axe completely determined. (x) Let us list all nontrivial commutator relations between our base vectors: [JZ, 3W] = J W [Jz, w ] = w [Jz, z] = z
[JX, 3W] = J W [Jx, w ] = w + z [JX, x ] = - x
[ J W , X]
=
Z
(xi) It is easy to check that the following is a realization of g as an subalgebra of r o (C 3 , TC6) 0 0 #z =
Z--~ jx
=
8 Oz 0
O
0 az 0 0 jw = i_---Z~z 0w 0 0 W= Ow Oz" The J-structure is correct for e H = (i, 0, - i ) .
118 (xii) Now let us investigate where g spans the whole tangent space. Clearly Ira(z) r 0 is a necessary condition. Furthermore g must span the tangent vector i~-;z. Observe t h a t
J X + R e ( w ) W + I m ( w ) J W = - ( z + Re(w) + z Ira(to)) 0 . Hence g spans the whole tangent space at a given point (z, to, z) in C"e iff Ira(z) ~ 0 and
1re(z) + Re(,,,) + Ira(x) Ira(w) # o. (xiii) Finally we show t h a t G / H is biholomorphic to
{(x, to, z) I Im(~) < o and Im(z - ~ )
< 0}.
Since (z, w, z) ~ (z,)~w, z + f ( z , w)) is biholomorphic for any holomorphic function f mad any ;~ E C* it suffices to observe t h a t
Ira(z) + Re(w) + I r a ( z ) Ira(w) = Im(z + iw + wire(z)) = Im((z + i w + ~ i z ) - ~ - ~ w
).
I"1 T h e c a s e direr(N) = 4 and dimz(N') = 2 Here we have direr(N(2)) = direr(Z) = 1. Denote the centralizer of s in g b y c. T h e n c is a five-dimensional ideal containing n . Let us denote the centralizer of n ' in n by a . Now a is a three-dimensional ideal with n ' C a C n . T h u s we have a full flag of canonical Lie subalgebras in g : zCn~CaCnCcCg. Let us choose Z E a. The assumption of non-triviality implies that the fibration of G / H induced by z + J s has H + as fibre. Consequently the Lie algebra z + J s is non-abelian mad we m a y wlog assume [JZ, Z] = Z. From [ J r , Z] = Z in t u r n it follows t h a t J Z ~ n , hence n is not J-stable. Therefore the space V = n N J n is a two-dimensional vector subspace of n . Next choo6e W E n ' rl J c . This is possible because dimlt(c) = 5 and Z ~ J c . Now let us investigate J W . There are three possibilities
(1) (2) (3)
sw c c \ n s w c ,, \ yw e a
Now let us s t a r t with the first possibility (which turns out to be impossible)
119 The subcase J W E c \ n
L e m m a 3. Under the above assumptions J W E n . Proof. Assume to the contrary thltt J W ~ n, i.e. J n ' f3 n = {0}. Choose X E a r J n . Since Z is central in < J W , W , J X , X > l the integrability conditions for J yield
[Jz, J w ] = J[Jz, w ] and [JZ, JX] = J [ J Z , X]. This implies [JZ, J W ] = 0 = [J:Z,W] because J n ' N n = {0}. Moreover [JZ, X] E a N J n = < X >R. Hence J X and X are contained in an a-weight space for the ad(JZ)-action. Assume that a = 0. Then the vector subspace < JZ, J W , J X , W , X > l is the 0-weight space and < Z >R the 1-weight space, which contradicts Z E n'. Hence a ~ 0. But now < JZ, J W , W >R equals the 0-weight space, both J X and X are contained in the a-weight space and Z is central in n . This contradicts W E n ~. [3 The subcase JW E n \ a For the second possibility we obtain Lemma
4. A s s u m e that J W E n but J W f~ a. Then there ezists a basis
< X, Z, W , J X , J Z , J W > such that the following are all non-trivial commutator relations between the basis vectors:
[JZ, zl = z [JZ, J W ] = 1 j W
[JZ, w ] = 1 [sx, x] = x 1
[JX, Wl =
w
[JX, JW] _- _ l j w [JW, X] = W [JW, w] = + Z
120
Proof. (i) Consider the real two-dimensional subspace (c n J c ) A a. It contains the Choose a transversal non-zero element X in this space such that X is contained in some ad(JZ)-weightspace. Then J X is contained in the same weight space. Now n = < J W , X, W , Z >It, hence X ~ g~. Therefore X and J X are contained in the 0-weight space of ad(JZ). (ii) Observe that n = < J W >It ~ a , a = < X >It ~ n ' , n ' = < W , Z >It and < Z >It= n (~). This implies in particular that [ J W , W ] is a real non-zero multiple of Z. Since Z is a 1-eigenvector for ad(JZ), it follows that W and J W are contained in a ~-weight space for ad(JZ). Hence
ad(JZ)-eigenvector W .
V0 = < J Z , J X , X > I~ = < J W , W >It V1 ~-< J Z >It 9
(iii) Next we show that all these weight spaces are actually eigenspaees. Observe that both X and W are eigenvectors, because ad(JZ) stabilizes a n V0 and a O 1~. From an integrability condition we obtain [JZ, J X ] =
J[JZ, X]
and
[JW, J W ] = J [ J Z , W]. Therefore J X and J W are also eigenvectors. (iv) Since [JW, W] is a non-zero multiple of Z and W is unique only up to real scalar multiplication, we assume that [ J W , W ] = + Z . Note that [ J W , X] E ~ , because J W E V89 and X E V0. Thus [ J W , X] is a multiple of W , which is non-zero because W E n ~. Hence wlog [ J W , X] = W . Since a = < X, W , Z >It is abelian this completes the determination of the structure of n . (v) Observe that V0 O g~ C 170 O n = < X >It. Therefore [JX, X] = ~ X with ~ E R . Furthermore W E 1~ n n' implies [JX, W ] E 1~ O n ~. Hence [JX, W ] = 7 W for some 7 E R . (vi) Now consider the integrability condition [JX, J W ] =
J[JX, W ]
+ J [ X , J W ] -t- IX, W ] .
=~/W
=-/W
=0
It follows that [JX, J W ] = ( 7 - 1 ) J W . Recall that J X E c, hence [JX, Z] = 0. Now let Xx denote the A-eigenspace for ad(JX). Then Xo = < J Z , J X , Z >It
Xa =
It
X~ = < W >It X ~ - t = < J W >It .
121 (vii) Recall that [X~, X - j] C X~+j. Hence [JW, W] = 4-Z implies that " r + 7 - 1 = 0, i.e. 7 --89 and 7 - 1 = - 8 9 Finally from [JW, X] - W it follows that - 89+ a - 89and therefore a = 1. This completes the proof, a In particular we proved that the J-structure on G is isomorphic to one of two uniquely determined structures. Therefore we can proceed by explicitly giving two different such J-structures and then conclude that (G, J) is isomorphic to one of thenL Let p : G L 2 ( R ) - . G L s ( R ) denote the usual irreducible representation given by
ed
Let B denote the Borel subgroup of upper triangular matrices. Then it is easy to check that the following is an isomorphism of g and b Kp (R s) as real Lie algebras: r
jz+
jx+
JW+6X+
W+
Z
+ 8)
,
9
Moreover for (z, w, z) = (i, 0,-t-i) the J-structures coincide. Hence X - (G, J) is biholomorphic to an open orbit of BK p(R a, +) in C s . The G = Btr (R 3, +)-action induces a linear map ~b from the Lie algebra g into the tangent space at a given point (z, w, z) G C~ 9 To determine the open G-orbit~ we have to analyse where the g-vector fields span the whole tangent space. Let A=
a+d
Then
I ~(A) =
2b 2d
,
G g.
2az + bw -k e (. + d)w + 2bz + f ] e T(.,.,,)(C~). 2dz + g
/
Since e, f, g E R, it is clear that the real tangent vectors are spanned. The question is whether all imaginary tangent vectors are spanned abso. Observe that a
det
(2/mz Imw
b
d
Imw 21mz
Imw
1
=
21mz(41mzlmz-(Imw)
~)
21mz
Hence (z, w, z) lies in an open orbit of B Kp(R a, +) iff Im z ~ 0 and 4 I m z I m z (Ira w) 2 ~ 0. As we have proved above under our present assumptions the manifold G / H is biholomorphic to the open orbit through either (i, 0, i) or (i, 0, - i ) . This suffices to deduce the following:
122
LemmA 5. Assume in addition to the basic assumptions that d i m l t ( N ) = 4. Assume further d i m a ( N ' ) = 2 and J a n a = {0}, where a denotes the centralizer of n' in n. Then G / H is bihoiomorphic to one of the two manifolds f~+, 12given by n + = {(z, w, z) I I m z > O, 4 I,n z I,,, z > (Ira ,,,)~}
f~- = { ( z , w , z ) l l m z > 0 , 4 I m z l m z < (1row) 2} R e m a r k . The manifold t2+ is a bounded homogeneous domain and moreover is the hermitian symmetric domain of type BDI(3, 2) = C I ( 2 ) . The subcase JW E a
L e m m A 6. Assume J W E a. Then G / H is biholomorphic to
{(x, w, z) ~ c ~ I Ira(x) < 0 and Im(z - ~w) < 0}. Remark.
This is the same manifold as in Lemma 11.3.2.
Proof. (i) Observe that J n n n = < J W , W >it and that [JZ, Y] = J[JZ, Y] E g~ n Jg~ c n n J n for all Y E c n J c . Thus ad(JZ) maps the four-dimensional space J c n c into the two-dimensional space J n n n . Hence the 0-weight space of ad(JZ) in J c n c is at least two-dimensional and has positive-dimensional intersection with n , because dimit(n n J c ) = 3. Hence we may choose a non-zero element X in this intersection. Then X E n , J X E c and both X and J X are contained in the 0-weight space. (ii) Note that ad(JZ) stabilizes h e n (a n J a ) = < W > 1 . Hence W and J W are eigenvectors. Since Z E nO) it follows that IX, W ] = ~Z with ~ # 0. Therefore < J W , W, Z > a is the 1-weight space and < J Z , J X , X >it the 0weight space. (iii) Since < X >It equals the intersection of n and the 0-weight space, we obtain [JX, X] = "yX for 7 E ~ . Furthermore, since [JZ, X] -'- 0 , it follows that [ J X , JZ] - 0. (iv) Consider now the integrability condition
[ J x , J w ] = J [ ~ x , w ] + J[X, JW] + [x, w ] . The left side of this equation is contained in a = < J W , W , Z > a and the right side in J n ~+ z --< J W , J Z , Z > a . Hence IX, W ] = ~Z [JX, W ] = ~ W + CZ IX, J W ] = ~ W - Cz [ J x , J w ] = (7 + ~ ) J w + ~z.
123 (v) Consider the Jacobi equation 0 = [SX, IX, SWll + [X, [SW, JXll + [ J W , [JX, Xl] = [ s x , ~ w - (z] + Ix, - ( , + ~ ) s w - ~Zl + [ s w , 7 x ] = ~(, - ~ - ~ - 7)w + r + ~ + ~ + 7)z Observe that p # O. It follows that p = - 7 and ((p + y/) = O. (vi) We exploit another Jacobi-equation 0 = [JX, [X, Wl] + IX, [W, JXll + [W, [JX, Xl] = [ J x , ~Zl + IX, - ~ w - Cz] + [ w , 7x] = - ~ ( 7 + ,~)z Since ~ ~ 0 it follows that 7 = - ~ = - P . Hence ((~ + ~) = 2r = 0 imphes r (vii) At this point we list all non-trivial commutator relations between our basis vectors
[jz, z] = z
[sx, s w ] = 2 ~ s w + ~z
IX, J W l = ~ w
[Jz, w ] = w
[Jx, w] = ~w
[x, w] = ~z
[Jz, SWl = s w
[JX, Xl = - ~ , x
(viii) By a coordinate change of type (z, w, z) ~-* (tz, sw, z) with s, t E ni we may wlog assume that p -- ~ = 1. (ix) One can verify that the following realizes g as a subalgebra of F o ( C s , TC s) where the J-structure coincides with the induced one at eH - (:7"~' ' 0, - 0 sz = -(z ~
cO
Oz + W-~w) cO cO
sx = z~-
JW - zs
w=-_ Ow
+ z~0
10 20z
1 cO
X ' - - -
v,~cOz
1 0 W = ~---(2~--~w+ 2z0~)
0 Oz (x) In this realization g generates the whole tangent space iff Ira(z) ~ 0 and Z"---
Ira(z) - 2 Re(w) Ira(z) # O, This completes the proof, since
~,~(~) - 2 ~ ( w ) rm(~) = I.~(~ + 2 . o I . , ( ~ ) ) = i , , ( ( ~ + , ~ ) - ~,o). I"1
124 4. T h e c a s e
dimlt(N) = 5
Here we assume t h a t diml(N) = 5. Hence N is of codimension one in G and therefore the vector subspace V = J n n n is four-dimensional. Consider now any fibration
G/H
1
'---, G/[-I
1
,--. Oil with dime(I/H) = 1. By non-triviality I / H is biholomorphic to H + . Therefore I N N is real one-dimensional. Thus N acts transitively on G/I which implies t h a t G/I = G / i . Furthermore N acts transitively and freely on the G/I
boundary. We know from L e m m a 11.1.5 that the dimension of the center n cannot exceed one. Now 1 _< diml(N') it, 2. n t = < Z > l t , 3. [W, J W ] = + Z , IX, J X ] = + / - Z and all other c o m m u t a t o r s of base vectors are zero. These CR-hypersurfaces m a y be realized in P3 as
s = {[xo : . . . : x3] 9 $1 xo # o} with
= {[:o : . . . : ~3] 9 P3 1 S~n(~o~l) i I ~ l ~ +7- Ix312 = o}
125 where the N-action is given by (a, ~ , 0 :
[~o : . . . : ~ ] ~--
[z0
: zx + a z o :F 2 i z u i q:
ilslUz0 -/+ 2iz3t-/+ iltlUzo
: zu + s=o +
tz0]
for a E R, s,t E C. Thus we have a N-equivariant CR-homomorphism r : S --* P.~. But this map r is not only N-equivariant. This particular quadric in P8 has the amazing property that every CR-automorphism of every open subset of S extends to a holomorphic automorphism of Pa (This was proven by Tanaka in [Tan]). Hence r : S --, P3 is G-equivariant. Our hope is now to find open G-orbits in Pa and to prove that one of them is biholomorphic to the open G-orbit G / H in G / H . B y Prop. 1.12(b)of Andreotti-Fredericks ([AF]) there exists s unique extension ~b of r to a biholomorphic map from an open neighbourhood U of S in G / H onto an open neighbourhood V of S in Ps. It is clear that ~ is locally equivariant i.e.
~(9(p)) = r if g and p are contained in a sufficiently small neighbourhood of e resp. S, because ~b is equivariant and S analytically Zariski-dense in U. Now for every point p E G//?/ there exists a g E G such that g(p) E U. Thus we can define a map ~ : G//?/---, P3 by
~(p) = r with 9p E G such that gp(p) E U. This is well-defined since ~b is locally equivariant. Observe that if p E G//~/ and gp E G such that 9p(P) E U then there exists an open neighbourhood W(p) of p such that g~(W) C U. Therefore it is clear that r is a holomorphic map. Thus it is clear that r : (~//7/--, Pa is a holomorphic G-equivariant map with discrete fibres. Hence G / H is a covering over an open G-orbit in Pa. Furthermore ~ is actually biholomorphic, because R acts in Pa as a unipotent group. This implies ~(r = r is biholomorphic to C"e . The next step is to determine the open G-orbits in Pa. Observe that
= {[~o: ~ :
~ : ~ 1 ~ P~ I~o # o)
is an open R-orbit. It is clear that r equals this N-orbit. Now there are exactly three G-orbits in (~//~. Therefore the open G-orbits are exactly the connected components of the complement of S in (~/t/. Hence we obtain the following L e m m a 1. Assume in addition to the basic assumptions that dima(N) = 5 and dimlt(N') = 1. Then G / H is biholomorphic to one of the following
f h = {[~o : ... : m3] ~ ]Ps I=o # o and Xm(~o=,) - Izul u -1=81 u > O} n2 = {[zo : . . . : ~s] ~ Ps I mo # o and Wn~(*oxx) + Iz212 - I~sl 2 > O} f h = {[xo : . . . : =a] E ~a I xo # 0 and Xm(~ozt) + Imulu + 1=31u > O)
126
F r o m this we deduce the following
P r o p l m i t i o n 2. Assume in addition to the basic assumptions that dimit(N) = 5
and dimit(N') = 1. Then X = G / H is biholomorphic to one of the following
(i) = {(x, w, z) 9 C e I I~1s + Iwl s + Izl 2 < 1}
_~ {(~, ~, z) 9 c e I Im z + I~1 s + Izl s > 1} (iO P3\ (~uL)
_~ { ( ~ , ~ , z ) 9 c 8 I z m ~ + I,,,Is + Izl s > 1}
(iiO
A = {(x,w,z) eCe l I m z - R e w I m z
>O}
Proof. This follows from the preceding Lelnma by appropriate changes of coordinates. (i) Note that f~l -- {[=o: Xl: x2: z3] [ Ixo[ s > Ix1[ s + Iz21~ + Iz3l s} because Im(~oxl)
= ~ l x o - i ~ l l s - I~o + i~llS).
Hence ~'~1~ ~Lt aJld consequently
n3 ~_ ~3 \ ( ~ u L). (ii) It is clear that ns
_~ {(~, w,
z) 9 c 3 I i m z + Iwl s - I=1 s > o}
Now t2s _~ A follows from I~1 s - Ixl s = R , ( ( ~
+
~). (~
-
~)).
[3 T h e s u b e a s e dimit(N') = 2 Assume d i m e ( N 0 = 2. Now N(2) = Z because dimit(N(S)) it= n ' + J n ' is ad(JZ)-stable due to Cot. 11.1.4. Hence there exists a read two-dimensionad J-stable vector subspace V of n which is invatiant under the ad(JZ)-a~tion and transversal to n ' + J n ' . Let us denote the centralizer of n ' in n by c. Obviously c is read four-dimensionaL Hence there is an one-dimensionad intersection of V and c. Choose a nonzero vector X E V f3 c. Consider the integrability condition [ J X , J W ] = J[JX, W ] + J [ X , J W ] + IX, W].
127 Since n' N J n ' = (0}, it follows that and
[JX, ~'W] = [X, W] = 0 [IX, W] = - I X , 1 w ] .
Furthermore [dX, W] = - [ X , dW] G nO) = < Z >R because W E n'. Let us now again consider the weight space decomposition of a d ( J Z ) . For some /~, ~ G lit we obtain g = Vo + V1 + V~ + g~ with V0 = < J Z >1 VI ---~< Z > l l V~ = < W , J W >~t Vx = < X, d X > l Hereby every V~ denotes a vector subspace of the p-weight space. (Sulmpace because a priori some of these four numbers may coincide). Now W G n' actually implies that W E [VA,VA]. Hence 2A = p. Moreover Z G n (~) implies that either Z E [VA,Vu] or [ J W , W] # 0. Consider now the Jacobi equality
0 = [ J w , [Jx, Xl] + [Jx, [x, JWl] + (x, [ J w , ~'Xl] = [ J w , ,,w] + 0 + 0. To fulfill this necessarily [JW, W] must equal zero. Thus A + p = 1. Hence Let us n o w list all non-trivial c o m m u t a t o r relations in ,~:
[Jx, x] = [Jx, Wl = -IX, g w ] = ~z Next consider a base change by scalar multiplication of X (reap. W ) with a non-zero real number s (reap. t). Observe that
[sJX, tW] = - [ s x , t J w ] = 7 a z [sdX, sX] = ( ~ W . Hence by appropriately choosing st and t we may wlog assume that ~ = 7 = 1. Thus the structure of the Lie algebra g is completely determined. L e m m a 3. This Lie algebra can be realized as a stJbalgebra of Fo(C3,TCe) in
128
the following way: JZ-
zO 30z
2wO
0 ZOz
3 #w
0 Oz 0 W-', Ow Z ~
JW=i
O
0
0 Oz J X = i ~0- x ~ - ~0-
0
The J-strlJctsre of g coincides with the induced one at e H = (0, O, - i ) . Proof. One can verify this by explicitly calculating the commutator relations of these vector fields on C"e . 13 L e m n m 4. The boundary between the two open G-orbits is defined by
f ( x , w, z) = - I m z - Ira( ) - I m z(-glm w - I m z) Proof. The algebra g realized on C a as described above spans the whole vector space at a given point iff it spans i ~ . L e m m a 5. Under these assumptions G / H is bihoiomorphic to
{(~, w, z) I im(z + ~w) < 0}. Proof. This can be achieved by a change of coordinates because w
Ira(z) + Ira(3 ) + Ira z(2Ira(w) - I r a ( z ) = I m ( z + ~ + W
= 1,,,((~ +-~ 4
z-
~2
2i--~-(3w - z))
Z(X
--
X ~
U)
"3~w)) + ~(:_~i_))"
13
T h e s u b c a s e diml(N') = 3
L e n n n a 6. There does not e~;ist a nontriviai case with dimlt(N') = 3.
129
Proof. (i) There exist vectors A , B E n such that n = < A , B >a + n ' . Hence n' = a + n (2) and consequently dimlt(N(2)) = 2. Since N(2) is not central we obtain dimlt(N(3)) = 1. Thus N(3) = Z. (ii) Since ~r, cannot ~ct transitively on N / H f3 N (see Cor. 10.2.2), it follows that d i m a ( J n ' + n ' ) = 4. Thus dima(n') = 3 and J s ~ n imply n' = 9 (n' f3 J n ' ) . Let us choose W E n (2) f3 J n ' and Z E s. Then we obtain n(2) = < W , Z >m and n' = < Z, W , J W >m. (iii) Observe that dimm(Jn f3 n) = 4. Hence we may choose X E n f3 J n such that n = < X, JX, W , J W , Z > a . Next consider the integrability condition
(*)
[JX, J W ] = J[JX, W] + J[X, J W ] + IX, W]
Observe that [JX, J W ] e < W , Z >a J[JX, W] E< JT. >a J[X, J W ] E< J W , J Z >~
[x,w] ~< z >. Therefore (*) implies that [JX, JW], [X, J W ] E< Z > a . In, n~ C R in contradiction to diraa(n(2)) = 2. {3
But this implies
5. Classification Here we want to summarize the results of this chapter. P r o p o s i t i o n 1. Let X = G / H be a three-dimensional soivmanifoid with dirnit(G) = 6. Furthermore ass,me that the anticanonical fbration has discrete fibres, that the algebra of g-vector fields on X is a totally real subalgebra
of vo(x~ ToX). Then the manifold X is biholomorphic to one of the following (i) a complex homogeneous manifold, i.e. a quotient of complez Lie groups, (ii) a direct product of lower-dimensional homogeneous soinmanifolds, (iii) n l = {(x, to, z) e C ~ I Xra z - Xra(~w) < 0} __ {(x, w, z) e C ~ I I r a z + Iwl 2 - Ixl 2 < 0
a~ = {(=, w, z) ~ c ~ I Ira ~ > o and Ira
z
-
Ira(~w) < 0}
(~) ~z8 = ~
= {(=, w, z) e ~
I Ira z + I~12 + Izl = < o}
~4 = Ps \ (L u ~ ) = ~ = {(=, ~, =) ~ ~
I Zm z + Iwl 2 + I=12 > O}
(~iO ~+ = { ( z , w , z ) ~ c ~ I Ira= > 0
and41ra=Iraz > (Ira w) 2}
130
(This is a hermitian symmetric domain). n, = {(z,w,z) E Cs I lmz > 0
and 4 I m z l m z < ( I m w ) 2}
Proof. (i) First let us assume t h a t the anticanonical fibration is injective. (ii) F r o m L e m m a 10.2.8 we know t h a t dims(N) >_3. For the case dimn(N) = 3 we proved t h a t X is biholomorphic to a direct product ( L e m m a 11.2.1). F r o m L e m m a 10.5.1 it follows t h a t if N is abelian and dims(N) > 3, then X is a direct product or complex-homogeneous. (iii) Hence there are two cases for dimz(N) = 4: Either dims(N') = 1 or dimm(N') = 2. If dimm(N') = 1 then X is a direct product or f/2, as we proved in L e m m a 11.3.2. For dimlt(N') = 2 we obtained t h a t X is either trivial (a direct product or complex-homogeneous) or one of the following: ft2, f~5, ft6. (iv) For dimm(N) = 5 we have the subcases 1 _< dimtt(N') < 3. If dimn(N') = 1, then N is a Heisenberg group and X is biholomorphic to f~l, Gs, fl4For dims(N 0 = 2 we deduced t h a t if X is non-trivial then X _'2 f~l- For dimlt(N I) = 3 it follows t h a t X is trivial, i.e. complex-homogeneous or a direct product. (v) If direr(N) = 6, then N is nilpotent and thus X is complex-homogeneous due to L e m m a 9.4.4 (vi) Now let us consider the case t h a t the G-anticanonicM fibration G/H ---, G/I is only almost injective, i.e. has discrete fibres. T h e n the G-anticanonicM fibration of G/I is injective. Hence G/I is one of the manifolds listed in the proposition, and G/H is a covering of one of the listed manifolds. But these manifolds are Ml simply-connected. Hence G/H is contained in the above list. [3
131
Chapter The
case
G solvable
I~ and
d i m R ( G ) ~> 6
1. T h e basic a s s u m p t i o n s In this section we consider the case that X = G / H is a solvmanifold and G of minimal dimension, but dimm(G) > dima(X). More precisely we assume the following B a s i c A s s u m p t i o n s . (i) X = G / H is a homogeneous complex manifold. (it) The group G is a solvable simply-connected real Lie group which is of minimal dimension among all real Lie groups acting holomozphically and transitively on X . Furthermore dima(G) > dima(X). (ill) X is not a direct product of lower-dimensional homogeneous manifolds and no complex Lie group acts transitively on X . (iv) The anticanonical fibration of G / H is in~ective and the representation of the Lie algebra g in the algebra of holomorphic vector fields on X = G / H is totally real. Since the anticanonical fibration is injective it follows that there is a complexification G / H ~ G / H . By Lemma 10.3.3 the above assumptions imply that any Lie subgroup I in G is closed, if i f / C I and ] / / ~ is connected. Hence there axe fibrations
G/H
i l G/J
~
G/~I
~
1 1 GIJ.
GI#
air
Due to Lemma 10.6.1 it follows from the basic assumptions that for any such fibration I / H ~_ H + , J / I ~_ C and J / H ~- C x H + . From Lenmla 10.3.1 we know that d i m c ( G ' / ( ~ ' N / ~ ) ) ~_ 2. Thus we may assume that there are fibrations as above with J C G'H. Furthermore either ( ~ / i f / _ C 3 or (~//?/~_ C2 x C*. In the latter case we first replace G / H by an open G-orbit in the universal cover ~//~/0 in order to obtain G/I/___ C 3 . (Later it turns out that this case does not
occur). Our strategy for the classificationis n o w to realize the Lie algebra g as a concrete Lie algebra of holomorphic vector fieldson G / ~ / ~ C e . First we define A := ( N n j)0 where N is the nilra~licM of G. The analysis of the realization
of this algebra on G / f / _ C 3 and the representation of g on G / J ~- C axe key points in the classification together with the study of the ad(H)-action on g for appropriately chosen H E g \ gl. We will start with a detailed discussion of the structure of A. In particular we show that A is an abelian normal Lie subgroup of G with real threedimensional orbits. Next we will distinguish the different possibilities for the representation of G on G / J . If G / J ~_ H + , then after a change of coordinates
132 g = < ~--~,z ~ >it. Otherwise either G' C A and consequently g is represented as < ~ > c or g is represented as < aza-az' a 8 c_' _ i~az _ > l for some a E C*. It is not possible, that g is represented as < zb-~,~ > c , because if the G'-orbits are one-codimensional in G / H then by the assumption of minimality (cf. Lemma 8.2.6), G' has to be of codimension one in G. Finally it will turn out that the only possible representation of g on G / J _~ C is < zb~,b-~z,is-~ > 1 . 2. T h e s t r u c t u r e o f A = (N f3 j)0
In this paragraph we establish some basic properties of the Lie subgroup A := (N n j)0. Let L denote the ineffectivityof the G-action on G/I. W e know (cf.L e m m a 8.2.1) that L acts almost effectively on I / H ~_ H + . Thus, since L C G is solvable, dims(L) _< 2. Next, we recall that due to Lemma 8.2.5 d i m l ( L ) c + < ~'~w'~z >o(=) + < ~ z 0
0
0
a
0
>c
a
0
F u r t h e r m o r e we can achieve that
gn
a a >,, o(GIH) C< ~'z
that the point e l is d noted (x, w) = (0, 0) and that either
g or
g.
Proof. (i) G / J equals either C or H + . In both cases the intersection of isotropy and niiradical is normal in G, thus A is normal in G. Now J / H ~_ C x H + implies that there are fibrations J / H C , j / K ~_ H + and j / H H + ) j / I ~_ C where J / H --, J / K is the bounded holomorphic reduction of J / H . Hence from A C N : it follows that A' is contained in K n I = H , and therefore ineffective on G / H , because A is normal in G. Thus A is abelian. (ii) Note that H ~ = (G ~f3 H) ~ by Lemma 8.2.6 and (G ~N H) ~ C (N f3 j)0 = A. Since d i m l ( H ) > 0 by the basic assumptions this implies d i m l ( A f3 H) > O. Due to Lemma 8.2.3 it follows that the A-orbits in G / H are not totally real. Observe that I / H ~-- H + implies I'H = ( A n I ) H , hence the A N / - o r b i t s
133 in I / H are real one-dimensional. Thus the A-orbits in J/H are at most real three-dimensional. Moreover it is clear that a two-dimensional A-orbit in J/H would be totally real. Hence the A-orbits in J/H are real three-dimensional. (iii) Choose coordinate functions (z, w, z) on G / H _ Cs such that z is a coordinate for Cr/J and (z, ta) are coordinates for G/]. (iv) Let io be a one-dimensional complex subalgebra of the center of ~ , such that i0 is normal in I~ and ix be a complex two-dimensional subalgebra of such that [ix,g~ C 10 a n d [ix,l~ C ix. Without loss of generality we m a y assume I = ZoH and ZxH C Y. Furthermore we m a y assume that Zx C (~' since dima(A) ~_ 3 implies that dirnc(G') >__3. (v) If dirna(L) = 1 we require that i0 = 1 + ft. This is possible, because any one-dimensional ideal is contained in the center of the nihradical. (vi) Assume that Z0 fl G is discrete. Then choose a vector W0 E i0 and set Wx = R e ( W o ) , where the real part is defined with respect to i~ = g ~ ig. Observe t h a t [Wx,g] C Re(i0) C < W 0 , W x > c Since dima Re(i0) = 2 > dima(l) and W0 is ineffective on G / i , W l cannot be ineffective on G / i . 0 Hence we obtain W0 = h(z, w)#~; and W l = 9(z)~-~ + f(z, w, z ~ for some f E O(z, to, z), h E O*(z, w), 9 E O*(z), where h and g are nowhere vanishing, because both < W0 ~ c and < W 0 , W l > c are normal in !~. An appropriate change of coordinates yield h ---- 1 and O -- 1. (vii) Let us now consider the case where io N g ~ {0}. Here we choose W0 E /-0Ng and W l , W 2 E Re(ix) such that < W 0 , W l , W 2 > a = R e ( i l ) 4 g and W l + s'W2 E ix. Now diml(L) ~ 1 implies that ezp(Re(ix)) has real threedimensional orbits in G/H. Hence
Wo = h(z,w)-~-cgzsad W , = fl(z)~---~r+ gl(z,w,z)~---z W2
=
I~(z
o
+ 92(=, w, z ) ~z
with h,9i and /~ holomorphic, h , / i nowhere vanishing and fl(z) and f2(z) real linear independent for all = E G/J. With a first change of coordinates we can achieve, that h - 1, i.e. W0 = b~," Now [ix, i0] = {0) implies that the functions gi actually do not depend on z. Another change of coordinates yields 8 i.e. in particular /1 - 1, W l _- ~-~, (viii) N o w / 2 - 1 is a holomorphic function on G / J - C such that /2(z) is not real for all z in an open G-orbit, i.e./2 - 1 maps C\]~ onto C \ R . Furthermore f2 - 1 is on C a nowhere vanishing function since ~ W l + s'Vr W0 ~ r is an ideal in g. These both facts are enough to prove t h a t / 2 - 1 is constant, because otherwise /2 - 1 would be surjective on C with the exception of at most one point, and thus the connectivity of f2(]~) = f2(C) would show t h a t / ~ - 1 must vanish somewhere. (*) Alternative proof: f2 - 1 equals ezp o g for some holomorphic function g, where f2(C \ ~ ) C C \ R implies R + 2~ri7L C g(]~). Now connectivity of g(R) and g(C) proves that by A1 -- H + and Lionville 9 is constant. Thus /2 is
134 constant, i.e. wlog. W~ = i b ~ + g(x, w)a2/, where [Wl, W21 0 implies that g depends only on z. (ix) Now b~z is central in g' and [b~g,g'] C < b~z > c . Hence any X 9 g' equals ofa-~ + f ( z $)~w+ (g(z) + ~tv8)~-~zfor some a, ~ 9 C and f, g 9 O(C). Furthermore ow 9 a and the eommutativity of a imply that any X 9 a equals f(z)b~+g(z)-~z for some f, g 9 O(C). Using the fact, that < ~ > c is normal in g, and [0%' g] C < ~ >C + < O >O(,) for any X 9 g we are now able to obtain "
x - (~ + ~
+ (f(x) + ~,o
-
+ (o(x) + h(x, w) + I , z ~ , ,
Lenuna 2. The group A contains the connected component H ~ of the isotropw group H. Proof. Observe that H ~ C G' by the minimality amumption (see Lemma 8.2.6) and therefore
H ~ C (H n G') ~ C (J n G') ~ c A [3 C o r o l l a r y 3. Under the basic asswmptions it foiio~ that dima(A) >_4. 3. T h e case g _~< 0-~ > c o n G / J Let 0 : g ---, a u t ( C ) denote the representation of g in the algebra of holomorphic vector fields on G / J ___C. In this paragraph we consider the case that 0(g) _~< ~ > c , i.e. g is represented on G / J as < b~s >C and consequently G / J = G / J _~ C. We will final]y see that this case does not occur, i.e. we will show that the assumption O(g) = < o > r leads to a contradiction. We start with L e m m - 1. There exist ~ 9 R*, fo,go 9 O(z) such that
0 0 0 0 v = ~ T ~ + ~Z~z + f~ 0)0-~ + ~176 9 g" Proof. (i) Observe that I / H ~_ H + implies that I' does not act transitively on I / H . Hence we may choose a vector field V in i \ (i' + h). Then ~b(V) = 0, because O(g) = < ~ > c implies t~at J is normal in G. Therefore
g =
,u~-~w+7Z~z0 + ~U~z0 + f0(z0)Ow + ,o(z0)~z
for ~,7, r / 9 C and f0,g0 9 O(z). Note that I / H ~_ H + even implies 7 9 R . (ii) Next we will show t h a t / ~ = 7- For this purpose assume the contrary. Then 0 ~ >o(s) is the direct sum of the /~-eigenspace < (7 - / ~ ) ~ - ~ < b-~, >o(s) of the ad(V)-action and the 7-eigenspace < o > o ( , ) . Changing coordinates we are able to achieve 17 = 0. Now the stability of a under ad(V) implies
135 that a = a0 ~ al with a0 C < ~; >o(z) and al C < a-~ > o ( , ) . Recall that the A-orbits are three-dimensionai and that a0 can be at most one-dimensional. Hence a0 = < ~ > l t . Observe that the G-orbit through a given point p ~ G / / t is open if and only if V[~ is contained in the span of the a-vector fields at p. Therefore the open orbits of G are just Ira(z) < 0 and Ira(z) > 0 which contradicts our basic assumption (iii) that G/H is not a direct product. (iii) Next we choose Y ~ h C a obtaining Y = Y~(z 0)~w+ 0 2 ( z 0 ~z with Y2(0) = 0 = g2(0). Now I / H ~- H + implies that Y vanishes on I / H . Since the ineffectivity L of the G-action on G / I must act almost effectively on the fibre I / H , this implies Y r 1, i.e. f2 ~ 0. Now
g ~ Iv, Y] = # Y - ~f~(, o)~,
O, E g for some f2 ~ const. This shows that because # = "r- Hence vlf2(z~# r/=O. o Lemmm 2. There ezist A, B E g sach that
a B=i 0
0
0
0
with ~., ~, 9 ,3t, ~,, ~ e C and IV, A] = IV, B] = 0.
Proof. (i) By Lemma 12.2.1 it is clear that
_ o__+ A
-
o.
B=i~z+.
o
o
0
8
o
o
,,w
0
f,(z ~)~w+
for some A~,/J~,~ E C and fi,g~ E O(z). Note that we can substitute A and B by the sum of A reap. B and a real multiple of V . Hence we may wlog assume that A~ E iR. (ii) Note that r = 0 and that ad(V) stabilizes ketch. Therefore it is possible to chooso A, B E g such that ~b(X) = o_, ~b(B) = i ~~ and that furthermore X and B are contained in the 0-weight space V0 of ad(V), i.e. the space
go = { x 9 g 13~v : od(V)N(X) = 0}. Observe that V0 63 a = {0} and that V0 is a Lie subalgebra of g. Since the assumption ~(g) = < b~ > c implies g' C a it follows that V0 63 g is an abelian Lie algebra. Hence IV, A] = IV, B] = 0. An explicit calculation shows that
136 these equalities imply that all the functions f , , f~, g4 and g~ vanish identically. [] Due to the minimality assumption it is cleat that
g = < V, A, B >It ~a. P r o p o s i t i o n 3. The asstmption ~b(g) _ < b~m>r leads to a contradiction.
Proof. Wlog we m a y assume that there is no ineffectivity on G / I , because otherwise g would contain a subalgebra go = < V , A , B , ~ q - . . , i ~ + ..,~ >l such that the corresponding Lie group Go acts transitively on G / H with discrete isotropy. Therefore g 9 b~-w,sb-o~ + f0(z a)~. Furthermore
[A, ~
0
= -~.b-j~ - ,~4~ ~ g
and
yield (remember A4 E z~) that a,,fo(z) = i04. Hence 04 r 0 because a N ia = {0} implies f0 ~ 0. Finally A4 + ~4 = 0. Obviously the function f0is now constant. Similar calculations for B yield a6f0(z) = iqb and ,%b +/J6 = 0. Observe that, since gO < b~ >O(a/H)= {0} and a is abelian, if )t4 = )t~ = 0 then iad(A)and ad(B) restricted to a would coincide, if A, = a~ = 0. Now g f3 h ~ {0} implies that there exists some z~-~ o + f l ( z ) ~ E g. Hence Aa = Ab = 0 would imply that a N ia is positive-dimensional. Thus we conclude that (Xa, A~) ~ (0, 0). Calculating three further relations (using -17a = iA4fo )
[A,[A, wd] = ~:(x~--j+ fl~)- ,A.~-j+ (f~' + z~.f~ - ,,7.~Oz, A
[A, [n, wd] = a . a , w ~ - (a, + i.~.)~--j + (i f~ 1 - (,n + i,74) + (a, + i a a ) f ~ , and These yield 2iAaf0 -- f l I + 2A4(f~ + if0)
(1)
ixj/0 = ill' + (~, + ix.)(fl + iIo
(2)
0 = --fi t + 2i~b(f~ + ifo)
(3)
Now from (1) and (3) it follows that f~ is constant, i.e. f~' = 0. Then (1) shows that f{ ~ 0 and (3) proves iA~fo = 0. Next we deduce from (2) that iAbfo = i)t,,ifo. Thus either f0 ~ 0 or A4 = ~b = 0, but both cases imply g N i g r {0} and therefore lead to a contradiction. []
137
G/J
4. T h e
8 z~'is 8 >m. Here we discuss the case that g is represented on G / J ~ C as < ~-is, We will finally see that this case is also impossible, i.e. the assumption
a
a
>1
leads to a contradiction of the basic assumptions. 0 z ~-~ a >1 imnliea that G/J "~ H + and in patThe m u m p t i o n ~(g) ___< Fi, . t i c u l ~ that J is normal in G . As usual we start by choosing a base for g .
Lemana 1. The Lie algebra g contains elements V, H, A smch that g = ( V, H, A >1 ~ , ~(H) = z ~ , ~(A) = ~ sad V E i \ i'. F , r t h e r m o r e lye can achieve that A E H-1 and V E Ho, where Hx denotes the weight space of A /or the ad(H)-action on g, i.e. H~ = {X E g I 3 N : (ad(H) - ~)N(X) = 0 ) . Proof. (i) Choose H, A, V E g such that ~b(H) = zb~, ~(A) = b~ and V E i \ i ~. Now V , A, H r a. Since h C a and dima(A/H ~ = 3 it follows that g = < V,H,A >1 ~a. (ii) Note that [za--~,b~-ffi]= - ~ . Hence wlog A E H - 1 . Furthermore a d ( H ) : ._~ gl, hence g = H0 + gt and we may assume V E H0. n Next we will express the vector fields H , A, V in the standard coordinates on (~//~ _~ C s . Due to Lemma 12.2.1 this yields
H = z~-~z+ ~hw-ff-~w+phZ~z+ ~l~U~z+ HA, A and
0__+ 0 = Oz ~ . u ~ + AA
V = ~
+~Z~z+~U~z+V~,
~ >o(a/:). Now we choose X E i' \ h , where p~ ~ 0 and HA,AA,VA E o ( a / ~ ) + a n < ~ > o ( a / ~ ) would imply the nonemptiness of a N / a by considering eigenspace decompositions we derive that Pa - ~h E Z and )~ =
138 pe E R . Now with ~h, ~ E R we can wlog replace H by H - , ~ V and thus achieve ~s = 0. Hence H can be written as
0
0
0
H = z~-~z+p~Z~z+ .~U~z+ Ha with ph G 7/.. If we now assume that g f3 o(u/j) is positive-dimensional, then we obtain the fact that we can define go as the sum of the 0-eigenspace of ad(H) and < A, b~, > a in order to construct a Lie subalgebra whose corresponding Lie subgroup Go C G acts transitively and with discrete isotropy on G / H . Then go would be a Lie algebra, because it is simply the sum of i (the ineffectivity on G / I ) and all a-eigenspaces for ad(H) with cK_< 0. There would exist an open orbit, because [H ,~-~J ~ = - ~ / h ~ , which implies that the 0-eigenspace contains some ~-~+ .., i~-~+ .., and it would be the whole G/H, because g o N j is normal in j . Thus we m a y conclude that gN < ~ > o ( u / ; ) = {0}. But in this case we may assume, that b-~,~s~" o + f0(z)~-fx E g and therefore [H, a-~ = - t I h ~ [H, i 0., + fo(z) O~ =
Eg
. 0 --,~h~
0
~ h / o T ~z E g
which shows that ~s = ph -- 0. Hence H = z ~ + H a . Now the ad(H)8 Znb-~ 8 >It and therefore coneigenspace for any n > 0 in a is exactly < xn b-/,, o 8 tained in the isotropy. Hence < b~, ~ > c f3a must be real three-dimensional, but this implies a f3 ia ~ {0} and yields therefore a contradiction. 5. g "--< b~r >r + < c~ze-~> a o n G / J
Now let ns ~ u m e that g is r e p r e ~ t e d on ( ~ / j u < ~ >c + < ~ z ~ >m with ~ E C*. Then choose A, B E g' such that A and B are represented on ~ / j as ~ and i ~ , and H in g such that H is r e p r i n t e d on G/~ as According to Lemma 8.2.6 n~nimality of g now implies
~.
g=a(~
< H,A,B
>It 9
We m a y assume that H E i. Written down in coordinates we obtain 0 0 0 0 H = ~ z ~z + ~w~-~w+ 7 z ~z + ""~z + HA A
0__+ 0 = 0z ~~ +AA
B =
+ ~'zz
o
+ BA
8 8 > o ( a / 1 ) . This yields H = Ho + H1 with Ho = with HA,AA, BA 6.< -~,-~ azb~, + ~ t t ~ + 7zi~;, and H , = , w ~ , + HA.
139 L e m n m 1. By a change of coordinate we may wiog assame that [H0, H1] = 0. Then the weight spaces of ad(H0) and ad(H) coincide. Proof. Let a and b be holomorphic functions. Then the change of coordinates given by z ~-* z , w ~ w + a(z), z ~-+ z + b(z) induces HA ~ HA + (~a(z) - a z a ' ( z ) ) a~-~+ ( T b ( z ) - azb'(z) + rla(z))~---~.
Since HA = gl(z a)--~, ~ + g 2 ( z ) ~ we can by this method achieve that HA = pz m O~ + v z " ~ w i t h f l - a m = 0 i f p ~ 0 andT-an=0ifv~0. Furthermore a change of coordinates of type (z, w, z) ~-* (z, w, z + Cw) induces Hx ~-* HI + C(7 -/~)w~-~. Thus we may achieve that ~ = 0 unless /~=7. An explicit calculation shows that after these changes of coordinates we obtain [H0, H1] = 0. Note that (ad(H1)) 3 (g) = 0. Now (ad(H) - )t)" (X) = 0 implies that (ad(H0) - )t) n+a (X) = 0. Hence the weight spaces of ad(H) and ad(H0) coincide, o Let us now discuss the weight spaces of ad(Ho). Observe that a 0 ad(H0) : ~z ~ -a0-zz -d(Ho) :
(an -
ad(Ho)
(an
:
-
0
ad(Ho) : U~z --+ (8 - 7 ) ~ z This immediately implies the following statement: L e n n n a 2. Let X - - a~%---4-bu~;+/(z)~-~+g(z)-~ with a , b E C a n d / , g G O(C). Assume that X is contained in a ]inite-dimensional ad(H)-stable vector space. Then both f and g are polynomials and moreover the ad(H)-weight space decomposition is already an ad(n0)-eigen space decomposition, i.e. (ad(H) - )t)n ( x ) = 0 implies ad(Ho)(X) = )tX.
Denote by V~ the )t-weight space of ad(H). Then we may assume that A and B are contained in V-a + V_~. Now observe that [A, f ( z ) - ~ + g(z)aa-;] = ] ' ( z ~ + ~ g ( z") - ~ wo where ~ is some polynomial. Together with [ B , / ( z ) ~ #)~z= i/'(z)~w + ... this implies that a contains some elements
From these informatious we will deduce some relations between a, 8, 7, 7.
140 L e n u n a 3. There ezist integral nwmbers k,p > 0 swch that
7= (p+t+DRea
Fwr~hermore a E R or k + p is odd. Moreover 7 E R and toe may tolog assume that ~ = 0
and ~ E g .
Proof. First note t h a t W 0 , W l E V_~+V_$. Since g was assumed to be totally real, it follows t h a t W l - i W 0 =/(z)-~/x is a non-zero element in V_# + V_$. Now [H0, z ' ~ - ( a k - 7 ) z k ~ implies t h a t there exists a n u m b e r k > 0 such t h a t a k -- 7 = - / ~ or a k - 7 = - ~ Next we choose a vector from i ' \ h obtaining V = f ( z o)--o~ ~ + g(z 0)~, with /(0) - 0 # g(0), i.e. V equals g(0 o)~, for z = 0. Now [H, V] = ( a z f ( z )
- ~f(z))~
+ ( a z g ' ( z ) - 7g(z) - ~ f ( z ) ) ~ z "
Thus [H, V] evaluated at z = 0 equals -'rg(0)b~z. Since I / H ~_ H + it follows that 7 E R. Our next step is to prove t h a t b~, E g . First we assume the contrary, i.e. g N = = {0}. T h e n we get / ( z o)~ + g(z)-o~* E g with f ( 0 ) = 0 ~ g(0) and f ~ 0. Since we m a y assume t h a t y(z)~-~w+ g(z o~-;~E V_-~ it follows t h a t there exists another integer n > 0 such t h a t - 7 = n a - / ~ E R . Now h a - 7 = --fl would lead to n = - k in contradiction to n > 0 and k > 0. Thus we have &k + ~ = 7 = - a n - / ~ , which implies t h a t a E JR, k = n and /~ = k a + 7But now by reconsidering the weight space decomposition of a d ( H ) we discover t h a t under these circumstances B - iA cannot be a nontrivial vector in 0
0
0
< ~ z >c + < Oz' O--w>o(a/.y) 9 Thus B = i A and we achieved a contradiction to g N ig = {0}. Hence o E a . Now bT, + ~--~w ~o fo(z)o,.,~.+o / l ( z ) ~ E a and a being totally real shows t h a t k > O. Since k ~ - 3' = -/~ or k a - 7 = -/~ and 7 E R it follows from k > 0 that /~ ~ 7. By the considerations in the proof of the preceding Lemana this implies t h a t wlog y? = O. F r o m the minimality assumption we deduce t h a t the vector subspace
0 W = =
fo(z) ~ and W l = i ~ fl(Z)~-~z cannot be a Lie subalgebra. Hence [X,B] or [H,X]+ReaX+Z aB or [H,B]+ReaB-I aXis not
with W o = ~ +
contained in the vector space W . Therefore t h a t m a - ~ equals - a or - & or - 2 Re a. either ( m + k ) a - 7 or m a + k ~ - 7- Thus t h a t (p + k ) a - 7 or p a + k 6 - 7 equals - a
there exists an integer m > On the other hand m a - / ~ there exists an integer p > or - & , since - 2 Re a + a
0 such equals 0 such = -&.
141
Hence T = (P + k + 1) Re ~ ~ E a t or k + p is odd. Obviously/r& + ~ = 7 in any case. Assume I m a if the sign -I- is actually a t&+#=7. 13 Lemnm
and (17 -I- k +/- 1) I m a = 0. T h e l~tter implies t h a t holds if ~ is real. We will prove now, t h a t it is true i~ O. T h e n p :t: k +/- 1 = 0 which is poesible only minus. Hence p a + k& = +/-1 which implies t h a t
4. L e t h be the integer gieen by the preceding l e m m a . Them k = 1.
Proof. O b s e r v e t h a t i~, ~ ~t because otherwise i a d ( A ) and a d ( B ) restricted to a would coincide and thus contradict a N / a = {0}. T h e following calculations are done modulo a because it is clear t h a t [H, A] = - Re ~ A - I m a B modulo a s a d we want to exploit this fact: [H,
A] ~
[H, B] -~ - i ~
0
_
-
a
0
rl~t~z-0
Rea ( - i ~ -
0
,.
. a
0
z.,
0
+
.
a
.
0
thus -
Re ~rh -Im
- Re ~t
~r#t = - r l a 6 k
+ I m ar h = --rlb&}
which m a y be reformulated as
9a(&k -
R e a ) - I m a g t = O,
r/t(&k - Re a ) + I m arl~ = O.
Now (.., t/,) ~ (0, O) since i.,, # Or. Hence 0=det~
/ a t - Rea Im a
-Ima ) "
&k - R e a
Thus (&k - R e a)2 + (Ira a ) 2 = 0
=~ &2k2 - 2(Re ~ ) ~ k + ~
= 0
:~ R e a ( k ~ - 2k + 1) = 0
and I m a(1 --/=~) = 0. F r o m b o t h these equations it follows t h a t either Re a = 0 = I m a or k = 1. But ~ was assumed to be a non-zero complex number. Hence k = 1. 13 Note t h a t this implies/3 = p Re a + a and 7 = (P + 2) Re a . Our next goal is to determine p. For this purpoee we need some upper hound for the dimension of g which will be deduced from the following result. Lemm,
5. Zf
+
e - the,
- i o) e at.
142
Proof. By calculating commutators with A and B we obtain [A, [A, )~za-~+ f2(z a)~z]] = (f~' - 2Aqe a)~z
(I)
[B, [B, Aza-~ + ~f2(za~z]] = ( - f ~ - 2iA~ a)~z
(2)
Hence -2Aqa - 2 i A r ~ = - 2 i A ( ~ t - i~,) E R. c3 C o r o l l a r y 6. d i m . ( . ) = 4 and d i m . ( g ) = 7. Thus
0 a =< W2,Wi,W0,~z >I
with W2 = Az~+lu(z)-~m. Observe that W2 6 V - , s e a since a - 3 = - v R e a . By the minimality assumption it follows that W2 is a multiple of one of the following elements of g: [A, B], [H, A] + Re a A + I m a B , [H, B] + Re a B I m a A . Since these three vectors are contained in the weight spaces for the weights a , & and 2 Re c~, it follows that p -- I or p -- 2. Moreover for p - I we obtain a 6 R . L e m m a 7. H I = 0, i.e. H = Ho = c~z~zz + (p Re a + a)
+ (p + 2) Re c~zaO-~-.
Proof. Firstly assume a E R . Then # = ( p + l ) a
and 7 = ( P + 2 ) a and = 0 implies HI = ,-,zp+X ~e -1. vz,+~. Notice that a, ~, 7 E R and all the ad(H0)-weight spaces of g are in fact eigen spaces for real eigen values. This implies that g is ad(H0)-stable. Hence g is also ad(Hx)-stable. Thus g contains the following commutators
[Hx, Ho]
[HI,A] = -p(p + l ) z ' ~ + (pga - v(p+ 2)) z ' + l ~ [HI, B] = - i p ( p + 1)z" ~
+ (P~lt - iv(p + 2)) Z'+1 ~--~.
By Lemma 12.5.1 this implies p = 0. But for p = 0 we have [H1, B] = i[H1, A]. Thus g f3 ig = {0} implies [Hx, A] : 0, i.e. v = 0. Hence Hx = 0 for a E R . Secondly assume a ~ R . Observe that for n E 1~1 we have a n - / ~ = a ( n 1 ) - p R e ~ and a n - 7 = c m - ( p + 2 ) Reef. With a ~ R andp>__ 1 it foilowsthat a n - ~ and a n - 7 are non-zero numbers for all n. Since Hx = gx(z)~-~w+g~(z)-~x with [H0, HI] - 0, [H0, z"b~] = ( c m - # ) ~ and [H0, z " ~ ] - ( c m - 7)-~, it follows that Hx = 0 for a ~ R . a Lemm
8. By changing the groap we may wiog assame that a E R .
143
Proof. Let I~I = Re a z ~ +
0 R e , u ~ w + Re,z~-~.
Then [~, HI = 0. Furthermore [I?l, X] = Re AX for all X E Vx. It follows that gx = < H > ~ g is a Lie Algebra of vector fields which are globally integrable on Ce . Furthermore both g and ~ = < I?I > + g ' are normal subedgebras of g l . Since Re 7 = 7 ~ 0 it is clear that G has an open orbit in Ce . Now all the groups G, Gt and G have open orbits in Ce . Since both G and G are normal subgroups of Gx all these open orbits must coincide. Thus G acts transitively on X = G / H . r3 Hence from now on we will assume that a E R .
L e m m a 9. The aboee defined number p equals 2. Proof. Since we know already that 1 < p a is a subalgebra of g. Now consider the fibration G / H --* G / I . Since I / H _ H + and H , ~ E go it follows that I f3 Go acts transitively on the fibre. On the other hand note that G' acts transitively on the base G / I , Go f3 G' has obviously an open orbit in G / I and Go I"1G' is a normal Lie subgroup of G ~. Thus Go acts transitively on both the fibre and the base of the fibration G / H --, G / I . It follows that Go acts transitively on G / H contrary to the assumption that no real six-dimeusioned Lie group acts transitively and holomorphically on X = G / H . Hence the assumption p ~ 2 leads to s contradiction, o Thus p = 2, ~ = 2 Re ~ + a and 7 = 4 Re a . Next we want to discuss the elements A and B of g. Note that A, B E V-a + V-a immediately implies the following statement: L e m m a 10. There ezist ~a,~t, ao,al,bo,bl E C such that
0 A = ~-~z+ ~ , ~ z~ + a o z P ~ +
B
ia +
a + .
p0 +
alzP+l ~ blzp+la_ az
A coordinate change of type (z, w, z) ~-, (z, w + Cz 2, z -I- ~z 3) now leaves H invariant and yields wlog a0 = 0 = a l . Now both H and A are invariant under changes of coordinates like (z, w, z) ~-~ (z, Cw, Cz). Thus we can wlog achieve that b0 - ~, because b0 = 0 would imply, that < H , A , B , b~- ~ ,0~0 l--- tj0 tz~Oj~,, ~0+ " f l ( z ) ~ > a would form a Lie subalgebra, whose corresponding Lie subgroup would act transitive on G / H with discrete isotropy.
144 So far by &ppropriate changes of coordinates we achieved 0 0 4z O H = z~-~z+ 3u~w + A
O+ = 0z .0
0 ~~ 0
with ~ - i~a E s~. Thus [H, A] = - A
120
O
and [H, B] = - B
while
[A, B] = x 0_~ + ( 3 b - ~1 , J z, 2 0~z := W2
[A,w~] = o0w + (6b - 2 ~ ~
O) (2)
:= Wo
[B, W2] = i 0 + (6hi - i~lo - ~ b ) z O := W :
(3)
[A, W0] = (6b - 3~~ z := Zo
(4)
[B, W l ] = ( - 6 b q- ~a - 2i~b~)~z
(5)
[A, W1] = (6bi - 2i~1o - ~ ; b ~ z
(6)
[B, W0] = (6bi - 2i~o - ~b~O z
(7)
Hence 6 b - $~a := t E R and from (7) we derive R ~ (6bi - 2 i ~
- ~b) = it - (gb - iga)
But both it a n d ~b - iga are totally imaginary. Thus [A, W:] = [B, W0] = 0 a n d 6b - 2~a + igb = 0 = it - (Tib -- iga) which implies 6b - 37~ = - ( e t a ~
+ i~b) = - 6 b + ~a - 2i~b
i.e.
[A, [A, W~]] = In, In, w~ll - t o- = z0 --
Oz
Hence with g = < H, A, B, Wo, W : , W2, Z0 >1 the structure of g as a real Lie algebra is completely described. The isotropy is determined, too: h = < W2 >1. It remains to study the G-left-invariant complex structure on the real homogeneous manifold G / H . Denote by z0 the point (0, 0, is) with s E R + 9 Obviously for any s > 0 the isotropy coincides, such that evaluation at (0, 0, is) with s E R + yields & by s parametrised series of equivalent left-invaxiant structures on the real manifold G / H , which are parmnetrized by s. We obtain JA=B,
JW0=W:andJZ0=tH
4s
"
145
Since the complex structure on the real manifold G / H depends only on t, we m a y now assume wlog t h a t - 1 2 b = t, - 2 q , = t and - 2 0 / t = t. Since s was arbitrary in R + we are furthermore able to achieve t = + 2 . Hence we obtain:
A
=a_+ 0z
s
=
0 U~z
.0
130
.
0
0 W2 = z0w 0
W0 = ~
0
~ z~z
Wl =i~'4"iZ~z Z0 = : F 2 ~ Obviously we found two possibly different open orbits of a real Lie group G in C s . Next, we calculate s boundary-defining function. Note t h a t z0 E C s is a point in an open orbit if and only if g spans the whole tangent space at z0. This is equivalent to the existence of a vector field X E g which evaluated at z0 equals some ab~s with Ira(a) # O. Since the A-orbits in (~//~r are at most real three-dimensional, X cannot be contained in a . Hence we m a y assume t h a t if X exists then it equals H - Re z A - I m
z B - 3 Re w W o - 3 I m ltoJW 1 -- 1 p t e ( z 2 ) W 0 _ 1 [ I r n ( z 2 ) W l
But this equals 1 s
_
I~
_
.8
Thus we can conclude t h a t the real-analytic function
f ( z , w, z) = Im(4z - ew - -~z I m z + 3 ~ z + e2z I m z) defines the boundary. This m a y be reduced to f ( z , to, z) = 4 I m z -- 4 I m ( w ~ ) -- -~ 1 I m z I m ( z a + 3~uz)
- 4 I m z - 4 I m ( w ~ ) + 2-~4z - ~)(z 3 + 3~z 2 - 3 ~ z - ~s = 4 Im z - 4 Im(w~) + = 4/m(z
1 4 + 2 z s ~ + 6 z 2 ~ 2 + 2 ~ S z + ~4) Im(~-~(z
+liz4)--4Im(.(w+~zS))+llzl4
146 Hence in new coordinates with z' = 4z 4- ~iz 4 , z' = ~ z ~ z 3) we obtain
and to' = -V~(4to 4-
f(~', to', z') = X,n ~' + Im(~'to') + I:I'" 6. T h e e l a - d f l c a t i o n We now want to state the final result of the above considerations. P r o p o s i t i o n 1. Let X = G / H be a tkree-dimensionai komogeneona complez manifold. A u a m r tkat X is neitker complez-komogeneons nor bikolomorpbic to a direct prodact of lower-dimensional komogeneoas complez manifolds. A ~ n m e fartker tkat G is solvable and of minimal dimensional among all Lie groapa acting kolomorphically and transitively on X . Finally aJsame tkat dims(G) > 6 and that tke anticanonical fibration of G / H kas discrete fibres. Tken X is bikoiomorphic to one of tke following two manifolds:
Q+ = {(~, to, z) e C s I I~n z + Im(~to) + Izl 4 > O}
~ - = {(~, w, z) e c ~ I I ~ 9 + Im(~to) + I~1' < 0) Finally we want to prove that these two manifolds are really different.
L e m m a 2. f~+ and f l - are not biholomorphic. Proof. We will prove this explicitly although this result can also be deduced from the Chern-Moser-theory of normal forms for CR-hypersurfaces (see [CM]). After an appropriate change of coordinates we have f~+ = {(z, w, z)lf+(z, w, z) = Re z 4- Re(z~) 4- I~14 > 0) and f~- = ((z, w, z ) l f - ( z , w, z) = Re z 4- Re(z6~) - I=l 4 > 0). (Observe that ( R e z ) 4 = Re(z 4) 4- 4Re(z~ s) 4- 6[z[4). Assume that there exist a biholomorphic map ~b0 : f l - --, f~+. Now ~b0 may be extended to a map ~b between the envelopes of holomorphy with ~b(0f~-) C Of/+ 9 Since the boundary is homogeneous we m a y assume ~b(0,0, 0) = (0, 0, 0). Now f + o ~b and f - define the same boundary. This fact yields
Re f + Re(gg) + Igl4 = ~(Re z + R e ( z ~ ) - Iz14),
(,)
where f, g, h are holomorphic functions with ~ = (g, h, f ) and ~ is a positive real-analytic function. Wlog we m a y assume ~(0, 0, 0) = 1. Next we will compare the coefficients of the power series of both sides of (*) and thereby obtain a contradiction. We restrict our attention to {z = 0}. The terms of degree (1,1) yield Re(glhl) = R e ( z ~ ) , where gl denotes the first degree term of g. Comparison of the terms of degree (2,1) shows now, that g2hl 4- h2jl = ~1,0 R e ( z ~ ) , i.e. g2hl + h2il = ~1,0 Re(gxhl)
147
Now # being biholomorphic implies that gl/hz is a non-constant meromorphic function. Hence ( g 2 - ~1,091) and ( h 2 - Ai,0ht) must vanish identically. Next we compare the terms of degree (2,2):
But g~h2 + h]~] equals I~1,01= Re(z~). Hence for w ~ 0 we obtain that Igl4 = - I z l 4 . This is a contradiction for z ~ 0 . 0 R e m a r k s . 1. One can prove the following (see e.g. [Do]): Assmne that a solvable Lie group G acts transitively on a manifold X as a group of isometries. Then there exist s Lie subgroup Go of G such that Go acts transitively on X and h u only discrete isotropy. Hence f~+ and f~- defined as above are examples for three-dimensional solvnumifolds without any G-in~riant metric for any transitive G-action. 2. By results of V. Ezhov (see [El) on the automorphisms of CR-hypersurfaces in Ce it follows that every automorphism of ~ fixing (0, 0, 0) is already linear. A linear automorphism of C 3 which stabilizes ~ is d the form ( z , w , z ) ~-~ (~z,~(w + isz),z + it) with s,t E R and ~ E C with IAI = 1. Therefore the whole group of all autornorphisms of f]+ reap. fi- is real eight-dimensional and generated by the al>o,ce described real seven-dimensional group G and the S 1-action given by (z, tv, z) ~ (Ax, ~t~, z).
148
Chapter 13 The
non-solvable
case with
/~ t r a n s i t i v e
1. I n t r o d u c t i o n In this chapter we assume the following Basic A s s n m p t i o n s . G always denotes a real non-solvable simply-connected Lie group and H denotes a closed Lie subgroup such that X = G / H is a nontrivial three-dimensional homogeneous complex manifold. Recall that non-trivial means that X is neither complex-homogeneous nor biholomorphic to a direct product of lower-dimensional homogeneous complex manifolds. We assume in this chapter that the G-anticanonical fibration of G / H is injective and that the representation of the Lie algebra g in Fo(X, ToX) is totally real. We furthermore assume that G is of minimal dimension among all Lie groups which act holomorphically and transitively on X . Finally, in this chapter we only consider the case where the radical/~ of (~ acts transitively on G//7/. Under these assumptions there are two cases: Either there exists a fibration G / H --* G / J with 0 _< dime(G/J) _< 3 or there doesn't. In both cases the nilpotent normal Lie subgroup A := (G~^GR) ~ plays a key role in the classification. In the latter case it is clear that A acts transitively (~//7/, because otherwise := A / t would induce a proper fibration. (Under the basic assumptions the radical /~ can't be central in G, therefore we have always dimc(A) >_ 1). Furthermore if A is non-abelian then I : - A~/~ induces a proper fibration, since the commutator group N ~ of a nilpotent group N never acts transitively on the positive-dimensional N-orbits (see Cor. 10.2.2). This yields A -~ (Cz, + ) . Furthermore =/:/tr A C GLs(C) K (C~ +) and we continue with a case-by-case check for the different possibilities of representations of S in GL3(C) -~ Aut(A). In the other case, where we have a proper fibration, we first prove that we can wlog assume that G / J ~- Ca \ R 2 and J / H -~ H + and then determine the structure of G in detail. Thereby we again use our result on the ineffectivity L of the G-action on G / I (Prop. 8.2.1). In particular we exploit the fact that L acts almost effectively on I / H if I / H ~_ Ca \~2 or I / H ~0 H + . This helps to deduce G = GL2(R)K //3 where //3 denotes the real three-dimensional Heisenberg group.
149 2. T h e case 0 C GL3(C) K (Ce, +) Lennna 1. Assume in addition to the basic assumptions that there is no flbration
1
al~
~
!
~li
with 0 < d i m c ( G / I ) < 3. Then the [ollo~ng statements hold. fi) The g~up A = ( ~ ' n i~) o is .~elian and thee-dimensional. F u r b e l o w ,
A acts trensitively on ~ / f l . (ii) The mani/old ~ / f l is biholomorphic to Ce. In appropr, ately chosen coordinates on G/I-I the group 0 acts by a~ine-linear transformations, i.e. 0 C GL,(~ K (Ce, +). (iii) The representation of S in Aut(A) ~_ GLs(C) is irreducible. (iv) Eithe~ 9 -'_ SLy(C) or ~ -_ SLy(C). Proof. (i) The center of the group G is discrete, since the anticanonical fibration is injective. Hence the group /] = ((~' f3/~)0 is positive-dimensional, because otherwise the radical /~ would be central in 0 . The A-orbits are closed by Lemma 2.2.13. Thus the assumption of the Lemma implies that /L acts transitively. Recall that by Corollary 2.2.12 the group A is nilpotent. Hence by Lemma 10.2.1 its commutator group/]' can't act transitively on G / H . Thus is abelian and therefore centralizes A f3 H . Since/[ acts transitively, it follows that .4 f3 f / is ineffective on 0//~r . Hence /~ f3 H is discrete, i.e. dime(A) = 3. (ii) Since the anticanonical fibration embeds G / f / equivariantly in some P , , it follows that Jt acts as a unipotent group. Hence 0//-~/= A/(A f3 f/) _ Ce (see e.g. [Hu], Prop.i.E.1). Thus the map /i --* G//?/ defined by a ~-, a/~ is biholomorphic. Now 0 = H ,< A. Choose h E /~/. Observe that hah-l~I = ha~l. Hence the /?/-action on 0//~"~'_~'A is given by the homomorphism/~ ---, Aut(A) = GL3(C). Thus ~ r GL3(C) K (Ce, +). (iii) Let SH be a maximal semisimple subgroup o f / f . Since f / ~ 0//1 and is solvable, it follows that SH is a maximal semisimple subgroup of 0 . Hence wlog S C / t . Thus S cannot centralize A. Now assume that the S-action on is reducible. Then A = A1 x A2 where dimc(J~')= i and the J~. are S-stable. Since dime(A1) = 1 it follows that S centralizes A1. Thus G' = S,~ A implies 0 " = S ,< A~. But now the radical of G" (i.e. A2) has two-dimensional orbits. This is contrary to the assumption that there is no proper fibration. (iv) Finally, A "~ C"e implies that either S _~ SLa(C) or S _ SL2(C).o L e m m a 2. By conjugation in 0 we may wlog assume that S C t t and moreover S . ~ = f t , where ~_, denotes the centralizer of S in G.
150
Proof. Recall that 6 =/:/K J] and wlog S C / ~ by the considerations in the proof of the above Lemma. Now let R~r denote the radical of/:/. T h e n / ? / = S K R~r and /~ = R ~ K A. Thus from A = ( 6 ' f3/~)0 it follows that ( 6 ' f3 R/t) ~ = {e}. Hence [S, R~] = {e}, i.e. R ~ C C. On the other hand the irreducibility of the representation of S in Aut(A) implies that A f3 C = {e}. Therefore R ~ = C. [3 T h e case $ ~_ SLs(C)
L e m m ~ 3. Auume in addition to the above aasmmpfions tAat S ~ SLs(C). Then G / H ~_ Cs \ R 3. Proof. By definition A is a canonical Lie subgroup of 6 . Hence & = a (B/a and by changing coordinates in A - (C6 , +) we may wlog assume that A = (R s, +) with A := (G' Cl R) ~ . Since S = S N G is a real form of SLs(C) and stabilizes A _ (R 3,+) in A - (C s , + ) , it follows that S ~- SL3R. Thus SL3(R) t< (R3, +) C G and GL3(R)K (R3,+). Observe that both SL3(R)K (Rs, +) and GLs(R) K (~3, +) have the same two orbits in C 3 , i.e. R 3 and C s \ R 3. Thus G / H ~_ C6 \ R s. n T h e case S __. SLy(C) Here we a~ume that S ~_ SL2(C). From the above Lemma we know that the representation p of S on A _ C"6 is irreducible. Note that S K (R 3, +) C G. Hence S = p(SL2(C)) rl GL3(R) and therefore S ~- SL2(R). Note that the (modulo conjugation unique) irreducible representation of SLy(C) in SL3(C) is given by
p:
--* c
2ae ad + be c~ cd
.
The centralizer of p(SL~(R)) in GLa(R) is just C=
,)
/
Hence
p(SL2(R)) K (R 3, +) C G C p(GL2(R))K (R s, +). Since (R s, +) C G it follows that the G-orbits are tube domains. Observe that p(GL2(R)) stabilizes both R 3 and I~ s. Therefore we discuss now the action of p(SL2(R)) resp. p(GL~(R)) on I~ s.
151 L e m m a 4. Let p denote tAe irreducible representation o/ GL2(R) in GL3(R). Choose coordinates zx, z2, zs such that p(SL2(R)) -
. d + ~,
o d - ~ , - 1 .,~d . , b, ~, d e ~
ed TAe, tAe ~s=ctio, D ( z z , z 2 , z3) = 4zzzs - z I is in~aria,t u#der tAr actios o/ p (SL2(R)). Proof. Consider the induced representation of sl2(R) on the tangent space and note that the image of sly(R) in the tangent space at the point (zz, z2, zs) is generated by (z2, 2zs, 0), (2zz, 0, - 2 z s ) and (0, 2zz, z2). All three of these tangent vectors are orthogonal to (4zs,-2z2, 4zz). Observe that (4x~, - 2 x 2 , 4 ~ ) = u ~ d ( 4 ~ s
-
~).
Hence the function D -- 4zzzs - z~ is inwriant under the p(SL~(R))-action.r~ R e m a r k . Observe that the irreducible representation of GL2(R) in GLs(R) may be obtained in the following way. Consider the natural action of GL2(R) on R 2 and the induced action on R[X, Y], the ring of polynomials on R 2 . Then the space of homogeneous polynomials of degree two, i.e. < X ~, XY, y 2 > a is stabilized by GL2(R). Moreover the GL2(R)-action is irreducible. Now the invariant function D is called the discriminant. It was first proven by Gaafl in 1801 that it is invariant under the GL2(R)-action. (For details see e.g. [Kr, p.g]). Note that from the representation of p(SL2(R)) in the tangent space it furthermore follows that all p(SL2(R))-orbits in R s except {(0, 0, 0)} are real two-dimensional. Observe that the sets
are connected for c ~ 0 and consist of two connectivity components for c > 0. In the latter case zz, zs > 0 holds for one connectivity component and zz < 0, zs < 0 holds for the other component. The p(SL2(R))-orbits in R s are precisely the connectivity components of the sets Ac for c ~ 0, the singular orbit (0, 0, 0) and the two connectivity components of A0 \ (0, 0, 0). C o r o l l a r y 5. The group p(SL~(R))K (Rs,+)
/las
no open orbit in C s .
Next let us consider the C-action. Observe that with g,x =
it follows that
A
152 L e n n n a 6. Let G denote the growp G = p(GL2(it) +) =
( a2 2ac c2
ab ad + bc cd
a d - b c > O and a, b,c, d E i t
9
Then there are the following G-orbits in It s .
z2, zs) < 0} z2, zs) f~s = {(zx, z~, z~) I D(zx, z2, z3) ~ = ( ( ~ , =~, =s) I D ( ~ , z2, =s) a s = {(zl, =~, zs) I D(=x , z~,=s) n~ = {(o, o, o))
> 0 and zl, z2 > 0} > 0 and zx, z2 < 0} = 0 and zl, =2 > 0}
= 0 a n d z l , z 2 < 0}
Proof. From D(g~(zl, z2, z3)) - ~2D(Zl, z2, zs) it follows that the connectivity components of {z E R31 D ( z ) > 0} and {z E Its l D ( z ) < 0), i.e. f~, for 1 < i < 3 are G-orbits. Furthermore note that G stabilizes (0, 0, 0) and that p(SL2(It)) already acts transitively on the connectivity components of {z E Ita I D ( z ) -- 0}. Hence the f~i for 4 < i < 6 are G - o r b i t s . o C o r o l l a r y 7. The group G = p(GL2(It))K ( i t 3 +) acting on C s has the following open orbits.
AI = {(Xl, x2,-s) c ~ 14 Xm(,l) Im(=s) - I~(~2) 2 < 0} A~ = {(Xl, =~,-s) ~ C~ 14 I~(~1) Im(xs) - I ~ ( ~ : ) ~ > 0, Im(=1), I~(=2) > 0} As = ( ( z l , z 2 , zs) E C s [ 4 I m ( z l ) I m ( z 3 ) - I m ( z ~ ) 2 > 0, Ira(z1), Ira(=2) < 0) L e m m a 8. The G-orbits A2 and As are bihoiomorphic to bownded domains. Moreover they are bihoiomorphic to each other. Proof. (i) The map ~ : (zx, za, z3) ~-, ( - z x , z 2 , - z 3 ) is a biholomorphic diffeomorphism between A1 and A2. (ii) Consider the map
r " (Zl, Z2, Z3) t-~ (:1, Z2 "~" 2Xl "~" 2:3, :3). We will show that r
+ xH + •
+.
Assume the contrary, i.e. (z2 + 2zl + 2z3) ~ H + . Then
- I~(~2) _> 2(I~(~x) + Im(~s)). Consequently
Im(=~) ~ >_ 4 Im(=~) 2 + s I~(~1) I m ( ~ ) + 4 Im(~s) ~
153
But (Zl, z2, z3) E A2 implies that I,~(z~) ~ < 4 Im(,~) I,,(:~). Hence the assumption (z2 + 2zl + 2z3) ~ H + leads to a contradiction. Thus A~ may be embedded in a bounded don~in, o R e m a r k . X _~ .42 or X _~ As would contradict the minimafity aussumption, since there exist real six-dimensional solvable Lie groups acting transitively on A2 and As (see Lemma 11.3.5). R e m a r k . Actually the open orbits A~ and As are hermitian symmetric dom&ins of type BDI(3, 2) = CI(2) in the notation of [Hell.
Lenxma 9. ~x(Ax) -- Z .
The G-orbit Ax is diffeomorphic to
3. T h e case dimc((~/I)
=
S 1 X R 5.
In particmlar,
1
Lenxnm 1. Assltme in addition to the basic asssmptions that there are fibrations
G/H ~
1
GIf
*-,
#/H
1
Oil
with dimc(GII) = 1. Then there are also ]ibrations
GIH
1
GIJ
,-,
~lfX
~
~1/
1
with d i m c ( G I J ) = 2.
Proof. Assume the contrary. (i) Assume furthermore that I / H ---- ~//~r. Then either G / H ~ G/I-[ is a holomorphic fibre bundle over the contractible Stein manifold H + or G / I = G / I . Hence G / H is either a direct product of G / I and [ / H or complexhomogeneous. Both contradicts the basic assumptions. (ii) Since R acts transitively on the homogeneous Riemann surface G / I , it follows that S acts trivially on G / [ . Denote the radical of I by R I. Note that by Lenuna 8.2.2 S acts almost effectively on I / H , while RI acts transitively on I / I / . In (i) we showed that [ / H ~ I / f [ . Thus it follows from the classification of homogeneous surfaces that I / H ~_ C2 \ R ~. Furthermore I is represented on I / H ~ C ~ \ R 2 either as SL2(R)K (R3,+) or GL2(R)K (R2,-t-). (iii) Let L denote the ineffectivity of G on G/I. Observe that L acts almost effectively on I / H . Since L is normal and S C L, it follows that either L _~ SL2(R)K (1~2 +) or L ~_ GL~(~)K (]R~,+). In both cases L' ~_ SL2 (R)K (R ~, +). Observe that G" C L and therefore L' = G " _~ SLy(R)K (R 2, +).
154 Note, that ad(S) stabilizes g ' " . Hence it stabilizes also a transversal subspace e. But now [s, c] C e f3 g " = {0} implies that c centralizes I. Observe that the centraliser of s in g " equals {0). Thus c equals the centralizer of s in g and is therefore a subalgebra of g. Hence G = (C x S) g RG .... (iv) Observe that St< RG,- = L' acts transitively on I / H and C acts transitively on G / I . Note that if a solvable Lie group C acts transitively on a Rienman surface then there is always a subgroup Co of C , such that Co acts transitively and almost freely. Now (Co • S)K P~,,, acts transitively on G / H . Hence by the assumption of minimMity it follows that dima(C) = 2. (v) Consider the natural homomorphism r : C x S --* Aut(RG,,). Since dimjt(C) = 2, #b(S) "" SL2(R) and Aut(Rao,) ~- GL~(R), it follows that @lc : C --* A u t ( R a , , ) has a positive-dimeusional kernel A. More precisely 1 d i ~ ( J / I I ) _> 1. Hence it remains to show that d i m c ( J / H ) ~ 2. Note that A C J . Hence the C-orbits in G / J are at most real one-dimensional, i.e. C cannot act transitively on G / J . Thus G " must act non-trivially on G / J . Furthermore 6 / / ? / = / ~ / ( / ~ N H) implies that R acts non-trivially on G / J . But there doesn't exist a homogeneous Riemann surface such that both G " and R act non-trivially. Hence dirnc(J/H) = 1. [3 4. T h e case dirnc(G/I) = 2
T ~ m m - 1, Asswme in addition to the basic asswmptions that there are J~brations
1
alI
!
Oil
with d i m c ( G / I ) = 2. Denote the ine~ectivity of the G-action on G / I by L. Then G / I ~- C r 2 and I / H ~- H + . Moreover d i m l ( L ) < 1 and G / L C GL2(R)K (R 2, + ) .
155
Proof. (i) First note that (~//~ is holomorphically separable, since/~ acts transitively and (~//~ is equivariantly embedded in P , . Hence any non-trivial orbit of a complex semisimple subgroup of G is at least two-dimensional. Therefore L cannot contain any semisimple subgroup. Thus L is solvable and S acts nontrivially on G / I . Recall that /~ acts transitively on G / I . It follows that either G / I -~ Cr \ R ~ or G / I ~_ Cr . But G / I ~_ Cr would imply that S sad hence S has a fixed point in G / I . This would imply that S fixes the fibre over this point in (~/f pointwise. Therefore it would follow that S has a fixed point in G/H. Then the r~dical R of G would act transitively on G / H contrary to the assumption of minimality. Hence G / I _ Cr \ R 2. Furthermore G / I ~_ Cr \ R 2 impfies that either G/L "" SLy(R),< (R 2, +) or G/L ~_ GL~(R),< (R 2, +). (ii) We know that G/i"L_~'C r . Thus the bundle G//~/ --~ (~/i is globally holomorphically trivial. Therefore i / / t = I / H would imply that G / H is biholomorphic to G / I • I / H . Hence either I / H ~_ C" or I / H ~_ H + . (iii) Assume that I / H _~ C*. Let L r denote the smallest complex subgroup of G containing L. Now L has a fixed point in I//~/, hence L c also fixes this point. Since L r is a normal subgroup of (~, it follows that dims(L) = dimc(L r = O. Thus diml(G/L) >_ 6 and therefore G/L ", GL~(R)K R 2. Next observe that the isotropy of the GL2(R)K RLaction on C~\R 2 is isomorphic to the real twodimensional B~el group. But I / H ", C* implies that I is abelian. Thus we obtain a contr~iiction. (iv) Note that direR(L) _< 1 follows from I / H ~_ H + by the Lemmata 8.2.4 and 8.2.5. r~ L e m m a 2. The above assumptions imply that dims(L) = 1 and G/L "" GL2(R) ,< R 2.
Proof. (i) Assume that G/L ~- SL2(R),< R 2. Then from dimm(G/L) = 5 it follows that dims(L) = 1. Denote the nilr~dical of G by N . Observe that both the abelian normal subgroup L and G' N R are contained in N . Hence N = R. Since L is one-dimensionai it follows that N and S centralize L. Thus L is central in G = S,< N contrary to the basic assumptions. Hence it follows that G/L ~_ GL2(R)K R 2. (ii) Assume that direR(L) = 0. Then it fonows that G ~ GL2(R)^K R ~. Hence R ~ is real two-dimensional. Note that /~ acts transitively on G / I . The /~-orbits are closed by Cot. 2.2.13, hence two-dimensional. This gives a fibration G/H
l
GIS
"--*
G/[-I
1
Let L j denote the ineffectivity of the G-action on G / J . Observe that R e C L j . Since/~ ~cts transitively on G//~/it follows that R is not contained in L j . Thus L~G - GL2(R)KR 2 implies that L = R ~. But there is no homogeneous Riemann
156 surface on which G / L "" GL2(]~) acts almost effective. This contradiction shows that dima(L) = 1.n
L e m m a 3. Let N denote the nilradicai of G. Then L C N and N I L ~(]~2, +). Moreover N is isomorphic to the three-dimensional Heisenberf gromp. Proof. (i) Since L is a normal abefian subgroup it follows that L C N .
Moreover
R ~ C N and N I L C NG/L. Thus N I L = (~t2, +). (ii) Assume that N is abellan. Observe that G ~ = S K N . T h e n G " = S g ( R 2, + ) and R n G " is a two-dimensional subgroup of G . Note that /~ n G " has closed orbits by Cot. 2.2.13. Consider the induced fibration
G/H
1
G/J
,--,
G/fI
1
'--* G / ( R n G")~
= G/JfI.
Let L j denote the ineffectivity of the G - a c t i o n on G / J . Note that dime(G/J) = 1. Since R acts transitively on ( ~ / H it follows that R acts non-trivially on G / J . Hence S C L j and therefore L~ = G " . Note that G " centralizes L. It follows that G / L j is non-abelian, because otherwise L would be central in the whole group G. Thus G/L$ is a real two-dimensional non-abelian Lie group. Hence G / J ~- H + . Furthermore from the G " - a c t i o n on 3 / H it follows that J / H is biholomorphic to either C 2 \ ~ 2 are a covering of C 2 \ R 2. But as a subset of ( ~ / H the manifold 3 / H is holomorphically separable, i.e. J / H ~_ Ca\R ~. Therefore G / 3 ~_ H + ~_ I / H and 3 / H ~- C2\R 2 ~- G / I . Hence G / H is biholomorphic to G / I x G / J contrary to the assumption of non-triviality. [3 Next we want to determine completely the Lie algebra structure of g . We already know that N I L is the three-dimensional Heisenberg group and G / L "~ GL2(R)K (R ~, + ) . Hence we obtain L e m m a 4. The Lie algebra g has s base < H, X, Y, V, A, B, C >It swch that
[H, X] = 2X [A, B] = C i v , A] = x IX, B] = A
[H, Y] = - 2 Y [H, A] = A Iv, n] = B [Y,A] = B
IX, Y] = H [H, B] = - B IV, C] = 2C
and 1 = < C > l . Furthermore every other commutator relation between the base vectors vanishes. Proof. (i) Choose a mammal semisimple subalgebra s in g and a base < H, X, Y > of s such that [H, X] = 2X, [H, Y] - - 2 Y , [X, Y] = H . (ii) Note that Ad(S) stabilizes g' n r . Hence it also st~bifizes a transversal s u b s p ~ e c in r . Now c N g ' = {0}, hence [s,c] -- {0). Observe that s stabilizes n ' = (g' • r ) ' . Thus there is a Ad(S)--st~d)le vector subspcce a of n such that n - a (~ n I .
157 (iii) Since G / L "" GL2(~t)K (R 2, +) we may choose vectors A, B E a, such that
[H, A] = A, [H, B] = - B , IX, B] -- A, [Y, A] - B (iv) Again from G / L ~_ GL2(R)K (R 2, +) we conclude that we can chooee V E c such that IV, A] = A and IV, B] = B. Since N is isomorphic to the threedimensional Heisenberg group, it follows that n' = < [A, B] > a . Now we define C = [A, B]. Note that IV, C] = 2C follow~ from
Iv, [A, B]] + [X, [B, Vl]+ []3, IV, X]l = 0 n
L e m m a S. s r : g - . r o ( C a \ ~ 2 , T ( C a \ R 2 ) ) denote the natwral representation o/ g on G / I ~ Ca \ R 2. Then in appropriate coordinates z, w on G / I the above chosen base vectors V, H, X, Y, A, B, C of g are represented on G / I as follows 0 ,-(v) = - ~ -
0
, ~0 8
8
~(Y) = - % - ~
a ~ ( x ) = -z~-g~
0 8
~(c) = o. Moreover i = < V + H, Y, C >a /or the base point eI = (i, 0). Proof. This follows immediately from the fact that G acts on G / I "" Ca \ R 2 as G / L ~ GL2(R)K (•2, + ) . o L e m m a 6. The complez str~cture on G / I "" C a \ R ~ indmces the following J-str~cture on g rood i with respect to the base point e l = (i, O) : J A = - X and J B = H. Furthermore h = < Y > a . Proof. The first assertion is obvious. To show that h = < Y >a note that I is solvable. Hence I / H ~_ H + implies that H is normal in I. Consider the commutator relations in i:
[V+H,Y]=-2Y,
[V+H,C]=+2C,
[Y,C]=0
158 It follows that < Y > x and < C >1 are the only one-dimensional ideals in g. Recall that L = < ezp(C) > acts almost effectively on I / H . Hence h = < Y > a . D
L e n u n a 7. Consider the fibre bundle G / H ~ G / I . Then for a b~e point appropriately chosen in the fibre over eI = (i, O) E G / I we obtain the following J-strwcture in g modulo h =< Y >n. JC=V+H J A -- - X + ~ ( V + H ) + -yC J B = H + / ~ ( V + H ) + 6C
with a, 8, 7, 6 E C. Proof. Recall that I / H _ H + and i / h --< C, V + H >R with IV + H, C] = 2C. It follows that for an appropriately chosen b&se point in H + ~_ I / H we obtain J C = V + H . The other relations follow from the above determined J--structure on g m o d i and the fact that i = h + < V + H , C >R .a L e m m A 8. From the integrability condition for the J-structure it follows that ~=0, 8=6=0 and 7 = 8 9
Proof. (i) Consider the following integrability condition. (All calculations are done modulo < Y >R= h ) . [JC, J B ]
-
9 2~C "
-
J [ J C , B] + J[C, J B ] + [C, B] "
0
-~JC
0
=~ - 2 6 C = 2 ~ J C
=~6=0=~ (ii) Next we discuss the following relation. [JA, J B ] 2(l+p)x+~(as--v~)c
= J [ J A , B ] + J [ A , J B ] + [A,B] -SA
-(l+~p),VA
c
Since ~ = 6 = 0, this implies 2X = - 2 ( - X + a(V + H) + 7C) + IC :~ 2X ----2X - 2c~(V -I- H ) + (I - 27)C :~-0
and
1 ~=7
v
Now we are in a position to give a concrete realization of G / H .
159
L e m n m 9. Consider i)
a
Y =-"~z
+ t o ~'z
a
x =-z~+x A = ~ -a
2a
2zF; a
B - - - a=
C = 2 ~--az" This realizes g as an algebra of globally intefrabic vector ~dds on C s where the above determined J-structure on g coincides with the J-structure indlced by the comp:cz ,t,~ct.,~ o. C / ~ for the ba,e point (=, to, z) = O, O , - 0 . To determine the open G-orbits it is necessary to analyse where the g-vector fields span the whole tangent space. From the above considerations we know that (z, to) E Ca \ R 2 is a necessary condition. Moreover g spans the whole tangent space iff (z, to) E C2 \ R s and simultanonsly i~-ij is spanned by g. L e m n m I0. The open G-orbits in Ca for the above representation are n2 = {(=, to, y) I Xm z + 2 Zm x Re to < O, (x, to) ~ Ca \ a s } . TheW are biho]omorphic to each other. Proof. (i) To determine at which points the tangent vector / ~ is spanned by the g-vector fields, it is convenient to change the base of g. Note that Wl = ilmz
o + ilmz~-~
Ws = -i Im ~
+ i Ira(to s
W s = i l m w ~ w + i(2P,eto l m z + Imz~az 0
W4 = - i I m z~-~w + i(Im(= 2)
2 Zm= Re z +y-~_. OZ
160 together with A = _~_a_ ~w 2z a_~_ az a B----~ ~z
spans g over R, since Wl = ~H-
V)- RezB- 1RezC
W~ = Y + RewB - IRe(w~)C WS = -2~H + V ) -
RewA+(RewRez+lRez)C
w 4 = x + Re ~A - ~ R ~ ( , 2) - 2(R~ ,)~)C. Therefore it is clear that ib~x is spanned by g-vector fields, iff either I m w Im z + Im z I m ( w 2) # 0
or 2 R e w ( I m z ) 2 + I m z l m z + I m w l m ( z ~) - 2Ira w l m z R e z # O,
i.e. either Imw(Imz + 2ImzRew)
# 0
or I m z ( I m z + 2 I m z R e w ) # O.
This proves that statement that the whole tangent space is spanned by g, iff Zmz + 2 Z m x R ~ w # 0, and (x,w) ~ C2\R 2. []
5. T h e classification Here we summarize the results of the preceding paragraphs. P r o p o s i t i o n 1. Let X -- G / H be a three-dimensional homogeneous complez manifold. Assume that X is neither complez-homogeneons nor a direct prodmct of lower-dimensional manifolds. Assnme that G is non-solvable, that the anticanonical fibration of G / H has discrete fibres and that the representation of the Lie algebra g in ro(X, ToX) is totally real. Fwrthermore asswme that G is of
161
minimal dimension among all Lie growps which act holomorphicallyl and transitively on X . Finally, assume that the radical it of G acts transitively on G / I f . Then X is biholomorphic to one of the following.
(0
c ~ \ at-"
(ii)
A = {(**, ~ , ~ ) e C ~ 1 4 Zm(.x) I m ( . ~ ) - t , , , ( ~ ) ~ < O}
(iii)
Q={(z,w,y)
l Imz+2ImzRew>O,(z,w)
E C a \ R 2}
and coverings of these manifolds. (Note that Ce \ ]is is simpllt-connected, but Proof. (i) First assume that the anticanonical fibration is injective. (ii) There ate two cases: Either there exist a fibration G / H --* G / I compatible with the complex structure and 0 _< dimc(G/I) _< 3 or there doesn't exist such a fibration. As we have seen in paragraph 2, in the second case either S ~- SLs(R) or S "~ SL2(R) and correspondingly G / H = C"e \ R s reap. G / H = A. (iii) If there exist a proper fibration we proved in the preceding paragraphs that under our assumptions it follows that G / I = C2 \ R 2, I / H ~- H + and G / H ~_ fl. (iv) If the anticanonical fibration G / H ---, G / H t is only assumed to have discrete fibres, then the anticanonical fibration of the base is still injective. Hence the base G / H I of the anticanonical fibration is bihotomorphic to C e , R s, A or f~ and therefore G / H is a covering over one of these manifolds. 13
162
Chapter 1~
dimc(d/
The
)= 1
Basic amsumptions In this chapter we assume that the orbits of the radical 1~ of G in {~//I are complex two-dimensional. As usual we assume that X = G / H is neither complex-homogeneons nor biholomorphic to a direct product of lower-dimensional manifolds. We furthermore assume that the anticanonical fibration of (~//~r is injective, that G is totally real in G and that G is of minimal dimension. By Lemma 2.2.14 the /~-orbits are closed, hence there exist fibrations
G/H
1
GIJ
~
d/H
~
dl~
1
From dimc(Rl(R 13 H)) = 2 it follows that (~//~/?/ ___ P l . We denote the connectivity component of the ineffectivity of the G-a~tion on G I J by L. There are three different cases for the structure of S: Either S ~-- SL2(R) or S ~- SU~ or dima(S) > 3. We will discuss these three cases separately. First we deduce some auxiliary results. 1. P r e p a r a t i o n s R e m a r k . Assume that V is an irreducible sl2(C)-module. Let < H, X, Y > denote a basis of s with [H, X] = 2X, [H, Y] = - 2 Y and [X, Y] = H . Then V has a base Z0, ..., Z , such that ~b(H)(Zk) ----( n
--
2k)Zt
~(x)(z~) = (. + I - t)z~_x
r
= (k + 1)Zt+1.
L e m m a 1. Assame that dima(S) = 3. Then there ezist 6brations
GIH
"-,
dlH
l l
.-,
01i
a/~
a13 ~ with dimc(G/I) = 2.
l 1
GI/= dlRH = P1
163
Proof. (i) From the basic assumptions it follows that R is not central in G. Hence A = (r f3/~)0 is s positive-dimensional group. The Lemma is obviously fulfilled if the A-orbits aren't complex two-dimensional. Therefore let us assume that acts transitively on the /~-orbits. Since A is enipotent, it follows that j / f I ~_ C2 . In particular it follows that H is connected. (ii) Note that G / J ~_ Pl implies that j = B K /~, where /~ denotes a Borel subgroup of S _~ $L3(C). Thus j is a solvable Lie subgroup of (~. Hence there exist a connected one-codimensional Lie subgroup I of ,f such that H C I C J . It remains to show that I is a closed subgroup. Observe that G/3 ~ PI implies that j is a connected Lie group with w1(J) --- Z. Hence by Lemma 2.2.5 any connected Lie subgroup of J is closed. [] 2. A~sume i= additiou to the basic arswmptions that tAere are fibratio~ GIH
1
~
G/~I
!
GII
61i
GII
GI/ = IRH
1
with dime(G/I) = 2 . Theu I / H ~ T./~I. Proof. (i) Note that the R-orbits in G / I are analytically Zariski-deuse in the /~-orbits. Thus it is not possible that J / I -~ C' and simultaneously J / I ~ C. Hence either J / I = J / i or J / I U+. (ii) Let G / I --, G / K denote the bounded holomorphic reduction of G / I . Observe that G / K is a bounded homogeneous domain (see [Gi]). Hence G / K is a contractible Stein manifold with injective anticanonical fibrstion. Thus there exist the complexiflcation G/ K ~ G/ K. (iii) Recall that either G / J "" H + or G / J ~ P1. Hence it is obvious that E l i = / ~ / i holds if J / I = J / i . Therefore assume J / I ~_ H + . Then it follows that G / I ~_ H + • H + reap. G / I ~_ H + x P1. Thus either I = K or K / I ~_ P1. In both cases it follows that K / I = / ~ / i and therefore K / H = K / t l . (iv) Note that G / H --~ G / K is a holomorphic fibre bundle, since K / H = K / / ~ . Since the base G / K is a contractible Stein manifold it follows that this bundle is trivial, i.e. G / H is biholomorphic to G / K x K / H . Thus G/H is biholomorphic to s direct product of lower-dimensional homogeneous manifolds if dimc(G/K) > 0 and complex-homogeneous if dime(G/K) = O.[]
T h e s t r u c t u r e o f ~l N/~ In order to understand the structure of the Lie group G we begin with the discussion of its normal unipotent Lie subgroup ,4 := ((~0 N/~)0 (The group A is unipotent due to Lemma 2.2.12).
164
L e m m a 3. Let G be a real solvable Lie frolp with one-dimensional commutator 9roup G ~. Let Z denote the center of G and assume that direR(G) >_3. Then diml(Z) > O. Proof. Let C denote the centralizer of G' in G. Observe that C is the kernel of the natural homomorphism G --* Aut(G') given by corrugation. Hence direR(C) >_ diml(G) - 1. Clearly the Lemma is fulfilled if G' is central, i.e. if direR(C) = dim~(G). Therefore let us m u m e that direR(C) = dims(G) - 1. Now cho~e Yt, Y~ arbitrarily in the Lie algebra c corresponding to C. Choose X E g such that g = c+ < X >R. Finally choose Z E g' such that Z ~ 0. Observe that [Y1, Y2] = a Z and IX, Z] = ~ with a,/~ E It, since g' = < Z >R. Note that ~ ~ 0, because X r c. Next consider the following Jacobi-identity.
IX, b',, Y2ll +
[Y2, Xll + [Y,,
= 0.
Since [e, gt] = 0, it follows that IX, [Y1, Y3]] = a/~Z = 0. Hence [Y1, Y2] = 0, i.e. c is abelian. Now consider the map ad(X) : e --* g'. Since dim,,(G) > 3 implies direR(C) >_ 2, this map has a positive-dimensional kernel. This kernel is central in g. n C o r o l l a r y 4. From the basic assumptions and dimlt(S) = 3 it follows that direR(A) > 2 with A = (G'N R) ~
Proof. Assume the contrary. Note that direR(S) = 3 implies direR(R) >_ 3. Hence from the above Lemma it follows that R has a positive-dimensional center. Moreover r decomposes into two transversal Ad(S)-stable subspaces: r = r rl g' 9 V . Obviously Is, V] C V lh g' = {0}. Moreover [8, g' lh r] = {0}, because d i m l ( R N G ~) < 1. Thus R commutes with S. Hence the center of R is central in G, contrary to the antieanonieal fibration being injective. 13 T h e .4-orbits are t w o - d l m e n s i o n a l
L e m m a 5. The group ,4 = (~' n R)O acts transitively on the [~-orbits. Proof. Assume the contrary. Since/] is a normal Lie subgroup of 0 , it follows that the/i-orbits are complex one-dimensional. Note that direR(A) >_ 2 by the above Corollary. Hence by Lemma 8.2.3 it follows that the A-orbits can't be real one-dimensional. Thus they are real two-dimensional. Now consider the fibration G / H "--, G/It
1
GII
l
O i l = 01,4
Note that any /-orbit in I/f-/ contains an A-orbit and is therefore open in ] / [ t . Since A is connected it follows that I / H = I / H . This contradicts Lemma 14.1.2. ta
165 2. T h e case S ~- SL2(R) In this paragraph we u s u m e in addition to the basic assumptions that S ~_ S L ~ ) . Note that this implies G / J "" H + . Hence a Borel group of S acts transitively on G / J . Exploiting this fact together with the minlmality assumption we deduce that dims(G) = 6 and that the complex structure on X is essentially a left-invariant complex structure on GL2(R)K (R 2, +). A discussion of such left-invanant structures with methods similar to the methods used in the solvable case then finally leads to the conclusion that there doesn't exist any such left-invariant complex structure on GL~(R)K (R 2, +). Therefore the basic assumptions imply that S ~ SLy(R).
L e m n m X. Assnme that S ~ SLy(R). Then the 9ronp R does not act transitivelll o n J / H , and the growp S has only three-dimensional orbits in G / H , i.e. the grotlp S f3 J has no fized poiut in J / H . Proof. The isotropy Sf3J is a maximal compact subgroup of S -~ SLy(R). The group S has a two-dimensional Borel subgroup B such that S = (S f3 J). B. Note that S f3 J _~ S t . The minimality assumption implies that B K R doesn't act transitively on G / H . Since B acts transitively on G / J it follows that R doesn't act transitively on J / H . Since J = (S [3 J) K R, this implies that every S f3 J-orbit in J / H is real one-dimensional. [] L e m m a 2. Under the above assumptions J / H ~_ C2 \ R 2 and fwrthermore G is as a real Lie group isomorphic to the wniversal cover of GL2(R) K (R 2, +).
Proof. (i) We have seen in the preceding paragraph that under the above assumptions there exist fibrations GIH
1 1
r
CJI[-I
1 1
GIz
OIt
GIJ
01] = OIRH = Pt
with dimc(G/J) < dimc(G/I) < dimc(G/H). Observe that J / i = C and j//~r _~ Cr because the unipotent group A = (G' N/~)0 acts transitively on the /~-orbits. (ii) Since the R-orbits ate analytically Zatiski-dense in the /~-orbits, it follows that either J / I _~ C or J / I _~ H + . In both cases any S 1-action on J / H has a fixed point, thus S f3 J fixes some point in J / H . Hence wlog S n J C I . Since all S f3 J-orbits in J / H ate real one-dimensional, this implies that there exist a fixpoint free S 1-action on I / H . Consequently I / H ~_ C*. (iii) Let Lt denote the ineffectivity of the G-action on G / I and L1c its complexiflcation in G (Caveat: This is not necessarily the ineffectivity of the G-action on G / i ) . Next observe that Lt has a fixed point in G//~/, because I / H ~- C*
166
and ]/~I = C. Thus the normal complex Lie subgroup L~ fixes a point in G / H . Since the G-action is almost effective, this implies dima(L1) = O. Therefore G acts almost effectively on G / I . (iv) Now we prove that J / I _~ C. For this assume J / I ~_ H + . This would imply that R acts almost effectively on J/1, hence diml(R) _< 2 and consequently dima(G) < 5 contrary to dima(G/H) = 6. (v) Now J / H is a real analytic C" -bundle over C. Since any R-orbit is analytically Zariski-dense in / ~ / f - / _ Ca , the real two-dimensional R-orbit in /~//~/is not a complex-analytic set, hence J / H is not Stein and therefore it follows due to the classification of homogeneous surfaces ([OR,Hu]) that J / H ~- Ca\R ~. (vi) Now J is s solvable group acting transitively on C"z \ R ~. Moreover both R and S N J act almost effectively on J / H ~- Ca \ R ~ (The R-action is almost effective by Lemma 8.2.1). Thus J is as s real Lie group isomorphic to the complex two-dimensional Borel group. In particular A = ( RnGV) ~ is isomorplfic to (]~2 + ) and both R and S are acting non-trivially on A by conjugation. Since A u t a , ( R 2) _ GL2(I~), this implies G ~- GLa(R)K (I~3, +). n L e t ~ - i n v a r i a n t s t r u c t u r e s o n GL2(R)K (R ~, + ) In this section we first assume that there exists s left-invariant complex structure on GL~(R) K (R 2, +) with d i m a ( r + J r ) = 4 = dima(r ~+ Jr~). We will analyse this structure with methods similar to those used in the solvable case. In particular we choose a base of g as canonical as possible and calculate explicitly using this base. This will finally lead to a contradiction. Therefore we obtain as a result that there doesn't exist such left-invariant complex structures on
GLa(R) K (R 2, +).
P r o p o s i t i o n 3. Under the basic assumptions S ~ SL2(I~).
Proof. This follows from the Lemma below, o LenaLma 4. There exist no iefl-invariant complex strscture on GLa(]R) K (R 2, +)
swch that dima(r + J r ) = 4 = d i m a ( r ~ + Jr~). Proof. Assume the contrary. (i) Note that dim,(r ~ + J r ~) = 4 > 3 = dima(r) implies that J r ~ cannot be contained in r . On the other hand dima(Jr ~N r) _> 1, because dima(Jr') = 2, dima(r) = 3 and both are contained in J r ~+ r j , which is real four-dimensional. Hence d i m l ( J r ~N r) = 1 and we may choose a non-zero element Z E J r ~ N r ) . Now Z E r s implies [JZ, Z] ~ 0, because r = < J Z >R ~ r ~. Thus we m a y wlog assume
[JZ, Z] = Z.
(ii) Clearly J Z ([ g'. Thus dima(r ~ N J g ' ) < 1. Hence dima(r ~N J g 0 = 1, because dima(G') = 5 implies dima(r j s J g ' ) _> 1. Thus we choose a non-zero element W E r s CI J g ' .
167 (iii) [JZ, Z]Z and J Z E r imply that ad(JZ)lr, =- id. Thus [JZ, W] = W . Next let us consider the following integrability condition: [JZ, JW]
= J[JZ, W ] +
J[Z, JW]
Em'lS'f
=JW
EJrSf<Jg,JW>
+[Z,W]. =0
It follows that [JZ, J W ] = 0 and [JW, Z] = W . (iv) Consider the linear map
ad(JZ) : g ---, r N g' = < Z, W >/t 9 It is surjective. Hence b := Ker(ad(JZ)) is real four-dimensional. Observe that b is the centralizer of J Z in g and therefore an algebra. Thus g = bK r ~. This implies in particular that b __ gl2(R ) and that b ~ is a maximal semisimple Lie subalgebra of g. Hence we define s := b'. (v) Let B be arbitrarily chosen in b f3 J b . Then [JZ, JB] = J [ J Z , B] + J[Z, JB] + [Z, B] =0
=0
EJr ~
Er ~
(Observe that J Z is central in b, therefore [JZ, JB] = 0 = [JZ, B] .) Since J r ~n r ~ = {0}, this implies that any B E b I"1J b cormnutes with Z. (vi) From the representation of SL2(R) in GL2(R)K (R 2, +) it is cleat that the linear map ad(Z) : g --~ 1" is zurjective and that the intersection of the kernel with a maximal semisimple zubalgebra of g is real one-dimensional. Therefore dimR(Jbf3z) _< 1. Conversely diml(JbNs) > 0, because diml(b)+dimm(s) = 7 > 6 = diml(G). Thus dimR(Jb N z) = 1 and we choose a non-zero element X E Jbnl. (vii) Now IX, Z] = 0. This implies that X is represented in sl2(R) - sl(r ~) as a unipotent element. In particular [X, W] : ~Z for some ~ E R \ {0}. (viii) Observe that X, J W E 9 JZ, J X ~ 9 and b = < X, J X , JZ, J W > s . Therefore 9 = < X, J X + p J Z , J W > s with p E R \ {0}. Recall that the choice of X and W was unique only modulo real scalar multiplication. Hence we may wlog assume that p = I and ~ = 1. (ix) Note that the structure of GL2(~t)K (R ~, +) implies that the Lie algebra homomorphism ad : b ---, gl(r') -~ gl2(R ) is bijective. Therefore in order to understand b we study the representation of b in glR(r' ). Now we are in a position to determine this representation
168
completely. We already know the following [JZ, Z] = [Jz, w ] = [X, Zl = Ix, w ] = [Jx, z] = [ a w , z] =
Z w 0 w 0 w
Observe that if V E g~ then trace(ad(V)lr, ) = 0. Therefore it follows that [JW, W] - 7Z for some 7 E R. Furthermore J X + J Z E g~ and [ J X + JZ, Z] = Z imply that [JX + JZ, W] = - W + ~Z for some ~ E R. Hence [Jx, w ] = - 2 w + ~ z . (x) Let p denote the Lie algebra homomorphism b --, glR(r~). Note that if A, B, C E b and p(C) = Lo(A), p(B)] then C = [A, B], because p is in~ective. Therefore we can calculate the commutator relations in b from the above deduced representation in glR(r~). These explicit calculations yield [X, J W ] = J X + J Z - ~X [JX, J W ] = ~(JX + JZ) - 2 J W + (47 - ~2)Z (xi) So far we haven't use the following integrability condition [JX, J W ] = J[JX, W] + J[X, J W ] + [X, W]. Now we will use this equation to obtain the desired contradiction. From the above equations it follows that [aX, J W l = ~(VX + JZ) - 2 J W + (47 - ~ ) Z [JX, W] = - 2 W + ~Z IX, J W ] = J X + J Z - ~X
Ix, w ] = z Hence 2 ~ J X + (47 - ~2 + 1 ) x + J Z = 0.
This equation is false for any ~, 7 E R. D 3. T h e case S ~- SU2 In this paragraph we assume that S ~- SU2. We will first discuss the structure of the radical/~ and its unipotent normal Lie subgroup A = (G~N/~)~ . We already know that A = (Gf N/~)0 acts transitively on the /~-orbitj and we will see that the SU2-representation in Aut(A) is non-trivial. Then we prove that A is ahelian, i.e. A ~ = (e). A key point in the proof of this is the fact that any totally real irreducible SU~-module is real odd-dimensional, while N / N ~ is always real even-dimensional, if N is a real Heisenberg group.
169 Thezeafter we determine the structure of (g, J ) in detail and finally realize it as an algebra of vector fields on Ps \ L. This leads to the conclusion that X is biholomorphic to t~,2 = ([z0 : . . . : z3] IIz012 + Izffil~ - Iz212 - I z 3 1 ~ > 0). This is a manifold which is aJso homogeneous under a number of other Lie groups, in particular it is an open orbit of SU2,2 in Ps. L e n n n a 1. Let V r be a complcz vector space. A u w m e that V c is SL2(C)-modwle and that SU2 stabilizes a totallll real vector sub9 Then d l m c ( V v ) is odd-dimensional.
an
irrcdncible V o] V c .
Proof. (i) Assume the contrary and choose m such that d i m c ( V c ) - 2m + 2. Let < H , X, Y > denote the usual base for sl2(C). Then ~fter appropriate conjugation in i we obtain 9 - 9 - < i l l , X - Y , i X + i Y > R . C h o o s e Z~ E V c as above. Note that n = 2m + 1. Then from a weight space decomposition for i H it follows that Y equals the direct sum of its intersections with the spaces V~ - < Z k , Z n - k >C. Hence YN < Zm,Zm+l >C is a real two-dimensional totally real subvectorspace of < Z,,, Zm+l > c - Thus there exists a ~ E C such that A ---- Zm "~-~Zm-I-1 ~ V. (ii) Note that [H, Zm] = Zm, [H, Zm+l] = - Z m + l , IX, Zm+l] = (m + l ) Z m and ~Y, Zm] = (m -~- 1)Zm+ 1. Hence
Jill, A] = iZm - ~iZm+l E V. Moreover IX - Y , A] = - ( m + 1)Zm+x + ~(m + 1)Zm - (m + 2)~Zm+2 + m Z , ~ - i =~
- ( m + 1)Z,~+I +/~(m + 1)Z,. E =~
1/
/~Zm - Z,.+I E V
(iii) Recall that we assume that V f3 < Zm, Zm+l > is totally real. Hence it follows that ~Zm - Zm+l is a real linear combination of Zm + ~Zm+l and iZm - ~ i Z m + l . Thus ~Re~i~Im~ = -1. Since ~ Re ~ - i/~ I m ~
=
I~1=, this
is the desired contradiction. D
Contradict the case G - R • S Here we assume that the Ad(S)-action on R ~ trivial, Leo G = S • R . We will see that this assumption leads to a contradiction. We consider G/H
1
GIJ
~
G/H
~
G I ] = ~ I R H ~_ P1
!
We recall that .4 = ((~' N/~)0 acts transitively on the /~-orbits.
170
I,enuna 2. From the asslmptions it follows that all the (~,-orb~ts in G/I~ are complez one-dimensional.
Prool (i) Since S is normal in G, any two S-orbits are biholomorphic. Note that ~/~r is the orbit of the unipotent group ((~' f3/~)0. Hence I / H ~ C 2 . Now from a homotopy sequence applied to the bundle (~/H --, (~/] it follows that lr2(G/H) _~ Z and lrl(G//f) = 1. Therefore S ~ SLy(C) cannot act transitively on (~//~r, because x2(S) = {e} = xx(S). Hence the S-orbits are at most twodimensional. (ii) Let us assume that the S-orbits are two-dimeusional. Note that S = ~ . Hence S is a normal linear algebraic subgroup of (~ and in particular the Sorbits are closed. Observe that Wl(G//~/) = {e) implies that H is connected. Since S is also connected, it follows that S H is a closed connected subgroup of G. Thus G / S / t is a simply-connected one-dimensional solv-manifold, i.e. ~/~/~r ,~ C. Hence from a homotopy sequence it follows that lr2(S/~r//~r) = Z and ~rl(SH/H) = (e}. Thus S H / H is biholomorphic to the two-dimensional aiTlne quadric. (iii) By assumption the R-orbits in G / H are two-dimensional. Therefore ~ H has one-dimensional orbits in this aifme quadric. Note that/~NS/~/is the radical of S/~/. But if a complex Lie group acts transitively on the affine quadric then its radical acts trivially. Hence we obtained a contradiction. Thus the S-orbits are one-dimensional. [3 P r o p o s i t i o n 3.
Under the basic assumptions the S-action on [i cannot be
trivial. Proof. (i) Assume the contrary, i.e. G = S • /~. Then from the above Lemma it follows that the S-orbits are complex one-dimensional. Hence the S-orbits in G / H are at most real two-dimensional. Consider the fibration
G/H
!
G/I
"---,
~
G/~I
1
G/Rft =PI
It follows that the radical R of G acts transitively on I / H . (ii) Now consider the following fibration G/H
l
'--*
G//~/
!
Note that G / J ~_ C 2 , i.e. G/J is a contractible Stein manifold. Therefore G / H --, (~/S/Y/ is a holomorphically trivial fibre bundle. Observe that J/f-[ Pt. Since S ~_ SU2, it follows that S acts transitively on J i l l . Thus J / H = J / H and hence G / H "" G / J • P1, i.e. G/H is a direct product in contradiction to the basic assumptions. This is the desired contradiction. [3
171
J / H 7~ Ca \ R 2 As an application of the above proposition we prove
L e m n m 4. Assume S ~-- SU2 and consider the fibrotions
G/H
!
1
Then itfolloutsfrom the basic assumptions that J/H ~ Ca \R 2. Proof. Assume the contrary, i.e. J/H ~_ C a \R 2. Then J is a solvable Lie group acting transitively on J/H ~_ Ca\R 2. Furthermore the R-action on J/H is almost effective by L e m m a 8.2.1. There is only one real solvable Lie group acting transitively and effectivelyon J / H , hence the representation of J on J / H is as a real Lie group isomorphic to the complex two-dimensional Iksrel group. This implies dima(/~) = 2 and 1 _< dima(R/R') _< 2. Since any non-trivial SU2-module is at least real three-dimensional, this yields a contradiction to the above proven fact that the representation of S ~- SUa in Aut(R) is non-trivial. Hence J / H ~_ Ca \ R 2. 13 T h e g r o u p A is abp-lian To prove that A = (G' f'l R)0 is abelian we assume the contrary and finally deduce a contradiction. A key point herein is the fact that any totally real irreducible SU2-module is real odd-dimensional, while for any real Heisenberg group N the quotient N / N ' is real even-dimensional. We will use this to achieve a contradiction to the minimality assumption. But first we will prove that the A-orbits in G / H are real three-dimensional.
L e m n m 5. Under the basic assmmptions the orbits of the gromp A = (G' CIR) ~ are real three-dimensional. Proof. (i) Observe that iJ has complex two-dimensional orbits. Hence it is clear that 2 < dima(A(=o)) < 4 for zo E G/H. (ii) Assume dima(A(zo)) = 2. Since the A-orbits are analytically Zariski-dense in the A-orbits, it follows that the A-orbit are totally real subsets of G / H . By Lemma 8.2.3 this implies that dims(A) = 2. This contradicts the fact that there is a non-trivial representation of S ~_ SU2 in Aut(R). Hence dimlt(A(zo)) >_3. (iii) Recall that A is nilpotent. Thus by Lemma 9.4.4 dima(A(zo)) = dima(A(zo)) would imply that A(z0) = ~J(z0) and therefore G/H = G / t / , since G / J = ~ / ~ r = P l . Hence dima(A(zo))= 4 is impossible. Therefore dima(A(z0))= 3.13
172 L e m m a 6. Assnme that A := (G'NR) ~ is non-abclian. Then there are fibrations
G I HI
~
OI l~l
GIX
= GI(G' n R)'H
1
!
with d i m e ( G / J ) = 1 and I / H = H + . Fwrthermore I/[-I ~ C ~- :f/I and either or J / I ~_ C. Moreover dims(G' N R)' = 1.
J/I ~- H +
Proof. (i) Recall that J] = ((~' N R) ~ acts transitively on the /~-orbits. Since Y] is nilpotent, it follows that the commutator _A' can't act transitively on the Aorbits. Hence dimc(A'/(.~l/f3/~)) = 1. Furthermore ]1/~/,~ C and J / i ~_ C, because Jl is unipotent. (ii) Either I / H = .T//?/~ C or I / H "~ H + . Assume I / H ~ C. Then by the same arguments as in the proof of Lemma 14.1.2 it follows that G / H is trivial contrary to the basic assumptions. (iii) Note that G ' N R is nilpotent and acts almost effectively on I / H ~_ H + . Hence d i m l ( G ~f3 R) ~ = 1. o
L e m m a 7. It follows that JII = Y / I
c.
Proof. As we have seen above the representation of S "" SU2 in Aut(R) is nontrivial. Hence dima(G' N R) __>3. By Lemma 8.2.5 the ineffectivity of G' N R on G / I cannot exceed dimension one. Thus the isotropy group of the G ~N R-action on G / I is not ineffective. By Lemma 8.2.3 this implies that the G ~ N R-orbits are not totally real, i.e. G' fl R acts transitively on J / I . Thus J / I ~ H + , because G ~N R is nilpotent. Hence J / I ~- C. n Corollary 8. Let B denote any Lie subgroup of A = ( G ' N R) ~ such that A ~ C B and dims(B) >_ 3. Then the asswmption dims(W) > 0 implies that B acts transitively on the A-orbits. Proof. Since I / H ~_ H + , the ineffectivity LB of the B-action on G / I acts ahnost effectively on I / H ~- H+. Since B is nilpotent, this implies dima(LB) 2 it follows that the B-orbits in G / I can't be real one-dimensional. Therefore they are real two-dimensional and consequently the B-orbits in G / H are real three-dimensional. Since B is a normal subgroup of A and dima(A/(A N H ) ) = 3, this suffices to prove that B acts transitively on the A-orbits. [3 L e m n m 9. Asswmr that A is a real nilpotent Lie group with one-dimensional commutator w Moreover asnme that A / A I is odd-dimensional. Then
d/mR(W) < where Z denotes the center of A.
di t(Z) < d/mR(a),
173
Proof. Assume that dima(A e) = dime(Z). Then A would be a Heisenberg group. But for a Heisenberg group A/A' is always even-dimensional, n L e m n m 10. Let Z denote the connectivity component of the center of the Lie group A = (G oA R) ~ . Them from the basic asswmptions it follows that Z - A'.
Proof. (i) Let 9 denote the Lie algebra corresponding to Z . First we will show that d i m e ( Z ) _< 2. For this purpose assume the contrary. Then from Cor. 14.3.8 it follows that Z acts transitively on the A-orbits. Hence A = Z - ( A A H ) s a d therefore ,4' C (A A H ) . This is the desired contradiction, since .4' acts freely on its orbits. (ii) Observe that from dime(Z) < 2 it follows that S ~ SU2 commutes with Z . Let sx denote the centralizer of 9 in a. Then a has s transversal Ad(S)-st&ble vector subspace V, i.e. a = V ~ s x with Is, V] = V. (iii) Note that a is a Ad(S)-stable sub 9 of r . Hence r = c ~ a with [s, e] C cAg' = cam = {0). Observe that g = g ' ~ e . Now S acts transitively on G / 3 s a d A has three-dimensional orbits in J / H . Hence G' has five-dimensional orbits in G / H . Therefore by L e m m a 8.2.6 the minimality assumption implies dime(c) = 1. Thus e is a Lie subalgebrs of g . Since c commutes with s, it follows that ad(c) stabilizes V. Hence go := s + c + V + a' is a Lie subalgebra ofg. (iv) Since S acts non-trivially on r , it follows that dims(V) >__3. Hence V + a ' generates a normal Lie subgroup At of A which acts transitively on A / ( A A H ) . Therefore Go acts transitively on G / H . Hence the assumption of minimality implies G = Go, i.e. A' = Z . [3 P r o p o s i t i o n 11. Under the basic assumptions (G' O R) ~ is abelian.
Proof. Assume the contrary. Define c sad V as in the proof of the above Lemma. Since V + a' = a is the algebra of a Heisenberg group, it follows that V is an real even-dimensional vector space. Now V is a direct sum of irreducible SU2modules. By L e m m a 14.3.1 each irreducible SU2-module is odd-dimensional. Hence V is the direct sum of at least two irreducible S-modules. Since each of these modules is at least real three-dimensional, it follows t h a t dime(V) >_ 6, i.e. dime(A) _> 7. Thus dima(A O h) _> 4. Now choose A1, A2 E a such that a = a ' ~ < A 1 , A 3 > a ~ h n a . Note that a n h is an abellan subalgebr&, because a' A h = { 0 ) . Hence b := {X E m n h I [X, Ax] = [X, Az] = 0} is central in a. Now from dima(A') = 1 sad dima(A n H ) _> 4 it follows that dime(B) >_ 2. This is the desired contradiction, since Z = A' .n
174
4. T h e e a s e A a b e l i a n So f~r we proved that A = (r f3/~)0 is abelian and acts transitively on the two-dimensional/~-orbits. Furthermore we have shown that S ~- SU2(C) acts non-trivially on A by conjugation. We already know that there are fibrations
G/H
l GII !
~
GIJ
611I
t !
GIJ =
= P,
We will discuss these fibrations now
L e n n n a 1. From the basic assumptio, it ]oliou~s that J / I ~ C, I / H ~ H + and J / H ~" H + • C . Proof. The group ~] is an abelian unipotent group acting transitively on C2 and A is a real subgroup having three-dimensional orbits. Hence J / H is an open submanifold of C 2 stable under a real three-dimensional group of translations. This implies J / H ~_ H + x C. We already know that / / H ~ I / H . Hence / / H - H + and J / I ~- C . [3 L e m m a 2. The radical R contains a real one-dimensional stJbgrowp C such that R = CKA and [S,C] = {e}. Moreoverwiof we may assume that Sf3J C H and C C I . Proof. (i) From S ~- SU2(C) and S / ( S f3 J ) _ Pl it follows that S N J _ S I . Note that any holomorphic S x-action on C or H + has a fixpoint. Thus we m a y wlog aasume that S N J C H . (ii) Hence from dimn(A(zo)) = 3 it follows that the orbits of G' = S K A &re real five-dimensional. Therefore direR(G) = diml(G') + 1 by L e m m a 8.2.6. (iii) Since Ad(S) stabilizes the subspace a of r , it also stabilizes a transversal subspace c in r . The space c is real one-dimensional, because g' = s + a. Hence c is a subalgebra. Note that c I'1 g' = {0} implies that Is, c] = {0}. (iv) Note that I is solvable. Hence I / H ~- H + implies that H is a normal subgroup of I , i.e. the S I"1J-action on I / H is trivial. Hence the S I"1J - a c t i o n on J / I must be non-trivial since there is a non-trivial S-action on A. Obviously S I"11, which is contained in H , fixes the point e I in J / I . All the other S f3 J orbits &re real one-dimensional. Since C commutes with S it follows that C stabilizes the same point el. Thus wlog C C I . 13 Next choose a non-zero element Co E e. Let /t = ( ~ a ga be the weight space decomposition for ad(C0). Note that a = ~ a I"1(Va + V~). Especially
175 a
=
al ~ a~ where a~ = & and
~1 = ( ~ V. aER
,i2 = i ~ Vo. ~,lot L e m m a 3. From the basic assumptions it follows fhst
a2 =
{0}.
Proof. (i) We consider the fibrations
Gin
~
Ol~z
GII
! 1
,-,
01?
GIJ
~
O/Y
1 !
Note that C stabilizes A as well as A13I and A13H. Moreover from I / H ~_ H + it follows that a2 13 i C h. Hence A2 cannot act transitively on the A-orbits. Conversely I - H C At 9H . (ii) Assume that dima(A2) > 0. By the minimality assumption it follows that (S x C)K A1 cannot act transitively on G/H. In particular A1 cannot act transitively on the A-orbits. Since the A-orbits are real three-dimensional, it follows that the Al-orbits are at mvst real two-dimensional. Note that the A113/-orbits in I / H ~_ H + are real one-dimensional. Hence it follows that the Al-orbits are totally real subsets of G / H . By Lemma 8.2.3 this implies that
dima(A1) 3. Observe that A2 cannot act transitively on the real three-dimensional A-orbits. Hence dima(A~/(A2 f3 H)) _< 2. By Lemma 8.2.3 it follows that the A2-orbits &re not totally real in G / H . But they are at m a t real two-dimensional. Thus they are complexanalytic subsets of G / H . (iv) It follows that there are fibrations G/H
1
G I I = GIA~H
~
t~/~f/
,-,
Oli = Olden
!
with I I H = i l l / . This contradicts the non-triviality of G / H by Lemma 14.1.2. [] C o r o l l a r y 4. The endomorphism Ad(C0) E GL(a) has onill real ei~envalaes
and is therefore triagonalizable. L e m m a 5. The representation of S in Aut(A1) is irredmcible.
176
Proof. (i) Note that "
= "1 =
n v.) aER
Define aa := a f3 Vs. Since C commutes with S, it is clear that the spaces s e are Ad(S)-stable. Next choose a E R such that [s, as] ~ {0}. We m a y wlog assmne that a = I, because we m a y substitute ax-C0 for Co. Next we choose b C al such that b is a non-trivial irreducible S-module. Note that dims(b) >_3. (ii) From dims(b) _> 3 it follows that either B acts transitively on the real three-dimensional A-orbits or the B-orbits are complex one-dimensional complexanalytic subsets of G / H . In the latter case we obtain a contradiction to the basic assumptions in the same way as in the proof of the above Lemma. Hence B acts transitively on the A-orbits. Thus (S x C) t< B acts transitively on G / H . By the minimality assumption it follows that G = (S x C) t< B, i.e. B : A. o
L e m m a 6. The Lie algebra g has a base < X, H, Y, V, Z 0 , . . . , Zn >C- For an appropriate choosen base point we obtain | = < V, H, X > c ~it and fwrthermore s ]a = < Z 0 , . . . , Z , - 2 > c . Proof. Note that G / I _ ]P1 Hence we may assume that ~ = < H, X > c ~f: 9 ft. Note t h a t g is a two-codimensional subalgebra of i. Since/tnl~ has codiraension two in it, it follows that i~ = , ~ N I ~
< H + AhX+
A 2 , V + As > c
with Ai E it Note that it is abelian. Hence the ad(Ai)-action on it is trivial. Thus it f3 !~ is stable both under the ad(H)- and under the ad(X)-action. This implies /i N i~ = < Z 0 , . . . , Z , - 2 > c [] 9
L e m m a ?. The algebra a is three-dimensional, i.e. n ~- 2.
Proof. (i) The algebra it< it has a real form at< a. By Lemma 14.3.1 it follows that a is odd-dimensional, hence n = 2m for some m > 1. Note that iH E i stabilizes a. Consider the weight space decomposition for ad(iH) of it. It follows that - =
- n (< z,.+,, z,._, >c). 0_C) is a totally real vector subspace of < Z m + i , Z , , - , > c , which is stable under the ad(iH)-action. For i ~ 0 the base vectors Zm+i and Zm-i are ad(iH)-eigenvectors corresponding to non-real complex eigenvalues (for i = 0 note that Zm is an 0-eigenvector of ad(iH)). Hence for i ~ 0 it follows that aN < Z m - / > c = {0}.
177 (iii) Recall that if31~ = < Z 0 , . . . , Z n - 2 > c . Consequently af31~ = {0} if n = 2 and anla = aN < Z2,...,Z,-2 >c for n = 2m _> 4. It follows that the A-orbits in G / H are real three-dln~nsional for n = 2 and real four-dimensional if n > 2. But the latter contradicts the basic assumptions since it would imply that G / H is trivial. Therefore n = 2. o
Lenxnm 8. We may wiow assmme that g = < i l l , X - Y, i X -~- iY, V , Z0 - Z2, Zx, iZ0 -~- iZ2 >n
Proof. We already know that g = s ~ ( g f 3 < V > c ) ( 9 ( g N i ) . Since S ~- SU2(C) we obtain s = < s~l, X - Y, i X + ,'Y > a after conjugation in s. Observe that a = a / 3 < Z0, Z2 > c + a N < Z1 > c 9 Note that we m a y substitute ~Zx for Z1 with ~ E C* without disturbing the Lie algebra structure of g . Therefore we m a y assume Zx E g. This implies [X - Y, Zx] = 2(Z0 - Z2) E g . Furthermore [ill, Zo - Zx] = 2(iZ0 + / Z z ) E g. Finally observe that [~V, ZI] = AZ1 implies that g f3 < V > c = < V > 1 . n
L e n n n a 9. We obtain i~ = < H + ~Zl + ~Z2, X + ~fZ1 -~- 6Z2, V "~"fZl -[" 7Z2, Z0 >C
with ~, ~, 7, 6, ~, 7 E C. Proof. This follows immediately from s f3 ]a = < Z0 > c in connection with s
= c
Lemmt
+ s
10. The Lie algebra strmcture of g implies that
=6--0 andS=-27=-27,. l.e.
I~--c. Proof. Consider the following commutator relations in I~. [H + a Z l + / ~ Z 2 , X + 7Z1 + 6Z2] = 2 X - 26Z2 - / ~ Z 1 - 2aZ0
(1)
[H + a Z l +/~Z2, V + cZl + 7Z2] = - ( 2 7 + ~)Z2 - aZ1
(2)
[ x + ~z~ + 6z2, v + ~z~ + 7z2] = - 6 z 2 + (,7 - ~)z~
(3)
It follows from (1) that 2X - 26Z2 - ~Z1 - 2aZ0 = 2(X + 7Z1 + 6Z2) - 2aZ0 and therefore 6 = 0 and 27 = - ~ . Furthermore from (2) it follows that a = 0 and 27 = - ~ . implies 7 = 7. Thus ~ = - 2 7 = - 2 7 . 13
(4) Finally (3)
178 L e m m a 11. Let [ denote any Lie s.balgebra of g with h C ~ and dimc([) -dimc(f ) + 1. Then
i = < H + / ~ Z 2 , X , V - ~Z2, Z0,Zl > c Proof. (i) Note that dirnc(i~ni) = 1. Hence 1 _< dirnc(|nit) _< 2. First assmne that f3 it is one-dimensional. This implies that | + it = g. Since it is abelian, it follows that | f3 it is stable under the Ad(G)-sction. In particular A d ( S ) must stabilize ~f3it contrary to the irreducibility of the S-action on it Thus it follows that dimc([ f3 it) = 2. (ii) Recall that X - ~ Z 1 E i~ C |. Hence | and especially ~f3it are ad(X)-stable. Therefore ~ n it = < Z0, Z1 > c . Thus
|=
c +!~ = < H + / ~ Z 2 , X , V - ~ Z 2 ,
Z0, Zl > c .
[] Lemrrm 12. We mall tvlog asswme that/5 = O. Proof. G/~f/ is a C-bundle over P1 with a non-trivial action of the radical /~ of G. Therefore the manifold G / f / is a line bundle over Pl. This implies that "" SL2(C) has a complex one-dimensional orbit in G / H . Now /~ = 0 iff the S-orbit through e / t is complex one-dimensional. Hence it is clear that by changing the base point z0 = e/~ we may wlog assume that/~ = 0. []
L e m m a 13. It follows that h = < iH > a if e j~ R and h = < ill, V -F eZl > a if 9 E R . Furthermore i = < ill, Z t , V > a . L e m m a 14. We may wiog asswme that 9 = -I-i. Proof. Recall that
i~=< H,X,V+
9
Z0 > c
a = < H , X , V , Zl, Z0 > c 9
Now da = < H, X, Zo > c is the maximal ideal of [ contained in !~, i.e. the ineffectivity algebra of the I-action on I / / f . The algebra | is represented on I / H as < V, Zl > c with IV, Z1] = Z1. In particular I / f / _ _ C and the vector fields V and g l are represented z~-;, and ~ for an appropriate coordinate z on ]/~f/-~ C. Note that V + eZl vanishes at - 9 since V + 9 = (z + 9o)~s" Consider the /-orbits in I//~/. From the above Lemma it follows that a point z G I / H ~0 C is on an o p e n / - o r b i t iff z E C \ R . Thus either I / H = H + or I / H = H - . Correspondingly we may choose -t-i as base point in I / H . Hence we may wlog assume that either 9 = i or e = - i . [3
179 Concrete re-!;~tion By giving a concrete reaIi=ation in Ps we prove that X _ 2,2. First we claim that G / f i _ P s \ L with L = {[z0 : . . . :z s] [z0 = Zl = 0}. We note that P s \ L m a y be covered by the two open subsets UI, U2 with
It, :={[zo:...:z3) lz,#o}
-- ~ .
We choose coordinates zl z2, z3 on [To by zi := ~J@ and coordinates t01, tv2, ws on U1 by wl := ~ , and tv~ := ~ for i = 2, 3. We will prove G / I / _ Ps \ L by giving explicitly a realization of the Lie algebra g on P8 as an algebra of globally integrsble vector fields. The following is a realization of g as an algebra of global vector fields on U0.
2=2~
2z1~+
H=
=2 8 X = -
8
8
8
1~-7= x - z1=2~-z2 - = l z 3 ~ . a + =2~.za
OZ1
OZ2
8
#
v = -=2~-~=2 - = ~ 3
8
Z0 = -zl~; 3 # Z1 = zlOz 2 Z: =
8 #zs
8 az2
On [To f3 U1 this coincides with the following rea/ization of g on U1 : 8 H = -2w1~-~wI
8 2wsaw s
x = a_+.2o___ Owl
y - _w1~01
8ws wlw303 _
v = -~3g~
0
0
3 - w2g~--~ 2
#
go
=
--
ZI
= -wl O-~--+ 8
aws
8ws 8 Z2 = +wlOw 2
8w2
_
8
_
8
~1~2F~-~2+ ~3F~2
180 Note that all these vector fields are globally integrable on 1~ . Furthermore for (=l,Z~,zs) = ( 0 , 0 , - e ) , i.e. [z0 : . . . : za] = [1 : 0 : 0 : -r the isotropy algebra is just I~ = < H , X , V + CZI, Z0>r Observe that a point in (~[H is contained in an open G-orbit iff it is contained in a real four-dimensional R-orbits. Thus to determine the open G-orbits it suffices to check where the radical r = < V , Z 0 - Z ~ , iZ0 + i Z a , Zt>R spans a real four-dimensional subspace of the tangent space. Observe that on [To 0 v
Z0
=
0 -
Zt = z~t~z~_ 0 Z~ --~--oza
Ozs 0 0z3
iZ0 + iZ2 -- i~z2 0 - izx Next we determine where these four vector fields are real linearly independant. --Z2
det
X2
--X3
~
-
Zl -1
-1 -Zl
-=1/
-i
-ix1
i$x
-- - 4 ( [ z t [ 2 + 1)/m(Z2Zl -k z3)
Hence a point (zt,z2,z3) E [TO lies in an open G-orbit iff
0 # -4(Ixlla + 1 ) I m ( z a ~ t + z3). Similar calculations on Ut yield that a point (to1, w2, w3) E [71 is contained in an open orbit if[ 0 # 4(Iwl[ a + 1) Irn(ws~l + w2). Note that
I m ( z 2 ~ t + z3) = ~i z2~l - z1~2 + z3 - zs) = ~z2~1
- zt~2 + z3~0 - zo~a) (Izl - iz21 ~ + Izo - iz312 - I z l + iz212 -Izo + izsI 2)
1
and
1
- 2ijz-~l z3~o - zoO3 + z2~] - z1~2)
1 = 8 1 ~ t j l z l - iz2[ ~ + Izo - izsl 2 - Izl + iz~l ~ - Izo + iz~l~).
181 Hence [z0 : . . . z3] 6 P3 is contained in an open G-orbit iff
Iz, - iz212 + Iz0 - izal 2 = Izx + iz212 - Izo + izsI 2. (Observe that [zo : . . . : zs] 6 L implies Iz, - iz2]~ + Izo - izsI ~ = Iz, + iz21~ Iz0 + iz312. ) This suffices to prove the following P r o p o s i t i o n 15. Assmme that X = G / H is a three-dimensional homogeneons complez manifold with almost injecti~e anticanonical fibration. Assume that the representation of g in Fo(X, T o X ) is totaiig real, that no complez Lie group acts transitieei~l on X and that X is not bikolomorpkic to a direct prodmct of lower-dimensional homogeneows manifolds. Assume that G is of minimal dimension among all Lie groups acting holomorphicall~l and transitively on X . Finally assnme that dims(S) = 3 and that R does not act transitively on C,/[I. Then X is biholomorphic to n2,2 = {[z0 : ... : z,] I lzol 2 + Iz, I2 - Iz212 + Iz312}
Proof. Since R does not act transitively on r and dims(S) = 3 it follows that the R-orbits in G / / ~ are complex two-dimensional. Hence the assumptions of the proposition imply the basic assumptions of this chapter. From proposition 14.2.3 we know that S ~ SL2(It). Hence S ~- SU2(C) and the proposition follows from the preceding considerations, o R e m a r k . The manifold G / H ~ fl~,2 is a C x H+-bundle over P1. The restriction of this bundle to P1 \ {z0} with z0 E P, is again a homogeneous manifold, but not via a subgroup of G. 5. T h e case dims(S) > 3 In this paragraph we discuss the case that dims(S) > 3. To be more precise: we assume the following B a s i c a s s - m p t i o n s . X = G / H is a three-dimensional homogeneous complex manifold. The group G is of minimal dimension among all Lie groups acting transitively on X . The anticanonical fibration is injective and g is represented in Fo(X, T o X ) as totally real subalgebra. The R-orbits in G//~r are complex two-dimensional and dims(S) > 3. Observe that d i m c ( R / ( R f3/t)) = 2 implies that the R-orbits are closed (see Lemma 2.2.14). Hence there are fibrations
G/H
1
G/J
~-, G / H L - , P N ~
l
r
We will discuss these fibrations in detail in order to understand G / H . Finally we will obtain the following result
182 P r o p o s i t i o n 1. Under the above basic assumptions G / H is biholomorphic to
~2,2 = {[z0 : ... : zs] IIz012 + Iz, l2 > Iz=l 2 + Iz312}.
For the proof we first discuss the structure of G in detail. We will find out that under these basic assumptions G and G / H are uniquely determined. Finally we will prove that this uniquely determined structure can be realised in IPa\ L. This lea& to G / H "" f12,3. But first we deduce the following auxiliary remark.
L e m n m 2. Under the basic asswmptions I / H is a proper swbset of ] / t l , i.e.
i/8 # i/U. Proof. Assume the contrary. There are two possible cases: Either G/1 ~_ H + or G / I = (~/] = Pl. The first case implies that G / H is a holomorphic fibre bundle over the contractible Stein manifold G/1. From this it follows that G / H is biholomorphic to the direct product of H + and I / H , contrary to the basic assumptions. The second case implies G / H = G / / t in contradiction to the basic assumptions, c3 The structure of We will prove the following
Lemr,~, 3. Under the basic asnmptions G/[I = tf k ~ H k
with k E H, i.e. G / t l is biholomorphic to a vector bundle of rank two over PI which is a direct smm of two copies of the k-tit power of the hyperplane bmndle over Pl. F~rthermore
where ~, -_ SLy(C) and pl : ~ --. GL(C2), ~ :~2 --" GL(C*+x) are e,'ed,cibte representations.
Proof. (i) Consider the fibration
and let L denote the ineffectivity of the G-action on the base. Let $2 denote a maximal connected semisimple subgroup of L. Then
~--(SL2(C~ x $2) K k and L = ~ , a K k .
183 (ii) By Lemma 8.2.2 it is clear that $2 acts almost effectively on /~/(/~ A [/). This implies that /~/(/~ A / t ) _ C2 and $2 -~ SL2(C). Furthermore it follows that L is represented in A u t o ( R / ( [ t A H)) as SL2(C) K (Cr +) or GL2(C) g (Cr +). Thus L G G' is represented as SL2(C) K (Cr +). (iii) It follows that ~ A ~' -- i l | i2 with Si acting only on /~. Furthermore dirnc(ia) = 2 and the St-action on i t - C 2 is non-trivial. (iv) The Sl-action o n / q is irreducible, because otherwise the G-action on G / H could not be almost effectively, n Lem~
4. Tkere is an abelian Lie subyronp C in f~ swcA tAaf =
•
K ( 6 ' n R)o.
Furtkermore dime(C) ~_ 1 and C acts on [g' n ~ as the group of complez scalar multiplication, if dime(C) = 1. Proof. The semisimple group S s t ~ i f i r ~ the vector aubspace i = g' A ~ in ~. Hence S also stabifizes a transversal subspace 6. Since ~ n g ' = {0} it follows that [i, ~] = {0}. Recall that the anticanonical fibration was assumed to have discrete fibres. Thus the center of g is trivial and in particular the morphism p : ~ --~ a u t ( g ' O ~) must be i~jective. Since the C-action commutes with the S-action, it is clear that C can act on g' A ~ only by complex scalar multiplication. Hence dime(C) _< 1. Finally dimr = 1 clearly implies that is an abelian Lie group, a L e m m a 5. Under the basic assumptions G = (S x C) K ( G ' n R )
with dimz(C) _4. (ii) It follows that diml(G' n R) _> 4. Note that A = (G' A R) ~ would act transitively on I / H , if its orbits are real four-dimensional. Therefore they are at most real three-dimensional. (iii) Recall that the A-orbits are analytically Zariski-dense in the ~]-orbits. It follows that they are either real three-dimensional or two-dimensional and totally real. But the latter case implies by Lemma 8.2.3 that diml(A) = 2. Hence they are real three-dimensional, n From this result we deduce
185 Lemn~
8. The manifold G / H is a C x H + -blndle over ~#1.
Proof. Recall that S ~_ SL2(C) s implies G / I = P l . Furthermore A = (G'f3R) ~ is abelian and has real three-dimensional orbits in A / ( A f3 t / ) . Thus I / H is a open subset of Ca invariant under RS-translations. Hence I / H ~_ C x H + . []
T h e r e p r e s e n t a t i o n o f S in A u t ( R ) We determined already the Lie group structure of C;-" (SLy(C) x SL2(C) • (~),< (Ca | C~+1). Now G is a real form of G with S ~_ SL2(C) s . We will prove that the existence of such a real form implies k - 1. Then we will determine the structure in detail and give a concrete realization. L e m m a 9. Let p denote the irreducible representation of S = $1 • $2 with S~ ~- SLy(C) in G L c ( C "a | C h+l) given as a tensor product of an irreducible representation of $1 in Ca and an irreducible representation of S~ in C t+l . Then p(SL2(C) R) stabilizes a totally real subspace of Ca | C TM iff lc = 1. Proof. Let V c denote the space Ca | C T M . Let r denote the corresponding Lie algebra homomorphism : t -. sic(re).
Choose bases < H~,Xi,Y~ > c in it as usual. Let s -- (aHx + ~H2 + bXx + bX2 + cY1 + c-Y2 [ a, b, c E C}. Choose E,,m E V c such that r ( H 1 ) ( E . , m ) = nE.,m and r ( H 2 ) ( E . , m ) = mE.,.~. Then a base of V c is given by
E-1,-t,
E1,-k,
E-1,2-I:,
E1,2-k,
E-1,4-t,
...
,E-I,t
E1,4-k,
...
, El,k.
Assume that V is a totally real subspace of V r with d i m a ( V ) = d i m c ( V C ) . Assume that V is stable under p(SL2(C)R). Since V is stable under ~'(H1 + H2), it follows that wlog E - 1 , - t E V. Then X l + X 2 E s wlog implies E 1 , - t + E - L 2 - t E V. And from iX1 - iX2 E s it follows that i E L _ t - i E - 1 , 2 - t E V. Since ill1 - ill2 E s, from E 1 , - t + E - 1 , 2 - t E V it follows that (k + 1)iE1,-t + (/c + 3 ) i E - 1 , 2 - t E V. Since V was assumed to be totally real in V c , we can conclude that It +
]:
t-
3] = [1: -1],
i.e. k + 1 = 3 - k. Therefore k = 1. rl Furthermore from this proof it follows that the ~-(s)-stable totally real subspace V of V c is unique. Hence it suffices to give one example fulfilling our assumptions to prove that G is isomorphic to this example. Therefore we obtain
186 L e m m a 10. The growp G ~ = S Jr A is isomorphic to
o~:{(~ -~ ~ : ~ ) 1 ~ , ~ o ~ . ,
~, _~t
with G ~ o = I ( A"
A ' B ) I A 1 A, E S L , ( C ) , B E C a x ' } A1 '
This enables us to deduce the following statement. L e m u m 11. Asstme G ~ G'. Then G is isomorphic to
o~_{(~ -~ ~:~)1~o~,~ ~ ,
~, _-~}
Proof. This follows immediately from the above proven fact that G = C x G I with direR(C) 3, this implies
V = { w ~ C ~ I ~'.w
~IR}.
Q
Now we are able to determine the dimension of the R-orbits explicitly. L e m m a 15. Let R =
Then dima(R([zo : zl : z2 : z3]) =
3 4
if zoO2 + zl~s = z2~o + zaz,1 if ZoO.2+ z1~.3 g z2~o + za~q
Proof. (i) First note that from the representation of G on Ps \ L it follows that the radical-fibration G//?/--, (~//~/7/ is given by [ZO : Zl : Z2 -" Z3] ~
[2'2 : "g3] ~ ~1"
(ii) Observe that
R=
A~I
a E C, p, q E R, A G R+
.
~-1 (iii) Note that
~al
q
z2Zl
~-1
zs
+~_az~z~+ qz3 || = ~z~ + A- l z3
J
~(aZ2z2+ qzs) za
189 (iv) Recall that a
Hence the above auxiliary L e m m a implies that [z0 : zl : z2 : z3] is contained in a real four-dimensional R-orbit iff
i.e. itt" zoO'2 + zl~'s - z2~'o - z3i'l = O. o Actually this result suffices to determine the open G-orbits in Ps \ L: L e m m a 16. A point zo E P3 \ L is contained in an open G-orbit i ~ it lies in a real four-dimensional R-orbit. Proof. T h e manifold G / / ~ = Ps \ L is a C 2-bundle over P l . We know that there are two open G-orbits realized as C• H + -bundles over P1 and a G-stable CR-hypersufface. Observe that any two R-orbits in the same G-orbit are conjugate, hence it is clear that the two open sets consisting of real four-dimensional R-orbits equal the open G-orbits. t3 Finally we observe that Z0Z2 Jr ~gl ~'3 -- Z2~'0 -- Z3Z1
: --~ IZ2 -- iZo [2 + [zs -
iZl 12 - [z2 -I- iz0 [2 - Iz3 -~- iZl [2)
This completes the proof of the proposition stated at the beginning of this paragraph.
190
Chapter 15 Holomorphic
flbrations
in the case dimR(S ) > 3
1. GenerAlities Basic A s s u m p t i o n s . Let G/H be a non-trivial homogeneous 3-fold. We assume that G = S K R is a Levi-Malcev decomposition with dima(S) > 3 and that G acts ahnost effectively on G/H. Let G/H -~ G/I denote the G-saticanonical fibration and G/H --+ G/J the "maximal holomorphic flbmtion" given by Prop. 9.3.4. We require that J/H or R be positive-dimensional. Furthermore we require that the ]~-orbits in ~/iF are at most complex onedimensional. 2. T h e s t r u c t u r e o f G/I We will start with
Lemnm 1. Let
! denote the G-anticanonical ]ibration of G/H. Assume that dimc(G]I) < 3. Then either d / i ~_ P2 and ~ ~_ SL3(C) or ~ / i ~- P1 • P1 and ~ ~_ SL2(C) •
SL2(C). Proof. (i) Let L denote the connectivity component of the ineffectivity of the Gaction on G/I. Furthermore let SL denote a maximal semisimple Lie subgroup of L. Note that by Lemma 8.2.2 the group SL acts almost effectively on I/H. Observe that I/H ~ is a complex Lie group with 1 _< dimc(I/H ~ a ) = 5. This is possible even if 3, and 6 are complex linear-independent, i.e. if C a / < ' r , 6 >z--- C" x C ' x C . It occurs if 6 = a a + b~ + c'r with (a, b) E Ca \ {(0, 0)} and c E JR. Next we will introduce appropriate coordinates on A "~ Ca. For this purpose let (zl, z2, zs) be coordinates on 2] such that
r = ~-l(e) = { ( m , . , 0)Ira,. ~ z} and
= {(0, 0, z)lz ~ C}. Define Ei := {(zl, z2, zs) E Ca Iz, = 0} for 1 < i < 3. Since both Ei and /~0 = Ri'li'
are two-dimensional vector subspaces of A _~ (Ca, + ) , it follows that for any i
198 either Ei = ~ 0 or the intersection Ei n ~ 0 is complex one-dimensional. Note that H n / ~ is discrete, because R is complex one-dimensional and I = R/:/. Thus ~ 0 # Ei for i = 1, 1, since /~ C Ei for i = 1, 2. Hence the intersection El n ~ 0 is complex one-dimensional for i = 1, 2 and
~ 0 = ( # n El) $ (/~ n E2). This suffices to prove the following
L e m m ~ 1. There exist A, p E C swch that /t = H~
= {(* + n, Y + m, Az + py)lz, Y e C, n, m e ~}.
R e m a r k . Note that we may apply an automorphism ~ E C* = Aut(C, +) on the rwclical R to obtain (A and ( P instead of A resp. p. Thus for (A, P) # (0, 0) only the value [A : p] E IPa(C) is characteristic for / t . Now we discuss the special c~se A = P = 0. L e m m a 2. Assume that p = A = O. Then the bundle G / H ~ G / I is globally
holomorphically trivial. Proof. The assumption A = p = 0 i m p l i e s / : / = S A I . It follows that d i m e ( S ) dimc(S n / : / ) = 2. Consequently direR(S/H) _< 4. Since S acts transitively on the simply-connected base G / I it follows that the S-orbits in G / H are realanalytic sections in the holomorphic principal bundle G / H -+ G / I . Note that by dimension reasons the i-vector fields span the whole tangent space of the S-orbits. Hence the S-orbits in G / H are complex-analytic sets and therefore holomorphic sections in the principal bundle G / H ~ G / I . But a holomorphic principal bundle with a holomorphic section is globally trivial, o C o n v e n t i o n . From now on we assume that (p, A) r (0, 0). So far we discussed mainly the complex groups, now we will establish the relation to the real form G of G. L e m m a 3. The isotropy group I of the real form S = S L 2 ( C ) It -
{(A,A)IA E
SL2(L~}
at to = ([1: 0], [0: 1]) on IP, x ~t equals
Define J := r - l ( I ] ( I n P ) ) . Then s = { ( . + re, v + - , . ) 1 - , u, z ~ c , m , .
e x,.
= {(*,~ + - , . ) l * , z e c , . e z } . Hence
(Jq n J) ~ = {(., ~, ~ . + u ~ l * ~ c}
= ~}
199 and
( ~ n J n,~) ~ = {(~, ~,o)1,~= + ~,~ = o}. Furthermore L e m m a 4. It follows that
Proof. Observe that
J = {(=,~+ t,z)l=,z ~ C,t ~z} and
Ar = {(= + . , y + m , ~ + ~w)l=,v ~ c , . , ~ ~ x } ___{(~, w, x(~ - . ) + v(w - m)l~, ~ e c , . , m r z } Assume that (z, ~ + k, z) = (v, w, X(v - n) + p(w - m). Then z = v, ~ + k = w and z = A(v - n) + p(w - m ) . Consequently
n J = ((=, ~ + t, x(= - . ) + ~(~ - m)l~ ~ c, t , . . , .
~ z}.
[] Obviously the Lie group H ~ must be closed in G. Hence we first determine for which p, )t the group
{(= + m , ~ + n , ~ +
,~I= r c , m , .
~ ~)
is a closed subgroup of A. For this purpose we use the following auxiliary result. L e m m A 5. Let A be a lattice and V a real vector sabspace of R m. Let < A >It denote the real vector subspace spanned by A . Then A + V is a closed Lie
s,,~gro,,p of (R", +) iff d i m l t ( V + < A >1) = rank(A) + direR(V) - rank(A n V). L e m m a 6. For any p, A E C D = {(= + m , ~ + n, ;~= + i , ~ ) l z 6 C , , - , n ~ Z} is a closed Lie st~bgroup in A = (Ce, +). Proof. (i) Observe that
/-t o = ( ~ n j)o = {(~, ~, x~ + ~,~1= ~ c} and I - / = H ~
(ii) Let (=, ~, X= + i'~) r :t~ ~ d ( , . , . , O) r r . A . u m e (z, ~, Xz + z~) = (m, n, 0).
200 Then z = $ = m = n and (A + p)z = 0. Therefore e if A # - p 7/. i f A = - p .
FnI-/~ Thus r+Ho~{R2+2~
-
R 2+7/-
2 ifA#-p
if~=-p.
(iii) The space t / + < r >1 is spanned by (1, 1 , A + p ) , ( i , - i , i ( A - p ) , (1,0,0) and (0, 1, 0). Hence
dim~t(~/+l)= (4
3
ifA#-p
if A = - p .
It follows that H is a closed subgroup for all p, A E C. ra
Corollary 7. From the proof of the above Lemma it furthermore follows that C* if A # --p
I/H ~
C
if~=-v.
Lemma 8. The Lie subgroup d n ~I has the following str~cture as an abstract real Lie group.
Jn~__
( ~ 2 x Z 2 if[#:AlEll~l(Q) R 2 x X 3 if[p:,~]~PI(Q)-
Proof. Note that J n H = ( J N/~)0 + A with A = {(0, t, An + vmlt, m , - r X } Observe that A_
X Z2
if [A : p] ~ ~I(Q) if [A: p] E PI(Q) 9
Prom
( ~ n 1) 0 = {(., ~, ~
+
~ ) I - ~ c}
it follows that
A n (J n/~)o = {(o, o, 0)}. This completes the proof. 13
Lemma 9. Let V denote the real vector sebspace of A spanned by J n ft . Then direR(V) =
{~
/ f ~ : ~] G " l ( ] l )
if ~ : ~1 r ~1(~)"
Proof. Observe that V = {(z, ~ + k, A(. - n) + ~,(~" - m)) Iz ~ C, k, m, n ~ R}.
201 Application of the real linear endomorphism
: (zl, ~ , xs) ~ (z~, x~ - ~ , ~s - ~
-/,~)
yields
~(v) = {(~, t,-:~. - ~,(~ + t))l~ e c, t, m,. ~ a}. Therefore
dimlt(~(V) ) diml (V )
I,5 if[p: A] r I~(R)"
[]
L e m m a 10.
The group Z~ = C n / t
is dose~ i~ [~ : ~] ~ (PI(C) \ Px(R)) U
~(Q). Lemma
11.
The group S acts transitively o . G / H i g I~1 # I.I-
Proof. Since S is normal in G it follows that S acts transitively on G / H iff S has an open orbit. Note that dima(S t'l H ) = d i m a ( / t t'l J f'l ~) and ( ~ n J n ~)0 = {(x, ~, o)1~ + . ~ = 0}. Consider the real linear map ~b : z s-+ Az+p~: from C - R ~ to C = R ~ . Observe that
rank(~/)
=
2 i f IAI # I~1 1 if I~1 = I~1 # 0 0
if~=p=0.
Hence S has an open orbit in G / H if I~1 #
I~1-
n
Since S ~ SL2(C) as a real Lie group, it follows that for Igl # IAI the universal covering G / H ~ of G / H yields an exceptional left-invariant complex structure on SL2(C). Now we want to discuss under which circumstances the group H is closed in (~. L e m m a 12. Under the above assumptions
i / f f ~_ C~ < ~, ~ >~ Proof. Observe that ]/[-I "~ A / [ f and = ~~
= {(x+ .,y+
m,~x +,y)lx, y ~ c,.,m
~ g}.
The vectors ( 1 , 0 , ~ ) , (0, 1,p) and (0,0, 1) form a base for Ca = A. Note that /~0 __< (1, O, A), (0, 1, p) > a . Now the Lemma follows from the fact that the lattice F is spanned over g by the vectors (1, O, A) - ~(0, O, 1) and (0, 1, p) - p(O, O, 1).n C o r o l l a r y 13. For p = 0 and -~ ~ Q, i.e. for [~ : ~] ~ PI(Q), the group ~r is closed and ] / t I ~-- C*.
202
For ~ 9 R \ Q the group H is not closed. For -~ ;~ 9 C \ R the group :-[ is closed and i/f'[ -" TI. Assume that /:/ is connected. Let H1 = /~ n G. following fibrations G/H ~
/ G/H
Then we obtain the
l G/H1
~
G/f-I
G/~
~
G/i
We will now investigate the structure of H1. In particular, we want to determine, whether G / H I is simply-connected, i.e. whether H1 = H ~ 9
S. Coverings Assume that H is a closed Lie subgroup of G. Then we have the following fibrations G/H o
G/H
/
1
G/H1 "~ G//~
where H1 : = / ? / A G . Although the g r o u p s / : / a n d G are by definition connected there intersection H1 may have several connectivity components. We will now discuss this problem in detail. The group HI is connected iff H A J = ( / ~ n J ) ~ ' . In general H 1 / H ~ ~_ ~ I / ( J n /~)I'. L e n u n a 1. (i) For [~ : #] ~ ~I(Q) it follows that H 1 / H ~ "~ Z . (ii) Assume [A: p] 9 IPt(Q). Let r, s 9 Z such that gcd(r, s) = 1 and [),: p] = [r : s]. Then H 1 / H ~ ~_ Z iff A = - p . For )~ ~ - # the yroup H I / H ~ is a finite cyclic froup of order [r + s[.
Proof. Observe that /~ n J ___{(x, ~ + t, ~(x - .) + ~,(~ - m))lx 9 c, t, m , . 9 ~}
and
( ~ n J ) ~ _- {(x, ~ + ~, x(x - . ) + , ( ~ - .))Ix 9 c, k , . 9 z}. Hence
/~ n J = Ao + (/~ n J ) ~ with Ao := {(0, 0, - # p ) I P E Z}. Thus
HI~It ~ ~_ fI n J/(( ft n J)~
__ Ao/(Ao n (/~ n J)~
Therefore we have to discuss under which circumstances (z, 9 + k, ~(z - n) + ~,(~ - n)) = (0, O, - ~ p ) .
203 This is equivalent to z = 0, k = 0 and n(~ + p) = pp. Hence (0, 0, - p p ) E (f! n J ) ~ iff
P--Y--P A+p e z. In particular it is clear that H 1 / H ~ ~_ Z if [A : p] r PI(Q) or A = - p and that H 1 / H ~ ~- Zir+, I otherwise, n 9. S n m m J , r y We now want to summarize the above results. P r o p o s i t i o n 1. Let X = G / H be a three-dimensional homogeneous complex manifold. Let G / H ~ G / I denote the G-anticanonical fibration. Assume that no complex Lie group acts transitively on X and that X is not biholomorphic to a direct product of lower-dimensional homogeneous manifolds. Furthermore we assume that dima(S) > 3 and that G acts almost effectively on X . Finally we require that at least one of the following is positive-dimensional: I / H , m = g f 3 i g or R. Then the manifold X is bihoiomorphic to one of the following manifold. Conversely all of these manifolds listed below are homogeneous complez manifolds.
(i) ~2,~ -- {[~o: z~: ~2: z~] I lzol 2 + I~12 > Iz312 + Izsl 2} (ii) The ,~trictio. to P~ • P ~ \ ~ f or ~ ( ~ \ ~ ( ~ t ) or ~ \ ~ of a line bundle over Pz • P1 resp. P~ generated by a positive divisor. (iii) The restriction to P2(C) \ P2(~t) or P2 \ ~ of any C"- or Tl-principai bundle over P2. (iv) Simply-connected principal bundles over ~1 • ~ z \ P f for any [~: p] ~ PI(C) arising as quotients X[~:~] = G / ( G N/~[~:~])o with G = {(z, A, B) E C x SL2(C) x SL2(C) I A = / ~ }
and
H[~:~] = {(~" + P~'
w
e-"
'
e-Y
I ~,~,z, ~ e C}.
(v) Quotients of the above simply-connected principal bundles by discrete subgroups of the principal structure group acting from the r~ght. Proof. This follows immediately from the preceding paragraplm with the sole exception of the case where the anticanonicM fibration has discrete fibres, m = {0} and the/~-orbits in G / / ~ are two-dimensional. In this exceptional case the proposition follows from Proposition 14.5.1. 13 We can state some further results about the structure of the manifolds
204 P r o p o s i t i o n 2. Let X = G / H = G / ( G n/~[~:,])o be defined as above. Let H1 := G N [/[:~:~]. Denote the G-anticanonical fibration of X by G/H
1 GIZ
~
Gli
,--, P,,.
(i) The subgroup S = SL2(C) R of G acts transitively on G I H iff ]A[ = I~1. Hence for Ihl = I~1 the manifold X[~:~] U bihoiomorphic to the Lie group SL2(C) equipped with a iefl-invariant complex stlctlre other than the usual one. (ii) The gro,p [z[~,:,,] is 9 closed ~ie s , bgroup of =
C
x
SL2(C)
x
SLy(C)
[~ : /~] ~ (PI(C) \ PI(~)) U ]~I(Q). (iii) If [/[~:~] is closed, then G/H1 ---, G / I is the restriction of a principal bundle C / H -~ G / i ~_ ~ • ~1
with fibre
In particular i / H --~ C" if [~: ~] ~ PI(Q).
U~/Ho = ~-o(G/H~) ~_
{
~ ~l,+ol
if [~: ~] r P~(Q) or ~ = -~, if p, : ~] = [r: ~] with r, s ~ Z, gcd(r,s) = 1 and r ~ - s .
In particular G / H t is simply-connected only if [A: p] = [r : 1 - r] with r E Z . (v) The G-anticanonical fibration G / H --* G / I has the fibre I/H...
c" C
if~#-~ if :~ = ~,.
In particular this yields homogeneous complex manifolds for which there does not exist any "eomplexification'. To be precise: P r o p o s i t i o n 3. Let X[~:~] be defined as in the preceding proposition with [~ : PI(R) \ PI(Q). Then X is a complex manifold homogeneous wnder a reai Lie group G such that there does not exist any eqeivariant holomorphic map with discrete fibres to any complex-homogeneous manifold.
~] ~
205 10. Lei%-invariant c o m p l e x s t r u c t u r e s o n r e d u c t i v e Lie g r o u p s A left-invariant complex structure on a real Lie group G is a complex structure such that the left multiplication L e : h ~ gh is a holomorphic map for all g G G. Hence a left-invatiant complex structure on a real six-dimensional Lie group yields a homogeneous threefold. In the classification of the non-trivial homogenous complex three-dimensional solv-manifolds we have seen that with only two exceptions for every three-dimensional solv-manifold there exists a real solvable Lie group which acts transitively and freely. Thus they arise from leftinvariant complex structures on real solvable Lie groups. Observe that there exist real even-dimensional Lie groups without any left-invariant complex structure. In contrast to this there are many homogeneous manifolds not coming from a left-invariant structure in the non-solvable case. And it is known (see [Mo]) that every real even-dimensional reductive Lie group posseses a left-invariant complex structure. A real six-dimeusional reductive Lie group G is one of the following: (i) isomorphic to SL2 (C) as a real Lie group, (ii) $1 x $2,
(i~) S l x A (iv) or A, where Si is a real form of SL2(C) and A an abelian real Lie group. In any case let S denote the maximal semisimple connected Lie subgroup of G and A the radical of G. Let G ~ G / I denote the G-anticanonical fibration. First let us discuss the cases (ii)-(iv). The anticanonical fibration has positive-dimensional fibres in the cases (iii) and (iv), because A is central in G . In case (ii) it has positive-dimensional fibres due to [Ma, Th.1]. Note that I is isomorphic to a complex Lie group, since G ---* G / I is the anticanonical fibration. Since G is not isomorphic to SL2(C) in the cases (ii) - (iv) it follows from dime(I) 0} for 1 -n,m'- ~tln,mW. The corresponding norm is 2
Let v E C " + = such that II~ll~,~ # 0. From the non-degenera~:y of this hermitian form it follows that we may construct an orthogonal basis ( ( v l , . . . , vn+m)) for C n+m with v = ~)1 and [iv, lib,m = =i=Hvi[~,,,~ . It is cleat that n of the norms *
o
[[vil[n m are positive and the other m of these values are negative. Xssurne that ( ( v l , . . . , V , + m ) ) and ( ( w l , . . . , W , + m ) ) are both orthogonal bases. Furthermore assume that II~,ll~,~ = IIw, ll.+~ 2 for ~ i. Then the linear endomorphism $ defined by ~b(vi) = tvi is an isometry. Hence we obtain the following Lemamm 2. Let v , w E C n+m such that I1~11.,~ 2 2 = Ilwll.,~. an i s o m e t ~ g such that g(v) = w . Now we are prepared to prove the following
Then there exists
210 Lemnxa 3. The group SUn,m has the following three orbits in
n+
= {[zo : . . . : z.+.,-a] I ~ l z ,
l' > Y~lz, l'}
i_n
Iz, I2 = ~
i 2 , 1 = 0 for sJl z E E . Since C s = E + E • this implies < ~, z > 2 , 1 - 0 for all z E C s . Hence ~ - 0. T h u s the restriction of < . , . >2,1 t o E is nondegenerate.G L e m r n ~ 5. Assnlne that Ilwl[~,l = o, i.e. that the restriction of 2,1 to
E • is deffenerate. Then E • C E and for an appropriately chosen base in E the restricted hermitian form is represented by the matriz
P r o 4 It is cleat t h a t IIw'll],l - 0 implies w' 9 E •177= E and therefore E • C E . Next choose a base ((Vl, v2, vs)) o f C"e such t h a t v2 9 E and v2 = w ' 9 E • . T h e n < . , . >2,t is represented b y a m a t r i x
A=~]t
=
q
with a, b 9 C and p, q 9 R . B y substituting a real multiple o f v2 for v2 we m a y wlog assume t h a t q = d : l . Now observe t h a t det(A) < 0 since the h e r m i t i a n f o r m < . , . >2,1 has signature (2, 1). B u t det(A) = -qlbl 2. Hence q > 0.[3 T h u s the following combinations m a y occur signature o f the restriction o f < . , . >2,1 t o L E E•
214 + + +
++ +0 +-
0 +
0
+0
0
0 -
++-
+ +
Obvously thin yields six SUa,l-stable subsets in Fl,a(3). But moreover from the Lemmata 16.2.3 and 16.3.1 it follows that SU2,1 already acts transitively on these subsets. This implies the following P r o p o s i t i o n 6. There are three open SU2,x-orbits in
Fx.~(3) = Q = if[z], [~]) I [z], [~] e P2(C) n d ; ' ~ = 0}. T h e s e are
F+++ = {([z], [~]) e O I I1=11]: > 0 ,,,,d Ilwll]: < 0} F~_ = if[z], [,,,]) 9 @ I I1"11]: > 0 ,,,a 11"1122,1> 0} F~._ = {([z], [,,,]) 9 @ I Ilzll],~ < 0 ,,rid I1"11~,, > 0} The orbits F++ and F+,_ are biholomorphic. (The indices at F denote the signatere of the restriction of < .,. >24 to L and E ). 4. T h e case S = Spa(C) a n d ~//2/~_ Pa(C)
(1)
(1)
First let us introduce some language, mc6tly in accordance with [Hel, p.444ff]. Let J and K denote the following matrices.
1
and K : =
1
-1
-1 -1
" 1
Now Spa(C) = {A E SL,t(C) I A ' - J "
A = J}.
Let SUe,2 be represented in SL4(C) by SU2,2 = { A E SL4(C) I A '
.
K . A = K}
Then the non-compact real forms of Spa(C) are given by Spa(R) = Spa(C) N GL4(R) and
Sp,,~
= Sn(C) n Su2,2
l~Lemitrk~ Since Spa(C) ~_ SOs(C) it is clear that these real forms correspond to real forms of SOs(C). Actually sp2(R ) _~ sos,~ and s p l : "- so4,1.
215 T h e case S ~- Sp2(R) In this paragraph we assume that S -~ Sp2(R) = Sp2(C) f3 GL4(R). Obviously Sp2(R) stabilizes the totally real submanifold Pa(R) in Pa(C). Moreover the following Lemma holds. Lemma
1. The real swbmanifold P3(R) of Ps(C) is a closed Sp2(R)-orbit.
Proof. Since S = Sp2(C) acts transitivelyon the 3-fold P3(C) it follows that the Lie algebra i spans the whole complex tangent space at any point of Ps(C). Hence s + is = i implies that any orbit of a real form S of S = Sp~(C) is at least real three-dimensional. This fact suffices to prove the Lemma, because P3(R) is a connected real three-dimensional Sp2(R)-stable subset of P3(C).E3 Next we will show that Sp2(R) does not act transitively on the complement P3(C) \ Pa(R). Actually it stabilizes a real hypersufface in P3(C). Let A E Sp2(R). Then A t 9J 9A = J a n d / ] = A. Hence ~lt.J.A=J.
It follows that Q = {[z] e ~ ( c )
I :J~
= 0}
= {[z] e P 3 ( c ) I z3~1 - z l ~ + z4~2 - z ~ 4 = 0} -
-
{[z]e P3(c) I z,n(zx~3 + z2~4) = 0}
is a closed Sp2(R)-stable subset of ~3(C). W e will see below that Sp2(R) acts transitivelyon the connectivity components of the complement of this Sp~ (R)stable set. A firststep in this direction is the following
L e m m a 2. L e t v e C 4 such that [v] ~ Pa(R). T h e n there crisis a g e S p 2 ( ~ ) and A, a, ~, 7 E R such that g(v) = (A, ai,/~i, 70Proof.
(i) Note that [v] ~ IFzR implies v = s + it with s, t E ]R4 and s, t ~ O. Since Sp2(R) acts transitivelyon P3(R) (see the above Lamina) it follows that there exists a gl e Sp2(R) such that [91(s)]= [I: 0: 0: 0] and consequently
9~ (,,) = ( ~ + ui, ,~i, ~i, -fi) for some A, 14,a,/~,7 E ~. (ii) Note that
a,b, ce ]~} C Sp2(R).
216
(iii) Consider
(x a, 1 c1 -a
,, 1
(
~ + (~ + a a + bE + cT)i \
=
7i
~i (~ - a/~)i
)
"
Note that [ v ] r Ps(R) in particular implies that [v] ~ [1 : 0 : 0 : 0], i.e. (c~, 8, 7) ~ (0, 0, 0). Hence there exist a, b, c E R such that
/,,: pi
=
ffi
9
for some ~,~',ff, T' E R . n L e m m a 3. The statement [A : ai : Bi : 7,] E Q hoids /or A, a, B, T E R and ~ ~ O iff B = O. Proof. Observe that Im(zl~.3 + z2~4) = -/~A for [z] = [~: ~i: ;~i: 7'1.u
C o r o l l a r y 4. The real qwadric Q is connected and its complement in P3(C) consists of two connectivity components. Proof. Since Sp2(R) is a connected group, it follows from the above Lemma that Q and its complement have at most one resp. two connectivity components. Furthermore from the definition of Q it follows that
P~(c) \ 0 = {[z] ~ P~(c) I I m ( z l ~ + z2~,) # 0} has at least two connectivity components.o Next we consider the representation of the Lie algebra s in the tangent bundle evaluated at [1 : ai :/~i : 7s~, i.e. the natural vector space homomorphism ~ : 9 ~ Ttl:~,:p,..,](P~(C3 )
Since the S-action is linear it follows that
ker§ = {A ~ ap2(R) I A . (1, ~i, ;~i, 7i) ~ ( 0 , ~I, ;~i, 7 0 ) c ) Note that
ap~(R) =
_A ~
A, B, C ~ gl2(R), S = S ' and C = C'
217 (see e.g. [Hel, p.446]). Define a linear map ~b : spz(]R ) --, C t by
Then
ke,r = r Lennna
5. For an~
7i))c.
(a,~,7) E Rs\{(0,0,0)}
there ezists an A E
that
Proof. Note
t h a t A E sp2(R ) implies
A =
k
-a
i
-b
for some a, b, c, d, e, f , g, h, k, ! E R . Consider
(i 'e d
!
k i
-a -b
This equation implies a = 0, c = - a , b
9
d
/
-7 i
0 -b
h = -8,
/
k = - 7 and
o,
The latter matrix equation is equivalent to
/~ 7 -7 But it is clear t h a t the matrix
-7
(i)
sp2(R) such
218 has rank three for all (a, 8, 7) E R s \ {(0, 0, 0)}, e.g. for a r 0 the submatrix
-8
-7
~
obviously has rank three. Hence it is dear that it is possible to choose the values b, d, e, f, g, i &ppmpriately.u Thus the vector subspace
V=
AEsp2(R) a.
od /3i
~ai = /)tSi/'AER
7i
\ )tTi /
is real one-codimensional in ker @ for any (a,/~, 7) E R s \ {(0, 0, 0)} We win discuss this vector space V. L e n u n a 6.
The
above defined vector space V has the foiio~ng dimension 7
dima(V)=
ira = 8='r =0
4 i f S - O and(o~,7)~(O,O ) 3 ~fS#o
Proof. Consider the equation d
f
k !
-a
i "
-b
8i
, -
~ 7i)
for A E R . It is equivalent to a = A , c a 0 ,
ha0,
~ai
ASi ~7i / k=0
and
--G
-b
7i
\ aTi /
The latter matrix equation is equivalent to
-8
-7
Ill
no,,
219 Denote -~ a ~ -2j3 -7 -j3 - 7 Observe that B contains the following submatrix
7
B =
a
Hence r . , , k ( B ) = 4 i ~ ~ # 0. Obviously , ' . n k ( B ) = 0 i ~ ~, = ~ = "r - 0.
Thus let us discuss the case ]~ = 0 and (a, 7) ~ (0, 0). Then
B
=
-a
7
a
-7
-7
a
This matrix has rank(B) = 3 for any (a, 7) ~ (0, 0). Together with the fact that A E V requires c - h = k -- 0 it follows that
cod/mR(V, sp2(R)) =
7 if/~ # 0 6 if/~ = 0 and (a, 7) ~ (0, 0) 3 ifct=~=7=O
This completes the proof, since diml(sp2(~)) = 10. n LemmaT.
Let zo = [l : ai : ~i : 74. Then
dim,,(S~(R)(~o)) -
i I ~ ; 0 ,n~ (~, ~) # (0, 0)
Proof. Note that by Le~urna 16.4.5 the vector space V has real c~lime~ion one in the isotropy algebra, if (~, ~, 7) r (0, 0, 0). For ~ = ]~ = 7 = 0 the space V equals the isotropy algebra. Hence this Lemma follows from the above Lemma. a Lem~
8.
The growp Sp2(R) has three orbits in Pa(C).
These are the two
connect~it~ component, of P3(C) \ Q, Q \ Ps(R) ~n~ Ps(R) with Q = ([~] ~ P~(C) I I ~ ( ~ : ~
+ ~4)
= 0}.
Proof. Let z0 E Pa(C). Then the above Lemma implies diml(Sp2(R)(zo)) =
6 5 3
ifz0 ~[ Q if z0 E Q \ Pa(R) if z0 E Pa(R).
This completes the proof, since Q \ Pa(R) is five-dimensional and connected, c]
220 P r o p o s i t i o n 9. The open SI~(R)-orbits in P3(C) are biholomorphic to
n~= = {[z0 : ... : z3] I Iz01~ + Izll 2 > Iz212 + Iz312}.
Proof. Note that 2 Im(zl~s + z2~4) = ]z, + iz3[ ~ + Iz2 + iz41 ~ - [zx - iz31 ~ - [z2 - iz41 ~.
Hence a linear automorphism of P3(C) maps Q on $2,~. o T h e case S _ SpI,I Recall that
su2 =
-el
Z2
~2
zl, z2 9 C and Izxl 2 + Iz~l 2
=
1
Consider the following embedding of SU~ • SU2 in SpLx.
SU2 x SU~ =
-~.2
wl
~,
--~2
w2
Izl[ 2 + Iz212 = 1
~1
IwI12 + Iw212- 1
Next let us discuss the map r : P3(C) ---, PI(R) given by
~([z0 : Zl: z2: z3]) = [Iz012 + Iz212 : Izll 2 + Iz3l 2] Note that the image of r is just
T(P3(C)) -- {[xo : 2:1]9 Pl(~)lXOZl ~ 0}. LemmA 10. The group SU2 • SU2 embedded in SpI,1 acts transitively on the ~bres of thc abow d ~ n e d map ~ : ~ ( C ) --* P~(R).
Proof. This follows immediately from the fact that the orbits of the usual SU2 (C)-action on C2 are exactly the spheres S, = {(z, w) I Izl 2 + I~1 ~ = r}. []
R e m a r k . Thus the map r restricted to S / H is actually the projection of S / H onto the double cosetspace SU2 x S u 2 \ S / H . This is not an equivariant fibration. Our next step is to show that the fibre z-*([1 : 1]) is actually stabilized by the whole group SpI,1. Recall that Sp,,, = Sp(2, C) f3 SU2,2. Hence for any A 9 Sp(2, 2) the equation ~t KA
= K
holds. It follows that
fJtA? K A v : ~tKv.
221
Thus the set {v 9 Pa(C) I~'Kv = 0} is invariant under the Spl,l-action. Since
V = {v 9 Ps(C) [ ~'Kv = 0} = {[~o : ~ : ~3: ~ ] I I=oI ~ + I~1 ~ = I~+1a + I~12}
= r-1([1 : 11)
we proved the following statement Lem
11. The real gnadric
r = {[zo: zl : za: z~] I1~ol a + Izal a = Izxl a + Izal a}
is
a closed
SpI,x-orbit in P3(C).
The complement of this real qusdric in Ps consists of two connectivity components, which are biholomorphic to each other. Our goal is to show that Spl,1 acts transitively on these open sets. For this purpose consider the matrices Ar for z 9 C defined by
V/1 + I~12
Z
l~l-~lxl ~
A,
+ Ixl 2
I
An explicit calculation shows that A~JA= = J and f ~ K A = = K , hence A= E S p l j . Note that Af([Z:0:0:0])=[
1~-+1zl2:~:0:0]
and therefore
,'oA.([I: O" O" 0]) = [1 + Izl~: Izl~]. Similarly r o A.([0
:
1 : 0: 0]) = [Izl 2 : 1 + Iz12].
This completes the proof for the following Lemma L e m m a 12. The group Spl,1 has ezactly three orbits in P3(C). These are
f~+ = ([zo: zz: z2: z3] I1~ol 2 + Iz212 > Izzl 2 + 1~31a} S = {[zo : zz : za : z3] I lzol a + Iz213 = I~zl a + I~sl 2} ~ - = {[zo: z l : z2: ~3] I lzol a + Iz212 < Izzl 2 +lz31 a} C o r o l l a r y 13. The Spl,1-orbits in P3(C) coincide with the SU2,2-orbits.
222 5. T h e c a s e 9 ~-
S05(C) and 9/[1 ~_Qa
First we introduce some notation. Define
1
j.,., :=
and
I
l
1 ] i I
I
"'~ In,m :--
I 1
-1 -1
I Define
SO.+,.(C) = {A 6 SL.+,.(C) [ A'A = 1} SU.,.,(C) = {A 9 SL.+m(C) [ i'I,,,,~A = I.,m} SO.,.,(R) = {A 9 SL.+m(R) [A'I.,,nA = I.,m GL.,m(R) = {A I ].,..A]~,i 9 GLn+m(R)} SO(n, m) = SOn+re(C) f3 GLn,m(~t) Note that SO(n, m) is a real form of SO.+m(C). Furthermore it's true that SO(n, m) = J.,.~ . SO.,m(R)" JEXm. Actually any real form of SO.+m(C) is conjugate to some SO(n, m), if n + m is odd (see e.g. [Ti3]). We are here discussing the case n + m = 5 where SO5(C) acts on the complex quadric
Q3 = {[,] 9 P,(c) I,', = 0}. Observe that SO,~,,n(R) C SUn,,~(C) and ].,,n 9 SU,~,,n(C). Hence
so(., ~) c su.,..(c). It follows that
SO(n,m)
stabilizes the real quadric
s.,~ = {[z] 9 P.+m-,
I :z.,,.z = 0}.
(see Lemma 16.2.3). This implies the following
223
Lennna 1. The following subsets of Q3 are SO(n, m)-stable (with m = 5 - n).
q*.,..
2 > o} = Q3 n n+,~ = {[z] 9 Q31 Ilzll.,~
0~
= {[=] 9 Q31 Ilzll,,,,,, = o} = 03 n s.,,.
2'
= {[d 9 Q3 I I1=11.,,. < o} = 03 n n;,,,, with I1=11.,., 2
=
~I,,mZ.
Lennna 2. The set O+,,, /a non-empty i # n > 2. Proof. First note t h a t [1 : i : 0 : 0 : 0] 9 Q + iff n _> 2. Hence O+,m is n o n - e m p t y if n _ > 2. T o discuss n = 1 let [z] = [z0 : . . . : z4] 9 Q . T h e n
zo2 = -(z~ -I-.-. "Fz~). Hence 4
4
Iz01' = IE:z l < I2 I= l i=1
i=1
Thus Ilzlh2,4 < 0. o 3. The set Qn,m is non-empty iff m >_ 2.
Corollary
N e x t we will p r o v e t h a t SO(n, m) acts transitively on Q+,m. p u r p o s e we need the following auxiliary result.
For this
Let z = z + iy E Ce with z, y E R 5 such that [z] K Q s . Ilzll 2 = Ildl 2 and z •
Lemma
4.
Then
Proof. Note t h a t z*z = 0 is equivalent to Tt
~(z,
+ ira) 2 = 0
i=I
Hence the L e m m a follows f r o m (zi + iy~) 2 = (z~ -- ~ ) + 2iziy~ .n
L e m m a 5. Let n >_ 2 and z E C s such that [z]EQs and Ilzll.,m 2 - 2. Then ezists a A E SO(n, m) such that A(1, i,O,O,O)= z. Proof. (i) Let z x = ( z 0 , . . . , z , - x ) and z u = ( z , , , . . . , z 4 ) . Let z i a n d ~ denote the real resp. i m a g i n a r y p a r t s of z i , i.e. z i = z ~ + i ~ . F u r t h e r m o r e let A' denote Jn,mAJl,,m a n d let e denote (1, I, O, 0, 0). T h e n A - e = z is equivalent to A ' . e = J~,,~z, because n >_ 2 implies t h a t J~lme = e. Observe t h a t J~lmz = (z 1 +
224 ill t, tP -- iz~). Since A' e G L s ( R ) the desired equation A'(1, i, 0, 0, 0) = J~,XmZ
,1 "/
implies
A I -.-
--Z 2
*
(ii) To find such a matrix A' in SO.,m(R) means to construct a base for R s which is orthonormal with respect to < .,. > . , , . and contains the vectors (z 1, tr~) and (yl, _z2). Hence such an A' exists iff 2
2
I1(~ x, 37)112,,,,,, = I1(~1, -= )11,,,,,, = 1 and
< (::, I?), (~', - ~ ) >.~,.,= o.1 (iii) Observethat [z] E Qs implies that
II,,Xll ~ + I1,,~112 =
I1r
+ IIz?ll ~
(1)
0.
(2).
and
zl'y 1 + z2'~
=
Note that equation (1) is equivalent to I~(x], I/2)ll~,,n = I1(~,-z2)l[~,,~ 9 Furthermore equation (2) is equivalent to < (z l, f ) , (y~, _r >n,m = O. From 2 Ilzll.,m = 2 it follows that I1=111~ + Ilulll ~ = 2 + I1=~11~ + I1~11 ~. This implies II(z x, ~1)11~, m + I1(~,-z2)ll~.~ = 2. Hence ~1
II(
1
2
, y )lt.,~ = I I ( ~ , - :
3
2
)It.,. = r
[]
Lem~J
6. Any open SO(k, m)-orbits in Q3 is biholomorphic to 42 Q,,,s-,, = Q8 n a~,5_,, = ([z] 9 P4 1 z'z = o, Ilzll.,5_. > o}
for some n with 2 < n < 4.
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[SD] [SJZ]
[SJ2]
[Sp] [Sd [st]
[stw] [Tan] [Til] [Ti2] [Ti3] [Ti4] [To]
Cn , in: Lie Group Reprentations III, Lecture Notes No.1077, Springer, 1984 Penney, R.: The structure of rational homogeneous domains in Cn , Purdue University Potters, J.: On almost homogeneous compact complex analytic surfaces. Invent. Math. 8, 244-266 (1969) Pyatetskii-Shapiro, I.I.: The Geometry and classification of bounded homogeneous domains. Russ. Math. Surveys 20, p.1 41965) Pyatetskii-Shapiro, I.I.: On a problem proposed by E. Caftan. Dokl. Akad. Nauk SSSR 124 (1959), 272-273 Raghunathan, M.S.: Discrete subgroups of Lie groups. Erg. Math. Grenzgeb. 68 , Springer (1972) Richthofer, W.: Homogene CR-Mannigfaltigkeiten. Dissertation, RuhrUniversitk't Bochum (1985) Sen~, J.: Alg~b~ de Lie semi-simples complexes. Benjamin Inc. New York. (1966) Sasaki, T.: Classification of left-invariant complex structures on GL2(R) and U(2). Kumamoto J.Sci.Math. 14 (1981), 115-123 Siu, Y.T.: Techniques of extensions of analytic objects. LN 8, Springer. Snow, D.: Stein quotients of connected complex Lie groups. Manuscripta Math. (1985) Snow, J.: Complex solv-manifolds of dimension two and three. Thesis, Notre Dame Univ., Indiana (1979) Snow, J.: On the classification of solv-manifolds in dimension two and three. Revue del'Institut E.Cartan 10, Nancy (1986) Springer, A.T.: Invariant Theory. Springer, Lecture Notes 585 , Heidelberg 1977 Steenrod, N.: The topology of fibre bundles. Princeton University Press, Princeton (1951) Steinsiek, M.: Uher homogen-rationale Mannigfaltigkeiten. Dissertation. Univ. Miinster 1981 Stewart, T.E.: On groups of diffeomorphisms. Proc. Am. Math. Soc., 11 (1960), 559-563 Tanaka, N.: On the pseudoconformal geometry of hypersurfaces of the space of n complex variables. J. Math. Soc. Japan 14, 397-429 (1962) Tits, J.: Lie'sche Gruppen und Algehren. Springer (1983) Tits, J.: Espaces homog~nes complexes compacts. Comm. bIath. Heir. 37, 111-120 (1962) Tits, J.: Tahellen zu den einfachen Liegruppen und ihren Darsteilungen. LN 40. Springer (1967) Tits, J.: Free subgroups in linear groups. Jourua/ofa/gebra 20, 250-270 (1972) Tolimieri, R.: Heisenberg manifolds and theta functions Trans.Am.Math. Soc. 239, 293-319 (1978)
228
[Va]
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[w3] [w4] [Wh] [we] [Wo]
[z]
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229
Subject Index In order to find information on a special manifold listed in the classification, the easiest way is to follow the tree diagrams on p. 21 and 86 displaying the structure of the classification. In the following subject index a page number is printed b o l d f a c e if a definition can be found st this page. If a page number is printed roman, this means that some relevant information on this subject (but not a definition) can be found at the given place. anticanonical fibration bounded homogeneous domains C2 -bundles over Pl C2 \ R2-bundles Cs \ R 3 canonical subgroups closedness conditions for subgroups closedness conditions for algebraic subgroups compact homogeneous manifold complex line reduction complexitlcation of a Lie group complexifcation of a manifold
98
5, 10, 92, 152 58 109 150 8Y 22 24 38 6 87 204 C R-hypersurfaces 124 discrete subgroups in semisimple groups 2, 38 functions associated to a divisor on IF2 \ Q1 54 Heisenberg group 4, 124 Heisenberg group is not a commutator group of a nilpotent group 105 Hopf surfaces 3 hypersurface separability 6 ineffectivity 88 ineffectivity for H +-bundles 90 Kobayashi-pseudometric 6 left-invariant complex structures 95, 205 line bundle over the alTme quadric 65 line bundles over ~2 \ Q1 53 line bundles over homogeneous-rational manifolds 51 line bundles over the affme quadric 52 maximal holomorphic fibration 10, 100 maximal subgroups in semisimple groups 46 meromorphic separability 5 minimMity conditions 88 nilpotent groups 105, 172 nilradical 106 principal bundles over homogeneous-rational manifolds 49 principal bundles over I)2(C) \ I)2(R) 195, 203
230 principal bundles over P~(C) \ pseudoconcave manifolds Q~\L radical-fibration representations of SL2(C) symmetric domains vector bundles or rank two over IP1 weight spaces Zariski-density of orbits of real forms
196, 203 8
80 25,191 58 5 62 96
87
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