L e t ]z be a compact complex manifold and let E-->E--> Y be an analytic vector bundle. Then H~ Horn (E, E)) is not only a vector space but also a finite dimen- sional algebra, the (associative) algebra of endomorphisms of E denoted a(E). As usual, E is t e r m e d indecomposable if no exact sequence 0--> E'--> E--> E"--> 0 splits analytically. Now a(E) is related to the indecomposability of E as follows: I f E is in- decomposable and ~ E a ( E ) , the characteristic equation of ~v has holomorphic, hence constant, coefficients. Thus the eigenvalues of q~ are constant, and, if ~ has two distinct eigenvalues, this would give a splitting E = E ' r E". Thus a(E) is an algebra consisting of multiples of the identity (1-dimensional subspace) plus a nilpotent ideal;
i.e., a(E) is a special algebra ([1]). The converse is clearly true and we h a v e P ~ O P O S I T I O ~ 8.1 (Atiyah.). E is indecomposable ~ a(E) is a special alqebra.
L e t now EQ-->Eq--->X be a homogeneous vector bundle over a C-space X = M / V = G/U. Then if ~:U--->GL(E ~) is reducible, E q is decomposable; on the other hand, m a y not be irreducible and E q m a y still be indecomposable (L(U(n)/Tn)). Thus we m a y a s k : i f r ~) is irreducible, then is E q indecomposable?
T H ~ O R ~ 8. I / Eq-->E~->X is a homogeneous vector bundle over a C.space X = G / U = M / V , then, if ~ i8 irreducible, ]~q is indecomposable.
Proo[. The proof is done in five steps.
(i) We shall show t h a t
Q(E "~
is a special algebra; b y Theorem 5, it will suffice to do this when X is K~hler. I f then 11 = 11 r fi0, Q i it = 0 and Q is the complexifica- tion of an irreducible representation of the compact group V. I f ~ is 1-dimensional, the result is trivial and thus we m a y assume t h a t r ~ = 0 where Z(~ ~ = center of G ~ Thus ~ is essentially the complexification of an irreducible representation of a c o m p a c t semi-simple group and we m a y use the theory of weights ([26]).H O M O G E N E O U S C O M P L E X M A N I F O L D S . I 1 4 7
(ii) We denote by V ~, V",... irreducible g-modules with highest weights 2, a . . . . and by E Q, E ~, ... irreducible 5~ with highest weights Q, v . . . Then, by Theo- rem K(w 2),
a(E ~) = ~ V -~ | H~ V ~ | H o m (E ~, Eq)) ~
= ~ V -~ | (H~ V~)| (E ~, EQ)) ~~
= ~ V-k| (E~| Horn (E q, Eq)) ~~
e D(g)
Now those ~ E a(E) such t h a t ~ EV ~ | (E ~ | Hom(E e, Eq))~*~-Hom (E q, Eq) ~0 are simply the multiples of the identity automorphism (Schur's Lemma). I t will suffice to show: if
kv-~i | (e~ k | gz) E V -~ | (E a | (E q, Eq)) ~~ (it ~ 0), then each gz is nilpotent. For this we m a y prove: if
~(e~ @ ga) E(E ~ | Horn (E q, Eq)) ;~ (~:~0),
k
then each g~ is nilpotent. We prove this latter statement assuming t h a t ~0 is semi- simple ((i) above).
(iii) Let E~= ~ E~t, B e = ~ Eq~
4 t E Z (D ~k ~ ~ (e)
be decompositions into weight spaces relative to a Cartan sub-algebra ~)~ 5 0 and let e~EE ~j, %~EE% be weight vectors. For an E~, there wiU be in general several e~j's If ~ E (E ~ | Horn (E ~ E~)) ~, then by Sehur's lemma
= ~. e 5 | g~ (g~ E H o m (E q, Eq)).
1
Since h o ~0 = 0, we have for h E [),
0 = ~ ()t~, h) e~ | g~ (%~) + e~. | Q (h) ~ (%~) - e~ | (e~, h~ ~ (%~), which implies that, for each ] and all k, either
(a~) 9r or
(a~) g~(%~) =# 0 and e(h)g~(ee,) = <e~ - ~, h> ~(e e~).
We m a y examine (a2), and, in this case, g~(%,)eE ~ - ~ and either
/ r - -
(:r g~(q~)-O for all k and some N) or (~) ~ = 0.
I t will suffice to examine (aa); i.e., go.
148 P H . A . G R I F F I T H S
(iv) Now e ~ o ~ = 0 (for all ~ELF) and thus 0 = e~ o ( ~ % | gj)
]
The terms with an e 0 (0-weight) occurring here are
5 2(e~)e_~ | g~ + 5 e o | Q(e~)g o - e o | goQ(e~)
and since the eAj are a basis for E ~, this t e r m m u s t = 0. Thus having picked out a particular t e r m e 0 | go, there exists e e E -~ (e m a y = 0 ) a n d g_~E Horn (E q, E q) such t h a t 2{e_~)e=e 0 and then
2(e~)e | g_~ + e o | Q(e~)g o - e o | go ~(e~) = 0; or (~5) g-~ = ~(e~)go - goe(e~) 9
(v) F r o m (~a) in (iii), it follows that, for ~g=0, (g_~)N= 0 (large N ) a n d thus the mapping go//: ~~ (E ~) defined b y go//(e~) = [go, e(e~)] ( = (a~)) gives a nilpotent repre- sentation of 5 ~ B u t then gH 0 = 0 and go e H o m (E ~, E~) ~ and either go = 0 or all g~ = 0 ((~5)); in either case, we are done.
(il) The Embedding Theorem for Homogeneous Bundles
L e t X be a complex manifold and L - - > L - - > X a holomorphic line bundle. F o r a suitable covering ( U ~ of X, we m a y take a local nowhere zero section ai of L[U~;
then a n y section of L] U~ is given b y Z - ~ ~ (z) a~ (z) (z e Ui). L e t H~ = H~ i:); if for each x E X , there exists a a E H~ such t h a t a(x):#0, t h e n we m a y classically define ~: X-->Pg_I(C), where N = d i m H ~ Indeed, if ~1 .. . . . ~N give a basis of H~ then E I U, is given b y the m a p p i n g Z - ~ [ ~ (z) . . . ~v(z)] where [$0 . . . ~N-1]
are homogeneous coordinates in PN-I(C). I f H-->PN-I(C) is the hyperplane bundle, then ~ - I ( H ) = L . The same r e m a r k s hold for a vector bundle, provided t h a t the global sections generate the fibre at each point.
Now if X = G / U is a C-space, then L - - > L - - > X is a homogeneous line bundle, a n d thus the canonical complex connection (w 5 and [11]) is defined in L.
THEOREM 8. Let L be such that there exists a non-zero a in H~ T h e n the above mappinfl ~ : X--> P N - I (C) is defined. Furthermore, ~ gives a projective embeddinq o / X i/
and only i / c I (L), when c o m p u t e d / r o m the canonical complex connexion, is positive de/inite.
Proo/. Since L is homogeneous, we m a y write L = L ~ for same holomorphic Q : U--> G L ( L ~) (dim L ~ = 1). L e t a : X--> L Q be such t h a t q(x) ~ 0; if x' E X , and if g E G
H O M O G E I ~ I E O U S C O M P L E X M A N I F O L D S . I 149 is such t h a t g(x) = x', t h e n (g o a) (x') = g (a(x)) 4= 0 in (LO)x. (Similarly, for h o m o g e n e o u s v e c t o r bundles if the global sections g e n e r a t e t h e fibre a t one point, t h e y do so a t all points.) F r o m this, t h e first s t a t e m e n t in t h e t h e o r e m is clear.
W e n o w define T : H ~ (Lq)-+Le-->0 as follows: for a E H ~ (Lq), t h e n v ( a ) = a(e) where we consider ~ as a h o l o m o r p h i c function f r o m G t o L ~ satisfying a(gu)=~(u -1) a(g) (g E G, u e U). Then, for u e U, a e H ~ (L~ Q(u) ~ (a) = T p* (u) a (p* = induced representa- tion on cohomology). I n d e e d , z ~* (u) ~ = (~* (u) a) (e) = a (u -1 e) = Q (u) a (e) = ~ (u) z (a).
Thus, if K q = k e r T, the sequence O-->Kq-->H~ is a n e x a c t sequence of U-modules. L e t G = GL(H ~ (L~)) a n d V = {~ e G I ?(K Q) ~_ K~ t h e n G~ V = PN-1 (C). N o w the holomorphic r e p r e s e n t a t i o n ~*:G-+G satisfies ~*(U)~_ V t h e induced m a p p i n g of G/U to G / V is just ~:G/U---->PN-I(C). If U ' = ( e * - I ) ( v ) , t h e n U'~_U a n d ~. is in- jective if a n d o n l y if U'=U. B u t ~:U--->GL(L ~) e x t e n d s to ~':U'-+GL(Lq), a n d t h u s t o p r o v e t h e T h e o r e m , we m u s t show:
LEMMA. Let X = M / V be a C-space, let Q:V--->GL(L Q) be a 1-dimensional repre- sentation. Let ~.~ be the curvature o/ the canonical complex connexion in the bundle L~ --> M / V. Then ~q = ( 2 ~ V - ~ - i ) -1 ~ represents Cl(L ~) and ~q is positive-de/inite i/and only i/ Q does not extend to a C-subgroup ~ ~ V.
Proo/. I n the n o t a t i o n s of w 1, ~o = (2 g V~-I) -1Z~z~ _ ~+ (~, a ) ~o~ A &~. Since H~ q) :~ O, we h a v e t h a t (~, ~)>~ 0 for ~ E ~ + - ~ + ; if, for some a, (~, ~ ) = 0, t h e n we m a y e x t e n d
t o ]? where ~0 = ~o r v~ 9 ~_~. If, conversely, we m a y e x t e n d ~ t o V, then, for some :r E ~.+ - ~F +, e~ ~ ~0 _ 50 a n d t h e n ~[e~, e_~] = ~(h~) = (~, a ) = 0. Q.E.D.
I ) E F I N I T I O I ~ 8.1. A holomorphic m a p p i n g ~:G/U-->PN_I(~) is called equi- variant if t h e r e exists ~*:G--->SL(N,~) such t h a t , for a n y g e G , ~*(g)Y(x)=~(gx)
(xeO/v).
P R O P O S I T I O N 8.1. Any mapping ~ :G/U--+P~-I(~) is analytically equivalent to an equivariant malaping.
Proo]. I f 5: is defined b y m e a n s of global sections of a h o m o g e n e o u s line b u n d l e L~-->G/U, t h e n ~* m a y be t a k e n to be t h e induced r e p r e s e n t a t i o n on H~ ~) a n d then, f r o m t h e proof of T h e o r e m 8, ~ is e q u i v a r i a n t . B u t a n y m a p p i n g ~ : G/U--> P~_~ (C) is b y global sections of ~-1 (H), a n d a n y line bundle is a n a l y t i c a l l y e q u i v a l e n t to a h o m o g e n e o u s line bundle. Q.E.D.
T h e following t h e o r e m w a s given w i t h o u t proof in w 1, where it was s t a t e d t h a t a differential-geometric proof could be given. Using t h e proof of T h e o r e m 8, we give a direct proof.
1 5 0 r H . A. GRIFFITHS
THEOREM. Let X = G / U be a C-space a~rl let Lq-->Lq--~ X be a homogeneous line bundle. Let ~ be as in the proo/ o/ Theorem 8 (~q = c 1 (LQ)), and assume that ~e has n - r negative eigenvalues and r O-eigenvalues (n = d i m cX). Then
H q ( X , s for q < n - r .
Proo/. B y t h e remarks in t h e proof of T h e o r e m 8, we m a y find a C-subgroup D V such t h a t ~ extends to ~" a n d furthermore d i m ~ / V = 2 r a n d ? I V is a C- space. T h e n we have a holomorphic fibering ~ / V - - > M / V - - > M / ? . As in w 2, there is an (Er~ such t h a t E :~ ~ H* ( M / V , C Q) a n d E~ 'q = H e (M/~ ?, F~ q) | Hq(?/V, ~). Thus E~ 'q = 0 (q * 0) a n d H p ( i / V , F~ q) = H p ( M / ? , i~ q) = 0 for 0 ~< p < n - r = dim c M / t 7, since Lq-->M/~ is a negative line bundle. Q.E.D.
Remark. This theorem seems to hold, in some extent, for general c o m p a c t , com- plex manifolds. We can prove it for r = n - 1, n - 2, a n d for all r p r o v i d e d t h a t we replace L b y L " for a suitable ~u > 0 .
(ill) Extrinsic Geometry of C-Spaces and a Geometric Proof of Rigidity in the Kiihler Case We shah n o w use t h e a b o v e proposition a b o u t e q u i v a r i a n t embeddings to h a v e a look at some h o m o g e n e o u s sub-manifolds of PN(C), L e t X a n d X ' be c o m p a c t homogeneous complex manifolds a n d write X = G/U, X ' = G'/U', where G, U, G', a n d U' are connected complex Lie groups. F u r t h e r m o r e , let Q:G--> G' be a holomorphie h o m o m o r p h i s m , a n d let / : X - - > X ' be a p r o p e r holomorphic mapping.
D E F I N I T I O N 8.2. T h e m a p p i n g / is said to be equivariant with respect to Q if, for a n y x E X a n d g E G ,
/(g. x) = e(g)"/(x)-(1) (8.1)
Since ] is proper, /(X) is a s u b - v a r i e t y of X ' , a n d equivarianee implies t h a t t(X) is in fact a non-singular sub-variety. T h u s we m a y define t h e normal bundle N r of ](X) in X ' . Indeed, we h a v e o v e r /(X) t h e e x a c t sequence
o -~ L~(~) -~ L~. [/(X) - ~ N~ - ~ 0. (8.2) W e r e m a r k n o w t h a t we m a y assume in t h e sequel t h a t / is a n embedding. I n fact, /(X) is clearly a homogeneous complex manifold, a n d t h e m a p p i n g /:X-->/(X) is a homogeneous fibration. More precisely, in t h e cases we shall be considering, G a n d G' will be semi-simple, ~ m a y be assumed faithful, a n d hence we m a y write / ( X ) = G/O for some analytic s u b g r o u p ~___ U.
(1) It is due to Blanchard that any surjective ] with connected fibres is equivariant.
H O M O G E N E O U S C O M P L E X M A N I F O L D S . I 151 Then / is just the projection in the fibration
O/u-a/u~e/O xL/(x).
All statements we shall make a b o u t injections /(X)-->G'/U' will " l i f t " to X = G/U.
We m a y describe an equivariant / as follows. L e t x o E X be the origin; t h e n a n y x E X m a y be written as gx o (g E G) and ](x) =/(gxo) = ~(g)/(Xo). I n particular, for u E U, ](Xo) =](uxo)= ~(u)/(xo) , which implies that, taking x'0 = / ( x 0) to be the origin in X ' , Q(U) c U'. Thus the equivariant mappings are given b y the representations ~ : G - + G ' such t h a t ~ ( U ) ~ U', and this m a p p i n g is an embedding if and only if Q(U)=
Q(G) n U'.
PROPOSITION 8.2. The normal bundle N t o/ an equivariant embedding is a homo- geneous vector bundle,
Proo/. Lf(x)~ L x is the homogeneous bundle obtained from the adjoint representa- tion Ad of U on ~/tt, and Lx, I f ( X ) = ] - l ( L x , ) is the homogeneous bundle obtained b y the representation A d o ~ of U on ~'/tt'. :Now since / is an embedding, the injec- tion Q :g-->g' induces an injection of U-modules g/lt-->g'/u' and we have the exact sequence of U-modules
0 -~ ~ / u --> ~ ' / u ' --> q --> O, (8.3) and Nf is just the homogeneous bundle obtained b y the action U on q. Q.E.D.
For a U-module r, we denote b y (~) the corresponding homogeneous bundle and b y Hq(r) the groups
Ha(X,
(~)). We have an exact diagram of U-modules0 0 0
O--->~/u-->~'/u'-> q ~ 0
t ~ t
0--> fi -+ g' ~ g ' / ~ - - - > O 0 --> u --> It' -->11'/1I--> 0
0 0 0
(8.4)
F r o m w IV, we get the two diagrams of M-modules:
152 P H . A. O R I F F I T H S
0
1'
O - - > g
i'
O - - > g
0
0 0
--> H 1 (1[') --> H I (ll'//u) --> 0
t t
-~ H~ -~ H~ -+ 0
---> g' -~ g ' / g --> 0
H ~ ') --> H ~ -+ 0
0 0
(8.5)
a n d 0 --> H * ( u ') - > H U ( u ' / U ) - ~ 0 H 1 (q) --> 0
0
(8.6)
W e w a n t to use (8.5) a n d (8.6) to o b t a i n geometric i n f o r m a t i o n a b o u t t h e position of /(X) in X ' . Clearly t h e k e y to t h e situation lies in t h e groups
Ha(If ~)
( q = 0 , 1 , 2 . . . . ), a n d we have, u n f o r t u n a t e l y , b e e n able to t r e a t these groups only in v e r y special cases.Case I. W e a s s u m e t h a t [ g , u ' ] _ ~ u '. T h e n H ~ ' a n d Hq(u')=O ( q > 0 ) . W e let t i c g' be t h e sub-space s p a n n e d b y g a n d u'.
P R O P O S I T I O N 8.3. G ' / ~ is a C.space and the injection / : G / U - - > G ' / U ' is the injection o/ a fibre in the homogeneous /ibration
a / v ~ a'/v' ~ a'/O.
(8.7)Proo/. T h e fact t h a t fi is a c o m p l e x sub-algebra of g' follows f r o m t h e relation [g, It'] _~ u'. T h e rest of t h e Proposition is t h e n clear.
Remarks. q = g'/fi a n d t h e n o r m a l bundle N I is j u s t t h e restriction of z~ -1 (Lc,/5) to a fibre in (8.7). This is t h e h o m o g e n e o u s bundle g i v e n b y t h e action of U - c O on g'/fi. :For e x a m p l e , if we let F(n) = U ( n ) / T n t h e n t h e inclusion U(n - 1) --> U(n) in- duces a fibering
F(n - 1) --->F(n) --> Pn-1 (C) as given in (8.7).
H O M O G E N E O U S C O M P L E X M A N I F O L D S . I 153 L e t Y be a compact non-singular sub-manifold of a complex manifold X, and let Ny be the normal bundle of Y in X. One m a y consider the de/ormations of Y in X; a 1-parameter family of such is given b y a family Yt(tEC, Itl< ~) of compact complex sub-manifolds of X such t h a t Y0 = Y. The manifolds Yt as a b s t r a c t manifolds will not in general have the same complex structure as t h a t on X, although t h e y are all differentially equivalent. I t was shown in [18] that, if H I ( Y , ~ y ) = 0 , t h e n a neighborhood of 0 in H~ ~ r ) parametrizes a complete local family of sub-manifolds varying Y c X ; we shall use this fact to give a geometric proof of the rigidity of K~hler C-spaces.
Case II. L e t X = G / U = M / V be a K~hler C-space. We shall prove:
THEOREM. The complex structure on X is locally rigid.
The proof is done in two steps:
(i) L e t /:X-->PN=PN(C) be an equivariant projective embedding of X (w (ii)), and let X t ( X o = X ) be a 1-parameter variation of the complex structure on X. Then we shall show:
PROPOSITION 8.4. There exist projective embeddings It: X---> PN such that the/amily (/t(Xt)) gives a variation o/ the sub-mani/old /(X) o/ PN.
PROPOSITION 8.5. For a suitable projective embedding/:X-->PN, there exists a complex curve gt C S L ( N + 1, C) = G' such that /t(Xt) =gt([o(Xo)).
Remark. Intuitively we shall show t h a t a n y deformation Xt of X can be "covered"
b y projective embeddings /t of Xt, and then we shall show t h a t the variations of the equivariantly embedded sub-manifold / ( X ) c P~ are given b y the orbits of /(X) under G', and this shows t h a t the manifolds all have the same complex structure.
Proo/ o/ Proposition 8.4. L e t X be a compact complex manifold with Hi(X, ~ ) = 0 = H2(X, s and let E-->E-->X be a positive line bundle such t h a t the global sec- tions of E give a projective embedding. I f Xt is a variation of X =X0, t h e n we know:
(i) H I ( X t , ~ t ) = 0 = H~(Xt, s (upper-semi-continuity, see [16]), (ii) H2(Xt, ~*) = H2(Xt, Z) b y (i),
(iii) there are positive line bundles Et---->Et-->Xt b y (ii) and since the Xt are differentiably equivalent,
(iv) dim H~ ~t) = dim H~ G0) (by upper-semi-continuity a n d since t h e sheaf Euler-charaeteristic I ( X , Et) is constant) and
1 5 4 P m A. GRIFFITHS
(v) the global sections of
Et-->Xt
give projective e m b e d d i n g s / t : X t - - > P N and the sub-manifolds/~(Xt)
give a deformation of /0(X0) b y (i)-(v) (for more details, see [16], w 13). Finally, b y w IV, if X is a K~hler C-space, thenHx(X, fl)=O=H~(X,O). Q.E.D.
Proo/ o/ Proposition 8.5.
We first r e m a r k that, for a suitable equivariant em- bedding / : X--> PN given b y global sections of a positive line bundle Ee --> E q --> G//U --- X, it will suffice to prove t h a tHq(X, ~f)=O (q>O)
and H~ )(g'=sl(N + i, c)).
This is so, since b y K o d a i r a ' s theorem, this will prove t h a t all variations of
/(X)
in PN are given locally b y the action ofG'= SL(N+
1, C) on/(X)
(the stability group beingQ(G)c_G').
F r o m (8.5) and (8.6), it will suffice to find aQ:U-->GL(E ~)
such t h a t ~ 6D~ (i.e., E p is positive) andHq(ll ') =0
for q = 0 , l, 2. We suspect t h a t this is in fact true forall Q 6D~
b u t we do not know a proof. However, ifg = ~ e ( e ~ ) , ~ ~
and if we take ~ = gl = 1 ~ a,
aGZ+-~F+
then we m a y use L e m m a 5.9 of [21] and a calculation just as in the proof of Theo- r e m 4 to prove t h a t
Hq(ll')=O
for all q. (Then the bundle E Q = K -1 where K is the canonical b u n d l e - - t h e embedding is classically called thecanonical embedding.)
We shall not go into the details now, and thus we conclude the proof of the theorem.Remarks.
(i) I n Case I above, the normal bundle is trivial;NI~-X•
andH~ ~f)=g/fi,
which is just as it should be.(ii) We m a y give in a n y case a geometric proof of the fact t h a t H ~ Indeed, we make a g-reductive decomposition g ' = g 9 ][ and in (8.5) it is seen t h a t
7(H~
L e t r = H ~ ') and let ~ = g e r _ c g '. Now H~ ') is a g-module and thus is a g-module; since r is a sub-algebra, ~ is a sub-algebra and in (8.5), k e r r = g'//~.Geometrically, this means t h a t S (with Lie algebra 3) is the stability group of the v a r i e t y
/(X) ~ IN.
Since G = a u t o m o r p h i s m group of X (see w IV), there is an analytic onto homomorphisma:S--> G
and if we let K = ker a, K is a closed analytic sub- group ofSL(N+
1, C) which leaves/(X)~PN
pointwise fixed. However, this is im- possible unless dim X = 0 or dim K = 0, provided t h a t/ ( X ) c PN
is in general position.Q.E.D.
HOMOGENEOUS COMPLEX ~IANIFOLDS. I 155
[14]. HII~ZEBI~UCH, F., Neue topologische Methoden in der algebraischen Geometric. Ergeb.
Math., 9 (1956).
[25]. ~'EYL, H., Theorie der Darstellung kontinuierlieher halb-einfacher Gruppen durch lineare Transformationen, I. Math. Z., 23 (1925), 271-309; I I - I I I , 24 (1925), 328-395.
[26]. - - - - , The Classical Groups. Princeton, 1939.
Received Jan. 19, 1962, in revised ]orm July 3, 1962 and Jan. 7, 1963