The Ricci tensor of an almost homogenous Kaehler manifold

32 Pages·0.32 MB·English

by  Andrea Spiro

THE RICCI TENSOR OF AN ALMOST
HOMOGENOUS KA¨HLER MANIFOLD
1
0
0
2 Andrea Spiro

n
a
J Abstract. WedetermineanexplicitexpressionfortheRiccitensorofaK-manifold,
thatisofacompactKa¨hlermanifoldM withvanishingfirstBettinumber,onwhich
1
a semisimplegroupG ofbiholomorphicisometriesactswithanorbitofcodimension
2
one. We also prove that the Ka¨hler form ω and the Ricci form ρ of M are uniquely
] determined by two special curves with values in g= Lie(G), say Zω,Zρ : R → g=
G Lie(g) and we showhow the curve Zρ is determinedby the curve Zω.
D These results are used in another work with F. Podesta`, where new examples of
non-homogeneouscompactKa¨hler-Einsteinmanifoldswith positivefirstChernclass
.
h are constructed.
t
a
m
[
1. Introduction.

1
v
2 The objectsof our study are the so-called K-manifolds, that is Ka¨hler manifolds
7 (M,J,g) with b (M) = 0 and which are acted on by a group G of biholomorphic
1
1
isometries, with regular orbits of codimension one. Note that since M is compact
1
C
0 and G has orbits of codimension one, the complexified group G acts naturally
1 on M as a group of biholomorphic transformations, with an open and dense orbit.
0
According to a terminology introduced by A. Huckleberry and D. Snow in [HS],
/
h M is almost-homogeneous with respect to the GC-action. By the results in [HS],
t C
a the subset S M of singular points for the G -action is either connected or with
m ⊂
exactly two connected components. If the first case occurs, we will say that M is a
: non-standard K-manifold; we will call it standard K-manifold in the other case.
v
i The aim of this paper is furnish an explicit expression for the Ricci curvature
X
tensorof a K-manifold, to beused for constructing(andpossibly classify) new fam-
r
a iliesofexamplesofnon-homogeneousK-manifoldwithspecialcurvatureconditions.
Asuccessfulapplicationofourresultsisgiven in [PS1], whereseveralnewexamples
of non-homogeneous compact Ka¨hler-Einstein manifolds with positive first Chern
class are found.
Note that explicit expressions for the Ricci tensor of standard K-manifolds can
be found also in [Sa], [KS], [PS] and [DW]. However our results can be applied to
any kind of K-manifold and hence they turn out to be particularly useful for the
1991 Mathematics Subject Classification. 53C55,53C25, 57S15.
Key words and phrases. Ka¨hler-Einsteinmetrics,cohomogeneityone actions.
Typesetby AMS-TEX
1
2 RICCI TENSOR OF AN ALMOST KA¨HLER HOMOGENEOUS MANIFOLD
non-standard cases (at this regard, see also [CG]). They can be resumed in the
following three facts.
Let g be the Lie algebra of the compact group G acting on the K-manifold
(M,J,g) with at least one orbit of codimension one. By a result of [PS1], we may
always assume that G is semisimple. Let also be the Cartan-Killing form of
B
g. Then for any x in the regular point set M , one can consider the following
reg
-orthogonal decomposition of g:
B
g =l+RZ +m , (1.1)
where l = g is the isotropy subalgebra, RZ +m is naturally identified with the
x
tangent space T (G/L) T (G x) of the G-orbit G/L= G x, and m is naturally
o x
≃ · ·
identified with the holomorphic subspace m
x
≃ D
= v T (G x) : Jv T (G x) . (1.2)
x x x
D { ∈ · ∈ · }
Noticethatforanypointx M the -orthogonaldecomposition(1.1)isuniquely
reg
∈ B
given; on the other hand, two distinct points x,x M may determine two
′ reg
∈
distinct decompositions of type (1.1).
OurfirstresultconsistsinprovingthatanyK-manifoldhasafamily ofsmooth
O
curves η : R M of the form
→
η =exp(itZ) x ,
t o
·
where Z g, x M is a regular point for the GC-action and the following
o
∈ ∈
properties are satisfied:
(1) η intersects any regular G-orbit;
t
(2) for any point η M , the tangent vector η is transversal to the regular
t ∈ reg t′
orbit G η ;
t
·
(3) any element g G which belongs to a stabilizer G , with η M ,
∈ ηt t ∈ reg
fixes pointwise the whole curve η; in particular, all regular orbits G η are
t
·
equivalent to the same homogeneous space G/L;
(4) thedecompositions(1.1)associatedwiththepointsη M donotdepend
t reg
∈
on t;
(5) thereexistsabasis f ,...f formsuchthatforanyη M thecomplex
1 n t reg
{ } ∈
structure J : m m, induced by the complex structure of T M, is of the
t → ηt
following form:
1
J f = λ (t)f , J f = f ; (1.3)
t 2j j 2j+1 t 2j+1 2j
−λ (t)
j
wherethefunctionλ (t)iseitheroneofthefunctions tanh(t), tanh(2t),
j
− −
coth(t) and coth(2t) or it is identically equal to 1.
− −
Wecallanysuch curveanoptimal transversal curve; thebasisforRZ+m ggiven
⊂
by (Z,f ,...,f ), where the f ’s verify (1.3), is called optimal basis associated
1 2n 1 i
−
with η. An explicit description of the optimal basis for any given semisimple Lie
group G is given in 3.
§
A. SPIRO 3
Noticethatthefamily ofoptimaltransversalcurvesdependsonlyontheaction
O
of the Lie group G. In particular it is totally independent on the choice of the G-
invariant Ka¨hler metric g. At the same time, the Killing fields, associated with the
elementsofanoptimalbasis,determinea1-parameterfamilyofholomorphicframes
atthepointsη M , which areorthogonalw.r.t. atleastoneG-invariant Ka¨hler
t reg
∈
metric g. It is also proved that, for all K-manifold M which do not belong to a
special class of non-standard K-manifold, those holomorphic frames are orthogonal
w.r.t. any G-invariant Ka¨hler metric g on M (see Corollary 4.2 for details). From
these remarks and the fact that η = JZˆ , where Z is the first element of any
t′ ηt
optimal basis, it may be inferred that any curve η is a reparameterizationof a
∈ O
normal geodesics of some (in most cases, any) G-invariant Ka¨hler metric on M.
Our second main result is the following. Let η be an optimal transversal curve
of a K-manifold, g = l + RZ + m the decomposition (1.1) associated with the
regular points η M and let ω and ρ be the Ka¨hler form and the Ricci form,
t reg
∈
respectively, associated with a given G-invariant Ka¨hler metric g on (M,J).
By a slight modificationof argumentsused in [PS], we show that there exist two
smooth curves
Z ,Z : R C (l)=z(l)+a , a =C (l) (RZ +m) , (1.4)
ω ρ g g
→ ∩
satisfyingthefollowingproperties(herez(l)denotesthecenteroflandC (l)denotes
g
the centralizer of l in g): for any η M and any two element X,Y g, with
t reg
∈ ∈
associated Killing fields Xˆ and Yˆ,
ω (Xˆ,Yˆ)= (Z (t),[X,Y]) , ρ (Xˆ,Yˆ) = (Z (t),[X,Y]) . (1.5)
ηt B ω ηt B ρ
We call such curves Z (t) and Z (t) the algebraic representatives of ω and ρ along
ω ρ
η. It is clear that the algebraic representatives determine uniquely the restrictions
ofω andρtothetangentspacesoftheregularorbits. ButthefollowingProposition
establishes a result which is somehow stronger.
Before stating the proposition, we recall that in [PS] the following fact was
established: if g = l + RZ + m is a decomposition of the form (1.1), then the
subalgebra a = C (l) (RZ + m) is either 1-dimensional or 3-dimensional and
g
∩
isomorphic with su . By virtue of this dichotomy, the two cases considered in the
2
following proposition are all possible cases.
Proposition 1.1. Let η be an optimal transversal curve of a K-manifold (M,J,g)
t
acted on by the compact semisimple Lie group G and let g = l+RZ +m be the
decomposition of the form (1.1) determined by the points η M . Let also
t reg
∈
Z : R C (l)= z(l)+a be the algebraic representative of the Ka¨hler form ω or of
g
→
the Ricci form ρ. Then:
(1) if a is 1-dimensional, then it is of the form a = RZ and there exists an
element I z(l) and a smooth function f : R R so that
∈ →
Z(t) =f(t)Z +I ; (1.6)
(2) if a is 3-dimensional, then it is of the form a = su = RZ +RX +RY,
2
with [Z,X] = Y and [X,Y]= Z and there exists an element I z(l), a real
∈
4 RICCI TENSOR OF AN ALMOST KA¨HLER HOMOGENEOUS MANIFOLD
number C and a smooth function f : R R so that
→
C
Z(t)= f(t)Z + X +I . (1.7)
D cosh(t)
Conversely, if Z : R C (l) is a curve in C (l) of the form (1.6) or (1.7), then
g g
→
there exists a unique closed J-invariant, G-invariant 2-form ̟ on the set of regular
points M , having Z(t) as algebraic representative.
reg
In particular, the Ka¨hler form ω and the Ricci form ρ are uniquely determined
by their algebraic representatives.
Using (1.5), Proposition 1.1 and some basic properties of the decomposition
g = l+RZ +m (see 5), it can be shown that the algebraic representatives Z (t)
ω
§
and Z (t) are uniquely determined by the values ω (Xˆ,JXˆ) = (Z (t),[X,J X])
ρ ηt B ω t
andρ (Xˆ,JXˆ) = (Z (t),[X,J X]),whereX mandJ isthecomplexstructure
ηt B ρ t ∈ t
on m induced by the complex structure of the tangent space T M.
ηt
Here comes our third main result. It consists in Theorem 5.1 and Proposition
5.2, where we give the explicit expression for the value r (X,X) = ρ (Xˆ,JXˆ)
ηt ηt
for any X m, only in terms of the algebraic representative Z (t) and of the Lie
ω
∈
brackets between X and the elements of the optimal basis in g. By the previous
discussion, this result furnishes a way to write down explicitly the Ricci tensor of
the Ka¨hler metric associated with Z (t).
ω
Acknowledgement. Many crucial ideas for this paper are the natural fruit of the
uncountable discussions that Fabio Podest`a and the author had since they began
working on cohomogeneity one Ka¨hler-Einstein manifolds. It is fair to say that
most of the credits should be shared with Fabio.
Notation. Throughout the paper, if G is a Lie group acting isometrically on a
Riemannian manifold M and if X g = Lie(G), we will adopt the symbol Xˆ to
∈
denote the Killing vector field on M corresponding to X.
The Lie algebra of a Lie group will be always denoted by the corresponding
gothic letter. For a group G and a Lie algebra g, Z(G) and z(g) denote the center
of G and of g, respectively. For any subset A of a group G or of a Lie algebra g,
C (A) and C (A) are the centralizer of A in G and g, respectively.
G g
Finally, for any subspace n g of a semisimple Lie algebra g, the symbol n
⊥
⊂
denotes the orthogonal complement of n in g w.r.t. the Cartan-Killing form .
B
2. Fundamentals of K-manifolds.
2.1 K-manifolds, KO-manifolds and KE-manifolds.
A K-manifold is a pair formed by a compact Ka¨hler manifold (M,J,g) and a
compact semisimple Lie group G acting almost effectively and isometrically (hence
biholomorphically) on M, such that:
i) b (M)= 0;
1
ii) G acts of cohomogeneity one with respect to the action of G, i.e. the regular
G-orbits are of codimension one in M.
A. SPIRO 5
In this paper, (M,J,g) will always denote a K-manifold of dimension 2n, acted on
by the compact semisimple Lie group G. We will denote by ω( , ) = g( ,J ) the
· · · ·
Ka¨hler fundamental form and by ρ= r( ,J ) the Ricci form of M.
· ·
For the general properties of cohomogeneity one manifolds and of K-manifolds,
see e.g. [AA], [AA1], [Br], [HS], [PS]. Here we only recall some properties, which
will be used in the paper.
If p M is a regular point, let us denote by L= G the corresponding isotropy
p
∈
subgroup. Since M is orientable, every regular orbit G p is orientable. Hence we
·
may consider a unit normal vector field ξ, defined on the subset of regular points
M , which is orthogonal to any regular orbit. It is known (see [AA1]) that any
reg
integralcurveofξ isageodesic. Anysuchgeodesicisusuallycallednormalgeodesic.
A normal geodesic γ through a point p verifies the following properties: it inter-
sects any G-orbit orthogonally; the isotropy subalgebra G at a regular point γ
γt t
is always equal G =L (see e.g. [AA], [AA1]). We formalize these two facts in the
p
following definition.
We call nice transversal curve through a point p M any curve η : R M
reg
∈ →
with p η(R) and such that:
∈
i) it intersects any regular orbit;
ii) for any η M
t reg
∈
η / T (G η ) ; (2.1)
t′ ∈ ηt · t
iii) for any η M , G =L =G .
t ∈ reg ηt p
The following property of K-manifold has been proved in [PS].
Proposition 2.1. Let (M,J,g) be a K-manifold acted on by the compact semisim-
ple Lie group G. Let also p M and L= G the isotropy subgroup at p. Then:
reg p
∈
(1) there exists an element Z (determined up to scaling) so that
RZ C (l) l , C (l+RZ) =z(l)+RZ ; (2.2)
g ⊥ g
∈ ∩
in particular, the connected subgroup K G with subalgebra k = l+RZ is
⊂
the isotropy subgroup of a flag manifold F =G/K;
(2) the dimension of a = Cg(l) l⊥ is either 1 or 3; in case dimRa = 3, then
∩
a is a subalgebra isomorphic to su and there exists a Cartan subalgebra
2
tC lC+aC gC so that aC = CH +CE +CE for some root α of the
α α α
roo⊂t system o⊂f (gC,tC). −
Notethatif forsomeregularpointpwehavethatdimRa =1(resp. dimRa =3),
then the same occurs at any other regular point. Therefore we may consider the
following definition.
Definition 2.2. Let (M,J,g) be a K-manifold and L =G the isotropy subgroup
p
of a regular point p. We say that M is a K-manifold with ordinary action (or
shortly, KO-manifold) if dimRa =dimR(Cg(l) l⊥) = 1.
∩
In all other cases, we say that M is with extra-ordinary action (or, shortly, KE-
manifold).
6 RICCI TENSOR OF AN ALMOST KA¨HLER HOMOGENEOUS MANIFOLD
Another useful property of K-manifolds is the following. It can be proved that
any K-manifold admitsexactlytwo singular orbits, at least one of which is complex
(see [PS1]). By the results in [HS], it also follows that if M is a K-manifold whose
singular orbitsare both complex, thenM admitsa G-equivariant blow-up M˜ along
the complex singular orbits, which is still a K-manifold and admits a holomorphic
fibration over a flag manifold G/K =GC/P, with standard fiber equal to CP1.
Several other important facts are related to the existence (or non-existence) of
two singular complex orbits (see [PS1] for a review of these properties). For this
reason, it is convenient to introduce the following definition.
Definition 2.3. We say that a K-manifold M, acted on by a compact semisimple
group G with cohomogeneity one, is standard if the action of G has two singular
complex orbits. We call it non-standard in all other cases.
2.2 The CR structure of the regular orbits of a K-manifold.
A CR structure of codimension r on a manifold N is a pair ( ,J) formed by a
D
distribution TN ofcodimensionr andasmoothfamilyJ ofcomplexstructures
D ⊂
J : on the spaces of the distribution.
x x x
ADCR→stDructure ( ,J) is called integrable if the distribution 10 TCN, given
by the J-eigenspacesD 10 C corresponding to the eigenvalueD+i,⊂verifies
Dx ⊂Dx
[ 10, 10] 10 .
D D ⊂ D
Note that a complex structure J on manifold N may be always considered as an
integrable CR structure of codimension zero.
Asmoothmapφ : N N betweentwoCRmanifolds(N, ,J)and(N , ,J )
′ ′ ′ ′
→ D D
is called CR map (or holomorphic map) if:
a) φ ( ) ;
′
∗ D ⊂ D
b) for any x N, φ J = J φ .
∈ ∗◦ x φ′(x)◦ ∗|Dx
A CR transformation of (N, ,J) is a diffeomorphism φ : N N which is also a
D →
CR map.
Any codimension one submanifold N M of a complex manifold (M,J) is
⊂
naturally endowed with an integrable CR structure of codimension one ( ,J),
D
which is called induced CR structure; it is defined by
= v T N : Jv T N J = J .
Dx { ∈ x ∈ x } x |Dx
It is clear that any regular orbit G/L = G x M of a K-manifold (M,J,g)
· ∈
has an induced CR structure ( ,J), which is invariant under the transitive action
D
of G. For this reason, several facts on the global structure of the regular orbits of
a K-manifolds can be detected using what is known on compact homogeneous CR
manifolds (see e.g. [AHR] and [AS]).
Here, we recall some of those facts, which will turn out to be crucial in the next
sections.
A. SPIRO 7
Let (G/L, ,J) be a homogeneous CR manifold of a compact semisimple Lie
D
groupG, withan integrableCRstructure( ,J)ofcodimensionone. If we consider
D
the -orthogonal decomposition g = l+n, where l = Lie(L), then the orthogonal
B
complement n is naturally identifiable with the tangent space T (G/L), o =eL, by
o
means of the map
φ : n T (G/L) , φ(X) =Xˆ .
o o
→ |
If we denote by m the subspace
m= φ 1( ) n ,
− o
D ⊂
we get the following orthogonal decomposition of g:
g =l+n =l+RZ +m . (2.3)
D
where Z (l + m) . Since the decomposition is ad -invariant, it follows that
⊥ l
D ∈
Z C (l).
g
D ∈
Using again the identification map φ : n T (G/L), we may consider the
o
→
complex structure
def
J : m m , J = φ (J ) . (2.4)
∗ o
→
Note that J is uniquely determined by the direct sum decomposition
mC = m10+m01 , m01 =m10 , (2.5)
where m10 and m01 are the J-eigenspaces with eigenvalues +i and i, respectively.
−
In all the following, (2.3) will be called the structural decomposition of g associ-
ated with ; the subspace m10 mC (respectively, m01 = m10) given (2.5) will be
D ⊂
called the holomorphic (resp. anti-holomorphic) subspace associated with ( ,J).
D
We recall that a G-invariant CR structure ( ,J) on G/L is integrable if and
only if the associated holomorphic subspace m10D mC is so that
⊂
lC+m10 is a subalgebra of gC . (2.6)
We now need to introduce a few concepts which are quite helpful in describing
the structure of a generic compact homogeneous CR manifold.
Definition 2.4. Let N = G/L be a homogeneousmanifold of a compact semisim-
ple Lie group G and ( ,J) a G-invariant, integrable CR structure of codimension
D
one on N.
We say that a CR manifold (N = G/L, ,J) is a Morimoto-Nagano space if
D
either G/L = S2n 1, n > 1, endowed with the standard CR structure of S2n 1
− −
⊂
CPn, or there exists a subgroup H G so that:
⊂
a) G/H is a compact rank one symmetric space (i.e. RPn =SO /SO Z ,
n+1 n 2
·
Sn = SO /SO , CPn = SU /SU , HPn = Sp /Sp or OP2 =
n+1 n n+1 n n+1 n
F /Spin );
4 9
b) G/L is a sphere bundle S(G/H) T(G/H) in the tangent space of G/H;
⊂
c) ( ,J) is the CR structure induced on G/L = S(G/H) by the G-invariant
D C C
complex structure of T(G/H)=G /H .
∼
8 RICCI TENSOR OF AN ALMOST KA¨HLER HOMOGENEOUS MANIFOLD
If a Morimoto-Naganospace is G-equivalent to a sphere S2n 1 we call it trivial; we
−
call it non-trivial in all other cases.
A G-equivariant holomorphic fibering
π : N =G/L =G/Q
→ F
of (N, ,J) ontoa non-trivialflag manifold( =G/Q,J )with invariant complex
D F F
structure J , is called CRF fibration. A CRF fibration π : G/L G/Q is called
F →
nice if the standard fiber is a non-trivial Morimoto-Nagano space; it is called very
nice if it is nice and there exists no other nice CRF fibrationπ : G/L G/Q with
′
→
standard fibers of smaller dimension.
The following Proposition gives necessary and sufficient conditions for the exis-
tence of a CRF fibration. The proof can be found in [AS].
Proposition 2.5. Let G/L be homogeneous CR manifold of a compact semisim-
ple Lie group G, with an integrable, codimension one G-invariant CR structure
( ,J). Let also g=l+RZ +m be the structural decomposition of g and m10 the
D D
holomorphic subspace, associated with ( ,J).
D
Then G/L admits a non-trivial CRF fibration if and only if there exists a proper
parabolic subalgebra p = r+n ( gC (here r is a reductive part and n the nilradical
of p) such that:
a) r = (p g)C ; b) lC+m01 p ; c) lC (r .
∩ ⊂
C
In this case, G/L admits a CRF fibration with basis G/Q= G /P, where Q is the
connected subgroup generated by q = r g and P is the parabolic subgroup of GC
∩
with Lie algebra p.
Let us go back to the regular orbits of a K-manifold (M,J,g) acted on by the
compact semisimple group G. We already pointed out that each regular orbit
(G/L = G x, ,J), endowed with the induced CR structure ( ,J), is a compact
· D D
homogeneous CR manifold. In the statement of the following Theorem we collect
the main results on the one-parameter family of compact homogeneous CR mani-
folds given by the regular orbits of a K-manifold, which is a direct consequenceTh.
3.1 in [PS1] (see also [HS] and [PS] Th.2.4).
Theorem 2.6. Let (M,J,g) be a K-manifold acted on by the compact semisimple
Lie group G.
(1) If M is standard, then there exists a flag manifold (G/K,J ) with a G-
o
invariant complex structure J , such that any regular orbit (G x = G/L,
o
·
,J) of M admits a CRF-fibration π : (G/L, ,J) (G/K,J ) onto
o
D D →
(G/K,J ) with standard fiber S1.
o
(2) If M is non-standard, then there exists a flag manifold (G/K,J ) with a
o
G-invariant complex structure J such that any regular orbit (G/L = G
o
·
x, ,J) admits a very nice CRF fibration π : (G/L, ,J) (G/K,J )
o
D D →
where the standard fiber K/L is a non-trivial Morimoto-Nagano space of
dimension dimK/L 3.
≥
A. SPIRO 9
Furthermore, if the last case occurs, then the fiber K/L of the CRF fibration π :
(G/L, ,J) (G/K,J ) has dimension 3 if and only if M is a non-standard KE-
o
D →
manifold and K/L is either S(RP2) T(RP2) = CP2 [z] : tz z = 0 or
⊂ \ { · }
S(CP1) T(CP1) =CP1 CP1 [z] =[w] .
⊂ × \{ }
3. The optimal transversal curves of a K-manifold.
3.1 Notation and preliminary facts.
C C
IfGisacompactsemisimpleLiegroupandt g isagivenCartansubalgebra,
⊂
we will use the following notation:
- is the Cartan-Killing form of g and for any subspace A g, A is the
⊥
B ⊂
-orthogonal complement to A;
- BR is the root system of (gC,tC);
C
- H t is the -dual element to the root α;
α
∈ B
- for any α,β R, the scalar product (α,β) is set to be equal to (α,β) =
∈
(H ,H );
α β
B
- E is the root vector with root α in the Chevalley normalization; in par-
α
ticular (E ,E ) = δ , [E ,E ] = H , [H ,E ] = (β,α)E and
α β αβ α α α α β β
B − −
[H ,E ] = (β,α)E ;
α β β
− − −
- for any root α,
1 i
F = (E E ) , G = (E +E ) ;
α α α α α α
√2 − − √2 −
note that for α,β R
∈
(F ,F ) = δ = (G ,G ) , (F ,G ) = (F ,H ) = (G ,H ) = 0 ;
α β αβ α β α β α β α β
B − B B B B
- the notation for the roots of a simple Lie algebra is the same of [GOV] and
[AS].
Recall that for any two rootsα,β, with β = α, in case [E ,E ] is non trivial then
α β
6 −
it is equal to [E ,E ]= N E where the coefficients N verify the following
α β α,β α+β α,β
conditions:
N = N , N = N . (3.1)
α,β β,α α,β α, β
− − − −
From (3.1) and the properties of root vectors in the Chevalley normalization, the
following well known properties can be derived:
(1) for any α,β R with α =β
∈ 6
[F ,F ],[G ,G ] span F ,γ R , [F ,G ] span G ,γ R ; (3.2)
α β α β γ α β γ
∈ { ∈ } ∈ { ∈ }
(2) for any H tC and any α,β R, (H,[F ,F ]) = (H,[G ,G ]) = 0 and
α β α β
∈ ∈ B B
(H,[F ,G ]) =iδ (H,H ) =δ α(iH) ; (3.3)
α β αβ α αβ
B B
10 RICCI TENSOR OF AN ALMOST KA¨HLER HOMOGENEOUS MANIFOLD
Finally, for whatconcernstheLie algebraof flag manifoldsandof CR manifolds,
we adopt the following notation.
Assume that G/K is a flag manifold with invariant complex structure J (for
definitions and basic facts, we refer to [Al], [AP], [BFR], [Ni]) and let π : G/L
→
G/K be a G-equivariant S1-bundle over G/K. In particular, let us assume that l
is a codimension one subalgebraof k. Recall that k =kss+z(k), with kss semisimple
part of k. Hence the semisimple part lss of l is equal to kss and k = l + RZ =
(kss+z(k) l)+RZ for some Z z(k).
LettC ∩kC bea Cartansubalg∈ebrafor gC containedin kC andR therootsystem
of (gC,tC)⊂. Then we will use the following notation:
- R = α R , E k ;
o α
{ ∈ ∈ }
- R = α R , E m ;
m α
- for any{α∈ R, we de∈note}by g(α)C =spanC E α,Hα and g(α) =g(α)C
∈ { ± } ∩
g;
- m(α) denotes the irreducible kC-submoduli of mC, with highest weight α
∈
R ;
m
- if m(α) and m(β) are equivalent as lC-moduli, we denote by m(α)+λm(β)
the irreducible lC-module with highest weight vector E +λE , α,β R ,
α β m
∈
λ C.
∈
3.2The structuraldecomposition g=l+RZ +mdetermined by theCR structure
D
of a regular orbit.
The main results of this subsection are given by the following two theorems on
thestructuraldecompositionof theregularorbitsof a K-manifolds. The first oneis
a straightforward consequence of definitions, Theorem 2.6 and the results in [PS].
Theorem 3.1. Let (M,J,g) be a standard K-manifold acted on by the compact
semisimple group G and let g=l+RZ +mand m10 be thestructural decomposition
D
and the holomorphic subspace, respectively, associated with the CR structure ( ,J)
D
of a regular orbit G/L=G p. Let also J :m m be the unique complex structure
on m, which determines the·decomposition mC→= m10+m10.
Then, k = l + RZ is the isotropy subalgebra of a flag manifold K, and the
D
complex structure J : m m is ad -invariant and corresponds to a G-invariant
k
→
complex structure J on G/K.
In particular, there exists a Cartan subalgebra tC kC and an ordering of the
⊂
associated root system R, so that m10 is generated by the corresponding positive root
vectors in mC =(k )C.
⊥
The following theorem describes the structural decomposition and the holomor-
phic subspace of a regular orbit of a non-standard K-manifold. Also this theorem
can be considered as a consequence of Theorem 2.6, but the proof is a little bit
more involved.