Homogeneous Kähler and Hamiltonian manifolds
aa r X i v : . [ m a t h . C V ] J un HOMOGENEOUS K ¨AHLER AND HAMILTONIAN MANIFOLDS
BRUCE GILLIGAN, CHRISTIAN MIEBACH, AND KARL OELJEKLAUS
Dedicated to Alan T. Huckleberry
Abstract.
We consider actions of reductive complex Lie groups G = K C on K¨ahlermanifolds X such that the K –action is Hamiltonian and prove then that the closuresof the G –orbits are complex-analytic in X . This is used to characterize reductivehomogeneous K¨ahler manifolds in terms of their isotropy subgroups. Moreover weshow that such manifolds admit K –moment maps if and only if their isotropy groupsare algebraic. Introduction
A reductive complex Lie group G is a complex Lie group admitting a compact realform K , i. e. G = K C . Equivalently a finite covering of G is of the form S × Z = S × ( C ∗ ) k , where S is a semisimple complex Lie group. It is well known that everycomplex reductive Lie group admits a unique structure as a linear algebraic group.Holomorphic or algebraic actions of reductive Lie groups appear frequently in complexand algebraic geometry and interesting connections arise between the structure of theorbits of such groups and the isotropy subgroups of the orbits.A result of this type was proved independently by Matshushima [19] and Onishchik[20]. They consider G a complex reductive Lie group and H a closed complex subgroupof G and show that G/H is Stein if and only if H is a reductive subgroup of G . In[1] Barth and Otte prove that the holomorphic separability of the homogeneous space G/H implies H is an algebraic subgroup of the reductive group G .In the case of semisimple actions, it is known that K¨ahler is equivalent to algebraicin the sense that S/H is K¨ahler if and only if H is an algebraic subgroup of the complexsemisimple Lie group S , see [2] and [3]. The simple example of an elliptic curve C ∗ / Z shows that this result does not hold in the reductive case. Instead homogeneous K¨ahlermanifolds X = G/H with G = S × ( C ∗ ) k reductive are characterized by the twoconditions S ∩ H is algebraic and SH ⊂ G is closed, as we shall prove. If, in additionto the existence of a K¨ahler form, there exists a K –moment map on X , then X is calleda Hamiltonian G –manifold. Huckleberry has conjectured that the isotropy groups ina Hamiltonian G –manifold are algebraic. In the present paper we prove that this isindeed the case.The moment map plays a decisive role in our proof which depends in an essentialway on the work of Heinzner-Migliorini-Polito [11]. In the third section of their paper Date : June 2, 2010.1991
Mathematics Subject Classification. they investigate the closure of certain orbits and prove the following: suppose T is analgebraic torus acting holomorphically on a complex space X such that the semistablequotient π : X → X//T exists. Let A be a subanalytic set in X such that π | A : A → X//T is proper. Then T · A is subanalytic in X . We use the moment map in orderto ensure the existence of the semistable T –quotient locally. This is sufficient to showthat the G –orbits are locally subanalytic and hence locally closed in the Hamiltonian G –manifold X . Moreover, we deduce from this fact that the closure of any G –orbit iscomplex-analytic in X . This generalizes previous work of [22] and [8] to non-compactK¨ahler manifolds.Our work was partially motivated by [17], where Margulis constructed discrete sub-groups Γ of SL(2 , R ) ⋉ R which are free groups generated by two elements. Thesegroups Γ can be divided into two non-empty classes depending on whether the inducedaction of Γ on R is properly discontinuous or not. The associated homogeneous com-plex manifolds (SL(2 , C ) ⋉ C ) / Γ are not K¨ahler in the “non properly discontinuous”case by Corollary 3.6. It seems to be a difficult problem to decide the K¨ahler questionfor these quotients in the “properly discontinuous” case.The paper is organized as follows. In section 2, the definitions of K –moment mapsand Hamiltonian actions are recalled. Furthermore two lemmata are proved for lateruse. The main result of section 3 is the analyticity of orbit closures. Since for G = S semisimple there always is a moment map, any semisimple Lie group action on a K¨ahlermanifold has locally closed orbits.In section 4 we prove that the reductive homogeneous manifold X = G/H is Hamil-tonian if and only if H is an algebraic subgroup of G and use this to give a new proofof the main results in [3] and [2]. Finally, in the last section this result is used in orderto prove our characterization of those closed complex subgroups H ⊂ G such that X = G/H admits a K¨ahler form.2.
Hamiltonian G –manifolds Let G = K C be a complex reductive Lie group with maximal compact subgroup K .Let X be a complex manifold endowed with a holomorphic G –action.We denote the Lie algebra of K by k . The group K acts via the coadjoint represen-tation on the dual k ∗ . In the following equivariance of a map with values in k ∗ is alwaysmeant with respect to the coadjoint action. If ξ ∈ k , we write ξ X for the holomorphicvector field on X whose flow is given by ( t, x ) exp( tξ ) · x . If ω is a K –invariantK¨ahler form on X , then the contracted form ι ξ X ω is closed for every ξ ∈ k . By defini-tion, a K –equivariant smooth map µ : X → k ∗ is a moment map for the K –action on X if for each ξ ∈ k the smooth function µ ξ ∈ C ∞ ( X ), µ ξ ( x ) := µ ( x ) ξ , verifies dµ ξ = ι ξ X ω .The K –action on X is called Hamiltonian if an equivariant moment map µ : X → k ∗ exists. Note that, if µ is a moment map and if λ ∈ k ∗ is a K –fixed point, then µ + λ isanother moment map on X . Definition 2.1.
We say that X is a Hamiltonian G –manifold if X admits a K –invariant K¨ahler form such that the K –action on X is Hamiltonian with equivariantmoment map µ : X → k ∗ . Remark. If G is semisimple, then every K¨ahler manifold X on which G acts holomor-phically is a Hamiltonian G –manifold which can be seen as follows. Let dk be the OMOGENEOUS K ¨AHLER AND HAMILTONIAN MANIFOLDS 3 normalized Haar measure of K . If ω is any K¨ahler form on X , then b ω := R K k ∗ ωdk is a K –invariant K¨ahler form on X . Since K is semisimple, there exists a uniqueequivariant moment map µ : X → k ∗ by Theorem 26.1 in [9].In this paper we will often use the following. Lemma 2.2.
Let X be a Hamiltonian G –manifold and let e G be a complex reductivesubgroup of G . Then every e G –stable complex submanifold e X of X is a Hamiltonian e G –manifold.Proof. We may assume without loss of generality that e G = e K C for some compactsubgroup e K ⊂ K . Composing the moment map µ : X → k ∗ with the orthogonalprojection onto e k ∗ we obtain a moment map for the e K –action on X . Restricting thismap to the K¨ahler manifold e X we see that the e K –action on e X is Hamiltonian. (cid:3) Example.
Let G → GL( V ) be a holomorphic representation of the complex reductivegroup G on a finite dimensional complex vector space V . Then each G –stable complexsubmanifold of V or of P ( V ) is a Hamiltonian G –manifold. In particular, if H is analgebraic subgroup of G , then the homogeneous space G/H is a quasi-projective variety(see e. g. Theorem 5.1 in [7]) and hence a Hamiltonian G –manifold.For later use we note the following Lemma 2.3.
Let ( X, ω ) be a Hamiltonian G -manifold with µ : X → k ∗ its momentmap and let p : e X → X be a topological covering. If the G –action lifts to e X , then ( e X, p ∗ ω ) is a Hamiltonian G –manifold with moment map p ∗ µ .Proof. We equip e X with the unique complex structure such that p is locally biholo-morphic. If the G –action lifts to e X , then G acts holomorphically on e X and p is G –equivariant. Consequently, p ∗ ω is a K –invariant K¨ahler form on e X .For ξ ∈ k let ξ e X and ξ X be the induced vector fields on e X and X , respectively. Since p is equivariant, we have p ∗ ξ e X = ξ X . Hence, we obtain d ( p ∗ µ ) ξ = dp ∗ µ ξ = p ∗ dµ ξ = p ∗ ι ξ X ω = ι ξ e X p ∗ ω, which shows that p ∗ µ is an equivariant moment map for the K –action on e X . (cid:3) Local closedness of G –orbits Let X be a Hamiltonian G –manifold where G = K C is a complex reductive group.Suppose that X is G –connected, i. e. that X/G is connected. In particular, X has onlyfinitely many connected components since this is true for G . We want to show thatthe topological closure of every G –orbit is complex-analytic in X .We fix a maximal torus T in K . Then T := T C is a maximal algebraic torus in G and the moment map µ : X → k ∗ induces by restriction a moment map µ T : X → t ∗ for the T –action on X . Since t is Abelian, for every λ ∈ t ∗ the shifted map µ T + λ isagain a moment map for T . Consequently, every x ∈ X lies in the zero fiber of somemoment map for the T –action on X which has the following consequences (see [10]). Theorem 3.1.
Let G be a complex reductive group and X be a G –connected Hamil-tonian G –manifold. BRUCE GILLIGAN, CHRISTIAN MIEBACH, AND KARL OELJEKLAUS (1) Every isotropy group T x is complex reductive and hence the connected componentof the identity ( T x ) is a subtorus of T .(2) For every x ∈ X there exists a complex submanifold S of X which contains x such that the map T × T x S → T · S , [ t, y ] t · y , is biholomorphic onto its openimage.(3) For λ ∈ t ∗ we define S λ := (cid:8) x ∈ X ; T · x ∩ µ − T ( λ ) = ∅ (cid:9) . Then S λ is a T –stableopen subset of X such that the semistable quotient (see [11] ) S λ → S λ //T exists.Moreover, the inclusion µ − T ( λ ) ֒ → S λ induces a homeomorphism µ − T ( λ ) /T ∼ = S λ //T .Remark. Properties (1) and (2) imply that if the T –action on X is known to be almostfree, then it is locally proper.For the following we have to review the definition of subanalytic sets. For moredetails we refer the reader to [4] and to [12].Let M be a real analytic manifold. A subset A ⊂ M is called semianalytic if everypoint in M has an open neighborhood Ω such that A ∩ Ω = S rk =1 T sl =1 A kl , where every A kl is either of the form { f kl = 0 } or { f kl > } for f kl ∈ C ω (Ω). A subset A ⊂ M is called subanalytic if every element of M admits an open neighborhood Ω such that A ∩ Ω is the image of a semianalytic set under a proper real analytic map. We notethat finite intersections and finite unions as well as topological closures of subanalyticsets are subanalytic. Finally we call a set A ⊂ M locally subanalytic if there are opensets U , . . . , U k ⊂ M such that A ⊂ U ∪ · · · ∪ U k and such that A ∩ U j is subanalyticin U j for every j . For later use we cite the following theorem of Hironaka ([12]). Theorem 3.2.
Let
Φ : M → N be a real analytic map between real analytic manifoldsand let A ⊂ M be subanalytic. If Φ | A : A → N is proper, then Φ( A ) is subanalytic in N . It is shown in [11] that, if the semistable quotient X → X//T exists globally, thenthe semistable quotient X → X//G exists. The first step in the proof of this theoremconsists in showing that the existence of
X//T implies that the G –orbits are subanalyticand thus locally closed in X . In our situation the semistable quotient of X with respectto T exists only locally (in the sense of Theorem 3.1(3)). As we will see this impliesthat the G –orbits in X are locally subanalytic which is sufficient for them to be locallyclosed.The following lemma is the essential ingredient in the proof of this statement. Lemma 3.3.
Let A ⊂ X be a compact subanalytic set. Then T · A is locally subanalyticin X .Proof. Since A is compact, we have A ⊂ S nk =1 S λ k . For every k = 1 , . . . , n let U k bean open subanalytic subset of S λ k such that U k ⊂ S λ k is compact and such that A ⊂ S nk =1 U k . Consequently, for every k the intersection A ∩ U k is a compact subanalyticsubset of S λ k . Since for each k the semistable quotient S λ k → S λ k //T exists, we concludefrom the proposition in Section 3 of [11] that T · ( A ∩ U k ) = ( T · A ) ∩ ( T · U k ) is subanalyticin S λ k . It follows that for every k the intersection ( T · A ) ∩ ( T · U k ) is subanalytic in theopen set T · U k ⊂ X . Since we have T · A = T · (cid:0)S nk =1 A ∩ U k (cid:1) = S nk =1 (cid:0) ( T · A ) ∩ ( T · U k ) (cid:1) ,we conclude that T · A is locally subanalytic in X . (cid:3) OMOGENEOUS K ¨AHLER AND HAMILTONIAN MANIFOLDS 5
Lemma 3.4.
Let A ⊂ X be (locally) subanalytic. Then K · A is (locally) subanalyticin X .Proof. Since K is compact, the real analytic map Φ : K × X → X , ( k, x ) k · x ,is proper: For every compact subset C ⊂ X the inverse image Φ − ( C ) is closed andcontained in K × ( K · C ), hence compact. We conclude that the restriction of Φ to K × A is proper. Therefore Hironaka’s theorem 3.2, [12] implies that Φ( K × A ) = K · A is subanalytic.If A is locally subanalytic, then A is covered by relatively compact subanalytic opensets U such that A ∩ U is subanalytic. Then it follows as above that K · ( A ∩ U ) issubanalytic, and consequently K · A is locally subanalytic. (cid:3) Now we are in a position to prove the main result of this section.
Theorem 3.5.
Suppose X is a G –connected Hamiltonian G –manifold, where G is acomplex reductive group. Then(1) every G –orbit is locally subanalytic and in particular locally closed in X ,(2) the boundary of every G –orbit contains only G –orbits of strictly smaller dimen-sion, and(3) the closure of every G –orbit is complex-analytic in X .Proof. For every x ∈ X the orbit K · x is a compact real analytic submanifold of X .By Lemma 3.3 the set T · ( K · x ) is locally subanalytic in X . Thus Lemma 3.4 impliesthat K · (cid:0) T · ( K · x ) (cid:1) is locally subanalytic as well. Because of G = KT K every G –orbitis locally subanalytic.In order to see that the G –orbits are locally closed, we take U ∪ · · · ∪ U k to be anopen covering of G · x such that for every j the intersection ( G · x ) ∩ U j is subanalyticin U j . Since the boundary of a subanalytic set is again subanalytic and of strictlysmaller dimension, we see that ( G · x ) ∩ U j contains an interior point of its closurein U j . Moving this point with the G –action it follows that ( G · x ) ∩ U j is open in itsclosure in U j . Consequently, G · x is locally closed.For the second claim it is sufficient to note that the dimension of an orbit G · x can bechecked in the intersection with an open set U such that ( G · x ) ∩ U is subanalytic in U .More precisely, let x, y ∈ X such that G · y ⊂ G · x . Since { x, y } is compact subanalytic,there are finitely many open sets U , . . . , U k such that ( G · x ) ∪ ( G · y ) ⊂ U ∪ · · · ∪ U k and such that (cid:0) ( G · x ) ∪ ( G · y ) (cid:1) ∩ U j is subanalytic in U j for every j . Suppose y ∈ U .Then ( G · y ) ∩ U lies in the closure of ( G · x ) ∩ U in U . After possibly shrinking U wemay assume that ( G · y ) ∩ U is subanalytic in U which implies that ( G · x ) ∩ U is alsosubanalytic in U . Hence, we obtain dim G · y = dim( G · y ) ∩ U < dim( G · x ) ∩ U =dim G · x as was to be shown.Finally let x ∈ X and E := { x ∈ X ; dim G · x < dim G · x } . The set E is complex-analytic and its complement Ω := X \ E is G –invariant. Since the boundary of G · x contains only orbits of strictly smaller dimension by the previous claim, the orbit G · x is closed in Ω and therefore a complex submanifold of Ω. We will show that G · x iscomplex-analytic in X by applying Bishop’s theorem ([5]). For this we must check thatevery point x ∈ E has an open neighborhood U ⊂ X such that U ∩ ( G · x ) has finitevolume with respect to some hermitian metric on X . Without loss of generality wemay assume that x ∈ G · x ∩ E holds. According to what we have already shown we BRUCE GILLIGAN, CHRISTIAN MIEBACH, AND KARL OELJEKLAUS find an open neighborhood U ⊂ X of x such that U ∩ G · x is subanalytic in U . Afterpossibly shrinking U we may assume that U is biholomorphic to the unit ball in C n .It is known (see the remark following Proposition 1.4 in [16]) that the 2 k –dimensionalHausdorff volume (where k := dim C G · x ) of U ∩ ( G · x ) is finite. Since U ∩ ( G · x )is also an immersed submanifold of U , the 2 k –dimensional Hausdorff volume coincideswith the geometric volume associated with the standard hermitian metric on C n (seepage 48 in [21]). This observation allows us to deduce from Bishop’s theorem that G · x is complex-analytic in X . (cid:3) Remark.
We restate the following fact which is shown in the third part of the proof andmight be of independent interest: Let E ⊂ B n be a complex-analytic subset. Supposethat A ⊂ B n \ E is complex-analytic and that A ⊂ B n is locally subanalytic and aninjectively immersed complex submanifold. Then the topological closure of A in B n iscomplex-analytic. Remark.
In [22] holomorphic actions of complex reductive groups G on compact K¨ahlermanifolds X are considered. Under the additional assumption that the G –action on X is projective it is shown that for every x ∈ X the closure G · x is complex-analytic in X . Sommese’s notion of projectivity of a G –action on X is equivalent to the fact that G acts trivially on the Albanese torus Alb( X ). Hence, by Proposition 1 on page 269in [13], G acts projectively on X if and only if X is a Hamiltonian G –manifold.In [8] some properties of algebraic group actions are extended to the more generalclass C of compact complex spaces that are the meromorphic images of compact K¨ahlerspaces and it is shown that the orbit closures are complex analytic in this setting.From the remark after Definition 2.1 we obtain the following Corollary 3.6.
Let G = S be a semisimple complex Lie group acting holomorphicallyon the K¨ahler manifold X . Then the S –orbits are locally closed in X . Homogeneous Hamiltonian G –manifolds Let G = K C be a connected complex reductive group and let H be a closed complexsubgroup of G . Suppose that the homogeneous space X = G/H admits a K –invariantK¨ahler form ω . We want to show that the existence of a K –equivariant moment map µ : X → k ∗ implies that H is an algebraic subgroup of G . Example. If G is Abelian, i. e. if G = ( C ∗ ) k , then the fact that G/H is K¨ahler does notimply that H is algebraic as the example of an elliptic curve C ∗ / Z shows. However, if G/H is a Hamiltonian G –manifold, then by Theorem 3.1(1) the group H is complexreductive and hence algebraic. Example.
Suppose that X = G/H is a Hamiltonian G –manifold with moment map µ : X → k ∗ . If µ − (0) = ∅ , then the semistable quotient X//G exists (and is a point)and thus X = G/H is Stein by [11]. In this case H is a reductive complex subgroup of G and hence is algebraic, see [19] and [20].We will need the following technical result. Lemma 4.1.
Let Γ be a discrete subgroup of G normalizing H such that X = ( G/H ) / Γ is a Hamiltonian G –manifold with moment map µ . Suppose that Γ acts by holomorphic OMOGENEOUS K ¨AHLER AND HAMILTONIAN MANIFOLDS 7 transformations on a complex manifold Y and that Y admits a Γ –invariant K¨ahlerform ω Y . Recall that the twisted product ( G/H ) × Γ Y is by definition the quotient of ( G/H ) × Y by the diagonal Γ –action γ · ( gH, y ) := ( γ · gH, γ · y ) . Then ( G/H ) × Γ Y is aHamiltonian G –manifold with moment map b µ : ( G/H ) × Γ Y → k ∗ , b µ [ gH, y ] := µ ( gH Γ) .Proof. Let p : G/H → ( G/H ) / Γ be the quotient map and let ω be a K –invariantK¨ahler form on ( G/H ) / Γ. Then p ∗ ω is a K – and Γ–invariant K¨ahler form on G/H and thus p ∗ ω + ω Y is a Γ–invariant K¨ahler form on ( G/H ) × Y . Hence, we see that b Y := ( G/H ) × Γ Y is K¨ahler.The map b µ is well-defined and K –equivariant. Let ξ b Y be the vector field on b Y inducedby ξ ∈ k . Let U ⊂ ( G/H ) / Γ be an open set such that the bundle q : b Y = ( G/H ) × Γ Y → ( G/H ) / Γ is trivial over U , i. e. such that q − ( U ) ∼ = U × Y . For every [ gH, y ] ∈ U the vector ξ b Y [ gH, y ] corresponds to (cid:0) ξ ( G/H ) / Γ (cid:0) p ( gH ) (cid:1) , (cid:1) ∈ T p ( gH ) ( G/H ) / Γ ⊕ T y Y .Moreover, we have d b µ ξ = dµ ξ in this trivialization. By construction of the K¨ahler formon b Y we conclude that b µ is a moment map for the K –action on b Y = ( G/H ) × Γ Y . (cid:3) We will first prove the algebraicity of H under the assumption that H is a discretesubgroup of G . In this case we write Γ instead of H . Proposition 4.2.
Let G be a connected complex reductive group and let Γ be a discretesubgroup of G such that X = G/ Γ is a Hamiltonian G –manifold. Then Γ is finite.Proof. Let us briefly recall the Jordan decomposition of elements in the affine algebraicgroup G = K C (see Chapter I.4 in [7]). Suppose that G is a subgroup of GL( N, C ).An element γ ∈ G is called semisimple if the matrix representing γ is diagonalizable,and unipotent if the matrix γ − I N is nilpotent. It can be shown that these notions donot depend on the chosen embedding G ֒ → GL( N, C ). Moreover, every element γ ∈ G has a (unique) Jordan decomposition γ = γ s γ u = γ u γ s in G , where γ s is semisimple and γ u is unipotent.Suppose there is an element γ ∈ Γ with γ u = e . Then there exists a nilpotentelement ξ ∈ g with γ u = exp( ξ ). Since the group exp( C ξ ) is closed in G , the sameholds for the cyclic group h γ i := { γ m ; m ∈ Z } ∼ = Z . Lemma 2.3 implies then that G/ h γ i is a Hamiltonian G –manifold. The group h γ i acts on C ∗ by γ m · z := e im z .Applying Lemma 4.1 we conclude that the twisted product G × h γ i C ∗ is a Hamiltonian G –manifold. Since the G –orbits in this twisted product intersect C ∗ in h γ i –orbits andsince these orbits are dense in the S –orbits in C ∗ , we arrive at a contradiction toTheorem 3.5. Consequently, every γ ∈ Γ must be semisimple.If γ = γ s , then the Zariski closure of the cyclic group generated by γ is either finiteor a complex torus T ∼ = ( C ∗ ) l for some l ≥
1. Assume that the latter holds. Then
T / (Γ ∩ T ) is a Hamiltonian T –manifold by Lemma 2.2, and consequently Γ ∩ T must befinite. Since h γ i is contained in Γ ∩ T , it follows that T is finite, a contradiction. Thusevery element of Γ is semisimple and generates a finite group. According to Lemma 2.1in [1] the group Γ is finite. (cid:3) Remark.
If the group G is semisimple, then every holomorphic G –manifold which ad-mits a K¨ahler form is Hamiltonian. Hence, we have given a new proof for Theorem 3.1in [3]. BRUCE GILLIGAN, CHRISTIAN MIEBACH, AND KARL OELJEKLAUS
Now we return to the general case that H is any closed complex subgroup of G suchthat X = G/H is a Hamiltonian G –manifold. The following theorem the proof of whichcan be found in [1] gives a necessary and sufficient condition for H to be algebraic. Theorem 4.3.
For h ∈ H let A ( h ) denote the Zariski closure of the cyclic groupgenerated by h in G . The group H is algebraic if and only if A ( h ) is contained in H for every h ∈ H . Using this result we now prepare the proof of our main theorem in this section.Let h ∈ H . In order to have better control over the group A ( h ) we follow closelyan idea which is described on page 107 in [1]. For this let h = h s h u be the Jordandecomposition of h in G . As we already noted above, if h is semisimple, then A ( h )is either finite or isomorphic to ( C ∗ ) l . In the first case we have A ( h ) ⊂ H . In thesecond case, X = G/H is a Hamiltonian A ( h )–manifold which implies that the orbit A ( h ) · eH ∼ = A ( h ) / (cid:0) A ( h ) ∩ H (cid:1) is Hamiltonian. Hence, A ( h ) ∩ H is algebraic whichyields A ( h ) ⊂ H .If h is unipotent, then there exists a simple three dimensional closed complex sub-group S of G containing h (see [14]). Again X = G/H is a Hamiltonian S –manifold.Hence the orbit S · eH ∼ = S/ ( S ∩ H ) is Hamiltonian and in particular K¨ahler. We haveto show that S ∩ H is algebraic in S . Then we have A ( h ) ⊂ S ∩ H ⊂ H , as was to beshown. Algebraicity of S ∩ H will be a consequence of the following lemma for whichwe give here a direct proof. Lemma 4.4.
Let H be a closed complex subgroup of S = SL(2 , C ) . If S/H is K¨ahler,then H is algebraic.Proof. Since every Lie subalgebra of s = sl (2 , C ) is conjugate to { } , to C ( ), to C ( − ), to a Borel subalgebra b , or to s , we conclude that the identity component H is automatically algebraic. Therefore it suffices to show that H has only finitelymany connected components since then H is the finite union of translates of H whichis algebraic.For H = S this is trivial. Since the normalizer of a Borel subgroup B of S coincideswith B , we see that H = B implies H = B , hence that H is algebraic in this case.If H is a maximal algebraic torus in S , then its normalizer in S has two connectedcomponents, thus H has at most two connected components as well.Suppose that H is unipotent. Then its normalizer is a Borel subgroup. If H hasinfinitely many connected components, we find an element h ∈ H \ H which generatesa closed infinite subgroup Γ of S . Then S/ (Γ H ) is K¨ahler (for it covers S/H ), andwe conclude from Lemma 4.1 that (
S/H ) × Γ C ∗ is a Hamiltonian S –manifold whereΓ acts on C ∗ by γ m · z := e im z . As above this contradicts Theorem 3.5.Since the case that H is trivial, i. e. that H is discrete, has already been treated,the proof is finished. (cid:3) Now suppose that h = h s h u with h s = e and h u = e . In this case there is asimple three dimensional closed complex subgroup S of the centralizer of A ( h s ) whichcontains A ( h u ). Then A ( h ) ⊂ S A ( h s ) and a finite covering of S A ( h s ) is isomorphic toSL(2 , C ) × ( C ∗ ) l . (We may suppose that A ( h s ) has positive dimension, if not, we areessentially in the previous case.) Moreover, there is a closed complex subgroup e H of OMOGENEOUS K ¨AHLER AND HAMILTONIAN MANIFOLDS 9 e G = SL(2 , C ) × ( C ∗ ) l containing the element h = (cid:0) ( ) , ( e a , . . . , e a l ) (cid:1) such that e G/ e H is Hamiltonian. We must show that A ( h ) = ( C ) × ( C ∗ ) l is contained in e H .In order to simplify the notation we will continue to write G and H instead of e G and e H . The following observation is central to our argument. Lemma 4.5.
We may assume without loss of generality that H ∩ ( C ∗ ) l = { e } .Proof. For this note that the action of ( C ∗ ) l on G/H is Hamiltonian. This implies that H ∩ ( C ∗ ) l is a central subtorus T of G . Consequently, G/H ∼ = ( G/T ) / ( H/T ). If
H/T is algebraic in
G/T , then H is algebraic in G . (cid:3) Let p and p denote the projections of G = SL(2 , C ) × ( C ∗ ) l onto SL(2 , C ) and ( C ∗ ) l ,respectively. Lemma 4.6.
The map p : G → SL(2 , C ) maps the group H isomorphically onto aclosed complex subgroup of SL(2 , C ) .Proof. We show first that p ( H ) is closed in SL(2 , C ). For this note that G/H is aHamiltonian ( C ∗ ) l –manifold. By Theorem 3.5 all ( C ∗ ) l –orbits are locally closed in G/H . Since ( C ∗ ) l is the center of G , we have ( C ∗ ) l · ( gH ) = g · (cid:0) ( C ∗ ) l · eH (cid:1) . Hence all( C ∗ ) l –orbits have the same dimension. This implies that all ( C ∗ ) l –orbits are closed in G/H . Consequently, ( C ∗ ) l H is closed in G which shows that p ( H ) = SL(2 , C ) ∩ ( C ∗ ) l H is closed.Since the restriction of p to the closed subgroup H of G is a surjective holomorphichomomorphism onto p ( H ) with kernel H ∩ ( C ∗ ) l = { e } , the claim follows. (cid:3) If p ( H ) = SL(2 , C ), then p : H ∼ = SL(2 , C ) → ( C ∗ ) l must be trivial. But thiscontradicts the fact that ( e a , . . . , e a l ) is contained in p ( H ). Therefore p ( H ) must bea proper closed subgroup of SL(2 , C ) which contains the element ( ). In particular,we conclude that H is solvable.There are essentially three possibilities. The image p ( H ) is a Borel subgroup ofSL(2 , C ) (which implies that H is a connected two-dimensional non-Abelian subgroupof G ), or p ( H ) = ( C ), or p ( H ) is discrete containing ( Z ). If p ( H ) is discrete,then H is discrete. We have already shown that H is finite in this case, hence algebraic. Remark.
Suppose that p ( H ) is the Borel subgroup of upper triangular matrices inSL(2 , C ). The map p | H : H → p ( H ) is a surjective homomorphism with kernel H ′ = H ∩ SL(2 , C ). Thus we have H ∩ SL(2 , C ) = ( C ).Suppose that p ( H ) is one-dimensional or a Borel subgroup. We know that p ( H )is a closed complex subgroup of ( C ∗ ) l containing ( e a , . . . , e a l ). Since dim p ( H ) =1, we conclude that p ( H ) = (cid:8) ( e ta , . . . , e ta l ); t ∈ C (cid:9) . Lemma 2.3 implies that if G/H is Hamiltonian, then the same holds for
G/H . The possibilities for H are H = (cid:8)(cid:0)(cid:0) e ta s e − ta (cid:1) , ( e ta , . . . , e ta l ) (cid:1) ; t, s ∈ C (cid:9) (if p ( H ) is a Borel subgroup) or H = { (( t ) , ( e ta , . . . , e ta l )) ; t ∈ C } . We have to show that in both cases G/H is notHamiltonian.Let us first consider the case that H = (cid:26)(cid:18)(cid:18) t (cid:19) , ( e ta , . . . , e ta l ) (cid:19) ; t ∈ C (cid:27) . Let T = ( C ∗ ) l − ×{ } ⊂ ( C ∗ ) l and e G := SL(2 , C ) × T . Then we have T ∩ p ( H ) =: Γ ∼ = Z and ( C ∗ ) l /p ( H ) ∼ = T /
Γ. Moreover,
G/H is a Hamiltonian e G –manifold and we have G/H ∼ = e G/ e H , where e H = e G ∩ H ∼ = Z . This contradicts our result in the discrete case.Hence, G/H cannot be Hamiltonian.Finally, suppose that H = (cid:26)(cid:18)(cid:18) e ta s e − ta (cid:19) , ( e ta , . . . , e ta l ) (cid:19) ; t, s ∈ C (cid:27) . Again we consider T = ( C ∗ ) l − × { } and e G = SL(2 , C ) × T . We have e H = e G ∩ H =Γ ⋉ ( C ) where Γ ∼ = Z . As in the discrete case we let Γ act on C ∗ by γ m · z := e im z andconsider the twisted product ( e G/ e H ) × Γ C ∗ . If G/H is Hamiltonian, then the sameholds for e G/ e H and thus for e G/ e H . Then ( e G/ e H ) × Γ C ∗ is Hamiltonian by Lemma 4.1.Since the Γ–orbits in C ∗ are not locally closed, this contradicts Theorem 3.5 and weconclude that G/H is not Hamiltonian.Summarizing our discussion in this section we proved the following.
Theorem 4.7.
Let G be a connected complex reductive group and let H be a closedcomplex subgroup. If X = G/H is a Hamiltonian G –manifold, then H is an algebraicsubgroup of G . In particular we obtain the following result which was originally proved in [3] and [2].
Corollary 4.8.
Let S be a connected semisimple complex Lie group and let H be aclosed complex subgroup of S . If S/H admits a K¨ahler form, then H is an algebraicsubgroup of S . Homogeneous K¨ahler manifolds
Let G = K C be a connected complex reductive Lie group. In this section we char-acterize those closed complex subgroups H of G for which X = G/H admits a K¨ahlerform.According to [7], Corollary I.2.3, the commutator group S := G ′ is a connectedalgebraic subgroup of G and, since G is reductive, S is semisimple. Let Z := Z ( G ) ∼ =( C ∗ ) k . Then G = SZ and S ∩ Z is finite. Theorem 5.1.
Let G be a reductive complex Lie group and H ⊂ G a closed complexsubgroup. Then the manifold X = G/H admits a K¨ahler form if and only if S ∩ H ⊂ S is algebraic and SH is closed in G .Proof. After replacing G by a finite cover we may assume that G = S × Z . Supposefirst that G/H is K¨ahler. Then S/ ( S ∩ H ) is also K¨ahler and hence S ∩ H ⊂ S isalgebraic by Corollary 4.8. By Theorem 3.5 all S –orbits in G/H are open in theirclosures and their boundaries only contain orbits of strictly smaller dimension. In viewof the reductive group structure of G the S –orbits in X all have the same dimension.This implies that every S –orbit in G/H is closed. Consequently, SH is closed in G .Now suppose that S ∩ H ⊂ S is algebraic and that SH is closed in G . Although it isnot used in the proof we remark that we may assume that H is solvable, since otherwiseone can divide by the (ineffective) semisimple factor of H . Consider the fibration X = G/H → G/SH.
OMOGENEOUS K ¨AHLER AND HAMILTONIAN MANIFOLDS 11
The base
G/SH is an Abelian complex Lie group and the fiber is
SH/H = S/ ( S ∩ H ).There is a subgroup ( C ∗ ) l ∼ = Z ⊂ Z ∼ = ( C ∗ ) k such that G := S × Z ⊂ G actstransitively on X and Z ∩ SH is discrete. With H := H ∩ G we have that X = G /H ,that SH is closed in G and that the base of the fibration X = G /H → G /SH = Z / ( Z ∩ SH )is a discrete quotient of Z . Furthermore S ∩ H = S ∩ H is an algebraic subgroup of S .Let Γ := Z ∩ SH and Γ ⊂ Z be a discrete subgroup such that Γ ∩ Γ = { e } andsuch that Γ := Γ + Γ is a discrete cocompact subgroup of Z . Since H ′ is contained in S ∩ H , we can define the closed complex subgroup H ⊂ G to be the group generatedby H and Γ ( S ∩ H ). One still has that H ∩ S = H ∩ S = H ∩ S is algebraic in S and that SH is closed in G . Hence X = G /H → G /H is a covering map andone sees that in order to finish the proof it is sufficient to show that the base G /H admits a K¨ahler form.So we may drop the indices and have to prove that a reductive quotient X = G/H of G = S × ( C ∗ ) k = S × Z by a closed complex solvable subgroup H with S ∩ H ⊂ S algebraic, SH ⊂ G closed and G/SH a compact torus, is K¨ahler.Let p : G → S be the projection onto S . Note that the algebraic Zariski closure H of H in G is the product b H × ( C ∗ ) k , where b H is the Zariski closure of the projectionp ( H ) of H in S . The commutator group H ′ of H is also the commutator group of H and of b H and is contained in H ∩ S . Therefore one gets a natural algebraic rightaction of H on the homogeneous manifold Y := S/ ( S ∩ H ) given by( ⋆ ) h (cid:0) s ( S ∩ H ) (cid:1) := s (cid:0) p ( h ) (cid:1) − ( S ∩ H ) . As a consequence we can equivariantly compactify the SH –manifold Y to an almosthomogeneous projective SH –manifold Y , see [18], Proposition 3.1, and [15], Proposi-tion 3.9.1. Since X = G/H is realizable as a quotient of the manifold S/ ( S ∩ H ) × ( C ∗ ) k by the natural action of H/ ( S ∩ H ), where the S –factor of the action is given by ( ⋆ ), wesee that X is an open set in a holomorphic fiber bundle X with a compact torus as baseand the simply connected projective manifold Y as fiber. Finally we apply Blanchard’stheorem, see [6], p. 192, to get a K¨ahler form on X and then, by restriction, on X also.The theorem is proved. (cid:3) References [1] Barth, W., Otte, M.: Invariante holomorphe Funktionen auf reduktiven Liegruppen, Math.Ann. 201, 97–112 (1973)[2] Berteloot, F.: Existence d’une structure k¨ahl´erienne sur les vari´et´es homog`enes semi-simples, C. R. Acad. Sci. Paris S´er. 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