We prove a relation between the $\bar\partial_M$ cohomology of a minimal
orbit $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a flag
manifold $G/Q$ and the Dolbeault cohomology of the Matsuki dual open orbit $X$
of the complexification $K$ of a maximal compact subgroup $K_0$ of $G_0$, under
the assumption that $M$ is Levi-flat.Comment: 7 pages, AMS-LaTeX to appear in Rend. Ist. Mat. Univ. Trieste v2:
  added reference in introduction + minor revisio

Altomani, Andrea

English

arXiv

We prove a relation between the $\bar{\partial}_{M}$ cohomology of
a minimal orbit of M of a real form of $\mathbf{G}_{0}$ of a complex
semisimple Lie group $\mathbf{G}$ in a flag manifold $\mathbf{G}/\mathbf{Q}$
and the Dolbeault cohomology of the Matsuki dual open orbit X of the
complexification $\mathbf{K}$ of a maximal compact subgroup $\mathbf{K}_{0}$
of $\mathbf{G}_{0}$, under the assumption that M is Levi-flat

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Rend. Istit. Mat. Univ. TriesteVol. XXXVII, 283–293 (2005)A Note on the CR Cohomology ofLevi-Flat Minimal Orbits inComplex Flag ManifoldsAndrea Altomani (∗)Summary. - We prove a relation between the ∂̄M cohomology of aminimal orbit M of a real form G0 of a complex semisimple Liegroup G in a flag manifold G/Q and the Dolbeault cohomologyof the Matsuki dual open orbit X of the complexification K of amaximal compact subgroup K0 of G0, under the assumption thatM is Levi-flat.1. IntroductionMany authors have studied the ∂̄M cohomology of CR manifolds (seee. g. [6, 7, 12] and references therein). In particular, since Andreottiand Fredricks [2] proved that every real analytic CR manifold Mcan be embedded in a complex manifold X, it is natural to try tofind relations between the ∂̄M cohomology of M and the Dolbeaultcohomology of X.In this paper we examine this problem for a specific class ofhomogeneous CR manifolds, namely minimal orbits in complex flagmanifolds that are Levi-flat.Given a (generalized) flag manifold Y = G/Q, with G a complexsemisimple Lie group and Q a parabolic subgroup of G, a real form(∗) Author’s address: Andrea Altomani, Dipartimento di Matematica II Univer-sità di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy,e-mail: altomani@mat.uniroma2.itKeywords: Cohomology, Cauchy-Riemann complex, Levi-flat, Homogeneous CRmanifold, Minimal orbit in a flag manifold.AMS Subject Classification: Primary: 32L10; Secondary: 14M15, 32V30, 57T15.284 ANDREA ALTOMANIG0 of G acts on Y with finitely many orbits. Among these there isexactly one orbit that is compact, the minimal orbit M = G0 ·o. LetK0 be a maximal compact subgroup of G0, and K its complexifica-tion. Then X = K ·o is an open dense complex submanifold of G/Qand contains M as an embedded submanifold. It is known that Mis a deformation retract of X, so Hp(M, C) = Hp(X, C) ([4], [8]).Let E be a K-homogeneous complex vector bundle on X andE|M its restriction to M . Under the additional assumption that Mis Levi-flat we prove that the restriction map from the Dolbeault co-homology Hp,q(X,E) to the ∂̄M cohomology Hp,q(M,E|M ) is con-tinuous, injective and has a dense range. More precisely we showthatHp,q(M,E|M ) = OM (M) ⊗OX(X) Hp,q(X,E)where OM (M) (resp. OX(X)) is the space of CR (resp. holomor-phic) functions on M (resp. X), and that the restriction map fromOX(X) to OM (M) is injective, continuous and has a dense range.2. Preliminaries on minimal orbits in complex flagmanifoldsLet G be a complex connected semisimple Lie group, with Lie al-gebra g, and let Q be a parabolic subgroup of G, with Lie algebraq. Then Q is the normalizer of q, Q = NG(q) and is connected.The coset space Y = G/Q is a compact complex manifold, called aflag manifold (it is a complex smooth projective variety). It is notrestrictive to assume that q does not contain any simple ideal of g.Let σ be an anti-holomorphic involution of G; we will also denoteby σ its differential at the identity and we will write x̄ = σ(x). Areal form of G is an open subgroup G0 of Gσ. It is a Lie subgroupand its Lie algebra g0 satisfies g0 = gσ and g = C ⊗ g0.Let K0 be a maximal compact subgroup of G0, and θ the cor-responding Cartan involution: K0 = Gθ0. Still denoting by θ thecomplexification of θ, there is exactly one open subgroup K of Gθsuch that K ∩ G0 = K0. Let k and k0 be the corresponding Liesubalgebras.The groups G0 and K act on Y via left multiplication. Thereis exactly one closed G0-orbit M ([1, 19]) and it is contained inCOHOMOLOGY OF LEVI FLAT MINIMAL ORBITS 285the unique open K orbit X ([3, 14]), we denote by j : M → X theinclusion. The open orbit X is dense in Y and is the dual orbit ofM , in the terminology of [14].The manifolds M and X do not depend on the choice of G andG0, but only on g0 and q ([1, 14]). So there is no loss of generalityassuming that G is simply connected, that G0 is connected and thatM and X are the orbits through the point o = eQ. We will writeM = M(g0, q).The isotropy subgroups at o of the actions of G0 and K are G+ =G0 ∩Q and L = K∩Q, with Lie algebras g+ = g0 ∩ q and l = k∩ q.Since M is compact, the action on M of maximal compact subgroupK0 is transitive: M = K0/K+, where K+ = K0 ∩ Q = G0 ∩ L andLie(K+) = k+ = k0 ∩ q = g0 ∩ l.In the language of [1] the pair (g0, q) is an effective parabolicminimal CR algebra and M is the associated minimal orbit. On Mthere is a natural CR structure induced by the inclusion into X.We recall that M is totally real if the partial complex structureis trivial. We give a more complete characterization of totally realminimal orbits:Theorem 2.1. The following are equivalent:1. M is totally real.2. q̄ = q.3. l = kC+.4. X is a Stein manifold.5. X is a smooth affine algebraic variety defined over R and Mis the set of its real points.Proof. (5) ⇒ (4) because every closed complex submanifold of acomplex vector space is Stein.(4) ⇒ (3) Since X is Stein, its covering X̃ = K/L0 is also Stein([17]). Furthermore K is a linear algebraic group that is the com-plexification of a maximal compact subgroup K0; the result thenfollows from Theorem 3 of [15] and Remark 2 thereafter.(3) ⇒ (5) If l is the complexification of k+, then L is the com-plexification of K+. Hence X = K/L is the complexification of286 ANDREA ALTOMANIM = K0/K+ in the sense of [9], and (5) follows from Theorem 3 ofthe same paper.(1) ⇔ (2) is easy, and proved in [1].(5) ⇒ (1) is obvious.(2) ⇒ (3) We have that k̄ = k, thus k∩ q = (k∩ q∩ g0)C = kC+.We denote by ON the sheaf of smooth CR functions on a CRmanifold N . If N is complex or real, then ON is the usual sheaf ofholomorphic or smooth (complex valued) functions. For every openset U ⊂ N , the space ON (U) is a Fréchet space (with the topologyof uniform convergence of all derivatives on compact sets).Corollary 2.2. If M is totally real then j∗(OX(X))is dense inOM (M) = C∞(M).Proof. Let X ⊂ CN be an embedding as in (5) of Theorem 2.1, sothat M = X ∩ RN . The restrictions of complex polynomials in CNare contained in OX(X) and dense in OM (M) (see e.g. [18]).3. Levi-flat orbits and the fundamental reductionIn this paper we consider Levi-flat minimal orbits. They are or-bits M = M(g0, q), where q′ = q + q̄ is a subalgebra (necessarilyparabolic) of g. Let Q′ = NG(q′), Y ′ = G/Q′, G′+ = G ∩ Q′,M ′ = M(g0, q′) = G0/G′+, K′+ = K0 ∩ Q′, L′ = K ∩ Q′ andX ′ = K/L′.From Theorem 2.1 we have that M ′ is totally real and X ′ is Stein.The inclusion Q → Q′ induces a fibrationπ : Y = G/Q −→ G/Q′ = Y ′ (1)with complex fiber F ≃ Q′/Q. This fibration is classically referredto as the Levi foliation, and is a special case of the fundamentalreduction of [1]. In fact Levi-flat minimal orbits are characterized bythe property that the fibers of the fundamental reduction are totallycomplex.We identify F with π−1(eQ).Lemma 3.1. π−1(M ′) = M , π−1(X ′) = X and F is a compact con-nected complex flag manifold.COHOMOLOGY OF LEVI FLAT MINIMAL ORBITS 287Proof. First we observe that F is connected because Q′ is connected.Let F ′ be the fiber of the restriction π|M : M → M′. Then F ′is totally complex and CR generic in F , hence an open subset ofF . Proposition 7.3 and Theorem 7.4 of [1] show that there exists aconnected real semisimple Lie group G′′0 acting on F′ with an openorbit, and that the Lie algebra of the isotropy is a t-subalgebra (i. e.contains a maximal triangular subalgebra) of g′′0. Hence a maximalcompact subgroup K′′0 of G′′0 has an open orbit, which is also closed.Since F ′ is open in F and F is connected, K′′0 is transitive on F , andF ′ = F , proving the first two statements.Furthermore the isotropy subgroup G′′+ for the action of G′′0 onF and the homogeneous complex structure are exactly those of atotally complex minimal orbit, hence by [1, § 10] F is a complex flagmanifold.The total space M is locally isomorphic to an open subset ofM ′ × F , hence to U × Rk, where U is open in Cn, for some integersn and k.For a Levi flat CR manifold N and a nonnegative integer p, let ΩpNbe the sheaf of p-forms that are CR (see [7] for precise definitions).They are ON -modules and Ω0N ≃ ON .Let Ap,qN be the sheaf of (complex valued) smooth (p, q)-formson N , ∂̄N the tagential Cauchy-Riemann operator and Zp,qN , (resp.Bp,qN ) the sheaf of closed (resp. exact) (p, q)-forms. As usual wedenote by Hp,q(N) = Zp,qN (N)/Bp,qN (N) the cohomology groups ofthe ∂̄M complex on smooth forms. The Poincaré lemma is valid for∂̄N (see [11]), thus the complex:0 → ΩpN → Ap,0N∂̄N−−→ . . .∂̄N−−→ Ap,qN∂̄N−−→ . . .is a fine resolution of ΩpN . This implies that Hp,q(N) ≃ Hq(ΩpN ).Let EN be a homogeneous CR vector bundle on N (i. e. a com-plex vector bundle with transition functions that are CR ) with fiberE, and let EN be the sheaf of its CR sections.We denote by EpN the bundle of CR , EN -valued, p-forms, withassociated sheaf of CR sections EpN = ΩpN ⊗ON EN .Let Ap,qN,EN = Ap,qN ⊗ON EN , denote by ∂̄EN the tangential Cauchy-Riemann operator on EN and let Zp,qN,EN(resp. Bp,qN,EN ) be the sheaf288 ANDREA ALTOMANIof ∂̄EN -closed (resp. exact) smooth forms with values in EN .Then Hp,q(N,EN ) = Zp,qN,EN(N)/Bp,qN,EN (N), but we also have:Hp,q(N,EN ) = Hq(EpN ) = H0,q(N,EpN ) = Z0,qN,EpN(N)/B0,qN,EpN(N).For any open set U ⊂ N , the spaces Ap,qN,EN (U) and Zp,qN,EN(U) areFréchet spaces with the topology of uniform convergence of all deriva-tives on compact sets. If Bp,qN,EN (N) is closed in Zp,qN,EN(N), thenHp,q(N,EN ), with the quotient topology, is also a Fréchet space.4. Statements and proofsLet EF be a L′-homogeneous holomorphic vector bundle on F . TheL′ action induces a natural L′ action on Ap,qF,EF , hence on Hp,q(F,EF ),because the action of L′ preserves closed and exact forms. Since F isa compact complex manifold, Hp,q(F,EF ) is finite dimensional andwe can construct the K-homogeneous holomorphic vector bundle onX ′:Hp,qX′ (F,EF ) = K ×L′ Hp,q(F,EF ).In a similar way we define a K0-homogeneous complex vectorbundle on M ′:Hp,qM ′(F,EF ) = K0 ×K′+ Hp,q(F,EF ).The following thorem has been proved by Le Potier ([13], see also[5]):Theorem 4.1. Let X, X ′ and F be as above, EX a K-homogeneousholomorphic vector bundle on X and EX |F its restriction to F . Thenthere exists a spectral sequence pEs,tr , converging to Hp,q(X,EX),withpEs,t2 =⊕iH i,s−i(X ′,Hp−i,t+iX′ (F,EX |F )).For p = 0 the spectral sequence collapses at r = 2 and, recallingthat X ′ is a Stein manifold, we obtain:H0,q(X,EX) = H0,0(X ′,H0,qX′ (F,EX |F )).Recalling that Hp,q(X,EX) = H0,q(X,EpX ) we finally obtain:COHOMOLOGY OF LEVI FLAT MINIMAL ORBITS 289Proposition 4.2. Let X, X ′ and F be as above, EX a K-homogeneousholomorphic vector bundle on X and EX |F its restriction to F .Then:Hp,q(X,EX) = H0,0(X ′,H0,qX′ (F,EpX |F ))as Fréchet spaces.A statement analogous to the last proposition holds for M :Proposition 4.3. Let M , M ′ and F be as above, EM a K0-homogeneousCR vector bundle on M and EM |F its restriction to F . Then:Hp,q(M,EM ) = H0,0(M ′,H0,qM ′(F,EpM |F ))as Fréchet spaces.Proof. Fix p, q, let ZM ′ = π∗(Z0,qM,EpM), BM ′ = π∗(B0,qM,EpM) and HM ′be the sheaf of sections of H0,qM ′(F,EpM |F ) We already know thatHp,q(M,EM ) ≃ ZM ′(M′)/BM ′(M′).We now define a map φ : ZM ′ → HM ′ as follows.Let U ⊂ M ′, x ∈ U , x = gK′+, g ∈ K0 and ξ ∈ ZM ′(U). Letξg = (g−1 · ξ)|F . This is a closed EpM -valued (0, q)-form on F , thatdetermines a cohomology class [ξg] in H0,q(F,EpM |F ). Then the classof (g, [ξg ]) in H0,qM ′(F,EpM |F ) does not depend on the particular choiceof g, but only on x, hence it defines a section sξ = φ(ξ) of HM ′ onU .The sheaves ZM ′ , BM ′ , HM ′ , ker φ are OM ′-modules and φ is amorphism of OM ′-modules. Since OM ′ is fine, to prove that φ(M′) iscontinuous, surjective and that its kernel is exactly BM ′(M′) it suficesto check that this is true locally, and this reduces to a straightforwardverification.We prove now the main theorem of this paper:Theorem 4.4. Let M and X be as above, EX a K-homogeneousholomorphic vector bundle over X. Then:Hp,q(M,EX |M ) ≃ OM (M) ⊗OX(X) Hp,q(X,EX).290 ANDREA ALTOMANIProof. Define M ′ and X ′ as above, fix integers p, q ≥ 0 and let HX′ =H0,qX′ (F,EpX |F ), HM ′ = H0,qM ′(F,EpX |F ) = HX′ |M ′ . By Proposi-tions 4.2 and 4.3, we have that Hp,q(M,EX |M ) = Γ(M′,HM ′) andHp,q(X,EX) = Γ(X′,HX′).Since dimR M′ = dimC(X′), a global section of HX′ that is zeroon M ′ must be zero on X ′, i. e. the restriction map Γ(X ′,HX′) →Γ(M ′,HM ′) is injective.On the other hand, X ′ is Stein, thus HX′ is generated at everypoint by its global sections. Together with the fact that HM ′ =HX′ |M ′ this implies thatΓ(M ′,HM ′) = OM ′(M′) ⊗OX′(X′) Γ(X′,HX′),where in the right hand side we implicitly identify global holomorphicsections on X ′ with their restrictions to M ′.The theorem follows from the observation that OM (M) ≃ OM ′(M′)and OX(X) ≃ OX′(X′) because the fiber F of π is a compact con-nected complex manifold.This, together with Corollary 2.2, implies the following.Corollary 4.5. With the same assumptions, the inclusion j : M →X induces a map:j∗ : Hp,q(X,EX) −→ Hp,q(M,EX |M )that is continuous, injective and has a dense range.5. An exampleLet G = SL(4, C), and Q be the parabolic subgroup of upper trian-gular matrices. We consider the real form G0 = SU(1, 3), identifiedwith the group of linear trasformations of C4 that leave invariant theHermitian form associated to the matrixB =(0 0 0 10 1 0 00 0 1 01 0 0 0).Then G/Q is the set of complete flags {ℓ1 ⊂ ℓ2 ⊂ ℓ3 ⊂ C4} andM = G0 · eQ is the submanifold {ℓ1 ⊂ ℓ2 ⊂ ℓ3 ⊂ C4 | ℓ3 = (ℓ1)⊥}.COHOMOLOGY OF LEVI FLAT MINIMAL ORBITS 291Let Q′ be the set of block upper triangular matrices of the formQ′ ={g =(∗ ∗ ∗ ∗0 ∗ ∗ ∗0 ∗ ∗ ∗0 0 0 ∗)| g ∈ G},so that M ′ is the totally real manifold {ℓ1 ⊂ ℓ3 ⊂ C4 | ℓ3 = (ℓ1)⊥}and M fibers over M ′ with typical fiber F isomorphic to CP1. Thefibration is given by(ℓ1, ℓ2, ℓ3) 7−→ (ℓ1, ℓ3). (*)Choose K to be the stabilizer in G of the subspaces V =Span(e1 − e4) and W = Span(e1 + e4, e2, e3) so that K is isomor-phic to S(GL(1, C) × GL(3, C)) and K0 to S(U(1) × U(3)). ThenX is the set of flags {ℓ1 ⊂ ℓ2 ⊂ ℓ3 ⊂ C4} in a generic positionwith respect to the subspaces V and W , and X ′ is the set of flags{ℓ1 ⊂ ℓ3 ⊂ C4} in a generic position with respect to V and W .The map from X to X ′ given by (*) is a fibration with typical fiberisomorphic to CP1 and X ′ is a Stein manifold.Finally let E = X × C be the trivial line bundle. According toPropositions 4.2 and 4.3 the cohomology of M and X is given byHp,q(X) = Hp,q(X,E) = H0,0(X ′,H0,qX′ (F,Ep|F )),Hp,q(M) = Hp,q(M,E|M ) = H0,0(M ′,H0,qM ′(F,Ep|F )).Recalling that Hp,q(F ) ≃ C if p = q = 0 or p = q = 1 and 0otherwise, we obtain:{Hp,q(X) ≃ OX(X) ≃ OX′(X′) if p = q = 0 or p = q = 1;Hp,q(X) = 0 otherwise;and analogously:{Hp,q(M) ≃ OM (M) ≃ OM ′(M′) if p = q = 0 or p = q = 1;Hp,q(M) = 0 otherwise;and it is clear thatHp,q(M) = OM (M) ⊗OX(X) Hp,q(X).292 ANDREA ALTOMANIReferences[1] A. Altomani, C. Medori, and M. 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A Note on the CR Cohomology of Levi-Flat Minimal Orbits in Complex Flag Manifolds

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A note on the CR cohomology of Levi-Flat minimal orbits in complex flag manifolds

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