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Let σ be an anti-holomorphic involution of G; we will also denote by σ its differential at the identity and we will writex ¯ = σ(x). A real form of G is an open σ σ subgroup G0 of G . It is a Lie subgroup and its Lie algebra g0 satisfies g0 = g and g = C ⊗ g0. Let K0 be a maximal compact subgroup of G0, and θ the corresponding Car- θ tan involution: K0 = G0. Still denoting by θ the complexification of θ, there is θ exactly one open subgroup K of G such that K ∩ G0 = K0. Let k and k0 be the corresponding Lie subalgebras. The groups G0 and K act on Y via left multiplication. There is exactly one closed G0-orbit M ([1, 17]) and it is contained in the unique open K orbit X ([3, 13]), we denote by j : M → X the inclusion. The open orbit X is dense in Y and is the dual orbit of M, in the terminology of [13]. The manifolds M and X do not depend on the choice of G and G0, but only on g0 and q ([1, 13]). So there is no loss of generality assuming that G is simply connected, that G0 is connected and that M and X are the orbits through the point o = eQ. We will write M = 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 subgroup K0 is transitive: M = K0/K+, where K+ = K0 ∩ Q = G0 ∩ L and Lie(K+)= k+ = k0 ∩ q = g0 ∩ l. In the language of [1] the pair (g0, q) is an effective parabolic minimal CR algebra and M is the associated minimal orbit. On M there is a natural CR structure induced by the inclusion into X. We recall that M is totally real if the partial complex structure is trivial. We give a more complete characterization of totally real minimal orbits:

Theorem 2.1. The following are equivalent: (1) M is totally real. (2) ¯q = q. C (3) l = k+. (4) X is a . (5) X is a smooth affine algebraic variety defined over R and M is the set of its real points.

Proof. (5) ⇒ (4) because every closed complex submanifold of a complex vector space is Stein. (4) ⇒ (3) Since X is Stein, its covering X˜ = K/L0 is also Stein ([15]). Fur- thermore K is a linear algebraic group that is the complexification of a maximal compact subgroup K0; the result then follows from Theorem 3 of [14] and Remark 2 thereafter. (3) ⇒ (5) If l is the complexification of k+, then L is the complexification of K+. Hence X = K/L is the complexification of M = K0/K+ in the sense of [9], and (5) follows from Theorem 3 of the same paper. (1) ⇔ (2) is easy, and proved in [1]. (5) ⇒ (1) is obvious. ¯ C C  (2) ⇒ (3) We have that k = k, thus k ∩ q = (k ∩ q ∩ g0) = k+.

We denote by ON the sheaf of smooth CR functions on a CR manifold N. If N is complex or real, then ON is the usual sheaf of holomorphic or smooth (complex valued) functions. For every open set U ⊂ N, the space ON (U) is a Fr´echet space (with the topology of uniform convergence of all derivatives on compact sets).

∗ ∞ Corollary 2.2. If M is totally real then j OX (X) is dense in OM (M)= C (M).  COHOMOLOGYOFLEVI-FLATMINIMALORBITS 3

Proof. Let X ⊂ CN be an embedding as in (5) of Theorem 2.1, so that M = X∩RN . N The restrictions of complex polynomials in C are contained in OX (X) and dense in OM (M) (see e.g. [16]). 

3. Levi-flat orbits and the fundamental reduction

In this paper we consider Levi-flat minimal orbits. They are orbits M = M(g0, q), ′ ′ ′ where q = q + q¯ is a subalgebra (necessarily parabolic) of g. Let Q = NG(q ), ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ Y = G/Q , G+ = G∩Q , M = M(g0, q )= G0/G+, K+ = K0 ∩Q , L = K∩Q and X′ = K/L′. From Theorem 2.1 we have that M ′ is totally real and X′ is Stein. The inclusion Q → Q′ induces a fibration ′ ′ (3.1) π : Y = G/Q −→ G/Q = Y with complex fiber F ≃ Q′/Q. This fibration is classically referred to as the Levi foliation, and is a special case of the fundamental reduction of [1]. In fact Levi-flat minimal orbits are characterized by the property that the fibers of the fundamental reduction are totally complex. We identify F with π−1(eQ). Lemma 3.1. π−1(M ′)= M, π−1(X′)= X and F is a compact connected complex flag manifold.

Proof. 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 of F . Proposition 7.3 and Theorem 7.4 of [1] show ′′ ′ that there exists a connected real semisimple Lie group G0 acting on F with an open orbit, and that the Lie algebra of the isotropy is a t-subalgebra (i. e. contains ′′ ′′ a maximal triangular subalgebra) of g0 . Hence a maximal compact subgroup K0 ′′ ′ of G0 has an open orbit, which is also closed. Since F is open in F and F is ′′ ′ connected, K0 is transitive on F , and F = F , proving the first two statements. ′′ ′′ Furthermore the isotropy subgroup G+ for the action of G0 on F and the ho- mogeneous complex structure are exactly those of a totally complex minimal orbit, hence by [1, § 10] F is a complex flag manifold. 

The total space M is locally isomorphic to an open subset of M ′ × F , hence to U × Rk, where U is open in Cn, for some integers n and k. p For a Levi flat CR manifold N and a nonnegative integer p, let ΩN be the sheaf of p-forms that are CR (see [7] for precise definitions). They are ON -modules and 0 ΩN ≃ ON . p,q ¯ Let AN be the sheaf of (complex valued) smooth (p, q)-forms on N, ∂N the p,q p,q tagential Cauchy-Riemann operator and ZN , (resp. BN ) the sheaf of closed (resp. p,q p,q p,q exact) (p, q)-forms. As usual we denote by H (N)= ZN (N)/BN (N) the coho- mology groups of the ∂¯M complex on smooth forms. The Poincar´elemma is valid for ∂¯N (see [10]), thus the complex:

p p,0 ∂¯N ∂¯N p,q ∂¯N 0 → ΩN → AN −−→ ... −−→AN −−→ ... p p,q q p is a fine resolution of ΩN . This implies that H (N) ≃ H (ΩN ). Let EN be a homogeneous CR vector bundle on N (i. e. a complex vector bundle with transition functions that are CR) with fiber E, and let EN be the sheaf of its CR sections. Ep E We denote by N the bundle of CR, N -valued, p-forms, with associated sheaf p p of CR sections EN =ΩN ⊗ON EN . 4 ANDREA ALTOMANI

p,q p,q O ¯E Let AN,EN = AN ⊗ N EN , denote by ∂ N the tangential Cauchy-Riemann p,q p,q E ¯E operator on N and let ZN,EN (resp. BN,EN ) be the sheaf of ∂ N -closed (resp. exact) smooth forms with values in EN . p,q E p,q p,q Then H (N, N )= ZN,EN (N)/BN,EN (N), but we also have: p,q E q p 0,q Ep 0,q 0,q H (N, N )= H (EN )= H (N, N )= Z Ep (N)/B Ep (N). N, N N, N p,q p,q For any open set U ⊂ N, the spaces AN,EN (U) and ZN,EN (U) are Fr´echet spaces with the topology of uniform convergence of all derivatives on compact sets. p,q p,q p,q E If BN,EN (N) is closed in ZN,EN (N), then H (N, N ), with the quotient topology, is also a Fr´echet space.

4. Statements and proofs ′ ′ Let EF be a L -homogeneous holomorphic vector bundle on F . The L action L′ p,q p,q E induces a natural action on AF,EF , hence on H (F, F ), because the action of L′ preserves closed and exact forms. Since F is a compact , p,q H (F, EF ) is finite dimensional and we can construct the K-homogeneous holo- morphic vector bundle on X′: p,q H E ′ p,q E X′ (F, F )= K ×L H (F, F ). ′ In a similar way we define a K0-homogeneous complex vector bundle on M : p,q p,q H ′ E K K′ E M (F, F )= 0 × + H (F, F ). The following thorem has been proved by Le Potier ([12], see also [5]): ′ Theorem 4.1. Let X, X and F be as above, EX a K-homogeneous holomorphic E vector bundle on X and X |F its restriction to F . Then there exists a spectral p s,t p,q E sequence Er , converging to H (X, X ), with p s,t i,s−i ′ Hp−i,t+i E  E2 = H X , X′ (F, X |F ) . i M  For p = 0 the spectral sequence collapses at r = 2 and, recalling that X′ is a Stein manifold, we obtain: 0,q E 0,0 ′ H0,q E H (X, X )= H X , X′ (F, X |F ) . p,q E 0,q Ep Recalling that H (X, X )= H (X, X ) we finally obtain: ′ Proposition 4.2. Let X, X and F be as above, EX a K-homogeneous holomor- E phic vector bundle on X and X |F its restriction to F . Then: p,q E 0,0 ′ H0,q Ep H (X, X )= H X , X′ (F, X |F )  as Fr´echet 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-homogeneous CR E vector bundle on M and M |F its restriction to F . Then: p,q E 0,0 ′ H0,q Ep H (M, M )= H M , M ′ (F, M |F ) as Fr´echet spaces.  0,q 0,q Proof. Fix p, q, let ZM ′ = π∗(Z Ep ), BM ′ = π∗(B Ep ) and HM ′ be the sheaf of M, M M, M 0,q p ′ ′ H E p,q E ′ ′ sections of M ′ (F, M |F ) We already know that H (M, M ) ≃ ZM (M )/BM (M ). We now define a map φ: ZM ′ → HM ′ as follows. ′ ′ − ′ 1 Let U ⊂ M , x ∈ U, x = gK+, g ∈ K0 and ξ ∈ ZM (U). Let ξg = (g · ξ)|F . Ep This is a closed M -valued (0, q)-form on F , that determines a cohomology class COHOMOLOGYOFLEVI-FLATMINIMALORBITS 5

0,q Ep H0,q Ep [ξg] in H (F, M |F ). Then the class of (g, [ξg]) in M ′ (F, M |F ) does not depend on the particular choice of g, but only on x, hence it defines a section sξ = φ(ξ) of HM ′ on U. The sheaves ZM ′ , BM ′ , HM ′ , ker φ are OM ′ -modules and φ is a morphism of ′ OM ′ -modules. Since OM ′ is fine, to prove that φ(M ) is continuous, surjective and ′ that its kernel is exactly BM ′ (M ) it sufices to check that this is true locally, and this reduces to a straightforward verification.  We prove now the main theorem of this paper:

Theorem 4.4. Let M and X be as above, EX a K-homogeneous holomorphic vector bundle over X. Then: p,q E p,q E . H (M, X |M ) ≃ OM (M) ⊗OX (X) H (X, X ) ′ ′ 0,q p H ′ H E Proof. Define M and X as above, fix integers p, q ≥ 0 and let X = X′ (F, X |F ), 0,q p H ′ H E H ′ M = M ′ (F, X |F ) = X |M ′ . By Propositions 4.2 and 4.3, we have that ′ ′ p,q E H ′ p,q E H ′ H (M, X |M )=Γ(M , M ) and H (X, X )=Γ(X , X ). ′ ′ ′ Since dimR M = dimC(X ), a global section of HX′ that is zero on 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 every point by its H ′ H ′ global sections. Together with the fact that M = X |M ′ this implies that ′ ′ ′ Γ(M , HM ′ )= OM ′ (M ) ⊗O ′ Γ(X , HX′ ), X′ (X ) where in the right hand side we implicitly identify global holomorphic sections 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 connected 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: ∗ p,q E p,q E j : H (X, X ) −→ H (M, X |M ) that is continuous, injective and has a dense range. 

5. An example Let G = SL(4, C), and Q be the parabolic subgroup of upper triangular ma- trices. We consider the real form G0 = SU(1, 3), identified with the group of linear trasformations of C4 that leave invariant the Hermitian form associated to the matrix 0001 0100 B = 0010 . 1000  1 2 3 4 Then G/Q is the set of complete flags {ℓ ⊂ ℓ ⊂ ℓ ⊂ C } and M = G0 · eQ is the submanifold {ℓ1 ⊂ ℓ2 ⊂ ℓ3 ⊂ C4 | ℓ3 = (ℓ1)⊥}. Let Q′ be the set of block upper triangular matrices of the form ∗ ∗ ∗ ∗ ′ 0 ∗ ∗ ∗ Q = g = 0 ∗ ∗ ∗ | g ∈ G , ∗   0 0 0   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. The fibration 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 isomorphic to S(GL(1, C) × GL(3, C)) and 1 2 3 4 K0 to S(U(1) × U(3)). Then X is the set of flags {ℓ ⊂ ℓ ⊂ ℓ ⊂ C } in a 6 ANDREA ALTOMANI generic position with 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 fiber isomorphic to CP1 and X′ is a Stein manifold. Finally let E = X × C be the trivial . According to Propositions 4.2 and 4.3 the cohomology of M and X is given by p,q p,q E 0,0 ′ H0,q Ep H (X)= H (X, )= H X , X′ (F, |F ) , p,q p,q E 0,0 ′ H0,q Ep H (M)= H (M, |M )= H M , M ′ (F, |F ) . Recalling that Hp,q(F ) ≃ C if p = q =0 or p =q = 1 and 0 otherwise, we obtain: p,q ′ H (X) ≃ OX (X) ≃ OX′ (X ) if p = q =0 or p = q = 1; p,q (H (X) = 0 otherwise; and analogously:

p,q ′ H (M) ≃ OM (M) ≃ OM ′ (M ) if p = q =0 or p = q = 1; p,q (H (M) = 0 otherwise; and it is clear that p,q p,q H (M)= OM (M) ⊗OX (X) H (X).

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(A. Altomani) Dipartimento di Matematica, II Universita` di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma (Italy) E-mail address: [email protected]