On parabolic CR manifolds Costantino Medori (joint work with A. - PowerPoint PPT Presentation

On parabolic CR manifolds Costantino Medori (joint work with A. Altomani and M. Nacinovich) Luxembourg, March 25, 2009 () On parabolic CR manifolds Luxembourg, March 25, 2009 1 / 23

On parabolic CR manifolds Given: a complex flag manifold F = G / Q ( G complex semisimple Lie group, Q parabolic subgroup) a real form G 0 of G we consider the natural action of G 0 on F . Definition A parabolic (CR) manifold is a G 0 -orbit in F We recall that (having fixed F and G 0 ): there exists a finite number of G 0 -orbits in F only one of them is closed (compact) open orbits are simply connected (see J.A.Wolf, 1969). () On parabolic CR manifolds Luxembourg, March 25, 2009 2 / 23

homogeneous CR structure on a given parabolic manifold, equivariant fibrations with complex manifolds as fibers, topological results, in particular regarding the fundamental group. () On parabolic CR manifolds Luxembourg, March 25, 2009 3 / 23

CR manifolds and maps Definition (CR manifold) M = M 2 n + k , real manifold H = H 0 , 1 ⊂ T C M, complex subbundle (of rank n) s.t. � H ∩ H = 0 , [Γ( H ) , Γ( H )] ⊂ Γ( H ) . The numbers n , k are the CR dimension and CR codimension, respectively. Remark (Homogeneous CR manifolds) An orbit M for a group of biholomorphisms of a complex manifold X is a CR manifold with CR structure � H p = T 0 , 1 p X ∩ T C p ∈ M . H = H p , p M , p ∈ X In particular a parabolic manifold M ⊂ F = G / Q is a CR manifold. () On parabolic CR manifolds Luxembourg, March 25, 2009 4 / 23

Definition (CR maps and fibrations) A CR map between two CR manifolds ( M , H ) and ( M ′ , H ′ ) is an f : M → M ′ df C ( H ) ⊂ H ′ . such that A CR map f is a CR fibration if f is submersion and df C ( H ) = H ′ . () On parabolic CR manifolds Luxembourg, March 25, 2009 5 / 23

Weakly nondegenerate CR manifolds Definition A CR manifold ( M , H ) is said to be weakly nondegenerate (briefly WND) in p ∈ M if ∀ Z ∈ Γ( H ) , Z p � = 0 , ∃ Z 1 . . . , Z ℓ ∈ Γ( H ) : [ Z 1 , . . . , [ Z ℓ , Z ] . . . ] p �∈ H + H . (The case k = 1 corresponds to a nondegenerate Levi-form) Criterion Let M be a homogeneous CR manifold. Then M is WD if and only if there exists a local CR fibration π : M → M ′ , with nontrivial complex fibers. () On parabolic CR manifolds Luxembourg, March 25, 2009 6 / 23

Example The Grassmannian Gr C (2 , 4) of 2-spaces of C 4 is a flag manifold F = SL (4 , C ) / Q where Q = { Z ∈ SL (4 , C ) | Z ( � e 1 , e 2 � C ) ⊆ � e 1 , e 2 � C } . We consider a Hermitian symmetric form on C 4 associated to the matrix   0 0 0 1 0 1 0 0   K =  .   0 0 1 0  1 0 0 0 Let G 0 = { Z ∈ SL (4 , C ) | KZ + Z ∗ K = 0 } ≃ SL (3 , 1). The parabolic manifolds are given by the sets of 2-spaces with signature (2 , 0), (1 , 1) and (1 , 0). The compact orbit is G 0 · ( � e 1 , e 2 � C ). It is a CR manifold of CR-dimension 3 and CR-codimension 1, with degenerate Levi form but WND. () On parabolic CR manifolds Luxembourg, March 25, 2009 7 / 23

CR algebras To a parabolic CR manifold immersed in a flag manifold M = G 0 / I 0 ֒ → F = G / Q we associate a pair of Lie algebras ( g 0 , q ) = ( Lie ( G 0 ) , Lie ( Q )) This is called a CR algebra. Note that q ⊂ g is a complex parabolic subalgebra. Such CR algebra is called a parabolic CR algebra. To given a parabolic CR algebra ( g 0 , q ), we associate a parabolic CR manifold M = M ( g 0 , q ) in the following way: - G is a connected Lie group with Lie algebra g , - Q and G 0 are analytic subgroups of G with Lie algebras q and g 0 , - F = G / Q , M = M ( g 0 , q ) is the orbit of G 0 in F = G / Q through the point o = eQ . () On parabolic CR manifolds Luxembourg, March 25, 2009 8 / 23

The isotropy subgroup of M ( g 0 , q ) and its isotropy subalgebra are I 0 = G 0 ∩ Q , i 0 = g 0 ∩ q . Moreover, dim CR M ( g 0 , q ) = dim C ( q / ( q ∩ ¯ q )) . Let M = M ( g 0 , q ) and M ′ = M ( g 0 , q ′ ) be parabolic CR manifolds with ′ := g 0 ∩ q . We have a G 0 -equivariant fibration F between them: i 0 ⊆ i 0 → M ′ = G 0 / I 0 F ′ M = G 0 / I 0 − Then: ⇒ q ⊂ q ′ F a CR map ⇐ ⇒ q ′ = q + ( q ′ ∩ ¯ q ′ ). F a CR fibration ⇐ () On parabolic CR manifolds Luxembourg, March 25, 2009 9 / 23

Standard parabolic subalgebras It is possible to choose h ⊂ q , CSA of g B = { α 1 , . . . , α ℓ } ⊂ R = R ( g , h ), basis of simple roots such that q = q S for a subset S ⊆ B , where � � = g α + h + q S g α α ∈R + α ∈R − supp ( α ) ∩S = ∅ � � = + h + g α g α α ∈R + supp ( α ) ∩S = ∅ supp ( α ) ∩S� = ∅ � �� � q r � �� � q n and supp ( α ) := { α j ∈ B | α = � j n j α j , n j � = 0 } . () On parabolic CR manifolds Luxembourg, March 25, 2009 10 / 23

Remark ∃ h 0 , CSA of g 0 : h 0 ⊂ i 0 := g 0 ∩ q . With compact parabolic manifolds, we can choose h 0 maximally non-compact. This is generally not possible. Remark Different choices of a root basis B (corresponding to a Borel subalgebra b ⊂ q ) are given by different choices of Weyl chambers C. () On parabolic CR manifolds Luxembourg, March 25, 2009 11 / 23

Weakening of the CR structure Let M = M ( g 0 , q ) ≃ G 0 / I 0 ֒ → F = G / Q Recall that: i 0 = g 0 ∩ q = g 0 ∩ ( q ∩ ¯ q ), dim CR M = dim q − dim( q ∩ ¯ q ). Definition Denote by q w ⊂ q , the minimal parabolic subalgebra such that: q w ∩ ¯ q w = q ∩ ¯ q . The parabolic manifold M w = M ( g 0 , q w ) is called the CR-weakening of M ( g 0 , q ) . Note that: F ′ = G / Q w . M w = M ( g 0 , q w ) ≃ G 0 / I 0 ֒ → (1) Then: M w is diffeomorphic to M as real manifold M w has a different (minimal) CR structure given by the immersion (1). () On parabolic CR manifolds Luxembourg, March 25, 2009 12 / 23

(Weakening of the CR structure) As q w ⊂ q , the natural fibration ≃ � M = M ( g 0 , q ) f : M w = M ( g 0 , q w ) CR is a CR map and a diffeomorphism. Proposition We have: q w = q n + q ∩ ¯ q For a suitable choice of a basis of simple roots B ⊂ R (corresponding to an S-fit Weyl chamber), we obtain: S ∗ = S ∪ { α ∈ B | ¯ q = q S , S ⊆ B = ⇒ q w = q S ∗ , α > 0 , supp (¯ α ) ∩S � = ∅} . Lemma M w is either weakly degenerate or real (i.e. with a trivial CR structure). () On parabolic CR manifolds Luxembourg, March 25, 2009 13 / 23

Reduction to WND manifolds Theorem (WND reduction) Let M = M ( g 0 , q ) be a parabolic CR manifold. Then there exists a G 0 -equivariant CR fibration → M ′ π : M − with a WND base M ′ = M ( g 0 , q ′ ) simply-connected complex fibers (eventually disconnected). The fibers are non trivial ⇐ ⇒ M is WD. Definition → M ′ is said WND reduction of M. The CR fibration π : M − () On parabolic CR manifolds Luxembourg, March 25, 2009 14 / 23

(Reduction to WND manifolds) Criterion A parabolic CR manifold M ( g 0 , q ) is WD if and only if there is a complex subalgebra q ′ of g such that q � q ′ ⊂ q + ¯ q . Proposition For a suitable choice of a basis of simple roots B ⊂ R (corresponding to a V-fit Weyl chamber), we obtain: q ′ = q S ′ , S ′ = { α ∈ S | ¯ q = q S , S ⊆ B = ⇒ α > 0 } . () On parabolic CR manifolds Luxembourg, March 25, 2009 15 / 23

Structure theorem Theorem (Structure theorem) Let M = M ( g 0 , q ) be a parabolic CR manifold. Then there exists a G 0 -equivariant fibration Ψ M − → M c with a real flag manifold M c = M ( g 0 , c ) as base space, simply-connected, complex fibers. Definition The parabolic manifold M c = M ( g 0 , c ) is called the real core of M. () On parabolic CR manifolds Luxembourg, March 25, 2009 16 / 23

� � � Construction M π (0) f − 1 (0) � M (0) M (0) w π (1) f − 1 (1) � M (1) M (1) w π (2) f − 1 ( r ) � M c · · · M (2) where the vertical maps are WND reductions (CR maps) the horizontal maps give the weakening of the CR structure (diffeomorphisms) Each manifold M ( j ) w is either WD or real. Ψ = f − 1 ( r ) ◦ π ( r ) ◦ · · · ◦ f − 1 (0) ◦ π (0) () On parabolic CR manifolds Luxembourg, March 25, 2009 17 / 23

Example F 7 d 1 ,..., d r := { ( ℓ 1 , . . . , ℓ r ) | ℓ 1 � ℓ 2 � · · · � ℓ r subspaces of C 7 , dim ℓ j = d j } . Let ( e 1 , . . . , e 7 ) be the standard basis of C 7 and ǫ 1 = e 1 + ie 7 , ǫ 2 = e 2 , ǫ 3 = e 3 + ie 6 , ǫ 4 = e 4 , ǫ 5 = e 5 , ǫ 6 = e 3 − ie 6 , ǫ 7 = e 1 − ie 7 Let G 0 = SL (7 , R ) and consider the parabolic manifold M = G 0 · γ ⊂ F 7 1 , 2 , 3 , 4 , 5 , 6 , 7 where γ = ( � ǫ 1 � , . . . , � ǫ 1 , . . . , ǫ 7 � ) ∈ F 7 1 , 2 , 3 , 4 , 5 , 6 , 7 The WND reduction of M is the G 0 -orbit M (0) = G 0 · γ 0 ⊂ F 7 2 , 4 through the flag γ 0 = ( � ǫ 1 , ǫ 2 � , � ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 � ) ∈ F 7 2 , 4 () On parabolic CR manifolds Luxembourg, March 25, 2009 18 / 23