Generalized pseudo Kaehler structures
aa r X i v : . [ m a t h . DG ] M a r Generalized pseudo-K¨ahler structures
Johann Davidov ∗ , Gueo Grantcharov, Oleg Mushkarov † ,Miroslav Yotov Abstract
In this paper we consider pseudo-bihermitian structures – pairsof complex structures compatible with a pseudo-Riemannian metric.We establish relations of these structures with generalized (pseudo-)K¨ahler geometry and holomorphic Poisson structures similar to thatin the positive definite case. We provide a list of compact complexsurfaces which could admit pseudo-bihermitian structures and giveexamples of such structures on some of them. We also consider a na-turally defined null plane distribution on a generalized pseudo-K¨ahler4-manifold and show that under a mild restriction it determines anEngel structure.
Bihermitian structures have recently received a serious attention due totheir relations to supersymmetric sigma models in theoretical physics andgeneralized geometry. However one of the reasons they were introducedin [3] was the observation that the self-dual component of the Weyl ten-sor of an oriented Riemannian 4-manifold determines a restriction on thenumber of (local) complex structures compatible with the metric and theorientation. The possibilities are 0, 1, 2, or ∞ , if we do not distinguishstructures differing by sign. The bihermitian structures thus arise naturallyon 4-manifolds with 2 different (up to sign) compatible complex structures. ∗ Partially supported by ”L.Karavelov” Civil Engineering Higher School, Sofia, Bul-garia under contract No 10/2009 † Partially supported by CNRS-BAS joint research project
Invariant metrics and com-plex geometry, 2008-2009 , N = (2 ,
2) supersymmetric sigmamodel [12, 18]. This interpretation brought an important new viewpointfor studying deformations of such structures and led to a number of newexamples [17, 14].On a pseudo-Riemannian 4-manifold of neutral signature (+ , + , − , − )there are analogs for most of the notions in the Riemannian case. In partic-ular, compatible complex structures and self-duality are well defined, unlikethe Lorentzian case. Many results in the neutral setting are similar to resultsin the Riemannian case but there are also important differences.In this note we develop the notion of a pseudo-bihermitian structurewhich was considered also in the physics literature [13]. We show that, inthe same way as in the Riemannian case, it can be related to (twisted)generalized pseudo-K¨ahler structures (Section 3) as well as to holomorphicPoisson structures (Section 4). In Section 5 we show that the 3-dimensionalcomplex flag manifold F l carries a generalized K¨ahler structure. We alsoprove that any holomorphic line bundle on
F l is a holomorphic Poissonmodule with respect to a Poisson structure of a special type. In Section 6we provide a list of all compact complex surfaces which might carry pseudo-bihermitian structures. It contains the list of bihermitian surfaces obtainedin [3]. In Section 7 we adapt a construction of [21, 16] to find examples ofpseudo-bihermitian structures, which are collected in Proposition 10. Notethat no Kodaira surface admits generalized K¨ahler structures [4, 5], but itadmits a generalized pseudo-K¨ahler structure.We consider also some other differences between the Riemannian andthe neutral setting. The first one is related to the basic observation that ona 4-dimensional vector space two complex structures J + and J − inducingthe same orientation are compatible with a positive-definite inner productiff J + J − + J − J + = 2 pId for a constant p with | p | <
1. The same holds forstructures compatible with a split-signature inner product, but this time | p | >
1. The difference appears when the above identities are consideredglobally on a 4-manifold. If p is a function with | p | < J + and J − . However we show in Section 7, Example 3 thatthere are compact 4-manifolds admitting two such structures J + and J − with | p | > Acknowledgements : The authors express their gratitude to V.Apostolovfor helpful discussions and comments on a preliminary version of this paper.Part of this work was done during the visit of the first and the third-namedauthors at the Abdus Salam School of Mathematical Sciences, GC Univer-sity Lahore, Pakistan and the second named author’s visit to the Instituteof Mathematics and Informatics at the Bulgarian Academy of Sciences. Theauthors thank the two institutions for their hospitality.
In this section we consider the indefinite analog of bihermitian structureson 4-manifolds. An almost para-hypercomplex structure on a smooth4-manifold M (also called an almost complex product [1] or a neutralalmost hypercomplex structure [11]) consists of three endomorphisms J , J , J of T M satisfying the relations J = − J = − J = − Id, J J = − J J = J (1)of the imaginary units of the paraquaternionic algebra (split quaternions).A metric g on M is called compatible with the structure { J , J , J } if g ( J X, J Y ) = − g ( J X, J Y ) = − g ( J X, J Y ) = g ( X, Y ) (2)(such a metric is necessarily of neutral signature (+ , + , − , − )). In this casewe say that { g, J , J , J } is an almost para-hyperhermitian structure .For any such a structure we define three 2-forms Ω i setting Ω i ( X, Y ) = g ( J i X, Y ), i = 1 , ,
3. If the Nijenhuis tensors of J , J , J vanish, thestructure { g, J , J , J } is called para-hyperhermitian and ( J , J , J ) is3alled para-hypercomplex . When additionally the 2-forms Ω i ( X, Y ) = g ( J i X, Y ) are closed, the para-hyperhermitian structure is called para-hyperk¨ahler (also called hypersymplectic [19] and neutral hyperk¨ahler [11]).Hypercomplex or para-hypercomplex structures can be obtained in thefollowing way. Consider a 4-manifold with two complex structures J + and J − such that J + J − + J − J + = 2 pId (3)for a function p .Suppose that | p | < J + , K = 12 p − p [ J + , J − ], S = − p − p ( J − + pJ + ) form an almost hypercomplex structure (cf.e.g.[13]). Thus the complex structures J + and J − are compatible with apositive definite metric.If | p | > J + , K = 12 p p − J + , J − ] , S = − p p − J − + pJ + )form an almost para-hypercomplex structure [13]. Hence by [9] there is alocally defined metric compatible with the structure { J + , K, S } . It is clearthat the structure J − is also compatible with this metric. Conversely, if thestructures J + and J − are compatible with a pseudo-Riemannian metric g ,so will be K and S , hence g is of neutral signature. Note that, unlike thepositive definite case, given J + and J − , such a metric may not exist globally(see Example 3 in Section 7).It follows from the above discussion that if | p | 6 = 1 at every point, then J + and J − yield the same orientation. This is a consequence from thewell-known fact that two non-collinear (almost) complex structures on a4-manifold both compatible with a pseudo-Riemannian metric determineopposite orientations exactly when they commute. Definition 1 If J + = ± J − are complex structures on a -manifold compat-ible with a pseudo-Riemannian metric g and if they yield the same orienta-tion, then ( g, J + , J − ) is said to be a pseudo-bihermitian structure. Such astructure is called strict if J + = ± J − at every point. g, J + , J − ) is a pseudo-bihermitian structure, then J + and J − satisfy identity (3) with p = − g ( J + , J − ).The following lemma is well-known in the positive definite case. For theneutral case it is stated in [13] and proved in [25] for generalized K¨ahlerstructures. For the sake of completeness we provide a new proof, whichworks both in the positive and neutral-signature cases. Lemma 1
Let J + and J − be complex structures on a -manifold such that J + J − + J − J + = 2 pId for p = const and | p | > . Then { J + , K, S } is apara-hypercomplex structure.Proof: We have to prove that the almost product structures K and S areintegrable. To do this we shall use a local neutral metric g compatiblewith the structure { J + , K, S } . Then J − is also compatible with g and p = − g ( J + , J − ). Denote by F ± the K¨ahler 2-form of ( g, J ± ). Then astandard formula for the Hermitian structure ( g, J ± ) gives: g (( ∇ X J ± )( Y ) , Z ) = ( ∇ X F ± )( Y, Z ) =12 ( dF ± ( J ± X, Y, J ± Z ) + dF ± ( J ± X, J ± Y, Z )) , (4)where ∇ is the Levi-Civita connection of g .Since the dimension of the manifold is four, there is a unique 1-form θ ± (the Lee form) such that dF ± = θ ± ∧ F ± . Then g (( ∇ X J ± )( Y ) , Z ) = g ( X, Z ) θ ± ( J ± Y ) − g ( J ± X, Z ) θ ± ( Y ) − g ( X, Y ) θ ± ( J ± Z ) − g ( J ± X, Y ) θ ± ( Z )It follows that2 X ( g ( J + , J − )) = 2 g ( ∇ X J + , J − ) + 2 g ( J + , ∇ X J − ) = − θ + ([ J + , J − ] X ) + θ − ([ J + , J − ] X )Thus 2 d ( g ( J + , J − )) = − ( θ + − θ − ) ◦ [ J + , J − ] (5)In view of the identity 2 p = − g ( J + , J − ), the condition p = const leads to θ + = θ − since [ J + , J − ] = 2 p p − K = 0 at every point. Then, using theidentity S = − p p − J − + pJ + ), we see that the fundamental 2-form5 S of S is a linear combination of F − and F + with constant coefficients.Hence dF S = θ + ∧ F S , so the Lee form of ( g, S ) is θ + . Let F K be thefundamental 2-form of ( g, K ) and denote its Lee form by θ K . Take a g -orthogonal basis of tangent vectors { E , E , E , E } with || E || = || E || =1, || E || = || E || = −
1. Set ε i = || E i || , i = 1 , , ,
4. Then the identities dF K = θ K ∧ F K and P i =1 ε i dF K ( E i , KE i , Z ) = 2 P i =1 ε i g (( ∇ E i K )( KE i ) , Z ) . give θ K ( Z ) = − X i =1 ε i [ g (( ∇ E i K )( E i ) , KZ ) . for any tangent vector Z . Since K = − J + S , we have θ K ( Z ) = − X i =1 ε i [ g (( ∇ E i J + )( SE i ) , J + SZ ) − X i =1 ε i [ g (( ∇ E i S )( E i ) , SZ ) . Using (4) and the fact that dF + = θ + ∧ F + one can easily see that thefirst term on the right-hand side vanishes. The second term is θ S ( Z ). Thus θ K = θ S = θ + , therefore the structures K and S are integrable [24]. q.e.d. Recall that a H -twisted generalized complex structure on a smooth manifold M is an endomorphism I of the bundle T M ⊕ T ∗ M satisfying the followingconditions:( a ) I = − Id ,( b ) I preserves the natural metric < X + ξ, Y + η > = ( ξ ( Y ) + η ( X )) , X, Y ∈ T M, ξ, η ∈ T ∗ M ( c ) the + i -eigensubbundle of I in ( T M ⊕ T ∗ M ) ⊗ C is involutive with respectto the H -twisted Courant bracket defined by[ X + ξ, Y + η ] H = [ X, Y ] + L X η − L Y ξ −
12 ( dı X η − dı Y ξ ) + ı Y ı X H, where H is a closed 3-form. 6he integrability condition ( c ) is equivalent to vanishing of the Nijenhuistensor N H ( A, B ) = [
A, B ] H − [ IA, IB ] H + I [ IA, B ] H + I [ A, IB ] H , A, B ∈ T M ⊕ T ∗ M. The space of 2-forms Ω ( M ) acts on T M ⊕ T ∗ M as e b ( X + ξ ) = X + ξ + ı X b for any b ∈ Ω ( M ). Then the Courant bracket satisfies [ e b ( A ) , e b ( B )] H =[ A, B ] H + db . In particular if I is a generalized complex structure, integrablewith respect to the H -twisted Courant bracket, then J = e − b Ie b is a gen-eralized complex structure, integrable with respect to the ( H − db )-twistedCourant bracket. So whenever H is exact, H = db for some 2-from b ,the structure I is called untwisted since the structure J is integrable withrespect to the Courant bracket with vanishing 3-form.Following M.Gualtieri [16, 18] we introduce the following: Definition 2
A (twisted) generalized pseudo-K¨ahler structure is a pair ofcommuting (twisted) generalized complex structures I , I : T M ⊕ T ∗ M → T M ⊕ T ∗ M , such that the ± -eigenspaces L ± of G = I I are transversalto T M and the canonical inner product on
T M ⊕ T ∗ M is non-degenerateon L ± . Using the same proof as in [16, 18], we have
Theorem 2 A H -twisted generalized pseudo-K¨ahler structure on a man-ifold M is equivalent to a quadruple ( g, J + , J − , b ) , where g is a pseudo-Riemannian metric, J + and J − are g -Hermitian complex structures, and b is a 2-form such that d + F + = − d − F − = H + db , where F ± is the K¨ahlerform of ( g, J ± ) and d ± is the imaginary part of the ∂ -operator of J ± . In this section we prove an indefinite analog of the well-known result [21]that a generalized K¨ahler manifold carries a holomorphic Poisson structure.In fact, we have the following slightly more general result.
Theorem 3
Let ( M, g ) be a pseudo-Riemannian manifold and let J + , J − be two complex structures on M compatible with g and such that d + F + = − d − F − . Then M admits a J + -holomorphic Poisson structure which van-ishes iff [ J + , J − ] = 0 . roof: Let Π be the bivector field on M determined by the endomorphism[ J + , J − ] − iJ + [ J + , J − ] of T C M and the complex bilinear extension of g .We shall prove that Π is a holomorphic Poisson field. To show that Π isholomorphic we shall use the Chern connection D + of the pseudo-Hermitianstructure ( g, J + ). It is defined by the identity g ( D + X Y, Z ) = g ( ∇ X Y, Z ) − dF + ( J ± X, Y, Z ), where ∇ is the Levi-Civita connection of g . As in thepositive case, D + is a Hermitian connection such that the restriction of its(0 ,
1) part on the holomorphic tangent bundle is the ∂ -operator of J + .In view of (4) and the identity d ± F ± ( X, Y, Z ) = − dF ± ( J ± X, J ± Y, J ± Z ),we have2 g (( D + X J − )( Y ) , Z ) = 2 g ( D + X J − Y, Z ) + 2 g ( D + X Y, J − Z ) =2 g (( ∇ X J − )( Y ) , Z ) − dF + ( J + X, J − Y, Z ) − dF + ( J + X, Y, J − Z ) = dF − ( J − X, Y, J − Z ) + dF − ( J − X, J − Y, Z ) − dF + ( J + X, J − Y, Z ) − dF + ( J + X, Y, J − Z ) = d − F − ( X, J − Y, Z ) + d − F − ( X, Y, J − Z )+ d + F + ( X, J + J − Y, J + Z ) + d + F + ( X, J + Y, J + J − Z )Thus2 g (( D + X J − )( Y ) , Z ) = − d + F + ( X, J − Y, Z ) − d + F + ( X, Y, J − Z )+ d + F + ( X, J + J − Y, J + Z ) + d + F + ( X, J + Y, J + J − Z )(6)The 3-form d + F + has no (3 ,
0) and (0 , d + F + ( A, B, C ) = d + F + ( J + A, J + B, C ) + d + F + ( J + A, B, J + C ) + d + F + ( A, J + B, J + C )Applying this identity to the last two terms in (6) we get2 g (( D + X J − )( Y ) , Z ) = − d + F + ( J + X, J − Y, J + Z ) − d + F + ( J + X, J + J − Y, Z ) − d + F + ( J + X, Y, J + J − Z ) − d + F + ( J + X, J + Y, J − Z ) (7)Set Q = [ J + , J − ]. Then, since D + J + = 0, we have2 g (( D + X Q )( Y ) , Z ) − g (( D + J + X Q )( Y ) , J + Z ) = − g (( D + X J − )( Y ) , J + Z ) − g (( D + X J − )( J + Y ) , Z ) − g (( D + J + X J − )( Y ) , Z ) + g (( D + J + X J − )( J + Y ) , J + Z )8pplying (6) to the first and the second term, and (7) to the third and thefourth term, we easily get g (( D + X Q )( Y ) , Z ) − g (( D + J + X Q )( Y ) , J + Z ) = 0 (8)As in [3] and [21], consider the form Ω( X, Y ) = g ( QX, Y ). The (1 , J + vanishes sinceΩ( J + X, J + Y ) = − g ( J J − J + X, Y ) − g ( J − J X, J + Y ) = g ( J − J + X, Y ) + g ( J − X, J + Y ) = − Ω( X, Y ) . Then the (0 , (0 , ( X, Y ) = Ω (0 , ( X (0 , , Y (0 , ) = Ω( X (0 , , Y (0 , ) = [Ω( X, Y ) − Ω( J + X, J + Y )] + i [Ω( J + X, Y ) + Ω(
X, J + Y )] = [Ω( X, Y ) + i Ω( X, J + Y )] = [ g ( QX, Y ) + ig ( QX, J + Y )] = g (Π , X ∧ Y )(9)It follows that Π is of type (2 ,
0) with respect to J + . Moreover, we have g ( D + X + iJ + X Π , Y ∧ Z ) = 2( D + X + iJ + X Ω (0 , )( Y, Z ) =[ g (( D + X Q )( Y ) , Z ) + ig (( D + X Q )( Y ) , J + Z )]+ i [ g (( D + J + X Q )( Y ) , Z ) + ig (( D + J + X Q )( Y ) , J + Z )] =[ g (( D + X Q )( Y ) , Z ) − g (( D + J + X Q )( Y ) , J + Z )]+ i [ g (( D + X Q )( Y ) , J + Z ) + g (( D + J + X Q )( Y ) , Z )]Hence, by (8), g ( D + X + iJ + X Π , Y ∧ Z ) = 0 for every X, Y, Z ∈ T M.
Thisshows that D + X + iJ + X Π = 0, therefore Π is a holomorphic section of theanti-canonical bundle Λ T (1 , M of ( M, J + ).To prove that the Schouten-Nijenhuis bracket [Π , Π] vanishes, we notefirst that it is enough to show that [ Re Π , Re Π] = 0. Indeed, since Πis holomorphic, it is easy to see in local holomorphic coordinates that[Π , Π] = 0. Note also that [ Re Π , Im Π] = [ Im Π , Re Π] since Re Π and Im Πare of degree 2. Thus, we have 0 = [Π , Π] = [ Re Π , Re Π] + [ Im Π , Im Π].Suppose that [ Re Π , Re Π] = 0. Then we get [ Im Π , Im Π] = 0, hence[Π , Π] = 2 i [ Re Π , Im Π]. Because [Π , Π] is of type (3,0) and purely imag-inary, we conclude that [Π , Π] = 0 9ccording to (9), the endomorphism Q of T M corresponds to the bivec-tor field Re Π via the metric g . Then, in view of [28, Proposition 1.9], theequality [ Re Π , Re Π] = 0 is equivalent to G g (( ∇ QX Q )( Y ) , Z ) = 0 , where G means the cyclic sum over X, Y, Z and ∇ is the Levi-Civita connec-tion of g . To prove the latter identity we use the fact that the Levi-Civitaconnection ∇ and the Chern connection D + of ( g, J + ) are related by g ( ∇ X Y, Z ) = g ( D + X Y, Z ) − d + F + ( X, J + Y, J + Z ) . Set P = J + J − + J − J + . Then, by (6) we have2 g (( ∇ X Q )( Y ) , Z ) = 2 g (( ∇ X QY, Z ) + 2 g ( ∇ X Y, QZ ) =2 g (( D + X Q )( Y ) , Z ) − d + F + ( X, J + QY, J + Z ) − d + F + ( X, J + Y, J + QZ ) = d + F + ( X, P Y, Z ) + d + F + ( X, Y, P Z )+2 d + F + ( X, J − Y, J + Z ) + 2 d + F + ( X, J + Y, J − Z ) . (10)Therefore2 G g (( ∇ QX Q )( Y ) , Z ) = G [ d + F + ( QX, P Y, Z ) + d + F + ( QX, Y, P Z )+2 d + F + ( QX, J − Y, J + Z ) + d + F + ( QX, J + Y, J − Z )]Using the skew-symmetry of d + F + , it is easy to see that G [ d + F + ( QX, P Y, Z ) + d + F + ( QX, Y, P Z )] =2 G [ d + F + ( J + J − X, J + J − Y, Z ) − d + F + ( J − J + X, J − J + Y, Z )] . We have d + F + = − d − F − , so d + F + is of type (2 ,
1) + (1 ,
2) for both J + and J − . Therefore d + F + ( A, B, C ) = G d + F + ( J + A, J + B, C ) = G d + F + ( J − A, J − B, C ) .
10t follows that G g (( ∇ QX Q )( Y ) , Z ) = G [ d + F + ( J + J − X, J + J − Y, Z ) − d + F + ( J − J + X, J − J + Y, Z )+ d + F + ( J + J − X, J − Y, J + Z ) − d + F + ( J − J + X, J − Y, J + Z )+ d + F + ( J + J − X, J + Y, J − Z ) − d + F + ( J − J + X, J + Y, J − Z )] = G [ d + F + ( J − X, J − Y, Z ) − d + F + ( J + X, J + Y, Z )+ d + F + ( J − X, J − Y, Z ) − d + F + ( J + X, Y, J + Z )+ d + F + ( J − X, Y, J − Z ) − d + F + ( J + X, J + Y, Z )] =3[ d + F + ( X, Y, Z ) − d + F + ( X, Y, Z )] = 0 . This proves that [ Re Π , Re Π] = 0 which implies, as we have mentioned,that [Π , Π] = 0, i.e. Π is a Poisson field.One can also prove that the field Π is Poisson using the fact that the2-vector corresponding to the endomorphism J + + J − is Poisson [26] and its(2 , q. e. d. A holomorphic Poisson structure on a complex surface is merely a holo-morphic section of its anti-canonical bundle. Using this fact N. Hitchin [21]proposed a simple way for constructing generalized K¨ahler structures onDel Pezzo surfaces. A different approach by M. Gualtieri [17] based on thenotion of generalized complex branes extends this construction to higher-dimensional Fano manifolds. Here, we state a modification of his resultwhich can be proved in the same way as [17, Theorem 7.1]
Theorem 4
Let L be a holomorphic line bundle on an n -dimensional com-pact complex manifold M with holomorphic Poisson structure σ such that c ( L ) n = 0 . Let ( g , J ) be a pseudo-K¨ahler structure with K¨ahler form F ∈ c ( L ) . Consider σ and F as homomorphisms σ : ( T C M ) ∗ → T C M and F : T C M → ( T C M ) ∗ , and suppose that the following conditions aresatisfied:(i) σ ◦ F = ∂X , for some (1 , vector field X , ;(ii) [ Re X , , Im σ ] = 0 for the Schouten-Nijenhuis bracket.Then the choice of a Hermitian structure on L with curvature F de-termines a family of generalized pseudo-K¨ahler structures ( g t , J t , J ) with t = φ ∗ t ( J ) for a 1-parameter group of diffeomorphisms φ t such that J t = J for t = 0 only at the poins of M where σ = 0 . Remark 1
Using Theorem 4 or the construction in [22], one expects toproduce examples of generalized pseudo-K¨ahler structures on ruled surfacesover a Riemann surface of genus greater than one. For example, consider aruled surface M over a curve C of genus g > V of degree deg ( V ) < − g . Its anti-canonical bundle hasa nowhere-vanishing holomorphic section s and the choice of a Hermitianmetric on it will produce a curvature 2-form F = dd c log | s | . Suppose that F is non-degenerate at each point. Then Theorem 4 and [22] produce gen-eralized pseudo-K¨ahler structure with non-trivial canonical bundle. Notethat when V = O ⊕ L is decomposable, the admissible metrics on M con-sidered in [2] define Hermitian metrics on the anti-canonical bundle of M which are candidates to provide such F . However one can check that noneof these metrics has a non-degenerate Ricci tensor. In case deg ( V ) > − g ,there are metrics with this property but there is no holomorphic Poissonstructure. So, it is an open question whether any ruled surface admits ageneralized pseudo-K¨ahler structure. Note that R. Goto [15] has recentlyconstructed positive definite generalized K¨ahler structures on some of thesesurfaces using more general deformations of K¨ahler-Poisson structures [14]than that considered in [17]. However his approach is based on ellipticmethods and can not be adapted directly to the pseudo-Riemannian case. Consider the complex flag manifold
F l = { ( L, V ) | ∈ L ⊂ V ⊂ C , dim L =1 , dim V = 2 } . It can be embedded into CP × CP as the quadric F l = { ( x , x , x ; y , y , y ) ∈ CP × CP | x y + x y + x y = 0 } . Let ω be theK¨ahler form of the standard K¨ahler structure on CP normalized so that ω be integral. Denote by p and p the projections of CP × CP onto thefirst and the second factor. Set ω = p ∗ ω and ω = p ∗ ω . The restrictions ofthese forms to F l will be denote by the same symbols.
Lemma 5
For any integers a and b with ab < and a + b = 0 , the form F = aω + bω is non-degenerate on F l . roof: Suppose that for such a and b the 2-form F = aω + bω is degenerateat some point of F l . The group U (3), embedded diagonally in U (3) × U (3),acts transitively and holomorphically on F l and F is invariant under thisaction. It follows that F degenerates at every point of F l . This implies thatthe top degree F vanishes since degF = 2. We have ω i = ( p ∗ i ω ) | F l = 0 for i = 1 ,
2. Therefore F = 3 ab ( a ω ∧ ω + b ω ∧ ω ). Let ψ : F l → F l be theholomorphic map induced by the map ψ ([ x ] , [ y ]) = ([ y ] , [ x ]) on CP × CP .It is clear that ψ ∗ ω = ω and ψ ∗ ω = ω . Therefore 0 = ψ ∗ ( aω ∧ ω + bω ∧ ω ) = aω ∧ ω + bω ∧ ω . Then ( a + b )( ω ∧ ω + ω ∧ ω ) = 0 andwe get the identity ( a + b )( ω + ω ) = 3( a + b )( ω ∧ ω + ω ∧ ω ) = 0.But the latter identity does not hold since ω + ω is the K¨ahler form of F l induced by the product of the Fubini-Studi forms on each factor of CP , acontradiction. q. e. d. Later in the paper we’ll need the following:
Lemma 6
Let U and V be commuting holomorphic vector fields on a com-plex manifold and ϕ a smooth function on the manifold. Then ( U ∧ V ) ◦ dd c ϕ = i∂ (( U ϕ ) V − ( V ϕ ) U ) . Proof . We use the identity d c = ( i∂ − i∂ ) and the fact that for any (0 , Z , [ U, Z ] (1 , = 0. Then we have2( dd c ϕ )( U, Z ) = iU ( ∂ϕ ( Z )) + iZ ( ∂ϕ ( U )) − i∂ϕ ([ U, Z ]) = iU Zϕ + iZU ϕ − [ U, Z ] ϕ = 2 iZU ϕ, so ı U dd c ϕ = i∂ ( U ϕ ) . From here we get( U ∧ V ) ◦ dd c ϕ = ı U dd c ϕ ⊗ V − ı V dd c ϕ ⊗ U = i∂ ( U ϕ ) ⊗ V − i∂ ( V ϕ ) ⊗ U = i∂ (( U ϕ ) V − ( V ϕ ) U ) . q. e. d. Now we are ready to prove the following:
Proposition 7
The flag manifold
F l admits a generalized pseudo-K¨ahlerstructure. roof: Take arbitrary integers a and b with ab < a + b = 0. Then F = aω + bω is non-degenerate by Lemma 5, so it determines a pseudo-K¨ahler metric on F l .Since the form F is integral, it determines a Hermitian holomorphicline bundle L on F l with curvature F . We have c ( L ) = 0 since c ( L ) is represented by the invariant form F on F l and the 2- form F is non-degenerateNow we want to define a holomorphic Poisson structure on F l as σ = Z ∧ Z for two commuting holomorphic vector fields Z and Z . Let Z and Z be the fields on CP × CP generated by the complex 1-parameter groups( x , x , x ; y , y , y ) → ( e t x , e − t x , x ; e − t y , e t y , y ) and ( x , x , x ; y , y , y ) → ( e t x , x , e − t x ; e − t y , y , e t y ), respectively. Clearly Z and Z are com-muting holomorphic vector fields tangent to F l . Then Z ∧ Z is a holo-morphic Poisson structure on F l . To show that
F l admits a generalizedpseudo-K¨ahler structure it remains only to check conditions ( i ) and ( ii ) inTheorem 4. Denote by X the holomorphic vector field on CP generatedby the group ( x , x , x ) → ( e t x , e − t x , x ). Then Z = ( X ◦ p , − X ◦ p ).Similarly Z = ( Y ◦ p , − Y ◦ p ) where Y is the vector field on CP generatedby the group ( x , x , x ) → ( e t x , x , e − t x ) The bi-vector filed τ = X ∧ Y isa holomorphic section of the anti-canonical bundle of CP . Set f = ln || τ || where the norm is taken with respect to metric yielded by the normalizedFubini-Study metric g of CP . We claim that, although f is defined onlyoutside of the zero set of τ , the functions Xf and Y f are globally definedand smooth. To check this we use the standard coordinates of C P . Forthe coordinates z = x x , z = x x , set g αβ = g ( ∂∂z α , ∂∂z β ) and G ( z ) = g g − | g | . Then || τ || = 4 | z z | G ( z ) and we have X = − z ∂∂z − z ∂∂z , Y = − z ∂∂z , τ = 2 z z ∂∂z ∧ ∂∂z ,Xf = − − z ln G ( z ) ∂z − z ln G ( z ) ∂z , Y f = − − z ln G ( z ) ∂z . (11)14n the coordinates u = x x , u = x x we have X = 2 u ∂∂u + u ∂∂u , Y = − u ∂∂u , τ = − u u ∂∂z ∧ ∂∂z ,Xf = 3 + 2 u ln G ( u ) ∂u + u ln G ( u ) ∂u , Y f = − − u ln G ( u ) ∂u . (12)Finally, in the coordinates v = x x , v = x x we have X = v ∂∂v − v ∂∂v , Y = v ∂∂v + v ∂∂v , τ = 2 v v ∂∂v ∧ ∂∂v ,Xf = v ln G ( v ) ∂v − v ln G ( v ) ∂v , Y f = 2 + v ln G ( v ) ∂v − v ln G ( v ) ∂v . (13)It follows from (11), (12), (13) that τ vanishes on the analytic set C = { [ x ] ∈ CP : x x x = 0 } and that Xf , Y f can be extended to smooth functionson a neighborhood of every point of C . Since CP \ C is dense, we see that Xf , Y f can be extended to unique smooth functions on the whole space CP . We shall denote the extensions by the same symbols. Identities (11),(12), (13) imply also that if ζ = ( ζ , ζ ) is a standard coordinate system of CP , we have dd c ln || τ || = dd c ln G ( ζ ) on CP \ C . Therefore dd c ln || τ || on CP \ C is the Ricci from of the standard K¨ahler structure on CP . Asit is well-known, the Ricci form of this structure is equal to 3 times theK¨ahler form. Thus, since we are working with the normalized K¨ahler form,we have dd c ln || τ || = 3 λω where λ > k = 1 , dd c (ln || τ || ◦ p k ) = 3 λp ∗ k ω = 3 λ ω k on the set M = { ( x , x , x ; y , y , y ) ∈ CP × CP | x x x y y y = 0 } . Thus on M we have( Z ∧ Z ) ◦ F = 13 λ ( Z ∧ Z )( a dd c (ln || τ || ◦ p ) + b dd c (ln || τ || ◦ p )) (14)It follows from (14) and Lemma 6 that if we set X , = i λ { [ a ( Xf ) ◦ p − b ( Xf ) ◦ p ] Z − [ a ( Y f ) ◦ p − b ( Y f ) ◦ p ] Z } where f = ln || τ || as above, we have ( Z ∧ Z ) ◦ F = ∂X , on the openset M . This identity holds everywhere since the vector field X , is smooth15n CP × CP and M is dense. Thus condition ( i ) of Theorem 4 is satisfiedfor σ = Z ∧ Z . To show that condition ( ii ) also holds, we note that[ X , , Z ∧ Z ] = − i λ { a ([ X, Y ] f ) ◦ p + b ([ X, Y ] f ) ◦ p } Z ∧ Z = 0since [ X, Y ] = 0. The function f is real-valued, so Xf = Xf , Y f = Y f and we have[ X , , Z ∧ Z ] = i λ { a ([ X, Y ] f ) ◦ p + b ([ X, Y ] f ) ◦ p } Z ∧ Z = 0 . Using the identities [
X, X ] = [
Y, Y ] = [
X, Y ] = [
X, Y ] = 0, it is easy to seethat [ X , , Z ∧ Z ] − [ X , , Z ∧ Z ] = 0 . It follows that [
Re X , , Im ( Z ∧ Z )] = 0. Then, by Theorem 4, the flagmanifold F l admits a generalized pseudo-K¨ahler structure. q. e. d.
Note that
F l admits also a usual generalized K¨ahler structure [14].
Corollary 8
Any holomorphic line bundle on the 3-dimensional flag man-ifold
F l carries a structure of a holomorphic Poisson module with respect tothe holomorphic Poisson structure U ∧ U defined by commuting holomor-phic vector fields U and U .Proof: First we notice that any two commuting vector fields on
F l span amaximal torus in the algebra sl (3 , C ) of the holomorphic vector fields on F l and all such tori are conjugate in the group of biholomorphisms. So, wemay assume that the vector fields U and U in the corollary coincide Z and Z defined in the proof of Proposition 7. Denote by K the canonicalbundle of CP . It is well known that every holomorphic line bundle over F l is of the form L mn = m p ∗ K + n p ∗ K where m, n ∈ Z . If we consider K with the metric induced by the normalized Fubini-Study metric of CP ,the curvature form of K with respect to its canonical connection is equalto the K¨ahler form ω . Therefore the form m p ∗ ω + n p ∗ ω = m ω + n ω represents the first Chern class of L mn . Denote this form by F and set σ = Z ∧ Z . We have seen above that there is a (1 , X , suchthat σ ◦ X , = ∂X , and [ X , , σ ] = 0. Now the Corollary follows from [17,Proposition 10] since the first Chern class of L mn coincides with its Atiyahclass. q. e. d. The four-dimensional case
In dimension four, a pseudo-hermitian metric is either positive (negative)definite or of signature (2,2). Using the results in Section 4 we shall provethe following:
Theorem 9
Let ( M, g, J + , J − ) be a compact pseudo-bihermitian 4-manifold. ( i ) If d + F + = − d − F − , then ( M, J + ) (and ( M, J − ) ) is one of the fol-lowing complex surfaces: a complex torus, a K3 surface, a primary Kodairasurface, a blow-up of a surface of class V II , a ruled surface described in[7] with χ ± τ divisible by 4, where χ and τ are the Euler characteristic andthe signature of M . ( ii ) If the bihermitian structure is strict, then ( M, J + ) (and ( M, J − ) )is one of the following: a complex torus, a K3 surface, a primary Kodairasurface, a properly elliptic surface of odd first Betti number, a Hopf surface,a minimal Inoue surface without curves.Proof: According to Theorem 3, under assumption ( i ) there is a non-zeroholomorphic section of the anti-canonical bundle of ( M, J + ). Such surfaceswith even first Betti number are described in [7] and they exhaust the firstfour cases in ( i ). The restriction on χ ± τ in the last case comes from Mat-sushita’s topological condition for existence of a split-signature metric [27].For the case of surfaces with odd first Betti number, we notice that theproof of Proposition 2.3 in [8] shows that either the Kodaira dimension of( M, J + ) (and ( M, J − )) is −∞ or its canonical bundle is holomorphicallytrivial. Then the Kodaira classification of minimal compact complex sur-faces [6] leads to the list in ( i ).Part ( ii ) follows from the fact that the canonical bundle is topologicallytrivial in the case of strictly pseudo-bihermitian surfaces, since the 2-formΩ (0 , given by (9) provides a non-vanishing section. So one can use thewell-known list of the surfaces with vanishing first Chern class [30] q. e. d. Remark 2 .Notice that, by [7, Lemma 2.1], if a compact complex surfaceis not minimal and has a nowhere-vanishing holomorphic section of theanti-canonical bundle, then its minimal model also admits such a section.Moreover, the dimension of the space of holomorphic sections decreases byat most one after a blow-up. It keeps the same dimension only if the blow-up is at a base point of the anti-canonical linear system. This leads to17dditional restrictions on the possible blow-ups of surfaces in case ( i ), butwe shall not discuss this question here. Remark 3 . There are generalized pseudo-K¨ahler manifolds (
M, g, J + , J − )so that J + and J − induce opposite orientations. In the four dimensionalcase such structures commute. In any dimension, for a generalized pseudo-K¨ahler manifold with commuting J + and J − , the same reasoning as in [5]shows that the holomorphic tangent bundle of ( M, J + ) splits into a sum oftwo holomorphic subbundles. Conversely, if the holomorphic tangent bundleof a compact complex surface ( M, J ) splits, then by [5] there is a generalized(pseudo) K¨ahler structure ( g, J + , J − ) such that J + = J and [ J + , J − ] = 0. It has been observed in [3, 16, 21] that one can explicitly define a gen-eralized K¨ahler structure by means of a hyperk¨ahler structure. Given apara-hyperk¨ahler structure, a similar construction can be applied to ob-tain a generalized pseudo-K¨ahler structure. Let { g, J , J , J } be a para-hyperk¨ahler structure on a 4-manifold M with J = − J = − J = − Id and J = J J . We would like to construct two commuting generalizedalmost complex structures I and I following [21]. To do this we needcomplex valued 2-forms β and β on M which satisfy( β − β ) = ( β − β ) = 0 , β = β , β = β (15)at every point. We set exp( β k ) = 1 + β k + β k , k = 1 ,
2, and ( X + ξ ) . exp( β k ) = ı X exp( β k ) + ξ ∧ exp( β k ) for X + ξ ∈ T M ⊕ T ∗ M (theClifford action of T M ⊕ T ∗ M on the forms). Then E k = { A ∈ ( T M ⊕ T ∗ M ) C | A. exp( β k ) = 0 } is the + i -eigenspace of a generalized almost com-plex structure I k . If β k is closed, I k is Courant integrable [16, 21]. It is shownin [21, Lemma 1] that I and I commute. Moreover, E ∩ E ⊕ E ∩ E isthe ( − I I and E ∩ E ⊕ E ∩ E is the (+1)-eigenspace.Note also that E ∩ E = { U − ı U β | U ∈ T C M, ı U β = ı U β } ([21]). Thus,for A = U − ı U β ∈ E ∩ E , B = V − ı V β ∈ E ∩ E , we have < A + A, B + B > = − Re { ( β − β )( U, V ) } = − Re { ( β − β )( U, V ) } = − Re { ( β − β )( U, V ) } (16)18ow, given a para-hyperk¨ahler structure { g, J , J , J } on a 4-manifold M ,set J + = J and J − = aJ + bJ + cJ where a, b, c are fixed numbers suchthat a − b − c = 1 and a = 1. Then J + and J − are complex structurescompatible with the metric g satisfying the identity J + J − + J − J + = − aId. (17)As in Section 2, set K = 12 √ a − J + , J − ] , S + = − √ a − J − − aJ + )Then { g, J + , K, S + } is a para-hyperhermitian structure with S + = J + K .Let F + ( X, Y ) = g ( J + X, Y ), F K ( X, Y ) = g ( KX, Y ) and ω ′ ( X, Y ) = g ( S + X, Y )be the corresponding fundamental 2-forms. Similarly, if S − = 1 √ a − J + − aJ − ) , then { g, J − , K, S − } is a para-hyperhermitian structure with S − = J − K . Wedenote the fundamental 2-forms of J − and S − by F − and ω ′′ , respectively.Set ω + = ω ′ + ω ′′ , ω − = ω ′ − ω ′′ . Then ω + ( X, Y ) = r a + 1 a − g ( J + X − J − X, Y ) = r a + 1 a − F + ( X, Y ) − F − ( X, Y )) ω − ( X, Y ) = r a − a + 1 g ( J + X + J − X, Y ) = r a − a + 1 ( F + ( X, Y ) + F − ( X, Y ))(18)In particular, the forms ω + and ω − are closed since F + and F − are so.Identity (10) implies that ∇ [ J + , J − ] = 0, thus ∇ K = 0. Therefore the form F K is also closed. Now, similar to [16] we set β = F K + iω + , β = − F K + iω − . Conditions (15) for these forms are equivalent to F K ω + = F K ω − = ω + ω − = ω + ω − − F K ) = 0 . (19)19et X be a tangent vector with g ( X, X ) = 1. Then { X, J + X, KX, S + X } is a g -orthonormal basis of tangent vectors. Using (17), (18) and the paraquater-nionic identities, it is easy to see that( ω + ∧ ω + )( X, J + X, KX, S + X ) = 4( a + 1) , ( ω − ∧ ω − )( X, J + X, KX, S + X ) = − a − , ( ω + ∧ ω − )( X, J + X, KX, S + X ) = 0 , ( F K ∧ ω ± )( X, J + X, KX, S + X ) = 0 . We also have ( F K ∧ F K )( X, J + X, KX, S + X ) = 2. It follows that identities(19) are satisfied.The identity β − β = 2 F K + i ( ω + − ω − ) implies that a vector U ∈ T C M satisfies ı U ( β − β ) = 0 if and only if √ a − KU + iJ + U − iaJ − U = 0 . (20)Thus E ∩ E = { U − ı U β | U ∈ T C M, U satisfies (20) } . Let L − be the( − I I acting on T M ⊕ T ∗ M . Any X + ξ ∈ L − can bewritten as X + ξ = U + U where U = ( X + iY ) ∈ E ∩ E , Y ∈ T M , and ξ = ı U β − ı U β . In this notation, (20) is equivalent to √ a − KX − J + Y + aJ − Y = 0 √ a − KY + J + X − aJ − X = 0 . (21)In fact, either of these identities is a consequence of the other one. For every V = ( Z + iT ) ∈ E ∩ E , we have Re ( β − β )( U, V ) = r a + 1 a − g ( J + X − J − X, T ) − g ( J + Y − J − Y, Z )] = − r a + 1 a − g ( X, J + T − J − T ) + g ( J + Y − J − Y, Z )] . Applying K to the second identity of (21) we get √ a − Y = S + X − aS − X .This gives √ a − J + Y = − KX + a √ a − X + a √ a − J + J − X, √ a − J − Y = aKX + 1 √ a − X + a √ a − J − J + X.
20t follows that √ a − J + Y − J − Y ) = ( a − a + 2) KX + a − √ a − X. Similarly, √ a − J + T − J − T ) = ( a − a + 2) KZ + a − √ a − Z. Then ( a − Re ( β − β )( U, V ) = − r a − a + 1 g ( X, Z ) . (22)Suppose that < X + ξ, A > = 0 for every A ∈ L − . Take any Z ∈ T M and set T = ( a − − / [ S + Z − aS − Z ] . Then V = ( Z + iT ) satisfies (20). Indeed we have √ a − KZ − J + T + aJ − T = √ a − KZ − √ a − − KZ − aJ + S − Z ) + a √ a − J − S + Z + aKZ ) =1 √ a − a KZ + a ( J + S − Z + J − S + Z )) = 1 √ a − a KZ − a [ J + , J − ] Z √ a − √ a − a KZ − a KZ ) = 0 . Moreover, √ a − KT + J + Z − aJ − Z = KS + Z − aKS − Z + J + Z − aJ − Z = 0 . Thus V ∈ E ∩ E and, by our assumption, (16) and (22), we have g ( X, Z ) =0. Since the latter identity holds for every Z , we conclude that X = 0.Then Y = ( a − − / [ S + X − aS − X ] = 0, hence U = 0, thus ξ = ı U β − ı U β = 0. This proves that the canonical inner product on T M ⊕ T ∗ M isnon-degenerate on L − . Moreover, the inclusion T M ∩ L − ⊂ E ∩ E andidentity (20) imply that T M ∩ L − = { } . Similar arguments show that themetric < . , . > is non-degenerate on the (+1)-eigenspace L + of I I and T M ∩ L + = { } . Thus I , I is a generalized pseudo-K¨ahler structure on M . 21e can deform this structure using arbitrary smooth function f on M .Let H t be the flow of the F K -Hamiltonian vector field ı df F K , so H ∗ t ( F K ) = F K . Define γ = F K + i ( ω ′ + H ∗ t ω ′′ ) , γ = − F K + i ( ω ′ − H ∗ t ω ′′ ) . Then γ − γ = 2 F K + 2 iH ∗ t ω ′′ = H ∗ t (2 F K + 2 iω ′′ ) = H ∗ t ( β − β ) and γ − γ = β − β . It follows that for small t , the forms γ and γ define ageneralized pseudo-K¨ahler structure .Finally, let us note that a generalized pseudo-K¨ahler structure can beexplicitly defined by means of the pseudo-K¨ahler structures ( g, J + ), ( g, J − )and [16, (6.14)]. Example 1.
The construction above can be applied to 4-tori and primaryKodaira surfaces since each of these surfaces admits a para-hyperk¨ahlerstructure (see, for example, [23, 24]). Recall that the Kodaira surfaces donot admit any (positive) generalized K¨ahler structure [4, 5].
Example 2.
Any para-hyperhermitian structure which is locally confor-mally para-hyperk¨ahler can be deformed as in [3] to obtain a strictly pseudo-bihermitian structure. The universal cover of the locally conformally para-hyperk¨ahler manifold M is globally conformally para-hyperk¨ahler. The de-formation is performed on its para-hyperk¨ahler structure such that H t isinvariant with respect to the fundamental group of M . Then one obtains ageneralized pseudo-K¨ahler structure which after a (global) conformal changedescends to a pseudo-bihermitian structure on the quotient. In particular,there are pseudo-bihermitian metrics on properly elliptic surfaces of oddfirst Betti number and the Inoue surfaces of type S + [10]. These surfacesdo not admit any (positive) bihermitian structure [4]. On the other hand thequaternionc Hopf surfaces admit both bihermitian and pseudo-bihermitianstructures since they have both hyperhermitian and para-hyperhermitianmetrics [10]. They also have bihermitian metrics arising from twisted gen-eralized K¨ahler structures [5], however it is not clear whether these surfacesadmit twisted generalized pseudo-K¨ahler structures. The same question isopen for K3 surfaces too.Notice that the above constructions produce ”complementary” examplesof bihermitian and pseudo-bihermitian structures on the surfaces in the listsin Theorem 9. We summarize the examples obtained so far in: Proposition 10
Generalized pseudo-K¨ahler structures exist on complex 2-tori and primary Kodaira surfaces. Pseudo-bihermitian structures exist also n the quaternionic Hopf surfaces, properly elliptic surfaces with odd firstBetti number and Inoue surfaces of type S + . Example 3.
Here we provide examples of complex structures J + and J − satisfying the relation (3) J + J − + J − J + = 2 pId for a nonconstant function p with | p | >
1, which are not compatible withany global neutral metric. Consider Example 1 above in the case of a com-plex torus which is a product of 2 elliptic curves. It admits a holomorphicinvolution φ without fixed points, such that the quotient is a smooth com-plex surface. This surface is called a hyperelliptic surface of type I a . Onecan check that the natural para-hypercomplex structure of the torus de-scends to a para-hypercomplex structure on the quotient, but it admits nocompatible para-hyperhermitian metrics [10]. In particular, one can fix apara-hyperk¨ahler family of φ -invariant complex structures on the torus andcan deform any two structures of this family via the procedure described inExample 2. The Hamiltonian deformations H t are defined by a single func-tion and if one chooses this function to be φ -invariant, then both ( J + ) t = J + and ( J − ) t are φ -invariant for all t . Since they satisfy the relation (3) forsmall t , they descend to structures which satisfy the same identity on thequotient hyperelliptic surface. Since | p | > t , K = 0everywhere. If there were a compatible metric, then the fundamental forms F K + iF J + K obtained as in the consideration above would provide a trivi-alization of the canonical bundle, which is an absurd because the canonicalbundle of a hyperelliptic surface is not topologically trivial. In this section we show that, under a mild restriction, a naturally de-fined null-plane distribution on a pseudo-bihermitian 4-manifold M de-termines a local Engel structure. Recall that an Engel structure is bydefinition a 2-dimensional distribution D on a 4-manifold M such that rank [ D , D ] = 3 and rank [ D , [ D , D ]] = 4 at each point of M . These struc-tures have been actively investigated recently (see the introduction in [29]for an overview). They admit canonical coordinates and are preserved by23mall C -deformations. The global existence of an oriented Engel structureon an oriented compact manifold leads to triviality of its tangent bundle.Moreover, Vogel [29] showed that the converse also holds - any paralellizable4-manifold admits such a structure.Let ( M, g, J + , J − ) be a pseudo-bihermitian 4-manifold with J + J − + J − J + = 2 pId where | p | >
1. Let F ± and θ ± be the K¨aher and the Lee formof ( g, J ± ), respectively. Suppose that the pseudo-bihermitian structure is de-fined by a (twisted) generalized pseudo-K¨ahler one. Then d + F + + d − F − = 0by Theorem 2 and taking the Hodge-dual 1-forms we get θ + + θ − = 0.If we set K = [ J + , J − ] / p p − K = Id and K = ± Id .Moreover, g ( KX, Y ) = − g ( X, KY ), in particular the eigenspaces of K consists of isotropic vectors. Lemma 11
For the endomorphism N ± = J + + ( p ± p p − J − of T M ,we have
Ker N ± = Im N ± = ∓ K .Proof: It is easy to see that N ± = 0 and Ker N + ∩ Ker N − = { } . Moreover, − KJ + = J + K = p p p − J + − p p − J − , − KJ − = J − K = 1 p p − J + − p p p − J − . (23)It follows that if K ± is the ± K , then K − ⊂ Ker N + and K + ⊂ Ker N − . Hence dim Ker N ± ≥
2. This implies the lemma since thekernels of N + and N − are transversal and the dimension of the ambientspace is 4. q. e. d. Denote the vector field dual to θ ± w.r.t. g by the same letter. Set X = ( J + + f J − ) θ + where f = p − p p − Y = θ + + Kθ + . Clearly X, Y ∈ KerN − . One can easily see that X and Y are isotropic. Assumethat | θ + | x = 0 at some point x ∈ M . Then the vector fields X and Y are linearly independent at x . Indeed, suppose that λX + µY = 0 at x for some constants λ and µ . Applying N − to both sides of this identity,we get µ ( N − + N − K ) θ + = 0 at x . Then, using (23), we compute easilythat µ ( J + θ + − f J − θ + ) x = 0. If J + θ + = f J − θ + at x , we would have | θ + | x = f ( x ) | θ + | x , hence | θ + | x = 0, a contradiction. Therefore µ = 0, thus λ ( J + θ + + f J − θ + ) x = 0. This implies λ = 0 since | θ + | x = 0.Now define a 2-plane in T x M setting D x = Span ( X, Y ) x . (24)24 heorem 12 Let ( M, g, J + , J − ) be a (twisted) generalized pseudo-K¨ahler -manifold with nowhere vanishing Lee forms θ + = − θ − and such that J + J − + J − J + = 2 pId with | p | > . Then the null distribution D defined by (24)is an Engel structure on an open subset of M or the flow of Y consists ofnull-geodesics.Proof: Set N = N − . Then D = Ker N = Im N . We are going to cal-culate N [ X, Y ] and show that it is proportional to
N J + Y . This will im-ply that [ X, Y ] ∈ Span ( X, Y, J + Y ), so rank [ D , D ] = 3. Then we willshow that [ Y, J + Y ] has vanishing J + X component iff ∇ Y Y = F Y for somesmooth function F . This proves that either the flow of Y is geodesic or rank [ D , [ D , D ]] = 4 on an open subset of M .For the Levi-Civita connection we have [3]:2( ∇ X J ± ) Y = g ( X, Y ) J ± θ ± + g ( J ± X, Y ) θ ± + θ ± ( J ± Y ) X − θ ± ( Y ) J ± X and therefore2( ∇ X N ) Y = g ( X, Y )( J + − f J − ) θ + + g (( J + − f J − ) X, Y ) θ + + θ + (( J + − f J − ) Y ) X − θ + ( Y )( J + − f J − ) X + 2 X ( f ) J − Y (25)since θ − = − θ + . Also p = − / g ( J + , J − ) = 1 / tr ( J + ◦ J − ) and we get by(5) that dp = 12 θ + ◦ [ J + , J − ] = p p − θ + ◦ K. (26)We have X, Y ∈ Ker N and g ( X, X ) = g ( Y, Y ) = 0 ,g ( X, Y ) = g ( X, J + X ) = g ( Y, J + Y ) = 0 ,g ( J + X, Y ) = ( f − | θ + | = 2( f p − | θ + | . (27)Then the vector fields X, Y, J + X, J + Y form a basis of the tangent space ateach point of M . We have also that g ( θ + , J + J − θ + ) = g ( θ + , J − J + θ + ) = − g ( J + θ + , J − θ + ) = p | θ + | Y = θ + + J + J − − pId + J + J − p p − θ + = − f θ + + J + J − θ + p p − . (28)25ow ( ∇ X N ) Y − ( ∇ Y N ) X = − N ∇ X Y + N ∇ Y X = − N [ X, Y ] since
N X = N Y = 0. To compute N [ X, Y ] we use the fact that J + X = − f J − X, J + Y = − f J − Y . Hence by (25) and (27) we have2 | θ + | − ( ∇ X N ) Y = 2( f − θ + − J + X = − f p p − Y since θ + ( J + Y ) = g ( J + Y, θ + ) = 0 by (28), θ + ( Y ) = | θ + | and, in view of (26)and Lemma 11, X ( f ) = f θ + ( KX ) = f θ + ( X ) = f g ( X, θ + ) = 0. Similarly2 | θ + | − ( ∇ Y N ) X = − f − θ + + 2( f p − Y − f J − X = 2( f p + f p p − − Y = 0since θ + ( X ) = 0, θ + ( J + X ) = g ( − θ + + f J + J − θ + , θ + ) = −| θ + | + f p | θ + | by (28) and Y ( f ) = f g ( Y, θ + ) = f | θ + | . So N [ X, Y ] = f p p − | θ + | Y .We can easily check that N J + + J + N = 2( pf − Id . Then N J + Y =2( pf − Y so [ X, Y ] ∈ Span ( X, Y, J + Y ). It follows from (27) that X, Y, J + Y are linearly independent at every point, hence rank [ D , D ] = 3. If [ Y, J + Y ]has nowhere-vanishing J + X -component, then rank [[ D , D ] , D ] = 4, so D isan Engel structure. To find the J + X -component of [ Y, J + Y ] we use that[ Y, J + Y ] = ∇ Y J + Y − ∇ J + Y Y . First observe that ( ∇ Y N ) Y = 0, so ∇ Y Y ∈ Span { X, Y } . We have also that 2( ∇ Y J + ) Y = − θ + ( Y ) J + Y = −| θ + | J + Y .Since ∇ Y J + Y = ( ∇ Y J + ) Y + J + ∇ Y Y , then ∇ Y J + Y ∈ Span { J + X, J + Y } .Moreover, the J + X -component of ∇ Y J + Y is equal to the J + X -componentof J + ∇ Y Y which is also the X -component of ∇ Y Y . On the other hand wehave 2( ∇ J + Y N ) Y = − θ + ( Y )( J + − f J − )( J + Y ) = 2 pf | θ + | Y since ( J + − f J − ) J + Y = J + ( J + + f J − ) Y − pf Y = − pf Y . So N ∇ J + Y Y = − pf | θ + | Y and ∇ J + Y Y ∈ Span { X, Y, J + Y } does not have J + X -component.Then [ Y, J + Y ] = ∇ Y J + Y − ∇ J + Y Y has nowhere-vanishing J + X -componentiff ∇ Y Y has nowhere-vanishing X -component.To finish the proof notice that ∇ Y Y ∈ Span { X, Y } and if its X -component vanishes locally, ∇ Y Y = F Y which in turn means that theflow of Y is geodesic. q.e.d. Note finally that if p = const , then θ + = θ − and the distribution D isintegrable. 26 eferences [1] A. Andrada, S. Salamon, Complex product structures on Lie algebras ,Forum Math. 17 (2005), no. 2, 261–295.[2] V.Apostolov, D.Calderbank, P.Gauduchon, C.Tonnesen-Friedmann,
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