Sasakian structures. A foliated approach

Recent renewed interest in Sasakian manifolds is due mainly to the fact that they can provide examples of generalized Einstein manifolds, manifolds which are of great interest in mathematical models of various aspects of physical phenomena. Sasakian manifolds are odd dimensional counterparts of K\"ahlerian manifolds to which they are closely related. The book of Ch. Boyer and K. Galicki, Sasakian Geometry is both the best introduction to the subject and at the same time it gathers state of the art information and results on these manifolds. However, although the authors are well aware that a Sasakian structure is a very special one-dimensional Riemannian foliation with K\"ahlerian transverse structure, they use this fact only in a few very special cases. The paper presents an approach to Sasakian manifolds on which the author gave several lectures, most recently at the Workshop on almost hermitian and contact geometry at the Banach Center in B\c{e}dlewo in October 2015 and at University of the Basque Country in February 2016. The first lectures on the topic the author gave at Universidad de Sevilla in October 1988 and then presented the consequence for the geometry of Sasakian manifolds, in particular the relationns between various curvatures and those of the transverse K\"ahler manifold. The results were published in several sections of \cite{WO_S} as well as in \cite{Wo_deb}. The most general theory of geometrical structures"adapted"to a foliation was presented in \cite{WO_T}, see also \cite{WO_S}. The paper concentrates on cohomological properties of Sasakian manifolds and of transversely holomorphic and K\"ahlerian foliations. These properties permit to formulate obstructions to the existence of Sasakian structures on compact manifolds. The presented results are due to the author as well as his former and present Ph.D. students.

Definition 5 A contact metric structure pg, ξ, η 1 , φq on the manifold M is called K-contact if its Reeb vector field ξ is a Killing vector field of the Riemannian metric g. Then pM, g, ξ, η 1 , φq is called a K-contact manifold.
And finally, the most complex structure considered is the Sasakian structure (manifold).
Definition 7 A K-contact manifold whose underlying almost contact structure is normal is called a Sasakian manifold.

Transverse properties of Sasakian manifolds
Let F be a foliation on a Riemannian m-manifold pM, gq. Then F is defined by a cocycle U " tU i , f i , g ij u i,jPI modeled on a 2q-manifold N 0 such that (1) tU i u iPI is an open covering of M, (2) f i : U i Ñ N 0 are submersions with connected fibres, (3) g ij : N 0 Ñ N 0 are local diffeomorphisms of N 0 with f i " g ij f j on U i X U j . The connected components of the trace of any leaf of F on U i consist of the fibres of f i . The open subsets N i " f i pU i q Ă N 0 form a q-manifold N U " >N i , which can be considered as a transverse manifold of the foliation F . The pseudogroup H U of local diffeomorphisms of N generated by g ij is called the holonomy pseudogroup of the foliated manifold pM, F q defined by the cocycle U. Different cocycles can define the same foliation, then we have two different transverse manifolds and two holonomy pseudogroups. In fact, these two holonomy pseudogroups are equivalent in the sense of Haefliger, cf. [14].
According to Haefliger, cf. [13], a transverse property of a foliated manifold is a property of foliations which which is shared by any two foliations with equivalent holonomy pseudogroup. For example, being Riemannian, transversely symplectic, transversely almost-complex, transversely Kähler, etc., is a transverse property. A Riemannian foliation, i.e., admitting a bundle-like metric, is defined by a cocycle U modelled on a Riemannian manifold whose local submersions are Riemannian submersions. Then the associated transverse manifold N U is Riemannian and the associated holonomy pseudogroup H U is a pseudogroup of local isometries. Any foliation defined by a cocycle V whose holonomy psudogroup H V is equivanet to H U is also Riemannian, as the equivalence of pseudogroups transports the Riemannian metric from N U to N V and ensures that the pseudogroup H V is a pseudogroup of local isometries of the transported metric. This metric can be lifted to a bundle-like metric (not unique) on the other foliated manifold making the second foliation Riemannian. The same procedure can be applied to any geometrical structure, for the discussion of this general procedure see [20,19].
The space of H U -invariant k-forms on the manifold N U can be identified with the space of foliated sections of the bundle Ź k NpM, F q˚which in turn is isomorphic to the space of k basic forms A k pM, F q " tα P A k pMq : i X α " i X dα " 0 f or all vectors X P F u The differential sends basic forms to basic forms and the cohomology of the complex pA˚pM, F q, dq is called the basic cohomology of the foliated manifold pM, F q. In the language of basic cohomology we can express a very important property of foliations: For the discussion the meaning and importance of the condition see [10] Let φ : U Ñ R pˆRq , φ " pφ 1 , φ 2 q " px 1 , ..., x p , y 1 , ..., y q q be an adapted chart on a foliated manifold pM, F q. Then on U the vector fields B Bx 1 , ... B Bxp span the bundle T F tangent to the leaves of the foliation F , the equivalence classes denoted byB By 1 , ...B Byq of B By 1 , ... B Byq span the normal bundle NpM, F q " T M{T F which is isomorphic to the subbundle T F K .
All the definitions of Section 1 have been formulated in a purely geometrical way without any reference to the characteristic foliation. Let us look at the transverse structure of the characteristic foliation.
The characteristic foliations of a contact manifold pM, rηsq is transversely symplectic as the 2-form dη is basic and defines a transverse symplectic form.
The basic cohomology class rdηs P H 2 pM, F q is in the kernel of the natural mapping H 2 pM, F q Ñ H 2 pMq, and rdηs n P H 2n pM, F q is in the kernel of the natural mapping H 2n pM, F q Ñ H 2n pMq. Therefore if transverse volume form rωs n defines a non-zero basic cohomology class, then this 2n-form is in the kernel the natural mapping H 2n pM, F q Ñ H 2n pMq, thus this mapping cannot be injective providing an obstruction to a transversely symplectic 1-dimensional foliation being the characteristic foliation of a contact structure.
For example, such a 1-dimensional foliation cannot admit a transverse foliation with a compact leaf, as then according to the result of Molino-Sergiescu this mapping should be injective, cf. Theorem 2 of [15].
In the case of an almost contact structure pξ, η, φq on a smooth manifold M, the following conditions are equivalent, cf. [12] or Lemma 6.3.3 of [2]: 1) there exists a Riemannian metric for which the orbits of ξ are geodesics; 2) L ξ η " 0 : The conditions (2) and (3) are evidently equivalent, and they just say that the 2-form dη is basic.
Therefore the characteristic foliation of a K-contact manifold is transversely almost-Kähler, and the characteristic foliation of a Sasakian manifold is transversely Kähler, [2] Theorem 7.1.3. However, even if the characteristic foliation of a K-contact manifold is transversely Kähler, it does not imply that the structure is Sasakian.

Transversely Kähler foliations
We have noticed that the characteristic foliation of a Sasakian manifold is transversely Kähler. In this section we will gather the results which are particular to transversely Kähler foliations and therefore are also true for the characteristic foliations of a Sasakian manifold. It will facilitate the search for characterizations of Sasakian manifolds and properties which can distinguish between K-contact and Sasakian manifolds. Let F be a foliation of dimension p and codimension 2q on a smooth manifold M of dimension m " p`2q. It is a transversely Kähler foliation if there is a cocycle U " tpU i , f i , g ij qu i,jPI defining the foliation F modelled on a Kähler manifold pN, g N , J N q such that the local diffeomorphisms g ij of N are Kähler isometries, or equivalently that the associated holonomy pseudogroup H U is a pseudogroup of Kähler isometries of a Kähler structure on the transverse manifold N U .
We assume that the foliation F is transversely holomorphic, of complex codimension q and that the manifold M is compact. Therefore on the normal bundle NpM, F q of the foliation F we have a foliated Kähler structure, i.e. a foliated Riemannian metricḡ and an endomorphismJ of the normal bundle such thatJ 2 "´Id (an almost complex structure in the normal bundle) compatible withḡ, i.e. for any X, Y P NpM, F qḡ pJX,JY q "ḡpX, Y q and satisfying the " integrability " condition: for any X and Y Then the fundamental Kähler 2-formΩpX, Y q "ḡpX,JY q is basic and correspond to the Kähler form of the transverse Kähler structure of the foliation. For any point x of M there exists an adapted chart px 1 , ..., x m´2q , z 1 , ..., z q q modelled on R m´2qˆC q defined on an open neighbourhood of x. Basic forms on pM, F q are in one-to-one correspondence with holonomy invariant (H-invariant) forms on the transverse manifold. Basic k-forms are just foliated sections of the kth exterior product of the conormal bunsle NpM, F q˚foliated by the natural lift of the foliation F , cf. [20]. If the foliation F is transversely holomorphic its normal bundle has a (almost) complex structure corresponding to the complex structure of the transverse manifold. Therefore any complex valued basic k-form can be represented as a sum of k-forms of pure type pr, sq corresponding to the decomposition of k-forms on the complex manifold N. It is possible as elements of the holonomy pseudogroup are bi-holomorphic local diffeomorphisms. This decomposition can be also obtained by considering the decomposition of sections of the complex bundle Λ k C NpM, F q˚. Then a basic k-form α is of pure type pr, sq if for any point of M there exists an adapted chart px 1 , ..., x m´2q , z 1 , ..., z q q such that Let us denote by A k C pM, F q the space of complex valued basic k-forms on the foliated manifold pM, F q, and by A r,s C pM, F q the space of complex valued basic forms of pure type pr, sq. Then andB of bidegree p1, 0q and p0, 1q, correspondingly, The basic cohomology of transversely holomorphic and transversely Kähler foliations was studied in depth by A. El Kacimi-Alaoui, cf. [9]. We recall some basic results from this paper.
We assume that the foliation is transversely Hermitian. The operator [11] using the transverse part of the bundle-like metric, and the corresponding standard * operator on the level of the transverse manifold, can be extended to an operator The normal part of the bundle-like metric g, or the corresponding transverse metric, defines a Riemannian (Hermitian) metric g k on the bundle Λ k C NpM, F q˚, and therefore we can define a scalar product on The operator δ : A k C pM, F q Ñ A k´1 pM, F q defined as δ "¯d¯is the adjoint operator of d with respect to the scalar product ă, ą.
The foliated Laplacian sends basic forms into basic forms, it is a self-adjoint foliated (transversely) elliptic operator, cf. [9].
We can also define basic Dolbeault cohomology of the foliated manifold pM, F q. For a fixed r, 0 ď r ď q, consider the differential complex: Its cohomology is called the basic Dolbeault cohomology of the foliated manifold pM, F q, and denoted H r,˚p M, F q " Σ q s"0 H r,s pM, F q. The operator¯induces an isomorphism¯: A r,s Ñ A q´r,q´s . Using the same procedure as for the operator δ we define an operatorδ by the formulaδ "´¯B¯.
The operatorδ is the adjoint ofB with respect to the just defined scalar product. Moreover, the operator ∆" "Bδ`δB is a self-adjoint foliated (transversely) elliptic operator.
In the case of transversely Kähler foliations we can say much more about the basic cohomology and operators just defined.
The Kähler form of the transverse manifold N corresponds to a basic p1, 1q-form on pM, F q which we call the (transverse) Kähler form of the foliated manifold. Using this form we define the L operator Its adjoint with respect to ă, ą is Λ "´¯L¯.
For transversely Kähler foliations on compact manifolds we have the following relations: ΛB´BΛ "´?´1δ, These identities permited A. ElKacimi Alaoui to prove the following theorem, cf. [9] Theorem 1 Let F be a transversely Kähler foliation on a compact manifold M. If F is homologically oriented, then i) a basic k-form α " Σ r`s"k α r,s , α r,s P A r,s is harmonic if and only if the forms α r,s are harmonic, thus

ii) the conjugation induces isomorphisms
H r,s pM, F q -H s,r pM, F q.
iii) for any 0 ď r ď q, the form ω r is harmonic, thus H r,r pM, F q ‰ 0.
The complex A˚" Σ r,s A r,s of complex valued basic forms can be filtered by The filtration is compatible with the bigradation of the complex. Therefore we can define the associated spectral sequence which is called the basic Frölicher spectral sequence of the transversely holomorphic foliation F , cf. [6] . It converges to the complex basic cohomology of the foliated manifold pM, F q.
The terms E r,s 1 are just the basic pr, sq-Dolbeault cohomology groups. If the foliation F is homologically oriented and transversely Kähler, then it is a simple consequence of Theorem 1 that the Dolbeault spectral sequence collapses at the first term, cf. Theorem 2 of [6]. Indeed, the Hodge theorem combined with the just metioned theorem ensures that E r,s 1 -H r,s pM, F q where H r,s pM, F q is the space of pr, sq-pure harmonic forms. As harmonic forms are closed the operator d 1 is trivial(vanishes).
In [6] the authors noticed that the so called dd c -lemma for Kähler manifolds is an algebraic consequence of several identities. These identities have their counterparts for the basic cohomology of the transversely Kähler foliation on a compact manifold so the dd c -lemma is also true for the basic cohomology of a transversely Kähler foliation on a compact manifold. On the other hand the proof of the formality of the cohomology of a compact Kähler manifold, [8]. Therefore retracing the steps of the original proof we obtain Theorem 2 Let F be a transversely Kähler foliation on a compact manifold M. If F is homologically oriented then the minimal model of the complex basic cohomology of F is formal and thus Massey products of complex valued basic forms vanish.

Obstructions to existence of Sasakian structures
We have remarked that the characteristic foliation of the Sasakian manifold is transversely Kähler. Therefore we have a 1-dimensional (tangentially) orientable foliation with a very sophisticated transverse structure. Moreover, the normality condition is not a transverse property as its formulation involves vectors tangent to leaves of the foliation, in particular the characteristic vector field ξ. The corresponding transverse property can be formulated as follows: LetJ : NpM, F q Ñ NpM, F q be the endomorphism of the normal bundle defined for any tangent vector X asJ pXq " φpXq whereX is the vector in the normal bundle corresponding to a tangent vectoc X. The endomorphism is well defined andJ 2 "´id. Therefore it is an almost complex structure in the normal bundle.
The vector field ξ acts on the normal bundle, and therefore on the endomorphismJ. It is a foliated endomorphism iff L ξJ " 0, i..e. iff L ξJ pXq " 0 " rξ, φpXqs´φprξ, Xsq for any foliated vection of the distribution D. ThenJ corresponds to an almost complex structure J on the transverse manifold of the characteristic foliation. The normality condition insures that J is integrable (i.e., the Nujenhuis tensor N J " 0,). However, the normality condition is stronger, the equality N J " 0 equivalent to NJ " 0, and thus to the fact that for any sections X, Y of D, N φ pX, Y q is a vector field tangent to the characteristic foliation F ξ , i.e., of the form hξ for some smooth function h on M, but not necessarily 2dηpX, Y q as requires the normality condition. Therefore having given a 1-dimensional foliation we can ask many questions like: Is this foliation Riemannian, (transversely) Hermitian, transversely symplectic, transversely holomorphic, transversely Kähler?
These questions are about the transverse structure of the foliation and can be answered in the language of transverse properties, so the basic cohomology can provide some obstructions to the existence of such structures.
It is not difficult to see that a 1-dimensional transversely Kähler foliation admits a contact metric structure in the sense that it is the characteristic foliation of this structure. If the manifold M is compact, the non-triviality of the basic cohomology ensures that one can modify the Riemannian metric to ensure that the foliation is Riemannian and minimal, i.e. generated by a Kiling vector field.
Let ξ be a non-vanishing vector field on the manifold M. Assume that the foliation F ξ generated by ξ is transversely Kähler. Therefore on the transverse manifold N of the foliated manifold pM, F ξ q there exists a holonomy invariant Kähler structure pĝ,Ĵq, i.e.ĝ is a Riemannian metric,Ĵ a complex structure, and for any tangent vectors X.Y of N gpĴX,ĴY q " gpX, Y q andΩpX, Y q "ĝpX,ĴY q is a closed 2´f orm.
We can lift the Riemannian metricĝ to a Riemannian metricḡ in the normal bundle by the formula ḡ y pX,Ȳ q "ĝ f i pyq pdf i pXq, df i pȲ qq where f i : U i Ñ N 0 is a submersion from a cocycle defining the foliation F ξ ,X,Ȳ P NpM, F ξ q y , and y P U i . The complex structure is lifted to an almost comlex structureJ in the normal bundle in a similar way: Then the associated 2-formΩpX,Ȳ q "ḡpX,JȲ q is (locally) the pull-back of the Kähler 2-formΩ, i.e.

fiΩ "Ω
for any submersion f i from the cocycle defining the foliation F ξ . Next choose a suplementary subbundle D to the foliation, which is isomorphic to the normal bundle as a vector bundle, and define the Riemannian metric g on M as follows: the subbundles T F ξ and D are orthogonal, gpξ, ξq " 1, and transportḡ via the isomorphism to D.
The tensor field φ is defined in a similar fashion: for vectors from the subbundle D we define φ as the pull-back ofJ via the isomorphism from the normal bundle, and φpξq " 0.
Then, obviously, the triple pξ, η, φq is an almost contact structure on the manifold M. Let X, Y be any vectors on M. Taking into account the splitting T F ξ ' D we can write X " a X ξ`X and Y " a Y ξ`Ȳ . Thus gpX, Y q " gpX,Ȳ q`gpa X ξ, a Y ξq " gpX,Ȳ q`a X a Y " gpX,Ȳ q`ηpXqηpY q " gpφpXq, φpȲ qq`ηpXqηpY q " gpφpXq, φpY qq`ηpXqηpY q as the metricḡ isJ-invariant. Therefore the quadruple pg, ξ, η, φq is an almost contact metric structure.
Assume that the vector field ξ is the characteristic vector field of a (strict) contact structure η on the manifold M. The 2-form dη is basic and defines a foliated symplectic form, so it projects to a symplectic formΩ on the transverse manifold N. Assume that the 2-formΩ is the Kähler form of the transverse Kähler structure pĝ,Ĵq. Take D " kerη. Then it is not difficult to verify that pg, ξ, η, φq is a contact metric structure, asΩpX, Y q "ΩpĴpXq,ĴpY qq andΩpX, Y q "ĝppX,ĴpY qq.
This equality when lifted to the foliated manifold pM, F ξ q reads gpX, φpY qq " dηpX, Y q The fact that the Kähler formΩ isĴ invariant on the foliated manifold pM, F ξ q reads as dηpφpXq, φpY qq " dηpX, Y q.
The condition dηpφpXq, Xq ą 0 for 0 ‰ X P D, translates itself on the level of the transverse manifold tô ΩpĴX, Xq ą 0 which follows immediately from the definition of the form Ω : ΩpĴX, Xq "ĝpĴpXq,ĴpXqq ą 0 provided that X ‰ 0. Therefore strict contact structure η whose characteristic foliation is transversely Kähler admits a contact metric structure whose 1-form is the contact form η.
If the manifold M is compact, the foliated symplectic form ω " dη is basic and defines a non-zero 2-basic cohomology class and the 2n basic form ω n defines a non-zero 2n-basic cohomology class, so the characteristic foliation is taut, cf. [18]. Therefore we can modify the bundle-like metric g along the tangent bundle to the characteristic foliation to a Riemannian metric g 1 making the characteristic foliation minimal, i.e. the tangent vector field of unit length in the metric g 1 is Killing. The modification of the metric did preserve the splitting of the tangent bundle. Therefore on the contact manifold M we have a K-contact structure pg 1 , ξ 1 , η 1 , φq whose characteristic foliation is the same F ξ .
These considerations can be summed up by the following statement No transverse property can distinguish K-contact manifolds from Sasakian manifolds. Transverse properties can only say that a given foliation is not transversely Kähler. However, the characteristic foliation of a K-contact manifold can be transversely Kähler without the structure itself being Sasakian.
Thus if we want to prove that a given K-contact structure on a compact manifold is not Sasakian (i.e. it is not normal) we have to look for some properties which are not transverse, e.g., it is useless to study properties of the basic cohomology of the characteristic foliation.
The Sasakian version of the Hard Lefschetz Theorem proved by B. Cappelletti Montana et al. provides precisely a true obstruction to "being Sasakian," cf. [3] Theorem 3 Let pM, g, ξ, η, φq be a compact connected Saasaki manifold of dimension m=2n+1. Then for any 0 ď p ď n the multiplication by the form η^dη p induces an isomorphism between H n´p pMq and H n`p`1 pMq.
To complement this result the authors constructed two nilmanifolds of dimension 5 and 7, respectively, which are K-contact but do not admit any Sasakian structure. To prove that they use the properties of the cohomology ring which can be drived from the Hard Lefschetz Theorem, cf. [4].
The theorem coupled with these examples demonstrates that the Hard Lefschetz property is an obstruction to being Sasakian for compact manifolds.

Other cohomology theories
In search for more cohomological obstructions one can turn to other cohomology theories which have been developed for complex manifolds, for the most recent and up-to-date information see [1]. The foliated versions of several of these cohomology theories have been defined and studied by P. Raźny, cf. [16], a Ph.D. student at the Jagiellonian University.

Basic Bott-Chern cohomology of foliations
Let M be a manifold of dimension m " p`2q, endowed with a transversely Hermitian (i.e. transversely holomorphic, posessing a tranverse Hermitian metric) foliation F of complex codimension q. We can define the basic de Rham complex (denoted ACpM, F q) as the subcomplex of the standard de Rham complex of M consisting of basic forms. As in the manifold case the transversly holomorphic structure induces a decomposition of the cotangent spaces into forms of type (0,1) and (1,0), cf. Section 3. The basic Bott-Chern cohomology of F is defined as H˚,B C pM, F q :" KerpBq X KerpBq ImpBBq Remark Complex conjugation induces an antilinear isomorphism: Raźny proves a decomposition theorem for basic Bott-Chern cohomology using the operator: where B˚andB˚are the adjoint operators to B andB, respectively, with respect to the Hermitian product, defined by the transverse Hermitian structure, as defined in [9]. He notices that the operator ∆ BC is transversely elliptic and self-adjoint. To prove ellipticity he uses the fact that the operator projects, on the local quotient manifold, to the manifold version of the ∆ BC operator, which is elliptic, cf. [17].
Theorem 4 (Decomposition of the basic Bott-Chern cohomology) If M is a compact manifold, endowed with a transversely Hermitian foliation F , then we have the following decomposition: A˚,˚pM, F q " Kerp∆ BC q ' ImpBBq ' pImpB˚q`ImpB˚qq In particular: H˚,B C pM, F q -Kerp∆ BC q and the dimension of H˚,B C pM, F q is finite.

Basic Aeppli cohomology of foliations
We define the basic Aeppli cohomology of F as: H˚,Å pM, F q :" KerpBBq ImpBq`ImpBq Remark As in the Bott-Chern case complex conjugation induces an antilinear isomorphism: A pM, F q Ñ H q,p A pM, F q To obtain a decomposition theorem for the basic Aeppli cohomology of F we define a basic self-adjoint, transversely elliptic differential operator ∆ A : ∆ A :" BB˚`BB˚`pBBq˚pBBq`pBBqpBBq˚`pBB˚q˚pBB˚q`pBB˚qpBB˚qå nd thus we have Theorem 5 (Decomposition of basic Aeppli cohomology) Let M be a compact manifold, endowed with a Hermitian foliation F . Then we have the following decomposition A˚,˚pM, F q " Kerp∆ A q ' pImpBq`ImpBqq ' ImppBBq˚q In particular there is an isomorphism: H˚,Å pM, F q -Kerp∆ A q and the dimension of H˚,Å pM, F q is finite.
A duality theorem for basic Bott-Chern and Aeppli cohomology is also true, but we have to assume that our foliation is homologically orientable. Remark The above condition guaranties, that the following equalities hold for basic r-forms: B˚" p´1q r˚B˚,B˚" p´1q r˚B˚ Corollary 6 If M is a compact manifold endowed with a Hermitian, homologicaly orientable foliation F , then the transverse star operator induces an isomorphism: The theorem below is the main result concerning the basic Bott-Chern and Aeppli cohomologies proved in [16]: Theorem 7 (Basic Frölicher-type inequality) Let M be a manifold of dimension n, endowed with a transversely holomorphic foliation F of complex codimension q. Let us assume that the basic Dolbeault cohomology of F are finitely dimensional. Then, for every k P N, the following inequality holds: ÿ p`q"k pdim C pH p,q BC pM, F qq`dim C pH p,q A pM, F qqq ě 2dim C pH k pM, F , Cqq Furthermore, the equality holds for every k P N, iff F satisfies the BB-lemma (i.e. it's basic Dolbeault double complex satisfies the BB-lemma).
In the case when F is a transversely Hermitian foliation on a closed manifold M, we get the following corollary: Corollary 8 Let F be a transversely Hermitian, homologicaly orientable foliation on a closed manifold M. Then for all k P N the following inequality holds: ÿ p`q"k pdim C pH p,q BC pM, F qq`dim C pH p,q A pM, F qqq ě 2dim C pH k pM, F , Cqq Furthermore, the equality holds for every k P N, iff F satisfies the BB-lemma (i.e. it's basic Dolbeault double complex satisfies the BB-lemma).
These properties of new basic cohomology theories provide new tools to distinguish various transverse structures of Riemannian and holomorphic foliations. They can be used to check whether a given Riemannian foliation admits a rich transverse holomorphic structure, in partticular whether it is transversely Kähler.