Deformation of Sasakian metrics

Deformations of the Reeb flow of a Sasakian manifold as transversely K\"ahler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely K\"ahler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.

of Kähler metrics on compact complex manifolds is one of fundamental tools in deformation theory of complex structures. A Sasakian metric is a geometric structure on an odd dimensional manifold M whose standard extension to the cone M × R >0 gives rise to a Kähler metric (see Definition 2.1). We present an example of a Sasakian manifold to show that Sasakian metrics may not have stability in a smooth family of transversely holomorphic Riemannian flows (see Section 8.2). In our examples of nonstable Sasakian metrics, it is easy to see that the nontriviality of the (0, 2)-component of the basic Euler class of flows gives an 1 obstruction for stability of the Sasakian metrics. Our main result shows that this obstruction is the unique obstruction. Precisely, a Sasakian metric is stable in a smooth family of small deformation of transversely holomorphic Riemannian flows, if the basic Euler class is of class (1,1). Our reference on Sasakian manifolds is Boyer and Galicki [5] (see Section 8.2 for deformation theory of Sasakian metrics). A 1-dimensional foliation is called a flow in this article in accordance with references. Our main result is stated as follows: Let T be an open neighborhood of 0 in R ℓ and {(F t , I t )} t∈T be a smooth family of transversely holomorphic Riemannian flows on a closed manifold M . Assume that (F 0 , I 0 ) has a compatible Sasakian metricg. For the existence of a smooth family of compatible Sasakian metrics, it is necessary that the basic Euler class of (F t , I t ) is of class (1, 1) (see Lemma 3.5).
The difficulty to prove Theorem 1.1 comes from the noncontinuous change of basic differential complexes of foliations. We cannot translate directly our problem to a smooth family of partial differential equations to solve and cannot apply the Hodge-de Rham-Kodaira theory to the families of Laplacians on the basic de Rham complexes to prove Theorem 1.1. To avoid this difficulty, we will apply results on deformation of Riemannian foliations in Nozawa [28] and [29]. This allows us to change problems on families of basic de Rham complexes to problems on families the de Rham complexes which is much easier. This is a generalization of the part of Proposition 2.4 of Boyer, Galicki and Nakamaye [6]. They deduced Theorem 1.2 from Kodaira-Baily vanishing theorem for complex orbifold in Baily [2] for the case where Sasakian manifolds are quasi-regular, that is, every orbit of the flow generated by the Reeb vector field is closed. Theorem 1.2 is a consequence of the following version of Kodaira-Akizuki-Nakano vanishing theorem for basic Dolbeault cohomology: Let (M, F ) be a closed manifold with a homologically orientable transversely Kähler foliation of complex codimension n. Let E be an F -fibered Hermitian holomorphic line bundle over (M, F ) (see Definitions 3.2 and 6.6). This is a generalization of Kodaira-Baily vanishing theorem [2] for the case where every leaf of F is compact. Note that H p,q b (M/F , E) may not be isomorphic to where Ω p b is the sheaf of basic holomorphic p-forms on (M, F ) as we will remark in the paragraph after Definition 6.10. Because of this difficulty, the classical argument to show Theorem 1.2 for Fano manifolds does not work for basic cohomology of transversely Kähler foliations as pointed out by Boyer, Galicki and Nakamaye before Proposition 2.4 of [6]. We show that a simple observation, Lemma 6.11, can be used to avoid this difficulty.
The basic cohomology of Riemannian foliations has certain aspects similar to those of the de Rham cohomology of manifolds. But some cohomology vanishing theorems fail to hold for basic cohomology of Riemannian foliations at least directly (see Min-Oo, Ruh and Tondeur [26]). One of the reason is that the formal adjoint operator of the differential d on the basic de Rham complex is not given by the conjugation by the basic Hodge star operator (see Proposition 3.6 of Kamber and Tondeur [23] for the difference). But, for homologically orientable Riemannian foliations, El Kacimi and Hector [13] and El Kacimi [12] provided a method to overcome this difficulty based on Molino's structure theory. We show that their method can be applied to show Theorem 1.3.
1.3. Stability of positive Sasakian metrics. We obtain the following corollary: Let T be an open neighborhood of 0 in R ℓ and {(F t , I t )} t∈T be a smooth family of transversely holomorphic Riemannian flows on a closed manifold M . Assume that (F 0 , I 0 ) has a compatible Sasakian metricg. Assume that the Sasakian metricg is positive. Hence the basic Euler class of (F t , I t ) is of degree (1, 1). Thus Theorem 1.1 implies the existence of a family of Sasakian metrics compatible to (F t , I t ).

Stability of K-contact structures in families of Riemannian flows.
Recall that a contact form η on a smooth manifold M is K-contact if there exists a metric on M preserved by the flow generated by the Reeb vector field of η. We state a K-contact variant of Theorem 1.1: Let M be a closed manifold. Let (g, η) be a K-contact structure on M . Let T be an open neighborhood of 0 in R ℓ . Let {F t } t∈T be a smooth family of Riemannian flows on M such that F 0 is the flow defined by the Reeb vector field of η.
There exists an open neighborhood V of 0 in T and a smooth family {η t } t∈U of K-contact structures on M such that F t is induced by the orbits of the Reeb vector field of η t for every t in U and η 0 = η.

1.5.
Moduli space of Sasakian metrics with a fixed transversely Kähler flow. At last, we show a result which describes the difference of moduli spaces of Sasakian metrics and their underlying transversely Kähler flows. Let M be a closed manifold. Let S be the set of Sasakian metrics on M . Let K be the isomorphism classes of transversely Kähler flows on M . There exists a natural map M : S −→ K which corresponds each Sasakian metric to the underlying transversely Kähler flow. We take a point k 0 on K. Let Diff 0 (k 0 ) be the identity component of the group of diffeomorphisms of M which preserves the transversely Kähler flow k 0 . We define Then we have Theorem 1.6. There exists a homeomorphism Note that Diff 0 (k 0 )/ Ham(k 0 ) is an abelian group (see the second paragraph of Section 7). Thus M −1 (k 0 )/ Diff 0 (k 0 ) is a quotient of a vector space by an abelian group. As a corollary, we have the following Corollary 1.7. Let M be a closed manifold whose first Betti number is zero. If the underlying transversely Kähler flows of two Sasakian metrics (g 1 , α 1 ) and (g 2 , α 2 ) are isomorphic, then (M, g 1 , α 1 ) and (M, g 2 , α 2 ) are isomorphic. We denote the tangent bundle of F by T F . By the integrability of F , the Lie bracket on C ∞ (T M ) induces the Lie derivative with respect to vector fields tangent to the leaves (5) for every nonnegative integer r and s.
For a transversely holomorphic foliation (F , I) on M , we call I the complex structure of (F , I). For a Riemannian foliation (F , g), we call g the transverse metric of (F , g). For a transversely Kähler foliation (F , I, g), we call the above 2-form ω the transverse Kähler form of (F , I, g).
We will regard the transverse Kähler form ω of a transversely Kähler flow as a 2-form on M by the injection We recall the definition of the isometricity of a Riemannian flow (F , g) on M . We recall also Definition 2.5 (Geometrically tautness). A foliated manifold (M, F ) is geometrically taut if there exists a Riemannian metricg on M such that every leaf of F is a minimal submanifold of (M,g). Such metricg is called a minimal metric on (M, F ).
By the following lemma due to Carrière in [8], the isometricity is equivalent to the geometrically tautness for oriented Riemannian flows: Lemma 2.6. Let (M, F ) be a closed manifold with an oriented Riemannian flow. Then a minimal metricg is Killing if and only ifg is bundle-like.
Recall that a metric g on (M, F ) is bundle-like if the metric induced on T M/T F from (T F ) ⊥ is a transverse metric. By a theorem of Molino and Sergiescu [27] or a theorem of Masa [25], the isometricity of Riemannian flows is equivalent to the following cohomological property: Definition 2.7 (Homological orientability). A Riemannian foliation of codimension n is homologically orientable if the basic cohomology group H n b (M/F ) of degree n is nontrivial where This terminology is due to El Kacimi [12]. We refer El Kacimi and Hector [13] for the basic cohomology of foliated manifolds.
By the last sentence of Section 6.4 of Blair [3] or Proposition 6.5.14 of Boyer and Galicki [5],  Indeed, let (M,g, η) be a Sasakian manifold determined by a pair of an integrable CR-structure (H, J) and a K-contact form η such that H = ker η. We write ξ for the Reeb vector field of η. Let F be the flow generated by ξ. Here (g, ξ) is a Killing pair on F , because ξ is the Reeb vector field of η which is K-contact by Lemma 2.8. Futaki, Ono and Wang proved that F has a transversely Kähler structure in Section 3 of [14].
We have the following characterization of Sasakian metrics in terms of flows with transverse structures: Lemma 2.10. A pair of a transversely Kähler flow (F , I, g) and a contact form η determines a Sasakian structure on M if dη = ω where ω is the transverse Kähler form of (F , I, g).
Proof. By Lemma 2.8, it is suffices to show that a pair of a transversely Kähler flow (F , I, g) and a contact form η determines a pair of a CR-structure and a contact form in Lemma 2.8. We put H = ker η.
We show that η is K-contact. Let ξ be the Reeb vector field of η. Then, η is tangent to ker ω = T F . Since the flow generated by ξ preserves the transverse metric of F and a orthogonal plane field H, the flow generated by ξ preserves a Riemannian metric on M . Hence η is K-contact.
We denote the restriction of the canonical projection T M −→ T M/T F to H by π. Putting J(X) = π −1 • I • π, we have a CR-structure (H, J) on M . We will show that (H, J) is integrable. For local sections X and Y of H, we have We will show that the Nijenhaus tensor vanishes for any local sections X and Y of H. Fix a point x on M . Take two vectors X 0 and Y 0 in H x .
We have an open neighborhood W x such that (W x , F | Wx , I| Wx ) is isomorphic to (R m × B n , F std , I std ) as transversely holomorphic foliations by a result of Gómez Mont [17]. Here (R m × B n , F std , I std ) is the standard transversely holomorphic foliation defined by a decomposition R m × B n = ⊔ z∈B n R m × {z}. We can assume that We can take linear vector fields X B n and Y B n on B n so that ( Since X B n and Y B n are linear, their flow preserves the complex structure I on B n . Thus we have Moreover we have We take basic sections X and Y of H| Wx so that (8) and (9), we have Hence we have It follows that N (X, Y ) is a local section of H. On the other hand, by the integrability of I, we have (iii) A smooth family of Riemannian flows on M is a pair of a smooth family of flows F t and a smooth metric g amb on the family of normal bundles of We refer El Kacimi and Hector [13] for basic forms and basic cohomology of Riemannian foliations.
Definition 3.1 (The basic Euler class of isometric Riemannian flows, Saralegui [32]). Let (g, ξ) be a Killing pair on F . We define a 1- [32] proved that the basic Euler class of F depends only on the smooth type of the flow F (see Royo Prieto [30] for basic Euler classes extended to general Riemannian flows).
If F is an isometric flow defined by fibers of a circle bundle, the basic cohomology of F coincides with the de Rham cohomology of the base manifold. In this case, the basic Euler class of F coincides with the Euler class of the circle bundle up to multiplication of real numbers.
3.2. F -fibered Hermitian vector bundles and basic Dolbeault cohomology. Let M be a smooth (2n + 1)-manifold. We recall the notion of F -fibered Hermitian vector bundles on foliated manifolds (M, F ). In this paragraph, we use the complex number field as the coefficient ring of differential forms.
Let (F , I) be a transversely holomorphic foliation of real dimension m and complex codimension n on a closed manifold M . Let B n be the unit ball in C n . Let (F std , I std ) be the standard transversely holomorphic foliation on as transversely holomorphic foliations (see Gómez Mont [17]). We take finite points {x j } so that M = ∪ j W xj . We denote the composite by φ j where the second map is the second projection.
An F -fibered vector bundle is holomorphic if the transition functions are pull back of holomorphic functions on W xj .
These notion are independent of the foliated atlas {W xj }. Basic (p, q)-forms with values in E are similarly defined. This F -fibered Hermitian vector bundle is Ffibered in the meaning of El Kacimi [12]. We denote the space of basic (p, q)-forms We remark about the cohomology of the sheaf Ω p b of basic holomorphic p-forms on (M, F ) with values in E. If the leaves of F are not closed, the sheaf of (p, q)forms with values in E may not be acyclic. Hence we may not have an isomorphism Here the situation is different from the case of complex manifolds or orbifolds where we always have The (0, 2)-component of the basic Euler class of transversely holomorphic flows. Let M be a closed smooth manifold. Let (F , g, I) be a transversely holomorphic Riemannian flow on M . We assume that (F , g) is isometric with a Killing pair (g, ξ). Let η be the characteristic form of F defined by η(X) = g(ξ, X). Then the basic cohomology class of dη is the basic Euler class of (M, F ). Note that the (0, 2)-component (dη) 0,2 of dη is ∂-closed, because ∂(dη) 0,2 = (ddη) 0,3 = 0. We show the following lemma in a way different from the argument of Saralegui to show the well-definedness of the basic Euler class in [32]. Proof. Let (g 1 , ξ 1 ) and (g 2 , ξ 2 ) be two Killing pairs on (M, F ). Let η j be the characteristic form of (M, F ,g j ) for j = 1 and 2. By Lemma 7.1, there exist a real number r and a diffeomorphism f of M which maps each leaf of F to itself, isotopic to the identity and satisfies Since f is isotopic to the identity as a diffeomorphism which maps each leaf of F to itself, we have Obviously if the basic Euler class of (F , g, I) is of degree (1, 1), then there exists a basic 1-form β such that d(η + β) is a basic (1, 1)-form on (F , I).
By Lemma 2.10, the underlying transversely holomorphic Riemannian flow of a Sasakian manifold is transversely Kähler. Moreover the basic cohomology class of the transverse Kähler form is the basic Euler class of the flow. Hence we have

Existence of extensions of Sasakian metrics
We prove Theorem 1.1. Let (M, F , I, g) be a closed manifold with a transversely holomorphic Riemannian flow. We assume that g is I-invariant. Let (g, ξ) be a Killing pair on F such that the transverse component ofg is equal to g. Let η be the characteristic form of (M, F ,g). Put H = ker η. We denote the restriction of the canonical projection The coefficient ring of differential forms is C in this section. By the decomposition . We can decompose the differential d as where the subscripts correspond to the triple grading of Ω • (M ). Note that the basic (p, q)-forms can be embedded to Ω p,q,0 . The restriction of d 0,1,0 and d 1,0,0 to basic Dolbeault complex are equal to ∂ and ∂, respectively. We consider a differential operator D = d − d 1,1,−1 . Here d 1,1,−1 is a differential operator of degree 0. Indeed, d 1,1,−1 is written as where X s is a section of H 1,0 and Y t is a section of H 0,1 for each s and t. Thus d 1,1,−1 satisfies d 1,1,−1 (f α) = f d 1,1,−1 α for any smooth function f on M , which means that d 1,1,−1 is a differential operator of degree 0. Note that d 1,1,−1 is the zero map on Ω j,k,0 . Hence the symbol of D is equal to the symbol of d. Let D * be the adjoint of D with respect to the inner product on Ω • (M ) defined byg. We put Then ∆ D is a self-adjoint strongly elliptic operator, because the symbol of ∆ is equal to the symbol of the Laplacian of d. Since g is I-invariant, the complex conjugation of the Hodge star operator * maps Ω j,k,h to Ω n−j,n−k,1−h where 2n is real codimension of F . We have D * = − * D * integrating the formula We show (i). By the Rummler's formula (see the second formula in the proof of Proposition 1 in Rummler [31] or Lemma 10.5.6 of Candel and Conlon [7]), ι ξ dζ is the basic mean curvature form of (M, F ) with respect to the characteristic form ζ. Thus ι ξ dζ is zero by the assumption (a). It follows that dζ is a basic form. Thus dζ is written as a sum of a (2, 0, 0)-form, a (1, 1, 0)-form and a (0, 2, 0)-form. Since dζ is of degree (1, 1, 0) by the assumption (b), we have dζ = d 1,1,0 ζ. Thus we have Dζ = 0.
We recall a result in Nozawa [29]:  Proof. We extend the smooth family of transverse metrics {g t } t∈T to a smooth family of Riemannian metrics {g ′t } t∈T on M so that g ′t is bundle-like with respect to F t and g ′0 =g. Let κ t be the mean curvature form of (M, F t , g ′t ). We define a 1-form η t by η t (Y ) = g ′t (ξ t , Y ).
We show that there exists a minimal metricĝ t on (M, F t ) such that the orthogonal plane field of T F t is a contact structure for t in V . Only in this paragraph, we will use a double grading of real de Rham complex Ω • (M ) determined by the splitting T M = (ker η t ) ⊕ T F t instead of the triple grading above. We put for each t. The differential d and its formal adjoint δ t are decomposed as where the indices correspond to the double grading of Ω • (M ). By Theorem 4.2, (M, F t ) is isometric. By a result of Carrière [8], the isometricity of (M, F t ) is equivalent to geometrically tautness of (M, F t ) (see Lemma 2.6). According to Proposition 4.3 and Equation 5.3 ofÁlvarez López [1], κ t is contained in the image of δ t 0,−1 +d t 1,0 : A t 1,1 ⊕A t 0,0 −→ A t 1,0 . By the continuity of δ t 0,−1 +d t 1,0 : A t 1,1 ⊕A t 0,0 −→ A t 1,0 , we can choose an element β t of A t 1,1 and a smooth function f t on M which satisfy κ t = δ t 0,−1 β t + d t 1,0 f t so that β t is sufficiently close to 0. We modify the orthogonal plane field (T F t ) ⊥ by β t and modify the metricg ′t | T F ⊗T F along leaves by f t to obtain a minimal metricĝ t according to Proposition 4.3 and Equation 5.3 ofÁlvarez López [1]. Since β t is close to 0, the orthogonal plane field H t of T F t with respect tog t 1 is close to (T F 0 ) ⊥ . Since the space of contact forms are open in Ω 1 (M ), a plane field sufficiently close to a contact structure on a closed manifold is a contact structure. Thus H t is a contact structure.ĝ t satisfies the condition.
We show that there exists a 1-form ζ t on (M, F t ) which satisfies the conditions (a), (b), (c) and (d) in Lemma 4.1 for t in V . Fix a point t on V . Let θ t be the characteristic form of (M, F t ,ĝ t ). Since the (0, 2)-component of the basic Euler class is trivial by the assumption, there exists a basic (0, 1)-form σ such that ∂σ = (dθ t ) 0,2 . We consider σ + σ as a real 1-form on M . By the Hodge decomposition of basic de Rham complex (see El Kacimi and Hector [13] orÁlvarez López [1]), we can take σ + σ in the image of the adjoint operator d * b : We put ζ t = θ t − (σ + σ). By the argument of the previous paragraph, ker θ t is a contact structure. We can take small σ so that ker ζ t is also a contact structure. Then dζ t is a transverse symplectic form on T M/T F t . The conditions (a), (b) and (d) are satisfied. We have (30) d(θ t − (σ + σ)) = (dθ t ) 1,1 − (∂σ + ∂σ).
Thus dζ t is a real closed basic (1, 1)-form. Since t is sufficiently close to 0, we can take σ sufficiently close to 0. Then ζ t is nondegenerate, because dθ t is nondegenerate. Thus ζ t satisfies the condition (c). We consider a differential operator ∆ D (t) on (M, F t ) as above using a splitting Recall that L t is given by the wedge product with the basic Euler class [dη] (see [32]). Since dθ t is a transverse symplectic form, [dθ t ] n is nontrivial in H 2n b (M/F t ; C). Thus [dθ t ] is nontrivial in H 2 b (M/F ; C). By the exact sequence (31), we have dim is constant with respect to t by the consequence of the previous paragraph, the projection F t : Ω 1 (M ) −→ H 1 D (t) maps a smooth family of 1-forms to a smooth family of 1-forms by Theorem 5 of Kodaira and Spencer [24]. Putting ̟ t = F t (η t ), we have a smooth family {̟ t } t∈V of ∆ D (t)harmonic 1-forms such that ζ 0 = F 0 (η 0 ) = η 0 . We show that d̟ t is a basic (1, 1)-form. By Lemma 4.1, ̟ is a sum of a basic 1-form and ζ t . Thus d̟ t is basic. By the D-closedness of ̟ t , we have d̟ t = d 1,1,0 (t)̟ t . Thus d̟ t is of degree (1, 1). Putting Re ̟ t = ̟ t +̟ t 2 , we have a smooth family Re ̟ t of real 1-forms such that the differential is a basic (1, 1)-form. Since dη 0 is nondegenerate, Re ̟ t is also nondegenerate for t in V . By Lemma 2.10, a pair of a transversely Kähler flow (F t , I t , d Re ̟ t ) and a contact form Re ̟ t determines a Sasakian metric on M . Theorem 1.1 directly follows from Proposition 4.3.

Stability of K-contact structures
Theorem 1.5 is proved by an argument analogous to the proof of Theorem 1.1. We describe the outline here. We use real de Rham complex here. We use the double grading Ω • (M ) = ⊕ j,k A t j,k of Ω • (M ) as (28) instead of the triple grading. We use the operatorD = d 1,0 +d 0,1 instead of D above. We put ∆D =D * D +DD * . This ∆D is self-adjoint strongly elliptic operator as D. The lemma corresponding to

Kodaira-Akizuki-Nakano vanishing theorem for transversely Kähler foliations
We prove Theorems 1.2 and 1.3. We recall basic notion of Lefschetz theory for basic forms. In this section, we follow the notation of the book by Huybrechts [22] basically. Let (F , I) be a complex codimension n transversely Kähler foliation with transverse Kähler form ω. At each point x on M , we have the Hodge star operator is defined by * b,E (α ⊗ s) = * b α ⊗ h(s) for sections of the form α ⊗ s where α is a basic (p, q)-form and s is a local holomorphic section of E. We define the basic Lefschetz operator Let∧ be the wedge product defined by the composite of (38) We assume that (M, F ) is homologically orientable in the sequel. We define an inner product on Ω • b (M/F , E) under the assumption of homologically orientability following the argument of El Kacimi and Hector in Section 4.5 of [13] (see also El Kacimi [12]). Let ρ : M 1 −→ M be the orthonormal frame bundle of the normal bundle of F . Let π : M 1 −→ W be the basic fibration. Let X 1 , X 2 , . . ., X n(n−1) 2 be the vector fields on M which generate the free action of SO(2n) on M 1 . Let θ i be the basic form which is the dual of X i . We define a n(n−1) is the integration along fibers of π defined under the assumption of the homologically orientability of F by Hector and El Kacimi (see Proposition 3.2 of [13]). This I commutes with d as shown there. If W is not orientable, the orientation cover of W can be used to define an inner product in the same equation as (39). In what follows, we also assume that the orientability of W for the simplicity.
Proof. By local computation, we have The argument in Proposition 4.6 of El Kacimi and Hector [13] or Section 3.2.4 of El Kacimi [12] imply that the third terms of the right-hand sides of (42) and (43) are zero. Composing I and integrating on W both sides of (42) and (43), we have (40) and (41).
We denote the formal adjoint of a differential operator D on E with respect to (·, ·) by D * .
As in Proposition 2.7.5 of El Kacimi [12], Ω • b (M/F , E), (·, ·) is isomorphic to the space of global sections of a Hermitian vector bundle E over W as Hermitian vector spaces. By Propositions 2.7.7 and 2.7.8 of El Kacimi [12], ∂ E induces a strongly elliptic differential operator on E. The formula (40) in Lemma 6.1 implies that the formal adjoint ∂ * E of ∂ E is given by ∂ * . Hence this argument of El Kacimi proves that the Hodge decomposition theorem holds for basic Dolbeault complex with inner product (·, ·): Hodge decompositions for ∂ and ∂ on Ω • p,q (M/F ) imply the following ∂∂-lemma as in the case of Kähler manifolds (see Corollary 3.2.10 of Huybrechts [22]): We put Λ = L * . We have The following proposition shows that the positivity of (E, h E ) is determined only by the basic first Chern class [F ∇ ] of E if we allow to change the metric. Proposition 6.7. If there exists a basic positive form ω such that is an F -fibered Hermitian holomorphic line bundle whose curvature form is ω. Proposition 6.7 follows from Corollary 6.3 as in the case of Kähler manifolds (see the last paragraph of the proof of Theorem 7.10 of Voisin [33]).
We define the curvature operator We recall the definition of basic (p, q)-forms with values in E on (M, F ). We use the notation in Section 3.2. Clearly the basic Lefschetz operator L and the basic Hodge star operator * b,E satisfy β for a differential form β on B n where L B n is a Lefschetz operator on B n with respect to ω and * E B n is a Hodge star operator on B n with values in E B n . We have the following formulas for any homologically orientable transversely Kähler foliations as well as the case of complex manifolds: We have the Nakano identity and Bochner-Kodaira-Nakano equality: Proof. We have by the classical Nakano identity on B n , which is proved by a local argument on B n where Λ B n = * E * B n L * E B n (see Lemma 5.2.3 of Huybrechts [22]). Here (i) is proved by pulling back (49) by φ j .
(ii) [15] If F ∇ has rank at least equal to s at every point on X, then We refer Boyer, Galicki and Nakamaye [6] for more detailed information on positive Sasakian manifolds.
Even though we have Hodge decomposition Theorem 6.2 and Serre duality Theorem 6.4, the classical argument for complex manifolds does not work to show our Theorem 1.2. This is because H p,q b (M/F ) may not be isomorphic to H q (M/F , Ω p b ). Indeed, the basic Dolbeault complex with values in E may not be an acyclic resolution when the leaves are not closed. We use Lemma 6.11 which allows us to avoid the sheaf cohomology of basic holomorphic forms.
Let K F be the canonical line bundle of (M, F ). We fix a transverse Hermitian metric on (T M/T F ) ⊗ C. Let h KF be the Hermitian metric on K F induced from the metric on (T M/T F ) ⊗ C. We recall a C-antilinear isomorphism ·). Note that the canonical line bundle (K F , h KF ) of F is an F -fibered Hermitian holomorphic line bundle. Hence we can apply the argument in this section. Let H p,q b be the space of basic harmonic (p, q)-forms. Let H p,q b (K F ) be the space of basic harmonic (p, q)-forms with values in K F . Let 1 KF be the identity in End(K F ) = C ∞ (K F ⊗ K * F ). We have natural maps (53) Ξ : Ω n,q b (M/F ) −→ Ω 0,q b (M/F , K F ) (54) Θ : Ω 0,q b (M/F ) −→ Ω n,q b (M/F , K * F ) defined by Ξ(α 1 ∧ α 2 ) = α 2 ⊗ α 1 and Θ(α 2 ) = 1 KF ∧ α 2 for α 1 in Ω n,0 b (M/F ) = C ∞ (K F ) and α 2 in Ω 0,q b (M/F ). Clearly both of Θ and Ξ are bijective. Lemma 6.11.
we have Ξ(ker ∂) = ker ∂ KF . We will show (58) Ξ(ker ∂ * ) = ker ∂ Let α be an element of ker ∂ * . By a result of Gómez Mont [17], for each point x on M , we can take an open neighborhood W of x such that (W, F | W , I| W ) is isomorphic to the standard transversely holomorphic foliation on R m × B n defined by a decomposition R m × B n = ⊔ z∈B n R m × {z} where B n is the unit ball of C n . Since that (58) is a local formula, we can compute on W . We can write α| W as α| W = j (s j ∧ β j ) where s j is a section of K F | W which satisfies h KF (s j , s j ) = 1 for each j. By the definition of * b and h(s j , s j ) = 1, we have for each j. Since h(s j , s j ) = 1, we have that s j ⊗ h(s j ) = 1 KF . Then we have Hence we have Ξ(ker ∂ * ) = ker ∂ * KF . We show (56) in a similar way. It is easy to see Θ(ker ∂) = ker ∂ K * F . We will show (62) Θ(ker ∂ * ) = ker ∂ * K * F . By a result of Gómez Mont [17], for each point x on M , we can take an open neighborhood W of x such that (W, F | W , I| W ) is isomorphic to the standard transversely holomorphic foliation on R m × B n defined by a decomposition R m × B n = ⊔ z∈B n R m × {z}. Since that (62) is a local formula, we can compute on W . Then we can take a section s of K F | W such that By (59) and (60), we have Thus Θ(ker ∂ * ) = ker ∂ * K * F follows from (57) and (64), because (65) . The proof is completed. Lemma 6.11 allows us to prove Theorem 1.2 by Theorem 1.3 without using the sheaf cohomology of basic holomorphic forms.
Proof of Theorem 1.2. The underlying transversely Kähler flow of a Sasakian manifold is isometric. Hence the flow is homologically orientable by a result of Molino and Sergiescu [27]. By the Kähler identity for homologically orientable transversely Kähler foliation by El Kacimi (see Section 3.4 of [12]), the complex conjugation gives an isomorphism By Lemma 6.11, it follows that The last term vanishes if K * F is positive and p > 0 by Theorem 1.3. We remark on the other possibility of the inner product on the basic de Rham complex.Álvarez López [1] proved the Hodge decomposition theorem for basic de Rham complex with respect to the inner product ·, · defined by the restriction of the usual inner product of the de Rham complex. Note that the formal adjoint of d with respect to ·, · is not given by the basic Hodge star operator as Lemma 6.5. The difference of the formal adjoint and − * b d * b is given by the mean curvature form of F (see Proposition 3.6 of Kamber and Tondeur [23] ). Hence Lemma 6.5 is not true in general for ·, · . To show Theorem 1.3 using ·, · , one can apply Masa's theorem in [25] for the existence of a minimal metric on homologically orientable foliations. Since the mean curvature form is zero for a minimal metric, the adjoint of d with respect to ·, · is given by its conjugation of d by the basic Hodge operator. Then Lemma 6.5 are true for ·, · . Then the rest of the argument is the same as above.

Moduli space of Sasakian metrics with a fixed underlying transverse Kähler flow
We will prove Theorem 1.6. Let M be a closed manifold with a Sasakian metric with contact form η. Let k 0 be the underlying transversely Kähler flow. We define Diff 0 (k 0 ) and Ham(k 0 ) as in Section 1.5. We describe the relation of Diff 0 (k 0 ) and Ham(k 0 ). Obviously we have an exact sequence closed, because f preserves the transverse Kähler form dη. We will use the leafwise cohomology and the spectral sequence of foliated manifolds (M, F ). A filtration of the de Rham complex of M is defined by F in a way similar to the Leray spectral sequence of fiber bundles. The k-th leafwise cohomology of (M, F ) is naturally identified with the E 0,k 1 -term of this spectral sequence. For spectral sequence of foliations, we refer El Kacimi and Hector [13] or Kamber and Tondeur [23] .
In the sequel, Diff 0 (M, F ) denotes the subgroup of Diff(M ) consisting of diffeomorphisms such that • f maps each leaf of F to itself and • f is isotopic to the identity through diffeomorphisms which map each leaf of F to itself. Note that Diff 0 (M, F ) is a subgroup of Ham(k 0 ). Lemma 7.1. Let (M, F ) be an isometric Riemannian flow. Let η j be the characteristic forms of (M, F ,g j ) for a Killing metricg j for j = 1 and 2. There exists a real number r and a diffeomorphism f of M in Diff 0 (M, F ) such that Proof. η 1 and η 2 are the characteristic forms of Killing metrics. Thus, by Corollary 4.7 of Kamber and Tondeur [23] , both of the leafwise cohomology classes of η 1 | T F and η 2 | T F generate the E 2 -term E 0,1 2 (F ) of dimension 1. Hence we have a real number r such that in the leafwise cohomology group H 1 (F ). By the leafwise version of the Moser's argument (see Hector, Macias and Saralegui [21]), we have a diffeomorphism f of M in Diff 0 (M, F ) such that  (73) f * η = η + dh.
Proof. We consider maps For a vector field Y on M , we have Hence (ii) is proved.
We take a large integer N so that Dφ t + 1 N dh ⊗ ξ is nondegenerate at each point of x for any t in [0, 1]. We define ζ j by for j = 0, 1, . . ., N . By ξh = 0, we have L ξ ζ j = 0 and ζ j (ξ) = 1 for each j. By (i) and (ii), we have a diffeomorphism f 1 Hence we can take f in the statement of (iii) as f = (f 1 N ) N . Let (M, η,g) be a closed Sasakian manifold. By Lemma 2.10, a Sasakian metric on M is determined by a contact form η and a transversely Kähler structure of the flow F defined by the flow generated by the Reeb vector field of η on M such that the differential of the contact form coincides with the transverse Kähler form. For a basic closed form β, we can construct a Sasakian metric σ β determined by a contact form η + β and the same transversely Kähler structure of F as (η,g). We consider the set of Sasakian metrics S 1 = {σ β | β is a basic harmonic 1-form on (M,g)}. Proof. Take a Sasakian metric (η 1 , g 1 ) whose underlying transversely Kähler flow is isomorphic to the transversely Kähler flow of (η, g). The isomorphism of transversely Kähler flows means dη = dη 1 and the transverse metrics induced by g and g 1 are equal. Let F be the common underlying flow of (η, g) and (η 1 , g 1 ). By Lemma 7.1, there exist a real number r and a diffeomorphism f 1 of M in Diff 0 (M, F ) such that  [13] orÁlvarez López [1]), there exists a basic harmonic 1-form β and a smooth basic function h 2 such that Let ξ be the common Reeb vector field of η and η 2 . By Lemma 7.2, we have a diffeomorphism f of M in Diff 0 (M, F ) such that Thus f * 2 η 2 is an element of S 1 . Since f * 2 η 2 is on the same orbit of the action of Diff 0 (M, F ) as η 1 , it follows that S 1 intersects with every orbit of the action of Diff 0 (M, F ). Proof. Assume that we have for two harmonic basic 1-forms β 1 , β 2 and an element f of Diff 0 (M, F ). We will show β 1 = β 2 .
Then η − f * η is basic. It follows that f satisfies Here f * β 2 is a basic harmonic 1-form which is cohomologous to β 2 , because f is isotopic to the identity and preserves the transverse metric of F . Since each basic cohomology class is represented by a unique harmonic form by the Hodge decomposition theorem (see El Kacimi and Hector [13] orÁlvarez López [1]), we have We take an isotopy {φ s } s∈[0,1] such that φ 0 = id, φ 1 = f and φ s is an element of Ham 0 (M, F ) for every s in [0, 1]. We put X s = d dt t=s φ t . Since η − φ * s η is exact, L Xs η = ι Xs dη + d(η(X s )) is exact.  is (h s + η(X s ))ξ, which is tangent to F . Thus φ s and φ ′ s induces the same map on the leaf space of (M, F ). 1] gives an isotopy • φ s is an element of Diff 0 (M, F ). We put ψ s = (φ ′ s ) −1 • φ s . We put Y s = d dt t=s ψ t . Since ψ s is an element of Diff 0 (M, F ), this Y s is tangent to F for each s. Hence we have (93) ι Ys dη = 0.
Since the right hand side is a harmonic form, we have β 1 = β 2 .
Theorem 1.6 follows from Propositions 7.3 and 7.4. At last, we remark that we obtain the following result as a consequence of Proposition 7.3:  r(x 1 , y 1 , x 2 , y 2 , · · · , x n , y n ) = x 2 1 + y 2 1 + x 2 2 + y 2 2 + · · · + x 2 n + y 2 n where (x 1 , y 1 , x 2 , y 2 , · · · , x n , y n ) is the standard coordinate on R 2n . Let S 2n−1 be a unit sphere of R 2n defined by r = 1. Let g std = n i=1 (dx i ⊗ dx i + dy i ⊗ dy i ) be the standard metric on R 2n . Define a 1-form η std = 1 2r(x1,··· ,yn) 2 n i=1 (x i dy i − y i dx i ) on R 2n − {0}. Then (S 2n−1 , g std | S 2n−1 , η std | S 2n−1 ) is a Sasakian manifold. The flow generated by the Reeb vector field of η std | S 2n−1 is given by the principal S 1 -action whose orbits are tangent to the fiber of Hopf fibration. The base space of the Hopf fibration is CP n−1 .
Indeed, R 2n − {0} = S 2n−1 × R >0 is a metric cone of S 2n−1 where the coordinate on the second component is given by a function r. The Kähler metric on R 2n − {0} is given by g std and the standard Kähler form ω std = n i=1 dx i ∧ dy i . To describe the deformation of transversely holomorphic flows, we use the result of Girbau, Haefliger and Sundararaman (Proposition 6.1 in [16]). They showed that the Kuranishi space of the deformation of the transversely holomorphic flow defined by fibers of a circle bundle over a complex manifold X is identified with an open neighborhood of 0 in H 0 (X, T 1,0 X), the space of holomorphic vector fields on X, if X satisfies In this example, we can apply Corollary 1.4, because H 0,2 (CP n−1 ) = 0. Hence, the Sasakian metric (g std , η std ) is stable. By the simply connectedness of S 2n−1 , we can apply also Corollary 1.7. Thus, the space of isomorphism classes of Sasakian metrics are identified with the isomorphism classes of the underlying transversely Kähler flows.
For description of universal families of deformation of the Hopf fibration as a transversely holomorphic flow, see Haefliger [19] and Duchamp and Kalka [11].
We can apply Corollaries 1.4 and 1.7 to the circle bundles associated to positive line bundles over Fano manifolds X which satisfies the condition (99) in a similar way.

8.2.
Circle bundles over complex tori. We present an example of a family of transversely Kähler flows in which the stability of Sasakian metrics does not hold.
Let X be an projective complex torus with a positive holomorphic line bundle E. We fix a Hermitian metric on E so that its curvature form is positive. Let M be the unit circle bundle of E. Then M has a Sasakian metric whose underlying transversely Kähler flow is defined by the fibers of the circle bundle.
It is well known that there exists a smooth family of complex tori {X t } t∈]−1,1[ and a dense subset K in ] − 1, 1[ such that X 0 = X and X t is not projective for every t in K.
We denote the total space of the family of complex tori by Υ. We fix a trivialization φ : Υ ∼ = X×] − 1, 1[ as a smooth fiber bundle over ] − 1, 1[. We pull back the complex Hermitian line bundle E on X to Υ by φ• pr 1 where pr 1 : X×]− 1, 1[−→ X is the first projection. We define M t as the unit circle bundle associated to the complex line bundle (φ * pr * 1 E)| X t −→ X t . Let F t be a flow on M t defined by the fibers of the circle bundle M t −→ X t . Kodaira's stability theorem implies the existence of Kähler metrics on X t for t sufficiently close to 0. Since the leaf space X t is Kähler, F t is a transversely Kähler flow for t sufficiently close to 0.
There exists a compatible Sasakian metric on M 0 by definition. But, for t in K, M t does not have any compatible Sasakian metric. Indeed, if M t has a compatible Sasakian metric, then X t must be projective by the theorem of Hatakeyama [20]. This is contradiction.
In this example, the basic Euler class of F t is the Euler class of circle bundles M t −→ X t and can be considered as an element of H 2 (X t ; Z). Clearly this class is of topological nature and independent of t. On the other hand, the Hodge decomposition H 2 (X t ; C) ∼ = H 2,0 (X t )⊕H 1,1 (X t )⊕H 0,2 (X t ) changes when t varies.