Deformations of Transverse Calabi Yau Structures on Foliated Manifolds

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1 Publ. RIMS Kyoto Univ. 46 (2010), DOI /PRIMS/11 Deformations of Transverse Calabi Yau Structures on Foliate Manifols by Takayuki Moriyama Abstract We evelop a eformation theory of transverse structures given by calibrations on foliate manifols, incluing transverse Calabi Yau, hyperkähler, G 2 an Spin(7) structures. We show that the eformation space of the transverse structures is smooth uner a cohomological assumption. As an application, we obtain unobstructe eformations of transverse Calabi Yau structures on foliate manifols. We also prove a Moser type stability result for transverse structures, which implies Moser s stability theorem for presymplectic forms Mathematics Subject Classification: Primary 58H15; Seconary 53C12. Keywors: foliation, transverse Calabi Yau structure, eformation theory. 1. Introuction Koaira an Spencer worke out the eformation theory of compact complex manifols [10]. They showe that there exists a family of eformations of complex structures parameterize by a smooth finite-imensional space which is versal, uner a cohomological assumption. Kuranishi prove a general theorem on the existence of a versal eformation space for a given complex structure, where the versal eformation space (Kuranishi space) is given by an analytic space which is not necessarily smooth [11]. Duchamp Kalka [4] an Gomez-Mont [7] showe a weak version of Kuranishi s theorem for eformations of transversely holomorphic foliations on compact manifols. Girbau, Haefliger an Sunararaman [6] constructe the Kuranishi space of eformations of transversely holomorphic foliations on compact manifols. In this paper, we introuce a ifferent eformation theory of transverse geometric structures on foliate manifols. We consier transverse structures efine in terms of systems of close forms calle transverse calibrations (see Def- Communicate by H. Nakajima. Receive December 17, Revise September 1, T. Moriyama: RIMS, Kyoto University, Kyoto , Japan; moriyama@kurims.kyoto-u.ac.jp c 2010 Research Institute for Mathematical Sciences, Kyoto University. All rights reserve.

2 336 T. Moriyama inition 1.1). In the theory of Riemannian holonomy groups, it is useful to characterize geometric structures as systems of close forms. From this point of view, McLean [12] prouce the eformation theory of compact special Lagrangian submanifols. Goto [12] iscusse a eformation space of calibrations efine by close ifferential forms. y consiering Calabi Yau, hyperkähler, G 2 an Spin(7) structures as calibrations, he showe that these eformations are unobstructe. We fix a foliation on a manifol an eform the transverse calibrations on it. One of the avantages of our approach is that we can use Hoge theory on a foliate manifol [5]. (Recently, this transverse version of Hoge theory plays an important role in Sasakian geometry [2].) As a result, in a suitable case such as transverse Calabi Yau, hyperkähler, G 2 an Spin(7) structures, we obtain a smooth eformation space of the structures. We can also generalize Moser s theorem in symplectic geometry to transverse calibrations, an prove Moser s stability theorem for presymplectic forms. (We refer to [3] for another generalization of Moser s theorem given by Crainic an Fernanes in Poisson geometry.) Let M be a compact manifol of imension p + q an F a foliation on M of coimension q. The foliation F is efine by ata {U i, f i, T, γ ij } consisting of an open covering {U i } i of M, a q-imensional transverse manifol T, submersions f i : U i T an iffeomorphisms γ ij : f i (U i U j ) f j (U i U j ) for U i U j satisfying f j = γ ij f i. A transverse structure on (M, F) is a geometric structure on T which is invariant uner γ ij. For example, a transverse Kähler structure is efine by a Kähler structure on T preserve by γ ij. A foliation F is calle transverse Kähler if there exists a transverse Kähler structure on (M, F). On the compact manifol M with a transverse Kähler foliation F, if the basic canonical line bunle is trivial, then there exists a transverse Calabi Yau structure on (M, F) by applying the basic version of Yau s theorem [5]. Alternative efinitions for such transverse structures can be given in terms of basic sections of basic bunles over (M, F) (see Section 2). In particular, any transverse Calabi Yau structure is characterize by a pair of close basic forms (see Definition 6.3). We iscuss the eformation space of transverse Calabi Yau structures on (M, F) by introucing Goto s eformation theory of calibrations to the geometry of foliations. Our iea is to consier basic ifferential forms on (M, F) instea of ifferential forms on M. Let W be a q-imensional vector space an p W the space of skew-symmetric tensors of the ual space W. Then the group G = GL(W ) acts iagonally on the irect sum l pi W. Let Φ W = (φ 1,..., φ l ) be an element of l pi W an O (= A O (W )) the G-orbit through Φ W with an isotropy group H, so O is the homogeneous space G/H. On the foliate manifol (M, F), we have a completely integrable istribution F of imension p an the quotient bunle Q = T M/F over M. Let A O (M, F) be the fiber bunle x M A O(Q x )

3 Transverse Calabi Yau Structures 337 an E O the set Γ(M, A O (M, F)) l pi of sections of A O(M, F) efine to be basic forms, where p i enotes the space of basic p i-forms on M. Definition 1.1. A system Φ of ifferential forms on (M, F) is calle a transverse calibration associate with the orbit O if Φ is an element of E O whose components are close as ifferential forms. Let M O (M, F) be the set of transverse calibrations associate with O. We can ientify M O (M, F) with the intersection E O H where H is the Hilbert space consisting of all close forms on M. If we consier E O as a Hilbert manifol, then the infinitesimal tangent space of M O (M, F) at Φ is regare as the intersection T Φ E O H. We investigate whether the infinitesimal tangent space is ientifie with the tangent space of actual eformations. Definition 1.2. A transverse calibration Φ M O (M, F) is unobstructe if for any α T Φ E O H, there exists an integral curve Φ t in M O (M, F) such that t Φ t = α. t=0 For Φ M O (M, F), we efine in Section 3.2 a complex ( Φ ) an maps p k : H k ( Φ ) i Hpi+k 1 (M) for each k 0, where H k (M) is the basic e Rham cohomology group on (M, F). We also efine the ellipticity of an orbit O (Definition 3.1). Then we obtain the main theorem: Theorem 1.1. Let O be an elliptic orbit. If Φ M O (M, F) an the map p 2 : H 2 ( Φ ) i Hpi+1 (M) is injective, then Φ is unobstructe. We can regar a transverse Calabi Yau structure on (M, F) as a transverse calibration associate with the orbit O CY of Calabi Yau structures. Then we obtain Theorem 1.2. Any transverse Calabi Yau structure on (M, F) is unobstructe. In Section 2, we prepare some results in foliate geometry an provie efinitions of basic objects, for example, basic vector bunles, basic sections, etc. In Section 3, we introuce transverse calibrations on (M, F). Each transverse calibration inuces a eformation complex. Then we see that the eformation complex is a subcomplex of the basic e Rham complex. In Section 4, we provie a sufficient conition for the transverse structures to have unobstructe eformations (Theorem 4.1). We consier a generalization of Moser s stability theorem (Theorem 5.1) in Section 5. As an application, we obtain Moser s stability theorem for presym-

4 338 T. Moriyama plectic forms an prove that the eformations of transverse Calabi Yau structures on (M, F) are unobstructe in the last section. 2. Preparations on foliate geometry In this section, we provie some results in foliate geometry. For these results an the notation, we refer to [1], [14] an [13]. We assume that M is a compact manifol of imension p + q an F is a foliation on M of coimension q. We enote by F a completely integrable istribution of imension p associate to the foliation F. Let Q be the normal bunle T M/F an π : T M Q the natural projection asic vector fiels an basic forms A vector fiel u Γ(T M) is foliate if [u, v] Γ(F ) for any v Γ(F ). We enote by Γ(M, F) the set of foliate vector fiels on (M, F). Let X(M, F) be the quotient space of Γ(M, F) by Γ(F): X(M, F) = Γ(M, F)/Γ(F ) We call an element u of X(M, F) a basic vector fiel on (M, F). A ifferential k-form φ k on M is a basic form on (M, F) if the interior prouct i v φ an the Lie erivative L v φ vanish for any v Γ(F ). Let k be the set of basic k-forms on (M, F): k = {φ k i v φ = L v φ = 0, v Γ(F )}. For a section u Γ(Q) an a basic k-form φ k, the interior prouct i uφ an the Lie erivative L u φ are efine by the (k 1)-form iũφ an the k-form Lũφ for a lift ũ Γ(T M) of u, respectively. If u is a basic vector fiel, then i u φ an L u φ are basic forms. We efine a foliate iffeomorphism to be a iffeomorphism f of M preserving the foliation F, i.e., f (F ) = F. We enote by Diff(M, F) the group of foliate iffeomorphisms: Diff(M, F) = {f Diff(M) f (F ) = F }. We can efine an action of Diff(M, F) on the space of basic forms by pullback. For u X(M, F), any lift ũ Γ(T M) of u inuces a one-parameter family of foliate iffeomorphisms f t. Then the Lie erivative L u φ for φ k may be regare as the limit t f t φ t= asic bunles an basic sections Let ι : P M be a principal fiber bunle an ω a connection form on P. The horizontal subbunle H is efine by the subbunle Ker ω of the tangent bunle T P.

5 Transverse Calabi Yau Structures 339 Then the erivative ι restricte to H is the isomorphism from H to T M. Hence we have the subbunle F = ι 1 (F ) of H over P. If F is integrable, then F inuces the foliation F on P. Definition 2.1. A principal fiber bunle P is foliate if there exists a connection form ω on P such that the bunle F is integrable. Moreover, if the form ω is basic with respect to the inuce foliation F, then the bunle P is calle basic. A vector bunle π : E M is calle foliate (resp. basic) if the associate principal bunle P E is a foliate (resp. basic) bunle. In the case π : E M is a foliate vector bunle, the bunle P E M amits a foliation F on the total space P E by the efinition. This foliation F inuces a foliation F E on E. In aition, if E is basic then there exists a connection of E whose connection form is basic. Such a connection is calle a basic connection on E. Definition 2.2. Let E be a basic vector bunle with a basic connection. A section s Γ(E) is calle basic if v s = 0 for any v Γ(F ). We enote by Γ (E) the set of basic sections of E. Observe that for a basic bunle E, the ual bunle E an the exterior powers E are also basic bunles. We consier a hermitian metric h on E as a section of a basic bunle. Then we call E a basic hermitian bunle if the hermitian metric h is basic Riemannian foliations We efine an action of Γ(F ) on any section u Γ(Q) as follows: L v u = π[ũ, v] Γ(Q) for any vector fiel v Γ(F ) where ũ Γ(T M) is a lift of u, i.e., a vector fiel u Γ(T M) with π(ũ) = u. This action is inepenent of the choice of lifts ũ Γ(T M) of u. Let g be a Riemannian metric on M. Then we have an orthogonal ecomposition T M = F g F an the isomorphism σ : Q F. We enote by S 2 Q the symmetric 2-tensors of Q. Set a metric g Q = σ g F Γ(S 2 Q ) for the inuce metric g F on F. Then the map σ : (Q, g Q ) (F, g F ) is an isometry. Let M be the Levi-Civita connection with respect to g. Then we introuce a connection on Q as follows: (2.1) v u = { L v u, v Γ(F ), π( M v ũ), v Γ(F ), for u Γ(Q), where ũ Γ(T M) is a lift of u. In general, the connection (2.1) is not necessarily basic.

6 340 T. Moriyama A foliation F is Riemannian if the ata {U i, f i, T, γ i,j } are such that T is a Riemannian manifol an each γ i,j is an isometry. A Riemannian metric g is calle bunle-like if L v g Q = 0 for any v Γ(F ) where the tensor L v g Q Γ(S 2 Q ) is efine by (2.2) (L v g Q )(u, w) = v(g Q (u, w)) g Q (L v u, w) g Q (u, L v w) for u, w Γ(Q). It turns out that F is a Riemannian foliation if an only if there exists a bunle-like Riemannian metric g on M. For a bunle-like metric g, the connection in (2.1) is basic. Hence Q is a basic vector bunle for a Riemannian foliation F. It is easy to see that any basic section of k Q is a basic k-form on M: k = Γ ( k Q ). The space Γ (Q) is nothing but X(M, F): X(M, F) = Γ (Q). So we also call an element s of Γ (Q) a basic vector fiel. Moreover, a basic vector fiel s Γ (Q) is ientifie with a foliate vector fiel u s = σ(s) Γ(F ) by the isomorphism σ. Therefore we have the following ientification: (2.3) Γ (Q) {u Γ(F ) [u, v] Γ(F ), v Γ(F )}. From now, we consier any basic section of Q as a vector fiel on M uner the ientification of (2.3). Then a basic vector fiel u Γ (Q) inuces a one-parameter family of foliate iffeomorphisms f t since a vector fiel u Γ(F ) yiels a oneparameter family of iffeomorphisms Transversely elliptic operators Let E be a basic bunle of rank N. A linear map D : Γ (E) Γ (E) is calle a basic ifferential operator of orer l if, in local coorinates (x 1,..., x p, y 1,..., y q ) for which F is given by the equations y 1 = = y q = 0, D has the following expression: D = s a s (y), s1 y 1... sq y q s l where s = (s 1,..., s q ) N q an each a s is an N N-matrix value basic function. We efine the principal symbol of D at z = (x, y) an the basic covector ξ Q z as the linear map σ(d)(z, ξ) : E z E z given by σ(d)(z, ξ)(η) = ξ s ξsq q a s (y)(η) for any η E z. s =l

7 Transverse Calabi Yau Structures 341 Definition 2.3. A basic ifferential operator D is transversely elliptic if σ(d)(z, ξ) is an isomorphism for every z M an ξ( 0) Q z. We suppose that E is a hermitian basic bunle with a hermitian metric h an l = 2l. Then a quaratic form A(D)(z, ξ) : E z C is given by A(D)(z, ξ)(η) = ( 1) l σ(d)(z, ξ)(η), η. Definition 2.4. A basic ifferential operator D is strongly transversely elliptic if A(D)(z, ξ) is positive efinite for every z M an every non-zero ξ Q z. Let {(E k, D k )} k=0,1,...,q be a family of hermitian basic bunles an basic ifferential operators of orer 1 with the ifferential complex (2.4) Dk 1 Γ (E k ) D k Γ (E k+1 ) D k+1 where D k : Γ (E k ) Γ (E k ) for k = 0, 1,..., q. We enote by σ k the principal symbol σ(d k )(z, ξ) of D k. Then the complex (2.4) is transversely elliptic if the symbol sequence σk 1 Ez k σ k E k+1 z σ k+1 is exact for any z an any non-zero ξ. Note that the complex (2.4) is transversely elliptic if an only if the basic operator L k = Dk D k + D k 1 Dk 1 is strongly transversely elliptic, where Dk is the formal ajoint operator. We have the Hoge theory for the transversely elliptic complex (2.4) with the cohomology H k(e ): Proposition 2.1 ([1, Theorem 15]). (i) The kernel H k of L k is finite-imensional an we have an orthogonal ecomposition Γ (E k ) = H k Im(D k 1) Im(D k ). (ii) The orthogonal projection Γ (E k ) H k H k(e ) to H k. inuces an isomorphism from 2.5. Transverse Riemannian structures A Riemannian foliation is characterize by the following structure: Definition 2.5. A symmetric 2-tensor g Γ( 2 Q ) is a transverse Riemannian structure on (M, F) if g is positive efinite on Q an satisfies L v g = 0 for any v Γ(F ) where L v g is efine by (2.2). A bunle-like metric g inuces a transverse Riemannian structure g Q on (M, F). Conversely, for a transverse Riemannian structure g, we can take a bunlelike metric g such that g Q = g. Given a transverse Riemannian structure g Q on

8 342 T. Moriyama (M, F), the complexification Q C is a basic hermitian bunle, an hence so is k Q C. Thus from Proposition 2.1 we have Proposition 2.2 ([1, Theorem 16]). (i) The kernel H k of the basic Laplacian + on k is finite-imensional an we have an orthogonal ecomposition k = Hk Im() Im( ). (ii) The orthogonal projection k Hk inuces an isomorphism from the basic e Rham cohomology H k(m) to Hk Transverse Kähler structures We can efine an action of Γ(F ) on any section J Γ(En(Q)) as follows: (L v J)(u) = L v (J(u)) J(L v u) for v Γ(F ) an u Γ(Q). If J Γ(En(Q)) is a complex structure of Q, i.e. J 2 = i Q, an if L v J = 0 for any v Γ(F ), then a tensor N J Γ( 2 Q Q) can be efine by N J (u, w) = [Ju, Jw] Q [u, w] Q J[u, Jw] Q J[Ju, w] Q for u, w Γ(Q), where [u, w] Q enotes the bracket π[ũ, w] for any lifts ũ an w. Definition 2.6. A section J Γ(En(Q)) is a transverse complex structure on (M, F) if J is a complex structure of Q, i.e. J 2 = i Q, such that L v J = 0 for any v Γ(F ) an N J = 0. A foliation F is transversely holomorphic if an only if there exists a transverse complex structure on (M, F). Thus we may regar a transverse complex structure as a generalization of complex structures on complex manifols. A transverse complex structure J on (M, F) give rises to the ecomposition k C = r+s=k in the same manner as in complex geometry. We enote by H r,s (M) the (r, s)- basic Dolbeault cohomology group. We provie the following remark about the integrability conition of a transverse complex structure. r,s Remark. Let J be a complex structure of Q such that L v J = 0 for any v Γ(F ). Then J is a transverse complex structure, i.e. N J = 0, if an only if ( 1,0 ) 2,0 1,1, which is equivalent to ( 0,1 ) 1,1 0,2, where enotes the exterior erivative. Definition 2.7. A pair of sections ( g, J) Γ( 2 Q ) Γ(En(Q)) is a transverse Kähler structure on (M, F) if g is a transverse Riemannian structure an J is a

9 Transverse Calabi Yau Structures 343 transverse complex structure on (M, F) such that for u, w Γ(Q). g(, J ) is a -close form an g(ju, Jw) = g(u, w) A transversely Kähler foliation F is efine by ata {U i, f i, T, γ ij } with a Kähler manifol T an local iffeomorphisms γ ij preserving the Kähler structure. We remark that there exists a transverse Kähler structure on (M, F) if an only if F is a transverse Kähler foliation. Given a transverse Kähler structure ( g, J), the bunles k C an r,s are all basic hermitian bunles. Then each basic Dolbeault cohomology group H r,s (M) is finite-imensional from Proposition 2.1. Moreover, the basic e Rham Hoge ecomposition hols: Proposition 2.3 ([1, Theorem 17]). Let F be a transverse Kähler foliation on M. Then there exists an isomorphism H k(m, C) = r+s=k Hr,s (M). 3. Transverse calibrations 3.1. Orbits in vector spaces Let W be a vector space of imension q. We choose l natural numbers (p 1,..., p l ) N l. We enote by ρ the representation of G = GL(W) on the space l pi W where each p i W is the space of skew-symmetric tensors of egree p i of the ual space W. We fix an element Φ W = (φ 1,..., φ l ) l pi W an enote by H the isotropy group of Φ W : H = {g G ρ g Φ W = Φ W } Then the G-orbit O = {ρ g Φ W l pi W g G} through Φ W is regare as the homogeneous space G/H. We enote by A O (W ) the G-orbit O through Φ W. For an element Φ 0 A O (W ), the tangent space T Φ0 A O (W ) is given by E 1 Φ 0 (W ) = {ˆρ ξ Φ 0 l pi W ξ g}, where g is the Lie algebra of G an ˆρ is the ifferential representation of g. We also efine vector spaces E 0 Φ 0 (W ) an E k Φ 0 (W ) by E 0 Φ 0 (W ) = {i v Φ 0 = (i v φ 1,..., i v φ l ) l pi 1 W v W }, E k Φ 0 (W ) = {α i v Φ 0 l pi+k 1 W α k W, v W }

10 344 T. Moriyama for integers k 2. Then we have a complex ( Φ0 ) 0 E 0 Φ 0 (W ) u E 1 Φ 0 (W ) u E 2 Φ 0 (W ) u for a form u W. Definition 3.1. An orbit O is elliptic if the complex ( Φ0 ) is exact for any nonzero element u W at E 1 Φ 0 (W ) an E 2 Φ 0 (W ). We give some examples of elliptic orbits. Now we assume that W is evenimensional, that is, q = 2n. Example 1. The set of all symplectic forms on W is an orbit O symp, which is isomorphic to the quotient space GL(2n, R)/Sp(2n, R). For any Φ 0 O symp, the complex ( Φ0 ) is 0 1 W u 2 W u 3 W for any element u W. Thus the orbit O symp is elliptic. u Example 2. A non-zero complex n-form Ω n C is calle an SL n (C) structure on W if the form Ω satisfies W C = Ker Ω Ker Ω, where Ker Ω enotes the space {v W C i v Ω = 0}. We remark that an SL n (C) structure Ω inuces a complex structure J Ω on W efine by { 1v for v Ker Ω, (3.1) J Ω (v) = 1v for v Ker Ω. Then Ω is an (n, 0)-form with respect to the complex structure J Ω. Let O SL be the set of SL n (C) structures on W. Then it turns out that O SL is an orbit such that For any Φ 0 O SL, the complex ( Φ0 ) is O SL = GL(2n, R)/SL(n, C). 0 n 1,0 W u n,0 W n 1,1 W u n,1 W n 1,2 W u for any u W. Here we regar the element u as an element of 1,0 W 0,1 W such that ū = u. So this orbit O SL is elliptic.

11 Transverse Calabi Yau Structures 345 Example 3. A pair (Ω, ω) n C 2 is calle a Calabi Yau structure on W if Ω is an SL n (C) structure an ω is a symplectic structure on W such that Ω ω = 0, Ω Ω = c n ω n 0, ω(, J Ω ) is positive efinite, where c n = 1 n! ( 1)n(n 1)/2 (2/ 1) n. Let O CY be the set of Calabi Yau structures on W. Then O CY is an elliptic orbit such that (cf. [8, Proposition 4.9]). O CY = GL(2n, R)/SU(n) 3.2. Transverse calibrations in foliate manifols Let M be a compact manifol of imension p + q an F a Riemannian foliation on M of coimension q. We consier the completely integrable istribution F associate to F an the quotient bunle Q = T M/F over M. For each x M, we ientify Q x with W = R q. Then, as in Section 3.1, we have an orbit A O (Q x ) = A O (W ) at x M for an orbit O. Note that the orbit A O (Q x ) pi i Q x oes not epen on the choice of the ientification h : Q x W. Then we can efine a G/H-bunle A O (M, F) by A O (M, F) = A O (Q x ) M. x M Since A O (M, F) i pi Q, we can consier the Lie erivative an the exterior erivative for any section of A O (M, F) as a ifferential form. We enote by E O (M, F) the space of sections of A O (M, F) which are basic forms: E O (M, F) = Γ(A O (M, F)). Let Ker Φ be the space {v T M i v Φ = 0} for Φ E O (M, F). Definition 3.2. A section Φ E O (M, F) is calle a transverse calibration associate with the orbit O if Φ is a close form such that Ker Φ = F. We enote by M O (M, F) the space of transverse calibrations associate with O. The group Diff(M, F) acts on M O (M, F) by pull-back. Given a transverse calibration Φ M O (M, F), we can consier the vector spaces EΦ k x (Q x ) at each point x M, an efine vector bunles EΦ(M, k F) = EΦ k x (Q x ) M x M

12 346 T. Moriyama for integers k 0. Each bunle EΦ k (M, F) is a basic bunle since its associate principal bunle is that of the normal bunle Q. It follows that a section i v Φ Γ(EΦ 0 (M, F)) is basic if an only if v Γ(Q) is a basic section from Ker Φ = F an L w (i v Φ) = i LwvΦ for any w Γ(F ). Hence we have Γ (E 0 Φ(M, F)) = {i v Φ l pi 1 Q v Γ (Q)}, Γ (E 1 Φ(M, F)) = {ˆρ ξ Φ l pi Q ξ Γ (En(Q))}. We introuce the grae vector spaces E Φ (M, F) = k Ek Φ (M, F). For simplicity, we shall enote by Γ (E k ) an Γ (E) the spaces Γ (EΦ k (M, F)) an Γ (E Φ (M, F)), respectively. Proposition 3.1. The moule Γ (E) is a ifferential grae moule in k ( pi+k 1 i ) with respect to the erivative, where is the exterior erivative restricte to the space of basic forms. Proof. We prove that a Γ (E k ) for all a Γ (E k 1 ). To show this, it is sufficient to prove that i v Φ Γ (E 1 ) for any element i v Φ Γ (E 0 ), since Γ (E) is generate by Γ (E 0 ). The basic vector fiel v inuces a one-parameter transformation {f t } such that each f t is an element of Diff(M, F). Then it follows from Φ = 0 that i v Φ = L v Φ = t f t Φ. t=0 Since f t Φ is in E O (M, F), t f t Φ t=0 lies in the tangent space of E O (M, F) at Φ, which is the space Γ (E 1 ). This implies that i v Φ = t f t Φ t=0 Γ (E 1 ). Thus we obtain a complex ( Φ ) 0 Γ (E 0 ) 0 Γ (E 1 ) 1 Γ (E 2 ) 2 where i = E i Rham complex: for each i. The complex ( Φ ) is a subcomplex of the basic e (3.2) 0 0 Γ (E 0 ) Γ (E 1 ) 1 Γ (E 2 ) i pi 1 i pi i pi+1 2 We enote by H k ( Φ ) the cohomology groups of the complex ( Φ ): H k ( Φ ) = {α Γ (E k ) k α = 0}/{ k 1 β Γ (E k ) β Γ (E k 1 )}. Then, for each k 0, we efine p k as the map in cohomology inuce by the chain map in (3.2): p k : H k ( Φ ) i Hpi+k 1 (M).

13 Transverse Calabi Yau Structures 347 If O is an elliptic orbit, then the complex ( Φ ) is transverse elliptic at Γ (E 1 ) an Γ (E 2 ). Hence the cohomology groups H k ( Φ ) for k = 1, 2 are finite-imensional from Proposition Deformations of transverse calibrations We suppose that F is a Riemannian foliation on a compact manifol M an M O (M, F) is the set of transverse calibrations associate with an orbit O on (M, F). Let H be the space of all close forms on M an let E be the space of sections Φ of E O (M, F) satisfying Ker Φ = F. The space M O (M, F) is the intersection H E. Hence we can regar the infinitesimal tangent space of M O (M, F) as H T Φ E (= H Γ (E 1 )). We will investigate whether the infinitesimal tangent space agrees with the tangent space of actual eformations. Definition 4.1. A transverse calibration Φ M O (M, F) is unobstructe if for any α H Γ (E 1 ) there exists a smooth curve Φ t in M O (M, F) such that t Φ t = α. t=0 We provie the following criterion for the unobstructeness of transverse calibrations: Theorem 4.1. Let O be an elliptic orbit. If p 2 : H 2 ( Φ ) i Hpi+1 (M) is injective for Φ M O (M, F), then Φ is unobstructe. Proof. We apply the argument of Section 2 in [8] to basic ifferential forms. Define a linear operator A 1 a : for a Γ (En(Q)) as the commutator [, ˆρ a ] = ˆρ a ˆρ a : A 1 a = [, ˆρ a ]. Inuctively, we efine linear operators A l a : for l 2 as A l a = [A l 1 a, ˆρ a ] for a Γ (En(Q)). Let (e Aa ) be the formal power series efine by (e Aa ) = l=0 1 l! (Al a). The action ρ e a of the exponential e a Γ (GL(Q)) is written as (4.1) ρ e a = 1 + ˆρ a + 1 2! ˆρ a ˆρ a + 1 3! ˆρ a ˆρ a ˆρ a +.

14 348 T. Moriyama Then (4.2) ρ e aρ e a = (e Aa ) for any section a Γ (En(Q)). We start to show the theorem. Suppose that α is in H Γ (E 1 ). Then there exists an element a 1 Γ (En(Q)) such that α = ˆρ a1 Φ by the efinition of Γ (E 1 ). Now we consier formal power series a(t) in t: a(t) = a 1 t + 1 2! a 2t ! a 3t 3 + Γ (En(Q))[[t]] where each a k is a basic section of En(Q). We efine an element Φ t of E as Φ t = ρ e a(t)φ for e a(t) Γ (GL(Q))[[t]]. Then Φ t satisfies t Φ t t=0 = α. Hence it is sufficient to show that there exists an element a(t) such that Φ t is a close form, that is, (4.3) ρ e a(t)φ = 0. Equation (4.3) is equivalent to (4.4) (e A a(t) )Φ = 0 by (4.2). Now we efine ((e A a(t) )Φ)[k] to be the k-th homogeneous part of (e A a(t) )Φ with respect to t. Then (4.4) reuces to the system of infinitely many equations (4.5) ((e A a(t) )Φ) [k] = 0. We etermine the coefficients a k by inuction on k, in orer to obtain a solution a(t). First, the element a 1 satisfies ˆρ a1 Φ = 0 since ˆρ a1 Φ = α H. It follows from the conition Φ = 0 that ((e A a(t) )Φ)[1] = Φ+[, ˆρ a1 ]Φ = 0. Next, we assume that a 1,..., a k 1 satisfy ((e A a(t) )Φ)[j] = 0 for j k 1. We efine a form Ob(a 1,..., a k 1 ) by ( k ) 1 Ob(a 1,..., a k 1 ) = l! (Al a(t))φ. [k] For simplicity, we enote by Ob k the obstruction form Ob(a 1,..., a k 1 ). Then Ob k is a section of Γ (E 2 ) an satisfies l=2 ((e A a(t) )Φ) [k] = Ob k + 1 k! ˆρ a k Φ.

15 Transverse Calabi Yau Structures 349 From the inuction hypothesis, we have (ρ e a(t)φ) [k] = ((e A a(t) )Φ)[k]. Thus Ob k is -exact an 2 -close. We can consier the cohomology class [Ob k ] in H 2 ( Φ ) an call it the k-th obstruction class. In the case of ((e A a(t) )Φ)[k] = 0, one has [Ob k ] = [ 1 k! ˆρ a k Φ] = 0 in H 2 ( Φ ). Conversely, if the class [Ob k ] is zero, then a k may be foun so that a(t) satisfies the equation ((e A a(t) )Φ)[k] = 0. In fact, by applying basic Hoge theory to the basic elliptic complex ( Φ ), we can take an element a k such that (4.6) 1 k! ˆρ a k Φ = 1G (Ob k ) where G an 1 are the Green s operator an the ajoint operator of 1, respectively. If the map p 2 : H 2 ( Φ ) i Hpi+1 (M) is injective, then the obstruction class [Ob k ] H 2 ( Φ ) vanishes for all k since Ob k is -exact. y the above argument, the elements a k efine by (4.6) satisfy (4.5) for all k 1. Hence we obtain a solution a(t) of (4.3) as a formal power series. Moreover, this formal power series converges uniformly in the Sobolev norm s. We iscuss the regularity of the solution a(t) constructe above. We recall that the element a 1 satisfies ˆρ a1 Φ = 0 as an initial conition an the coefficients a k for k 2 are efine by (4.6). Hence the power series a(t) satisfies 0 ˆρ a(t) Φ = 0 ˆρ a1 Φ since 0 ˆρ ak Φ = 0 for k 2. Moreover, we have 1 ˆρ a(t) Φ + l ˆρl a(t) Φ = 0 l=2 from ρ e a(t)φ = 0 an (4.1). Thus ˆρ a(t) Φ is weak solution of the basic elliptic ifferential equation ( ) ˆρ a(t) Φ l ˆρl a(t) Φ = 0 0 ˆρ a1 Φ l=2 where is the basic Laplace operator Elliptic regularity implies that the solution ˆρ a(t) Φ is smooth for a smooth element ˆρ a1 Φ. Hence the form ρ e a(t)φ is also smooth. 5. A generalization of Moser s stability theorem Let M be a compact manifol of imension p + q an F a Riemannian foliation on M of coimension q. Let M O (M, F) be the set of transverse calibrations associate with an orbit O on (M, F).

16 350 T. Moriyama We fix a transverse calibration Φ M O (M, F). Let EΦ k enote the vector bunle EΦ k (M, F) efine in Section 3.2. We efine a vector bunle Ek, Φ to be the orthogonal complement of EΦ k in pi+k 1 i Q with respect to the metric inuce by the transverse Riemannian structure on (M, F). We can ientify Γ (E k, Φ ) with the quotient pi+k 1 i /Γ (EΦ k ) since each vector bunle Ek, Φ is isomorphic to the quotient bunle pi+k 1 i Q /EΦ k. Then the basic exterior erivative on pi+k 1 i inuces a ifferential operator : Γ (E k, Φ ) Γ (E k+1, Φ ), an we obtain the ifferential complex ( Φ ) 0 Γ (E 0, Φ ) Γ (E 1, Φ ) Γ (E 2, Φ ) For simplicity, we shall enote by E k, the bunle E k, Φ. The complex ( Φ ) is relate to the basic e Rham complex as follows: (5.1) i pi 1 i pi i pi+1 0 Γ (E 0, ) Γ (E 1, ) Γ (E 2, ) where the vertical arrows are epimorphisms. We enote by H k ( Φ ) the cohomology groups of the complex ( Φ ): H k ( Φ) = {α Γ (E k, ) α = 0}/{ β Γ (E k, ) β Γ (E k 1, )}. Then, for each k 0, we efine p k, as the map in cohomology inuce by the chain map in (5.1): p k, : i Hpi+k 1 (M) H k ( Φ ). We see that there is a relation between the complex ( Φ ) an the complex ( Φ) efine in Section 3.2. y the construction of E k, we obtain the commutative iagram Γ (E 0 ) 0 Γ (E 1 ) 1 Γ (E 2 ) 2 i pi 1 i pi i pi+1 0 Γ (E 0, ) Γ (E 1, ) Γ (E 2, ) where the columns are exact. y the stanar construction of a long exact sequence in cohomology from a short exact sequence of chain complexes, we can easily show

17 Transverse Calabi Yau Structures 351 Lemma 5.1. If the complex ( Φ ) is transversely elliptic at Γ (E k ), then the complex ( Φ ) is also transversely elliptic at Γ (E k 1, ). Next, we provie a sufficient conition for the map p 1, to be surjective. Lemma 5.2. If the map p 2 : H 2 ( Φ ) i Hpi+1 (M) is injective, then the map p 1, : i Hpi (M) H1 ( Φ ) is surjective. Proof. We efine an injection i an a surjection π as follows: 0 Γ (E k ) i pi+k 1 π i Γ (E k, ) 0 for each k 0. Now we consier the iagram Γ (E 1 ) 1 Γ (E 2 ) 2 Γ (E 3 ) i pi i pi+1 Γ (E 1, ) Γ (E 2, ) 0 0 i pi+2 Suppose [φ ] H 1 ( Φ ) where φ is an element of Γ (E 1, ) with φ = 0. Then we can choose an element φ pi i such that π(φ ) = φ. This φ satisfies π( φ ) = π(φ ) = 0. From the exactness of Γ (E 2 ) pi+1 i Γ (E 2, ), there exists an element ψ Γ (E 2 ) with i(ψ) = φ. Then ψ is 2 -close, an the class [ψ] H 2 ( Φ ) is in the kernel of p 2. So we can take an element φ Γ (E 1 ) with 1 φ = ψ since p 2 is injective. Now we efine φ by φ i(φ ) pi+1 i. Then it is easy to see that φ = 0 an π(φ) = φ. Thus we obtain the class [φ] i Hpi (M) satisfying p1, ([φ]) = [φ ], an this finishes the proof. We now show that the imension of H 1 ( Φ ) is invariant uner eformations of Φ M O (M, F): Proposition 5.1. Let O be an elliptic orbit an {Φ t } t [0,1] a smooth family of M O (M, F). If the maps p 1 an p 2 are injective for each t [0, 1], then im H 1 ( Φt ) is inepenent of t [0, 1]. Proof. The complex ( Φt ) is transversely elliptic at EΦ 1 t by the efinition of the elliptic orbit O. The complex ( Φ t ) is also transversely elliptic at E 1, Φ t by Lemma 5.1.

18 352 T. Moriyama Therefore, each of the cohomology groups H 1 ( Φt ) an H 1 ( Φ t ) is the kernel of a transversely elliptic operator. Hence the imensions im H 1 ( Φt ) an im H 1 ( Φ t ) are upper semicontinuous in t. We have the exact sequence 0 H 1 ( Φt ) i Hpi (M) H1 ( Φ t ) 0 for each t [0, 1] by using Lemma 5.2 an the assumptions on p 1 an p 2. This sequence yiels (5.2) im H 1 ( Φt ) = i im H pi (M) im H1 ( Φ t ) for each t [0, 1]. From (5.2) an the upper semicontinuity of im H 1 ( Φ t ) it follows that im H 1 ( Φt ) is lower semicontinuous. On the other han, it is upper semicontinuous. Therefore it is inepenent of t in a small interval, an hence in the whole [0, 1] by compactness. This completes the proof. Let Diff 0 (M, F) enote the ientity component of Diff(M, F). We provie a generalization of Moser s stability theorem: Theorem 5.1. Let O be an elliptic orbit an {Φ t } t [0,1] a smooth family of M O (M, F) with the same basic cohomology class [Φ t ] = [Φ 0 ] i Hpi (M). If the maps p 1 an p 2 are injective for each t [0, 1], then there exists a smooth family {f t } t [0,1] of Diff 0 (M, F) such that f 0 = i M an ft Φ t = Φ 0. Proof. y assumption we may choose an element A t pi i with smooth epenence on t such that Φ t Φ 0 = A t for each t [0, 1]. It suffices to fin a smooth family f t Diff 0 (M, F) such that (5.3) t (f t Φ t ) = 0. Note that (ft 1 ) f t Φ t is the infinitesimal transformation L vt Φ t for some basic vector fiel v t. So (5.3) is equivalent to L vt Φ t + A t = 0. Hence the proof reuces to fining a smooth family of basic vector fiels v t Γ (Q) with L vt Φ t = A t. Now the class [A t ] of H 1 ( Φt ) vanishes since p 1 : H 1 ( Φt ) i Hpi (M) is injective. Thus we can efine an element t of Γ (EΦ 0 t ) as t = 0G Φt A t

19 Transverse Calabi Yau Structures 353 where 0 an G Φt enote the ajoint operator of 0 an Green s operator at Γ (EΦ 1 t ) of the complex ( Φt ), respectively. Note that im H 1 ( Φt ) is inepenent of t from Proposition 5.1, so the families A t = Φ t an G Φt are smooth in t [0, 1]. Hence there exists a smooth family v t Γ (Q) such that t = i vt Φ t by the efinition of Γ (EΦ 0 t ). Then L vt Φ t = t = A t. We thus obtain a smooth family {f t } t [0,1] in Diff 0 (M, F) satisfying (5.3), which completes the proof. 6. Applications We assume that M is a compact manifol with a Riemannian foliation F of coimension 2n. Let F enote the integrable istribution inuce by F Transverse symplectic structures A transverse symplectic foliation F on M is characterize by the following structure. Definition 6.1. A real 2-form ω 2 is calle a transverse symplectic structure on (M, F) if ω is a basic form on (M, F) such that ω = 0 an ω n 0. Let M symp (M, F) be the set of transverse symplectic structures on (M, F). A transverse symplectic structure ω is a transverse calibration associate with the orbit O symp. Hence we can regar M symp (M, F) as the set of transverse calibrations associate with O symp. Lemma 6.1. For any ω M symp (M, F), the maps p k : H k ( ω ) H k+1 (M) are injective for k 1. Proof. For a transverse symplectic structure ω M symp (M, F), it is known that Γ (Eω) k = k+1 for k 0. Therefore we obtain the cohomology group H k ( ω ) = H k+1 (M) for each k 1, so each pk : H k ( ω ) H k+1 (M) is an isomorphism. Observe that a transverse symplectic structure ω on (M, F) is a presymplectic form on M such that Ker ω = F. y applying Theorem 5.1 to the elliptic orbit O symp, we obtain Moser s stability theorem for presymplectic forms: Theorem 6.1. Let {ω t } t [0,1] be a smooth family of presymplectic forms on M with Ker ω t = F an the same basic cohomology class [ω t ] = [ω 0 ] H 2 (M) for all t [0, 1]. Then there exists a smooth family {f t } t [0,1] in Diff 0 (M, F) such that f 0 = i M an ft ω t = ω 0.

20 354 T. Moriyama 6.2. Transverse SL n (C) structures Definition 6.2. A nowhere vanishing complex n-form Ω n C is calle a transverse SL n (C) structure on (M, F) if Ω is a basic form such that Ω = 0 an where Ker Ω = {v T M C i v Ω = 0}. Q C = Ker Ω/F Ker Ω/F A transverse SL n (C) structure Ω inuces a complex structure J Ω of Q such that Ω is an (n, 0)-basic form on (M, F) (see Example 2). Then we can check that θ 2,0 1,1 for any θ 1,0 because (θ) Ω = 0. It follows from the Remark before Definition 2.7 that J Ω is a transverse complex structure on (M, F). Hence (F, J Ω ) is a transverse holomorphic foliation on M. Let M SL (M, F) be the space of transverse SL n (C) structures on (M, F). Any element Ω M SL (M, F) inuces a transverse calibration associate with the orbit O SL, an conversely. Thus, we can ientify M SL (M, F) with the set M OSL (M, F) of transverse calibrations associate with the orbit O SL. We recall that the orbit O SL is elliptic. For Ω M SL (M, F) the complex ( Ω ) is 0 n 1,0 0 n,0 n 1,1 1 n,1 n 1,2 2. Unfortunately, the maps p 1 an p 2 are not always injective for Ω M SL (M, F). However, we obtain Proposition 6.1. If (F, J Ω ) is a transverse Kähler foliation, then Ω is unobstructe. Proof. We suppose that Ω M SL (M, F) is such that (F, J Ω ) is a transverse Kähler foliation on M. y moifying the argument of Proposition 4.4 in [8], we obtain H 1 ( Ω ) = H n,0 (M) Hn 1,1 (M), H 2 ( Ω ) = H n,1 (M) Hn 1,2 (M). The maps p 1 an p 2 are injective by Proposition 2.3. Applying Theorem 4.1 to the elliptic orbit O SL completes the proof Transverse Calabi Yau structures Definition 6.3. A pair (Ω, ω) n C 2 is calle a transverse Calabi Yau structure on (M, F) if Ω is a transverse SL n (C) structure an ω is a transverse symplectic structure on (M, F) such that Ω ω = 0, Ω Ω = c n ω n 0, ω(, J Ω ) is positive efinite on Q, where c n = 1 n! ( 1)n(n 1)/2 (2/ 1) n.

21 Transverse Calabi Yau Structures 355 We enote by M CY (M, F) the space of transverse Calabi Yau structures on (M, F). Any element Φ = (Ω, ω) of M CY (M, F) is a transverse calibration associate with the elliptic orbit O CY. Proposition 6.2. The maps p 1 an p 2 are injective for any Φ M CY (M, F). Proof. Given Φ M CY (M, F), by aapting the computation of cohomology groups in [8, Theorem 4.8] to basic forms we obtain H 1 ( Φ ) = H n,0 (M) Hn 1,1 (M) P 1,1,R, H 2 ( Φ ) = H n,1 (M) Hn 1,2 (M) (H 2,1 (M) H1,2 (M)) R where (H 2,1 (M) H1,2 (M)) R an P 1,1,R enote the real part of H2,1 (M) H1,2 (M) an the space of real harmonic an primitive basic (1, 1)-forms, respectively. Hence the maps p 1 : H 1 ( Φ ) H n (M, C) H 2 (M), p 2 : H 2 ( Φ ) H n+1 (M, C) H3 (M) are injective from Proposition 2.3 an the Lefschetz ecomposition of basic ifferential forms (cf. [5, Section 3.4.7]). We obtain the following results: Theorem 6.2. Any element of M CY (M, F) is unobstructe. Proof. This follows immeiately from Theorem 4.1 an Proposition 6.2. Theorem 6.3. Let {(Ω t, ω t )} t [0,1] be a smooth family of transverse Calabi Yau structures on (M, F) with the same basic cohomology class ([Ω t ], [ω t ]) = ([Ω 0 ], [ω 0 ]) H n(m, C) H2 (M) for any t [0, 1]. Then there exists a smooth family {f t } t [0,1] in Diff 0 (M, F) such that f 0 = i M an (ft Ω t, ft ω t ) = (Ω 0, ω 0 ). Proof. Apply Theorem 5.1 an Proposition 6.2. Remark. We can also obtain Moser s stability theorems an unobstructe eformations of transverse hyperkähler, G 2 an Spin(7) structures by applying Theorems 4.1 an 5.1 to these structures (cf. Sections 5, 6 an 7 in [8]) Linear foliations on tori Let T 2n+1 be the real torus R 2n+1 /Z 2n+1 of imension 2n + 1. We take a local coorinate (x 1,..., x n, y 1,..., y n, t) on T 2n+1. Then a foliation F (λ,µ) is inuce by the vector fiel n ξ = λ i + µ i x i y i t

22 356 T. Moriyama for (λ, µ) = (λ 1,..., λ n, µ 1,..., µ n ) R 2n. The foliation F (λ,µ) is calle a linear foliation on T 2n+1. Note that F (λ,µ) is a Riemannian foliation with respect to the stanar flat metric on T 2n+1. We efine z i to be the complex function z i = x i + λ i t + 1(y i + µ i t) for i = 1,..., n. Then (z 1,..., z n ) becomes a transverse coorinate on (T 2n+1, F (λ,µ) ). Now we efine a pair (Ω, ω) of forms as Ω = z 1 z n, ω = 1 2 n z i z i. Then it is easy to check that (Ω, ω) is a transverse Calabi Yau structure on (T 2n+1, F (λ,µ) ). In particular, the n-form Ω is a transverse SL n (C) structure so that (F (λ,µ), J Ω ) is a transverse Kähler foliation. Hence, it follows from Proposition 6.1 that Ω M SL (T 2n+1, F (λ,µ) ) is unobstructe an H 1 ( Ω ) equals H n,0 (T 2n+1 ) H n 1,1 (T 2n+1 ). The vector space H p,q (T 2n+1 ) is generate by the classes of the wege proucts z i1 z ip z j1 z jq, an thus we obtain ( )( ) im C H p,q n n (T 2n+1 ) =. p q The first cohomology group H 1 ( Ω ) can be regare as a parameter space of eformations of the transverse SL n (C) structure Ω, an has the imension im C H 1 ( Ω ) = im C (H n,0 (T 2n+1 ) H n 1,1 (T 2n+1 )) = 1 + n 2. Next, we consier the eformations of transverse Calabi Yau structures on (T 2n+1, F (λ,µ) ). y Theorem 6.2, any element Φ M CY (T 2n+1, F (λ,µ) ) is unobstructe an H 1 ( Φ ) is equal to H n,0 (T 2n+1 ) H n 1,1 (T 2n+1 ) P 1,1,R. It follows that im R P 1,1,R = im C P 1,1 = n2 1 from the basic Lefschetz ecomposition H 1,1 (T 2n+1 ) = P 1,1 + C ω. Hence, any element Φ M CY (T 2n+1, F (λ,µ) ) essentially has a eformation space parameterize by H 1 ( Φ ) whose imension is im R H 1 ( Φ ) = 2(1 + n 2 ) + n 2 1 = 3n Acknowlegements The author is very grateful to Professor R. Goto for his avice. He also thanks the referee for his careful reaing of the paper an useful comments.

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