THE MODULI SPACE OF TRANSVERSE CALABI YAU STRUCTURES ON FOLIATED MANIFOLDS

Transcription

1 Moriyama, T. Osaka J. Math. 48 (211), THE MODULI SPACE OF TRANSVERSE CALAI YAU STRUCTURES ON FOLIATED MANIFOLDS TAKAYUKI MORIYAMA (Received August 21, 29, revised December 4, 29) Abstract In this paper, we develop a moduli theory of transverse structures given by calibrations on foliated manifolds, including transverse Calabi Yau structures. We show that the moduli space of the transverse structures is a smooth manifold of finite dimension under a cohomological assumption. We also prove a local Torelli type theorem. If the foliation is taut, we can construct a Riemannian metric on the set of transverse Riemannian structures. This metric induces a distance on the moduli space of the transverse structures given by a calibration. As an application, we show the moduli space of transverse Calabi Yau structures is a Hausdorff and smooth manifold of finite dimension. 1. Introduction Kodaira and Spencer introduced the deformation theory of compact complex manifolds [12]. They showed that there exists a deformation of complex structures parameterized by a smooth finite dimensional space which is versal, under a cohomological assumption. Kuranishi proved a general theorem on the existence of a versal deformation space for any given complex structure, where the versal deformation space (Kuranishi space) is given by an analytic space which is not necessarily smooth [13]. ogomolov, Tian and Todorov proved that the Kuranishi space of Calabi Yau structures is smooth by using the Kodaira Spencer Kuranishi theory [1], [14] and [16]. Goto provided a deformation theory of Calabi Yau, hyperkähler, G 2 and Spin(7) structures by a method which is different from the deformation theory of complex manifolds [1]. He considered these structures as systems of closed differential forms (called calibrations), and showed that deformation spaces are smooth and moduli spaces become smooth manifolds under a cohomological condition. In the geometry of holomorphic foliations, the theory of deformations was initiated by Kodaira and Spencer. Duchamp Kalka [4] and Gómez-Mont [9] showed a weak version of Kuranishi s theorem for deformations of transversely holomorphic foliations on compact manifolds. Girbau, Haefliger and Sundararaman constructed the Kuranishi space of deformations of transversely holomorphic foliations on compact manifolds [8]. In a 21 Mathematics Subject Classification. Primary 53C12; Secondary 35C38.

2 384 T. MORIYAMA previous paper [18], we provided a deformation theory of transverse geometric structures other than transversely holomorphic structures. We considered transverse geometric structures defined in terms of closed forms and called such closed forms transverse calibrations. The transverse calibrations include transverse Calabi Yau, hyperkähler, G 2 and Spin(7) structures as examples. y modifying Goto s deformation theory, we obtained the deformation theory of transverse calibrations. We fixed a foliation on a manifold and deformed the transverse calibrations on it. One of the advantage of our approach was that we could use the Hodge theory on a foliated manifold [6]. As a result, we obtained a generalization of Moser s theorem and a smooth deformation space of transverse calibrations. El Kacimi-Alaoui, Guasp and Nicolau give a deformation theory of transversely homogeneous foliations defined by systems of 1-forms, which are not transverse calibrations [7]. In this paper, we discuss the moduli space of transverse calibrations and provide a criterion for the moduli space to be a Hausdorff and smooth manifold of finite dimension. If the foliation is taut, then we can construct a Riemannian metric on the set of transverse Riemannian structures. This result is a generalization of Ebin s results in Riemannian geometry to effect that there exists a Riemannian metric on the set of Riemannian structures on a closed manifold [5]. The metric on the set of transverse Riemannian structures induces a distance on the moduli space of transverse calibrations. As a result, the moduli space becomes Hausdorff. Let M be a closed manifold of dimension (p q) and F a foliation on M of codimension q. The foliation F is defined by data U i, f i, T, i j consisting of an open covering U i i of M, a q-dimensional transverse manifold T, submersions f i Ï U i T and diffeomorphisms i j Ï f i (U i U j ) f j (U i U j ) for U i U j satisfying f j i j Æ f i. A transverse structure on (M, F) is a geometric structure on T which is invariant by i j. For example, a transverse Kähler structure is defined by a Kähler structure on T preserved by i j. A foliation F is called transverse Kähler if there exists a transverse Kähler structure on (M, F). On a closed manifold M with a transverse Kähler foliation F, if the basic canonical line bundle is trivial, then there exists a transverse Calabi Yau structure on (M, F) by applying the basic version of Yau s theorem [6]. Remark that we can give alternative definitions for such transverse structures in terms of basic sections of basic bundles over (M, F) (see Section 2). In particular, any transverse Calabi Yau structure is characterized by a pair of two closed basic forms (see Definition 6.3). We apply Goto s method to transverse structures on a foliated manifold (M, F). Our idea is to consider basic differential forms on (M,F) instead of differential forms on M. Let W be a q-dimensional vector space and Îp W the space of skew-symmetric tensor of the dual space W. Then the group G GL(W ) acts on diagonally the direct sum Ä l pi i1î Ä W l. Let W ( 1,, l ) be an element of pi i1î W and O (A O (W )) the G-orbit through W with an isotropy group H, so O is the homogeneous space GH. On the foliated manifold (M, F), we have a completely integrable distribution

3 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 385 F of dimension Ë p and the quotient bundle Q T MF over M. Let A O (M, F) be a fiber bundle x¾m A O(Q x ) and E O the set ¼(M, Î Ä l Î A O (M, F)) pi i1 of sections p of A O (M, F) which are basic forms, where i denotes the space of basic p i-forms on M. DEFINITION 1.1. A system of differential forms on (M, F) is called a transverse calibration associated with the orbit O if is an element of E O whose components are closed as differential forms. Let ÉM O (M,F) be the set of transverse calibrations associated with O. We denote by Diff(M, F) the group of diffeomorphisms preserving the foliation F. We define M O (M, F) to be the quotient of ÉM O (M, F) divided by the action of Diff (M, F): M O (M, F) ÉM O (M, F)Diff (M, F) where Diff (M,F) denotes the identity component of Diff(M,F). The set M O (M,F) is the moduli space of transverse calibrations associated with O. We also give definitions of an orbit O being elliptic (Definition 3.1), metrical (Definition 3.4) and topological (Definition 3.6). We can consider the map PÏ É Ä ÉM O (M,F) i H p i (M) which is defined by corresponding to the basic de Rham cohomology class []. This map ÉP induces a map P Ï M O (M, F) Å i H p i (M) since Diff (M, F) acts trivially on the basic de Rham cohomology groups. The map P is called a period map. We assume that M is a closed oriented manifold and F is a Riemannian foliation. Then we can show the local Torelli type theorem: Theorem 1.2. injective. If O is elliptic and topological, then the period map P is locally We can also prove Theorem 1.3. We suppose that F is taut. If an orbit O is elliptic, metrical and topological, then the moduli space M O (M, F) is a Hausdorff and smooth manifold of finite dimension. We can regard a transverse Calabi Yau structure on (M, F) as a transverse calibration associated with the orbit O CY of Calabi Yau structures. Then we obtain Theorem 1.4. The moduli space of transverse Calabi Yau structures on (M, F) is a Hausdorff and smooth manifold of finite dimension if F is taut.

4 386 T. MORIYAMA This paper is organized as follows. In Section 2, we prepare some definitions and results in foliated geometry. In Section 3, we introduce transverse calibrations on (M, F). Each transverse calibration induces a deformation complex. Then we see that the deformation complex is a subcomplex of the basic de Rham complex. In Section 4, we construct a Riemannian metric on the set of transverse Riemannian structures on (M, F). In Section 5, we provide a sufficient condition for the moduli space to be a Hausdorff and smooth manifold (Theorem 5.6) and also show the local Torelli type theorem (Theorem 5.5). In the last section, as an application of Theorem 5.6, we prove that the moduli space of transverse Calabi Yau structures on (M, F) is a Hausdorff and smooth manifold (Theorem 6.5). We study some examples of transverse Calabi Yau structures and compute the dimension of their moduli spaces. 2. Preparations on foliated geometry In this section, we will give a brief review of some elementary results in foliated geometry. For much of this material, we refer to [6], [15] and [17]. We assume that M is a closed manifold of dimension (p q) and F is a foliation on M of codimension q. We denote by F a completely integrable distribution of dimension p associated to the foliation F asic vector fields and basic forms. A vector field u ¾ ¼(T M) is foliated if [u, Ú] ¾ ¼(F) for any Ú ¾ ¼(F). We denote by ¼(M, F) the set of foliated vector fields on (M, F). Let X(M, F) be the quotient space of ¼(M, F) by ¼(F): X(M, F) ¼(M, F)¼(F). We call an element u of X(M, F) a basic vector field on (M, F). A differential k-form ¾ Îk on M is a basic form on (M,F) Î if the interior product i Ú and the Lie derivative L Ú vanish for any Ú ¾ ¼(F). Let k be the set of basic k-forms on (M, F): k Ò k ¾ Ó i Ú L Ú, Ú ¾ ¼(F). For a section u ¾ ¼(T MF) and a basic k-form ¾ Î k, the interior product i u and the Lie derivative L u are defined by the (k 1)-form i Éu and the k-form L Éu for a lift Éu ¾ ¼(T M) of u, respectively. If u is a basic vector field, then i u and L u are basic forms. We define a foliated diffeomorphism f as a diffeomorphism f of M preserving the foliation F, i.e., f (F) F. We denote by Diff(M, F) the group of foliated diffeomorphisms: Diff(M, F) f ¾ Diff(M) f (F) F.

5 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 387 Î We can define an action of Diff(M, F) on the space of basic forms by pull-back. For u ¾ X(M, F), any lift Éu ¾ ¼(T M) of u induces a one parameter family of foliated diffeomorphisms f t. Then the Lie derivative L u for ¾ Î k limit (ddt) f t t by the one-parameter family f t. may be regarded as the 2.2. asic bundles and basic sections. Let Ï P M be a principal fiber bundle and a connection form on P. The horizontal subbundle H is defined by the subbundle Ker of the tangent bundle T P. Then the derivative restricted to H is the isomorphism from H to T M. Hence we have the subbundle F É 1 (F) of H over P. If F É is integrable, then F É induces the foliation ÉF on P. DEFINITION 2.1. A principal fiber bundle P is foliated if there exists a connection form on P such that the bundle F É is integrable. Moreover, if the form is basic with respect to the induced foliation ÉF, then the bundle P is called basic. A vector bundle Ï E M is called foliated (resp. basic) if the associated principal bundle P E is a foliated (resp. basic) bundle. In the case Ï E M is a foliated vector bundle, the bundle P E M admits a foliation ÉF on the total space P E by the definition. This foliation ÉF induces a foliation ÉF E on E. In addition, if E is basic then there exists a connection Ö of E whose connection form is basic. Such a connection Ö is called a basic connection on E. DEFINITION 2.2. Let E be a basic vector bundle with a basic connection Ö. A section s ¾ ¼(E) is called basic if Ö Ú s for any Ú ¾ ¼(F). We denote by ¼ (E) the set of basic sections of E. Remark that for a basic bundle E, the dual bundle E, exterior powers Îk E and symmetric covariant tensors S k E are also basic bundles, where k is non-negative integer. We consider a hermitian metric h on E as the section of a basic bundle. Then we call E a basic Hermitian bundle if the hermitian metric h is basic Riemannian foliations. Let Q be the normal bundle T MF and Ï T M Q the natural projection. We define an action of ¼(F) on any section u ¾ ¼(Q) as follows: L Ú u [ Éu, Ú] ¾ ¼(Q) for any vector field Ú ¾ ¼(F) where Éu ¾ ¼(T M) is a lift of u, i.e., a vector field u ¾ ¼(T M) with ( Éu) u. This action is independent of the choice of lifts Éu ¾ ¼(T M) of u. Let g be a Riemannian metric on M. Then we have an orthogonal decomposition T M F g F and the isomorphism Ï Q F. Set a metric g Q g F for the induced metric g F on F. Then the map Ï (Q, g Q ) (F, g F ) is an isometry. Let

6 388 T. MORIYAMA Ö M be the Levi-Civita connection with respect to g. Then we introduce a connection Ö on Q as follows: L (1) Ö Ú u Ú u, Ú ¾ ¼(F), (Ö M Ú Éu), Ú ¾ ¼(F ) for u ¾ ¼(Q), where Éu ¾ ¼(T M) is a lift of u. In general, the connection (1) is not necessary basic. A foliation F is Riemannian if the date U i, f i, T, i, j satisfies that T is a Riemannian manifold and each i, j is an isometry. A Riemannian metric g is called bundle like if L Ú g Q for any Ú ¾ ¼(F) where the tensor L Ú g Q ¾ ¼(S 2 Q ) is defined by (2) (L Ú g Q )(u, Û) Ú(g Q (u, Û)) g Q (L Ú u, Û) g Q (u, L Ú Û) for u, Û ¾ ¼(Q). It turns out that F is a Riemannian foliation if and only if there exists a bundle like Riemannian metric g on M. For a bundle like metric g, the connection Ö in (1) is basic. Hence Q is a basic vector bundle for a Riemannian foliation F. It is easy to see that any basic section of Îk Q is a basic k-form on M: k ¼ k Q. The space ¼ (Q) is nothing but X(M, F): X(M, F) ¼ (Q). So we also call an element s of ¼ (Q) a basic vector field. Moreover, a basic vector field s ¾ ¼ (Q) is identified with a foliated vector field u s (s) ¾ ¼(F ) by the isomorphism. Therefore we have the following identification: (3) ¼ (Q) ³ u ¾ ¼(F ) [u, Ú] ¾ ¼(F), Ú ¾ ¼(F). From now, we consider any basic section of Q as a vector field on M under the identification of (3). Then a basic vector field u ¾ ¼ (Q) induces a one parameter family of foliated diffeomorphisms f t since a vector field u ¾ ¼(F ) associates a one parameter family of diffeomorphisms Transversely elliptic operators. Let E be a basic bundle of rank N. A linear map D Ï ¼ (E) ¼ (E) is called a basic differential operator of order l if, in local coordinates (x 1,, x p, y 1,, y q ) for which F is given by the equations dy 1 dy q, D has the following expression: s D a s (y) s 1 y1 s q yq sl

7 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 389 where s (s 1,, s q ) ¾ Æ q and each a s is an N N-matrix valued basic function. We define the principal symbol of D at z (x, y) and the basic covector ¾ Q z as the linear map (D)(z, )Ï E z E z given by (D)(z, )() sl s 1 1 s q q a s (y)() for any ¾ E z. DEFINITION 2.3. A basic differential operator D is transversely elliptic if (D)(z, ) is an isomorphism for every z ¾ M and ( ) ¾ Q z. We suppose that E is a Hermitian basic bundle with a hermitian metric h and l 2l ¼. Then a quadratic form A(D)(z, )Ï E z is given by A(D)(z, )() ( 1) l¼ (D)(z, )(),. DEFINITION 2.4. A basic differential operator D is strongly transversely elliptic if A(D)(z, ) is positive definite for every z ¾ M and every non-zero ¾ Q z. Let (E k, D k )k,1,,q be a family of Hermitian basic bundles and basic differential operators of order 1 with the differential complex (4) D k 1 ¼ (E k ) D k ¼ (E k1 ) D k1 where D k Ï ¼ (E k ) ¼ (E k ) for k, 1,, q. We denote by k the principal symbol (D k )(z, ) of D k. Then the complex (4) is transversely elliptic if the symbol sequence k 1 E k z k E k1 z k1 is exact for any z and any non-zero. Remark that the complex (4) is transversely elliptic if and only if the basic operator L k D k D k D k 1 D k 1 is strongly transversely elliptic, where D k is the formal adjoint operator. We have the Hodge theory for the transversely elliptic complex (4) with the cohomology H k (E ): Proposition 2.5 ([6, Theorem 2.8.7]). (i) The kernel H k of L k is finite dimensional and we have an orthogonal decomposition ¼ (E k ) H k Im(D k 1 ) Im(D k ). (ii) The orthogonal projection ¼ (E k ) H k induces an isomorphism from H k (E ) to H k.

8 39 T. MORIYAMA 2.5. Transverse Riemannian structures. A Riemannian foliation is characterized by the following structure: DEFINITION 2.6. A symmetric 2-tensor Ég ¾ ¼(S 2 Q ) is a transverse Riemannian structure on (M,F) if Ég is positive definite on Q and L Ú Ég for any Ú ¾ ¼(F) where L Ú Ég is defined by (2). A bundle like metric g induces a transverse Riemannian structure g Q on (M, F). Conversely, for a transverse Riemannian structure Ég, we can take a bundle like metric g such that g Q Ég. Given a transverse Riemannian structure g Q on (M, F), then the complexification Q Å is a basic hermitian bundle, and so Îk Q Å is. Hence from Proposition 2.5 we have Proposition Î 2.7 ([6, Theorem 3.2.5]). (i) The kernel H k of the basic Laplacian dd d k d on is finite dimensional and we have an orthogonal decomposition k Hk Im(d) Im(d ). Î k (ii) The orthogonal projection Hk de Rham cohomology H k (M) to Hk. induces an isomorphism from the basic 2.6. Transverse Kähler structures. We can associate an action of ¼(F) to any section J ¾ ¼(End(Q)) as follows: (L Ú J)(u) L Ú (J(u)) J(L Ú u) for Ú ¾ ¼(F) and u ¾ ¼(Q). If J ¾ ¼(End(Q)) is a complex structure of Q, i.e. J 2 id Q, and satisfies that L Ú J for any Ú ¾ ¼(F), then a tensor N J ¾ ¼ Æ2 Q Å Q can be defined by N J (u, Û) [Ju, JÛ] Q [u, Û] Q J[u, JÛ] Q J[Ju, Û] Q for u, Û ¾ ¼(Q), where [u, Û] Q denotes the bracket [ Éu, ÉÛ] for each lift Éu and ÉÛ. DEFINITION 2.8. A section J ¾ ¼(End(Q)) is a transverse complex structure on (M, F) if J is a complex structure of Q, i.e. J 2 id Q, such that L Ú J for any Ú ¾ ¼(F) and N J. A foliation F is transversely holomorphic if and only if there exists a transverse complex structure on (M, F). Thus we may regard a transverse complex structure as a generalization of complex structures on complex manifolds. A transverse complex structure J on (M, F) give rises to the decomposition Î k Å Ä rsk Î r,s in the

9 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 391 same manner as complex geometry. We denote by H r,s (M) the (r, s)-basic Dolbeault cohomology group. We provide the following remark about the integrability condition of transverse complex structure. Let J be a complex structure of Q such that L Ú J for any Ú ¾ REMARK 2.9. Î 1, ¼(F). Then J is a transverse complex structure, i.e. N J, if and only if d Î Î 2, Î1,1, which is equivalent to d,1 Î 1,1 Î,2, where d denotes the exterior derivative. DEFINITION 2.1. A pair of sections ( Ég, J) ¾ ¼(S 2 Q ) ¼(End(Q)) is a transverse Kähler structure on (M, F) if Ég is a transverse Riemannian structure and J is a transverse complex structure on (M, F) satisfying for u, Û ¾ ¼(Q). Ég(, J ) is a d-closed form Ég(Ju, JÛ) Ég(u, Û) A transversely Kähler foliation F is defined by date U i, f i, T, i, j with a Kähler manifold T and local diffeomorphisms i, j preserving the Kähler structure. We remark that there exists a transverse Kähler structure on (M,F) if and only if F is a transverse Î Î k Kähler foliation. Given a transverse Kähler structure ( Ég, J), then Å and r,s are all basic hermitian bundles. Then Proposition 2.5 implies that each basic Dolbeault cohomology group H r,s (M) is finite dimensional. Moreover, the basic de Rham Hodge decomposition holds: Proposition 2.11 ([6, Theorem 3.4.6]). on M. Then there exists an isomorphism 3. Transverse calibrations Å H k (M, ) rsk Let F be a transverse Kähler foliation H r,s (M) Orbits in vector spaces. Let W be a vector space of dimension q. We Î pi W is the space of skew-symmetric tensor of degree p i of the dual space W. denote by the representation of G GL(W ) on the space Ä l i1 Î pi W where each We fix an element W ( 1,, l ) ¾ Ä l i1 Î pi W and denote by H the isotropy group of W : H g ¾ G g W W.

10 392 T. MORIYAMA The G-orbit space O Ä l Î g W ¾ pi i1 W g ¾ G through W is regarded as the homogeneous space GH. We denote by A O (W ) the G-orbit space O: lå µ A O (W ) g W ¾ i1 pi W g ¾ G For an element ¾ A O (W ), the tangent space T A O (W ) is given by E 1 (W ) Ç ¾ lå i1 pi W ¾ g where g is the Lie algebra of G and Ç is the differential representation of g. We also define vector spaces E (W ) and E k (W ) by E (W ) E k (W ) i Ú (i Ú 1,, i Ú l ) ¾ «i Ú ¾ lå i1 lå i1 pi k 1 W for integers k 2, respectively. Then we have a complex µ pi 1 W. Ú ¾ W «¾ k W, Ú ¾ W µ, µ ( ) E (W ) u E 1 (W ) u E 2 (W ) u for a form u ¾ W. DEFINITION 3.1. An orbit O is elliptic if the complex ( ) is exact for any nonzero element u ¾ W at E 1 (W ) and E 2 (W ). We give some examples of elliptic orbits. Now we assume that W is even dimensional, that is q 2n. EXAMPLE 3.2. The set of all symplectic forms on W is an orbit space O symp, which is isomorphic to the quotient space GL(2n, Ê)Sp(2n, Ê). For any ¾ O symp, the complex ( ) is 1 W u 2 W u 3 W u for any element u ¾ W. Thus the orbit O symp is elliptic.

11 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 393 EXAMPLE 3.3. A non-zero complex n-form Đ ¾ În Å is called an SL n () structure on W if the form Đ satisfies that W Å Ker Đ Ker Đ. where Ker Đ denotes the space Ú ¾ W Å i Ú Đ. We remark that an SL n () structure Đ induces a complex structure J Đ on W defined by Ô 1Ú for Ú ¾ Ker Đ, (5) J Đ (Ú) Ô 1Ú for Ú ¾ Ker Đ. Then Đ is an (n, )-form with respect to the complex structure J Đ. Let O SL be the set of SL n () structures on W. Then it turns out that O SL is an orbit space such that For any ¾ O SL, the complex ( ) is O SL GL(2n, Ê)SL(n, ). n 1, W u n, W n 1,1 W u n,1 W n 1,2 W u for any u ¾ W. Here we regard the element u as an element of Î 1, W Î,1 W such that Æu u. So this orbit O SL is elliptic. An orbit O is metrical if the isotropy group H is a subgroup DEFINITION 3.4. of the orthonormal group O(W ) with respect to a metric g W on W. The above two examples O symp and O SL are not metrical. However, we may have an example of an elliptic and metrical orbit: EXAMPLE 3.5. is called a Calabi Yau structure on W if Đ is an SL n () structure and is a symplectic structure on W such that A pair (Đ, ) ¾ Î n Å Î2 Đ ÆĐ, Đ ÆĐ c n n, (, J Đ ) is positive definite where c n (1n!)( 1) n(n 1)2 (2 Ô 1) n. Let O CY be the set of Calabi Yau structures on W. Then O CY is an elliptic orbit such that O CY GL(2n, Ê)SU(n) ([1, Proposition 4.9]). Thus the orbit O CY is metrical.

12 394 T. MORIYAMA 3.2. Transverse calibrations in foliated manifolds. Let M be a closed manifold of dimension p q and F a Riemannian foliation on M of codimension q. We consider the completely integrable distribution F associated to F and the quotient bundle Q T MF over M. For each x ¾ M, we identify Q x with W Ê q. Then, as in Section 3.1, we have Ä an orbit A O (Q x ) A O (W ) at x ¾ M for an orbit O. Note that Î the orbit A O (Q x ) pi i Q x does not depend on the choice of the identification h Ï Q x ³ W. Then we can define GH-bundle A O (M, F) by A O (M, F) x¾m A O (Q x ) M. Since A O (M, F) Ä i Î pi Q, we can consider the Lie derivative and the exterior derivative for any section of A O (M,F) as a differential form. We denote by E O (M,F) the space of sections of A O (M, F) which are basic forms: E O (M, F) ¼(A O (M, F)) Å i pi Q x. Let Ker be a space Ú ¾ T M i Ú for ¾ E O (M, F). DEFINITION 3.6. A section ¾ E O (M, F) is called a transverse calibration associated with the orbit O if is a closed form such that Ker F. We denote by ÉM O (M, F) the space of transverse calibrations associated with O. The group Diff(M,F) acts on ÉM O (M,F) by pull-back. Given a transverse calibration ¾ ÉM O (M,F), we can consider the vector spaces E k x (Q x ) at each point x ¾ M, and define vector bundles E k (M, F) x¾m E k x (Q x ) M for integers k. Each bundle E k (M,F) is a basic bundle since its associated principal bundle is that of the normal bundle Q. It follows that a section i Ú ¾ ¼(E (M, F)) is basic if and only if Ú ¾ ¼(Q) is a basic section since Ker F and L Û (i Ú ) i L Û Ú for any Û ¾ ¼(F). Hence we have lå µ ¼ (E (M, F)) pi 1 i Ú ¾ Q Ú ¾ ¼ (Q), i1 lå µ ¼ (E 1 (M, F)) pi Ç ¾ Q ¾ ¼ (End(Q)). i1 We introduce the graded vector spaces E (M,F) Ä k E k (M,F). For simplicity, we shall denote by E k and E the spaces E k (M, F) and E (M, F), respectively.

13 à à à MODULI OF TRANSVERSE CALAI YAU STRUCTURES 395 Ä k Ä i Î pi k 1 Proposition 3.7. The module ¼ (E) is a differential graded module in with respect to the derivative d, where d is the exterior derivative d restricted to the space of the basic forms. Proof. We prove that d a ¾ ¼ (E k ) for all a ¾ ¼ (E k 1 ). To show this, it is sufficient to prove that d i Ú ¾ ¼ (E 1 ) for any element i Ú ¾ ¼ (E ), since ¼ (E) is generated by ¼ (E ). The basic vector field Ú induces a one-parameter transformation f t such that each f t is an element of Diff(M, F). Then it follows from d that di Ú L Ú d dt f t t. The right hand side (ddt) f t t is contained in the tangent space of E O (M, F) at since f t is in E O (M, F). Recall that the tangent space of E O (M, F) at is the space ¼ (E 1 ). This implies that di Ú (ddt) f t t ¾ ¼ (E 1 ). Thus we obtain a complex ( ) where d i d E i complex: ¼ (E ) d ¼ (E 1 ) d 1 ¼ (E 2 ) d 2 for each i. The complex ( ) is a subcomplex of the basic de Rham à ¼ (E ) d à ¼ (E 1 ) d 1 à ¼ (E 2 ) d 2 Ã Ã Ä i Î pi 1 d Ã Ä i Î pi d Ã Ä i Î pi 1 d Ã. We denote by H k ( ) the cohomology groups of the complex ( ): H k ( ) «¾ ¼ (E k ) d k «d k 1 ¾ ¼ (E k ) ¾ ¼ (E k 1 ). Then we can obtain a map p k Ï H k ( ) Å i H p i k 1 (M) for each k. are in- DEFINITION 3.8. A section ¾ E O (M, F) is topological if p 1 and p2 jective. An orbit O is topological if any ¾ E O (M, F) is topological. REMARK 3.9. If O is an elliptic orbit, then the complex ( ) is transverse elliptic at ¼ (E 1 ) and ¼ (E 2 ), and the operators ½ k d k 1d k 1 d k d k are strongly transversely elliptic for k 1, 2.

14 396 T. MORIYAMA 4. Riemannian metrics on the set of transverse Riemannian structures We assume that M is a closed oriented manifold of dimension m ( p q) and F is a Riemannian foliation of codimension q. Let M(M, F) be the set of transverse Riemannian structures on (M, F). We denote by S 2 Q the bundle of symmetric covariant 2-tensors on Q Completions of M(M, F) and Diff(M, F). At first, we may regard M(M, F) as a Fréchet manifold which is an open subset of the Fréchet space ¼ (S 2 Q ). Now we consider the completion ¼ s (S 2 Q ) of ¼(S 2 Q ) with respect to the Sobolev norm, s. This space ¼ s (S 2 Q ) is a anach space (in fact, a Hilbert space), and ¼ s (S 2 Q ) C k ¼(S 2 Q ) for s k m2. From now, we assume that s 1 m2. We define ¼ s (S2 Q ) ¼ s (S 2 Q ) C 1 ¼ (S 2 Q ) where C 1 ¼ (S 2 Q ) denotes the set u ¾ C 1 ¼(S 2 Q ) L Ú u, Ú ¾ ¼(F). Then the vector space ¼ s (S2 Q ) is a closed subspace of ¼ s (S 2 Q ), so it is a anach space. We introduce the set M s (M, F) Ég ¾ ¼s (S2 Q ) Ég Ï positive definite. Then the set M s (M, F) is an open subset of the anach space ¼s (S2 Q ), and so a anach manifold. Secondly, we study the properties of the set Diff(M,F). In [19], Omori shows that Diff(M,F) is an IL-Lie group with the model ¼(M,F), ¼ s (M,F), s 1. Then we may obtain a anach manifold Diff s (M, F) with the model ¼ s (M, F) for each s 1. The group Diff(M, F) acts on M(M, F) by pull-back. This action naturally extends that of Diff s1 (M, F) on M s (M, F). Then we prove that Proposition 4.1. The action of Diff s1 (M, F) on M s (M, F) is continuous. Proof. Let M(M) be the set of Riemannian metrics on M. The group Diff(M) acts on M(M) by pull-back. We use the fact that the action of Diff(M) on M(M) can be extended to continuous one of Diff s1 (M) on ¼ s (S 2 T M), which is proved by Ebin in [5]. Let A É denote the extended action, that is, the continuous map (6) É AÏ Diff s1 (M) ¼ s (S 2 T M) ¼ s (S 2 T M). Now the inclusions Diff s1 (M,F) Diff s1 (M) and ¼ s (S2 Q ) ¼ s (S 2 T M) are continuous. Hence we have a continuous map Diff s1 (M, F) ¼ s (S2 Q ) ¼ s (S 2 T M)

15 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 397 by restricting the map É A to Diff s1 (M, F) and ¼ s (S2 Q ). The image of this map is in ¼ s (S2 Q ) and the topology of ¼ s (S2 Q ) coincides with the relative topology as a subspace of ¼ s (S 2 T M). Hence we obtain a continuous map AÏ Diff s1 (M, F) ¼ s (S2 Q ) ¼ s (S2 Q ). This map A induces the continuous action of Diff s1 (M, F) on M s (M, F). This finishes the proof. For any element ¾ ¼ s (S2 Q ), we define a map by A ( ) A(, ). Then we have the A Ï Diff s1 (M, F) ¼ s (S2 Q ) Proposition 4.2. If is a smooth element of ¼ (S 2 Q ), then A Ï Diff s1 (M,F) ¼ s (S2 Q ) is a smooth map. Proof. The map É A given by (6) induces the smooth map ÉA Ï Diff s1 (M) ¼ s (S 2 T M) for a smooth element ¾ ¼(S 2 T M) (cf. [5, p. 18]). Since any smooth element of ¼ (S 2 Q ) can be regarded as smooth one of ¼(S 2 T M), the map A É is smooth for any element ¾ ¼ (S 2 Q ). y restricting A É to Diff s1 (M, F), we consider the map ÉA Diff s1 (M,F) Ï Diff s1 (M, F) ¼ s (S 2 T M). Then this map A É Diff s1 (M,F) is smooth since Diffs1 (M, F) is a anach submanifold of Diff s1 (M). The image of A É Diff s1 (M,F) is in ¼ (S 2 Q ) which is a anach subspace of ¼ s (S 2 T M). Thus we can get a smooth map (7) A É Diff s1 (M,F) Ï Diff s1 (M, F) ¼ (S 2 Q ). The smooth map (7) is nothing but the map A, which completes the proof Riemannian structures on M s (M, F). We assume the s 1 m2. We recall that M s (M, F) is a anach manifold whose tangent space is identified with ¼ s (S2 Q ). Each element ¾M s (M,F) induces a metric, on S 2 Q and a transverse volume form on (M, F). For elements «and of ¼ s (S2 Q ) ( T M), we

16 398 T. MORIYAMA obtain the basic q-form «, on (M, F). To integrate this form, we need a volume form along the foliation. Fix a bundle like metric g on (M, F). Then a characteristic form F is defined by F (X 1,, X p ) det(g(x i, e j ) i, j ) for X i ¾ ¼(T M), where e j j1,, p is an orthonormal basis of F with respect to g. Now we define the Riemannian structure (, ) on M s (M, F) as follows. («, ) M «, F for any «, ¾ ¼ s (S2 Q ). The bilinear form (, ) is positive definite and smooth for ¾ M s (M, F). However, for any s the space ¼s (S2 Q ) is not complete under (, ). We denote the inner product (, ) by (, ). We can find a unique affine connection Ö on M s (M, F) by a similar argument in p. 19 of [5]. Then the connection Ö associates an isomorphism D s Ï J s (S2 Q ) så i S i Q Å S 2 Q where J s (S2 Q ) is a basic jet bundle ([6, Ä Theorem 2.3.6]). For ¾ M s (M, F), we s have the positive definite bilinear form on i Si Q Å S 2 Q induced by (, ). Hence, under the isomorphism D s, we obtain a positive definite bilinear form (, ) s on ¼s (S2 Q ) ( T M s (M, F)). Then the space ¼s (S2 Q ) is complete under (, ) s for each ¾ M s (M, F) (cf. [5, p. 21]). Thus we obtain the Riemannian metric (, )s on M s (M, F) Diff s1 (M,F)-invariant Riemannian structures. Let Diff s1 (M,F) be the identity component of Diff s1 (M, F). In previous section, we construct a Riemannian structure (, ) s on M s (M, F). In general, the structure (, )s is not Diff s1 (M, F)- invariant. We will show that this structure (, ) s is Diff s1 (M, F)-invariant if F is a taut foliation. DEFINITION 4.3. A foliation F is called taut if there exists a Riemannian metric g on M such that each leaf of F is a minimal submanifold of (M, g). In this case, the Poincare duality holds on basic de Rham cohomology groups, i.e., if F is taut then there Ê exists a non-degenerate Î pairing: H Î r (M) Å H q r (M) Ê induced by the integral M «r F for «¾ and ¾ q r ([15, Corollary 7.58]). It implies that (8) M d«f

17 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 399 Î q 1 for any «¾. We can prove Proposition 4.4. invariant. IfF is taut, then the Riemannian structure (, ) s is Diff s1 (M,F)- Proof. At first, we Î check that (, ) is Diff s1 (M, F)-invariant. For any ¾ Diff s1 q (M, F) and ¾, we have (9) K (d) d K () d K () where K is the homotopy operator associated Î to ¾ Diff s1 (M, F). Remark that the q form K () is basic for any basic form ¾. It follows from (8) and (9) that M ( ) F M M ( d K ()) F F. It gives rise that ( «, ) M M M («, ). «, F («, ) F «, F Hence preserves the structure (, ) on M s (M, F). Secondly, we consider the metric (, ) s for s. The connection Ö in Subsection 4.2 satisfies (1) (Ö X Y ) Ö X Y Diff s1 (M, F)-invariant. Equa- for vector fields X, Y ¾ TM s (M, F) since (, ) is tion (1) gives rise to Æ D s D s Æ. Thus the action of commutes with the isomorphism D s. y the definition of (, ) s, the metric (, ) s is Diff s1 (M,F)-invariant since (, ) is Diff s1 (M, F)-invariant. Hence we finish the proof.

18 4 T. MORIYAMA 5. Moduli spaces of transverse calibrations In this section, we provide a sufficiently condition for a moduli space M O (M, F) of transverse calibrations to be a Hausdorff and smooth manifold. Moreover, we show a local Torelli type theorem for transverse calibrations. We assume that the manifold M is closed oriented and F is a Riemannian foliation Local coordinates of M O (M,F). Let be an element of M O (M,F). We suppose that O is an elliptic orbit and p 2 is injective. We consider a formal power series a(t) in t: a(t) a 1 t 1 2! a 2t 2 1 3! a 3t 3 ¾ ¼ (End(Q))[[t]] where each a k is a basic section of End(Q). Then we obtain a formal power series e a(t) exp a(t) ¾ ¼ (GL(Q))[[t]]. Let À 1 ( ) be ½ 1-harmonic elements «¾ ¼ (E 1 ) ½1«where ½1 is the operator d d d 1 d 1. From Theorem 4.2 in [18], for an element Ç a1 ¾ À 1 ( ) there exists a smooth form e a(t) ¾ ÉM O (M, F). Hence we have a map É Ï À 1 ( ) ÉM O (M, F) Ç a1 e a(1) where e a(1) is the value of the e a(t) at t 1. We denote by the projection: ÉM O (M, F) M O (M, F) ÉM O (M, F)Diff (M, F) and consider the composition map Æ É Ï À 1 ( ) M O (M, F). The map maps the origin of À 1 ( ) to the class of in M O (M, F). Ä Proposition 5.1. If p 1 Ï H 1 ( ) i H p i (M) is injective, then there exists an open neighborhood S of the origin in À 1 ( ) such that S Ï S M O (M,F) is injective. Ä Proof. We define a map PÏ M O (M,F) i H p i (M) by P()[]¾Ä i H p i (M) for any ¾M O (M, F) and consider the composition P Æ Ï À 1 ( ) Å i H p i (M).

19 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 41 Then the differential of PÆ at the origin is given by the map p 1. Since p 1 is injective, there exists a Ä small neighbourhood S À 1 ( ) of the origin such that the restriction P Æ S Ï S i H p i (M) is injective. Hence SÏ S M O (M,F) is also injective. Ä Î Let E s be the set C 1 ¼(A H (M, F)) ¼ s1 pi i. Then Diff s1 (M, F) acts on the set E s by the pull-back. If we give a vector field ¾ ¼ s1 (Q) and the diffeomorphism f ¾ Diff s1 (M, F) associated by. Then, for e a ¾ E s, there exists a section b ¾ ¼ (End(Q)) such that (11) f e a e b since the set E s is Diff s1 (M,F)-invariant. For a d-closed element e a, we can choose such that Ç b is in À 1 ( ): Lemma 5.2. If e a is an element of M O (M, F) with a s for sufficiently small, then there exists a C ½ -vector field ¾ ¼ (Q) satisfying Ç b ¾ À 1 ( ). Proof. We assume that a vector field ¾ ¼ s1 (Q) and the diffeomorphism f ¾ Diff s1 (M,F) are given as in (11) for e a ¾ M O (M,F). It is sufficient to show that there exists a vector field satisfying (12) d Ç b. È Note that Ç b e b k2 (1k!) Çk b and f e a e a L e a Ç a di ÉW (, a) where ÉW (, a) is the higher order term with respect to and a. Therefore we obtain (13) Ç b Ç a di W (, a) where W (, a) is defined by the higher term that W (, a) is an element of E 1 and satisfies ÉW (, a) È k2 (1k!) Çk b. We remark (14) W ( 2, a) W ( 1, a) s 2 1 s1 for sufficiently small 1, 2, a and a positive constant 1 (see [1, Lemma 3.3]). We choose a vector field such that the harmonic part of i ¾ E vanishes. Then it follows from equation (13) that d Ç b is equivalent to (15) i G d Ç a G d W (, a) where G is Green s operator of the complex ( ). Now we can take 1 ¾ ¼ (Q) satisfying i 1 d G Ç a.

20 42 T. MORIYAMA Inductively, we define k ¾ ¼ (Q) for k 2 as follows i k G d Ç a G d W ( k 1, a). From the estimate k s1 Ci k s1 for a constant C, it follows that k1 k s1 CW ( k1, a) W ( k, a) s. y taking a sufficiently small 1 in (14), we have k1 k s1 k k 1 s1. Therefore the sequence k k converges uniformly to a vector field ½ ¾ ¼ s1 (Q) with respect to the norm s1. It turns out that ½ satisfies equation (15). Hence ½ is in ¼ s1 (Q) and satisfies (12). This completes the proof. From this Lemma 5.2, we immediately show the following proposition. Proposition 5.3. There exists an open neighborhood U of () in M O (M, F) such that S Ï S U is surjective for a small open neighbourhood S S of the origin in À 1 ( ). Proof. If we define an open neighbourhood U of () by U ( e a ¾ ÉM O (M, F) a s ) for a small constant as in Lemma 5.2, then for any e a ¾ U, there exists an element Ç b ¾ À 1 ( ) such that É( Ç b ) e a. Hence is surjective Distance on M O (M, F). We assume that the orbit O is elliptic and topological. We construct a distance on the moduli space M O (M, F). Proposition 5.4. We suppose that F is taut. If the orbit O is metrical, then there exists a distance on M O (M, F). Proof. Since O is metrical, for each ¾ E s there exists a metric Ä g on Q. The Î metric g induces an L 2 -metric on the tangent space T E s ¼ s pi i. Hence we obtain a Riemannian metric (, ) s on E s which is Diff s1 (M, F)-invariant by using the same argument in Proposition 4.4. Then ÉM s O (M, F) H T E s admits a distance d É given by the Riemannian structure of E s. Now we define a function d Ï M O (M, F) M O (M, F) Ê by d((), ( ¼ )) inf f,g¾diff (M,F) Éd( f, g ¼ )

21 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 43 for, ¼ ¾ ÉM O (M, F), where is the projection ÉM O (M, F) M O (M, F). We remark that d((), ( ¼ )) inf f ¾Diff É (M,F) d(, f ¼ ). We shall see that d is a distance on M O (M, F). The triangle inequality holds: d((), ( ¼¼ )) inf f,g Éd( f, g ¼¼ ) inf f,g ( É d( f, ¼ ) É d( ¼, g ¼¼ )) inf f É d( f, ¼ ) inf g Éd( ¼, g ¼¼ ) d((), ( ¼ )) d(( ¼ ), ( ¼¼ )). To show the positivity of d, we suppose that d((), ( ¼ )) for, ¼ ¾M O (M,F), that is, inf f ¾Diff (M,F) É d(, f ¼ ). Then there exists a sequence f j j¾æ of Diff (M,F) such that Éd(, f j ¼ ), ( j ½). This implies that (16) f j ¼ L 2, ( j ½) since É d is locally equivalent to the L 2 -metric induced by g. It follows from (16) that (17) [ f j ¼ ] ¾ Å i H p i (M), ( j ½) where [ f j ¼ ] is the basic cohomology class of f j ¼. Since Diff (M,F) acts trivially on the basic cohomology group H p i (M), the cohomology class [ f j ¼ ] is [] [ ¼ ], and independent of j. Hence, Ä it follows from (17) that [] [ ¼ ] must be zero, so we obtain [] [ ¼ ] ¾ i H p i (M). We may assume that (¼ ) is included in an open set U given as in Proposition 5.3. Remark Ä that the period map P U restricted to U is injective since P Æ S Ï S i H p i (M) is injective and S Ï S U is isomorphic (see Propositions 5.1 and 5.3). Now we have P(()) [] [ ¼ ] P(( ¼ )). Hence () ( ¼ ) ¾ M O (M, F). Thus d is a distance on M O (M, F) Main theorems. We recall that the period map P Ï M O (M, F) lå i1 H p i (M) is induced by taking the basic de Rham cohomology class []. We can show the local Torelli type theorem:

22 44 T. MORIYAMA Theorem 5.5. injective. If O is elliptic and topological, then the period map P is locally Proof. It follows from Propositions 5.1 and 5.3 that a small open set of M O (M,F) is isomorphic to an open neighbourhood S of origin in À 1 ( ) by the map Ï À 1 ( ) M O (M, F). Thus it is sufficient to show that the composition map P Æ S is injective for a small open set S. However, as in the proof of Proposition 5.3, there exists a small open set S such that P Æ S is injective, and the proof is finished. We prove the main theorem: Theorem 5.6. We suppose that F is taut. If O is metrical, elliptic and topological, then the moduli space M O (M, F) is a Hausdorff and smooth manifold. Proof. If O is elliptic and topological, Propositions 5.1 and 5.3 implies that the moduli space M O (M, F) admits local coordinates given by the open neighbourhood of the origin in H 1 ( ). The dimension dim H 1 ( ) is not independent of in a connected component of M O (M, F) by Proposition 5.1 in [18]. Thus M O (M, F) is a manifold. In addition, if O is metrical then M O (M, F) has a distance as in Proposition 5.4. Hence M O (M, F) is Hausdorff. 6. The moduli space of transverse Calabi Yau structures In this section, we will show the moduli space of transverse Calabi Yau structures is a Hausdorff and smooth manifold (Theorem 6.5). We assume that M is a closed oriented manifold and F is a Riemannian and taut foliation of codimension 2n. Let F denote the integrable distribution induced by the foliation F Transverse SL n () structures. DEFINITION 6.1. A nowhere vanishing complex n-form Đ ¾ În Å is called a transverse SL n () structure on (M, F) if Đ is a basic form such that dđ and Q Å Ker ĐF Ker ĐF where Ker ĐF is the space Ú ¾ Q Å i Ú Đ. A transverse SL n () structure Đ induces a complex structure J Đ on Q such that Đ is Î an (n, )-basic form on (M, Î F) (see Example 3.3). Then we can check that 2, d ¾ Î1,1 1, for any ¾ because of (d) Đ. It follows from Remark 2.9 that J Đ is a transverse complex structure on (M,F). Hence (F, J Đ ) is a transverse holomorphic foliation on M. Let ÉM SL (M, F) be the space of transverse SL n ()

23 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 45 structures on (M, F). Any element Đ ¾ ÉM SL (M, F) induces a transverse calibration associated with the orbit O SL, and converse is true. Thus, we can identify ÉM SL (M, F) with the set ÉM OSL (M, F) of transverse calibrations associated with the orbit O SL. We recall that the orbit O SL is elliptic. For Đ ¾ ÉM SL (M, F), the complex ( Đ ) is n 1, d n, n 1,1 d 1 n,1 n 1,2 d 2. Unfortunately, the maps p 1 Đ Ï H 1 ( Đ ) H n (M, ), p 2 Đ Ï H 2 ( Đ ) H n1 (M, ) are not always injective for Đ ¾ ÉM SL (M, F). However, we obtain Proposition 6.2. If (F, J Đ ) is a transverse Kähler foliation, then the element Đ ¾ ÉM SL (M, F) is topological. Moreover, the period map P is injective on a neighbourhood of the equivalent class of Đ in M SL (M, F). Proof. We suppose that Đ ¾ ÉM SL (M, F) satisfies (F, J Đ ) is a transverse Kähler foliation on M. y modifying the argument of Proposition 4.4 in [1], we obtain H 1 ( Đ ) H n, H 2 ( Đ ) H n,1 n 1,1 (M) H (M), n 1,2 (M) H (M). The maps p 1 and Đ p2 are injective by Proposition 2.11, so Đ is topological. Moreover, Đ it follows from Propositions 5.1 and 5.3 that there exists an open neighbourhood U Đ ¾ M SL (M,F) of (Đ) such that the period map P UĐ restricted to U Đ is injective. Hence we finish the proof Transverse Calabi Yau structures. We say that a real 2-form ¾ Î2 is a transverse symplectic structure on (M, F) if is a basic form on (M, F) such that d and n. Î n DEFINITION 6.3. A pair (Đ, ) ¾ Å Î2 is called a transverse Calabi Yau structure on (M, F) if Đ is a transverse SL n () structure and is a transverse symplectic structure on (M, F) such that Đ ÆĐ, Đ ÆĐ c n n, where c n (1n!)( 1) n(n 1)2 (2 Ô 1) n. (, J Đ ) is positive definite on Q

24 46 T. MORIYAMA We denote by ÉM CY (M,F) the set of transverse Calabi Yau structures on (M,F). Any structure (Đ, ) ¾ ÉM CY (M, F) is a transverse calibration associated with the orbit O CY. Proposition 6.4. The orbit O CY is metrical, elliptic and topological. Proof. It suffices to show that O CY is topological. Given a structure (Đ, ) ¾ ÉM CY (M, F), then, by repeating a similar argument of the computation of cohomology groups ([1, Theorem 4.8]) to basic forms we obtain H 1 ( ) H n, n 1,1 (M) H (M) È 1,1,,Ê H 2 ( ) H n,1 n 1,2 (M) H (M) (H 2,1 1,2 (M) H (M)) Ê where (H 2,1 1,2 (M) H (M)) Ê and È 1,1 2,1 1,2 denote the real part of H,Ê (M) H (M) and the space of real harmonic and primitive basic (1, 1)-forms, respectively. We refer to Section in [6] for the Lefschetz decomposition theorem for a transverse Kähler foliation. Hence the maps p 1 Ï H 1 ( ) H n (M, ) H 2 (M), p 2 Ï H 2 ( ) H n1 (M, ) H 3 (M) are injective from Proposition 2.11 and the Lefschetz decomposition on basic differential forms. We obtain the following results: Theorem 6.5. The moduli space M CY (M,F) is a Hausdorff and smooth manifold of dimension dim Ê (H n, n 1,1 (M) H (M) È 1,1 ).,Ê Proof. It immediately follows from Theorem 5.6 and Proposition 6.4 that M CY (M, F) is a Hausdorff and smooth manifold. The dimension of M CY (M, F) is dim H 1 ( ) since M CY (M, F) is locally diffeomorphic to an open subset of H 1 ( ). In the proof of Proposition 6.4, we showed that H 1 ( ) is equal to H n, n 1,1 (M) H (M) È 1,1. Hence this ends the proof.,ê 6.3. Examples Linear foliations on tori. Let T 2n1 be the real torus Ê 2n1 2n1 of dimension 2n 1. We take a local coordinate (x 1,, x n, y 1,, y n, t) on T 2n1, then a foliation F (,) is induced by the vector field n i1 i i x i y i t

25 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 47 for (, ) ( 1,, n, 1,, n ) ¾ Ê 2n. The foliation F (,) is called a linear foliation on T 2n1. Note that F (,) is a taut foliation with respect to the standard flat metric on T 2n1. We define z i by complex functions z i x i i t Ô 1(y i i t) for i 1,, n, then (z 1,, z n ) is a transverse coordinate on (T 2n1, F (,) ). Now we define a pair (Đ, ) of forms as Đ dz 1 dz n, Ô 1 n dz i dæz i, 2 i1 then it is easy to see that (Đ, ) is a transverse Calabi Yau structure on (T 2n1,F (,) ). We start to compute the dimension of the moduli space M CY (T 2n1, F (,) ). The vector space À p,q (T 2n1 ) is generated by wedge products dz i1 dz i p dæz j1 Æz jq, and thus we obtain dim H p,q n n (T 2n1 ). p q It follows that dim Ê È 1,1,Ê dim È 1,1 n 2 1 from H 1,1 (T 2n1 ) È 1,1. The moduli space M CY (T 2n1, F (,) ) is a smooth manifold of dimension dim Ê (H n, (T 2n1 ) H n 1,1 (T 2n1 ) È 1,1 ) by Theorem 6.5.,Ê Hence we can see that dimm CY (T 2n1, F (,) ) 2(1 n 2 ) n 2 1 3n 2 1. We refer to [18] for deformations of transverse SL n () and Calabi Yau structures on (T 2n1, F (,) ) Null-Sasakian structures. A (2n 1)-dimensional Riemannian manifold (M, g) is a DEFINITION 6.6. Sasakian manifold if the metric cone (C(M), Æg) (Ê M, dr 2 r 2 g) is Kähler. Under the identification M with r 1 C(M), we can obtain a vector field and a 1-form over M defined by J r, ( ) g(, ) M

26 48 T. MORIYAMA where J is the complex structure on (C(M), Æg). Then the vector field is a Killing vector field such that each integral curve is geodesic. Thus induces a taut foliation F on M. The vector field and the foliation F are called a Reeb field Î and a Reeb 2 foliation, respectively. The 1-form satisfies that () 1 and d ¾. We may consider the distribution D over M defined by Ker. Then D is the 2n-dimensional distribution satisfying the orthogonal decomposition T M D F where F is the trivial line bundle generated by. We define a section of End(T M) by setting D J D and F. The data (,,, g) is called a Sasakian structure on M. Under the identification D with the quotient bundle Q T MF, the basic form d and the section induces a transverse Kähler structure on (M, F ). We denote by Ric Q the transverse Ricci tensor of the transverse Riemannian metric g Q g D. Then the transverse Ricci form Q is defined by Q (, ) Ric Q (, ). The form Q is a basic closed (1, 1)-form on (M,F ), and defines the basic cohomology class [ Q ] ¾ H 1,1 (M). The basic class [(12) Q] is called the basic first Chern class and is denoted by c 1 (M). DEFINITION 6.7. A Sasakian structure (,,, g) is a null-sasakian structure on M if c 1 (M). We say that (M,,,, g) is a null-sasakian manifold if (,,, g) is a null-sasakian structure on M. The class c 1 (M) is independent of the choice of a Sasakian structure whose Reeb foliation is F. y the transverse version of Yau s theorem proved by El Kacimi-Alaoui [6], there exists a transverse Calabi Yau structure (Đ, ) on a null-sasakian manifold (M,,,, g). Let us start to compute the dimension of the moduli space M CY (M,F ) on a null- Sasakian manifold (M,,,, g). We remark an important property of Sasakian structures that, on a compact Sasakian manifold M, a k-form is harmonic if and only if it is primitive and basic for 1 k n ([2, Proposition ]): (18) H k (M, ) È k for 1 k n. On a null-sasakian manifold M of 5-dimension (the case of n 2), we will see that dimm CY (M, F ) is given by the etti number of M. It is obvious that dim H 2, dim H,2 1 since Đ is the basic holomorphic (2, )-form. We have a decomposition (19) È 2 È 2, È 1,1 È,2 H 2, È 1,1 H,2.

27 MODULI OF TRANSVERSE CALAI YAU STRUCTURES 49 Equations (18) and (19) give rise to (2) dim È 1,1 dim È 2 2 dim H 2, b 2 2 where b 2 is the second etti number dim H 2 (M, ). It follows that dim H 1,1 dim È 1,1 1 b2 1 from equation (2) and the basic Lefschetz decomposition H 1,1 È 1,1. Thus we obtain (21) dim (H 2, H 1,1 ) 1 b2 1 b 2. We remark that H 2, H 1,1 can be regarded as the tangent space of deformations of transverse SL n () structures on (M, F ) (cf. [18]). The moduli space M CY (M, F ) is a smooth manifold of dimension dim Ê (H 2, 1,1 1,1 (M) H (M) È ). Hence equations,ê (2) and (21) yield (22) dimm CY (M, F ) 2b 2 b 2 2 3b 2 2. On a null-sasakian manifold M of 7-dimension (the case of n 3), we will find the dimension of the deformation space of transverse SL n () structures is given by etti numbers of M. Now we consider the basic Hodge decompositions È 3 È 3, È 2,1 È 1,2 È,3, È 1 È 1, È,1. Then it follows from (18) and dim È 3, dim H 3, 1 that (23) 2 dim È 2,1 b 3 2, 2 dim È 1, b 1 where b 1 and b 3 are etti numbers dim H 1 (M, ) and dim H 3 (M, ), respectively. y equation (23) and the basic Lefschetz decomposition H 2,1 È 2,1, we have and dim H 2,1 1 2 (b3 2) 1 2 b1 1 2 (b3 b 1 2), È 1, (24) dim (H 3, H 2,1 ) (b3 b 1 2) 1 2 (b3 b 1 ). The vector space H 3, H 2,1 can be identified with the tangent space of deformations of transverse SL n () structures on (M, F ). We also have that dim È 1,1 dim H 1,1 1

28 41 T. MORIYAMA by the relation H 1,1 È 1,1. Hence the manifold M CY(M,F ) has the dimension (25) dimm CY (M, F ) (b 3 b 1 ) dim È 1,1 b 3 b 1 dim H 1, Calabi Yau orbifolds. The geometry of Riemannian foliations is related to that of orbifolds, and so our method of transverse calibrations is useful for geometric structures on orbifolds. We will see this phenomenon on examples. We refer to [2] and [11] for some facts of Sasakian links and the notation of orbifolds, respectively. DEFINITION 6.8. A singular real manifold X of dimension m is an orbifold if singularities are locally isomorphic to quotient singularities Ê m G for finite subgroups G GL(m, Ê) such that each group G is small, that is, for any 1 ¾ G the subspace V Ê m fixed by has codimension at least two. We can also define a complex orbifold in a similar way. Any compact complex orbifold X is a leaf space of a Riemannian and transversely holomorphic foliation F on a smooth compact manifold É X (cf. [6, 4]). Therefore, we can regard geometric structures on an orbifold X as transverse geometric structures on a smooth foliated manifold ( É X, F). DEFINITION 6.9. Let X be an orbifold of dimension 2n. A pair (Đ, ) is a Calabi Yau structure on X if (Đ, ) is a Calabi Yau structure on the non-singular set of X in the sense of Definition 6.3, and wherever X is locally isomorphic to Ê 2n G, (Đ, ) is the quotient of a G-invariant Calabi Yau structure defined near in Ê 2n. We say that (X, Đ, ) is a Calabi Yau orbifold if (Đ, ) is a Calabi Yau structure on X. We denote by M orb CY (X) the moduli space of Calabi Yau structures on X, then M orb CY (X) is a smooth manifold by Theorem 6.5. Any Calabi Yau structure on X corresponds to a transverse Calabi Yau structure on ( X, É F). Thus the moduli space M CY ( X, É F) can be identified with M orb CY (X). We can easily compute the dimension of M CY ( X, É F) in a special case. We consider the -action on n1 defined by (z,, z n ) ( Û z,, Û n z n ) where ¾ and w (Û,, Û n ) ¾ ( ) n1. Let (w) denote this -action. The weighted projective space P(w) is defined as the quo- DEFINITION 6.1. tient ( n1 Ò ) (w). The weighted projective space P(w) is a complex orbifold. However, it is not a Calabi Yau orbifold. To obtain a Calabi Yau orbifold, we consider hypersurfaces on