Cohomologies of Sasakian groups and Sasakian solvmanifolds
aa r X i v : . [ m a t h . DG ] O c t COHOMOLOGIES OF SASAKIAN GROUPS AND SASAKIANSOLVMANIFOLDS
HISASHI KASUYA
Abstract.
We show certain symmetry of the dimensions of cohomologies of the funda-mental groups of compact Sasakian manifolds by using the Hodge theory of twisted basiccohomology. As applications, we show that the polycyclic fundamental groups of compactSasakian manifolds are virtually nilpotent and Sasakian solvmanifolds are finite quotients ofHeisenberg nilmanifolds. Introduction
The purpose of this paper is to study the fundamental groups of compact Sasakian manifolds.Sasakian manifolds constitute an odd-dimensional counterpart of the class of K¨ahler manifolds.A Riemannian manifold (
M, g ) is a Sasakian manifold if the cone metric on the cone manifold M × R + is a K¨ahler metric. The fundamental groups of compact K¨ahler manifolds satisfyvarious properties. Since the first cohomology of manifolds and the first group cohomology oftheir fundamental groups are isomorphic, the first group cohomology of the fundamental groupof a compact K¨ahler manifold was studied precisely by using the Hodge theory. In this paper,we study the first cohomologies of the fundamental groups of compact Sasakian manifolds for1-dimensional representations by using the Hodge theory of the basic cohomology.Let Γ be a group. We denote by C (Γ) the space of characters Γ → GL ( C ) which can befactored as Γ → H (Γ , Z ) / (torsion) → GL ( C ) . For ρ ∈ C (Γ), we denote by H ∗ (Γ , C ρ ) the group cohomology of Γ with values in the moduleassociated with the 1-dimensional representation ρ . Considering the exponential map C → C ∗ = GL ( C ), we have the surjective map E : H (Γ , C ) → C (Γ). We define the ”real” action of R ∗ on C (Γ) such that for f + √− f ∈ H (Γ , C ) with f , f ∈ H ( M, R ), the action is givenby t · E ( f + √− f ) = E ( tf + √− f )for t ∈ R ∗ .In this paper we prove the ” R ∗ -symmetry” of cohomologies of the fundamental group of acompact Sasakian manifold. Theorem 1.1.
Let M be a compact Sasakian manifold. Then for each ρ ∈ C ( π ( M )) and t ∈ R ∗ , we have dim H ∗ ( π ( M ) , C ρ ) = dim H ∗ ( π ( M ) , C t · ρ ) . For a group Γ, we consider the set J k (Γ) = { ρ ∈ C (Γ) | dim H (Γ , C ρ ) ≥ k } . Mathematics Subject Classification.
Key words and phrases.
Sasakian group, Hodge theory, solvmanifold, basic cohomology.
For C ( π ( M )) ∋ ρ = E ( f + √− f ) with f , f ∈ H ( M, R ), E ( f + √− f ) is fixed by the R ∗ -action if and only if f = 0 (equivalently ρ is unitary). Hence we have the following corollary. Corollary 1.2.
Let M be a compact Sasakian manifold. If there exists a non-unitary character ρ : π ( M ) → GL ( C ) satisfying H ( π ( M ) , C ρ ) ≥ k , then the set J k ( π ( M )) is an infinite set. A group Γ is polycyclic if it admits a sequenceΓ = Γ ⊃ Γ ⊃ · · · ⊃ Γ k = { e } of subgroups such that each Γ i is normal in Γ i − and Γ i − / Γ i is cyclic. By Corollary 1.2, weprove the following result which is analogous to the Arapura-Nori’s result in [2]. Corollary 1.3.
Let M be a compact Sasakian manifold. Suppose that the fundamental group π ( M ) is polycyclic. Then π ( M ) is virtually nilpotent i.e., it admits a nilpotent subgroup offinite index.Remark . In [7], Chen proves that the solvable fundamental group of a compact Sasakianmanifold is virtually nilpotent. Chen’s result is based on Campana’s results on K¨ahler orbifolds[5]. Campana uses algebraic and complex analytic geometrical techniques. In this paper, weonly use standard differential geometrical techniques on transversely K¨ahler foliations. Thisgives a technical advantage. In Chen’s proof, we need the existence of quasi-regular Sasakianstructure on a Sasakian manifold. On the other hand, in this paper, we do not use quasi-regularity.We consider nilmanifolds and solvmanifolds. Solvmanifolds (resp. nilmanifolds) are compacthomogeneous spaces of solvable (resp. nilpotent) Lie groups. It is known that every nilmanifoldcan be represented by G/ Γ such that G is a simply connected nilpotent Lie group and Γ is alattice in G (see [15]).In [6], it is proved that a compact 2 n +1-dimensional nilmanifold admits a Sasakian structureif and only if it is a Heisenberg nilmanifold H n +1 / Γ where H n +1 is the (2 n + 1)-dimensionalHeisenberg Lie group and Γ is its lattice.By Corollary 1.3, we can easily extend the result in [6] for solvmanifolds as in [10]. Corollary 1.4.
A compact n + 1 -dimensional solvmanifold admitting a Sasakian structure isa finite quotient of Heisenberg nilmanifold.Proof. It is known that the fundamental group of a compact solvmanifold is a torsion-freepolycyclic group and solvmanifolds with isomorphic fundamental groups are diffeomorphic (see[15]). Hence, by Corollary 1.3, we can easily show that a compact 2 n + 1-dimensional solvman-ifold admitting a Sasakian structure is a finite quotient of a Sasakian nilmanifold. Thus theCorollary follows from the result in [6]. (cid:3) In particular, we have the following result.
Corollary 1.5.
Let G be a n +1 -dimensional simply connected solvable Lie group with a lattice Γ . We assume that G is completely solvable (i.e., for any g ∈ G , all eigenvalues of the adjointoperator Ad g are real). Then the compact solvmanifold G/ Γ admits a Sasakian structure if andonly if it is a Heisenberg nilmanifold.Proof. By the Saito’s rigidity theorem in [16], if a simply connected completely solvable Liegroup contains a nilpotent lattice, then it is nilpotent. Hence, if G/ Γ admits a Sasakianstructure, then by Corollary 1.3 we can easily show that G is nilpotent. Thus the Corollaryfollows from the result in [6]. (cid:3) OHOMOLOGIES OF SASAKIAN GROUPS AND SASAKIAN SOLVMANIFOLDS 3
Acknowledgements.
The author would like to thank Xiaoyang Chen for introducing to his results. This researchis supported by JSPS Research Fellowships for Young Scientists.2.
Preliminary
Let M be a compact smooth manifold and A ∗ ( M ) the de Rham complex of M . For a C -valued closed 1-form φ ∈ A r ( M ) ⊗ C , we consider the operator φ ∧ : A r ( M ) ⊗ C → A r +1 ( M ) ⊗ C of left-multiplication. Define d φ = d + φ ∧ . Then we have d φ d φ = 0 and hence ( A ∗ ( M ) ⊗ C , d φ )is a cochain complex. We denote by H ∗ ( M, φ ) the cohomology of ( A ∗ ( M ) ⊗ C , d φ ). Thecochain complex ( A ∗ ( M ) ⊗ C , d φ ) is considered as the de Rham complex with values in thetopologically trivial flat bundle M × C with the connection form φ . Hence the structure ofthe cochain complex ( A ∗ ( M ) ⊗ C , d φ ) is determined by the character ρ φ : π ( M ) → GL ( C )given by ρ φ ( γ ) = exp (cid:16)R γ φ (cid:17) . We have an isomorphism H ( M, φ ) ∼ = H ∗ ( π ( M ) , C ρ φ ). Themap H ( M, C ) ∋ [ φ ] ρ φ ∈ C ( π ( M )) is identified with the map E as in Introduction. Hencethe action of R ∗ on C ( π ( M )) as in Introduction is given by t · ρ φ = ρ t Re φ + √− φ for t ∈ R ∗ and [ φ ] ∈ H ( M, C ). 3. Proof of Theorem 1.1
Basic cohomology.
Let M be a compact (2 n + 1)-dimensional Sasakian manifold with aSasakian metric g and η the contact structure associated with the Sasakian structure. Take ξ theReeb vector field. Let A ∗ ( M ) be the de Rham complex of M . A differential form α ∈ A ∗ ( M )is basic if ι ξ α = 0 and ι ξ dα = 0. Denote by A ∗ B ( M ) the differential graded algebra of thebasic differential forms on M and denote by H ∗ B ( M, R ) (resp. H ∗ B ( M, C )) the cohomology of A ∗ B ( M ) (resp. A ∗ B ( M ) ⊗ C ). Then it is known that the inclusion A ∗ B ( M ) ⊂ A ∗ ( M ) induces acohomology isomorphism H B ( M, R ) ∼ = H ( M, R ) and H B ( M, R ) ∼ = H ( M, R ) (see [4]).Consider the Hodge star operator ∗ : A r ( M ) → A n +1 − r ( M ) for the Sasakian metric g . Wedefine the transverse Hodge star operator ∗ T : A rB ( M ) → A n − rB ( M ) as ∗ T ( α ) = ∗ ( η ∧ α ).Then we have ∗ T α = ( − r α . Restricting the scalar product h , i : A ∗ ( M ) × A ∗ ( M ) → R onthe basic forms A ∗ B ( M ) we consider formal adjoint δ B : A rB ( M ) → A r − B ( M ) of the differential d on A ∗ B ( M ). Then we have δ B = − ∗ T d ∗ T . Consider the basic Laplacian ∆ B = dδ B + δ B d .A basic form α ∈ A rB ( M ) is harmonic if ∆ B α = 0. Denote H rB ( M ) = ker ∆ B | ArB ( M ) . Then wehave the Hodge decomposition (see [11], [9] and [18]) A rB ( M ) = H rB ( M ) ⊕ im d | A r − B ( M ) ⊕ im δ B | Ar +1 B ( M ) . We have the transverse complex structure on ker ι ξ ⊂ V T M ∗ and we obtain the bi-grading A rB ( M ) ⊗ C = L p + q = r A p,qB ( M ) with the bi-differential d = ∂ B + ¯ ∂ B . We consider the formaladjoint ∂ ∗ B : A p,qB ( M ) → A p − ,qB ( M ) and ¯ ∂ ∗ B : A p,qB ( M ) → A p,q − B ( M ) of ∂ B and ¯ ∂ B respectivelyfor the restricted Hermitian inner product on A ∗ B ( M ) ⊗ C . Then we have ∂ ∗ B = − ∗ T ¯ ∂ B ∗ T and¯ ∂ ∗ B = − ∗ T ∂ B ∗ T . Denote ∆ ′ = ∂ B ∂ ∗ B + ∂ ∗ B ∂ B and ∆ ′′ = ¯ ∂ B ¯ ∂ ∗ B + ¯ ∂ ∗ B ¯ ∂ B .Let ω = dη . Then ω gives a transverse K¨ahler structure. We can take a complex coordinatesystem ( z , . . . , z n ) which is transverse to ξ such that ω is a Kahler form on ( z , . . . , z n ). Definethe operator L : A p,qB ( M ) → A p +1 ,q +1 B ( M ) by Lα = ω ∧ α and consider the formal adjointΛ : A p,qB ( M ) → A p − ,q − B ( M ) of L . We have Λ = − ∗ T L ∗ T . By using the transverse K¨ahlergeometry, we obtain the following K¨ahler identity. HISASHI KASUYA
Lemma 3.1. Λ ∂ B − ∂ B Λ = √− ∂ ∗ B and Λ ¯ ∂ B − ¯ ∂ B Λ = −√− ∂ ∗ B . This implies ∆ B = 2∆ ′ B = 2∆ ′′ B and hence we have the Hodge structure H rB ( M ) ⊗ C = M p + q = r H p,qB ( M ) and H p,qB ( M ) = H q,pB ( M )where H p,qB ( M ) = ker ∆ ′ B | Ap,qB ( M ) = ker ∆ ′′ B | Ap,qB ( M ) .3.2. Twisted basic cohomology.
Let φ ∈ A B ( M ) ⊗ C be a closed basic 1-form. Then( A ∗ B ( M ) ⊗ C , d φ ) is a cochain complex. Denote by H ∗ B ( M, φ ) the cohomology of this complex.It is known that there exists a sub-torus
T ⊂
Isom(
M, g ) such that A ∗ B ( M ) ⊗ C ⊂ ( A ∗ ( M ) ⊗ C ) T and we have the exact sequence of complexes0 / / A ∗ B ( M ) ⊗ C / / ( A ∗ ( M ) ⊗ C ) T / / A ∗− B ( M ) ⊗ C / / d (see [4, Section 7.2.1]). We can say that this is also exact for twisteddifferential d φ . Hence, taking the long exact sequence, we have the exact sequence0 / / H B ( M, φ ) / / H ( M, φ ) / / H B ( M, φ ) . We can easily check H B ( M, φ ) = H ( M, φ ) = 0and hence we have:
Lemma 3.2. H B ( M, φ ) ∼ = H ( M, φ ) . Consider the formal adjoint ( φ ∧ ) ∗ B : A rB ( M ) ⊗ C → A r − B ( M ) ⊗ C of the operator φ ∧ for therestricted Hermitian inner product on A ∗ B ( M ) ⊗ C . Then we have( φ ∧ ) ∗ B = ∗ T ( ¯ φ ∧ ) ∗ T (see the proof of [14, Corollary 2.3]). Taking the formal adjoint ( φ ∧ ) ∗ : A r ( M ) ⊗ C → A r − ( M ) ⊗ C on the usual de Rham complex A ∗ ( M ) ⊗ C , we have( φ ∧ ) ∗| A ∗ ( M ) ⊗ C = ( φ ∧ ) ∗ B . Let δ B,φ = δ B + ( φ ∧ ) ∗ , ∆ B,φ = d φ δ B,φ + δ B,φ d φ and ker ∆ B,φ | Ar ( M ) ⊗ C = H rB ( M, φ ). As in [11],[9] and [18], we have the Hodge decomposition A rB ( M ) ⊗ C = H rB ( M, φ ) ⊕ im d φ | ArB ( M ) ⊗ C ⊕ im δ φ | ArB ( M ) ⊗ C . Consider the double complex ( A ∗ , ∗ B ( M ) , ∂ B , ¯ ∂ B ). For a (1 , θ , considering theoperators θ ∧ : A p,qB ( M ) → A p +1 ,qB ( M ) and Λ : A p,qB ( M ) → A p − ,q − B ( M ), as the local argumentfor the K¨ahler identities, see, e.g., [19, Lemma 6.6], we have the following identity. Lemma 3.3. Λ( θ ∧ ) − ( θ ∧ )Λ = −√− ∗ T ( θ ∧ ) ∗ T and Λ(¯ θ ∧ ) − (¯ θ ∧ )Λ = √− ∗ T (¯ θ ∧ ) ∗ T . Let φ ∈ H B ( M ) ⊗ C . By H B ( M ) = H , B ( M ) ⊕H , B ( M ), we can take unique θ , θ ∈ H , B ( M )such that φ = θ + ¯ θ + θ − ¯ θ . Define ∂ B,θ ,θ = ∂ B + θ ∧ +¯ θ ∧ and ¯ ∂ B,θ ,θ = ¯ ∂ B − ¯ θ ∧ + θ ∧ . OHOMOLOGIES OF SASAKIAN GROUPS AND SASAKIAN SOLVMANIFOLDS 5 By H p,qB ( M ) = ker ∆ ′ B | Ap,qB ( M ) = ker ∆ ′′ B | Ap,qB ( M ) , ( A ∗ B ( M ) ⊗ C , ∂ B,θ ,θ ) and ( A ∗ B ( M ) ⊗ C , ¯ ∂ B,θ ,θ )are cochain complexes. Denote by H ∗ B ( M, θ , θ ) the cohomology of ( A ∗ B ( M ) ⊗ C , ¯ ∂ B,θ ,θ ). Assimilar to [13, Lemma 2.1], we obtain the following lemma. Lemma 3.4.
For any t ∈ R ∗ , we have dim H ∗ B ( M, θ , θ ) = dim H ∗ B ( M, tθ , θ ) . Consider the formal adjoint ∂ ∗ B,θ ,θ = ∂ ∗ B + ( θ ∧ ) ∗ + (¯ θ ∧ ) ∗ and ¯ ∂ ∗ B,θ ,θ = ¯ ∂ ∗ B − (¯ θ ∧ ) ∗ + ( θ ∧ ) ∗ . Since we have ( θ i ∧ ) ∗ = ∗ T (¯ θ i ∧ ) ∗ T , by the Lemma 3.1 and 3.3, we obtain the following K¨ahleridentity (cf. [1, Section 1.2], [17, Page 15]). Lemma 3.5. Λ ∂ B,θ ,θ − ∂ B,θ ,θ Λ = √− ∂ ∗ B,θ ,θ and Λ ¯ ∂ B,θ ,θ − ¯ ∂ B,θ ,θ Λ = −√− ∂ ∗ B,θ ,θ . Define∆ ′ B,θ ,θ = ∂ B,θ ,θ ∂ ∗ B,θ ,θ + ∂ ∗ B,θ ,θ ∂ B,θ ,θ and ∆ ′′ B,θ ,θ = ¯ ∂ B,θ ,θ ¯ ∂ ∗ B,θ ,θ + ¯ ∂ ∗ B,θ ,θ ¯ ∂ B,θ ,θ . Then Lemma 3.5 implies ∆
B,φ = 2∆ ′′ B,θ ,θ = 2∆ ′ B,θ ,θ . Denote H rB ( M, θ , θ ) = ∆ ′′ B,θ ,θ | A rB ( M ) ⊗ C .We have the Hodge decomposition A rB ( M ) ⊗ C = H rB ( M, θ , θ ) ⊕ im ¯ ∂ B,θ ,θ | A r − B ( M ) ⊗ C ⊕ im ¯ ∂ ∗ B,θ ,θ | A r +1 B ( M ) ⊗ C . Hence we obtain an isomorphism H rB ( M, φ ) ∼ = H rB ( M, θ , θ ) . By Lemma 3.2 and 3.4, we have the following result.
Theorem 3.6.
For any t ∈ R ∗ , we have the equation dim H ( M, φ ) = dim H ( M, tθ + t ¯ θ + θ − ¯ θ ) . Since we have isomorphisms H ( M, C ) ∼ = H B ( M, C ) ∼ = H B ( M ) ⊗ C , we obtain Theorem 1.1.4. Proof of Corollary 1.3
Let G be a simply connected solvable Lie group and N be the nilradical (i.e. maximalconnected nilpotent normal subgroup) of G . Denote by g the Lie algebra of G . Then we cantake a simply connected nilpotent subgroup C ⊂ G such that G = C · N (see [8, Proposition3.3]). Since C is nilpotent, the mapΦ : C ∋ c (Ad c ) s ∈ Aut( g )is a diagonalizable representation where (Ad c ) s is the semi-simple part of the adjoint operatorAd c (see [12]). We take a diagonalization Φ = diag( α , . . . , α n ) where α , . . . , α n are C -valuedcharacters of C . Since the adjoint representation Ad on the nilradical N is unipotent and wehave an isomorphism G/N ∼ = C/C ∩ N , α , . . . , α n are considered as characters of G . For each g ∈ G , α i ( g ) is an eigenvalue of Ad g .Suppose G admits a lattice Γ. We consider the solvmanifold G/ Γ. G/ Γ is an asphericalmanifold with the fundamental group Γ. In [13], the set J (Γ) was studied. The author provedthat J (Γ) is a finite set ([13, Corollary 5.9]) and if one of the characters α , . . . , α n is non-unitary, then there exists a non-unitary character ρ ∈ C (Γ) of Γ such that ρ ∈ J (Γ) ([13,Corollary 5.10]). We suppose that Γ can be the fundamental group of a compact Sasakianmanifold. Then by Corollary 1.2, characters α , . . . , α n are all unitary. Hence, for any g ∈ G , HISASHI KASUYA all eigenvalues of Ad g are unitary. In this case, a lattice Γ of G is virtually nilpotent (see [3,Chapter IV. 5]). It is known that every polycyclic group contains a lattice of some simplyconnected solvable Lie group as a finite index normal subgroup (see [15, Theorem 4.28]). HenceCorollary 1.3 follows. References [1] D. Angella, H. Kasuya, Hodge theory for twisted differentials. Complex Manifolds (2014), 64–85.[2] D. Arapura, M. Nori, Solvable fundamental groups of algebraic varieties and K¨ahler manifolds, CompositioMath. (1999), no. 2, 173–188.[3] L. Auslander, An exposition of the structure of solvmanifolds, I. Algebraic theory., Bull. Amer. Math. Soc. (1973), no. 2, 227–261.[4] C. P. Boyer, K. Galicki, Sasakian geometry. Oxford Mathematical Monographs. Oxford University Press,Oxford, 2008.[5] F. Campana, Quotients r´esoluble ou nilpotents des groupes de K¨ahler orbifoldes, Manus. Math. (12)(2011), 117–150.[6] B. Cappelletti-Montano, A. De Nicola, J. C. Marrero, I. Yudin, Sasakian nilmanifolds. Int Math Res Notices(2014) doi: 10.1093/imrn/rnu144[7] X. Chen On the fundamental groups of compact Sasakian manifolds. Math. Res. Lett. (2013), no. 1,27–39.[8] K. Dekimpe, Semi-simple splittings for solvable Lie groups and polynomial structures, Forum Math. (2000), no. 1, 77–96.[9] A. El Kacimi-Alaoui, Op´erateurs transversalement elliptiques sur un feuilletage riemannien et applications.Compositio Math. (1990), no. 1, 57–106.[10] K. Hasegawa, A note on compact solvmanifolds with K¨ahler structures. Osaka J. Math. (2006), no. 1,131–135.[11] F. W. Kamber, P. Tondeur, de Rham-Hodge theory for Riemannian foliations. Math. Ann. (1987), no.3, 415–431.[12] H. Kasuya, de Rham and Dolbeault cohomology of solvmanifolds with local systems. Math. Res. Lett. (2014), no. 4, 781–805.[13] H. Kasuya, Flat bundles and Hyper-Hodge decomposition on solvmanifolds, arXiv:1309.4264v1[math.DG] . to appear in Int. Math. Res. Not. IMRN[14] E. Park, K. Richardson, The basic Laplacian of a Riemannian foliation. Amer. J. Math. (1996), no. 6,1249–1275.[15] M. S. Raghnathan, Discrete subgroups of Lie Groups, Springer-verlag, New York, 1972.[16] M. Saitˆo, Sur certains groupes de Lie r´esolubles II, Sci. Papers Coll. Gen. Ed. Univ. Tokyo (1957) 157–168.[17] C. T. Simpson, Higgs bundles and local systems. Inst. Hautes ´Etudes Sci. Publ. Math.
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