Basic Kirwan injectivity and its applications
aa r X i v : . [ m a t h . S G ] F e b BASIC KIRWAN INJECTIVITY AND ITS APPLICATIONS
YI LIN AND XIANGDONG YANG
Abstract.
Consider the Hamiltonian action of a torus on a transversely symplectic foliationthat is also Riemannian. When the transverse hard Lefschetz property is satisfied, weestablish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltoniantorus actions on transversely K¨ahler foliations. Among other things, we prove a foliatedversion of the Carrell-Liberman theorem. As an immediate consequence, this confirms aconjecture raised by Battaglia and Zaffran on the basic Hodge numbers of symplectic toricquasifolds. As an aside, we also present a symplectic approach to the calculation of basicBetti numbers of symplectic toric quasifolds. Introduction
Reinhart [19] introduced the notion of basic cohomology as a cohomology theory for theleaf space of a foliation. It turns out to be a very useful tool in the study of Riemannianfoliations. Killing foliation is an important class of Riemannian foliations, and is known topossess a type of “ internal ” symmetry given by the transverse action of their structural Liealgebras. In order to study this important type of symmetries, Goertsches and T¨oben [9]proposed the notion of equivariant basic cohomology . Among other things, they proved aBorel type localization theorem for the transverse actions of the structural Lie algebras onKilling foliations. Lin and Sjamaar [15] generalized their result to the transverse isometricaction of an arbitrary Lie algebra on a general Riemannian foliation.In a slightly different direction, Lin and Sjamaar [13] considered Hamiltonian action of acompact Lie group on a transversely symplectic foliation. They discovered that when theaction is clean , components of a moment map must be Morse-Bott functions, and extendedAtiyah-Guillemin-Sternberg-Kirwan convexity theorem to clean Hamiltonian actions. Linand Yang [14] studied Hamiltonian actions on a transversely symplectic foliation from Hodgetheoretic viewpoint, and established in this setup the equivariant formality result for theequivariant basic cohomology. In the current paper, building on the equivariant formalityresult in [14] and the Borel localization result in [15], we establish the following foliatedversion of the Kirwan injectivity theorem in symplectic geometry.
Theorem 1.1.
Let ( M, F ) be a transversely symplectic foliation on a closed manifold M thatis also Riemannian. Suppose that ( M, F ) satisfies the transverse hard Lefschetz property, thatthere is a Hamiltonian action of a compact torus G on M . Let X be the fixed-leaf set of M Date : February 20, 2019.2010
Mathematics Subject Classification.
Key words and phrases. transversely symplectic foliations, basic cohomology, Kirwan injectivity.The second author was partially supported by the National Natural Science Foundation of China (grantNo. 11701051) and the China Scholarship Council/Cornell University Joint Scholarship Program. and i : X ֒ → M the inclusion map. Then the localization homomorphism in equivariant basiccohomology i ∗ : H G ( M, F ) → H G ( X, F ) is injective. To demonstrate the usefulness of the Kirwan injectivity theorem in foliation theory, weapply Theorem 1.1 to study Hamiltonian torus actions on transversely K¨ahler foliations.When the action preserves the transversely holomorphic structure, we obtain a foliationanalogue of the Carrell-Lieberman theorem [6] and establish the vanishing of certain basicHodge numbers. When the Hamiltonian action is clean and when the fixed leaf set consistsentirely of isolated closed leaves, we prove that the odd dimensional basic Betti number mustvanish, and that the basic Euler characteristic number is given by the total number of thevertices in the moment polytope.Using a variation of the Delzant construction, out of any non-rational simple polytopePrato [18, Section 3] constructed a foliation analogue of a symplectic toric manifold, calleda symplectic toric quasifold. From our viewpoint, Prato’s toric quasifolds provide a rich andinteresting class of Hamiltonian torus actions on transversely K¨ahler foliations to which ourmain results apply very well. Indeed, our foliation analogue of the Carrell-Liberman theoremprovides immediately a positive answer to a conjecture raised by Battaglia and Zaffran [3],which asserts that the basic Hodge numbers of toric quasifolds are concentrated on the maindiagonal. Our method also enables us to completely determine the basic Betti numbers ofsymplectic toric quasifolds, and show that they are given by the components of the h -vectorsof the moment polytope.Historically the study of toric varieties involves two important viewpoints: symplectic vs.algebraic. Similarly, there are two different approaches to toric quasifolds from symplectic andcomplex geometries respectively. Battaglia and Zaffran [3] first noted that toric quasifoldscan be realized as the leaf spaces of holomorphic foliations on compact LVMB complexmanifolds, and calculated the basic Betti numbers of a class of holomorphic foliations onLVMB manifolds constructed using shellable simplicial fans. Lin and Sjamaar [13] thenobserved that toric quasifolds can also be realized as the leaf spaces of transversely symplecticfoliations. More recently, Ishida [11] showed that the basic Hodge numbers of holomorphicfoliations on certain complex manifolds with maximal torus actions concentrated on the maindiagonal, which appears to be the complex geometric counterpart of our Corollary 6.2 in thispaper. Nevertheless, we believe that our work from symplectic viewpoint complements theexisting works arising from the world of complex geometry. To our best knowledge, oursymplectic approach to basic Betti numbers of toric quasifolds appears to be new even forsymplectic toric manifolds (cf. [5, Theorem I.3.6]). Furthermore, as shown in the exampleconstructed in the proof of Theorem 5.8, our main result on the vanishing of basic Hodgenumbers applies to a wider class of transversely K¨ahler foliations which may not admit alarge symmetry from a torus of greatest possible dimension.The rest of this paper is organized as follows. Section 2 recalls the definitions of transversegeometric structures on foliations. Section 3 reviews the equivariant basic cohomology andestablishes the Kirwan injectivity theorem. Section 4 collects some useful facts on clean ASIC KIRWAN INJECTIVITY 3
Hamiltonian torus actions. Section 5 proves an analogue of the Carrell-Liberman theorem ina foliated setting, as well as a result on the basic Betti numbers and basic Euler characteristicnumbers of transversely symplectic foliations with Hamiltonian torus actions. Section 6applies the main results to determine the basic Hodge numbers and basic Betti numbers ofsymplectic toric quasifolds.
Acknowledgements.
The first author would like to thank the School of Mathematics ofSichuan University for hosting his research visit during the summer of 2017 and 2018 whenhe was working on things related to this project. The second author would like to thankthe Department of Mathematics of Cornell University for providing him an excellent workingenvironment since he joined Cornell University as a visiting scholar in September 2017. Heis also indebted to the China Scholarship Council for the financial support during his visit.Part of this project grows out of the first author’s joint work with Professor Reyer Sjamaar.Both authors are grateful to him for many useful discussions.2.
Transverse geometric structures on foliations
Let F be a foliation on a smooth manifold M , and let T F be the tangent bundle of thefoliation. Throughout this paper we denote by X ( F ) ⊂ X ( M ) the subspace of vector fieldstangent to the leaves of F . We say that a vector field X ∈ X ( M ) is foliate , if [ X, Y ] ∈ X ( F )for all Y ∈ X ( F ). We will denote by X ( M, F ) the space of foliate vector fields on ( M, F ).Clearly we have that X ( F ) ⊂ X ( M, F ). In this context, a transverse vector field is anequivalent class in the quotient space X ( M, F ) / X ( F ). The space of transverse vector fields,denoted by X ( M/ F ), forms a Lie algebra with a Lie bracket inherited from the natural oneon X ( M, F ). The space of basic forms on M is defined to beΩ( M, F ) = (cid:8) α ∈ Ω( M ) | ι ( X ) α = L ( X ) α = 0 , for X ∈ X ( F ) (cid:9) . Since the exterior differential operator d preserves basic forms, we obtain a sub-complex { Ω ∗ ( M, F ) , d } of the usual de Rham complex, called the basic de Rham complex . The asso-ciated cohomology H ∗ ( M, F ) is called the basic cohomology .Let Q = T M/T F be the normal bundle of the foliation. A moment’s consideration showsthat for any foliate vector field X , and for any ( r, s )-type tensor σ ∈ C ∞ ( Q ∗ ⊗ · · · ⊗ Q ∗ | {z } r ⊗ Q ⊗ · · · ⊗ Q | {z } s ) , the Lie derivative L ( X ) σ is well defined. Definition 2.1. A transverse Riemannian metric on a foliation ( M, F ) is a Riemannianmetric g on the normal bundle Q of the foliation, such that L ( X ) g = 0, for X ∈ X ( F ).We say that F is a Riemannian foliation if there exists a transverse Riemannian metric on( M, F ).Let q be the co-dimension of a Riemannian foliation F on a compact manifold M . If H q ( M, F ) = R , we say that F is homologically orientable . The following result [20] usesthe notion of Molino sheaf to characterize when a Riemannian foliation is homologicallyorientable. We refer the interested readers to [17] and [15] for a detailed account on Molinosheaf. YI LIN AND XIANGDONG YANG
Lemma 2.2.
Let F be a Riemannian foliation on a compact manifold. Then F is homologi-cally orientable if and only if the top wedge power of its Molino sheaf has a nowhere vanishingglobal section. Let ( M, F ) be a Riemannian foliation with a transverse metric g , and let X be a transversevector field. Define L ( X ) g = L ( X ) g , where X is a foliate vector field that represents X . Thenit is straightforward to check that the definition of the Lie derivative L ( X ) g does not dependon the choice of X . A transverse vector field X is said to be transversely Killing if L ( X ) g = 0.Suppose that both X and Y are transversely Killing. Then it follows easily from the Cartanidentities that [ X, Y ] is also transversely Killing. In other words, the space of transverselyKilling vector fields, which we denote by Iso( M/ F ), forms a Lie subalgebra of X ( M/ F ). Definition 2.3. A transverse almost complex structure J on ( M, F ) is an almost complexstructure J : T M/T
F →
T M/T F such that L ( X ) J = 0, for X ∈ X ( F ). A transversealmost complex structure J on ( M, F ) is said to be integrable , if for every p ∈ M , thereexists an open neighborhood U of p , such that for any two transverse vector fields X and Y on U with respect to the foliation F | U , the Nijenhaus tensor N J ( X, Y ) = [ J X, J Y ] − J [ J X, Y ] − J [ X, J Y ] − [ X, Y ]vanishes. An integrable transverse almost complex structure is also called a transverse com-plex structure . The foliation F is said to be transversely holomorphic if there is a transversecomplex structure J on ( M, F ). Definition 2.4.
Let F be a foliation on a smooth manifold M . We say that F is a transverselysymplectic foliation , if there exists a closed 2-form ω , called a transversely symplectic form ,such that for each x ∈ M , the kernel of ω x coincides with T x F . Definition 2.5. A transversely K¨ahler structure on ( M, F ) consists of a transverse complexstructure J and a transverse Riemannian metric g , such that the tensor field ω defined by ω ( X, Y ) = g ( X, J Y ) is transversely symplectic when considered as a 2-form on M given bythe injection ∧ Q ∗ → ∧ T ∗ M. The 2-form ω will be called a transverse K¨ahler form . Wesay that F is a transversely K¨ahler foliation if there exists a transverse K¨ahler structure on( M, J ).Applying the basic Hodge theory developed in [12], the following foliated version of the¯ ∂∂ -lemma was shown in [7, Lemma 1]. Theorem 2.6.
Suppose that F is a homologically orientable transversely K¨ahler foliation ona compact manifold M . Then on the space of basic forms Ω( M, F ) the following ¯ ∂∂ -lemmaholds. ker ¯ ∂ ∩ im ∂ = im ¯ ∂ ∩ ker ∂ = im ¯ ∂∂. Kirwan injectivity theorem for the Equivariant basic cohomology
In this section, we prove a foliated version of the Kirwan injectivity theorem, i.e., Theorem1.1. We begin with a review of the notion of transverse Lie algebra actions, and the associatednotion of equivariant basic cohomology.
ASIC KIRWAN INJECTIVITY 5
Definition 3.1. A transverse action of a Lie algebra g on a foliated manifold ( M, F ) is aLie algebra homomorphism g → X ( M/ F ) . (3.1)A transverse action of g on ( M, F ) is said to be isometric, if the image of the map (3.1) liesinside Iso( M/ F ).Suppose that there is a transverse action of a Lie algebra g on a foliated manifold ( M, F ).For all ξ ∈ g , we will denote by ξ M the transverse vector that is the image of ξ under (3.1),and by ξ M the foliate vector that represents ξ M . For α ∈ Ω( M, F ), define ι ( ξ ) α = ι ( ξ M ) α, L ( ξ ) α = L ( ξ M ) α. Since α is basic, the contraction and Lie derivative operations defined above do not dependon the choices of representatives of the transverse vector field ξ M . Goertsches and Toben [9,Proposition 3.12] observed that they obey the usual rules of E. Cartan’s differential calculus,namely [ L ( ξ ) , L ( η )] = L ([ ξ, η ]) etc. To put it another way, a transverse g -action equipsthe basic de Rham complex Ω( M, F ) with the structure of a g ⋆ -algebra in the sense of [10,Chapter 2]. Therefore there is a well-defined Cartan model of the g ⋆ -algebra Ω( M, F ) givenby Ω g ( M, F ) := [ S g ∗ ⊗ Ω( M, F )] g . An element of Ω g ( M, F ) can be naturally identified with an equivariant polynomial map from g to Ω( M, F ), and is called an equivariant basic differential form .The equivariant basic Cartan complex has a bigrading given byΩ i,j g ( M, F ) = [ S i g ∗ ⊗ Ω j − i ( M, F )] g ;moreover, it is equipped with the vertical differential 1 ⊗ d , which we abbreviate to d , andthe horizontal differential d ′ , which is defined by( d ′ α )( ξ ) = − ι ( ξ ) α ( ξ ) , ∀ ξ ∈ g . As a single complex, Ω g ( M, F ) has a grading given byΩ k g ( M, F ) = M i + j = k Ω i,j g ( M, F ) , and a total differential d g = d + d ′ , which is called the equivariant exterior differential. Theequivariant basic de Rham cohomology H g ( M, F ) of the transverse g -action on ( M, F ) isdefined to be the total cohomology of the Cartan complex { Ω g ( M, F ) , d g } .Now let G be a compact connected Lie group with Lie algebra g . A transverse action of G on a foliated manifold ( M, F ) is simply a transverse action of its Lie algebra g on ( M, F ).The equivariant basic cohomology of a transverse G -action on ( M, F ) is defined to be H G ( M, F ) := H g ( M, F ) . We say that the action of a Lie group G on a foliated manifold ( M, F ) is foliate , if theaction preserves the foliation structure. Suppose that there is a foliate G -action on a foliated YI LIN AND XIANGDONG YANG manifold ( M, F ). Then we have the following commutative diagram g ●●●●●●●●●● / / X ( M, F ) pr (cid:15) (cid:15) X ( M/ F ) . Here the horizontal map is induced by the infinitesimal action of g on M , and the verticalmap is the natural projection. Thus a foliate action of G naturally induces a transverse actionof G . As transverse vector fields are not genuine vector fields on M , the converse may notbe true. Definition 3.2. ([15]) Consider the action of a Lie group G with Lie algebra g on a foliatedmanifold M with a transversely symplectic foliation ( F , ω ). We say that the action of G is Hamiltonian , if there exists a G -equivariant map Φ : M → g ∗ , called the moment map, suchthat ι ( ξ M ) ω = d h Φ , ξ i , for all ξ ∈ g . Here ξ M is the fundamental vector field on M generated by ξ , and h· , ·i denotes the dualpairing between g ∗ and g . It is easy to check that a Hamiltonian action must automaticallybe foliate. Definition 3.3.
A foliate G -action on ( M, F , J ) is holomorphic , if the induced G -action onthe normal bundle of the foliation T M/T F preserves the transverse complex structure J .We recall the following equivariant formality result proved in [14, Theorem 1.1]. Theorem 3.4 (Equivariant Formality [14, Theorem 1.1]) . Consider the Hamiltonian actionof a compact group G on a transversely symplectic foliation ( F , ω ) over a compact manifold M . Suppose that ( M, F , ω ) satisfies the transverse hard Lefschetz property. Then there is acanonical ( S g ∗ ) G -module isomorphism from the equivariant basic cohomology H G ( M, F ) to ( S g ∗ ) G ⊗ H ( M, F ) . We are in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
By assumption, there is a transverse Riemannian metric g on ( M, F ).By averaging over the compact torus G we get another transverse Riemannian metric g that is invariant under the action of G . Thus without loss of generality we may assume thatthe induced transverse action of G is isometric. Then by [15, Proposition 5.2.3] connectedcomponents of the fixed-leaf set X are F -saturated submanifolds of M that are invariantunder the action of G . By [15, Theorem 5.3.1], the kernel of the restriction homomorphism i ∗ : H G ( M, F ) → H G ( X, F ) (3.2)is a S g ∗ -torsion submodule. However, by Theorem 3.4 H G ( M, F ) is a free S g ∗ -module.Therefore the map (3.2) must be injective. (cid:3) According to El Kacimi-Alaoui [12, Section 3.4.7], any homologically orientable trans-versely K¨ahler foliation on closed manifolds must satisfy the transverse hard Lefschetz prop-erty. As an immediate consequence of Theorem 1.1 we have the following result.
ASIC KIRWAN INJECTIVITY 7
Corollary 3.5.
Let F be a homologically orientable transversely K¨ahler foliation on a closedmanifold M . Suppose that there is a Hamiltonian action of a compact torus G on ( M, F ) that is also holomorphic. Then the restriction homomorphism i ∗ : H G ( M, F ) → H G ( X, F ) is injective, where X is the fixed-leaf set. Clean Hamiltonian torus actions on transversely symplectic foliations
In this section we collect some useful facts on the image of a moment map of a cleanHamiltonian torus action, which we need later in this paper. We recall the definition of clean group actions on foliated manifolds.
Definition 4.1. ([13, Section 2.6]) Consider the foliate action of a Lie group G on a foliatedmanifold ( M, F ). We say that the G -action on M is clean, if there exists an immersedconnected normal Lie subgroup N of G , called the null subgroup , such that T x ( N · x ) = T x ( G · x ) ∩ T x F , for all x ∈ M. Definition 4.2.
Suppose that F is a transversely symplectic foliation on a manifold M , andthat there is a clean foliate action of a compact torus T with Lie algebra t on ( M, F ). Forall x ∈ M , set t ¯ x = { ζ ∈ t | ζ M ( x ) ∈ T x F } . We say that ξ ∈ t is generic , if ξ / ∈ [ t ¯ x = t t ¯ x . Remark . Suppose that N is the null subgroup of T , and that n = Lie( N ). Since theaction of T is assumed to be clean, it is straightforward to check that t ¯ x = t x + n . If M is compact, then there are only finitely many distinct isotropy subalgebras t x , and so onlyfinitely many Lie subalgebras t ¯ x .Throughout the rest of this section, we assume that there is an effective and clean Hamil-tonian action of a compact torus T on a transversely symplectic foliation F over a compactmanifold M , that the null group is N with a Lie algebra n , and that Φ : M → t ∗ is a momentmap, where t = Lie( T ). Lemma 4.4.
For any generic element ξ ∈ t ∗ , the critical subset Crit (Φ ξ ) of Φ ξ coincideswith X , the set of fixed leaves under the action of T .Proof. It suffices to show that Crit (Φ ξ ) ⊂ X . Suppose that x ∈ M is a critical point of Φ ξ .Then it follows from the Hamiltonian equation ι ( ξ M ) ω = d Φ ξ that ξ M is tangent to the leafat x . This shows that ξ M ∈ t x . Since ξ is generic, we must have that t ¯ x = t . Thus for all ζ ∈ t , the vector field ζ M must be tangent to the leaf at x . It follows that the leaf through x is invariant under the action of T . This completes the proof of Lemma 4.4. (cid:3) Lemma 4.5.
The image ∆ = Φ( M ) of the moment map is a convex polytope, called the moment polytope . Moreover, for any face F ⊂ ∆ , the set Φ − ( F ) is a connected F -saturated T -invariant submanifold of M to which the restriction of F is transversely symplectic. YI LIN AND XIANGDONG YANG
Proof.
The first assertion was shown in [13, Theorem 3.5.1]. Now let h F = { ξ ∈ t | h x, ξ i = max y ∈ ∆ h y, ξ i , for all x ∈ F } , where h· , ·i dentoes the dual pairing between t ∗ and t . Then it is easy to see that h is aLie subalgebra of Lie( T ) that contains n , and that Φ − ( F ) is a disjoint union of connectedcomponents of M [ h F ] = { x ∈ M | ξ M ( x ) ∈ T x F , for all ξ ∈ h F } , where ξ M denotes the fundamental vector field on M generated by ξ ∈ h F . It follows from[13, Proposition 3.4.4] that each connected component of Φ − ( F ) must be a F -saturatedsubmanifold of M to which the restriction of F is transversely symplectic. To finish the proofof Lemma 4.5, it suffices to show that Φ − ( F ) is connected. By mathematical induction onthe co-dimension of F in ∆, we may assume that F is a facet of ∆. In this case, h F / n is onedimensional. Choose a vector ζ ∈ h F such that its image under the quotient map h F → h F / n is not zero. Then Φ − ( F ) is precisely the maximum critical submanifold of Φ ζ . Note that by[13, Theorem 3.4.5] Φ ζ is a Morse-Bott function with even indices. It follows from a classicalresult of Atiyah [2, Lemma 2.1] that Φ − ( F ) must be connected. (cid:3) Let ξ ∈ t be a generic element. Then [13, Theorem 3.5.1], together with Lemma 4.4,implies that there is a one-to-one correspondence between the connected components of crit-ical submanifold of the Morse-Bott function Φ ξ and the vertices of the moment polytope∆ = Φ( M ). Definition 4.6.
Fix a generic element ξ ∈ t . We will say that a vertex λ ∈ ∆ has index l relative to the Morse-Bott function Φ ξ , if the corresponding critical submanifold Φ − ( λ ) hasMorse index l . The index of the vertex λ will be denoted by ind ξ ( λ ). Lemma 4.7.
Let ξ ∈ t be a generic element, and let F be a non-empty face of the momentpolytope ∆ . Then there is a unique vertex of F , denoted by λ F , such that ind ξ ( λ F ) < ind ξ ( ν ) , for all vertices ν = λ F ∈ F. (4.1) The point λ F will be called the vertex of F with the minimal index.Proof. Note that by Lemma 4.5 the set Y = Φ − ( F ) is a connected F -saturated T -invariantsubmanifold of M to which the restriction of F is transversely symplectic. Clearly, the actionof T on ( Y, F | Y ) is also clean and Hamiltonian. Thus by [13, Theorem 3.4.5], the functionΦ ξ | Y is also a Morse-Bott function with even indexes. As a result, the restriction of Φ ξ to Y must have a unique local minimum on Y , see [2, Lemma 2.1]. It follows from Lemma 4.4 and[13, Theorem 3.5.1] that this local minimum coincides with Φ − ( λ F ) for some vertex λ F ∈ F .Finally, (4.1) follows from the uniqueness of the local minimum. (cid:3) Basic Hodge numbers and basic Betti numbers
A foliated Carrell-Lieberman-type theorem.
Let J be a transversely holomorphicstructure on a foliation ( M, F ). Then J induces a direct sum decomposition of the complexof basic forms Ω( M, F ) = M p,q ≥ Ω p,q ( M, F ) , ASIC KIRWAN INJECTIVITY 9 where by definition a basic form α ∈ Ω p,q ( M, F ) if and only if J α = ( √− p − q α . Let α be abasic form of ( p, q )-type. Define ¯ ∂α to be the component of dα that lies in Ω p,q +1 ( M, F ), and ∂α the component of dα that lies in Ω p +1 ,q ( M, F ). Then the exterior differential d naturallysplits as d = ¯ ∂ + ∂ .Now suppose that there is a foliate action of a compact Lie group G with Lie algebra g on( M, F ) which preserves the transversely holomorphic structure J . Since the action of G isholomorphic, Ω p,q ( M, F ) is a G -module for all ( p, q ). Thus the spaceΩ p,q g ( M ) := M p ′ + i = pq ′ + i = q (cid:0) S i ( g ∗ ) ⊗ Ω p ′ q ′ ( M, F ) (cid:1) g (5.1)is well defined for all ( p, q ).For any ξ ∈ g , denote by ξ , M and ξ , M respectively the (1 ,
0) and (0 ,
1) components of thetransverse vector field on M induced by ξ . Then on the basic Cartan complex Ω g ( M, F ) theoperator d ′ splits as d ′ = d ′ , + d ′ , , where( d ′ , α )( ξ ) = ι ( ξ , M )( α ( ξ )) and ( d ′ , α )( ξ ) = ι ( ξ , M )( α ( ξ )) , for all α ∈ Ω g ( M, F ) and for all ξ ∈ g . Also note that on the space of equivariant basicforms Ω g ( M, F ) the operator 1 ⊗ d splits as 1 ⊗ d = 1 ⊗ ¯ ∂ + 1 ⊗ ∂ . For brevity, we will alsoabbreviate 1 ⊗ ¯ ∂ to ¯ ∂ and 1 ⊗ ∂ to ∂ . Thus the equivariant exterior differential d g splits as d g = ¯ ∂ g + ∂ g , where ∂ g = ∂ + d ′ , , ¯ ∂ g = ¯ ∂ + d ′ , . It is straightforward to check that¯ ∂ g = 0 , ∂ g = 0 , and ∂ g ¯ ∂ g + ¯ ∂ g ∂ g = 0 . Definition 5.1.
The equivariant basic Dolbeault cohomology of M , denoted by H p, ∗ g ( M, F ),is defined to be the cohomology of the differential complex { Ω p, ∗ g ( M, F ) , ¯ ∂ g } .The following result generalizes the earlier result [8, Theorem 7.5] in two ways: it is validfor a transversely K¨ahler foliation which may not be Killing, and it applies to a more generaltype of symmetry which may not come from the structural Lie algebra of a Killing foliaiton. Theorem 5.2.
Let G be a compact torus, and let F be a homologically orientable transverselyK¨ahler foliation on a closed manifold M . Suppose that there is a Hamiltonian action of G on ( M, F ) which is also holomorphic. Then each connected component of the fixed-leaf set X is a G -invariant F -saturated transversely K¨ahler submanifold of M ; furthermore, H p,q ( M, F ) = 0 for | p − q | > codim C ( F | X ) , where codim C ( F | X ) is the maximum of the finite set { codim C ( F | X i ) | X i is a connected component of X } . Proof.
Equip the equivariant basic Cartan complex { Ω g ( M, F ) , d g } with the bi-grading givenby Ω k g ( M, F ) = M p + q = k Ω p,q g ( M, F ) , together with the vertical differential ¯ ∂ g and the horizontal differential ∂ g . It was shown in [8,Theorem 7.1] that the spectral sequence of the double complex { Ω • , • g ( M, F ) , ¯ ∂ g , ∂ g } relativeto the horizontal filtration degenerates at the E term.Since the connected components of fixed-leaf set X is a F -saturated closed submanifoldof M , the normal bundle T X/T ( F | X ) is a subbundle of the pullback of Q = T M/T F to X . Let J : Q → Q be the transversely holomorphic structure on ( M, F ). Applying [15,Proposition 5.2.1], it is straightforward to check that T X/T ( F | X ) is an invariant subbundleof J : Q | X → Q | X ; moreover, a routine check shows that J : T X/T ( F | X ) → T X/T ( F | X )is a transversely holomorphic structure, and that X inherits the structure of a transverselyK¨ahler foliation from that of M . As a result, Ω g ( X, F | X ) admits a bi-grading as given in(5.1).Let X i be a connected component of X . Since X i is a F -saturated submanifold of M , theMolino sheaf of the Riemannian foliation F | X i is just the pullback of the Molino sheaf of F under the inclusion map X i ֒ → M . By Lemma 2.2, the top wedge power of the Molino sheafof ( M, F ) has a nowhere vanishing global section. It follows that the top wedge power ofthe Molino sheaf of ( X i , F | X i ) must also admit a nowhere vanishing global section. Thus byLemma 2.2, the foliation F | X i must be homologically orientable as well.Observe that the induced transverse G -action on each X i is trivial , which implies thatΩ( X i , F | X i ) is a trivial g ⋆ -module. Applying Theorem 2.6, an easy calculation shows thatthe spectral sequence associated to the double complex { Ω ∗ , ∗ ( X i , F | X i ) , ¯ ∂ g , ∂ g } relative to thehorizontal filtration also degenerates at the E term.By Theorem 1.1, the restriction homomorphism i ∗ : H G ( M, F ) → H G ( X, F | X )is injective. It then follows from simple homological algebra considerations that the homo-morphism i ∗ : E p,q (Ω g ( M, F )) → E p,q (Ω g ( X, F | X ))is also injective. Since E p,q (Ω g ( M, F )) = H p,q g ( M, F ) and E p,q (Ω g ( X, F | X )) = H p,q g ( X, F | X ),we conclude that the restriction homomorphism i ∗ : H p,q g ( M, F ) → H p,q g ( X, F | X ) = M ≤ j ≤ min { p,q } S j ( g ∗ ) ⊗ H p − j,q − j ( X, F | X ) (5.2)is injective.By our assumption, for all | p − q | > codim C ( F | X ), we have that Ω p − j,q − j ( X, F | X ) = 0,which implies that H p − j,q − j ( X, F ) = 0. Thus by the injectivity of (5.2) we must have that H p,q g ( M, F ) = 0. It follows from [8, Theorem 7.1] again that H p,q ( M, F ) = 0. This completesthe proof. (cid:3) The following result is an easy consequence of Theorem 5.2.
Corollary 5.3.
Under the same assumptions as in Theorem 5.2, if F has only finitely manyleaves that are invariant under the action of G , then H p,q ( M, F ) = 0 for any p = q . ASIC KIRWAN INJECTIVITY 11
Basic Betti numbers.
In this subsection we apply Theorem 1.1 to study the basicBetti numbers of a Hamiltonian transversely symplectic foliation. Throughout this sectionwe will make the following assumptions.( A1 ) F is a transversely symplectic foliation of co-dimension q on a compact manifold M which satisfies the transverse hard Lefshetz property.( A2 ) There is a clean Hamiltonian action of a compact torus T on ( M, F ) with the momentmap Φ : M → t ∗ , which has only finitely many leaves invariant under the action of T .( A3 ) F is a homologically orientable Riemannian foliation on M .We will also need the following Morse inequality for basic cohomologies established in [1,Theorem A]. Theorem 5.4.
Suppose that f : M → R is a basic Morse-Bott function on a Riemannianfoliation ( M, F ) of co-dimension q , that the critical submanifolds of f are isolated closedleaves, and that M is compact. Let b j ( M, F ) = dim R H j ( M, F ) , and let v j ( f ) be the numberof critical submanifolds of f which has Morse index j . Then we have the inequalities b ( M, F ) ≤ ν ( f ) ,b ( M, F ) − b ( M, F ) ≤ ν ( f ) − ν ( f ) ,b ( M, F ) − b ( M, F ) + b ( M, F ) ≤ ν ( f ) − ν ( f ) + ν ( f ) , etc. Moreover, we also have the following equality q X j =0 ( − j b j ( M, F ) = q X j =0 ( − j ν j ( f ) . Proposition 5.5.
Let f = Φ ξ for a generic element ξ ∈ t . Under the assumptions ( A1 ),( A2 ) and ( A3 ), f must be a perfect basic Morse-Bott function, that is, b j ( M, F ) = ν j ( f ) ,for all ≤ j ≤ q .Proof. It suffices to show that b k − ( M, F ) = 0, for any 1 ≤ k ≤ [ q +12 ]. Let X be the set offixed leaves as before. Note that the restriction homomorphism (3.2) maps H odd T ( M, F ) into H odd T ( X, F ), which by assumption ( A2 ) must vanish. It follows immediately from Theorem1.1 that H odd T ( M, F ) = 0. However, by Theorem 3.4 we have that H odd T ( M, F ) = S t ∗ ⊗ H odd ( M, F ). Therefore all the odd dimensional basic Betti numbers must be zero, fromwhich Proposition 5.5 follows. (cid:3) Proposition 5.5 has the following easy consequence.
Corollary 5.6.
Consider the clean Hamiltonian action of a compact torus T on a transverselysymplectic foliation F . Suppose that it satisfies the assumption ( A1 ), ( A2 ) and ( A3 ). Thenwe have that (i) b i +1 ( M, F ) = 0 ; (ii) b i ( M, F ) coincides with the total number of vertices with index i in the momentpolytope ∆ ; (iii) the basic Euler characteristic number χ ( M, F ) , given by χ ( M, F ) = codim( F ) X i =0 ( − i b i ( M, F ) , equals the total number of vertices in the moment polytope ∆ . An example that is not a toric quasifold.
In this subsection, we present an exampleof a clean Hamiltonian torus action on a transversely K¨ahler foliation with isolated fixedleaves. In this example, the leaf space of the transversely K¨ahler foliation possesses a much“smaller” symmetry than that of a toric quasifold. We begin with a simple observation thatwe will use when counting the dimension of the fixed-leaf set in the proof of Theorem 5.8.
Lemma 5.7.
Let λ < λ be two real scalars, let H λ be the set of × Hermitian matriceswhose spectrum is given by { λ , λ } , and let f be a function given by f : H λ → R , a a a a ! a µ + a µ , where µ and µ are fixed real constants. Suppose that c ∈ R is a regular value of f . Then f − ( c ) is a one dimensional submanifold of H λ .Proof. It follows easily from the observation that H λ is diffeomorphic to the complete flagmanifold U (2) / ( U (1) × U (1)). (cid:3) Theorem 5.8.
For n ≥ , there exists a clean Hamiltonian action of an n dimensional torus T on a homologically orientable transversely K¨ahler foliation F which has dimension one andco-dimension n − n − , such that there are only finitely many isolated closed leaves invariantunder the action of T .Proof. For n ≥
4, let U ( n ) be the unitary group of degree n , let H be the space of n × n Hermitian matrices, and let H λ be the space of n × n Hermitian matrices whose spectrumis given by { λ , · · · , λ n } , where we assume that λ < · · · < λ n . By assumption H λ is diffeo-morphic to the complete flag manifold U ( n ) / U (1) × · · · × U (1) | {z } n , and therefore has dimension n − n .Clearly, the map H → u ( n ) , X i X, identifies H with the Lie algebra u ( n ) of U ( n ). Use the inner product( · , · ) : u ( n ) × u ( n ) → R , ( A, B ) tr( A t B )to identify u ( n ) with u ( n ) ∗ . Then we get a natural identification of H λ with the coadjointorbit of U ( n ) through a diagonal matrix in u ( n ) ∗ whose entries on the main diagonal aregiven by ( i λ , · · · , i λ n ).So H λ is a symplectic manifold equipped with the canonical Konstant-Krillov K¨ahler 2-form ω λ . Let T be the maximal torus of U ( n ), and let t be the Lie algebra T . It is well-knownthat T is n dimensional, and that the conjugate action of T on H λ is Hamiltonian, with amoment map Φ : H λ → t ∗ ∼ = ( R n ) ∗ (5.3) ASIC KIRWAN INJECTIVITY 13 which associates to each symmetric Hermitian matrix in H λ its diagonal entries. Let n ⊂ t be spanned by a vector ξ ∈ t that generates a flow whose closure in T is of dimension ≥ N be the connected immersed normal subgroup of T whose Lie algebra is n , and letΦ N : H λ → n ∗ be the composition of (5.3) with the natural projection t ∗ → n ∗ . Then Φ N is a moment map for the action of N on H λ . Choose a regular value a ∈ n ∗ ∼ = R of Φ N .Let M = Φ − N ( a ), and let σ be the restriction of ω λ to M . Note that M is invariant underthe action of T on H λ , and that by [16, Proposition 6.2] the locally free action of N on M generates a transversely K¨ahler foliation F of co-dimension n − n −
2. An easy applicationof Proposition A.1 shows that F is homologically orientable. Clearly, the induced T actionon M is clean and is Hamiltonian with respect to the transversely K¨ahler two form σ . Itthen follows from [13, Theorem 3.5.1] that the set of fixed leaves X is non-empty.We claim that any leaf in X is isolated. To see this, first observe that a Hermitian matrix A ∈ X if and only if the T -orbit through A is one dimensional. In other words, a Hermitianmatrix A ∈ X if and only if the isotropy subgroup of T at A is n − M iscompact, there are only finitely many n − H , · · · , H r . It isstraightforward to check that X = S ri =1 M H i , where M H i denotes the fixed point set of H i ,1 ≤ i ≤ r . Therefore to establish our claim, it suffices to show that for an n − H of T , its fixed point submanifold must be one dimensional.Since T is the maximal torus of the unitary group U ( n ), T = (cid:8) diag( e π i θ , e π i θ , · · · e π i θ n ) | ≤ θ j ≤ , ≤ j ≤ n (cid:9) ∼ = S × S × · · · × S | {z } n . (5.4)For 1 ≤ j ≤ n , let ϑ j : H → S be the weight given by the composition of the inclusion map H ֒ → T with the projection from T to the j -th factor on the right hand side of (5.4). Thengiven A = ( a ij ) ∈ M H , we have that for all g ∈ H , gAg − = a ϑ ( g ) a ϑ ( g ) · · · ϑ ( g ) a n ϑ n ( g ) ϑ ( g ) a ϑ ( g ) a · · · ϑ ( g ) a n ϑ n ( g ) · · · · · · · · · · · · ϑ n ( g ) a n ϑ ( g ) ϑ n ( g ) a n ϑ ( g ) · · · a nn = a a · · · a n a a · · · a n · · · · · · · · · · · · a n a n · · · a nn . (5.5)However, since dim( H ) = n −
1, there is at most one pair ( i, j ) with 1 ≤ i < j ≤ n −
1, suchthat ϑ i ( g ) = ϑ j ( g ) for all g ∈ H . If this is the case, then it follows from (5.5) and Lemma5.7 that the fixed point set of H is one dimensional. Otherwise A will be an isolated fixedpoint of H . However, note that the T -orbit through A is one dimensional and lies inside M H .It follows that the fixed point set of H must be one dimensional. This finishes the proof ofTheorem 5.8. (cid:3) Remark . a) In the above example, the null subgroup N of T is one dimensional.Thus the quasi-torus T /N (in the sense of [18, Definition 2.1]) is n − F is n − n −
2. Clearly, n − < n − n − for n ≥ h p,q ( M, F ) =0 for all p = q , and b i +1 ( M, F ) = 0 for all i . Also note that by Proposition A.1 thefoliation F in this example is Killing. However, the dimension of the structural Lie algebra of the Killing foliaiton is dim( N ) −
1, which is in general much smaller than thedimension of the Lie algebra of the quasi-torus
T /N . In addition, unless the closure of N equals T , leaves invariant under the transverse action of the structural Lie algebraare not isolated in this example . Thus one can not deduce the same results on thebasic Hodge numbers and basic Betti numbers of ( M, F ) from [8, Theorem 7.5].6. Main example: Symplectic toric quasifolds
In this section, we apply the tools we developed in Section 5 to determine the basic Hodgenumbers and basic Betti numbers of a symplectic toric quasifold.6.1.
Symplectic toric quasifolds I: basic Hodge numbers.
We begin with a quickreview of the definition of a symplectic toric quasifold. Let V be an m dimensional vectorspace, and ∆ ⊂ V ∗ an m dimensional simple convex polytope given by∆ = d \ j =1 (cid:8) µ ∈ V ∗ | h µ, X j i ≥ λ j (cid:9) , where X j ∈ V and λ j ’s are real numbers. Here ∆ being simple means that at any vertexthere are exactly m edge vectors. Now consider the standard action of torus T d on C d givenby ( e πiθ , · · · , e πiθ d ) · ( z , · · · , z d ) = ( e πiθ z , · · · , e πiθ d z d ) . This action is Hamiltonian relative to the standard symplectic form ω = i P dj =1 dz j ∧ dz j on C d , with a moment mapΦ : C d → ( R d ) ∗ , ( z , · · · , z d )
7→ − · d X j =1 | z j | · e ∗ j + λ, (6.1)where λ ∈ ( R d ) ∗ is a constant vector, and e ∗ , · · · , e ∗ d are the dual basis of the standard basis e , · · · , e d in R d . Let n be the kernel of the surjective map given by π : R d → R m , e j X j , (6.2)and let N be the connected immersed normal subgroup of T d whose Lie algebra is n . Thenthe induced action of N on C d is also Hamiltonian with a moment map Φ N : C d → n ∗ givenby the composition of the map (6.1) with the natural projection map ( R d ) ∗ → n ∗ .It is straightforward to check that M = Φ − N (0) is compact, and that the infinitesimal actionof n on M is free and generates a transversely symplectic foliation F . The leaf space of thetransversely symplectic foliation F is defined to be a symplectic toric quasifold. Clearly, theinduced action of T d on the transversely symplectic foliation ( M, F ) is Hamiltonian. Indeed,the restriction of Φ to M is a moment map for the induced action of T d on ( M, F ), whichby abuse of notation we also denote by Φ. Let π ∗ : ( R m ) ∗ → ( R d ) ∗ be the dual of the map(6.2). Then the image of M under the moment map Φ is given by π ∗ (∆). Lemma 6.1.
The foliation F is transversely K¨ahler and Killing. As a result, F must behomologically orientable. ASIC KIRWAN INJECTIVITY 15
Proof.
The assertion that F is transversely K¨ahler follows from [16, Proposition 6.2]. Anapplication of Proposition A.1 to the pair ( N, N ) implies that F is a Killing foliation. So itfollows from Lemma 2.2 that F is also homologically orientable. (cid:3) An easy application of the presymplectic slice theorem [13, Theorem C.5.1] shows thatthe Hamiltonian T d -action on ( M, F ) has only finitely many fixed leaves. As an immediateconsequence of Corollary 5.3 we have the following result on the basic Hodge numbers of asymplectic toric quasifold. Corollary 6.2.
Let ( M, F ) be the transversely K¨ahler foliation constructed out of a non-rational simple polytope ∆ as above. Then we have that h p,q ( M, F ) = 0 when p = q .Remark . Corollary 6.2 provides a positive answer to a conjecture raised by Battagaliaand Zaffran in [3, Page 11812]. We refer to [16, Sec. 6] for a different approach outlined thereusing the compatibility of the Kirwan map with the Hodge structures.6.2.
Symplectic toric quasifolds II: basic Betti numbers.
We will keep the same as-sumptions and notations as in Section 6.1. It is straightforward to check that the action of T on ( M, F ) is clean; moreover, the null subgroup is precisely the connected immersed normalsubgroup N as constructed in Section 6.1. The following result is a refinement of Corollary5.6 in the case of symplectic toric quasifolds. Theorem 6.4.
For the transversely K¨ahler foliation ( M, F ) constructed in Section 6.1 wehave that (i) b k +1 ( M, F ) = 0 ; (ii) b k ( M, F ) = h k , where ( h , · · · , h m ) is the h -vector of the m -dimensional simplepolytope ∆ .Proof. Assertion (i) is an immediate consequence of Proposition 5.5. To show Assertion (ii),we first note that since ∆ is a simple polytope, its h -vector is given by h k = k X i =0 ( − k − i m − im − k ! f m − i , (6.3)where f j is the number of j dimensional faces in ∆ (cf. [21, Definition 8.18]). Fix a genericelement ξ ∈ t . We divide the rest of the proof into two steps. Step 1 . Let F ⊂ ∆ be a non-empty m − i dimensional face, and let λ F be the unique vertexof F with the minimal index. We will show that ind ξ ( λ F ) ≤ i . Choose x ∈ Φ − ( λ F ). By thepresymplectic slice theorem [13, Theorem C.5.1], we may assume that x has a T d -invariantopen neighborhood in M given by M = T d /H × H S , where H is the isotropy subgroup of T d at x , S = T x M/T x F is a symplectic vector space. Since all fixed leaves are isolated inour case, a simple dimension count shows that dim( S ) = 2dim( H ) = 2 m . Moreover, let h = Lie( H ), let t = Lie( T d ), and identify t ∗ with ( t / h ) ∗ × h ∗ . Then the restriction of Φ to M is given by Φ([ g, v ]) = λ − m X j =1 ( x j + y j ) · w j , for all g ∈ T d and v ∈ S, where ( x , y , · · · , x m , y m ) are Darbourx coordinates of v on S , and w , · · · , w m ∈ h ∗ ⊂ t ∗ ∼ =( R d ) ∗ are weight vectors. It follows that there is a neighborhood U of the vertex λ F ∈ ( R d ) ∗ such that U ∩ ∆ is the convex cone with vertex λ F and edge vectors w , · · · , w m , and suchthat Φ ξ ([ g, v ]) = h λ, ξ i − · m X j =1 h w j , ξ i · ( x j + y j ) . (6.4)Since x is a local minimum of the restriction of Φ ξ to Φ − ( F ), and since F is m − i dimensional,(6.4) implies that ind ξ ( λ F ) is at most 2 i . Step 2 . We deduce that b k ( M, F ) = h k from (6.3). In what follows, for a finite set E wewill denote by (cid:12)(cid:12) E (cid:12)(cid:12) its cardinality. For all 0 ≤ k ≤ m , let A k = (cid:8) F | F is an m − k dimensional face of ∆ such that ind ξ ( λ F ) = 2 k (cid:9) ,B k = (cid:8) F | F is an m − k dimensional face of ∆ such that ind ξ ( λ F ) < k (cid:9) . By definition, we have that f m − k = (cid:12)(cid:12) A k (cid:12)(cid:12) + (cid:12)(cid:12) B k (cid:12)(cid:12) . Let F and F be two different faces in A k . Since ∆ is simple, it follows easily from (6.4) that λ F and λ F must be two differentvertices of ∆ with index 2 k . Conversely, for any vertex ν ∈ ∆ with index 2 k , there is an m − k dimensional face F of ∆ such that ν = λ F . So by Corollary 5.6 b k ( M, F ) = (cid:12)(cid:12) A k (cid:12)(cid:12) . (6.5)In order to calculate (cid:12)(cid:12) B k (cid:12)(cid:12) , we introduce a partition of the set B k as follows. Let { σ , · · · , σ p } be the collection of all m − k + 1 dimensional faces of ∆. Define B jk = (cid:8) F ∈ B k | F ⊂ σ j , λ F = λ σ j (cid:9) , for all 1 ≤ j ≤ p. By definition we have that B k = S pj =1 B jk , and that B i k ∩ · · · ∩ B i s k = (cid:8) F ∈ B k | F ⊂ σ i j , λ F = λ σ ij , for all 1 ≤ j ≤ s (cid:9) , (6.6)where 1 ≤ i < · · · < i s ≤ p . Since the moment polytope ∆ is simple, for any sequence ofintegers 1 ≤ i < · · · < i s ≤ p , the set (6.6) is either empty or has exactly one element. If(6.6) is non-empty, then there exists a unique m − k + s dimensional face τ , such that for theunique face F ∈ B i k ∩ · · · ∩ B i s k , λ F = λ τ . Conversely, let τ be an m − k + s dimensional faceof ∆. Then τ has m − k + s edge vectors at its vertex with the minimal index λ τ , any choiceof m − k edge vectors from which uniquely determines a face F and a sequence of integers1 ≤ i · · · < i s ≤ k such that F ∈ B i k ∩ · · · ∩ B i s k . It follows that X ≤ i < ···
Inclusion-Exclusion Principle in elementary combinatorics (cid:12)(cid:12) B k (cid:12)(cid:12) = p X i =1 (cid:12)(cid:12) B ik (cid:12)(cid:12) + · · · + ( − s X ≤ i < ···
Combining (6.3), (6.5), and (6.7) we conclude that b k ( M, F ) = h k . This completes theproof of Theorem 6.4. (cid:3) Appendix A. A class of Riemannian foliations that are Killing
The following result provides a class of useful examples of Killing foliations. To prove itwe have to use Molino’s structure theory of a Riemannian foliaiton, for which we refer to [17]for a detailed account.
Proposition A.1.
Suppose that there is a compact Lie group G acting on a closed manifold M , that H is a dense immersed connected subgroup of G , and that the induced H action on M is locally free. Then the infinitesimal action of the Lie algebra h of H on M generates aRiemannian foliation F which is Killing. Moreover, the action of G is foliate with respect tothe foliation F .Proof. The first assertion that F is Riemannian is shown in [14, Lemma 5.1]. We need onlyto show that F is a Killing foliation, and that the action of G is foliate.Since H is a dense immersed subgroup of G , [4, Chapter 3, §
9, no. 2, Proposition 5]implies that h must be an ideal of G , from which it follows that the action of G must befoliate; moreover, it also implies that g / h is an abelian Lie algebra, where g = Lie( G ). Forany ξ ∈ g / h , let ξ ∈ g be a representative of ξ , let ξ M be the fundamental vector field on M induced by ξ , and let ξ M be the corresponding transverse vector field. Then we have awell-defined Lie algebra homomorphism g / h → X ( M/ F ) , ξ ξ M . To prove that F is Killing, a simple dimension count shows that it suffices to show that forall ξ ∈ g / h , and for all η ∈ X ( M/ F ), we have that [ ξ M , η ] = 0. To show this, using Molino’sstructure theory of Riemannian foliations we may assume without loss of generality that thespace of leaf closures N = M/ F is a manifold, and that there is a projection map ρ : M → N .Moreover, since G is compact, we may also assume that there is a bundle-like metric on M that is invariant under the action of G .Let ζ , · · · , ζ r be a basis in g / h , and let q ′ = dim( N ). For any vector field Y on N , denoteby Y M the unique G -invariant lifting of Y to M that is everywhere orthogonal to the G -orbits,and by Y M the corresponding transverse vector on M . Then the G -invariance implies that[ ξ M , Y M ] = 0, for all ξ ∈ g / h . Note that locally a transverse vector field η is always given as η = r X i =1 a i ζ i,M + q ′ X i =1 b i Y i,M , where Y i ’s are vector fields on N , and a i ’s and b i ’s are F -basic functions on M . Since F -basicfunctions are also F -basic, and since that g / h is commutative, a straightforward calculationshows that [ ξ M , η ] = 0, for all ξ ∈ g / h . This completes the proof of Proposition A.1. (cid:3) References [1] J. Alvarez L´opez,
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Y. Lin, Department of Mathematical Sciences, Georgia Southern University, Statesboro,GA, 30460 USA
E-mail address : [email protected] X. Yang, Department of Mathematics, Chongqing University, Chongqing 401331 ChinaCurrent address: Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
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