aa r X i v : . [ m a t h . C V ] S e p TORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS
HIROAKI ISHIDAA
BSTRACT . In this paper, we show the convexity of the image of a moment map on atransverse symplectic manifold equipped with a torus action under a certain condition. Wealso study properties of moment maps in the case of transverse K¨ahler manifolds. As anapplication, we give a positive answer to the conjecture posed by Cupit-Foutou and Zaffran.
1. I
NTRODUCTION
In [10], a family of complex manifolds (say LV manifolds here) which includes classicalHopf manifolds and Calabi-Eckmann manifolds are constructed by L´opez de Medrano andVerjovsky. Most of LV manifolds are non-K¨ahler; it is shown that an LV manifold admitsa K¨ahler form if and only if it is a compact complex torus of complex dimension 1. In [9],as a contrast, it is shown by Loeb and Nicolau that each LV manifold carries a transverseK¨ahler vector field. In [11], Meersseman generalizes the construction of LV manifolds andgives a new family of complex manifolds which are known as LVM manifolds. As wellas LV manifolds, an LVM manifold admits a K¨ahler form if and only if it is a compactcomplex torus. He also constructs a foliation F on each LVM manifold and shows that F is transverse K¨ahler. In [4], Bosio generalizes LVM manifolds and now they are known asLVMB manifolds. As well as LVM manifolds, if an LVMB manifold is K¨ahler then it is acompact complex torus. The first example of LVMB manifold which is not biholomorphicto any LVM manifold is given by Cupit-Foutou and Zaffran in [6]. In particular, the familyof LVMB manifolds properly contains the family of LVM manifolds. In [3], Battisti givesan explanation of the difference between LVM manifolds and LVMB manifolds in termsof toric geometry.In [6], Cupit-Foutou and Zaffran also construct a foliation F on each LVMB manifold,as a natural generalization of the case of LVM manifolds. They show that, under an as-sumption, if the foliation F on an LVMB manifold M is transverse K¨ahler then M is anLVM manifold. From this point of view, they give the following conjecture which is themotivation of this paper: Date : April 9, 2018.2010
Mathematics Subject Classification.
Primary 53D20, Secondary 14M25, 32M05, 57S25.
Key words and phrases.
Torus action, complex manifold, toric variety, moment-angle manifold, LVMmanifold, LVMB manifold, non-K¨ahler manifold, transverse K¨ahler form, moment map.The author was supported by JSPS Research Fellowships for Young Scientists.
Conjecture 1.1.
An LVMB manifold is an LVM manifold if and only if the foliation F istransverse K¨ahler. Our approach to Conjecture 1.1 uses techniques of Hamiltonian torus actions on sym-plectic manifolds. Especially, (an analogue of) the convexity theorem plays an importantrole. The convexity theorem shown by Atiyah, Guillemin and Sternberg in [1] and [7]states that if a compact torus G acts on a compact connected symplectic manifold M inHamiltonian fashion then the image of a corresponding moment map is a convex polytope.In this paper, first we show the following: Theorem 1.2 (see also Theorem 2.7) . Let M be a compact connected manifold equippedwith an action of a compact torus G. Let g ′ be a subspace of the Lie algebra g of G suchthat the action of g ′ is local free. Let F g ′ be the foliation on M whose leaves are g ′ -orbits.Let w be a G-invariant transverse symplectic form on M with respect to F g ′ . If there existsa moment map F : M → g ∗ with respect to w , then the image of M by F is the convexhull of the image of common critical points of h v for v ∈ g , where h v : M → R is given byh v ( x ) = h F ( x ) , v i . Example 1.3.
The setting of Theorem 1.2 naturally appears in symplectic manifolds. Let N be a compact connected symplectic manifold equipped with an effective Hamiltonianaction of a compact torus G . Let g ′ be any subspace of g and let i : g ′ → g denote theinclusion. Let c ∈ ( g ′ ) ∗ be a regular value of i ∗ ◦ F : N → ( g ′ ) ∗ and let M : = ( i ∗ ◦ F ) − ( c ) .Then, g ′ acts on M local freely and the restriction w | M of the symplectic form on N is atransverse symplectic form on M with respect to F g ′ . The image F ( M ) coincides with ( i ∗ ) − ( c ) ∩ F ( N ) , and it is a convex polytope.For a connected complex manifold M equipped with an effective action of a compacttorus G which preserves the complex structure J on M , the subspace g J : = { v ∈ g | X v = − JX v ′ , ∃ v ′ ∈ g } of g acts on M local freely (see Proposition 3.3). The foliation F g J whose leaves are g J -orbits coincides with the foliation which has been considered in [6] in case of LVMBmanifolds and the foliation which has been considered in [13] in case of moment-anglemanifolds equipped with complex-analytic structures. F g J gives a lower bound of G -invariant foliations that admit G -invariant transverse K¨ahler forms such that there existmoment maps with respect to the forms (see Proposition 4.2).An effective action of a compact torus G on a connected manifold M is said to be maxi-mal if there exists a point x ∈ M such that dim G + dim G x = dim M (see [8] for detail). If M is a compact connected complex manifold equipped with a maximal action of a compacttorus G preserving the complex structure J on M , one can associate a complete fan q ( D ) in g / g J with M . On the other hand, if there exists a G -invariant transverse K¨ahler form on M with respect to F g J and if there exists a moment map F : M → g ∗ with respect to the form,then one can find a lift e F : M → ( g / g J ) ∗ of F . The following is the main theorem in thispaper: ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 3
Theorem 1.4 (see also Theorem 5.7) . Let M be a compact connected complex manifoldequipped with a maximal action of a compact torus G which preserves the complex struc-ture J on M. If F g J is transverse K¨ahler, then a moment map F with respect to a G-invariant transverse K¨ahler form exists and e F ( M ) is a convex polytope normal to q ( D ) .Conversely, if q ( D ) is polytopal, then F g J is transverse K¨ahler. As an application of Theorem 1.4, we show that Conjecture 1.1 is true.This paper is organized as follows. In Section 2, we investigate Hamiltonian functionsfor almost periodic vector fields on transverse symplectic manifold and show the convexityof the image of a moment map with an almost same argument as Atiyah. In Section 3, weconstruct a G -invariant foliation F g J on a complex manifold equipped with an action ofa compact torus G . In Section 4, we show that the foliation F g J is a lower bound of G -invariant foliations that admit moment maps. In Section 5, we consider the case of maximaltorus action and give a proof of the conjecture posed by Cupit-Foutou and Zaffran. Convention and notation.
For a smooth manifold M and a smooth foliation F on M ,we denote by T F the subbundle of the tangent bundle T M consisting of vectors tangentto leaves of F . Let G be a compact torus acting on M smoothly. We say that F is G-invariant if T F is a G -equivariant subbundle of T M . We denote by g the Lie algebra of G . Through the exponential map, g also acts on M . For a subspace g ′ that acts on M localfreely, we denote by F g ′ the foliation on M whose leaves are g ′ -orbits. For a point x ∈ M ,we denote by G x the isotropy subgroup of G at x . We also denote by g x the Lie algebra of G x . Remark that g x is not the isotropy subgroup of g at x . For v ∈ g , we denote by X v thefundamental vector field generated by v on M . For a vector field X on M , we denote by X x the value of X at x . For the fundamental vector field, we denote by ( X v ) x the value of X v at x . For a differential form w , we denote by w x the value of w at x . We denote by L X w theLie derivative for w along X and by i X w the interior product of X and w . We identify R with the Lie algebra of S by the differential of the map t e p √− t . For a : G → S , wedenote by d a the differential at the unit of G and d a is regarded as an element in g ∗ . Wedenote by H F ( M ) the first basic cohomology group with coefficients in R .2. T HE CONVEXITY THEOREM
Let M be a smooth manifold and let F be a smooth foliation on M . A transversesymplectic form w with respect to F is a closed 2-form on M whose kernel coincides with T F . Let w be a transverse symplectic form on M with respect to F . Let a compact torus G act on M effectively. We assume that the action of G preserves w (and hence, F is G -invariant). In this case, by Cartan formula we have that0 = L X v w = d i X v w + i X v d w = d i X v w for v ∈ g . We say that a smooth map F : M → g ∗ is a moment map if the function h v : M → R given by h F ( x ) , v i = h v ( x ) satisfies that dh v = − i X v w . A moment map F with respect to w exists if and only if i X v w is exact for any v . In particular, the obstruction for the existenceof moment map sits in H F ( M ) . The purpose in this section is to show the convexity of the H. ISHIDA image of a moment map under certain conditions by an almost same argument as Atiyah(see [1]).Let { y a : U a → V a } a be foliation charts of ( M , F ) . The local leaf space U a / F isdiffeomorphic to an open subset of R dim M − dim F . The quotient map p a : U a → U a / F is a fiber bundle whose fibers are diffeomorphic to open balls of dimension dim F . Thetransition functions y ab : y b ( U a ∩ U b ) → y a ( U a ∩ U b ) can be written as y ab ( x , z ) = ( y T ab ( x ) , y F ab ( x , z )) ∈ y a ( U a ∩ U b ) ⊆ R dim M − dim F × R dim F for ( x , z ) ∈ y b ( U a ∩ U b ) ⊆ R dim M − dim F × R dim F .Let x ∈ M and let G x denote the isotropy subgroup at x of G . Let y : U → V be a localfoliation chart on an open neighborhood at x . The local leaf space U / F is diffeomorphicto an open subset of R dim M − dim F . Let p : U → U / F be the quotient map. Since G x is compact, the intersection U ′ : = T g ∈ G x g ( U ) is a G x -invariant open neighborhood at x .Since F is G -invariant, p ( U ′ ) is a G x -manifold of dimension dim M − dim F . Let w be a G -invariant transverse symplectic form on M with respect to F and suppose that dim F = ℓ and dim M = n + ℓ . w descends to a symplectic form w on p ( U ′ ) . By equivariant Darbouxtheorem, there exist G x -invariant open subsets U x of p ( U ′ ) and V x of T x M / T x F , a G x -equivariant diffeomorphism j : U x → V x and a basis ( x , . . . , x n , y , . . . , y n ) of ( T x M / T x F ) ∗ such that ( j − ) ∗ w = n (cid:229) i = dx i ∧ dy i and X v = p n (cid:229) i = h d a i , v i (cid:18) x i ¶¶ y i − y i ¶¶ x i (cid:19) for v ∈ g x , where a , . . . , a n ∈ Hom ( G x , S ) are weights at 0 ∈ T x M / T x F . Remark that y | p − ( U x ) : p − ( U x ) → y ( p − ( U x )) is a local foliation chart near x . We state this fact as alemma for later use. Lemma 2.1.
Let M be a smooth manifold of dimension n + ℓ equipped with an action of acompact torus G. Let F be a G-invariant smooth foliation on M of dimension ℓ and let w be a G-invariant transverse symplectic form on M with respcet to F . Then, for any x ∈ M,there exist • a local foliation chart y x : f U x → e V x on an open neighborhood U x at x such that f U x / F carries the action of G x , • a G x -invariant open neighborhood V x at of T x M / T x F , • a G x -equivariant diffeomorphism j x : f U x / F → V x , and • a basis ( x , . . . , x n , y , . . . , y n ) of ( T x M / T x F ) ∗ such that ( j − x ) ∗ w = (cid:229) ni = dx i ∧ dy i andX v = p n (cid:229) i = h d a i , v i (cid:18) x i ¶¶ y i − y i ¶¶ x i (cid:19) ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 5 for v ∈ g x , where a , . . . , a n ∈ Hom ( G x , S ) are weights at ∈ T x M / T x F . Let M , G , F , w be as Lemma 2.1. Let v ∈ g and suppose that there exists a smoothfunction h v : M → R such that − i X v w = dh v . Since w is transverse symplectic with respectto F , a point x ∈ M is a critical point of h v if and only if ( X v ) x ∈ T x F . Because of this,unfortunately, we can not deduce the property that h v is non-degenerate for general F , notlike symplectic case. Let g ′ be a subspace of g such that g ′ act on M local freely. Since G is abelian, F g ′ is a G -invariant foliation. Lemma 2.2.
Let M be a smooth manifold equipped with an action of a compact torusG. Let g ′ be a subspace of g such that g ′ acts on M local freely. Let w be a G-invarianttransverse symplectic form on M with respect to F g ′ . Let v ∈ g and suppose that thereexists a smooth function h v : M → R such that dh v = − i X v w . Then, h v is a non-degeneratefunction and the index of each critical submanifold is even.Proof. Let x ∈ M be a critical point of h v . Then, ( X v ) x ∈ T x F g ′ implies that there exist v x ∈ g x and v ′ ∈ g ′ such that v = v x + v ′ . Since i X v ′ w =
0, we have that i X v w = i X vx w . Since i X v w is basic for F g ′ , so is h v . Let y x : f U x → e V x , V x , j x , ( x , . . . , x n , y , . . . , y n ) be as Lemma2.1. Since h v is basic for F g ′ , h v descends to a smooth function h v : f U x / F g ′ → R such that p ∗ h v = h v , where p : f U x → f U x / F g ′ denote the quotient map. By definition of F g ′ , p sendsthe fundamental vector field X v generated by v on f U x to the fundamental vector field X v x generated by v x on f U x / F g ′ . Also, since j x is G x -equivariant, j x sends X v x on f U x / F g ′ to X v x on T x M / T x F g ′ .Therefore dh v = − i X v w = − i X vx w = p ∗ ( − i X vx w ) = p ∗ ◦ j ∗ x ( − i X vx ( j − x ) ∗ w ) . On the other hand, dh v = d ( p ∗ h v ) = p ∗ ( dh v ) = p ∗ ◦ ( j − x ) ∗ ( d (( j − x ) ∗ h v )) . Since p ∗ is injective and j x is a diffeomorphism, we have that d (( j − x ) ∗ h v ) = − i X vx ( j − x ) ∗ w .Let a , . . . , a n ∈ Hom ( G x , S ) be the weights at the origin in T x M / T x F g ′ . Then, X v x on T x M / T x F g ′ can be represented as X v x = p n (cid:229) i = h d a i , v x i (cid:18) x i ¶¶ y i − y i ¶¶ x i (cid:19) with the coordinates ( x , . . . , x n , y , . . . , y n ) . Therefore − i X vx ( j − x ) ∗ w = p n (cid:229) i = h d a i , v x i ( x i dx i + y i dy i ) and hence(2.1) ( j − x ) ∗ h v = ( j − x ) ∗ h v ( ) + p n (cid:229) i = h d a i , v x i ( x i + y i ) . H. ISHIDA
Therefore ( j − x ) ∗ h v is nondegenerate at 0 and the index at 0 is twice as many as the numberof a i such that h d a i , v x i <
0. Since j x ◦ p : f U x → V x is a fiber bundle, h v is nondegenerateat x and the index at x is twice as many as the number of a i such that h d a i , v x i <
0, provingthe lemma. (cid:3)
Remark . In the proof of Lemma 2.2, it follows from (2.1) that x attains a local minimumof h v if and only if h d a i , v x i ≥ a i . Remark . We are not sure whether Lemma 2.2 holds even if we replace F g ′ to any G -invariant foliation F or not.The following is the key of the convexity theorem. Lemma 2.5 ([1, Lemma 2.1]) . Let f : N → R be a non-degenerate function (in the senseof Bott) on the compact connected manifold N, and assume that neither f or − f has acritical manifold of index . Then f − ( c ) is connected (or empty) for every c ∈ R . By Lemmas 2.2 and 2.5, if M is compact then the level set h − v ( c ) is connected unlessempty. Moreover, if c is a regular value then h − v ( c ) is a connected submanifold of M .Since G is abelian, dh v = − i X v w is G -invariant. Therefore h v is also G -invariant. Therefore h − v ( c ) is G -invariant for c ∈ R . Lemma 2.6.
Let M be a compact connected manifold equipped with an action of a compacttorus G. Let g ′ be a subspace of g such that g ′ acts on M local freely. Let w be a G-invarianttransverse symplectic form on M with respect to F g ′ . Let v , . . . , v k ∈ g and suppose thatthere exists a smooth function h v i such that dh v i = − i X vi w for i = , . . . , k. Let c ∈ R k bea regular value of h = ( h v , . . . , h v k ) : M → R k . Then, g ′′ = g ′ + R v + · · · + R v k acts onh − ( c ) local freely and w | h − ( c ) is a transverse symplectic form with respect to the foliation F g ′′ .Proof. Let x ∈ h − ( c ) . Since c is a regular value, ( − i X vi w ) x is linearly independent for all i . This together with that the action of g ′ is local free yields that ( X v ′′ ) x = v ′′ = v ′′ ∈ g ′′ . Therefore the action of g ′′ is local free. T x ( h − ( c )) is given by ker ( dh ) x = ( T x F g ′′ ) ⊥ , where ( T x F g ′′ ) ⊥ denotes the annihilator of T x F g ′′ with respect to w . Therefore w x descends to a symplectic form on T x ( h − ( c )) / T x F g ′′ .It turns out that Y x ∈ ker ( w | h − ( c ) ) x if and only if Y x ∈ T x F g ′′ . Therefore w | h − ( c ) is a trans-verse symplectic form with respect to F g ′′ , proving the lemma. (cid:3) Now we are in a position to prove the convexity theorem.
Theorem 2.7.
Let M be a compact connected manifold equipped with an action of a com-pact torus G. Let g ′ be a subspace of g such that g ′ acts on M local freely. Let w be aG-invariant transverse symplectic form on M with respect to F g ′ . Let v , . . . , v k ∈ g andsuppose that there exists a smooth function h v i such that dh v i = − i X vi w for i = , . . . , k. Puth = ( h , . . . , h k ) : M → R k . Then the followings hold: ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 7 ( A k ) For c ∈ R k , the fiber h − ( c ) is connected unless empty. ( B k ) h ( M ) is convex. ( C k ) If Z , . . . , Z N are the connected components of the set of common critical points ofh v i , then h ( Z j ) is a point c j and h ( M ) is a convex hull of c , . . . , c N .Proof. The proof consists of following steps.Step 1. ( A k ) implies ( B k + ) .Step 2. ( A k ) holds by induction on k .Step 3. ( B k ) implies ( C k ) .Remark that it follows from the connectedness of M that ( B ) holds because h ( M ) is aclosed interval in R .Step 1. Assume that ( A k ) holds. Let p : R k + → R k be any linear projection given by p ( e i ) = (cid:229) kj = a i j e j for i = , . . . , k +
1. The composition h ′ : = p ◦ h : M → R k satisfies theassumption of the theorem. Namely, j -th component of h ′ is a smooth function (cid:229) k + i = a i j h v i ,but d k + (cid:229) i = a i j h v i ! = − i X ( (cid:229) k + i = ai jvi ) w . Therefore each fiber of h ′ is connected unless empty. Let x , y ∈ h ( M ) and assume that p is surjective and p ( x ) = p ( y ) = c . Since the fiber p − ( c ) is a line in R k + , it sufficesto see that h ( M ) ∩ p − ( c ) is connected. Since h ′ = p ◦ h , we have that h ( M ) ∩ p − ( c ) = h ( h ′− ( c )) . Since h is continuous and h ′− ( c ) is connected, h ( M ) ∩ p − ( c ) is connected,proving that ( A k ) implies ( B k + ) .Step 2. It follows from Lemmas 2.2 and 2.5 that ( A ) holds. Assume that ( A k ) holds.Let v , . . . , v k + ∈ g and assume that there exists a smooth function h v i : M → R such that dh v i = − i X vi w for i = , . . . , k +
1. Let h = ( h v , . . . , h v k + ) and let c = ( c , . . . , c k + ) be apoint in R k + . We want to show that h − ( c ) = h − v ( c ) ∩ · · · ∩ h − v k + ( c k + ) is connectedunless empty. If h has no regular value, then one of dh v i is a linear combination of theothers. By assumption that ( A k ) holds, we are done. Assume that h has a regular value.Then, the set of regular values is dense in h ( M ) . By continuity, we only need to showthat h − ( c ) is connected for any regular value c . Then, N : = h − v ( c ) ∩ · · · ∩ h − v k ( c k ) isa connected submanifold by ( A k ) . Moreover, it follows from Lemma 2.6 that w | N is atransverse symplectic form on N with respect to F g ′′ on N , where g ′′ = g + R v + · · · + R v k . The function h v k + | N satisfies that dh v k + | N = − i X vk + w | N . Therefore by Lemmas 2.2and 2.5, ( h v k + | N ) − ( c k + ) is connected. Therefore h − ( c ) = h − v ( c ) ∩ · · · ∩ h − v k + ( c k + ) isconnected, proving that ( A k ) holds for all k .Step 3. The former assertion that states that h ( Z j ) is a point c j is obvious. Let H be theclosure of exp ( g ′′ ) in G . Let x be a common critical point of h v , . . . , h v k . Since ( dh v i ) x =( − i X vi w ) x =
0, we have that ( X v i ) x ∈ T x F g ′ for i = , . . . , k . Therefore there exists v i , x ∈ h x and v ′ i ∈ g ′ such that v i = v i , x + v ′ i for i = , . . . , k . Let H x denote the identity component of H x . Then, { exp ( t v , x ) · · · · · exp ( t k v k , x ) | t i ∈ R } is dense in H x and exp ( h x + g ′ ) is dense in H. ISHIDA H . Conversely, for a subtorus H ′ of H , if exp ( h ′ + g ′ ) is dense in H , then each fixed point x ∈ M H ′ is a common critical point of h v , . . . , h v k . Let u ′ ∈ g ′ and v : = (cid:229) ki = a i v i + u ′ ∈ g ′′ such that { exp ( tv ) | t ∈ R } is dense in H . Put h v : = (cid:229) ki = a i h v i . We claim that eachcritical point of h v is a common critical point of h v , . . . , h v k . Let x be a critical point of h v . Since ( dh v ) x = ( − i X v w ) x =
0, there exists v x ∈ h x and v ′ ∈ g ′ such that v = v x + v ′ .Since { exp ( tv ) | t ∈ R } is dense in H , we have that { exp ( tv x ) | t ∈ R } is also dense in H x .By definition of v and v x , the closure of exp ( h x + g ′ ) is H . Therefore the critical point x of h v is a common critical point of h v , . . . , h v k . In particular, h v takes the minimum valuein a common critical point of h v , . . . , h v k . It turns out that the linear form a : = (cid:229) ki = a i e ∗ i restricted to h ( M ) takes the minimum value at one of c j ’s. Therefore(2.2) h ( M ) ⊆ \ ( a ,..., a k ) ∈ A { y = ( y , . . . , y k ) ∈ R k | h a , y i ≥ min ( h a , c j i | j = , . . . , N ) } , where A : = { ( a , . . . , a k ) | { exp ( tv ) | t ∈ R } is dense in H } . Since A is dense in R k , the right hand side of (2.2) is the convex hull of c j ’s. It follows from ( B k ) and c j ∈ h ( M ) for all j that h ( M ) is the convex full of c j ’s, proving the theorem. (cid:3)
3. H
OLOMORPHIC FOLIATIONS FROM TORUS ACTIONS
Let M be a complex manifold and let G be a compact torus acting on M as holomorphictransformations. In this section, we define a subspace g J of g which acts on M local freelyby using the complex structure J on M and the action of G . We begin with the followinglemmas. Lemma 3.1.
Let M be a complex manifold equipped with an action of a compact torusG which acts as holomorphic transformations. For x ∈ M, there exists G x -invariant openneighborhoods U at x ∈ M and V at ∈ T x M such that U and V are G x -equivariantlybiholomorphic.Proof. Let U be an open neighborhood at x and let j : U → V be a local holomor-phic coordinate centered at x , where V is an open subset of C n . Since G x is compact,the intersection T g ∈ G x g ( U ) is a G x -invariant open neighborhood at x . By restricting thedomain of definition, we may assume that U is G x -invariant. Through the differential ( d j ) x : T x M → T C n = C n , we identify C n with T x M . Then we have a biholomorphism ( d j ) − x ◦ j : U → ( d j ) − x ( V ) ⊆ T x M . By averaging on G x , we have a G x -equivariantholomorphic map j ′ : = Z g ∈ G x ( dg ) x ◦ (( d j ) − x ◦ j ) ◦ g − dg : U → T x M . j ′ is no longer injective, but, ( d j ′ ) x = id T x M . Therefore, it follows from the implicit func-tion theorem that there exists an open subset U of U such that j ′ | U : U → j ′ ( U ) is biholo-morphic. As before, we may assume that U is G x -invariant and then V : = j ′ ( U ) is also ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 9 G x -invariant. Therefore there exists a G x -equivariant biholomorphism j ′ : U → V ⊆ T x M ,proving the lemma. (cid:3) Lemma 3.2.
Let M be a connected complex manifold with the complex structure J. Let Xbe a nonzero almost periodic vector field on M whose flows preserve J. If X vanishes at apoint x ∈ M, then JX is not almost periodic.Proof.
Assume that JX is almost periodic. Since X is an infinitesimal automorphism of J , [ X , JX ] =
0. Therefore we may assume that a compact torus G acts on M effectivelyand as holomorphic transformations and there exist v , v ′ ∈ g such that X = X v , JX = X v ′ and the subgroup { exp ( sv ) exp ( tv ′ ) | s , t ∈ R } is dense in G . Since X x = ( JX ) x = x is a G -fixed point. Let a , . . . , a n ∈ Hom ( G , S ) be the weights of the G -representation T x M .By Lemma 3.1, there exists an equivariant biholomorphic map U → V ⊆ T x M , where U isan open neighborhood at x . Combining with the decomposition of T x M into 1-dimensionalrepresentations of weights a , . . . , a n , we have a local coordinate ( z , . . . , z n ) : U → C n such that z i ( g · p ) = a i ( g ) z i ( p ) for p ∈ U . Let x i and y i denote the real and imaginary partof z i , respectively. Then, through the local coordinate ( z , . . . , z n ) we can represent X and JX as X = p n (cid:229) i = h d a i , v i (cid:18) − y i ¶¶ x i + x i ¶¶ y i (cid:19) and JX = p n (cid:229) i = h d a i , v ′ i (cid:18) − y i ¶¶ x i + x i ¶¶ y i (cid:19) . On the other hand, J is represented as J = n (cid:229) i = (cid:18) ¶¶ y i ⊗ dx i − ¶¶ x i ⊗ dy i (cid:19) . Therefore0 = X + J X = p n (cid:229) i = (cid:18) h d a i , v i (cid:18) − y i ¶¶ x i + x i ¶¶ y i (cid:19) + h d a i , v ′ i (cid:18) − x i ¶¶ x i − y i ¶¶ y i (cid:19)(cid:19) = p n (cid:229) i = (cid:18) h d a i , − y i v − x i v ′ i ¶¶ x i + h d a i , x i v − y i v ′ i ¶¶ y i (cid:19) . Therefore, by substituting e , 0 < | e | << x i and y i , we have that0 = h d a i , − e v − e v ′ i = − e h d a i , v + v ′ i and 0 = h d a i , e v − e v ′ i = e h d a i , v − v ′ i for all i = , . . . , n . Thus we have h d a i , v i = i = , . . . , n . Since the action of G on M is effective and M is connected, d a i ∈ g ∗ for i = , . . . , n spans g ∗ . This together with the fact that h d a i , v i = i shows that v =
0. This contradicts the assumption that X = X v is nonzero and hence JX is not almost periodic, as required. (cid:3) Proposition 3.3.
Let M be a connected complex manifold with the complex structure J. LetG be a compact torus acting on M effectively and as holomorphic transformations. Define g J : = { v ∈ g | there exists v ′ ∈ g such that X v = − JX v ′ } . Then, (1) g J is a Lie subalgebra of g . (2) g J has the complex structure J which satisfies X J ( v ) = JX v . (3) g J acts on M holomorphically and local freely.Proof. Part (1) follows from the fact that G is commutative. For Part (2), let v ∈ g J . Assumethat v ′ , v ′′ ∈ g J satisfy that X v = − JX v ′ = − JX v ′′ . Then, X v ′ = X v ′′ . It follows from theeffectiveness of the G -action that v ′ = v ′′ . Therefore for v ∈ g J , there exists unique J ( v ) ∈ g J such that X J ( v ) = JX v . The map J : g J → g J is linear and J = −
1, proving Part (2).Part (3) follows from Part (2) and Lemma 3.2. The proposition is proved. (cid:3)
Let M be a connected complex manifold with the complex structure J and let a compacttorus G act on M effectively and as holomorphic transformations. By Proposition 3.3, g J acts on M holomorphically and local freely. Therefore we have a holomorphic foliation F g J whose leaves are g J -orbits.4. T ORUS INVARIANT TRANSVERSE
K ¨
AHLER FOLIATIONS
A transverse K¨ahler form is a special kind of transverse symplectic form. Let M bea complex manifold with the complex structure J . Let F be a holomorphic foliation on M . A real 2-form w on M is called transverse K¨ahler with respect to F if the followingconditions are satisfied:(1) w is transverse symplectic with respect to F .(2) w is of type ( , ) . Namely, For Y x , Z x ∈ T x M , w x ( JY x , JZ x ) = w x ( Y x , Z x ) .(3) w is positive. Namely, w x ( Y x , JY x ) ≥ Y x ∈ T x M .The conditions (1) and (3) imply that w x ( Y x , JY x ) = Y x ∈ T x F . For aholomorphic foliation F on M , if a transverse K¨ahler form w exists, we say that F is transverse K¨ahler . Proposition 4.1.
Let M be a complex manifold with the complex structure J. Let G bea compact torus acting on M as holomorphic transformations. Let F be a G-invariantfoliation and let w be a transverse K¨ahler form with respect to F . Then, Z g ∈ G g ∗ w dgis a transverse K¨ahler form with respect to F and invariant under the G-action on M. ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 11
Proof.
For short, denote w ′ = Z g ∈ G g ∗ w dg . Since w is closed, so is w ′ . Since G acts on M preserving the complex structure J , w ′ is apositive ( , ) -form. It remains to show that ker w ′ x = T x F for all x ∈ M . By definition, for Y x ∈ T x M , w ′ x ( Y x , JY x ) = Z g ∈ G ( g ∗ w ) x ( Y x , JY x ) dg = Z g ∈ G w g · x (( dg ) x ( Y x ) , ( dg ) x ( JY x )) dg = Z g ∈ G w g · x (( dg ) x ( Y x ) , J (( dg ) x ( Y x ))) dg because J is G -invariant. Since w g · x (( dg ) x ( Y x ) , J (( dg ) x ( Y x ))) ≥ ( dg ) x ( Y x ) ∈ T g · x F , it follows from the G -invariance of F that w ′ x ( Y x , JY x ) = Y x ∈ T x F , proving the proposition. (cid:3) Thanks to Proposition 4.1, if a G -invariant foliation F is transverse K¨ahler, we mayalways assume that the transverse K¨ahler form with respect to F is G -invariant withoutloss of generality.For foliations F and F on a smooth manifold M , we denote by F ⊆ F if T F ⊆ T F . Our next purpose is to give a lower bound of G -invariant transverse K¨ahler foliationsthat admit moment maps. Proposition 4.2.
Let M be a connected complex manifold with the complex structure J.Let a compact torus G act on M effectively and as holomorphic transformations. Let F be a G-invariant holomophic foliation and let w be a G-invariant transverse K¨ahler formwith respect to F . If there exists a moment map with respect to w , then F g J ⊆ F .Proof. Let F : M → g ∗ be a moment map. We denote by h v the smooth function given by h v ( x ) = h F ( x ) , v i for v ∈ g and x ∈ M . Since h v is G -invariant for v ∈ g , we have that(4.1) 0 = L X v h v = i X v dh v = − i X v i X v w = w ( X v , X v ) for any v , v ∈ g . Assume that v ∈ g J . Then,0 ≤ w ( X v , JX v ) = w ( X v , X J ( v ) ) = ( X v ) x ∈ T x F for all x and hence F g J ⊆ F , as required. (cid:3) Theorem 4.3.
Let M, J, G, F , w be as Proposition 4.2. Let q : g → g / g J be the quotientmap. Assume that there exists a moment map F : M → g ∗ with respect to w . Then, thereexist c ∈ g ∗ and a smooth map e F : M → ( g / g J ) ∗ such that F + c = q ∗ ◦ e F .Proof. For v ∈ g , h v denotes the smooth function given by h v ( x ) = h F ( x ) , v i . It followsfrom Proposition 4.2 that dh v = − i X v w = v ∈ g J . It turns out that h v is constant on M . Let i : g J → g denote the inclusion. Then, there exists c ∈ g ∗ J such that i ∗ ◦ F ( x ) = c forany x ∈ M . The sequences 0 / / g J i / / g q / / g / g J / / g ∗ J o o g ∗ i ∗ o o ( g / g J ) ∗ q ∗ o o o o are exact. Since i ∗ is surjective, there exists c ∈ g ∗ such that i ∗ ( c ) = c . In particular, i ∗ ( F ( x ) − c ) = x ∈ M . Therefore, there uniquely exists e F ( x ) ∈ ( g / g J ) ∗ such that q ∗ ( e F ( x )) = F ( x ) − c for all x ∈ M . The smoothness is obvious. The theorem is proved. (cid:3) We call e F : M → ( g / g J ) ∗ a lifted moment map . As a corollary of Theorems 2.7 and 4.3,we have the following. Corollary 4.4.
Let M be a compact connected complex manifold. Let a compact torusG act on M effectively and preserving the complex structure J on M. Assume that F g J istransverse K¨ahler and there exists a moment map F : M → g ∗ with respect to a G-invarianttransverse K¨ahler form. Then, the image of M by a lifted moment map e F : M → ( g / g J ) ∗ isa convex polytope in ( g / g J ) ∗ .
5. T
HE EXTREME CASE
In this section, we consider the extreme case. First we recall the notion of maximal torusaction introduced in [8]. Let M be a connected smooth manifold equipped with an effectiveaction of a compact torus G . Then, for any point x , we have that dim G x + dim G ≤ dim M .The G -action on M is maximal if there exists a point x ∈ M such thatdim G + dim G x = dim M . Any compact connected complex manifold M equipped with a maximal action of a com-pact torus G which preserves the complex structure can be described with a fan D in g anda complex subspace h of g C . Theorem 5.1 (see [8]) . Let M be a compact connected complex manifold M equippedwith a maximal action of a compact torus G which preserves the complex structure J.Then, there exists a nonsingular fan D in g and a complex subspace h such that M is G-equivariantly biholomorphic to X ( D ) / H, where X ( D ) denotes the toric variety associatedwith D and H : = exp ( h ) ⊆ G C y X ( D ) . We shall recall how to deduce D and h from M briefly. Each connected component ofthe set of fixed points of a circle subgroup of G is a closed complex submanifold of M . Ifsuch a submanifold has complex codimension one, then we call it a characteristic subman-ifold of M . The number of characteristic submanifolds is at most finite. Let N , . . . , N k becharacteristic submanifolds of M . Each characteristic submanifold N i is fixed by a circle ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 13 subgroup G i of G by definition. To each characteristic submanifold N i , we assign a groupisomorphism l i : S → G i ⊆ G such that ( l i ( g )) ∗ ( x ) = g x for all g ∈ S and x ∈ T M | N i / T N i . We can think of l ∈ Hom ( S , G ) as a vector in g by d l ( ) ∈ g . We have a collection D ofcones D : = ( pos ( l i | i ∈ I ) | \ i ∈ I N i = /0 ) , where pos ( l i | i ∈ I ) is the cone spanned by l i for i ∈ I . It has been shown that D is anonsingular fan in g with respect to the lattice Hom ( S , G ) . Since the action of G preservesthe complex structure J on M , it extends to a holomorphic action of G C on M . Then thecomplex subspace h of g C = g ⊗ C = g ⊗ + g ⊗ √− G C -action on M . Namely, h = { u ⊗ + v ⊗ √− ∈ g C | X u + JX v = } . The pair of D and h satisfies the followings.(1) The restriction p | h of the projection p : g C → g ⊗ ∼ = g is injective.(2) The quotient map q : g → g / p ( h ) sends D to a complete fan q ( D ) in g / p ( h ) .Conversely, if D and h satisfy the conditions (1) and (2), then the quotient X ( D ) / H is acompact connected complex manifold and the action of G on X ( D ) descends to a maximalaction on X ( D ) / H . Proposition 5.2.
Let M be a compact connected complex manifold M equipped with anaction of a compact torus G which preserves the complex structure J. Let h be the Liealgebra of global stabilizers of the G C -action on M. Then, p ( h ) = g J .Proof. This follows from the definitions of h and g J immediately. (cid:3) Lemma 5.3.
Let D , h , q be as above and let J denote the complex structure on X ( D ) / H.Assume that F g J is a transverse K¨ahler foliation on X ( D ) / H and let w be a G-invarianttransverse K¨ahler form with respect to F g J . In addition, assume that there exists a momentmap F : X ( D ) / H → g ∗ with respect to w . Then, the image of X ( D ) / H by a lifted momentmap e F is a convex polytope and e F ( X ( D ) / H ) is a normal polytope of q ( D ) . Before the proof of Lemma 5.3, we shall recall notions of normal fan and normal poly-tope. Let P be an n -dimensional polytope in a vector space V ∗ of dimension n . For a vector a ∈ V , we put F v : = { a ∈ P | h a , v i ≤ h a ′ , v i for all a ′ ∈ P } . F v is a face of P that attains the minimum value of v . For a face F of P , the (inner) normalcone s F of F is given by s F : = { v ∈ V | F v ⊆ F } . Its relative interior is given by { v ∈ V | F v = F } . The (inner) normal fan of P is thecorrection D P : = { s F } F of cones s F for the faces F of P . Conversely, for given fan D in V , a polytope P in V ∗ whose (inner) normal fan coincides with D is called an (inner) normalpolytope of D . If such a polytope P exists for D , then D is said to be polytopal .Now assume that D is a nonsingular fan in g . We also prepare several notations ofsubmanifolds. For a cone s ∈ D , we denote by X s the closed toric subvariety of X ( D ) corresponds to s . If l , . . . , l k ∈ Hom ( S , G ) be the primitive generators of 1-cones of D , s can be written as s = pos ( l i | i ∈ I ) for some I ⊆ { , . . . , k } and ( l i ) i ∈ I is a part of Z -basisof Hom ( S , G ) . More precisely, each point of X s is fixed by a subtorus G s of G , and ( l i ) i ∈ I is a Z -basis of Hom ( S , G s ) . The image Y s of X s by the quotient map X ( D ) → X ( D ) / H is a closed submanifold of X ( D ) / H because X s and Y s both are connected components ofthe set of fixed points by the G s -actions. If we denote by ( a Ii ) i ∈ I the dual basis of ( l i ) i ∈ I ,the set of nonzero weights of T x Y s coincides with ( a Ii ) i ∈ I for all x ∈ Y s . Proof of Lemma 5.3.
We may assume that q ∗ ◦ e F = F without loss of generality. For v ∈ g ,define h v : X ( D ) / H → g ∗ by h v ( x ) = h F ( x ) , v i . Then, h v ( x ) = h e F ( x ) , q ( v ) i . We shall seethat each connected component of the set of critical points of h v is one of Y s for some s ∈ D .Let x ∈ X ( D ) / H . Since dh v = − i X v w , x is a critical point of h v if and only if ( X v ) x ∈ T x F g J ,in particular, v ∈ g x + g J . Therefore, the set of critical points is [ s ; v ∈ g s + g J Y s . Assume that h v takes the minimum value a v on Y s . Let v s ∈ g s such that q ( v ) = q ( v s ) .Then, h d a Ii , v s i > i ∈ I , where ( l i ) i ∈ I is the set of primitive generators of s (seeRemark 2.3). Therefore v s sits in the relative interior of s . In particular, q ( v ) sits in therelative interior of q ( s ) . The converse is also true; if q ( v ) sits in the relative interior of q ( s ) , then h v takes the minimum value a v on Y s .By Corollary 4.4, e F ( X ( D ) / H ) is a convex polytope P in ( g / g J ) ∗ . We claim that, for each s , the image of Y s by e F is a face of P . Let v ∈ g such that q ( v ) sits in the relative interiorof s . Then, h v takes the minimum value a v on Y s . But h v ( x ) = h e F ( x ) , q ( v ) i implies that x attains the minimum value a v of h v if and only if e F ( x ) attains the minimum value a v of q ( v ) | P . Since h − v ( a v ) = Y s , we have that ( q ( v ) | P ) − ( a v ) = e F ( Y s ) . Since ( q ( v ) | P )( a ) ≥ a v for all a ∈ P , e F ( Y s ) is a face of P that is given by P ∩ H q ( v ) , a v , where H q ( v ) , a v is thehyperplane in ( g / g J ) ∗ defined as H q ( v ) , a v : = { a ∈ ( g / g J ) ∗ | h a , q ( v ) i = a v } .Conversely, if a face F of P is given by P ∩ H q ( v ) , a v , then F is the image of Y s by e F ,where s is the cone such that q ( v ) sits in the relative interior of q ( s ) . It turns out that foreach face F of P there exists a cone s such that the inner normal cone of F coincides with q ( s ) . Hence P is a normal polytope of q ( D ) , as required. (cid:3) Now we consider the obstruction for the existence of a moment map in case of LVMBmanifold with indispensable integer 0. Let S be an abstact simplicial complex on { , , . . . , m } (a singleton { i } does not need to be a member of S ). Let G = ( S ) m . Then g = R m and Z m is identified with Hom ( S , G ) . G acts on C P m via ( g , . . . , g m ) · [ z , z , . . . , z m ] : = ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 15 [ z , g z , . . . , g m z m ] for ( g , . . . , g m ) ∈ G and [ z , z , . . . , z m ] ∈ C P m . Put e : = − e − · · · − e m and define D : = { pos ( e i | i ∈ I ) | I ∈ S } . D is a nonsingular fan in R m and the toric variety X ( D ) associated with D S is given by X ( D S ) = [ I ∈ S U I , where U I = { [ z ] = [ z , . . . , z m ] ∈ C P m | z j = j / ∈ I } . Let h ⊆ C m such that D : = D S and h satisfy the conditions (1) and (2). We call the manifold X ( D S ) / H an LVMB manifold . Moreover, if q ( D S ) is polytopal, we call it an LVM manifold ,due to [3, Theorems 2.2 and 3.10].If an integer i satisfies that { i } / ∈ S , we say that i is indispensable , according to theliterature of LVMB manifolds (see [4], [11] and [12]). In case when 0 is indispensable, [ z ] ∈ X ( D S ) implies that z =
0. Therefore we can think of X ( D S ) as an open subset of C m via the map [ z , . . . , z m ] (cid:18) z z , . . . , z m z (cid:19) . Namely, X ( D S ) = [ I ∈ S U ′ I , where U ′ I = { z = ( z , . . . , z m ) ∈ C m | z j = j / ∈ I } . In case when 0 is indispensable, we call X ( D S ) / H an LVMB manifold with indispensableinteger X ( D S ) / H withrespect to F g J vanish for shellable S . Therefore for shellable S , there exists a momentmap for any transverse K¨ahler form on X ( D S ) / H with respect to F g J . We can avoid theassumption on S for the vanishing of first basic cohomology groups with a straightforwardcomputation. Lemma 5.4.
Let X ( D S ) / H be an LVMB manifold with indispensable integer . Let J bethe complex structure on X ( D S ) / H. Then, H F g J ( X ( D S ) / H ) = .Proof. Assume that, { i } is a member of S for i = , . . . , r but not for i = r + , . . . , m . Then, X ( D S ) = X ( D S ′ ) × ( C \ { } ) m − r , where S ′ is an abstract simplicial complex on { , . . . , r } such that if I ∈ S then I ∈ S ′ . X ( D S ′ ) is a complement of coordinate subspaces of realcodimension ≥ C r . Thus, X ( D S ′ ) is simply connected. Let c i : S → X ( D S ) be thecurve defined by c i ( t ) = ( , . . . , | {z } i − , t , , . . . , | {z } m − i ) ∈ X ( D S ) ⊆ C m for i = , . . . , m . c i is null-homologous for i = , . . . , r and the homology classes [ c i ] deter-mined by c i for i = r + , . . . , m form a basis of H ( X ( D S )) .Let b be a 1-form on X ( D S ) / H . b is closed and basic for F g J if and only if d b = i X v b = v ∈ g J . Let p : X ( D S ) → X ( D S ) / H be the quotient map. p ∗ b is a 1-formbasic for F h . That is, i X u p ∗ b = i X u d p ∗ b = u ∈ h . Therefore we need to showthat a closed 1-form g on X ( D S ) satisfying • i X u g = u ∈ h , • i X v g = v ∈ g J .is exact. Let g be such a 1-form on X ( D S ) . By averaging g with the action of G , we mayassume that g is G -invariant without loss of generality. Since g is real, we can represent g = m (cid:229) i = f i dz i + f i dz i with smooth functions f i : X ( D S ) → C . Let v = ( v , . . . , v m ) ∈ g = R m . Then X v can berepresented as(5.1) X v = p m (cid:229) i = √− v i (cid:18) z i ¶¶ z i − z i ¶¶ z i (cid:19) . Since g is G -invariant, we have that there exists a i ∈ R such that 2 p √− ( z i f i − z i f i ) = a i for i = , . . . , m . Since H ( X ( D S )) is generated by c i for i = r + , . . . , m and the Kroneckerpairing is given by h [ c i ] , [ g ] i = a i , it suffices to show that a i = i = r + , . . . , m .If u = ( u , + √− u , √− , . . . , u m , + √− u m , √− ) ∈ C m , X u can be represented as(5.2) X u = p m (cid:229) i = (cid:18) √− u i , (cid:18) z i ¶¶ z i − z i ¶¶ z i (cid:19) − u i , √− (cid:18) z i ¶¶ z i + z i ¶¶ z i (cid:19)(cid:19) . Since p ( h ) = g J by Proposition 5.2, it follows from (5.1) and (5.2) that the conditions i X v g = v ∈ g J and i X u g = u ∈ h are equivalent to(5.3) m (cid:229) i = v i z i f i = v = ( v , . . . , v m ) ∈ g J . Assume that { , . . . , n } ∈ S is a maximal simplex. Then, q ( e ) , . . . , q ( e n ) form a basis of g / g J . Let a , . . . , a n be the dual basis of q ( e ) , . . . , q ( e n ) . Then, we have a basis e j − n (cid:229) i = h a i , q ( e j ) i e i for j = n + , . . . , m of g J = ker q . Therefore we have that (5.3) is equivalent to(5.4) z j f j − n (cid:229) i = h a i , q ( e j ) i z i f i = j = n + , . . . , m . ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 17
The Kronecker pairing h [ c i ] , [ g ] i = i = , . . . , r because c i for i = , . . . , r is null-homologous. Therefore a i = p √− ( z i f i − z i f i ) = i = , . . . , n . This together with(5.4) yields that a j = j = n + , . . . , m . Therefore g is exact, proving the lemma. (cid:3) Corollary 5.5.
Assume that F g J on X ( D S ) / H is transverse K¨ahler. Then, the complete fanq ( D ) in g / g J is polytopal. Namely, X ( D S ) / H is an LVM manifold.Proof.
Let w be a transverse K¨ahler form on X ( D S ) / H with respect to F g J . Since F g J is G -invariant, we may assume that w is G -invariant by Proposition 4.1. The closed 1-form − i X v w is exact for all v ∈ g by Lemma 5.4. Therefore there exists a moment mapon X ( D S ) / H with respect to w . Let e F : X ( D S ) / H → ( g / g J ) ∗ be a lifted moment map. ByLemma 5.3, the image of X ( D S ) / H by e F is a normal polytope of q ( D ) . Therefore q ( D ) ispolytopal, as required. (cid:3) Conversely, we can construct a transverse K¨ahler form on X ( D S ) / H with respect to F g J from a normal polytope P of q ( D ) . Essentially, this fact has been shown in [9] and [11].But, the “language” in this paper is slightly different from them. For reader’s convenience,we give a brief explanation of the construction of a transverse K¨ahler form without a proof.Let P be a normal polytope of q ( D S ) represented as P = { a ∈ ( g / g J ) ∗ | h a , q ( e i ) i ≥ a i } . The map q ∗ : ( g / g J ) ∗ → g ∗ is an injective map. We consider the embedding P → g ∗ givenby a m (cid:229) i = ( h a , q ( e i ) i − a i ) e ∗ i = q ∗ ( a ) − m (cid:229) i = a i e ∗ i , where e ∗ i denotes the i -th dual basis vector of the standard basis e , . . . , e m of g = R m .Let i : g J → g be the inclusion and consider the dual map i ∗ : g ∗ → g ∗ J . The image ofembedded P is the point i ∗ ( (cid:229) mi = − a i e ∗ i ) = : b . X ( D S ) is an open subset of C m . So X ( D S ) has the standard K¨ahler form w st = √− m (cid:229) i = dz i ∧ dz i . G acts on X ( D S ) preserving w st . The map F : X ( D S ) → g ∗ given by F ( z , . . . , z m ) = p m (cid:229) i = | z i | e ∗ i is a moment map with respect to w st . For the compostion i ∗ ◦ F : X ( D S ) → g ∗ J , the value b ∈ g ∗ J is a regular value and ( i ∗ ◦ F ) − ( b ) = : Z P is a smooth manifold equipped withan action of G and the G -invariant transverse symplectic form w : = w st | Z P with respectto F g J . Each orbit of H intersects with Z P at exactly one point in Z P , and hence theinclusion Z P → X ( D S ) induces an equivariant diffeomorphism j : Z P → X ( D S ) / H . Theform ( j − ) ∗ w on X ( D S ) / H is what we wanted. The image of a lifted moment map isnothing but P up to translations. The construction above and Corollary 5.5 yields the following.
Theorem 5.6.
The holomorphic foliation F g J on an LVMB manifold X ( D S ) / H with indis-pensable integer is transverse K¨ahler with respect to F g J if and only if X ( D S ) / H is anLVM manifold with indispensable integer . We give remarks on the foliation F g J and equivariant holomorphic principal bundles.Let M and M are complex manifolds with the complex structures J and J , respec-tively. Assume that compact tori G and G act on M and M respectively. If we havean equivariant principal holomorphic bundle p : M → M , it is easy to see that T x F g J =( d p ) − x ( T p ( x ) F g J ) for all x ∈ M . Therefore we can obtain every basic form for F g J from a basic form for F g J by the pull-back operator p ∗ . Moreover, every basic form for F g J is also basic for the action of ker a (that is, invariant under the action of ker a and theinterior product with fundamental vector fields generated by the action of ker a vanishes).Therefore there exists the inverse operator ( p ∗ ) − of p ∗ defined for basic forms for F g J .In particular, F g J on M is transverse K¨ahler if and only if so is F g J on M . Also, thereexists a moment map F : M → g ∗ with respect to a G invariant transverse K¨ahler form w if and only if there exists a moment map F : M → g ∗ with respect to ( p ∗ ) − w .It has been shown in [8] that a compact connected complex manifold M equipped with amaximal action of a compact torus G is obtained as a quotient of an LVMB manifold withindispensable integer 0. Therefore, we can characterize the manifold with a maximal torusactions which admits a transverse K¨ahler form with respect to F g J . Theorem 5.7.
Let M, G, J, D , h be as Theorem 5.1. Then, the followings are equivalent: (1) F g J on M is transverse K¨ahler. (2) q ( D ) is polytopal.In this case, for any G-invariant transverse K¨ahler form w , there exists a moment map F : M → g ∗ with respect to w and the image of M by a lifted moment map e F : M → ( g / g J ) ∗ is an inner normal polytope of q ( D ) . As a corollary, we show that the conjecture posed in [6] holds.
Corollary 5.8.
For an LVMB manifold M, the holomorphic foliation F g J is transverseK¨ahler if and only if M is an LVM manifold. R EFERENCES [1] M. F. Atiyah,
Convexity and commuting Hamiltonians , Bull. London Math. Soc. (1982), no. 1, 1–15,DOI 10.1112/blms/14.1.1. MR642416 (83e:53037)[2] Fiammetta Battaglia and Dan Zaffran, Foliations Modeling Nonrational Simplicial Toric Varieties , Int.Math. Res. Notices, first published online: February 24, 2015, DOI 10.1093/imrn/rnv035, (to appearin print), available at http://imrn.oxfordjournals.org/content/early/2015/02/24/imrn.rnv035.full.pdf+html .[3] L. Battisti,
LVMB manifolds and quotients of toric varieties , Math. Z. (2013), no. 1-2, 549–568,DOI 10.1007/s00209-013-1147-8. MR3101820
ORUS INVARIANT TRANSVERSE K ¨AHLER FOLIATIONS 19 [4] Fr´ed´eric Bosio,
Vari´et´es complexes compactes: une g´en´eralisation de la construction de Meerssemanet L´opez de Medrano-Verjovsky , Ann. Inst. Fourier (Grenoble) (2001), no. 5, 1259–1297 (French,with English and French summaries). MR1860666 (2002i:32015)[5] Fr´ed´eric Bosio and Laurent Meersseman, Real quadrics in C n , complex manifolds and convex polytopes ,Acta Math. (2006), no. 1, 53–127, DOI 10.1007/s11511-006-0008-2. MR2285318 (2007j:32037)[6] St´ephanie Cupit-Foutou and Dan Zaffran, Non-K¨ahler manifolds and GIT-quotients , Math. Z. (2007), no. 4, 783–797, DOI 10.1007/s00209-007-0144-1. MR2342553 (2008g:32031)[7] V. Guillemin and S. Sternberg,
Convexity properties of the moment mapping , Invent. Math. (1982),no. 3, 491–513, DOI 10.1007/BF01398933. MR664117 (83m:58037)[8] Hiroaki Ishida, Complex manifolds with maximal torus actions , available at http://arxiv.org/abs/1302.0633v3 .[9] J. J. Loeb and M. Nicolau,
On the complex geometry of a class of non-K¨ahlerian manifolds , Israel J.Math. (1999), 371–379, DOI 10.1007/BF02808191. MR1750427 (2001b:32034)[10] Santiago L´opez de Medrano and Alberto Verjovsky,
A new family of complex, compact, non-symplecticmanifolds , Bol. Soc. Brasil. Mat. (N.S.) (1997), no. 2, 253–269, DOI 10.1007/BF01233394.MR1479504 (98g:32047)[11] Laurent Meersseman, A new geometric construction of compact complex manifolds in any dimension ,Math. Ann. (2000), no. 1, 79–115, DOI 10.1007/s002080050360. MR1760670 (2001i:32029)[12] Laurent Meersseman and Alberto Verjovsky,
Holomorphic principal bundles over projective toricvarieties , J. Reine Angew. Math. (2004), 57–96, DOI 10.1515/crll.2004.054. MR2076120(2005e:14080)[13] Taras Panov, Yury Ustinovsky, and Misha Verbitsky,
Complex geometry of moment-angle manifolds ,available at http://arxiv.org/abs/1308.2818 .D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE , G
RADUATE S CHOOL OF S CIENCE AND E NGINEERING , K
AGOSHIMA U NIVERSITY
E-mail address ::