On displaceability of pre-Lagrangian fibers in contact toric manifolds
OON DISPLACEABILITY OF PRE-LAGRANGIAN TORIC FIBERS INCONTACT TORIC MANIFOLDS
A. MARINKOVI ´C, M. PABINIAK
Abstract.
In this note we analyze displaceability of pre-Lagrangian toric fibers in con-tact toric manifolds. While every symplectic toric manifold contains at least one non-displaceable Lagrangian toric fiber and infinitely many displaceable ones, we show thatthis is not the case for contact toric manifolds. More precisely, we prove that for thecontact toric manifolds S d − ( d ≥
2) and T k × S d + k − ( d ≥
1) all pre-Lagrangian toricfibers are displaceable, and that for all contact toric manifolds for which the toric actionis free, except possibly non-trivial T -bundles over S , all pre-Lagrangian toric fibers arenon-displaceable. Moreover we also prove that if for a compact connected contact toricmanifold all but finitely many pre-Lagrangian toric fibers are non-displaceable then theaction is necessarily free. On the other hand, as we will discuss, displaceability of allpre-Lagrangian toric fibers seems to be related to the non-orderability of the underlyingcontact manifolds. Introduction
One of the questions of great importance in symplectic geometry is whether a givenLagrangian submanifold of a symplectic manifold can be displaced off itself via a Hamil-tonian isotopy. If the symplectic manifold is toric, i.e. it can be equipped with an effectiveHamiltonian action of a torus of dimension equal to half of the dimension of the manifold,then every generic toric orbit (i.e. of maximal dimension) is a Lagrangian submanifold. Itis usually called a Lagrangian toric fiber as it is a fiber of the moment map. Displaceabilityproperties of Lagrangian toric fibers in symplectic toric manifolds have been extensivelystudied. In particular the following two important results have been proved(A) any compact connected symplectic toric manifold contains a non-displaceable La-grangian toric fiber ([18], [19], [17], [24], [32]), whereas(B) any compact connected symplectic toric manifold has uncountably many displace-able Lagrangian toric fibers ([33], [1]).The goal of this paper is to study the analogue of the above questions in the setting ofcontact toric manifolds.The question of displaceability of Lagrangians in symplectic manifolds can be translatedto the contact setting in two different ways. Given a pre-Lagrangian L d in a contact manifold( V d − , ξ ) one can ask if there exists a contact isotopy ϕ t of ( V d − , ξ ) which displaces L d .Or, given a Legendrian submanifold N d − of ( V d − , ξ ) one can ask if there exists a contactisotopy ϕ t of ( V d − , ξ ) such that there are no Reeb chords between N d − and ϕ ( N d − ).In this work we concentrate on the first question as this is the direct translation of the a r X i v : . [ m a t h . S G ] O c t A. MARINKOVI ´C, M. PABINIAK problem of displaceability of generic torus orbits: if ( V d − , ξ ) is a contact toric manifold ,then the generic orbits of the toric action are pre-Lagrangian submanifolds and are fibersof the moment map (with the exception of ( T , ker α k ), k >
1, described in Section 2, whereeach fiber consists of k orbits). For a precise definition of pre-Lagrangians, moment mapsin the contact setting and other background information we direct the reader to Section2. We show that displaceability properties of pre-Lagrangian toric fibers in contact toricmanifolds are very different from those of Lagrangian toric fibers in the symplectic setting.None of the above itemized statements holds when translated to the contact toric setting.First we observe that contact toric manifolds may have all pre-Lagrangian toric fibersdisplaceable, i.e. the contact version of (A) does not hold. Proposition 1.1.
Every pre-Lagrangian toric fiber in the standard contact toric sphere S d − , d ≥ , is displaceable. In fact all closed proper subset of ( S d − , ξ st ) are displaceable (Proposition 4.1). It is alsoworth observing that there exists a contact isotopy that simultaneously displaces all thesepre-Lagrangian toric fibers (Remark 4.2). Moreover, the standard contact sphere is not theonly instance of this phenomenon. Theorem 1.2.
Every pre-Lagrangian toric fiber in T k × S d + k − , d ≥ , is displaceable. This phenomenon seems to be related to the notion of orderability introduced by Eliash-berg and Polterovich in [16] and recalled here in Section 2. The contact manifolds S d − and T k × S d + k − ( d ≥
2) are known to be not orderable [15]. To the best of our knowledge,at the time of writing these are all contact toric manifolds which are proved to be non-orderable. Orderability of T k × S k +1 remains an open question, even for k = 1. The factthat Theorem 1.2 holds also for d = 1 provides a slight indication that T k × S k +1 mightalso be non-orderable.A connection between orderability and non-displaceability of certain subsets of a contactmanifold has already been studied. It was proved by Eliashberg and Polterovich that acontact manifold is orderable if it contains a pair with the stable intersection property(Theorem 2.3.A in [16]), that is a pair ( L, K ), where L is a pre-Lagrangian and K iseither a pre-Lagrangian or a Legendrian, such that the stabilizations of K and L cannotbe displaced from each other in V × T ∗ S , the stabilization of V . (For precise definitionssee Section 2.2 in [16].) If a pre-Lagrangian L paired with itself has stable intersectionproperty, i.e. if it is stably non-displaceable, then it is non-displaceable. Moreover, Bormanand Zapolsky in [9] proved that any compact contact toric manifold admitting a monotonequasimorphism with the vanishing property is orderable and contains a non-displaceablepre-Lagrangian toric fiber. Theorems 1.1 and 1.2 imply that neither S d − , with d ≥ T k × S d + k − , with d ≥
1, can be equipped with a monotone quasimorphism with thevanishing property . Based on our work and the above results, we conjecture Here we would like to mention another result related to the non-existence of monotone quasimorphismfor the standard sphere: Fraser, Polterovich and Rosen in [20] proved that every conjugation invariant normon the identity component of the universal cover of the contactomorphism group of S d − , d ≥
2, must bebounded and discrete, hence equivalent to the trivial norm.
N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 3
Conjecture 1.3.
If a compact contact toric manifold is not orderable then all its pre-Lagrangian toric fibers are displaceable.
Note that the compactness assumption is crucial. The contact toric manifold ( S × R d , ker( dθ − (cid:80) y j dx j )) is orderable ([34]), even though all pre-Lagrangian toric fibers aredisplaceable (see Example 3.4). Moreover, ( R d +1 , ker( dz − (cid:80) y j dx j )) is orderable ([7]),while every bounded subset of it is displaceable by a translation in the z direction.Another difference in rigidity properties of symplectic and contact manifolds is that thecontact version of (B) does not hold: a contact toric manifold does not need to contain anydisplaceable pre-Lagrangian toric fiber. Proposition 1.4.
Every pre-Lagrangian toric fiber in the contact toric manifold T d × S d − ,d ≥ , (co-sphere bundle of T d ), is non-displaceable. This follows from results of Eliashberg, Hofer and Salamon from [14] where they usedLagrangian Floer homology in the symplectization of contact manifolds to analyze dis-placeability of graphs of non-vanishing 1–forms in cosphere bundles, (see Section 5). Infact, using Example 2.4.A of [16] one can show a stronger result, namely that any pre-Lagrangian toric fiber in T d × S d − , d ≥
2, is stably non-displaceable. Due to Theorem2.3.A of [16] it follows that T d × S d − , d ≥
2, is orderable.
Proposition 1.5.
Every orbit of the torus action in the contact toric manifold ( T = S θ ) × T θ ,θ ) , ker(cos(2 πkθ ) dθ + sin(2 πkθ ) dθ )) , k ≥ , is non-displaceable. In particular, everypre-Lagrangian toric fiber of T with one of the above contact forms is non-displaceable. It is interesting to observe that for contact toric manifolds from Propositions 1.4 and 1.5the toric action is free. We believe that the non-existence of displaceable pre-Lagrangiantoric fibers is related to the fact that the given toric action is free. A toric action on acompact symplectic toric manifold is never free (it is free only on the pre-image of theinterior of the moment map image). In fact even a Hamiltonian circle action on a compactsymplectic toric manifold is never free: compactness implies that the Hamiltonian momentmap must attain its extrema, and the non-degeneracy of the symplectic form implies thatthe Hamiltonian vector field vanishes at these points, thus these points must be fixed underthe circle action. In the contact setting, the Hamiltonian vector field only needs to be inthe kernel of dα at the points where the moment map attains its extrema.All contact toric manifolds admitting a free toric action are listed in Lerman’s Classifi-cation Theorem (Theorem 2.18 in [28]) which we recall in Section 2. The only contact toricmanifolds with a free toric action that are not included in the statements of Propositions1.4 and 1.5 are the non-trivial T -bundles over S . It would be interesting to see if in thesecases all pre-Lagrangian toric fibers are non-displaceable. Finally, in Section 5.3 we provethe following.
Theorem 1.6.
Every compact connected contact toric manifold for which the toric actionis not free contains uncountably many displaceable pre-Lagrangian toric fibers.
Organization.
Section 2 contains background material on contact manifolds and toricactions. In Section 3 we present some methods of displacing pre-Lagrangian toric fibers by
A. MARINKOVI ´C, M. PABINIAK analyzing prequantization maps, contact reduction and contact cutting. In Section 4 weapply these tools to displace all pre-Lagrangian toric fibers of the non-orderable contacttoric manifolds of Propositions 1.1 and 1.2. Section 5 is devoted to study non-displaceabilityof pre-Lagrangian toric fibers of contact toric manifolds with free toric action. There weprove Propositions 1.4, 1.5 and Theorem 1.6.2.
Background on contact manifolds and torus actions.
Contact manifolds and pre-Lagrangian submanifolds.
Let ( V d − , ξ ) be a coori-ented contact manifold. Recall that the symplectization of ( V d − , ξ ) is the symplecticmanifold SV = { ( p, η p ) ∈ T ∗ V | ker η p = ξ p and η p agrees with the coorientation of ξ p } with the symplectic form dλ | SV , where λ is the canonical Liouville 1-form on T ∗ V. Thestandard R + -action on SV defined by t · ( p, η p ) (cid:55)→ ( p, tη p ) makes SV a principal R + -bundleover V . Any contact form for ξ is a section of this bundle. We denote by π : SV → V theprojection π ( p, η p ) = p. A submanifold L d ⊂ ( V d − , ξ ) is said to be a pre-Lagrangian if it is the diffeomorphic image under π of some Lagrangian submanifold (cid:101) L ⊂ SV.
Thenotion of pre-Lagrangian submanifold and the related displaceability problems have beenfirst studied in [14].We will analyze the problem of displaceability of pre-Lagrangians under contact iso-topies. In symplectic topology every (possibly time-dependent) function h t : M (cid:55)→ R on asymplectic manifold ( M, ω ) induces a symplectic isotopy ϕ t , which is defined to be the flowof the vector field X h t determined by the relation X h t (cid:121) ω = dh t . The isotopy { ϕ t } is said tobe a Hamiltonian isotopy, with Hamiltonian function h t . A Lagrangian L in a symplecticmanifold ( M, ω ) is said to be non-displaceable if ϕ ( L ) ∩ L (cid:54) = ∅ for every Hamiltonian iso-topy { ϕ t } t ∈ [0 , on M with ϕ = id . In the contact setting, every (possibly time-dependent)function h t : V (cid:55)→ R on a contact manifold ( V, ξ = ker α ) induces a contact isotopy ϕ t ,which is defined to be the flow of the vector field X h t determined by the relation(1) α ( X h t ) = h t and X h t (cid:121) dα = dh t ( R α ) α − dh t . Here R α denotes the Reeb vector field associated to a contact form α, that is, the uniquevector field such that R α (cid:121) dα = 0 and α ( R α ) = 1 . Note that { ϕ t } depends on the choice of a contact form α . The function h t is then saidto be the Hamiltonian function of the contact isotopy { ϕ t } with respect to the contactform α . Contrary to the symplectic case, any contact isotopy is induced by a Hamiltonianfunction (which is defined uniquely by h t = α ( X h t ), where X t is the vector field generatingthe isotopy). Translating the notion of non-displaceable Lagrangian fiber in a symplecticmanifold to the contact setting we obtain the following definition. A pre-Lagrangian L ⊂ V is called non-displaceable if ϕ ( L ) ∩ L (cid:54) = ∅ for every contact isotopy { ϕ t } t ∈ [0 , on V , with ϕ = id . Otherwise, it is called displaceable. N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 5
Contact toric manifolds.
A co-oriented contact manifold ( V d − , ξ ) with an effec-tive action of the torus T d that preserves the contact structure ξ is called a contact toricmanifold . As T d is compact one can always choose a T d -invariant contact form α for ξ .For such a T d -invariant contact form α , the α -moment map µ α : V → ( t d ) ∗ is defined by (cid:104) µ α ( p ) , X (cid:105) = α p ( X p ) , where X is the vector field on V generated by X ∈ t d and (cid:104)· , ·(cid:105) denotes the natural pairingbetween t d and ( t d ) ∗ . If V is equipped with a contact toric action of T d , the lift of this actionto T ∗ V is symplectic and keeps SV invariant, making SV a symplectic toric manifold. Thecorresponding moment map Φ : SV → ( t d ) ∗ , called the contact moment map , is givenby (cid:104) Φ( p, η p ) , X (cid:105) = η p ( X p )and does not depend on the choice of a contact form. The moment cone is the image ofthe contact moment map. One can identify the Lie algebra t d of T d with R d by fixing asplitting of T d into a product of circles and an identification Lie ( S ) ∼ = R , and view themoment maps µ α and Φ as maps to ( R d ) ∗ . We use the convention S ∼ = R / Z . Note thatswitching to a different convention would change µ α ( V ), but not the moment cone Φ( SV ),as the cone is invariant under rescaling in the radial direction. Changing the splitting of T d , i.e. reparametrizing the action, would result in applying a GL ( d, Z ) transformation to µ α ( V ) and Φ( SV ) (see Example 3.1).For any T d -invariant contact form α it holds that Φ ◦ α = µ α (Proposition 2.8 in [28])and, for any c ∈ µ α ( V ) ∩ Int Φ( SV ) , the restriction of the projection π : SV → V mapsdiffeomorphically Φ − ( c ) onto µ − α ( c ). Therefore generic fibers of µ α are pre-Lagrangians.We call them the pre-Lagrangian toric fibers . Connected components of the fibers of µ α are T d -orbits (see Lemma 3.16 in [28]). If dim V > µ α are connected. In fact the only contact toric manifolds with disconnectedpre-Lagrangian fibers are ( T , ker α k ) described below, with k >
1. (Each fiber has exactly k connected components).The classification of compact contact toric manifolds initiated by Banyaga-Molino [5],[6] and Boyer-Galicki [11] was concluded by Lerman in [28]. Compact connected contacttoric manifolds ( V, ξ ) are classified as follows (Theorem 2.18 in [28]): • Suppose dim V = 3 and the torus action is free. Then V is diffeomorphic to T = S θ ) × T θ ,θ ) with the contact form α k = cos( kt ) dθ + sin( kt ) dθ , for some k ≥ R . Note that T is the cosphere bundle of T and the standardcontact structure of the cosphere bundle corresponds to k = 1. • Suppose dim V = 3 and the torus action is not free. Then V is diffeomorphic to S , S × S or a lens space. There are various possible toric actions and various possiblecontact structures (including overtwisted ones). For details see [28]. • Suppose dim
V > V is a principal T d -bundle over S d − , and the moment cone is the whole R d . Each such principal bundle has a unique T d -invariant contact structure making it a contact toric manifold. Since principal A. MARINKOVI ´C, M. PABINIAK T d -bundles over a manifold are in one-to-one correspondence with the second co-homology classes of the manifold with coefficients in Z d and since H ( S d − , Z d ) = 0for d − (cid:54) = 2 , it follows that when dim V = 2 d − > , V must be the trivial bundle T d × S d − . • Suppose dim
V > GL (2 d − , Z ) transfor-mations (corresponding to changing a splitting of a torus into a product of circles).When the moment cone is strictly convex then V is of Reeb type, i.e. V admitsa contact form whose Reeb vector field generates a circle subaction of the toricaction. Otherwise, i.e. when the moment cone is convex but not strictly convex, V is T k × S d + k − . Orderability.
Eliashberg and Polterovich in [16] defined a relation (cid:22) on the universalcover of the identity component of the contactomorphism group, (cid:93)
Cont ( V, ξ ): two elements[ { φ t } ] , [ { ψ t } ] ∈ (cid:93) Cont ( V, ξ ) satisfy [ { φ t } ] (cid:22) [ { ψ t } ] if [ { ψ t } ] ◦ [ { φ t } ] − can be represented bya non-negative contact isotopy, i.e. a contact isotopy that moves every point of V in a direc-tion positively transverse or tangent to ξ (equivalently, a contact isotopy that is generatedby a non-negative contact Hamiltonian). This relation is always reflexive and transitive. Ifit is also anti-symmetric then it defines a bi-invariant partial order on (cid:93) Cont ( V, ξ ) and thecontact manifold (
V, ξ ) is called orderable . Equivalently, according to Proposition 2.1.A in[16], a contact manifold is orderable if there are no contractible loops of contactomorphismsgenerated by a strictly positive contact Hamiltonian. Eliashberg, Kim and Polterovich inTheorem 1.16 in [15] showed that the ideal contact boundary of a product M × C d of aLiouville manifold M and ( C d , ω = i (cid:80) dz ∧ d ¯ z ) is not orderable for d ≥
2. Two importantcases which we consider here are when M is a point and when M = T ∗ S k . In these casesthe ideal contact boundary of M stabilized d times is, respectively, the standard contactspheres S d − and the manifolds T k × S d + k − with the contact form which is described indetail in Section 4.2. Hence, S d − and T k × S d + k − are non-orderable if d ≥ . In Section4 we prove that all their pre-Lagrangian toric fibers are displaceable.3.
Methods for displacing.
In this section we describe some methods of displacing pre-Lagrangian toric fibers incontact toric manifolds. All these methods are obtained by a similar pattern: we look atvarious ways of constructing a new manifold from a given one (prequantization, contactreduction, contact cut) and deduce relations between displaceability properties of corre-sponding subsets of the manifold we started with and of the manifold we constructed.3.1.
Prequantization of a symplectic toric manifold and displaceability.
Let (
M, ω )be a symplectic manifold such that the cohomology class [ ω ] is integral. The prequanti-zation of ( M, ω ) is the principal S − bundle π : V → M with Euler class [ ω ]. There is aconnection 1-form α on V such that π ∗ ω = dα , which is also a contact form on V , and thus( V, ξ = ker α ) is a contact manifold. This construction is due to Boothby-Wang [8] (see also[21, Section 7.2.]). The orbits of the Reeb vector field R α associated to the contact 1-form N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 7 α are the fibers of the bundle. If V is the prequantization of ( M, ω ) then for the subgroup Z k = { e πi l/k ; l = 0 , . . . , k − } ⊂ S the quotient submanifold V / Z k is the prequantizationof ( M, kω ).If ( M d , ω ) is a compact symplectic toric manifold then the T d -action lifts to a contact T d -action on V . Together with the S -action given by the Reeb flow this gives a T d +1 -action on the prequantization ( V d +1 , ξ ) making it a contact toric manifold (see [29]). Thepre-image under π of a Lagrangian toric fiber in M is a pre-Lagrangian toric fiber in V. Moreover, the image of the contact moment map Φ : SV → ( R d ) ∗ × ( R ) ∗ is a cone over themoment map image ∆ of M , namely it is C = { a ( x, ∈ ( R d ) ∗ × ( R ) ∗ | x ∈ ∆ , a ∈ R > } . Symplectic reduction of SV with respect to the circle { } × S ⊂ T d +1 taken at level 1gives back the symplectic toric manifold ( M, ω ) (Lemma 3.7 in [29]).
Example . The standard contact sphere, ( S d − , ker α st ), α st = i (cid:80) di =1 ( z i d ¯ z i − z i dz i ), isthe prequantization of the complex projective space ( CP d − , π ω F S ) as the first Chern classof the Hopf fibration S d − → CP d − is π ω F S . Performing the above construction for thestandard T d − -action on ( CP d − , ω F S ), one equips the prequantization space S d − with atoric T d -action ( t , . . . , t d ) ∗ ( z , . . . , z d − ) = ( t d z , t d t z , . . . , t d t d − z d − )(the Reeb flow gives the diagonal circle action). The corresponding moment cone is spannedby the directions e + e d , e + e d , . . . , e d − + e d , e d , where e i , i = 1 , . . . d are the coordinate axesin R d . Observe that the resulting action differs by a reparametrization from the standardaction of T d on S d − induced from the T d -action on C d , where each circle in T d rotates thecorresponding copy of C with speed 1 (see Section 4.1). This is why the above moment conediffers by a GL ( d, Z )-transformation from the moment cone of the standard action of T d on S d − (which is spanned by the directions e , . . . , e d ; Section 4.1). Furthermore, the realprojective space RP d − = S d − / Z , and more generally the lens spaces L d − p = S d − / Z p ,p ∈ N , with contact forms induced by α st are prequantizations of ( CP d − , pπ ω F S ). Hence,they are also contact toric manifolds, and their moment cones are spanned by the directions pe + e d , pe + e d , . . . , pe d − + e d , e d . We now explain a connection between non-displaceability in a symplectic toric manifold(
M, ω ) and in its prequantization (
V, ξ ). Note that if L (cid:48) ⊂ M is a Lagrangian submanifoldthen π − ( L (cid:48) ) ⊂ V is a pre-Lagrangian submanifold. By Ham ( M, ω ) we denote the groupof Hamiltonian diffeomorphisms on (
M, ω ) and by
Cont ( V, ξ ) the identity component ofthe contactomorphisms group of (
V, ξ ) . Lemma 3.2. ( Lifting property for prequantization ) Let ϕ ∈ Ham ( M, ω ) . Then, there is (cid:101) ϕ ∈ Cont ( V, ξ ) such that π ◦ (cid:101) ϕ = ϕ ◦ π .Proof. Let ϕ t be a Hamiltonian isotopy such that ϕ = ϕ and let h t : M → R be the cor-responding time dependent Hamiltonian function. Let (cid:101) ϕ t be the contact isotopy generatedby (cid:101) h t = h t ◦ π . Then (cid:101) ϕ = (cid:101) ϕ has the desired properties. A. MARINKOVI ´C, M. PABINIAK (cid:3)
What follows is an analogue of the result of Abreu and Macarini on preserving non-displaceability under symplectic reduction (Proposition 3.2 in [2]). The proof immediatelyfollows from Lemma 3.2.
Proposition 3.3.
Let ( V, ξ ) be a contact toric manifold which is the prequantization of asymplectic toric manifold ( M, ω ) . If a Lagrangian toric fiber L ⊂ M is displaceable then sois the pre-Lagrangian toric fiber π − ( L ) ⊂ V. Example . The contact toric manifold ( S × R d , ker( dθ − (cid:80) dj =1 y j dx j )) is the prequanti-zation of the symplectic toric manifold ( R d , (cid:80) dj =1 dx j ∧ dy j ) (with the standard T d -actionon R d ∼ = C d ). Every Lagrangian toric fiber in R d is displaceable by Hamiltonian isotopies:for example, by an appropriate translation in one of the coordinates. Thus, by Proposition3.3, every pre-Lagrangian toric fiber in S × R d is also displaceable.3.2. The method of probes.
The method of probes was introduced by McDuff [33] andserves to displace some Lagrangian toric fibers in symplectic toric manifolds. We brieflyrecall it here, since the results of Section 4.2 will be proved by combining Proposition 3.3with displaceability results in symplectic toric manifolds obtained via this method.Let ∆ = µ ( M ) ⊂ R n be the Delzant polytope corresponding to some symplectic toricmanifold M n with moment map µ . Take any facet F of ∆ ⊂ R n and denote its inwardnormal by η F . An integral vector λ ∈ Z n is called integrally transverse to F if |(cid:104) λ, η F (cid:105)| = 1.The probe p F,λ ( w ) with direction λ ∈ Z n and initial point w ∈ F is the half open linesegment consisting of w and all the points in Int ∆ that lie on the ray from w in the direction λ . McDuff in Lemma 2.4 of [33, Lemma 2.4] proved that if w ∈ Int F and u ∈ Int ∆ isany point which lies on the probe p F,λ ( w ), less than halfway along it, then the Lagrangiantoric fiber in M corresponding to u is displaceable. Moreover, this fiber can be displacedby an isotopy of M supported in a compact subset of µ − ( p F,λ ( w )).Using this method McDuff [33] and Abreu-Borman-McDuff [1] showed that any compactsymplectic toric manifold contains uncountably many displaceable Lagrangian toric fibers.Combining this with Proposition 3.3 we see that any contact toric manifold ( V, ξ ) whichis the prequantization of a compact symplectic toric manifold contains uncountably manydisplaceable pre-Lagrangian toric fibers.In particular this implies that a contact toric manifold for which all pre-Lagrangian toricfibers are non-displaceable (for instance, T k × S d + k − as we will see in 4.2) cannot be theprequantization of a compact symplectic toric manifold. Remark . According to the definition given by Entov and Polterovich in [17], a La-grangian toric fiber in a symplectic toric manifold is a stem if any other Lagrangian toricfiber is displaceable. Translating to the contact setting one obtains the following definition:a pre-Lagrangian toric fiber is a (contact) stem if every other pre-Lagrangian toric fiber isdisplaceable. From Proposition 3.3 we see that if L ⊂ M is a stem, then π − ( L ) ⊂ V is astem. While in the symplectic setting a stem, if it exists, is unique and non-displaceable([17]), in the contact setting this is not necessarily true. In S d − and T k × S d + k − , d ≥ N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 9 every pre-Lagrangian toric fiber is a stem and none of them is non-displaceable (see Section4). However, as was proved by Borman and Zapolsky, in the family of contact toric man-ifolds admitting a monotone quasimorphism with the vanishing property stems behave asin the symplectic case: if a stem exists then it is unique and non-displaceable (see Corollary1.19 and Corollary 1.11 in [9]). The prequantization of any even monotone symplectic toricmanifold (with the symplectic form scaled appropriately) admits such a quasimorphism(see Theorem 1.3. in [9]). (cid:5)
Example . The central fiber in CP d − , the Clifford torus T CP = { [ z , . . . , z d ] ∈ CP d − | | z | = · · · = | z d | } is proved to be non-displaceable (Cho-Poddar [12]). On the other hand, using the methodof probes, it can be shown that all other Lagrangian toric fibers in CP d − are displaceable.Therefore the Clifford torus is a stem. By Remark 3.5 the pre-images of the Clifford torusin S d − , RP d − and L d − p = S d − / Z p (under the prequantization map) are also stems.However their displaceability properties are very different. The real projective space admitsa monotone quasimorphism with the vanishing property and therefore its stem is non-displaceable [9]. It is expected that all lens spaces also admit such quasimorphism ([25] inpreparation). That would imply that the preimage of the Clifford torus in any L d − p is alsonon-displaceable. In the case of S d − , the preimage of the Clifford torus is displaceableas shown in Proposition 1.1. Absence of a non-displaceable pre-Lagrangian toric fiber alsoimplies that S d − does not admit a monotone quasimorphism with the vanishing property(see Theorem 1.14. in [9]).3.3. Contact reduction of a contact toric manifold and displaceability.
In thisSection we explain how one can use contact reduction to deduce certain results about(non)-displaceability. This is done for the sake of completeness, as in this article we arenot applying this method.We first recall the notion of a contact reduction (for more details see [4] or Section 7.7.in [21]). Suppose that a compact Lie group G acts on a contact manifold ( (cid:101) V , (cid:101) ξ = ker (cid:101) α )preserving the contact form (cid:101) α and let µ G be the corresponding moment map. Assumemoreover that 0 ∈ g ∗ is a regular value and that G acts freely and properly on the level µ G − (0) . Let ρ : µ G − (0) → µ G − (0) /G be the quotient map. The contact form (cid:101) α naturallyinduces a contact form α on V such that ρ ∗ α = (cid:101) α on µ G − (0) . Moreover, if (cid:101) V d +1 is acontact toric manifold with a toric T d +1 -action, and G = S is a subgroup of T d +1 , thenthe residual torus T d +1 /G acts on the reduced space V , turning it into a contact toricmanifold.The results of Abreu and Macarini ([2]), establishing a connection between rigidity ofLagrangian toric fibers in a symplectic manifold and in its reduction, can easily be trans-lated to the contact setting. Proposition 3.8 follows immediately from the following lemma,whose proof is omitted since it is analogous to the one in the symplectic case (see Section3 in [2]). Lemma 3.7. ( Lifting property for contact reduction ) Let ϕ ∈ Cont ( V, ξ ) . Then, there is (cid:101) ϕ ∈ Cont ( (cid:101) V , (cid:101) ξ ) such that ρ ◦ (cid:101) ϕ = ϕ ◦ ρ . Proposition 3.8. If L ⊂ V is a displaceable pre-Lagrangian toric fiber, then (cid:101) L = ρ − ( L ) ⊂ (cid:101) V is a displaceable pre-Lagrangian toric fiber. Contact cuts and displaceability.
The procedure of contact cutting was definedby Lerman in [27] who also proved the following result.
Theorem 3.9 (Theorem 2.11 in [27]) . Let ( V, α ) be a contact manifold with an action of S preserving α and let µ α denote the corresponding α -moment map. Suppose that S actsfreely on the zero level set µ − α (0) . Then the cut manifold , defined as V [0 , ∞ ) = µ − α ([0 , ∞ )) / ∼ , where x ∼ x if and only if µ α ( x ) = µ α ( x ) = 0 and the points x and x are in thesame circle orbit, is naturally a contact manifold. Moreover, the natural embedding of thereduced space V = µ − α (0) / S into V [0 , ∞ ) is contact and the complement V [0 , ∞ ) \ V iscontactomorphic to the open subset { x ∈ V ; µ α ( x ) > } of ( V, α ) . If the above S is a subcircle of a torus T (dim V +1) acting on V in a Hamiltonian way(thus giving V a structure of a contact toric manifolds) then the T (dim V +1) action re-stricts to an action on the cut manifold V [0 , ∞ ) turning it also into a contact toric man-ifold. The moment cone corresponding to V [0 , ∞ ) is the intersection of the cone of V in R (dim V +1) = Lie ( T (dim V +1) ) ∗ with the half space { x ∈ R (dim V +1) ; (cid:104) x, ξ (cid:105) ≥ } where ξ ∈ Lie ( T (dim V +1) ) is the infinitesimal generator of the chosen S . The following easyobservation will be used repetitively throughout Section 4.2. Lemma 3.10.
Any contactomorphism of V [0 , ∞ ) , compactly supported in V [0 , ∞ ) \ V , can beextended to a contactomorphism of the whole V . Therefore, if L ⊂ µ − α ((0 , ∞ )) ⊂ V is apre-Lagrangian displaceable in V [0 , ∞ ) by a contact isotopy supported in V [0 , ∞ ) \ V , then itis displaceable in V .Example . The standard T action on the contact toric manifold V := S × S ⊂ S × R × C comes from the standard S actions on S and on C . The associated momentcone is presented on the left picture in Figure 1. (This space is a representative of the family T k × S d + k − discussed in Section 4.2.) Performing a contact cut with respect to the circle S = { ( t, t ) ∈ T } we obtain a contact toric manifold V [0 , ∞ ) whose moment cone is presentedon the right picture. This cone differs from the moment cone of S with the standard contactstructure, viewed as the prequantization of ( CP , π ω F S ), by a GL (2 , Z ) transformation (seeExample 3.1). Therefore V [0 , ∞ ) is contactomorphic to ( S , ξ st ) and the toric action differs(from the action on S viewed as the prequantization of CP ) just by a reparametrizationof the torus. In particular the torus orbits remain unchanged. Using Proposition 3.3 wecan to lift isotopies of CP obtained by the method of probes (Section 3.2) to isotopiesof ( S , ξ st ). These isotopies displace pre-Lagrangians toric fibers corresponding to the rayscontained in the green region in Figure 2. Moreover, as they are supported in V [0 , ∞ ) \ V , N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 11 S × S , S Figure 1.
The contact cut of S × S with respect to S = { ( t, t ) ∈ T } andthe resulting in S .they can be extended to isotopies of S × S . Therefore all pre-Lagrangian toric fibers in S × S which map to the green region are displaceable. Figure 2.
Pre-Lagrangians displaceable in S by isotopies coming from theprobes methods.4. Displaceability of fibers in non-orderable manifolds
In this Section we prove that for all contact toric manifolds that are presently known tobe not orderable all pre-Lagrangian toric fibers are displaceable. This observation suggestthat the lack of orderability of a contact toric manifold implies displaceability of its toricfibers. Our proof is done on a case by case basis. Therefore it, unfortunately, does notexplicitly show how the lack of orderability could affect the displaceability of the fibers.To the best of our knowledge the only contact toric manifolds which are known to benon-orderable are: • Contact spheres ( S d − , ξ st ), with d ≥
2, equipped with the standard contact struc-ture ξ st = T S d − ∩ J ( T S d − ). • Contact toric manifolds T k × S d + k − , with d ≥ k >
0, with contact structuredescribed explicitly in Section 4.2.
The first ones are ideal contact boundaries of C d , i.e. the d -stabilization of a point. Thesecond ones are ideal contact boundaries of ( T ∗ S ) k × C d i.e. the d -stabilization of the Liou-ville manifolds ( T ∗ S ) k . They are not orderable by a result of Eliashberg-Kim-Polterovich(Theorem 1.16 in [15]). It remains an open question whether T k × S k +1 are orderable, evenin the case k = 1.4.1. Contact sphere.
The standard contact sphere ( S d − , ξ st ) can be presented in com-plex coordinates as S d − = { ( z , . . . , z d ) ∈ C d | (cid:80) dj =1 | z j | = 1 } with ξ st given as the kernelof α st := i (cid:80) dj =1 ( z j d ¯ z j − z j dz j ) . The standard T d -action on S d − defined by( t , . . . , t d ) ∗ ( z , . . . , z d ) (cid:55)−→ ( t z , . . . , t d z d )makes the sphere ( S d − , ξ st ) a contact toric manifold. The moment map with respect to α st is µ α st ( z , . . . , z d ) = π ( | z | , . . . , | z d | ) . Thus, the moment cone is R d ≥ . A pre-Lagrangiantoric fiber in ( S d − , ξ st ) is any submanifold(2) L c ,...,c d = { ( z , . . . , z d ) ∈ S d − | | z j | = c j , j = 1 , . . . , d } where c j ∈ (0 ,
1) are constants such that (cid:80) dj =1 c j = 1 . We will now prove Proposition 1.1,i.e. the fact that all pre-Lagrangian toric fibers are displaceable. In fact a stronger result istrue. We are grateful to Patrick Massot and the referee who independently pointed out tous that, as long as one does not require the displacing isotopy to displace all pre-Lagrangiantoric fibers simultaneously, one can construct an isotopy displacing one particular fiber inthe following way. (Our original idea, providing one isotopy simultaneously displacing allthe fibers, is explained in Remark 4.2.)
Proposition 4.1.
Every closed proper subset of ( S d − , ξ st ) , with d ≥ , is displaceable.Proof. Let L ⊂ S d − be a closed proper subset and let p ∈ ( S d − \ L ) be any point. Thekey point in this proof is the observation that ( S d − \ { p } , ξ st ) is contactomorphic to R d − with its standard contact structure ξ = ker( dx d + (cid:80) d − j =1 x j · dy j ) (by Proposition 2.1.8 andExample 2.1.3 in [21]). Denote this contactomorphism by ψ : ( S d − \ { p } , ξ st ) → ( R d − , ξ ) . Observe that any translation in the x d direction is a contactomorphism of ( R d − , ξ ) . As L is compact, there exists R > { φ t } t ∈ [0 , , with φ t being the translation in x d direction by tR , displaces ψ ( L ). Let h t denote the contactHamiltonian generating the isotopy φ t . Let δ be a cut-off function on R d − which is 1on a neighborhood of ∪ t ∈ [0 , φ t ( ψ ( L )), and 0 outside of a compact set. Then the isotopygenerated by the contact Hamiltonian δh t displaces ψ ( L ). Moreover, as it is compactlysupported in R d − = ψ ( S d − \{ p } ), the isotopy ψ − ◦ φ t ◦ ψ of S d − \{ p } can be extendedto an isotopy of the whole sphere S d − , displacing L . (cid:3) Remark . In fact all pre-Lagrangian toric fibers can be displaced using only one contactisotopy. In the proof of their non-orderability result, Eliashberg, Kim and Polterovich builda positive contractible loop as a composition of certain contactomorphisms. One of thesebuilding blocks is an isotopy called a distinguished contact isotopy. In the case of S d − they N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 13 give an explicit formula for this isotopy. Moreover Giroux in [23] gives another formula for acontact isotopy of S d − playing the same role in the construction of a positive contractibleloop. This contact isotopy displaces all pre-Lagrangian toric fibers in S d − , d ≥
2, as wenow explain. Consider the map τ t : B d → C d , t ≥ τ t ( z , . . . , z d ) = 1cosh t + z sinh t (sinh t + z cosh t, z , . . . , z d ) , where B d is the unit ball and cosh t = e t + e − t , sinh t = e t − e − t ([23]). We first observe thatthe map τ t is well defined because cosh t + z sinh t (cid:54) = 0 for every | z | ≤ t ≥ . Indeed, if t = 0 the expression is equal to 1. Suppose there is some t > z = x + iy such that cosh t + z sinh t = 0 . That would imply that cosh t + x sinh t = 0 and y sinh t = 0 . Consider the function f ( x ) = cosh t + x sinh t on the domain | x | ≤ , for a fixed constant t > . We have f ( −
1) = e − t > f (cid:48) ( x ) = cosh t > , hence f ( x ) > | x | ≤ . Bya straightforward calculation we check that τ t is a complex automorphism of the unit ball( B d , i (cid:80) dz j ∧ d ¯ z j ). Since every complex automorphism of the unit ball B d ⊂ C d restrictsto a contactomorphism of its boundary, i.e. the standard contact sphere ( S d − , ξ st ), itfollows that τ t is a contactomorphism on S d − for every t ≥ . Since τ = id , the map τ t isa well defined contact isotopy starting at the identity. Take any pre-Lagrangian toric fiber L = L c ,...,c d given by (2). We claim that for t > L ∩ τ t ( L ) = ∅ . It is enough to show that there exists t > z with | z | = c ∈ (0 , | cosh t + z sinh t | (cid:54) = 1, as this implies that the norm of the secondcoordinate changes after applying τ t , and thus such τ t displaces L . Write z = x + iy, hence x + y = c , and consider the function f t ( x ) = | cosh t + z sinh t | −
1, for | x | ≤ c . Notethat f t ( x ) = | cosh t + z sinh t | − t + x sinh t ) + ( y sinh t ) −
1= cosh t + 2 x cosh t sinh t + c sinh t − t + 2 x cosh t sinh t + c sinh t = sinh t (2 x cosh t + (1 + c ) sinh t )Since sinh t > t >
0, it is enough to show that for t big enough the expression(2 x cosh t +(1+ c ) sinh t ) is positive for all | x | ≤ c <
1. Take t > t > − x c . Such t always exists since tanh t tends to infinity when t is large. For this choice of t wehave that f t ( x ) > | x | ≤ c <
1, proving that τ t displaces L . (cid:5) Remark . Note that we showed L ∩ τ t ( L ) = ∅ by comparing the norm of the secondcoordinate, thus we used the fact that d ≥
2. In the case d = 1 the map τ is a well definedcontactomorphism of a circle. However as the only pre-Lagrangian of S is the whole S , itis trivially non-displaceable. (cid:5) Displaceability of fibers in T k × S d + k − . Consider the contact toric manifold T k × S d + k − described as { ( e πiθ , . . . , e πiθ k , x , . . . , x k , z , . . . , z d ) ∈ T k × R k × C d ; k (cid:88) l =1 x l + d (cid:88) j =1 | z j | = 1 } with the contact structure given by the kernel of the form β k = k (cid:88) l =1 x l dθ l + i d (cid:88) j =1 ( z j d ¯ z j − z j dz j ) . This is the ideal contact boundary of T ∗ T k × C d , and thus is non-orderable if d ≥ T d + k is given by( t , . . . , t d , s , . . . , s k ) ∗ ( e πiθ , . . . , e πiθ k , x , . . . , x k , z , . . . , z d ) =( s e πiθ , . . . , s k e πiθ k , x , . . . , x k , t z , . . . , t d z d ) . The corresponding β k -moment map, µ β k : T k × S d + k − → Lie ( T d + k ) ∗ = R d + k is µ β k ( e πiθ , . . . , e πiθ k , x , . . . , x k , z , . . . , z d ) = π ( | z | , . . . , | z d | , x , . . . , x k ) , thus the moment cone is ( R ≥ ) d × R k . Pre-Lagrangian toric fibers in T k × S d + k − aresubmanifolds L = L c ,...,c d given by { ( e πiθ , . . . , e πiθ k , x , . . . , x k , z , . . . , z d ) ∈ T k × S d + k − ; | z j | = c j , j = 1 , . . . , d } for some x , . . . , x k ∈ R and c , . . . , c d > (cid:80) kl =1 x l + (cid:80) dj =1 c j = 1, and thus theycorrespond to the rays { t ( c , . . . , c d , x , . . . , x k ); t ∈ R + } in( R > ) d × R k ⊂ Cone ( µ β k ( T k × S d + k − )) . In this subsection we show that all pre-Lagrangian toric fibers in ( T k × S d + k − , ker β k ), d ≥ k ≥
1, are displaceable. To construct a contact isotopy displacing a given pre-Lagrangian toric fiber L of T k × S d + k − we will look at a manifold obtained via contactcutting T k × S d + k − with respect to appropriate circles. Proof of Theorem 1.2 with d ≥ . Case k = 1 . We start by analyzing S × S d . The cone corresponding to this contact toricmanifold is C = C S × S d = R d ≥ × R , and its outward normals are − e , . . . , − e d . Performa contact cut with respect to the diagonal circle in T d +1 (see Section 3.4). Denote themoment map for that circle by µ + , i.e. µ + ( θ, x, z , . . . , z d ) (cid:55)→ π ( x + (cid:80) dj =1 | z j | ), and theresulting cut manifold by M + := ( µ + ) − ([0 , ∞ )) / ∼ . It is a contact toric manifold corresponding to the convex cone, C = C +1 , obtained from C by cutting it with the hyperplane perpendicular to the vector − (cid:80) d +1 l =1 e l . This means that C is spanned by the directions: e d +1 and e i − e d +1 for i = 1 , . . . , d . In fact it is the contacttoric sphere S d +1 . Indeed, as we have already seen in Example 3.1 contact sphere S d +1 ,viewed as the prequantization of ( CP d , π ω F S ), is the contact toric manifold corresponding
N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 15 to the cone, which we call C , whose edges are in the directions e i + e d +1 , for i = 1 , . . . , d .The GL ( d + 1 , Z ) transformation given by the following lower triangular matrix (only thelast row and the diagonal have non-zero entries) maps the cone C to the cone C . A = . . .
12 2 . . . . Note that d = 1 is exactly the case described in Example 3.11 and on Figure 1.To construct isotopies displacing pre-Lagrangian toric fibers in S d +1 we will use Lemma3.2 and isotopies displacing Lagrangian toric fibers in ( CP d , π ω F S ) obtained via McDuff’smethod of probes (see Example 3.2). The point of using these particular isotopies of S d +1 (instead of, for example, isotopies from Proposition 4.1) is that we know their support: itis far from the “cut” and therefore these isotopies of S d +1 are extendable to isotopies of S × S d .Recall that ( CP d , π ω F S ) is a symplectic toric manifold whose moment map image is the d -dimensional simplex of size 1, ∆ d (1). For each i = 1 , . . . , d let F i denote the facet of ∆ d (1)whose outward normal is − e i . The vector e i can be used as a direction of a probe. This waywe can displace all Lagrangian toric fibers in CP d corresponding to points x ∈ ∆ d (1) ⊂ R d such that x = 12 a i e i + (cid:88) l (cid:54) = i a l e l , with a , . . . , a d > , e i , and { e l } l (cid:54) = i ) by anisotopy with support contained in the preimage of Int ∆ d (1) ∪ Int F i . Lifting this isotopyto the sphere we can displace any pre-Lagrangian toric fiber corresponding to a ray through y ∈ C ⊂ R d +1 such that y = a i (2 e d +1 + e i ) + a d +1 e d +1 + (cid:88) l (cid:54) = i, d +1 a l ( e l + e d +1 ) , with a , . . . , a d +1 > e + e d +1 , . . . , e i − + e d +1 , e i + 2 e d +1 , e i +1 + e d +1 , . . . , e d + e d +1 , e d +1 )with an isotopy supported on the preimage of Int C ∪ Int (cid:101) F i , where (cid:101) F i is the facet of C with normal − e i . (Figure 2 in Example 3.11 presents this set for the case d = 1.) The map A − , mapping the cone C to the cone C of the cut space M + , maps this set to the set ofpoints y = a i e i + a d +1 e d +1 + (cid:88) l (cid:54) = i, d +1 a l ( e l − e d +1 ) , with a , . . . , a d +1 > . These are the points y = ( y , . . . , y d +1 ) ∈ C with y l > l ∈ { , . . . , d } , and y d +1 > − (cid:80) l (cid:54) = i, d +1 y l . In particular this includes( R > ) d × R ≥ , if d > , and ( R > ) × R > , if d = 1 . Note that all these isotopies are supported away from the facet of C with normal − (cid:80) d +1 l =1 e l .Therefore, by Lemma 3.10, these isotopies can be extended to isotopies of the whole S × S d .We now repeat the whole construction but starting with contact cutting with respect tothe circle ( t, . . . , t, t − ) ∈ T d × S . The moment map for that circle is µ − ( θ, x, z , . . . , z d ) (cid:55)→ π ( − x + d (cid:88) i =1 | z i | ) . Thus the cut manifold M − := ( µ − ) − ([0 , ∞ )) / ∼ is a contact toric manifold correspondingto the cone, C − , spanned by the edges − e d +1 and e i + e d +1 for i = 1 , . . . , d . The resultingcut space is again a toric contact sphere S d +1 . Repeating the whole process we constructisotopies of S × S d that displace, in particular, pre-Lagrangian toric fibers correspondingto the rays in ( R > ) d × R ≤ if d ≥
2, and to the rays in R > × R < if d = 1.Putting these two steps together we are able to displace all pre-Lagrangian toric fibersof S × S d if d ≥
2, and all fibers in S × S apart from the fiber corresponding to the ray { ( y , y ∈ R ≥ } if d = 1. It will be shown later that this fiber is also displaceable. General case.
To obtain a sphere as a cut manifold of T k × S d + k − and displacecertain fibers as we did above for the case k = 1, we need to cut T k × S d + k − k times in2 k different ways. To keep a record of the cuts we introduce the following notation. Forany j ∈ { , . . . , k } and any ε j ∈ {− , } , let S j,ε j denote the circle subgroup of T d × T k generated by ( t, . . . , t ; 1 , . . . , , t ε j , , . . . , , where t ε j is at the position d + j . For any ε = ( ε , . . . , ε k ) ∈ {− , } k let M ε denote thecontact toric manifold obtained from T k × S d + k − by consecutive performing k contactcuts, with respect to circles S ,ε , ..., S k,ε k . We denote the corresponding moment cone by C ε . Observe that C ε is the cone in R d + k with ( d + k ) facets whose outward normals are − e , . . . , − e d , η ε j := − d (cid:88) i =1 e i − ε j e d + j , j = 1 , . . . , k. The edges of this cone have directions ε j e d + j , j = 1 , . . . , k, and e i − k (cid:88) j =1 ε j e d + j , i = 1 , . . . , d. The facets of C ε created by cutting are the ones with normals η ε j , j = 1 , . . . , k. Thus thereis a contactomorphism between the preimage in M ε of C ε \ { facets with normals η ε j , j =1 , . . . , k } , which we denote by U ε , and an open subset of T k × S d + k − . Any contact isotopyof M ε compactly supported in U ε can be extended to a contact isotopy of the whole T k × S d + k − . N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 17
Note that the space M ε is contactomorphic to the standard contact sphere S d +2 k − and the action differs only by a reparametrization of the torus. Indeed, the GL ( d + k, Z )transformation given by the following matrix A ε = . . . ε . . . . . .1 . . . ε k − k . . . k ε . . . ε k − ε k maps e i − k (cid:88) j =1 ε j e d + j (cid:55)→ e i + e d + k , i = 1 , . . . , dε j e d + j (cid:55)→ e d + j + e d + k , j = 1 , . . . , k − ,ε k e d + k (cid:55)→ e d + k . Therefore A ε maps the cone C ε to the cone, that we denote by C ε , of the contact toricsphere S d +2 k − viewed as the prequantization of ( CP d + k − , π ω F S ).Observe that the facet of C ε with normal η ε j , j = 1 , . . . , k −
1, is the cone with theedges in the directions ε l e d + l , l ∈ { , . . . , k } \ { j } and e i − (cid:80) kj =1 ε j e d + j , i = 1 , . . . , d. Suchfacet is mapped by A ε to the facet of C ε which is the cone with the edges in the directions e d + l + e d + k , l ∈ { , . . . , k } \ { j, k } , e d + k and e i + e d + k for i = 1 , . . . , d , i.e. the facet of C ε with normal − e d + j . The facet of C ε with normal η ε k , i.e. the cone whose edges are in thedirections ε l e d + l , l ∈ { , . . . , k − } and e i − (cid:80) kj =1 ε j e d + j , i = 1 , . . . , d, is mapped by A ε to thefacet of C ε which is the cone with the edges in the directions e d + l + e d + k , l ∈ { , . . . , k − } ,and e i + e d + k for i = 1 , . . . , d , i.e. the facet of C ε with normal − (cid:80) d + kl =1 e l . Therefore thecontactomorphism mentioned above is between an open subset U ε of T k × S d + k − and thepreimage in S d +2 k − of the set C ε \ { facet with normal ( − d + k (cid:88) l =1 e l ) and facets with normals − e d + j , j = 1 , . . . , k − } . Similarly to the case of S × S d analyzed before, we displace certain pre-Lagrangiantoric fibers in M ε by isotopies that are the lifts of isotopies of CP d + k − obtained via themethod of probes. For i = 1 , . . . , d let F i denote the facet of ∆ d + k − (1), the moment coneof ( CP d + k − , π ω F S ), with normal − e i . As we have already observed, the vector e i can beused as a direction of a probe, producing an isotopy of CP d + k − displacing Lagrangian toricfibers corresponding to x = 12 a i e i + (cid:88) l (cid:54) = i a l e l , with a , . . . , a d + k − ≥ , d + k − (cid:88) l =1 a l < , by an isotopy with support contained in the preimage of Int ∆ d + k − (1) ∪ Int F i . Liftingthis isotopy to the sphere we can displace any pre-Lagrangian toric fiber corresponding toa ray through y ∈ C ε ⊂ R d + k such that y = a i (2 e d + k + e i ) + a d + k e d + k + (cid:88) l (cid:54) = i, d + k a l ( e l + e d + k ) , with a , . . . , a d + k ≥ , with an isotopy supported in the preimage of Int C ε ∪ Int (cid:101) F i , where (cid:101) F i is the facet of C ε with normal − e i . The image of this set under the linear map( A ε ) − = − ε . . . − ε ε − ε . . . − ε . . . . . . ε k − − ε k . . . − ε k − ε k . . . − ε k ε k , mapping the cone C ε to the cone C ε , is the set of points y = a i ( e i − k − (cid:88) j =1 ε j e d + j ) + a d + k ε k e d + k + (cid:88) l ∈ [ d ] \{ i } a l ( e l − k (cid:88) j =1 ε j e d + j ) + k − (cid:88) l =1 a d + l ( ε l e d + l ) , with a , . . . , a d + k >
0. Here we used the symbol [ d ] to denote the set { , . . . , d } . These arethe points y = ( y , . . . , y d + k ) ∈ C ε with coordinates y = ( a , . . . , a d , ε ( a d +1 − (cid:88) l ∈ [ d ] a l ) , . . . , ε k − ( a d + k − − (cid:88) l ∈ [ d ] a l ) , ε k ( a d + k − (cid:88) l ∈ [ d ] \{ i } a l )) , that is, the points y = ( y , . . . , y d + k ) ∈ C ε with y l > l ∈ [ d ], and ε j y d + j + (cid:88) l ∈ [ d ] y l = a d + j > , for all j = 1 , . . . , k − ,ε j y d + k + (cid:88) l ∈ [ d ] \{ i } y l = a d + k > . This set contains, in particular, the set( R > ) d × ε R ≥ × . . . × ε k R ≥ , if d ≥
2, and if d = 1 the set R > × ε R ≥ × . . . × ε k R > . Here ε j R ≥ denotes the set { x ∈ R | ε j x ≥ } . The isotopies are supported only in thepreimage of Int C ε ∪ Int of the facet with normal ( − e i )and thus can be extended to isotopies of the whole T k × S d + k − . Therefore the correspond-ing fibers are displaceable in T k × S d + k − . N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 19
The above procedure can be performed for each ε ∈ {− , } k . In this way we can displaceall pre-Lagrangian toric fibers in T k × S d + k − corresponding to the rays in ( R > ) d × R k if d ≥
2, and to the rays in R > × R k − × R (cid:54) =0 if d = 1. In particular we displace all pre-Lagrangian toric fibers in T k × S d + k − for d ≥
2, and all apart from the fibers correspondingto the rays { ( y , . . . , y k , } in R > × R k if d = 1.Note that while mapping the cone C ε to the cone C ε , we made a choice to map thedirection of ε d + k e d + k to e d + k . The direction e d + k is special while viewing the sphere as theprequantization of the projective space, as it corresponds to the direction of the fibers ofprequantization map (Reeb direction). We could have made a different choice and requirethat A ε maps the direction of ε d + j e d + j to e d + k for some j = 1 , . . . , k −
1. Repeating thewhole procedure with this choice allows us to displace the fibers corresponding to the rays y = ( y , . . . , y k ) ∈ R > × R k with y j +1 (cid:54) = 0. Still, this method does not displace thepre-Lagrangian fiber corresponding to the ray y = ( y , , . . . , ∈ R > × R k . Recall thatin the case k = 1 and d = 1 we have also displaced all the fibers apart from the fibercorresponding to the ray y = ( y , ∈ R > × R . Performing a similar procedure with theuse of lens spaces instead of spheres allows us to displace these special pre-Lagrangian toricfibers, as we now explain. Special fibers of the case d = 1 . Let k ≥
1. Fix a prime number p and considerthe prequantization of ( CP k , pπ ω F S ), i.e. the lens space L k +1 p = S k +1 / Z k with the contactstructure induced from the standard contact structure on the sphere. This is a contacttoric manifold whose contact moment cone, that we denote C , is the convex cone in R k +1 spanned by the directions η l = pe l + e k +1 , for l = 1 , . . . , k , and η k +1 = e k +1 . Applying the GL ( k + 1 , Z ) transformation ( A ε ) − for ε = (1 , . . . ,
1) =: we obtain a convex cone, C ,whose edges are in the directions( A ) − ( pe + e k +1 ) = pe − k (cid:88) l =2 pe l + ( − p + 1) e k +1 , ( A ) − ( e k +1 ) = e k +1 , ( A ) − ( pe j + e k +1 ) = pe j + ( − p + 1) e k +1 , j = 2 , . . . , k. The outward normals of this cone are η := − e , η j := − e − e j , for j = 2 , . . . , k, and η k +1 := ( k − ( k + 1) p ) e + (cid:32) (1 − p ) k − (cid:88) j =2 e j (cid:33) − pe k +1 =( k − ( k + 1) p, − p, . . . , − p, − p ) . Observe that this is the cone one obtains from R ≥ × R k , i.e. the moment cone of T k × S k +1 ,after k contact cuts with respect to the circles generated by the vectors − η j for j = 2 , . . . , k ,and − η k +1 . Similarly as above one shows that any contact isotopy of L k +1 p compactlysupported in the preimage of C \ { facets with normals η , . . . , η k +1 } = ( A ) − ( C \ { facets with normals − e , . . . , − e k , ( k (cid:88) j =1 e j ) − pe k +1 } )can be extended to an isotopy of T k × S k +1 . Observe that, using the method of probes,for probes in ( CP k , pπ ω F S ) supported in the facet with the outward normal − e one candisplace all the fibers corresponding to the points in C which are contained in a convexcone, denoted by C disp , spanned by the directions pe + 2 e k +1 , pe j + e k +1 , j = 2 , . . . , k, and e k +1 . Note that ( A ) − ( C disp ) is the convex cone spanned by the directions pe − p k (cid:88) l =2 e l + ( − p + 2) e k +1 , pe j + ( − p + 1) e k +1 , j = 2 , . . . , k, and e k +1 . Therefore pre-Lagrangian toric fibers corresponding to the rays in ( A ε ) − ( C disp ) ⊂ R ≥ × R k are displaceable. Note that ( A ε ) − ( C disp ) contains the set { ( y , , . . . , } . Indeed, given any( y , , . . . , ∈ R > × R k note that y p ( pe − p k (cid:88) l =2 e l + ( − p + 2) e k +1 ) + y p k (cid:88) j +2 pe j + ( − p + 1) e k +1 + ( k + 1) y p ( p − e k +1 = ( y , , . . . , . (cid:3) Example . To illustrate this idea better we take a closer look at the case when k = 2.Then the cone C is spanned by the directions η = pe + e , η = pe + e , and η = e .The matrix ( A ) − = − − − maps the cone C to a cone, called C , spanned by the directions( p, − p, − p + 1) , (0 , p, − p + 1) , (0 , , . The outward normals of C are η = ( − , , , η = ( − , − , , η = (2 − p, − p, − p ) . One obtains C from R ≥ × R , i.e. the moment cone of T × S , by performing twocontact cuts with respect to the circles generated by the elements − η = (1 , ,
0) and − η = (3 p − , p − , p ) in the Lie algebra of T . Any contact isotopy of L p compactlysupported in the preimage of C \ { facets with normals ( − , − , , (2 − p, − p, − p ) } = ( A ) − ( C \ { facets with normals (0 , − , , (1 , , − p ) } ) N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 21 can be extended to an isotopy of T × S . Viewing L p as the prequantization of ( CP , pπ ω F S )one construct isotopies, with compact supports contained in the above sets, displacing pre-Lagrangian toric fibers of L p corresponding to the rays in the interior of the cone C disp spanned by the directions ( p, , , (0 , p, , and (0 , , . Then the set ( A ) − ( C disp ) is acone spanned by the directions ( p, − p, − p + 2), (0 , p, − p + 1), and (0 , ,
1) and it containsthe set R > × { } × { } as given any ( y, ,
0) we have that yp ( p, − p, − p + 2) + yp (0 , p, − p + 1) + 3 yp (3 p − , ,
1) = ( y, , . Free torus actions and non-displaceability of pre-Lagrangian toricfibers.
In this section we prove that in all contact toric manifolds with free toric action, exceptpossibly for non-trivial principal T -bundles over S , all pre-Lagrangian toric fibers arenon-displaceable. Moreover we show that all contact toric manifolds whose toric action isnot free have uncountably many displaceable pre-Lagrangian toric fibers. Note that whenthe toric action is free then every orbit is a pre-Lagrangian. From Lerman’s classification ofcontact toric manifolds, recalled here on page 5, it follows that the contact toric manifoldsequipped with a free toric action are ( T , ker α k ), k ≥
1, and principal T d bundles over S d − with the unique T d -invariant contact structures. Note that if d (cid:54) = 3 such bundle mustbe trivial. Indeed, principal T d -bundles over a manifold are classified by homotopy classesof maps from this manifold to the classifying space B T d = ( CP ∞ ) d . As ( CP ∞ ) d is anEilenberg-MacLane space K ( Z d , Z d . Therefore principal T d bundles over S d − are classified by H ( S d − ; Z d ). This groupis trivial for d (cid:54) = 3 and equal to Z for d = 3. Therefore compact connected contact toricmanifolds with a free toric action are: • In dimension 3: ( T , ker α k ) , k ≥ • In dimension 5: a Z collection of principal T -bundles over S , each with a unique T -invariant contact structure. • In dimension greater than 5: T d × S d − , d ≥
4, with a unique T d -invariant contactstructure.Below we show that all pre-Lagrangian toric fibers in ( T , ker α k ) and in the trivialprincipal bundles T d × S d − , with d ≥
3, i.e. in the cosphere bundles of tori, are non-displaceable. So far we were unable to prove non-displaceability of orbits in non-trivialprincipal T bundles over S . These contact toric structures are explicitely described in[31] where it is shown that each of them is contactomorphic to one of T × L k , k ∈ N , ( L = S ) with the unique contact toric structure (up to a reparametrization of the torus)and that the contact toric structure on T × L k = T × ( S / Z k ) is induced by the diagonal Z k -action on S -factor of T × S . Thus, in order to prove that all pre-Lagrangian toric fibersin T × L k , k > , are non-displaceable it would be enough to prove that all pre-Lagrangiantoric fibers in T × S are non-displaceable. Non-displaceability in ( T , ker α k ) . In this section we prove Proposition 1.5, i.e.the fact that every pre-Lagrangian toric orbit in the contact toric manifold ( T = S θ ) × T θ ,θ ) , ker α k ) , with k ≥
1, where α k = cos(2 πkθ ) dθ + sin(2 πkθ ) dθ , is non-displaceable.We are grateful to Patrick Massot and the anonymous referee for explaining to us how theproof follows from the classification of tight contact structures on T × [0 , . The toric T -action on ( T , ker α k ) is given by ( t , t ) ∗ ( θ, θ , θ ) = ( θ, θ + t , θ + t ) . The moment map with respect to the contact form α k is given by µ k ( θ, θ , θ ) =(cos(2 πkθ ) , sin(2 πkθ )) . Therefore each fiber of µ k has k connected components. The mani-folds ( T , ker α k ) with k > Proof of Proposition 1.5.
Let L θ = { θ } × T be a toric orbit, i.e. a connected componentof some pre-Lagrangian toric fiber in ( T , ker α k ). Suppose that there exists a contactisotopy { φ t } t ∈ [0 , such that φ is the identity and φ ( L θ ) ∩ L θ = ∅ . Then one can lift thisisotopy to an isotopy of ( R × T , ker(cos(2 πθ ) dθ + sin(2 πθ ) dθ )) using the covering map R × T → S × T given by ( θ, θ , θ ) (cid:55)→ ( k θ, θ , θ ), which is a local contactomorphism.Let { Φ t } t ∈ [0 , denote the lift of { φ t } t ∈ [0 , to an isotopy of R × T . Then Φ t displaces L = { kθ } × T . As L is compact, one can construct a smooth “cut-off” function, i.e. afunction with compact support, which is 1 on a neighborhood of ∪ t ∈ [0 , Φ t ( L ). Multiplyingthe contact Hamiltonian generating Φ t by this cut-off function, we can modify Φ t to anisotopy with compact support contained in ( kθ − a, kθ + a ) × T for some integer a > t .Let N denote the region in [ kθ − a, kθ + a ] × T which is bounded by the disjoint pre-Lagrangian 2-tori L and L := Φ ( L ). The two boundary components of N are boundaryparallel in the (irreducible) manifold [ kθ − a, kθ + a ] × T , and therefore N is diffeomorphicto [0 , × T .Let ξ N denote the contact structure on N induced by the universally tight contact struc-ture of the ambient space, R × T . (The universal cover of R × T is ( R , ker(cos(2 πθ ) dθ +sin(2 πθ ) dθ )), which is contactomorphic to R with the standard contact structure andthus is tight by a result of Bennequin.) Thus the contact structure ξ N on N is also univer-sally tight, and ( N, ξ N ) is contactomorphic to one of the structures in Giroux’s classification(Theorem 1.5 [22]; We remark here that a classification of tight contact structures on athickened torus was also obtained independently by Honda, [26]). The characteristic folia-tions on both (pre-Lagrangian) boundary components of N are linear foliations of the tori,and both with the same direction. By results in [22] such contact manifolds are classifiedby their π -torsion as we explain in the next paragraph.In [22] Giroux analyses the set SCT ( I × T ) of tight contact structures on I × T withgiven characteristic foliations on the boundary. The connected components of this set are,by a theorem of Gray, the isotopy classes (for isotopies fixing the boundary) of contactstructures on I × T . Theorem 1.5 in [22] states in particular that if the characteristic N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 23 foliation on the boundary is topologically linearizable then the map ( χ ∂ , τ π ) from SCT ( I × T ) to H ( I × T ; Z ) × N , sending a tight contact structure to the pair (its relative Euler class,its π -torsion), has connected fibers. Moreover the theorem specifies the image of ( χ ∂ , τ π ) onuniversally tight contact structures. For universally tight contact structures with identicallinear characteristic foliations on the boundary the image is { } × N ⊂ H ( I × T ; Z ) × N .Indeed, foliations on both boundary components generate the same element σ ∈ H ( T ; R ).Thus, using the notation from [22], B = { σ } and X u = { } . Therefore in this case therelative Euler class vanishes and the (isotopy classes of) contact structures are classifiedby their π -torsion.Therefore the contact manifold ( N, ξ N ) is contactomorphic to [ kθ, kθ + b ] × T with thecontact structure given by ker(cos(2 πθ ) dθ + sin(2 πθ ) dθ ), for some b ∈ Z > . It followsthat the isotopy Φ t induces a contactomorphism from ([ kθ − a, kθ ] × T , ker(cos(2 πθ ) dθ +sin(2 πθ ) dθ )) to ([ kθ − a, kθ + b ] × T , ker(cos(2 πθ ) dθ + sin(2 πθ ) dθ )). These manifoldshave different π -torsions (by Proposition 3.42 of [22]) and thus such a contactomorphismcontradicts the aforementioned classification of tight contact structures on [0 , × T (The-orem 1.5 [22]). This proves that L θ must be non-displaceable. (cid:3) Non-displaceability in cosphere bundles of tori.
Let N be a smooth manifoldand P + T ∗ N the corresponding cosphere bundle. The Liouville form on T ∗ N descends to acontact form on P + T ∗ N making it a contact manifold. Its symplectization S ( P + T ∗ N ) is T ∗ N without the zero section. Since the graph of any closed 1-form on N is a Lagrangiansubmanifold of T ∗ N, it follows that the graph (cid:101) L α of a nowhere vanishing closed 1-form α is a Lagrangian submanifold in the symplectization. If π : S ( P + T ∗ N ) → P + T ∗ N isthe projection then L α = π ( (cid:101) L α ) is a pre-Lagrangian submanifold in P + T ∗ N . Thus toevery nowhere vanishing closed 1-form on N corresponds a pre-Lagrangian submanifold in P + T ∗ N. If N is a torus T d , one obtains as its cosphere bundle the contact manifold P + T ∗ T d ∼ = T d × S d − = { ( e πiθ . . . , e πiθ d , x , . . . , x d ) ∈ T d × R d | d (cid:88) i =1 x i = 1 } with the contact structure given by the kernel of the 1-form (cid:80) di =1 x i dθ i . The T d -action on( T d × S d − , ker( (cid:80) di =1 x i dθ i )) defined by( t , . . . , t d ) ∗ ( e πiθ , . . . , e πiθ d , x , . . . , x d ) → ( t e πiθ , . . . , t d e πiθ d , x , . . . , x d ) , is an effective action by contactomorphisms and thus it turns T d × S d − into a contacttoric manifold. The moment map, µ : T d × S d − → ( R d ) ∗ , associated to the contact form (cid:80) di =1 x i dθ i is given by µ ( e πiθ . . . , e πiθ d , x , . . . , x d ) = π ( x , . . . , x d ) . Hence, every pre-Lagrangian toric fiber in T d × S d − is of the form T d × { p } , for some pointp ∈ S d − . All these orbits are non-displaceable as we now explain. The characteristic foliation on the boundary T is called topologically linearizable if all the orbits aredense or all the orbits are closed. Proof of Proposition 1.4.
Take any pre-Lagrangian toric fiber T d × { p } , p = ( p , . . . , p d ) ∈ S d − , and observe that T d × { p } is the pre-Lagrangian corresponding to the graph of thenowhere vanishing closed 1-form (cid:80) di =1 p i dθ i on T d . If all p j ’s are rational then T d × { p } isfoliated by ( d − T d × { p } : indeed, Corollary 2.5.2 from [14] says that if Λ is any of these Legendriansand { ϕ t } is any contact isotopy, with ϕ = id and ϕ (Λ) transverse to T d × { p } , thenthe number of intersection points ϕ (Λ) ∩ ( T d × { p } ) is at least 2 d − . In particular, T d ×{ p } is non-displaceable. As the set of points p with rational coefficients is dense in S d − ,and displaceability is an open property, we deduce that all pre-Lagrangian toric fibers T d × { p } are non-displaceable. Alternatively, instead of [14], one could use Example 2.4.Afrom [16] showing that these fibers are not only non-displaceable, but also stably non-displaceable. (cid:3) Displaceability of pre-Lagrangian toric fibers when a toric action is notfree.
The goal of this Section is to prove Theorem 1.6, i.e. to show that every compactconnected contact toric manifold for which the toric action is not free contains uncountablymany displaceable pre-Lagrangian toric fibers. The idea of the proof is the following. Weconsider separately manifolds of Reeb and not of Reeb type. We show that the first onesare prequantizations of symplectic toric orbifolds (Lemma 5.1 below), and thus one candisplace their fibers using the methods of Section 3.1 (Corollary 5.2 below). The secondones are either overtwisted contact toric manifolds of dimension 3 or T k × S d + k − with d ≥ , k ≥
1. In Proposition 5.3 below we analyze the 3-dimensional overtwisted caseand displace uncountably many fibers using the method of contact cuts. All the fibers of T k × S d + k − , d ≥ , k ≥
1, are displaceable by Theorem 1.2, already proved in Section 4.2.5.3.1.
Contact toric manifolds of Reeb type.
Let ( V d − , ξ ) be a contact toric manifold and α a T d -invariant contact form for ξ , giving rise to the α -moment map µ α . Then the flowof the Reeb vector field R α preserves the level sets of µ α , as for any point p ∈ V and anyvector X ∈ Lie ( T d ), we have (cid:104) dµ α ( R α ( p )) , X (cid:105) = dα ( R α ( p ) , X ) = 0 , implying that dµ α ( R α ( p )) = 0 for all p ∈ V (Lemma 7.7.4 [21]). Thus the Reeb orbit iscontained in the T d -orbit. If in addition the Reeb vector field corresponds to some η ∈ t d in the Lie algebra of T d then V is called a contact toric manifold of Reeb type . For thesemanifolds one can always perturb a contact form (keeping same contact structure) so thatan integral multiple of the associated Reeb vector field corresponds to a lattice element η ∈ t d Z and thus the Reeb flow generates an S -action which is a subaction of the actionof T d ([11]). In that case the image of the contact moment map is a good strictly convexcone (Definition 2.17 in [28]), namely a cone over a convex polytope (Theorem 4.9 in [11]).Moreover, such contact manifolds can be obtained as contact reductions of the standardcontact sphere (Theorem 5.1 in [11]).Let V be any contact toric manifold of Reeb type and C the corresponding good strictlyconvex cone. Denote by v , . . . , v N ∈ Z d the primitive inward vectors normal to the facets N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 25 of the cone C. That is C = N (cid:92) i =1 { x = ( x , . . . , x d ) ∈ ( R d ) ∗ |(cid:104) x, v i (cid:105) ≥ } . For any R = (cid:80) Ni =1 a i v i with a , . . . , a N ∈ R > there exists a T d -invariant contact form α such that R = R α is the Reeb vector field corresponding to it (Proposition 2.19 in [3]). Takeany such R in the lattice of the Lie algebra of T , i.e. R = (cid:80) Ni =1 a i v i ∈ t d Z , a , . . . , a N ∈ R > .The corresponding Reeb flow generates an S -action on V and one can perform symplecticreduction of the symplectization SV with respect to the lift of that S -action. The resultof this reduction is a symplectic orbifold M and V is a prequantization of M (Theorem2.7 in [10]; see also Lemma 3.7 in [29]). Recall that symplectic toric orbifolds are classifiedby rational and simple polytopes with positive integral labels attached to each facet ([30]).The points in the preimage of a facet with label m have neighborhoods modeled on C d / Z m .The polytope corresponding to M is the intersection of the moment cone C for V with thehyperplane perpendicular to R . Note that this intersection is always a compact polytope.Indeed, if the hyperplane is given by h := { x ∈ ( R d ) ∗ | (cid:104) x, R (cid:105) = c } then any x ∈ C ∩ h must satisfy a (cid:104) x, v (cid:105) + · · · + a N (cid:104) x, v N (cid:105) = c, (cid:104) x, v (cid:105) ≥ , . . . , (cid:104) x, v N (cid:105) ≥ . As a j > v , . . . , v N generate the whole R d (due to strict convexity), the abovecondition implies that 0 ≤ (cid:104) x, v j (cid:105) ≤ ca j and C ∩ h is compact (it is empty if c < M are intersections of the facets of C with the hyperplane h . Thelabel of a facet corresponding to the facet of C with inward normal v j is the index of thelattice generated by v j and R inside span R ( v j , R ) ∩ Z d , where Z d is the lattice of R d = t d . Ifthe collection { v j , R } can be completed to a Z basis of Z d , the label on the correspondingfacet is 1. Lemma 5.1. If V is a contact toric manifold of Reeb type then V can be presented as aprequantization of some symplectic toric orbifold having at least one label equal to . Proof.
Let C be a good, strictly convex cone corresponding to V , with inward normals v , . . . , v N ∈ R d . Suppose the facets corresponding to normals v , . . . , v d − intersect in anedge. The assumption that the cone is good implies that the vectors v , . . . , v d − forma Z -basis of Z d − = span R ( v , . . . , v d − ) ∩ t Z . Without loss of generality we can assumethat v = (1 , , . . . , v d − = (0 , . . . , , , a j > j = 1 , . . . , N , so that R = (cid:80) Nj =1 a j v j = ( c , . . . , c d − ,
1) or R = ( c , . . . , c d − , −
1) with each c j ∈ Z . In both cases the collection { v , . . . , v d − , R } forms a Z -basis of Z d . This impliesthat V is a prequantization of a symplectic toric orbifold obtained as a reduction of SV bythe circle action generated by R , and that facets of this orbifold corresponding to normals v , . . . , v d − have labels 1.For each v j let ( v j, , . . . , v j,d ) denote its coordinates. Suppose that at least one of v d,d , . . . , v N,d is positive. With this assumption, we find constants a j > R = (cid:80) Nj =1 a j v j =( c , . . . , c d − , v d,d , . . . , v N,d were non positive we would similarly find a j >
0, so that R = ( c , . . . , c d − , − ∈ Z d .) Note that at least one of v d,d , . . . , v N,d is non-zero as v , . . . , v N form a basis of R d . Take any a d , . . . , a N ∈ R + such that (cid:80) Nj = d a j v j,d = 1 (soalso (cid:80) Nj =1 a j v j,d = 1). For a j , j = 1 , . . . , d −
1, take any positive real number such that a j + (cid:80) Nl = d a l v l,j is an integer. For example one can take a j = − (cid:80) Nl = d a l v l,j if this sum wasnon-positive, and a j = (cid:108)(cid:80) Nl = d a l v l,j (cid:109) − (cid:80) Nl = d a l v l,j if this sum is positive. In both cases (cid:80) Nl =1 v l,j = a j + (cid:80) Nl = d a l v l,j is an integer. (cid:3) Corollary 5.2.
Any contact toric manifold V of Reeb type contains uncountably manydisplaceable pre-Lagrangian toric fibers.Proof. The proof uses McDuff’s method of probes for displacing Lagrangian fibers in sym-plectic toric manifolds ([33], [1]), recalled here in Section 3.2. Though this method wasdeveloped for symplectic toric manifolds, it can be generalized to symplectic toric orbifoldsas long as the probe starts at a facet with label 1. This is because the isotopy constructedvia the method of probes is supported only in a small neighborhood of the preimage of aprobe. A probe is a half open interval almost entirely contained in the interior of a momentpolytope: only the starting point of a probe lies on the interior of a facet. Therefore if onlythis facet has label 1, a small enough neighborhood of the preimage of a probe containsno orbifold points. Using Lemma 5.1, the method of probes and Proposition 3.3 we candisplace uncountably many pre-Lagrangian toric fibers. (cid:3)
Toric contact manifolds not of Reeb type, with toric action that is not free.
Dimen-sion . We start by analyzing 3-dimensional manifolds separately as these are not, in general,uniquely determined by their moment cones.
Proposition 5.3.
Let V be a -dimensional contact toric manifold not of Reeb type andsuch that the toric action is not free. Then V contains uncountably many displaceablepre-Lagrangian toric fibers.Proof. It was shown by Lerman in Section 6.2 of [28] (proof of Theorem 2.18 (2)) that 3-dimensional contact toric manifold for which the toric action is not free are diffeomorphicto lens spaces (including S and S × S ), and are classified by pairs of real numbers ( t , t )such that 0 ≤ t < π, t < t , and tan t i ∈ Q or cos t i = 0, for each i = 1 ,
2. Thelast condition means that (cos t i , sin t i ) ∈ R lies on a ray through some ( m i , n i ) ∈ Z .Moreover, the assumption of being not of Reeb type implies that the moment cone is notstrictly convex and thus t + π ≤ t . The manifold corresponding to ( t , t ), denoted hereby M t ,t , is obtained from R × S × S with the contact form α = cos t dθ + sin t dθ at ( t, θ , θ ) ∈ R × S × S by contact cutting performed twice. Namely, if ( m i , n i ) ∈ Z denotes the primitive integral vector in the direction (cos t i , sin t i ), i = 1 ,
2, then M t ,t = [ t , t ] × S × S / ∼ , where the relation ∼ denotes the following identifications on the boundary of [ t , t ] × S × S :on { t } × S × S we identify the orbits of a circle action generated by − n ∂∂θ + m ∂∂θ , N DISPLACEABILITY IN CONTACT TORIC MANIFOLDS 27 and on { t } × S × S we identify the orbits of a circle action generated by n ∂∂θ − m ∂∂θ .Topologically this is equivalent to gluing two solid tori along their boundaries by variousautomorphisms of the boundary therefore M t ,t is a lens space. The moment cone of M t ,t is the cone in R over the image of the interval [ t , t ] ⊂ R under the map t (cid:55)→ (cos t, sin t ) ∈ R . Hence it is the whole R whenever t − t ≥ π .Take any t (cid:48) > t such that t (cid:48) − t < π and (cos t (cid:48) , sin t (cid:48) ) ∈ R lies on a ray throughsome ( m (cid:48) , n (cid:48) ) ∈ Z . It is obvious from the construction that M t ,t (cid:48) can be obtained from M t ,t by a contact cut. Therefore any isotopy of M t ,t (cid:48) whose support is compact andcontained in ([ t , t (cid:48) ) × S × S / ∼ ) ⊂ M t ,t (cid:48) , can be extended to an isotopy of M t ,t .Observe that the moment cone of M t ,t (cid:48) is strictly convex, thus M t ,t (cid:48) is a prequantizationspace. Similarly to what was done in Section 4.2, one can displace uncountably many pre-Lagrangian toric fibers of M t ,t (cid:48) by isotopies which are lifts of isotopies obtained via theprobes method. If a probe is based on a facet “far from the cut”, i.e. such that it’s preimageunder prequantization map is disjoint from the set { t (cid:48) } × S × S / ∼ , then such isotopy hasa compact support contained in ([ t , t (cid:48) ) × S × S / ∼ ) and can be extended to an isotopyof M t ,t . (cid:3) Dimension greater than . We now analyze the contact toric manifolds not of Reeb type, whose action is not free,and whose dimension is greater than 3. These are determined by their (not strictly convex)good moment cones. Let k be the dimension of the maximal linear subspace contained inthe good moment cone, and ( d + k ), d ≥
1, be the dimension of the torus acting. (When d = 0 the manifold is necessarily a T k bundle over S k − and the toric action is free). Thensuch a cone is isomorphic to the cone of T k × S d + k − and therefore the manifold mustbe equivariantly contactomorphic to T k × S d + k − with the unique T d + k invariant contactstructure, described in Section 4.2, (Theorem 2.18 in [28]; see also the proof of Theorem1.3 in Section 7 of [28]). Thus, by Theorem 1.2 (Section 4.2) all pre-Lagrangian toric fibersin T k × S d + k − are displaceable. This concludes the proof of Theorem 1.6. Acknowledgments
The authors are very grateful to Patrick Massot and to the anonymous referee whoindependently pointed out to us that the proof in Remark 4.2 can be simplified to theproof of Proposition 1.1, and they explained to us how to prove Proposition 1.5. Ad-ditionally we would like to thank the referee for his/her comments that helped us im-prove the exposition. Moreover we thank Miguel Abreu and Strom Borman for help-ful comments on the first version of the paper and Sheila Sandon, Maia Fraser andRoger Casals for useful discussions. The authors were supported by the Funda¸c˜ao para aCiˆencia e a Tecnologia (FCT, Portugal): fellowships SFRH/BD/77639/2011 (Marinkovi´c)and SFRH/BPD/87791/2012 (Pabiniak); projects PTDC/MAT/117762/2010 (Marinkovi´cand Pabiniak) and EXCL/MAT-GEO/0222/2012 (Pabiniak).
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