The symplectic structure of a toric conic transform

Suppose that a compact $r$-dimensional torus $T^r$ acts in a holomorphic and Hamiltonian manner on polarized complex $d$-dimensional projective manifold $M$, with nowhere vanishing moment map $\Phi$. Assuming that $\Phi$ is transverse to the ray through a given weight $\boldsymbol{\nu}$, associated to these data there is a complex $(d-r+1)$-dimensional polarized projective orbifold $\hat{M}_{\boldsymbol{\nu}}$ (referred to as the $\boldsymbol{\nu}$-th \textit{conic transform} of $M$). Namely, $\hat{M}_{\boldsymbol{\nu}}$ is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polarization of $M$. With the aim to clarify the geometric significance of this construction, we consider the special case where $M$ is toric, and show that $\hat{M}_{\boldsymbol{\nu}}$ is itself a K\"{a}hler toric obifold, whose moment polytope is obtained from the one of $M$ by a certain"transform", operation (depending on $\Phi$ and $\boldsymbol{\nu}$).


Introduction
Consider a d-dimensional connected projective manifold M, with complex structure J, and let (A, h) be a positive holomorphic line bundle on M. Thus A is ample, h is a Hermitian metric on it, and the unique covariant derivative ∇ on A compatible with both the metric and the complex structure has curvature Θ = −2 ı ω, with ω ∈ Ω 2 (M) a Kähler form on (M, J).
We shall denote by A ∨ the dual line bundle to A, and by X ⊂ A ∨ the unit circle bundle, with bundle projection π : X → M. Then ∇ corresponds to a connection 1-form α ∈ Ω 1 (X), which is a contact form on X and satisfies dα = 2 π * (ω), . (1) Let T r be an r-dimensional compact torus, with Lie algebra t r and coalgebra t r∨ .Furthermore, let µ M : T r ×M → M a holomorphic and Hamiltonian action of T r on (M, 2 ω, J), with moment map Φ : M → t r∨ (see e.g.[GS3] for general background on Hamiltonian actions and moment maps).
Any ξ ∈ t r thus determines a Hamiltonian vector field ξ M on M. As is well-known [Ko], Φ determines a natural lift of ξ M to a contact vector field ξ X = ξ Φ X on (X, α), given by (2) here V ♯ is the horizontal lift to X, with respect to α, of a vector field V on M, and ∂ θ is the generator of the structure S 1 -action on X, given by counterclockwise fiber rotation.The flow of ξ X preserves the contact and CR structures of X, and the flows of ξ X and ξ ′ X commute, for any ξ, ξ ′ ∈ t r .We shall make the stronger hypothesis that µ M lifts to a metric preserving line bundle action on A, and the induced action µ X : T r × X → X has the correspondence ξ → ξ X in (2) as its differential.We shall say that µ X is the (contact and CR) lift of the homolomorphic Hamiltonian action (µ M , Φ).
For example, when r = 1 and µ M is trivial, Φ : M → t 1 ∨ is constant; choosing Φ = ı in (2) yields the circle action ρ X generated by −∂ θ , thus given by clockwise fiber rotation.If ∂ S 1 θ is the standard generator of the Lie algebra of S 1 , ∂ θ = (∂ S 1 θ ) X in (2) is the vector field on X generating the structure S 1 -action given by counter-clockise fiber rotation, while −∂ θ is the vector field generating ρ X (clockwise fiber rotation); we parametrize S 1 by θ → e ı θ .
Let us fix a non-zero weight ν ∈ t r∨ .The results below rest on the following Basic Assumption on (Φ, ν), henceforth referred to a BA 1.1.
Under these circumstances, a polarized Kähler orbifold ( M ν , ω ν , J ν ) can be constructed from the previous data, by taking a suitable quotient by a locally free action of T r of a locus in X defined by (Φ, ν); here ω ν and J ν denote the (orbifold) symplectic and complex structures on M ν [P4].We refer to [P4] (where M ν is denoted N ν and ω ν by η ν ) for a discussion of the relevance of this geometric construction in geometric quantization; it generalizes the one of weighted projective spaces as quotients of an odd-dimensional sphere.Here we aim to clarify the relation between the symplectic structures of M and M ν in the toric setting: as we shall see, assuming that M is a toric manifold, ( M ν , 2 ω ν ) is a toric symplectic orbifold, and its marked moment polytope ∆ ν can be explicitly recovered from the moment polytope ∆ of M (by [LT] toric symplectic orbifolds are classified by marked convex rational simple polytopes).
Before stating the result precisely, let us briefly recall the geometric construction in point, referring to [P1] and [P4] for details.Let us set (3) Then, assuming BA 1.1, the following holds: 1. M ν ⊆ M is a connected and compact (real) submanifold, of codimension r − 1; 2. µ X is locally free on X ν .
We may and will assume without loss that µ X is generically free on X (and X ν ).Then the quotient (denoted N ν in [P4]) is naturally a (d − r + 1)-dimensional complex orbifold, and comes equipped with a Kähler structure ( M ν , ω ν , J ν ) induced by (M, ω, J); here T r acts on X ν by the restriction of µ X to X ν .We shall call M ν the ν-th conic transform of M; it depends on µ X , hence on Φ.We are interested in clarifying the geometry of M ν in the toric setting, thus assuming that M be toric, with structure action γ M : T d × M → M and moment polytope ∆ ⊆ t ∨ .
Let us briefly recall the Delzant construction of M from ∆; obviously with no pretense of exhaustiveness, we refer to [D], [G2], and [LT] for more complete discussions.Since M is smooth, ∆ is a Delzant polytope [G2].We shall denote by F (∆) the collection of all faces of ∆, by F l (∆) ⊆ F (∆) the subset of codimension-l faces, and specifically by G(∆) = F 1 (∆) the subset of facets.If G(∆) = {F 1 , . . ., F k }, then for every j = 1, . . ., k there exists unique with υ j primitive, such that and for every j = 1, . . ., k the relative interior of F j (open facet) is Let us set Then M can be regarded as symplectic reduction of (C k , 2 ω 0 ) under the standard action of T k , as follows.Denote the general element of T k by e ı ϑ = e ı ϑ 1 , . . ., e ı ϑ k ; for e ı ϑ ∈ T k and z = ( is then Hamiltonian on C k , 2 ω 0 , with moment map where (e j ) k j=1 is the canonical basis of R k and (e * j ) k j=1 is the dual basis.The linear map R k → R d such that e j → u j induces a short exact sequence of tori 0 → N → T k → T d → 0; hence Γ restricts to a Hamiltonian action of N on C k , 2 ω 0 , with a moment map Ψ N λ naturally induced from Ψ λ .Given that ∆ is Delzant, N acts freely on Z ∆ := Ψ N λ −1 (0); then M = Z ∆ /N, with its symplectic structure 2 ω, is the Marsden-Weinstein reduction of C k , 2 ω 0 for the action of N. By the arguments of [GS1], the standard complex structure J 0 of C k descends to a compatible complex structure J on M, whence (M, ω, J) is a Kähler manifold.
Furthermore, Γ descends to a holomorphic and Hamiltonian action of In addition, if λ ∈ Z k this construction can be extended by the arguments of [GS1] so as to obtain an induced toric positive line bundle (A, h) on M, with curvature Θ = −2 ı ω ( §); that (A, h) is toric means that γ M lifts to a metric preserving line bundle action of T d on A. Hence by restriction we obtain a contact CR action γ X : T d × X → X lifting γ M , where X ⊂ A ∨ is the unit circle bundle.
In addition, we suppose given an effective holomorphic and Hamiltonian action µ M : T r × M → M of an r-dimensional compact torus T r on M, with moment map Φ : M → t r∨ satisfying BA 1.1 for a certain ν ∈ t r∨ , and commuting with γ M .Thus µ M factors through an injective group homomorphism T r → T k , hence we may assume wihout loss of generality that T r T d and that µ M is the retriction of γ M to T r ; therefore, letting ι : t r ֒→ t d be the Lie algebra inclusion, Let us assume that (µ M , Φ) lifts to µ X : T r × X → X according to the previous procedure.While µ M is the restriction of γ M to T r , µ X is the restriction of γ X only if δ = 0 in (11).
If µ X exists, we can consider the conic transform M ν with respect to µ X ; as mentioned, ( M ν , 2 ω ν ) turns out to be a symplectic toric orbifold.Furthermore, its associated marked convex rational simple polytope ( ∆ ν , s ν ) is obtained by applying a suitable 'transform' to ∆ (depending on ν).
Since the situation is at its simplest when r = 1, we shall describe this case first.Thus µ M : T 1 × M → M is a Hamiltonian action on (M, 2 ω), with a nowhere vanishing moment map Φ : M → t 1 ∨ ; the primitive integral weight ν ∈ t 1 ∨ is uniquely determined by the condition that M ν = ∅.Then M ν = M, X ν = X, µ X is locally free, and M ν = X/T 1 (the quotient is with respect to µ X ).
We shall see that ( M ν , 2 ω ν ) is a toric symplectic orbifold, and that ∆ ν is its moment polytope.To complete the combinatorial description of ( M ν , 2 ω ν ) following [LT], we need to specify the corresponding marking of ∆ ν , that is, the assignment to each of its facets F j of an appropriate integer s j ≥ 1.We shall denote the marking by s ν = (s j ) k j=1 ∈ N k , and the marked polytope by the pair ( ∆ ν , s ν ).
We premise a further piece of notation.Given a rank-r integral lattice L ⊂ V in a real vector space, and a basis (ℓ 1 , . . ., ℓ r ) of L, if ℓ ∈ L we shall denote by (l) the greatest common divisor of the coefficients of ℓ in the given basis, that is, The definition is well-posed, since (ℓ) is independent of the choice of a basis of L. Furthermore, the following holds: 1. ℓ is primitive in L if and only if (ℓ) = 1; 2. if T is a (real) torus and ξ = 0 ∈ L = L(T ), then e ϑ ξ = 1 ∈ T if and only if e ı ϑ is a (ξ)-th root of unity.
The following consequence generalizes to conic transforms a well-known property of weighted projective spaces [Ka].
Corollary 1.1.Under the previous assumptions (thus with r = 1), Let us now consider a general r ≤ d.Let ν ⊥ t r be the kernel of ν, and T r−1 ν ⊥ T r the corresponding subtorus.Under Basic Assumption 1.1, T r−1 ν ⊥ acts locally freely on M ν ; then M ν := M ν /T r−1 ν ⊥ , the Marsden-Weinstein reduction of M with respect T r−1 ν ⊥ , is a Kähler orbifold.The transversality requirement in Basic Assumption 1.1 can be conveniently reformulated in a transversality condition between ∆ + δ and ν ⊥ 0 ⊆ t d ∨ (the annhilator of ν ⊥ ), see §3.4.We shall for simplicity require that T r−1 ν ⊥ acts freely on M ν , which amounts to ∆ ′ ν := (∆ + δ) ∩ ν ⊥ 0 being a Delzant polytope (see §3.5).Then M ν is naturally a toric Kähler manifold, acted upon by the quotient torus T d−r+1 q := T d /T r−1 ν ⊥ ; the associated moment polytope ∆ ν can be identified with ∆ ′ ν under the natural isomorphism between t d−r+1 q (the Lie algebra of T d−r+1 q ) and ν ⊥ 0 .The general case can then be reduced to the case r = 1, with M replaced by M ν .
We have an analogue of Corollary 1.1, linking the cohomology groups of the symplectic reduction M ν and of the conic reduction M ν .By the theory of [Ki], H l (M ν , Q) is tightly related to the equivariant cohomology of M for the action of T r−1 ν ⊥ .Corollary 1.2.Under the hypothesis of Theorem 1.2, 2 The case r = 1

Preliminaries
Before embarking on the proof of Theorem 1.1, we need to recall some basic constructions from toric geometry, referring to [D], [G2] and [G1] for details.We premise a digression on the geometric relation between ∆ and ∆ ν .

The transform of a polytope
Although not logically necessary, it is suggestive to describe the passage from ∆ to ∆ ν in terms of a general 'transform' operation on rational polytopes in a finite-dimensional real vector space with a full-rank lattice L, depending on the datum of a decomposition of L as the product of an oriented rank-1 sublattice and a complementary sublattice.Let V be a d-dimensional real vector space, L ⊂ V a full-rank lattice, V ∨ the dual vector space, and L ∨ the dual lattice.Suppose that ∆ ⊂ V ∨ is d-dimensional rational simple convex polytope (terminology as in [LT]).This means that there exist primitive v i ∈ L and and that exactly d facets of ∆ meet at each of its vertexes.In addition, we shall say that ∆ is integral if λ j ∈ Z for every j.Suppose given: 1. a primitive lattice vector v = 0 ∈ L; Then we may uniquely extend δ to δ ∈ L ∨ ∩ span(v * ) ⊆ V ∨ so that δ = δ v * with δ ∈ Z (a different choice of δ would result in a translation of the transormed polytope).By ( 24), ∆ + δ lies in the open half-space Let us determine ρ(∆+ δ).For each j, we can write uniquely v j = v ′ j +ρ j v where v ′ j ∈ L ′ and ρ j ∈ Z. Hence δ(v j ) = δ ρ j .We have Since ̺ = ̺ −1 , by ( 25) and ( 26) we have Thus ∆ is the convex polytope obtained from ∆ by replacing each primitive normal vector v j = v ′ j + ρ j v with the integral vector v j := v ′ j − (λ j + δ ρ j ) v, and each λ j with −ρ j .Clearly, ∆ is rational; it is not claimed that each v j be primitive, hence neither that ∆ be integral.Furthemore, (27) shows that, if F j is the facet of ∆ normal to v j , then F j := ρ(F j + δ) is the facet of ∆ normal to v j ; this correspondence passes to intersection of facets, i.e. faces.Thus we have a bijection between the set of faces of each given dimension of ∆ and ∆, hence in particular between the families of their respective vertexes.In particular, the vertexes of ∆ are the images by ̺ of the vertexes of ∆, and furthermore ∆ is simple since so is ∆.

The toric line bundle A and its circle bundle
Let us review the construction of the positive toric line bundle (A, h) on M from the Delzant polytope ∆ λ , for λ ∈ Z k , based on pairing the Delzant construction of M as a symplectic quotient of C k with the construction of a polarization on the quotient in [GS1].Consider the trivial line bundle L := C k × C, and define a Hermitian metric κ on L by setting In the following we shall implicitly identify Y and C k × S 1 .In terms of the previous diffeomorphism, the unique compatible connection 1-form is Applying this with f = 1 we obtain that, letting Θ 0 be the curvature of the unique compatible connection on L, where ω 0 = (ı/2) k j=1 dz j ∧ dz j is the standard symplectic form on C k .
Given that λ ∈ Z k , the Hamiltonian action (Γ C k , Ψ λ ) (see ( 9) and ( 10)) This is the restriction a similarly defined metric preserving linearization Γ L ∨ λ : As in [GS1], we can take the quotient and obtain a positive line bundle (A, h) on M = Z ∆ /N, by setting with associated unit circle bundle X ⊂ A ∨ given by

The complexification N and its stable locus
Besides representing M as a Marsden-Weinstein reduction for the quotient of N, it is useful to consider its parallel description as a GIT quotient for the action of the complexification N ([G2], [G1]).In the following, for every compact group T , T is the complexification of T .Every face F ∈ F (∆) of codimension c F of ∆ is uniquely an intersection of facets; hence there exists a unique increasing multi-index The following holds: 5. M = C ∆ / N as a complex manifold; 6. for every We have the following ( [D], [G2], [G1]).
Lemma 2.1.For any Similar statements hold in the complexifications.
Let us denote by P : Z ∆ → M and by P :

The lifted action of T d on X
By passing to the quotient, Γ C d and Γ Y λ determine corresponding actions given that Γ Y λ is the contact and CR lift of (Γ C k , Ψ λ ), γ X is the contact and CR lift of (γ M , Ψ). Given Since ρ F (e ı e j ) = e υ j for j ∈ I F , for x = (m, ℓ) ∈ X m we have where We can reformulate this as follows.
Lemma 2.2.Suppose that F is a face of ∆, that m ∈ F 0 , and that x ∈ X m .Then for every t ∈ T d m we have γ 2.1.5The lifted action of T 1 on X We have remarked that 11).Since however both µ X and the restriction of γ X to T 1 lift µ M , there is a character χ = χ δ : Let us make χ explicit.Since ν is primitive, the map e ı ϑ ∈ S 1 → e ϑ υ ∈ T 1 is an isomorphism of Lie groups.
Proof.Recall that, by choice of ν, ν = ν * ∈ t 1 ∨ is the dual basis to ν, and so δ = δ ν, where δ = δ( ν).Let us write ν Φ X and ν Ψ X for the vector field on X induced by ν under µ X and γ X , respectively.In view of (2), we obtain Hence for every x ∈ X and e Given that ν ∈ L(T 1 ), (39) implies ρ X e 2π ı δ = id X .Since ρ X is free, this implies δ ∈ Z. Since (39) holds for any ϑ, the second claim follows as well.

M ν and its Kähler structure
By assumption, µ X : T 1 × X → X lifts µ M : T 1 × M → M; since Φ ν > 0, µ X is locally free by (2), and given that µ M is holomorphic µ X preserves the CR structure of X. Hence the quotient M ν := X/µ X is a d-dimensional complex orbifold with complex structure J ν ([P2], [P4]).Furthermore, µ X is effective, hence generically free; therefore the projection π ν : X → M ν is a principal V -bundle with structure group T 1 .
We shall now see that ( M ν , J ν ) carries a Kähler structure ω ν , naturally induced from ω. Aside from slight changes in notation, the discussion is close to the ones in §2 of [P2] and §5.3 of [P4], so we'll be rather sketchy.To lighten notation, we shall adopt the following conventions.
4. we shall generally omit symbols of pull-backs for functions, and denote by the same symbol a function f : M → C and its pull-back π * (f ) : X → C; 5. similarly, if f is invariant and hence π * (f ) descends to M ν , we shall also denote by f : M ν → C the descended function.
Let the invariant differential 1-form α ν ∈ Ω 1 (X) be defined by Then ι(ν X ) α ν = −1, hence α ν is a connection 1-form for π ν .Hence there is a unique orbifold 2-form ω ν on M ν such that d α ν = 2 π * ν ( ω ν ).Since by ( 40) ker(α) = ker( α ν ), π and π ν share the same horizontal bundle, i.e., H x (π) = H x ( π ν ) for every x ∈ X.On the other hand, since Φ ν > 0 by (2) we have νX (x) ∈ H(π) x at every x ∈ X. Hence we can split the tangent bundle T X of X in the two alternative ways: In particular, if m ← x → m then there are complex linear isomorphisms where the latter denotes the uniformizing tangent space of M ν at m. Since 2.1.7Horizontal and contact lifts with respect to π ν Since ν is primitive, the map e ı ϑ ∈ S 1 → e ı ϑ ν ∈ T 1 is an isomorphism of Lie groups.Composing the latter with the effective action µ X , we obtain an effective action of S 1 on X, which is free on a dense invariant subset.Therefore, there exists a dense (and smooth) open subset M ′ ν ⊆ M ν over which π ν restricts to principal S 1 -bundle.Let us set X ′ := π −1 ν ( M ′ ν ).Given a smooth orbifold vector field υ on M ν , we shall say that a (smooth) vector field on X is the horizontal lift of υ (with respect to π ν ) if it is horizontal (i.e., tangent to H(π) = H( π ν )) and π ν -related to υ over M ′ ν .
Proposition 2.1.Any smooth orbifold vector field υ on M ν has a unique horizontal lift to X.
We shall denote the horizontal lift in Proposition (2.1) by υ ♭ .
Proof.Any two horizontal lifts of υ clearly coincide on X ′ , hence everywhere in X.As to existence, obviously the horizontal lift exists over the smooth locus (i.e. on X ′ ), so the point is to see that it has a smooth extension over the singular locus.Suppose m = π ν (x) ∈ M ν , and let F 1 ⊂ X be a slice for µ X through x.Thus F 1 uniformizes an open neighbourhood of m, and υ corresponds to a vector field v 1 on F 1 , invariant under the action of the stabilizer subgroup T x F 1 .Hence we can push forward v 1 (or, more precisely, (0, v 1 )) under the local diffeomorphism T 1 x × F 1 → U 1 , and obtain a smooth vector field 1 with respect to π ν , that is, its projection on H x ( π ν ) along span(ν X ) in (41).Then υ 1 is a smooth vector field on U 1 , horizontal and π ν -related to υ on U 1 ∩ X ′ .
Another such vector field υ 2 similarly constructed on an invariant open set U 2 will necessarily coincide with υ 1 on U 1 ∩ U 2 ∩ X ′ , whence on all of U 1 ∩ U 2 if the latter is non-empty.Hence by glueing these local constructions we obtain the desired lift.
Suppose that f is a C ∞ real function on M ν , and let υ f be its Hamiltonian orbifold vector field with respect to 2 ω ν .Let us define Proposition 2.2.υ c f is a contact vector field on (X, αν ).If in addition the flow of υ f is holomorphic on ( M ν , J ν ), then the flow of υ c f preserves the CR structure of X.
Proof.We have (writing f for π * ν (f )) This proves the first statement.On the other hand, the flow of υ c f preserves the horizontal tangent bundle and covers a holomorphic flow on M ν ; the second statement then follows in view of the unitary isomorphisms (42).
By the same principle, we can consider lifts of Hamiltonian actions for π ν just as one does for π.Suppose given a holomorphic and Hamiltonian action ς Mν of a compact and connected Lie group G on ( M ν , 2 ω ν ) (in the orbifold sense, see [LT]), with moment map Λ : M ν → g ∨ .Thus any ξ ∈ g determines an induced Hamiltonian (orbifold) vector field ξ Mν on M ν .Applying (43) with υ = ξ Mν , thus setting ξ X := ξ c Mν , we associate a contact and CR vector field on X to each ξ ∈ g.A standard argument shows that this assignment defines an infinitesimal contact and CR action of g on X.If this infinitesimal action is the differential of a Lie group action ς X of G on X, we shall call the latter the (contact and CR) lift of (ς Mν , Λ).
When G acts on both M and M ν , we have in principle two lifts in the picture and two different meanings for ξ X .We will clarify this point in the following section.

Transfering Hamiltonian actions from M to M ν
Suppose that G is a connected compact Lie group and let Ξ M : G × M → M be a holomorphic and Hamiltonian action, with moment map Υ : M → g ∨ .Assume the following: 1. (Ξ M , Υ) lifts to the contact CR action Ξ X : G × X → X; 2. Ξ M and µ M commute.
Then one can see that Ξ X commutes with µ X ; therefore Ξ X descends to an action Ξ Mν : G × M ν → M ν .
Suppose m = π ν (x) ∈ M ν .Choose a slice F ⊂ X at x for µ X , and view it as the uniformizing open set of an open neighbourhood of m in M ν .We obtain a local action Ξ F of G on F as follows.For any y in a neighbourhood F ′ ⊆ F of x and g in a neighbourhood G ′ ⊂ G of the identity e G , there exists a unique s(g, y) ∈ T 1 such that µ X s(g,y) • Ξ X g (y) ∈ F .Let us set If g 1 , g 2 ∈ G ′ are sufficiently close to the identity, Given ξ ∈ g, the induced vector field ξ F on F may be computed by considering the restricted local action of the 1-parameter subgroup τ → e τ ξ ∈ G, hence by differentiating at τ = 0 the path Ξ F e τ ξ (y) = µ X s(e τ ξ ,y) • µ X e τ ξ (y).We conclude the following.
Here ξ X is as in (2), with Υ in place of Φ.At any y ∈ F , we have a direct sum decomposition T y X = T y F ⊕ span νX (y) .Thus Lemma 2.4 may be reformulated as follows.
Corollary 2.1.For any y ∈ F and ξ ∈ g, ξ F (y) is the projection of ξ X (y) on T y F along span νX (y) .
By the commutativity of µ X and Ξ X , the stabilizer subgroup of x in T 1 acts on F preserving the previous direct sum of vector bundles on F .It follows that ξ F is an invariant vector field on F , and the collection of all such is the induced vector field ξ Mν on M ν .
If y ∈ F as above, ι ξ F (y) (dα F ) y is the restriction to T y F ⊂ T y X of ι ξ F (y) (d α ν ) y .On the other hand, by Lemma 2.4 we have We have used that both ξ X and νX are contact vector fields for α ν .
To prove the last statement of Proposition 2.3, we need to verify that ξ X = ξ c Mν for every ξ ∈ g.Since both ξ X and ξ c Mν lift ξ X = ξ c Mν under π ν , it suffices to show that the coefficient of ξ X along νX is Υ ξ .Therefore, the equality implies that ξ ♯ M − Υ ξ ν♯ M is the horizontal lift (with respect to π ν ) of ξ Mν , and that ξ X = ξ c Mν .
2.1.9The torus T d and its action on M ν As in the Introduction, let T d−1 c T d be a complementary subtorus to T 1 .Let us define a new torus Since ρ X (the action of S 1 on X with generator −∂ θ ) and γ X (the contact CR action of T d on X) commute, the restriction of γ X to T d−1 c and ρ X may be combined to yield a new action Since furthermore β X commutes with µ X : T 1 × X → X, it descends to an action In fact, ( 47) is the contact and CR lift of a Hamiltonian action is Hamiltonian, with moment map Ψ ′ .On the other, hand, with the usual identification of the Lie algebra and coalgebra of S 1 with ı R, ρ X is the contact lift of the the trivial action of S 1 on (M, 2 ω) with costant moment map ı.Therefore, β X is the contact lift of the Hamiltonian action In view of Proposition 2.3, we conclude the following.
Proposition 2.4.β Mν in ( 48) is Hamiltonian, with moment map We now argue that β Mν can be complexified to a holomorphic action of T d , the complexification of T d , on M ν .More generally, for any compact Lie group G we shall denote its complexification by G.
To this end, we consider the complement of the zero section A ∨ 0 ⊂ A ∨ , and observe that all the actions involved on X uniquely extend to complexified actions on In view of the discussion in §2, §3, and §5 of [P4] (applied with r = 1), under Basic Assuption 1.1 the following holds: 2. µ A ∨ 0 is proper and locally free; 3. there is a natural biholomorphism where the former quotient is taken with respect to µ X , and the latter with respect to µ A ∨ 0 .
Since β A ∨ 0 commutes with µ A ∨ 0 , it descends to the quotient and we conclude the following.
Proposition 2.5.β Mν admits a unique holomorphic extension β Mν : We aim to relate the stabilizer of m ∈ M under γ M to the stabilizer of m ∈ M ν under under β Mν if m ← x → m.More generally, we can consider the same issue for the complexified actions γ M : T d × M → M and β Mν ; by Proposition 1.6 of [Sj], the stabilizer of m under γ M is the complexification of the stabilizer under γ M .
There is a dense open subset M 0 ⊂ M where γ M is free; then γ M is free and transitive on M 0 .Let us consider the corresponding open set Let us set X 0 := π −1 (M 0 ).
Proposition 2.6.Under the previous assumptions, the following holds.

β
Mν is free and transitive on M 0 .
In particular, M 0 is smooth.
Proof.Suppose m ← x → m with m ∈ M 0 , and that (t, e ıθ ) ∈ T d stabilizes m.Hence there exits h ∈ T 1 such that With χ as in (37) and Lemma 2.3, we conclude that and T 1 are complementary subtori we conclude t = h = 1.Hence e ı θ = 1 as well.
This proves the first statement; that β Mν is free on M 0 then follows either by a similar argument using complexifications, or else by appealing to Proposition 1.6 of [Sj].
There exists t ′ ∈ T d such that γ M t ′ (m 1 ) = m 2 .We can factor t ′ uniquely as t ′ = t h, where t ∈ T d−1 c and h ∈ T 1 .Lifting first this relation to X, and then descending to M ν , this means that for some e ıθ ∈ S 1 we have where t := t, χ(h) −1 e −ı θ ∈ T d , and χ is as in (37).Finally, since µ X lifts the restriction of γ M to T 1 , and on the other hand γ M is free on M 0 , it follows that µ X is free on X 0 ; the third statement follows.
Corollary 2.2.M ν (with the Hamiltonian action ( β Mν , Ξ)) is a symplectic toric orbifold and a complex toric variety.Let T 1 x T 1 be the stabilizer of x is a finite subgroup of T 1 m .Proof.Since µ X is locally free, T 1 x is a discrete subgroup of T 1 , hence finite.Furthermore, since µ X lifts µ M , which is the restriction of γ M to T 1 , we also have In particular, T d m and T d m are tori of the same dimension.Proof.For every m ∈ M, there is a character δ m : where k := (t, δ m (k) −1 χ(h) −1 ) ∈ T d .Hence we obtain a Lie group homomorphism Let us set Lemma 2.6.ker(ψ m ) = T 1 x .Proof.By (37) and ( 52), we have for Let us prove that ψ m is surjective.Suppose (t,

The polytope ∆ ν
By Proposition 2.4 and Corollary 2.2, M ν is a Kähler toric orbifold, and its associated convex rational simple polytope ( We aim to describe the faces of ∆ ν in terms of the faces of ∆; to this end, we premise a few remarks.
Suppose that R and S are µ M -invariant, and that m ∈ R ∩ S. Hence there exist m 1 ∈ R, m 2 ∈ S and The stabilizer subgroup of m is then the 1-paramenter subgroup τ → e τ υ j , where υ j is as in (7).Hence µ M e τ υ j (m) = m for every τ ∈ R, and by Lemma 2.2 In view of ( 13), (37), and Lemma 2.3, (57) may be rewritten as follows where β X was introduced in (47).Passing to the quotient, we can reformulate (58) in terms of β Mν : Given Proposition 2.9, the facets of ∆ ν are defined by equations of the form

Proof of Theorem 1.1
We can now combine the previous results to a proof of Theorem 1.1.Let us premise a piece of notation.We shall denote by d S 1 θ the dual basis in Lie(S 1 ) Proof of Theorem 1.1.We have ∆ ν = Ξ M ν .Assume ρ ∈ ∆ ν , and choose a triple m ← x → m with ρ = Ξ( m).Then Ψ(m), υ j ≥ λ j for every j = 1, . . ., k.With υ j as in (13) this yields for every j In view of Proposition 2.9 and (60), dividing (61) by Φ ν (m) > 0, one gets Hence, every ρ ∈ ∆ ν satisfies ρ, υ j ≥ −ρ j for every j.Furthermore, the previous argument also shows that the inequalities are all strict if and only if ρ ∈ M 0 (notation as in ( 50)), and that on the other hand equality holds for exactly one j if and only if ρ belongs to the corresponding facet Fj .
Since we know already that ∆ ν is a rational convex polytope and the F j 's are its facets, we conclude that and in particular that each υ j is inward-pointing.
The previous discussion completes the proof that shape of ∆ ν is as claimed in the statement of Theorem 1.1, except that ∆ ν is realized in the Lie coalgebra of T d rather than T d .To obtain the corresponding statement of Theorem 1.1 we need only compose with the isomorphism given by the product of the identity and e ϑ ν → e ı ϑ .
It remains to determine the marking of ∆ ν , that is, the assignment to each facet F j of the order s j ≥ 1 of the structure group G j of an arbitrary m ∈ M 0 j .By construction, given m ← x → m, up to isomorphism G j may be identified with the stabilizer subgroup T 1 x T 1 of x under µ X .Now if µ X e ϑ ν (x) = x then µ M e ϑ ν (m) = m by equivariance of π.Since m ∈ M 0 j , this means that for some (unique) e ı ϑ ′ ∈ S 1 e ϑ ν = e ϑ ′ υ j = e ϑ ′ (υ ′ j +ρ j ν) (notation as in ( 13)).
In particular, since ν is primitive, we see from ( 63) that e ı ϑ = e ı ρ j ϑ ′ ∈ S 1 .Let us distinguish the following cases, depending on the relation between ν and υ j .
Proof of Corollary 1.1.Since J and J ν are torus invariant complex structures on M and M ν , respectively, by Theorem 9.1 of [LT] both M and M ν have structures of complex toric varieties (of course in the case of M this is our starting assumption); furthermore, the corresponding fans Fan(M) and Fan( M ν ) are defined by their respective polytopes, ∆ and ∆ ν .Since ∆ and ∆ ν are simple and compact, Fan(M) and Fan( M ν ) are simplicial and complete.
Hence the Betti numbers β j and β j of M and M ν are determined by the collection of the all the numbers d r and d r of r-dimensional cones in Fan(M) and Fan( M ν ), respectively ( §4.5 of [F]).Thus it suffices to prove that d r = d r for any r.
On the other hand, in order to determine the fan Fan Γ associated to a polytope Γ we may assume without loss that Γ contains the origin in its interior; in this case, furthermore, the cones in Fan Γ are the cones over the faces of the polar polytope Γ 0 to Γ ( §1.5 of [F]).Hence we need to show the polar polytopes of (suitable translates of) ∆ and ∆ ν share the same number of faces in each dimension.However, for any d-dimensional polytope Γ in a d-dimensional real vector space, containing the origin in its interior, there is an order-reversing bijection between the faces of Γ and those of Γ 0 , with corresponding faces F and F * having dimensions adding up to d − 1 (loc.cit.).Thus the statement follows from Proposition 2.8.st ×S 1 on M ν given by the product of ϕ Mν = ψ Mν and ρ Mν , where the latter is the action of S 1 on M ν obtained by descending ρ X .Let us adopts the previous choices of Hamiltonian structures (for the second factor, we use the same Hamiltonian struture as in the proof of Theorem 1.1, see Propositions 2.3 and 2.4).The corresponding moment maps then differ by a translation by a constant in t d−1 c ∨ × {0}; hence so do the corresponding moment polytopes, say ∆ ψ ν and ∆ ϕ ν .The previous considerations may be extended to the case where δ = 0, and therefore ϕ Mν = ψ Mν .In fact, if δ = 0, then ϕ X and ψ X differ by the composition of a morphism T d−1 st → T 1 , say of the form e ı ϑ → e ı a,ϑ ν where a ∈ Z d−1 , with γ X .Hence, passing to the quotient, in view of (37) the induced actions ϕ Mν and ψ Mν will now differ by the composition of a character T d−1 st → S 1 of the form e ı ϑ → e −ı δ a,ϑ with ρ Mν .Identifying the coalgebra of T d−1 st with ı R d−1 , the corresponding moment maps Φ ψ and Φ ϕ for ψ Mν and ϕ Mν are related by a relation of the form Φ ψ = Φ ϕ −δ a Γ, where Γ : It follows that the two cones are related by a transformation in ı R d × ı R of the form ı (x, y) → ı (x − y δ a, y), followed perhaps by a translation.

The case of arbitrary r
We shall now now remove the restriction that r = 1, and allow any value 1 ≤ r ≤ d.Before dealing directly with the geometric situation, we shall dwell on some handy technical results.

Preliminaries on transversality of polytopes
Definition 3.1.Let V be a finite dimensional real vector space, Γ ⊂ V a convex polytope, W ⊆ V an affine subspace.We shall say that Γ and W meet transversely, of that they are transverse to each other, if W is transverse to the relative interior F 0 of every face F of Γ.
In the hypothesis of Definition 3.1, let us set Γ W := Γ ∩ W . Clearly, Γ W is a convex polytope in W .
Let F (Γ) be the collection of faces of Γ and G(Γ) = {F 1 , . . ., F s } ⊆ F (Γ) be the subset of its facets.For each j = 1, . . ., s let ℓ j ∈ V ∨ be an inward normal covector to F j , so that for certain λ j ∈ R; the j-th facet is thus If L ∈ F (Γ), there exists a unique subset I L ⊆ {1, . . ., s} such that L = j∈I L F j .We are interested in simple polytopes (meaning that exactly n facets of Γ meet at each vertex, where n = dim R (V ) [LT]; if Γ is simple, then every codimension-k face L ∈ F (Γ) is the intersection of exactly k facets, that is, |I L | equals the codimension of L.
Proposition 3.1.In the setting of Definition 3.1, suppose that Γ and W meet transversely.The following holds.
More precisely, regarding 1. we shall show that for any p ∈ W ∩ F 0 and any open neighborhood W ′ of p in W one has W ′ ∩ L 0 = ∅.Similarly, regarding 2. we shall show that for any p ∈ Γ W and any open neighborhood Proof of 1.Since W ⊆ V is an affine subspace, it is a translate of a vector subspace W ⊆ V .Suppose p ∈ F 0 ∩ W . Then W = p + W , and by transversality the map ρ : (w, q) ∈ W × F 0 → w + q ∈ V is submersive, hence open.We have p = ρ(0, p), hence the image of an arbitrary small neighborhood of (0, p) in W × F 0 contains an open neighborhood of p.
Since p ∈ F ⊆ L, we can find points p ′ ∈ L 0 arbitrarily close to p.For any such p ′ , therefore, there exist w ∈ W , w ∼ 0, and q ∈ F 0 , q ∼ p, such that p ′ = w + q.
Claim 3.1.With the previous choices, w + p ∈ W ∩ L 0 .Proof of Claim 3.1.Clearly, w + p ∈ W by construction.Let us prove that w + p ∈ L 0 , i.e. that ℓ j (w + p) = λ j if j ∈ I L and ℓ j (w + p) > λ j if j ∈ I L .
Thus for every p ∈ W ∩ F 0 we have found points p + w arbitrarily close to p in W ∩ L 0 , and this completes the proof of 1.. Proof of 2. This is a slight modification of the previous argument.Suppose p ∈ Γ ∩ W .If p ∈ Γ 0 , there is nothing to prove.Otherwise, p ∈ F 0 for some face F ∈ F (Γ).We can find p ′ ∈ Γ 0 arbitrarily close to p, and therefore -by the previous considerations -for any such p ′ there exist w ∈ W with w ∼ 0 and q ∈ F 0 with q ∼ p such that p ′ = w + q.
Proof of 1.Let F j be the j-th facet of Γ as in (69), and suppose F j ∩ W = ∅.Then F 0 j ∩ W = ∅ by Corollary 3.1.Since W meets F 0 j transversely by assumption, F 0 j ∩ W has codimension one in W and ℓ j restricts to a non-constant affine linear functional on W . Thus, if p ∈ F 0 j then every neighborhood of p in W intersects both Γ 0 and Γ c .It follows that F j ∩ W is a facet of Γ W .
Conversely, let F be a facet of Γ W , and let p ∈ F 0 .Since p ∈ Γ 0 (for else p ∈ Γ 0 W ), there exists j ∈ {1, . . ., s} such that p ∈ F j , and therefore F 0 j ∩W = ∅.By the above F j ∩W is a facet of Γ W . Since a small neighborhood of p in W meets no facet of Γ W other that F , we may slightly perturb p in F 0 and assume that p ∈ F 0 ∩ F 0 j and therefore F 0 ∩ W ′ = F 0 j ∩ W ′ for some open neighborhood W ′ of p in W .This forces F = F j ∩ W .
Proof of 2. Since every face is the intersection of the facets containing it, by 1. we have Proof of 3. and 4. If F is a codimension-k face of Γ such that F ∩W = ∅, let us choose p ∈ F 0 ∩ W = ∅.Since Γ is simple, I F = {i 1 , . . ., i k } ⊆ {1, . . ., s} and there is a neighborhood W ′ of p in W such that By transversality, F 0 ∩ W has codimension k in W and furthemore each F i j is a facet of Γ W by 1.. Thus F W := F ∩ W is a face of Γ W , since it is a non-empty intersection of facets, and it has codimension-k in W , since it has a non-empty open subset which has codimension k.Furthermore, it is given by the intersection of the k facets F i j ∩ Γ W of Γ W .It then follows that Conversely, suppose that K ⊂ Γ W is a face, and suppose p ∈ K 0 .Since p ∈ Γ 0 , there exists a unique face F of Γ such that p ∈ F 0 .By the previous discussion, F W = F ∩ W is also a face of Γ W , and p ∈ F 0 ∩ W = F 0 W . Since distinct faces of Γ W have disjoint relative interiors, K = F W .
Proof of 5.By 3., every codimension-k face of Γ W is the intersection of k facets, and this means that Γ W is simple.
and the corresponding statement holds for the stabilizer in the complexification,T k F T k ; 3. C ∆ := F ∈F (∆) O F isthe open subset of stable points for the action of T k on C k with the given linearization or, equivalently, the T k -saturation of Z ∆ ; 4. N acts freely and properly on C ∆ ; Let us consider a general triple m ← x → m, and denote by T d m T d the stabilizer of m for γ M , and by T d m T d the stabilizer of m for β Mν .We want to describe the relation between T d m and T d m.
It is in order to briefly digress on how ∆ ν in Theorem 1.1 depends on the choice of T d−1 c T d .Suppose first that δ = 0 in (11), so that µ X is the restriction of γX to T 1 .Let S d−1 c , T d−1 c T d be different complementary subtorii to T 1 , so that T d ∼ = S d−1 c × T 1 ∼ = T d−1 c × T 1 ; thus projecting onto T d−1 c along T 1 yields an isomorphism P : S d−1 c ∼ = T d−1 c .Let us choose an isomorphism of the standard torus T d−1 st = (S 1 ) d−1 with S d−1 c, so that (composing with P )T d−1 st ∼ = S d−1 c ∼ = T d−1 c .We obtain actions ϕ X and ψ X of T d−1 st on X, by composing the previous isomorphisms with the restrictions of γX to S d−1 c and T d−1 c respectively; then ϕ X = ψ X (unless S d−1 c = T d−1 c) and, by construction, the two actions differ by the composition of a character T d−1 st → T 1 with µ X .On the other hand, since ϕ X and ψ X commute with µ X , they descend to symplectic actions ϕ Mν and ψ Mν of T d−1 st on ( M ν , 2 ω ν ); in fact, by the previous remark and the construction of M ν as a quotient, ϕ Mν = ψ Mν .Thus ϕ X and ψ X are different contact lifts to X of the same symplectic action of T d−1 st on M ν ; hence they correspond to different Hamiltonian structures for the latter action, whose moment maps differ by a translation in t d−1 st ∨ .Let us consider the action of T d := T d−1

Corollary 3. 1 .
Under the hypothesis of Proposition 3.1, if L is a face of Γ andΓ W ∩ L = ∅, then Γ W ∩ L 0 = ∅.Proof.If p ∈ W ∩ L, there is a face F ∈ F (Γ) with F ⊆ L and p ∈ W ∩ F 0 .Hence W ∩ L 0 = ∅ by Proposition 3.1.Proposition 3.2.Under the hypothesis of Proposition 3.1, the following holds:1.the facets of Γ W are the non-empty intersections of W with the facets of Γ;2.Γ 0 W = W ∩ Γ 0 ;3.if Γ is simple, then the codimension-k faces of Γ W are the non-empty intersections of W with the codimension-k faces of Γ;4. if F is a face of Γ such that F W := F ∩W = ∅, then the relative interior of F W is F 0 W = F 0 ∩ W ;