TORIC CONTACT GEOMETRY IN ARBITRARY CODIMENSION

VESTISLAV APOSTOLOV, DAVID M. J. CALDERBANK, PAUL GAUDUCHON, AND EVELINE LEGENDRE

Abstract. We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

Introduction

Toric contact geometry lies at the interface of of two important themes in contemporary geometry. The first is the use of toric methods, beginning with toric varieties and toric symplectic manifolds, to provide large classes of geometric objects which can be understood in an essentially combinatorial way. The second is a growing interest in geometries where the tangent bundle is not completely reducible, but is instead equipped with a distribution (subbundle) or filtration. The simplest nontrivial example is a (2m + 1)-manifold N equipped with a corank one distribution 𝒟 ≤ T N which is locally the kernel of a contact form, i.e., a 1-form θ such that θ ∧dθ^{∧m} is a nonvanishing volume form. The restriction of dθ to 𝒟 defines a symplectic form on 𝒟, and thus (N, 𝒟) may be viewed as an odd-dimensional analogue of a symplectic manifold called a contact manifold.

Any contact manifold has a canonical symplectization: the total space of the annihilator 𝒟^{0} ≤ T^{∗}M of 𝒟 inherits an exact 2-form Ω^{𝒟} from the tautological symplectic structure on T^{∗}M, and Ω^{𝒟} is nondegenerate, hence symplectic, on the complement of the zero section. It follows that contact manifolds, like symplectic manifolds, have no local invariants, and it is natural to extend symplectic techniques to contact manifolds. In particular, a contact (2m + 1)-manifold N is toric if its symplectization is, which means that N has an effective action of an (m + 1)-torus preserving the contact distribution 𝒟, and such toric contact manifolds have been extensively studied.

Our interest here is in extending this theory to higher codimension, more precisely, to distributions 𝒟 of corank ℓ > 1 in T N. Such distributions arise naturally in CR geometry [12], for example on real submanifolds of codimension ℓ > 1 in a complex manifold, and in subriemannian geometry [25]. However, it is not immediately clear how best to generalize the nondegeneracy condition, as the obstruction to integrability of 𝒟 is now a 2-form on 𝒟 with values in T N/𝒟, called the Levi form. We suggest that the most natural nondegeneracy requirement for a contact manifold (N, 𝒟) in codimension ℓ is, as in codimension one, the local existence of a contact form, which is now a 1-form θ with 𝒟 ≤ ker θ and dθ|_{𝒟} nondegenerate. This means equivalently that the Levi form of 𝒟 has a nondegenerate component at every point, which is not as weak as it may seem, because nondegeneracy is a (fibrewise Zariski) open condition. Thus any contact manifold has a symplectization U_{𝒟} ⊆ 𝒟^{0} (with symplectic form induced from T^{∗}N) which meets each fibre of 𝒟^{0} → N in a nonempty Zariski open subset. The complement of U_{𝒟} in 𝒟^{0} is then, fibrewise, the cone over a projective variety (algebraically, a projective hypersurface of degree m, where 𝒟 has rank 2m) in P(𝒟), which we call the degeneracy variety of (N, 𝒟). In particular, for ℓ > 1, a contact manifold (N, 𝒟) has local invariants except in some low rank cases, and local classification would involve understanding the deformation theory of arbitrary projective hypersurfaces.

Date: September 18, 2017.

V.A. is supported in part by IMI/BAS and an NSERC discovery grant. E.L. is partially supported by France ANR project EMARKS No ANR-14-CE25-0010. The authors are grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, the London Mathematical Society, and the Labex CIMI (Toulouse) for hospitality and financial support. They would also like to thank D. Dragnev for pointing out to them the references [6, 11], and R. Ponge for drawing their attention to fat distributions in subriemannian geometry.

Despite these difficulties, toric contact geometry in higher codimension turns out to be surprisingly tractable. A contact manifold (N, 𝒟) of dimension 2m+ℓ and codimension ℓ has a symplectization U_{𝒟} of dimension 2(m + ℓ), hence is toric if it has an effective action of an (m + ℓ)-torus preserving the contact distribution 𝒟. In order to avoid technical difficulties appearing already in codimension 1, we also require that there is an ℓ-subtorus whose orbits are transverse to the contact distribution. In this setting, we find that despite the presence of local invariants, toric contact geometry is essentially encoded in a natural generalization of the momentum polytope of a toric symplectic manifold, namely a kind of polytope in the grassmannian of ℓ-dimensional subspaces of the dual of the Lie algebra of the torus (which is just the projectivization when ℓ = 1). Thus the theory still has a strong combinatorial flavour, and the local invariants arise simply because polytopes in a grassmannian are more flexible than polytopes in an affine or projective space.

In more detail, the contents and main results of the paper are as follows. After defining contact manifolds and their symplectizations in §1.1 (and giving some simple examples, including products of odd dimensional spheres), we begin the study of contact actions in §1.2. Here we introduce as Condition 1, the transversality property that we use to avoid some technicalities in the theory. We also note that a local contact action satisfying this condition induces a family of 2-forms on 𝒟 which we call Levi structures. In a companion paper [1], we use nondegenerate Levi structures, in the CR context, to construct Kähler metrics on quotients by transverse actions. While we do not pursue this angle in the present paper, our approach is in part motivated by it.

We begin our study of contact actions of a torus by introducing the local theory in §2.1. Here Levi structures are induced by Levi pairs (𝔤, λ), where 𝔤 is a subalgebra of the infinitesimal torus 𝔱_{N} acting on the contact manifold N and λ ∈ 𝔤^{∗} picks out a component of the Levi form. We introduce fundamental methodology, Stratagem 1, for parametrizing such pairs (𝔤, λ) by maps to the quotients 𝔨 = 𝔱_{N}/𝔤 and 𝔥 = 𝔱_{N}/ ker λ. This stratagem is closely linked to the affine geometry of momentum maps for hamiltonian torus actions, which we review in Appendix A. We thus see that a Levi pair provides an affine slice of the momentum map for the hamiltonian 𝔱_{N} action on the symplectization U_{𝒟} of N.

The central idea of the paper, the grassmannian momentum map of Definition 7, is a way to package these slices together in a single compact object (analogous to the polytope of a compact toric symplectic manifold or orbifold) in the grassmannian Gr_{ℓ}(𝔱_{N}^{∗}) of ℓ-dimensional subspaces of 𝔱_{N}^{∗}, which we call the grassmannian image of N.

When the 𝔱_{N} action integrates to an effective contact action of a torus 𝕋_{N} on N, properties of the grassmannian image can be derived from fairly standard properties of the 𝕋_{N} action on N and U_{𝒟}. In §2.2, we apply the standard theory of symplectic slices to obtain a local model for these actions. Then in §2.3, we obtain a connectedness and convexity result for the slices of 𝔱_{N} action on U_{𝒟} given by a Levi pair. This is essentially a special case of a convexity result for transverse symplectic foliations obtained independently by Ishida [17], and follows easily from the methods of Atiyah [2].

These preparations lead us to our main results on toric contact manifolds. In section §3.1, we show that the quotient of N by 𝕋_{N} is a simple polytope in the sense that it is at least a manifold with corners. We improve this in §3.2, where we show that any Levi pair (𝔤, λ) provides a diffeomorphism of the quotient N/𝕋_{N} with a simple polytopes in the affine subspace of 𝔱_{N}^{∗} defined by (𝔤, λ). This shows that the grassmannian momentum map induces an embedding of N/𝕋_{N} into Gr_{ℓ}(𝔱_{N}^{∗}) whose image is polyhedral submanifold with corners as in Definition 14.

We would like to prove that toric contact manifolds are classified by their grassmannian image, with a suitable labelling of the codimension one faces. This is true locally by the local models obtained in §2.2 using symplectic slices. However, as in toric symplectic geometry [24] and toric contact geometry in codimension one [22], there remains a local-to-global question governed by the cohomology of a sheaf. Thus we have the following result.

Theorem. The grassmannian image Ξ of a compact toric contact manifold of Reeb type (N, 𝒟, K) is Delzant and polyhedral of Reeb type in Gr_{ℓ}(𝔱_{N}^{∗}), and there is a sheaf 𝔠𝔬𝔫^{𝕋}(𝒟) on Ξ such that toric contact manifolds of Reeb type with grassmannian image Ξ are parametrized up to isomorphism by H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)).

The terminology in this theorem is defined in §3.3, where we prove that the grassmannian image has these properties, and establish the uniqueness, modulo H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)), of (N, 𝒟, K) given Ξ. Then in §3.4, we prove that any such Ξ arises in this way.

In contrast to the symplectic and codimension one cases, 𝔠𝔬𝔫^{𝕋}(𝒟) is the sheaf of solutions of a linear partial differential equation (for infinitesimal contactomorphisms), which is overdetermined and typically not involutive. We show that H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)) vanishes in special cases, such as when (N, 𝒟, K) is a product of codimension one toric contact manifolds. However, it remains an open question whether it vanishes in general.

1. Contact geometry in arbitrary codimension

1.1. Levi nondegenerate distributions and symplectization. Let N be smooth manifold of real dimension 2m+ℓ equipped with a rank 2m vector distribution 𝒟 ≤ T N. Let 𝒟^{0} be the annihilator of 𝒟, which is a rank ℓ subbundle of T^{∗}N. We thus have dual short exact sequences of vector bundles

0 → 𝒟 → T N ⟶^{q_{D}} T N/𝒟 → 0
0 → 𝒟^{0} → T^{∗}N → 𝒟^{∗} → 0,

where the transpose of the quotient q_{𝒟}: T N → T N/𝒟 identifies 𝒟^{0} canonically with (T N/𝒟)^{∗}: thus we identify an element or section α of (T N/𝒟)^{∗} with α ∘ q_{𝒟} in 𝒟^{0}.

Definition 1. The Levi form L_{𝒟}: ∧^{2}𝒟 → T N/𝒟 of 𝒟 ≤ T N is defined, using sections X, Y ∈ Γ(𝒟), by the tensorial expression

(1)  L_{𝒟}(X, Y) = −q_{𝒟}([X, Y]).

The nondegeneracy locus of 𝒟 is the open subset

U_{𝒟} = {α ∈ 𝒟^{0} ≅ (T N/𝒟)^{∗} | α ∘ L_{𝒟} is nondegenerate}

of 𝒟^{0}. We say 𝒟 is Levi nondegenerate, and (N, 𝒟) is contact of rank m and codimension ℓ, if U_{𝒟} has nonempty intersection with each fibre of 𝒟^{0} over N. A (local) section of U_{𝒟} is called a (local) contact form on N. We often describe U_{𝒟} by its complement, which is (the bundle of cones over) the degeneracy variety

V_{𝒟} = {[α] ∈ P(𝒟^{0}) | α ∘ L_{𝒟} is degenerate},

viewed as a bundle of submanifolds of the fibrewise projectivization P(𝒟^{0}) of 𝒟^{0}.

Our normalization of the Levi form is chosen so that for any section α of 𝒟^{0}, the restriction of dα to ∧^{2}𝒟 ≤ ∧^{2}T N is α ∘ L_{𝒟}. A Levi nondegenerate distribution 𝒟 has local frames α_{1}, … α_{ℓ} for 𝒟^{0} in U_{𝒟}, and hence each fibre of the degeneracy variety V_{𝒟} is the projective hypersurface of degree m where (∑_{i=1}^{ℓ} t_{i}α_{i}) ∘ L_{𝒟} degenerates (i.e., the determinant, a homogeneous degree m polynomial in the ℓ variables t_{1}, … t_{ℓ}, vanishes).

Examples 1. (i) The projectivization P(W) = W^{×}/ℝ^{×} ≅ ℝP^{2m+1} of a symplectic vector space (W, ω) (with dim W = 2m + 2) is contact of codimension one, where 𝒟_{ξ} = Hom(ξ, ξ^{⊥}/ξ) ≤ Hom(ξ, W/ξ) = T_{ξ}P(W) and ξ^{⊥} is the orthogonal space to ξ with respect to ω; thus U_{𝒟} = 𝒟^{0} \ 0 and the degeneracy variety V_{𝒟} is empty. The odd dimensional sphere 𝕊^{2m+1} ≅ W^{×}/ℝ^{+} (the space of rays in W) is also contact, being a double cover of P(W). (ii) If (N_{i}, 𝒟_{i}) are contact manifolds, with codimensions ℓ_{i}, for i ∈ {1, … n}, then so is (∏_{i=1}^{n} N_{i}, 𝒟_{1} ⊕ ⋯ ⊕ 𝒟_{n}), with codimension ℓ = ℓ_{1} + ⋯ + ℓ_{n} and U_{𝒟} = ∏_{i=1}^{n} U_{𝒟_{i}}. In particular, the product of n = ℓ codimension one spheres 𝕊^{2m_{1}+1} × ⋯ × 𝕊^{2m_{ℓ}+1} is a contact manifold with codimension ℓ, where each fibre of U_{𝒟} is the disjoint union of the 2^{ℓ} open ℓ-quadrants (simple cones) spanned by ±η_{1}, …, ±η_{ℓ}. Each fibre of the degeneracy variety V_{𝒟} is thus an union of ℓ-hyperplanes (i.e., the facets of an (ℓ − 1)- simplex) with multiplicities m_{i}. (iii) Another special case, studied in [6, 11], is that of ℓ-contact manifolds. In the terminology of this paper, these are contact manifolds (N, 𝒟) of rank m and codimension ℓ for which the Levi form L_{𝒟}: ∧^{2}𝒟 → T N/𝒟 has rank one (but this component is nondegenerate) at every point. Equivalently, each fibre of the degeneracy variety V_{𝒟} is a single hyperplane of multiplicity m. (iv) The case that the degeneracy variety V_{𝒟} is empty (i.e., in each fibre, it has no real points) has been studied in subriemannian geometry as a natural generalization of the codimension one case [25]. This condition means equivalently that for any z ∈ N and any nonzero X ∈ T_{z}N, L_{𝒟}(X, ·): 𝒟_{z} → T_{z}N/𝒟_{z} is surjective. A distribution with this property is called fat.

Definition 2. A diffeomorphism Ψ between contact manifolds (N, 𝒟) and (N′, 𝒟′) is called a contactomorphism if Ψ_{∗}(𝒟) = Ψ^{∗}𝒟′ ≤ Ψ^{∗}T N′. We denote by 𝔠𝔬𝔫(N, 𝒟) the Lie algebra of infinitesimal contactomorphisms of (N, 𝒟), i.e., the space of vector fields X on N such that

L_{X}Γ(𝒟) ⊆ Γ(𝒟).

We now show that U_{𝒟} ⊆ 𝒟^{0} ≤ T^{∗}N (or any connected component of U_{𝒟}) provides a “symplectization” of N, to which contactomorphisms lift as hamiltonian vector fields. For this recall that the tautological 1-form τ on T^{∗}N is defined by τ_{α} = α ∘ p_{∗}, where α ∈ T^{∗}N and p_{∗} is the differential of the projection p: T^{∗}N → N. The latter fits into an exact sequence

(2)  0 → p^{∗}T^{∗}N → T (T^{∗}N) →^{p_{∗}} p^{∗}T N → 0,

where (the image of) p^{∗}T^{∗}N is the vertical bundle of T^{∗}N. The canonical symplectic form on T^{∗}N is then Ω = dτ. For any local section α of T^{∗}N, α^{∗}τ = α and hence α^{∗}Ω = α^{∗}dτ = d(α^{∗}τ) = dα. Since τ vanishes on p^{∗}T^{∗}N, so does Ω, and the induced pairing between p^{∗}T^{∗}N and p^{∗}T N the natural one: for any 1-form α and any lift X̃ of a vector field X on N, Ω(p^{∗}α, X̃) = d(τ(X̃))(p^{∗}α) = α(X), since τ(X̃): T^{∗}N → ℝ is evaluation at X, which is linear on the fibres of p.

Any vector field X on N admits a natural lift to T^{∗}N, whose local flow is the natural lift of the local flow of X. For further use, we recall the following well-known facts. Proposition 1. The natural lift of any vector field X on N to T N is the unique lift X̃ with LX̃τ = 0; it is hamiltonian with respect to Ω, with momentum map α ↦ α(X), and is given by X̃_{α} = α_{∗}(X) − p^{∗}L_{X}α for any extension of α ∈ T^{∗}N to a local section. Proof. The natural lift of any diffeomorphism of N to T^{∗}N preserves τ (by its tautological construction), and hence the natural lift of X preserves τ. On the other hand for any vector field X̃ on T^{∗}N preserving τ, 0 = LX̃τ = ιX̃Ω + d(τ(X̃)), so X̃ has hamiltonian τ(X̃). If X̃ is a lift of X then τ(X̃)(α) = α(X), which determines X̃ uniquely from X. For the final formula, observe that α^{∗}(ιX̃Ω) = −d(α(X)) = ι_{X}dα − L_{X}α = α^{∗}(ι_{α_{∗}(X)}Ω−ι_{p^{∗}L_{X}α}Ω), since α^{∗}(ι_{α_{∗}(X)}Ω) = ι_{X}(α^{∗}Ω) = ι_{X}dα; from this the result follows by nondegeneracy of Ω. □

Let τ^{𝒟} be the pullback of τ to 𝒟^{0} ≤ T^{∗}N, so that Ω^{𝒟} := dτ^{𝒟} is the pullback of Ω. Henceforth, p will denote the bundle projection of 𝒟^{0} rather than T^{∗}N above, and we now have an exact sequence

(3)  0 → p^{∗}𝒟^{0} → T 𝒟^{0} →^{p_{∗}} p^{∗}T N → 0.

Proposition 2. For a contact manifold (N, 𝒟), U_{𝒟} is the open subset of 𝒟^{0} on which Ω^{𝒟} is nondegenerate, hence a symplectic form. Also, for any contact form α: N → U_{𝒟}, T N = 𝒟 ⊕ 𝒦^{α}, where 𝒦^{α} is the distribution orthogonal to 𝒟 via dα. The natural lift of any X ∈ 𝔠𝔬𝔫(N, 𝒟) to T^{∗}N is tangent to 𝒟^{0}, and its restriction defines an isomorphism between 𝔠𝔬𝔫(N, 𝒟) and the space of projectable vector fields X̃ on 𝒟^{0} with LX̃τ^{𝒟} = 0.

Proof. The restriction of Ω_{α}^{𝒟} to V_{α}(𝒟^{0}) × T_{z}N descends to the canonical nondegenerate pairing between V_{α}(𝒟^{0}) ≅ 𝒟_{z}^{0} and T_{z}N/𝒟_{z}. Hence Ω_{α} is nondegenerate if and only if α^{∗}Ω^{𝒟} = dα is nondegenerate on 𝒟 for any extension of α to a local section. But dα|_{𝒟} = α ∘ L_{𝒟}, which proves the first part. The second part is immediate from the nondegeneracy of dα on 𝒟.

For the last part, observe that the local flow of X ∈ 𝔠𝔬𝔫(N, 𝒟) preserves 𝒟, hence its natural lift to T^{∗}N preserves 𝒟^{0}, i.e., the induced vector field is tangent to 𝒟^{0}. The restriction X̃ to 𝒟^{0} is clearly a lift of X, hence projectable, and LX̃τ^{𝒟} = 0 since τ^{𝒟} is the pullback of τ. Conversely if a lift X̃ of X to 𝒟 satisfies LX̃τ^{𝒟} = 0 then for any section Y of 𝒟, any lift Ỹ of Y to 𝒟^{0}, and any α ∈ 𝒟^{0}, 0 = (LX̃τ^{𝒟})_{α}(Ỹ) = −τ_{α}^{𝒟}([X̃, Ỹ]) = α(p_{∗}[X̃, Ỹ]) = α([p_{∗}X̃, p_{∗}Ỹ]) = α(L_{X}Y). Hence L_{X}Y is a section of 𝒟, i.e., X ∈ 𝔠𝔬𝔫(N, 𝒟). Furthermore, if f_{X}: 𝒟^{0} → ℝ is defined by f_{X}(α) = α(X) then df_{X} + ιX̃Ω^{𝒟} = 0, which determines X̃ uniquely from X on U_{𝒟}, hence on 𝒟^{0}, since U_{𝒟} is dense in 𝒟^{0}. □

We refer to 𝒦^{α} as the Reeb distribution of α.

If we identify 𝒟^{0} with (T N/𝒟)^{∗}, then the function f_{X}: 𝒟^{0} → ℝ with f_{X}(α) = α(X) is the fibrewise linear form defined by q_{𝒟}(X) ∈ T N/𝒟. Since f_{X} is a hamiltonian for the lift X̃ on U_{𝒟}, it follows that X̃ is in fact uniquely determined by q_{𝒟}(X).

Remark 1. The degeneracy variety is a local contact invariant of a contact manifold (N, 𝒟). For example 𝕊^{1}×𝕊^{5} and 𝕊^{3}×𝕊^{3} cannot be even locally contactomorphic because the degeneracy variety of the latter has two points in each fibre, whereas the former has only one (with multiplicity two). Also the lift X̃ to 𝒟^{0}, for any X ∈ 𝔠𝔬𝔫(N, 𝒟) must preserve (the cone over) the degeneracy variety. Thus although 𝔠𝔬𝔫(N, 𝒟) may be identified with a linear subspace of Γ(T N/𝒟), this linear subspace is typically small.

1.2. Local contact actions and transversality. Effective actions of Lie groups by contactomorphisms on (N, 𝒟) may be described locally as follows.

Definition 3. A (local, effective) contact action of a Lie algebra 𝔤 on a contact manifold (N, 𝒟) is a Lie algebra monomorphism K: 𝔤 → 𝔠𝔬𝔫(N, 𝒟). For v ∈ 𝔤, we write K_{v} for the induced vector field K(v), and we define κ^{𝔤}: N × 𝔤 → T N by κ^{𝔤}(z, v) = K_{v,z}. Let 𝒦^{𝔤} ≤ T N be the image of κ^{𝔤}, i.e., 𝒦^{𝔤}_{z} := span{K_{v,z} | v ∈ 𝔤}.

Observation 1. Let K: 𝔤 → 𝔠𝔬𝔫(N, 𝒟) be a local contact action of 𝔤 on (N, 𝒟) and define μ_{𝔤}: 𝒟^{0} → 𝔤^{∗} by ⟨μ_{𝔤}(α), v⟩ = α(K_{v}) for α ∈ 𝒟^{0} and v ∈ 𝔤. Then the lift of K to T^{∗}N preserves 𝒟^{0}, and the induced local action K̃ is hamiltonian on U_{𝒟} with momentum map μ_{𝔤}|_{U_{𝒟}} ; in particular ⟨dμ_{𝔤}(K̃_{v}), w⟩ = −⟨μ_{𝔤}, [v, w]_{𝔤}⟩ for all v, w ∈ 𝔤.

This is immediate from the discussion at the end of §1.1.

Example 2. Let G be a Lie group of dimension ℓ with Lie algebra 𝔤 and let π: N → M be a principal G-bundle with connection η: T N → 𝔤, where dim M = 2m. Then 𝒟 := ker η is a rank 2m distribution on N, G acts on (N, 𝒟) by contactomorphisms, and η induces a bundle isomorphism of T N/𝒟 with N × 𝔤. In this trivialization, the Levi form of 𝒟 is dη + \frac{1}{2}[η ∧ η]_{𝔤}, the pullback to N of the curvature F^{η} of η.

Suppose K: 𝔤 → 𝔠𝔬𝔫(N, 𝒟) is a local contact action where (N, 𝒟) is contact of codimension ℓ = dim 𝔤. Abstracting the local geometry of Example 2, we say the action of 𝔤 is transversal if the following condition holds.

Condition 1. At every point of N, 𝒟 + 𝒦^{𝔤} = T N. Equivalently:

(i) rank 𝒦^{𝔤} = ℓ everywhere on N; (ii) 𝒟 ∩ 𝒦^{𝔤} is the zero section of T N (and thus T N = 𝒟 ⊕ 𝒦^{𝔤}).

The composite q_{𝒟} ∘ κ^{𝔤}: N × 𝔤 → 𝒦^{𝔤} → T N/𝒟 is a bundle isomorphism and so there is a canonically defined 1-form η^{𝔤}: T N → 𝔤, characterized by

ker η^{𝔤} = 𝒟 and ∀ v ∈ 𝔤, η^{𝔤}(K_{v}) = v.

We also denote by η^{𝔤} the induced map from T N/𝒟 to 𝔤. We can now verify the usual properties of a connection 1-form.

Lemma 1. Let η = η^{𝔤} denote the 𝔤-valued 1-form of a contact action on N satisfying Condition 1. Then for any v ∈ 𝔤, L_{K_{v}}η + [v, η]_{𝔤} = 0, and dη + \frac{1}{2}[η ∧ η]_{𝔤} = η ∘ L_{𝒟}, where L_{𝒟} is extended by zero from 𝒟 to T N = 𝒟 ⊕ 𝒦^{𝔤}.

Proof. For any v ∈ 𝔤, K_{v} preserves 𝒟 and 𝒦^{𝔤}, hence also κ^{𝔤} ∘ η: T N → 𝒦^{𝔤}. Thus

0 = L_{v}(K ∘ η)(X) = (L_{v}K)(η(X)) + K(L_{K_{v}}η)(X).

Since K is a morphism of Lie algebras, (L_{K_{v}}K)(w) = [K_{v}, K_{w}] = K([v, w]_{𝔤}), so (L_{K_{v}}η)(X) + [v, η(X)]_{𝔤} = 0 by the injectivity of K. Now for X, Y ∈ Γ(𝒟) and v, w ∈ 𝔤, dη(X + K_{v}, Y + K_{w}) = −η([X + K_{v}, Y + K_{w}]) = −η([X, Y]) − [η(K_{v}), η(K_{w})]_{𝔤} = η(L_{𝒟}(X + K_{v}, Y + K_{w})) − [η(X + K_{v}), η(Y + K_{w})]_{𝔤}. □

Since K: 𝔤 → 𝔠𝔬𝔫(N, 𝒟) is a Lie algebra morphism, 𝒦^{𝔤} is an integrable distribution, equal to 𝒦^{α} for any contact form α such that α(K_{v}) is constant for all v ∈ 𝔤. For any λ ∈ 𝔤^{∗}, define η^{λ} = η^{𝔤,λ}: N → 𝒟^{0} by η_{z}^{λ}(X) = ⟨η_{z}(X), λ⟩, so that η^{λ}(K_{v}) = λ(v) and dη^{λ}|_{𝒟} = ⟨dη|_{𝒟}, λ⟩ = η^{λ} ∘ L_{𝒟} is the λ-component of the Levi form of 𝒟.

Definition 4. For a contact action K: 𝔤 → 𝔠𝔬𝔫(N, 𝒟) satisfying Condition 1, we refer to 𝒦^{𝔤} and its integral submanifolds as the associated Reeb distribution and Reeb foliation transverse to 𝒟. For λ ∈ [𝔤, 𝔤]^{0} ≤ 𝔤^{∗}, we call (𝒟, dη^{λ}|_{𝒟}) the induced Levi structure; it is said to be nondegenerate dη^{λ}|_{𝒟} is, i.e., η^{λ} is a contact form (U_{𝒟}-valued).

2. Contact torus actions

2.1. Abelian local contact actions. Suppose that K: 𝔱_{N} → 𝔠𝔬𝔫(N, 𝒟) is a local contact action of an abelian Lie algebra 𝔱_{N} on a contact manifold (N, 𝒟). Let

κ := κ^{𝔱_{N}}: N × 𝔱_{N} → T N,  with  κ(z, v) = K_{v,z}, and
μ := μ_{𝔱_{N}}: 𝒟^{0} → 𝔱_{N}^{∗},  with  ⟨μ(α), v⟩ = α(K_{v}),

so that (p, μ): 𝒟^{0} → N × 𝔱_{N}^{∗} is the pointwise transpose of q_{𝒟} ∘ κ: N × 𝔱_{N} → T N/𝒟. By Observation 1, K lifts to a hamiltonian action on U_{𝒟} with momentum map μ|_{U_{𝒟}} . The following is the main tool in our analysis of abelian local contact actions.

Definition 5. An ℓ-dimensional subalgebra ι_{𝔤}: 𝔤 ↪ 𝔱_{N} and an element λ ∈ 𝔤^{∗} \ 0 together form a Levi pair (𝔤, λ) for K if:

• 𝔤 acts transversally on N via K, i.e., 𝒦^{𝔤} := span{K_{v,z} | v ∈ 𝔤} satisfies Condition 1. Let η: T N → 𝔤 be the connection 1-form of 𝔤; we say (𝔤, λ) is nondegenerate if:

• η^{λ} = ⟨η, λ⟩ is a contact form, i.e., (𝒟, dη^{λ}|_{𝒟}) is a nondegenerate Levi structure. Thus (p, μ_{𝔤}): 𝒟^{0} → N × 𝔤^{∗}, with μ_{𝔤} := ι_{𝔤}^{⊤} ∘ μ, is an isomorphism. We say (N, 𝒟, K) is Reeb type if it admits a nondegenerate Levi pair.

Let (𝔤, λ) be a Levi pair. For any v ∈ 𝔱_{N}, (dη^{λ})(K_{v}, ·) = −d(η^{λ}(K_{v})). We may thus view η^{λ}(K_{v}) = ⟨μ(η^{λ}), v⟩ as the “horizontal momentum” of K_{v} with respect to the Levi structure (𝒟, dη^{λ}|_{𝒟}). Observe that if v ∈ 𝔤, η_{z}^{λ}(K_{v}) = ⟨v, λ⟩, which vanishes for v ∈ ker λ ≤ 𝔤. Hence z ↦ μ(η_{z}^{λ}) ∈ 𝔱_{N}^{∗} takes values in (ker λ)^{0} ≅ (𝔱_{N}/ ker λ)^{∗}.

Stratagem 1. For any pair (𝔤, λ) with 𝔤 ≤ 𝔱_{N} and λ ∈ 𝔤^{∗} \ 0, the quotient 𝔱_{N}/ ker λ is an extension by ℝ of the quotient 𝔱_{N}/𝔤. To allow (𝔤, λ) to vary, it is convenient to fix this extension 𝔥 → 𝔨 (where 𝔥 and 𝔨 are abelian Lie algebras of dimensions n − ℓ + 1 and n − ℓ for n = dim 𝔱_{N}); then the commutative diagram

[figure: 0; 𝔤; ι_{𝔤}; 𝔱_{N}; u; 𝔨; 0; 0; ℝ; λ; ε; 𝔥; L; d; 𝔨; ‖; 0.]

of short exact sequences associates pairs (𝔤, λ), with 𝔱_{N}/ ker λ ≅ 𝔥, to surjective linear maps L: 𝔱_{N} → 𝔥 (thus 𝔤 is the kernel of u := d ∘ L, and λ is induced by L|_{𝔤}).

Let 𝒜 ⊆ 𝔥^{∗} be the affine subspace (ε^{⊤})^{−1}(1) modelled on 𝔨^{∗}; then 𝔥 may be identified with the affine linear functions ℓ: 𝒜 → ℝ, whence dℓ ∈ 𝔨 is the linear part of ℓ ∈ 𝔥.

If (𝔤, λ), defined by L: 𝔱_{N} → 𝔥, is a Levi pair, then the map μ^{λ}: N → 𝒜 ⊆ 𝔥^{∗}, determined uniquely by the formula

(4)  ⟨μ^{λ}(z), L(v)⟩ = η_{z}^{λ}(K_{v})

for all z ∈ N and v ∈ 𝔱_{N}, will be called the horizontal (natural) momentum map of (𝒟, dη^{λ}|_{𝒟}) (cf. Appendix A). Equivalently the diagram

[figure: N; η^{λ}; U_{𝒟}; 𝔥^{∗}; μ^{λ}; L^{⊤}; 𝔱_{N}^{∗}; μ]

commutes, i.e., L^{⊤} ∘ μ^{λ} = μ ∘ η^{λ}.

Lemma 2. Let (N, 𝒟, K) be a contact manifold with an abelian local contact action. Then the following are equivalent.

(i) (N, 𝒟, K) admits a nondegenerate Levi pair in a neighbourhood of any point. (ii) (p, μ): 𝒟^{0} → N × 𝔱_{N}^{∗} is injective. (iii) q_{𝒟} ∘ κ: N × 𝔱_{N} → T N/𝒟 is surjective.

Proof. The last two conditions are equivalent because (p, μ) is the transpose of q_{𝒟} ∘ κ. Over any open neighbourhood where there is a nondegenerate Levi pair, (p, μ_{𝔤}) is an isomorphism and hence (p, μ) is injective. Conversely, if q_{𝒟} ∘ κ surjects, a complement to the kernel at a point gives a locally transversal subalgebra ι_{𝔤}: 𝔤 ↪ 𝔱_{N}, hence also a Levi pair: U_{𝒟} is open with nonempty fibres, so we can find λ ∈ 𝔤^{∗} such that η^{λ} is locally a contact form. □

Remark 2. If N is compact, {λ ∈ 𝔤^{∗} \ 0 | η^{λ} ∈ Γ(U_{𝒟})} is an open cone C_{𝔤} ⊆ 𝔤^{∗}.

Definition 6. (N, 𝒟, K) is locally Reeb type if the conditions of Lemma 2 hold.

For a vector space 𝔱 of dimension n ≥ ℓ, we denote by Gr_{ℓ}(𝔱^{∗}) the grassmannian of ℓdimensional subspaces of 𝔱^{∗}. This is a manifold of dimension ℓ(n−ℓ) with distinguished affine charts defined as follows. Let ι_{𝔤}: 𝔤 ↪ 𝔱 be an ℓ-dimensional subspace of 𝔱 and let Gr_{ℓ}^{𝔤}(𝔱^{∗}) = {ξ ∈ Gr_{ℓ}(𝔱^{∗}) | ι_{𝔤}^{⊤}(ξ) = 𝔤^{∗}} be set of subspaces ξ ≤ 𝔱^{∗} complementary to the kernel of ι_{𝔤}^{⊤}: 𝔱^{∗} → 𝔤^{∗}. Then Gr_{ℓ}^{𝔤}(𝔱^{∗}) is an open subset of Gr_{ℓ}(𝔱^{∗}), and is isomorphic to the affine space {θ ∈ Hom(𝔤^{∗}, 𝔱^{∗}) | ι_{𝔤}^{⊤} ∘ θ = Id_{𝔤^{∗}}} via the map sending θ to im θ. The inverse sends ξ ∈ Gr_{ℓ}^{𝔤}(𝔱^{∗}) to ψ_{𝔤}(ξ): 𝔤^{∗} → 𝔱^{∗} where

(5)  for λ ∈ 𝔤^{∗}, ⟨ψ_{𝔤}(ξ), λ⟩ ∈ 𝔱^{∗} is the unique point in (ι_{𝔤}^{⊤})^{−1}(λ) ∩ ξ.

Definition 7. Let 𝒦 = 𝒦^{𝔱_{N}} = im κ, E := im(p, μ) ≤ N × 𝔱_{N}^{∗}, and Θ := E^{0} = ker(q_{𝒟} ∘ κ) = κ^{−1}(𝒟) ≤ N × 𝔱_{N}. If (N, 𝒟, K) is locally Reeb type, 𝒦 ∩ 𝒟 has codimension ℓ in 𝒦 and E is a rank ℓ subbundle of N × 𝔱_{N}^{∗} (with E^{∗} ≅ (N × 𝔱_{N})/Θ ≅ T N/𝒟). Thus E defines a smooth map N → Gr_{ℓ}(𝔱_{N}^{∗}); z ↦ E_{z} which we call the grassmannian momentum map of (N, 𝒟, K), and also denote by E.

Remark 3. If (𝔤, λ) is a nondegenerate Levi pair (over an open subset of N) then E takes values in the affine chart Gr_{ℓ}^{𝔤}(𝔱_{N}^{∗}) (on that open subset) and ⟨ψ_{𝔤}∘E, λ⟩ = μ∘η^{λ} = L^{⊤}∘μ^{λ}.

Example 3. Let us revisit Examples 1.

(i) The codimension one case is particularly straightforward as Gr_{1}(𝔱_{N}^{∗}) is just the projectivization of 𝔱_{N}^{∗} and the grassmannian momentum map E: N → P(𝔱_{N}^{∗}) is just the projectivization of the momentum map μ on U_{𝒟}, i.e., E_{p(α)} = [μ(α)]. In particular, if N = 𝕊^{2m+1} = W^{×}/ℝ^{+} and 𝔱_{N} is the Lie algebra of the diagonal (m + 1)-torus with respect to a basis of W, then im μ is the union of the standard quadrant in 𝔱_{N}^{∗} and its negation. Hence im E is the projectivization of this quadrant, the standard simplex.

(ii) Let N = 𝕊^{2m_{1}+1} × ⋯ × 𝕊^{2m_{ℓ}+1} be a product of ℓ contact spheres 𝕊^{2m_{i}+1} = W_{i}^{×}/ℝ^{+}, with the product local contact action of an abelian Lie algebra 𝔱_{N} = ⨁_{i=1}^{ℓ} 𝔱_{i}, where 𝔱_{i} has dimension m_{i} + 1 and acts diagonally on W_{i} as in (i). Then, via the natural embedding of ∏_{i=1}^{ℓ} P(𝔱_{i}^{∗}) into Gr_{ℓ}(𝔱_{N}^{∗}), sending (U_{1}, … U_{ℓ}) to U_{1} ⊕ ⋯ ⊕ U_{ℓ}, the image of E is a product of simplices in Gr_{ℓ}(𝔱_{N}^{∗}).

Proposition 3. Let K: 𝔱_{N} → 𝔠𝔬𝔫(N, 𝒟) be an abelian local contact action. If K is locally Reeb type, the lift K̃ of K to 𝒟^{0} → N is transverse to the fibres, i.e., if K_{v,z} = 0 (for v ∈ 𝔱_{N}, z ∈ N) then K̃_{v} is identically zero along 𝒟_{z}^{0} = p^{−1}(z).

Proof. Since (p, μ): 𝒟^{0} → N × 𝔱_{N}^{∗} is a monomorphism of vector bundles, it is an immersion. However, for any v ∈ 𝔱_{N}, dμ(K̃_{v}) = 0 by equivariance of μ, so if K_{v,z} = p_{∗}(K̃_{v})_{z} = 0, then d(p, μ)(K̃_{v}) = 0 along p^{−1}(z), hence K̃_{v} = 0 along p^{−1}(z). □

Hence for any α ∈ 𝒟^{0}, the infinitesimal stabilizer 𝔰𝔱𝔞𝔟_{𝔱_{N}} (α) = 𝔰𝔱𝔞𝔟_{𝔱_{N}} (p(α)). Since ker κ ≤ ker q_{𝒟} ∘ κ = Θ, for any z ∈ N, 𝔰𝔱𝔞𝔟_{𝔱_{N}} (z) ≤ Θ_{z} and hence E_{z} ≤ 𝔰𝔱𝔞𝔟_{𝔱_{N}} (z)^{0}.

2.2. Symplectic slices for torus actions. Let (N, 𝒟) be a contact manifold of rank m and codimension ℓ, and let 𝕋_{N} = 𝔱_{N}/2πΛ be a (real) torus with (abelian) Lie algebra 𝔱_{N} = Λ ⊗_{ℤ} ℝ, where Λ is the lattice of circle subgroups of 𝕋_{N}.

Definition 8. A contact torus action of 𝕋_{N} on N is a local contact action K: 𝔱_{N} → 𝔠𝔬𝔫(N, 𝒟) which integrates to an effective (i.e., faithful) action of 𝕋_{N}.

We recall the construction of a symplectic slice along an 𝕋_{N}-orbit 𝕋_{N} · α for α ∈ U_{𝒟}. The isotropy representation of Stab_{𝕋_{N}} (α) on T_{α}U_{𝒟} induces an effective linear action, symplectic with respect to ω = Ω_{α}^{𝒟}, of any subtorus H ≤ Stab_{𝕋_{N}} (α). Since 𝕋_{N} · α is isotropic with respect to Ω^{𝒟}, the isotropy representation has an H-invariant filtration

(6)  0 ≤ T_{α}(𝕋_{N} · α) ≤ T_{α}(𝕋_{N} · α)^{⊥} ≤ T_{α}U_{𝒟},

where ^{⊥} denotes the orthogonal space with respect to ω = Ω_{α}^{𝒟}.

Remark 4. There is also [2, 10, 24] an H-invariant ω-compatible complex structure on T_{α}U_{𝒟} and a decomposition (T_{α}U_{𝒟}, ω) = ⨁_{i=0}^{k}(W_{i}, ω_{i}) into H-invariant ω-orthogonal symplectic subspaces, so the induced representation of H (or its Lie algebra 𝔥) on W_{i} has weight β_{i} ∈ 𝔥^{∗} for i ∈ {0, … k}. We assume β_{0} = 0 and β_{i} ≠ 0 for i > 0, so W_{0} ≤ T_{α}U_{𝒟} is the tangent space to the fixed point set of H at α. Since the H-action is effective, the weights β_{1}, … β_{k} span the weight lattice of 𝔥^{∗}, dual to the lattice Λ ∩ 𝔥. Note that T_{α}(𝕋_{N} · α) ≤ W_{0} and T_{α}(𝕋_{N} · α)^{⊥} + W_{0} = T_{α}U_{𝒟}.

When H is the identity component H_{α} of Stab_{𝕋_{N}} (α) and 𝔥_{α} := 𝔰𝔱𝔞𝔟_{𝔱_{N}} (α),

T_{α}(𝕋_{N} · α) ≅ 𝔱_{N}/𝔥_{α} and T_{α}U_{𝒟}/T_{α}(𝕋_{N} · α)^{⊥} ≅ (𝔱_{N}/𝔥_{α})^{∗} ≅ 𝔥_{α}^{0} ≤ 𝔱_{N}^{∗}.

Definition 9. The middle composition factor in (6),

W^{α} := T_{α}(𝕋_{N} · α)^{⊥}/T_{α}(𝕋_{N} · α),

is called the symplectic isotropy representation. By Remark 4, W^{α} ≅ W_{0}^{α} ⊕ ⨁_{i=1}^{k} W_{i}, where W_{0}^{α} = (T_{α}(𝕋_{N} · α)^{⊥} ∩ W_{0})/T_{α}(𝕋_{N} · α) and W_{i} are the nonzero weight spaces of H_{α} (with weights β_{i} ∈ 𝔥_{α}^{∗}). The momentum map μ_{W} : W^{α} → 𝔥_{α}^{∗} of the H_{α}-action is

(7)  μ_{W} (w_{0}, w_{1}, … w_{k}) = ∑_{i=1}^{k} |w_{i}|^{2}β_{i}

for all w_{0} ∈ W_{0}^{α} and (w_{1}, … w_{k}) ∈ ⨁_{i=1}^{k} W_{i}. The image of μ_{W} is thus the convex cone C^{α} in 𝔥_{α}^{∗} generated by the nonzero weights β_{1}, … β_{k} of W^{α}.

To construct a model for a neighbourhood of 𝕋_{N} · α, observe that T^{∗}𝕋_{N} × W^{α} ≅ 𝕋_{N} × 𝔱_{N}^{∗} × W^{α} is a symplectic manifold with commuting hamiltonian actions of 𝕋_{N} (the right action on T^{∗}𝕋_{N}) and Stab_{𝕋_{N}} (α) (the diagonal left action), where the latter has momentum map (g, ξ, v) ↦ μ_{W} (v) − ξ|_{𝔥_{α}}. By the orbit-stabilizer theorem 𝕋_{N} is a principal Stab_{𝕋_{N}} (α)-bundle over 𝕋_{N} · α ≅ 𝕋_{N}/Stab_{𝕋_{N}} (α), and if we choose a splitting χ: 𝔱_{N} → 𝔥_{α}, we may identify the symplectic quotient of T^{∗}𝕋_{N} × W^{α} by Stab_{𝕋_{N}} (α) at the zero momentum level (ξ|_{𝔥_{α}} = μ_{W} (v)) with the associated vector bundle

(8)  Y := 𝕋_{N} ×_{Stab_{𝕋_{N}} (α)} (W^{α} ⊕ 𝔥_{α}^{0})

over 𝕋_{N}/Stab_{𝕋_{N}} (α). The induced action of 𝕋_{N} is hamiltonian with momentum map

(9)  μ_{χ}([g, v + ξ]) = μ(α) + χ^{⊤}(μ_{W} (v)) + ξ,

where g ∈ 𝕋_{N}, v ∈ W^{α} and ξ ∈ 𝔥_{α}^{0}. The Symplectic Slice Theorem [3, 19, 21, 22, 24] now asserts the following.

Lemma 3. For any splitting χ, there is a 𝕋_{N}-invariant neighbourhood U of the orbit 𝕋_{N} · α ⊆ U_{𝒟} and a 𝕋_{N}-equivariant symplectomorphism Ψ_{χ} from (U, Ω) to a neighbourhood of the zero section of (Y, Ω_{χ}) such that μ|_{U} = μ_{χ} ∘ Ψ_{χ}.

Thus μ − μ(α) maps U to a neighbourhood of 0 ∈ 𝔥_{α}^{0} ⊕ χ^{⊤}(C^{α}).

2.3. Orbit stratification and convexity. The theory of effective proper abelian group actions [3, 14] implies that for H ≤ 𝕋_{N},

N_{(H)} = {z ∈ N | H = Stab_{𝕋_{N}} (z)} ≤ N^{H} = {z ∈ N | H ≤ Stab_{𝕋_{N}} (z)}

is an open submanifold of a closed submanifold of N, and if N_{(H)} is nonempty (for which H must be a closed subgroup) then it is dense in the fixed point set N^{H} of H.

Definition 10. The connected components of N_{(H)}, for H ≤ 𝕋_{N}, and their closures (which, if nonempty, are connected components of N^{H}) are called the open and closed orbit strata of (N, K). (Thus the open orbit strata partition N.) The combinatorics Φ_{N} of (N, K) is the poset of closed orbit strata, partially ordered by inclusion.

Proposition 4. Let K be a contact torus action of 𝕋_{N} on a contact manifold (N, 𝒟) of codimension ℓ with locally Reeb type, let i: N′ → N be the inclusion of a closed orbit stratum N′ ∈ Φ_{N}, let 𝒟′ = T N′ ∩ 𝒟, and let H be the kernel of the induced 𝕋_{N}-action on N′. Then (N′, 𝒟′) is a contact manifold of codimension ℓ, with U_{𝒟′} ≅ i^{∗}U_{𝒟}, and K induces a contact torus action of 𝕋_{N}/H on N′ which is locally Reeb type.

Proof. The local Reeb type condition means that 𝒟 ∩ 𝒦 has codimension ℓ in 𝒦 (the tangent distribution to the 𝕋_{N} orbits). Since i^{∗}𝒦 ≤ T N′, 𝒟′ has codimension ℓ in T N′, and i_{∗}^{⊤}: i^{∗}T^{∗}N → T^{∗}N′ restricts to an isomorphism i^{∗}𝒟^{0} → (𝒟′)^{0}, which identifies U_{𝒟′} with i^{∗}U_{𝒟} (since the induced map (𝒟′)^{0} → 𝒟^{0} pulls the tautological 1-form τ^{𝒟} back to the corresponding τ^{𝒟′}). Hence (N′, 𝒟′) is contact and the rest is immediate. □

Note that H = Stab_{𝕋_{N}} (z) for any z in the open orbit stratum corresponding to N′. If a contact torus action K of 𝕋_{N} on N has locally Reeb type, Proposition 3 implies

that the hamiltonian 𝕋_{N}-action on U_{𝒟} is transverse to the fibres of p: 𝒟^{0} → N. Thus for any subtorus H ≤ 𝕋_{N}, the fixed point set of H in U_{𝒟} is p^{−1}(N^{H}).

Recall that the critical submanifolds of a smooth function f: N → ℝ are the connected components of the zero-set of df; then f is called a Morse–Bott function if along any critical submanifold, its transverse hessian is nondegenerate.

Lemma 4. Suppose (N, 𝒟, K) has Reeb type. Then for any nondegenerate Levi pair (𝔤, λ) and any v ∈ 𝔱_{N}, f := η^{λ}(K_{v}) is a Morse–Bott function on N whose critical submanifolds all have even index.

Proof cf. [2]. Since d(η(K_{v})) = −(dη)(K_{v}, ·) and dη^{λ} has kernel 𝒦^{𝔤}, a point z ∈ N is critical for f iff K_{v,z} ∈ 𝒦^{𝔤}, in which case d(η(K_{v}))_{z} = 0. This holds iff ∃ w ∈ 𝔤 such that K_{v−w} vanishes at z, and since η^{λ}(K_{v−w}) = f −⟨w, λ⟩ has the same critical submanifolds as f, we may assume w = 0. Let Z be the critical submanifold of f containing z; since η(K_{v})_{z} = 0 and d(η(K_{v}))|_{Z} = 0, it follows that K_{v} vanishes along Z and hence Z is a component of the fixed point set N^{H} of the subtorus H of 𝕋_{N} generated by exp(K_{v}). By Remark 4, with α = η_{z}^{λ}, the tangent space T_{α}U_{𝒟} decomposes into symplectic weight spaces and the zero weight space is the tangent space to p^{−1}(Z). Hence the normal bundle to Z in N is isomorphic to the sum of the nonzero weight spaces. The transverse hessian of f at z is the hermitian form (w_{1}, … w_{k}) ↦ ∑_{i} β_{i}(v)|w_{i}|^{2}, which is nondegenerate of even index because K_{v} generates H. □

Proposition 5. Let (N, 𝒟) be a compact connected contact manifold of rank m and codimension ℓ with a contact action of a torus 𝕋_{N} with Lie algebra 𝔱_{N}. Assume that the action is Reeb type with Levi pair (𝔤, λ).

For any v_{1}, … v_{k} ∈ 𝔱_{N}, the map f: N → ℝ^{k} with components f_{i} = η^{λ}(K_{v_{i}}) satisfies (A) all fibres f^{−1}(x) are empty or connected; (B) the image f(N) is convex.

Further, if Z_{j} : j ∈ I are the connected components of the set of common critical points of f_{i}, then f(Z_{j}) = {p_{j}} is a single point and f(N) is the convex hull of {p_{j} | j ∈ I}.

The proof is essentially identical to the proof of the symplectic convexity theorem by M. Atiyah [2] (see also [15]): the key ingredient is Lemma 4, which makes essential use of Condition 1 (the transversality property of the subalgebra 𝔤). Without this assumption, convexity may fail even in codimension one [20], although in that case, the conditions needed for convexity are well understood [9, 21, 22, 23]. A similar convexity result for transverse symplectic foliations has been obtained recently by Ishida [17].

3. Toric contact manifolds

3.1. The quotient manifold with corners. If (N, 𝒟) is contact of rank m, then rank 𝒦 ∩ 𝒟 ≤ m, and hence dim 𝔱_{N} ≤ m + ℓ. On any open set where rank 𝒦 = m + ℓ, (N, 𝒟, K) is locally Reeb type.

Definition 11. A compact contact manifold (N, 𝒟) of rank m and codimension ℓ with a contact torus action K of 𝕋_{N} is toric if it is locally Reeb type and dim 𝕋_{N} = m + ℓ.

When (N, 𝒟, K) is toric, U_{𝒟} is a (noncompact) toric symplectic manifold under the lifted hamiltonian 𝕋_{N}-action K̃ of Observation 1.

Lemma 5. Let (N, 𝒟, K) be a toric contact manifold of codimension ℓ. Then its closed orbit strata are toric contact of codimension ℓ, and for any z ∈ N, its stabilizer H := Stab_{𝕋_{N}} (z) is connected (i.e., a subtorus). For any α ∈ p^{−1}(z), H = Stab_{𝕋_{N}} (α) and the normal bundle in N to the fixed point set N^{H} is isomorphic to the symplectic isotropy representation of H, which is a direct sum of 2-dimensional symplectic subrepresentations whose weights form a basis for the weight lattice of 𝔰𝔱𝔞𝔟_{𝔱_{N}} (z)^{∗}.

Proof cf. [10, 24]. Let m be the rank of N, z ∈ N, α ∈ p^{−1}(z) and dim 𝔰𝔱𝔞𝔟_{𝔱_{N}} (z) = k. Thus 𝕋_{N}/H has dimension m + ℓ − k, U_{𝒟} has dimension 2(m + ℓ) and the symplectic isotropy representation W^{α} has dimension 2(m + ℓ) − 2(m + ℓ − k) = 2k. The nonzero weights span 𝔰𝔱𝔞𝔟_{𝔱_{N}} (α)^{∗}, so they are linearly independent with 2-dimensional weight spaces, and W_{0}^{α} = 0. Hence the open orbit stratum through z has dimension 2(m−k)+ℓ and its closure is toric by Proposition 4. Since maximal tori in Sp(2k) are maximal closed abelian subgroups, H is connected, and is therefore equal to Stab_{𝕋_{N}} (α). □

Thus inverse images under p of orbit strata in N are orbit strata in U_{𝒟}, and Lemma 3 specializes as follows.

Proposition 6. For any α ∈ U_{𝒟}, there is a 𝕋_{N}-invariant neighbourhood U of the orbit 𝕋_{N} · α ⊆ U_{𝒟} on which image of the momentum map μ|_{U} − μ(α) is a neighbourhood of 0 in the product of a quadrant with 𝔰𝔱𝔞𝔟_{𝔱_{N}} (α)^{0} in 𝔱_{N}^{∗}, it is a submersion over the interior of this cone, and the fibres (in U) are 𝕋_{N}-orbits.

Karshon and Lerman [19] observe that these charts make U_{𝒟}/𝕋_{N} into a manifold with corners. Recall (see e.g. [18]) that an m-dimensional manifold with corners is a Hausdorff topological space Q equipped with a subsheaf of the continuous functions which is locally isomorphic (as a space with a sheaf of rings) to a quadrant in a (finite dimensional, real) vector space, hence also to [0, ∞)^{m} ⊆ ℝ^{m}. Here, a function f on a subset B of vector space is smooth iff every v ∈ B has an open neighbourhood U such that f|_{B∩U} is the restriction of a smooth function on U. Any x ∈ Q has a neighbourhood isomorphic to a neighbourhood of the origin in the product of a quadrant and a vector space, i.e., to [0, ∞)^{k} × ℝ^{m−k} ⊆ ℝ^{m} for a well-defined k ∈ ℕ called the depth of x. The subspace Q_{k} of all points with depth k is a smooth manifold of dimension m − k. The closures in Q of the connected components of Q_{k} are called the (m − k)-dimensional faces of Q. The combinatorics Φ_{Q} of Q is the poset of all faces, ordered by inclusion. Faces of depth 1 are called facets.

In our situation, the momentum map μ also maps the fibre p^{−1}(y), for y ∈ N, to the linear subspace E_{y} ≤ 𝔰𝔱𝔞𝔟_{𝔱_{N}} (y)^{0} ≤ 𝔱_{N}^{∗}. Since Stab_{𝔱_{N}} (y) fixes this fibre pointwise, Proposition 6 implies that μ(U) is foliated by its intersection with the linear subspaces E_{p(β)} : β ∈ U. On the other hand μ(U) is an open neighbourhood of z = μ(α), and shrinking U if necessary, we may assume that the foliation of μ(U) is regular, so that the leaf space is isomorphic to an open neighbourhood of zero in the product of a quadrant with 𝔰𝔱𝔞𝔟_{𝔱_{N}} (α)^{0}/E_{z}.

Corollary 1. The orbit space N/𝕋_{N} has the structure of a manifold with corners such that the quotient map induces a poset isomorphism from Φ_{N} to Φ_{N/𝕋_{N}}.

Remark 5. We shall also need to know that the symplectic slices in Lemma 3 may be chosen compatible with the foliation of U_{𝒟} over M. For this recall that the Symplectic Slice Theorem proceeds by constructing a differentiable slice to 𝕋_{N} · α in U_{𝒟} and then using the Equivariant Relative Darboux Theorem to standardize the symplectic form on Y = 𝕋_{N} ×_{Stab_{𝕋_{N}} (α)} (W^{α} ⊕ 𝔥_{α}^{0}). Now although the 𝕋_{N} orbit of the fibre over z ∈ N is μ_{χ}^{−1}(E_{z}), with E_{z} ≤ 𝔥_{α}^{0}, there is no reason to suppose a priori that the foliation of this orbit over M is compatible with the fibration over 𝕋_{N} · α ≅ 𝕋_{N}/Stab_{𝕋_{N}} (α).

To establish this, we use the transitive action of equivariant symplectomorphisms on angular coordinates. Since we have not found a local proof of this in the literature, we sketch it here, using arguments from [26]. First, we recall from Lemma 5 that W^{α} is a direct sum of 2-dimensional weight spaces W_{1}, … W_{k} where 0 ≤ k = dim 𝔥_{α} ≤ m. Compatible complex structures on each W_{j} yield 𝕋_{N}-invariant angular coordinates θ_{1}, … θ_{k} on Y^{0} = {[g, w_{1} + ⋯ w_{k} + ξ] ∈ Y | w_{i} ≠ 0}. The level surfaces of θ_{1}, … θ_{k} form a 𝕋_{N}-invariant coisotropic foliation of Y^{0}. Extending by angular coordinates on 𝕋_{N} · α (pulled back to Y^{0}, we obtain an 𝕋_{N}-invariant lagrangian foliation F_{0} of Y^{0}, singular along the special orbits w_{i} = 0. The fibres of U_{𝒟} → M need not lie in these leaves, but they do lie in the coisotropic foliation. Thus the coisotropic foliation contains an 𝕋_{N}- invariant lagrangian foliation F_{1} containing the fibres of U_{𝒟} and tangent to F_{0} along the singular orbits. On a neighbourhood of the zero section of Y, there is therefore an 𝕋_{N}-equivariant diffeomorphism, equal to the identity along the zero section, pulling back F_{1} to F_{0}. The pullback of Ω_{χ} is thus a 𝕋_{N}-invariant symplectic form Ω_{1} agreeing with Ω_{χ} along the zero section, and F_{0} is lagrangian for both Ω_{1} and Ω_{χ}.

We now apply the Equivariant Moser Lemma, using the observation that for equivariant differential forms on the total space of an equivariant vector bundle, the homotopy operator can be defined by

I(β) = ∫_{0}^{1} σ_{t}^{∗}(ι_{X_{t}}β)dt,

where σ_{t} denotes fibrewise scalar-multiplication by t and X_{t} is its derivative (the radial vector field). Now, following [26], we set β = Ω_{1} − Ω_{χ} and Ω_{t} = Ω_{χ} + tβ, noting that Ω_{t} is symplectic for t ∈ [0, 1] and that β = dα with α = I(β), so we may define Y_{t} by α = −ι_{Y_{t}}Ω_{t}; the local flow ϕ_{t} then satisfies that ϕ_{t}^{∗}Ω_{t} is independent of t, so that ϕ_{1}^{∗}Ω_{1} = ϕ_{0}^{∗}Ω_{0} = Ω_{χ}. However for any ξ tangent to F_{0},

Ω_{t}(Y_{t}, ξ) = −α(ξ) = − ∫_{0}^{1} σ_{t}^{∗}(Ω_{1} − Ω_{χ})(X_{t}, ξ) = 0

since X_{t} is radial, hence also tangent to F_{0}, and F_{0} is isotropic with respect to both Ω_{1} and Ω_{χ}. Indeed it is maximal isotropic, and so it follows that Y_{t} is tangent to F_{0} for all t. Hence ϕ_{t} preserves F_{0}. We now have a symplectic slice for U_{𝒟} along 𝕋_{N} · α in which the fibre over z is contained in F_{0} ∩ μ_{χ}^{−1}(E_{z}), hence in a fibre of Y over 𝕋_{N} · α.

3.2. The toric symplectic cone and Levi pairs. We combine the local theory of the previous section with Proposition 5 to establish global properties of toric contact manifolds of Reeb type. We use the setting of Strategem 1 in which pairs (𝔤, λ) are given by linear maps L: 𝔱_{N} → 𝔥 with d ∘ L = u: 𝔱_{N} → 𝔨. Recall that a convex polytope Δ in the m-dimensional affine space 𝒜 = (ε^{⊤})^{−1}(1) ⊆ 𝔥^{∗} is a subset of the form

Δ := {ξ ∈ 𝒜 | ∀s ∈ 𝒮, L_{s}(ξ) ≥ 0}

where 𝒮 is a finite set, and L_{s} ∈ 𝔥 (an affine function on 𝒜) for each s ∈ 𝒮.

Definition 12. Given Δ ⊆ 𝒜 as above, and ξ ∈ 𝒜, let S_{ξ} = {s ∈ 𝒮 | L_{s}(ξ) = 0} and Φ_{Δ} = {S_{ξ} ⊆ 𝒮 | ξ ∈ Δ} with the induced partial ordering. We assume ∅ ∈ Φ_{Δ} (so Δ ⊆ 𝒜 has nonempty interior) and all singletons {s} : s ∈ 𝒮 belong to Φ_{Δ} (otherwise we may discard s without changing Δ). The poset Φ_{Δ} is then called the combinatorics of Δ; it is isomorphic to the poset of closed faces of F of Δ via the map sending S ∈ Φ_{Δ} to F_{S} = {ξ ∈ Δ | S ⊆ S_{ξ}} = {ξ ∈ Δ | ∀s ∈ S, L_{s}(ξ) = 0}—in particular S_{∅} = Δ, and s ∈ 𝒮 may be identified with the facet (codimension one face) F_{s} := F_{{s}}. Any closed face is the intersection of the facets containing it: F_{S} = ⋂_{s∈S} F_{s}. We say Δ is simple if every vertex is m-valent (or equivalently Φ_{Δ} is a simplicial set: if S′ ⊆ S ∈ Φ_{Δ} then S′ ∈ Φ_{Δ}). In this case Δ is a manifold with corners.

Recall also from Definition 10 that the combinatorics of N is the poset Φ_{N} of closed orbit strata of N. The analogue of a facet is a closed orbit stratum stabilized by a circle, so we take 𝒮 to be (in bijection with) this subset of Φ_{N}, i.e., for each s ∈ 𝒮, N_{s} ∈ Φ_{N} is a connected component of N^{H_{s}} where H_{s} = {exp(te_{s}) : t ∈ ℝ} ≤ 𝕋_{N} is a circle with primitive generators ±e_{s} ∈ Λ ⊆ 𝔱_{N}.

Definition 13. Let (N, 𝒟, K) be a connected toric contact manifold. For any nondegenerate Levi pair (𝔤, λ), we denote by U^{(𝔤,λ)} the connected component of U_{𝒟} containing η^{λ}(N). We say U ⊆ U_{𝒟} is a Reeb component of U_{𝒟} if U = U^{(𝔤,λ)} for some nondegenerate Levi pair (𝔤, λ).

Clearly a Reeb component exists if and only if the contact action has Reeb type.

Theorem 1. Suppose (N, 𝒟, K) is a (compact, connected) toric contact manifold, and let N_{s} : s ∈ 𝒮 index the closed orbit strata stabilized by a circle.

(i) For any Reeb component U of U_{𝒟}, the signs of the primitive generators e_{s} may be chosen uniquely so that for all s ∈ 𝒮, ⟨μ, e_{s}⟩ ≥ 0 on U. Furthermore, for any α ∈ U, the e_{s} with ⟨μ(α), e_{s}⟩ = 0 are linearly independent elements of E_{p(α)}^{0} ≤ 𝔱_{N}.

(ii) If U = U^{(𝔤,λ)} for a nondegenerate Levi pair (𝔤, λ), then the image of the horizontal momentum map μ^{λ}: N → 𝒜 ⊆ 𝔥^{∗} is the compact simple convex polytope Δ = Δ_{𝔤,λ} in 𝒜 defined by the affine functions L_{s} := L(e_{s}). Furthermore the fibres of μ^{λ} are 𝕋_{N}-orbits, and if μ^{λ}(z) is in the interior of a face F then the annihilator of the tangent space to F at z is u(𝔰𝔱𝔞𝔟_{𝔱_{N}} (z)) ≤ 𝔨 ≅ T_{μ^{λ}(z)}^{∗}𝒜.

In particular, μ^{λ} induces a poset isomorphism of Φ_{N} with Φ_{Δ}, sending N_{s} to F_{s} for all s ∈ 𝒮 (so both Φ_{N} and Φ_{Δ} are simplicial sets), and the restriction of μ^{λ} to an orbit stratum is a submersion over the interior of the corresponding face.

Proof. First note that by Lemma 5, the orbit strata have connected stabilizers, so N_{s} : s ∈ 𝒮 are the maximal proper strata in Φ_{N}; furthermore ⟨μ(N_{s}), e_{s}⟩ = 0.

Proposition 6 shows that the connected components of the fibres of μ: U_{𝒟} → 𝔱_{N}^{∗} are 𝕋_{N} orbits, and if α ∈ N_{s} with stabilizer <exp(te_{s})>, then μ maps a neighbourhood of 𝕋_{N} · α to a half space bounded by ⟨μ, e_{s}⟩ = 0.

Given a nondegenerate Levi pair (𝔤, λ), Proposition 5 implies, by choosing a basis for a complement to the image of ker λ in 𝔱_{N}, that the image Δ of μ^{λ} is the convex hull of the points μ^{λ}(z) where 𝒦_{z}^{𝔤} = 𝒦_{z}, and that the fibres of μ^{λ} are connected. Since L^{⊤} ∘μ^{λ} = μ∘η^{λ} is the restriction of μ to η^{λ}(N) = μ_{𝔤}^{−1}(λ), the fibres of μ^{λ} are 𝕋_{N}-orbits and we may choose the signs of e_{s} so that L_{s} := L(e_{s}) ≥ 0 on Δ.

Since μ maps the fibres of U_{𝒟} ⊆ 𝒟^{0} → N linearly onto 𝔱_{N}^{∗} (sending 𝒟_{z}^{0} to E_{z} ≤ 𝔱_{N}^{∗}) these signs ensure ⟨μ, e_{s}⟩ ≥ 0 on the Reeb component containing η^{λ}(N), proving (i).

The explicit local description of Proposition 6 now shows that Δ is the compact convex polytope on which L_{s} ≥ 0 and that this polytope is simple with the stated face and submersive properties, proving (ii). The last part now follows. □

We may assume, if necessary, that the primitive generators e_{s} : s ∈ 𝒮 span 𝔱_{N}: otherwise, there is a nontrivial subtorus H of 𝕋_{N} acting freely on N and transversely to 𝒟. Thus N is a principal H-bundle over a toric contact manifold (N/H, 𝒟/H). Conversely, any principal torus bundle with connection over a toric contact manifold is toric contact as well. There is no reason to suppose that e_{s} : s ∈ 𝒮 are linearly dependent, or even distinct, but the case that they form a basis is of particular interest.

Example 4. Theorem 1 associates to every Reeb component of a toric contact manifold, a simple convex polytope

Δ := {ξ ∈ 𝒜 = (ε^{⊤})^{−1}(1) ⊆ 𝔥^{∗} | ∀s ∈ 𝒮, L_{s}(ξ) ≥ 0}

which is labelled in the sense that the affine functions L_{s} ∈ 𝔥, parametrized by the set 𝒮 of facets of Δ, are given, whereas in general they are only determined by Δ up to rescaling by a positive real number. Let ℤ_{𝒮} be the free abelian group generated by 𝒮, let 𝔱_{𝒮} = ℤ_{𝒮} ⊗_{ℤ} ℝ and ℂ_{𝒮} = ℤ_{𝒮} ⊗_{ℤ} ℂ be the corresponding free vector spaces over ℝ and ℂ, and let 𝕋_{𝒮} = 𝔱_{𝒮}/2πℤ_{𝒮}. Denote the generators of ℤ_{𝒮} ⊆ 𝔱_{𝒮} ⊆ ℂ_{𝒮} by e_{s} : s ∈ 𝒮. The affine functions L_{s} are encoded by the induced linear map L: 𝔱_{𝒮} → 𝔥 with L(e_{s}) = L_{s}.

Conversely, given a simple labelled polytope (Δ, L), we now exhibit many toric contact manifolds with a Reeb component inducing these data. For this, we first observe that 𝕋_{𝒮} and 𝕋_{𝒮}^{ℂ} = ℂ_{𝒮}/2πℤ_{𝒮} ≅ ℂ_{𝒮}^{×} act diagonally on ℂ_{𝒮}, via [∑_{s} t_{s}e_{s}] · (∑_{s} z_{s}e_{s}) = ∑_{s} exp(it_{s})z_{s}e_{s}. The action of 𝕋_{𝒮} on ℂ_{𝒮} is hamiltonian with respect to the standard symplectic form ω_{𝒮} on 𝒮 and has a momentum map σ: ℂ_{𝒮} → 𝔱_{𝒮}^{∗} defined by

(10)  ⟨σ(z), e_{s}⟩ = σ_{s}(z) = \frac{1}{2}|z_{s}|^{2},

where z_{s}: ℂ_{𝒮} → ℂ denote the standard (linear) complex coordinates on ℂ_{𝒮}. We define 𝔤 = ker(d ∘ L: 𝔱_{𝒮} → 𝔨) and let ι_{𝔤}: 𝔤 ↪ 𝔱_{𝒮} denote the inclusion. Since L ∘ ι_{𝔤} takes values in ker d there is a unique λ ∈ 𝔤^{∗} with L ∘ ι_{𝔤} = ε ∘ λ. We now define

N := (F ∘ σ)^{−1}(0) with F = ι_{𝔤}^{⊤} − λ: 𝔱_{𝒮}^{∗} → 𝔤^{∗}.

Since Δ is simple, λ is a regular value of ι_{𝔤}^{⊤} ∘σ which is a momentum map for the action of 𝔤. Thus we get that 𝒦^{𝔤} has full rank everywhere on N. Denote by τ_{𝒮}, a primitive of ω_{𝒮} such that τ_{𝒮}(K_{w}) = ⟨σ, w⟩ for every w ∈ 𝔱_{𝒮} (for e.g. take τ_{𝒮} = ∑_{s∈𝒮} d^{c}σ_{s}). By definition, N = {z ∈ ℂ_{𝒮} | ⟨σ_{z}, w⟩ = ⟨λ, w⟩ ∀w ∈ 𝔤}. Hence, on N, for any w ∈ 𝔤 we have τ_{𝒮}(K_{w}) = ⟨λ, w⟩ and 𝒦^{ker λ} ⊆ ker τ_{𝒮}. Observe that the convexity of Δ implies that 𝔤 ∩ (ℝ_{>0})_{𝒮} is not empty. In particular, there exists w ∈ 𝔤 such that τ_{𝒮}(K_{w}) = ∑_{s∈𝒮} w_{s}|z_{s}|^{2} ≠ 0 which ensures that the rank of (ker τ_{𝒮}) ∩ T N is 2m + ℓ − 1. Any complementary bundle 𝒟 of 𝒦^{ker λ} in (ker τ_{𝒮}) ∩ T N is contact and admits (𝔤, λ) as a Levi pair. Indeed, for any local sections X, Y ∈ Γ(𝒟), where 𝒟 ⊆ ker τ_{𝒮} we have

ω(X, Y) = dτ_{𝒮}(X, Y) = −τ_{𝒮}([X, Y]),

whereas ω is nondegenerate on T N/𝒦^{𝔤} (by symplectic reduction) and thus on 𝒟.

If 𝒟 is 𝕋_{𝒮}-invariant we get a contact toric manifold (N, 𝒟, K) together with a Levi pair (𝔤, λ) for which the pull back of τ_{𝒮} is the contact form η^{λ}. Recall that the momentum map μ of 𝕋_{𝒮} at α ∈ 𝒟^{0} is defined by ⟨μ_{α}, v⟩ = α(K_{v}) for all v ∈ 𝔱_{𝒮}^{∗}. Thus ⟨L^{⊤} ∘ μ^{λ}, v⟩ = ⟨μ ∘ τ_{𝒮}, v⟩ = ∑_{s∈𝒮} v_{s}|z_{s}|^{2} and L^{⊤}(im μ^{λ}) = (ι_{𝔤}^{⊤})^{−1}(λ) ∩ im σ coincides with L^{⊤}(Δ). Such contact distributions exist—for instance 𝒟 = (ker τ_{𝒮}) ∩ ( ⋂_{w∈ker λ} ker(d^{c}⟨σ, w⟩)) works.

In this example, there is a close connection with the Delzant–Lerman–Tolman construction of toric symplectic orbifolds. More generally, we have the following.

Remark 6. If (𝔤, λ) be a nondegenerate Levi pair for (N, 𝒟, K) then there is a subtorus G ≤ 𝕋_{N} with Lie algebra 𝔤 if and only if the lattice of circle subgroups in 𝔱_{N} is mapped, via u, to a lattice Λ in 𝔨 containing span_{ℤ}{u_{s} = u(e_{s}) | s ∈ 𝒮}. In this case, the quotient (N/G, dη^{λ}|_{𝒟}, 𝕋_{N}/G) is a toric symplectic orbifold, and μ^{λ} descends to a momentum map on N/G. If [z] ∈ N/G such that μ^{λ}(z) ∈ F_{S} for some S ⊆ 𝒮 in Φ_{Δ}, then the stabilizer of z in 𝕋_{N} = 𝔱_{N}/2πΛ is connected and 𝔰𝔱𝔞𝔟_{𝔱_{N}} (z) = span_{ℝ}{e_{s} | s ∈ S}. An element [v] = [∑_{s∈S} t_{s}e_{s}] of the stabilizer of z in 𝕋_{N} is in G iff ∑_{s∈S} t_{s}u_{s} ∈ Λ, and is zero iff t_{s} ∈ ℤ for all s ∈ S. Hence the orbifold structure group of [z] (the stabilizer of z in G) is isomorphic to (Λ ∩ span_{ℝ}{u_{s} | s ∈ S})/span_{ℤ}{u_{s} | s ∈ S}.

The Delzant–Lerman–Tolman correspondence then associates the symplectic toric orbifold N/G with (Δ, L). However, if 𝔤 is not the Lie algebra of a closed subtorus of 𝕋_{N}, the image of Λ, the lattice of circle subgroups in 𝔱_{N}, is not sent to a lattice in 𝔨 via u and there is no preferred lattice in 𝔨. In particular, there is no reason for the polytope Δ to be rational with respect to any lattice.

3.3. The grassmannian image. Let (N, 𝒟, K) be a (compact, connected) codimension ℓ toric contact manifold of Reeb type, and let N/𝕋_{N}. Corollary 1 and Theorem 1 imply that N/𝕋_{N} is a compact connected manifold with corners of dimension m, diffeomorphic to a simple polytope Δ. However, Example 4 shows that even the labelled polytope (Δ, L) is insufficient data to recover (N, 𝒟, K).

To remedy this we consider instead the grassmannian momentum map

E: N → Gr_{ℓ}(𝔱_{N}^{∗})
z ↦ E_{z} = μ(𝒟_{z}^{0})

introduced in Definition 7: recall that if (N, 𝒟, K) is locally Reeb type, then μ maps the fibres of 𝒟^{0} to ℓ-dimensional linear subspaces of 𝔱_{N}^{∗}. If (𝔤, λ) is a nondegenerate Levi pair, then by Remark 3, E takes values in Gr_{ℓ}(𝔱_{N}^{∗}) and ⟨ψ_{𝔤} ∘ E, λ⟩ = L^{⊤} ∘ μ^{λ}. Since the latter is an orbit map and E is 𝕋_{N}-invariant, E induces an embedding of N/𝕋_{N} into Gr_{ℓ}(𝔱_{N}^{∗}). This is the grassmannian image that we will use to characterize (N, 𝒟, K).

Definition 14. Let 𝔱 be the Lie algebra of a compact (m + ℓ)-torus 𝕋 = 𝔱/2πΛ. We say that a manifold with corners Ξ ⊆ Gr_{ℓ}(𝔱^{∗}), with facets Ξ_{s} : s ∈ 𝒮, is

• polyhedral, if for all s ∈ 𝒮, there is a 1-dimensional subspace E_{s} ≤ 𝔱 such that the facet Ξ_{s} lies in the subgrassmannian Gr_{ℓ}(E_{s}^{0}) ⊆ Gr_{ℓ}(𝔱^{∗});

• labelled by e = (e_{s})_{s∈𝒮} if for all s ∈ 𝒮, e_{s} ∈ 𝔱 \ 0 and Ξ_{s} ⊆ Gr_{ℓ}(e_{s}^{0}). A labelling e of Ξ determines a (possibly empty) cone

(11)  C_{e} := {x ∈ 𝔱^{∗} | ∀ s ∈ 𝒮, ⟨x, e_{s}⟩ ≥ 0 and S_{x} ∈ Φ_{Ξ}},

where S_{x} = {s ∈ 𝒮 | ⟨x, e_{s}⟩ = 0}. A labelled polyhedral manifold with corners (Ξ, e) is • rational if e_{s} ∈ Λ for all s ∈ 𝒮;

• Delzant if for each S ∈ Φ_{Ξ}, (e_{s})_{s∈S} is a ℤ-basis for Λ ∩ span_{ℝ}{e_{s} | s ∈ S}.

• of Reeb type if there is an ℓ-dimensional subspace ι_{𝔤}: 𝔤 ↪ 𝔱 and λ ∈ 𝔤^{∗} such that Ξ ⊆ Gr_{ℓ}^{𝔤}(𝔱^{∗}) and ⟨ψ_{𝔤}, λ⟩: Gr_{ℓ}^{𝔤}(𝔱^{∗}) → 𝔱^{∗} maps Ξ bijectively to the convex polytope

(12)  Δ_{𝔤,λ} := (ι_{𝔤}^{⊤})^{−1}(λ) ∩ C_{e}.

Note that the Delzant and Reeb type conditions for (𝔤, λ) determine the labelling e.

Corollary 2. The grassmannian image Ξ of a compact toric contact manifold of Reeb type (N, 𝒟, K) is polyhedral in Gr_{ℓ}(𝔱_{N}^{∗}), and any Reeb component U ⊆ U_{𝒟} induces a labelling e such that (Ξ, e) is Delzant of Reeb type.

We would like to show that the grassmannian image classifies toric contact manifolds of Reeb type, following a well-known line of argument [19, 24] in the symplectic and codimension one case, which builds on [16]. However, in higher codimension, local invariants obstruct a straightforward description of 𝔠𝔬𝔫(N, 𝒟), and so the method yields less explicit conclusions. Let Con_{0}^{𝕋}(𝒟) be the sheaf over N/𝕋_{N} of 𝕋_{N}-equivariant contactomorphisms of N preserving the orbits (i.e., inducing the identity on N/𝕋_{N}) and let 𝔠𝔬𝔫^{𝕋}(𝒟) be the sheaf over N/𝕋_{N} of 𝕋_{N}-invariant contact vector fields on (N, 𝒟).

Note that if a contact vector field X is 𝕋_{N}-invariant, then so is the hamiltonian f_{X} for its lift X̃ to U_{𝒟}, i.e., Ω^{𝒟}(X̃, K̃_{v}) = −df_{X}(K̃_{v}) = 0 for any v ∈ 𝔱_{N}. Since the latter span a lagrangian subspace on a dense open set, X̃_{α} = K̃_{ṽ(α),α} for some 𝔱_{N}-valued function of the form ṽ = p^{∗}v (since X̃ is 𝔱_{N}-invariant). Hence for any z ∈ N, X_{z} = K_{v(z),z} is tangent to the 𝕋_{N}-orbit through z.

Lemma 6. There is an exact sequence of sheaves over N/𝕋_{N}:

(13)  0 → \underline{2πΛ} → 𝔠𝔬𝔫^{𝕋}(𝒟) → Con_{0}^{𝕋}(𝒟) → 0,

where the first nontrivial map is the inclusion of the locally constant sheaf associated to 2πΛ ⊆ 𝔱_{N} into 𝔠𝔬𝔫^{𝕋}(𝒟) and the second is the time 1 flow of a contact vector field.

Proof. Exactness at the first two steps is straightforward, following [19, 24]; surjectivity of the second map is less so. However, given a contactomorphism ϕ in Con_{0}^{𝕋}(U, 𝒟) over a 𝕋_{N}-invariant open subset U of N, the lift ϕ̃ to p^{−1}(U) preserves τ^{𝒟}, hence the momentum map μ, hence the 𝕋_{N}-orbits. Thus [16, Theorem 3.1] shows that for U sufficiently small, ϕ̃(α) = exp(ṽ(α)) · α for a smooth 𝕋_{N}-invariant function ṽ: p^{−1}(U) → 𝔱_{N}. Since p(ϕ̃(α)) = ϕ(p(α)), exp(ṽ(α)) · p(α) = ϕ(p(α)) and so we may assume ṽ = p^{∗}(v) for a smooth 𝕋_{N}-invariant function v: U → 𝔱_{N}. Thus ϕ̃ = ψ_{1}, where ψ_{t} is the flow of the projectable vector field X̃_{α} = K̃_{v(p(α)),α} (this flow exists because X̃ is tangent to the compact 𝕋_{N} orbits).

We now show that X̃ is the lift to p^{−1}(U) of a infinitesimal contactomorphism. First note that since the 𝕋_{N}-orbits are isotropic, Ω^{𝒟}(X̃, K_{w}) = 0 for all w ∈ 𝔱_{N} and hence LX̃τ^{𝒟} = ιX̃Ω^{𝒟} + dιX̃τ^{𝒟} is a basic 1-form with respect to the 𝕋_{N} action. Since ψ_{1} = ψ̃ and ψ_{0} = Id preserve τ^{𝒟},

0 = ψ_{1}^{∗}τ^{𝒟} − ψ_{0}^{∗}τ^{𝒟} = ∫_{0}^{1} \frac{d}{dt}ψ_{t}^{∗}τ^{𝒟} dt = ∫_{0}^{1} ψ_{t}^{∗}(LX̃τ^{𝒟}) dt = ∫_{0}^{1} LX̃τ^{𝒟} dt = LX̃τ^{𝒟},

where in the penultimate step, we have used that ψ_{t} induces the identity on the p^{−1}(U)/𝕋_{N}, and so preserves basic 1-forms. □

Identifying N/𝕋_{N} with Ξ via the grassmannian momentum map, we may view (13) as an exact sequence of sheaves on Ξ.

Proposition 7. Let (N, 𝒟, K) be a toric contact manifold of Reeb type with grassmannian image Ξ. Then toric contact manifolds of Reeb type with the same grassmannian image are parametrized up to isomorphism by H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)).

Proof. Suppose N′ has the same grassmannian image. This induces a diffeomorphism between N/𝕋 and N′/𝕋. It follows from Remark 5 that N and N′ are locally isomorphic by a fibre preserving contactomorphism. Thus a standard argument [16, 19, 24] shows that N′ determines and is determined up to isomorphism by an element of H^{1}(Ξ, Con_{0}^{𝕋}(𝒟)). However, since N has Reeb type, Ξ is diffeomorphic to a simple convex polytope, and hence is contractible, so H^{i}(Ξ, \underline{2πΛ}) = 0 for i ≥ 1. Thus by the long exact sequence associated to (13), H^{1}(Ξ, Con_{0}^{𝕋}(𝒟)) ≅ H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)). □

In order to understand the sheaf 𝔠𝔬𝔫^{𝕋}(𝒟), it is convenient to introduce a transversal subalgebra 𝔤 ≤ 𝔱_{N} which is the Lie algebra of a subtorus G ≤ 𝕋_{N} and has C_{𝔤} ⊆ 𝔤^{∗} nonempty. Thus N is a principal G-bundle over a compact orbifold M = N/G, and 𝒟 defines a 𝕋_{M} := 𝕋_{N}/G-invariant principal G-connection η on N → M, cf. Example 2.

Definition 15. A toric ℓ-symplectic manifold (or orbifold) is a 2m-manifold (or orbifold) M with an effective action of an m-torus 𝕋_{M} together with a 𝕋_{M}-invariant principal G-bundle π: N → M with connection η, for G an ℓ-torus, such that for some λ ∈ 𝔤^{∗}, ⟨ω, λ⟩ is nondegenerate, where ω ∈ Ω^{2}(M, 𝔤) is the curvature of η.

In this setting, (N, 𝒟 = ker η) is a toric contact manifold (or orbifold) of dimension ℓ under an extension 𝕋_{N} of 𝕋_{M} by G, the induced action of 𝔤 is transversal, and ⟨Ω, λ⟩ is nondegenerate (hence a symplectic form on M) iff λ ∈ C_{𝔤}, i.e., (𝔤, λ) is a Levi pair.

Proposition 8. Let M be a toric ℓ-symplectic manifold with associated toric contact manifold (N, 𝒟, 𝕋_{N}). For Z ∈ 𝔠𝔬𝔫^{𝕋}(𝒟), viewed as a sheaf on N, we may write η(Z) = π^{∗}f_{Z} and this defines an isomorphism between 𝔠𝔬𝔫^{𝕋}(𝒟) and the sheaf of 𝕋_{M}-invariant 𝔤-valued functions f on M such that df = −ι_{X}ω for some vector field X.

Proof. For Z ∈ 𝔠𝔬𝔫^{𝕋}(𝒟), η(Z) is G-invariant, hence of the form π^{∗}f_{Z} for some 𝔤-valued function f_{Z} on M, which is 𝕋_{M}-invariant because η(Z) is 𝕋_{N}-invariant. However, since Z is contact 0 = L_{Z}η = ι_{Z}dη + π^{∗}df_{Z}, so if f_{Z} = 0 on an open subset of M, ι_{Z}dη = 0 on its inverse image in N, and hence Z = 0 (since dη is nondegenerate on the kernel of η). Since ι_{Z}dη is G-invariant, and vanishes on generators of the G-action, it is basic. However dη = π^{∗}ω, so Z is projectable, to a vector field X with ι_{X}ω = −df_{Z}. Conversely, for any 𝕋_{M}-invariant 𝔤-valued functions f with df = −ι_{X}ω for some vector field X, we may take Z = X^{H} +⟨π^{∗}f, K^{𝔤}⟩, where X^{H} is the horizontal lift of X to 𝒟 and K^{𝔤}: 𝔤 → 𝔠𝔬𝔫^{𝕋}(𝒟) generate the infinitesimal action of 𝔤 on N. Then Z is 𝕋_{N}-invariant with η(Z) = π^{∗}f, and L_{Z}η = ι_{Z}dη + π^{∗}df_{Z} = π^{∗}(ι_{X}ω + df) = 0, so Z is contact. □

Note that the vector field X here is ℓ-hamiltonian, in the sense that it has a hamiltonian with respect to every (nondegenerate) component of ω. When ℓ = 1, or more generally, when the Levi form has rank one image (the ℓ-contact manifolds of [6, 11]— cf. Examples 1 (iii)), this places no condition on f, so 𝔠𝔬𝔫^{𝕋}(𝒟) is isomorphic to the sheaf of functions on M/𝕋_{M} = Ξ, which is a fine sheaf, so H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)) = 0.

We also have the following generalization.

Proposition 9. If N is a product of codimension one contact manifolds of Reeb type, then H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)) = 0.

Proof. We take G to be the product of transverse actions on each factor, so that M = N/G is a product of symplectic manifolds (M_{i}, ω_{i}) : i ∈ {1, … ℓ} and ω = ω_{1}⊕⋯⊕ω_{ℓ} ∈ Ω^{2}(M, ℝ^{ℓ}). Now for any vector field X on M, −ι_{X}ω = (α_{1}, … α_{ℓ}) where α_{i} is a section of the pullback of T^{∗}M_{i} to M, viewed as a subbundle of T^{∗}M. For this to be of the form (df_{1}, … df_{ℓ}) for functions f_{1}, … f_{ℓ} on M, we must have df_{i}(X_{j}) = 0 whenever X_{j} is tangent to M_{j} for j ≠ i. Hence f_{1}, … f_{ℓ} are locally pullbacks from M_{1}, … M_{ℓ}, and are otherwise constrained only by 𝕋_{N}-invariance. Thus 𝔠𝔬𝔫^{𝕋}(𝒟) is a product of pullbacks of fine sheaves, so has H^{1}(Ξ, 𝔠𝔬𝔫^{𝕋}(𝒟)) = 0. □

This result applies in particular to products of spheres.

3.4. Existence. It remains to construct a toric contact manifold of Reeb type from a Delzant, Reeb type, labelled polyhedral manifold with corners (Ξ, e) in Gr_{ℓ}(𝔱^{∗}).

Theorem 2. Let 𝔱 be the Lie algebra of a (ℓ + m)-dimensional torus 𝕋 = 𝔱/2πΛ and (Ξ, e) be a Delzant, Reeb type, labelled polyhedral manifold with corners in Gr_{ℓ}(𝔱^{∗}). There is a codimension ℓ compact toric contact manifold with a Reeb component whose associated labelled manifold with corners is (Ξ, e).

Proof. The Reeb type condition implies there exist ι_{𝔤}: 𝔤 ↪ 𝔱 and λ ∈ 𝔤^{∗} such that ⟨ψ_{𝔤}, λ⟩: Ξ → Δ_{𝔤,λ} is a diffeomorphism, with Δ_{𝔤,λ} defined by (12). This condition is open in λ (by compactness of Ξ, cf. Remark 2), so for λ′ in a contractible open neighbourhood N of λ ∈ 𝔤^{∗}, ⟨ψ_{𝔤}, λ′⟩ is a diffeomorphism from Ξ to Δ_{𝔤,λ}. Thus

V := (ι_{𝔤}^{⊤})^{−1}(N) ∩ C_{e}

(with C_{e} defined by (11)) is foliated by its intersections with ξ ∈ Ξ, i.e., it has a foliation F with leaf space Ξ and leaves F_{ξ} := V ∩ ξ for ξ ∈ Ξ. Note that F is compatible with the face decomposition of V: for any S ∈ Φ_{Ξ} and any ξ ∈ Ξ_{S}, F_{ξ} ⊆ V_{S} := {x ∈ V | ∀ s ∈ S, ⟨x, e_{s}⟩ = 0}. Let Ṽ be open in 𝔱^{∗} with V = Ṽ ∩ C_{e}.

We now use a variant of the Delzant construction, cf. [19], to build a toric symplectic 2(m + ℓ)-manifold (M, ω) under 𝕋. First, since Ṽ × 𝕋 ↪ 𝔱^{∗} × 𝕋 ≃ T^{∗}𝕋, it is a toric symplectic manifold with momentum map

x: Ṽ × 𝕋 → 𝔱^{∗}

induced by the embedding of the first factor into 𝔱^{∗}. It has 𝕋-invariant symplectic form ω̃ = ⟨dx ∧ dθ⟩, where θ: 𝕋 → 𝔱/2πΛ are the tautological angular coordinates on 𝕋.

Next, as in Example 4, let ℤ_{𝒮} be the free abelian group generated by 𝒮, let 𝔱_{𝒮} = ℤ_{𝒮} ⊗_{ℤ} ℝ and ℂ_{𝒮} = ℤ_{𝒮} ⊗_{ℤ} ℂ be the corresponding free vector spaces over ℝ and ℂ, and let 𝕋_{𝒮} = 𝔱_{𝒮}/2πℤ_{𝒮}. Since (Ξ, e) is rational, the map s ↦ e_{s} ∈ Λ induces a group homomorphism ℤ_{𝒮} → Λ and hence also homomorphisms π: 𝕋_{𝒮} → 𝕋 and π_{∗}: ℝ_{𝒮} → 𝔱.

We now consider the action of γ ∈ 𝕋_{𝒮} on Ṽ × 𝕋 × ℂ_{𝒮} by

(14)  γ · (x, [θ], z) = (x, π(γ) · [θ], γ · z)

where 𝕋 acts on 𝕋 and 𝕋_{𝒮} on ℂ_{𝒮} in the standard ways. This action is Hamiltonian for the symplectic form ω̃ ⊕ (−ω_{std}) on (Ṽ × 𝕋) × ℂ_{𝒮} with momentum map

\begin{matrix}ϕ: Ṽ × 𝕋 × ℂ_{𝒮} & → ℝ_{𝒮}^{∗} ≅ ℝ^{𝒮} \\ (x, [θ], z) & ↦ (l_{s}(x) − \frac{1}{2}|z_{s}|^{2})_{s∈𝒮}. \end{matrix} ; (15)

where l_{s}(x) := ⟨x, e_{s}⟩ are the linear maps defining C_{e}. We now show that 𝕋_{𝒮} acts freely on ϕ^{−1}(0). Indeed, if x ∈ Ṽ\V then ϕ(x, [θ], z) < 0 and so

(16)  ϕ^{−1}(0) = ⋃_{x∈V} ({x} × 𝕋 × 𝕋_{𝒮}(x))

where 𝕋_{𝒮}(x) := ∏_{s∈𝒮} 𝕊_{l_{s}(x)}^{1} and 𝕊_{l_{s}(x)}^{1} = \{z ∈ ℂ | \frac{1}{2}|z|^{2} = l_{s}(x)\} is a circle of radius \sqrt{2l_{s}(x)} ≥ 0. Hence if γ ∈ Stab_{𝕋_{𝒮}}(x, [θ], z) then for all s ∈ 𝒮, either γ_{s} = 0 or l_{s}(x) = 0. Now for x ∈ V, S_{x} := {s ∈ 𝒮 | l_{s}(x) = 0} is in Φ_{Ξ} and so e_{s} : s ∈ S_{x} are linearly independent. Indeed by the Delzant condition, they belong to a basis for Λ, and so if π(γ) = 0 then γ = 0.

Consequently, the symplectic reduction of Ṽ × 𝕋 × ℂ_{𝒮} at 0 with respect to the action (14) of 𝕋_{𝒮} is a smooth symplectic manifold (M, ω). Observe that the natural action of 𝕋 on the second factor of Ṽ × 𝕋 × ℂ_{𝒮} commutes with the action (14) of 𝕋_{𝒮} and descends to an effective action on M, whose momentum map μ: M → 𝔱^{∗} is determined by μ ∘ q = x|_{ϕ^{−1}(0)} where q: ϕ^{−1}(0) → M is the quotient by 𝕋_{𝒮}.

The foliation F of V induces a 𝕋_{𝒮}-invariant isotropic foliation of ϕ^{−1}(0) ⊆ V ×𝕋×ℂ_{𝒮}, which descends to a 𝕋-invariant isotropic foliation G of M. Note that N_{𝔤,λ} := μ^{−1}(Δ_{𝔤,λ}) is a global transversal submanifold, i.e., meets each leaf of G in a single point. Moreover, N_{𝔤,λ} is smooth, compact (recall that μ is proper) and 𝕋-invariant. Thus the space of leaves N := M/G is 𝕋-equivariantly diffeomorphic to N_{𝔤,λ}. We denote by f: M → N the quotient map with G_{z} = f^{−1}(z). The momentum map μ: M → V is an orbit map for 𝕋, and it descends to a 𝕋-invariant map E: N → Ξ, hence a rank ℓ subbundle E ≤ N × 𝔱^{∗} over N with fibre E_{z} = E(z) such that, for z ∈ N, we have μ(G_{z}) = F_{E_{z}}.

On the other hand, the symplectic quotient construction provides action angle coordinates on M̊ := μ^{−1}(V̊), that is a splitting of the sequence

0 → M̊ × 𝔱 ⟶^{κ} T M̊ ⟶^{dμ} M̊ × 𝔱^{∗} → 0

which identifies

(17)  T M̊ ≃ M̊ × (𝔱 ⊕ 𝔱^{∗})

such that the symplectic structure ω restricted on M̊ coincides with the fibrewise pairing on the left side. Now observe that via the identification (17), the pullback of E^{0} ⊕ 𝔱^{∗} ≤ N × (𝔱 ⊕ 𝔱^{∗}) on M, that is f^{∗}(E^{0} ⊕ 𝔱^{∗}) → M, coincides over M̊ with T G^{⊥_{ω}}. Hence T G^{⊥_{ω}} is projectable on N̊ := f(M̊). However a distribution containing T G is (locally) projectable if and only if it is preserved by Lie derivatives along vector fields in T G. Hence T G^{⊥_{ω}} is projectable on M by continuity.

We now show that 𝒟 := f_{∗}(T G^{⊥_{ω}}) ≤ T N, a rank 2m distribution on N, is contact. If we denote v = x\frac{∂}{∂x} ∈ T V the vector field induce from the ℝ^{∗}-action of dilation, v is obviously tangent to each leaf of F and thus defines a vector field Y on M such that L_{Y} ω = dι_{Y} ω = ω. In particular, ι_{Y} ω is a primitive 1-form whose kernel contains T G^{⊥_{ω}}, and by using the same argument as in Example 4, we conclude that 𝒟 is contact. □

Remark 7. If we begin the construction with (Ξ, e) rational instead of Delzant, we obtain instead a toric contact orbifold.

Appendix A. Affine geometry of natural momentum maps

Let (M, ω) be a connected symplectic manifold; then the symplectic gradient grad_{ω} f of a smooth function f is the unique vector field X with df = −i_{X}ω. The image of grad_{ω} is the Lie subalgebra 𝔥𝔞𝔪(M, ω) ≤ C^{∞}(M, T M) of hamiltonian vector fields, and there is an exact sequence

0 → ℝ → C^{∞}(M, ℝ) ⟶^{grad_{ω}} 𝔥𝔞𝔪(M, ω) → 0

of Lie algebras, where C^{∞}(M, ℝ) (the space of smooth functions) is a Lie algebra under Poisson bracket, and ℝ is included as the (central) ideal of constant functions. For any (local) action of a Lie algebra 𝔨 on M by hamiltonian vector fields, i.e., any Lie algebra morphism 𝔨 → 𝔥𝔞𝔪(M, ω), this exact sequence pulls back to a central extension

0 → ℝ →^{ε} 𝔥 → 𝔨 → 0

of Lie algebras—if 𝔨 is a Lie subalgebra of 𝔥𝔞𝔪(M, ω), then 𝔥 ≤ C^{∞}(M, ℝ) is the space of hamiltonian generators f of the action (i.e., with grad_{ω} f ∈ 𝔨). Such an extension has an interpretation in affine geometry: dual to the inclusion ε: ℝ → 𝔥, we have a surjection ε^{⊤}: 𝔥^{∗} → ℝ and hence an exact sequence

0 → 𝔨^{∗} → 𝔥^{∗} →^{ε^{⊤}} ℝ → 0.

The inverse image of 1 ∈ ℝ under ε^{⊤} is an affine subspace 𝒜 := (ε^{⊤})^{−1}(1) of 𝔥^{∗}, modelled on 𝔨^{∗}. The space of affine functions f: 𝒜 → ℝ is canonically isomorphic to 𝔥: the projection of f to 𝔨, viewed as a linear form on 𝔨^{∗}, is its derivative df (at every point of 𝒜), and the constant functions are ε(c), c ∈ ℝ.

Definition 16. Let 𝔨 ↪ 𝔥𝔞𝔪(M, ω) be a (local, effective) action by hamiltonian vector fields, and identify the extension 𝔥 of 𝔨 with its image in C^{∞}(M, ℝ). The natural momentum map μ: M → 𝒜 ⊆ 𝔥^{∗} is defined by ⟨μ(x), f⟩ = f(x) for x ∈ M and f ∈ 𝔥.

A (local, effective) action by hamiltonian vector fields is called a hamiltonian action if the extension 𝔥 → 𝔨 has a (Lie algebra) splitting. However, if the splitting is not unique, it can be more convenient to work with the natural momentum map rather than its projection onto 𝔨^{∗} using a splitting. If 𝔨 is abelian, then the action is hamiltonian iff 𝔥 is also abelian, i.e., the hamiltonian generators Poisson commute.

References

[1] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and E. Legendre, Levi–Kähler reduction of CR structures, products of spheres, and toric geometry, Preprint (2017), arXiv:1708.05253.

[2] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1–15.

[3] M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress Math. 93, 1991.

[4] A. Banyaga and P. Molino, Géométrie des formes de contact complètement intégrables de type toriques, Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1991–1992 (Montpellier), 1–25, Univ. Montpellier II, Montpellier, 1993.

[5] A. Banyaga and P. Molino, Complete integrability in contact geometry, Penn State preprint PM 197 (1996).

[6] P. Bolle, A contact condition for p-codimensional submanifolds of a symplectic manifold (2 ≤ p ≤ n), Math. Z. 227 (1998), 211–230.

[7] C. P. Boyer and K. Galicki, A note on toric contact geometry, J. Geom. Phys. 35 (2000), 288–298.

[8] C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.

[9] R. Chiang and Y. Karshon, Convexity package for momentum maps on contact manifolds, Algebraic and Geometric Topology 10 (2010), 925–977.

[10] T. Delzant, Hamiltoniens périodiques et image convexe de l’application moment, Bull. Soc. Math. France 116 (1988), 315–339.

[11] D. Dragnev, Symplectic rigidity, symplectic fixed points, and global perturbations of Hamiltonian systems, Comm. Pure Appl. Math. LXI (2008), 346–370.

[12] S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics 246, Springer, Berlin, 2007.

[13] H. Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Phil. Soc. 121 (1997), no. 3, 455–464.

[14] V. Ginzburg, V. Guillemin and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs 98, Amer. Math. Soc., Providence, 2002.

[15] V. Guillemin, S. Sternberg, Convexity properties of the moment mapping, Inventiones mathematicae 67 (1982), 491–513.

[16] A. Haefliger and É. Salem, Actions of tori on orbifolds, Ann. Global Anal. Geom. 9 (1991), 37–59.

[17] H. Ishida, Torus invariant transverse Kähler foliations, arXiv:1505.06035.

[18] D. Joyce, On manifolds with corners, arXiv/DG.0910.3518

[19] Y. Karshon and E. Lerman, Non-compact symplectic toric manifolds, preprint (2009), arXiv: 0907.2891.

[20] E. Lerman, Contact cuts, Israel J. Math. 124 (2001), 77–92.

[21] E. Lerman, A convexity theorem for torus actions on contact manifolds, Illinois J. Math. 46 (2002), 171–184.

[22] E. Lerman, Contact toric manifolds, J. Symplectic Geom. 1 (2002), 785–828.

[23] E. Lerman, Geodesic flows and contact toric manifolds, Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001), 175–225, Adv. Courses Math. CRM Barcelona, Birkhauser, Basel, 2003.

[24] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), 4201–4230.

[25] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, 2006.

[26] A. Weinstein, Symplectic manifolds and their lagrangian submanifolds, Advances in Mathematics 6 (1971), 329–346.

[27] C. Willett, Contact reduction, Trans. Amer. Math. Soc. 354 (2002), 4245–4260.

Vestislav Apostolov, Département de Mathématiques, UQAM, C.P. 8888, Succursale Centre-ville, Montréal (Québec), H3C 3P8, Canada

  E-mail address: apostolov.vestislav@uqam.ca

David M. J. Calderbank, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

  E-mail address: D.M.J.Calderbank@bath.ac.uk

Paul Gauduchon, Centre de Mathématiques, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France

  E-mail address: pg@math.polytechnique.fr

Eveline Legendre, Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France

  E-mail address: eveline.legendre@math.univ-toulouse.fr