LEVI–KÄHLER REDUCTION OF CR STRUCTURES, PRODUCTS OF SPHERES, AND TORIC GEOMETRY

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

Abstract. We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi–Kähler quotient, for constructing Kähler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi–Kähler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi–Kähler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author [10].

Introduction

In recent years there has been considerable interest in the interaction between Kähler geometry and its odd-dimensional younger cousin, Sasaki geometry. On the one hand, ideas in Kähler geometry, such as toric methods or extremal metrics, have led to the development of analogues in Sasaki geometry. On the other hand, Sasaki manifolds have a canonical 1-dimensional foliation generated by the Reeb vector field, which both provides a construction of Kähler metrics on the leaf space when the latter is a manifold or orbifold, as well as a “horizontal” generalization of such quotients when it is not.

Our thesis herein is that these ideas need not be limited to 1-dimensional foliations. Indeed, any Sasaki manifold has an underlying codimension one CR structure, whereas CR manifolds arise naturally in arbitrary codimension. This prompts us to introduce transverse “Reeb foliations” on arbitrary CR manifolds (N, 𝒟, J) (a theory of such foliations has recently been developed in [42], but here we focus on the horizontal Kähler geometry of (𝒟, J)). However, whereas in codimension one, the exterior derivative of the contact form equips the horizontal distribution 𝒟 with a nondegenerate 2-form (which, together with the complex structure J on 𝒟, defines the horizontal Kähler structure on 𝒟), in higher codimension, the non-integrability of 𝒟 is measure by a 2-form on 𝒟 with values in T N/𝒟, called the Levi form L_{𝒟}. In order to construct a Kähler metric on the leaf space of the Reeb foliation, we therefore need also to choose a nondegenerate component of L_{𝒟}. This construction, which we call a Levi–Kähler quotient, is our main topic of study.

In particular, the Levi form of 𝒟 must have a nondegenerate component. Rank 2m distributions 𝒟 of this type, on manifolds N of dimension 2m + ℓ, were studied in a companion paper [7], to which the present work may be viewed as a sequel, although we do not here rely upon knowledge of that paper, or its main results. Indeed, whereas in [7] we study the general theory of toric contact manifolds in higher codimension, the applications in Kähler geometry we develop herein use only the simplest examples: toric CR submanifolds of flat space, and in particular, products of spheres.

Date: September 18, 2017.

V.A. is supported in part by 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.

Our prime motivation is the construction of interesting toric Kähler orbifolds M, which are Kähler orbifolds of real dimension 2m admitting a isometric hamiltonian action of a real m-torus 𝔱/2πΛ. As a symplectic orbifold [12, 26, 41] or complex toric variety [23, 30], M is classified respectively by the image Δ of its momentum map, which is a convex polytope in an m-dimensional real affine space 𝒜 modelled on 𝔱^{∗}, or by the underlying fan Y in 𝔱 of normal rays to the facets (codimension one faces) of Δ. Explicitly, Δ is an intersection of half-spaces L_{s} ≥ 0, where s ∈ 𝒮 indexes the facets of Δ, and L_{s} ∈ 𝔥, the (m + 1)-dimensional vector space of affine functions on 𝒜; the normals of Y are then the linear parts u_{s} ∈ 𝔱 of L_{s} for s ∈ 𝒮. It is convenient to encode these data in linear maps L: ℝ_{𝒮} → 𝔥 and u: ℝ_{𝒮} → 𝔱, where ℝ_{𝒮} is the standard real vector space with a basis e_{s} : s ∈ 𝒮, L(e_{s}) = L_{s} and u(e_{s}) = u_{s}. The kernel 𝔤 of u is the Lie algebra of a subtorus G of ℝ_{𝒮}/2πℤ_{𝒮}, and we let λ ∈ 𝔤^{∗} be the restriction of L to 𝔤 (which takes values in the constant affine functions, i.e., ℝ). Equipping the complexification ℂ_{𝒮} of ℝ_{𝒮} with its standard flat Kähler structure, M is then equivariantly symplectomorphic to the symplectic quotient of ℂ_{𝒮} by G at momentum level λ [26, 41], and equivariantly biholomorphic to a GIT quotient of ℂ_{𝒮} by the complexification of G [22].

However, M need not be isometric to the Kähler quotient of ℂ_{𝒮} by G, and the Kähler quotient metric, known as the Guillemin metric [34], has not been found to be particularly interesting, except in the simplest cases. We are thus motivated to make the following observations. First, the level set μ_{𝔤}^{−1}(λ) of the momentum map of G is a toric CR submanifold N_{𝔤,λ} of ℂ_{𝒮}, in fact an intersection of quadric hypersurfaces. Secondly, since G preserves N_{𝔤,λ} with orbits transverse to the CR distribution 𝒟, we may regard the Levi form as a 𝔤-valued 2-form on 𝒟. We find that the λ component is nondegenerate and induces a Kähler metric on M ≅ N_{𝔤,λ}/G—in other words, the data (𝔤, λ) is exactly what we need to define a Levi–Kähler quotient of N_{𝔤,λ}. While this metric does not seem to be particularly interesting either, there is a third observation that proves to be decisive: we do not need to use the same pair (𝔤, λ) to define the toric CR submanifold N as we use to take the Levi–Kähler quotient. Indeed, (𝔤, λ) determines a Levi–Kähler quotient of N_{𝔤_{o},λ_{o}} biholomorphic to M provided only that the data (𝔤_{o}, λ_{o}) arises from a polytope Δ_{o} with the same combinatorial type as Δ.

We take advantage of these observations by using the principle that a Levi–Kähler quotient of a CR manifold N can be expected to have nice curvature properties if N does. The simplest examples, in codimension one, are round CR (2m + 1)-spheres, which are the toric CR submanifolds associated to m-simplices, and are circle bundles over complex projective spaces. Products of such spheres provide examples in higher codimension. Thus if Δ is a polytope with the same combinatorial type as a product of simplices, we obtain a distinguished toric Kähler metric on the toric symplectic orbifold associated to Δ as a Levi–Kähler quotient of a suitable product of spheres.

The main results come in three courses, which we serve up in Sections 2, 3 and 4, after presenting some background and preliminary results in Section 1. The preliminary material reviews the notion of a CR (2m + ℓ)-manifold of codimension ℓ and studies local CR torus actions transverse to the CR distribution. These have associated Kähler cones of dimension 2(m + ℓ) and so may be viewed as a natural generalization of Sasaki structures. Such a transverse local action of an ℓ-dimensional abelian Lie algebra 𝔤, together with an element λ ∈ 𝔤^{∗} is called a positive Levi pair if it defines a horizontal Kähler structure on the CR distribution. We obtain from this the Levi–Kähler quotient construction when the action of G integrates to an action of a Lie group G.

Section 2 presents our general results on toric CR submanifolds of ℂ_{𝒮} and their Levi– Kähler quotients, thus establishing the observations made above. To do this, we first review the elements of toric geometry and combinatorics, and make precise the notion of combinatorial type. Then we show in Theorem 2 that if (𝔤, λ) is a positive Levi pair associated to polytope Δ and N is a toric submanifold with the same combinatorial type, then the image of the momentum map of the horizontal Levi–Kähler structure (or the Levi–Kähler quotient when that exists) has image Δ. Here we need a small part of the toric contact theory developed in [7], which we summarize in Theorem 1. In the case that N is toric intersection of quadric hypersurfaces contained in a round hypersphere, the Levi–Kähler structure can be made explicit, as we show in Theorem 3.

In Section 3, we study the construction of toric Kähler metrics as Levi–Kähler quotients of products of spheres. Building on Theorem 2, we characterize such quotients in Theorem 4 as those associated to a polytope with the combinatorial type of a product of simplicies. In turn, Theorem 5 builds on Theorem 3 by giving explicit formulae for the Levi–Kähler quotients of products of spheres and their symplectic and Kähler potentials. In the remainder of the section, we explore relations between this construction and other explicit methods in Kähler geometry. The explicit form of certain Levi–Kähler quotients of products of 3-spheres motivates a new ansatz for toric Kähler metrics whose Delzant polytope is projectively equivalent to a cube, extending the ambitoric ansatz of Segre type [4, 5] to arbitrary dimension.

Another source of toric Kähler orbifolds M whose Delzant polytope has the combinatorics of a product of simplices can be obtained from the generalized Calabi construction, where both the base and the fibre are toric orbifolds with Delzant polytopes having the combinatorics of product of simplices (see [9]). This includes the complex Hirzebruch surfaces, or more generally any holomorphic projective bundle over a projective space, and, inductively, toric fibrations where the base and the fibre are one of the mentioned smooth complex manifolds. We show that in this setting, the Kähler metric corresponding to the Levi–Kahler quotient of the product of spheres associated to M is obtained from the generalized Calabi construction, where the metrics on the base and on the fibre are themselves Levi–Kahler quotients of product of spheres.

The final tranche of results concern curvature properties of a Levi–Kähler quotient M of a product of spheres N. The product structure on N induces distributions on M and we show that the curvature of M has vanishing Bochner component on each such distribution, simply because CR spheres have vanishing Chern–Moser tensor. For 3-dimensional CR manifolds, the vanishing of the Chern–Moser tensor is automatic. We compute instead the scalar curvature of a Levi–Kähler quotient of a product of 3-spheres and observe that when the polytope is projectively equivalent to a cube, the Levi–Kähler quotient is extremal in a weighted sense that was introduced (in a special case) in [10].

More precisely, given a “conformal dimension” p ∈ ℝ and a positive function w on a compact symplectic orbifold (M, ω) whose hamiltonian vector field is quasiperiodic (i.e., it belongs to the Lie algebra of a torus 𝕋 in Ham(M, ω)), we can generalize the approach of Donaldson [28] and Fujiki [31] to Calabi’s extremal Kähler metrics [20] by using w^{−(p−1)} as a weight for the formal Frechét symplectic structure on the space of 𝕋-invariant compatible complex structures. Then the action of Ham^{𝕋}(M, ω) on this space is hamiltonian, and if we weight the inner product on its Lie algebra of by w^{−(p+1)} then the momentum map at J may be identified with a modification s_{J,w,p} of the scalar curvature of g_{J} = ω(·, J·), which is what the scalar curvature of the conformally related metric w^{2}g_{J} would be if M had dimension p.

If the polytope Δ of M is projectively equivalent to a cube, then opposite facets of Δ meet in coplanar lines, so there is a unique affine function w up to scale which is positive on Δ and vanishes on the intersection of opposite facets. We prove in Theorem 6 that the unique (up to scale) toric metric on (M, ω) for which s_{J,w,m+2} is an affine function is the one arising as a Levi–Kähler quotient of a product of 3-spheres.

The Levi–Kähler quotient metrics of products of ℓ odd dimensional spheres do not lead to new extremal Kähler metrics (in the classical sense) unless ℓ ≤ 2. In the case ℓ = 1, the quotient of a sphere are Bryant’s Bochner-flat Kähler metrics on weighted projective spaces [18], which are extremal. We end the paper with some new extremal examples obtained as the quotient of a product of two spheres.

1. Levi–Kähler quotients of CR manifolds

1.1. CR structures of arbitrary codimension.

Definition 1. A CR structure (𝒟, J) of rank m and codimension ℓ on a real (2m + ℓ)- dimensional manifold N is a real rank 2m distribution 𝒟 ⊆ T N equipped with an almost complex structure J: 𝒟 → 𝒟, which satisfies the integrability conditions

\begin{aligned}[X, Y] − [JX, JY] & ∈ Γ(𝒟), \\ [X, JY] + [JX, Y] & = J([X, Y] − [JX, JY]), ∀ X, Y ∈ Γ(𝒟), \end{aligned} ; (1)

or, equivalently,

[Γ(𝒟^{1,0}), Γ(𝒟^{1,0})] ⊆ Γ(𝒟^{1,0}),

where 𝒟^{1,0} ⊆ T N ⊗ ℂ is the subbundle of (1, 0) vectors in 𝒟 ⊗ ℂ.

(N, 𝒟, J) is then called a CR manifold (of codimension ℓ).

The underlying rank 2m distribution 𝒟 on N may be viewed as a codimension ℓ generalization of a contact structure on N [7]. The fundamental invariant of 𝒟 is its Levi form L_{𝒟}: ∧^{2}𝒟 → T N/𝒟, defined, via X, Y ∈ Γ(𝒟), by the tensorial expression

(2)  L_{𝒟}(X, Y) = −q_{𝒟}([X, Y])

where q_{𝒟}: T N → T N/𝒟 is the quotient map. The transpose of q_{𝒟} identifies (T N/𝒟)^{∗} canonically with the annihilator 𝒟^{0} of 𝒟, which is a rank ℓ subbundle of T^{∗}N. The normalization convention for L_{𝒟} 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}.

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}. If U_{𝒟} ∩ 𝒟_{z}^{0} is nonempty then, since nondegeneracy is an open condition, 𝒟_{z}^{0} has a basis α_{1}, … α_{ℓ} in U_{𝒟} and so U_{𝒟} ∩ 𝒟_{z}^{0} is the complement of the set where (∑_{i=1}^{ℓ} t_{i}α_{i}) ∘ L_{𝒟} degenerates, which is the cone over a projective hypersurface V_{𝒟,z} of degree m (the zero set of a homogeneous degree m polynomial in the ℓ variables t_{1}, … t_{ℓ}). In [7], V_{𝒟} ⊆ P(𝒟^{0}) is called the degeneracy variety of 𝒟. Therein it is shown that U_{𝒟} is a canonical “symplectization” of (N, 𝒟): U_{𝒟} is the open subset of 𝒟^{0} over which the pullback of the canonical symplectic form Ω on T^{∗}N to 𝒟^{0} is nondegenerate.

Definition 2. Let (N, 𝒟, J) be a CR manifold (of codimension ℓ). We say 𝒟 is Levi nondegenerate if U_{𝒟} has nonempty intersection with each fibre of p: 𝒟^{0} → N. A (local) section of U_{𝒟} is called a (local) contact form on N.

Note that the Levi form L_{𝒟} satisfies

L_{𝒟}(X, Y) = −\frac{1}{2}q_{𝒟}([X, Y] + [JX, JY])

and hence is J-invariant or “type (1,1)” on 𝒟. It follows that h_{𝒟}(X, Y) := L_{𝒟}(X, JY) is a section of S^{2}𝒟^{∗} ⊗ T N/𝒟. We say (N, 𝒟, J) is Levi definite if at each z ∈ N there exists α ∈ 𝒟_{z}^{0} such that α ∘ h_{𝒟} ∈ S^{2}𝒟_{z}^{∗} is positive definite.

Clearly Levi definite CR manifolds are Levi nondegenerate: more generally U_{𝒟}^{+} := {α ∈ 𝒟^{0} | α ∘ h_{𝒟} is positive definite} is an open and closed submanifold of U_{𝒟}.

Examples 1. (i) A maximally real codimension ℓ submanifold of ℂ^{m+ℓ} is a smooth submanifold N ⊆ ℂ^{m+ℓ} for which 𝒟 := T N ∩ JT N, where J is the standard complex structure of ℂ^{m+ℓ}, has rank 2m (i.e., corank ℓ in T N). Then (N, 𝒟), with the induced action of J on 𝒟, is a CR manifold of rank m and codimension ℓ. A model example, in codimension one, is the unit sphere 𝕊^{2m+1} in ℂ^{m+1}. (ii) If (N_{i}, 𝒟_{i}, J_{i}) are CR manifolds, with codimensions ℓ_{i}, for i ∈ {1, … n}, then so is (∏_{i=1}^{n} N_{i}, 𝒟_{1} ⊕ ⋯ ⊕ 𝒟_{n}, J_{1} ⊕ ⋯ ⊕ J_{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 CR spheres 𝕊^{2m_{1}+1} × ⋯ × 𝕊^{2m_{ℓ}+1} is a CR manifold with codimension ℓ.

Remark 1. The Levi form of a CR manifold (N, 𝒟, J) is traditionally defined to be the hermitian form h_{𝒟} + iL_{𝒟}: 𝒟 × 𝒟 → ℂ ⊗ T N/𝒟; however it is uniquely determined (given J) by its real or imaginary part, and the imaginary part is an invariant of the underlying (real) distribution 𝒟. The rank is usually called the CR dimension.

Levi nondegeneracy implies that X ↦ L_{𝒟}(X, ·) is an injective bundle homomorphism from 𝒟 to Hom(𝒟, T N/𝒟). This condition, together with the assumption that L_{𝒟} is surjective onto T N/𝒟, appears in the study [13] of CR automorphisms of real quadrics N_{σ} := {(z, w) ∈ ℂ^{m+ℓ} | ℑ(w) = ℑσ(z, z)}, where σ: ℂ^{m} × ℂ^{m} → ℂ^{ℓ} is hermitian and ℑ denotes the imaginary part. Such quadrics are homogeneous CR manifolds of rank m and codimension ℓ, with Levi form isomorphic to σ (or ℑσ in our sense).

Levi definiteness extends the codimension one notion of strict pseudoconvexity.

1.2. Local CR actions, generalized Sasaki structures and Kähler cones.

Definition 3. Let (N, 𝒟, J) be a CR manifold; then the space 𝔠𝔯(N, 𝒟, J) of CR vector fields is the Lie subalgebra of vector fields X on N such that

(3)  L_{X}Γ(𝒟) ⊆ Γ(𝒟) and L_{X}J = 0.

A (local, effective) CR action of a Lie algebra 𝔤 on (N, 𝒟, J) is a Lie algebra monomorphism K: 𝔤 → 𝔠𝔯(N, 𝒟, J). 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}. Since K: 𝔤 → 𝔠𝔯(N, 𝒟, J) is a Lie algebra morphism, 𝒦^{𝔤} is an integrable distribution.

Example 2. Let π: N → M be a principal G-bundle with connection η: T N → 𝔤, where dim G = ℓ and dim M = 2m. Then 𝒟 := ker η is a rank 2m distribution on N, 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. If M has an (integrable) complex structure J for which F^{η} is J-invariant, the horizontal lift of J to 𝒟 equips N with a G-invariant CR structure. For G abelian, this is the principal torus bundle construction of [43].

Suppose K: 𝔤 → 𝔠𝔯(N, 𝒟, J) is a local CR action, where (N, 𝒟, J) is CR 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. 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.

If K integrates to an action of a connected Lie group G on N, then Condition 1(i) implies that the G-action is locally free, so that M := N/G is a compact orbifold. Condition 1(ii) then ensures that 𝒟 is isomorphic to the pullback of T M to N, and hence G-invariant data on 𝒟 descend to M. Invariant components of the Levi form provide examples of such data.

In codimension one, a transversal CR action is essentially a CR Reeb vector field, or equivalently, a compatible Sasaki structure, which makes the symplectic cone Kähler.

To generalize this to arbitrary codimension, note that the total space 𝒟^{0} of p: 𝒟^{0} → N inherits from T^{∗}N a tautological 1-form τ: using the exact sequence

(4)  0 → p^{∗}𝒟^{0} → T 𝒟^{0} →^{p_{∗}} p^{∗}T N → 0,

τ_{α} = α ∘ p_{∗}: T_{α}𝒟^{0} → ℝ for any α ∈ 𝒟^{0}. We set Ω^{𝒟} = dτ; this is the pullback of the tautological symplectic form on T^{∗}N to 𝒟^{0}.

Any X ∈ Γ(T N) has a lift to a hamiltonian vector field X̃ on T^{∗}N with p_{∗}(X̃) = X and hamiltonian f_{X} = τ(X̃), i.e., f_{X}(α) = α(X); furthermore {f_{X}, f_{Y} } = f_{[X,Y]}. (Explicitly, df_{X} = −Ω^{𝒟}(X̃, ·), where X̃_{α} = α_{∗}(X_{z}) − (L_{X}α)_{z} for any extension of α ∈ T_{z}^{∗}N to a local section.) If X ∈ 𝔠𝔯(N, 𝒟, J), then X̃ is tangent to 𝒟^{0} ⊆ T^{∗}N.

Observation 1. Let K: 𝔤 → 𝔠𝔯(N, 𝒟, J) be a local CR 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. Now (p, μ_{𝔤}): 𝒟^{0} → N × 𝔤^{∗} is a bundle isomorphism with inverse ψ_{𝔤}(z, λ) := ⟨η_{z}, λ⟩: if α = ⟨η_{z}, λ⟩ for some λ ∈ 𝔤^{∗} and z ∈ N, then μ_{𝔤}(α) = λ.

Lemma 1. Let τ be the tautological 1-form on 𝒟^{0}. Then

(5)  (ψ_{𝔤}^{∗}τ)_{(z,λ)}(X + a) = ⟨η(X), λ⟩,

and hence

(6)  (ψ_{𝔤}^{∗}Ω^{𝒟})_{(z,λ)}(X + a, Y + b) = ⟨a, η(Y)⟩ − ⟨b, η(X)⟩ + ⟨dη(X, Y), λ⟩.

Proof. Since τ_{α}(Z) = α(p_{∗}(Z)), (ψ_{𝔤}^{∗}τ)_{(z,λ)}(X + a) = τ_{⟨η_{z},λ⟩}((ψ_{𝔤})_{∗}(X + a)) = ⟨η(X), λ⟩. Hence ψ_{𝔤}^{∗}τ = ⟨p_{2}, p_{1}^{∗}η⟩, where p_{1} and p_{2} are the first and second projections of N × 𝔤^{∗}. Now ψ_{𝔤}^{∗}Ω^{𝒟} = ψ_{𝔤}^{∗}dτ = d(ψ_{𝔤}^{∗}τ) = ⟨dp_{2} ∧ p_{1}^{∗}η⟩ + ⟨p_{2}, p_{1}^{∗}dη⟩, which yields (6) □

For (z, v) ∈ N × 𝔤, let J_{(z,v)} be the complex structure on T_{(z,v)}(N × 𝔤) = T_{z}N ⊕ 𝔤 defined by

J_{(z,v)}(X + w) = JX^{𝒟} + K_{w,z} − η(X)

where X^{𝒟} denotes the 𝒟_{z}-component of X ∈ T_{z}N = 𝒟_{z} ⊕ (𝒦^{𝔤})_{z}.

Lemma 2. The almost complex structure J is integrable if and only if 𝔤 is abelian.

Proof. Let X, Y be vector fields on N with η(X) and η(Y) constant and let u, v ∈ 𝔤^{∗}; as vector fields on N × 𝔤, these are constant in the 𝔤 direction. First observe that

N_{J} (X, Y) = [X, J Y] + [J X, Y] − J ([X, Y] − [J X, J Y])
= [X, JY^{𝒟}] + [JX^{𝒟}, Y] − J[X, Y]^{𝒟} + η([X, Y]) + J [JX^{𝒟}, JY^{𝒟}] = 0,

since [X, Y]^{𝒟} = [X^{𝒟}, Y^{𝒟}] + [K_{η(X)}, Y^{𝒟}] + [X^{𝒟}, K_{η(Y)}]. Next

J N_{J} (u, Y) = J ([u, J Y] + [J u, Y]) + [u, Y] − [J u, J Y]
= J[K_{u}, Y]^{𝒟} − η([K_{u}, Y]) − [K_{u}, JY^{𝒟} − η(Y)] = 0.

Finally,

J N_{J} (u, v) = J ([u, J v] + [J u, v]) + [u, v] − [J u, J v]
= [K_{u}, K_{v}] = K_{[u,v]},

which vanishes for all u, v ∈ 𝔤 iff 𝔤 is abelian. □

Using η to identify 𝒦^{𝔤} with N × 𝔤, we observe that

(7)  T_{(z,λ)}(N × 𝔤^{∗}) ≅ 𝒟_{z} ⊕ 𝔤 ⊕ 𝔤^{∗} and T_{(z,v)}(N × 𝔤) ≅ 𝒟_{z} ⊕ 𝔤 ⊕ 𝔤;

in these terms, (ψ_{𝔤}^{∗}Ω^{𝒟})_{(z,λ)} is the sum of ⟨dη_{z}, λ⟩ and the standard symplectic structure on 𝔤 ⊕ 𝔤^{∗}, while J_{(z,v)} is the sum of the complex structure J on 𝒟_{z} and the standard complex structure on 𝔤 ⊕ 𝔤. Thus if 𝔤 is abelian and we identify N × 𝔤 with N × 𝔤^{∗} using a symmetric positive definite bilinear form ζ on 𝔤, we obtain a Kähler structure on the open subset of (z, λ) ∈ N × 𝔤^{∗} with ψ_{𝔤}(z, λ) ∈ U_{𝒟}^{+}.

Proposition 1. Let (N, 𝒟, J) be a codimension ℓ CR manifold, and 𝔤 a transversal CR action of an (ℓ-dimensional) abelian Lie algebra 𝔤 with a positive definite inner product. Then U_{𝒟}^{+} ⊆ N × 𝔤^{∗} has a canonical Kähler metric on which 𝔤 defines a local isometric hamiltonian action whose momentum map is the projection p_{2} : N × 𝔤^{∗} → 𝔤^{∗}.

Proof. Lemmas 1 and 2 imply that U_{𝒟}^{+} is Kähler. The hamiltonian vector field generated by the component ⟨p_{2}, v⟩ of p_{2} is the pullback of K_{v}, which clearly preserves the complex structure J, hence the metric h on N × 𝔤^{∗}. □

Definition 4. Let (N, 𝒟, J) be Levi definite, and let 𝔤 be an (ℓ-dimensional) abelian Lie algebra. Then a 𝔤-Sasaki structure on N is a transversal CR action of 𝔤 together with a positive definite inner product ζ; given such an action, we say (N, 𝒟, J, 𝔤, ζ) is a codimension ℓ Sasaki manifold with generalized Kähler cone U_{𝒟}^{+}.

1.3. CR torus actions and Levi–Kähler quotients.

Definition 5. A local CR action K: 𝔱_{N} → 𝔠𝔯(N, 𝒟, J) of an abelian Lie algebra 𝔱_{N} on a CR manifold (N, 𝒟, J) is called a local CR torus action, and is said to be a CR torus action of 𝕋_{N} = 𝔱_{N}/2πΛ_{N} if it integrates to an effective (i.e., faithful) action of 𝕋_{N}. If K: 𝔤 → 𝔠𝔯(N, 𝒟, J) satisfies Condition 1, we refer to 𝒦^{𝔤} and its integral submanifolds as the associated Reeb distribution and Reeb foliation transverse to 𝒟.

Given a local CR torus action K: 𝔱_{N} → 𝔠𝔯(N, 𝒟, J), 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.

Definition 6. An ℓ-dimensional subalgebra ι: 𝔤 ↪ 𝔱_{N} and an element λ ∈ 𝔤^{∗} \ 0 together form a Levi pair (𝔤, λ) for a local CR torus action 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, η^{λ} := ⟨η, λ⟩, and h_{𝒟,λ} := dη^{λ}|_{𝒟}(·, J·). Then (𝒟, h_{𝒟,λ}, J) is called a Levi structure and we say that (g, λ) or (𝒟, h_{𝒟,λ}, J) is • nondegenerate if η^{λ} is a contact form, i.e., h_{𝒟,λ} is nondegenerate on 𝒟;

• positive if h_{𝒟,λ} is positive definite on 𝒟. We say (N, 𝒟, J, K) is Reeb type if it admits a nondegenerate Levi pair, and if positive, we say that (N, 𝒟, J, 𝔤, λ) or (N, 𝒟, J, h_{𝒟,λ}) is Levi–Kähler.

If K is a CR torus action of 𝕋_{N} and 𝔤 is the Lie algebra of a closed subgroup G of 𝕋_{N}, then M/G, with the Kähler metric induced by (h_{𝒟,λ}, J, dη^{λ}|_{𝒟}) is called the Levi–Kähler quotient of (N, 𝒟, J) by (𝔤, λ).

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

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 → 𝔱 (where 𝔥 and 𝔱 are abelian Lie algebras of dimensions m + 1 and m); then the commutative diagram

[figure: 0; 𝔤; ι; 𝔱_{N}; u_{N}; 𝔱; 0; 0; ℝ; λ; ε; 𝔥; L_{N}; d; 𝔱; ‖; 0.]

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

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

By Observation 1, K lifts to a hamiltonian action on U_{𝒟} with momentum map μ|_{U_{𝒟}} . If (𝔤, λ), defined by L_{N}: 𝔱_{N} → 𝔥, is a Levi pair, then the map μ^{λ}: N → 𝒜 ⊆ 𝔥^{∗}, determined uniquely by the formula

(8)  ⟨μ^{λ}(z), L_{N}(v)⟩ = η_{z}^{λ}(K_{v})

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

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

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

Observation 2. Let (N, 𝒟, J, K) be a CR manifold of Reeb type, and let (𝔤, λ) be a nondegenerate Levi pair, where 𝔤 is the Lie algebra of a subtorus G of 𝕋_{N}. Then M := N/G, equipped with the 2-form induced by dη^{λ}|_{𝒟}, is the symplectic quotient μ_{𝔤}^{−1}(λ)/G of U_{𝒟} by the lifted G-action; it is therefore a compact symplectic orbifold with a hamiltonian action of 𝕋 = 𝕋_{N}/G whose momentum map is induced by the Ginvariant map μ^{λ}: N → 𝒜 ⊆ 𝔥^{∗} defined in (8).

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 rank 𝒟 = 2m, then rank 𝒦 ∩ 𝒟 ≤ m, and hence dim 𝔱_{N} ≤ m + ℓ.

Proposition 2. Let K: 𝔱_{N} → 𝔠𝔯(N, 𝒟, J) be a local CR torus action. Then 𝒦 ∩ 𝒟 is an integrable distribution, i.e., L_{𝒟}(X, Y) = 0 for all X, Y ∈ 𝒦 ∩ 𝒟.

Proof. For any v, w ∈ 𝔱_{N} and any section α of 𝒟^{0}, dα(K_{v}, K_{w}) = (L_{K_{v}}α)(K_{w}) − (L_{K_{w}}α)(K_{v}). Hence if X = ∑_{i} f_{i}K_{v_{i}} and Y = ∑_{j} g_{j}K_{w_{j}} are sections of 𝒦 ∩ 𝒟 (for functions f_{i} and g_{j} on N), then

(α ∘ L_{𝒟})(X, Y) = dα(X, Y) = ∑_{i} f_{i}(L_{K_{v_{i}}}α)(Y) − ∑_{j} g_{j}(L_{K_{w_{j}}} α)(X) = 0

since K_{v} preserves 𝒟^{0} for any v ∈ 𝔱_{N}. □

Remark 2. If (𝔤, λ) is a Levi pair then (p, μ_{𝔤}): 𝒟^{0} → N × 𝔤^{∗}, with μ_{𝔤} := ι^{⊤} ∘ μ, is an isomorphism. In particular, (p, μ) injects, i.e., E is a rank ℓ subbundle of N × 𝔱_{N}^{∗} (with E^{∗} ≅ (N × 𝔱_{N})/Θ ≅ T N/𝒟). Equivalently, the transpose q_{𝒟} ∘ κ surjects (pointwise), i.e., 𝒦 ∩ 𝒟 has codimension ℓ in 𝒦. Conversely, this suffices for the local existence of a transversal subalgebra ι: 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.

Definition 7. We say (N, 𝒟, J, K) is locally Reeb type if q_{𝒟} ∘ κ surjects and toric if it is locally Reeb type with dim 𝔱_{N} = m + ℓ.

On any open set where rank 𝒦 = m + ℓ, (N, 𝒟, K) is locally Reeb type.

Proposition 3. Let (N, 𝒟, J) be a toric CR manifold under 𝕋_{N} and let N^{o} be the open subset on which the action of 𝕋_{N} is free, and U_{𝒟}^{o} its inverse image in U_{𝒟} ⊆ 𝒟^{0}.

Then U_{𝒟} is a toric symplectic manifold with momentum map μ: U_{𝒟} → 𝔱_{N}^{∗} defined by ⟨μ(α), v⟩ = α(K_{v}), where α ∈ U_{𝒟} ⊆ T^{∗}N and K_{v} := K(v) for v ∈ 𝔱_{N}. Further, there are angular coordinates φ: U_{𝒟}^{o} → 𝕋_{N} = 𝔱_{N}/2πΛ_{N}, unique up to an additive constant, with Ω^{𝒟} = ⟨dμ ∧ dφ⟩ and ker dφ = p_{∗}^{−1}(J(𝒦 ∩ 𝒟)) (restricting p: 𝒟^{0} → N to U_{𝒟}^{o}).

Proof. The first part is immediate from Observation 1. By Proposition 2, 𝒦 ∩ 𝒟 is an integrable rank m subbundle of 𝒟, hence so is J(𝒦 ∩ 𝒟) by the J-invariance of the Levi form, and T N^{o} = 𝒦 ⊕ J(𝒦 ∩ 𝒟). The 1-form β: T N^{o} → 𝔱_{N} defined by ker β = J(𝒦 ∩𝒟) and β(K_{v}) = v is therefore closed. It is not exact, but by definition of N^{o}, the local primitives fit together to give a primitive with values in 𝔱_{N}/2πΛ_{N} unique up to a constant, whose pullback to U_{𝒟}^{o} yields φ. □

In terms of the short exact sequence (4) restricted to U_{𝒟}^{o}, we have

[figure: 0; p^{∗}𝒟^{0}; ker dφ; p^{∗}J(𝒦 ∩ 𝒟); 0; 0; p^{∗}𝒟^{0}; ‖; T U_{𝒟}^{o}; p_{∗}; p^{∗}T N^{o}; 0,]

where for any α ∈ U_{𝒟}^{o}, (dμ)_{α}(ker dφ) = 𝔱_{N}^{∗}, and (dμ)_{α}(p^{∗}𝒟^{0}) = μ(𝒟_{p(α)}^{0}) = E_{p(α)}. This identifies (dμ)_{α}(p^{∗}J(𝒦 ∩ 𝒟)) with Θ_{p(α)}^{∗}.

A subalgebra ι: 𝔤 ↪ 𝔱_{N} satisfying Condition 1 splits the exact sequence

[figure: 0; E; N^{o} × 𝔱_{N}^{∗}; Θ^{∗}; 0]

i.e., ker ι^{⊤} ≅ 𝔱^{∗} is transverse to E_{z} for all z ∈ N^{o}. Thus μ(U_{𝒟}^{o}) ⊆ 𝔱_{N}^{∗} is foliated by its intersection with the m-dimensional family E_{z} of ℓ-dimensional linear subspaces of 𝔱_{N}^{∗}.

Remark 3. The Levi–Kähler quotient of N by (𝔤, λ) is the Kähler quotient of any 𝕋_{N}-invariant, Ω^{𝒟} compatible metric ĝ on U_{𝒟}^{+} whose pullback to μ_{𝔤}^{−1}(λ) ≅ N is the orthogonal sum of a metric on 𝒦^{𝔤} and the metric h_{𝒟,λ} on 𝒟. We may assume ĝ has angular coordinates dφ on U_{𝒟}^{o,+} = U_{𝒟}^{o} ∩ U_{𝒟}^{+}, so that it is determined uniquely there by the induced 𝕋_{N}-invariant metric on ker dφ ≅ N^{o} ×𝔱_{N}^{∗}, which descends to a metric G on μ(U_{𝒟}^{o,+}) ⊆ 𝔱_{N}^{∗}. Since h_{𝒟,cλ} = c h_{𝒟,λ} for c ∈ ℝ^{+}, we assume G is homogeneous of degree 1 on 𝔱_{N}^{∗}, i.e., as an S^{2}𝔱_{N}-valued function on μ(U_{𝒟}^{o,+}), it is homogeneous of degree −1. Examples of such metrics include the generalized Kähler cone metrics.

If (𝔤, λ) is given by L_{N}: 𝔱_{N} → 𝔥, then the Levi–Kähler quotient metric depends only on the pullback of G to L_{N}^{⊤}(𝒜) = (ι^{⊤})^{−1}(λ), an affine subspace transverse to E.

2. Levi–Kähler reduction in toric geometry

2.1. Polytopes, fans, combinatorics, and toric contact manifolds. Suppose that (N, 𝒟, J, K) is a toric CR manifold of rank m and codimension ℓ, under a (real) torus 𝕋_{N} = 𝔱_{N}/2πΛ_{N} with (abelian) Lie algebra 𝔱_{N} = Λ_{N} ⊗_{ℤ} ℝ.

The theory of effective actions of tori [12, 33] implies that for any subtorus 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 then N_{(H)} is dense in N^{H}. The connected components of N_{(H)} and their closures in N are called open and closed orbit strata of (N, K). Let Φ_{N} be the set of closed orbit strata, partially ordered by inclusion, and let N_{s} : s ∈ 𝒮 index the closed orbit strata stabilized by a circle. We refer to Φ_{N} as the combinatorics of N; it is a “poset over 𝒮”.

Definition 8. A poset (partially ordered set) over a set 𝒮 is a set Φ equipped with a partial ordering (reflexive antisymmetric transitive relation) and a map 𝒮 → Φ. A morphism Φ → Φ′ of posets over 𝒮 is an order preserving map whose composite with the map 𝒮 → Φ is the map 𝒮 → Φ′. We say Φ and Φ′ have the same combinatorial type if they are isomorphic as posets over 𝒮. The combinatorics arising in toric geometry are typically isomorphic to subposets of the power set P (𝒮) or its opposite P (𝒮)^{op}, which are posets over 𝒮 under inclusion or reverse inclusion respectively, with the map from 𝒮 being the singleton map s ↦ {s}.

To illustrate this, we start, as in Stratagem 1, with an exact sequence

0 → ℝ →^{ε} 𝔥 →^{d} 𝔱 → 0

of vector spaces, viewed as an extension of abelian Lie algebras with dim 𝔱 = m, and let 𝒜 := (ε^{⊤})^{−1}(1) be the corresponding m-dimensional affine subspace of 𝔥^{∗}. Recall that a convex polytope Δ in 𝒜 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.

Remark 4. The combinatorics Φ_{Δ} of Δ is the poset over 𝒮 of closed faces of Δ. More precisely, for ξ ∈ 𝒜, let S_{ξ} = {s ∈ 𝒮 | L_{s}(ξ) = 0}; we assume that Δ ⊆ 𝒜 has nonempty interior Δ̊ (so for ξ ∈ Δ̊, S_{ξ} = ∅) and that for any s ∈ 𝒮, there exists ξ ∈ Δ with S_{ξ} = {s} (otherwise we may discard s without changing Δ). The map sending S ⊆ 𝒮 to F_{S} := {ξ ∈ Δ | S ⊆ S_{ξ}} = {ξ ∈ Δ | ∀s ∈ S, L_{s}(ξ) = 0} restricts to an isomorphism from {S_{ξ} ∈ P (𝒮)^{op} | ξ ∈ Δ} to Φ_{Δ} over 𝒮. Any closed face is thus the intersection of the facets F_{s} := F_{{s}} containing it: F_{S} = ⋂_{s∈S} F_{s}. We assume that the empty face is an element of Φ_{Δ}, so that F_{S} ∈ Φ_{Δ} for all S ∈ P (𝒮).

Given a compact convex polytope Δ ⊆ 𝒜, the positive span ℝ^{+}Δ is a cone in 𝔥^{∗}; the dual cone to Δ is

Δ^{∗} := {L ∈ 𝔥 | ∀ξ ∈ Δ, L(ξ) ≥ 0},

and its projection onto 𝔱 defines a decomposition Y of 𝔱, called the associated (complete) fan, into a union of polyhedral cones

C_{S} := {dL ∈ 𝔱 | ∀ξ ∈ Δ, L(ξ) ≥ 0 with equality for all ξ ∈ F_{S}} = span{u_{s} | s ∈ S}

corresponding to the faces F_{S} of Δ. These cones form a poset Φ_{Y} over 𝒮 under inclusion, and a (complete) fan Y is uniquely determined by its combinatorics Φ_{Y} and its rays (one dimensional cones) C_{s} := C_{{s}}. When Y is constructed from Δ as above then Φ_{Y} is canonically isomorphic to Φ_{Δ}^{op} over 𝒮, and in particular there is a canonical bijection between the facets of Δ and the rays C_{s} := C_{{s}} of Y.

The rays of Y determine u_{s} : s ∈ 𝒮 up to positive scale, and similarly Δ determines L_{s} : s ∈ 𝒮 up to positive scale. Given a choice of these scales, we say Δ is a labelled polytope with affine normals L_{s} ∈ 𝔥 and inward normals u_{s} := dL_{s} ∈ 𝔱 for s ∈ 𝒮, and that Y is a labelled fan with generators u_{s} ∈ 𝔱 for s ∈ 𝒮.

Definition 9. A (complete) fan Y is simplicial if the rays in any cone are linearly independent. A (compact) convex polytope Δ is simple if its fan is simplicial. In terms of a labelling, this means for all S ∈ Φ_{Y}, u_{s} : s ∈ S is linearly independent, or (for polytopes) ∀ξ ∈ Δ, B_{ξ} := (u_{s} : s ∈ S_{ξ}) is linearly independent. This condition only depends on the vertices ξ of Δ, where it means that B_{ξ} is a basis; in particular, each vertex is m-valent.

Returning to (N, 𝒟, J, K), the underlying contact manifold (N, 𝒟) is toric under (K, 𝕋_{N}) in the sense of [7], where the following result is established.

Theorem 1. Let (N, 𝒟, K) be a (compact, connected) toric contact manifold under 𝕋_{N}. Then the stabilizers in 𝕋_{N} of points in N are connected (i.e., subtori) and the fibres of the momentum map μ on U_{𝒟} are 𝕋_{N}-orbits. For any nondegenerate Levi pair (𝔤, λ) the signs of the primitive generators v_{s} ∈ 𝔱_{N} of the circle stabilizers of N_{s} : s ∈ 𝒮 may be chosen uniquely such that the image of the horizontal momentum map μ^{λ}: N → 𝒜 ⊆ 𝔥^{∗} is the compact simple convex polytope Δ in A defined by the affine functions L_{s} := L_{N}(v_{s}) and μ^{λ} is a submersion over the interior of each face.

In particular, μ^{λ} induces a poset isomorphism over 𝒮 of Φ_{N} with Φ_{Δ}.

Corollary 1. N and Δ (i.e., Φ_{N} and Φ_{Δ}) have the same combinatorial type.

Remark 5. The primitive generators v_{s} : s ∈ 𝒮 need not be linearly independent in 𝔱_{N}, nor even distinct, although after taking a quotient of N by a subtorus acting freely, we may assume that they span.

Suppose now that G is a closed subgroup of 𝕋_{N} with Lie algebra 𝔤. Such a subgroup exists if and only if the lattice of circle subgroups Λ_{N} in 𝔱_{N} is mapped to a (rank m) lattice Λ in 𝔱, which holds if and only if u_{s} : s ∈ 𝒮 span a lattice in 𝔱.

Definition 10. If Λ ⊆ 𝔱 is a lattice, then a polytope Δ or fan Y labelled by L_{s} (with u_{s} = dL_{s}) or u_{s} : s ∈ 𝒮 is rational with respect to Λ if for all s ∈ 𝒮, u_{s} ∈ Λ.

We are now ready to study the Levi–Kähler quotient N/G, which is a compact toric Kähler orbifold M of real dimension 2m under an m-torus 𝕋 with Lie algebra 𝔱 and hamiltonian generators 𝔥. Indeed, with respect to the symplectic form on N/G induced by dη^{λ}|_{𝒟}, μ^{λ}: N → 𝒜 ⊆ 𝔥^{∗} descends to a (natural) momentum map for the action of 𝕋 = 𝕋_{N}/G on N/G, whose image is the the rational simple convex polytope Δ ⊆ 𝒜. Rational Delzant theory [26, 41] asserts that any such toric Kähler orbifold M is determined up to symplectomorphism or biholomorphism by its labelled polytope or fan. The construction of M from these data is relevant here, so we review it now.

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} and 𝕋_{𝒮}^{ℂ} = ℂ_{𝒮}/2πℤ_{𝒮} ≅ ℂ_{𝒮}^{×} be the corresponding real and complex tori. Denote the generators of ℤ_{𝒮} ⊆ 𝔱_{𝒮} ⊆ ℂ_{𝒮} by e_{s} : s ∈ 𝒮, and observe that 𝕋_{𝒮} and 𝕋_{𝒮}^{ℂ} 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

(9)  ⟨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}.

The labellings s → L_{s} and s ↦ u_{s} of Δ and Y induce, and are defined by (without loss, surjective) linear maps L: 𝔱_{𝒮} → 𝔥 and u: 𝔱_{𝒮} → 𝔱 with L(e_{s}) = L_{s} and u(e_{s}) = u_{s} for all s ∈ 𝒮. Let 𝔤̃ be the kernel of u; then L determines a linear form λ̃ ∈ 𝔤̃^{∗} completing the following diagram:

(10)  [figure: 0; 𝔤̃; ι̃; 𝔱_{𝒮}; u; 𝔱; 0; 0; ℝ; λ̃; ε; 𝔥; L; d; 𝔱; ‖; 0.]

When Δ (or Y) is rational, then u maps ℤ_{𝒮} to Λ and hence defines a map from 𝕋_{𝒮} = 𝔱_{𝒮}/2πℤ_{𝒮} to 𝔱/2πΛ whose kernel is a closed subgroup G̃ of 𝕋_{𝒮} with Lie algebra 𝔤̃.

The combinatorics Φ_{Y} of Y (or Δ) define an open subset ℂ_{𝒮}^{∘} ⊆ ℂ_{𝒮} as the union of z ∈ ℂ_{𝒮} for which S_{z} := {s ∈ 𝒮 | z_{s} = 0} is in Φ_{Y}. In other words, ℂ_{𝒮}^{∘} is the union of the 𝕋_{𝒮}^{ℂ}-orbits ℂ_{𝒮,S} := {z ∈ ℂ_{𝒮} | z_{s} = 0 iff s ∈ S} over S ∈ Φ_{Y}. Thus the set of 𝕋_{𝒮}^{ℂ} orbits in ℂ_{𝒮}^{∘} is isomorphic to Φ_{Y}, and S ⊆ S′ iff the closure of ℂ_{𝒮,S} contains ℂ_{𝒮,S′}.

Lemma 3. Let (Y, u) be a simplicial fan with combinatorics Φ_{Y}. Then 𝔤̃ ⊆ 𝔱_{𝒮} acts locally freely on ℂ_{𝒮}^{∘}. If in addition (Y, u) is rational, and G̃ is the corresponding closed subgroup of 𝕋_{𝒮}, then for any S ∈ Φ_{Y}, the stabilizer in G̃ of any z ∈ ℂ_{𝒮,S} is StabG̃(z) ≅ (Λ ∩ span_{ℝ}{u_{s} | s ∈ S})/span_{ℤ}{u_{s} | s ∈ S}.

Proof. Let S ∈ Φ_{Y} and z ∈ ℂ_{𝒮,S}, so that z_{s} = 0 iff s ∈ 𝒮. Then any element of the stabilizer of z in 𝔱_{𝒮} has the form v = ∑_{s∈S} t_{s}e_{s}, which belongs to 𝔤̃ iff ∑_{s∈S} t_{s}u_{s} = 0. However, since (Y, u) is simplicial, u_{s} : s ∈ S is linearly independent, hence v = 0.

For the second part, an element [v] = [∑_{s∈S} t_{s}e_{s}] of the stabilizer of z in 𝕋_{𝒮} = 𝔱_{𝒮}/2πℤ_{𝒮} is in G̃ iff ∑_{s∈S} t_{s}u_{s} ∈ Λ, and is the identity element iff t_{s} ∈ ℤ for all s ∈ S. The result follows. □

The Delzant–Lerman–Tolman correspondence and the relation between symplectic and complex (GIT) quotients [26, 41] now assert that:

(i) as a complete toric variety, M is a complex (GIT) quotient ℂ_{𝒮}^{∘}/G̃^{ℂ} of ℂ_{𝒮} by G̃^{ℂ}; (ii) as a compact toric symplectic orbifold, M is a symplectic quotient Ñ/G̃—where Ñ = (ι̃^{⊤}σ)^{−1}(λ̃)—of ℂ_{𝒮} by G̃ at momentum level λ̃ ∈ 𝔤̃^{∗}. Here the labellings of the polytope and fan are related to the orbifold structure groups of M by Lemma 3. Note that the complex structure in (i) is biholomorophic to the complex structure of the Kähler quotient in (ii), which is the quotient of the induced CR structure on Ñ ⊆ ℂ_{𝒮}. However, M is not typically isometric to the Kähler quotient of ℂ_{𝒮} by G, which is called the Guillemin metric [34, 1, 2] of Δ.

In particular, the Levi–Kähler quotient (N/G, dη^{λ}|_{𝒟}, J|_{𝒟}) is symplectomorphic to the toric symplectic orbifold obtained from (Δ, L) by the Delzant–Lerman–Tolman construction, but also biholomorphic to the Levi–Kähler quotient of the toric CR submanifold Ñ of flat space by (𝔤̃, λ̃). In general N and Ñ have different dimensions, but there is a map from Λ to Λ_{N} sending e_{s} to v_{s}, and if the latter form a basis for Λ_{N}, then we may identify 𝕋_{N} with 𝕋_{𝒮}, L_{N} with L, and hence (𝔤̃, λ̃) with (𝔤, λ). This motivates the study of Levi–Kähler quotients of toric CR submanifolds of flat space, which will occupy us for the remainder of the paper.

2.2. Toric CR submanifolds of flat space. Let 𝒮 be a d element set, and define ℤ_{𝒮}, 𝔱_{𝒮}, ℂ_{𝒮}, 𝕋_{𝒮} and 𝕋_{𝒮}^{ℂ} as in §2.1. Let K: 𝔱_{𝒮} → 𝔥𝔞𝔪(ℂ_{𝒮}, ω_{𝒮}) be the (local) hamiltonian action, and let ϑ: ℂ_{𝒮}^{×} → 𝕋_{𝒮} be angular coordinates conjugate to the momentum components (9) on the open set ℂ_{𝒮}^{×} of ℂ_{𝒮} where z_{s} ≠ 0 for all s ∈ 𝒮; thus dϑ: T ℂ_{𝒮}^{×} → 𝔱_{𝒮} satisfies dϑ(K_{v}) = v and dϑ(JK_{v}) = 0 for all v ∈ 𝔱_{𝒮}. The flat Kähler metric in action-angle coordinates on ℂ_{𝒮}^{×} is then

(11) g_{𝒮} = ∑_{s∈𝒮}(\frac{dσ_{s}^{2}}{2σ_{s}} + 2σ_{s}dϑ_{s}^{2}), ω_{𝒮} = ∑_{s∈𝒮} dσ_{s} ∧ dϑ_{s}, d^{c}σ_{s} := Jdσ_{s} = 2σ_{s}dϑ_{s}.

In particular, the metric H(v, w) := g_{𝒮}(K_{v}, K_{w}) on the 𝕋_{𝒮}-orbits is given by the smooth function H = 2δσ: ℂ_{𝒮} → S^{2}𝔱_{𝒮}^{∗}, where δ: 𝔱_{𝒮}^{∗} → S^{2}𝔱_{𝒮}^{∗} ⊆ 𝔱_{𝒮}^{∗} ⊗ 𝔱_{𝒮}^{∗} is the coproduct dual to componentwise multiplication in 𝔱_{𝒮}; thus if we write v = ∑_{s∈𝒮} v_{s}e_{s} and w = ∑_{s∈𝒮} w_{s}e_{s} then H_{z}(v, w) = ∑_{s∈𝒮} 2σ_{s}(z)v_{s}w_{s}, which is positive definite for z ∈ ℂ_{𝒮}^{×}. Note the following crucial property of the flat Kähler metric on ℂ_{𝒮}: d^{c}σ_{s}(K_{v}) = 2σ_{s}v_{s} for all v ∈ 𝔱_{𝒮}, i.e.,

(12)  d^{c}σ(K_{v}) = H(v) = 2(δσ)(v),

where we use the natural inclusion S^{2}𝔱_{𝒮}^{∗} ⊆ Hom(𝔱_{𝒮}, 𝔱_{𝒮}^{∗}) to evaluate H = 2δσ on v. We now restrict attention to Levi–Kähler quotients in the following setting.

Definition 11. A toric CR submanifold of ℂ_{𝒮} is a compact connected CR submanifold (N, 𝒟, J) which is invariant and locally Reeb type under the action of 𝕋_{𝒮}.

We assume that for any S ⊆ 𝒮, the intersection of N with 𝕋_{𝒮}^{ℂ} orbit ℂ_{𝒮,S} is connected; these intersections are then the orbit strata, and the combinatorics Φ_{N} of N may be identified with the poset of S ⊆ 𝒮 such that N ∩ ℂ_{𝒮,S} is nonempty. We also assume that for all s ∈ 𝒮, {s} ∈ Φ_{N}, i.e., N_{s} := N ∩ ℂ_{𝒮,{s}} = {z ∈ N | z_{s} = 0} is nonempty, with generic stabilizer <exp(te_{s})> (if this did not hold, the <exp(te_{s})> circle action on N would be free, and we could take a quotient).

We refer to a Levi–Kähler quotient M of a toric codimension ℓ CR submanifold (N, 𝒟, J) in ℂ_{𝒮} by a positive Levi pair (𝔤, λ), where 𝔤 is the Lie algebra of an ℓdimensional subtorus G ⊆ 𝕋_{𝒮}, as a (codimension ℓ) Levi–Kähler reduction of ℂ_{𝒮}.

The data (N, 𝒟, J) and (𝔤, λ) are linked by Condition 1, which may be viewed as a constraint on (N, 𝒟, J) given (𝔤, λ) or vice versa. We specify the choice of (𝔤, λ) as in §1.3 via a surjective linear map L: 𝔱_{𝒮} → 𝔥, or equivalently, an indexed family L_{s} : s ∈ 𝒮 of vectors in 𝔥 which span (where L_{s} = L(e_{s})). In other words, for toric CR submanifolds (N, 𝒟, J) of ℂ_{𝒮}, a pair (𝔤, λ) is associated canonically, via the set-up (10), with a (not necessarily compact or nonempty) convex polytope Δ_{𝔤,λ} = {ξ ∈ 𝒜 | ∀ s ∈ 𝒮, L_{s}(ξ) ≥ 0} labelled (formally) by L_{s} : s ∈ 𝒮 (although some facets F_{s} could be empty a priori). We denote the combinatorics of Δ_{𝔤,λ} by Φ_{𝔤,λ}.

Lemma 4. Let N be a toric CR submanifold of ℂ_{𝒮} satisfying Condition 1 relative to (𝔤, λ). Then there is a smooth pointwise surjective function χ_{N,𝔤}: N → Hom(𝔱_{𝒮}^{∗}, 𝔤) such that for all z ∈ N, η_{z} = χ_{N,𝔤}(z) ∘ d^{c}σ_{z} and dη_{z}|_{𝒟_{z}} = χ_{N,𝔤}(z) ∘ dd^{c}σ_{z}|_{𝒟_{z}}.

Proof. The CR submanifold N may be written (at least locally, and in our examples globally) N = (F ∘ σ)^{−1}(0) where F: 𝔱_{𝒮}^{∗} → W is a smooth function with values in an ℓdimensional vector space W, for which 0 is a regular value. Hence T_{z}N = ker dF_{σ(z)}∘dσ_{z} and so 𝒟 is the kernel of the pullback ν of dF ∘d^{c}σ to N, with ν_{z}(K_{v}) = dF_{σ(z)}(H_{z}(v)) for v ∈ 𝔱_{𝒮} and z ∈ N. By Condition 1, dF ∘ H ∘ ι: N → Hom(𝔤, W) is a pointwise isomorphism, and η = (dF ∘ H ∘ ι)^{−1}ν.

We may now set χ_{N,𝔤} = (dF ∘ H ∘ ι)^{−1} ∘ dF; this formula may only be valid locally, but the result is independent of the choice of F: 𝔱_{𝒮}^{∗} → W, so χ_{N,𝔤}, with η = χ_{N,𝔤} ∘ d^{c}σ is globally defined. Since ν|_{𝒟} = 0, dη|_{𝒟} = (dF ∘ H ∘ ι)^{−1}dν|_{𝒟}. Now d(dF ∘ d^{c}σ) = (Hess F)(dσ ∧ d^{c}σ) + dF ∘ dd^{c}σ. Pulling back to N and restricting to 𝒟, the first term vanishes (since F ∘ σ is constant on N). Hence dη|_{𝒟} = χ_{N,𝔤} ∘ dd^{c}σ|_{𝒟}. □

The characteristic function of (N, 𝔤, λ) is the (nowhere vanishing) function χ = χ_{N,𝔤,λ}: N → 𝔱_{𝒮} with χ_{N,𝔤,λ}(z) = λ ∘ χ_{N,𝔤}(z). Hence η^{λ} := ⟨η, λ⟩ = ⟨d^{c}σ, χ_{N,𝔤,λ}⟩ and the horizontal momentum map μ^{λ}: N → A ⊆ 𝔥^{∗} satisfies:

(13)  ⟨μ^{λ}(z), L(v)⟩ = η_{z}^{λ}(K_{v}) = H_{z}(v, χ_{N,𝔤,λ}(z)) = 2⟨δσ(z), v ⊗ χ_{N,𝔤,λ}(z)⟩.

Thus η^{λ} = ∑_{s∈𝒮} χ_{s} d^{c}σ_{s} and ⟨μ^{λ}, L(v)⟩ = ∑_{s∈𝒮} 2σ_{s}χ_{s}v_{s}, i.e., L_{s}(μ^{λ}) = 2σ_{s}χ_{s}. Since dd^{c}σ_{s} = 2dσ_{s} ∧ dϑ_{s}, the induced metric on 𝒟 (over ℂ_{𝒮}^{×} ∩ N) is

(14)  h_{𝒟,λ} = ∑_{s∈𝒮} \frac{L_{s}(μ^{λ})}{σ_{s}}(\frac{dσ_{s}^{2}}{2σ_{s}} + 2σ_{s}dϑ_{s}^{2})|_{𝒟} = ∑_{s∈𝒮} 2χ_{s}(\frac{dσ_{s}^{2}}{2σ_{s}} + 2σ_{s}dϑ_{s}^{2})|_{𝒟}.

Theorem 1 shows that if (𝔤, λ) is a nondegenerate Levi pair, then the image of the horizontal momentum map μ^{λ} is the compact simple convex polytope Δ, defined by ±L_{s} : s ∈ 𝒮 for some choice of signs, and also that Δ has the same combinatorial type as N. Now if Δ = Δ_{𝔤,λ} (i.e., all signs are positive) then equation (14) shows that (𝔤, λ) is a positive Levi pair. This motivates the introduction of the following constraint.

Condition 2. Δ_{𝔤,λ} is a compact convex polytope with the same combinatorial type as N (as a subset of P (𝒮)).

Theorem 2. Let N be a toric submanifold of ℂ_{𝒮} and suppose (𝔤, λ) is a Levi pair. Then im μ^{λ} = Δ_{𝔤,λ} if and only if (𝔤, λ) is a positive Levi pair satisfying Condition 2.

Proof. If im μ^{λ} = Δ_{𝔤,λ}, then (14), applied to each orbit stratum, shows that h_{𝒟,λ} is positive definite over the interior of each face of Δ_{𝔤,λ}, hence everywhere. Thus (𝔤, λ) is a positive Levi pair. Under this assumption, Theorem 1 shows that im μ^{λ} = Δ_{𝔤,λ} if and only if Condition 2 holds. □

2.3. Levi–Kähler reduction for quadrics. We now specialize to the case that N is an intersection of quadrics. For N to be toric with codimension ℓ, it is then the level set of an ℓ-dimensional family of components of σ: ℂ_{𝒮} → 𝔱_{𝒮}^{∗}, hence of the form (F ∘ σ)^{−1}(0), with F = ι_{o}^{⊤} − λ_{o}: 𝔱_{𝒮}^{∗} → 𝔤_{o}^{∗}, where ι_{o}: 𝔤_{o} ↪ 𝔱_{𝒮} is an inclusion of an ℓ-dimensional subspace, and λ_{o} ∈ 𝔤_{o}^{∗} is in the image of the positive quadrant of 𝔱_{𝒮}^{∗}.

Thus N = μ_{o}^{−1}(λ_{o}), where μ_{o} = ι_{o}^{⊤}σ, is defined by the same sort of data (𝔤_{o}, λ_{o}) as the data (𝔤, λ) which determines the Levi–Kähler structure on N. These data may therefore be fixed in the same way as (𝔤, λ) using a diagram of linear maps

[figure: 0; 𝔤_{o}; ι_{o}; 𝔱_{𝒮}; u^{o}; 𝔱; 0; 0; ℝ; λ_{o}; ε; 𝔥; L^{o}; d; 𝔱; ‖; 0.]

We write N_{𝔤_{o},λ_{o}} or N_{L^{o}} for the CR submanifold corresponding to these data. We shall assume that Δ_{𝔤_{o},λ_{o}} is a compact convex polytope, so that it satisfies Condition 2: the image of σ: N → 𝔱_{𝒮}^{∗} thus lies in the nonnegative quadrant of 𝔱_{𝒮}^{o} := {ξ ∈ 𝔱_{𝒮}^{∗} | ∀ v ∈ 𝔤_{o}, ξ_{s}v_{s} = 0 ∀ s ∈ 𝒮 ⇒ v = 0}, and u_{s}^{o} : s ∈ 𝒮 are the normals of a complete fan.

Since F is affine linear, dF is constant, equal to ι_{o}^{⊤}, and so ν_{z}(K_{v}) = ι_{o}^{⊤}(H_{z}(v)). Hence E_{z} = im(H_{z} ∘ ι_{o}: 𝔤_{o} → 𝔱_{𝒮}^{∗}) and so E = σ^{∗}E^{o} where E^{o} → 𝔱_{𝒮}^{o} has fibre E_{ξ}^{o} = {(ξ_{s}v_{s})_{s∈𝒮} ∈ 𝔱_{𝒮}^{∗} | v ∈ 𝔤_{o}}.

We can satisfy Condition 1 by letting (𝔤, λ) equal (𝔤_{o}, λ_{o}).

Proposition 4. On N_{𝔤_{0},λ_{0}}, (𝔤_{o}, λ_{o}) is a positive Levi pair.

Proof. Since u_{s}^{o} : s ∈ 𝒮 are the normals of a complete fan, {α ∈ 𝔱^{∗} | ∀ s ∈ 𝒮, α(u_{s}^{o}) ≥ 0} = {0}. Hence (u^{o})^{⊤}(𝔱^{∗}) meets the positive quadrant of 𝔱_{𝒮}^{∗} only at 0, so the image in 𝔤_{o}^{∗} of this positive quadrant is a strictly convex cone C whose dual cone C_{∗} is the intersection of 𝔤_{o} with the inverse image (under ι_{o}) of the positive quadrant in 𝔱_{𝒮}. Since H_{z} is diagonal and positive definite, it maps the positive quadrant of 𝔱_{𝒮} onto the positive quadrant of 𝔱_{𝒮}^{∗}. Thus ι_{o}^{⊤} ∘ H_{z} ∘ ι_{o} maps C_{∗} onto C. Since λ_{o} ∈ C, χ^{o}(z) := ι_{o}(ι_{o}^{⊤} ∘ H_{z} ∘ ι_{o})^{−1}(λ_{o}) has positive components, and hence h_{𝒟,λ_{o}} is positive definite. □

If the fan associated to (𝔤_{o}, λ_{o}) is rational, (M, J) is the underlying complex orbifold of the Delzant–Guillemin Kähler quotient of ℂ_{𝒮} by (𝔤_{o}, λ_{o}). Its Kähler form belongs to the same Kähler class as the Levi–Kähler quotient, but will not be the same in general.

Remark 6. By continuity, we also obtain a positive definite metric for (𝔤, λ) in an open neighbourhood of (𝔤_{o}, λ_{o}). In particular, we can fix 𝔤 = 𝔤_{o} and vary λ to obtain an ℓdimensional family of Levi–Kähler quotients on the same complex orbifold. As H_{dR}^{2}(M) is ℓ-dimensional, it is natural to ask whether λ effectively parametrizes the Kähler cone of (M, J).

The characteristic function χ = χ_{N,𝔤,λ} of N = N_{𝔤_{o},λ_{o}} with respect to an arbitrary Levi pair (𝔤, λ) is given by χ(z) = (ι^{⊤} ∘ H_{z} ∘ ι_{o})^{−1}(λ) ∈ 𝔤_{o}, where we tacitly omit the inclusion ι_{o}: 𝔤_{o} ⊆ 𝔱_{𝒮}.

Remark 7. The function χ: N → 𝔤_{o} is determined by H(χ)|_{𝔤} = λ; since H = 2δσ, it is linear in σ, which implies χ is a rational function of σ. Now for any v ∈ 𝔤_{o} and z ∈ N, \frac{1}{2}H(∑_{s∈𝒮} e_{s}) = ∑_{s∈𝒮}⟨σ_{s}(z), v⟩ = λ_{o}(v), and so the characteristic function χ^{o} of the canonical Levi pair (𝔤_{o}, λ_{o}) is characterized by ι_{o}^{⊤}H_{z}(χ^{o}(z) − \frac{1}{2} ∑_{s∈𝒮} e_{s}) = 0.

Proposition 5. If ∑_{s∈𝒮} u_{s}^{o} = 0 then 2χ_{s}^{o} = 1 for all s ∈ 𝒮.

Proof. Since ∑_{s∈𝒮} e_{s} ∈ 𝔤_{o}, χ^{o}(z) = \frac{1}{2} ∑_{s∈𝒮} e_{s} for all z ∈ N by the characterization. □

If this assumption holds, we say N is spherical: N is then contained in a hypersphere in ℂ_{𝒮}. (The equivariant topology of such manifolds has been studied in [15], but here we focus on the geometry of their Levi–Kähler quotients.) In the spherical case, σ_{s} = ⟨μ^{λ_{o}}, L_{s}^{o}⟩, and the canonical Levi–Kähler quotient metric agrees with the Delzant–Guillemin Kähler quotient. In particular, the reduced metric on 𝒜 ⊆ 𝔥^{∗} is ∑_{s∈𝒮}(dL_{s}^{o})^{2}/2L_{s}^{o}, which is the pullback by (L^{o})^{⊤} of the metric h_{o} = ∑_{s∈𝒮} dζ_{s}^{2}/2ζ_{s} on 𝔱_{𝒮}^{∗} , where we write ζ_{s} for the linear function ζ_{s}(ξ) = ξ_{s} on 𝔱_{𝒮}^{∗} corresponding to e_{s} ∈ 𝔱_{𝒮} (thus dζ_{s} is e_{s}, viewed as a constant 1-form).

In order to compute the reduced metric for any Levi pair (𝔤, λ), we observe that the bundle E^{o} ⊆ T 𝔱_{𝒮}^{o} = 𝔱_{𝒮}^{o}×𝔱_{𝒮}^{∗} (with E = σ^{∗}E^{o}) is the orthogonal complement to 𝔱_{𝒮}^{o}×(u^{o})^{⊤}(𝔱^{∗}) with respect to h_{o}:

∑_{s∈𝒮} \frac{dζ_{s}((ξ_{s}v_{s})_{s∈𝒮})dζ_{s}(w ∘ u^{o})}{ζ_{s}(ξ)} = ∑_{s∈𝒮} v_{s}(w ∘ u^{o})_{s} = w(u^{o}(v)) = 0.

Hence by Remark 3, we have the following result.

Theorem 3. Let (𝔤, λ) be a positive Levi pair on a spherical quadric N = N_{𝔤_{o},λ_{o}}. Then the reduced metric of the Levi–Kähler structure is the pullback by L^{⊤}: Δ_{𝔤,λ} → 𝔱_{𝒮}^{o} of the restriction of h_{o} to 𝔱_{𝒮}^{o} × (u^{o})^{⊤}(𝔱^{∗}) ⊆ T 𝔱_{𝒮}^{o} (extended by zero on E^{o} ⊆ T 𝔱_{𝒮}^{o}).

Example 3. The weighted projective space ℂP_{a}^{m} of weight a = (a_{0}, …, a_{m}) ∈ ℕ^{m+1} has the structure of a toric symplectic orbifold whose Delzant polytope is a labelled simplex (Δ, u). The corresponding momentum level set N_{λ} ⊆ ℂ^{m+1} is CR G-equivariantly isometric to the sphere 𝕊^{2m+1} ⊆ ℂ^{m+1}, acted by the a-weighted diagonal 𝕊^{1} action. By a result of S. Webster [44], Levi–Kähler reduction defines on ℂP_{a}^{m} a homothety class of Bochner-flat Kähler metrics [18, 25], which are extremal Kähler metrics [20]. The Bochner-flat metric coincides with the Guillemin symplectic-Kähler reduction if and only if a = (1, …, 1), i.e., only on ℂP^{m}, see e.g., [2]. Thus one can obtain Levi–Kähler quotients of the same (flat) CR structure on 𝕊^{2m+1} ⊆ ℂ^{m+1} on any labelled rational simplex, by varying the subgroup G ≅ 𝕊^{1} within a fixed maximal torus 𝕋^{m+1} in the group Aut_{CR}(𝕊^{2m+1}) = PU(m + 1, 1) of CR transformations of 𝕊^{2m+1}.

Locally, the construction is defined by a one-dimensional subspace 𝔤 ⊆ 𝔱_{𝒮}, generated by a non-zero element v ∈ 𝔱_{𝒮}, with corresponding vector field K_{v} transverse to the CR distribution on 𝕊^{2m+1}, and the choice of a contact form η^{v} with η^{v}(K_{v}) = 1 and ker η^{v} = 𝒟. In this case (K_{v}, η_{v}, 𝒟, J) defines a Sasaki structure compatible with the standard CR structure (𝒟, J) on 𝕊^{2m+1}, see e.g. [17]. The horizontal Kähler geometry (dη^{v}, 𝒟, J) may be described by a compatible toric metric over a (perhaps not rational) labelled simplex (see [37]). The fact that 𝔤 is the Lie algebra of a subgroup G ≤ 𝕋^{m+1} implies a rationality condition on 𝔤, hence on the corresponding labelled simplex.

3. Levi–Kähler reduction for products of spheres

Our main motivation for the study of toric Levi–Kähler quotients is the construction of Kähler metrics on toric varieties with “nice” curvature properties. For this we observe that CR submanifolds of ℂ_{𝒮} have local invariants, and so one approach to constructing Levi–Kähler quotients with nice curvature is to start from a nice CR submanifold N of ℂ_{𝒮}. In particular, when N is a product of spheres, it is flat as a CR manifold. Hence we might hope that Levi–Kähler quotients of products of spheres have interesting curvature properties.

3.1. Products of simplices and products of spheres. We specialize the set-up of §2 as follows. Fix positive integers ℓ and m_{1}, m_{2}, … m_{ℓ}, and let I = {1, 2, … ℓ}, I_{i} = {0, 1, … m_{i}}, 𝒮 = {(i, r) | i ∈ I and r ∈ I_{i}}. Let m = ∑_{i∈I} m_{i} and d = m + ℓ as usual. Thus ℂ_{𝒮} ≅ ℂ^{m_{1}+1} × ℂ^{m_{2}+1} × ⋯ × ℂ^{m_{ℓ}+1} ≅ ℂ^{d} and 𝔱_{𝒮} has a natural subspace 𝔤_{o} = {x ∈ 𝔱_{𝒮} | x_{iq} = x_{ir} for all i ∈ I and q, r ∈ I_{i}}. We denote by x_{i} the common value of the x_{ir} and thus identify 𝔤_{o} with ℝ^{ℓ}. On 𝔤_{o} we have a natural linear form λ_{o} sending (x_{1}, x_{2}, … x_{ℓ}) to x_{1} + x_{2} + ⋯ + x_{ℓ} ∈ ℝ, and we let L^{o}: 𝔱_{𝒮} → 𝔥 = 𝔱_{𝒮}/ ker λ_{o} and u^{o}: 𝔱_{𝒮} → 𝔱 = 𝔱_{𝒮}/𝔤_{o} be the quotient maps.

Under the canonical identification of 𝔱_{𝒮}^{∗} with 𝔱_{𝒮}, 𝔥^{∗} is isomorphic to the subspace of ξ = (ξ_{s})_{s∈𝒮} such that ∑_{r∈I_{i}} ξ_{ir} is independent of i, this constant being the natural projection 𝔥^{∗} → ℝ. Hence 𝔱^{∗} is a product (over i ∈ I) of the codimension one linear subspaces of ℝ^{m_{i}+1} where the coordinate sum is zero, and A is the corresponding product of affine subspaces A_{i} where the coordinate sum is one.

Notation 1 (The faces of Σ). The polytope Σ in 𝔥^{∗} defined by L^{o} is a product of simplices Σ_{i} in the affine spaces A_{i}. In the following, we sometimes write i(s) and r(s) for the components of s ∈ 𝒮.

• The facets of Σ_{i} are F_{r}^{i} = {ξ_{ir} = 0} ∩ Σ_{i} for r ∈ I_{i}.

• The vertices of Σ_{i} may also be indexed by r ∈ I_{i}: we let p_{r}^{i} be the unique vertex of Σ_{i} that is not in F_{r}^{i}.

• The vertices of Σ are thus indexed by (r_{1}, … r_{ℓ}) ∈ I_{1} × ⋯ × I_{ℓ}:

p_{(r_{1},...r_{ℓ})} = (p_{r_{1}}^{1}, … p_{r_{ℓ}}^{ℓ}).

• Each facet F_{r}^{i} (for r ∈ I_{i}) of the simplex Σ_{i} determines a facet

F_{ir} = Σ_{1} × ⋯ × Σ_{i−1} × F_{r}^{i} × Σ_{i+1} × ⋯ × Σ_{ℓ}

of Σ and a corresponding inward normal u_{ir}^{o} = dL_{ir}^{o}.

The corresponding CR submanifold of ℂ_{𝒮} is

N = N_{L^{o}} = {z ∈ ℂ_{𝒮} | ∑_{r∈I_{i}} σ_{ir} = 1 for all i ∈ I},

where σ_{ir} = \frac{1}{2}|z_{ir}|^{2}. Thus N ≅ 𝕊^{2m_{1}+1} × ⋯ × 𝕊^{2m_{ℓ}+1}. As in §2.3, N is the level set, at the regular value λ_{o}, of the momentum map μ_{o} = ι_{o}^{⊤}σ. Thus

μ_{o}(z) = (σ^{1}(z), … σ^{ℓ}(z)), where σ^{i}(z) = ∑_{r∈I_{i}} σ_{ir}(z) = ∑_{r∈ I_{i}} \frac{1}{2}|z_{ir}|^{2},

and we denote z = (z^{1}, … z^{ℓ}) with z^{i} = (z_{i0}, …, z_{im_{i}}) the linear coordinates of ℂ_{𝒮}. These data are associated to the Delzant construction for the product Σ = Σ_{1} × ⋯ × Σ_{ℓ} ⊆ A of standard Delzant simplices Σ_{i} ⊆ A_{i}. More specifically, 𝔤_{o} is the Lie algebra of a subtorus G_{o} of 𝕋_{𝒮}, which acts freely on N preserving the CR structure (𝒟, J), with quotient space (M, J) = ℂP^{m_{1}} × ⋯ × ℂP^{m_{ℓ}}. The Lie algebra 𝔤_{o} of G_{o} defines a Reeb foliation on N ⊆ ℂ_{𝒮} with induced horizontal Levi structure consisting of scales of product of Fubini–Study metrics.

Theorem 4. Let N = 𝕊^{2m_{1}+1} × ⋯ × 𝕊^{2m_{ℓ}+1} ⊆ ℂ_{𝒮} be a product of standard CR spheres. Then for a pair (𝔤, λ), defined by L: 𝔱_{𝒮} → 𝔥, with associated polytope Δ_{𝔤,λ} ⊆ 𝒜, the following are equivalent.

(i) (𝔤, λ) is a positive Levi pair, i.e., defines a Levi–Kähler structure on N. (ii) (𝔤, λ) is a Levi pair (i.e., 𝔤 satisfies Condition 1) whose horizontal momentum map μ^{λ}: N → 𝒜 has im μ^{λ} = Δ_{𝔤,λ}. (iii) (𝔤, λ) satisfies Condition 2, i.e., Δ_{𝔤,λ} is a compact convex polytope with the same combinatorial type as Σ.

The proof makes use a couple of Lemmas.

Lemma 5. If (𝔤, λ) satisfies Condition 2, then it satisfies Condition 1.

Proof. Condition 1(i) holds since it only depends on the combinatorics of Δ.

Suppose that 𝔤 does not satisfy Condition 1(ii). Then there exist z ∈ N and v = (x_{s}) ∈ 𝔱_{𝒮} \ 0 such that K_{v}(z) ∈ 𝒟 = ⋂_{i∈I} ker d^{c}σ^{i} and v ∈ 𝔤, that is

(15)  ∑_{s∈𝒮} x_{s}u_{s} = 0 ∈ 𝔱 and ∑_{r∈I_{i}} x_{ir}σ_{ir} = 0  for i ∈ I,

where σ_{s} = \frac{1}{2}|z_{s}|^{2}. As equations on v = (x_{s}) for fixed z, this system is a linear map from 𝔱_{𝒮} to ℝ^{ℓ} ⊕ 𝔱, which both have dimension d. We may write the d × d-matrix A = A_{z} of this linear map as follows: for j ∈ I the j-th row is σ_{s}δ_{i(s)j}, while the lower part B of the matrix is the m × d-matrix, whose s-th column is u_{s} for s ∈ 𝒮 (written with respect to some basis of 𝔱). We compute the determinant of A = A_{z} by expanding along the first ℓ rows. The nonzero terms are all obtained by choosing, for each j ∈ I, r_{j} ∈ I_{j} to obtain a minor

±σ_{1r_{1}}σ_{2r_{2}} ⋯ σ_{ℓr_{ℓ}} det B_{(r_{1},...r_{ℓ})},

where B_{(r_{1},...r_{ℓ})} is the submatrix of B obtained by removing the columns u_{1r_{1}}, … u_{ℓr_{ℓ}}. Up to an overall sign (depending on |I_{j}|) each such minor contributes to det A with sign (−1)^{∑_{j∈I} r_{j}}. Hence to show det A ≠ 0, it suffices to show that (for a fixed basis of 𝔱) (−1)^{∑_{j∈I} r_{j}} det B_{(r_{1},...r_{ℓ})} > 0 for all (r_{1}, … r_{ℓ}) ∈ I_{1} × ⋯ × I_{ℓ}, because σ_{s} ≥ 0 and the products ∏_{j∈I} σ_{jr_{j}} do not all vanish at the same time.

Since Δ has the same combibatorial type as Σ, we know that the colomns of B_{(r_{1},...r_{ℓ})} are inward normals of the facets meeting at the vertex p_{(r_{1},...r_{ℓ})}, see Notation 1. These form a basis by the Delzant condition on Δ, and so it suffices to show that (−1)^{∑_{j=1}^{l} r_{j}} times the wedge product of the columns of B_{(r_{1},...r_{ℓ})} has sign independent of (r_{1}, … r_{ℓ}). This will hold for Δ if it holds for Σ, so it suffices to check that for each j ∈ I, (−1)^{r_{j}}u_{j0}^{o} ∧ ⋯ û_{jr_{j}}^{o} ∧ ⋯ u_{jm_{j}}^{o} (with the u_{jr_{j}}^{o} factor omitted) is independent of r_{j} ∈ I_{j}. Since ∑_{r∈I_{j}} u_{jr}^{o} = 0, this is a triviality. □

Lemma 6. Suppose (𝔤, λ) satisfies Condition 1 and let χ: N → 𝔤_{o} ≅ ℝ^{ℓ} be the characteristic function of (N, 𝔤, λ). Then (𝔤, λ) is a positive Levi pair if and only if χ_{1}, … χ_{ℓ} are positive.

Proof. First observe that since 𝒟 = ⋂_{i∈I} ker d^{c}σ^{i}, we have

(16)  η^{λ} = ∑_{i∈I} χ_{i} d^{c}σ^{i}, dη^{λ}|_{𝒟} = ∑_{i∈I} χ_{i} dd^{c}σ^{i}|_{𝒟}.

Moreover, 𝒟 splits as a sum 𝒟 = ⨁_{i∈I} 𝒟_{i} where 𝒟_{i} is tangent the i-th sphere in the product N = ∏_{i∈I} 𝕊^{2m_{i}+1} (that is 𝒟_{i} = T 𝕊^{2m_{i}+1} ∩ JT 𝕊^{2m_{i}+1}). For i ∈ I, dd^{c}σ^{i} is nondegenerate on 𝒟_{i}, and if j ≠ i, 𝒟_{j} ⊆ ker dd^{c}σ^{i}. Thus dη^{λ}|_{𝒟} defines a positive definite metric iff χ_{i} > 0 for all i ∈ I. □

Before proving the theorem we need a bit more notation. For each j ∈ I, we define

C_{j}^{±} := {ξ ∈ 𝒜 | ∀ r ∈ I_{j}, ±L_{jr}(ξ) ≥ 0}

where C_{j}^{−} is potentially empty but C_{j}^{+} is not, and Δ = ⋂_{j∈I} C_{j}^{+}.

Proof of Theorem 4. By Lemma 5, we may assume Condition 1 holds. Then the formula (13) for the induced momentum map μ^{λ} here reduces to

(17)  L_{ir}(μ^{λ}(z)) = 2χ_{i}(z)σ_{ir}(z)

where L_{ir} = L(e_{ir}) for the standard basis e_{ir} of 𝔱_{𝒮}.

If (𝔤, λ) is a positive Levi pair, the functions χ_{i} are positive by Lemma 6 and then equation (17) and Theorem 1 imply that im μ^{λ} = Δ_{𝔤,λ}; thus (i)⇒(ii).

Now (ii)⇒(i)&(iii) by Theorem 2, which also shows (i)⇒(iii).

Finally, to prove (iii)⇒(i), it suffices, by Lemma 6, to show that the functions χ_{i} are positive. First note the following consequences of equation (17).

(a) im μ^{λ} contains all the vertices of Δ = Δ_{𝔤,λ}. Indeed, each z ∈ N having only one nonzero coordinates in each spherical factor is sent to a vertex of Δ. Moreover, on the vertices of Δ, L_{s′} ≥ 0 for each s′ ∈ 𝒮; thus equation (17) implies that χ_{i}(z) ≥ 0 for any z ∈ N such that μ^{λ}(z) is a vertex of Δ.

(b) If L_{ir}(μ^{λ}(z)) ≥ 0 (resp. L_{ir}(μ^{λ}(z)) ≤ 0) then for all q ∈ I_{i}, L_{iq}(μ^{λ}(z)) ≥ 0 (resp.

L_{iq}(μ^{λ}(z)) ≤ 0). That is μ^{λ}(z) ∈ ⋂_{j∈I}(C_{j}^{+} ∪ C_{j}^{−}). Thanks to the statement (a) above, it is sufficient to prove that none of the χ_{i}’s vanishes on N. From statement (b) we have the following inclusion

im μ^{λ} ⊆ ⋂_{j∈I}(C_{j}^{+} ∪ C_{j}^{−}) = ⋃_{J ⊆I} Δ_{J} ,

where Δ_{J} = (⋂_{j∈J} C_{j}^{−}) ∩ (⋂_{j∈J^{c}} C_{j}^{+}) with J^{c} := I\J (so Δ_{∅} = Δ).

Statement (a) implies that Δ ∩ im μ^{λ} is not empty, but this image is connected and for J nonempty, Δ_{J} does not meet Δ. Hence im μ^{λ} is contained in Δ. However, if χ_{i}(z) = 0 for some i ∈ I and z ∈ N, then L_{ir}(μ^{λ}(z)) = 0 for all r ∈ I_{i}, contradicting the combinatorial type of Δ. □

Corollary 2. Any toric symplectic orbifold whose rational Delzant polytope (Δ, L) has the combinatorics of a product of simplexes admits a compatible toric Kahler metric h_{L} which is a Levi–Kähler reduction of a product of spheres.

We now give a closed formula for the symplectic potential of h_{L}.

Theorem 5. Let N = 𝕊^{2m_{1}+1} × ⋯ × 𝕊^{2m_{ℓ}+1} ⊆ ℂ_{𝒮} be a product of standard CR spheres, and suppose that the kernel 𝔤 of u = d ∘ L satisfies Condition 1. Then

G_{L} = \frac{1}{2} ∑_{i∈I} ∑_{r∈I_{i}∪\{∞\}} L_{ir} log |L_{ir}| = \frac{1}{2} ∑_{i∈I} ∑_{r∈I_{i}} L_{ir} log |\frac{L_{ir}}{L_{i∞}} |

is a symplectic potential for the Levi–Kähler metric, where L_{ir} ∈ 𝔥 is viewed as a linear function on 𝔥^{∗}, hence an affine function on 𝒜 ⊆ 𝔥^{∗}, and L_{i∞} = − ∑_{r∈I_{i}} L_{ir}.

Equivalently, the reduced metric on the image of the horizontal momentum map μ^{λ} is given by

h_{L}^{red} = \frac{1}{2} ∑_{i∈I} ∑_{r∈I_{i}∪\{∞\}} \frac{dL_{ir}^{2}}{L_{ir}} = \frac{1}{2} ∑_{i∈I} ∑_{0≤r<s≤m_{i}} \frac{L_{ir}L_{is}}{∑_{t=0}^{m_{i}} L_{it}} (\frac{dL_{ir}}{L_{ir}} − \frac{dL_{is}}{L_{is}} )^{2}.

Proof. The hessian of the stated potential G_{L} evaluates readily to the stated reduced metric, which we can compute in two ways. Using Theorem 3, we decompose

h_{o} := ∑_{s∈𝒮} \frac{dζ_{s}^{2}}{2ζ_{s}} = \frac{1}{2} ∑_{i∈I}(∑_{r∈I_{i}} \frac{dζ_{ir}^{2}}{ζ_{ir}} − \frac{(∑_{r∈I_{i}} dζ_{ir})^{2}}{∑_{r∈I_{i}} ζ_{ir}} ) + \frac{1}{2} ∑_{i∈I} \frac{(∑_{r∈I_{i}} dζ_{ir})^{2}}{∑_{r∈I_{i}} ζ_{ir}}

into components orthogonal and parallel to E^{o}, and the reduced metric is the pullback of the first term by L. Alternatively, by (14), the horizontal metric on J(𝒦 ∩ 𝒟) is

\begin{aligned}∑_{i∈I} 2χ_{i} ∑_{r∈I_{i}} \frac{dσ_{ir}^{2}}{2σ_{ir}} |_{𝒟} & = σ^{∗}[\frac{1}{2} ∑_{i∈I} ρ_{i}(∑_{r∈I_{i}} \frac{dζ_{ir}^{2}}{ζ_{ir}} − \frac{(∑_{r∈I_{i}} dζ_{ir})^{2}}{∑_{r∈I_{i}} ζ_{ir}} )]|_{𝒟} \\ & = \frac{1}{2} ∑_{i∈I} σ^{∗}(∑_{r∈I_{i}} \frac{d(ρ_{i}ζ_{ir})^{2}}{ρ_{i}ζ_{ir}} − \frac{(∑_{r∈I_{i}} d(ρ_{i}ζ_{ir}))^{2}}{∑_{r∈I_{i}} ρ_{i}ζ_{ir}} )|_{𝒟} \end{aligned}

where σ^{∗}ρ_{i} = 2χ_{i} and we use that ∑_{r∈I_{i}} σ_{ir} is constant on N, so that σ^{∗}(∑_{r∈I_{i}} dζ_{i}) = 0, and then exploit rescaling invariance. The result is the pullback by μ^{λ} of the stated reduced metric, since L_{ir}(μ^{λ}) = 2χ_{i}σ_{ir} by (17). □

As shown by Guillemin [34], a Kähler potential may be computed as a Legendre transform of the symplectic potential G_{L} with respect to some basepoint p ∈ A:

H_{L} = ⟨μ^{λ} − μ^{λ}(p), dG_{L}⟩ − G_{L}

= ∑_{i∈I} ∑_{r∈I_{i}∪\{∞\}}(\frac{1}{2}(⟨μ^{λ} − μ^{λ}(p), dL_{ir}⟩ − L_{ir}) log |L_{ir}| + \frac{1}{2}⟨μ^{λ} − μ^{λ}(p), dL_{ir}⟩)

= ∑_{i∈I} ∑_{r∈I_{i}∪\{∞\}} \frac{1}{2}L_{ir}(p) log |L_{ir}|,  since ∑_{r∈I_{i}∪\{∞\}} L_{ir} = 0.

3.2. Products of 3-spheres. As a special case of Theorem 4, consider an ℓ-fold product N = 𝕊^{3} × ⋯ × 𝕊^{3} of 3-spheres as a codimension ℓ submanifold of ℂ^{2ℓ} ≅ ℂ^{ℓ} ⊗ ℂ^{2} with momentum coordinates σ_{ir}(z) (i ∈ I = {1, … ℓ}, r ∈ {0, 1}). Thus 𝔱_{𝒮} = ℝ^{2ℓ} is the Lie algebra of 𝕋_{𝒮} = 𝕋^{2ℓ} acting diagonally on ℂ^{2ℓ}. We study the Levi–Kähler metric on the open subset where the quotient torus 𝕋 acts freely, ignoring rationality conditions.

3.2.1. Geometry of the Levi–Kähler metric. Consider, for any ℓ-dimensional subspace 𝔤 ⊆ 𝔱_{𝒮} = ℝ^{2ℓ}, the integrable distribution 𝒦^{𝔤} = span{K_{v} | v ∈ 𝔤} on N. Then, around each point of N such that 𝒦^{𝔤} is transversal to 𝒟, the local quotient space M of leaves of 𝒦^{𝔤} has induced complex structure J. We will further assume that λ ∈ 𝔤^{∗} is such that dη^{λ} induces a Kähler metric (h_{L}, J, ω_{L}) on M.

The reduced metric of Theorem 5 specializes to

h_{L}^{red} = \frac{1}{2} ∑_{i∈I} \frac{L_{i0}L_{i1}}{L_{i0} + L_{i1}} (\frac{dL_{i0}}{L_{i0}} − \frac{dL_{i1}}{L_{i1}} )^{2}, i.e., (μ^{λ})^{∗}h_{L}^{red} = ∑_{i∈I} \frac{χ_{i} dσ_{i}^{2}}{σ_{i}(1 − σ_{i})},

where L_{ir}(μ^{λ}) = 2χ_{i}σ_{ir}, 2χ_{i} = −L_{i∞}(μ^{λ}), σ_{i} := σ_{i0} = −L_{i0}(μ^{λ})/L_{i∞}(μ^{λ}), L_{i0} + L_{i1} = −L_{i∞} and hence σ_{i1} = 1 − σ_{i} on N. In other words, the characteristic functions χ_{i} and the orthogonal coordinates σ_{i} are affine and birational functions (respectively) of the momentum coordinates μ^{λ}. Thus the momentum images of the coordinate hypersurfaces with σ_{i} constant are hyperplanes in 𝒜 through the codimension two affine subspace where L_{i0}(μ^{λ}) = L_{i1}(μ^{λ}) = 0, and σ_{i} is inverse to the unique affine coordinate on this pencil of hyperplanes sending 0, 1 and ∞ to the facets L_{i0}(μ^{λ}) = 0, L_{i1}(μ^{λ}) = 0 (of Δ_{𝔤,λ}) and the “characteristic hyperplane” χ_{i} = −\frac{1}{2}L_{i∞}(μ^{λ}) = 0 (respectively). These pencils introduce a factorization structure (in the sense of [5]) which is adapted to the class of polytopes in 𝒜 with the combinatorics of the product of intervals.

To allow for more general coordinates, set H = {0, 1, ∞} (so that ∑_{r∈H} L_{ir} = 0) and introduce an arbitrary affine coordinate ξ_{i} = −N_{i0}(μ^{λ})/N_{i∞}(μ^{λ}) on the pencil which takes the values α_{ir} at the points [L_{ir}], meaning

σ_{i} = \frac{(ξ_{i} − α_{i0})(α_{i1} − α_{i∞})}{(ξ_{i} − α_{i∞})(α_{i1} − α_{i0})}, 1 − σ_{i} = \frac{(ξ_{i} − α_{i1})(α_{i0} − α_{i∞})}{(ξ_{i} − α_{i∞})(α_{i0} − α_{i1})}, L_{ir} = \frac{2(ξ_{i} − α_{ir})N_{i∞}}{A′_{i}(α_{ir})}

for affine functions N_{ir} of μ^{λ} with ∑_{r∈H} N_{ir} = 0, and A_{i}(y) = a_{i} ∏_{r∈H}(y − α_{ir}). Note that we also allow α_{i∞} = ∞, in which case A_{i} is of degree 2. (This latter case can be derived from the generic case deg A_{i} = 3 by letting α_{i∞} = 1/ε and taking a limit of εA_{i} as ε → 0.) Thus

dσ_{i} = \frac{(α_{i1} − α_{i∞})(α_{i0} − α_{i∞})}{(ξ_{i} − α_{i∞})(ξ_{i} − α_{i∞})^{2}}, \frac{χ_{i} dσ_{i}^{2}}{σ_{i}(1 − σ_{i})} = \frac{L_{i∞}(μ^{λ}) A′_{i}(α_{i∞})dξ_{i}^{2}}{2(ξ_{i} − α_{i∞})A_{i}(ξ_{i})} = N_{i∞}(μ^{λ}) \frac{dξ_{i}^{2}}{A_{i}(ξ_{i})}.

On the other hand,

dξ_{i} = \frac{N_{i0}(μ^{λ})dN_{i∞} − N_{i∞}(μ^{λ})dN_{i0}}{N_{i∞}(μ^{λ})^{2}} ∘ dμ^{λ} = −\frac{(ξ_{i}dN_{i∞} + dN_{i0}) ∘ dμ^{λ}}{N_{i∞}(μ^{λ})}
∴ dμ^{λ} = − ∑_{j∈I} N_{j∞}(μ^{λ})dξ_{j} ⊗ Q_{j}, where Q_{i}: M → 𝔱^{∗} satisfy
∑_{i∈I}(ξ_{i}dN_{i∞} + dN_{i0}) ⊗ Q_{i} = Id_{𝔱} and ⟨(ξ_{i}dN_{i∞} + dN_{i0}), Q_{j}⟩ = δ_{ij}.

Hence dQ_{i} = − ∑_{j}⟨Q_{i}, dN_{j∞}⟩dξ_{j} ⊗ Q_{j} and the Levi–Kähler metric is

\begin{aligned}h_{L} & = ∑_{i∈I} N_{i∞}(μ^{λ})(\frac{dξ_{i}^{2}}{A_{i}(ξ_{i})} + A_{i}(ξ_{i})θ_{i}^{2}), \\ ω_{L} & = ⟨dμ^{λ} ∧ dt⟩ = − ∑_{i∈I} N_{i∞}(μ^{λ})dξ_{i} ∧ θ_{i}, \end{aligned} ; (18)

with θ_{i} = ⟨Q_{i}, dt⟩ for angular coordinates t on M such that dt^{(1,0)} is holomorphic:

dd^{c}t = d(∑_{i∈I} \frac{ξ_{i}dN_{i∞} + dN_{i0}}{A_{i}(ξ_{i})} dξ_{i}) = 0.

The canonical affine coordinates may be obtained by setting A_{i}(t) = 2t(1 − t)(1 − εt) in the limit ε → 0; we then have N_{ir} = −L_{ir} and ξ_{i} = σ_{i} so that N_{i∞}(μ^{λ}) = 2χ_{i} and

\begin{aligned}h_{L} & = ∑_{i∈I} 2χ_{i} (\frac{dσ_{i}^{2}}{2σ_{i}(1 − σ_{i})} + 2σ_{i}(1 − σ_{i})θ_{i}^{2}), \\ ω_{L} & = − ∑_{i∈I} 2χ_{i} dσ_{i} ∧ θ_{i}. \end{aligned} ; (19)

3.2.2. Explicit normal form. Recall that 𝔤_{o} ⊆ 𝔱_{𝒮} has a canonical basis e_{i} := e_{i0} + e_{i1} for i ∈ I. We obtain a similar basis w_{i} : i ∈ I for 𝔤 by observing that the vectors e_{i0} in 𝔱_{𝒮} project to 𝔱 = 𝔱_{𝒮}/𝔤_{o} ≅ 𝔱_{𝒮}/𝔤 to the normals u_{i0} : i ∈ I at a vertex of Δ_{𝔤,λ}. Hence 𝔤 is transversal to span{e_{i0} | i ∈ I} and there are canonical w_{i} ∈ 𝔤 of the form

(20)  w_{i} = e_{i} + ∑_{j∈I} C_{ji}e_{j0} = e_{i0} + e_{i1} + ∑_{j∈I} C_{ji}e_{j0}

for an ℓ × ℓ matrix of real numbers C. We denote by K_{i}^{o} = K_{e_{i}} and K_{i} = K_{w_{i}} the induced vector fields in 𝒦^{𝔤_{o}} and 𝒦^{𝔤} respectively. We shall also write K_{ir} as a shorthand for K_{e_{ir}} = ∂/∂ϑ_{ir}. Thus

d^{c}σ^{i}(K_{j}^{o}) = 2(σ_{i0}dϑ_{i0} + σ_{i1}dϑ_{i1})(K_{j0} + K_{j1}) = 2δ_{ij}
d^{c}σ^{i}(K_{j}) = 2(σ_{i0}dϑ_{i0} + σ_{i1}dϑ_{i1})(K_{j}^{o} + ∑_{k}C_{kj}K_{k0}) = 2(δ_{ij} + σ_{i}C_{ij}).

on N, since σ_{i0} + σ_{i1} = 1. Now if η_{i} : i ∈ I are the 1-forms on N defined by η_{i}(K_{j}) = δ_{ij} and ⋂_{i∈I} ker η_{i} = 𝒟 = ⋂_{i∈I} ker d^{c}σ^{i}, we may write

(21)  \frac{1}{2}d^{c}σ^{i} = ∑_{j∈I} P_{ij}η_{j} where P_{ij} := \frac{1}{2}d^{c}σ^{i}(K_{j}) = δ_{ij} + σ_{i}C_{ij}.

We have noted that e_{i0} : i ∈ I project onto a bases for 𝔱 = 𝔱_{𝒮}/𝔤_{o} ≅ 𝔱_{𝒮}/𝔤. In order to compute the toral part of the quotient metric, we need to project the corresponding vector fields K_{i0} onto 𝒟. We thus define projections

X_{i}^{o} = K_{i0} − ∑_{j∈I} \frac{1}{2}d^{c}σ^{j}(K_{i0})K_{j}^{o} = K_{i0} − σ_{i}(K_{i0} + K_{i1}) = (1 − σ_{i})K_{i0} − σ_{i}K_{i1},
X_{i} = K_{i0} − ∑_{j∈I} η_{j}(K_{i0})K_{j},

along 𝒦^{𝔤_{o}} and 𝒦^{𝔤} respectively. Let I_{σ} = diag(σ_{1}, … σ_{ℓ}).

Lemma 7. X_{i}^{o} = ∑_{j∈I} X_{j}P̃_{ji} with P̃_{ji} = δ_{ji} + C_{ji}σ_{i}, i.e., P I_{σ} = I_{σ}P̃.

Proof. Since ∑_{k∈I} P_{jk}η_{k}(K_{i0}) = \frac{1}{2}d^{c}σ^{j}(K_{i0}) = σ_{i}δ_{ij}, it follows from the relation between P and P̃ that \frac{1}{2}d^{c}σ^{j}(K_{i0}) = ∑_{k∈I} η_{j}(K_{k0})P̃_{ki}. We now have

X_{i}^{o} = K_{i0} − ∑_{j∈I} \frac{1}{2}d^{c}σ^{j}(K_{i0})(K_{j} − ∑_{k∈I} C_{kj}K_{k0})

= K_{i0} − ∑_{j,k∈I}(η_{j}(K_{k0})P̃_{ki}K_{j} − σ_{j}δ_{ij}C_{kj}K_{k0})

= ∑_{k∈I} K_{k0}(δ_{ki} + C_{ki}σ_{i}) − ∑_{j,k∈I} η_{j}(K_{k0})K_{j}P̃_{ki} = ∑_{k∈I} X_{k}P̃_{ki}

as required. □

Writing λ = (c_{1}, … c_{ℓ}) with respect to the dual basis to w_{i} : i ∈ I, η^{λ} = ∑_{i∈I} c_{i}η_{i}. The corresponding momentum coordinates μ_{i} : i ∈ I are given by (cf. (13))

(22)  μ_{i} = ⟨μ^{λ}, L_{i0}⟩ = ∑_{j∈I} c_{j}η_{j}(K_{i0}) = ∑_{j∈I} c_{j}σ_{j}Q̃_{ji} = ∑_{j∈I} c_{j}Q_{ji}σ_{i} = 2χ_{i}σ_{i},

where Q̃ = P̃^{−1}, Q = P^{−1} and 2χ_{i} = ∑_{j∈I} c_{j}Q_{ji}. If we rewrite (22) as

c_{j}σ_{j} = ∑_{i∈I} μ_{i}P̃_{ij} = μ_{j} + ∑_{i∈I} μ_{i}C_{ij}σ_{j},

then we can specialize (18) with N_{jr} = L_{jr} (up to overall scale) using

L_{j0}(μ^{λ}) = μ_{j}, L_{j∞}(μ^{λ}) = −(L_{j0} + L_{j1})(μ^{λ}) = ∑_{i∈I} μ_{i}C_{ij} − c_{j}.

We may also compute directly that the toral part of the metric on 𝒟 is

h_{L}^{tor} := ∑_{i,j,k∈I} c_{i}dη_{i}(X_{j}, JX_{k}) dt_{j} dt_{k} = ∑_{i,j,k,p∈I} \frac{1}{2}c_{i}Q_{ip} dd^{c}σ^{p}(X_{j}, JX_{k}) dt_{j} dt_{k}
= ∑_{i,j,k,p,q,r∈I} 2c_{i}Q_{ip} (σ_{p}dϑ_{p0}^{2} + (1 − σ_{p})dϑ_{p1}^{2})(X_{q}^{o}Q̃_{qj}, X_{r}^{o}Q̃_{rk}) dt_{j} dt_{k}

= ∑_{i∈I} 4χ_{i}σ_{i}(1 − σ_{i})(∑_{j∈I} Q̃_{ij} dt_{j})^{2},

(23)

where we use σ_{i}(1 − σ_{i})^{2} + (1 − σ_{i})σ_{i}^{2} = σ_{i}(1 − σ_{i}). This agrees with (18) since

dμ_{i} = ∑_{j} c_{j}(dσ_{j} Q̃_{ji} + σ_{j} dQ̃_{ji}) = ∑_{j,k,l} c_{j}(Q_{jk}P_{kl} − σ_{j}Q̃_{jk}C_{kl}) dσ_{l} Q̃_{li}

= ∑_{j,k,l} c_{j}Q_{jk}(P_{kl} − σ_{k}C_{kl}) dσ_{l} Q̃_{li} = ∑_{k} 2χ_{k} dσ_{k} Q̃_{ki}.

3.3. Projective cubes. For Levi–Kähler quotients of a ℓ-fold product of 3-spheres, the polytope Δ_{𝔤,λ} is an ℓ-cuboid, i.e., it has the combinatorics of a product of m = ℓ intervals (an m-cube). Such a polytope is projectively equivalent to a cube if the intersections of pairs of opposite facets lie in a hyperplane: transforming this hyperplane to infinity, opposite facets become parallel and meet the hyperplane at infinity in the facets of an (m − 1)-simplex, and all simplices are projectively equivalent. Projective equivalence to an m-cube is automatic when m = 2, but is restrictive for m ≥ 3 (when m = 3, opposite facets of a generic cuboid meet in skew lines, not coplanar lines).

The assumption of projective equivalence to an m-cube simplifies the previous analysis, because we may take N_{i∞} to be the equation of the hyperplane common to the pencils spanned by opposite facets, independent of i ∈ I = {1, … m}. Concretely, let b_{j} ∈ ℝ for 0 ≤ j ≤ m and μ_{j} : 0 ≤ j ≤ m be affine coordinates on the affine space 𝒜 = {(μ_{0}, μ_{1}, … μ_{m}) : ∑_{j=0}^{m} b_{j}μ_{j} = 1}.

We now set N_{i∞} = μ_{0}, N_{i0} = −μ_{i} and N_{i1} = μ_{i} − μ_{0} for 1 ≤ i ≠ m. We thus have

(24)  ξ_{i} = \frac{μ_{i}}{μ_{0}}  and hence b_{0} + ∑_{i=1}^{n} b_{i}ξ_{i} = ∑_{i=0}^{m} \frac{b_{i}μ_{i}}{μ_{0}} = \frac{1}{μ_{0}} ,

so that the inverse transformation is

(25)  μ_{0} = \frac{1}{b_{0} + b_{1}ξ_{1} + ⋯ + b_{m}ξ_{m}} , μ_{i} = ξ_{i}μ_{0} = \frac{ξ_{i}}{b_{0} + b_{1}ξ_{1} + ⋯ + b_{m}ξ_{m}} .

Differentiating μ_{i}, we then have for the symplectic form

(26)  ω_{L} = ∑_{i=1}^{m} dμ_{i} ∧ dt_{i} = μ_{0} ∑_{i=1}^{m} dξ_{i} ∧ θ_{i},

where

(27)  θ_{i} = dt_{i} − μ_{0}b_{i} ∑_{j=1}^{m} ξ_{j}dt_{j} = dt_{i} − b_{i} ∑_{j=1}^{m} μ_{j}dt_{j}.

In particular dθ_{i} = −b_{i}ω_{L}. Letting

(28)  Jdξ_{i} := A_{i}(ξ_{i})θ_{i}, 1 ≤ i ≤ m,

J defines an integrable almost complex structure and the t_{i} are pluriharmonic. We thus have the following diagonal form of (h_{L}, ω_{L})

\begin{aligned}h_{L} & = \frac{1}{b_{0} + b_{1}ξ_{1} + ⋯ + b_{m}ξ_{m}} ∑_{i=1}^{m}(\frac{dξ_{i}^{2}}{A_{i}(ξ_{i})} + A_{i}(ξ_{i})θ_{i}^{2}) \\ ω_{L} & = \frac{1}{b_{0} + b_{1}ξ_{1} + ⋯ + b_{m}ξ_{m}} ∑_{i=1}^{m} dξ_{i} ∧ θ_{i}, \end{aligned} ; (29)

where the 1-forms θ_{i} are given by (27).

The product of intervals ξ_{i} ∈ [α_{i0}, α_{i1}], 1 ≤ i ≤ m transforms to the compact convex polytope Δ determined by the hyperplanes (ξ_{i} − α_{ir})μ_{0} = 0, (for r ∈ {0, 1}, 1 ≤ i ≤ n). As before, we set

A_{i}(y) := a_{i} ∏_{r∈H}(y − α_{ir}), L_{ir}(μ) := \frac{2(ξ_{i} − α_{ir})μ_{0}}{A′_{i}(α_{ir})} = \frac{2(μ_{i} − α_{ir}μ_{0})}{A′_{i}(α_{ir})},

where r ∈ H = {0, 1, ∞}. Note that L_{ir} ≥ 0 on Δ for 1 ≤ i ≤ m and r ∈ {0, 1}, and that ∑_{r∈H} L_{ir} = 0. We can compute a Kähler potential from the symplectic potential G_{L} by Legendre transform based at μ_{j} = 0 to get

(30)  H_{L} = ∑_{i,r} \frac{log |L_{ir}|}{A′_{i}(α_{ir})} = ∑_{i,r} \frac{log |ξ_{i} − α_{ir}|}{A′_{i}(α_{ir})} + const = ∑_{i=1}^{m} ∫ ^{ξ_{i}} \frac{ds}{A_{i}(s)}.

3.4. Levi–Kähler metrics of convex quadrilaterals. We now specialize to the case m = 2, i.e.,

N = 𝕊^{3} × 𝕊^{3} = {z ∈ ℂ^{4} ≅ ℂ^{2} ⊗ ℂ^{2} | (σ_{10} + σ_{11})(z) = 1, (σ_{20} + σ_{21})(z) = 1}.

By Theorem 4, the compact Kähler 4-orbifolds (M, h_{L}, ω_{L}) obtained as a Levi–Kähler quotient of 𝕊^{3} × 𝕊^{3} by an abelian subgroup G ⊆ 𝕋^{4} are the compact toric 4-orbifolds whose rational Delzant polytope is a quadrilateral. Note that from its very construction, h_{L} is compatible with a second complex structure J̃ on M, coming from the quotient of the product CR structure (𝒟, J_{1}−J_{2}) on N = 𝕊^{3}×𝕊^{3}, where (𝒟, J) = (𝒟_{1}, J_{1})⊕(𝒟_{2}, J_{2}) is the direct sum of the CR distributions of each 𝕊^{3} factor. Thus J = J_{1} + J_{2}, and J̃ = J_{1} − J_{2} defines a second CR structure on N, associated to the same distribution 𝒟 ⊆ T N, which commutes with J and induces the opposite orientation on 𝒟. In the terminology of [4], h_{L} is Kähler with respect to J and ambihermitian with respect to commuting complex structures (J, J̃) on M. We are going to show that (h_{L}, J̃) is, in fact, conformal to another 𝕋-invariant Kähler metric (h̃_{L}, J̃) (which induces the opposite orientation of (M, J)), i.e., that (h_{L}, J) and (h̃_{L}, J̃) define an ambitoric structure on M in the sense of [4]. These structures have been extensively studied and classified, both locally [4] and globally [5].

As any quadrilateral is a projective cube, the general form of the Levi–Kähler metric h_{L} is described by (29) but we shall also describe below how this form is derived from the choice of the subgroup G. Following the notation in §3.2, we specialize (23) to ℓ = 2, and set

C = (\begin{matrix}α & γ \\ β & δ \end{matrix}) ,

so that

A = (\begin{matrix}1 + ασ_{1} & γσ_{1} \\ βσ_{2} & 1 + δσ_{2} \end{matrix}) and Ã = (\begin{matrix}1 + ασ_{1} & γσ_{2} \\ βσ_{1} & 1 + δσ_{2} \end{matrix});

hence

B = \frac{1}{Z} (\begin{matrix}1 + δσ_{2} & −γσ_{1} \\ −βσ_{2} & 1 + ασ_{1} \end{matrix}) and B̃ = \frac{1}{Z} (\begin{matrix}1 + δσ_{2} & −γσ_{2} \\ −βσ_{1} & 1 + ασ_{1} \end{matrix}) ,

where Z = (1 + ασ_{1})(1 + δσ_{2}) − βγσ_{1}σ_{2} = 1 + ασ_{1} + δσ_{2} + (αδ − βγ)σ_{1}σ_{2}. Then

(31)  μ_{1} = \frac{c_{1}σ_{1}(1 + δσ_{2}) − c_{2}βσ_{1}σ_{2}}{Z} , μ_{2} = \frac{−c_{1}γσ_{1}σ_{2} + c_{2}(1 + ασ_{1})σ_{2}}{Z} ,

while the toral part of the metric on 𝒟 is

(32) h_{L}^{tor} = σ_{1}(1 − σ_{1})((1 + δσ_{2})c_{1} − βσ_{2}c_{2})((1 + δσ_{2})dt_{1} − γσ_{2}dt_{2})^{2}/Z^{3}

+ σ_{2}(1 − σ_{2})((1 + ασ_{1})c_{2} − γσ_{1}c_{1})((1 + ασ_{1})dt_{2} − βσ_{1}dt_{1})^{2}/Z^{3}.

We now transform this expression into the ansatz (29) for projective cubes (all quadrilaterals are projectively equivalent). To do this we first find the base loci of the families of lines with σ_{1} or σ_{2} constant. We find that for σ_{1} = c_{2}/(c_{1}γ − c_{2}α), 1 + ασ_{1} = c_{1}γ/(c_{1}γ − c_{2}α) and hence (μ_{1}, μ_{2}) = (c_{2}/γ, 0), independent of σ_{2}. Similarly for σ_{2} = c_{1}/(c_{2}β − c_{1}δ), (μ_{1}, μ_{2}) = (0, c_{1}/β), independent of σ_{1}. These coordinates are examples of Segre factorization structures, as defined in [5]. We transform the coordinate singularity to infinity by setting

ξ_{1} = σ_{1}Δ_{1}, Δ_{1} = \frac{1}{c_{2}(1 + ασ_{1}) − c_{1}γσ_{1}} = \frac{1 + (c_{1}γ − c_{2}α)ξ_{1}}{c_{2}} ,
ξ_{2} = σ_{2}Δ_{2}, Δ_{2} = \frac{1}{c_{1}(1 + δσ_{2}) − c_{2}βσ_{2}} = \frac{1 + (c_{2}β − c_{1}δ)ξ_{2}}{c_{1}}

We then compute that 1 + c_{1}γξ_{1} + c_{2}βξ_{2} = c_{1}c_{2}Δ_{1}Δ_{2}Z and

Δ_{1} + αξ_{1} = (1 + c_{1}γξ_{1})/c_{2}, Δ_{2} + δξ_{2} = (1 + c_{2}βξ_{2})/c_{1},
(1 + δσ_{2})dt_{1} − γσ_{2}dt_{2} = ((1 + c_{2}βξ_{2})dt_{1} − c_{1}γξ_{2}dt_{2})/(c_{1}Δ_{2})
(1 + ασ_{1})dt_{2} − βσ_{1}dt_{1} = ((1 + c_{1}γξ_{1})dt_{2} − c_{2}βξ_{1}dt_{1})/(c_{2}Δ_{1}),

so that, setting μ_{0} = 1/(1 + c_{1}γξ_{1} + c_{2}βξ_{2}), we have

h_{L}^{tor} =c_{1}c_{2}^{3}μ_{0}Δ_{1}ξ_{1}(Δ_{1} − ξ_{1})(dt_{1} − μ_{0}c_{1}γ(ξ_{1}dt_{1} + ξ_{2}dt_{2}))^{2}
+ c_{1}^{3}c_{2}μ_{0}Δ_{2}ξ_{2}(Δ_{2} − ξ_{2})(dt_{2} − μ_{0}c_{2}β(ξ_{1}dt_{1} + ξ_{2}dt_{2}))^{2},

(33)

μ_{1} = \frac{σ_{1}}{Δ_{2}Z} = \frac{c_{1}c_{2}ξ_{1}}{1 + c_{1}γξ_{1} + c_{2}βξ_{2}} , μ_{2} = \frac{σ_{2}}{Δ_{1}Z} = \frac{c_{1}c_{2}ξ_{2}}{1 + c_{1}γξ_{1} + c_{2}βξ_{2}} (34)

This has the form (29) with A_{1}(ξ_{1}) = c_{1}c_{2}^{3}Δ_{1}ξ_{1}(Δ_{1}−ξ_{1}) and A_{2}(ξ_{2}) = c_{1}^{3}c_{2}Δ_{2}ξ_{2}(Δ_{2}−ξ_{2}).

We now relate the Levi–Kähler metrics to the local forms of ambitoric metrics studied in [4]. The results depend crucially on whether β = 0 (when the curves ξ_{1} =constant pass through the point at infinity), γ = 0 (when the curves ξ_{2} =constant pass through the point at infinity), or both (when we have a product structure). We break this down into three cases as follows.

3.4.1. The product case. This is the case when β = 0, γ = 0. Letting x = ξ_{1}, y = ξ_{2} the metric becomes

h_{L} = \frac{dx^{2}}{A(x)} + \frac{dy^{2}}{B(y)} + A(x)dt_{1}^{2} + B(y)dt_{2}^{2}

with A(x), B(y) positive valued polynomials of degree 2 or 3, i.e., a local product of extremal toric Riemann surfaces. The construction yields (up to an equivariant isometry corresponding to affine transformations of x and y) all such products for which A(x) and B(y) are of degree 2 or 3 with distinct real roots.

3.4.2. The Calabi case. Without loss of generality, this is the case β = 0, γ ≠ 0, so that the curves ξ_{1} =constant pass through the point at infinity. We now let x = 1+c_{1}γξ_{1}, y = −c_{1}γξ_{2} so that the metric becomes

(35)  h_{L} = \frac{1}{x}(\frac{dy^{2}}{B(y)} + B(y)dt_{2}^{2} + \frac{dx^{2}}{A(x)} + \frac{A(x)}{x^{2}}(dt_{1} + y dt_{2})^{2})

i.e., given by the Calabi construction with respect to the variables x̄ = 1/x, y (see e.g. [3, 36]) starting from a toric extremal Riemann surface (Σ, g_{Σ} = \frac{dy^{2}}{B(y)} + B(y)dt_{2}^{2}), and taking an extremal toric metric on the fibre associated to the profile function Θ(x̄) := Ā(x̄)/x̄ with Ā(x̄) = x̄^{4}A(1/x̄). Once again, up to affine changes of x and y, one covers all toric metrics of Calabi type for which the functions A(x), B(y) are polynomials of degree 2 or 3 with distinct real roots.

3.4.3. The negative orthotoric case. This is the generic case when βγ ≠ 0. We can therefore let x = 1 + c_{1}γξ_{,}y = −c_{2}βξ_{2} so that the metric becomes

(36)  h_{L} = \frac{1}{x − y} (\frac{dx^{2}}{A(x)} + \frac{dy^{2}}{B(y)}) + \frac{A(x)(dθ_{1} + ydθ_{2})^{2} + B(y)(dθ_{1} + xdθ_{2})^{2}}{(x − y)^{3}} ,

where A(x) and B(y) are both polynomials of degree 2 or 3 with distinct real roots. (In terms of [4], the conformal oppositely oriented Kähler metric (h̃_{L}, J̃) is orthotoric.)

A case by case inspection shows that the conformal factor w such that the oppositely oriented Kähler metric is h̃_{L} = (1/w^{2})h_{L}, where w = 1, 1/x or 1/(x − y), in the above three cases, respectively. We observe that w is an affine function in the momenta with respect to ω_{L}, which vanishes at the (possibly infinite) intersection points of the pair of opposite facets of (Δ, L). We summarize the discussion as follows.

Proposition 6. Let (h_{L}, ω_{L}) be a Levi–Kähler quotient of 𝕊^{3} × 𝕊^{3} ⊆ ℂ^{2} × ℂ^{2}, corresponding to a subspace 𝔤 ⊆ 𝔱_{𝒮}, and some λ ∈ 𝔤^{∗}. Then (h_{L}, ω_{L}) is ambitoric in the sense of [4], and is either a product, of Calabi-type or conformal and oppositely oriented to an orthotoric metric, depending on whether 𝔤 intersects nontrivially two, one or zero of the 2-dimensional subspaces (𝔱_{1}, 𝔱_{2}), where 𝔱_{i} = ℝ^{2} ⊆ ℝ^{4} is the Lie algebra of the 2-torus 𝕋_{i}^{2} ⊆ 𝕋^{4} naturally acting on the i-th factor (i = 1, 2) of ℂ^{4} = ℂ^{2} × ℂ^{2}. Furthermore, in all cases, h_{L} is expressed in terms of two arbitrary polynomials of degree 2 or 3, each with real distinct roots, whereas the oppositely oriented Kähler metric is h̃_{L} = (1/w^{2})h_{L} for a positive affine function w on the quadrilateral Δ_{𝔤,λ}, vanishing at the intersection points of its opposite facets. Conversely, any ambitoric metric of the above mentioned types determined by two polynomials of degree 2 or 3 with distinct real roots^{1} arises as a Levi–Kähler quotient of 𝕊^{3} × 𝕊^{3}.

1This constraint comes from the fact that 𝔤 is a subspace of the Cartan subalgebra 𝔱_{𝒮} consisting of diagonal elements of the Lie algebra 𝔰𝔲(1, 2) ⊕ 𝔰𝔲(1, 2) of the CR automorphisms of 𝕊^{3} × 𝕊^{3}.

3.5. Toric bundles and Levi–Kähler quotients of products of spheres. The Calabi and the product cases appearing in the analysis in the previous subsection have a natural generalization to higher dimensions in the framework of semisimple rigid toric bundle construction of [8, 9], where it is also referred to as the generalized Calabi construction. Let us first recall briefly the setting of these works.

Let π: M → S be a bundle of toric kählerian manifolds or orbifolds of the form M = P ×_{T} V, for an ℓ-torus T, a principal T -bundle P over a kählerian manifold S of dimension 2d, and a toric 2ℓ-manifold (or orbifold) V with Delzant polytope Δ ⊆ 𝔱^{∗}; thus M has dimension 2m = 2(d + ℓ). We let F_{i} (i = 1, … n) denote the co-dimension one faces of Δ and u_{i} the primitive inward normals (with respect to a lattice Λ). Let be the dimension of M. Let θ ∈ Ω^{1}(M, 𝔱) the connection 1-form induced by a principal T -connection on P, with curvature Ω ∈ Ω^{1}(S, 𝔱). Suppose that Ω^{0} is a closed 2-form on S. Then the rigid toric bundle construction on M is a Kähler metric of the form

(37)  g = g_{0} + ⟨x, g_{Ω}⟩ + ⟨dx, (H^{V})^{−1}, dx⟩ + ⟨θ, H^{V} , θ⟩,
ω = Ω_{0} + ⟨x, Ω⟩ + ⟨dx ∧ θ⟩,  dθ = Ω,

where:

• x ∈ C^{∞}(M, 𝔱^{∗}) is the momentum map of the T action with image Δ;

• H^{V} ∈ C^{∞}(Δ, S^{2}𝔱^{∗}) is a matrix valued function which, firstly, satisfies the boundary conditions that on any co-dimension one face F_{i}, there is a function h_{i} with

∑_{t} H_{st}^{V}(x)(u_{i})_{t} = 0,  ∑_{t} \frac{∂H_{st}^{V}}{∂x_{r}} (x)(u_{i})_{t} = (u_{i})_{r}h_{i}(x)_{s}

and ⟨h_{i}(x), u_{i}⟩ := ∑_{s} h_{i}(x)_{s}(u_{i})_{s} = 2 for all x ∈ F_{i}; secondly the inverse (H^{V})^{−1} ∈ C^{∞}(Δ, S^{2}𝔱) of H^{V} is the hessian of a function G_{V} on Δ; thirdly, H^{V} induces a positive definite metric on the interior of each face F of Δ (as an element of S^{2}(𝔱/𝔱_{F})^{∗}, where 𝔱_{F} is the isotropy algebra of F);

• the metric g_{0} + ⟨x, g_{Ω}⟩ associated to Ω^{0} + ⟨x, Ω⟩ via the complex structure on S is positive definite for all x ∈ Δ ⊆ 𝔱^{∗}.

Throughout, angle brackets denote natural contractions of 𝔱 with 𝔱^{∗}, and we omit pullbacks by x and π. In particular x itself will denote the standard 𝔱^{∗}-valued coordinate on Δ, as well as its pullback to M.

The function G_{V} is called a symplectic potential for H^{V} and is determined up to an affine function on 𝔱^{∗}. According to [2, Thm. 2] and [8, Rem. 4.2], the boundary and positivity conditions above can be equivalently formulated in terms of G_{V}, by requiring that G_{V} is smooth and strictly convex on the interior Δ̊ of Δ, such that

G_{V} − \frac{1}{2} ∑_{i=1}^{k} L_{i} log L_{i} ∈ C^{∞}(Δ)

det(Hess G_{V}) ∏_{i=1}^{k} L_{k} ∈ C^{∞}(Δ) and is strictly positive, (38)

where L_{i} = ⟨u_{i}, x⟩ − v_{i}, i = 1, … k are the labels defining Δ.

Let X be a holomorphic vector field os S which is hamiltonian with respect to Ω_{0} + ⟨x, Ω⟩ for all x ∈ Δ. Thus −ı_{X}(Ω_{0}+⟨x, Ω⟩) = df_{0}+⟨x, df_{Ω}⟩ for functions f_{0} ∈ C^{∞}(S, ℝ) and f_{Ω} ∈ C^{∞}(S, 𝔱). Generalizing an observation from the proof of [9, Lemma 5], any such X can be lifted to a hamiltonian Killing vector field of (M, g, ω):

(39)  X̂ = X^{H} + ⟨f_{Ω}, K⟩,

where K := grad_{ω} x ∈ C^{∞}(M, T M) ⊗ 𝔱^{∗} is the family of hamiltonian vector fields generated by the principal T bundle P, and X^{H} denotes the horizontal lift of X (to the kernel of θ). Indeed, −ıX̂ω = −ı_{X}(Ω_{0} + ⟨x, Ω⟩) + ⟨f_{Ω}, dx⟩ = d(f_{0} + ⟨x, f_{Ω}⟩), so X̂ has hamiltonian f_{0} + ⟨x, f_{Ω}⟩ (omitting pullbacks of f_{0} and f_{Ω} to M).

Suppose now that the family g_{0} + ⟨x, g_{Ω}⟩ of Kähler metrics on S is toric with respect to a fixed torus action of a torus T_{S} with Lie algebra 𝔱_{S}. For each fixed x, the momentum map of T_{S} may be written ξ_{0} +⟨x, ξ_{Ω}⟩, where ξ ∈ C^{∞}(S, 𝔱_{S}^{∗}) and ξ_{Ω} ∈ C^{∞}(S, 𝔱_{S}^{∗} ⊗𝔱), and pulling back these functions to M, ξ_{0} + ⟨x, ξ_{Ω}⟩ is the momentum map for the T_{S} action on M defined by lifting the generators to M using (39). Since these lifts commute with K, M is toric under the combined action of T × T_{S}.

Lemma 8. Let (M, g, ω) be a Kähler manifold or orbifold given by (37) with fibrewise symplectic potential G_{V}, and suppose that (S, Ω_{0} + ⟨x, Ω⟩) is toric with respect to a fixed action of T_{S} for all x ∈ Δ. Then, (M, g, ω) is toric and with respect lifted T × T_{S} action and has a symplectic potential

(40)  G_{M} = G_{V} + G_{0} + ⟨x, G_{Ω}⟩,

where for eached fixed x, G_{0} + ⟨x, G_{Ω}⟩ is a symplectic potential for g_{0} + ⟨x, g_{Ω}⟩ on S.

Proof. The torus action T on (M, g, ω) is rigid, meaning that the metric on the torus orbits depends only on the value of the momentum map x. Hence for each fixed ξ ∈ 𝔱_{S}, the restriction of G_{M} to 𝔱^{∗} + ξ ≅ 𝔱^{∗} differs from G_{V} by an affine function of x, hence has the form (40) for functions G_{0} and G_{Ω} of ξ ∈ 𝔱_{S}. By construction, for each fixed x ∈ Δ, (S, g_{0} + ⟨x, g_{Ω}⟩) is the Kähler quotient of M by T at momentum level x. Hence by [19], the restriction of G_{M} to x + 𝔱_{S}^{∗} ≅ 𝔱_{S}^{∗} is a symplectic potential for g_{0} + ⟨x, g_{Ω}⟩, hence so is G_{0} + ⟨x, G_{Ω}⟩. □

A rigid toric bundle metric (37) is semisimple if S is a product of Kähler manifolds (S_{j}, ω_{j}) and there exist c_{j} ∈ ℝ and p_{j} ∈ 𝔱 (for j ∈ {1, 2, … N}) such that Ω_{0} + ⟨x, Ω⟩ = ∑_{j=1}^{N}(c_{j} + ⟨p_{j}, x⟩)ω_{j}. If each (S_{j}, ω_{j}) is toric under a torus T_{j} with Lie algebra 𝔱_{j}, then (S, Ω_{0} + ⟨x, Ω⟩) is toric for all x under the product action of T_{S} = ∏_{j=1}^{N} T_{j} with 𝔱_{S} = ⨁_{j=1}^{N} 𝔱_{j}. We refer to this special case as the toric generalized Calabi ansatz.

Proposition 7. Suppose that (g, ω) is a Kähler metric obtained by the toric generalized Calabi ansatz, where the fibre V is a compact toric orbifold with labelled polytope (Δ, L_{1}, …, L_{k}), and each factor (S_{j}, ω_{j}) of the base S = ∏_{j=1}^{N} S_{j} is a compact toric orbifold with labelled Delzant polytope (Δ_{j}, L_{1}^{j}, …, L_{k_{j}}^{j}), respectively. Then (g, ω) is a toric Kähler metric on the compact symplectic orbifold with labelled Delzant polytope

Δ̂ = {L_{i}(x) ≥ 0, L̂_{r_{j}}^{j} := (⟨p_{j}, x) + c_{j})L_{r_{j}}^{j} ≥ 0}.

Moreover, if the fibrewise toric metric determined by G^{V} and the Kähler metrics ω_{j} are all Levi–Kähler quotients of product spheres, then the resulting metric (37) is a Levi–Kähler quotient of the overall product of spheres.

Proof. We check that G_{M} satisfies the conditions (38). Lemma 8 and the fact that ⟨p_{j}, x⟩ + c_{j} is strictly positive on Δ imply that G_{M} differs by a smooth function

G_{M}^{0} := \frac{1}{2}( ∑_{i=1}^{k} L_{i} log L_{i} + ∑_{j=1}^{N} ( ∑_{r_{j}=1}^{k_{j}} L̂_{r_{j}}^{j} log L̂_{r_{j}}^{j})).

It remains to see that det(Hess(G_{M}))(∏_{i=1}^{k} L_{i})(∏_{j,r_{j}} L̂_{r_{j}}^{j}) is smooth and positive on Δ̂. The determinant det(Hess(G_{M})^{−1}) is, up to a positive scale, the norm with respect to g of the wedge product of the Killing vector fields (K_{1}, … K_{ℓ}, K̂_{r_{j}}^{j}) for j = 1, … N, r_{j} = 1, … d_{j}. Using (39) and the specific form (37) of the metric g, one sees that

det(Hess(G_{M})^{−1}) = C( ∏_{j=1}^{N}(⟨p_{j}, x⟩ + c_{j})^{d_{j}}) det(Hess(G_{V})^{−1}) det(Hess(G_{j})^{−1}),

where C > 0 is constant. Using the compactification criteria (38) for each G_{V} and G_{j}, and the fact that Δ̂ is a simple polytope, near any point y on a face F̂ ⊆ Δ̂ we have:

\begin{aligned}det(Hess(G_{M})^{−1}) & = δ(∏_{i=1}^{k} L_{k})(∏_{j=1}^{N}(⟨p_{j}, x⟩ + c_{j})^{d_{j}}( ∏_{r_{j}=1}^{k_{j}} L_{r_{j}}^{j})) \\ & = δ′(∏_{i=1}^{k} L_{k}) ∏_{j=1}^{N}( ∏_{r_{j}=1}^{k_{j}} L̂_{r_{j}}^{j}), \end{aligned} ; (41)

where δ and δ′ are smooth positive functions around y ∈ Δ̂.

For the second part, we assume by Theorem 5 that G_{V} = \frac{1}{2} ∑_{ir} L_{ir} log L_{ir} and G_{j} = \frac{1}{2} ∑_{q_{j}r_{j}} L_{q_{j}r_{j}}^{j} log L_{q_{j}r_{j}}^{j} with ∑_{r} L_{ir} = 0 and ∑_{r_{j}} L_{q_{j}r_{j}}^{j} = 0. Then, by Lemma 8,

G_{M} = \frac{1}{2}( ∑_{ir} L_{ir} log L_{ir} + ∑_{j=1}^{N}(⟨p_{j}, x⟩ + c_{j})( ∑_{q_{j}r_{j}} L_{q_{j}r_{j}}^{j} log L_{q_{j}r_{j}}))

= \frac{1}{2}( ∑_{ir} L_{ir} log L_{ir} + ∑_{j=1}^{N} ( ∑_{q_{j}r_{j}} L̂_{q_{j}r_{j}}^{j} log L_{q_{j}r_{j}}))

= \frac{1}{2}( ∑_{ir} L_{ir} log L_{ir} + ∑_{j=1}^{N} ( ∑_{q_{j}r_{j}} (L̂_{q_{j}r_{j}}^{j} log L̂_{q_{j}r_{j}} − ∑_{q_{j}r_{j}} L_{q_{j}p_{j}} log(⟨p_{j}, x⟩ + c_{j}))))

= \frac{1}{2}( ∑_{ir} L_{ir} log L_{ir} + ∑_{j=1}^{N} ( ∑_{q_{j}r_{j}} L̂_{q_{j}r_{j}}^{j} log L̂_{q_{j}r_{j}}))

We conclude by using Theorem 5 again. □

Remark 8. The toric Calabi construction provides a practical method for constructing new toric metrics from old ones. Suppose (g_{j}, ω_{j}) and (g^{V} , ω_{V}) are toric Kähler metrics on Δ̊_{j} × 𝕋^{d_{j}} and Δ̊ × 𝕋^{ℓ} respectively, for labelled (simple convex, compact) polytopes Δ_{j} = {x^{j} ∈ ℝ^{d_{j}} : L_{r_{j}}^{j}(x^{j}) ≥ 0, r_{j} = 1, … k_{j}} (j = 1, … N) and Δ = {x ∈ ℝ^{ℓ} : L_{i}(x) ≥ 0, i = 1, … N}. Let p_{j} = (p_{j1}, …, p_{jℓ}), θ = (θ_{1}, …, θ_{ℓ}) with

θ_{i} = dt_{i} + ∑_{j=1}^{N} p_{ji}( ∑_{r_{j}=1}^{d_{j}} x_{r_{j}}^{j}dt_{r_{j}}^{j}),

where the affine functions ∑_{i=1}^{ℓ} p_{ji}x_{i}+c_{j} are positive on Δ. Then, as in Propostion 7, we get a toric Kähler metric on the interior of Δ̂ times 𝕋^{ℓ}×𝕋^{d_{1}} ×⋯×𝕋^{d_{N}}, whose symplectic potential G_{M} is given by Lemma 8. Taking all the affine functions L_{r_{j}}^{j} and L_{i}, and (∑_{i=1}^{ℓ} p_{ji}x_{i} + c_{j}) be all with rational coefficients, one gets a rational labelled polytope Δ̂, and if the ingredient metrics (g_{j}, ω_{j}) and (g^{V} , ω_{V}) satisfty the Abreu boundary conditions (38) for the corresponding labellings L_{r_{j}}^{j} and L_{i}, then so does the metric given (locally) by the toric generalized Calabi Ansatz, with the labelling defined in Proposition 7. Hence the metric compactifies on the compact toric orbifold M given by Δ̂ and these labels.

4. Curvature of Levi–Kähler quotients of products of spheres

4.1. Bochner condition. Recall (see e.g. [4, 18]) that if R ∈ ∧^{1,1}ℂ^{m∗}⊗𝔲(m) is a formal Kähler curvature tensor, i.e., R_{u,v}(w)+R_{v,w}(u)+R_{w,u}(v) = 0, then the Bochner part of R is its orthogonal projection B(R) onto the U(m)-submodule of formal Kähler curvature tensors with vanishing Ricci trace. The Bochner tensor B^{g} of a Kähler manifold is then the (pointwise) Bochner part B(R^{g}) of its Riemannian curvature R^{g} ∈ Γ(∧^{1,1}T^{∗}M ⊗ 𝔲(T M)). One can extend this definition to more general Hermitian curvature tensors R ∈ ∧^{2}ℂ^{m∗} ⊗ 𝔲(m) (which do not a priori satisfy the Bianchi identity) where one still denotes by B(R) the orthogonal projection of an element R onto the U(m)-submodule of formal Kähler curvature tensors with vanishing Ricci trace.

In our language, S. Webster [44] showed that in codimension one the Bochner tensor of a Levi–Kähler quotient M of CR manifold N pulls back to the Chern–Moser tensor of N, which vanishes when the N is locally CR diffeomorphic to a standard CR sphere (𝕊^{2m+1}, 𝒟, J). In particular, every Levi–Kähler quotient (M^{2m}, g, J) of 𝕊^{2m+1} is Bochner-flat. We generalize this to arbitrary codimension, by describing the curvature of Kähler metrics arising as Levi–Kähler quotients of a product of CR-spheres.

Let N = 𝕊^{2m_{1}+1} × ⋯ × 𝕊^{2m_{ℓ}+1} ⊆ ℂ_{𝒮} be a product of standard CR spheres and (𝒟 = ⨁_{i∈I} 𝒟_{i}, J = ⨁_{i∈I} J_{i}) be the product CR structure and denote by N_{i} ≅ 𝕊^{2m_{i}+1} the i-th factor of N, with projection p_{i}: N → N_{i}. The bundle p_{i}^{∗}T N_{i} is identified with the subbundle E^{i} := ⋂_{j≠i} ker(p_{j∗}) of T N via the restriction p_{i∗}: E_{z}^{i} ⟶^{≃} T_{p_{i}(z)}N_{i}. We denote the projection r_{i}: T N → E^{i}.

Let (𝔤, λ) be a positive Levi pair (corresponding to L) and assume that 𝔤 is the Lie algebra of a subtorus G of 𝕋_{𝒮}; denote by M = N/G the Levi–Kähler quotient and by π: N → M the quotient map. The global assumption on 𝔤 is made purely to make statements about M rather than local quotients. In addition to a Kähler structure (ǧ = h_{L}, J, ω_{L}), M inherits of a ǧ-orthogonal splitting of its tangent space, namely

(42)  T M = ⨁_{i∈I} 𝒟̌_{i}

where 𝒟̌_{i} = π_{∗}(𝒟_{i}). We denote by ˇ∇^{i} the connection on 𝒟̌_{i} induced by the Levi– Civita connection ˇ∇ of ǧ, by R ˇ∇^{i} the corresponding curvature tensor (a section of ∧^{2}T^{∗}M ⊗𝔤𝔩(𝒟̌_{i}, J), where 𝔤𝔩(𝒟̌_{i}, J) denotes the bundle of J-commuting endomorphisms of 𝒟̌_{i}), and by B^{i} := B^{i}(R ˇ∇^{i}) the Bochner projection of R ˇ∇^{i}|𝒟̌_{i} ∈ ∧^{2}𝒟̌_{i}^{∗} ⊗ 𝔤𝔩(𝒟_{i}, J).

Proposition 8. For each i ∈ I, B^{i} = 0.

We prove this result using the observation (see [24] and [44]) that the Chern–Moser tensor of (N_{i}, J_{i}, 𝒟_{i}) may be computed from the horizontal part of the curvature of the Tanaka connection (see e.g. [14]) associated to any contact form α compatible with the (codimension 1) CR structure (𝒟, J). The Chern–Moser tensor does not depend upon the chosen compatible contact structure (it is a CR invariant). If the Reeb vector field V of α (determined by α(V) = 1 and L_{V} α = 0) is a transverse CR vector field, the horizontal part of the Tanaka connection (i.e., its restriction on 𝒟) is the pullback to N of the Levi-Civita connection of the Levi–Kähler quotient (M, g, J, ω) of N by V using the identification 𝒟 ≅ π^{∗}T M.

In our situation, for each w ∈ ∏_{j≠i} N_{j}, there is an embedding ι_{w}: N_{i} ↪ N. We shall (slightly abusively) still denote by 𝒟_{i} the pull-back bundle ι_{w}^{∗}𝒟_{i} and by J_{i} the corresponding (standard) CR-structure on (N_{i}, 𝒟_{i}). Recall that N is endowed with a 1-form η^{λ} with dη^{λ} = π^{∗}ω_{L}, where λ ∈ 𝔤^{∗} is the value defining the Levi–Kähler quoting ω_{L}. The pull-back α_{w,i} := ι_{w}^{∗}η^{λ} thus defines a 1-form on N_{i}, and the next Lemma implies that α_{w,i} is a contact 1-form compatible with the CR structure (N_{i}, J_{i}, 𝒟_{i}).

Lemma 9. For any i ∈ {1, … ℓ} and z ∈ N, the subspace of E_{z}^{i} defined by

ℛ_{z}^{𝔤,i} := r_{i}(K_{η(E_{z}^{i})})

has dimension 1 and is transverse to 𝒟_{i}. In particular, ℛ^{𝔤,i} → N is a real line bundle and there exists a unique vector field V_{i} ∈ Γ(ℛ^{𝔤,i}) such that η^{λ}(V_{i}) = 1. Furthermore, for each w ∈ ∏_{j≠i} N_{j}, α_{w,i} is a contact form on N_{i}, which induces ι_{w,i}^{∗}∘π^{∗}ω_{L} as a transversal symplectic structure and the vector field V_{w,i} on (N_{i}, 𝒟_{i}) defined by ι_{w∗}V_{w,i} = V_{i} is a Reeb vector field for α_{w,i}.

Proof. Since η(E_{z}^{i}) = η((E^{i}/𝒟_{i})_{z}), K_{η(E_{z}^{i})} ⊆ 𝒦^{𝔤} is at most 1-dimensional. Since ∑_{i=1}^{ℓ} K_{η(E_{z}^{i})} = K_{η(T_{z}N)} = 𝒦_{z}^{𝔤} is ℓ-dimensional, it follows that K_{η(E_{z}^{i})} is 1-dimensional and dim(ℛ_{z}^{𝔤,i}) ≤ 1. As 𝒦^{𝔤} is transversal to 𝒟 ((𝔤, λ) being Levi pair), each X ∈ E_{z}^{i} decomposes as X = K_{η(X)} + X^{𝒟}; applying r_{i} (and using r_{i}(X) = X, r_{i}(𝒟) = 𝒟_{i}), we obtain X = r_{i}(K_{η(X)}) + X^{𝒟_{i}}, showing E^{i} = ℛ^{𝔤,i} ⊕ 𝒟_{i}.

For any Y, Z ∈ 𝒟_{i}, dα_{w,i}(Y, Z) = −α_{w,i}([Y, Z]) = −η^{λ}([Y, Z]) = π^{∗}ω_{L}(Y, Z). In order to show that V_{w,i} is Reeb field for α_{w,i} we need to check that for any Z ∈ 𝒟_{i}, dα_{w,i}(V_{w,i}, Z) = dη^{λ}(V_{i}, Z) = 0 (where we have used that E^{i} is integrable and ι_{w}^{∗}(E_{i}) ≅ T N_{i}). Writing V_{i} = r_{i}(K_{η(X)}) = X−X^{𝒟_{i}} for X ∈ E^{i}, and decomposing X = K_{η(X)}+X^{𝒟}, we have

dη^{λ}(V_{i}, Z) = dη^{λ}(X − X^{𝒟_{i}}, Z) = dη^{λ}(K_{η(X)}, Z) + ∑_{j≠i} dη^{λ}(X^{𝒟_{j}}, Z) = 0,

where (in the last equality) the first term vanishes because η^{λ} is 𝔤-invariant, whereas the second term vanishes because for j ≠ i, 𝒟_{i} is dη^{λ}-orthogonal to 𝒟_{j}, see (42). □

We define a partial connection ∇^{i}: Γ(E^{i})×Γ(E^{i}) → Γ(E^{i}) on the involutive subbundle E^{i} ⊆ T N, which pulls back by ι_{w} to the Tanaka connection of α_{w,i}, as follows:

• ∇^{i} preserves 𝒟_{i};

• V_{i}, J_{i} and ω are ∇^{i} parallel;

• For any X, Y ∈ Γ(𝒟_{i}), the torsion T^{∇^{i}} satisfies T^{∇^{i}}(X, Y) = ω(X, Y)V_{i} and

T^{∇^{i}}(V_{i}, J_{i}X) = −J_{i}T^{∇^{i}}(V_{i}, X).

In particular, ∇^{i} satisfies

g(∇_{X}^{i}Y, Z) = ǧ( ˇ∇X̌ Y̌ , Ž) = ǧ( ˇ∇i ; X̌ Y̌ , Ž)

We next show that ∇^{i} can be extended to a full connection ∇ on T N preserving 𝒟, and such that:

• ∇|_{𝒟} = π^{∗} ˇ∇;

• ι_{w}^{∗}(r_{i} ∘ ∇) = ∇^{i} is the Tanaka connection on (N_{i}, α_{i}, 𝒟_{i}, J_{i}).

The first condition tells us that the torsion of X, Y ∈ Γ(𝒟) is the vertical part of −[X, Y], that is

(43)  T^{∇}(X, Y) = −K_{η([X,Y])}.

Hence for X, Y ∈ Γ(𝒟_{i}), r_{i}(T^{∇}(X, Y)) = −r_{i}(K_{η([X,Y])}) = ω(X, Y)V_{i} = T^{∇^{i}}(X, Y).

Let ℛ^{𝔤} := ⨁_{i=1}^{ℓ} ℛ^{𝔤,i} be the rank ℓ subbundle of T N over N which is everywhere transverse to 𝒟. We extend the endomorphism J of 𝒟 by zero on ℛ^{𝔤}, and define a linear connection ∇ on

T N = ⨁_{i∈I} E^{i} = ⨁_{i∈I}(𝒟_{i} ⊕ ℛ^{𝔤,i}) = 𝒟 ⊕ ℛ^{𝔤}

such that (i) ∇ agrees with the pullback connection π^{∗}∇^{g} on 𝒟 ≅ π^{∗}T M, (ii) ∇V_{i} = 0, (iii) ∇_{s}X = [s, X] + ∇_{X}s − \frac{1}{2}J(L_{s}J)X for each section s ∈ Γ(ℛ^{𝔤}) and X ∈ Γ(𝒟).

Lemma 10. Let ∇ be a connection on N satisfying the conditions (i)–(iii) above. Then, ι_{w}^{∗}(r_{i} ∘ ∇) is the Tanaka connection ∇^{i} on (N_{i}, α_{i}, 𝒟_{i}, J_{i}).

Proof. We first show that ∇_{s}JX = J∇_{s}X for any X ∈ Γ(T N) and s ∈ Γ(ℛ^{𝔤}). It is clear that ∇_{s}Jt = J∇_{s}t = 0 for s, t ∈ Γ(ℛ^{𝔤}) and when X ∈ Γ(𝒟), we have J∇_{X}s = 0 as well as the decomposition

[s, X] = [s, X]^{𝒟} − ∇_{X}s

with respect to the splitting T N = 𝒟 ⊕ ℛ^{𝔤}. Using this two facts and condition (iii), a straightforward calculation shows ∇_{s}JX = J∇_{s}X. Together with this last identity, conditions (i)–(ii) ensure that ι_{w}^{∗}(r_{i} ∘ ∇) satisfies the first two properties of the Tanaka connection of (N_{i}, 𝒟_{i}, J_{i}, α_{i}). It remains to check the torsion property. From equation (43) it follows that ι_{w}^{∗}(r_{i} ∘ ∇) has the same torsion on 𝒟_{i}. Moreover, using again condition (iii) we get

T^{∇}(s, JX) = −\frac{1}{2}J(L_{s}J)JX = \frac{1}{2}J^{2}(L_{s}J)X = −JT^{∇}(s, X)

for any X ∈ Γ(T N) and s ∈ Γ(ℛ^{𝔤}). □

Proof of Proposition 8. To compare the curvatures ∇^{i} and ˇ∇^{i}, we notice that

g(∇_{[X,Y]}^{i}Z, T) = g(∇_{[X,Y]^{𝒟_{i}}}^{i} Z, T) − ω_{L}(X̌, Y̌)g(∇_{V_{i}}^{i}Z, T)
= g(∇_{[X,Y]^{𝒟}} Z, T) − ω_{L}(X̌, Y̌)g(∇_{V_{i}^{𝒦}} Z, T)
= ǧ( ˇ∇_{[X̌,Y̌]}^{i}Ž, Ť) + ω_{L}(X̌, Y̌)Ǎ^{i}(Ž, Ť),

where to go from first line to the third we have decomposed ω(X, Y)V_{i} = ω(X, Y)V_{i}^{𝒦^{𝔤}} − ∑_{j≠i}[X, Y]^{𝒟_{j}} (with V_{i}^{𝒦^{𝔤}} being the projection of V_{i} to 𝒦^{𝔤}), and we view Ǎ^{i}(Ž, Ť) = −g(∇_{V_{i}^{𝒦^{𝔤}}} Z, T) as a (0, 2)-tensor on M, since pullbacks to N of smooth functions on M are 𝒦^{𝔤}-invariant. It then follows that

g(R_{X,Y}^{∇^{i}} Z, T) = ǧ(R ˇ∇^{i} ; X̌,Y̌ Ž, Ť) + ω_{L}(X̌, Y̌)Ǎ^{i}(Ž, Ť).

Applying the projection B^{i} to the both sides, and using that B^{i}(R^{∇^{i}}) equals the Chern– Moser tensor of N_{i} (which is zero) as well as B^{i}(ω_{L} ⊗ A^{i}) = 0 (as B^{i} projects onto the space of ω_{L}-primitive Kähler curvature tensors), we obtain B^{i} = B^{i}(R ˇ∇^{i}) = 0. □

4.2. Curvature for Levi–Kähler quotients of products of 3-spheres. We next use the explicit form (18) of the Kähler metric h_{L} in order to compute the its scalar curvature. Up to a factor of −1/2, a Ricci potential is given by the log ratio of the symplectic volume form to a holomorphic volume form. Using dt^{(1,0)} to compute the latter, we readily obtain

∑_{i∈I} log A_{i}(ξ_{i}) + ∑_{i∈I} log N_{i∞}(μ^{λ}) − 2 log ⋀_{i∈I}(ξ_{i}dN_{i∞} + dN_{i0}).

The derivatives of the first two terms are straightforward to compute, using that d(N_{j∞}(μ^{λ})) = − ∑_{i∈I} N_{i∞}(μ^{λ})⟨Q_{i}, dN_{j∞}⟩dξ_{i}. For the third, observe by Cramer’s rule that its exterior derivative is ∑_{i∈I} 2a_{ii} dξ_{i} where the coefficients a_{ij} solve the linear system ∑_{j∈I}(ξ_{j}dN_{j∞} + dN_{j0})a_{ij} = dN_{i∞}, i.e., a_{ij} = ⟨Q_{j}, dN_{i∞}⟩. Thus

ρ_{L} = \frac{1}{2}(d ∑_{i∈I} α_{i}θ_{i}), (44)

where

α_{i} = A′_{i}(ξ_{i}) − ∑_{j∈I} \frac{⟨Q_{i}, dN_{j∞}⟩N_{i∞}(μ^{λ})}{N_{j∞}(μ^{λ})}A_{i}(ξ_{i}) − 2⟨Q_{i}, dN_{i∞}⟩A_{i}(ξ_{i})

and θ_{i} = ⟨Q_{i}, dt⟩. We are interested in the scalar curvature only, defined by

s_{L} = \frac{2mρ_{L} ∧ ω^{m−1}}{ω^{m}} , \frac{ω^{m−1}}{ω^{m}} = −\frac{1}{m} ∑_{j∈I} \frac{∏_{k≠j} dξ_{k} ∧ θ_{k}}{N_{j∞}(μ^{λ}) dξ_{1} ∧ θ_{1} ∧ ⋯ ∧ dξ_{m} ∧ θ_{m}} .

Straightforward computation using (44) then yields

s_{L} = − ∑_{i∈I} \frac{A′′_{i}(ξ_{i})}{N_{i∞}(μ^{λ})} + ∑_{i∈I} 2A′_{i}(ξ_{i})(Q_{ii} + ∑_{j∈I} Q_{ij})

− ∑_{i∈I} A_{i}(ξ_{i})N_{i∞}(μ^{λ})(2\frac{⟨Q_{i}, dN_{j∞}⟩^{2}}{N_{i∞}(μ^{λ})^{2}} + ∑_{j∈I}(4Q_{ii}Q_{ij} − Q_{ij}^{2}) + ∑_{j,k} Q_{ij}Q_{ik}).

(45)

where Q_{ij} = ⟨Q_{i}, dN_{j∞}⟩/N_{j∞}(μ^{λ}).

We now specialize to the case of projective cubes as in Section 3.3, where N_{i∞}(μ^{λ}) = μ_{0} = 1/(b_{0} + b_{1}ξ_{1} + ⋯ b_{m}ξ_{m}), independent of 1 ≤ i ≤ m. Using

dμ_{0} = ⟨dN_{j∞}, dμ⟩ = − ∑_{i=1}^{m} μ_{0}dξ_{i}⟨Q_{i}, dN_{j∞}⟩,

we obtain immediately that Q_{ij} = b_{i}. The Ricci potential specializes to give

(46)  μ_{0}^{m+2} ∏_{i=1}^{m} A_{i}(ξ_{i}),

as may be verified directly using (27) and (29), while the scalar curvature reduces to

(47)  s_{L} = − ∑_{i∈I} \frac{A′′_{i}(ξ_{i})}{μ_{0}} + ∑_{i∈I} 2(m + 1)b_{i}A′_{i}(ξ_{i}) − ∑_{i∈I}(m + 1)(m + 2)μ_{0}b_{i}^{2}A_{i}(ξ_{i}).

4.3. Projective cubes and (w, p)-extremality. A labelled cuboid (Δ, L) is a labelled Delzant polytope which has the combinatorics of an m-cube. By Theorem 4, any labelled cuboid (Δ, L) admits a compatible toric metric h_{L}, which is Levi–Kähler quotient metric of an m-fold product of 3-spheres. We consider here the case that the cuboid Δ is a projective cube, i.e., the intersections of pairs of opposite facets lie in a hyperplane. For any labelled projective cube (Δ, L), h_{L} is given by (29), and we provide here a characterization of this toric metric in terms of the toric geometry of (Δ, L), as developed in [2, 34, 29].

The starting point of our approach is based on a recent observation in [10], which in turn extends the formal GIT framework of [28, 31] (realizing the scalar curvature of a Kähler metric as a momentum map under the action of the group of hamiltonian transformations) to a larger family of related GIT problems. Let (M, ω) be a compact symplectic manifold (or orbifold) and Ham(M, ω) the group of hamiltonian transformations. Fix a torus 𝕋 ≤ Ham(M, ω) and a positive hamiltonian w > 0 with grad_{ω} w ∈ 𝔱 := Lie(𝕋). Let C^{𝕋}(M, ω) be the space of 𝕋-invariant, ω-compatible complex structures on (M, ω) and Ham^{𝕋}(M, ω) the subgroup of 𝕋-equivariant hamiltonian transformation, acting naturally on C^{𝕋}(M, ω). The Lie algebra of Ham^{𝕋}(M, ω) is identified with the space C_{0}^{∞}(M)^{𝕋} of smooth, 𝕋-invariant functions of integral zero, endowed with the Ham^{𝕋}(M, ω) bi-invariant inner product

(48)  ⟨h_{1}, h_{2}⟩_{w,p} := ∫_{M} h_{1}h_{2}w^{−(p+1)}v_{ω},

where p is a real constant (which we call the conformal dimension) and v_{ω} = ω^{m}/m! is the volume form of ω. The space C^{𝕋}(M, ω) carries a formal Fréchet Kähler structure, (J, Ω^{w,p}), defined by

J_{J}(J̇) = JJ̇, Ω_{J}^{w,p}(J̇_{1}, J̇_{2}) = \frac{1}{2} ∫_{M} tr(JJ̇_{1}J̇_{2})w^{−(p−1)}v_{ω},

where the tangent space of C^{𝕋}(M, ω) at J is identified to be the Fréchet space of smooth sections J̇ of End(T M) satisfying

J̇J + JJ̇ = 0, ω(J̇·, ·) + ω(·, J̇·) = 0.

The formal complex structure J is the same as the one in [28, 31] whereas the modified formal symplectic form Ω^{w,p} stays closed (as can easily be checked).

In the following, we denote by g_{J} the Kähler metric corresponding to J ∈ C^{𝕋}(M, ω) and by s_{J} and Δ_{J} the corresponding scalar curvature and Laplace operator. We then have a straightforward (mutatis mutandis) generalization of [10, Thm. 1] (which corresponds to p = 2m).

Lemma 11. The action of Ham^{𝕋}(M, ω) on (C^{𝕋}(M, ω), J, Ω^{w,p}) is hamiltonian with a momentum map μ: C^{𝕋}(M, ω) → (C_{0}^{∞}(M)^{𝕋})^{∗}, whose value at J is identified with the ⟨·, ·⟩_{w,p}-dual of

(49)  s_{J,w,p} := w^{2}s_{J} − 2(p − 1)wΔ_{J}w − p(p − 1)g_{J}^{−1}(dw, dw).

Note that s_{J,w,p} is the trace of the conformal modification

(50)  ρ_{J,w,p} := w^{2}ρ_{J} + (p − 1)wdd^{c}w − \frac{1}{2}p(p − 1)dw ∧ d^{c}w

of the Ricci form.

Remark 9. As in [10], one can extend the definition of (J, Ω^{w,p}) on C^{𝕋}(M, ω) to the larger Frechét space AK^{𝕋}(M, ω) of 𝕋-invariant ω-compatible almost-Kähler structures J. Then, using the formulae in [32, Ch. 8] (see also [38, Lemma 2.1 & Prop. 3.1]), the momentum map for the action of Ham^{𝕋}(M, ω) on AK^{𝕋}(M, ω) is still given by Lemma 11, except that in (49) we must take s_{J} to be the Hermitian scalar curvature of g_{J} (the trace of the Ricci form of the canonical Hermitian connection, see e.g. [38]).

Using the contractibility of AK_{ω}^{G}, we obtain, as in [10], a generalized Futaki invariant 𝔉_{ω,w,p}^{𝕋}: 𝔱 → ℝ of (M, ω, 𝕋, w, p): for any vector field H ∈ 𝔱 with a hamiltonian h,

(51)  𝔉_{ω,w,p}^{𝕋}(H) := ∫_{M} s̊_{J,w,p}hw^{−(p+1)}v_{ω}

is independent of the choice of J ∈ AK_{ω}^{G}, where s̊_{J,w,p} is the L^{2}-projection of s_{J,w,p} onto functions with integral zero with respect to the volume form w^{−(p+1)}v_{ω}.

Specializing to the case of a toric manifold (or orbifold) (M, ω, 𝕋), the above formalism allows one to extend the theory of extremal toric metrics from [29] to the (w, p)-extremal toric case (the case p = 2m is developed in detail in [10]). In particular, we have the following result:

Proposition 9. Let (M, ω, 𝕋) be a compact toric orbifold with labelled Delzant polytope (Δ, L) in ℝ^{m} and w a positive affine-linear function on Δ. Then,

(a) There exists at most one (up to equivariant isometry) compatible toric metric g_{J} on (M, ω, 𝕋), for which s_{J,w,p} is an affine-linear function.

(b) The affine-linear function in (a) is uniquely determined by (Δ, L, w, p).

Definition 12. The (unique) compatible toric metric satisfying the condition (a) of Proposition 9 is called the (w, p)-extremal metric of (M, ω, 𝕋).

Theorem 6. Suppose (Δ, L) is a labelled projective cube in ℝ^{m}, corresponding to a compact toric orbifold (M, ω, 𝕋) and let h_{L} be the Levi–Kähler quotient metric defined by (29). Then h_{L} is the (w, m + 2)-extremal metric of (Δ, L), where w is the unique up to scale positive affine-linear function on ℝ^{m}, vanishing on the hyperplane containing the intersections of opposite facets of Δ.

Proof. We take w = μ_{0} and apply (50) with p = m + 2, ρ_{J} = ρ_{L} to obtain

ρ_{L,μ_{0}} = μ_{0}^{2}ρ_{J} + (m + 1)(μ_{0}dd^{c}μ_{0} − \frac{1}{2}(m + 2)dμ_{0} ∧ d^{c}μ_{0})

= −\frac{1}{2}μ_{0}^{2}dd^{c} log(μ_{0}^{m+2} ∏_{i} A_{i}(ξ_{i})) + (m + 1)(μ_{0}^{2}dd^{c} log μ_{0} − \frac{m}{2}dμ_{0} ∧ d^{c}μ_{0})

= \frac{1}{2}(−μ_{0}^{2}dd^{c} log ∏_{i} A_{i}(ξ_{i}) + mμ_{0}^{2}dd^{c} log μ_{0} − m(m + 1)dμ_{0} ∧ d^{c}μ_{0}).

(52)

We now compute

dd^{c} log ∏_{j} A_{i}(ξ_{i}) = ∑_{i} d(A′_{i}(ξ_{i})θ_{i}) = ∑_{i} A′′_{i}(ξ_{i})dξ_{i} ∧ θ_{i} − ∑_{i,j} b_{i}μ_{0}A′_{i}(ξ_{i})dξ_{j} ∧ θ_{j}

dd^{c} log μ_{0} = − ∑_{i} d(b_{i}μ_{0}A_{i}(ξ_{i})θ_{i})

= − ∑_{i} b_{i}μ_{0}A′_{i}(ξ_{i})dξ_{i} ∧ θ_{i} + ∑_{i,j} A_{i}(ξ_{i})(b_{i}b_{j}μ_{0}^{2}dξ_{j} ∧ θ_{i} + b_{i}^{2}μ_{0}^{2}dξ_{j} ∧ θ_{j})

dμ_{0} ∧ d^{c}μ_{0} = ∑_{i,j} b_{i}^{2}μ_{0}^{4}A_{i}(ξ_{i})dξ_{i} ∧ θ_{j}.

It follows that

s_{J,μ_{0}} = \frac{2mρ_{L,μ_{0}} ∧ ω^{m−1}}{ω^{m}} = −\frac{∑_{i=1}^{m} A′′_{i}(ξ_{i})}{b_{0} + b_{1}ξ_{1} + ⋯ + b_{m}ξ_{m}} ,

which (as deg A_{j}(ξ) ≤ 3) is an affine-linear function in momenta. □

Remark 10. We notice that for m = 2, m + 2 = 2m and s_{J,w,4} computes the scalar curvature of the conformal oppositely oriented metric h̃_{L} = (1/w^{2})h_{L} of Proposition 6, i.e., h_{L} is w-extremal in the sense of [10].

4.4. Extremal Levi–Kähler quotients. Formula (47) shows that for ℓ = m > 2, the Levi–Kähler metric h_{L} associated to a projective cube cannot be extremal unless b_{i} = 0, i = 1, … m, i.e., M is the product of weighted projective lines. However, we show below that when ℓ = 2, Levi–Kähler quotients of a product of two spheres 𝕊^{2m_{1}+1} × 𝕊^{2m_{2}+1} can provide new examples of extremal Kähler orbifolds.

4.4.1. Extremal Levi–Kähler quotients of 𝕊^{3} × 𝕊^{3}. As any quadrilateral is a projective cube, h_{L} is (w, 4)-extremal by Theorem 6. Furthermore, s_{J,w,4} is the scalar curvature of the conformal metric h̃_{L} = (1/w^{2})h_{L}, see [10]. By Proposition 6, (h_{L}, J) is either a product, of Calabi type, or a regular ambitoric Kähler metric of Segre type. We can then use [4, 36] to characterize the extremality of h_{L} as follows.

Proposition 10. Let (M, ω) be a compact toric 4-orbifold whose rational Delzant polytope is a labelled quadrilateral (Δ, L) and h_{L} the corresponding Levi–Kähler metric. If (Δ, L) is a parallelogram, then (M, ω_{L}, h_{L}) is an extremal toric orbifold which is the Kähler product of two extremal weighted projective lines; otherwise h_{L} is extremal if and only if the oppositely oriented ambitoric metric h̃_{L} = (1/w^{2})h_{L} has constant scalar curvature, or equivalently, h̃_{L} is a conformally Kähler, Einstein–Maxwell metric in the sense [10], where w is a positive affine linear function on Δ, determined up to positive scale by the property that it vanishes where the opposite sides of Δ intersect.

Proof. The product case follows from Proposition 6 (see Section 3.4.1). When h_{L} is of Calabi-type, i.e. given by (35) for polynomials A(x) and B(y) of degree 2 or 3, the negative ambitoric metric h̃_{L} = x^{2}h_{L} is also of Calabi-type with respect to the variables (x, y) and functions A(x) and B(y). It follows from [3, 36] that h_{L} is extremal if and only if h̃_{L} is extremal if and only if B(y) has degree 2 and (xA(x))′′(0) = −B′′(0). As deg A ≤ 3, this is precisely the condition that h̃_{L} is of constant scalar curvature (see [3, Prop. 14]). The case when h_{L} is regular ambitoric (i.e., negative orthotoric) is treated similarly, using the local form (36) and [3, Prop. 11]. Finally, as h̃_{L} = (1/w^{2})h_{L} with w being a Killing potential with respect to h_{L}, the scalar curvature h̃_{L} is constant if and only if h̃_{L} defines a conformally Kähler, Einstein–Maxwell metric, see [10]. □

Remark 11. The metric h_{L} is extremal (and hence h̃_{L} is Einstein–Maxwell) if and only if the affine function defined by (Δ, L) in Proposition 9(b) is constant. By an observation originating in [36], for a given quadrilateral Δ this places two linear constraints on the labels L, see also [10]. Thus, the above characterization for the extremality of h_{L} lead to the following useful observation: given a compact convex quadrilateral Δ which is not a parallelogram, there is a two-parameter family of inward normals to the faces, such that the corresponding Levi–Kähler metric is extremal.

4.4.2. CSC Levi–Kähler quotients of 𝕊^{5} × 𝕊^{3}. We discuss here examples of Levi–Kähler quotients of constant scalar curvature (CSC), obtained from the generalized Calabi construction in §3.5, where the base is S = ℂP^{1} equipped with a Fubini–Study metric ω_{S} and the fibre V is a toric orbifold with Delzant image a simplex in ℝ^{2}. By Proposition 7, the resulting 6-dimensional orbifold (M, g, ω) is obtained as a Levi–Kähler quotient of 𝕊^{5}×𝕊^{3} as soon as the fibrewise metric is Bochner–flat (which is the condition to be Levi– Kähler in the case of one factor). As the extremality condition is difficult to characterize in general, we shall use the hamiltonian 2-form ansatz with ℓ = 2 and N = 1 from [6, 8], which in turn is a special case of the generalized Calabi ansatz [8, §4]. We briefly recall the construction below and invite the Reader to consult [6, 8] for further details.

Let (S, g_{S}, ω_{S}) be a compact Riemann orbi-surface and η a real constant. We build a Kähler metric (g, ω) with a hamiltonian 2-form of order 2 and constant root η, defined on an orbifold fibration over M → S, with fibres isomorphic to an orbifold quotient of a weighted projective plane. The Kähler metric (g, ω) is written on a dense subset M̊ ⊆ M as follows (see [8]):

\begin{aligned}g & = (η − ξ_{1})(η − ξ_{2})g_{S} + \frac{(ξ_{1} − ξ_{2})p_{c}(ξ_{1})}{F (ξ_{1})}dξ_{1}^{2} + \frac{(ξ_{2} − ξ_{1})p_{c}(ξ_{2})}{F (ξ_{2})}dξ_{2}^{2} \\ & + \frac{F (ξ_{1})}{(ξ_{1} − ξ_{2})p_{c}(ξ_{1})}(θ_{1} + ξ_{1}θ_{2})^{2} + \frac{F (ξ_{2})}{(ξ_{2} − ξ_{1})p_{c}(ξ_{2})}(θ_{1} + ξ_{2}θ_{2})^{2} \\ ω & = (η − ξ_{1})(η − ξ_{2})ω_{S} + dσ_{1} ∧ θ_{1} + dσ_{2} ∧ θ_{2}, \\ dθ_{1} & = −ηω_{S}, dθ_{2} = ω_{S} p_{c}(t) = (t − η), σ_{1} = ξ_{1} + ξ_{2}, σ_{2} = ξ_{1}ξ_{2}. \end{aligned} ; (53)

Here ξ_{1} ∈ [−1, β], ξ_{2} ∈ [β, 1] (for some |β| < 1) are the orthotoric coordinates on the fibre, |η| > 1, and F (x) is a smooth function which satisfies the positivity and boundary conditions

• F (x)/p_{c}(x) > 0 on (β, 1); F (x)/p_{c}(x) < 0 on (−1, β);

• F (±1) = F (β) = 0.

It is easy to see that (53) is a special case of (37), with N = 1, (x_{1}, x_{2}) = (σ_{1}, σ_{2}), p_{11} = −η, p_{12} = 1, c_{1} = η^{2}, and a toric orbifold fibre whose Delzant polytope Δ is the the image of [−1, β] × [β, 1] under the map (σ_{1}, σ_{2}) = (ξ_{1}, ξ_{2}, ξ_{1}ξ_{2}) and labelling

L_{−1} = −c_{−1}(σ_{1} + σ_{2} + 1), L_{+1} = −c_{1}(−σ_{1} + σ_{2} + 1), L_{β} = −c_{β}(−βσ_{1} + σ_{2} + β^{2}),

where c_{±1}(F/p_{c})′(±1) = 2 = c_{β}(F/p_{c})′(β), see [8, Prop. 9]. Here we assume the usual rationality condition for the simplex (Δ, L), which certainly holds for β, η, c_{±1}, c_{β} ∈ ℚ.

We recall from [6] that the metric (53) is extremal if and only if the scalar curvature of g_{S} is constant s and F (x) is a polynomial of degree at most 5 satisfying

(54)  F′′(η) = −s.

The metric g is of constant scalar curvature if, furthermore, the degree of F (x) is at most 4. Likewise, by the same result, the fibrewise orthotoric metric

\begin{aligned}g^{V} & = \frac{(ξ_{1} − ξ_{2})p_{c}(ξ_{1})}{F (ξ_{1})}dξ_{1}^{2} + \frac{(ξ_{2} − ξ_{1})p_{c}(ξ_{2})}{F (ξ_{2})}dξ_{2}^{2} \\ & + \frac{F (ξ_{1})}{(ξ_{1} − ξ_{2})p_{c}(ξ_{1})}(dt_{1} + ξ_{1}dt_{2})^{2} + \frac{F (ξ_{2})}{(ξ_{2} − ξ_{1})p_{c}(ξ_{2})}(dt_{1} + ξ_{2}dt_{2})^{2} \end{aligned} ; (55)

is Bochner–flat if and only if F (x)/p_{c}(x) is a polynomial of degree at most 4. By Proposition 7, taking in (29) (S, ω_{S}) to be (an isometric orbifold quotient of) ℂP^{1} endowed with a Fubini–Study metric and F (x) = −c(x^{2} − 1)(x − β)(x − η)(x − γ) (resp. F (x) = −c(x^{2} −1)(x−β)(x−η)), one gets an ansatz for extremal (resp. constant scalar curvature) toric orbifolds, which are also Levi–Kähler quotients of 𝕊^{5} × 𝕊^{3}.

We further specialize to the constant scalar curvature case, i.e. F (x) = −c(x^{2}−1)(x− β)(x−η) for a non-zero positive constant c. Then, (54) reduces to 2c(3η^{2}−2βη−1) = s, whereas the positivity conditions for F (x) imply η < −1. Together with Scal_{S} > 0, these are the only constraints, subject to a rationality condition which is trivially solved by taking β, η, c rational. For instance, letting β = 1/n, η = −n, c = 2/(3n^{2} + 1), s = 4 gives rise to a CSC Levi–Kähler quotient orbifold, which is not a product.

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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