THE CR GEOMETRY OF WEIGHTED EXTREMAL KÄHLER AND SASAKI METRICS

VESTISLAV APOSTOLOV AND DAVID M. J. CALDERBANK

Abstract. We establish an equivalence between conformally Einstein–Maxwell Kähler 4-manifolds recently studied in [5, 10, 35, 44, 48, 49, 50] and extremal Kähler 4-manifolds in the sense of Calabi [20] with nowhere vanishing scalar curvature. The corresponding pairs of Kähler metrics arise as transversal Kähler structures of Sasaki metrics compatible with the same CR structure and having commuting Sasaki–Reeb vector fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kähler metric [7, 11, 45, 46, 47], illuminating and uniting several explicit constructions in Kähler and Sasaki geometry. It also leads to new existence and non-existence results for extremal Sasaki metrics, suggesting a link between the notions of relative weighted K-stability of a polarized variety introduced in [11, 47], and relative K-stability of the Kähler cone corresponding to a Sasaki polarization, studied in [18, 26].

Introduction

The famous Calabi problem [20], which seeks the existence of canonical Kähler metrics, is a central and very active topic of current research in Kähler geometry. As a candidate for a canonical metric on a complex manifold (M, J), Calabi proposed a notion of extremal Kähler metric g, meaning that its scalar curvature Scal(g) is a Killing potential, i.e., the vector field J grad_{g} Scal(g) is a Killing vector field for g. Examples include constant scalar curvature (CSC) Kähler metrics on (M, J), and hence also Kähler–Einstein metrics.

More recently, in real dimension 4, another natural generalization of CSC Kähler metrics has been studied [5, 48, 49, 50]: Kähler metrics g admitting a positive Killing potential f for which the scalar curvature of the conformal metric g̃ = (1/f^{2})g is a constant c, i.e.,

(1)  Scal(g̃) = f^{2}Scal(g) − 6fΔ_{g}f − 12|df|_{g}^{2} = c,

where Δ_{g} is the riemannian laplacian and |·|_{g} the norm defined by g. The metric g̃ satisfies a riemannian analogue of the Einstein–Maxwell equations with cosmological constant in generally relativity [59, 29], and thus we say g is a conformally Einstein–Maxwell Kähler metric. Many explicit examples of such metrics have been exhibited [5, 6, 10, 35, 44, 49, 50], and they have a striking resemblance to similar explicit examples of extremal Kähler metrics, see e.g. [10, Prop. 3]. Elucidating the connection suggested by these examples was the main motivation for this article, and our main result implies in particular an equivalence between the classes of conformally Einstein–Maxwell Kähler 4-manifolds and extremal Kähler 4-manifolds of nowhere zero scalar curvature. Our approach was suggested in part by [6, App. C], which implies that both kinds of metric can arise as quotients of a common strictly pseudo-convex CR 5-manifold (N, 𝒟, J) of Sasaki type. It also generalizes to complex manifolds (M, J) of real dimension 2m, using the notion of a weighted extremal metric [7, 10, 45], as we now explain.

Let (g, ω) be a Kähler metric on (M, J), f a function on M, and ν ∈ ℝ a real number (which we call the weight). Then the (f, ν) scalar curvature of g is defined to be

(2)  Scal_{f,ν}(g) := f^{2}Scal(g) − 2(ν − 1)fΔ_{g}f − ν(ν − 1)|df|_{g}^{2},

VA was supported in part by an NSERC Discovery Grant. The authors are grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Science where some of this work was conducted.

Definition 1. Let (g, ω) be a Kähler metric on (M, J) let f be a Killing potential for g and let ν ∈ ℝ. We say that g is (f, ν)-extremal if its (f, ν) scalar curvature, Scal_{f,ν}(g), given by (2), is also a Killing potential.

When M is compact and f > 0 on M, the (f, ν) scalar curvature (2) is the momentum map associated to a formal GIT problem on the space K_{ω}(M)^{𝕋} of 𝕋-invariant ω-compatible Kähler metrics for any torus 𝕋 in the group Ham(M, ω) of hamiltonian symplectomorphisms of (M, ω) which contains the flow of K [7, 10, 45]. This is similar to the framework found by Donaldson [33] and Fujiki [34] for Calabi extremal Kähler metrics. Indeed, the latter can be recovered from the weighted generalization by setting f ≡ 1.

For m = 2 and ν = 4, (2) reduces to (1), so that Definition 1 includes the conformally Einstein–Maxwell Kähler metrics already discussed. This case was extended to the weight ν = 2m (for any m) in [10], where it was noted that (2) then computes the scalar curvature of the hermitian metric (1/f^{2})g. Thus examples of (f, 2m)-extremal metrics include the conformally Einstein Kähler metrics studied in [31, 32]. The weight most relevant here is instead ν = m+2, which first appeared in [7], where it was discovered that certain quotients, of an m-fold product 𝕊^{3}×⋯×𝕊^{3} of CR 3-spheres by an m-torus, are (f, m+2)-extremal for a suitable f. However, an intrinsic geometric interpretation of (f, m + 2)-extremality with m > 2 has so far been lacking. Our main result rectifies this by providing an interpretation in CR geometry, whose basic notions we now recall (see also Section 1).

Let (N, 𝒟) be a contact (2m + 1)-manifold and denote by L_{𝒟}: 𝒟 × 𝒟 → T N/𝒟 the Levi form of 𝒟, defined, via local sections X, Y ∈ C_{N}^{∞}(𝒟), by the tensorial expression L_{𝒟}(X, Y) = −η_{𝒟}([X, Y]), where η_{𝒟}: T N → T N/𝒟 is the quotient map. A contact vector field is a vector field X such that L_{X}(C_{N}^{∞}(𝒟)) ⊆ C_{N}^{∞}(𝒟). We make fundamental use of the following basic fact in the theory of contact manifolds (see e.g. [13]).

Lemma 1. The map X ↦ η_{𝒟}(X) from contact vector fields to sections of T N/𝒟 is a linear isomorphism, whose inverse ξ ↦ X_{ξ} is a first order linear differential operator.

There is thus a contact Lie algebra 𝔠𝔬𝔫(N, 𝒟) of sections ξ of T N/𝒟 under the Jacobi bracket

[ξ, χ] := η_{𝒟}([X_{ξ}, X_{χ}]) = L_{X_{ξ}}χ = −L_{X_{χ}}ξ.

Now suppose J ∈ End(𝒟) is a CR structure on (N, 𝒟); then we obtain a second order linear differential operator ξ ↦ L_{X_{ξ}}J on 𝔠𝔬𝔫(N, 𝒟). Its kernel

𝔠𝔯(N, 𝒟, J) := {ξ ∈ 𝔠𝔬𝔫(N, 𝒟) : L_{X_{ξ}}J = 0}

is a Lie subalgebra of 𝔠𝔬𝔫(N, 𝒟), whose elements ξ correspond to CR vector fields X_{ξ} on N. If moreover (𝒟, J) is strictly pseudo-convex then T N/𝒟 has an orientation whose positive sections χ are those for which χ^{−1}L_{𝒟}(·, J·) is positive definite. Note that χ^{−1}L_{𝒟} = dη_{χ}|_{𝒟} where η_{χ} := χ^{−1}η_{𝒟} is the contact form defined by χ. We let 𝔠𝔬𝔫_{+}(N, 𝒟) ⊆ 𝔠𝔬𝔫(N, 𝒟) be the open cone of positive sections χ of T N/𝒟. We then have the following fundamental definitions (see e.g. [14]).

Definition 2. Let (N, 𝒟, J) be a strictly pseudo-convex CR manifold. Then the Sasaki cone of (N, 𝒟, J) is 𝔠𝔯_{+}(N, 𝒟, J) := 𝔠𝔯(N, 𝒟, J) ∩ 𝔠𝔬𝔫_{+}(N, 𝒟). If 𝔠𝔯_{+}(N, 𝒟, J) is nonempty then (N, 𝒟, J) is said to be of Sasaki type, an element χ ∈ 𝔠𝔯_{+}(N, 𝒟, J) is called a Sasaki structure on (N, 𝒟, J), with Sasaki–Reeb vector field X_{χ}, and (N, 𝒟, J, χ) is called a Sasaki manifold. We say χ is quasi-regular if the flow of X_{χ} generates an 𝕊^{1} action on N, and moreover regular if this action is free.

The following well-known construction provides a standard way (see e.g. [13]) to extend geometric notions on Kähler manifolds to Sasaki manifolds.

Example 1. Let (M, J, g, ω) be a Kähler manifold such that [ω/2π] is an integral de Rham class. Then there is a principal 𝕊^{1}-bundle π: N → M with a connection 1-form η satisfying dη = π^{∗}ω. Thus (N, 𝒟, J, χ) is a Sasaki manifold, where 𝒟 = ker η ⩽ T N, J is the pullback of the complex structure on T M to 𝒟 ≅ π^{∗}T M and χ is the image in T N/𝒟 of the generator X_{χ} of the 𝕊^{1} action (with η(X_{χ}) = 1, so η = η_{χ}).

Conversely, if χ ∈ 𝔠𝔯_{+}(N, 𝒟, J) is (quasi-)regular, then N is a principal 𝕊^{1}-bundle (or orbibundle) π: N → M over a Kähler manifold (or orbifold) M. Irrespective of regularity, this correspondence between Kähler geometry and Sasaki geometry holds locally: any point of a Sasaki manifold (N, 𝒟, J, χ) has a neighbourhood in which the leaf space M of the flow of X_{χ} is a manifold and has a Kähler structure (g, J, ω) induced, using the identification 𝒟 ≅ π^{∗}T M, by the transversal Kähler structure (g_{χ}, J, ω_{χ}) on 𝒟, where ω_{χ} := dη_{χ}|_{𝒟} and g_{χ} := ω_{χ}(·, J·). Indeed g_{χ}, J, and ω_{χ} are all X_{χ}-invariant, so they all descend to M, and we refer to (M, g, J, ω) as a Sasaki–Reeb quotient of (N, 𝒟, J, χ).

For χ ∈ 𝔠𝔯(N, 𝒟, J), we set

𝔠𝔬𝔫^{χ} := {ξ ∈ 𝔠𝔬𝔫(N, 𝒟) | [χ, ξ] = 0},
𝔠𝔯^{χ} := 𝔠𝔬𝔫^{χ} ∩ 𝔠𝔯(N, 𝒟, J) and 𝔠𝔯_{+}^{χ} := 𝔠𝔬𝔫^{χ} ∩ 𝔠𝔯_{+}(N, 𝒟, J).

If in addition χ ∈ 𝔠𝔯_{+}(N, 𝒟, J), then

C_{N}^{∞}(ℝ)^{χ} := {f ∈ C_{N}^{∞}(ℝ) : df(X_{χ}) = 0}

is a Lie algebra under the transversal Poisson bracket {f_{1}, f_{2}} := −ω_{χ}^{−1}(df_{1}|_{𝒟}, df_{2}|_{𝒟}), and we have the following elementary but central lemma.

Lemma 2. The map f ↦ ξ = fχ is a Lie algebra isomorphism from C_{N}^{∞}(ℝ)^{χ} to 𝔠𝔬𝔫^{χ}, and ξ ∈ 𝔠𝔯^{χ} if and only if f is a transversal Killing potential for (g_{χ}, ω_{χ}), i.e., −ω_{χ}^{−1}(df) is a transversal Killing vector field for g_{χ}.

Thus we obtain elements of 𝔠𝔯^{χ} as pullbacks of Killing potentials from (local) Sasaki– Reeb quotients of N by χ. The Levi-Civita connection on Sasaki–Reeb quotients pulls back to a connection ∇^{χ} on 𝒟 preserving (g_{χ}, J, ω_{χ}), which turns out to be (see e.g. [27, §4]) the so-called Tanaka–Webster connection [63] of (𝒟, J, χ). Thus the scalar curvature of Sasaki–Reeb quotients pulls back to the Tanaka–Webster scalar curvature Scal(g_{χ}) of ∇^{χ}, and hence (N, 𝒟, J, χ) is CSC, i.e., Scal(g_{χ}) is constant, if and only if its Sasaki–Reeb quotients are. We may define (weighted) extremal Sasaki structures similarly.

Definition 3. Let (N, 𝒟, J, χ) be a Sasaki manifold and ξ = fχ ∈ 𝔠𝔯^{χ}. The (ξ, ν) scalar curvature Scal_{ξ,ν}(g_{χ}) of χ is the function induced on N by the (f, ν) scalar curvature (2) on Sasaki–Reeb quotients. We say that χ is (ξ, ν)-extremal if Scal_{ξ,ν}(g_{χ})χ ∈ 𝔠𝔯(N, 𝒟, J). For constant f, this reduces to extremality of χ in the sense of [14].

Example 2. Lemma 2 shows that any quasi-regular Sasaki manifold over an (f, ν)-extremal orbifold (M, J, g, ω) is (ξ, ν)-extremal, with ξ = fχ, cf. Example 1 in the regular case.

For (f, ν)-extremal metrics, we have noted that the weight ν = 2m has a special interpretation in conformal geometry. The next lemma provides an analogous interpretation in CR geometry mentioned above of the weight ν = m + 2 for (ξ, ν)-extremal metrics.

Lemma 3. For any ξ ∈ 𝔠𝔯(N, 𝒟, J), σ(ξ) := Scal_{ξ,m+2}(g_{χ}) χ ∈ 𝔠𝔬𝔫(N, 𝒟) is independent of χ ∈ 𝔠𝔯_{+}^{ξ}. Hence ξ ↦ σ(ξ) is a second order quadratic differential operator, with σ(ξ) = Scal(g_{ξ}) ξ in the case that ξ ∈ 𝔠𝔯_{+}(N, 𝒟, J).

We now emphasise a key feature of Sasaki geometry, which was used in [25, 37, 52, 57] to construct CSC Sasaki manifolds from Kähler manifolds which are not necessarily CSC (see also [15, 35]). Namely, 𝔠𝔯_{+}(N, 𝒟, J) is open in 𝔠𝔯(N, 𝒟, J), so if (N, 𝒟, J) is of Sasaki type, and dim 𝔠𝔯(N, 𝒟, J) ⩾ 2, we obtain a family of Sasaki structures χ on N, inducing transversal Kähler structures on (𝒟, J). With this in mind, the following is our main result, which is an immediate consequence of Lemma 3, but has many ramifications.

Theorem 1. Let (N, 𝒟, J) be a CR (2m + 1)-manifold with Sasaki cone 𝔠𝔯_{+}(N, 𝒟, J). Then, for any χ, ξ ∈ 𝔠𝔯_{+}(N, 𝒟, J) with [χ, ξ] = 0, (N, 𝒟, J, χ) is (ξ, m + 2)-extremal if and only if (N, 𝒟, J, ξ) is extremal.

Together with Example 2, this result shows that the constructions of (f, m+2)-extremal Kähler metrics available in [5, 49, 50, 35, 44, 45] yield many new extremal Sasaki metrics.

We further observe that since L_{X_{ξ}}(Scal(g_{ξ})) = 0, [ξ, σ(ξ)] = 0. Hence the equivalent conclusions of Theorem 1 correspond to the occurrence of σ(ξ) in 𝔠𝔯^{ξ}; we then have ±σ(ξ) ∈ 𝔠𝔯_{+}^{ξ} if and only if the scalar curvature Scal(g_{ξ}) is everywhere positive or everywhere negative, in which case we can take χ = ±σ(ξ) in Theorem 1 to obtain Scal_{ξ,m+2}(g_{χ}) = ±1.

Corollary 1. Let (N, 𝒟, J, ξ) be an extremal Sasaki (2m + 1)-manifold. Then the 𝔠𝔯_{+}^{χ}- family of (ξ, m + 2)-extremal Sasaki structures on (N, 𝒟, J) contains a Sasaki structure χ := ±σ(ξ) of constant nonzero (ξ, m + 2) scalar curvature if and only if the extremal Sasaki structure ξ has nowhere zero scalar curvature. Thus there is an equivalence between Sasaki manifolds (N, 𝒟, J, χ) of constant nonzero (ξ, m + 2) scalar curvature and extremal Sasaki manifolds (N, 𝒟, J, ξ) with nowhere zero scalar curvature.

The proofs of Lemmas 1–3 are straightforward rephrasings of standard results in CR geometry, but for the convenience of the reader, we indicate their proofs in Section 1. In Section 2, we define, on a compact contact manifold of Sasaki type, a formal GIT setting for the search for (ξ, ν)-extremal Sasaki structures, extending the picture in [40] and providing a conceptual explanation for the key Lemma 3 above. Then in the rest of the paper we return to Kähler geometry and applications of Theorem 1, which gives a way of relating different Kähler geometries locally or (under suitable rationality conditions) globally. We formalize this as follows.

Definition 4. Let (M, g, J, ω) be a Kähler manifold and f a positive Killing potential. We say that (M̃, g̃, J̃, ω̃) is a CR twist of M by f, or an f-twist for short, if it is a Sasaki–Reeb quotient by the Sasaki structure fχ on the Sasaki manifold (N, 𝒟, J, χ) corresponding (over any open subset where [ω/2π] is integral) to M via Example 1.

A CR f-twist can be seen as a special case of the twist construction of Swann [60] (see also [43, 58]) which has been used to study different geometric structures. In these terms, Theorem 1 shows that any extremal Kähler metric can (locally) be obtained from a (f, m + 2)-extremal Kähler metric, via a CR f-twist, while Corollary 1 establishes an equivalence between Kähler metrics of constant (f, m + 2) scalar curvature and extremal Kähler metrics of nonvanishing scalar curvature.

In real dimension 2m = 4, the latter reduces to the equivalence between conformally Einstein–Maxwell Kähler metrics (g, J, ω, f) and extremal metrics (g̃, J̃, ω̃) of nonvanishing scalar curvature alluded to above: the extremal Kähler 4-manifold is obtained as the Sasaki–Reeb quotient with respect to an extremal Sasaki structure ξ of (N, 𝒟, J), whereas (g, J, ω, f) is obtained as the quotient with respect to the Sasaki structure defined by the scalar curvature of ξ. Thus our correspondence gives a conceptual explanation and generalization of [10, Prop. 3], and can be used to obtain new examples of extremal Sasaki and Kähler metrics from the known conformally Einstein–Maxwell Kähler ones.

More generally, as any CR twist of an extremal Kähler metric is (f, m + 2)-extremal for some f, one of the main theses of this paper is that one can reduce the search for extremal Kähler metrics to the search of (f, m + 2)-extremal Kähler metrics on simpler Kähler manifolds. We explore this idea in the remainder of the paper.

As a warm-up, in Section 3, we consider the simplest examples: the Bochner-flat Sasaki– Reeb quotients of CR spheres [19, 63] and products. Then, in Section 4, we turn our attention to toric geometry. While toric Kähler and Sasaki geometries have been well-studied, to apply our theory, we develop a CR-invariant viewpoint, building on [51, 52, 54, 56]. In the toric case, CR-invariance corresponds to projective invariance on the image of the momentum map, and as an interesting side benefit, we give a manifestly projectively invariant treatment of the Legendre transform, by relating it to a particular case of a Bernstein– Gelfand–Gelfand resolution [12, 53], as constructed in [21, 22]. Returning the main line of the paper, we then obtain explicit descriptions of the CR f-twists of toric manifolds and of toric bundles given by the generalized Calabi ansatz, showing that the latter are CR f-twists of a product metric. In Section 5, we recast, in terms of the general correspondence herein, some of the explicit families of (f, m + 2)-extremal Kähler metrics, including those obtained by the regular ambitoric ansatz in [5] and by an ansatz in [7]. This leads both to a higher dimensional extension of the regular ambitoric ansatz [5] and to a complete classification of the (f, m + 2)-extremal Kähler metrics obtained by this ansatz. In the final Section 6, we turn to global considerations. We define the Calabi problem for (ξ, ν)-extremal Sasaki metrics, which naturally generalizes the existence problem of extremal Sasaki metrics in a given Sasaki polarization [14], recently studied in many places [16, 17, 18, 26, 52, 56, 57, 62]. We end by illustrating how the (non)existence of (f, ν)-extremal Kähler metrics in a given integral Kähler class of a geometrically ruled complex surface (as studied in [11, 44, 47]) leads both to existence and non-existence results for extremal Sasaki metrics compatible with (possibly irregular) Sasaki–Reeb vector fields on the corresponding contact manifolds. In particular, we establish the following Yau–Tian–Donaldson type correspondence.

Theorem 2. Let (M, J) = P (O ⊕ L) → B be a compact ruled complex surface over a Riemann surface B, L a polarization of (M, J), and ω ∈ 2πc_{1}(L) an 𝕊^{1}-invariant Kähler metric with respect to the circle action by scalar multiplication in O. Let (N, 𝒟, J, χ) be the regular Sasaki manifold over (M, J, ω) given by Example 1, ξ ∈ 𝔠𝔯_{+}(N, 𝒟, J) the lift of the generator of the 𝕊^{1}-action on M and Ẑ_{ξ} the induced holomorphic vector field on L. Then N admits a X_{χ}-invariant, 𝒟-compatible CR structure which is extremal Sasaki with respect to ξ if and only if (M, L, Ẑ_{ξ}) is analytically relatively (Ẑ_{ξ}, 4) K-stable with respect to admissible test-configurations in the sense of [11].

This suggests a link between the weighted K-stability of [11, 47] for a smooth polarized variety, and K-stability of the Kähler cone of a Sasaki polarization, studied in [18, 26].

1. Proofs of Lemmas 1–3

Let (N, 𝒟) be a contact (2m + 1)-manifold, i.e., 𝒟 ⩽ T N is a rank 2n distribution on N, with quotient map η_{𝒟}: T N → T N/𝒟, whose Levi form L_{𝒟}(X, Y) = −η_{𝒟}(X, Y) (X, Y ∈ C_{N}^{∞}(𝒟)) is nondegenerate at each point of N. A CR structure on (N, 𝒟) is a complex structure J on 𝒟 such that the subbundle 𝒟^{(1,0)} of (1, 0)-vectors in 𝒟 ⊗ ℂ is closed under Lie bracket. This implies in particular that J is an almost CR structure, i.e., a complex structure on 𝒟 such that L_{𝒟} has type (1, 1) with respect to J. We say that (𝒟, J) is strictly pseudo-convex if L_{𝒟} is definite (with respect to J) at each point of N, i.e., L_{𝒟}(·, J·) is a definite bundle metric on 𝒟. It then follows that (N, 𝒟) is co-oriented, i.e., T N/𝒟 is an oriented real line bundle.

Proof of Lemma 1. We show that any section ξ of T N/𝒟 has a unique lift to a contact vector field X_{ξ} (with η_{𝒟}(X_{ξ}) = ξ). On the open subset where ξ is nonzero, the contact condition implies that X_{ξ} is the Reeb vector field of the contact form η_{ξ} := ξ^{−1}η_{𝒟}, which is characterized by η_{ξ}(X_{ξ}) = 1 and dη_{ξ}(X_{ξ}, ·) = 0. Now suppose ξ = fχ with χ nonvanishing and f a smooth function. Then dη_{χ} = df ∧ η_{ξ} + dη_{ξ} and the characterization of Reeb vector fields gives

(3)  X_{ξ} = fX_{χ} − (dη_{χ}|_{𝒟})^{−1}(df|_{𝒟})

where f is nonzero, but this formula extends X_{ξ} smoothly over the zeroset of f. This also shows ξ ↦ X_{ξ} is a first order differential operator. Since a contact vector field in 𝒟 is necessarily zero by the nondegeneracy of the Levi form, the lift is unique. □

Proof of Lemma 2. Clearly L_{X_{χ}}(fχ) = df(X_{χ})χ, and if f, h ∈ C_{M}^{∞}(ℝ)^{χ} then (3) implies

[fχ, hχ] = dh(X_{fχ}) = −(ω_{χ}|_{𝒟})^{−1}(df|_{𝒟}, dh|_{𝒟})

so the first part is immediate. We may thus suppose that f is the pullback of a smooth function, also denoted f, on a Sasaki–Reeb quotient (M, g, J, ω), with symplectic gradient K. Then the same formula (3) shows that X_{ξ} is a lift of K to N. Furthermore, since X_{ξ} is contact and X_{χ}-invariant, L_{X_{ξ}}J is horizontal and X_{χ}-invariant, hence vanishes iff its pushforward to M vanishes, which holds iff L_{K}J = 0 on M, i.e., f is a Killing potential. □

Proof of Lemma 3. We let ξ = fχ with χ ∈ 𝔠𝔯_{+}^{ξ} and expand the definition of the (ξ, m + 2) scalar curvature of g_{χ} on N to obtain

(4) Scal_{ξ,m+2}(g_{χ}) χ = (f^{2}Scal(g_{χ}) − 2(m + 1)fΔ_{g_{χ}}f − (m + 1)(m + 2)|df|_{𝒟}|_{g_{χ}}^{2}) χ

= (fScal(g_{χ}) − 2(m + 1)Δ_{g_{χ}}f − \frac{(m + 1)(m + 2)}{f}|df|_{𝒟}|_{g_{χ}}^{2}) ξ. (5)

where (5) holds on the open subset U where ξ is nonzero. Now, as noted already, Scal(g_{ξ}) and Scal(g_{χ}) are the Tanaka–Webster scalar curvatures of the Tanaka–Webster connections induced by ξ and χ [27, §4], and a straightforward but tedious computation of the change of the Tanaka–Webster scalar curvature under a change of connection, which can be found e.g. in [42, (2.9)], shows that on U, (5) computes Scal(g_{ξ})ξ, independently of f. However, on any open subset where ξ = 0, f = 0 and hence the right hand side of (4) is zero. Thus Scal_{ξ,m+2}(g_{χ}) χ is independent of χ on a dense open subset, hence everywhere. The equality (4) now shows that this is a second order quadratic differential operator in ξ. □

2. Formal GIT picture for weighted extremal Sasaki metrics

Let (N, 𝒟) be a compact co-oriented contact (2m + 1)-manifold (or orbifold), and fix a torus 𝕋 in its group Con(N, 𝒟) of contact transformations. As explained in the introduction, we tacitly identify the Lie algebra of Con(N, 𝒟) with the space 𝔠𝔬𝔫(N, 𝒟) of smooth sections of T N/𝒟 and denote by 𝔠𝔬𝔫_{+}(N, 𝒟) the open cone of positive sections in of T N/𝒟 with respect to its orientation. Let Con(N, 𝒟)^{𝕋} denote the group of 𝕋-equivariant contact transformations, with Lie algebra identified with the space 𝔠𝔬𝔫(N, 𝒟)^{𝕋} of 𝕋-invariant sections of T N/𝒟. Thus, the Lie algebra 𝔱 of 𝕋 as a linear subspace of 𝔠𝔬𝔫(N, 𝒟)^{𝕋}.

Now observe that vol_{𝒟} := η_{𝒟}∧L_{𝒟}^{∧m} is a well-defined section of ∧^{2m+1}T^{∗}N ⊗(T N/𝒟)^{m+1}: indeed for any nonvanishing section χ of T N/𝒟, χ^{−m−1}vol_{𝒟} = η_{χ} ∧ dη_{χ}^{∧m}. Fix two such sections χ, ξ ∈ 𝔱∩𝔠𝔬𝔫_{+}(N, 𝒟) and ν ∈ ℝ. Then 𝔠𝔬𝔫(N, 𝒟)^{𝕋} has a bi-invariant inner product

⟨ξ_{1}, ξ_{2}⟩_{ξ,χ,ν} := ∫_{N}(ξ_{1}/χ)(ξ_{2}/χ)(ξ/χ)^{−ν−1} η_{χ} ∧ dη_{χ}^{∧m} = ∫_{N} ξ_{1} ξ_{2} ξ^{−ν−1}χ^{ν−2−m} vol_{𝒟}.

Let AC_{+}(N, 𝒟)^{𝕋} be the space of 𝕋-invariant almost CR structures on (N, 𝒟), such that L_{𝒟} is of type (1, 1) and positive definite with respect to J and the given orientation on T N/𝒟. We denote by C_{+}(N, 𝒟)^{𝕋} ⊆ AC_{+}(N, 𝒟)^{𝕋} the subset of 𝕋-invariant compatible CR structures on (N, 𝒟). Notice that Con(N, 𝒟)^{𝕋} acts naturally on AC_{+}(N, 𝒟)^{𝕋} (preserving C_{+}(N, 𝒟)^{𝕋}) and the tangent space of AC_{+}(N, 𝒟)^{𝕋} at J is identified with the Fréchet space of smooth sections J̇ of End(𝒟) satisfying

J̇J + JJ̇ = 0, L_{𝒟}(J̇·, ·) + L_{𝒟}(·, J̇·) = 0,

so AC_{+}(N, 𝒟)^{𝕋} has a formal Fréchet Kähler structure (J, Ω^{ξ,χ,ν}) defined by J_{J}(J̇) := JJ̇ and

Ω_{J}^{ξ,χ,ν}(J̇_{1}, J̇_{2}) := \frac{1}{2} ∫_{N} tr(JJ̇_{1}J̇_{2}) (ξ/χ)^{−ν+1} η_{χ} ∧ dη_{χ}^{∧m} = \frac{1}{2} ∫_{N} tr(JJ̇_{1}J̇_{2}) ξ^{−ν+1}χ^{ν−2−m} vol_{𝒟}.

To see this, we can take χ ∈ 𝔱∩𝔠𝔬𝔫_{+}(N, 𝒟) to be quasi-regular with a global quotient (M, ω). Then our set-up reduces to the formal GIT picture for (f, ν)-extremal ω-compatible, 𝕋/𝕊_{χ}^{1}- invariant almost-Kähler metrics on the symplectic orbifold (M, ω), discussed in [7, 10, 45]. The momentum map for the action of Ham(M, ω)^{𝕋} at a compatible Kähler structure is identified with the (f, ν) scalar curvature, showing that for a CR structure J ∈ C_{+}(N, 𝒟)^{𝕋}, the corresponding momentum map for the action of Con(N, 𝒟)^{𝕋} on AC_{+}(N, 𝒟)^{𝕋} is Scal(g_{ξ})ξ (where we multiply by ξ to obtain an element of 𝔠𝔬𝔫(N, 𝒟)^{𝕋}).

We now notice that for ν = m + 2, the bi-invariant inner product on 𝔠𝔬𝔫(N, 𝒟)^{𝕋} and the formal Kähler structure on AC_{+}(N, 𝒟)^{𝕋} are independent of χ. In this case, our setting reduces to the formal GIT picture for extremal Sasaki metrics on (N, η_{ξ}) discussed in [40], where the momentum map for the action of Con(N, 𝒟)^{𝕋} is the Tanaka–Webster scalar curvature Scal(g_{ξ})ξ (the multiplication by ξ is implicit in [40] through the identification of the Lie algebra with smooth functions).

Hence this provides another explanation as to why the weight m + 2 is special and the transversal (ξ, m + 2) scalar curvature Scal_{ξ,m+2}(g_{χ})χ of g_{χ} is independent of χ and equal to the Tanaka–Webster scalar curvature of g_{ξ}, viewed as an element of 𝔠𝔬𝔫(N, 𝒟)^{𝕋}.

3. Basic Examples

3.1. Bochner-flat (f, m + 2)-extremal metrics. Let us now consider the standard CR sphere 𝕊^{2m+1} ⊆ ℂ^{m+1}, m ⩾ 2, with 𝒟 = T 𝕊^{2m+1} ∩ J(T 𝕊^{2m+1}), J induced by the standard complex structure on ℂ^{m+1}, and 𝔠𝔯(𝕊^{2m+1}, 𝒟, J) ≅ 𝔰𝔲(1, m+1). By a result of Webster [63], for any χ ∈ 𝔠𝔯_{+}(𝕊^{2m+1}, 𝒟, J), the transversal Kähler structure (g_{χ}, J, ω_{χ}) is Bochner-flat, and thus extremal (see [19]), and any Bochner-flat Kähler manifold (M, g, J, ω) is (locally) obtained as a χ-reduction of (𝕊^{2m+1}, 𝒟, J) for some such χ. It then follows from Theorem 1 that for any ξ ∈ 𝔠𝔯_{+}^{χ}, (𝒟, J, χ) is a (ξ, m + 2)-extremal, and hence by Lemma 2 (see Example 2), we have the following observation.

Proposition 1. Let (M, g, J, ω) be a Bochner-flat Kähler 2m-manifold and f > 0 a Killing potential. Then (g, ω) is (f, m + 2)-extremal.

To obtain global examples, we let χ_{w} ∈ 𝔠𝔯_{+}(𝕊^{2m+1}, 𝒟, J) correspond to the weighted Hopf fibration 𝕊^{2m+1} → ℂP_{w}^{m}, realizing (𝕊^{2m+1}, 𝒟, J, χ_{w}) as a quasi-regular Sasaki manifold over the Bochner-flat weighted projective space (ℂP_{w}^{m}, J, g, ω) (see [19, 28]). Thus we have the following higher dimensional extension of [10, Prop. 5].

Corollary 2. The Bochner-flat metric on ℂP_{w}^{m} is (f, m + 2)-extremal for any positive Killing potential f.

3.2. Flat (f, ν)-extremal metrics. The conclusion of Proposition 1 can be strengthened for flat Kähler metrics.

Proposition 2. Let (V, g_{V} , ω_{V}) be a flat Kähler manifold and f > 0 a Killing potential on V. Then, for any scalar-flat Kähler manifold (B, g_{B}, ω_{B}) the Kähler product (M, g, ω) of (V, g_{V} , ω_{V}) and (B, g_{B}, ω_{B}) is (f, ν)-extremal for any ν.

Proof. As B is scalar-flat, (2) implies that the (f, ν) scalar curvature of V × B equals the (f, ν) scalar curvature of M. Thus, we need to establish the claim on M := V. As g is a flat metric, (2) reduces to

(6)  −2(ν − 1)fΔ_{g}f − ν(ν − 1)|df|_{g}^{2}

so it suffices to show that each of the two terms in (6) is a Killing potential for g. For the first term, using that g is Ricci-flat and f is a Killing potential, the Bochner identity shows that Δ_{g}f is a constant, and thus fΔ_{g}f is a Killing potential of g. For the second term, using that g is Bochner-flat and Proposition 1, it follows that (6) with ν = m + 2 gives rise to a Killing potential, and hence |df|_{g}^{2} is a Killing potential for g. □

3.3. (f, m + 2)-extremal products. As noted in the proof of Proposition 2, the Kähler product of a scalar-flat, Kähler 2(m−ℓ)-manifold (B, g_{B}, ω_{B}) with a (f, ν)-extremal Kähler 2ℓ-manifold (V, g_{V} , ω_{V}) gives rise to a (f, ν)-extremal 2m-manifold (M, g, ω). In [11, 45], for any given ν, large families of (f, ν)-extremal Hodge (i.e., compact, integral) Kähler manifolds (V, g_{V} , ω_{V}) of dimension 2ℓ are constructed. Taking such a (V, g_{V} , ω_{V} , f) with ν = m + 2 (m ⩾ ℓ) and considering the Kähler product of (V, g_{V} , ω_{V}) with a scalar-flat Hodge Kähler 2(m − ℓ)-manifold (B, g_{B}, ω_{B}), we obtain a compact (f, m + 2)-extremal Kähler 2m-manifold (M, g, ω, J), which gives rise to a compact extremal Sasaki (2m+1)-manifold (N, 𝒟, J, ξ) via Example 1. Notice that the extremal Sasaki manifold thus obtained is not in general quasi-regular, but when it is (which places a rationality condition on the positive Killing potential f of g_{V}), the resulting extremal Kähler orbifold is not in general a product, even though (M, g, ω) is. We detail and generalize this observation below in the setting of toric bundles.

4. Toric geometry and toric bundles

4.1. Toric contact manifolds. Applications of Theorem 1 depend in particular on the existence of independent commuting elements ξ, χ ∈ 𝔠𝔯(N, 𝒟, J) ⩽ 𝔠𝔬𝔫(N, 𝒟). The maximal dimension of an abelian subalgebra of 𝔠𝔬𝔫(N, 𝒟) is m + 1 (assuming N is connected of dimension 2m + 1). Let us therefore consider the case that we have such an (m + 1)- dimensional abelian subalgebra 𝔥 ↪ 𝔠𝔬𝔫(N, 𝒟); a ↦ ξ_{a}, in which case (N, 𝒟, 𝔥) is said to be toric. We usually assume that the corresponding contact vector fields generate an effective contact action of a real (m + 1)-torus 𝕋^{m+1}, whose Lie algebra is thus canonically isomorphic to 𝔥. Hence we have an integral lattice Λ ⊆ 𝔥 with 𝕋^{m+1} ≅ 𝔥/2πΛ, and, on the dense open set N^{∘} where the 𝕋^{m+1}-action is free, angle coordinates t: N^{∘} → 𝔥/2πΛ.

We also assume that the tautological bundle homomorphism

N × 𝔥 → T N/𝒟
(p, a) ↦ ξ_{a}(p)

is surjective (as it is in the Sasaki case), so that its transpose (T N/𝒟)^{∗} → N × 𝔥^{∗} is injective. We thus obtain a momentum map μ̄: N → P(𝔥^{∗}), where μ̄(p) is the image of (T N/𝒟)_{p}^{∗} in 𝔥^{∗}, for any p ∈ N. Hence μ̄^{∗}O_{𝔥^{∗}}(−1) ≅ (T N/𝒟)^{∗}, where O_{𝔥^{∗}}(−1) is the tautological line bundle over the projective space P(𝔥^{∗}), with fibre O_{𝔥^{∗}}(−1)_{τ} = τ ⩽ 𝔥^{∗}.

Remark 1. Alternatively observe (cf. [54]) that the annihilator 𝒟^{0} ⩽ T^{∗}N of 𝒟 inherits from T^{∗}N a closed 2-form which is nondegenerate on the complement of the zero section, and any contact vector field on N lifts to a hamiltonian vector field on 𝒟^{0}. The restriction to 𝔥 of the momentum map of this action is μ̃: 𝒟^{0} → 𝔥^{∗} with ⟨μ̃(α), a⟩ = α(ξ_{a}) for α ∈ 𝒟^{0} and a ∈ 𝔥, where angle brackets denote contraction of 𝔥^{∗} with 𝔥. Using the natural duality 𝒟^{0} ≅ (T N/𝒟)^{∗}, (p, μ̃(α)) is the element of μ̄^{∗}O_{𝔥^{∗}}(−1) ⩽ N × 𝔥^{∗} corresponding to α ∈ 𝒟_{p}^{0}.

Herein, we generally work instead with the momentum section μ̂: N → 𝔥^{∗} ⊗ (T N/𝒟) defined by ⟨a, μ̂(p)⟩ = ξ_{a}(p) ∈ (T N/𝒟)_{p} for a ∈ 𝔥 and p ∈ N. If z: P(𝔥^{∗}) → 𝔥^{∗} ⊗ O_{𝔥^{∗}}(1) denotes the tautological section, with O_{𝔥^{∗}}(1) := O_{𝔥^{∗}}(−1)^{∗} and ⟨a, z(τ)⟩ = ⟨a, ·⟩|_{τ} for a ∈ 𝔥 and τ ∈ P(𝔥^{∗}), then under the isomorphism μ̄^{∗}O_{𝔥^{∗}}(1) ≅ T N/𝒟, μ̂ = μ̄^{∗}z.

Any nonzero ε ∈ 𝔥 defines an affine chart

A := {p ∈ 𝔥^{∗}|⟨ε, p⟩ = 1} ↪ P(𝔥^{∗})

and if U ⊆ P(𝔥^{∗}) is an open subset of the image of this chart, then ⟨ε, z⟩ restricts to a trivialization of O_{𝔥^{∗}}(1)|_{U}. In this trivialization, z is the affine lift of U to A ⊆ 𝔥^{∗}, i.e., we have z: U → 𝔥^{∗} with ⟨ε, z⟩ = 1. Hence ⟨ε, dz⟩ = 0, i.e.,

dz: T U → U × ε^{0}, where ε^{0} = {p ∈ 𝔥^{∗}|⟨ε, p⟩ = 0},

is the trivialization of T U in this affine chart. Thus 𝔥 is naturally identified with the space Aff (U) of affine functions on U: a ∈ 𝔥 defines the affine function ⟨a, z⟩ on U, with ε corresponding to the constant function 1. If 𝔱 = 𝔥/ span(ε), there is a short exact sequence

(7)  0 → ℝ ⟶^{ε} 𝔥 ⟶^{δ} 𝔱 → 0,

where the duality 𝔱^{∗} ≅ ε^{0} identifies 𝔱 with T_{p}^{∗}U for any p ∈ U, while the quotient map δ sends an affine function to its linear part (the constant value of its derivative).

Now μ̄: N → P(𝔥^{∗}) takes values in the affine chart defined by ε iff χ := ξ_{ε} ∈ 𝔠𝔬𝔫(N, 𝒟) is a nonvanishing section of T N/𝒟. In this trivialization the momentum section becomes a function μ̂: N → 𝔥^{∗} with ⟨ε, μ̂⟩ = 1. The transversal Kähler form dη_{χ}|_{𝒟} descends to a symplectic form ω on any quotient of M of N by χ. If χ is quasi-regular, we may take (M, ω) to be the global quotient, which is then a toric symplectic 2m-orbifold under the hamiltonian action of the real m-torus 𝕋^{m} = 𝕋^{m+1}/𝕊_{χ}^{1} where 𝕊_{χ}^{1} is the circle action generated by X_{χ}.

Concretely, we can choose a basis e_{0}, … e_{m} for 𝔥 such that ε = e_{0}, introduce coordinates z_{j} = ⟨e_{j}, z⟩ on U with z_{0} = 1, and write ⟨a, z⟩ = a_{0} + a_{1}z_{1} + ⋯ a_{n}z_{n}. We then have coordinates μ̂ = (μ̂_{0}, μ̂_{1}, … μ̂_{m}) and t = (t_{0}, t_{1}, … t_{m}) on N^{∘} with μ̂_{0} = 1 and

(8)  η_{χ} = ⟨μ̂, dt⟩ = dt_{0} + ∑_{j=1}^{m} μ̂_{j}dt_{j}.

Hence dη_{χ} = ⟨dμ̂ ∧ dt⟩, and since dμ̂ takes values in 𝔱^{∗}, ⟨dμ̂ ∧ dt⟩ depends only on δ(dt), which descends to a 𝔱-valued 1-form on M. If we denote by μ the map M → A induced by the momentum section on N in the affine chart A, we may therefore write ω = ⟨dμ ∧ dt⟩ on the image M^{∘} of N^{∘}. The isomorphism f ↦ fχ of Lemma 2 identifies 𝔥 with an abelian Lie subalgebra of C_{M}^{∞}(ℝ) under the Poisson bracket induced by ω. Evidently, χ = ξ_{ε} corresponds to f ≡ 1 on M. More generally, for any a ∈ 𝔥 the function f_{a} = f_{a}(μ) ∈ C_{M}^{∞}(ℝ) corresponding to ξ_{a} ∈ 𝔠𝔬𝔫(N, 𝒟) satisfies f_{a}(μ(p)) = ⟨a, μ(p)⟩ for all p ∈ M, i.e., f_{a}(μ) = ⟨a, μ⟩ is the pullback by μ of the affine function f_{a}(z) = ⟨a, z⟩ on A. Pulling back by μ, we may thus reinterpret (7) on M: in particular, we may view δ as the restriction to 𝔥 ↪ C_{M}^{∞}(ℝ) of the symplectic gradient, and 𝔱 as its image in Ham(M, ω).

4.2. Toric CR manifolds and their Sasaki–Reeb quotients. Thus far we have only considered the toric contact geometry of N and induced toric symplectic geometry on M. According to [39], on the dense open subset M^{∘}, any toric almost Kähler structure may be written in momentum–angle coordinates (μ, t) as:

(9) g = ⟨dμ, G(μ), dμ⟩ + ⟨dt, H(μ), dt⟩,  Jdt = −⟨G(μ), dμ⟩, ω = ⟨dμ ∧ dt⟩,  Jdμ = ⟨H(μ), dt⟩,

where H is a smooth positive definite S^{2}𝔱^{∗}-valued function on the momentum image Δ^{∘} := μ(M^{∘}) and G = H^{−1} is its pointwise inverse, a smooth S^{2}𝔱-valued function. This local expression makes sense in the affine setting, where Δ^{∘} lies in an affine space A ⊆ 𝔥^{∗}, modelled on 𝔱^{∗}: G is a metric on T Δ^{∘} ≅ Δ^{∘} × 𝔱^{∗}, and H the inverse metric on T^{∗}Δ^{∘}.

A metric of the form (9) is Kähler, i.e., J is integrable, if and only if ⟨dG ∧ dz⟩ = 0, which in affine coordinates z = (1, z_{1}, … z_{m}) on A reads

∂G_{ij}/∂z_{k} = ∂G_{ik}/∂z_{j}

for all i, j, k ∈ {1, … m}. Since G is symmetric, this is the integrability condition to write G = Hess(u) for a smooth strictly convex function u defined on the momentum image Δ^{∘}, which is called a symplectic potential. When M is a compact manifold (or orbifold), Delzant theory [30, 55] implies that Δ^{∘} is the interior of a (rational) Delzant polytope Δ ⊆ 𝔱^{∗}, and u satisfies the Abreu boundary conditions [1] on ∂Δ.

The theory of symplectic potentials in toric Kähler geometry thus relies upon a locally exact complex of linear differential operators

(10)  C_{A}^{∞}(ℝ) ⟶^{Hess} C_{A}^{∞}(S^{2}𝔱) ⟶^{D} C_{A}^{∞}(∧^{2}𝔱 ⊙ 𝔱)

where D(G) = ⟨dG ∧ dz⟩ and ∧^{2}𝔱 ⊙ 𝔱 denotes the alternating-free tensors in ∧^{2}𝔱 ⊗ 𝔱 (the kernel of the projection, alternation, to ∧^{3}𝔱). This complex is invariant under affine transformations by construction, but can actually be made projectively invariant. To do this, observe that the kernel of the hessian consists of affine functions, which on a domain U ⊆ P(𝔥^{∗}) in projective space are not naturally ordinary functions, but sections of O_{𝔥^{∗}}(1), as we discussed above. Also the cotangent space is naturally T^{∗}U ≅ O_{𝔥^{∗}}(−1)^{0} ⊗O_{𝔥^{∗}}(−1) ⩽ 𝔥 ⊗ O_{𝔥^{∗}}(−1). With these modifications, we obtain a locally exact complex of projectively invariant linear differential operators, beginning

(11) 0 → 𝔥 ⟶^{⟨z,·⟩} C_{U}^{∞}(O_{𝔥^{∗}}(1)) ⟶^{Hess} C_{U}^{∞}(S^{2}T^{∗}U ⊗ O_{𝔥^{∗}}(1)) ⟶^{D}

C_{U}^{∞}(∧^{2}T^{∗}U ⊙ T^{∗}U ⊗ O_{𝔥^{∗}}(1)) → ⋯,

and which reduces to (10) in any affine chart.

The complex (11) is a simple example of a Bernstein–Gelfand–Gelfand resolution [12, 53]. Without wishing to dwell on the general machinery, we observe that the construction of this resolution in [21, 22] gives a manifestly invariant construction of the projective hessian. The main idea is to relate (11) to the 𝔥-valued de Rham complex

0 → 𝔥 → C_{U}^{∞}(𝔥) ⟶^{d} C_{U}^{∞}(T^{∗}U ⊗ 𝔥) → ⋯

using the following construction.

Lemma 4. For any u ∈ C_{U}^{∞}(O_{𝔥^{∗}}(1)) there is a unique L(u) ∈ C_{U}^{∞}(𝔥) with ⟨z, L(u)⟩ = u (i.e., L(u) is a lift of u) and ⟨z, dL(u)⟩ = 0. Furthermore, in any local affine chart with z_{0} = 1 and coordinates z_{1}, … z_{m}, L(u) = (u_{0}, u_{1}, … u_{m}) where

−u_{0} = z_{1} \frac{∂u}{∂z_{1}} + ⋯ + z_{m} \frac{∂u}{∂z_{m}} − u

is the Legendre transform of u, and u_{j} = ∂u/∂z_{j} for j ∈ {1, … m}.

Proof. In local affine coordinates with z_{0} = 1, L(u) = (u_{0}, u_{1}, … u_{m}) is a lift of u iff u = u_{0}+z_{1}u_{1}+⋯+z_{m}u_{m}, and we require in addition 0 = ⟨z, dL(u)⟩ = du_{0}+z_{1}du_{1}+⋯+z_{m}du_{m}. Thus du = u_{1}dz_{1} + ⋯ + u_{m}dz_{m}, forcing u_{j} = ∂u/∂z_{j}, which in turn determines u_{0}. □

This observation has some interesting consequences.

• L: C_{U}^{∞}(O_{𝔥^{∗}}(1)) → C_{U}^{∞}(𝔥) is a first order projectively invariant linear differential operator. Furthermore L(u) is constant if and only if u is an affine section of O_{𝔥^{∗}}(1).

• This differential lift L(u) of u is a “universal Legendre transform” in the sense that for any ε ∈ 𝔥 and p ∈ 𝔥^{∗} with ⟨ε, p⟩ = 1, −⟨L(u), p⟩ is the Legendre transformation of u in the affine chart defined by ε with basepoint p (it thus depends only on p, not ε). Furthermore, the projection of L(u) onto 𝔱 = 𝔥/ span(ε) gives the conjugate coordinates (the components of du in this affine chart, which depend only on ε, not p).

• Any u ∈ C_{U}^{∞}(O_{𝔥^{∗}}(1)) defines a congruence of affine hyperplanes {y ∈ 𝔥 : ⟨z(p), y⟩ = u(p)} in 𝔥 parametrized by p ∈ U. The lift L(u) is the envelope of this hyperplane congruence (a classical view on the Legendre transformation): ⟨z, L(u)⟩ = u and ⟨z, dL(u)⟩ = 0.

To complete the construction of the projectively invariant hessian, it remains to observe that since ⟨z, dL(u)⟩ = 0, dL(u) is a section of T^{∗}U ⊗ O_{𝔥^{∗}}(−1)^{0}; hence ⟨dz ⊗ dL(u)⟩ is a section of S^{2}T^{∗}U ⊗ O_{𝔥^{∗}}(1), since dz is well-defined modulo O_{𝔥^{∗}}(−1) and we have 0 = d⟨z, dL(u)⟩ = ⟨dz∧dL(u)⟩. In affine coordinates, dL(u) = (du_{0}, d(∂u/∂z_{1}), … d(∂u/∂z_{m})) and projecting away from ε = (1, 0, … 0) gives the usual hessian of u.

To apply this to a toric contact manifold (N, 𝒟, 𝔥) with μ̄(N^{∘}) = Δ^{∘}, it is convenient to view the projectively invariant hessian of u ∈ C_{Δ^{∘}}^{∞}(O_{𝔥^{∗}}(1)) as the section G = Hess(u) of S^{2}O_{𝔥^{∗}}(−1)^{0} ⊗ O_{𝔥^{∗}}(−1) ⩽ S^{2}𝔥 ⊗ O_{𝔥^{∗}}(−1) with ⟨G(z), dz⟩ = dL(u). Then we may define a CR structure J on N^{∘} by Jdt|_{𝒟} = −μ̄^{∗}dL(u)|_{𝒟} = −⟨G(μ̂), dμ̂⟩|_{𝒟}. For any χ ∈ 𝔠𝔯(N, 𝒟, J), this reduces to the toric Kähler structure defined by u on local Sasaki–Reeb quotients.

Example 3. If m = 1 and u = u(z_{1}) is a symplectic potential in the affine chart z = (1, z_{1}) then the differential lift of u to 𝔥 is L(u) = (u(z_{1}) − z_{1}u′(z_{1}), u′(z_{1})) with ⟨(1, z_{1}), L(u)⟩ = u(z_{1}) and dL(u) = u′′(z_{1})(−z_{1}, 1) dz_{1}. Thus Jdt_{0}|_{𝒟} = u′′(μ_{1})μ_{1} dμ_{1}|_{𝒟} and Jdt_{1}|_{𝒟} = −u′′(μ_{1}) dμ_{1}|_{𝒟}, in accordance with (dt_{0} + μ_{1}dt_{1})|_{𝒟} = η_{χ}|_{𝒟} = 0.

Remark 2. A key feature of our approach is that we avoid considering compatible complex structures on the symplectic cone in 𝒟^{0} over N: such structures induce not only a CR structure J on N, but also a preferred Sasaki structure χ, a choice we wish to decouple. However, it is straightforward to compare our approach with works such as [2, 52, 57] which use the symplectic cone. First, sections of O_{𝔥^{∗}}(1) over U ⊆ P(𝔥^{∗}) correspond bijectively to homogeneous functions of degree 1 on the inverse image of U in 𝔥^{∗} \ {0} which contains the momentum image Ũ of the symplectic cone. Thus a symplectic potential in our sense induces an ordinary function u on Ũ, homogeneous of degree 1. However, the hessian of any such function is degenerate in radial directions, so does not define a metric on the symplectic cone. To get around this, we exploit the fact that symplectic potentials are not well-defined: for any a ∈ 𝔥, we can add the linear form ⟨a, z⟩|_{Ũ} to u without changing its hessian. Hence symplectic potentials are really elements of the quotient of C∞ ; Ũ(ℝ) by 𝔥.

To be concrete, if f_{1}, … f_{k} are linear forms on Ũ corresponding to a_{(1)}, … a_{(k)} ∈ 𝔥, then u = ∑_{j=1}^{k} f_{j} log |f_{j}| is a function on Ũ with u(λp) = λu(p) + log |λ| ∑_{j=1}^{k} f_{j}. Hence it is homogeneous of degree 1 modulo 𝔥, but only strictly homogeneous of degree 1 if ∑_{j=1}^{k} a_{(j)} = 0. Its hessian is G̃ = ∑_{j=1}^{k} a_{(j)}^{2}/f_{j} with ⟨z, G̃⟩ = ∑_{j=1}^{k} a_{(j)}, which is constant.

We can interpret this in our formalism by modifying the differential lift: for a ∈ 𝔥 and u ∈ C_{Δ^{∘}}^{∞}(O_{𝔥^{∗}}(1)), we define L_{a}(u) by ⟨z, L_{a}(u)⟩ = u and ⟨z, dL_{a}(u)⟩ = 2a. Assuming Hess(u) is nondegenerate and ⟨a, z⟩ is nonvanishing, dL_{a}(u) is nondegenerate, and defines the metric on the symplectic cone corresponding to the CR structure defined by u and the Sasaki structure ξ_{a}.

4.3. The CR twist of a toric manifold. Suppose (M, g, J, ω) is given by (9) and (N, 𝒟, J, χ) is a (local) Sasaki (2m+1)-manifold over M corresponding to an extension (7) of 𝔱 by ℝ. We can suppose we are in the affine picture with ⟨ε, μ⟩ = 1 on M, where χ = ξ_{ε}, and introduce coordinates z_{j} = ⟨e_{j}, z⟩ with e_{0} = ε so that the induced contact form on N is given by (8), where μ̂ is the pullback of μ to N.

For any a ∈ 𝔥, the affine function f_{a}(z) = ∑_{j=0}^{m} a_{j}z_{j} is positive on the momentum image of M if and only if f_{a}(μ) is a positive Killing potential on (M, g, J, ω) if and only if ξ_{a} = f_{a}(μ̂)χ is a Sasaki structure on (N, 𝒟, J). A CR f_{a}-twist of (M, g, J, ω) is then the induced toric Kähler metric on any Sasaki–Reeb quotient of (N, 𝒟, J) by ξ_{a}, which in turn has the form (9) for suitable coordinates μ̃, t̃ and a symplectic potential ũ(μ̃) with G̃ = Hess(ũ). The new affine chart Ã on P(𝔥^{∗}) has tautological affine coordinate z̃ with 1 = ⟨a, z̃⟩ = f_{a}(z)⟨ε, z̃⟩ and hence z̃ = z/f_{a}(z).

To obtain explicit momentum-angle coordinates on M̃, we need to extend ẽ_{0} := a to a basis ẽ_{0}, ẽ_{1}, … ẽ_{m} of 𝔥. One approach (see e.g. [52]) is to assume a_{0} ≠ 0 (which we can arrange by a translation of z) so that we may take ẽ_{j} = e_{j} for j ∈ {1, … m}. Since u transforms as a section of O_{𝔥^{∗}}(1), we have the following result.

Lemma 5. Any CR f_{a}-twist (g̃, ω̃) of the toric Kähler metric (9) with respect to the positive affine function f_{a}(z) = a_{0} + a_{1}z_{1} + ⋯ + a_{m}z_{m} with a_{0} ≠ 0 is a toric Kähler metric of the form (9) on Δ̃^{∘} × ℝ^{m} with respect to momentum-angle coordinates μ̃, t̃ and symplectic potential ũ given by ⟨μ̃, a⟩ = 1,

(12)  μ̃_{j} = \frac{μ_{j}}{f_{a}(z)}, t̃_{j} := t_{j} − \frac{a_{j}}{a_{0}} t_{0}, j ∈ \{1, … m\} and ũ(z̃) = \frac{u(z)}{f_{a}(z)}.

As shown in [51, 52], when M is compact, the rescaling z ↦ z̃ sends the polytope Δ in A to a polytope Δ̃ ⊆ Ã, and we have a compact CR f-twist (M̃, g̃, ω̃) provided this polytope is rational.

Example 4. We illustrate the CR twist in the simple case of a toric Riemann surface metric

(13)  g_{V} = \frac{dμ_{1}^{2}}{A(μ_{1})} + A(μ_{1}) dt_{1}^{2},  ω_{V} = dμ_{1} ∧ dt_{1},

with profile function A. This is a Sasaki–Reeb quotient of a Sasaki 3-manifold (N, 𝒟, J, χ) with

η_{χ} = dt_{0} + μ_{1}dt_{1}, Jdt|_{𝒟} = (Jdt_{0}|_{𝒟}, Jdt_{1}|_{𝒟}) = −\frac{(−μ_{1}, 1)}{A(μ_{1})} dμ_{1}|_{𝒟},

using the affine chart z = (1, z_{1}) as in Example 3. Thus A(z_{1}) = 1/u′′(z_{1}) for a symplectic potential u = u(z_{1}), and there are straightforward integral formulae

L(u)(z_{1}) = ∫ ^{z_{1}} \frac{(−x, 1)}{A(x)}dx, u(z_{1}) = ⟨L(u), (1, z_{1})⟩ = ∫ ^{z_{1}} \frac{z_{1} − x}{A(x)}dx,

for u and its differential lift (i.e., projective Legendre transformation) L(u). Any CR f_{a}- twist, with f_{a}(z) = a_{0} + a_{1}z_{1}, is then a Sasaki–Reeb quotient of N by the Sasaki structure ξ = f_{a}(μ)χ with contact form η_{ξ} = η_{χ}/f_{a}(μ). If a ≠ 0, then as in Lemma 5 we may set t_{0} = a_{0}t̃_{0} and t_{1} = t̃_{1} + a_{1}t̃_{0}, so that η_{ξ} = dt̃_{0} + μ̃_{1}dt̃_{1}, with μ̃_{1} = μ_{1}/(a_{0} + a_{1}μ_{1}), has Sasaki–Reeb field X_{ξ} = ∂/∂t̃_{0}. We then compute

Jdt̃|_{𝒟} = (Jdt̃_{0}|_{𝒟}, Jdt̃_{1}|_{𝒟}) = −\frac{(−μ̃_{1}, 1)}{Ã(μ̃_{1})} dμ̃_{1}|_{𝒟}

z̃_{1} = \frac{z_{1}}{a_{0} + a_{1}z_{1}} and Ã(z̃_{1}) = \frac{a_{0}^{2} A(z_{1})}{(a_{0} + a_{1}z_{1})^{3}} . with

However, other choices can be convenient: for example if the momentum image μ(V) is [−1, 1] and f_{a} ≠ 0 on [−1, 1] (i.e., |a_{0}| > |a_{1}|) then we can instead preserve [−1, 1] with

z̃_{1} = \frac{a_{0}z_{1} + a_{1}}{a_{0} + a_{1}z_{1}} and Ã(z̃_{1}) = \frac{(a_{0}^{2} − a_{1}^{2})^{2}}{(a_{0} + a_{1}z_{1})^{3}} A(z_{1}).

4.4. The generalized Calabi ansatz. We now discuss CR twists of Kähler metrics on certain toric fibre bundles π: M → B over a Kähler base manifold (B, g_{B}, ω_{B}). We follow the approach in [9], to which we refer the reader for further details, recalling here only the special case in which we are interested.

Let (V, g_{V} , ω_{V} , 𝕋^{ℓ}) be a toric Kähler 2ℓ-manifold and let π: P → B be a principal 𝕋^{ℓ}- bundle over a Kähler 2d-manifold (B, g_{B}, ω_{B}), equipped with a connection 1-form θ ∈ Ω^{1}(P, 𝔱) (𝔱 being the Lie algebra of 𝕋^{ℓ}) such that

(14)  dθ = ζ ⊗ ω_{B}, ζ ∈ 𝔱.

As before, we assume 𝔱 = 𝔥/ span(ε), where 𝔥 may be identified with the space of affine functions on the image Δ ⊆ 𝔥^{∗} of the momentum map of V. Using (7), we choose a ∈ 𝔥 such that δa = ζ and the affine function f_{a}(z) := ⟨a, z⟩ is positive on Δ.

Given these data, we can construct a Kähler 2m-manifold (M, g, ω), with m = ℓ + d,

M = P ×_{𝕋^{ℓ}} V ⟶^{π} B, and

\begin{aligned}g & = f_{a}(μ)π^{∗}g_{B} + ⟨dμ, G(μ), dμ⟩ + ⟨θ, H(μ), θ⟩, \\ ω & = f_{a}(μ)π^{∗}ω_{B} + ⟨dμ ∧ θ⟩, dθ = δa ⊗ ω_{B}, \end{aligned} ; (15)

where G and H are determined by the toric Kähler metric on V in momentum–angle coordinates (9). We refer to (15) as the generalized Calabi ansatz, with data (V, g_{V} , ω_{V}), (B, g_{B}, ω_{B}) and a ∈ 𝔥. For any b ∈ 𝔥, the affine function f_{b}(z) = ⟨b, z⟩ pulls back to Killing potentials for both (g_{V} , ω_{V}) and (g, ω), and their CR f_{b}-twists are related as follows.

Proposition 3. Let (M, g, ω) be given by the generalized Calabi ansatz for data (V, g_{V} , ω_{V}), (B, g_{B}, ω_{B}) and a ∈ 𝔥. Then for b ∈ 𝔥 with f_{b} > 0 on M, the generalized Calabi ansatz, with data a CR f_{b}-twist (V_{b}, g_{b}, ω_{b}) of V, (B, g_{B}, ω_{B}) and a ∈ 𝔥, is a CR f_{b}-twist of M. In particular, the Kähler product of (B, g_{B}, ω_{B}) and (V_{a}, g_{a}, ω_{a}) is a CR f_{a}-twist of M.

Proof. It is enough to prove the result, for arbitrary V and B, in the case a = ε, with (M, g, ω) being the Kähler product of (B, g_{B}, ω_{B}) and (V, g_{V} , ω_{V}). Indeed, we may then recover the Kähler metric (15), associated to a given (B, g_{B}, ω_{B}), (V, g_{V} , ω_{V}) and a, as a CR twist of the Kähler product with (B, g_{B}, ω_{B}) of a CR f_{a}-twist (V_{a}, g_{a}, ω_{a}) of (V, g_{V} , ω_{V}), by taking b = a.

The Sasaki structure (N, 𝒟, J, χ = ξ_{ε}) associated to the Kähler product of B and V is (locally) defined by the contact form

η_{χ} = ⟨μ̂, dt + θ_{B} ⊗ ε⟩ = ∑_{k=0}^{m} μ̂_{j}dt_{j} + θ_{B}

where θ_{B} is a (local) 1-form on B with dθ_{B} = ω_{B}, and the second expression uses a basis of 𝔥 (for which we may assume e_{0} = ε so that μ̂_{0} ≡ 1 and X_{χ} = ∂/∂t_{0}). The CR structure is determined from dη_{χ} = π^{∗}(ω_{B} + ω_{V}) and g_{χ} = π^{∗}(g_{B} + g_{V})|_{𝒟}.

Now let f_{b}(z) = ∑_{k=0}^{m} b_{k}z_{k} be a positive affine function defining new affine coordinates z̃_{j} = z_{j}/f_{b}(z) on P(𝔥^{∗}). The symplectic form ω̃ on any Sasaki–Reeb quotient M̃ of N by ξ_{b} pulls back to d(η_{χ}/f_{b}(μ̂)) = d⟨μ̂, dt + θ_{B} ⊗ ε⟩ and hence is given by

ω̃ = ⟨μ̃, dθ⟩ + ⟨dμ̃ ∧ θ⟩ = ⟨μ̃, ε⟩ω_{B} + ⟨dμ̃ ∧ θ⟩,

where μ̃ is the pullback to M̃ of z̃ on P(𝔥^{∗}) and θ ∈ Ω^{1}(Ñ, 𝔱̃), with 𝔱̃ = 𝔥/ span(b), pulls back to dt + θ_{B} ⊗ ε mod b on N. The complex structure on B is unaffected by the CR twist, while consideration of the action of the CR structure on dμ̃|_{𝒟} allows us to identify the toric fibres of M̃ over B with the CR f_{b}-twist of V, as in Lemma 5. As in that Lemma, we can make the momentum–angle coordinates more explicit in a basis of 𝔥 with e_{0} = ε and b_{0} ≠ 0. In any case, the result now follows. □

Corollary 3. If (B, g_{B}, ω_{B}) is an extremal Kähler manifold and (V, g_{V} , ω_{V}) is an (f_{a}, ℓ+2)- extremal Kähler manifold then (g, ω) given by (15) is (f_{a}, m + 2)-extremal. In particular, a ℂP^{ℓ}-bundle (M, J) = P (L_{0} ⊕ ⋯ ⊕ L_{ℓ}) → B over an extremal Hodge Kähler manifold (B, g_{B}, ω_{B}) has a natural 1-parameter family of (f_{a}, m + 2)-extremal Kähler metrics.

Proof. The CR f_{a}-twist of the Kähler metric (15) defined by Proposition 3 is the Kähler product of (B, g_{B}, ω_{B}) with the CR f_{a}-twist (V_{a}, g_{a}, ω_{a}) of (V, g_{V} , ω_{V}). As (V, g_{V} , ω_{V}) is (ℓ + 2, f_{a})-extremal, (V_{a}, g_{a}, ω_{a}) is extremal by Theorem 1. It follows that the CR f_{a}-twist of (15) is extremal, so by Theorem 1 again, we conclude that (15) is (f_{a}, m + 2)-extremal.

In the special case M = P (L_{0} ⊕ ⋯ ⊕ L_{ℓ}) → B, we can apply the above construction with (V, g_{V} , ω_{V}) = (ℂP^{ℓ}, g_{F S}, ω_{F S}), where (g_{F S}, ω_{F S}) is a Fubini–Study metric on ℂP^{ℓ}. By Corollary 2, g_{F S} is (f_{a}, ℓ + 2)-extremal, so the claim follows. □

4.5. The Calabi ansatz. We now specialize to the case that (V, g_{V} , ω_{V}) in the generalized Calabi ansatz is a toric 2-manifold or orbifold (13); this is the original Calabi ansatz when V = ℂP^{1}, and (15) reduces to

(16) g = (a_{0} + a_{1}μ_{1})g_{B} + \frac{dμ_{1}^{2}}{A(μ_{1})} + A(μ_{1})θ^{2},  ω = (a_{0} + a_{1}μ_{1})ω_{B} + dμ_{1} ∧ θ,

where dθ = a_{1}ω_{B}. By Proposition 3, (M, g, ω) has a CR f_{a}-twist (M̃, g̃, ω̃) given by the the Kähler product of (B, g_{B}, ω_{B}) and (Ṽ , gṼ , ωṼ), where the latter is a CR f_{a}-twist of (V, g_{V} , ω_{V}) as in Example 4. Furthermore, by Theorem 1, for any b ∈ 𝔥, g is (f_{b}, m + 2)- extremal if and only if g̃ is (f̃_{b}, m + 2)-extremal, where f_{b}(μ_{1}) and f̃_{b}(μ̃_{1}) are the Killing potentials induced by b on M and M̃ respectively.

If a and b are linearly independent then f̃_{b} is nonconstant and, up to homothety, we may assume that f̃_{b}(z̃_{1}) = z̃_{1} + b̃_{0}. Then g̃ is (f̃_{b}, m + 2)-extremal iff g_{B} has constant scalar curvature s_{B} and gṼ has profile function Ã with

(17)  Ã(z̃_{1} − b̃_{0}) = p_{0}z̃_{1}^{m+2} + p_{1}z̃_{1}^{m+1} − s_{B}z̃_{1}^{m} + p_{3}z̃_{1} + p_{4}.

If (B, g_{B}, ω_{B}) is a CSC Hodge Kähler manifold (where we may assume without loss that [ω_{B}/2π] is primitive) then this picture globalizes in a couple of ways as follows.

First, we may start from a weighted projective line Ṽ = ℂP_{w}^{1}, where w = (w_{−}, w_{+}) is a pair of positive integers. We equip ℂP_{w}^{1} with the toric symplectic structure ω_{w} induced by the quasi-regular Sasaki structure (𝕊^{3}, 𝒟, J, χ_{w}) on the 3-sphere 𝕊^{3} ⊆ ℂ^{2}. By (7), the rational Delzant polytope [30, 55] of (ℂP_{w}^{1}, ω_{w}) in the affine chart defined by χ_{w} is given by {(z_{0}, z_{1}) : z_{i} ⩾ 0, w_{−}z_{0} + w_{+}z_{1} = 1}, but we use instead the parametrization z_{0} = (1 + z̃_{1})/(2w_{+}), z_{1} = (1 − z̃_{1})/(2w_{−}) to realize this rational Delzant polytope as the interval [−1, 1] with inward normals 1/(2w_{+}) and −1/(2w_{−}). As explained in [13], for any positive integer k, the product Kähler orbifold (B, g_{B}, ω_{B}) × (ℂP_{w}^{1}, kg_{w}, kω_{w}) gives rise to a compact quasi-regular extremal Sasaki orbifold (N_{w,k}, 𝒟, J, χ_{w,k}), which is the Sasaki join of the regular Sasaki manifold (N_{B}, 𝒟_{B}, J_{B}, χ_{B}) associated to (B, g_{B}, ω_{B}) and (𝕊^{3}, 𝒟, J, \frac{1}{k}χ_{w}). There are well-understood conditions in terms of the integers (w_{+}, w_{−}, k) ensuring that N_{w,k} is a smooth manifold, see [13]. Now any Ã given by (17) with |b̃_{0}| > 1, which satisfies the well-known positivity and boundary conditions

(18)  Ã(z̃_{1}) > 0 on (−1, 1), Ã(±1) = 0 and Ã′(±1) = ∓4w_{∓}/k,

gives rise to a toric, kω_{w}-compatible Kähler metric g̃_{w} on ℂP_{w}^{1}, such that the product metric g_{B} +g̃_{w} is (f̃_{b}, m+2)-extremal. We thus get a new Sasaki structure (N_{w,k}, 𝒟, J_{w}, ξ_{b}) which is extremal by Theorem 1. Note that ξ_{b} is not quasi-regular if b̃_{0} is irrational.

For a fixed a_{0}, the endpoint conditions (18) determine the unknown coefficients p_{0}, p_{1}, p_{3} and p_{4} of a polynomial Ã satisfying (17), and it remains to examine the positivity condition for Ã. This is therefore an effective tool for generating compact examples of extremal Sasaki metrics, providing an explanatory framework for the constructions in [16, 17].

Secondly, we may begin instead with M = P(O ⊕ L) where L is a holomorphic line bundle over B such that c_{1}(L) = ℓ[ω_{B}/2π] for ℓ ∈ ℤ^{+} (and O denotes the trivial line bundle). Then (16) defines a Kähler metric on M such that the 𝕊^{1}-action induced by scalar multiplication in O is isometric and hamiltonian with momentum map μ_{1} and momentum image μ_{1}(M) = [−1, 1] ⊂ ℝ if and only if A(z_{1}) is a smooth function on [−1, 1] satisfying the boundary conditions

(19)  A(±1) = 0, A′(±1) = ∓2,

and the positivity condition

(20)  A(z_{1}) > 0 on (−1, 1),

and a_{0} +a_{1}z_{1} is positive on [−1, 1] with |a_{1}| = ℓ (and we may assume a_{1} = ℓ by replacing z_{1} with −z_{1} if necessary). Here θ is the connection form associated to a principal 𝕊^{1}-connection on the unit circle bundle in M → B and

(21)  [ω/2π] = c_{1}(O_{O⊕L}(2)) + (a_{0} + a_{1})c_{1}(π^{∗}L).

For any positive integers k, n such that n/k > ℓ, L_{k,n} := O_{O⊕L}(k)⊗π^{∗}L^{n/ℓ} is a polarization on M, c_{1}(L_{k,n}) being homothetic to a Kähler class of the form (21) with a_{0} = (2n/k) − ℓ and a_{1} = ℓ. We thus let (N_{k,n}, 𝒟, J, χ) be the smooth Sasaki manifold corresponding to the Kähler manifold (M, \frac{k}{2}g, \frac{k}{2}ω) via Example 1, where (g, ω) is given by (16) (with a_{0} = (2n/k) − ℓ and a_{1} = ℓ). Up to a covering, (N_{k,n}, 𝒟, χ) is determined by the ratio n/k, so we assume henceforth that k and n are coprime positive integers. In [17, (37)], the contact manifold (N_{k,n}, 𝒟) is identified with the Sasaki join (N_{w,k}, 𝒟) constructed over B × ℂP_{w}^{1} above, with weights w_{+} = n, w_{−} = n − kℓ. The theory of CR twists further identifies the CR structure J on (N_{k,n}, 𝒟) induced by (16) with the CR structure J_{w} on (N_{w,k}, 𝒟) induced by

g̃ = g_{B} + \frac{dμ̃_{1}^{2}}{Ã(μ̃_{1})} + Ã(μ̃_{1})dt^{2},  ω̃ = ω_{B} + dμ̃_{1} ∧ dt,
where  Ã(z̃_{1}) = \frac{(a_{0}^{2} − a_{1}^{2})^{2}A(z_{1})}{(a_{0} + a_{1}z_{1})^{3}}, z_{1} = \frac{a_{0}z̃_{1} − a_{1}}{a_{0} − a_{1}z̃_{1}} , a_{0} = \frac{2n}{k} − ℓ and a_{1} = ℓ.

5. Separable toric geometries

5.1. Regular ambitoric structures. In [5, 6], the following 4-dimensional geometric structure was studied.

Definition 5. An ambikähler structure on a real 4-manifold or orbifold M consists of a pair of Kähler metrics (g_{−}, J_{−}, ω_{−}) and (g_{+}, J_{+}, ω_{+}) such that

• g_{−} and g_{+} are conformally equivalent;

• J_{−} and J_{+} have opposite orientations. The structure is said to be ambitoric if in addition there is a 2-dimensional subspace 𝔱 of vector fields on M, linearly independent on a dense open set, whose elements are hamiltonian and Poisson-commuting Killing vector fields with respect to both (g_{−}, ω_{−}) and (g_{+}, ω_{+})—i.e., both Kähler structures are locally toric.

It was shown in [5] that any ambitoric structure is locally either a product, of Calabi type, or a regular ambitoric structure given by the following ansatz. Let q(x) = q_{0}+2q_{1}x+q_{2}x^{2} be a quadratic polynomial and let M be a 4-manifold or orbifold with real-valued functions (x_{1}, x_{2}, τ_{0}, τ_{1}, τ_{2}) such that x_{1} > x_{2}, 2q_{1}τ_{1} = q_{0}τ_{2} + q_{2}τ_{0}, and their exterior derivatives span each cotangent space. Let 𝔱 be the 2-dimensional space of vector fields K on M with dx_{1}(K) = 0 = dx_{2}(K) and dτ_{j}(K) constant, and let A(x) and B(x) be positive functions on open neighbourhoods of the images of x_{1} and x_{2} in ℝ, on whose product f_{q}(x_{1}, x_{2}) := q_{0} + q_{1}(x_{1} + x_{2}) + q_{2}x_{1}x_{2} is positive. Then M is ambitoric with

g_{±} = (\frac{x_{1} − x_{2}}{f_{q}(x_{1}, x_{2})})^{±1}(\frac{dx_{1}^{2}}{A(x_{1})} + \frac{dx_{2}^{2}}{B(x_{2})} + A(x_{1})α_{1}^{2} + B(x_{2})α_{2}^{2}),

ω_{±} = (\frac{x_{1} − x_{2}}{f_{q}(x_{1}, x_{2})})^{±1}(dx_{1} ∧ α_{1} ± dx_{2} ∧ α_{2}),  \begin{aligned}J_{±}dx_{1} & = A(x_{1})α_{1}, \\ J_{±}dx_{2} & = ±B(x_{2})α_{2}, \end{aligned}

α_{1} = \frac{dτ_{0} + 2x_{2}dτ_{1} + x_{2}^{2}dτ_{2}}{(x_{1} − x_{2})f_{q}(x_{1}, x_{2})} ,  α_{2} = \frac{dτ_{0} + 2x_{1}dτ_{1} + x_{1}^{2}dτ_{2}}{(x_{1} − x_{2})f_{q}(x_{1}, x_{2})}

(22)

There is a gauge freedom to make a simultaneous projective transformation of the coordinates x_{1}, x_{2}, with q transforming as a quadratic polynomial, and A, B as quartics [5]. If q has repeated roots, we may use this freedom to set q = 1, and then g_{+} is a 2-dimensional orthotoric metric, as studied in [3, 4]. We then refer to g_{−} as a negative orthotoric metric.

Ambitoric structures are examples of separable toric geometries, i.e., they admit separable coordinates x_{1}, … x_{m} in which the metric is determined by m functions of 1 variable (and some explicit data, such as q here). We now explore CR twists for some separable toric geometries. While we could simply apply the general approach given in Lemma 5, this is not expedient for a couple of reasons. On a practical level, we would need to compute: the transformation from separable coordinates to momenta, the symplectic potential, its CR twist in terms of the new momenta, and finally the transformation from these momenta back to separable coordinates. This is rather involved, and unnecessarily so, because whereas a CR twist involves a change of momentum coordinates due to the change of affine chart, the separable coordinates remain fixed. We illustrate this first in the simplest separable situation: Kähler products of toric Riemann surfaces.

5.2. The CR twisted toric product ansatz. A Kähler product of toric Riemann surfaces has a Kähler metric of the form:

(23)  g = ∑_{i=1}^{m}(\frac{dx_{i}^{2}}{A_{i}(x_{i})} + A_{i}(x_{i})dt_{i}^{2}), ω = ∑_{i=1}^{m} dx_{i} ∧ dt_{i}, Jdx_{i} = A_{i}(x_{i})dt_{i},

where A_{1}, … A_{m} are arbitrary functions of 1 variable. In this case the separable coordinates and momenta coincide: the toric Killing potentials have the form

f_{b}(x_{1}, … x_{m}) = b_{0} + b_{1}x_{1} + ⋯ + b_{m}x_{m}.

It is straightforward to compute the CR structure associated to (g, ω, J) as in Example 1. Denoting by t_{i}, x_{i} also their pullbacks to N, we have 𝒟 = ker η and J: 𝒟^{∗} → 𝒟^{∗} given by

η = dt_{0} + ∑_{i=1}^{m} x_{i}dt_{i}, J(dx_{i}|_{𝒟}) = A_{i}(x_{i})dt_{i}|_{𝒟}.

We now lift f_{b}(x_{1}, … x_{m}) to a new Sasaki structure ξ_{b} on N and compute the new Sasaki– Reeb quotient. The new contact form is η_{b} := η_{ξ_{b}} = η/f_{b}, with

dη_{b} = ∑_{i=1}^{m} dx_{i} ∧ \frac{∂η_{b}}{∂x_{i}} = \frac{1}{f_{b}(x_{1}, … x_{m})} ∑_{i=1}^{m} dx_{i} ∧ (dt_{i} − b_{i}η_{b}).

Since J(dx_{i}|_{𝒟}) = A_{i}(x_{i})(dt_{i} − b_{i}η_{b})|_{𝒟}, the Sasaki–Reeb quotient is given by the following toric ansatz, originally proposed in [7], which we refer to here as a twisted toric product:

(24) \begin{matrix}g_{b} = \frac{1}{f_{b}(x_{1}, … x_{m})} ∑_{i=1}^{m}(\frac{dx_{i}^{2}}{A_{i}(x_{i})} + A_{i}(x_{i})α_{i}^{2}),  ω_{b} = \frac{1}{f_{b}(x_{1}, … x_{m})} ∑_{i=1}^{m} dx_{i} ∧ α_{i}, \\ J_{b}dx_{i} = α_{i},  dα_{i} = −b_{i}ω,  f_{b}(x_{1}, … x_{m}) = b_{0} + b_{1}x_{1} + ⋯ + b_{m}x_{m}. \end{matrix}

For b_{0} ≠ 0, we may obtain more explicit angle coordinates by setting τ_{i} = t_{i} − b_{i}t_{0}/b_{0} and α_{i} = dτ_{i} − (b_{i}/f_{b}) ∑_{j=1}^{m} x_{j}dτ_{j}. We may take as momenta

μ_{0} = \frac{1}{f_{b}(x_{1}, … x_{m})},  μ_{j} = \frac{x_{j}}{f_{b}(x_{1}, … x_{m})},  j ∈ \{1, … m\}.

Hence the momentum coordinates and separable coordinates no longer agree. The original product metric (23) is a CR twist of (24) by μ_{0} = f_{b}^{−1} = 1/f_{b}. It was shown in [7] that when m = 2, this construction unifies the ambitoric product, Calabi and negative orthotoric ansatz of [5] in a single family.

It was also shown in [7] that a twisted toric product metric (24) is (f_{b}^{−1}, m + 2)-extremal if and only if A_{j} is a cubic polynomial for all j ∈ {1, … m}. We see here that this follows straightforwardly from Theorem 1, as (24) is (f_{b}^{−1}, m + 2)-extremal if and only if (23) is extremal, and a toric Kähler product is extremal if and only if the factors are, meaning that each A_{i} is a cubic. In this case, we may also identify the CR manifold which has these metrics as its Sasaki–Reeb quotients. Indeed, straightforward computation shows that the Cartan tensor of a Sasaki 3-manifold vanishes precisely when the (transversal, i.e., Tanaka–Webster) scalar curvature is transversally holomorphic (see e.g. [41]). It then follows from [24] that the (f_{b}^{−1}, m + 2)-extremal metrics given by (24) are obtained as Sasaki–Reeb quotients with respect to the CR structure of a (local) Sasaki join [13] of m copies of the standard CR structure (𝒟_{0}, J_{0}) on the 3-sphere 𝕊^{3} ⊆ ℂ^{2}, with respect to a (local) Sasaki structure on each factor.

We next consider the extremality condition for the Kähler metrics (24), using again Theorem 1 to infer that (24) is extremal if and only if the product metric (23) is (f_{b}, m+2)- extremal. We thus have

(25) ∑_{j=1}^{m}(−f_{b}^{2}A′′_{j}(x_{j}) + 2(m + 1)f_{b} \frac{∂f_{b}}{∂x_{j}} A′_{j}(x_{j}) − (m + 1)(m + 2)(\frac{∂f_{b}}{∂x_{j}} )^{2}A_{j}(x_{j}))

= − ∑_{j=1}^{m} f_{b}^{m+3} \frac{∂^{2}}{∂x_{j}^{2}} (\frac{A_{j}(x_{j})}{f_{b}^{m+1}}) = Scal_{f_{b},m+2}(g) = c_{0} + c_{1}x_{1} + ⋯ + c_{m}x_{m},

where c_{0}, c_{1}, … c_{m} are some real constants. For m = 1 we get that A_{1} must be a polynomial of degree ⩽ 3 and for m = 2 (24) is given by the ambitoric product, Calabi or negative orthotoric ansatz of [5], and the extremality condition (25) can be solved [5] in terms of two polynomials A_{1} and A_{2} of degree ⩽ 4. We thus assume from now on that m ⩾ 3.

Proposition 4. For m ⩾ 3 the Kähler metric (24) is extremal if and only if it is a product of extremal Riemann surfaces, or is given by the Calabi ansatz over a product of m−1 CSC Riemann surfaces, or is the Kähler product of a scalar-flat product of Riemann surfaces with a product of flat Riemann surfaces, as in Proposition 2.

Proof. Differentiating (25) (m + 1) times with respect to x_{j} yields

f_{b}^{2}A_{j}^{(m+3)}(x_{j}) = 0,

showing that each A_{j} must be a polynomial of degree ⩽ m + 2. Thus, both sides of (25) are polynomials in x_{i}, so we may compare coefficients. Taking two derivatives in x_{j} gives

0 = f_{b}^{m} \frac{∂^{2}}{∂x_{j}^{2}} (\frac{A_{j}^{(2)}(x_{j})}{f_{b}^{m−1}}) = f_{b}^{2}A_{j}^{(4)}(x_{j}) − 2(m − 1)b_{j}f_{b}A_{j}^{(3)}(x_{j}) + m(m − 1)b_{j}^{2}A_{j}^{(2)}(x_{j}).

If b_{i} ≠ 0 for some i ≠ j, the vanishing of the polynomial coefficients containing x_{i}^{2} in the above relation show that A_{j} has degree ⩽ 3; if furthermore b_{j} ≠ 0, then the coefficients containing x_{i} show that A_{j} has degree ⩽ 2. Substituting back in (25) and comparing coefficients, this yields the following three possibilities for the solutions to (25) with m ⩾ 3:

• f_{b}(x_{1}, … x_{m}) = b_{0}. Then the A_{j} are polynomials of degree ⩽ 3 and the corresponding extremal metric (24) is a product of extremal Riemann surfaces;

• f_{b}(x_{1}, … x_{m}) = b_{0} + b_{j}x_{j} with b_{j} ≠ 0. Then (24) is given by the Calabi ansatz over the product of (m − 1) Riemann surfaces indexed by i : i ≠ j. In particular, for each i : i ≠ j, A_{i} is a polynomial of degree ⩽ 2 whereas A_{j} is a polynomial of degree ⩽ m + 2, as described by (17).

• there are 0 ≠ j_{1} ≠ j_{j} ≠ 0 with b_{j_{1}} ≠ 0 ≠ b_{j_{2}}. Then for each j with b_{j} ≠ 0, A_{j} is a polynomial of degree ⩽ 1, and for each i with b_{i} = 0, A_{i} is a polynomial of degree ⩽ 2 with ∑_{i:b_{i}=0} A′′_{i} = 0. Thus, in this case, g is the product metric of a scalar-flat product of (CSC) Riemann surfaces (indexed by {i : b_{i} = 0}) with a product of flat Riemann surfaces (indexed by {j : b_{j} ≠ 0}). □

5.3. CR twists of positive regular ambitoric structures. We return now to regular ambitoric structures (22), for which it was shown in [5] that g_{+} is extremal if and only if g_{−} is extremal if and only if

(26)  A = pq + P,  B = pq − P,

where p is a quadratic polynomial orthogonal to q, and P is polynomial of degree ⩽ 4. Note that the orthogonality condition ⟨p, q⟩ := p_{0}q_{2} − p_{1}q_{1} + p_{2}q_{0} = 0 means that the roots of p and the roots of q harmonically separate each other.

When q has distinct roots, the positive and negative structures are equivalent, cf. [6, Remark 5], while in the case of repeated roots the negative orthotoric structures are twisted toric products [7], as noted above. Hence we only need to consider the positive ambitoric metrics. The CR structure associated to (g_{+}, J_{+}, ω_{+}) in (22) was computed in [6, App. C], which implies that 𝒟 = ker η and J: 𝒟^{∗} → 𝒟^{∗} are given by

(27) η = \frac{dt_{0} + (x_{1} + x_{2})dt_{1} + x_{1}x_{2} dt_{2}}{x_{1} − x_{2}} ,  J(dx_{1}|_{𝒟}) = A(x_{1})\frac{dt_{0} + 2x_{2}dt_{1} + x_{2}^{2}dt_{2}}{(x_{1} − x_{2})^{2}} |_{𝒟}, J(dx_{2}|_{𝒟}) = B(x_{2})\frac{dt_{0} + 2x_{1}dt_{1} + x_{1}^{2}dt_{2}}{(x_{1} − x_{2})^{2}} |_{𝒟},

independently of q, while the toric Killing potentials of (g_{+}, J_{+}, ω_{+}) have the form f_{w}/f_{q}, where w is quadratic polynomial, and lift to Sasaki structures ξ_{w} := f_{w}(x_{1}, x_{2})χ/(x_{1} − x_{2}), where χ ∈ 𝔠𝔬𝔫_{+}(N, 𝒟) with η = χ^{−1}η_{𝒟}.

The CR structures arising from an extremal positive ambitoric metric thus have A and B of degree ⩽ 4, such that the roots of A+B have harmonic cross-ratio (i.e., in {−1, 1/2, 2}). We may then write

(28) A = p_{1}p_{2} + P, B = p_{1}p_{2} − P, with deg P ⩽ 4, deg p_{j} ⩽ 2, ⟨p_{1}, p_{2}⟩ = 0.

Here we have renamed the quadratics compared to (26) so that we are free to use q to define an arbitrary Sasaki–Reeb quotient of (27). Indeed for any quadratic q, the Sasaki structure ξ_{q} is (f_{p_{j}}χ, 4)-extremal for j ∈ {1, 2} by Theorem 1 and is extremal if q = p_{1} or q = p_{2}. We obtain in particular a result of [10], as any Sasaki–Reeb quotient by ξ_{q}, given explicitly by (22), subject to (28), is (f_{p_{j}}/f_{q}, 4)-extremal for j ∈ {1, 2}, i.e., the scalar curvature of g̃_{j} = (f_{q}/f_{p_{j}})^{2}g_{+} is a Killing potential of g_{+}; in fact one can compute [5, 10]

(29)  Scal(g̃_{j}) = −\frac{f_{w}}{f_{q}} with w := \{p_{j}, (p_{j}, P)^{(2)}\},

where the Poisson bracket is given by {p, r} := p′ r − p r′ and

(p, P)^{(2)} := p P′′ − 3p′ P + 6p′′ P

is a transvectant of p and P. Special choices of q give special metrics in this family of Sasaki–Reeb quotients [5, 10].

• If q = p_{j}, then g̃_{j} = g_{+}, recovering the case that g_{+} is extremal.

• If ⟨q, p_{j}⟩ = 0, then g_{−} is also (f_{p_{j}}/f_{q}, 4)-extremal, and g̃_{j} has diagonal Ricci tensor; if in addition ⟨(p_{j}, P)^{(2)}, q⟩ = 0 then {p_{j}, (p_{j}, P)^{(2)}} is a multiple of q; hence g̃_{j} is CSC, so g_{+} is conformally Einstein–Maxwell (in fact to a riemannian Plebański–Demiański metric [5, 29, 59]).

• Combining these observations, if say q = p_{1}, then ⟨q, p_{2}⟩ = 0 so g_{+} = g̃_{1} is extremal, while g̃_{2} has diagonal Ricci tensor; if in addition ⟨(p_{2}, P)^{(2)}, q⟩ = 0 then g̃_{2} is Einstein.

• Finally, taking q = 1, we obtain an orthotoric metric in the family.

Corollary 4. Any regular positive ambitoric Kähler metric (g_{+}, ω_{+}, J_{+}) given by (22) can be obtained as a f_{q}-twist of an orthotoric metric.

5.4. The CR twisted orthotoric ansatz. Corollary 4 immediately suggests a higher dimensional extension of the positive regular ambitoric ansatz (22). For this, we start with an orthotoric 2m-manifold M with Kähler structure [4]:

g = ∑_{j=1}^{m}(\frac{Δ_{j}}{A_{j}(x_{j})} dx_{j}^{2} + \frac{A_{j}(x_{j})}{Δ_{j}}(∑_{r=1}^{m} σ_{r−1}(x̂_{j}) dt_{r})^{2}),

ω = ∑_{j=1}^{m} dx_{j} ∧ (∑_{r=1}^{m} σ_{r−1}(x̂_{j})dt_{r}) = ∑_{r=1}^{m} dμ_{r} ∧ dt_{r},

Jdx_{j} = \frac{A_{j}(x_{j})}{Δ_{j}} ∑_{r=1}^{m} σ_{r−1}(x̂_{j}) dt_{r},  Jdt_{r} = (−1)^{r} ∑_{j=1}^{m} \frac{x_{j}^{m−r}}{A_{j}(x_{j})} dx_{j},

(30)

where each A_{j} is a smooth function of 1 variable, μ_{r} = σ_{r}(x_{1}, … x_{m}) are the momentum coordinates (σ_{r} being the r-th elementary symmetric function with σ_{0} = 1) x̂_{j} = (x_{k} : k ≠ j), and Δ_{j} = ∏_{k≠j}(x_{j} − x_{k}). The separable coordinates x := (x_{1}, … x_{m}) are called orthotoric and have a natural gauge freedom under simultaneous affine changes x̃_{j} = ax_{j}+b. The toric Killing potentials in orthotoric coordinates are

(31)  f_{q}(x) = q_{0} + q_{1}μ_{1} + ⋯ + q_{m}μ_{m} with μ_{r} = σ_{r}(x),

which can be viewed as the polarized form of a degree ⩽ m polynomial

(32)  q(x) := f_{q}(x, … x) = ∑_{j=0}^{m} \binom{j}{m}q_{j}x^{j}.

As usual, to write down the CR structure (N, 𝒟, J) over M, it is convenient to view dt = (dt_{0}, dt_{1}, … dt_{m}) as a 1-form with values in the Lie algebra 𝔥 ≅ ℝ^{m+1} of Killing potentials (with basis σ_{0}, σ_{1}, … σ_{m}). Then 𝒟 = ker η and J: 𝒟^{∗} → 𝒟^{∗} are given by

(33)  η = dt(x) := ∑_{r=0}^{m} μ_{r}dt_{r},  J(dx_{j}|_{𝒟}) = \frac{A_{j}(x_{j})}{Δ_{j}} \frac{∂(dt(x))}{∂x_{j}} |_{𝒟}

with μ_{0} = 1 (omitting pullbacks to N).

Proposition 5. Let (g, ω, J) be the orthotoric Kähler metric (30) and let f_{q} be a positive function of the form (31). Then a CR f_{q}-twist of (g, ω, J) has toric Kähler metric

g_{q} = ∑_{j=1}^{m}(\frac{Δ_{j}}{A_{j}(x_{j})f_{q}(x)} dx_{j}^{2} + \frac{A_{j}(x_{j})f_{q}(x)}{Δ_{j}}(\frac{∂}{∂x_{j}} \frac{dt(x)}{f_{q}(x)})^{2}),

ω_{q} = ∑_{j=1}^{m} dx_{j} ∧ (\frac{∂}{∂x_{j}} \frac{dt(x)}{f_{q}(x)}),  J_{q}dx_{j} = \frac{A_{j}(x_{j})f_{q}(x)}{Δ_{j}}(\frac{∂}{∂x_{j}} \frac{dt(x)}{f_{q}(x)}).

(34)

Proof. The CR f_{q}-twist is the Sasaki–Reeb quotient of (33) by the Sasaki structure ξ_{q} = f_{q}χ with contact form η/f_{q}(x) (i.e., χ is the Sasaki structure with contact form η = χ^{−1}η_{𝒟}), and

d(\frac{η}{f_{q}(x)}) = ∑_{j=1}^{m} dx_{j} ∧ \frac{∂}{∂x_{j}} \frac{dt(x)}{f_{q}(x)}.

Now we observe that

J(dx_{j}|_{𝒟}) = \frac{A_{j}(x_{j})}{Δ_{j}} \frac{∂(dt(x))}{∂x_{j}} |_{𝒟} = \frac{A_{j}(x_{j})}{Δ_{j}}f_{q}(x)\frac{∂}{∂x_{j}} \frac{dt(x)}{f_{q}(x)}|_{𝒟}

and the 1-forms dx_{j} and \frac{∂}{∂x_{j}} \frac{dt(x)}{f_{q}(x)} are basic with respect to ξ_{q}—in the latter case because

(\frac{dt(x)}{f_{q}(x)})(X_{ξ_{q}}) = 1. Hence the transversal Kähler structure of ξ_{q} is the pullback of (34). □

We now turn to the extremality condition of the Kähler metrics given by (34). By Theorem 1, such a metric is extremal iff the orthotoric metric (30) is (f_{q}, m + 2)-extremal, a condition studied in [11, App. A]. Using standard formulae for the scalar curvature and laplacian of an orthotoric metric [4], this condition is

(35) ∑_{j=1}^{m}(−f_{q}(x)^{2} \frac{A′′_{j}(x_{j})}{Δ_{j}} + 2(m + 1)f_{q}(x)\frac{∂f_{q}}{∂x_{j}} \frac{A′_{j}(x_{j})}{Δ_{j}} − (m + 1)(m + 2)(\frac{∂f_{q}}{∂x_{j}} )^{2} \frac{A_{j}(x_{j})}{Δ_{j}})

= − ∑_{j=1}^{m} \frac{f_{q}(x)^{m+3}}{Δ_{j}} \frac{∂^{2}}{∂x_{j}^{2}} (\frac{A_{j}(x_{j})}{f_{q}(x)^{m+1}} ) = Scal_{f_{q},m+2}(g) = ∑_{k=0}^{m} c_{k}μ_{k}.

for some constants c_{0}, … c_{m}. If we let A_{j}(x) = P (x) for a j-independent polynomial P of degree ⩽ m + 2, the metric (30) is Bochner-flat by [4], and we obtain (for any q) a solution of (35) by Proposition 1. When q = 1, (35) describes the extremality condition for (30), which is studied in [4], where the solutions are given as A_{j}(x) = P (x) + p_{j1}x + p_{j0} for a j-independent polynomial P of degree ⩽ m + 2 and arbitrary real constants p_{j1}, p_{j0} (j ∈ {1, … m}). Another special case is q(x) = x^{m}, i.e., f_{q} = σ_{m}, which is studied in [11, Prop. A2], where the solutions are given as A_{j}(x) = P (x) + p_{j1}x^{m+1} + p_{j0}x^{m+2} for a jindependent polynomial P of degree ⩽ m+2 and arbitrary real constants p_{j1}, p_{j0}. However, we notice that in this case the corresponding extremal Kähler metric g_{q} given by (34) is orthotoric with respect to the variables x̃_{j} = 1/x_{j} and functions Ã_{j}(x̃_{j}) = x̃_{j}^{m+2}A_{j}(1/x̃_{j}), so the corresponding extremal Kähler metrics are not new. More generally, we may extend arguments from [4, Lemma 6] and [11, Prop. A2] as follows.

Proposition 6. Let m ⩾ 3. Then the orthotoric Kähler metric (30) is (f_{q}, m + 2)- extremal for some positive f_{q} in the form (31) if and only if either all A_{j}(x) are equal to a j-independent polynomial of degree ⩽ m + 2 or else the polynomial q has a root of multiplicity m (possibly at infinity) so that, up to a simultaneous affine transformation of the x_{j} in (30), we may assume that either q(x) = 1 or q(x) = x^{m}. Then, A_{j}(x) are the solutions described in [4, Prop. 17] and [11, Prop. A2], respectively. In particular, each extremal Kähler metric of the form (34) is either Bochner-flat or orthotoric with respect to suitable variables.

Proof. Multiplying (35) by Δ := ∏_{j<k}(x_{j} − x_{k}), we get the relation

f_{q}(x)^{m+3} ∑_{j=1}^{m} ±Δ(x̂_{j})\frac{∂^{2}}{∂x_{j}^{2}} (\frac{A_{j}(x_{j})}{f_{q}(x)^{m+1}} ) = Δ (∑_{k=0}^{m} c_{k}σ_{k}(x)),

where Δ(x̂_{j}) = ∏_{i<k≠j}(x_{i} −x_{k}) and the signs ± are left unspecified. The right hand side in the above equality is a polynomial of degree ⩽ m in each variable x_{j}, Δ(x̂_{j}) is a polynomial of degree (m − 2) in any x_{i}, i ≠ j and of degree 0 in x_{j}, whereas f_{q}(x) is a polynomial of degree ⩽ 1 in each x_{j}. It follows that for m ⩾ 2,

0 = \frac{∂^{m+1}}{∂x_{j}^{m+1}} (f_{q}(x)^{m+3} \frac{∂^{2}}{∂x_{j}^{2}} (\frac{A_{j}(x_{j})}{f_{q}(x)^{m+1}} )) = f_{q}(x)^{2}A_{j}^{(m+3)}(x_{j}),

showing that each A_{j} must be a polynomial of degree ⩽ m + 2.

Now let k ≠ j be fixed indices. Multiplying (35) by x_{j} − x_{k} and letting x_{j} = x = x_{k} leads to the vanishing of

(36) (f_{0} +xf_{1})^{2}P′′_{jk}(x)+2(f_{0} +xf_{1})(f_{1} +xf_{2})x^{m+2}(\frac{P′_{jk}(x)}{x^{m+1}})′ +(f_{1} +xf_{2})^{2}x^{m+3}(\frac{P_{jk}(x)}{x^{m+1}})′′,

where P_{jk}(x) = A_{j}(x) − A_{k}(x). Here each f_{k} = ∑_{r=0}^{m} q_{r+k}σ̂_{r}, with σ̂_{r} denoting the rth elementary symmetric function of the variables x_{i} : i ≠ j, k (and letting σ̂_{r} = 0 for r > m−2), is a polynomial of degree ⩽ 1 in each x_{i} : i ≠ j, k. Equivalently, f_{k} : k ∈ {0, 1, 2} can be viewed as affine functions in the variables σ̂_{1}, … σ̂_{m−2} and thus (36) can be viewed as a polynomial of degree ⩽ 2 in σ̂_{1}, … σ̂_{m−2}. By making an simultaneous affine change of the variables x_{j} in (30) if necessary (which preserves the orthotoric structure of the metric, see [4]), we can assume without loss that q_{0} ≠ 0, i.e., f_{0} ≠ 0. We thus consider the following three cases. Case 1. f_{0}, f_{1}, f_{2} are linearly independent affine functions of σ̂_{1}, … σ̂_{m−2}. Then, using f_{0}, f_{1}, f_{2} as independent variables, and considering the coefficients of f_{2}^{2}, f_{0}f_{1} and f_{0}^{2} in (36) yields that P_{jk}(x) must belong to the common kernel of the ODEs P′′(x) = 0, (P′(x)/x^{m+1})′ = 0 and (P (x)/x^{m+1})′′ = 0. The latter is trivial, thus showing that P_{jk}(x) ≡ 0 in this case, i.e., A_{j}(x) = P (x) must be a j-independent function. Case 2. f_{0}, f_{1}, f_{2} span a 2-dimensional subspace of affine functions of σ̂_{1}, … σ̂_{m−2}. In this case, (36) is a polynomial of degree 2 in two independent variables in the span of f_{0}, f_{1}, f_{2}, which places three relations involving P′′(x), (P′(x)/x^{m+1})′ and (P (x)/x^{m+1})′′. Using their functional independence, we conclude again that P′′(x) = 0, (P′(x)/x^{m+1})′ = 0 and (P (x)/x^{m+1})′′ = 0, i.e., P_{jk}(x) = A_{j}(x) − A_{k}(x) = 0 so that A_{j}(x) = P (x) is a j-independent function. Case 3. f_{0}, f_{1}, f_{2} span a 1-dimensional subspace of affine functions in σ̂_{1}, … σ̂_{m−2}. Thus, in this case, f_{1} = λ_{1}f_{0} and f_{2} = λ_{2}f_{0} for some λ_{1}, λ_{2} ∈ ℝ. The first identity means q_{r+1} = λ_{1}q_{r} for r ∈ {0, … m − 1} whereas the second identity is equivalent to q_{r+2} = λ_{2}q_{r} for r ∈ {0, … m − 2}. As we have assumed q_{0} ≠ 0, we conclude that λ_{2} = λ_{1}^{2} and then q_{r} = λ_{1}^{r}q_{0}, i.e., q(x) = q_{0}(1 + λ_{1}x)^{m}. It thus follows that either q(x) = 1 (i.e., λ_{1} = 0) and then (35) describes the extremal Kähler condition of an orthotoric metric, which has been analysed in [4, Prop. 15]. Otherwise, by making a simultaneous affine change of x_{j} in (30), we can assume q(x) = x^{m} and (35) then reduces to finding (σ_{m}, m + 2)-extremal metrics, which has been accomplished in [11, Prop. A2].

To summarize, we have proven that one of the following holds:

• q = 1 and A_{j}(x) = P (x)+p_{j1}x+p_{j0}, for a j-independent polynomial P of degree ⩽ m+2; • up to a simultaneous affine transformation of the x_{j} in (30), q(x) = x^{m} and A_{j}(x) =

P (x) + p_{j1}x^{m+1} + p_{j0}x^{m+2} for a (j-independent) polynomial P of degree ⩽ m + 2;

• A_{j}(x) = P (x) are all equal to a polynomial P of degree ⩽ m + 2. In the third case, the orthotoric Kähler metric (30) is Bochner-flat (see e.g. [4, Prop. 17] or [19]) and any q provides a solution to (35) (see Proposition 1). By result of Webster [63], any CR q-twist of g is again a Bochner-flat Kähler metric, which completes the proof. □

Remark 3. Similar arguments yield a classification of (f_{q}, ν)-extremal orthotoric metrics, where ν ∈ ℝ \ {1, … m + 2} and m ⩾ 3. Indeed, as shown in [11], in this case we have to consider the equation

(37)  − ∑_{j=1}^{m} \frac{f_{q}(x)^{ν+1}}{Δ_{j}} \frac{∂^{2}}{∂x_{j}^{2}} (\frac{A_{j}(x_{j})}{f_{q}(x)^{ν−1}} ) = Scal_{f_{q},ν}(g) = ∑_{k=0}^{m} c_{k}μ_{k}.

Multiplying by x_{j} − x_{k} and letting x_{j} = x = x_{k} leads again to the conclusion that one of the following three cases occurs: (1) q = 1 and A_{j}(x) = P (x) + p_{j1}x + p_{j0} for a polynomial P of degree ⩽ m by the classification in [4, Prop. 15], or, (2) up to a simultaneous affine change of the variables x_{j} in (30), q = x^{m} and A_{j}(x) = P (x) + p_{j1}x^{ν−1} + p_{j0}x^{ν} according to [11, Prop. A2], or (3) A_{j}(x) = P (x) are j-independent. In the third case, multiplying (37) with Δ leads to the equation

0 = \frac{∂^{m+1}}{∂x_{j}^{m+1}} (f_{q}(x)^{ν+1} \frac{∂^{2}}{∂x_{j}^{2}} (\frac{P (x_{j})}{f_{q}(x)^{ν−1}} )) = f_{q}(x)^{ν−m} \frac{∂^{2}}{∂x_{j}^{2}} (\frac{P^{(m+1)}(x_{j})}{f_{q}(x)^{ν−m−2}} ).

Letting x_{j} = x, f_{k} = ∑_{r=0}^{m} q_{r}σ̂_{r−k} where σ̂_{r} denotes the r-th elementary symmetric function of x_{i}, i ≠ j with σ̂_{r} = 0 for r ⩾ m, the above conditions reduce to the vanishing of

(f_{0}+xf_{1})^{2}P^{(m+3)}(x)−2(ν−m−2)f_{1}(f_{0}+xf_{1})P^{(m+2)}(x)+(ν−m−2)(ν−m−1)f_{1}^{2}P^{(m+1)}(x).

If f_{0} and f_{1} are linearly independent affine functions of σ̂_{1}, … σ̂_{m−1} (and as ν ≠ m+1, m+2 by assumption), this implies P^{(m+1)}(x) = 0, i.e., P must be a polynomial of degree ⩽ m and the metric (30) is flat (see [4, Prop. 17]). These are precisely the solutions described in [11, Prop. A1]. Otherwise, either f_{0} = 0 (i.e., q = x^{m}) or f_{1} = λf_{0} (i.e., q(x) = q_{0}(1+λx)^{m}), so we are again in a situation covered by [11, Prop. A2] and [4, Prop. 15].

6. The Calabi problem and non-existence results

6.1. The Calabi problem for (f, ν)-extremal Kähler metrics. Let (M, J) be a compact connected complex manifold of real dimension 2m, and Ω ∈ H^{2}(M, ℝ) a Kähler class. As observed in [10, 36, 45], many features of the theory of extremal Kähler metrics extend naturally to the (f, ν)-extremal case. In particular, one can formulate a weighted version of the Calabi problem [20] which seeks an (f, ν)-extremal Kähler metric (g, ω) with ω ∈ Ω. We pin down the function f indirectly by fixing first a quasi-periodic holomorphic vector field with zeroes K generating a torus 𝕋 ⩽ Aut^{r}(M, J) inside the reduced group of automorphisms of (M, J) (see e.g. [38]), and secondly, a real constant κ > 0 such that for any 𝕋-invariant Kähler metric (g, ω) with ω ∈ Ω, the Killing potential f of K with respect to g, normalized by ∫_{M} fω^{m}/m! = κ, is positive on M, see [10, Lemma 1].

Problem 1. Is there a 𝕋-invariant Kähler metric (g̃, ω̃) on (M, J) with ω̃ ∈ Ω, which is (f̃, ν)-extremal where f̃ is the Kähler potential of K with respect to g̃ determined by ∫_{M} f̃ω̃^{m}/m! = κ? We refer to such metrics as (K, κ, m + 2)-extremal.

Remark 4. It is easy to check that if (g, ω) and (g̃, ω̃) are two 𝕋-invariant Kähler metrics in Ω with Kähler forms ω̃ = ω + dd^{c}φ for a 𝕋-invariant smooth function φ on M, then the corresponding κ-normalized Killing potentials f̃ and f of K are related by

(38)  f̃ = f + d^{c}φ(K).

Indeed, the proof of [10, Lemma 1] shows that f̃ = f ∘ Φ for some diffeomorphism Φ of M. It thus follows that if f is the κ-normalized Killing potential of K with respect to g, then f̃ is the unique Killing potential of K with respect to g̃ such that f̃(M) = f(M). The latter property holds for f̃ defined by (38), as can be seen by considering points of minima and maxima for f and f̃ (at which K = J grad_{g} f = J gradg̃ f̃ vanishes).

6.2. The Calabi problem for (ξ, ν)-extremal Sasaki metrics. Let N be a compact connected (2m + 1)-manifold. Following [13, 14], one can extend the Calabi problem to an analogous problem in Sasaki geometry by fixing a nowhere zero vector field X as the candidate for the Sasaki–Reeb vector field of the extremal Sasaki structure on N, together with a complex structure J_{X} on the quotient bundle 𝒟_{X} of T N by the span of X. If (𝒟, J, χ) is a Sasaki structure with X_{χ} = X then 𝒟 is transverse to X and so the projection onto 𝒟_{X} is a bundle isomorphism, and we can require in addition that this isomorphism intertwines J and J_{X}. The corresponding Sasaki structures on N are completely determined by their contact distributions, or equivalently, their contact forms η, with η(X) = 1 and dη(X, ·) = 0.

Definition 6. [14] The subspace S(X, J_{X}) of Ω^{1}(N) whose elements η are contact forms of Sasaki structures compatible with (X, J_{X}) is called a Sasaki polarization of (N, 𝒟, J, X). We also fix a torus 𝕋 in the automorphism group Aut(N, X, J_{X}) and let S(X, J_{X})^{𝕋} denote the 𝕋-invariant elements of S(X, J_{X}).

One can now imitate constructions in Kähler geometry by using the basic de Rham complex Ω_{X}^{•}(N) = {α ∈ Ω^{•}(N) : ı_{X}α = 0 = L_{X}α}, with differential d_{X} given by restriction of d (which evidently preserves basic forms). Since {α ∈ ∧^{k}T^{∗}N : ı_{X}α = 0} is naturally isomorphic to ∧^{k}𝒟_{X}^{∗}, J_{X}^{∗} is also well defined on Ω_{X}^{•}(N), yielding a twisted differential d_{X}^{c}. The same holds for the subspace Ω_{X}^{•}(N)^{𝕋} of 𝕋-invariant basic forms. Following [14, Lemma 3.1] and [13, Prop. 7.5.7], S(X, J_{X})^{𝕋} is an open subset of an affine space modelled on C_{N,0}^{∞}(ℝ)^{𝕋} × Ω_{X,cl}^{1}(N)^{𝕋}, where C_{N,0}^{∞}(ℝ)^{𝕋} denotes the quotient by constants of the space of smooth 𝕋-invariant functions on N, and Ω_{X,cl}^{1}(N)^{𝕋} denotes the basic 𝕋-invariant closed 1-forms on N. Indeed, for any two elements η, η̃ ∈ S(X, J_{X})^{𝕋}, η̃ − η is basic and so

(39)  η̃ = η + d_{X}^{c}φ + α

for a 𝕋-invariant smooth function φ, and a basic 𝕋-invariant closed 1-form α. It follows from (39) that the induced Kähler forms on local quotients (M, J) of N by X are linked by ω̃ = ω + dd^{c}φ, i.e., belong to the same Kähler class. Moreover any K in the Lie algebra of 𝕋 is CR for both CR structures (𝒟, J) and (𝒟̃, J̃) induced by η and η̃, and hence induces on any such M a Killing vector field, also denoted K, for both ω and ω̃, with respective Killing potentials pulling back to f = η(K) and f̃ = η̃(K) = f + d^{c}φ(K) + α(K). Notice that by the 𝕋-invariance and closedness of α, the term α(K) is a constant.

Lemma 6. Let (N, 𝒟, J) be a compact CR manifold and χ, ξ ∈ 𝔠𝔯_{+}(𝒟, J) with [ξ, χ] = 0. Let X = X_{χ}, K = X_{ξ}. Then for any η̃ ∈ S(X, J_{X}) with L_{K} η̃ = 0, K is a Sasaki–Reeb vector field for the induced CR structure (𝒟̃, J̃).

Proof. As K is a CR vector field by construction, we need to check that f̃:= η̃(K) > 0. We let η ∈ S(X, J_{X}) be the contact form of (𝒟, χ). Since K is contact with respect to 𝒟̃ := ker η̃, we have K = f̃X − (dη̃|_{𝒟̃})^{−1}(df̃|_{𝒟̃}) and hence

η(K) = f̃− (dη̃|_{𝒟̃})^{−1}(df̃|_{𝒟̃}, η|_{𝒟̃})

Evaluating this relation at a global minimum p of f̃ we obtain f̃(p) = η(K)(p) > 0. □

We now specialize the above set-up. First, we fix a nowhere zero vector K and let 𝕋 be the torus in Aut(N, X, J_{X}) generated by X and K. In addition, following [37], we fix a basepoint η ∈ S(X, J_{X})^{𝕋} with corresponding Sasaki structure (𝒟, J, χ), and restrict attention to the affine slice of S(X, J_{X})^{𝕋} consisting of contact forms η̃ related to η by (39) with α = 0. We may thus identify this slice with

(40)  Ξ(X, J_{X})^{𝕋} := {φ ∈ C_{N,0}^{∞}(ℝ)^{𝕋} | η_{φ} := η + d_{X}^{c}φ is a contact form}

We also write (𝒟_{φ}, J_{φ}, χ_{φ}) for the Sasaki structure induced by η_{φ} for φ ∈ Ξ_{K}(X, J_{X})^{𝕋}, and let ξ_{φ} = η_{𝒟_{φ}}(K) and ξ = η_{𝒟}(K). In view of Lemmas 2 and 6, we now have an analogue of Problem 1 for (ξ, ν)-extremal Sasaki metrics.

Problem 2. Given a compact CR manifold (N, 𝒟, J) of Sasaki type and χ, ξ ∈ 𝔠𝔯_{+}(N, 𝒟, J) with [χ, ξ] = 0, is there φ ∈ Ξ(X, J_{X})^{𝕋} such that (𝒟_{φ}, J_{φ}, χ_{φ}) is (ξ_{φ}, ν)-extremal?

In the case χ = ξ, Problem 2 reduces to the search for extremal Sasaki metrics in a given Sasaki polarization, see [14], which has been studied in many places, see e.g. [14, 18, 26, 52, 56, 57, 62]. We notice that Problems 1 and 2 are naturally linked in the regular case, via Examples 1, 2 and Remark 4. Indeed, the parametrization (40) implies that for any φ ∈ Ξ(X, J_{X})^{𝕋}, the Kähler potentials f̃ = η_{φ}(K) and f = η(K) of K are linked on a Sasaki–Reeb quotient (M, J) via f̃ = f + d^{c}φ(K), which is consistent with (38).

Remarks 5. (i) By the equivariant Gray–Moser theorem and Lemma 6 above, for each φ ∈ Ξ(X, J_{X})^{𝕋} the corresponding contact form η_{φ} ∈ S(X, J_{X})^{𝕋} is equivalent to η by an identity component 𝕋-equivariant diffeomorphism Φ of N. Pulling back J_{φ} by Φ gives a CR structure J_{φ,η} in the space C_{+}(N, 𝒟)^{𝕋} of 𝕋-invariant 𝒟-compatible CR structures on (N, 𝒟) introduced in Section 2, where 𝕋 ⩽ Con(N, 𝒟) is the torus generated by the CR vector fields X_{χ} and X_{ξ}. Thus, Problem 2 can be viewed as a special case of the problem of finding critical points in C_{+}(N, 𝒟)^{𝕋} for the square norm of the momentum map of the Con(N, 𝒟)^{𝕋}-action on AC_{+}(N, 𝒟)^{𝕋} defined in Section 2. (ii) In view of Theorem 1, one may ask whether, given (N, 𝒟, J, χ, ξ) as in Problem 2, there exists a (ξ_{φ}, m + 2)-extremal solution φ ∈ Ξ(X, J_{X})^{𝕋} iff there exists an extremal Sasaki structure with contact form η_{φ} ∈ S(K, J_{K})^{𝕋}. This would be useful for studying irregular extremal Sasaki structures by taking X quasi-regular and K irregular. However, it is not clear how to relate the spaces Ξ(X, J_{X})^{𝕋} and S(K, J_{K})^{𝕋}, since J_{φ} is the lift of J_{K} to 𝒟_{φ} and its projection onto T N/ span(X) does not agree with J_{X} in general, and so the pullback J_{φ,η} = Φ^{∗}J_{φ} with Φ^{∗}η_{φ} = η need not descend to a complex structure in the same Teichmüller class as J on the quotient of N by X.

6.3. Extremal Sasaki structures from ruled complex surfaces. We now specialize to geometrically ruled complex surfaces and the regular Sasaki manifolds they define. Let (M, J) = π: P(O ⊕ L) → B be the underlying complex manifold of a projective ℂP^{1}- bundle over a compact Riemann surface B, where L is a holomorphic line bundle over B of positive degree ℓ. Let K be the generator of the holomorphic 𝕊^{1}-action on (M, J) induced by scalar multiplication in O. We denote by (g_{B}, ω_{B}) the Kähler metric on B of constant scalar curvature 4(1 − g), where g denotes the genus of B. It is well-known (see e.g. [8]) that the Kähler cone of (M, J) can be parametrized up to homothety by the cohomology classes of Kähler metrics (g, ω) given by the Calabi ansatz (16) as described in Section 4.5, with a_{1} = ℓ. For convenience, in this section, we let a denote the real constant a_{0}/ℓ and write z for z_{1} and μ for μ_{1}, so that a_{0} + a_{1}μ_{1} = ℓ(μ + a).

For each b ∈ ℝ with |b| > 1, f_{b} := μ + b is a positive Killing potential for K on (M, g, ω). The existence of a (f_{b}, 4)-extremal Kähler metric of the form (16) on M (up to homothety) is studied in [11, 44, 50]. It is shown there (see e.g. [11, Thm. 1]) that such a metric must be obtained from a smooth function A(z) = P_{a,b}(z)/(z + a), where P_{a,b} is a polynomial of degree ⩽ 4 uniquely determined from (19) in terms of a, b and ℓ; thus(M, g, J, ω) is (f_{b}, 4)-extremal iff P_{a,b}(z) satisfies the positivity condition (20). Conversely, we have the following result.

Proposition 7. Let M = P (O ⊕ L) → ℂP^{1} be a ruled surface and Ω = λ[ω] a Kähler class on M for λ > 0, a > 1. Let |b| > 1, f_{b} = μ + b and κ = \frac{λ^{3}}{2} ∫_{M} f_{b} ω^{2}. If the polynomial P_{a,b}(z) is not positive on (−1, 1), Ω contains no (K, κ, 4)-extremal Kähler metrics.

Proof. The proof follows from a slight modification of the arguments in [46, Cor. 1], taking into account the recent result [47, Cor. 1]. Indeed, suppose for contradiction that P_{a,b}(p_{0}) = 0 for some p_{0} ∈ (−1, 1), and that [ω] admits a (K, κ, 4)-extremal Kähler metric.

Consider first the case that P_{a,b}(z) is negative somewhere on (−1, 1). If the Kähler class [ω/2π] is rational (which is equivalent to a ∈ ℚ), we derive a contradiction by [47, Cor. 1] (which implies that the relative weighted Mabuchi functional must be bounded from below) and [11, Prop. 2.7] (which concludes otherwise). If the Kähler class [ω/2π] is not rational, we can approximate it with rational classes [ω̃/2π] of the form (21) by taking rational values ã close to a, and still ensure that P_{ã,b}(z) is negative somewhere on (−1, 1). Furthermore, by the openness of weighted extremal classes established in [45, Thm. 2], we can assume that [ω̃] admits a (K, κ̃, 4)-extremal Kähler metric with κ̃ = \frac{1}{2} ∫_{M} f_{b} ω̃^{2}. We get a contradiction as before.

It thus remains to consider the case when p_{0} ∈ (−1, 1) is a double root of the quadratic P_{a,b}(z)/(1 − z)^{2}. In this case, we prove that there exists a sequence b̃_{i} converging to b, such that P_{a,b̃_{i}}(p_{0}) < 0; by the openness result in [45, Thm. 2], we can then find b̃ with P_{a,b̃}(p_{0}) < 0 and such that the Kähler class [ω] admits a (K, κ̃, 4)-extremal Kähler metric with κ̃ = \frac{1}{2} ∫_{M} fb̃ ω^{2}, a situation we have already ruled out.

In order to find a sequence as above, it is enough to show that \frac{∂P_{a,b}}{∂b} (p_{0}) ≠ 0 at each double root p_{0} ∈ (−1, 1). The remainder of the proof establishes this technical fact.

If a = b, we get the natural solution of Corollary 3, in which case P_{a,b}(z) > 0 on (−1, 1). We can thus assume that a ≠ b, and then, adapting [44, (11) & (31)] to our notation, the polynomial P_{a,b} is given by

(41)  \frac{2P_{a,b}(z)}{1 − z^{2}} = 2(z + a) + (1 − z^{2})\frac{3c_{a,b} + a + s}{3c_{a,b}^{2} − 1},

where s = 2(1 − g)/ℓ and c_{a,b} = (ab − 1)/(a − b).

Suppose that \frac{∂P_{a,b}}{∂b} (p_{0}) = 0 for p_{0} ≠ ±1. We compute that this is equivalent to

2c_{a,b} s + 3c_{a,b}^{2} + 2a c_{a,b} + 1 = 0.

Since c_{a,b} ≠ 0, we may solve for s and substitute back in (41) to obtain

\frac{2P_{a,b}(z)}{1 − z^{2}} = \frac{1 + 4a c_{a,b} + 4c_{a,b} z − z^{2}}{2c_{a,b}} .

Now if P_{a,b} has a double root at p_{0} ≠ ±1, we must have p_{0} = 2c_{a,b} = 2(ab − 1)/(a − b), the critical point of this expression. However, (ab − 1)^{2} − (a − b)^{2} = (a^{2} − 1)(b^{2} − 1) > 0 for |b| > 1 and a > 1, so |p_{0}| > 2 and hence p_{0} ∉ (−1, 1). □

It is shown in [11, Prop. 2.12] that for a rational Kähler class of the form (21), and for z ∈ (−1, 1) ∩ ℚ, P_{a,b}(z) computes a weighted notion of the relative Donaldson–Futaki invariant associated to the polarized variety (M, L_{k,n}), where L_{k,n} is a polarization on M corresponding to a = a_{0}/ℓ = (2n/kℓ) − 1 as explained in Section 4.5. This motivates the following definition.

Definition 7. [11] Let (M, L_{k,n}) be a polarized ruled surface as above, and Ẑ_{b} the quasiperiodic (real) holomorphic vector field on L_{k,n}, given by the lift of K with respect to the potential f_{b}. We say that (M, L_{k,n}, Ẑ_{b}) is analytically relatively (Ẑ_{b}, 4) K-stable with respect to admissible test configurations if P_{a,b}(z) > 0 on (−1, 1).

Thus, Proposition 7 implies that that (M, J) admits a (K, κ, 4)-extremal Kähler metric in 2πc_{1}(L_{k,n}) iff (M, L_{k,n}, Ẑ_{b}) is analytically relatively (Ẑ_{b}, 4) K-stable with respect to admissible test configurations.

6.4. Proof of Theorem 2. Let (N_{k,n}, 𝒟, χ) be a compact regular contact manifold over a ruled surface M constructed in Section 4.5, and ξ_{b} ∈ 𝔠𝔬𝔫(N_{k,n}, 𝒟) the contact lift of the generator K of the 𝕊^{1}-action on M via potential μ + b. We denote by 𝕋 ⩽ 𝔠𝔬𝔫(N_{k,n}, 𝒟) the torus generated by χ and ξ_{b}, and let P_{a,b} be the polynomial corresponding to the Kähler class (21) with a_{0} = (2n/k) − ℓ and a_{1} = ℓ. We need to show that there exists a (ξ_{b}, 4)-extremal CR structure J ∈ C_{+}(N_{k,n}, 𝒟)^{𝕋} if and only if P_{a,b}(z) > 0 on (−1, 1).

Existence follows from Theorem 1 and the fact that when P_{a,b}(z) > 0 on (−1, 1), A(z) = P_{a,b}(z)/(z + a) in (16) defines an (f_{b}, 4)-extremal Kähler metric on M (see [11, Thm. 1]).

We now establish the non-existence claim. Suppose that P_{a,b}(z) has a zero on (−1, 1). Denote by J the CR-structure in C_{+}(N_{k,n}, 𝒟)^{𝕋} induced by a Kähler metric (g, ω) on M of the form (16) and suppose for contradiction that there exists a 𝒟-compatible CR structure J′ ∈ C_{+}(N_{k,n}, 𝒟)^{𝕋} such that (𝒟, J′, ξ_{b}) is an extremal Sasaki structure and P_{a,b}(z) has a zero on (−1, 1). As J′ is X_{ξ_{b}}-invariant, on M we obtain another ω-compatible complex structure J′ which is invariant under the 𝕊^{1}-action generated by K. As any compact Kähler 4-manifold admitiing a holomorphic vector field with zeroes must be a rational or a ruled surface [23], (M, J′) must be a ruled surface too, i.e., (M, J′) = P (O ⊕ L′) → B′ with B′ having the same genus as B. Intersection properties of the zero set of K (which equals the zero and infinity sections in either case) yield that deg(L) = deg(L′). Since J and J′ are compatible with the same symplectic form ω, the corresponding class [ω] is of the form (21) on either surface, with the same parameter a = a_{0}/ℓ > 1; furthermore, the Killing potential of K induced by ξ_{b} in both cases is f_{b} = μ + b for the same b > 1. It thus follows that the respective polynomials associated to [ω] on each ruled surface (M, J) and (M, J′) coincide; we denote them by P_{a,b}.

Now, by Theorem 1, the Kähler class [ω] on (M, J′) admits a (K, κ, 4)-extremal Kähler metric which, by Proposition 7, forces P_{a,b}(z) > 0 on (−1, 1), a contradiction. □

Remarks 6. (i) Theorem 2 yields a (ξ_{b}, 4)-extremal Sasaki metric on the Sasaki join (N_{w,k}, 𝒟) over B × ℂP_{w}^{1} (with weights w_{+} = n, w_{−} = n − kℓ, where a = (2n/kℓ) − 1) if P_{a,b}(z) > 0 on (−1, 1). This always happens when B has genus 0 or 1, or when a is sufficiently large (see e.g. [11, Thm. 1]). These extremal Sasaki structures are not new (see [16, 17]) but we have seen in Section 4.5 that they can equivalently be obtained from a (f̃_{b}, 4)-extremal product Kähler metric g_{B} + pg_{w} on B × ℂP_{w}^{1}. Taking the limit b → ∞ also yields the extremal Kähler metrics on the ruled surfaces constructed in [20, 61] as a CR twist of a product metric on B × ℂP_{w}^{1}. (ii) To the best of our knowledge, the non-existence result obtained via Theorem 2 is new, at least for irrational values of b (in which case ξ_{b} generates 𝕋, and thus is not quasi-regular). To construct specific examples, following [44], on any ruled surface M over a curve of genus ⩾ 2 as above, there exists an explicit a_{0}(M) > 1 such that for any a ∈ (1, a_{0}(M)] the polynomial P_{a,b_{a}}(z) has a zero on (−1, 1), where b = b_{a} > 1 is the unique solution of a = \frac{1+b^{2}}{2b} satisfying |b| > 1. Taking coprime positive integers k, n with a = (2n/kℓ) − 1 ⩽ a_{0}(M), we obtain a contact manifold (N_{k,n}, 𝒟) which admits no (ξ_{b}, 4)-extremal CR structure J ∈ C_{+}(N_{k,n}, 𝒟)^{𝕋}.

Acknowledgements. We would like to thank Paul Gauduchon, Gideon Maschler and Christina Tønnesen-Friedman for stimulating conversations. We are grateful to Abdellah Lahdili for his useful comments on the manuscript, and to Roland Puček for discussions which helped to clarify some of the toric constructions in this paper.

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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