WEIGHTED K-STABILITY OF POLARIZED VARIETIES AND EXTREMALITY OF SASAKI MANIFOLDS

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

Abstract. We use the correspondence between extremal Sasaki structures and weighted extremal Kähler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors in [1], and the theory of weighted K-stability (introduced by Lahdili in [40]) in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus and smooth equivariant test configurations with reduced central fibre is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the corresponding polarized affine cone (introduced by Collins–Székelyhidi in [18]), and prove that they agree on the class of test configurations we consider. As a byproduct, we strengthen the obstruction to the existence of a scalar-flat Kähler cone metric found in [18] from the K-semistability to the K-stability on these test configurations. We use our approach to give a characterization of the existence of a compatible extremal Sasaki structure on a principal 𝕊^{1}-bundle over an admissible ruled manifold in the sense of [4], expressed in terms of the positivity of a single polynomial of one variable over a given interval.

Introduction

The famous Calabi problem [15], which seeks the existence of canonical Kähler metrics in a given Kähler class ζ ∈ H^{2}(M, ℝ) on a compact complex manifold (M, J), is a central and very active topic of current research in Kähler geometry. As a candidate for a canonical metric 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.

There is an odd-dimensional analogue of the Calabi problem—and a corresponding notion of extremal Sasaki structures (called canonical in [11])—associated to compact, strictly pseudoconvex CR manifolds (N, 𝒟, J) of Sasaki type. This may be motivated using the construction of regular Sasaki structures on the total space N of the principal 𝕊^{1}-bundle over a Hodge manifold (M, J, ζ) determined by the given integral Kähler class ζ/2π ∈ H^{2}(M, ℤ): any Kähler metric ω ∈ ζ on M induces a connection 1-form η on N satisfying dη = π^{∗}ω, and hence a CR structure (𝒟, J) with 𝒟 = ker η ⊆ T N and J the pullback of the complex structure on M to 𝒟 ≅ π^{∗}T M. In this case, the vector field K induced by the fibrewise 𝕊^{1}-action on N is called regular Sasaki–Reeb vector field. The Calabi problem on (M, J, ζ) translates to the search for CR structures (𝒟, J) on N, compatible with the induced transversal holomorphic structure J^{K} on the vector bundle T N/⟨K⟩ → N, such that (𝒟, J, K) is a Sasaki structure corresponding to an extremal, CSC or Kähler–Einstein metric ω ∈ ζ on M. This leads respectively to notions of extremal, CSC and Einstein Sasaki structures on N.

The above construction can be generalized by relaxing the assumption that the 𝕊^{1}-action on N generated by the vector field K is free. If it is instead required only to be locally free, then N can be realized as a principal 𝕊^{1}-orbibundle over a kählerian orbifold (M, J, ζ), and K is then referred to as a quasiregular Sasaki–Reeb vector field on N. One can also weaken the assumption that K generates an 𝕊^{1}-action, and require only that the closure of the subgroup generated by the vector field K is a compact torus 𝕋 in the diffeomorphism group of N preserving the given transversal holomorphic structure J^{K}; if dim 𝕋 > 1, K is called an irregular Sasaki–Reeb vector field.

Date: January 14, 2021.

Irrespective of regularity, the correspondence between Kähler geometry and Sasaki geometry holds locally, allowing one to extend the notions of extremal, CSC and Einstein Sasaki structures: indeed, any point of a Sasaki manifold (N, 𝒟, J, K) has a neighbourhood in which the leaf space M of the flow of K is a manifold and has an induced Kähler structure (g, J, ω). We refer to (M, g, J, ω) as a Sasaki–Reeb quotient of (N, 𝒟, J, K), and say that (N, 𝒟, J, K) is extremal, CSC or Sasaki–Einstein if any local Sasaki–Reeb quotient is extremal, CSC, or Kähler–Einstein manifold, respectively.

The study of irregular extremal Sasaki metrics is more delicate than the regular case (which reduces to the Calabi problem in the Kähler context). One approach for dealing with the issue of irregularity stems from the seminal work of Martelli–Sparks–Yau [47], who showed that the existence of a Sasaki–Einstein metric compatible with a given transversal holomorphic structure (K, J^{K}) on N can be equivalently studied using strictly plurisubharmonic smooth positive functions on the (noncompact) smooth manifold V = ℝ^{+} × N equipped with a conical complex structure J inducing the transversal holomorphic structure (K, J^{K}); such functions define Kähler metrics on (V, J) which are Ricci-flat if and only if the induced Sasaki structure on N is Einstein. Furthermore, they observed that (V, J, K) can be given the structure of an affine cone embedded in ℂ^{k} for some k ≫ 0, with K being in the Lie algebra of a maximal compact algebraic torus 𝕋 ⩽ GL(k, ℂ) preserving (V, J). The framework of [47] can be adapted [36, 18, 14] to recast, more generally, the search for compatible extremal Sasaki structures on (N, K, J^{K}).

Using the algebraic nature of (V, J, K), Collins–Székelyhidi [18] introduced a suitable notion of K-stability of (V, J, K, 𝕋), and showed that the K-semistability of (V, J, K, 𝕋) is a necessary condition for the existence of a compatible CSC Sasaki structure on (N, K, J^{K}). A corresponding notion of relative K-stability for (V, J, K, 𝕋), obstructing the existence of extremal Sasaki structures on (N, K, J^{K}), has been introduced in [14], where the relative K-semistability of (V, J, K, 𝕋) with respect to 𝕋-equivariant test configurations of affine cones in ℂ^{k} is shown to be a necessary condition for the existence of a compatible extremal Sasaki structure on (N, K, J^{K}). These works suggest a general Yau–Tian– Donaldson correspondence—analogous to the projective Kähler case [25, 55, 61]—for the existence of compatible extremal Sasaki structures on (N, K, J^{K}) expressed in terms of a suitable notion of K-stability of the affine complex cone (V, J, K). Such a correspondence has only been established in the Sasaki–Einstein case in [19], where Collins and Székelyhidi refine their original notion of K-stability by considering compatible special test configurations of (V, J, K, 𝕋), show that the existence of a compatible Sasaki–Einstein metric implies that (V, J, K, 𝕋) is K-stable (not merely K-semistable) for such test configurations and, conversely, prove that the K-stability of (V, J, K, 𝕋) with respect to special test configurations yields the existence of a Sasaki–Einstein metric on (N, K, J^{K}). None of these refinements seem to be available in the general CSC and extremal cases.

Thus motivated, in this paper we propose and develop a new approach to the existence problem of extremal Sasaki structures and the corresponding notion of (relative) K-stability. It uses the key feature (see [17, 29, 42, 46]) that one can construct (even irregular) CSC Sasaki manifolds from Kähler manifolds which are not necessarily CSC. This follows from the observation that the space of Sasaki–Reeb vector fields K (i.e., the vector fields giving rise to transversal holomorphic structures of Sasaki type) is open in the Lie algebra 𝔱 of a compact torus 𝕋 inside the group of CR transformation of (N, 𝒟, J). In particular, one can consider two different Sasaki–Reeb vector fields X, K ∈ 𝔱, such that X is quasiregular even if K is not. We will focus in this paper on the case that X is a regular Sasaki–Reeb vector field, although some of our considerations can be extended to the case of quasiregular X, with the obvious technical difficulties coming from dealing with orbifolds rather than smooth manifolds.

A correspondence between Sasaki structures. Suppose therefore that (N, 𝒟, J, X) is a regular compact Sasaki manifold over a smooth polarized Kähler manifold (M, J, L), with induced Sasaki–Reeb quotient Kähler structure (g, ω) on M, ω ∈ ζ = 2πc_{1}(L), and suppose also that K is another Sasaki–Reeb vector field on (N, 𝒟, J) commuting with X. Then K projects to a Killing vector field on (M, J, g, ω), denoted by Ǩ, which (by the Sasaki–Reeb condition) has a positive Killing potential f with respect to g. Furthermore, as observed in [1], (𝒟, J, K) is an extremal Sasaki structure if and only if (g, ω) satisfies the condition that (m + 2, f)-weighted scalar curvature

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

of g is a Killing potential; here m is the complex dimension of M, and Scal(g) and Δ_{g} are the scalar curvature and riemannian Laplacian of g. We refer herein to such an (m + 2, f)- extremal Kähler metric as a K-extremal Kähler metric: see §3.2–§3.3 for more details.

Following the approach in [5], one can extend the Calabi problem to this (more general) weighted setting, and relate it to the extremal Sasaki problem as follows.

Proposition 1 (see Proposition 3.7). Let (N, 𝒟, J, X) be a compact quasiregular Sasaki manifold, and (M, J, L) the corresponding smooth polarized Kähler orbifold. Suppose that K is a Sasaki–Reeb vector field on (N, 𝒟, J) which commutes with X. Then N admits an extremal Sasaki structure compatible with the transversal holomorphic structure (K, J^{K}) determined by (K, 𝒟, J) if and only if (M, J) admits a K-extremal Kähler metric in the Kähler class 2πc_{1}^{orb}(L).

A weighted K-stability theorem. The class of K-extremal Kähler metrics is a special case of a more general notion of weighted extremal Kähler metrics studied recently by Lahdili [38, 39, 40]. In particular, it follows from these works that one can reduce the search for K-extremal metrics on a smooth polarized compact Kähler manifold (M, J, L) by fixing a maximal compact torus 𝕋 ⩽ Aut(M, L), such that K belongs to Lie(𝕋), and considering 𝕋-invariant Kähler metrics in ζ = 2πc_{1}(L). Furthermore, for any smooth 𝕋equivariant polarized test configuration (ℳ , ℒ) of (M, L), Lahdili [40] associates a notion of weighted Donaldson–Futaki invariant ℱ_{K}^{ext}(ℳ , ℒ) with respect to the Sasaki–Reeb vector field K, and proves that if (M, J) admits a K-extremal Kähler metric in ζ and the smooth test configuration (ℳ , ℒ) has a reduced central fibre, then ℱ_{K}^{ext}(ℳ , ℒ) ≥ 0. This leads to a notion of a relative weighted K-semistabilty of (M, J, L, 𝕋, K) which obstructs the existence of K-extremal Kähler metrics in 2πc_{1}(L).

Our first main result extends this weighted K-semistability to weighted K-stability.

Theorem 1 (see Theorem 5.5). Let (N, 𝒟, J, X) be a compact regular Sasaki manifold, and (M, J, L) the corresponding smooth polarized Kähler manifold. Suppose that K is a Sasaki–Reeb vector field on (N, 𝒟, J) which commutes with X, such that N admits an extremal Sasaki structure compatible with the transversal holomorphic structure (K, J^{K}) determined by (K, 𝒟, J). Let 𝕋 be a maximal torus in the group Aut(M, L) of automorphisms of the polarized variety (M, J, L), such that X, K belong to Lie(𝕋). Then, for any 𝕋-equivariant smooth polarized test configuration (ℳ , ℒ) associated to (M, J, L, 𝕋), which has reduced central fibre and is not a product test configuration, we have ℱ_{K}^{ext}(ℳ , ℒ) > 0, i.e., (M, J, L, K, 𝕋) is relative weighted K-stable on such test configurations.

By Proposition 1, the existence of an extremal Sasaki structure on (N, K, J^{K}) translates to the existence of a 𝕋-invariant K-extremal metric (g, ω) on M with ω ∈ 2πc_{1}(L). In this setting it is well-known since the seminal work by Tian [55] in the positive Kähler– Einstein case (see also [8, 33, 52] for unweighted CSC and extremal Kähler metrics) that a K-stability result like Theorem 1 may be deduced by establishing a suitable properness property for an appropriate Mabuchi energy functional. Furthermore, there is an axiomatic framework [21] for establishing such properness properties, which allows us to focus on the specificities of our problem.

In our setting, we denote the relevant functional by M_{Ǩ,κ}, where the subscripts Ǩ and κ stand respectively for the holomorphic vector field on (M, J) induced by K, and a positive real constant, determined by K, see §3.3. This functional is a weighted version of the relative Mabuchi energy introduced in [39] defined on the space of 𝕋-invariant hermitian products on L with positive curvature. To prove the relative weighted K-stability of (M, J, L, K, 𝕋), it suffices show that M_{Ǩ,κ} is proper with respect to the natural action of the complexified torus 𝕋_{ℂ} on that space.

In order to apply the axiomatic framework of [21], we need to extend M_{Ǩ,κ} to the d_{1}- completion of the space of 𝕋-invariant relative Kähler potentials on M. The existence of such an extension is not obvious because, in contrast to the CSC Kähler case, the definition of M_{Ǩ,κ} involves the κ-normalized potential of Ǩ with respect to a metric ω ∈ c_{1}(L), which has no natural interpretation in the space of weakly regular relative Kähler potentials.

The key for bypassing this difficulty is also the key to the proof of Proposition 1. Namely, we make use of a bijection Θ̂ (see §2.3, and in particular equation (19)) between (suitably normalized) Sasaki structures on (N, K, J^{K}) and (N, X, J^{X}) such that corresponding Sasaki structures have the same underlying CR structure. This map comes from a correspondence between their respective spaces of (radial) Kähler potentials on the cone (V, J, K) that we establish (see Corollary 2.15) building on the work of He–Sun [36]. While we avoid the noncompact cone in the notion of stability we use, (V, J, K) thus plays an essential role in establishing properties of the map Θ̂.

First, after identifying the normalized Sasaki structures on (N, K, J^{K}) with 𝕋-invariant hermitian forms on L with positive curvature, we show (see Lemma 4.4) that

M_{Ǩ,κ} ∘ Θ̂ = M^{K}

M^{K} is the relative Mabuchi energy acting on the space of (normalized) Sasaki structures in (N, K, J^{K}), studied in [34, 35, 56]. Secondly, in Lemma 2.17 we compute the derivative of Θ̂, which enables us to show (see Lemma 4.6) that it is bilipschitz with respect to d_{1}. Thirdly, Θ̂ is equivariant for the natural action of the complexified torus 𝕋_{ℂ} induced by K and X on the space of 𝕋-invariant hermitian products on L (similar to the setting in [30]). This is different than the action of complex automorphisms on the space of relative Kähler potentials considered in [8, 21, 33, 52], but has the practical advantage to induce also a natural action on the space of (normalized) Sasaki structures in (N, K, J^{K}).

These tools allow us to define a notion of properness for M_{Ǩ,κ} with respect to the natural action of 𝕋_{ℂ} which directly translates via Θ̂ to a corresponding notion of properness for M^{K} (see Corollary 4.11). This allows us to use the extension of M^{K} to the d_{1}-completion and the regularity of its minimizers (obtained in [34, 35]) to show the properness of M^{K} (see Theorem 4.12) and hence of M_{Ǩ,κ}. A key result for this to work is the transitivity of the action of 𝕋_{ℂ} on the space of 𝕋-invariant K-extremal Kähler metrics in 2πc_{1}(L), which has been obtained in [41] using the approach in [16]. Once the 𝕋_{ℂ}-properness of M_{Ǩ,κ} is established, the proof of Theorem 1 essentially follows from the arguments in [8, 52].

Since the complex cone (V, J, K) plays such an important role in the proof, we spend some time in Section 2 developing its properties, following [14, 18, 36, 47]. In particular, in §2.2, we detail the relation (see [14, 18] and Lemma 2.13) between various known points of view on the Reeb cone 𝔱_{+} in the Lie algebra 𝔱 of 𝕋, establishing that 𝔱_{+} consists of vector fields K whose cones (V, J, K) admit a radial Kähler potential.

Weighted K-stability versus K-stability of affine cones. Theorem 1 provides in particular an obstruction to the existence of CSC Sasaki structures compatible with a transversal holomorphic structure (K, J^{K}) on N, expressed in terms of the weighted Kstability of a fixed regular quotient (M, J, L, 𝕋) with respect to suitable test configurations (ℳ , ℒ). We first remark that the assumptions that ℳ is smooth and its central fibre is reduced are needed in the proof of Theorem 1 (as in [40]) both to define ℱ_{K}^{ext}(ℳ , ℒ) and to relate it to the slope of the corresponding Mabuchi energy M_{Ǩ,κ}. However, it is shown in [23, 51] that in the CSC case, restricting to such test configurations of (M, J, L) does not affect the corresponding (unweighted) K-stability condition. We expect that these assumptions on (ℳ , ℒ) will not limit the applicability of Theorem 1 either.

Secondly, we need to compare the weighted K-stability in Theorem 1 to the (relative) K-stability of the corresponding affine cone (V, J, K, 𝕋) in the sense of [18]. To do this, we use the fact that the complex manifold (V, J) can be identified with (L^{∗})^{×} (the total space of the dual bundle of L with the zero section removed) whereas for any test configuration (ℳ , ℒ) as in Theorem 1, the corresponding complex cone 𝒱 = (ℒ^{∗})^{×} gives rise to a 𝕋-equivariant smooth test configuration of the polarized affine cone (V, J, K) in the sense of [18, 45]. We shall refer to such test configurations of (V, J, K) as smooth 𝕋-equivariant regular test configurations (where we have also assumed that the central fibre of ℳ is reduced). We then have the following result.

Theorem 2 (see Corollary 6.10). Let (V, J, K) be the polarized complex cone associated to a compact Sasaki manifold (N, 𝒟, J, K), 𝕋 ⩽ Aut(V, J)^{K} a maximal torus and assume that the Lie algebra of 𝕋 contains a regular Sasaki–Reeb vector field X. If (V, J, K) admits a scalar-flat Kähler cone metric, then for any 𝕋-equivariant regular test configuration (𝒱 , K) of (V, J, K), which is not obtained from a product test configuration, the Donaldson–Futaki invariant of [18] associated to (𝒱 , K) is strictly positive.

This theorem follows readily from Theorem 1 once we establish that the Donaldson– Futaki invariant of [18] associated to (ℒ^{∗})^{×} is a positive multiple of the corresponding weighted Donaldson–Futaki invariant ℱ_{K}(ℳ , ℒ). To establish such a relationship, we propose in Definition 6.7 a differential geometric definition of a Donaldson–Futaki invariant for any smooth test configuration (𝒱 , K) (which itself is a complex affine cone) associated to the affine cone (V, J, K, 𝕋) of (N, K, J^{K}). We refer to this quantity as the global Futaki invariant and believe it is of independent interest. It is expressed as an integral formula on the Sasaki manifold (𝒩 , 𝒟, η_{𝒟}^{K}) associated to 𝒱 as

𝒢ℱ _{K}(𝒱 ) = \frac{c_{K}}{(m + 1)!} ∫_{𝒩} η_{𝒟}^{K} ∧ (dη_{𝒟}^{K})^{m+1} + \frac{2}{m!} ∫_{𝒩} (π^{∗}ρ_{o} − ρ) ∧ η_{𝒟}^{K} ∧ (dη_{𝒟}^{K})^{m},

where π: 𝒱 → ℂP^{1} is the equivariant map of the test configuration, c_{K} a normalizing constant, ρ is any representative of the basic first Chern class 2πc_{1}^{B}, and ρ_{o} ∈ 2πc_{1}(ℂP^{1}). We justify this definition in Proposition 6.8 by proving that the Donaldson–Futaki invariant [18] of any 𝕋-equivariant regular test configuration 𝒱 for (V, J, K) is a constant positive multiple of its global Futaki invariant. Then in Proposition 6.9, we prove that, up to a factor of 2π, the weighted Futaki invariant of 𝕋-equivariant smooth polarized test configuration (ℳ , ℒ) for (M, J, L, 𝕋) agrees with the global Futaki invariant of its cone 𝒱 .

Another interesting aspect of our approach is that it allows for the consideration (as in [23, 40, 51]) of Kähler 𝕋-equivariant smooth test configurations (ℳ , 𝒜) associated to the regular quotient (M, J, L), i.e., allowing both non-projective total spaces ℳ and transcendental Kähler classes 𝒜 on ℳ. Notice that in this more general framework, the weighted K-semistability of (M, J, L, K, 𝕋) still holds by the results in [40]. The novel implication of this to the theory of extremal Sasaki structures is that a smooth Kähler 𝕋equivariant test configuration (ℳ , 𝒜) for which 𝒜 is not an integral class does not define a test configuration of the polarized corresponding affine cone (V, J, K). This suggests (see Conjecture 5.8) that Theorem 1 extends to 𝕋-equivariant smooth Kähler test configurations (ℳ , 𝒜) associated to (M, J, L, 𝕋).

A Yau–Tian–Donaldson correspondence on admissible Sasaki manifolds. The importance of extending the K-stability criterion to Kähler test configurations goes back to an example from [4] which suggests that polarized test configurations are in general insufficient to detect the non-existence of extremal Kähler metrics on (M, J, L). In the final section of the article, we obtain Sasaki analogues of some results in [4]. To this end, we let (g, J, ω) be a Kähler metric on the total space M of a ℙ^{1}-bundle over the product of CSC polarized Kähler manifolds, obtained by the Calabi construction (also called admissible), and endow M with a fibrewise 𝕊^{1}-action which gives rise to a Killing vector field Ǩ on (M, J, ω). Furthermore, assuming also that ω ∈ 2πc_{1}(L) for a polarization L on M, we let K be the vector field on L defined by Ǩ and a positive Killing potential for Ǩ. In this setting, the search for K-extremal Kähler metrics in ζ given by the Calabi ansatz has been recently studied in [6]. The authors show there that (similarly to the extremal case studied in [4]), there exists a polynomial P (z) of degree ≤ m + 2 which is positive on the interval (−1, 1) if and only if (M, J) admits a K-extremal Kähler metric in ζ, and in this case, the extremal Kähler metric is given by the Calabi ansatz. On the other hand, the computations from [6, 40] also show that for each z_{0} ∈ (−1, 1), P (z_{0}) computes (up to a positive multiple) the relative weighted Donaldson–Futaki invariant of a smooth Kähler test configuration (ℳ , 𝒜_{z_{0}}) associated to (M, J, ζ); furthermore, (ℳ , 𝒜_{z_{0}}) is a polarized test configuration precisely when z_{0} ∈ (−1, 1) ∩ ℚ. Thus, the weighted K-stability established in Theorem 1 only yields that P (z) must be positive on (−1, 1) ∩ ℚ should a K-extremal metric exist, whereas Conjecture 5.8 mentioned above would imply the positivity of P (z) everywhere on (−1, 1), i.e., the existence of a K-extremal metric of Calabi type. Therefore the following result provides evidence for Conjecture 5.8.

Theorem 3 (see Corollary 7.3). Let (N, 𝒟, J, X) be the regular Sasaki manifold corresponding to an admissible (polarized) Kähler manifold (M, J, ω) and K be a Sasaki–Reeb vector field corresponding to a lift (which need not be quasiregular) of the generator of fibrewise 𝕊^{1}-action on M. Then N admits an extremal Sasaki structure in (K, J^{K}) if and only if the corresponding polynomial P (z) is positive on (−1, 1). In this case, the extremal Sasaki structure corresponds to an K-extremal Kähler metric on (M, J, ζ) of Calabi type.

We notice that, as in [4], Theorem 3 provides examples of Sasaki manifolds (N, K, J^{K}) (obtained as 𝕊^{1}-bundles over admissible manifolds), which do not admit extremal Sasaki structures for the reason that the corresponding polynomial P (z) has a unique double irrational root in (−1, 1). Our theory shows that one way to detect the non-existence in this case is to consider Kähler test configurations on the regular quotient.

1. Preliminaries on Sasaki geometry

1.1. Sasaki structures and transversal Kähler structures. Let (N, 𝒟) be a contact (2m + 1)-manifold and denote the Levi form of 𝒟 by L_{𝒟}: 𝒟 × 𝒟 → T N/𝒟; the latter is 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 on N such that L_{X}(C_{N}^{∞}(𝒟)) ⊆ C_{N}^{∞}(𝒟). We denote by 𝔠𝔬𝔫(N, 𝒟) the Lie algebra of contact vector fields in C_{N}^{∞}(T N).

We recall the following basic fact in the theory of contact manifolds (see e.g. [11]).

Lemma 1.1. The map X ↦ η_{𝒟}(X) from contact vector fields to sections of T N/𝒟 is a linear isomorphism, whose inverse σ ↦ X_{σ} is a first order linear differential operator.

In particular, for any vector field X on N, there is a unique contact vector field X_{τ} (namely the one with τ = η_{𝒟}(X)) such that X − X_{τ} is a section of 𝒟.

Now suppose J ∈ End(𝒟) is a CR structure on (N, 𝒟), i.e., a (fibrewise) complex structure on 𝒟 such that 𝒟^{(1,0)} := {X − iJX | X ∈ 𝒟} is closed under Lie bracket in T N ⊗ ℂ; then we denote by

𝔠𝔯(N, 𝒟, J) := {X ∈ 𝔠𝔬𝔫(N, 𝒟) | L_{X}J = 0}

the Lie subalgebra of 𝔠𝔬𝔫(N, 𝒟), whose elements correspond to CR vector fields X on (N, 𝒟, J). If moreover (𝒟, J) is strictly pseudoconvex then T N/𝒟 has a canonical orientation (and so (N, 𝒟) is co-oriented): the positive sections τ are those for which τ^{−1}L_{𝒟}(·, J·) is positive definite. Note that τ^{−1}L_{𝒟} = dη_{𝒟}^{X}|_{𝒟} where η_{𝒟}^{X} := τ^{−1}η_{𝒟} is the contact form defined by τ = η_{𝒟}(X). We let 𝔠𝔬𝔫_{+}(N, 𝒟) denote the space Reeb vector fields of (N, 𝒟) for which η_{𝒟}(X) is positive with respect to the chosen orientation. We then have the following fundamental definitions (see e.g. [12]).

Definition 1.2. Let (N, 𝒟, J) be a strictly pseudoconvex CR manifold. Then the Sasaki– Reeb 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 X ∈ 𝔠𝔯_{+}(N, 𝒟, J) is called Sasaki–Reeb vector field or a Sasaki structure on (N, 𝒟, J), and (N, 𝒟, J, X) is called a Sasaki manifold. We say X is quasiregular if the flow of X 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. [11]) to extend geometric notions on Kähler manifolds to Sasaki manifolds.

Example 1.3. 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, X) 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 X is the generator of the 𝕊^{1} action (with η(X) = 1, so η = η_{𝒟}^{X}).

Conversely, if X ∈ 𝔠𝔯_{+}(N, 𝒟, J) is (quasi)regular Sasaki–Reeb vector field on (N, 𝒟, J), 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, X) has a neighbourhood in which the leaf space M of the flow of X is a manifold and has a Kähler structure (g, J, ω) induced, using the local identification of 𝒟 and π^{∗}T M, by the transversal Kähler structure (g_{X}, J, ω_{X}) on 𝒟, where ω_{X} := dη_{𝒟}^{X}|_{𝒟} and g_{X} := ω_{X}(·, J·). Indeed g_{X}, J, and ω_{X} are all X-invariant, so they all descend to M, and we refer to (M, g, J, ω) as a Sasaki–Reeb quotient of (N, 𝒟, J, X).

For X ∈ 𝔠𝔯(N, 𝒟, J) we set

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

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

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

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

Lemma 1.4. The map K ↦ f = η_{𝒟}^{X}(K) is a Lie algebra isomorphism from 𝔠𝔬𝔫(N, 𝒟, J)^{X} to C_{N}^{∞}(ℝ)^{X}, and K ∈ 𝔠𝔯(N, 𝒟, J)^{X} if and only if f is a transversal Killing potential for (g_{X}, ω_{X}), i.e., −ω_{X}^{−1}(df) is a transversal Killing vector field for g_{X}.

1.2. Transversal holomorphic structures. Let N be a connected (2m + 1)-manifold, let X be a nowhere zero vector field, and let

β_{X}: T N → 𝒟_{X} := T N/ℝ·X

be the quotient of T N by the span of X. Thus 𝒟_{X} is everywhere locally canonically isomorphic to the pullback of the tangent bundle of local quotients of N by X.

Definition 1.5. A transversal holomorphic structure on (N, X) is a complex structure J^{X} on 𝒟_{X} which is everywhere locally the pullback of a complex structure (i.e., an integrable almost complex structure) on a local quotient of N by X.

Any Sasaki structure (𝒟, J, X) on N induces a transversal holomorphic structure on (N, X): since 𝒟 is transverse to X, β_{X}|_{𝒟} is a bundle isomorphism 𝒟 → 𝒟_{X}, and the complex structure on 𝒟_{X} induced by J is X-invariant and integrable because J is.

Definition 1.6. [12] A transversal holomorphic structure J^{X} on (N, X) has Sasaki type if N admits a Sasaki structure (𝒟, J, X) which is compatible with (X, J^{X}), i.e.,

β_{X}|_{𝒟}: 𝒟 → 𝒟_{X} intertwines J and J^{X}.

Such compatible Sasaki structures on (N, X, J^{X}) are completely determined by their contact forms or, equivalently, by the corresponding contact distributions: we let S(X, J^{X}) ⩽ Ω^{1}(N) be the subspace of contact forms of Sasaki structures compatible with (X, J^{X}), and write η_{𝒟}^{X} for the unique element of S(X, J^{X}) with ker η_{𝒟}^{X} = 𝒟.

As is well-known, the geometry of local quotients of N may be described 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 preserves basic forms); note that Ω_{X}^{0}(N) = C_{N}^{∞}(ℝ)^{X}—we also denote the closed and exact forms in Ω_{X}^{•}(N) by Ω_{X,cl}^{•}(N) and Ω_{X,ex}^{•}(N) respectively. Any vector field Y with L_{X}Y = fX is projectable onto local quotients, and hence induces Lie derivatives on both basic forms and X-invariant sections of 𝒟_{X} (i.e., Z ∈ C_{N}^{∞}(𝒟_{X}) with L_{X}Z = 0); these Lie derivatives depend on Y only via the X-invariant section β_{X}Y of 𝒟_{X}, hence may be denoted L_{β_{X}Y}.

A complex structure J^{X} on 𝒟_{X} is thus a transversal holomorphic structure if and only if L_{X}J^{X} = 0 and for any X-invariant section Z of 𝒟, L_{J^{X}Z}J^{X} = J^{X} ∘ L_{Z}J^{X}. Note that {α ∈ ∧^{k}T^{∗}N | ı_{X}α = 0} is naturally isomorphic to ∧^{k}𝒟_{X}^{∗}, so the X-invariance of J^{X} implies that it induces an operator on Ω_{X}^{•}(N), as in complex geometry. We thus obtain a twisted exterior derivative d_{X}^{c} = J^{X} ∘ d_{X} ∘ (J^{X})^{−1} on Ω_{X}^{•}(N). Also, the integrability of J^{X} means equivalently that the basic (1, 0)-forms generate a differential ideal.

If (N, X, J^{X}) is compact of Sasaki type, then Ω_{X}^{•}(N) has a Hodge decomposition with respect to the induced transversal metric of any η ∈ S(X, J^{X}), and the d_{X}d_{X}^{c}-lemma and transversal Kähler identities are satisfied [26], with the following consequences, cf. [11, 12].

Lemma 1.7. A 1-form γ on N is basic with J^{X}-invariant exterior derivative if and only if γ = d_{X}^{c}φ + α for a basic function φ ∈ Ω_{X}^{0}(N) and a closed basic 1-form α ∈ Ω_{X,cl}^{1}(N); further, α is uniquely determined by γ, as is φ up to an additive constant. In particular:

(i) S(X, J^{X}) is an open subset of an affine space with translation group Ω_{X}^{0}(N)/ℝ × Ω_{X,cl}^{1}(N), where Ω_{X}^{0}(N)/ℝ denotes the quotient of Ω_{X}^{0}(N) by constants; furthermore this open subset is Ω_{X,cl}^{1}(N)-invariant and convex. (ii) if Y is a projectable vector field, and η ∈ S(X, J^{X}), then L_{Y} η is tangent to S(X, J^{X}) (at η) if and only if

(1)  Y = JX_{φ} + Y^{H} + X_{f} + hX

where X_{φ} and X_{f} are contact vector fields with respect to η, J is the CR structure induced by η with JX = 0, φ, f and h are basic functions uniquely defined (by Y and η) up to φ ↦ φ + b, f ↦ f + c and h ↦ h − c for constants b, c ∈ ℝ, and α^{H} = ι_{Y^{H}}dη is a uniquely defined basic harmonic 1-form with respect to the transversal metric induced by η. Then L_{Y} η = d_{X}^{c}φ + α^{H} + dh, so any tangent vector to S(X, J^{X}) arises in this way.

Proof. If γ = d_{X}^{c}φ + α (as stated), it is basic with d_{X}γ = d_{X}d_{X}^{c}φ, which is J^{X}-invariant because J^{X}d_{X}d_{X}^{c}φ = −d_{X}^{c}d_{X}φ = d_{X}d_{X}^{c}φ. Conversely if d_{X}γ is J^{X}-invariant, γ is d_{X}d_{X}^{c}- closed, so γ = d_{X}^{c}φ + α (as stated) by the d_{X}d_{X}^{c}-lemma.

(i) Note that η ∈ Ω^{1}(N) is in S(X, J^{X}) if and only if η(X) = 1, L_{X}η = 0 and dη ∈ Ω_{X}^{2}(N) is J^{X}-invariant and J^{X}-positive. Thus if also η_{0} ∈ S(X, J^{X}), η − η_{0} is basic with J^{X}- invariant exterior derivative, hence

(2)  η = η_{0} + d_{X}^{c}φ + α

with φ ∈ Ω_{X}^{0}(N) and α ∈ Ω_{X,cl}^{1}(N). Conversely, given η_{0} ∈ S(X, J^{X}), such an η is in S(X, J^{X}) if dη is J^{X}-positive—a convex open condition depending only on η mod Ω_{X,cl}^{1}(N). (ii) For any basic function h, L_{hX}η = d(η(hX)) = dh, which is tangent to S(X, J^{X}). Hence if L_{Y} η is tangent to S(X, J^{X}), then Y = Y_{0} + η(Y)X with η(Y_{0}) = 0 and L_{Y_{0}}η = ι_{Y_{0}}dη tangent to S(X, J^{X}), i.e., of the form d^{c}φ+α_{0}. We write α_{0} = α^{H} −df with f a basic function and α^{H} a basic harmonic form with respect to the transversal metric induced by η. Thus Y_{0} = JX_{φ} + Y^{H} + X_{f} − fX with ι_{Y^{H}}dη = α^{H}, and Y has the stated form with h = η(Y) − f. Clearly any such Y has

L_{Y} η = dι_{Y} η + ι_{Y} dη = d(f + h) + d^{c}φ + α^{H} − df = d^{c}φ + α^{H} + dh,

which is tangent to S(X, J^{X}), and the uniqueness claims follow straightforwardly using the Hodge decomposition. □

Remark 1.8. Lemma 1.7(i) shows that S(X, J^{X}) is connected. It follows that for any η, η_{0} ∈ S(X, J^{X}), the Gray–Moser theorem implies that there exists Φ ∈ Diff_{0}(N), the identity component of the diffeomorphism group of N, with Φ^{∗}η = η_{0}. Lemma 1.7(ii) is an infinitesimal version of this fact: any element of T_{η}S(X, J^{X}) is of the form L_{Y} η with Y as in (1) uniquely determined up to a contact vector field of 𝒟 = ker η. By Lemma 1.1 we can fix the choice of Y with L_{Y} η = d^{c}φ + α^{H} + dh by subtracting the contact vector field X_{η(Y)} = X_{f+h} to obtain a section

Z = JX_{φ} + Y^{H} − X_{h} + hX

of 𝒟 = ker η. This yields a bijection between T_{η}S(X, J^{X}) and sections Z of 𝒟 such that the basic 2-form dL_{Z}η = L_{Z}dη is J^{X}-invariant.

Definition 1.9. The diffeomorphism group Diff (N) acts naturally on pairs (X, J^{X}), and we define the automorphism group Aut(N, X, J^{X}) to be the stabilizer of (X, J^{X}); its formal Lie algebra 𝔞𝔲𝔱(N, X, J^{X}) consists of X-invariant vector fields Y with L_{β_{X}Y} J^{X} = 0.

The automorphism group is infinite dimensional: its formal Lie algebra contains an infinite dimensional abelian ideal 𝔬(N, X) of X-invariant vector fields in the span of X; however, the quotient 𝔥_{0}(N, X, J^{X}) := 𝔞𝔲𝔱(N, X, J^{X})/𝔬(N, X) may be identified with the space of holomorphic sections of 𝒟_{X} and hence is a finite dimensional complex Lie algebra if N is compact [12]. In this case, we let O(N, X) denote the abelian normal subgroup of Aut(N, X, J^{X}) generated by the exponential image of 𝔬(N, X). However, the quotient group need not be well behaved: for example if 𝕋 is a torus in Aut(N, X, J^{X}) containing X, then O(N, X) ∩ 𝕋 is the group generated by X, which is not closed if X is irregular.

By construction, Aut(N, X, J^{X}) acts on S(X, J^{X}) and hence elements Y ∈ 𝔞𝔲𝔱(N, X, J^{X}) have the form (1) with respect to any η ∈ S(X, J^{X}) and induce vector fields η ↦ L_{Y} η on S(X, J^{X}). In these terms, elements of 𝔥_{0}(N, X, J^{X}) have representatives in 𝒟 of the form Y = JX_{φ} + Y^{H} + X_{f} − fX, and the complex structure on 𝔥_{0}(N, X, J^{X}) (coming from the action of J^{X} on sections of 𝒟_{X}) sends such a representative to JX_{f} + JY^{H} − X_{φ} + φX.

Remark 1.10. An element hX of 𝔬(N, X) induces the constant vector field η ↦ dh on S(X, J^{X}) and so if N is compact, elements of S(X, J^{X}) related by an exact basic 1form dh ∈ Ω_{X,ex}^{1}(N) induce isomorphic Sasaki structures via the action of O(N, X), and S(X, J^{X})/O(N, X) = S(X, J^{X})/Ω_{X,ex}^{1}(N).

On the other hand, elements of S(X, J^{X}) related by a closed basic 1-form induce the same transversal Kähler geometry (of local quotients): the transversal Kähler form dη ∈ Ω_{X}^{2}(N) depends only on η+Ω_{X,cl}^{1}(N) ∈ S(X, J^{X})/Ω_{X,cl}^{1}(N). Since any two such forms are related by d_{X}d_{X}^{c}φ for φ ∈ Ω_{X}^{0}(N), the quotient S(X, J^{X})/Ω_{X,cl}^{1}(N) may be viewed as the transversal Kähler class of (N, X, J^{X}).

This paper is concerned primarily with notions that depend only on the transversal Kähler geometry (of local quotients) and so it is desirable to fix the irrelevant translational gauge freedom by Ω_{X,cl}^{1}(N), and hence identify the transversal Kähler class with a subspace Ξ of S(X, J^{X}). This prompts the following definition.

Definition 1.11. A marking of a transversal holomorphic manifold (N, X, J^{X}) of Sasaki type is an affine slice Ξ ⊆ S(X, J^{X}) to the action of Ω_{X,cl}^{1}(N), i.e., a subspace of the form

(3)  Ξ = Ξ(X, J^{X}, η_{0}) := {η_{φ} ∈ S(X, J^{X}) | φ ∈ Ω_{X}^{0}(N)}

for some basepoint η_{0} ∈ S(X, J^{X}), where η_{φ} := η_{0} + d_{X}^{c}φ. We then say (N, X, J^{X}, Ξ) is marked. Elements of

(4)  Ξ̂(X, J^{X}, η_{0}) := {φ ∈ Ω_{X}^{0}(N) | η_{φ} ∈ Ξ(X, J^{X}, η_{0})}

will be called Sasaki potentials with respect to η_{0}.

For any marking Ξ, we have Ξ = Ξ(X, J^{X}, η_{0}) for any η_{0} ∈ Ξ. However, analogously to Kähler potentials, the space Ξ̂(X, J^{X}, η_{0}) of Sasaki potentials (which is an extension of Ξ by ℝ since N is connected) depends upon the choice of basepoint η_{0}.

Remark 1.12. The automorphism group Aut(N, X, J^{X}) does not preserve a marking Ξ. A more invariant object is the image of Ξ in S(X, J^{X})/Ω_{X,ex}^{1}(N): the fibres of the projection of S(X, J^{X})/Ω_{X,ex}^{1}(N) to the transversal Kähler class S(X, J^{X})/Ω_{X,cl}^{1}(N) are affine spaces modelled on the basic de Rham cohomology group H_{X}^{1}(N, ℝ), and the identity component Aut_{0}(N, X, J^{X}) acts trivially on basic de Rham cohomology, hence it preserves the image of the marking.

1.3. The Reeb cone of a transversal torus. Suppose N is compact with transversal holomorphic structure (X, J^{X}); then the automorphism group of any compatible Sasaki structure is compact. Hence if (X, J^{X}) has Sasaki type, there is a torus 𝕋 ⩽ Aut(N, X, J^{X}) whose Lie algebra 𝔱 contains X, and we now fix such a 𝕋. The considerations of the previous subsection apply equally to the space Ω_{X}^{•}(N)^{𝕋} of 𝕋-invariant basic forms, and the space S(X, J^{X})^{𝕋} of 𝕋-invariant elements of S(X, J^{X}). This space is acted upon by 𝕋-equivariant automorphisms, i.e., by the centralizer Aut(N, X, J^{X})^{𝕋} of 𝕋 in Aut(N, X, J^{X}). Also, for any basepoint η_{0} ∈ S(X, J^{X})^{𝕋}, we define

(5)  Ξ(X, J^{X}, η_{0})^{𝕋} = Ξ(X, J^{X}, η_{0}) ∩ S(X, J^{X})^{𝕋}

and a corresponding space Ξ̂(X, J^{X}, η_{0})^{𝕋} = Ξ̂(X, J^{X}, η_{0}) ∩ C_{N}^{∞}(ℝ)^{𝕋} of 𝕋-invariant Sasaki potentials with respect to η_{0}. We refer to a subspace of S(X, J^{X})^{𝕋} of the form (5) as a (𝕋-invariant) marking, with the 𝕋-invariance generally being understood from the context.

Any K ∈ 𝔱 is a CR vector field for the CR structure induced by any η ∈ S(X, J^{X})^{𝕋}. Furthermore, we have the following observation.

Lemma 1.13. Suppose that N is compact and K ∈ 𝔱 is a Sasaki–Reeb vector field with respect to the CR structure induced by some η_{0} ∈ S(X, J^{X})^{𝕋}. Then K is a Sasaki–Reeb vector field with respect to the CR structure induced by any η ∈ S(X, J^{X})^{𝕋}.

Proof. By assumption η_{0}(K) > 0 on N, and we need to show that f := η(K) > 0. As K is contact with respect to 𝒟 := ker η, we have K = fX − (dη|_{𝒟})^{−1}(df|_{𝒟}) and hence

η_{0}(K) = f − (dη|_{𝒟})^{−1}(df|_{𝒟}, η_{0}|_{𝒟}).

Evaluating this equation at a global minimum p of f we obtain f(p) = η_{0}(K)(p) > 0. □

Definition 1.14. The Reeb cone of (X, J^{X}, 𝕋) is the cone [47]

(6)  𝔱_{+}^{X} := {K ∈ 𝔱 | η(K) > 0}

in 𝔱 determined by some (and hence any) η ∈ S(X, J^{X})^{𝕋}. Given any such η, its contact momentum map μ_{η} : N → 𝔱^{∗} is defined [44] by ⟨μ_{η}, K⟩ = η(K).

Since ⟨μ_{η}, X⟩ ≡ 1, the image of μ_{η} in 𝔱^{∗} does not contain 0. Furthermore, if N is compact, results of [27, 44] therefore imply that Σ := ℝ^{+} · μ_{η}(N) ⊆ 𝔱^{∗} is a strictly convex polyhedral cone (with the vertex 0 ∈ 𝔱^{∗} removed) called the momentum cone of η. Moreover, as highlighted in [47], its dual cone Σ^{∗} coincides with the Reeb cone (6):

\begin{aligned}Σ^{∗} & := \{K ∈ 𝔱 | ∀ x ∈ Σ, ⟨x, K⟩ > 0 \} = \{K ∈ 𝔱 | ⟨μ_{η}, K⟩ > 0 on N\} \\ & = \{K ∈ 𝔱 | η(K) > 0 on N\} = 𝔱_{+}^{X}. \end{aligned} ; (7)

The image of μ_{η} itself is the transversal slice

(8)  P_{X} := μ_{η}(N) = {x ∈ Σ | ⟨x, X⟩ = 1}

of the momentum cone; it is thus a compact convex polytope which, following [42], we call the transversal polytope of (N, X, J^{X}, 𝕋)—in [46], it is instead called the characteristic polytope. We thus obtain the following immediate consequences of Lemma 1.13, Definition 1.14 and the duality (7).

Lemma 1.15. Let (N, X, J^{X}, 𝕋) be a compact of Sasaki type. Then all η ∈ S(X, J^{X})^{𝕋} have the same momentum cone Σ and the same transversal polytope P_{X}. Furthermore, the duality (7) identifies 𝔱 with the space of affine-linear functions Aff(P_{X}, ℝ) on the transversal polytope P_{X}, with 𝔱_{+}^{X} being the subset of strictly positive such functions.

Remarks 1.16. The fact that P_{X} ⊆ 𝔱^{∗} is independent of the choice of η ∈ S(X, J^{X})^{𝕋} may also be deduced from the 𝕋-equivariant Gray–Moser isotopy argument as in Remark 1.8.

When X is quasiregular, P_{X} is the (natural) momentum polytope of the Sasaki–Reeb quotient (M, J, ω) with respect to the induced action of 𝕋/𝕊_{X}^{1} where 𝕊_{X}^{1} ⩽ 𝕋 denotes the 𝕊^{1}-action generated by X.

More generally, the above discussion may be reinterpreted in the transversal Kähler geometry of (N, X, J^{X}, 𝕋), where we have in particular the following observation.

Lemma 1.17. Suppose that N is compact, and that η_{0}, η ∈ S(X, J^{X})^{𝕋} are related by (2); then the induced transversal Kähler forms are related by ω = ω_{0} + d_{X}d_{X}^{c}φ, i.e., they belong to the same transversal Kähler class. Moreover any K ∈ 𝔱 induces a transversal Killing field, denoted Ǩ, for both ω_{0} and ω, whose respective transversal Killing potentials f_{0} := η_{0}(K) and f := η(K) are related by

(9)  f = f_{0} + d^{c}φ(Ǩ).

Proof. The first part is immediate from (2), and this also implies f = f_{0} + d^{c}φ(K) + α(K). Since α is closed with L_{K}α = 0, the term α(K) is a constant. Using that α is basic with respect to X, the argument of Lemma 1.13 shows that f_{0} +d^{c}φ(K) and f agree at a global minimum of f, so α(K) = 0 and (9) holds. □

2. Sasaki structures via complex cones

There is a well-known link between Sasaki structures and conical Kähler metrics, which is used in many places, see e.g. [11, 13, 29, 47].

Let (N, 𝒟, J, X) be a Sasaki manifold and ℝ^{+} the multiplicative group of positive real numbers; then V = ℝ^{+} × N has a Kähler structure (Ĵ, ω̂, ĝ) defined as follows. Let r: V → ℝ^{+} and π: V → N denote the projections and let Z ∈ C_{V}^{∞}(T V) be the generator of the natural ℝ^{+} action on V; then we have direct sum decompositions

T V ≅ ℝ·Z ⊕ π^{∗}T N = ℝ·Z ⊕ π^{∗}(ℝ·X) ⊕ π^{∗}𝒟

and we let Ĵ be the unique almost complex structure on V which restricts to π^{∗}J on π^{∗}𝒟 and sends Z to π^{∗}X. Since J induces a transversal holomorphic structure with respect to X on N and dη_{𝒟}^{X} ∈ Ω_{X}^{(1,1)}(N), Ĵ is integrable: indeed, using dr(Z) = r, we have d^{c}r = r π^{∗}η_{𝒟}^{X} and so dr + iJr π^{∗}η_{𝒟}^{X} is a (1, 0)-form on V, as are pullbacks by π of basic (1, 0)-forms on N, and together these generate a differential ideal. We now set

ω̂ := \frac{1}{2}d(r^{2}π^{∗}η_{𝒟}^{X}) = rdr ∧ π^{∗}η_{𝒟}^{X} + \frac{1}{2}r^{2}π^{∗}dη_{𝒟}^{X},
ĝ := −ω̂Ĵ = dr^{2} + r^{2} π^{∗}g_{N}, where g_{N} := (η_{𝒟}^{X})^{2} + \frac{1}{2}dη_{𝒟}^{X}(·, J·),

and these evidently define a Kähler metric on V compatible with Ĵ.

Remarks 2.1. (V, Z, Ĵ, ω̂, ĝ) is sometimes called the Kähler cone of (N, 𝒟, J, X) and g_{N} its Sasaki metric, due to the following more general considerations.

(i) Let V be any smooth manifold equipped with a vector field Z; then a riemannian metric ĝ or symplectic form ω̂ is conical with respect to Z if L_{Z} ĝ = 2ĝ or L_{Z} ω̂ = 2ω̂ respectively. If in addition Z generates a free proper action of ℝ^{+} (so that V/ℝ^{+} is a manifold) then (V, Z, ĝ) or (V, Z, ω̂) is called a riemannian or symplectic cone respectively. (ii) For any riemannian manifold (N, g_{N}), (ℝ^{+} ×N, Z, dr^{2} +r^{2}π^{∗}g_{N}) is a riemannian cone, where Z generates the ℝ^{+} action and r, π are the projections (as above). Conversely, if (V, Z̃, ĝ) is a riemannian cone and r̃^{2} := ĝ(Z̃, Z̃), there is a unique metric g_{N} on N = V/ℝ^{+} such that the quotient π̃ is a riemannian submersion from (V, r̃^{−2}ĝ) to (N, g_{N}); then (r̃, π̃) defines an isomorphism of riemannian cones (an ℝ^{+}-equivariant isometry) from (V, Z̃, ĝ) to (ℝ^{+} × N, Z, dr^{2} + r^{2}π^{∗}g_{N}). In particular, the riemannian cone of a Sasaki metric is Kähler and this is often used as a definition of Sasaki manifolds—see e.g. [47]. (iii) Any co-oriented contact manifold (N, 𝒟) has a symplectization (𝒟_{+}^{0}, Ω), where 𝒟_{+}^{0} ⊆ T^{∗}N is the component of the annihilator of 𝒟 ⊆ T N distinguished by the co-orientation, and Ω is the pullback to 𝒟_{+}^{0} of the canonical Liouville symplectic form on T^{∗}N. Then (see e.g. [44]) (𝒟_{+}^{0}, Z, Ω) is a symplectic cone, where Z is the generator of ℝ^{+} action (λ, p) ↦ λ^{2}p, for λ ∈ ℝ^{+} and p ∈ 𝒟_{+}^{0} (thus Z is twice the generator of the natural action by scalar multiplication). Conversely any symplectic cone (V, Z̃, ω̂) arises in this way up to isomorphism (ℝ^{+}-equivariant symplectomorphism).

In this section, which is a synthesis of approaches in He–Sun [36], Collins–Székelyhidi [18] and Boyer–van Coevering [14], we study the cone V from a complex viewpoint. In contrast to symplectic and riemannian cones, (V, Z, Ĵ) cannot be constructed using only the transversal holomorphic structure of N, but this proves to be fortuitous for our purposes, as the additional data needed is a marking in the sense of Definition 1.11.

2.1. Complex cones and cone potentials. In order to characterize complex cones (V, Z, Ĵ) and their compatible Kähler cone metrics (ĝ, ω̂), a key fact is that for any such cone metric, the positive function r defined by

r^{2} := ĝ(Z, Z) = ω̂(Z, ĴZ)

satisfies

(10)  L_{X}r = 0, L_{Z}r = r, ω̂ = \frac{1}{4}dd^{c}(r^{2}),

where X = ĴZ. Thus the function r suffices to define ω̂, and hence ĝ = −ω̂Ĵ, which will be a compatible Kähler metric on (V, Ĵ) provided dd^{c}(r^{2}) > 0, i.e., r^{2} is (strictly) plurisubharmonic. Henceforth we drop the hat on J and describe compatible cone metrics purely in terms of such functions r. We also take the vector field X (not Z) as primitive.

Definition 2.2. On a complex manifold (V, J) with a holomorphic vector field X, we let

(11)  R_{X}(V, J) := {r ∈ C_{V}^{∞}(ℝ^{+}) : L_{X}r = 0, L_{−JX}r = r, dd^{c}r > 0}

be the space of (radial) cone potentials (X-invariant plurisubharmonic functions of homogeneity 1 with respect to −JX). If (V, J, X) admits a surjective cone potential r: V → ℝ^{+}, then X is called a polarization of (V, J) and (V, J, X) is called a polarized complex cone.

Example 2.3. Let (N, 𝒟_{0}, J_{N}, X) be a regular Sasaki manifold as in Example 1.3, and let (M, J_{M}, ω_{0}) be its Sasaki–Reeb quotient, a compact Kähler manifold. Associated to the principal 𝕊^{1}-bundle π: N → M is a holomorphic line bundle π : L → M, endowed with a hermitian product h, such that Chern curvature 2-form of (L, h) is ω_{0}. It this situation, N can be identified with the subset of unit vectors in the dual hermitian line bundle (L^{∗}, h^{−1}), with its induced hypersurface CR structure and Sasaki–Reeb vector field X = JZ, where Z generates the natural ℝ^{+}-action on the bundle L^{∗}. Furthermore, the conical Kähler metric associated to the Sasaki manifold N is isomorphic (ℝ^{+}-equivariantly isometric) to ((L^{∗})^{×}, Z) with the metric (ω̂ = \frac{1}{4}dd^{c}(r_{h}^{2}), ĝ = −ω̂J), where V = (L^{∗})^{×} is the complex manifold obtained from the total space of L^{∗} by removing the zero section, and r_{h} is the norm function on L^{∗} defined by the hermitian metric h^{−1}. Thus, in this case, the polarized complex cone is ((L^{∗})^{×}, X) where X is the generator of the fibrewise 𝕊^{1}-action on L^{∗}. Furthermore, we can identify the space R_{X}(V, J) with the space of hermitian metrics on the ample line bundle L whose curvature form ω is positive definite, whereas the space Ξ̂(X, J^{X}, η_{0}^{X}), where η_{0}^{X} is the contact form of (𝒟_{0}, X), is isomorphic to the space {f ∈ C_{M}^{∞}(ℝ) | ω = ω_{0} + dd^{c}f > 0} of relative Kähler potentials on (M, J, ω).

We want to show more generally that R_{X}(V, J) is a basepoint free version of the space Ξ̂(X, J^{X}, η_{0}^{X}) of Sasaki potentials (4). We begin with some straightforward observations.

Lemma 2.4. Let (V, J, X) be a polarized complex cone. Then:

(i) Z := −JX generates a free proper action of ℝ^{+} and we let N^{X} = V/ exp(−JX) be the quotient manifold with quotient map π_{X}: V → N^{X}; (ii) X descends to a vector field on N^{X}, which we also denote by X, and J to a transversal holomorphic structure J^{X} on 𝒟_{X} = T N^{X}/ℝ·X; (iii) for all r ∈ R_{X}(V, J) and c ∈ ℝ^{+}, N^{X} is diffeomorphic to r^{−1}(c), so any such r: V → ℝ^{+} is surjective and (r, π_{X}): V → ℝ^{+} × N^{X} is a diffeomorphism; (iv) N^{X} is compact if and only if some, hence any, cone potential r: V → ℝ^{+} is proper. Conversely if (V, J) is a complex manifold with a holomorphic vector field X such that −JX generates a proper action of ℝ^{+}, then any cone potential is surjective, and so (V, J, X) is a polarized complex cone if and only if R_{X}(V, J) is nonempty.

Proof. (i) and (iii) follow because −JX is transverse to the level surfaces of any cone potential r, (ii) because X, Z are holomorphic with [X, Z] = 0, (iv) by definition, and the converse by the equivariance of cone potentials with respect to the ℝ^{+} action. □

For any r ∈ C_{V}^{∞}(ℝ^{+}), we set N_{r} := r^{−1}(1) and η̃_{r} := d^{c}r/r.

Lemma 2.5. Let (V, J) be a complex manifold with a holomorphic vector field X and a function r ∈ C_{V}^{∞}(ℝ^{+}) such that L_{X}r = 0 and L_{−JX}r = r. Then η̃_{r} is the pullback π_{X}^{∗}η_{r} of a 1-form η_{r} on N^{X} with η_{r}(X) = 1 and L_{X}η_{r} = 0, hence also ι_{X}dη_{r} = 0. Further, for any λ ∈ ℝ^{+} the following are equivalent:

(i) r is plurisubharmonic, i.e., is in R_{X}(V, J); (ii) r^{λ} is plurisubharmonic, hence is in R_{X/λ}(V, J); (iii) η_{r} ∈ S(X, J^{X}). Thus r ↦ r^{λ} is a bijection from R_{X}(V, J) to R_{X/λ}(V, J), and if r ∈ R_{X}(V, J), π_{X}|_{N_{r}} is a CR isomorphism from N_{r} (with the hypersurface CR structure) to N^{X} (with the CR structure induced by J^{X} and 𝒟_{r} = ker η_{r}, i.e., by η_{r} ∈ S(X, J^{X})).

Proof. Set ρ = log r. For the first part, we compute L_{X}ρ = 0 and L_{−JX}ρ = 1, from which it follows that η̃_{r} is basic with respect to the holomorphic vector field Z = −JX. Hence it is a pullback as stated, and the properties of η_{r} follow straightforwardly. For the equivalence, observe that the function r^{λ} = e^{λρ} satisfies L_{X/λ}(r^{λ}) = 0, L_{−JX/λ}(r^{λ}) = r^{λ}, and

dd^{c}(r^{λ}) = λ d(r^{λ}η̃_{r}) = λr^{λ}(λ(dr/r) ∧ π_{X}^{∗}η_{r} + π_{X}^{∗}dη_{r})

This is positive if and only if dη_{r} is positive on ker η_{r}. Thus (ii) is equivalent to (iii), hence also (i) by taking λ = 1.

The bijection follows immediately, and the last part follows because T N_{r} = ker dr|_{N_{r}} = ker dρ|_{N_{r}}, so the hypersurface CR structure has contact distribution ker η̃_{r}|_{N_{r}} with the complex structure induced by J, hence is π_{X}-related to 𝒟_{r} with the complex structure induced by J^{X}. □

Corollary 2.6. If (V, J, X) is a polarized complex cone, then the induced transversal holomorphic structure (X, J^{X}) on N^{X} has Sasaki type.

Remark 2.7. For any cone potential r ∈ R_{X}(V, J), and any a, λ ∈ ℝ^{+}, dd^{c}(ar^{λ}) is thus a Kähler metric on (V, J). To obtain a conical Kähler metric of homogeneity 2 with respect to Z = −JX, we take λ = 2 and (conventionally) c = 1/4 so that the Kähler form is

ω_{r} := \frac{1}{4}dd^{c}(r^{2}) = \frac{1}{2}d(r^{2}η̃_{r}) = r dr ∧ η̃_{r} + \frac{1}{2}r^{2}dη̃_{r},

and \frac{1}{2}r^{2} = \frac{1}{2}ω_{r}(X, JX) is a Killing potential for X. However, we generally find it simpler to work directly with the cone potential r than a conventional choice of conical metric.

Lemma 2.8. Let (V, J, X) be a polarized complex cone, r_{0} ∈ R_{X}(V, J), r ∈ C_{V}^{∞}(ℝ^{+}), and η_{0} := η_{r_{0}} ∈ S(X, J^{X}). Then r ∈ R_{X}(V, J) if and only if there exists a Sasaki potential φ ∈ Ξ̂(X, J^{X}, η_{0}) such that r = exp(π_{X}^{∗}φ)r_{0}, and then η_{r} = η_{0} + d^{c}φ.

Proof. We may write r = exp(φ̃)r_{0}, for some φ̃ ∈ C_{V}^{∞}(ℝ); then L_{−JX}r = r and L_{X}r = 0 hold if and only if φ̃ = π_{X}^{∗}φ with L_{X}φ = 0. We then have η_{r} = η_{0} + d^{c}φ and the result follows by Lemma 2.5. □

Note that φ in Lemma 2.8 is determined by π_{X}^{∗}φ = log r − log r_{0}. Thus the above observations yield the following crucial descriptions of R_{X}(V, J).

Proposition 2.9. Let (V, J, X) be a polarized complex cone. Then:

(i) the image of r ↦ η_{r}: R_{X}(V, J) → S(X, J^{X}) is a marking Ξ of (N, X, J^{X}) in the sense of Definition 1.11, and η_{r} = η_{r_{0}} if and only if r = e^{a}r_{0} with a ∈ ℝ constant; (ii) for any r_{0} ∈ R_{X}(V, J), we have Ξ = Ξ(X, J^{X}, η_{0}) with η_{0} = η_{r_{0}}, and the map

(12)  r ∈ R_{X}(V, J) ↦ φ ∈ C_{N}^{∞}(ℝ) with π_{X}^{∗}φ = log r − log r_{0}

is a bijection from R_{X}(V, J) to the space Ξ̂(X, J^{X}, η_{0}) of Sasaki potentials (4) on N^{X}; (iii) r ↦ N_{r} is a bijection from R_{X}(V, J) to the set of strictly pseudoconvex (images of ) X-invariant sections of π_{X}.

Proof. (i)–(ii) are immediate from Lemma 2.8 and the connectedness of V.

(iii) By Lemma 2.5, for any r ∈ R_{X}(V, J), N_{r} is strictly pseudoconvex and is the image of an X-invariant section of π_{X}. Conversely given such a section, the condition that r = 1 on the image and that L_{−JX}r = r determine that r = e^{ρ} where ρ(y) is the time for the flow of JX starting at y ∈ V to meet the section. Since the section is X-invariant and X is holomorphic, r so defined is a smooth function on V with L_{X}r = 0 and L_{−JX}r = r; now dd^{c}r is positive at r = 1 by pseudoconvexity of the image of the section, hence it is positive on V, since it is homogeneous of degree 1 with respect to Z = −JX. □

Part (i) of this result shows that there is a canonical bijection up to isomorphism^{1} between polarized complex cones (V, J, X) and marked transversal holomorphic manifolds (N, X, J^{X}, Ξ), where an isomorphism of polarized complex cones is an ℝ^{+}-equivariant biholomorphism of the corresponding manifolds, and an isomorphism of marked transversal holomorphic manifolds is a diffeomorphism of corresponding manifolds intertwining their transversal holomorphic structures and markings.

Part (ii) shows that R_{X}(V, J) is an (infinite dimensional) affine manifold with a global affine chart Ξ̂(X, J^{X}, η_{0})—indeed the space {ρ = log r | r ∈ R_{X}(V, J)} of logarithmic cone potentials is an affine space modelled on Ξ̂(X, J^{X}, η_{0}).

2.2. Reeb cones revisited. Let (V, J, X) be a polarized complex cone with corresponding marked transversal holomorphic manifold (N^{X}, X, J^{X}, Ξ), assumed compact. In particular (X, J^{X}) has Sasaki type (Corollary 2.6) and so, as in §1.3, we may fix a compact torus 𝕋 ⩽ Aut(N, X, J^{X}), with X ∈ 𝔱 = Lie(𝕋), which preserves some compatible Sasaki structure η_{0} ∈ Ξ. Thus X, J^{X}, and Ξ ≅ Ξ(X, J^{X}, η_{0}) are 𝕋-invariant, hence there is an induced holomorphic action of 𝕋 on (V, J, X). We denote by Ξ^{𝕋} the space of 𝕋-invariant elements of Ξ (thus Ξ^{𝕋} ≅ Ξ(X, J^{X}, η_{0})^{𝕋} for any η_{0} ∈ Ξ^{𝕋}) and by R_{X}(V, J)^{𝕋} the space of 𝕋-invariant cone potentials in R_{X}(V, J). The considerations of the previous subsection apply mutatis mutandis in the 𝕋-invariant context to show that

(i) r ↦ η_{r} is a surjection with 1-dimensional fibres from R_{X}(V, J)^{𝕋} to Ξ^{𝕋} ⊆ S(X, J^{X})^{𝕋}; (ii) for any η_{0} ∈ Ξ^{𝕋}, R_{X}(V, J)^{𝕋} is isomorphic to the space Ξ̂(X, J^{X}, η_{0})^{𝕋} of 𝕋-invariant Sasaki potentials; (iii) r ↦ N_{r} identifies R_{X}(V, J)^{𝕋} with the space of 𝕋-invariant sections of π_{X}: V → N^{X}.

Let 𝔱_{+}^{X} ⊆ 𝔱 the Reeb cone of 𝕋 (Definition 1.14). An important observation in [14, 18] is that 𝔱_{+}^{X} can be defined intrinsically on (V, J, 𝕋), and indeed can be identified with

(13)  𝔱_{+} := {K ∈ 𝔱 | K is a polarization of (V, J)}.

1It is only an equivalence of categories if one works with the quotient groupoid of ℝ^{+}-equivariant isomorphisms of polarized complex cones by the action of ℝ^{+}.

To establish this, it is useful to reformulate the definition of 𝔱_{+}^{X} on V (see e.g. [36]).

Lemma 2.10. For any r ∈ R_{X}(V, J)^{𝕋},

(14)  𝔱_{+}^{X} = {K ∈ 𝔱 | L_{−JK}r > 0} = {K ∈ 𝔱 | JK is transverse to N_{r} := r^{−1}(1)}.

Consequently, (V, J) has a proper holomorphic action of a complex torus 𝕋_{ℂ} whose real part coincides with 𝕋. Furthermore 𝔱_{+}^{X} ⊆ 𝔱_{+}, and K ∈ 𝔱_{+}^{X} if and only if X ∈ 𝔱_{+}^{K}.

Proof. Since L_{−JX}r = (d^{c}r)(X) = r(π_{X}^{∗}η_{r})(X), the first equality in 14 is immediate, and the second follows because π_{X}^{∗}η_{r} is a pullback and N_{r} is a section of π_{X}.

It follows that for any K ∈ 𝔱_{+}, JK is a complete vector field on (V, J). As 𝔱_{+} is open in 𝔱, this gives the second claim. Now for K ∈ 𝔱_{+}^{X}, N_{r} is a 𝕋-invariant section of π_{K} which yields a surjective cone potential r̃ ∈ R_{K}(V, J) by Proposition 2.9(iii) and Lemma 2.4, so K ∈ 𝔱_{+}. As Nr̃ := r̃^{−1}(1) = N_{r} is a section of π_{X}, X ∈ 𝔱_{+}^{K}; the converse holds symmetrically. □

Remark 2.11. It follows straightforwardly (cf. [36, Lemma 2.3]) that 𝔱_{+}^{X} is a connected component of 𝔱_{+}. Indeed, for K ∈ 𝔱_{+}, the condition to be in 𝔱_{+}^{X} is open because N_{r} is compact. Suppose now that K is a limit point of 𝔱_{+}^{X} which is not in 𝔱_{+}^{X}. Then π_{X}^{∗}η_{r}(K) ⩾ 0 on N_{r}, but ∃ p ∈ N_{r} with η_{r}(K)(π_{X}(p)) = 0. Since this is a minimum d(η_{r}(K))_{π_{X}(p)} = 0 and hence dη_{r}(K, ·)_{π_{X}(p)} = 0 as K ∈ 𝔱 preserves η_{r}. If K ∈ 𝔱_{+}, with surjective cone potential r̃ ∈ R_{K}(V, J), then

dr̃ ∧ π_{K}^{∗}(ηr̃ ∧ dη ∧m ; r̃) = f dr ∧ π_{X}^{∗}(η_{r} ∧ dη_{r}^{∧m})

for some function f on V. Contracting with K and evaluating at p then yields (dη∧m ; r̃)_{π_{K}(p)} = 0, which contradicts Lemma 2.5. Thus 𝔱_{+}^{X} is both open and closed in 𝔱_{+}.

To prove 𝔱_{+} ⊆ 𝔱_{+}^{X}, we proceed indirectly, following [14, 18], by introducing another equivalent definition of the Reeb cone. To explain this, we first observe if Λ ⩽ 𝔱 is the lattice of circle subgroups of 𝕋, then (ℝ^{+} · Λ) ∩ 𝔱_{+} is dense in 𝔱_{+}, and we may thus assume X is quasiregular (𝔱_{+}^{X} varies continuously with X by (14)—indeed it is locally constant in X by Remark 2.11). Then, similarly to the regular case (Example 2.3), V can be identified with the space of nonzero vectors in the dual of an orbi-ample orbiline bundle L over the compact Kähler orbifold (M, J_{M}, ω_{0}) which is the Sasaki–Reeb quotient of N by the circle action generated by X. Thus V has a natural one point compactification V̂ = V ∪ {0} which the singular space obtained by blowing down the zero section in the total space of L^{∗}. Since cone potentials correspond to norms of hermitian metrics on L, for any r ∈ R_{X}(V, J), the apex 0 of the cone V̂ is characterized as the limit of points p ∈ V with r(p) → 0. Let

(15)  ℋ ≅ ⨁_{k∈ℕ} H^{0}(M, L^{k})

be the space of continuous complex-valued functions of V̂ which are holomorphic on V ≅ (L^{∗})^{×} and polynomial on each fibre of (L^{∗})^{×} over M. Here H^{0}(M, L^{k}) is the space of holomorphic sections s of L^{k}, which define fibrewise polynomial functions f_{s} on L^{∗} by f_{s}(p) = ⟨s, p^{k}⟩. Now 𝕋_{ℂ} acts on ℋ and for α ∈ 𝔱^{∗} we let

ℋ_{α} := {f ∈ ℋ | ∀ K ∈ 𝔱, L_{−JK}f = α(K)f}

be the α-weight space and

Γ := {α ∈ 𝔱^{∗} | ℋ_{α} ≠ 0}

be the set of (integral) weights of the action. Thus there is a weight space decomposition

(16)  ℋ ≅ ⨁_{α∈Γ} ℋ_{α},

where the degree k component of (15) is the direct sum of the weight spaces ℋ_{α} with α(X) = k. The key fact we need is that (by ampleness of L) ℋ separates points of V̂: in particular for any p ∈ V̂ there exists f ∈ ℋ with f(0) = 0 and f(p) ≠ 0.

Remark 2.12. In fact orbifold versions of the Kodaira embedding theorem (see [50]) embed V̂ as an affine variety in ℂ^{N} which is a cone with the singular apex at the origin. The functions in ℋ are then the regular functions on V̂, which separate points by definition.

The incisive idea in [18] is to define the Reeb cone in 𝔱 as dual to the cone generated by Γ in 𝔱^{∗}, i.e., to set

𝔱_{+}^{Γ} = {K ∈ 𝔱 | ∀α ∈ Γ \ {0}, α(K) > 0}.

Now [18, Prop. 2.1] asserts that 𝔱_{+}^{Γ} = 𝔱_{+}^{X}, while [14, Prop. 2] asserts that 𝔱_{+}^{Γ} = 𝔱_{+}, which are the results we need. However, [14] refers to [18] for the inclusion 𝔱_{+} ⊆ 𝔱_{+}^{Γ}, while we have been unable to verify the details of the argument in [18] that 𝔱_{+}^{Γ} ⊆ 𝔱_{+}^{X}. Since these claims are crucial for our work, for the convenience of the reader we present proofs here.

Lemma 2.13. If X ∈ 𝔱_{+} is quasiregular then 𝔱_{+} ⊆ 𝔱_{+}^{Γ} and 𝔱_{+}^{Γ} ⊆ 𝔱_{+}^{X}.

Proof. For 𝔱_{+} ⊆ 𝔱_{+}^{Γ}, we follow the main ideas in the proof of [18, Prop. 2.1]. Let K ∈ 𝔱_{+} and r ∈ R_{K}(V, J). As noted already, L_{−JK}r = r implies r is equivariant with respect to the ℝ^{+} action generated by −JK. Thus under the flow Φ_{t}^{−JK} of −JK, for any p ∈ V,

r(Φ_{t}^{−JK}(p)) → 0 as t → 0

and so Φ_{t}^{−JK}(p) → 0, the apex of V̂. Now let α ∈ Γ \ {0} and f ∈ ℋ_{α} \ {0}. Then L_{−JK}f = α(K)f, f(0) = 0, and there exists p ∈ V with f(p) ≠ 0. Since

\frac{d}{dt}f(Φ_{t}^{−JK}(p)) = α(K)f(Φ_{t}^{−JK}(p))

and f(Φ_{t}^{−JK}(p)) → f(0) = 0 as t → 0, we must have α(K) > 0. Thus K ∈ 𝔱_{+}^{Γ}.

For 𝔱_{+}^{Γ} ⊆ 𝔱_{+}^{X}, we give a novel argument. Let K ∈ 𝔱_{+}^{Γ} and r ∈ R_{X}(V, J). By (7), it suffices to show that K is in the dual of the momentum cone of the contact momentum map of η_{r} on N^{X} (Definition 1.14). For this, we first let p ∈ N_{r} have maximal stabilizer with respect to the 𝕋-action, i.e., 𝔱 = 𝔰𝔱𝔞𝔟_{𝕋}(p) ⊕ ℝ·X. Thus {Y_{p} | Y ∈ 𝔱} = ℝ·X_{p} and K_{p} − η_{r}(K)_{π_{X}(p)}X_{p} ∈ 𝔰𝔱𝔞𝔟_{𝕋}(p). Since ℋ separates points, there exists f ∈ ℋ_{α} with α ∈ Γ \ {0} and f(p) ≠ 0. Let Φ_{t}^{−JK} be the flow of −JK as before and observe that

f(Φ_{t}^{−JK}(p)) = f(Φ_{t}^{−η_{r}(K)_{π_{X} (p)}JX}(p))

so that

α(K)f(p) = η_{r}(K)_{π_{X}(p)}α(X)f(p).

Since α(X) > 0 and α(K) > 0 by assumption, we have η_{r}(K)_{π_{X}(p)} > 0. We conclude that ⟨ξ, K⟩ > 0 at points ξ = μ_{η_{r}}(p) on the edges of the momentum cone. Since the momentum cone is a strictly convex polyhedral cone, the result follows. □

Since 𝔱_{+}^{X} ⊆ 𝔱_{+} by Lemma 2.10, it follows that 𝔱_{+}^{X} = 𝔱_{+}^{Γ} = 𝔱_{+}. Henceforth we drop superscripts and refer to 𝔱_{+} (with these interchangeable definitions) as the Reeb cone of (V, J). We also note that the momentum cone of any η ∈ S(X, J^{X}) coincides with the weight cone generated by Γ.

Remark 2.14. For any r ∈ R_{X}(V, J)^{𝕋}, ω̂ := \frac{1}{4}dd^{c}(r^{2}) is a 𝕋-invariant exact symplectic form (see Lemma 2.5), so the 𝕋-action on (V, ω̂) is hamiltonian with momentum map μ_{r}: V → 𝔱^{∗} defined by

(17)  ⟨μ_{r}, K⟩ = \frac{1}{4}d^{c}(r^{2})(K) = \frac{1}{2}rd^{c}r(Y).

for K ∈ 𝔱. It is in fact the unique momentum map which is homogeneous of order 2 with respect to the ℝ^{+}-action generated by −JX. In our setting, for p ∈ N_{r},

μ_{r}(p) = \frac{1}{2}μ_{η_{r}}(π_{X}(p)).

In particular d^{c}r(K) = r so \frac{1}{2}r^{2} is a Killing potential for K. Hence the flow of −JK is the gradient flow of the positive function \frac{1}{2}r^{2} with respect to the Kähler metric ĝ = −ω̂J.

Observe that ⟨μ_{r}(p), X⟩ = 1/2 for any r ∈ R_{X}(V, J)^{𝕋} and for any p ∈ N_{r} thanks to the homogeneity condition (10), so N_{r} has momentum image (1/2)P_{X}.

2.3. Change of polarization. The 𝕋-invariant version of Proposition 2.9(iii) and the final part of Lemma 2.10 have the following crucial corollary, which develops an idea used by He–Sun in the proof of [36, Lemma 2.2].

Corollary 2.15. Let (V, J, X) be the polarized complex cone of a compact 𝕋-invariant Sasaki manifold (N, 𝒟, J, X) with X ∈ 𝔱, and let K ∈ 𝔱_{+}. Then there is a bijection

\begin{matrix}Θ: R_{X}(V, J)^{𝕋} & → R_{K}(V, J)^{𝕋} \\ r & ↦  r̃ \end{matrix} ; (18)

between 𝕋-invariant cone potentials with respect to X and such potentials with respect to K, characterized uniquely by Θ(r) = r̃ if and only if Nr̃ := r̃^{−1}(1) is equal to N_{r} := r^{−1}(1).

Fixing base points r_{0} ∈ R_{X}(V, J)^{𝕋} and r̃_{0} := Θ(r_{0}) ∈ R_{K}(V, J)^{𝕋}, we also have a bijection

Θ̂: Ξ̂(X, J^{X}, η_{r_{0}})^{𝕋} → Ξ̂(K, J^{K}, ηr̃_{0})^{𝕋}
φ ↦ ψ with Θ(e^{φ}r_{0}) = e^{ψ}r̃_{0}

(19)

such that φ and Θ̂(φ) induce Sasaki structures with the same underlying CR structure.

Remark 2.16. From the contact viewpoint (see Remark A.2), the images of Ξ(X, J^{X}, η_{r_{0}})^{𝕋} and Ξ(K, J^{K}, ηr̃_{0})^{𝕋} in C_{+}(N, 𝒟_{0})^{𝕋} are both formal complexifications of the same orbit of the group Con(N, 𝒟_{0})^{𝕋} on this space, which does not depend on the choice X or K of Reeb vector field. Hence we expect these images to coincide up to isomorphism. Thus Corollary 2.15 provides an explicit realisation of this correspondence on the level of potentials.

We now compute the derivatives of Θ and Θ̂. To do this, we note that T_{r}R_{X}(V, J)^{𝕋} ≅ C_{N^{X}}^{∞}(ℝ)^{𝕋}, which is isomorphic to C_{N_{r}}^{∞}(ℝ)^{𝕋} using pullback by Ψ_{r}^{X} := π_{X}|_{N_{r}}: N_{r} ≅ N^{X}.

Lemma 2.17. Suppose r_{t} is a smooth curve in R_{X}(V, J)^{𝕋} with derivative ṙ_{t} ∈ C_{N^{X}}^{∞}(ℝ)^{𝕋}. Then r̃_{t} := Θ(r_{t}) is a smooth curve in R_{K}(V, J)^{𝕋} with derivative

˙r̃_{t} = (Ψ_{r_{t}}^{X} ∘ (Ψ_{r_{t}}^{K})^{−1})^{∗}(\frac{ṙ_{t}}{η_{r_{t}}(K)}); in addition ηr̃_{t} = (Ψ_{r_{t}}^{X} ∘ (Ψ_{r_{t}}^{K})^{−1})^{∗}(\frac{η_{r_{t}}}{η_{r_{t}}(K)}).

Proof. We write r_{t} = e^{π_{X}^{∗}φ_{t}}r_{0} and r̃_{t} = e^{π_{K}^{∗}ψ_{t}} r̃_{0} with φ_{t} a smooth curve in Ξ̂(X, J^{X}, η_{r_{0}})^{𝕋} and ψ_{t} = Θ̂(φ_{t}) a curve in Ξ̂(K, J^{K}, ηr̃_{0})^{𝕋}. Let Π_{X} = (r_{0}, π_{X}): V ≅ ℝ^{+} × N^{X} be the product structure on V induced by X and r_{0} and similarly Π_{K} = (r_{0}, π_{K}): V ≅ ℝ^{+} × N^{K}. Since Nr̃_{t} = N_{r_{t}}, we have

(Π_{K} ∘ Π_{X}^{−1})(e^{−φ_{t}(x)}, x) = (e^{−ψ_{t}(Ψ_{t}(x))}, Ψ_{t}(x)),

where Ψ_{t} = Ψ_{r_{t}}^{K} ∘ (Ψ_{r_{t}}^{X})^{−1} : N^{X} → N^{K}. Since Π_{K} ∘ Π_{X}^{−1} is a diffeomorphism, Ψ_{t} is a smooth curve of diffeomorphisms, hence ψ_{t} and r̃_{t} are also smooth curves.

To compute the derivative, by translation of the parameter t, it suffices to relate the functions Ψ_{X}^{∗} ṙ_{0} and Ψ_{K}^{∗} ˙r̃_{0} in C_{N_{0}}^{∞}(ℝ)^{𝕋} where N_{0} := N_{r_{0}} = Nr̃_{0}, Ψ_{X} = Ψ_{r_{0}}^{X} and Ψ_{K} = ΨK ; r̃_{0}. For any p ∈ N_{0}, we let

Φ_{t}(p) := Φ_{ψ_{t}(π_{K}(p))}^{−JK}(p)

where Φ_{s}^{−JK} stands for the flow of −JK on V. Now Φ_{t}(p) ∈ N_{r_{t}} = Nr̃_{t} and so, for any p ∈ N_{0}, we have

(20)  Φ_{φ_{t}(π_{X}(Φ_{t}(p)))}^{JX}(Φ_{t}(p)) ∈ N_{0},

where Φ_{s}^{JX} is the flow of JX on V. Taking derivative at t = 0 in (20), we thus obtain an element in T_{p}N_{0}. To compute this derivative, we use that

\frac{d}{dt}Φ_{t}(p)|_{t=0} = −ψ̇_{0}(π_{K}(p))JK_{p},

φ_{0}(π_{X}(q)) = 0, Φ_{0}^{JX} = Id_{T V}

and get

\frac{d}{dt}(Φ_{φ_{t}(π_{X}(Φ_{t}(p)))}^{JX}(Φ_{t}(p)))|_{t=0} = φ̇_{0}(π_{X}(p))JX_{p} − ψ̇_{0}(π_{K}(p))JK_{p}.

As (𝒟_{0})_{p} = T_{p}V ∩ JT_{p}V, and K_{p} and X_{p} are in T_{p}N_{0}, we conclude that

φ̇_{0}(π_{X}(p))X_{p} − ψ̇_{0}(π_{K}(p))K_{p} ∈ (𝒟_{0})_{p},

which, after acting with η_{r_{0}}, gives

ψ̇_{0} ∘ π_{K} = \frac{φ̇_{0} ∘ π_{X}}{η_{r_{0}}(K)}

on N_{0}. This means equivalently that Ψ_{K}^{∗} ˙r̃_{0} = Ψ_{X}^{∗}(ṙ_{0}/η_{r_{0}}(K)) as required.

The formula for ηr̃_{t} follows because by Lemma 2.5, Ψ_{r_{t}}^{X} ∘ (Ψ_{r_{t}}^{K})^{−1} is a CR isomorphism between the CR structures induced by ηr̃_{t} and η_{r_{t}} on N^{K} and N^{X} respectively □

It immediately follows that Θ̂ is smooth, and its derivative is given as follows.

Corollary 2.18. Let φ ∈ Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} and ψ := Θ̂(φ) ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} induce CR structures (𝒟_{φ}, J_{φ}) and (𝒟_{ψ}, J_{ψ}) on N^{X} and N^{K} with contact forms η_{φ}^{X} = η_{0}^{X} + d^{c}φ and η_{ψ}^{K} = η_{0}^{K} + d^{c}ψ respectively. Then there is a CR isomorphism Ψ_{φ}: (N^{X}, 𝒟_{φ}, J_{φ}) → (N^{K}, 𝒟_{ψ}, J_{ψ}) such that for any φ̇ ∈ C_{N^{X}}^{∞}(ℝ)^{𝕋},

ψ̇ := dΘ̂_{φ}(φ̇) = Ψ_{φ}^{∗}(\frac{φ̇}{η_{φ}^{X}(K)}) and η_{ψ}^{K} = Ψ_{φ}^{∗}(\frac{η_{φ}^{X}}{η_{φ}^{X}(K)}).

3. Extremal Sasaki metrics and weighted extremal Kähler metrics

3.1. Extremal Sasaki metrics. Let (N, 𝒟, J, X) be a Sasaki manifold. The Levi-Civita connection of (local) Sasaki–Reeb quotients by X pulls back to a connection ∇^{X} on 𝒟, preserving the transversal Kähler structure (g_{X}, J, ω_{X}), which turns out to be (see e.g. [22, §4]) the Tanaka–Webster connection [60] of (𝒟, J, X). Thus the scalar curvature of Sasaki– Reeb quotients pulls back to the Tanaka–Webster scalar curvature Scal(g_{X}) of ∇^{X}.

Definition 3.1. [12] (N, 𝒟, J, X) is a CSC or extremal Sasaki structure if Scal(g_{X}) is constant or is a transversal Killing potential, respectively. Equivalently any Sasaki–Reeb quotient with respect to X is a CSC or extremal Kähler metric, respectively.

The Kähler geometry on Sasaki–Reeb quotients with respect to X ∈ 𝔠𝔯_{+}(N, 𝒟, J), induced by an extremal Sasaki structure (N, 𝒟, J, K) with respect to a possibly different Sasaki–Reeb vector field K ∈ 𝔠𝔯_{+}(N, 𝒟, J)^{X}, was studied in [1].

Lemma 3.2. [1] Let (M, J, g, ω) be the Kähler orbifold corresponding to the Sasaki–Reeb quotient of (N, 𝒟, J) with respect to a quasiregular X ∈ 𝔠𝔯_{+}(N, 𝒟, J), and f = η_{𝒟}^{X}(K) > 0 the induced positive Killing potential on (M, J, ω) by K ∈ 𝔠𝔯_{+}(N, 𝒟, J)^{X} via Lemma 1.4. Then (𝒟, J, K) is an extremal Sasaki structure on N if and only if the smooth function

(21)  Scal_{f}(g) := f^{2}Scal(g) − 2(m + 1)fΔ_{g}f − (m + 2)(m + 1))|df|_{g}^{2}

is a Killing potential on (M, J, ω).

When K = X (i.e. f = 1) we recover Definition 3.1. More generally, Lemma 3.2 allows us to reduce the study of extremal Sasaki metrics (with respect to a possibly irregular Sasaki–Reeb field K) in terms of a fixed Kähler manifold or orbifold M, obtained from a fixed regular or quasiregular Sasaki–Reeb field X ∈ 𝔠𝔯_{+}(N, 𝒟, J)^{K}. This is the point of view we take in this article.

Definition 3.3. Let (g, ω) be a Kähler metric on (M, J) and f be a Killing potential for g. We say that g is f-extremal if its f-scalar curvature Scal_{f}(g), given by (21), is also a Killing potential.

Lemma 3.2 motivates the following definition.

Definition 3.4. [1] Let (N, 𝒟, J, X) be a Sasaki manifold, K ∈ 𝔠𝔯_{+}(N, 𝒟, J)^{X} and f = η_{𝒟}^{X}(K) > 0. The K-scalar curvature Scal_{K}(g_{X}) of (N, 𝒟, J, X) is the function induced on N by (21) on any Sasaki–Reeb quotient by X. We say X is K-extremal if Scal_{K}(g_{X}) is a transversal Killing potential of (g_{X}, J, ω_{X}).

Thus Lemma 3.2 asserts that for K ∈ 𝔠𝔯_{+}(N, 𝒟, J)^{X}, X is K-extremal if and only if (𝒟, J, K) is extremal as in Definition 3.1. In particular, if K = λX with λ > 0 constant, K-extremality reduces to extremality.

3.2. The Calabi problem for f-extremal Kähler metrics. As observed in [5, 28, 38], many features of the theory of extremal Kähler metrics extend naturally to the f-extremal case. In particular, one can formulate a weighted version of the Calabi problem [15] which seeks an f-extremal Kähler metric (g, ω) with ω ∈ ζ, a Kähler class. In this setting, we pin down the function f indirectly by fixing first a quasiperiodic holomorphic vector field Ǩ in the Lie algebra 𝔱̌ of a (compact) torus 𝕋̌ ⩽ Aut_{r}(M, J) inside the reduced group of automorphisms of (M, J) (see e.g. [31]), and secondly, a real constant κ > 0 such that for any 𝕋̌-invariant Kähler metric (g, ω) with ω ∈ ζ, the Killing potential f of Ǩ with respect to g, normalized by ∫_{M} fω^{m}/m! = κ, is positive on M, see [5, Lemma 1].

Problem 3.5. Is there a 𝕋̌-invariant Kähler metric (g, ω) on (M, J) with ω ∈ ζ, which is f-extremal, where f is the Kähler potential of Ǩ with respect to g determined by ∫_{M} fω^{m}/m! = κ? We refer to such metrics as (Ǩ, κ)-extremal.

As observed in [40], if (g_{0}, ω_{0}) and (g, ω) are two 𝕋̌-invariant Kähler metrics in ζ with Kähler forms ω = ω_{0} + dd^{c}φ for a 𝕋̌-invariant smooth function φ on M, then the corresponding κ-normalized Killing potentials f and f_{0} of Ǩ are related by

(22)  f = f_{0} + d^{c}φ(Ǩ).

Another useful observation from [28, 38, 40] is that when M is compact, any (Ǩ, κ)- extremal metric on (M, J) is invariant under a maximal torus in Aut_{r}(M, J), so we can assume without loss of generality that 𝕋̌ itself is a maximal torus inside Aut_{r}(M, J).

3.3. The Calabi problem for K-extremal Sasaki metrics. In the case when (M, J, ω_{0}) is obtained as a Sasaki–Reeb quotient of a compact regular 𝕋-invariant Sasaki manifold (N, 𝒟_{0}, J, X) with Sasaki–Reeb vector field X ∈ 𝔱_{+} and corresponding contact 1-form η_{0} := η_{𝒟_{0}}^{X}, any other vector field K ∈ 𝔱_{+} pins down a positive Killing potential f_{0} = η_{0}(K) for the induced Killing vector field

Ǩ := K + ⟨X⟩ ∈ Lie(𝕋̌) = 𝔱/ℝ·X

on (M, J, ω_{0}), where 𝕋̌ = 𝕋/𝕊_{X}^{1} is the induced isometric torus action on M with 𝕊_{X}^{1} ⩽ 𝕋 being the 𝕊^{1}-action induced by X. By Lemma 1.17, any other compatible Sasaki structure η = η_{0} + α + d^{c}φ ∈ S(X, J^{X})^{𝕋} for α ∈ Ω_{X,cl}^{1}(N)^{𝕋} and φ ∈ Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋}—see (2)—gives rise to a Killing potential f := η^{X}(K) of Ǩ with respect to the induced Kähler structure ω_{φ} = ω_{0} +dd^{c}φ on (M, J), and f and f_{0} are related by (9), i.e., satisfy (22). It thus follows that the Killing potentials f and f_{0} are κ-normalized with

κ := \frac{1}{(2π)m!} ∫_{N} η^{X}(K)η^{X} ∧ (dη^{X})^{m}.

In particular, κ does not depend on η^{X} ∈ S(X, J^{X})^{𝕋}, a fact that can be also deduced from the 𝕋-equivariant Gray–Moser argument, see Remark 1.8. This is consistent with Lemma 1.15 in which K is seen as a positive affine-linear function on the natural moment polytope P_{X} ⊆ 𝔱̌^{∗} associated to (N, 𝒟_{0}, J, X).

Equivalently, we can think in terms of the induced polarization L of (M, J), as in Example 2.3. In this situation, the action of 𝕋 on the cone (V, J) corresponds to 𝕋 ⩽ Aut(M, L), and 𝕋̌ = 𝕋/𝕊^{1} is the induced action on (M, J); furthermore, for a fixed K ∈ 𝔱, and any 𝕋-invariant hermitian product h on L whose curvature form is a 𝕋̌-invariant Kähler metric ω ∈ ζ = 2πc_{1}(L), there is a naturally defined Killing potential f of Ǩ. Furthermore, for any other 𝕋-invariant hermitian product e^{φ}h on L, the corresponding Killing potential for Ǩ with respect to ω_{φ} ∈ ζ is given by (22).

Definition 3.6. Since K ∈ 𝔱 determines Ǩ and κ in both the regular Sasaki and polarized cases, we shall also refer to a (Ǩ, κ)-extremal Kähler metric (g_{φ}, ω_{φ}) on (M, J) in the Kähler class ζ = 2πc_{1}(L) as K-extremal.

Thus K-extremal metrics on (M, J, ζ) correspond to K-extremal Sasaki structures on (N, X, J^{X}). We can sharpen this observation as follows.

Proposition 3.7. Let (N, 𝒟_{0}, J) be a compact CR manifold of Sasaki type and X, K ∈ 𝔠𝔯_{+}(N, 𝒟_{0}, J) with [X, K] = 0. Suppose X is quasiregular with Sasaki–Reeb quotient the compact Kähler orbifold (M, J, ω). Then there is an extremal Sasaki structure in S(K, J^{K}) if and only if there is a K-extremal Kähler metric in the Kähler class [ω].

Proof. Let 𝕋 ⩽ Aut(N, 𝒟_{0}, J) ⩽ Aut(N, X, J^{X}) be the (compact) torus generated by X and K. Then there is K-extremal Kähler metric in the Kähler class [ω] if and only if there is a K-extremal Sasaki structure η^{X} in S(X, J^{X})^{𝕋}. Since K-extremality of η^{X} depends only on the transversal geometry, we may translate η^{X} by any closed basic 1-form α, and hence this holds if and only if there is a K-extremal Sasaki structure in Ξ(X, J^{X}, η_{𝒟_{0}}^{X})^{𝕋}. By Lemma 3.2 and the bijection Θ̂ of (19), this holds if and only if there is an extremal Sasaki structure in Ξ(K, J^{K}, η_{𝒟_{0}}^{K})^{𝕋}.

Hence it only remains to show that if there is an extremal Sasaki structure η in S(K, J^{K}), then there is also one in Ξ(K, J^{K}, η_{𝒟_{0}}^{K})^{𝕋}. Since (again) the extremality condition depends only on the transversal geometry, we may assume η ∈ Ξ(K, J^{K}, η_{𝒟_{0}}^{K}). Now by [12, Theorem 4.8] η is invariant under a maximal torus 𝕋_{max} ⩽ Aut(N, 𝒟, J)^{K} ⩽ Aut(V, J)^{K} (where 𝒟 = ker η). Note that 𝕋_{max} is also maximal in Aut(V, J)^{K}: any torus containing 𝕋_{max} defines an abelian subalgebra of 𝔞𝔲𝔱(V, J)^{K}, which descends to an abelian subalgebra of 𝔞𝔲𝔱(V, J)^{K}/<K, JK> ⩽ 𝔥_{0}(N, K, J^{K}), and this cannot be larger than the Lie algebra of 𝕋_{max} by [12]. Since 𝔥_{0}(N, K, J^{K}) is a finite dimensional Lie algebra, it follows that Aut(V, J)^{K} is a finite dimensional complex Lie group. Hence 𝕋_{max} is conjugate to a maximal torus containing 𝕋, and apply the conjugating automorphism to η, we obtain an extremal Sasaki structure in Ξ(K, J^{K}, η_{𝒟_{0}}^{K})^{𝕋}. □

4. Properness of the weighted Mabuchi energy

In this section, we consider a compact Sasaki manifold (N, 𝒟, J, K), with corresponding contact form η_{0}^{K}, and fix a maximal compact torus 𝕋 ⩽ Aut(N, 𝒟, J) such that K ∈ 𝔱. By [12, Theorem 4.8] (see the proof of Proposition 3.7), the search for an extremal Sasaki structure in S(K, J^{K}) can be reduced to the search of such structures in the space Ξ(K, J^{K}, η_{0}^{K})^{𝕋} of η_{0}^{K}-normalized, 𝕋-invariant Sasaki structures, see Definition 1.11. We denote by K_{ext} ∈ 𝔱 the extremal vector field associated to (K, 𝕋), see Definition B.1 and [35], which is well-defined irrespective of the existence of an extremal Sasaki structures in Ξ(K, J^{K}, η_{0}^{K})^{𝕋}, but which will coincide with the vector field defined by the transversal scalar curvature of any extremal Sasaki structure in that space, should it exist.

We fix a quasiregular Sasaki–Reeb vector field X ∈ 𝔱_{+} in the Reeb cone of (N, K, J^{K}, 𝕋). We consider the induced action of 𝕋 and of its complexification 𝔾 = 𝕋_{ℂ} on the polarized complex cone (V, J, K) (see Lemma 2.10) and identify the polarized complex cone of (N, 𝒟, J, X) with (V, J, X), see Section 2. The important feature of this setting is that 𝔾 naturally acts on the spaces R_{X}(V, J)^{𝕋} and R_{K}(V, J)^{𝕋} of 𝕋-invariant cone potentials of (V, J, X) and (V, J, K), respectively. Using the basepoints r_{0}^{X} ∈ R_{X}(V, J)^{𝕋} and r_{0}^{K} ∈ R_{K}(V, J)^{𝕋} corresponding to the contact forms η_{0}^{X}, η_{0}^{K}, there is an induced action of 𝔾 on the corresponding spaces Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} and Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} of 𝕋-invariant Sasaki potentials.

Definition 4.1. (see e.g. [34, 35]) The (relative) Mabuchi energy is the functional

M^{K}: Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} → ℝ

characterized by

(dM^{K})_{ψ}(ψ̇) = ∫_{N} ψ̇(Scal(g_{ψ}) − η_{ψ}(K_{ext})) η_{ψ} ∧ dη_{ψ}^{m},
M^{K}(0) = 0,

where for any ψ ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}, η_{ψ} := η_{0}^{K} + d_{K}^{c}ψ stands for the corresponding contact form in Ξ(K, J^{K}, η_{0}^{K})^{𝕋} whereas Scal(g_{ψ}) denotes the corresponding transversal scalar curvature.

Definition 4.2. The K-twisted (relative) Mabuchi energy is the functional

M_{K}^{X}: Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} → ℝ

characterized by

(dM_{K}^{X})_{φ}(φ̇) = ∫_{N} φ̇(Scal_{K}(g_{φ}) − η_{φ}(K_{ext})) (η_{φ}(K))^{−m−3}η_{φ} ∧ dη_{φ}^{m},
M_{K}^{X}(0) = 0.

Both functionals are invariant under additive constants, i.e., they descend to the spaces Ξ(K, J^{K}, η_{0}^{K})^{𝕋} and Ξ(X, J^{X}, η_{0}^{X})^{𝕋}, respectively.

Remark 4.3. As X ∈ 𝔱_{+} is quasiregular, M_{K}^{X} gives rise to a functional M_{Ǩ,κ} acting on the space of 𝕋̌-invariant (where we recall 𝕋̌ = 𝕋/𝕊_{X}^{1}) ω_{0}-relative Kähler potentials on the Sasaki–Reeb quotient (M, J, ω_{0}), defined by

(dM_{Ǩ,κ})_{φ}(φ̇) = (2π) ∫_{M} φ̇(Scal_{f_{φ}}(ω_{φ}))^{⊥}f_{φ}^{−m−3}ω_{φ}^{m}, M_{Ǩ,κ}(0) = 0,

where ω_{φ} = ω_{0} + dd^{c}φ is the Kähler metric corresponding to φ, f_{φ} is the κ-normalized Killing potential of Ǩ with respect to ω_{φ}, and (Scal_{f_{φ}}(ω_{φ}))^{⊥} denotes the orthogonal projection of the corresponding f_{φ}-scalar curvature to the orthogonal complement of the space of ω_{φ}-Killing potentials of elements of Lie(𝕋̌), with respect to global L^{2}-product (ϕ_{1}, ϕ_{2})_{f_{φ}} = ∫_{M} ϕ_{1}ϕ_{2}f_{φ}^{−m−3}ω_{φ}^{m}. This is, up to the multiplicative factor 2π, the relative weighted Mabuchi energy introduced in [39]. Equivalently, in terms of the momentum polytope P_{X} ⊆ 𝔱̌^{∗} of (M, ω_{0}, 𝕋̌), M_{K}^{X} is 2π times the Mabuchi functional M_{v,w} defined in [40], where the weights v(x) = (ℓ_{K}(x))^{−1−m} and w(x) = ℓ_{ext}(x)ℓ_{K}(x)^{−3−m} are determined by the affine-linear functions ℓ_{K}(x) and ℓ_{ext}(x) on 𝔱̌^{∗}, corresponding to K, K_{ext} ∈ 𝔱, see Lemma 1.15.

Lemma 4.4. Let Θ̂: Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} → Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} be the bijection defined by (19); then M^{K} ∘ Θ̂ = M_{K}^{X}.

Proof. We check M^{K} ∘ Θ̂ satisfies the characterization of M_{K}^{X}. Clearly M^{K}(Θ̂(0)) = M^{K}(0) = 0 and if φ ∈ Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} with ψ = Θ̂(φ) then d(M^{K} ∘Θ̂)_{φ} = (dM^{K})_{ψ}∘(dΘ̂)_{φ}. We now apply Corollary 2.18. As Ψ_{φ} is a CR diffeomorphism between the CR structures induced by φ and ψ, (46) implies that Ψ_{φ}^{∗}(\frac{Scal_{K}(g_{φ})}{η_{φ}(K)}) = Scal(g_{ψ}), and the result follows. □

We now recall the definition of the d_{1}-distance on the space Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}, originally due to Darvas [20] in the Kähler case, and extended by He–Li [35] to the Sasaki case.

Definition 4.5. For ψ_{0}, ψ_{1} ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}, we let

d_{1}(ψ_{0}, ψ_{1}) := inf_{ψ_{t}} { ∫_{0}^{1} ( ∫_{N} |ψ̇_{t}|η_{ψ_{t}} ∧ dη_{ψ_{t}}^{m})dt},

where the infimum is taken over all smooth paths ψ_{t}, t ∈ [0, 1], connecting ψ_{0} and ψ_{1} inside Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}.

The fact that d_{1} is a distance is not obvious, and is one of the main results of [35]. We define the d_{1}-distance on the space Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} by the same formula. When X is regular, d_{1} is, up to a factor 2π, the distance defined in [20] on the space of relative Kähler potentials on (M, J). We now show that Θ̂ is bilipschitz with respect to d_{1}.

Lemma 4.6. There exists Λ ∈ ℝ^{+}, depending only on (N, 𝒟, J, K, X), such that

\frac{1}{Λ}d_{1}(φ_{0}, φ_{1}) ≤ d_{1}(Θ̂(φ_{0}), Θ̂(φ_{1})) ≤ Λd_{1}(φ_{0}, φ_{1}).

Proof. Let φ_{t} be a smooth path connecting φ_{0} and φ_{1} in Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}. By Corollary 2.18, we have

∫_{0}^{1} ( ∫_{N} |ψ̇_{t}|η_{ψ_{t}} ∧ dη_{ψ_{t}}^{m})dt = ∫_{0}^{1} ( ∫_{N} \frac{|φ̇_{t}|}{(η_{φ_{t}}(K))^{m+2}} η_{φ_{t}} ∧ dη_{φ_{t}}^{m})dt

The key point is that the positive functions f_{t} = η_{φ_{t}}(K) define κ-normalized Kähler potentials on M with respect to the Kähler metrics ω_{φ_{t}}, i.e., they are related by the formula (22). It thus follows (see e.g. [40, Lemma 1] for a general statement) that f_{t}(N) = [a, b] is a fixed positive interval (independent of φ_{0}, φ_{1} and φ_{t}), i.e., we have the uniform estimates

\frac{1}{b^{m+2}} d_{1}(φ_{0}, φ_{1}) ≤ d_{1}(ψ_{0}, ψ_{1}) ≤ \frac{1}{a^{m+2}} d_{1}(φ_{0}, φ_{1}).  □

We next introduce a suitable notion of properness of M^{K} (resp. M_{K}^{X}), analogous to the corresponding notions in Kähler geometry, see [21, 54, 62]. To this end, we consider the action of 𝔾 := 𝕋_{ℂ} on the spaces Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} and Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} as explained in the beginning of the section.

Lemma 4.7. 𝔾 acts by d_{1} isometries on Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} and Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}.

Proof. For any smooth curve r(t) of cone potentials on (V, J, X), and any g ∈ 𝔾, let (g^{∗}r)(t) = r(t) ∘ g be the corresponding 𝔾 action on curves. We compute the d_{1}-lengths of r(t) and (g^{∗}r)(t):

∫_{0}^{1} ∫_{N} |\frac{d(g^{∗}r)(t)}{dt}|(η_{(g^{∗}r)(t)} ∧ (dη_{(g^{∗}r)(t)})^{m})dt = ∫_{0}^{1} ∫_{N} |g^{∗}ṙ(t)|g^{∗}(η_{r(t)} ∧ (dη_{r(t)})^{m})dt

= ∫_{0}^{1} ∫_{N} |ṙ(t)|η_{r(t)}(dη_{r(t)})^{m}dt. □

Lemma 4.8. Θ̂: Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} → Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} is 𝔾-equivariant.

Proof. Obvious from the definition of Θ̂. □

The following crucial result is a refinement (in the regular case) of the uniqueness result of [57] (which in turn is a Sasaki version of [7]).

Lemma 4.9. Suppose (N, 𝒟, J, 𝕋, X, K) is as above with X regular. Then any two extremal Sasaki structures ψ, ψ̃ ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} are isometric under the action of 𝔾.

Proof. Using Proposition 2.15 and the fact that Θ̂ is 𝔾-equivariant (Lemma 4.8), it is enough to establish the result for the corresponding K-extremal Sasaki structures in S(X, J^{X})^{𝕋}. As X is regular, it is enough to show that the induced 𝕋̌-invariant (K, κ)- extremal Kähler metrics ω and ω̃ on (M, J, ζ, 𝕋̌ = 𝕋/𝕊_{X}^{1}) are isometric by an element of 𝕋̌_{ℂ}. Indeed, as 𝕋̌ is a maximal torus in the reduced group of automorphisms Aut_{r}(M, J), and the induced extremal vector field Ǩ_{ext} := J grad_{g} Scal_{f}(g) is central in the Lie algebra of Killing fields of (M, J, g, ω), we must have Ǩ_{ext} ∈ 𝔱̌; it then follows by [40, Theorem B1] (see also [28, 38]) that the group Aut𝕋̌ ; r(M, J) of reduced automorphisms of (M, J), commuting with the 𝕋̌-action, is a reductive Lie group. As 𝕋̌ is simultaneously central and maximal, it follows that the identity component of Aut𝕋̌ ; r(M, J) is abelian and equal to 𝕋̌_{ℂ}.

The result now follows from the uniqueness result in [41], established in the more general context of (v, w)-extremal Kähler metrics and building on an argument in [16], which implies that any two 𝕋̌-invariant (Ǩ, κ)-extremal Kähler metrics in ζ are isometric by an element in the connected component of the identity of the group Aut𝕋̌ ; r(M, J). □

With this understood, for any ψ_{0}, ψ_{1} ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}, we let

d_{1}^{𝔾}(ψ_{0}, ψ_{1}) := inf_{g_{0},g_{1}∈𝔾}{d_{1}(g_{0} · ψ_{0}, g_{1} · ψ_{1})} = inf_{g∈𝔾}{d_{1}(ψ_{0}, g · ψ_{1})}.

and define similarly d_{1}^{𝔾}(φ_{0}, φ_{1}) for φ_{0}, φ_{1} ∈ Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋}.

Definition 4.10. M^{K} (resp. M_{K}^{X}) is 𝔾-proper if there exist constants Λ > 0 and C, such that for any ψ ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} we have M^{K}(ψ) ≥ Λd_{1}^{𝔾}(0, ψ) − C (resp. for any φ ∈ Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} we have M_{K}^{X}(φ) ≥ Λd_{1}^{𝔾}(0, φ) − C).

As Θ̂(0) = 0 by the definition of Θ̂, Lemmas 4.4 and 4.6 yield

Corollary 4.11. M^{K} is 𝔾-proper if and only if M_{K}^{X} is 𝔾-proper.

The main result in this section extends one implication in [34, Theorem 1] and [35, Theorem 5.1].

Theorem 4.12. Let (N, 𝒟, J) be a compact CR manifold of Sasaki type, 𝕋 ⩽ Aut(N, 𝒟, J) a maximal torus, and X, K ∈ 𝔱_{+} with X regular. If there exists an extremal Sasaki structure η_{ψ} associated to ψ ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}, then M^{K} is 𝔾-proper.

Proof. One can directly check that in the setting above, all the conditions of the general existence/properness principle of [21, Section 3] hold true with

(R, d, F, G) = (Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋}, d_{1}, M^{K}, 𝔾).

Indeed, for showing the conditions (A1)–(A4) of [21], we notice that the d_{1} completion of Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} and the extension of M^{K} is defined and proved in [35] whereas the fact that the 𝔾-action is isometric is established in Lemma 4.7 above. For proving the conditions (P1)–(P7) of [21], we notice that: (P1) follows from [35, Theorem 6.7] and the fact that the (relative) Mabuchi energy M^{K} differs from the functional K in [35] by an affine-linear function on bounded geodesics, see [9, Prop. 10] or [41] for a direct argument with respect to M_{K}^{X} on the regular quotient; (P2) follows from [35, Theorem 6.6]; (P3) is the regularity result [34, Theorem 5.4] which can be extended from the CSC to the extremal case by the argument in [33]. (P4) and (P5) are established in Lemmas 4.7 and 4.9 above. (P6) follows from the general principle in [23, Section 6] (see in particular [23, Prop. 6.8]) whereas (P7) is automatically satisfied as it can be checked easily, see also [35, Prop. 5.1]. □

Remark 4.13. An alternative way to establish Theorem 4.12 is to compare directly the notion of 𝔾-properness of M^{K} with the properness notion used in [34, Theorem 1] and [35, Theorem 5.1]. W. He considers in [34] the identity component G := Aut_{0}(N, K, J^{K}) of the group of diffeomorphisms of N preserving K and the transversal holomorphic structure J^{K}, which naturally acts on the space S(K, J^{K}) of compatible contact forms by pullback. There is an induced action of G on the quotient space S(K, J^{K})/Ω_{X,cl}^{1}(N) and hence, by using the base point η_{0}^{K}, on the marking Ξ(K, J^{K}, η_{0}^{K}). As G does not act on the space of η_{0}^{K}-relative Sasaki potentials Ξ̂(K, J^{K}, η_{0}^{K}), a slice Ξ̂_{0}(K, J^{K}, η_{0}^{K}) := (I^{K})^{−1}(0) is introduced, where I^{K} : Ξ̂(K, J^{K}, η_{0}^{K}) → ℝ is a Sasaki version of the Aubin–Mabuchi functional defined by

\begin{aligned}(dI^{K})_{ψ}(ψ̇) & = ∫_{N} ψ̇ η_{ψ} ∧ dη_{ψ}^{m}, \\ I^{K}(0) & = 0. \end{aligned}

Unlike M^{K}, I^{K} is not invariant but is equivariant under additive real constants and thus determines a unique representative ψ_{0} ∈ Ξ̂_{0}(K, J^{K}, η_{0}^{K}) of η_{ψ} ∈ Ξ(K, J^{K}, η_{0}^{K}). This leads to an action of G on the slice Ξ̂_{0}(K, J^{K}, η_{0}^{K}), which we denote by [g] · ψ_{0}, where [g] ∈ [G], the effective quotient of G. One direction of the statements in [34, Theorem 1] and [35, Theorem 5.1] then yields that if there exists an extremal Sasaki metric in S(K, J^{K})^{𝕋}, the Mabuchi energy M^{K} is proper on Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} with respect to [G], i.e., satisfies

(23)  M^{K}(ψ) ≥ Λd_{1}^{[G]}(0, ψ_{0}) − C,

where d_{1}^{[G]}(0, ψ_{0}) := inf_{g∈G}{d_{1}(0, [g] · ψ_{0})}.

Now 𝔾 also acts on S(K, J^{K})^{𝕋}, hence on Ξ̂_{0}(K, J^{K}, η_{0}^{K}), and we let [𝔾] denote the effective quotient. Although 𝔾 is not a subgroup of G, [𝔾] is a subgroup of [G] and the 𝔾-orbits of induced Sasaki structures are inside the [G]-orbits on Ξ̂_{0}(K, J^{K}, η_{0}^{K}). On the other hand, the action of 𝔾 on Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} includes translations by constants, so we conclude that for any ψ ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋} with projection ψ_{0} to Ξ̂_{0}(K, J^{K}, η_{0}^{K}) we have

d_{1}^{[𝔾]}(0, ψ_{0}) := inf_{g∈𝔾}{d_{1}(0, [g] · ψ_{0})} ≥ d_{1}^{[G]}(0, ψ_{0}), d_{1}^{𝔾}(0, ψ) = d_{1}^{𝔾}(0, ψ_{0}) ≤ d_{1}^{[𝔾]}(0, ψ_{0}).

It thus follows that the conclusion of Theorem 4.12 will hold true if one can improve (23) by replacing G with [𝔾] for functions ψ ∈ Ξ̂(K, J^{K}, η_{0}^{K})^{𝕋}. An inspection of the arguments in [34] (which in turn are an application of the general principle of [21]) reveals that the only missing ingredient to do so is proving that the [𝔾]-action is transitive on the space of smooth extremal Sasaki potentials ψ_{0} ∈ Ξ̂_{0}(K, J^{K}, η_{0}^{K})^{𝕋}. This property is established in Lemma 4.9 above.

Note that this argument yields the stronger notion of d_{1}^{[𝔾]}-properness of M^{K}, should an extremal Sasaki metric in S(K, J^{K}) exist. However, as the map Θ̂ does not respect the corresponding slices Ξ̂_{0}(K, J^{K}, η_{0}^{K})^{𝕋} = (I^{K})^{−1}(0) and Ξ̂_{0}(X, J^{X}, η_{0}^{X})^{𝕋} = (I^{X})^{−1}(0), it is not immediately clear whether or not the energy M_{K}^{X} is also d_{1}^{[𝔾]}-proper. The somewhat weaker but more natural notion of d_{1}^{𝔾}-properness we use above remedies this, see Corollary 4.11.

5. Weighted K-stability of regular Sasaki–Reeb quotients

In this section we relate the existence of extremal Sasaki structures on a compact Sasaki manifold of regular type with a special case of the weighted K-stability of a polarized variety, defined in [40]. We thus assume that (N, 𝒟, J) is a compact CR manifold of Sasaki type, 𝕋 ⩽ Aut(N, 𝒟, J) a torus, and X ∈ 𝔱_{+} a regular Sasaki–Reeb vector field. The Sasaki–Reeb quotient (M, J, L) is a smooth projective polarized variety, as in Example 1.3. We denote by 𝕋 the induced torus action on L, and by 𝕋̌ = 𝕋/𝕊_{X}^{1} the torus action on (M, J).

5.1. Polarized test configurations. We first recall the notion of a (compactified) polarized test configuration associated to (M, J, L, 𝕋), going back to [24, 55].

Definition 5.1. A 𝕋-equivariant polarized test configuration of exponent s ∈ ℕ associated to (M, J, L, 𝕋) is a normal polarized variety (ℳ , ℒ) endowed an action 𝕋 ⩽ Aut(ℳ , ℒ), and

• a (flat) surjective morphism π: ℳ → ℂP^{1} such that 𝕋̌ preserves each fibre M_{t} := π^{−1}(t) and, for t ≠ 0, (M_{t}, L_{t} := ℒ_{|M_{t}}) is 𝕋-equivariantly isomorphic, as a polarized variety, to (M, L, 𝕋);

• a ℂ^{×}-action ρ on (ℳ , ℒ) commuting with 𝕋 and covering the usual ℂ^{×}-action on ℂP^{1};

• an isomorphism of polarized varieties

λ: (M × (ℂP^{1} \ {0}), L^{s} ⊗ O_{ℂP^{1}}(s)) ≅ (ℳ \ M_{0}, ℒ),

which is equivariant with respect to the actions of 𝕋 × 𝕊_{0}^{1} on (M × (ℂP^{1} \ {0}), L^{s} ⊗ O_{ℂP^{1}}(s)) and 𝕋 × 𝕊_{ρ}^{1} on (ℳ \ M_{0}, ℒ ), where 𝕊_{0}^{1} stands for the standard 𝕊^{1}-action on O(1) → ℂP^{1} and 𝕊_{ρ}^{1} is the 𝕊^{1}-action induced by ρ.

The test configuration (ℳ , ℒ) as above is called product if (ℳ \M_{∞}, π) is 𝕋̌-equivariantly isomorphic (as morphism of complex varieties) to (M × (ℂP^{1} \ {∞}), π_{ℂP^{1}}).

Remark 5.2. Polarized test configurations are often introduced (see e.g. [48, 59]) only by the corresponding requirements for the induced torus actions on ℳ, M and ℂP^{1}, i.e., are independent of the chosen lifts of these actions to ℒ and L. In particular, as O(1) is trivial over ℂP^{1} \ {0}, the third condition reduces to the existence of a biholomorphism λ̌: M × (ℂP^{1} \ {0}) → ℳ \ M_{0} which is equivariant with respect to the induced actions

of 𝕋̌ × 𝕊̌_{0}^{1} on M × (ℂP^{1} \ {0}) and 𝕋̌ × 𝕊1 ; ρ̌ on ℳ \ M_{0}. It is not hard to see that given such a biholomorphism λ̌ (and assuming L and ℒ are very ample) one can always lift ρ̌, 𝕋̌ and λ̌ to obtain a polarized test configuration in the sense of Definition 5.1.

We shall further assume that the 𝕋-equivariant polarized test configuration (ℳ , ℒ , π, ρ) for (M, J, L) as above has a smooth total space ℳ, and take s = 1 for simplicity. We consider the associated polarized complex cones (𝒱 , X) and (V, X), where 𝒱 = (ℒ^{∗})^{×}, V = (L^{∗})^{×} and X denotes the regular Sasaki–Reeb vector field corresponding to the fibrewise 𝕊^{1}-actions on ℒ and L (see Example 2.3). According to Lemma 6.5, the induced momentum polytopes for the 𝕋̌-actions on ℳ and M are equal to the transversal polytope P_{X} defined by (8). By Lemma 1.15, K ∈ 𝔱_{+} is determined by an affine-linear function ℓ_{K}(x) on 𝔱̌^{∗}, which is positive on P_{X}, whereas the extremal vector field K_{ext} gives rise to an affine-linear function ℓ_{ext}(x) on 𝔱̌^{∗}.

For a 𝕋̌-invariant Kähler metric Ω̃ ∈ 2πc_{1}(ℒ), we let mΩ̃ denote the 𝕋̌-momentum map sending ℳ to P_{X}, and set

ℱ_{K}^{ext}(ℳ , ℒ) := − ∫_{ℳ} (Scal_{f̃,m+2}(Ω̃) − ℓ_{ext}(mΩ̃))f̃^{−3−m} \frac{Ω̃^{m+1}}{(m + 1)!}

+ 2 ∫_{ℳ} f̃^{−1−m}(π^{∗}ω_{F S}) ∧ \frac{Ω̃^{m}}{m!} ,

(24)

where ω_{FS} is a Fubini–Study metric with Ric(ω_{FS}) = ω_{FS}, f̃ = ℓ_{K}(mΩ̃) is a Killing potential of the Killing vector field induced by K on ℳ, and we have set

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

as in [2].

Formula (24) is in fact the weighted Futaki invariant of Lahdilli [40] with weights v(x) = ℓ_{K}(x)^{−m−1} and w(x) = ℓ_{ext}(x)ℓ_{K}^{−m−3}(x); it follows that it is independent of the choice of Ω̃ ∈ 2πc_{1}(ℒ). We also notice that the contact Futaki invariant associated to (N, 𝒟, K) (see Definition B.1) vanishes if and only if K_{ext} = c_{K}K, i.e., ℓ_{ext}(x) = c_{K}ℓ_{K}(x), in which case we have ℱ_{K}^{ext}(ℳ , ℒ) = ℱ_{K}(ℳ , ℒ), where we have set

\begin{aligned}ℱ_{K}(ℳ , ℒ) := & − ∫_{ℳ} (Scal_{f̃,m+2}(Ω̃) − c_{K} f̃)f̃^{−3−m} \frac{Ω̃^{m+1}}{(m + 1)!} \\ & + 2 ∫_{ℳ} f̃^{−1−m}(π^{∗}ω_{F S}) ∧ \frac{Ω̃^{m}}{m!} . \end{aligned} ; (25)

Proposition 3.7 and [40, Theorem 2] then yield the following necessary condition for the existence of extremal Sasaki structures in S(K, J^{K}), which we refer to as relative Kweighted K-semistability with respect to 𝕋.

Theorem 5.3. Let (N, 𝒟, J, X, K) be as in Theorem 4.12. If there exists an extremal Sasaki structure in S(K, J^{K}), then we have ℱ_{K}^{ext}(ℳ , ℒ) ≥ 0 for any 𝕋-equivariant smooth polarized test configuration with reduced central fibre (ℳ , ℒ) associated to (M, J, L).

We can sharpen this to a positivity result as follows.

Lemma 5.4. Let (M, J, L) a smooth polarized complex manifold, 𝕋 be a maximal torus in Aut(M, L), and ω_{0} ∈ 2πc_{1}(L) a 𝕋̌-invariant Kähler metric, where 𝕋̌ ⩽ Aut_{r}(M, J) is the induced torus action on M. Suppose that M_{Ǩ,κ} is 𝔾-proper with respect to the action of 𝔾 = 𝕋_{ℂ} on 𝕋̌-invariant ω_{0}-relative Kähler potentials, induced by its natural action on hermitian metrics on L. Then, for any 𝕋-equivariant smooth polarized test configuration (ℳ , ℒ) associated to (M, J, L), which has reduced central fibre (M_{0}, L_{0}) and is not a product, we have

ℱ_{K}^{ext}(ℳ , ℒ) > 0.

Proof. The 𝔾-properness implies that M_{Ǩ,κ} is bounded from below on the space of 𝕋̌- invariant ω_{0}-relative Kähler potentials, so that by [40, Thm. 7], we have

ℱ ext ; Ǩ(ℳ , ℒ) ≥ 0.

We thus have to establish that if ℱ_{K}^{ext}(ℳ , ℒ) = 0 for a smooth polarized test configuration (ℳ , ℒ , 𝕋) with reduced central fibre associated to (M, J, L, 𝕋), then (ℳ , ℒ , 𝕋) must be a ℂ^{×}-equivariant compactification at infinity of the product ℂ × M. To this end, we use a slight modification of the arguments from [8, 52]: suppose that ℱ_{K}^{ext}(ℳ , ℒ) = 0 and denote by u_{t} the smooth ray of 𝕋̌-invariant ω_{0}-relative Kähler potentials on (M, J, ω, 𝕋̌) with u_{0} = 0 (defined by the ℂ^{×}-action on 𝕋-invariant hermitian products on ℒ → ℳ on the test configuration (ℳ , ℒ , 𝕋)). Using that

0 = ℱ_{K}^{ext}(ℳ , ℒ) = lim_{t→+∞} \frac{M_{Ǩ,κ}(u_{t})}{t}

(see again [40, Thm. 7]) and the 𝔾-properness of M_{Ǩ,κ}, we conclude that

(26)  0 ≤ d_{1}^{𝔾}(0, u_{t}) ≤ o(t).

To any test configuration (ℳ , ℒ) as above, and a 𝕋̌-invariant ω_{0}-relative Kähler potential φ_{0}, Phong–Sturm [30] associate a C^{1,¯1} geodesic ray φ_{t} emanating from φ_{0}. A key estimate in [8] reads

|φ_{t} − u_{t}| ≤ C

which implies d_{1}(u_{t}, φ_{t}) ≤ C. As 𝔾 acts by d_{1}-isometries (Lemma 4.7) and using the triangle inequality, we have

d_{1}^{𝔾}(0, φ_{t}) − C ≤ d_{1}^{𝔾}(0, u_{t}).

and also

(27)  0 ≤ d_{1}^{𝔾}(φ_{0}, φ_{t}) ≤ o(t).

In our case, we can take φ_{0} = 0 (as we consider the space 𝕋-invariant hermitian metrics h = e^{−2φ}h_{0} on L with 𝕋 being a maximal torus in Aut(M, L)). In [21] and [8] the authors rather consider φ_{0} to be the normalized potential of a CSC metric in order to insure that it is fixed under a maximal compact subgroup of K ⩽ Aut_{r}(M, J), but we do not need this assumption as we apply [21, Prop. 6.8] with K := 𝕋 and G := 𝕋_{ℂ} = 𝔾 acting on the space 𝕋̌-invariant ω_{0}-relative Kähler potentials. One property needed for this result that we did not find readily available in the literature is that for a vector field Y ∈ 𝔱 on L, the flow Φ_{t}^{−JY} · φ generates a smooth geodesic in the space of 𝕋̌-invariant potentials. This is classically known to be true for the induced action [Φ_{t}^{−JY}]·φ on normalized relative Kähler potentials with respect to the Aubin–Mabuchi functional I, i.e., such that I([Φ_{t}^{−JY}]·φ) = 0

(see e.g. [31]), but the argument in [8, Lemma 4.2] actually shows that Φ_{t}^{−JY} · φ differs from [Φ_{t}^{−JY}] · φ by an affine linear function in t.

Now, the arguments in [8, Lemma 4.1] and [52, Prop. 4.10] (we have an o(t) bound for d_{1}^{𝔾}(φ_{0}, φ_{t}) as in [52]) extend tautologically in our situation, and show that φ_{t} is induced by the flow of a vector field JY on L with Y ∈ 𝔱. Finally, the fact that (ℳ , ℒ) is a product test configuration follows from [8, Prop. 4.3] or [52, Thm. A.6], noting that M_{Ǩ,κ} is 𝔾-invariant by its construction, see Remark 4.3. □

This lemma now leads to the following main result, which is an improvement of Theorem 5.3 to the following statement, which we call relative K-weighted K-stability with respect to 𝕋.

Theorem 5.5. Let (N, 𝒟, J, X) be a compact regular Sasaki manifold, 𝕋 a maximal torus in Aut(N, 𝒟, J), and K ∈ 𝔱_{+} a Sasaki–Reeb vector field such that there exists an extremal Sasaki structure in S(K, J^{K}). Denote by (M, J, L, 𝕋) the smooth polarized variety obtained as Sasaki–Reeb quotient with respect to X, with the induced action of 𝕋 on L. Then, for any 𝕋-equivariant smooth polarized test configuration (ℳ , ℒ) associated to (M, J, L), which has reduced central fibre (M_{0}, L_{0}) and is not a product, we have ℱ_{K}^{ext}(ℳ , ℒ) > 0.

Proof. By Theorem 4.12, the (modified) Mabuchi energy M^{K} is 𝔾-proper and by Corollary 4.11, the corresponding K-twisted modified Mabuchi energy M_{K}^{X} is also 𝔾-proper. Recall that 𝔾-action was introduced by the natural action of 𝕋_{ℂ} on the space of cone potentials R_{X}(V, J), Equivalently, in terms of the smooth polarized variety (M, J, L, 𝕋), the group 𝔾 = 𝕋_{ℂ} acts naturally on the space of 𝕋-invariant hermitian metrics h on L and, therefore, by considering the corresponding curvatures, on the space of 𝕋̌-invariant Kähler metrics ω ∈ 2πc_{1}(L). Furthermore, by introducing a basepoint h_{0} (with curvature ω_{0}), 𝔾 acts also on the space of 𝕋̌-invariant ω_{0}-relative Kähler potentials (which is naturally identified with the space Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} of 𝕋-invariant Sasaki potentials). As we recalled in Remark 4.3, the K-twisted Mabuchi energy M_{K}^{X} defined on the space Ξ̂(X, J^{X}, η_{0}^{X})^{𝕋} corresponds to the relative (Ǩ, κ)-Mabuchi energy M_{Ǩ,κ} acting on the space of 𝕋̌-invariant ω_{0}-relative Kähler potentials φ of Kähler metrics ω_{φ} = ω_{0} + dd^{c}φ ∈ 2πc_{1}(L).

By Theorem 4.12, M_{Ǩ,κ} is 𝔾-proper in the sense that for any 𝕋̌-invariant ω_{0}-relative Kähler potential φ

M_{K,κ}(φ) ≥ Λd_{1}^{𝔾}(0, φ) − C,

where we recall d_{1}^{𝔾} := inf_{g∈𝔾} d_{1}(0, g · φ). The result now follows from Lemma 5.4. □

Remark 5.6. One can extend the conclusions of Theorems 4.12 and 5.5 to smooth polarized test configurations with reduced central fibre, which are 𝕋-equivariant with respect to the torus 𝕋 ⩽ Aut(M, L) generated by the Sasaki–Reeb vector fields X, K, and the CR vector field K_{ext} of Definition B.1. To this end, one needs to consider instead of 𝔾 = 𝕋_{ℂ}, the action of the centralizer G_{𝕋} of 𝕋 in Aut(M, L) (which also acts on the spaces R_{X}(V, J) and R_{K}(V, J)), and use the uniqueness of 𝕋-invariant K-extremal metrics modulo G_{𝕋} (which is established [41]) in order to obtain the properness of M_{Ǩ,κ} with respect to G_{𝕋}.

5.2. Transcendental Kähler test configurations. A key feature of Lahdilli’s construction [40] is that it allows one to extend the conclusion of Theorem 5.3 to a more general class of smooth test configurations, the Kähler test configurations introduced in [23, 51], which are defined as follows

Definition 5.7. A 𝕋̌-equivariant Kähler test configuration associated to the Kähler manifold (M, J) with a fixed Kähler class ζ is a smooth Kähler manifold ℳ, endowed a Kähler class 𝒜 and a torus 𝕋̌ ⩽ Aut_{r}(ℳ) in its reduced group of automorphisms, and

• a surjective holomorphic map π: ℳ → ℂP^{1} such that 𝕋̌ preserves each fibre M_{t} := π^{−1}(t) and, for t ≠ 0, (M_{t}, ζ_{t} := 𝒜_{|M_{t}}) is 𝕋̌-equivariantly isomorphic to (M, ζ, 𝕋̌);

• a ℂ^{×}-action ρ commuting with 𝕋̌ and covering the usual ℂ^{×}-action ρ_{0} on ℂP^{1};

• a biholomorphism

λ: (M × (ℂP^{1} \ {0}), ζ) ≅ (ℳ \ M_{0}, 𝒜),

which is equivariant with respect to the actions of 𝕋̌×𝕊_{ρ_{0}}^{1} on M ×(ℂP^{1} \{0}) and 𝕋̌×𝕊_{ρ}^{1} on ℳ \ M_{0}.

Even in the case when ζ = 2πc_{1}(L) for some polarization L of (M, J), and ℳ is a smooth projective manifold, the Kähler class 𝒜 on ℳ need not be integral, thus allowing for a stronger K-stability condition. Indeed, following [40], if we have a smooth, 𝕋̌-equivariant test configuration (ℳ , 𝒜) associated to (M, J, L) as above, we can still use formula (24) in order to introduce the numerical invariant ℱ_{K}^{ext}(ℳ , 𝒜) (which again is independent of the choice of a 𝕋̌-invariant Kähler metric Ω̃ ∈ 𝒜). Furthermore, according to [40, Theorem 2], ℱ_{K}^{ext}(ℳ , 𝒜) ≥ 0 for any 𝕋̌-equivariant smooth Kähler test configuration with reduced central fibre (ℳ , 𝒜) of (M, J, 2πc_{1}(L)), should an extremal Sasaki structure exist in S(K, J^{K}), i.e., Theorem 5.3 holds true even for smooth Kähler test configurations with reduced central fibre associated to (M, J, 2πc_{1}(L), 𝕋̌).

We expect that the following will hold true.

Conjecture 5.8. In the setting of Theorem 5.5, for any 𝕋̌-equivariant smooth Kähler test configuration (ℳ , 𝒜) associated to (M, J, 2πc_{1}(L)), which has a reduced central fibre and is not a product, we have ℱ_{K}^{ext}(ℳ , 𝒜) > 0.

Remark 5.9. In the light of recent results in the (unweighted) Kähler setting [52], and the discussion at the end of Remark 4.13, the main missing point for establishing the above conjecture is the improvement of the d_{1}^{𝔾}-properness of M_{K}^{X} to the stronger notion of d_{1}^{[𝔾]}-properness.

5.3. Transcendental Kähler classes. We make here a few comments about the obstruction theory for the existence of a (Ǩ, κ)-extremal Kähler metric on a general compact Kähler manifold (M, J, ζ). When ζ = 2π(L) for an ample line bundle, and taking 𝕋̌ ⩽ Aut_{r}(M, J) be a maximal torus with Ǩ ∈ 𝔱̌, this is covered by Theorem 5.5 and Conjecture 5.8 (by using Proposition 3.7). Following [40], one can extend the weighted K-stability notion to the general Kähler case. To this end, one fixes a momentum polytope P ⊆ 𝔱̌^{∗} for the 𝕋̌-action on (M, J, ζ), which in turn gives rise to a positive affine-linear function ℓ(x) on P, such that the κ-normalized Killing potential f of Ǩ with respect to a 𝕋̌-invariant Kähler metric ω ∈ ζ is written as f = ℓ(m_{ω}), where the momentum map m_{ω} of (M, ω, 𝕋̌) is normalized by m_{ω}(M) = P. Furthermore, as shown in [40], the L^{2} projection Scal_{f}(g)^{ext} of the f-scalar curvature Scal_{f}(g) of ω to the space of Killing potentials with respect to ω of elements of 𝔱̌^{∗}, by using the weighted product (ϕ_{1}, ϕ_{2})_{f} = ∫_{M} ϕ_{1}ϕ_{2}f^{−m−3}ω^{m}/m! can be written as Scal_{f}(g)^{ext} = ℓ_{ext}(m_{ω}) for an affine-linear function on P, independent of ω ∈ ζ. For any smooth 𝕋̌-equivariant Kähler test configuration (ℳ , 𝒜) associated to (M, ζ, 𝕋̌), and 𝕋̌-invariant Kähler metric Ω̃ ∈ 𝒜, one can still define a numerical invariant via formula (24), where the momentum map mΩ̃ is normalized by the condition mΩ̃(ℳ) = P, and f̃ = ℓ(mΩ̃). This expression is independent of the 𝕋̌-invariant Kähler metric Ω̃ ∈ 𝒜 according to [40]; furthermore, it is not hard to see that the numerical invariant does not depend on the chosen momentum polytope P for (M, ζ, 𝕋̌): it merely depends upon the data (Ǩ, κ, 𝕋̌) so we shall denote it by ℱ_{(Ǩ,κ)}^{ext}(ℳ , 𝒜). In this more general Kähler setting, [41, Theorem 2] implies that if ζ admits a (Ǩ, κ)-extremal Kähler metric, then ℱ_{Ǩ,κ}^{ext}(ℳ , 𝒜) ≥ 0 for any smooth 𝕋̌-equivariant test configuration with reduced central fibre associated to (M, J, ζ, 𝕋̌), and we can extend Conjecture 5.8 to a relative (Ǩ, κ)-weighted K-stability statement.

6. Sasaki K-stability versus weighted K-stability

6.1. Sasaki K-stability. Given a complex polarized cone (V, J, K, 𝕋) as defined in Definition 2.2, it is known that (see e.g. the proof of [58, Theorem 3.1]) there exists a 𝕋equivariant holomorphic embedding V ↪ ℂ^{N} sending V biholomorphically to V̂\{0}, where V̂ ⊂ ℂ^{N} is a 𝕋_{ℂ}-homogeneous affine variety, and 𝕋_{ℂ} ⩽ GL(ℂ^{N}) is a diagonal torus whose real form is identified with 𝕋. When K is quasiregular, one can further identify the space ℋ of holomorphic functions on V introduced in §2.2 with the space of regular function functions on V̂ (see Remark 2.12), i.e., ℋ = ℂ[x_{1}, …, x_{N}]/I for a 𝕋_{ℂ}-homogeneous ideal I. Thus, in the decomposition (16), Γ ⊆ 𝔱^{∗} is the set of weights of the 𝕋_{ℂ}-action on the space of regular functions. Given a polarization K ∈ 𝔱_{+}, the pair (V̂, K) is referred to as a polarized affine cone.

In this latter setting, Collins and Székelyhidi [18] introduced the 𝕋-equivariant index character

F (K, t) := ∑_{α∈Γ} e^{−tα(K)}dim ℋ_{α}, Re(t) > 0,

and prove (cf. [18, Prop. 4.3]) that F (K, t) is a well-defined function in t, which has meromorphic extension to a neighbourhood of the origin, with Laurent expansion

F (K, t) = \frac{a_{0}(K)m!}{t^{m+1}} + \frac{a_{1}(K)(m − 1)!}{t^{m}} + O(t^{1−m}).

According to [18, Theorem 4.10], the coefficients a_{0}(K) > 0 and a_{1}(K) are smooth functions on 𝔱_{+}. With these ingredients, the following definition is proposed in [18].

Definition 6.1. Given a 𝕋_{ℂ}-homogeneous affine variety V̂ polarized by K ∈ 𝔱_{+}, and Y ∈ 𝔱, the K-Donaldson–Futaki invariant is defined to be the quantity

DF_{V̂ ,K}(Y) := \frac{a_{0}(K)}{m}D_{Y} (a_{1}(K)/a_{0}(K)) + \frac{a_{1}(K)D_{Y} a_{0}(K)}{m(m + 1)a_{0}(K)},

where D_{Y} denotes the derivative in the direction of Y on 𝔱.

If we assume, furthermore, that (V, J) is smooth and K is primitive in the lattice of circle subgroups of 𝕋, then the quotient of V by the subgroup ℂ_{K}^{×} generated by K is a polarized orbifold (M_{K}, L_{K}) whose generic isotropy groups are trivial, and for which the orbifold Riemann–Roch formula of [50, (2.17)] gives

(28)  dim_{ℂ} H^{0}(M_{K}, L_{K}^{k}) = a_{0}(K)k^{n} + (a_{1}(K) + ρ(K))k^{n−1} + ⋯

where ρ(K) is a periodic function in k with average 0, and, up to multiplicative positive dimensional constants, a_{0}(K) and a_{1}(K) are respectively the total volume and total scalar curvature of the polarized orbifold (M_{K}, L_{K}). It follows that, up to a positive dimensional constant, DF_{V̂ ,K}(Y) coincides with the Futaki invariant of the real holomorphic vector field induced by Y on the polarized orbifold (M_{K}, L_{K}), see [18, Props. 4.3 & 4.13 and Theorem 4.14].

Remark 6.2. In the notation of Appendix B, there is thus a dimensional constant c(m) > 0 such that

DF_{V̂ ,K}(Y) = c(m)Fut_{K}(Y),

where Fut_{K}(Y) denotes the contact Futaki invariant Fut_{K}(Y) introduced in Definition B.1. Indeed by Remark B.2, DF_{V̂ ,K}(Y) and Fut_{K}(Y) agree up to a positive dimensional constant when K is a quasiregular. However they are both continuous and homogeneous of order −m in K (where m + 1 is the complex dimension of V̂) and the quasiregular directions are dense in 𝔱_{+}.

Following [18], Definition 6.1 can be used to assign a numerical invariant to any 𝕋equivariant test configuration associated to the polarized affine cone (V̂, K), as follows. Let

w = (w_{1}, …, w_{N}), w_{i} ∈ ℤ

be a set of integers and denote by ρ_{w}: ℂ^{×} → GL(ℂ^{N}) the weighted ℂ^{×}-action defined by w. Taking the flat limit across 0 ∈ ℂ^{×} of the ℂ^{×} orbit of V̂ under the action of ρ_{w}, we obtain a flat family of affine schemes over ℂ, with central fibre C_{0}. Besides the 𝕋-action on V̂_{0}, the central fibre inherits the ℂ^{×}-action ρ_{w}, and we denote by Y_{w} the generator of the corresponding 𝕊^{1}-action on ℂ^{k} whose flow preserves V̂_{0}. It is shown in [18, Section 5] that for a fixed K ∈ 𝔱_{+}, Definition 6.1 can be applied on the scheme V̂_{0} and gives a necessary condition for the existence of a CSC Sasaki structures (known as K-semistability).

Theorem 6.3. [18] Let (N, 𝒟, J) be a compact CR manifold of Sasaki type, 𝕋 a torus in Aut(N, 𝒟, J), and K ∈ 𝔱_{+} a Sasaki–Reeb vector field such that N admits a CSC Sasaki structure in S(K, J^{K}). Then, for any 𝕋-equivariant test-configuration of the associated polarized affine cone (V̂, K) with central fibre V̂_{0}, DF_{V̂_{0},K}(Y_{w}) ≥ 0.

6.2. Test configurations of polarized complex cones. The Definition 5.1 of a 𝕋equivariant polarized test configuration of a regular Sasaki–Reeb quotient (M, J, L) provides a natural link to the theory of test configurations of affine cones described above. Indeed, given a 𝕋-equivariant polarized test configuration (ℳ , ℒ , π, ρ) for (M, J, L) as in Definition 5.1, and assuming that s = 1 and ℳ is smooth for simplicity, we consider the corresponding complex cones 𝒱 := (ℒ^{∗})^{×} and V := (L^{∗})^{×} and denote by K̃ and K the corresponding (regular) cone polarizations, see Examples 1.3 and 2.3. We thus get the following objects:

(i) a flat surjective morphism π: 𝒱 → ℂP^{1} such that 𝕋 preserves each fibre V_{t} := π^{−1}(t) and, for t ≠ 0, (V_{t}, K̃_{|_{V_{t}}}) is 𝕋-equivariantly isomorphic, as a polarized complex cone, to (V, K); (ii) a ℂ^{×}-action ρ on 𝒱 commuting with 𝕋 and covering the usual ℂ^{×}-action on ℂP^{1}; (iii) a biholomorphism

(29)  λ: V × (ℂP^{1} \ {0}) ≅ 𝒱 \ V_{0},

which is equivariant with respect to the actions of 𝕋 × 𝕊_{0}^{1} on V × (ℂP^{1} \ {0}) and 𝕋 × 𝕊_{ρ}^{1} on 𝒱 \ V_{0}, where 𝕊_{0}^{1} stands for the standard 𝕊^{1}-action on O(1) → ℂP^{1}.

This prompts the following

Definition 6.4. Let (V, J, K) be a smooth polarized complex cone endowed with a holomorphic action of a compact torus 𝕋, and K ∈ 𝔱_{+} a Sasaki–Reeb polarization. A smooth 𝕋-equivariant test configuration of (V, J, K) is a smooth polarized complex cone (𝒱 , 𝒥 , K̃), endowed with a 𝕋-action such that K̃ ∈ 𝔱̃^{+}, satisfying the conditions (i)–(iii) above.

Note that Definition 6.4 does not require K to be regular nor the existence of a regular Sasaki–Reeb vector field X commuting with K. To simplify the notation, we shall at times omit the complex structures J and 𝒥.

Lemma 6.5. Let (𝒱 , K̃) be a 𝕋-equivariant smooth test configuration of polarized cones associated to (V, K). Then, the reduced Sasaki–Reeb cone 𝔱_{+} of (V, K) coincides with the reduced Sasaki–Reeb cone 𝔱̃_{+} of (𝒱 , K̃). Moreover, K̃ = K ∈ 𝔱_{+}.

Proof. For t ≠ 0 we denote ι_{t}: V → V_{t} a 𝕋 × ℝ^{+}-equivariant biholomorphism given by the condition (i). By Lemma 1.15 and (7), the Reeb cone 𝔱̃_{+} ⊆ 𝔱 is independent of the choice of a compatible 𝕋-invariant cone potential r̃ ∈ RK̃(𝒱 )^{𝕋}. The same argument implies that the moment cone Σ_{t} associated to (V, K, ι_{t}^{∗}r̃) does not depend on t. We denote this cone by Σ. Using (17) we define the cone momentum maps μ^{t}: V → 𝔱^{∗} and μ̃: 𝒱 → 𝔱^{∗} associated respectively to the cones (V, K, ι_{t}^{∗}r̃) and (𝒱 , K̃, r̃). Since condition (i) ensures that orbits of 𝕋 lie in the fibre V_{t} of π we have, for p ∈ V_{t},

⟨μ_{p}^{t}, Y ⟩ = \frac{1}{4}ι_{t}^{∗}d^{c}(r̃^{2})(Y) = \frac{1}{4}d^{c}(r̃^{2})(Y) = ⟨μ̃_{p}, Y ⟩.

In particular, μ̃(𝒱 \V_{0}) ⊆ Σ and thus μ̃(𝒱 ) = Σ because they are both strictly convex polyhedral cones as recalled in §1.3. The relation (7) concludes the proof of the first claim whereas the condition (i) yields the second claim. □

Consequently, we will identify the Sasaki–Reeb vector fields K̃ and K and the reduced Sasaki–Reeb cones 𝔱_{+} and 𝔱̃_{+} in Definition 6.4 and in what follows.

Example 6.6. (Product test configuration of polarized cones) A particular class of smooth test configurations of polarized complex cones are the product test configurations. These are defined (similarly to Definition 5.1) by requiring that the holomorphic surjection (𝒱 \ V_{∞}, π) is 𝕋-equivariantly isomorphic to the product (V × (ℂP^{1} \ {∞}), π_{ℂP^{1}}). In the regular case, examples are obtained by taking the polarized complex cones over product polarized test configurations of the regular quotient in the sense of [10, Example 2.8].

6.3. The global Futaki invariant of a smooth test configuration of polarized complex cones.

Definition 6.7. Given a smooth 𝕋-equivariant test configuration (𝒱 , K) of the polarized complex cone (V, K) with K ∈ 𝔱_{+} as in Definition 6.4, and a cone potential r̃ ∈ R_{K}(𝒱 )^{𝕋}, we let

\begin{aligned}𝒢ℱ _{K}(𝒱 ) := & − \frac{1}{(m + 1)!} ∫_{𝒩} (Scal(g̃_{K}) − c_{K})η̃_{𝒟}^{K} ∧ (dη̃_{𝒟}^{K})^{m+1} \\ & + \frac{2}{m!} ∫_{𝒩} (π^{∗}ω_{F S}) ∧ η̃_{𝒟}^{K} ∧ (dη̃_{𝒟}^{K})^{m}, \end{aligned} ; (30)

where (𝒩 , 𝒟, 𝒥 , K) is the induced Sasaki manifold with contact form η̃_{𝒟}^{K}, c_{K} is the constant (48) of N, and ω_{FS} is the Fubini–Study metric on ℂP^{1} satisfying Ric(ω_{F S}) = ω_{FS}. We shall refer to (30) as the global Futaki invariant of (𝒱 , K).

Note that, as shown for instance in [29], the first line of (30) is independent of the choice of a cone potential r̃ ∈ R_{K}(𝒱 )^{𝕋} on (𝒱 , K) and so is the second line by a similar argument.

Proposition 6.8. Let (𝒱 , K) be a smooth 𝕋-invariant test configuration of polarized complex cones associated to (V, K) with K ∈ 𝔱_{+}. Consider the corresponding polarized affine cone (𝒱̂, K) with central fibre the polarized affine variety (V̂_{0}, K) and associated Donaldson–Futaki invariant DF_{V̂_{0},K}(Y_{w}). Then

(31)  DF_{V̂_{0},K}(Y_{w}) = λ(m)𝒢ℱ _{K}(𝒱 ).

for a dimensional constant λ(m) > 0.

Proof. As both functionals in (31) depend continuously on K ∈ 𝔱_{+} (see [18, Theorem 3]) and are homogeneous of degree −(m + 1) it will be enough to prove (31) for K primitive in the lattice of circle subgroups of 𝕋 (since the set of such vectors is dense in 𝔱_{+}/ℝ^{+}).

Assume that K is a primitive element in the lattice of circle subgroups of 𝕋, and denote ℂ_{K}^{×} ⩽ 𝕋_{ℂ} the subgroup generated by K. The ℂ_{K}^{×} GIT reduction of 𝒱 can be identified (cf. [49, 50]) with the compact polarized orbifold (ℳ , ℒ) associated to the grading induced by ℂ_{K}^{×}-action on the coordinates ring of the affine cone 𝒱̂ ⊆ ℂ^{N}, which also gives an orbifold embedding of (ℳ , ℒ) into a weighted projective space. By construction, ℳ ≃ 𝒱̂\{0}/ℂ_{K}^{×} ≃ 𝒱 /ℂ_{K}^{×}, so that ℳ has only cyclic orbifold singularities. The fact that K is primitive is used to determine the orbifold structure of the quotient such that the generic isotropy groups are trivial. Observe that the 𝕋_{ℂ}-invariant holomorphic map π: 𝒱 → ℂP^{1} descends to the quotient π̌: ℳ → ℂP^{1} as a ℂ_{ρ}^{×}-equivariant surjective orbifold holomorphic map. Then one can define a complex orbifold M := V_{1}/ℂ_{K}^{×} ≃ V/ℂ_{K}^{×} with orbiline bundle L = ℒ_{|_{M}} (the restriction of L to the orbifold charts is taken so that we get a locally ample orbiline bundle in the sense of [50]). This orbiline bundle is in fact ample (in the sense of [50]) because ((L^{∗})^{×}, K) ≃ (V_{1}, K) ⊆ 𝒱̂ and sections of ℒ restrict to section of L over M. Finally, we observe that (ℳ , ℒ) defines a test configuration for (M, L) in the sense of [50], as the map λ in (29) (which is 𝕋 × ℂ_{ρ}^{×} equivariant) descends to a (𝕋/𝕊_{K}^{1}) × ℂ_{ρ}^{×}-equivariant map.

Using Chern–Weil theory on orbifolds (in particular that c_{1}^{orb}(L) = \frac{1}{2π}[ω_{K}] for a 2-form ω_{K} which pulls back as the differential of the connection 1-form η_{𝒟}^{K} on the 𝕊^{1}- bundle N → M), we have

\begin{aligned}\frac{1}{(2π)}𝒢ℱ _{K}(𝒱 ) & = \frac{2(2π)^{m+1}}{m!} \{(𝒦_{ℳ /ℂP^{1}}^{orb} · ℒ^{m})_{ℳ} + \frac{c_{K}}{2(m + 1)}(ℒ^{m+1})_{ℳ} \} \\ & = \frac{2(2π)^{m+1}}{m!} \{(𝒦_{ℳ /ℂP^{1}}^{orb} · ℒ^{m})_{ℳ} \\ & − \frac{m}{m + 1}(\frac{(K_{M}^{orb} · L^{m−1})_{M}}{(L^{m})_{M}} )(ℒ^{m+1})_{ℳ} \} \end{aligned} ; (32)

where 𝒦 ^{orb} and K^{orb} stand for the (relative) canonical orbi-bundles on ℳ and M, respectively, and the products are taken by using integration of forms over orbifolds. Notice that up to a positive dimensional constant, (32) is an orbifold version of the intersection formula for the Donaldson–Futaki invariant found by Odaka [48] and Wang [59] in the settings of projective schemes.

We next argue that, similarly to the conclusions of [48, 59], (32) computes, up to a positive dimensional constant, the orbifold version of the Donaldson–Futaki invariant associated to the central fibre (M_{0} := π̌^{−1}(0), L_{0} := ℒ_{M_{0}}) of (ℳ , ℒ), defined in [50, Def. 6.4]. This follows from the considerations in [25, pp. 315] ^{2}, namely that for k ≫ 0 we have exact sequences

\begin{aligned}0 & → H^{0}(ℳ , ℒ^{k} ⊗ π̌^{∗}O(−1)) → H^{0}(ℳ , ℒ^{k}) → H^{0}(M_{0}, L_{0}^{k}) → 0 \\ 0 & → H^{0}(ℳ , ℒ^{k} ⊗ π̌^{∗}O(−1)) → H^{0}(ℳ , ℒ^{k}) → H^{0}(M, L^{k}) → 0 \end{aligned}

given by multiplication by the sections of O(−1) defining M_{0} and M_{∞}, respectively. This implies that d_{k} := dim_{ℂ} H^{0}(M_{0}, L_{0}^{k}) = dim_{ℂ} H^{0}(M, L^{k}) whereas the weight w_{k} for the induced ℂ^{×}-action on H^{0}(M_{0}, L_{0}^{k}) is

w_{k} = dim_{ℂ} H^{0}(ℳ , ℒ^{k} ⊗ π̌^{∗}O(−1)_{ℂP^{1}}) = dim_{ℂ}(ℳ , ℒ^{k}) − dim_{ℂ} H^{0}(M, L^{k}).

Now the claim follows from the definition [50, Def. 6.4] and an easy computation using the orbifold Riemann–Roch formula (28).

To conclude the proof of the proposition, we use that DF_{V̂_{0},K}(Y_{w}) too coincides with the Ross–Thomas orbifold Futaki invariant on (M_{0}, L_{0}), see [18, Def. 5.2] and the discussion following it. □

6.4. Sasaki K-stability versus weighted K-stability. Our main observation is the following result which links the relative Sasaki K-stability notion of [18] with the relative weighted K-stability of [40].

Proposition 6.9. Let (N, 𝒟, J, X) be a compact regular Sasaki manifold, and (M, J, L) the corresponding smooth polarized Kähler manifold. Let 𝕋 ⩽ Aut(N, 𝒟, J) be a compact torus with X ∈ 𝔱_{+} and K ∈ 𝔱_{+} another Sasaki–Reeb vector field on (N, 𝒟, J). Then, for any 𝕋-equivariant smooth polarized test configuration (ℳ , ℒ) associated to (M, J, L, 𝕋) we have

(33)  ℱ_{K}(ℳ , ℒ) = \frac{1}{2π} 𝒢ℱ _{K}(𝒱 ),

where ℱ_{K}(ℳ , ℒ) is introduced by (25).

2We are grateful to X. Wang for pointing out to us this reference and to J. Ross for answering our questions.

Proof. The definition (21) can be rewritten as

Scal_{f̃,m+2}(Ω̃)f̃^{−m−3} = (Scal_{f̃,m+3}(Ω̃) + 2f̃ΔΩ̃f̃+ 2(m + 2)|df̃|2 Ω̃)f̃^{−m−3}
= Scal_{f̃,m+3}(Ω̃)f̃^{−m−3} + 2δΩ̃(f̃^{−m−2}df).

Hence by Stokes’ Theorem

(34)  ∫_{ℳ} Scal_{f̃,m+2}(Ω̃)f̃^{−m−3}Ω̃^{m+1} = ∫_{ℳ} Scal_{f̃,m+3}(Ω̃)f̃^{−m−3}Ω̃^{m+1},

so that (25) is equivalently given by

(35) ℱ_{K}(ℳ , ℒ) = − ∫_{ℳ} (Scal f̃(Ω̃) − c_{K} f̃)f̃^{−3−m} \frac{Ω̃^{m+1}}{(m + 1)!} + 2 ∫_{ℳ} f̃^{−1−m}π^{∗}ω_{F S} ∧ \frac{ω^{m}}{m!} ,

where Scal f̃(Ω̃) = Scal_{f̃,m+1}(Ω̃) is f̃-scalar curvature of the Kähler manifold (ℳ , Ω̃). Formula (46) tells us that on the Sasaki manifold (𝒩 , 𝒟, 𝒥 , X) we have

η̃_{𝒟}^{X} = f̃η̃_{𝒟}^{K}, f̃ = η̃_{𝒟}^{X}(K), Scal_{K}(g̃_{X}) = Scal f̃(Ω̃) = f̃Scal(g̃_{K})

and we compute

ℱ_{K}(ℳ , ℒ) = − ∫_{ℳ} (Scal f̃(Ω̃) − c_{K} f̃)f̃^{−3−m} \frac{Ω̃^{m+1}}{(m + 1)!} + 2 ∫_{ℳ} f̃^{−1−m}π^{∗}ω_{F S} ∧ \frac{ω^{m}}{m!}
= − \frac{1}{2π(m + 1)!} ∫_{𝒩} (Scal_{K}(g̃_{X}) − c_{K}η̃_{𝒟}^{X}(K))η̃_{𝒟}^{X}(K)^{−3−m}η̃_{𝒟}^{X} ∧ (dη̃_{𝒟}^{X})^{m+1}

+ \frac{2}{(2π)m!} ∫_{𝒩} π^{∗}(ω_{F S}) ∧ η̃_{𝒟}^{X}(K)^{−1−m}η̃_{𝒟}^{X} ∧ (dη̃_{𝒟}^{X})^{m}

=\frac{1}{2π} 𝒢ℱ (𝒱 ).
□

As a direct consequence of Propositions 6.8 & 6.9 and Theorem 5.5 we obtain the following ramification of the result in [18].

Corollary 6.10. Let (V, K) be the polarized complex cone associated to a compact Sasaki manifold (N, 𝒟, J, K), 𝕋 ⩽ Aut(V, J)^{K} a maximal torus with K ∈ 𝔱_{+} and assume that there exists a regular Sasaki–Reeb vector field X ∈ 𝔱_{+}. If (V, K) admits a scalar-flat Kähler cone metric, then for any smooth 𝕋-equivariant test configuration (𝒱 , K) of (V, K), obtained from a smooth 𝕋-equivariant polarized test configuration (ℳ , ℒ) associated to the regular quotient (M, J, L, 𝕋), which has reduced central fibre and is not a product, we have DF_{V̂_{0},K}(Y_{w}) > 0.

Proof. It is well-known (see e.g. [11]) that a scalar-flat Kähler cone metric on (V, J, K) corresponds to a CSC Sasaki structure in S(K, J^{K}), such that the constant c_{K} defined by (48) equals m(m + 1). Thus, we are under the assumptions of Theorem 5.5 with ℓ_{ext}(x) = c_{K}ℓ_{K}(x), i.e., ℱ_{K}^{ext}(ℳ , ℒ) = ℱ_{K}(ℳ , ℒ) > 0. By Proposition 6.9 we have 𝒢ℱ _{K}(𝒱 ) > 0 whereas Proposition 6.8 yields DF_{V̂_{0},K}(Y_{w}) > 0. □

7. Applications

In this section we solve the existence problem of extremal Sasaki metrics on a special class of CR manifolds of Sasaki type, extending Theorem 2 in [1] to arbitrary dimension.

7.1. The Calabi construction on admissible manifolds. This is the familiar construction of special Kähler metrics on ℙ^{1} bundles, going back to Calabi [15] and studied in many other places.

We essentially follow the notation of [4] and consider (M, J) = ℙ(O ⊕ L) → B to be the total space of a ℙ^{1}-bundle over the product B = B_{1} × ⋯ × B_{k} of compact CSC Kähler manifolds (B_{j}, J_{j}, g_{j}, ω_{j}) satisfying the Hodge condition ω_{j} = 2πc_{1}(L_{j}) for a polarization L_{j} on B_{j}, and L = ⨂_{j=1}^{k} L_{j}^{p_{j}} with p_{j} ∈ ℤ being the induced line bundle on B. We fix real constants c_{j} ∈ ℝ such that the affine linear functions p_{j}z + c_{j} > 0 on the interval P = [−1, 1] ⊆ ℝ (equivalently, we require c_{j} > |p_{j}|).

Then, as shown for instance in [4], any smooth function Θ(z) on [−1, 1] satisfying the following boundary conditions

(36)  Θ(±1) = 0, Θ′(±1) = ∓2,

and

(37)  Θ(z) > 0 on (−1, 1).

gives rise to a Kähler metric on M of the form

\begin{aligned}g & = ∑_{j=1}^{k}(p_{j}z + c_{j})π^{∗}g_{j} + \frac{dz^{2}}{Θ(z)} + Θ(z)θ^{2}, \\ ω & = ∑_{j=1}^{k}(p_{j}z + c_{j})π^{∗}ω_{j} + dz ∧ θ, dθ = ∑_{j=1}^{k} p_{j}π^{∗}ω_{j}, \end{aligned} ; (38)

which is isometric to a Kähler metric in the same Kähler class ζ_{c} of (M, J) (here c = (c_{1}, …, c_{k}) stand for the fixed real constants in the definition (38)). Notice that in (38), Θ̂ is a connection 1-form on the principal 𝕊^{1}-bundle P over B, corresponding to line bundle L^{∗}, and M = ℙ^{1} ×_{𝕊^{1}} P → B is the corresponding fibre bundle associated to the toric variety (ℙ^{1}, O(2)). The induced 𝕊^{1}-action on M corresponds to fibrewise rotations in the trivial factor O of O ⊕ L, and (38) is 𝕊^{1}-invariant with m_{ω} = z being the corresponding momentum map with image P = [−1, 1] ⊆ ℝ. For any real constants b > |a|, f(z) = az +b is a positive Killing potential with respect to (38) for the Killing vector field aǨ where K is the generator of the 𝕊^{1}-action, and the constant b is essentially the normalization constant κ > 0 of §3.2. The f-scalar curvature of the Kähler metric (38) is computed in [6] to be

(39)  Scal_{f}(g) = −\frac{(f(z)^{−m−1}p_{c}(z)Θ(z))′′}{p_{c}(z)f(z)^{−m−3}} + f^{2}(z) ∑_{j=1}^{k} \frac{Scal_{j}}{p_{j}z + c_{j}} ,

where Scal_{j} stands for the constant scalar curvature of (B_{j}, g_{j}) and

p_{c}(z) := ∏_{j=1}^{k}(p_{j}z + c_{j})^{dim_{ℂ} B_{j}}

is a polynomial of degree (m − 1). By [6, Lemma 2.4], (38) gives rise to an f-extremal Kähler metric if and only if the smooth function Θ(z) on [−1, 1] satisfies (36), (37) and the condition

(40) (f(z)^{−m−1}p_{c}(z)Θ(z))′′ = p_{c}(z)f(z)^{−1−m}( ∑_{j=1}^{k} \frac{Scal_{j}}{p_{j}z + c_{j}} ) − (Az + B)f(z)^{−m−3}p_{c}(z)

for some (unknown) real constants A and B. The general solution of (40) thus depends on 4 real constants (including A and B) and the 4 boundary conditions in (36) determine these constants uniquely, as it follows from the arguments of [6, Prop. 2.2]. We denote by ℓ_{a,b,c}^{ext}(z) = A_{a,b,c}z + B_{a,b,c} the corresponding affine-linear function and by Θ_{a,b,c}^{ext}(z) the corresponding solution of (40), verifying (36) but possibly not (37).

Up to a suitable renormalization, this is the setting of [4, 6], except that we allowed p_{j} = 0 (which introduces a reducible manifold (M, J) on which nontrivial solutions of the f-extremal equation can exist when f ≠ const), and did not introduce here blow-downs. Indeed, when p_{j} ≠ 0, in order to get to the notation of [4], one simply needs to replace ω_{j} by (the possibly negative definite) CSC Kähler metric p_{j}ω_{j} and thus letting in (38) p_{j} = 1 and c_{j} = \frac{1}{x_{j}} , 0 < |x_{j}| < 1.

Definition 7.1. [4] A Kähler manifold (M, J) obtained as above is called admissible. The Kähler metric on it and the corresponding Kähler class ζ_{c} defined by (38) for some choice of c = (c_{1}, …, c_{k}) with c_{j} > |p_{j}| and smooth function Θ(z) are called admissible.

Our main result relevant to the admissible setting is the following explicit Yau–Tian– Donaldson type correspondence. Theorem 7.2. For an admissible Kähler manifold (M, J, ζ_{c}) as above, there exists an 𝕊^{1}- invariant (aǨ, b)-extremal Kähler metric in the Kähler class ζ_{c} if and only if the function Θ_{a,b,c}^{ext}(z) is positive on (−1, 1). In this case, (M, J) admits an admissible (aǨ, b)-extremal Kähler metric of the form (38).

Theorem 7.2 is established in various special cases, but to the best of our knowledge was open in the above generality prior to this work. Indeed, in the non-weighted case (which correspond to (a, b) = (0, 1)) Theorem 7.2 was first proved in [53] in the case of polarized ruled surfaces, and in [4] in general, whereas on a ruled complex surface and arbitrary (a, b) with |a| < b it is established in [1, Prop. 7]. Many other special cases are discussed in [6].

It is not hard to see that when the real constants c_{1}, …, c_{k} in (38) are all rational, the corresponding Kähler class ζ_{c} on (M, J) belongs to H^{2}(M, ℚ). Thus, for a suitable homothety factor λ_{c} > 0, a Kähler metric (λ_{c}g, λ_{c}ω) given by (38) defines a smooth CR manifold (N, 𝒟, J) of Sasaki type (see Example 1.3), endowed with a 2-dimensional torus 𝕋 ⩽ Aut(N, 𝒟, J) and a regular Sasaki–Reeb element X ∈ 𝔱_{+} generating an 𝕊_{X}^{1} ⩽ 𝕋, such that M is the Sasaki–Reeb quotient with respect to X and has the induced 𝕋̌ = 𝕋/𝕊_{X}^{1} = 𝕊1 ; Ǩ action generated by Ǩ. In other words, the choice of a positive Killing potential f = az + b, b > |a] with respect to the Kähler metric (λ_{c}g, λ_{c}ω) defines an element K_{a,b} ∈ 𝔱_{+}, inducing aǨ on M. In this setting, Proposition 3.7 and Theorem 7.2 yield the following generalization of [1, Theorem 2]. Corollary 7.3. (N, 𝒟, J) admits an extremal Sasaki metric in S(K_{a,b}, J^{K_{a,b}}) if and only if the corresponding smooth function Θ_{a,b,c}^{ext}(z) > 0 on (−1, 1).

Notice that in Theorem 7.2, the usually more difficult part, namely establishing the existence of an (aǨ, b)-extremal Kähler metric under a suitable positivity algebraic condition, is trivial because if Θ_{a,b,c}^{ext}(z) verifies (37), then letting Θ(z) = Θ_{a,b,c}^{ext}(z) in (38) defines an admissible (aK, b)-extremal Kähler in ζ_{c}. In the other direction, Lahdili shows in [40, Theorem 8] that for each z_{0} ∈ (−1, 1), one can find an 𝕊^{1}-invariant smooth test configuration ℳ which, as a complex manifold, is the degeneration to the normal cone [50] of (M, J) with respect to the infinity section B_{∞} := ℙ(O ⊕0) ⊆ M, together with a Kähler class 𝒜_{z_{0}}, such that the corresponding relative (Ǩ, κ)-weighted Donaldson–Futaki invariant introduced in §5.3 satisfy

ℱ_{aǨ,b}^{ext}(ℳ , 𝒜_{z_{0}}) = C(z_{0})Θ_{a,b,c}^{ext}(z_{0})

for a positive function C(z) on (−1, 1). Thus, Theorem 5.3 yields (see [40, Theorem 8]) that Θ_{a,b,c}^{ext}(z) ≥ 0 on (−1, 1) and the main difficulty in the proof of Theorem 7.2 is to establish the strict inequality in Theorem 5.3 for the test configurations (ℳ , 𝒜_{z_{0}}) as above. In particular, Conjecture 5.8 would imply Theorem 7.2 for c rational.

7.2. The case of arbitrary weights. Before giving the proof of Theorem 7.2, we will need to recall the more general setting of (v, w)-extremal Kähler metrics introduced in [40]. We limit the discussion to the ℙ^{1}-fibre bundle case, referring the Reader to [40] for the most general setting. Following [40, Section 10], let v(z) and w(z) be two positive smooth functions on [−1, 1], and let us look, more generally, for Kähler metrics on M = ℙ(O ⊕ L) → B of the form (38), satisfying the equation

(41)  (v(z)p_{c}(z)Θ(z))′′ = p_{c}(z)v(z)( ∑_{j=1}^{k} \frac{Scal_{j}}{p_{j}z + c_{j}} ) − (Az + B)w(z)p_{c}(z),

for some (unknown) real constants A, B. If we set v(z) = (az + b)^{−m−1} and w(z) = (az + b)^{−3−m} in (41), we obtain (40). For general positive weight functions (v, w) defined over P = [−1, 1], (41) describes a (v, w)-extremal Kähler metric in the Kähler class ζ_{c} in the sense of [40], which is given by the Calabi construction (38).

Definition 7.4. [40] Let v(z), w(z) be given positive smooth functions on [−1, 1] and ω̃ ∈ ζ_{c} an 𝕊^{1}-invariant Kähler metric with corresponding normalized momentum map mω̃: M → [−1, 1]. We say that ω̃ is (v, w)-extremal if it satisfies

(42) Scal_{v}(g̃) := v(mω̃)Scal(g̃) + 2Δg̃v(mω̃) + v′′(mω̃)g̃(K, K) = (Amω̃ + B)w(mω̃).

One can argue again that for v(z), w(z) and c = (c_{1}, …, c_{k}) fixed, (41) admits a unique smooth solution Θ_{v,w,c}^{ext}(z), satisfying (36) but possibly not (37). Notice that the constants A, B can be determined by integrating (41) on [−1, 1] against the affine-linear functions 1 and z. Integration by parts and using (36) yields a linear system for A, B, independent of Θ. This leads to the notion of the extremal affine-linear function ℓ_{v,w,c}^{ext}(z) := A_{v,w,c}z+B_{v,w,c} associated to (v, w, ζ_{c}).

By establishing a suitable notion of relative (v, w)-K-stability, the following result is obtained in [40, Theorem 8].

Theorem 7.5. [40] If Θ_{v,w,c}^{ext}(z_{0}) < 0 for some z_{0} ∈ (−1, 1) then (M, J) admits no (v, w)- extremal Kähler metric in ζ_{c}.

We shall next improve the above statement under a technical assumption allowing to deduce Theorem 7.2.

Theorem 7.6. If Θ_{v,w,c}^{ext}(z) vanishes at z_{0} ∈ (−1, 1) but ℓ_{v,w,c}^{ext}(z_{0}) ≠ 0, then there is no (v, w)-extremal metric in ζ_{c}.

Proof. Suppose Θ_{v,w,c}^{ext}(z_{0}) = 0 for some z_{0} ∈ (−1, 1) and ℓ_{v,w,c}^{ext}(z_{0}) ≠ 0, where ℓ_{v,w,c}^{ext}(z) = Az + B is the extremal affine-linear function. Assume for contradiction that ζ_{c} admits a (v, w)-extremal Kähler metric. By Theorem 7.5, Θ_{v,w,c}^{ext}(z) ≥ 0 on (−1, 1). The idea is to violate Theorem 7.5 by varying the weights (v, w).

Consider the smooth family of functions

Θ_{t}(z) := Θ_{v,w,c}^{ext}(z) − t(1 − z^{2})^{2}(Az + B)^{3}.

By its very construction, Θ_{t}(z) satisfies (36) for each t, Θ_{0}(z) = Θ_{v,w,c}^{ext}(z) and either for all t > 0 or for all t < 0 we have Θ_{t}(z_{0}) < 0. Suppose that Θ_{t}(z_{0}) < 0 for t > 0.

Notice that Θ_{t}(x) solves (41) with respect to the weight functions v(x) and

w_{t}(z) = w(z) + tw̃(z), w̃(z) := \frac{(v(z)p(z)(1 − z^{2})^{2}(Az + B)^{3})′′}{(Az + B)p(z)} .

As w̃(z) is a smooth function on [−1, 1], w_{t}(z) is positive and smooth on [−1, 1] for t small enough. It follows that Θ_{t}(z) = Θ_{v,w_{t},c}^{ext}(z) is the unique solution of (41) for the weights (v, w_{t}), satisfying (36). Using Theorem 7.5 again, we conclude that for small positive values of t, there is no (v, w_{t})-extremal Kähler metric in ζ_{c}. This contradicts Theorem B2 in [40]. □

7.3. Proof of Theorem 7.2.

Proof. As we have already explained, we are in the special case of Theorem 7.6 with v(z) = (az +b)^{−1−m} and w(z) = (az +b)^{−3−m} (b > |a|). Accordingly, it is enough to assume that (M, J) admits an (aK, b)-extremal Kähler metric in ζ_{c} but Θ_{a,b,c}^{ext}(z_{0}) = ℓ_{a,b,c}^{ext}(z_{0}) = 0 for some z_{0} ∈ (−1, 1), and derive a contradiction. To this end, we shall use a modification of the arguments from [4].

We start with the preliminary observation that Θ_{a,b,c}^{ext}(z) and ℓ_{a,b,c}^{ext}(z) are rational functions of z whose coefficients are rational functions of (a, b, c). Indeed, letting P_{a,b,c}(z) := p_{c}(z)Θ_{a,b,c}^{ext}(z) and multiplying both sides of (41) with (az + b)^{m+3}, it follows that P_{a,b,c}^{ext}(z) solves the ODE

\begin{matrix}(az + b)^{2}F′′(z) − 2a(m + 1)(az + b)F′(z) + a^{2}(m + 1)(m + 2)F (z) \\ = p_{c}(z)(az + b)^{2}( ∑_{j=1}^{k} \frac{Scal_{j}}{p_{j}z + c_{j}} ) − (Az + B)p_{c}(z). \end{matrix} ; (43)

where A and B are real constants (equal to A_{a,b,c} and B_{a,b,c} for the specific solution P_{a,b,c}(z)). When a ≠ 0, using that the monomials (az+b)^{s}, s = 0, …, m are eigenfunctions for nonzero eigenvalues of the linear differential operator in (43), we see that the general solution of (43) is a polynomial of degree ≤ m + 2 of the form

F (z) = ∑_{s=0}^{m+2} p_{s}(az + b)^{s}

whose coefficients p_{0}, …, p_{m} are determined from a, b, c and the two real constants A, B, whereas the coefficients p_{m+2} and p_{m+1} are the 2 free constants of integration. Imposing the boundary conditions (see (36))

(44)  F (±1) = 0, F′(±1) = ∓2p_{c}(±1)

leads to a linear system for (p_{m+2}, p_{m+1}, A, B). Under the geometric constraints |a| < b and c_{j} > |p_{j}| (and using that the setting is invariant under a homothety of (a, b)) it follows from the arguments in [6, Prop. 2.2] that the system admits a unique solution. This conclusion holds as long as the corresponding 4×4 system is nondegenerate. In summary, for the values of (a, b, c) such that a < |b|, a ≠ 0, c_{j} > |p_{j}|, P_{a,b,c}(z) is a polynomial of degree ≤ m + 2, whose coefficients are rational functions of a, b, c, and, similarly, ℓ_{a,b,c}^{ext}(z) = A_{a,b,c}z +B_{a,b,c} is an affine-linear function whose coefficients are rational functions of (a, b, c). The definition of P_{a,b,c}(z) and ℓ_{a,b,c}^{ext}(z) can be also extended for a = 0, b > 0 and c_{j} > |p_{j}| (e.g. by the arguments in the proof of [6, Prop. 2.2]) and for values (a, b, c) in a maximal connected component P of the geometric solutions (i.e., satisfying a < |b|, c_{j} > |p_{j}|).

We next observe that under the assumption made, for all (ã, b̃, c̃) sufficiently close to the initial value (a, b, c), (M, J) admits a (ãǨ, b̃)-extremal Kähler metric in ζc̃, Θ_{ã,b̃,c̃}^{ext}(z) ≥ 0 on (−1, 1) and Θ_{ã,b̃,c̃}^{ext}(z̃_{0}) = 0 = ℓ_{ã,b̃,c̃}^{ext}(z̃_{0}) for some z̃_{0} ∈ (−1, 1). (This implies that

−\frac{B_{ã,b̃,c̃}}{A_{ã,b̃,c̃}} is a double root, possibly at infinity, of the degree (m + 2) polynomial P_{ã,b̃,c̃}(z).) Indeed, otherwise, by the openness result [40, Theorem B2], we can find a sequence lim_{k→∞}(a_{k}, b_{k}, c_{k}) = (a, b, c) such that (M, J) admits an (a_{k}Ǩ, b_{k})-extremal Kähler metric in ζ_{c_{k}} (and therefore Θ_{a_{k},b_{k},c_{k}}^{ext}(z) ≥ 0 on (−1, 1) by Theorem 7.5) but Θ_{a_{k},b_{k},c_{k}}^{ext}(z) and ℓ_{a_{j},b_{j},c_{j}}^{ext}(z) do not having a common zero in (−1, 1). By Theorem 7.6, Θ_{a_{k},b_{k},c_{k}}^{ext}(z) > 0 on (−1, 1), i.e., there exist an admissible (a_{k}Ǩ, b_{k})-extremal Kähler metrics (g_{k}, ω) in ζ_{c_{k}} of the form (38). The latter are, by the results in [3], precisely the Kähler metrics on (M, J) admitting a hamiltonian 2-form of order 1 with associated Killing vector field Ǩ. Using the uniqueness result of [41] (see also Lemma 4.9 in this paper), we conclude as in the proof of Theorem 2 in [4] that the initial (aǨ, b)-extremal metric in ζ_{c} must also admit a hamiltonian 2-form of order 1 associated to Ǩ, a contradiction.

The conclusion, which will are going to use in order to find a contradiction, is that (by analytic continuation) for all (ã, b̃, c̃) ∈ P, P_{ã,b̃,c̃}(z) is a polynomial of degree ≤ (m + 2),

which has a double root at −\frac{B_{ã,b̃,c̃}}{A_{ã,b̃,c̃}}, i.e.,

(45)  (A_{ã,b̃,c̃})^{m+2}P_{ã,b̃,c̃}( − \frac{B_{ã,b̃,c̃}}{A_{ã,b̃,c̃}}) = 0,  (A_{ã,b̃,c̃})^{m+1}P′_{ã,b̃,c̃}( − \frac{B_{ã,b̃,c̃}}{A_{ã,b̃,c̃}}) = 0.

We thus reduced the proof to test (45) in the extended domain of definition P of (a, b, c) and show that (45) cannot hold for all such values of the parameters. We are going to specialize (45) for a = 1 = b. In this case, the system (44) for the general solution of (44) is still nondegenerate. Indeed, the boundary conditions at z = −1 of the general solution hold if and only if A_{1,1,c} = B_{1,1,c} = −2m(m + 1) which in turn identifies all but the coefficients p_{m+2} and p_{m+1} of the general solutions of (43). The boundary conditions at z = 1 thus fix the constants p_{m+2} and p_{m+1}. In particular, (1, 1, c) ∈ P. Notice that by (44), z = −1 is a single zero of P_{1,1,c}(z), which shows that the second equality in (45) does not hold. □

Appendix A. Holomorphic versus contact viewpoint

For any compact co-oriented contact (2m + 1)-manifold (N, 𝒟), let C_{+}(N, 𝒟)^{𝕋} be the space of 𝕋-invariant CR structures J on 𝒟, and let

𝔠𝔬𝔫_{+}(N, 𝒟) = {X ∈ 𝔠𝔬𝔫(N, 𝒟) | η_{𝒟}(X) > 0

(using the chosen orientation of T N/𝒟). Given a (compact, real) torus 𝕋 in Con(N, 𝒟) let Con(N, 𝒟)^{𝕋} denote the group of 𝕋-equivariant contact transformations, with Lie algebra 𝔠𝔬𝔫(N, 𝒟)^{𝕋} (in which the Lie algebra 𝔱 of 𝕋 is central).

Let (N, 𝒟_{0}, J_{0}, X) be a fixed Sasaki manifold invariant under a torus 𝕋, let η_{0} := η_{𝒟_{0}}^{X} be its contact form, and denote the induced transversal holomorphic structure on 𝒟_{X} by J_{0}^{X}. We consider the following two ways of parametrizing a family of CR structures on N.

• The contact viewpoint refers to the space C_{0}(X, 𝒟_{0})^{𝕋} ⊆ C_{+}(N, 𝒟_{0})^{𝕋} of 𝕋-invariant CR structures J on 𝒟_{0} such that the induced transversal holomorphic structure J^{X} is the pushforward of J_{0}^{X} by some Φ ∈ Diff_{0}(N)^{𝕋}. Note that J is uniquely determined by J^{X} (via the given fixed 𝒟_{0} and X), and it is sometimes convenient to view T_{J}C_{+}(N, 𝒟_{0})^{𝕋} as a space of endomorphisms of 𝒟_{X} rather than 𝒟_{0}.

• The (transversal) holomorphic viewpoint refers to the space S(X, J_{0}^{X})^{𝕋} of 𝕋-invariant contact forms η = η_{𝒟}^{X} which are compatible with the transversal holomorphic structure (X, J_{0}^{X}), and hence induce a CR structure (𝒟, J) where J is the lift of J_{0}^{X} to 𝒟. Note that the groups Con_{0}(N, 𝒟_{0})^{𝕋} := Con(N, 𝒟_{0})∩Diff_{0}(N)^{𝕋} and Aut_{0}(N, X, J_{0}^{X})^{𝕋} :=

Aut(N, X, J_{0}^{X})∩Diff_{0}(N)^{𝕋} act on C_{0}(X, 𝒟_{0})^{𝕋} and S(X, J_{0}^{X})^{𝕋} respectively by pushforward. Imitating arguments in [31], we have the following Sasaki analogue of a fundamental result in Kähler geometry.

Proposition A.1. The relation ∼ between C_{0}(X, 𝒟_{0})^{𝕋} and S(X, J_{0}^{X})^{𝕋}, defined by J ∼ η if and only if there exists Φ ∈ Diff_{0}(N)^{𝕋} such that (𝒟_{0}, J^{X}) = Φ · (𝒟, J_{0}^{X}), where 𝒟 = ker η, induces respectively Con_{0}(N, 𝒟_{0})^{𝕋} and Aut_{0}(N, X, J_{0}^{X})^{𝕋} invariant maps

p: C_{0}(X, 𝒟_{0})^{𝕋} → S(X, J_{0}^{X})^{𝕋}/Aut_{0}(N, X, J_{0}^{X})^{𝕋} and
q: S(X, J_{0}^{X})^{𝕋} → C_{0}(X, 𝒟_{0})^{𝕋}/Con_{0}(N, 𝒟_{0})^{𝕋},

which define a bijection

C_{0}(X, 𝒟_{0})^{𝕋}/Con_{0}(N, 𝒟_{0})^{𝕋} ≅ S(X, J_{0}^{X})^{𝕋}/Aut_{0}(N, X, J_{0}^{X})^{𝕋}.

The derivative of p at J, with J^{X} = Φ · J_{0}^{X}, is given by dp_{J}(−L_{Y} J^{X}) = Φ^{∗}(L_{Y} η_{0}) modulo 𝔞𝔲𝔱(N, X, J_{0}^{X})^{𝕋}, and the derivative of q at η = Ψ · η_{0}^{X} is given by dq_{η}(−L_{Y} η) = Ψ^{∗}(L_{Y} J_{0}^{X}) modulo 𝔠𝔬𝔫(N, 𝒟_{0})^{𝕋}.

Proof. Suppose J ∼ η; then J̃ ∼ η if and only if there exists Φ ∈ Diff_{0}(N)^{𝕋} with

Φ · (𝒟_{0}, J^{X}) = (𝒟_{0}, J̃^{X}), i.e., Φ ∈ Con_{0}(N, 𝒟_{0})^{𝕋} and Φ · J = J̃.

Similarly, J ∼ η̃, with ker η̃ = 𝒟̃ if and only if there exists Ψ ∈ Diff_{0}(N)^{𝕋} with

Ψ · (𝒟, J_{0}^{X}) = (𝒟̃, J_{0}^{X}), i.e., Ψ ∈ Aut_{0}(N, X, J_{0}^{X})^{𝕋} and Ψ · η = η̃

(since Ψ preserves X). Hence the relation J ∼ η induces invariant and uniquely defined, but perhaps only partially defined, p and q. However, p is everywhere defined by definition of C_{0}(X, 𝒟_{0})^{𝕋}, while q is everywhere defined by the equivariant Gray–Moser theorem (cf. Remark 1.8). Thus p and q are invariant maps as stated, and descend to mutually inverse maps on respective quotients by construction.

To compute dp_{J}(−L_{Y} J^{X}), let Φ_{t} be a curve in Diff_{0}(N)^{𝕋} with Φ_{0} = Φ and \frac{d}{dt}Φ_{t}Φ^{−1}|_{t=0} = Y and let J_{t} be the curve in C_{0}(X, 𝒟_{0})^{𝕋} with J_{t}^{X} = Φ_{t} · J_{0}^{X}; thus J_{t} passes through J at t = 0 with derivative (the lift to 𝒟_{0} of) −L_{Y} J^{X}. Then p(J_{t}) = Φ_{t}^{−1} · η_{0} = Φ^{∗}(Φ_{t}Φ^{−1})^{∗}η_{0} modulo Aut_{0}(N, X, J_{0}^{X})^{𝕋} and so

dp_{J}(−L_{Y} J^{X}) = Φ^{∗}(\frac{d}{dt}(Φ_{t}Φ^{−1})^{∗}η_{0}|_{t=0}) = Φ^{∗}(L_{Y} η_{0})

modulo 𝔞𝔲𝔱(N, X, J_{0}^{X})^{𝕋}. The derivative of q is computed similarly, or by using the inverse relationship between p and q. □

Remark A.2. We may describe the derivatives of p and q more explicitly using Lemma 1.7.

(i) If L_{Y} J^{X} is tangent to C_{0}(X, 𝒟_{0})^{𝕋} at J^{X}, its lift to 𝒟_{0} is skew with respect to dη_{0}. However, this lift differs from L_{Y} J by a 1-form with values in the span of X (on which dη_{0} vanishes), so L_{Y} J is skew with respect to dη_{0}. Since J is also skew with respect to dη_{0} it follows that J is skew with respect to d(L_{Y} η_{0}) = L_{Y} dη_{0}; hence d(L_{Y} η_{0}) is J^{X}-invariant, so L_{Y} η_{0} is tangent to S(X, J^{X})^{𝕋} at η_{0} hence of the form d_{X}^{J}φ + α by Lemma 1.7.

Hence, modulo 𝔞𝔲𝔱(N, X, J_{0}^{X})^{𝕋}, dp_{J}(−L_{Y} J) ≡ Φ^{∗}(L_{Y} η_{0}) = L_{Φ^{∗}Y} η = d_{X}^{J_{0}}ψ + β, with ψ = Φ^{∗}φ and β = Φ^{∗}α. Since 𝔬(N, X)^{𝕋} ⩽ 𝔞𝔲𝔱(N, X, J_{0}^{X})^{𝕋} induces exact forms df for 𝕋-invariant functions f, we only need to determine β, and hence α, modulo such exact forms. Thus we may take d^{J}φ + α = ι_{Y} dη_{0}, since this differs from L_{Y} η_{0} by such an exact form. Thus dp_{J}(−L_{Y} J) = d_{X}^{J_{0}}ψ + β = Φ^{∗}ι_{Y} dη_{0} modulo 𝔞𝔲𝔱(N, X, J_{0}^{X})^{𝕋}. (ii) Conversely the derivative of q at η = Ψ·η_{0} = Φ^{∗}η_{0}, with Ψ = Φ^{−1}, sends d_{X}^{J_{0}}(Φ^{∗}φ)+Φ^{∗}α to −L_{Y} J^{X} modulo 𝔠𝔬𝔫_{0}(N, 𝒟_{0})^{𝕋}, where ι_{Y} dη_{0} = d_{X}^{J}φ + α. We check that this has tangent vectors of the form df in its kernel, since they correspond to elements of 𝔠𝔬𝔫_{0}(N, 𝒟_{0})^{𝕋}. On the other hand, applied to a tangent vector of the form d_{X}^{J_{0}}(Φ^{∗}φ), we obtain instead −L_{Y} J^{X} where ι_{Y} dη_{0} = d_{X}^{J}φ modulo 𝔠𝔬𝔫_{0}(N, 𝒟_{0})^{𝕋}. Thus Y = −JX_{φ} modulo the span of X, where X_{φ} is the contact vector field of φ with respect to η_{0}, and we conclude that dq_{η}(d_{X}^{J_{0}}(Φ^{∗}φ)) = −L_{Y} J^{X} = L_{J^{X}X_{φ}}J^{X} = J^{X} ∘ L_{X_{φ}}J^{X} modulo 𝔠𝔬𝔫_{0}(N, 𝒟_{0})^{𝕋}.

Thus the image of the marking Ξ(X, J^{X}, η_{0}) under q may be interpreted as (the quotient by Con_{0}(N, 𝒟_{0})^{𝕋} of) an orbit for a complexification of Con_{0}(N, 𝒟_{0})^{𝕋}, even though such a complexification does not exist.

Appendix B. The contact Futaki invariant

Let (N, 𝒟) be a compact co-oriented contact (2m+1)-manifold and 𝕋 a fixed compact real torus in its group Con(N, 𝒟) of contact transformations. The group Con(N, 𝒟)^{𝕋} (with Lie algebra 𝔠𝔬𝔫(N, 𝒟)^{𝕋}) of 𝕋-equivariant contact transformations naturally acts on the space C_{+}(N, 𝒟)^{𝕋} of 𝕋-invariant CR structures on 𝒟, compatible with the fixed orientation on T N/𝒟 (which we assume is nonempty). Using Lemma 1.1, we identify 𝔠𝔬𝔫(N, 𝒟)^{𝕋} with the space C_{N}^{∞}(T N/𝒟)^{𝕋} of smooth 𝕋-invariant sections of T N/𝒟. Fix K ∈ 𝔱 ∩ 𝔠𝔬𝔫_{+}(N, 𝒟) where 𝔱 is the Lie algebra of 𝕋. It is shown in [1] that for any CR structure J ∈ C_{+}(N, 𝒟)^{𝕋} and any X ∈ 𝔱 ∩ 𝔠𝔬𝔫_{+}(N, 𝒟),

(46)  Scal_{K}(g_{X})τ = Scal(g_{K})σ =: Scal_{σ}(J),

where σ := η_{𝒟}(K), τ := η_{𝒟}(X). It follows that Scal_{σ}(J) is a well-defined element of C_{N}^{∞}(T N/𝒟)^{𝕋}; furthermore, following [1], the linear map

\begin{matrix}μ_{σ}(J): C_{N}^{∞}(T N/𝒟)^{𝕋} & → ℝ \\ ρ & ↦ ⟨⟨Scal_{σ}(J), ρ⟩⟩_{σ} \end{matrix} ; (47)

where for σ_{1}, σ_{2} ∈ C_{N}^{∞}(T N/𝒟)^{𝕋}

⟨⟨σ_{1}, σ_{2}⟩⟩_{σ} := ∫_{N} σ_{1}σ_{2}σ^{−m−3}vol_{𝒟}

with vol_{𝒟} := η_{𝒟} ∧ L_{𝒟}^{∧m} denoting the section of ∧^{2m+1}T^{∗}N ⊗ (T N/𝒟)^{m+1}, can be identified with the momentum map for the action of Con(N, 𝒟)^{𝕋} on the space C_{+}(N, 𝒟)^{𝕋}, with respect to a formal symplectic structure constructed in [32].

This provides a conceptual explanation for Lemma 3.2. Notice that for a J ∈ C_{+}(N, 𝒟)^{𝕋}, (𝒟, J, K) is extremal Sasaki if and only if the contact vector field X_{Scal_{σ}(J)} ∈ 𝔠𝔯(N, 𝒟, J) whereas (𝒟, J, K) is CSC Sasaki if and only if

X_{Scal_{σ}(J)} = c_{K}K

where c_{K} is the constant

(48)  c_{K} := \frac{⟨⟨Scal_{σ}(J), σ⟩⟩_{σ}}{⟨⟨σ, σ⟩⟩_{σ}} = \frac{∫_{N} Scal(g_{K})η_{𝒟}^{K} ∧ (dη_{𝒟}^{K})^{m}}{∫_{N} η_{𝒟}^{K} ∧ (dη_{𝒟}^{K})^{m}} .

We now use the above setting to give a definition of a contact Futaki invariant Fut_{K}: 𝔱 → ℝ associated to (N, 𝒟, 𝕋), which obstructs the existence of a CSC Sasaki structure in C_{+}(N, 𝒟)^{𝕋}, thus providing a contact analogue of the construction in [43]. Compared to the constructions in [12, 29], using Remark 1.8, Fut_{K} coincides with the restriction to the Lie algebra 𝔱 ⩽ 𝔞𝔲𝔱(N, K, J^{K}) of the invariant defined in these references with respect to the transversal holomorphic structure (N, K, J^{K}).

The point is that the flow of a vector field Y ∈ 𝔱 ⩽ 𝔠𝔬𝔫(N, 𝒟)^{𝕋} acts trivially on the space C_{+}(N, 𝒟)^{𝕋} and, therefore, by the momentum map interpretation of (47), we have that the quantity

⟨⟨Scal_{σ}(J), ρ⟩⟩_{σ}

where ρ = η_{𝒟}(Y) ∈ C_{N}^{∞}(T N/𝒟) is independent of J ∈ C_{+}(N, 𝒟)^{𝕋}. In particular, the constant c_{K} defined by (48) and the ⟨⟨·, ·⟩⟩_{σ}-orthogonal projection σ_{ext} of Scal_{σ}(J) to the space C_{N}^{∞}(T N/𝒟) are independent of J. We denote by K_{ext} ∈ 𝔱 the vector field corresponding to σ_{ext} via Lemma 1.1, and for any K_{1}, K_{2} ∈ 𝔠𝔬𝔫(N, 𝒟)^{𝕋} with corresponding sections σ_{1}, σ_{2} ∈ C_{N}^{∞}(T N/𝒟)^{𝕋}, we let

⟨⟨K_{1}, K_{2}⟩⟩_{K} := ⟨⟨σ_{1}, σ_{2}⟩⟩_{σ} = ∫_{N} η_{𝒟}^{K}(K_{1})η_{𝒟}^{K}(K_{2})η_{𝒟}^{K} ∧ (dη_{𝒟}^{K})^{m}.

The above prompts the following definition.

Definition B.1. Given a compact Sasaki manifold (N, 𝒟, J, K), a compact torus 𝕋 ⊆ Aut(N, 𝒟, J) with Lie algebra 𝔱 ⊆ 𝔠𝔬𝔫(N, 𝒟) such that K ∈ 𝔱, we define a linear map Fut_{K}: 𝔱 → ℝ by

Fut_{K}(Z) := ⟨⟨X_{Scal_{σ}(J)} − c_{K}K, Z⟩⟩_{K},

where the constant c_{K} is given by (48). We also write

Fut_{K}(Z) = ⟨⟨K_{ext} − c_{K}K, Z⟩⟩_{K}

where K_{ext} ∈ 𝔱 is the ⟨⟨·, ·⟩⟩_{K}-orthogonal projection of X_{Scal_{σ}(J)} to 𝔱, called extremal vector field of (K, 𝕋) (cf. [35] in the case when 𝕋 is maximal and [42] in the toric case).

As Fut_{K} is independent of J ∈ C_{+}(N, 𝒟)^{𝕋}, it follows that the condition Fut_{K} ≡ 0 (or equivalently K_{ext} = c_{K}K) is necessary for finding a CR structure J ∈ C_{+}(N, 𝒟)^{𝕋} which is CSC with respect to K.

Remark B.2. In the case when X is quasiregular, the Futaki invariant Fut_{K} can be rewritten in terms of the corresponding compact Kähler orbifold (M, J, ω) as

Fut_{K}(Z) = (2π) ∫_{M}(Scal_{f_{K}}(g) − c_{K}f_{K})f_{Z}f_{K}^{−m−3}ω^{m}

where f_{Z} = η_{𝒟}^{X}(Z) and f_{K} = η_{𝒟}^{X}(K) are the induced Killing potentials on M. This is, up to a positive dimensional constant, the Futaki invariant of [40], associated to the Kähler class ζ = [ω] on (M, J), the induced torus action 𝕋̌ with momentum polytope P_{X}, and the weight functions v(x) = ℓ_{K}(x)^{−m−1} and w(x) = ℓ_{K}(x)^{−m−2} (where ℓ_{K}(x) is the positive affine-linear function over P_{X} corresponding to K, see Lemma 1.15).

Acknowledgements

VA was supported in part by an NSERC Discovery Grant and is grateful to the University of Bath and the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences for their support and hospitality. EL was supported by the French ANR-14-CE25-0010 and is grateful to the UQAM and the UMI-CNRS in Montreal for their support and hospitality. We thank R. Dervan, A. Lahdili, J. Ross, Z. Sjöström Dyrefelt and X. Wang for valuable discussions.

References

[1] V. Apostolov and D. M. J. Calderbank, The CR geometry of weighted extremal Kähler and Sasaki metrics, arXiv:1810.10618, to appear in Math. Ann., https://doi.org/10.1007/s00208-020-02095-1.

[2] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and E. Legendre, Levi–Kähler reduction of CR structures, product of spheres, and toric geometry, arXiv:1708.05253, to appear in Math. Res. Lett.

[3] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and C. W. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler geometry II Global Classification, J. Differential Geom. 68 (2004), 277–345.

[4] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and C. W. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler geometry III Extremal metrics and stability, Invent. Math. 173 (2008), 547–601.

[5] V. Apostolov and G. Maschler, Conformally Kähler, Einstein–Maxwell geometry, J. Eur. Math. Soc. 21 (2019), 1319–1360.

[6] V. Apostolov, G. Maschler and C. W. Tønnesen-Friedman, Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles, arXiv:1808.02813, to appear in Comm. Anal. Geom.

[7] R. Berman and B. Berndtsson, Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics. J. Amer. Math. Soc. 30 (2017), 1165–1196.

[8] R. Berman, T. Darvas, and C. Lu, Regularity of weak minimizers of the K-energy and applications to properness and K-stability, Ann. Sci. Ec. Norm. Super. 53 (2020), 267–289.

[9] B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math. 200 (2015) 149–200.

[10] S. Boucksom, T. Hisamoto and M. Jonsson, Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier 67 (2017), 743–841.

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

[12] C. P. Boyer, K. Galicki and S. Simanca, Canonical Sasakian metrics, Comm. Math. Phys. 279 (2008), 705–733.

[13] C. P. Boyer, H. Huang, E. Legendre, An application of the Duistertmaat–Heckman theorem and its extensions in Sasaki geometry, Geometry & Topology, 22 (2018), 4205–4234.

[14] C. P. Boyer and C. van Coevering, Relative K-stability and extremal Sasaki metrics, Math. Res. Lett. 28 (2018), 1–19.

[15] E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, pp. 259–290, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, N.J., 1982.

[16] X. X. Chen, M. Paun and Yu Zeng, On deformation of extremal metrics, arXiv:1506.01290.

[17] K. Cho, A. Futaki and H. Ono, Uniqueness and examples of compact toric Sasaki–Einstein metrics, Comm. Math. Phys. 277 (2008), 439–458.

[18] T. Collins and G. Székelyhidi, K-semistability for irregular Sasaki manifolds, J. Differential Geom. 109 (2018), 81–109.

[19] T. Collins and G. Székelyhidi, Sasaki-Einstein metrics and K-stability, Geom. Topol. 23 (2019), 1339– 1413.

[20] T. Darvas, The Mabuchi geometry of finite energy class, Adv. Math. 285 (2015), 182–219.

[21] T. Darvas and Y. Rubinstein, Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc. 3 (2017), 347–387.

[22] L. David, Weyl connections and curvature properties of CR manifolds, Ann. Global Anal. Geom. 26 (2004), 59–72.

[23] R. Dervan and J. Ross, K-stability for Kähler manifolds, Math. Res. Lett. 24 (2017), 689–739.

[24] S. K. Donaldson, Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medallists lectures, 384–403, World Sci. Ser. 20th Century Math., 5, World Sci. Publishing, River Edge, NJ, 1997.

[25] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Diff. Geom. 62 (2002), 289–349.

[26] A. El Kacimi-Alaoui Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), 57–106.

[27] S. Falcão B. de Moraes and C. Tomei, Moment maps on symplectic cones, Pacific J. Math. 181 (1997), 357–375.

[28] A. Futaki and H. Ono, Conformally Einstein–Maxwell Kähler metrics and structure of the automorphism group, Math. Z. 292 (2019), 571–589.

[29] A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki– Einstein manifolds, J. Differential Geom. 83 (2009), 585–635.

[30] D. H. Phong and J. Sturm, Test configurations for K-stability and geodesic rays, J. Symp. Geom. 5 (2007), 221–247.

[31] P. Gauduchon, Calabi’s extremal metrics: An elementary introduction, Lecture Notes.

[32] W. He, On the transverse scalar curvature of a compact Sasaki manifold, Complex Manifolds 1 (2014), 52–63.

[33] W. He, On Calabi’s extremal metric and properness, Trans. Amer. Math. Soc. 372 (2019), 5595–5619.

[34] W. He, Scalar curvature and properness on Sasaki manifolds, arXiv:1802.03841.

[35] W. He and J. Li, Geometric pluripotential theory on Sasaki manifolds, arxiv:1803.0068, to appear in J. Geom. Anal. https://doi.org/10.1007/s12220-019-00257-5.

[36] W. He and S. Sun, Frankel conjecture and Sasaki geometry, Adv. Math. 291 (2016), 912–960.

[37] F.C. Kirwan, Cohomology of Quotients in Complex and Algebraic Geometry, Mathematical Notes 31, Princeton University Press, Princeton N. J., 1984.

[38] A. Lahdili, Automorphisms and deformations of conformally Kähler, Einstein–Maxwell metrics, J. Geom. Anal. 29 (2019), 542–568.

[39] A. Lahdili, Conformally Kähler, Einstein–Maxwell metrics and boundedness of the modified Mabuchi functional, arXiv:1710.00235, to appear in Int. Math. Res. Notices.

[40] A. Lahdili, Kähler metrics with weighted constant scalar curvature and weighted K-stability, Proc. London Math. Soc. 119 (2019), 1065–114.

[41] A. Lahdili, Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics, arXiv: 2007.01345.

[42] E. Legendre, Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics, Compos. Math. 147 (2011), 1613–1634.

[43] M. Lejmi, Extremal almost-Kähler metrics, Internat. J. Math. 21 (2010), 1639–1662.

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

[45] C. Li and C. Xu, Stability of Valuations: Higher Rational Rank, Peking Math J. 1 (2018), 1–79.

[46] D. Martelli, J. Sparks and S.-T. Yau, The geometric dual of a-maximisation for toric Sasaki–Einstein manifolds, Comm. Math. Phys. 268 (2006), 39–65.

[47] D. Martelli, J. Sparks and S.-T. Yau, Sasaki–Einstein manifolds and volume minimisation, Comm. Math. Phys. 280 (2008), 611–673.

[48] Y. Odaka, A genealization of the Ross–Thomas slope theory, Osaka J. Math. 50 (2013), 171–185.

[49] M. Reid, Graded rings and varieties in weighted projective space, lecture notes, 2002, URL:http://homepages.warwick.ac.uk/∼masda/surf/more/grad.pdf.

[50] J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 (2011), 109–159.

[51] Z. Sjöström Dyrefelt, K-semistability of cscK manifolds with transendental cohomology class, J. Geom. Anal. 28 (2018), 2927–2960.

[52] Z. Sjöström Dyrefelt (with an Appendix by R. Dervan), On K-polystability of CSCK manifolds with transcendental cohomology class, Int. Math. Res. Notices 2020 (2020), 2769–2817.

[53] G. Székelyhidi, Extremal metrics and K-stability, Bull. London Math. Soc. 39 (2007), 76–84.

[54] G. Tian, The K-energy on hypersurfaces and stability, Comm. Anal. Geom. 2 (1994), 239–265.

[55] G. Tian, Kähler–Einstein metrics with positive scalar curvature, Inventiones Math. 130 (1997), 1–37.

[56] C. van Coevering, Stability of Sasaki-extremal metrics under complex deformations, Int. Math. Res. Notices 24 (2013), 5527–5570.

[57] C. van Coevering, Monge-Ampère operators, energy functionals, and uniqueness of Sasaki-extremal metrics, arXiv:1511.09167.

[58] C. van Coevering, Examples of asymptotically conical Ricci-flat Kähler manifolds, Math. Z. 267 (2011), 465–496.

[59] X. Wang, Height and GIT weight, Math. Res. Lett. 18 (2012), 909–926.

[60] S. M. Webster, On the pseudo-conformal geometry of a Kähler manifold, Math. Z. 157 (1977), 265– 270.

[61] S. T. Yau, Open problems in Geometry. In: “Differential Geometry: Partial Differential Equations on Manifolds”. Proc. Sympos. Pure Math., vol. 54, pp. 1–28. AMS, Los Angeles, CA (1990).

[62] B. Zhou and X. Zhu, Relative K-stability and modified K-energy on toric manifolds, Adv. Math. 219 (2008), 1327–1362.

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

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

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