Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature conditions, including constant scalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitons and their weighted extensions. Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any non-product equivariant smooth K\"ahler test configuration with reduced central fibre, a property also known as weighted K-polystability on such test configurations. For a class of fibre-bundles, we use our result in conjunction with the recent results of Chen-Cheng, He, and Han-Li in order to characterize the existence of extremal Kahler metrics and Calabi-Yau cones associated to the total space, in terms of the coercivity of the weighted Mabuchi energy of the fibre. In particular, this yields an existence result for Sasaki-Einstein metrics on Fano toric fibrations, extending the results of Futaki-Ono-Wang in the toric Fano case, and of Mabuchi-Nakagawa in the case of Fano projective line bundles.


Introduction
This paper is concerned with the existence and obstruction theory of a class of special Kähler metrics, called weighted constant scalar curvature metrics, which were introduced by the third named author in [56], giving a vast extension of the notion of Kähler metrics of constant scalar curvature (cscK for short), and providing a unification for a number of related notions of Kähler metrics satisfying special curvature conditions.0.1.The weighted cscK problem.Let X be a smooth compact complex m-dimensional manifold with a given deRham cohomology class α ∈ H 1,1 (X, R) of Kähler metrics, and let T ⊂ Aut r (X) denote a fixed compact torus in the reduced group Aut r (X) of automorphisms of X, i.e. the connected subgroup of automorphisms of X generated by the Lie algebra of real holomorphic vector fields with zeros, see e.g.[41].It is well-known that T acts in a hamiltonian way with respect to any T-invariant Kähler metric ω ∈ α, and the corresponding momentum map µ ω sends X onto a compact convex polytope ∆ ⊂ t * in the dual vector space t * of the Lie algebra t of T (cf.[9,45]).Furthermore, up to translations, ∆ is independent of the choice of ω ∈ α.We shall further fix ∆, giving rise to a normalization of the corresponding momentum maps {µ ω , ω ∈ α}.
Following [56], let v(µ) > 0 and w(µ) be smooth functions defined on ∆.One can then consider the following condition for T-invariant Kähler metrics ω in α (and fixed polytope ∆), called (v, w)-cscK metric: (1) Scal v (ω) = w(µ ω ), V.A. was supported in part by an NSERC Discovery Grant and a Connect Talent Grant of the Région Pays de la Loire.S.J. was supported by PhD fellowships of the UQAM and the University of Toulouse III -Paul Sabatier.A.L. was supported by a postdoctoral fellowship of Aarhus University.We are very grateful to the anonymous referee for their careful reading of the manuscript and many valuable suggestions which substantially improved our work.We thank D. Calderbank, T. Darvas, R. Dervan, E. Inoue, E. Legendre, C. Li and G. Tian for their interest and comments on the manuscript.V.A. thanks P. Gauduchon for sharing with him his unpublished notes [42], and C. Tønnesen-Friedman for bringing the reference [43] to our attention.
where the so-called v-scalar curvature of ω is introduced by (

2)
Scal v (ω) := v(µ ω )Scal (ω) + 2∆ ω v(µ ω ) + g ω , µ * ω (Hess(v)) , with Scal (ω) being the usual scalar curvature of the riemannian metric g ω associated to ω, ∆ ω the Laplace operator of g ω , and where the contraction •, • is taken between the smooth t * ⊗ t * -valued function g ω on X (the restriction of the riemannian metric g ω to t ⊂ C ∞ (X, T X)) and the smooth t⊗t-valued function µ ω * (Hess(v)) on X (given by the pull-back by µ ω of Hess(v) ∈ C ∞ (∆, t⊗t)).The relevance of (1) to various geometric conditions is discussed in detail in [56], but we mention below a few special cases which partly motivate the study in this article.
• v = 1 and w is a constant: this is the familiar cscK problem; • v = 1 and w = with an affine-linear function on t * : (1) then describes an extremal Kähler metric in the sense of Calabi [21]; • v = e , w = 2( + a)e where is an affine-linear function on t * and a is a constant correspond to the so-called µ-cscK [51], extending the notion of Kähler-Ricci solitons [72] defined when X is Fano and α = 2πc 1 (X).
, where is a positive affine-linear function on ∆, m is the complex dimension of X, a is a constant, and L is a polarization of X: (1) then describes a scalar flat cone Kähler metric on the affine cone (L −1 ) × polarized by the lift of ξ = d to L −1 via , see [2,3].In general, the problem of finding a T-invariant Kähler metric ω ∈ α solving (1) is obstructed in a similar way that the cscK problem is obstructed by the vanishing of the Futaki invariant: for any T-invariant Kähler metric ω ∈ α and any affine-linear function on t * , one must have (3) Fut v,w ( ) := should a solution to (1) exists.In [56], an unobstructed modification of ( 1) is proposed, extending Calabi's notion [21] of extremal Kähler metrics.To this end, suppose that v, w 0 > 0 are given positive smooth functions on ∆.One can then find a unique affine-linear function ext v,w 0 (µ) on t * , called the extremal function, such that (3) holds for the weights (v, w) = (v, ext v,w 0 w 0 ).In this case, a solution of the (v, w)-cscK problem (1) is referred to as (v, w 0 )-extremal Kähler metric.We emphasize that (v, w 0 )-extremal Kähler metrics are (v, w)-cscK metrics with a special property of the weight function w, namely, w = w 0 with w 0 > 0 on ∆ and affine-linear.In particular, (v, w)-cscK metrics with w = 0 on ∆ are (v, w)-extremal with ext v,w = sign(w | ∆ ) and (v, 0)-cscK metrics are (v, w)-extremal with ext v,w = 0 for any w > 0. It follows that all the above listed special cases are examples of (v, w)-extremal Kähler metrics, and thus the setup of (v, w)-extremal Kähler metrics allows one to study these cases altogether.0.2.Relation to v-solitons.Motivated by works of T. Mabuchi [59,60] and consequent work by Berman-Witt-Nystrom [13], Y. Han and C. Li [46] have recently introduced and studied the general notion of a weighted v-soliton on a smooth Fano variety X, as follows.In the setup explained above, we let α = 2πc 1 (X) and consider the natural action of T on K −1 X , which fixes the momentum polytope ∆ of (X, α, T) and normalizes the momentum map µ ω for any T-invariant Kähler metric ω ∈ α.For a (smooth) positive weight function v(µ) on ∆, one defines a v-soliton as a T-invariant Kähler metric ω ∈ α, such that (4) where ρ ω denotes the Ricci form of ω.Notice that when v(µ) = e µ,ξ for some ξ ∈ t, one gets the well-studied class of Kähler-Ricci solitons [72] whereas the case when v(µ) is a positive affine-linear function on ∆ corresponds to the Mabuchi solitons studied in [59,60].As we shall see below other choices for v are also geometrically meaningful.We make the following useful observation.
We use the above result in order to make connection with the recent paper [46] (where the authors obtain a complete Yau-Tian-Donaldson type correspondence for the existence of v-Ricci solitons) which will play an important role in our present study of (v, w)-cscK metrics.
We also notice that v-solitons can be viewed as (v, w)-cscK metrics for different choices of weights.This is for instance the case when v(µ) = (µ) −(m+2) , where (µ) = ξ, µ + a is a positive affine-linear on ∆.Whereas Proposition 1 identifies the v-soliton as a (v, w)-cscK metric with v = −(m+2) , w = 2 −(m+3) (−2 + (m + 2)a) , we also observe that Proposition 2. Let (X, T) be a smooth Fano variety and (µ) = ( ξ, µ +a) a positive affine-linear function on its canonical polytope ∆.A T-invariant Kähler metric ω ∈ 2πc 1 (X) is an −(m+2)soliton if and only if the lift ξ of ξ = d to K X via is the Reeb vector field of a Sasaki-Einstein structure defined on the unit circle bundle N ⊂ K X with respect to the hermitian metric on K X with curvature −ω.The latter condition is also equivalent to ω be a ( −m−1 , 2ma −m−2 )-cscK metric.0.3.Main results.Similarly to the usual cscK case, it is shown in [56] that the solutions of (1) can be characterized as minimizers of a functional M v,w defined on the space of T-invariant Kähler metrics in α, extending the Mabuchi energy to the weighted setting (see Section 1 below for the precise definition).After the deep works [15,23], it is now well-understood that the coercivity of the Mabuchi energy is equivalent to the existence of a cscK metric in a given cohomology class.Noting that, by the results in [56], any (v, w)-extremal metric is invariant under a maximal compact torus in Aut r (X), our first main result is an extension of one direction of the correspondence in the cscK case to the weighted setting.
Theorem 1. Suppose T ⊂ Aut r (X) is a maximal torus in the reduced group of automorphisms of X, and ω 0 ∈ α a T-invariant (v, w 0 )-extremal Kähler metric.Then the weighted Mabuchi energy M v,w (with w = ext v,w 0 w 0 ) is coercive relative to the complex torus T C , in the sense of [29], i.e. there exist positive real constants λ, δ such that for any T-invariant Kähler metric ω ∈ α, where J denotes the Aubin functional on the space Kähler metrics, see Definition 3.1 below.
Our proof of Theorem 1 also adapts to the case when the torus T ⊂ Aut r (X) is not necessarily maximal, but instead of T C one takes the infimum of J(σ * ω) over Ĝ := Aut T r (X), the connected component of the identity of the centralizer of T in Aut r (X) (which by [56] is a reductive group if X admits a (v, w 0 )-extremal T-invariant Kähler metric, see Remark 7.7 for more details).Furthermore, we can also consider any reductive connected subgroup group G = K C ⊂ Ĝ with a compact form K containing T, and restrict M v,w to the space of K-invariant Kähler metrics in α as in [46]. 1  As noticed in [15] (in the polarized case) and in [66] (in the more general Kähler case), the coercivity of the Mabuchi energy yields a sharp estimate of the sign of the Donaldson-Futaki invariant of a T-equivariant test configuration.In our weighted setting, we consider T-equivariant (compactified) Kähler test configurations (X , A ) associated to (X, α, T), which have smooth total space.To any such test-configuration one can associate a weighted Donaldson-Futaki invariant by the formula (cf.[56]) where Ω ∈ A , ω ∈ α are T-invariant Kähler forms respectively on X and X, with respective ∆normalized momentum maps µ Ω , µ ω , and Scal v (Ω) is the v-scalar curvature of Ω defined by (2).In the above formula, for any 2-form ψ we use the convention ψ [k] := ψ k k! so that F v,w (X , A ) extends to the weighted setting the expression [63,73] of the Donaldson-Futaki invariant of (X , A ) in terms of intersection numbers.
Corollary 1.Under the hypotheses of Theorem 1, for any T-equivariant smooth Kähler test configuration (X , A ) of (X, α, T) which has a reduced central fibre, we have the inequality F v,w (X , A ) ≥ 0, with equality if and only if (X , A ) is a product test configuration.Furthermore, if α = 2πc 1 (L) corresponds to a polarization L of X and (X , L , T) is a T-equivariant smooth polarized test configuration of (X, L) as above, we have the inequality F v,w (X , A ) ≥ λ J NA T C (X , A ), where A = 2πc 1 (L ), λ > 0 is the constant appearing in Theorem 1, and J NA T C (X , A ) is the T Crelative non-Archimedean J-functional of the test configuration introduced in [50,58], see (20).
Corollary 1 improves the (T-equivariant) (v, w)-K-semistability established in [57,Thm. 2] to (T-equivariant) (v, w)-K-polystability on the test configurations as above, and, in the projective case, further to T C -uniform (v, w)-K-stability in the sense of [50,58].As we already mentioned, the fist part of Corollary 1 was proved in [68,15,66] in the cscK case (v = 1 and w is a constant), and in [34,69,66] in the unweighted extremal case (v = 1 = w 0 ).We however notice that in the extremal case our proof uses directly the coercivity of the relative Mabuchi energy (which follows from Theorem 1) whereas the proofs in [34,69] and [66] are based respectively on the Arezzo-Pacard existence results of extremal metrics on blow-ups [8], and on the coercivity of the unweighted Mabuchi energy M 1,c established in [15,66].The T C -uniform (v, w)-K-stability statement in the second part of Corollary 1 is established in the cscK case in [50,58], and in the case of a v-soliton in [46].Our proof of Corollary 1 in the general weighted case follows easily from Theorem 1 by the established techniques in the cscK case, see Section 4.
Another notable special case where our results apply is when α = c 1 (L) for an ample line bundle L over X, and v = −m−1 , w 0 = −m−3 for a positive affine-linear function on ∆.It is observed in [2] that in this case a (v, w 0 )-extremal Kähler metric in α describes an extremal Sasaki metric on the total space N of the unit circle bundle in L −1 with respect to the hermitian metric with curvature −ω, and Reeb vector field corresponding to the lift of d to L −1 via .In this special case, the first part of Corollary 1 above was obtained in [3] for polarized test configurations (see Theorem 1, Conjecture 5.8 and Remark 5.9 in [3]), by using the results in [49] which establish an analogue of Theorem 1 in the Sasaki case.Thus, our proofs of Theorem 1 and Corollary 1 presented in this paper allow one to recast and further generalize [3, Thm.1] entirely within the framework of weighted Kähler geometry of X. 0.4.Method of proof.We now discuss briefly the method of proof of Theorem 1 above.It is an application of the general coercivity principle [29,Thm. 3.4], see Section 3 below.The latter is used in the cscK case in [15], and our approach is mainly inspired by these two references.Noting that in the weighted extremal case M v,w is G-invariant and G := T C is reductive, by the results of [29], in order to obtain Theorem 1, one needs to accomplish the following steps: (1) extend M v,w to the space E 1 (X, ω 0 ) of ω 0 -relative pluri-subharmonic functions of maximal mass and finite energy; (2) show that the extension is convex and continuous along weak d 1 -geodesics in E 1 (X, ω 0 ); (3) establish a compactness result for the extension of M v,w , and (4) show the uniqueness modulo the action of G (and in particular the regularity) of the weak minimizers of M v,w , under the assumption that a (v, w 0 )-extremal metric exists.The steps (1), ( 2) and (3) in the unweighted cscK case are obtained in [14] and follow from the Chen-Tian formula of M 1,1 .The analogous formula for M v,w is obtained in [56], but the presence of weights does not allow for a straightforward generalization of the arguments in [14].Similar difficulty arises in [13], in the framework of v-solitons on a Fano variety, where the authors were able to obtain a suitable extension of the weighted Ding functional to the space E 1 (X, ω 0 ).The latter functional has milder dependence on the weights than the weighted Mabuchi functional we consider.Indeed, the arguments of [13] yield the existence of a continuous extension to E 1 (X, ω 0 ) of one of the three terms in the Chen-Tian decomposition of M v,w , which depend on the weight w.Building on [13], Han-Li [46] proposed a new approach to the extension problem in the case of v-solitons, based on an idea going back to Donaldson [36] (see in particular the proof of Proposition 3 in [36]), which amounts to consider suitable fibre-bundles Y over a cscK bases B and fibre X, and show that the weighted quantities on X correspond to the restrictions of unweighted quantities on the total space Y .This is the semi-simple principal (X, T)-fibration construction which we review in the next subsection.Going further than [46], we express in general the scalar curvature of a bundlecompatible Kähler metric on Y in terms of the weighted scalar curvature of X, and show that the usual (unweighted) Mabuchi energy on Y restricts to a suitably weighted Mabuchi energy on X.It thus follows that at least for suitable polynomial weights v, the remaining terms of the Chen-Tian decomposition of M v,w can be extended to E 1 (X, ω 0 ) simply by restricting to the fibres the corresponding (unweighted) extension of the Mabuchi energy of Y .The final crucial observation for obtaining the extension for any weights is that M v,w depends linearly and continuously on (v, w), so that one can further use (as in [46]) the Stone-Weierstrass approximation theorem over C 0 (∆).With this in place, and using the weighted analogue of the uniqueness [11] achieved in [57], we can adapt the arguments from [15].0.5.Applications to the semi-simple principal fibration construction.We briefly review here the semi-simple principal bundle construction, which is a key tool in our proof of Theorem 1, but also provides a framework for further geometric applications of our results, extending the setting of the generalized Calabi construction in [7].
We denote by T a compact r-dimensional torus with Lie algebra t and lattice Λ ⊂ t of generators of S 1 -subgroups, i.e.
which is a product of compact cscK Hodge Kähler 2n a -manifolds (B a , ω Ba ), a = 1, . . ., k.We then consider a principal T-bundle π : P → B endowed with a connection 1-form θ ∈ Ω 1 (P, t) with curvature For any smooth compact Kähler 2m-manifold (X, ω X , T), endowed with a hamiltonian isometric action of the torus T as in the setup above, we can construct the principal (X, T)-fibration where the T-action on the product is σ(x, p) = (σ −1 x, σp), x ∈ X, p ∈ P, σ ∈ T. Using the chosen connection on P , the almost complex structures on X and B lift to define a CR structure on the product X × P , and thus endow Y with the structure of a 2(m + n)-dimensional smooth complex manifold.Furthermore, Y comes equipped with an induced holomorphic fibration π : Y → B, with smooth complex fibres X, and induced fibre-wise T-action.Fixing constants c a ∈ R such that for each a = 1, . . ., k, the affine linear function p a , µ + c a on t * is strictly positive on the momentum image ∆ of X, one can define a lifted Kähler metric ω Y on Y which, pulled-back to X × P , has the form where •, • stands for the natural pairing of t and t * : thus p a , µ ω is a smooth function and dµ ω ∧ θ is a 2-form on X × P .As we show in Section 5 below, when ω X varies in a given Kähler class of X, the corresponding Kähler metric ω Y will vary in a fixed Kähler class on Y .We also notice that when (X, ω X , T) is a smooth toric Kähler manifold, the setup above reduces to the theory of semi-simple rigid toric fibrations studied in [4,5,7].Inspired by the results in the latter works, we show that the scalar curvature of ω Y can be expressed in terms of the p-weighted scalar curvature of (X, ω X ), where the weight function p(µ) is a polynomial depending on the fixed data (p a , c a , n a ) of the construction.With this observation in mind, we show that (similarly to the case of semi-simple rigid toric fibrations recently studied in [53]) the recent results [23,48] can be used to obtain a converse of Theorem 1 in the case of a semi-simple principal fibrations.
Theorem 2. Suppose Y is a semi-simple principal (X, T)-fibration, with a Kähler metric ω Y induced by a T-invariant Kähler metric ω X on X.We suppose, moreover, that T is a maximal torus in the reduced group of automorphisms Aut r (X).Then, the following conditions are equivalent (i) Y admits an extremal Kähler metric in the Kähler class [ω Y ]; (ii) X admits a T-invariant (p, w)-cscK metric in the Kähler class [ω X ], with weights where ext is an affine-linear function determined by the condition (3); (iii) The weighted Mabuchi energy M X p, w of (X, [ω X ], T) is coercive with respect to T C , where p, w are the weights defined in (ii).
Compared to the general setting of [35], the semi-simple principal (X, T)-fibration (trivially) satisfy the condition of optimal symplectic connection.Accordingly, one can conclude by [35] that (Y, [ω Y ]) admits an extremal Kähler metric, provided that (X, ω X ) is cscK, and if we take large enough constants c a .As a matter of fact, the conclusion also follows under the more general assumption that (X, ω X ) is extremal, by the proof of [5,Thm. 3].The novelty of Theorem 2 is therefore in the fact that it gives a precise condition (in terms of X) for the existence of an extremal Kähler metric in a given Kähler class [ω Y ], also revealing that (X, [ω X ]) needs not to be extremal in general.We finally note that in the case of toric fibre, [53] provides a further equivalence with a certain weighted notion of uniform K-stability of the corresponding Delzant polytope.
If all the factors (B a , ω Ba ) of the base are positive Kähler-Einstein manifolds, and the fibre (X, T) is a smooth Fano variety, the semi-simple principal (X, T)-fibration construction can produce a smooth Fano variety Y for suitable choice of the principal T-bundle over B (see Lemma 5.11 below).In this case, combining [46,Thm. 3.5] with the results in this paper, we get Theorem 3. Suppose Y is a Fano semi-simple principal (X, T)-fibration, obtained from the product of positive Kähler-Einstein Hodge manifolds (B a , ω Ba ) and a smooth Fano fibre (X, T) via Lemma 5.11.Suppose also that T is a maximal torus in the automorphism group Aut(X).Then Y admits a v-soliton in 2πc 1 (Y ), provided that the weighted Mabuchi functional M X pv, w of (X, T, 2πc 1 (X)) is coercive with respect to T C , where p is the weight defined in Theorem 2-(ii) and w = 2pv m + d log v, µ + d log p, µ .
If, furthermore, the fibre (X, T) is a smooth toric Fano variety, then the latter condition is equivalent to the vanishing of the Futaki invariant (3) associated to the weights (pv, w) on X.In particular, any Fano semi-simple principal (X, T)-fibration with smooth toric Fano fibre (X, T) admits a Kähler-Ricci soliton, and the corresponding affine cone (K Y ) × admits a Calabi-Yau cone metric, given by a Sasaki-Einstein structure on a unit circle bundle associated to the canonical bundle K Y .
The existence of a Kähler-Ricci soliton in the above setting is essentially known even though we didn't find it explicitly stated in the literature.In the toric case (i.e. when Y = X and B is a point) the result follows by [74] (see also [30]), and for P 1 -bundles by [54,26,6].In the general case, the result can be obtained from [67], which in turn extend [74] to the framework of multiplicity-free manifolds, but the arguments can be also adapted to the case of semi-simple principal (X, T)-fibrations (see [7,Rem.7] and [37]).Our approach, however, builds on the idea of [30].There are also related existence results for Kähler-Ricci solitons on spherical manifolds, see [32,31].On the other hand, the existence of Sasaki-Einstein metrics seems to be new in the above stated generality.Indeed, in the toric case the claim follows from [40], and there are known existence results [43,20,61] on P 1 -bundles.We expect our arguments to extend to spherical manifolds too.0.6.Structure of the paper.In Section 1, we recall the setup of weighted cscK metrics and state the main results we shall need from [56,57].In Section 2, we recall the notion of v-solitons from [59,46], and establish the equivalences stated in Propositions 1 and 2. Sections 3 and 4 review and recast in the weighted setting respectively the coercivity principle of [29] and its application to stability [15,66], thus outlining the main steps needed for the proofs of Theorem 1 and deriving Corollary 1 from the latter.In Section 5, we introduce the semi-simple principal (X, T)-fibration construction, and establish the main geometric properties allowing us to extend the results from [7].In Section 6, we use an idea from [46] in order to define an extension of the weighted Mabuchi energy to the space E 1 (X, ω 0 ), and show its convexity and compactness properties.In Section 7, we extend the arguments of [15] to show that weak minimizers of the weighted Mabuchi energy are smooth.Here, we complete the proof of Theorem 1.In Section 8, we detail the proofs of Theorems 2 and 3.In the Appendix, we present some technical computational results, detailing the linearization of the scalar and the twisted scalar curvature of a semi-simple principal (X, T)-fibre and re-casting the weighted Futaki invariant (3), which are needed for the proofs of Theorem 2 and 3.

Preliminaries on the weighted cscK problem
We recall the setup from [56].Let X be a smooth compact, connected Kähler manifold of (real) dimension 2m, and let be the space of ω 0 -relative smooth Kähler potentials on X.We let T ⊂ Aut r (X) be a fixed compact torus in the reduced group of automorphisms of X, i.e. the connected closed subgroup Aut r (X) of the group of complex automorphisms Aut(X), whose Lie algebra is the space of holomorphic vector fields of X with zeros (see e.g.[41]).Equivalently, Aut r (X) is the connected component of the identity of the kernel of the natural group homomorphism from Aut(X) to the Albanese torus, and is known to be isomorphic to the linear algebraic group in the Chevalleytype decomposition of Aut(X), cf.[38].We denote by C ∞ T (X) the space of T-invariant smooth functions on X and introduce the space It is well-known that the action of T on (X, ω 0 ) is hamiltonian, and we let µ 0 : X → t * be a momentum map, where t is the Lie algebra of T and t * the dual vector space.By the convexity theorem [9,45], the image ∆ := µ 0 (X) ⊂ t * is a compact convex polytope.For any ϕ ∈ K T (X, ω 0 ), the smooth t * -valued function (5) µ ϕ = µ 0 + d c ϕ is the T-momentum map of (X, ω ϕ ), normalized by the condition µ ϕ (X) = ∆.In the above formula, d c ϕ is viewed as a smooth t * -valued function via the identity d c ϕ, ξ := d c ϕ(ξ) for any ξ ∈ t ⊂ C ∞ (X, T X).
1.1.The (v, w)-constant scalar curvature Kähler metrics.Following [56], let v(µ) > 0 and w(µ) be smooth functions on ∆.One can then consider the condition (1) for a T-invariant Kähler metric ω ϕ in α (and the fixed polytope ∆), called (v, w)-cscK metric.We thus want to solve the following PDE for ϕ ∈ K T (X, ω 0 ): As we explained in the Introduction, the problem of finding ω ϕ ∈ α solving (6) is obstructed by the condition (3), and in the case when v, w 0 are positive weights, this can be resolved (similarly to the approach in [21]) by finding a unique affine-linear function ext v,w 0 (µ) on t * , called the extremal function, such that for any ω ϕ Geometrically, the above condition means that the weighted cscK problem with weights (v, w) = (v, ext v,w 0 w 0 ) is unobstructed in terms of (3), and a solution ω ϕ of the (v, ext v,w 0 w 0 )-cscK problem is referred to as (v, w 0 )-extremal metric.
1.2.The weighted Mabuchi energy.Definition 1.1.[56] Let v, w be weight functions on ∆ with v(µ) > 0. The weighted Mabuchi energy M v,w on K T (X, ω 0 ) is defined by Remark 1.2.It follows from the above definition and the results in [56] that for a constant c, M v,w (ϕ + c) = M v,w (ϕ) if and only if v, w satisfy the integral relation Furthermore, by the results in [56], ( 7) is a necessary condition for the existence of a solution of ( 6) and it is incorporated in the definition of M v,w given in [56], via the constant c v,w (α) in front of w, but we do not assume a priori this condition in the current article.It is however automatically satisfied if α admits a T-invariant (v, w)-cscK metric, or if we consider the weights (v, w) = (v, ext v,w 0 w 0 ) corresponding to (v, w 0 )-extremal Kähler metrics.In these cases, we shall write M v,w (ω ϕ ) to emphasize that the weighted Mabuchi functional acts of the space of T-invariant The following result is established in [57], generalizing [11] to arbitrary weights v > 0, w.
1.3.The automorphism group of a (v, w 0 )-extremal Kähler manifold.In what follows we will consider connected Lie groups.We recall that we have set Aut r (X) to be the connected component of the identity of the kernel of the Albanese homomorphism and, similarly, we denote by Aut T r (X) the connected component of the identity of the centralizer of the torus T in Aut r (X).We shall use the following result, established in [56, Thm.B.1] (cf.also [39]) and [57, Rem.2]: Proposition 1.4.If (X, α, T) admits a (v, w 0 )-extremal Kähler metric ω, then the connected component of the identity Aut T r (X) of the subgroup of T-commuting automorphisms in Aut r (X) is reductive, and ω is invariant under the action of a maximal compact connected subgroup of Aut T r (X).In particular, the isometry group of (X, ω) contains a maximal torus T max ⊂ Aut r (X) with T ⊂ T max .If, furthermore, T = T max , then Aut T r (X) = T C .Because of this result, we shall often assume (without loss of generality for solving (6)) that T = T max ⊂ Aut r (X) and thus Aut T r (X) = T C .
1.4.Uniqueness of the (v, w 0 )-extremal Kähler metrics.Another key result in the theory is the extension in [57] of the uniqueness results [11,24] to the weighted setting.

v-solitons as weighted cscK metrics
We review here the definition of v-solitons on a Fano manifold, following [13,46], and discuss their link with (v, w)-cscK metrics.
We thus suppose throughout this section that that X is a smooth Fano manifold, α := 2πc 1 (X) and T ⊂ Aut(X) a fixed compact torus.(We recall here that on a Fano manifold, Aut r (X) coincides with the connected component of the identity of the full automorphism group.)We further consider the natural action of T on the anti-canonical bundle K −1 X of X, which normalizes the momentum map µ ω of each T-invariant Kähler metric ω ∈ α, and fixes the momentum image ∆.We shall sometimes refer to this normalization as the canonical normalization of ∆.In this setup, we recall Definition 2.1.Let v > 0 be a positive smooth weight function on ∆.A v-soliton on X is a T-invariant Kähler metric ω ∈ 2πc 1 (X) which satisfies the relation (4): In the special case v = e ξ,µ we obtain a Kähler-Ricci soliton in the sense of [72].
Proof.We start by showing that (4) implies that ω is (v, w)-cscK with the weight w specified in the Lemma.Taking the trace in (4) with respect to ω gives where (ξ i ) i=1,••• ,r is a basis of t and v ,i denotes the partial derivative in direction of ξ i .On the other hand, by taking the interior product of (4) with ξ i and using that ξ i is Killing with respect to ω, we get where µ ξ ω := µ ω , ξ is the momentum of ξ.It follows that for some constant c.As we consider the canonical normalization of µ ω (corresponding to the natural lifted T-action on K −1 X ), one can see that c = 0. Indeed, the infinitesimal actions A i of the elements of the basis (ξ i ) i on smooth sections of K −1 X are given by A i (s) := L ξ i s.We denote by H g the induced hermitian metric on K −1 X through the Riemannian metric g ω of ω (so that H g has curvature ρ ω ) and by H = v(µ ω )H g the induced hermitian metric with curvature ω (by using (4)); comparing the actions of the corresponding Chern connections, ∇ g ξ i and We thus deduce (9).Now letting c = 0 in (9), multiplying it by v ,i (µω) v(µω) , and taking the sum over i, give which substituting back in (8) yields Now we show the converse.To this end, let ω ∈ 2πc 1 (X) be a T-invariant Kähler metric, v > 0 a positive smooth function on the canonically normalized polytope ∆ and w = 2(m + d log, µ )v the weight defined in Lemma 2.2.Let h ∈ C ∞ T (X) be an ω-relative Ricci potential, i.e.
Taking the trace with respect to ω and the interior product with ξ ∈ t in the above identity we get ( 11) , where we have used the canonical normalization of µ ω to determine the additive constant in the second inequality (as we did for ( 9)).Similar computations as in the first part of the proof (using ( 11)) give where ∆ ω,v := 1 v(µω) δ ω v(µ ω )d is the weighted Laplacian, see Appendix A. Using the second equality in (8), we compute Substituting back in (12) we obtain It follows that if ω is (v, w)-cscK then h = log v(µ ω ) + c by the maximum principle, showing that ω satisfies (4).
Remark 2.3.Using the second relation in (11) it follows that under the canonical normalization of µ ω we have ( 14) This is precisely the normalization of µ ω used in [72, Sect.2].
Proof.The proof is similar to the one of Lemma 2.2.
If ω is a v-soliton with v := −(m+2) , specializing ( 8) and ( 9) to the specific choice of v, and letting f := (µ ω ) = µ ξ ω + a, we get the identities Multiplying by f 2 the first equality and taking the sum with the second equality multiplied by mf gives The RHS is the (m + 2, f )-scalar curvature (see [2]) and it is straightforward check that the above equality is equivalent with the condition that ω is an ( −(m+1) , 2ma −(m+2] )-cscK metric.
In the other direction, for any Multiplying the first identity by f 2 and summing with the second identity multiplied by mf gives If we suppose that (15) holds, we conclude again by the maximum principle that ((m+2) log f +h) must be constant.
Lemma 2.6.On a Fano manifold (X, T), a T-invariant Kähler metric ω ∈ 2πc 1 (X) is a −(m+2)soliton with respect to a positive affine linear function = ξ, µ + a if and only if the lift ξ of the vector field ξ to K X , via the hermitian connection ∇ h with curvature −ω and the ω-momentum (µ ω ) of ξ, is a Reeb vector of a Sasaki-Einstein (transversal) structure of transversal scalar curvature 2am, defined on the unit circle bundle N of (K X , h).
Remark 2.7.The correspondence in Lemma 2.6 is, in fact, local and can be deduced directly from the relation between the transversal Ricci tensors of the two Sasaki structures on the CR manifold N ⊂ K X , respectively defined by ξ and the regular Reeb vector field χ (cf.[52,42]).
Proofs of Propositions 1 and 2. Propositions 1 and 2 from the introduction follow directly from Lemmas 2.2, 2.4 and 2.6 above.
3. The coercivity principle: Plan of proof of Theorem 1 We consider the following general setup, based on the results of [27,29,71].As before, we let T ⊂ Aut r (X) be a fixed connected compact torus in the reduced group of automorphisms of X, and denote by G = T C ⊂ Aut r (X) the corresponding complex torus.
By the above remark, for any Kähler metric ω ϕ in the Kähler class [ω 0 ], there exists a uniquely determined ω 0 -relative potential ϕ ∈ K(X, ω 0 ) satisfying We shall denote by K(X, ω 0 ) (resp.KT (X, ω 0 )) the subspaces of normalized ω 0 -relative Kähler potentials satisfying the above equality.We notice that the group G = T C naturally acts on the space of Kähler metrics in [ω 0 ], preserving the subspace of T-invariant K ȧhler metrics.This induces an action [G] on the spaces K(X, ω 0 ) and KT (X, ω 0 ), such that We introduce the G-relative distance on K(X, ω 0 ) and KT (X, ω 0 ) by 1 (ϕ 0 , ϕ 1 ) and thus d It is some times more natural to introduce G-coercivity in terms of the functional J, via the following result Proposition 3.4.[29] F is G-coercive if and only if there exist uniform positive constants (λ , δ ) such that then it is bounded below by (17).
Following [27], one can consider the metric completion (E 1 (X, ω 0 ), d 1 ) of (K(X, ω 0 ), d 1 ), which can be characterized by a suitable continuously embedded subspace in L 1 (X, ω 0 ); similarly we let (E 1 T (X, ω 0 ), d 1 ) be the metric completion of (K T (X, ω 0 ), d 1 ) which, again by the results in [29], can be viewed as the closed subspace of T-invariant elements of E 1 (X, ω 0 ).It will be important for us that (E 1 T (X, ω 0 ), d 1 ) is a geodesic space, i.e. each two elements , called a weak geodesic, obtained as the limit of C 1, 1-geodesics between elements of K T (X, ω 0 ), see [22,27].The latter object is a curve [22,16,25] and the proof of Proposition 5.8 below for more details about the weak C 1, 1-geodesics).
In [29,Thm. 3.4], the following general principle is established.Theorem 3.6 (Coercivity Principle).Let F : K T (X, ω 0 ) → R be a lower semicontinuous (lsc) functional with respect to d 1 , and F and, for some C > 0, d 1 (0, ψ j ) ≤ C, then there exists a ψ ∈ E 1 T (X, ω 0 ) and a subsequence {ψ T (X, ω 0 ), d 1 ).Then, the following two conditions are equivalent: The above result provides a clear framework for achieving the proof of Theorem 1: we need to find a suitable largest lsc extension of the weighted Mabuchi functional M v,w to the space E 1 T (X, ω 0 ), and show it satisfies the properties (i)-(iv).Notice that the invariance of M v,w under the action of G = T C is equivalent to the necessary condition (3) for the existence of a (v, w)-cscK metric whereas (iii) will follow from Theorem 1.5 once the regularity condition (ii) is established.Furthermore, the property (i) is proved in [57, Thm.1], so the core of our arguments is to define the extension of M v,w to E 1 T (X, ω 0 ) and establish the properties (ii) and (iv).These steps will be respectively detailed in Theorems 6.1, 7.1 and 6.17 below.

K-stability via coercivity: Deriving Corollary 1 from Theorem 1
We consider the following general setup, based on the results of [10,15,66,71,18,50,58] which deal with the K-polystability and uniform K-stability in the unweighted cscK case.Let T ⊂ Aut r (X) be a connected compact torus in the reduced group of automorphisms of X. Definition 4.1.A T-equivariant Kähler test configuration (X , A ) associated to (X, α, T) is a normal compact Kähler space X endowed with • a flat morphism π : X → P 1 ; • a C * -action ρ covering the standard C * -action on P 1 , and a T-action commuting with ρ and preserving π; ) is a smooth polarized variety and α = 2πc 1 (L), a polarized test configuration is a normal polarized variety (X , L ) such that for some r ∈ N * , (X , 1 r 2πc 1 (L )) defines a Kähler test configuration of (X, α) and, under Π 0 , (X, L | X×{τ } ) ∼ = (X, L r ).
4.1.Non-Archimedean functionals.We recall that any T×S 1 -invariant Kähler metric Ω ∈ A on X gives rise to a smooth ray of T-invariant Kähler metrics ω t ∈ α on X defined by Definition 4.2.Let F be a functional defined on the space of T-invariant Kähler metrics on X in the class α.We say that F admits a non-Archimedean version F NA , defined on a subclass C of T-equivariant Kähler test configurations (X , A ) associated to (X, α, T), if for any (X , A ) ∈ C, and any induced smooth ray of T-invariant Kähler metrics ω t ∈ α on X, the slope lim t→∞ F(ωt) t is well-defined and given by a quantity F NA (X , A ) which is independent of the choice of the T × S 1 -invariant Kähler form Ω ∈ A .
We give below two key examples of non-Archimedean versions of known functionals.The first one is established in the polarized case in [18] and in the generality we consider in [33,65]: where Π is the morphism (19) and α denotes both the Kähler class on X and its pull back to X × P 1 .
The above expression generalizes to dominating smooth test configurations which are only relatively nef (in the terminology of [66]), thus also providing a non-Archimedean version of J for any Kähler test configuration: indeed, by the equivariant Hironaka resolution, any Tequivariant test configuration can be dominated by a smooth relatively nef Kähler dominating test configuration, and the computation of J NA on the latter does not depend on the choice made.
The non-Archimedean functional J NA defined above is always non-negative and equals to zero precisely when (X , A ) is the trivial test configuration.The latter statement is established in [18,Thm. 7.9] in the polarized case, and follows from the results in [66] in the Kähler case: see in particular [66,Lemma 4.8] with G trivial and recall that the J-norm is Lipschitz equivalent to the d 1 -distance, so that the unique weak geodesic ray associated to a test configuration with vanishing J NA -norm must be constant, and hence the test configuration must be trivial by [66,Cor. 3.12].Thus, J NA can be thought of as a "norm" on the space of Kähler test configurations.
In order to obtain a norm which is zero for more general product test configurations, in [34,50,58] the authors consider smooth rays ωt ∈ α of T-invariant Kähler metrics on X which are obtained by composing an induced ray ω t from a T × S 1 -invariant Kähler metric Ω ∈ A on X with the flow of a vector field Jξ, where ξ ∈ t, i.e. ωt = exp(tJξ) * (ω t ).They show that the slope is well-defined and independent of the choice of induced ray ω t .We notice that when ξ ∈ 2πΛ is a lattice element (or more generally is rational), ξ induces an C * -action ρ ξ on X and ωt is an induced smooth ray from another Kähler test configuration (X ξ , A ξ ), called the ξ-twist of (X , A ), obtained from X by composing the initial C * -action ρ with ρ ξ , and compactifying trivially at infinity.(For instance, the product test configurations are precisely the ξ-twists of the trivial test configuration.)In this case, J NA (X ξ , A ξ ) is just the non-Archimedean J-functional computed as in Example 4.3 on (X ξ , A ξ ).For a general ξ, the quantity (X ξ , A ξ ) in the notation is not a test configuration in the usual sense (it is sometimes refereed to as a R-test configuration) but the value J NA (X ξ , A ξ ) can be obtained as a continuous extension of the corresponding quantity for rational ξ's.Following [50,58], we let A key observation [18,50,58] in the polarized case is that the equality in (20) on the class of T-equivariant polarized test configuration of (X, L, T).
Remark 4.6.In the unweighted case (i.e ), F v,w (X , A ) admits an equivalent expression in terms of intersection cohomology numbers on X , see [63,73].This allows one to extend the definition of the (unweighted) Donaldson-Futaki invariant to any normal Kähler test configuration.For arbitrary weight functions v > 0 and w, we don't have as yet a general definition for F v,w but ( 21) can be readily extended to orbifold test configurations.We also notice that the assumption on the central fibre in Example 4.4 is necessary in order to ensure the equality F v,w = M NA v,w (see [65] for a general formula of the non-Archimedean version of the unweighted Mabuchi energy).It will be interesting to obtain a non-Archimedean version of M v,w for any orbifold T-equivariant Kähler test configuration.4.2.F NA -K-stability.Definition 4.7.Let F be a functional defined on the space of T-invariant Kähler metrics on X in the Kähler class α, and suppose F admits a non-Archimedean version F NA (X , A ) (see Definition 4.2), defined on a class C of T-equivariant Kähler test configurations (X , A ) associated to (X, α, T).We suppose that C contains the product test configurations.We say that: is T-equivariant F NA -K-semistable, and, furthermore, F NA (X , A ) = 0 if and only if (X , A ) is a product test configuration.
(iii) (X, α, T) is T C -uniform F NA -K-stable (on test configurations of the class C) if there exists a uniform positive constant λ > 0 such that for any test configuration (X , A ) ∈ C K-semistable; furthermore both (ii) and (iii) imply (i) and, in the polarized case, (iii) implies (ii) by the results in [18,50,58].Theorem 4.9.[15,50,58,66] Suppose F is a functional defined on the space of T-invariant Kähler metrics in α, which is T-relatively T C -proper.Suppose, furthermore, that F admits a non-Archimedean version F NA defined for a class C of T-equivariant Kähler test configurations of (X, α, T).
Proof.For the first part, we follow [66] with some minor modifications.We want to show that if We fix a T × S 1 -invariant Kähler form Ω ∈ A and let ω t be the corresponding ray of smooth Tinvariant Kähler forms in α, and ψ t ∈ K T (X, ω 0 ) the normalized smooth ray of Kähler potentials satisfying I(ψ t ) = 0.According to [65], the Kähler test configuration (X , A ) also determines a unique C 1, 1 weak geodesic ray ϕ t in K 1, 1(X, ω 0 ), emanating from ψ 0 .Furthermore, ϕ t is invariant under T (by its uniqueness) provided that we have ψ 0 ∈ K T (X, ω 0 ).According to [66,Prop.4.2], we can consider instead of A the relative Kähler class ) (for a constant c determined from A and where [X 0 ] denotes the divisor corresponding to the central fibre X 0 of X ), such that the C 1, 1 weak geodesic ray ϕ c t corresponding to (X , A c ) is the projection of ϕ t to the slice K 1, 1 T (X, ω 0 )∩I −1 (0).Notice that the smooth (1, 1)-form Ω−cπ * ω FS ∈ A c defines the same smooth ray ω t of T-invariant Kähler metrics, and thus the same ray of smooth potentials The key point is that (17) and lim t→∞ We can now apply the arguments in the proof of the implication '(2) ⇒ (5)' of [66,Thm. 4.4], by replacing the Mabuchi energy with the abstract functional F and the group Aut 0 (X) with T C , noting that in our T-relative situation instead of the cscK potential ψ 0 in [66,Prop. 4.10] we can take any Kähler potential in K T (X, ω 0 ) (as ω ψ 0 is T-invariant and T C is reductive).We thus deduce the implication (5) of [66], namely, that the geodesic ray ϕ c t associated to (X , A c ) is given by the ω 0 -relative Kähler potentials of exp(tJξ) * (ω ψ 0 ) in I −1 (0), where ξ is a vector field in the Lie algebra of T; it follows from [66, Thm.A.6] that (X , A c ) and hence also (X , A ) is a product test configuration.
The second part follows immediately from (18) and Example 4.3-bis.
We next apply Theorem 4.9 to F = M v,w and showing that if M v,w is bounded below on K T (X, ω 0 ) (in particular if M v,w is T-relatively T Cproper), then the relation (7) holds and M v,w defines a functional on the space of T-invariant Kähler metrics in α (see Remark 1.2).In this case, Example 4.4 tels us that F v,w (X , A ) defines a non-Archimedean version of M v,w .We can now apply Theorem 4.9.

Semi-simple principal fibrations
Let (X, ω) be a compact Kähler 2m-manifold, endowed with a hamiltonian isometric action of an r-dimensional torus T. As T will act on various spaces, we shall use at times upper and under scripts to emphasize the space, on which T acts.For instance, T X will denote the T-action on X.Let t be the Lie algebra of T and Λ ⊂ t the lattice of generators of circle groups in T (i.e.T = t/2πΛ).We denote by µ ω : X → ∆ ⊂ t * the normalized T X -momentum map of ω, i.e. whose image is a fixed compact convex polytope ∆ ⊂ t * .
Let B = B 1 × • • • × B k be a 2n-dimensional cscK manifold, where each (B a , ω Ba ), a = 1, . . ., k is a compact cscK Hodge Kähler 2n a -manifold (i.e. 1 2π [ω Ba ] ∈ H 2 (B a , Z)), and π B : P → B a principal T-bundle endowed with a connection 1-form θ ∈ Ω 1 (P, t) with curvature (22) dθ Remark 5.1.The principle T-bundle P above can be described in terms of r complex line bundles over B as follows.Fixing a lattice basis {ξ 1 , . . ., ξ r } of t, and writing p a = r i=1 p ai ξ i , p ai ∈ Z, a = 1, . . .k, (22) yields that P is the (fiber-wise) product of r principle U(1)-bundles P i → B, where each P i is associated to a complex line bundle Fixing a connection 1-form θ on P as in (22) amounts to introducing a hermitian metric h * i on each L * i , with curvature − r a=1 p ai π * B (ω Ba ), and identifying P i ⊂ L * i with the corresponding unitary S 1 -bundle.
Let D = ann(θ) ⊂ T P be the horizontal distribution defined by θ, leading to a splitting where t P denotes the Lie algebra of T P inside C ∞ (P, T P ), corresponding to the T-action T P on P .The lift J B of the integrable almost complex structure of B to D gives rise to a CR structure (D, J B ) on P (of co-dimension r).
We further let Z := X × P and consider the induced T-action, denoted T Z , generated by (−ξ X i + ξ P i ) for any basis of Λ as above.We thus define It follows that Y is a 2(m + n)-dimensional smooth manifold, and π Y : Z = X × P → Y is a principal T-bundle over Y whereas π B : P → B defines a fibration π B : Y → B with smooth fibres X, as summarized in the diagram below.
The T X -action on the factor X in Z = X × P descends to a T-action on Y , denoted T Y , which preserves each fibre (and thus coincides with the action of T X ).Notice that the 1-form θ also defines a connection 1-form on Z with horizontal distribution H : (23) T giving rise to an induced CR structure (H , J = J X ⊕ J B ) of co-dimension r on Z, which is clearly invariant under the T Z -action, and therefore defines a T Y -invariant complex structure J Y on Y .
We now consider Kähler metrics on Y , compatible with the fibre-bundle construction of the above form.To simplify the notation, we denote by ω a := ω Ba the (fixed) cscK metric on each factor B a , by ω a T-invariant Kähler structure in the class α on X, and by ω the resulting Kähler structure on Y , which is defined in terms of a basic 2-form on Z = X × P , depending on k real constants c a ∈ R (which will be fixed) such that for each a = 1, . . ., k, the affine linear function p a , µ + c a on t * is strictly positive on the momentum image ∆: In the above expression, •, • stands for the natural pairing between t and t * : thus p a , µ ω is a smooth function, µ ω , θ is a 1-form, and dµ ω ∧ θ is a 2-form on Z.One can directly check from the above expression that ω is closed, T Z -basic, and is positive definite on (H , J X ⊕ J B ), so it is the pullback of a Kähler form on Y .We shall tacitly identify in the sequel the Kähler form on Y with its pullback (24) on Z = X × P .Notice that ω is T Y -invariant and µ ω , seen as a smooth T Z -invariant function on Z, is the ∆-normalized momentum map.
Remark 5.2.The horizontal part ωh : −→ X ×B is invariant and basic with respect to the action T P on the factor P , and thus induces a Hermitian (non-Kähler in general) metric on X × B = X × k a=1 B a , given by which is an instance of warped geometry.On can thus think of (X × B, ωh ) and (Y, ω) as being related by the twist construction of [70] applied to (Z, ω, T Z ) and (Z, ω, T P ).
Definition 5.3.The Kähler manifold (Y, T Y ) constructed as above will be called a semi-simple (X, T)-principal fibration associated to the Kähler manifold (X, T) and the product cscK manifold The T Y -invariant Kähler metric ω on Y constructed from a T X -invariant Kähler metric ω on X (and fixed cscK metrics ω a on B a ) will be called bundle-compatible.
Remark 5.4.In the case when (X, T, ω) is a toric Kähler manifold, a semi-simple (X, T)-principal fibration endowed with a bundle-compatible Kähler metric is an example of a semi-simple rigid toric fibration in the sense of [7], and is thus described by the generalized Calabi construction with a global product structure on the base and no blow-downs.
5.1.The space of functions.The above bundle construction gives rise to a natural embedding of the space C ∞ T (X) of T X -invariant smooth functions on X to the space C ∞ T (Y ) of T Y -invariant smooth functions on Y : for any ϕ ∈ C ∞ T (X) we consider the induced function on Z = X × P , which is clearly T Z -invariant, and thus descends to a smooth T Y -invariant function on Y .We shall tacitly identify ϕ and its image in . Notice that the above embedding is closed in the Fréchet topology, as we can identify a smooth T X -invariant function on X with a smooth T Y -invariant function ϕ on Y , which has the property More generally, for any T Y -invariant smooth function ψ ∈ C ∞ T (Y ) its lift π * Y ψ to Z = X × P a smooth function which is both T Z and T X -invariant, or equivalently T X and T P invariant.It thus follows that π * Y ψ can be equivalently viewed as a T X -invariant smooth function on X × B, i.e. we have an identification In particular, for any fixed point x ∈ X, we shall denote by ψ x ∈ C ∞ (B) the induced smooth function on B, and for any fixed point b ∈ B by ψ b ∈ C ∞ T (X) the induced function on X.We thus have the identification The space of bundle-compatible Kähler metrics.We shall next use the construction of (24) in order to identify the space K T (X, ω 0 ) of T X -invariant ω 0 -relative Kähler potentials on X as a subset of the space K T (Y, ω0 ) of T Y -invariant ω0 -relative Kähler potentials on Y .Lemma 5.5.Let ω ϕ = ω 0 + d X d c X ϕ be an T X -invariant Kähler form on X in the Kähler class α = [ω 0 ], where ϕ ∈ K T (X, ω 0 ) is a T X -invariant smooth function on X. Denote by µ ϕ the momentum map of T X with respect to ω ϕ , satisfying the normalization µ ϕ (X) = ∆, and by ωϕ the induced Kähler metric on Y , via (24).Then, Y ϕ, where ϕ stands for the induced smooth function on Y .
Proof.Recall that µ ϕ = µ 0 + d c ϕ (see ( 5)).By (24), the pullback of ωϕ to Z = X × P is , for any T X -invariant smooth function ϕ on X.To this end, let us choose a basis {ξ 1 , . . ., ξ r } of t, with dual basis {ξ 1 , . . ., ξ r } of t * , and write d c X ϕ = r j=1 (d c X ϕ)(ξ X j )ξ j and θ = r j=1 θ j ξ j for 1-forms θ j on Z such that θ j is zero on H and θ j (ξ P i ) = θ j (−ξ X i + ξ P i ) = δ ij .Thus, ( 26) is equivalent to Evaluating the RHS of the above equality on the generators (−ξ X j + ξ P j ) of t Z , we see that it is a π Y -basic 1-form on Z, and thus is the pullback of a 1-form on Y via π Y .The claim follows easily.
Thus, Lemma 5.5 defines an embedding K T (X, ω 0 ) ⊂ K T (Y, ω0 ) and we have also identified in Sect.5.1 a natural embedding of the space of T X -invariant functions on X into the space of T Y -invariant functions on Y , through their pull-backs to Z = X × P .
Letting θ := r j=1 θ j ⊗ ξ P j be the decomposition of the connection 1-form θ on P in a basis {ξ 1 , . . ., ξ r } of the lattice Λ ⊂ t, and θ ∧r := θ 1 ∧ • • • ∧ θ r , it follows from (24) and Lemma 5.5 that for any ϕ ∈ K T (X, ω 0 ) ⊂ K T (Y, ω0 ), the measure ω[m+n] ϕ on Y is the push-forward of the measure on Z: Corollary 5.6.There exists an embedding K T (X, ω 0 ) ⊂ K T (Y, ω0 ) such that, for any smooth curve , where p(µ) is the positive weight function on ∆ defined in (28), L X 1,p is the p(µ)-weighted length function on K T (X, ω 0 ) given by and L Y 1 is the length function on K T (Y, ω0 ) corresponding to the weight p = 1.In particular, for any , where d X 1,p is the induced distance via the length functional L X 1,p .Proof.A direct consequence of (29).
Lemma 5.7.Let ϕ be a smooth T X -invariant function on X, also considered as a smooth T Yinvariant function on Y , and ω be an T X -invariant Kähler metric on X with ω the corresponding T Y -invariant Kähler metric on Y given by (24).Then Proof.We use that Proof.Let ϕ 0 , ϕ 1 ∈ K T (X, ω 0 ).If ϕ 0 and ϕ 1 can be connected with a smooth geodesic ϕ t , i.e. with a smooth curve in K T (X, ω 0 ) such that (30) φ = ||d φ|| 2 ωϕ , then, by Lemma 5.7, it follows that ϕ t is also a smooth geodesic in In general, by the results in [22], ϕ 0 , ϕ 1 can be connected only with a weak C 1, 1-geodesic in K 1, 1 T (X, ω 0 ), where K 1, 1(X, ω 0 ) stand for the space of C 1 (X) functions ϕ on X such that ω 0 + dd c ϕ ≥ 0 and has bounded coefficients as a (1, 1)-current.More precisely, letting Σ := {1 < z < e} ⊂ C, it is shown in [22] that there exists a unique weak solution (i.e. a positive (1, 1)-current in the sense of Bedford-Taylor) of the homogeneous Monge-Ampère equation It was later shown in [25] that Φ is actually of regularity C 1,1 (X×Σ).Note that, by the uniqueness, Φ is T-invariant as soon as ϕ 0 and ϕ 1 are.The link with ( 30) is (see [64]) that if Φ were actually smooth, we can recover the smooth geodesic ϕ t joining ϕ 0 and ϕ 1 by letting t := log |z| and ϕ t (x) := Φ(x, log |z|).In the general case, the curve ϕ t of (weak) ω 0 -relative pluri-subharmonic potentials (of regularity C 1,1 (X × [0, 1])) is referred to as the weak C 1, 1-geodesic joining ϕ 0 and ϕ 1 .
We are thus going to check that any weak C 1, 1-geodesic on X (invariant under T X ) defines, via Lemma 5.5, a C 1, 1-geodesic on Y .To this end, we need to show that Φ satisfies (32) (π Y Φ ≥ 0, the regularity statements being automatically satisfied on Y . By the results in [22] and [16], Φ can be approximated as ε → 0, both it the weak sense of currents and in C 1,α (X × Σ) (for a fixed α ∈ (0, 1)), by smooth functions Ψ ε (x, z) on X × Σ which solve By the uniqueness of the smooth solution of (33) (and using that both ϕ 0 , ϕ 1 are T X -invariant), we have that Ψ ε (x, z) is a T X -invariant smooth function on X for any z ∈ Σ; furthermore, the positivity condition on the second line yields that Ψ ε (x, z) ∈ K T (X, ω 0 ) for any z ∈ Σ.We can then also see Ψ ε (x, z), via its pull-back to X ×P ×Σ, as a T Y -invariant smooth function on Y ×Σ; the arguments in the proof of Lemma 5.5 yield that π Furthermore, by the same proof, we have the following equality of volume forms on X × P × Σ: where, we recall, p(µ i with respect to a basis {ξ 1 , . . ., ξ r } of Λ ⊂ t), and, for any fixed z ∈ Σ, µ Ψ ε denotes the normalized T X -momentum map (5) of ω 0 +d X d c X Ψ ε .Notice that, as p is uniformly bounded on ∆ by positive constants, it follows by (34) weakly (as measures on Z × Σ).The push-forward measure of (π , so we obtain on Y : Furthermore, using the C 1,α -convergence of Ψ ε to Φ, we get the weak convergences (of positive (1, 1)-currents): Thus, (32) follows.
Proof.We apply the arguments in the proof [4,Prop. 7] to both (X, T X ) and (Y, T Y ) to compute the corresponding scalar curvatures, and compare the results.
On X, we consider the open dense subset X ⊂ X of stable points of the T X -action, and take the quotient π S : X → S := X/T C X under the induced complexified action T C X ∼ = (C * ) r (thus S is a complex 2(m − r)-dimensional orbifold). 2 Consider the point-wise ω-orthogonal and T-invariant decomposition and write the Kähler structure (g, J, ω) on X as where, for a fixed basis {ξ 1 , . . ., ξ r } of t, the 1-forms η j on X are defined by (η j ) ).We next fix a local volume form Vol S on S in some holomorphic coordinates, and write pointwisely (36) ω for some positive (locally defined) smooth function Q on X (where both ω and π * S (Vol S ) are seen as sections of ∧ m−1 H * ).According to [4, Prop.7], we have that (37) κ ω [m]  .
We can now make a similar argument on Y , noting that the Kähler reduction of Y by the induced T Y -action is S × B; taking a local volume form in holomorphic coordinates on S × B of the form Vol S ∧ Vol B 1 ∧ • • • ∧ Vol B k , and using (24), we see that a Ricci potential on Y (when pulled back to X × P ) is written as where Vol Ba is a Ricci potential of (B a , ω a ) and Thus, we obtain as functions on X×P .Introducing a basis (ξ i ) i of Λ and writing the connection 1-form θ ∈ Ω 1 (P, t) as θ = r j=1 θ j ⊗ ξ P j (where the 1-forms θ j on P are such that θ j is zero on D and θ j (ξ P i ) = δ ij ), we compute for the scalar curvature of ω 2 Our argument is actually local, around each point in X, so one can assume without loss that S is smooth.
By ( 26) and ( 38), the pullback of d Y d c Y κ to X × P is given by, where in the last equality we used ( 22) and we have denoted by p a the induced vector field on X by the element p a ∈ t.We shall compute the term d c X κ(p a ) on X: using (37) we get (41) Taking the wedge product of both sides of (36) with gives Applying the Lie derivative L Jpa to the above equality yields where we used that L Jpa η j is a basic form (since (L Jpa η j )(ξ i ) = −η j ([Jp a , ξ i ]) = 0).We thus get Using the above equation in (40), we continue the computation Recall that ( 27) on Z we have ω [na] a ∧ θ ∧r .Similarly, by (24), Using ( 39), ( 43), ( 27) and ( 44), we obtain On the other hand, using a basis (ξ i ) of t with a dual basis (ξ i ) of t * , we compute Comparing the above expression with (45), we obtain ( 46) Using that (as functions on X × P ) µ ω = µ ω and g ω (ξ i , ξ j ) = g ω(ξ i , ξ j ) (see the proof Lemma 5.7), we further compute from ( 46) where for passing to the last line we used the identity ∆ we further get Scal pv (ω).
The expression (35) follows from the above formulae.
Lemma 5.10.The restriction of the weighted Mabuchi energy M Y v,w on Y to the subspace K T (X, ω 0 ) ⊂ K T (Y, ω0 ) is equal to CM X pv, w, where p, w, w are given in Lemma 5.9 and C = Vol(B, ω B ).
Proof.A direct corollary of Lemma 5.9 and Definition 1.1.
We now specialize to the case when each (B a , ω a ) is a Hodge Kähler-Einstein manifold with positive scalar curvature s a = 2n a k a , where k a ∈ N. Equivalently, 2πc 1 (B a ) = k a [ω a ] for a positive integer k a and an integral Kähler class 1 2π [ω a ].Notice that k a must be a positive divisor of the Fano index Ind(B a ) of B a , which yields the a priori bound 1 ≤ k a ≤ Ind(B a ).We also assume that (X, T) is Fano, with canonically normalized momentum polytope ∆.We then have Lemma 5.11.In the setting above, if the affine linear functions ( p a , µ + k a ) > 0 on ∆, then the bundle-compatible Kähler metric ω on Y corresponding to the constants c a = k a belongs to deRham class 2πc 1 (Y ).Furthermore, ω is a v-soliton if and only if ω is a pv-soliton.
Proof.By using (38) and rearranging the terms in (40), we have the following relation (written on Z): where ρ ω, ρ ω and ρ a respectively denote the Ricci forms of (Y, ω), (X, ω) and (B a , ω a ), pulled back to Z, and µ ρω := d c X κ is the "momentum map" with respect to the Ricci form ρ ω .As in (42), we have X h for some T-invariant smooth function on X; by using that the momentum polytope ∆ is canonically normalized, we have (see (11)) µ ρω −µ ω = d c h.A closer look at the proof of Lemma 5.5 and the relation (48) The claim follows from the above.
Remark 5.12.Lemma 5.11 provides a useful way to construct semi-simple (X, T)-principal Fano fibrations.Indeed, for given positive Hodge Kähler-Einstein manifolds (B a , ω a ) as above, with corresponding integer constants k a , and a given Fano manifold (X, T) with associated canonical polytope ∆, one can try to find the possible principal T-bundles P over B = k a=1 B a , for which the corresponding semi-simple (X, T)-principal fibration is Fano.Such principal T-bundles P are in correspondence with the choice of lattice elements p a ∈ Λ ⊂ t and Lemma 5.11 tels us that for a set of elements p a to determine a Fano semi-simple (X, T)-principal fibration Y , it is sufficient to check that all the affine linear functions For instance, if we take B = B 1 = P 1 with a Fubini-Study metric ω 1 of scalar curvature 4 (so that k 1 = 2 and ω 1 is primitive) and (X, T) = (P 1 , S 1 ) with canonical polytope ∆ = [−1, 1], then the possible Fano (P 1 , S 1 )-principal fibrations will correspond to p 1 ∈ Z such that p 1 µ + 2 > 0 on [−1, 1], i.e. p 1 = ±1, 0 are the only possible values.This gives rise to the Fano surfaces 1)) and P 1 × P 1 .In general, the isomorphism class of the principal T-bundle P over B, and hence also the semi-simple (X, T)-principal Fano fibration constructed as above, is encoded by the Hodge classes The a priori bounds 1 ≤ k a ≤ Ind(B a ) for k a show that for given base B = k a=1 B a and fibre (X, T), there are only a finite number of semi-simple (X, T)-principal Fano fibrations constructed this way.
Remark 5.13.The relationship between the Ricci potentials h and h established in the proof of Lemma 5.11 and (29) yield, via Remark 2.3, that if the momentum map µ ω of (X, ω, T X ) is canonically normalized, then the momentum map µ ω = µ ω of the corresponding bundle-compatible Kähler metric ω on (Y, T Y ) is also canonically normalized.
We end up this section with the following straightforward extension of [7,Lemma 5].
Lemma 5.14.Suppose Y is a semi-simple principal (X, T)-fibration over B, such that T is a maximal torus in the reduced group of automorphisms Aut r (X).Let ω be a bundle-compatible Kähler metric on Y corresponding to a T-invariant Kähler metric ω on X, and K B ⊂ Aut r (B) be a maximal compact torus in the reduced group of automorphisms of B which (without loss by Lichnerowicz-Matsushima theorem) belongs to the isometry group of ω B .Then ω is invariant under the action of a maximal torus K Y ⊂ Aut r (Y ), and we have an exact sequence of Lie algebras Furthermore, for any positive weight functions v, w 0 defined on ∆ ⊂ t * , there exists a unique affine-linear function ext v,w 0 on t * such that, when pulled-back to the dual Lie algebra k * Y of K Y , (v, w 0 ext v,w 0 ) satisfy (3) with respect to ω on Y , and any affine-linear function on k * Y .Proof.The proof of the above result is not materially different than the proof of [7, Lemma 5] (which is made in the case when (X, T) is toric and v = w 0 = 1).We only give a sketch.A Killing potential f for a Killing vector field K ∈ k B := Lie(K B ) is of the form f = k a=1 f a , where f a is a Killing potential of (B a , ω a ).Letting K be the horizontal lift of K to P (using the t P -valued connection 1-form θ), one can check that the vector field on P K = K + k a=1 f a ξ P pa is a CR vector field on (P, D, J B ), hence also on (Z, H , J B ⊕ J X ).Furthermore, a direct verification in (24) reveals that (49) ı showing that K also preserves ω.We thus obtain a lift kB of the Lie algebra k B = Lie(T B ) to Z, which clearly commutes with the action T Z , and preserves both the CR structure of (Z, H ) and the 2-form ω.The Lie algebra k Y of K Y is then induced by t X ⊕ kB ⊂ T Z, which descend to an abelian Lie algebra of Killing fields on Y .The maximality of K Y ⊂ Aut r (Y ) and the exactness of the sequence follow from the maximality of each K B ⊂ Aut r (B) and T ⊂ Aut r (X), and the fact that (recall that Y is a locally trivial X-fibre bundle and therefore the fibres have trivial normal bundle) any holomorphic vector field on Y projects under π B to a holomorphic vector field on B.
For the final claim in Lemma 5.14, notice that by (49) the Killing potentials of all lifted Killing vector fields K from B are of the form k a=1 ( p a , µ ω + c a )f a .Thus, by Lemma 5.9 and using ( 27), the integral condition ( 3) on (Y, ω) will be zero for any such Killing potential, as soon as we normalize Ba f a ω na a = 0 and assume ext v,w 0 ∈ Aff (t * X ).On the other hand, examining ( 3) on (Y, ω) for the Killing potentials (µ ω), ∈ Aff (t * ) reduces (again by Lemma 5.9 and ( 29)) to an integral relation on (X, ω) which defines a unique element ext v,w 0 ∈ Aff (t * ).

Weighted functionals and distances and their extensions
Let ω 0 a T-invariant Kähler metric in the Kähler class α.We denote by PSH T (X, ω 0 ) the space of T-invariant ω 0 -pluri-subharmonic functions in L 1 (X, ω 0 ), and define the class of potentials of full volume by According to [27], the d 1 -completion of K T (X, ω 0 ) can be identified with the subspace of potentials of finite energy, i.e.
Our main result in this section will be the existence of a lsc extension of the weighted Mabuchi functional (defined in Definition 1.1 on the space K T (X, ω 0 )) to a functional on the space E 1 T (X, ω 0 ).Our starting point is that the weighted Mabuchi energy M v,w admits a weighted Chen-Tian decomposition [56,Thm. 5] into energy and entropy parts as follows: where ρ ω 0 is the Ricci form of ω 0 and the functionals I w and I ρω 0 v are introduced in Definition 6.2 below.We want to show the following Theorem 6.1.For smooth weight functions v(µ), w(µ) such that v(µ) > 0 on ∆, the weighted Mabuchi energy M v,w : K T (X, ω 0 ) → R extends using (50) to the largest d 1 lsc functional M v,w : E 1 T (X, ω 0 ) → R∪{∞} which is convex along the finite energy geodesics of E T (X, ω 0 ).Additionally, the extended weighted Mabuchi energy M v,w is linear in v, w, uniformly continuous in w in the C 0 (∆) topology and continuous with respect to v in the C 1 (∆) topology.
The above result is well-known for the unweighted case, by the work [14], and we will follow a similar path to get an extension in the weighted case.The proof of Theorem 6.1 will be given at the end of the section, and we detail below the definition and extension of each component of (50).
6.1.The weighted Aubin-Mabuchi functionals.Definition 6.2.[56] For a smooth weight function v(µ) on ∆, we let I v denote the functional on K T (X, ω 0 ), defined by ϕ , I v (0) = 0, and let J v := X ϕv(µ 0 )ω [m] 0 − I v (ϕ).Furthermore, for a fixed T-invariant closed (1, 1)-form ρ on X with momentum µ ρ : X → t * , we define the ρ-twisted Aubin-Mabuchi functional For v ≡ 1, we let I 1 = I, J 1 = J and I ρ v = I ρ , and notice that I, J are the functionals introduced in Definition 3.1 Remark 6.3.It follows from the above definition and the results in [56] that for any weight v(x) and a constant c, J v (ϕ + c) = J v (ϕ), allowing one to see J v as a functional on the space of Tinvariant Kähler metrics in the Kähler class α = [ω 0 ], and motivates the notation J v (ω ϕ ).Notice also that I v , J v I ρ v are linear in v.In the case when v > 0, J v is non-negative (see Lemma 6.4 below), whereas I v is monotone in the sense that for any ϕ 0 , ϕ 1 ∈ K T (X, ω 0 ) with ϕ 1 (x) ≥ ϕ 0 (x) The above inequality follows by Definition 6.2 and integrating the derivative of I v along the path tϕ 1 + (1 − t)ϕ 0 ∈ K T (X, ω 0 ) and integrating by parts.
Proof.The first relation follows from Lemma 6.4 above whereas the second inequality follows from the first and Definition 6.2.
Lemma 6.6.The restrictions of ) are respectively equal to CI X p and CJ X p , where p(µ) is the weight function defined in Lemma 5.9 and C = Vol(B, ω B ). Furthermore, if ρ is a Kähler form on Y , induced by a Kähler form ρ on X using (24), then the restriction of (I ρ 1 ) Y to the subspace Proof.The first part follows from the definition of The claim follows as (I ρ p ) X (0) = 0 = (I ρ) Y (0).
Lemma 6.8.For any weight v > 0, there exists uniform constant C = C(X, ω 0 , v) > 0 such that where Proof.The relation (53) follows from the fact that v(µ) is positive and uniformly bounded on ∆.This yields that d 1,v is a distance, as d 1 is a distance according to [27].
Lemma 6.9.For any smooth weight v > 0 we have Proof.For any smooth curve ϕ t between ϕ 1 and ϕ 2 , using Definition 6.2, we have The claim follows from the above and Lemma 6.8.

6.3.
Extensions to E 1 T (X, ω 0 ).Lemma 6.10.For any smooth weight v, the functionals I v and J v continuously extend to the space E 1 T (X, ω 0 ).Furthermore, for any ψ ∈ E 1 T (X, ω 0 ), the extended functionals are linear and uniform continuous in v, in the topology C 0 (∆).
Proof.I v is d 1 -Lipschitz by Lemma 6.9; for J v we get from Definition 6.2 Combining the above inequality with Lemma 6.9 and [27, Cor.5.7], there exists a uniform positive constant C = C(X, ω 0 , v) and, for any fixed positive real number R > 0, an increasing continuous function F R : R + → R + , F (0) = 0, defined in terms of (X, ω 0 , R), such that for any ϕ 0 , ϕ 1 ∈ K T (X, ω 0 ) with d 1 (0, ϕ i ) ≤ R, we have showing that J v is locally uniform continuous on (K T (X, ω 0 ), d 1 ) and thus extends continuously to (E 1 T (X, ω 0 ), d 1 ).The v-linearity of I v and J v is clear by continuity, see Remark 6.3.The continuity with respect to v follows from the continuous extensions of the inequalities in Lemma 6.5, noting that we have already shown that J v , J w , J, I v , I w all extend continuously, whereas || • || L 1 (X,ω 0 ) extends continuously by [27,Thm. 5.8].
Corollary 6.11.The metric completion of (K T (X, ω 0 Proof.Similarly to [29, Lemma 5.2], one can show that I v is linear along finite energy geodesics. As Lemma 6.12.Let v be a smooth weight function and ρ a T-invariant closed (1, 1)-form.The functional T (X, ω 0 ).Furthermore, the extended functional is linear and uniformly continuous in v, in the C 1 (∆) topology.
Following Berman-Witt-Nyström [13] and the recent work of Han-Li [46], we now define the extension of weighted Monge-Ampère measures to the space E T (X, ω 0 ).By the Riesz representation theorem, MA X v (ϕ) is a well-defined Radon measure.Remark 6.14.Notice that for any ϕ ∈ E T (X, ω 0 ), the measure MA v (ϕ) is absolutely continuous with respect to MA(ϕ) since v is bounded on ∆.In particular, for any positive weight v, we have that Lemma 6.15.Let v be a positive weight function and Proof.Let v(µ) be a polynomial of the form p(µ) := k a=1 ( p a , µ + c a ) na , ϕ j ∈ K T (X, ω 0 ), and f any continuous T-invariant function on X.We then have by the construction in Section 5 It follows that for each ϕ j ∈ E 1 T (X, ω 0 ) (using an approximation with a decreasing sequence of smooth potentials [17]), we have Using (56), we conclude that MA X p (ϕ j ) → MA X p (ϕ) weakly as j → ∞.For an arbitrary weight function v ∈ C 0 (∆), we take a sequence of polynomials p i of the above form converging to v in C 0 (∆).For any continuous function f on X, using (57), we have where we used the existence of the weak limits MA p i (ϕ j ) → MA p i (ϕ) and MA(ϕ j ) → MA(ϕ) as j → ∞ (by [27, Thm.5]).Taking the limit i → ∞ in the above inequality, we obtain For a finite measure χ on X we define the entropy of χ with respect to ω [m] by In the following lemma we show that the elements of E 1 T (X, ω 0 ) can be approximated in the d 1 distance by smooth potentials with converging entropy of the corresponding weighted Monge-Ampère measures.This is the weighted analogue of [15,Lemma 3.1].
It remains to show that M v,w : E 1 T (X, ω 0 ) → R ∪ {∞} is linear and continuous in v, w.For smooth potentials ϕ ∈ K T (X, ω 0 ), we have (58) Ent(ω which is manifestly linear in v.For ϕ ∈ E 1 T (X, ω 0 ), the above expression is still linear in v by Proposition 6.13.Substituting back in (50), and using Lemma 6.10 and 6.12, it follows that M v,w : E 1 T (X, ω 0 ) → R ∪ {∞} is linear in v, w.From these two lemmas, we know that I ρ v : E 1 T (X, ω 0 ) → R and I w : E 1 T (X, ω 0 ) → R are uniformly continuous in v, w.For the remaining entropy part, we notice that if which can be obtained again by approximating ϕ with a monotone sequence of smooth relative potentials and using Proposition 6.13.It follows that is uniformly continuous with respect to v for the weak topology on the space of measures.Since the entropy χ → Ent(ω m 0 , χ) is lsc on the space of finite measures with respect to the weak convergence of measures [11,Prop. 3.1], the term Ent(ω ) is lsc with respect to v. The linearity with respect to v in the RHS of (58) shows that Ent(ω We derive the following weighted version of the key compactness result from [12,14]: Theorem 6.17.Any sequence Proof.From the formula (50) and Lemmas 6.9 and 6.12, we see that Ent(ω ) is uniformly bounded under the hypotheses in the Corollary.We conclude using [46,Lemma 2.16].

Regularity of the weak minimizers of the weighted Mabuchi energy
In this section, we establish the regularity of the weak minimizers of M v,w .
The proof of this result, which is an adaptation of the arguments in [15], will occupy the reminder of the section.Definition 7.2.Let v(µ) > 0, w(µ) be smooth weight functions on ∆ and ρ > 0 a T-invariant Kähler form on X.We let and M ρ v,w := M v,w + I ρ , where I ρ is introduced via Lemma 6.12 and v = 1.By [29, Lemma 5.2] and Theorem 6.1, the set M v,w (when non-empty) is totally geodesic with respect to the finite energy geodesics of E 1 T (X, ω 0 ).Furthermore, if there exists a ψ ρ ∈ M v,w such that I ρ (ψ ρ ) = inf ψ∈Mv,w I ρ (ψ), then ψ ρ is unique by the strict convexity of I ρ established in [15,Prop. 4.5].Furthermore, by Theorem 6.1, the functional M ρ v,w : E 1 T (X, ω 0 ) → R ∪ {∞} will be also strictly convex along finite energy geodesics, showing the uniqueness of an element ϕ) (assuming that such minimizer ψ exists).
We then have the following weighted version of the continuity method of [15,Prop. 3.1]: Proposition 7.3.Let v > 0, w be smooth weight functions on ∆.Suppose that M v,w is nonempty, and ϕ ∈ K T (X, ω 0 ) ∩ I −1 (0).For any λ > 0, there exists a unique minimizer Proof.The proof follows by a straightforward adaptation of the arguments in [15,Prop. 3.1].
We next need a weighted analogue of [15,Lemma 3.3].
Proof.Using Theorem 6.1 and the fact that I ρ is d 1 -continuous (see [15] or Lemma 6.12), for any t ∈ [0, 1] there exists a sequence ( By the proof of [57, Cor.1], we get lim According to [15,Lemma 3.4], we can use the dominated convergence theorem on the RHS of the above inequality to conclude.
The last step is to establish a weighted version of [15,Prop. 3.2.].
Remark 7.6.The arguments in the proofs of Proposition 7.5 and Theorem 7.1 extend if we remove the maximality assumption for T ⊂ Aut r (X), but replace the group G = T C with the connected component of the identity Ĝ = Aut T r (X) of the centralizer of T in Aut r (X).The key points are that Ĝ is reductive (see Proposition 1.4), and Ĝ acts transitively on the space of T-invariant (v, w 0 )-extremal Kähler metrics (see Theorem 1.5).
Proof of Theorem 1.We apply the Coercivity Principle of [29], see Theorem 3.6.By Theorem 6.1, the extension of the weighted Mabuchi energy M v,w to the space E 1 T (X, ω 0 ) satisfies the hypotheses of Theorem 3.6 (the invariance of M v,w under the action of G = T C is equivalent to the necessary condition (3) for the existence of a (v, w)-cscK metric).We thus need to ensure that M v,w further satisfies the properties (i)-(iv) of Theorem 3.6.Theorem 6.1 also yields the convexity property (i) whereas the regularity property (ii) is established in Theorem 7.1.This last result also yields the uniqueness property (iii), via Theorem 1.5.Finally, the compactness property (iv) is established in Theorem 6.17.
Remark 7.7.By virtue of Theorem 1.5 and Remark 7.6, the conclusion of Theorem 1 holds true if one drops the assumption that T ⊂ Aut r (X) is a maximal torus, but instead of T C one considers the larger reductive group Ĝ = Aut T r (X) (see Proposition 1.4).

Proofs of Theorems 2 and 3
Proof of Theorem 2. The implication (ii) ⇒ (i) follows from Lemma 5.9 whereas (ii) ⇒ (iii) is established in Theorem 1.We shall prove below (iii) ⇒ (ii) and (i) ⇒ (ii).The arguments are very similar to the ones in the proof of [53,Thm. 1] where the case when (X, T) is toric is studied.The main idea is to show that on a semi-simple principal (X, T)-fibration, the continuity path used by Chen-Cheng [23] in the cscK-case and its modification by He [48] to the extremal case, can be adapted to bundle-compatible construction.We sketch the proof below for Reader's convenience.
Proof of (iii) ⇒ (ii).We shall work on Y .Let ω0 be a bundle-compatible Kähler metric on Y , corresponding to a T X -invariant Kähler metric ω 0 on X.By Lemma 5.14, ω0 is invariant under a maximal torus K Y ⊂ Aut r (Y ) (containing T Y ), and by this lemma and Lemma 5.10, the extremal affine-linear function corresponding to K Y is the pull-back to the vector space k * Y = (Lie(K Y )) * of the extremal affine-linear function ext (µ) on t defined in Theorem 2-(ii).Furthermore, by Lemma 5.10, we have that the restriction of M Y 1, ext to the subspace K T (X, ω 0 ) ⊂ K K Y (Y, ω0 ) (see Corollary 5.6 and Lemma 5.14) is a positive multiple of M X p, w, where the weights are the one defined in Theorem 2-(ii).In this setup, the main ingredients of the proof are as follows.
Step 1.Following [23,47,48], one considers the continuity path ϕ t ∈ K K Y (Y, ω0 ), determined by the solution of the PDE (66) t where ρ is a suitable (fixed) K Y -invariant Kähler metric on Y in the class [ω 0 ].By [23,48], there exits ρ ∈ [ω 0 ] and a t 0 ∈ (0, 1), such that a solution ϕ t of (66) exits for t in the interval [t 0 , 1); furthermore, the solution ϕ t (y) is smooth as a function on [t 0 , 1) × Y .The main observation of [53] is that, with a suitable choice for ρ, the path (66) can in fact be reduced to a continuity path on X.To see this, we observe that, by [48,Prop. 3.1], one can take ρ in (66) to be of the form ρ = ω0 + 1 r 0 dd c f with r 0 large enough, where f is the smooth function on Y with zero mean with respect to ω0 , which solves the Laplace equation By Lemmas 5.9 and A.3, f ∈ C ∞ T (X), whereas by Lemma 5.5 ρ is bundle-compatible, i.e.
where ρ = ω 0 + 1 r 0 dd c f is a T-invariant Kähler metric on X, see (24).Using Lemma 5.9 and that both ωϕ and ρ are of the form (24), we get a path of PDE's on X of the form (67) t Scal p (ω ϕt ) − w(µ ωϕ t ) = (1 − t)H(ϕ t ), t ∈ (t 0 , 1), where ϕ t ∈ K T (X, ω 0 ) and H(ϕ t ) := tr ωϕ t (ρ) − (n + m) is manifestly a second order differential operator on X for ϕ t ∈ K T (X, ω 0 ) ⊂ K K Y (Y, ω0 ).It follows that the solution ϕ t , t ∈ [t 0 , 1) of (66) will actually belong to K T (X, ω 0 ) ⊂ K K Y (Y, ω0 ).This last point is a consequence of the implicit function theorem (used in [47,48] to establish the openness) which can be applied directly to (67); to find the linearization of (67), we use [47] that the linearization of H(ϕ) on Y is the operator H ρ ωϕ,1 (see Definition A.1) so that, by virtue of Lemma A.3, the linearization of H(ϕ) when restricted to K T (X, ω 0 ) ⊂ K K Y (Y, ω0 ) is given by the p-weighted operator H ρ ωϕ,p introduced in Appendix A. Similar argument allows us to identify the linearization of Scal p (ω ϕ ) (see also [56,Lemma B1]).We refer the Reader to [53,Sect. 6] for further details.
Step 2. The next ingredient is a deep result from [23] with a complement in [48], showing that if M Y 1, ext is G-coercive along the continuity path ϕ t with respect to a reductive subgroup G ⊂ Aut r (Y ) containing the torus generated by the extremal vector field ξ Y ext = d ext ∈ t Y in its center, then there exists a subsequence of times j → 1 and elements σ j ∈ G, such that σ * j (ω ϕ j ) converges in C ∞ (Y ) to an extremal Kähler metric ω1 .In our case, assuming (iii), we have that Y -coercive (see Lemmas 6.6, 6.4 and Proposition 3.4).We can thus find σ j ∈ T C Y and ϕ j as above.The Kähler metrics σ * j (ω ϕ j ) are bundle-compatible in the sense of Definition 5.3, and thus are of the form σ . By Lemma 5.9, the corresponding Kähler metric ω 1 on X is then (p, w)-cscK.
Proof of (i) ⇒ (ii).The proof is very similar to the proof of (iii) ⇒ (ii).As in the Step 1 of the latter, we consider the continuity path (66) which defines potentials ϕ t ∈ K T (X, ω 0 ) ⊂ K K Y (Y, ω0 ) for t ∈ [t 0 , 1).We can assume without loss [21] that Y admits a K Y -invariant extremal Kähler metric in [ω 0 ], where K Y ⊂ Aut r (Y ) is the maximal torus given by Lemma 5.14.This implies that M Y 1, ext is G-coercive for G = K C Y .Indeed, this can be justified for instance by applying Theorem 1 and Proposition 3.4 in the case (v, w) = (1, ext ).As in the Step 2 of the proof of (iii) ⇒ (ii), we use [23,48] and the G-coercivity of M Y 1, ext along the path in order to find a sub-sequence of times j → 1 and elements σ j ∈ G, such that σ * j (ω ϕ j ) converges in C ∞ (Y ) to a K Y -invariant extremal Kähler metric ω1 ∈ [ω 0 ].However, unlike the proof of (iii) ⇒ (ii), in general σ * j (ω ϕ j ) and hence ω1 are not bundle-compatible, as σ j can act non-trivially on B (see the proof of Lemma 5.14).We thus need to modify slightly the argument in order to show that ω1 still induces a (p, w)-cscK metric on any given fibre X b = π where for the equalities on the second line we have used that the K Y -extremal function ext ∈ Aff (t * X ) (see Lemma 5.14).Thus ω 1 (b) is a (v, w)-cscK metric on X.
Proof of Theorem 3. In [46], Han-Li introduced a functional M HL v : K T (X, ω 0 ) → R whose critical points are the v-solitons, see [46,Lemma 4.4].A careful inspection using (50) shows that M HL v (ω) = M v,w (ω) − X log(v(µ ω ))v(µ ω )ω [m] , where w is the weight function defined in Proposition 1.Thus, the difference of the two functionals is a constant independent of the choice of a T-invariant Kähler metric ω ∈ 2πc 1 (X), see e.g.[56].Thus, by [46,Thm. 3.5] applied to (X, 2πc 1 (X), T) (and weights pv, w), the T C -coercivity of M X pv, w is equivalent with the existence of a vp-soliton on X.By Lemma 5.11, this implies that Y admits a (bundle-compatible) v-soliton.
By [46,Thm. 1.7], the T C -coercivity of M X pv, w is also equivalent to the uniform vp-K-stability on T-equivariant special test configurations.When (X, T) is a toric Fano variety, the only such test configurations are the product test configurations, and thus by [56,Prop.3],the condition is reduced to verifying (3) on X with respect to the weights (pv, w).
The convexity and properness of the above functional follow by the arguments in [72, Lemma 2.2], but under our toric assumption these can also be seen directly by rewriting the RHS in (68) as an integral over the Delzant polytope: ξ → (2π) m ∆ e ξ,µ p(µ)dµ.
The properness of the latter follows by the fact that the origin is in the interior of ∆ (by the canonical normalization condition of ∆, see Remark 2.3).Let ξ 0 ∈ t be the unique critical point of (68).We have that X ζ, µ ω e ξ 0 ,µω p(µ ω )ω [m] = 0, which is precisely the condition Fut v, w = 0 according to Lemma 2.1 in the Appendix 2.
The existence of a Sasaki-Einstein structure follows by a similar argument: By Proposition 2, Lemma 5.11 and Proposition 1 in that order, we want to find ξ 0 ∈ t such that (3) holds true for the weights given as in Proposition 1, with v(µ) = p(µ)( ξ 0 , µ + a) −(m+n+2) .(This will be enough to conclude the existence of a pv-soliton on the toric Fano manifold (X, T) and hence a v-soliton on Y by the general arguments evoked above.)We argue based on [62] who introduced the volume functional on the space of normalized positive affine-linear functions on ∆.Strictly speaking, the functional in [62,Sect.3] is introduced on the principle S 1 -bundle N over (X, ω) (which admits a natural strictly pseudo-convex CR structure (D, J) coming from X), and is then defined as the Sasaki volume of a (D, J)-compatible normalized Sasaki-Reeb vector field ξ on N ; using the point of view of [3] (see in particular Lemma 1.4), the volume functional can also be written on X, noting that positive affine-linear functions ξ = ξ, µ + a over ∆ are in bijection with Sasaki-Reeb vector fields ξ on (N, D, J), and the normalization condition used in [62] is equivalent to requiring ξ (0) = a = 1.Specifically, in our toric weighted setting, we let ξ → X ( ξ, µ ω + 1) −(m+n+1) p(µ ω )ω [m] = (2π) m ∆ ( ξ, µ + 1) −(m+n+1) p(µ)dµ, which is defined for ξ ∈ t such that ( ξ, µ + 1) > 0 on ∆; the properness of the functional follows by the fact that a canonically normalized Delzant polytope of a Fano toric manifold is determined by ∆ = {µ : L j (µ) ≥ 0} where the affine-linear functions L j (µ) satisfy L j (0) = 1, see e.g.[1,Sect. 7.4].The unique critical point ξ 0 ∈ t of the above convex functional then satisfies X ζ, µ ω ( ξ 0 , µ ω + 1) −(m+n+2) p(µ ω )ω [m] = 0, ζ ∈ t, which, by Lemma 2.1, is precisely the condition (3) for the weight functions considered.This concludes the proof of Theorem 3.

Appendix A. Weighted differential operators
Let (X, ω, T) be as in Section 1 and v > 0 be a positive smooth weight function defined over the polytope ∆.We denote by ∇ ω the Levi-Civita connection of the Riemannian metric g ω , and by δ ω the formal adjoint of ∇ ω .We define the following weighted differential operators which are self-adjoint with respect to the volume form v(µ ω )ω [m] on X.
We now specialize to the case when (Y, ω, T Y ) is a semi-simple principal (X, ω, T X )-fibration over B, as in Section 5. We then denote by ∆ Y ω , L Y ω and (H ρ ω) Y the corresponding unweighted operators on (Y, ω), where the Kähler form ρ in the definition of H ρ ω is bundle-compatible, i.e. given by ( 24) for a T X -invariant Kähler form ρ on X.We further let ∆ Ba ωa denote the Laplacian on (B a , ω a ), and ∆ B x and L B x respectively the Laplacian and Lichnerowicz operators on B with respect to the Kähler metric ω B (x) := k a=1 ( p a , µ ω (x) + c a )ω a .We thus have the following result.
Lemma A.3.Let ψ be a T Y -invariant smooth function on Y , seen as a T X -invariant function on X × B via (25), and ω a bundle-compatible T Y -invariant Kähler metric on Y associated to a T X -invariant Kähler metric ω on X.We then have (77) where ρ ω,p := ρ ω − 1 2 d X d c X log p(µ ω ) is the Ricci form of the weighted volume form p(µ ω )ω [m] .Using integration by parts, for any T Y -invariant smooth test function φ on Y , seen as a T X and T P -invariant function on Z = X × P via (25), we have From the above formula, using ( 44), ( 73) and (77), we compute (after some straightforward but long algebraic manipulations and integration by parts over X and B)  .

Weighted Futaki invariants
On a smooth Fano manifold (X, T) as in the setting and notation of Section 2, we further relate the weighted Futaki obstruction Fut v,w = 0 (see (3)) with weights v(µ), w(µ) as in Proposition 1 with the Futaki-type obstructions studied by Tian-Zhu [72] in the case of Kähler-Ricci solitons (i.e. when v = e ξ,µ ):

Example 4 . 3 .
The functional J introduced in Definition 3.1 admits a non-Archimedean version defined, up to a positive dimensional multiplicative constant, on the class of smooth T-equivariant dominating Kähler test configurations (X , L ) by

Definition 4 . 10 .
Let F NA = F v,w , where F v,w is defined on any smooth T-equivariant test configuration via the formula (21), see Definition 4.5.We then refer to the F NA -K-stability notions introduced in the Definition 4.7 (i)-(iii) respectively as T-equivariant (v, w)-K-semistability, Tequivariant (v, w)-K-polystability, and T C -uniform (v, w)-K-stability on T-invariant dominating smooth Kähler test configurations with reduced central fibre.Proof of Corollary 1 modulo Theorem 1.By the definition of M v,w (see Definition 1.1), we have

(
p a , µ + c a ) na , n a = dim C (B a )is a positive polynomial on ∆, determined by the semi-simple (X, T)-principal fibration Y and the given bundle-compatible Kähler class on it.It thus follows that any T X -invariant integrable function f on X defines an integrable T Y -invariant function on Y and, for any ϕ ∈ K T (X, ω 0 ) ⊂ K T (Y, ω0 ), we have

Proposition 5 . 8 .
m+n] ∧ θ ∧r (where the RHS are written on X × P ) together with d X ϕ = d Y ϕ and(27).The embedding in Corollary 5.6 is totally geodesic with respect to the weak C 1, 1 geodesics.
holds if and only if (X , L ) is a product test configuration.Furthermore, according to [50, Thm.B] and [58, Thm.3.14], we have Example 4.3.-bis.In the polarized case, the quantity J NA T C (X , A ) introduced in (20) defines a non-Archimedean version of the functional second example is established in [56, Thm.7]: Example 4.4.Consider the weighted Mabuchi functional M v,w introduced in Definition 1.1 and assume that the relation (7) holds, see Remark 1.2.Then M v,w admits a non-Archimedean version defined on smooth T-equivariant Kähler test configurations with reduced central fibre, given by the formula