An effective weighted K-stability condition for polytopes and semisimple principal toric fibrations

. The second author has shown that existence of extremal Kähler metrics on semisimple principal toric ﬁbrations is equivalent to a notion of weighted uniform K-stability, read oﬀ from the moment polytope. The purpose of this article is to prove various suﬃcient conditions of weighted uniform K-stability which can be checked eﬀectively and explore the low dimensional new examples of extremal Kähler metrics it provides.


Introduction
Calabi's work has been extremely influential in Kähler geometry, his name being still associated to some of the most fundamental objects of interest.The present article is motivated by two of these, Calabi's extremal Kähler metrics and Calabi's ansatz.
Extremal Kähler metrics provide a natural notion of canonical Kähler metrics in a given Kähler class on a compact Kähler manifold X: they are the metrics that achieve the minimum of the L 2 -norm of the scalar curvature.Kähler metrics with constant scalar curvature (cscK metrics for short) are special cases of such metrics, but Calabi showed in [10] that there may exist extremal Kähler metrics when there exists no cscK metrics at all, by exhibiting extremal Kähler metrics on Hirzebruch surfaces.In order to show this, Calabi relied on the simple yet powerful idea that one should search for extremal Kähler metrics among those Kähler metrics that behave well with respect to the geometry of the manifold.
This was not a new idea of course.Matsushima showed for example [36] that cscK metrics must behave well with respect to biholomorphism.More precisely, the automorphism group of X must be the complexification of the isometry group of the cscK metric, if it exists.This is preventing Hirzebruch surfaces from admitting cscK metrics as their automorphism group is non-reductive.
Calabi went further and restricted to metrics that respect the structure of P 1 -bundles of Hirzebruch surfaces.He was then able to translate, for such metrics, the extremal property into a simple ODE and to solve it, showing the existence of extremal Kähler metrics.His construction was later referred to as Calabi's ansatz, used in various situations and generalized in various directions.It would be easy to fill pages with a bibliographical review of these, but it is not the purpose of this introduction.We only stress that a common theme is usually the desire to get explicit existence results or criterions.An influential illustration is [5], where a variant of Calabi's ansatz was used to show that on various P 1 -bundles, existence of extremal Kähler metrics reduces to checking the positivity of a polynomial on [−1, 1], the so-called extremal polynomial.In the series of papers leading to [5] (see also [25]), the general idea of Calabi's ansatz was actually pushed way further, allowing for example to consider certain fibrations with toric fiber.
The interest for such fibrations was significantly renewed last year, when the second author proved in [26], using the breakthrough results of Chen and Cheng [14], that a uniform version of the Yau-Tian-Donaldson conjecture holds for semisimple principal toric fibrations, a very large class of toric fibrations.While it allows to translate the question of existence of extremal Kähler metrics on such manifolds into a question of convex geometry on their moment polytopes, it is not yet an explicitly checkable criterion, as the conditions to check still form an infinite dimensional space.Motivated by the more practical philosophy behind Calabi's ansatz, we prove in the present paper various sufficient conditions of existence of extremal Kähler metrics which may be easily checked.Our approach is based on an initial idea by Zhou and Zhu [41], exploited in greater generality by the first author in [15].
The previous paragraphs are meant as an introduction to our results, and it should be stressed that it presents as such a biased and very incomplete historical account of the study of extremal Kähler metrics on manifolds with large symmetry.We refer to Székelyhidi's book [39], Gauduchon's lecture notes [20] for a general introduction to extremal Kähler metrics, and to Donaldson's remarkable survey [18] and Apostolov's lecture notes [2] for some of the more specific aspects of manifolds with large symmetry.More recent developments very related to our work will be discussed at the beginning of Section 3.
Let us now highlight in the remainder of this introduction our main results.For this, a few notations are needed, and the full details will be given in Section 3. Semisimple principal toric fibrations are certain holomorphic fiber bundles π : Y B where the basis B = a B a is a product of Hodge manifolds (B a , ω a ) with constant scalar curvature s a , and where the fiber X is toric under a compact torus T. They are constructed from certain types of principal T-bundles, essentially determined by the data of a tuple (p a ) of one-parameter subgroups of T. In this paper, a one-parameter subgroup p a : S 1  T of T will be identified with the element of the Lie algebra of T determined by the image of 1 ∈ R = Lie(S 1 ) under the differential of p a at the neutral element.In particular, it defines a linear function on the dual of the Lie algebra of T. On such manifolds, a Kähler class is called compatible if it decomposes as the sum of a relative Kähler class induced by a Kähler class [ω X ] on X, and a sum of real multiples c a π * [ω a ] of the pull-backs of the Kähler classes [ω a ].An admissible Kähler class contains admissible Kähler metrics, that behave well with respect to the fibration structure.
Theorem 1.1.Assume that Y is a semisimple principal toric fibration, that the toric fiber X is Fano equipped with the Kähler class [ω X ] = t2πc 1 (X), and let [ω Y ] be an admissible Kähler class.Assume that for all a, 2 dim(B a )c a ≥ ts a and that at every vertex x of the moment polytope P of (X, [ω X ]), where l ext is the extremal affine function.Then there exists an extremal Kähler metric in [ω Y ].
Recall that, when a maximal torus of automorphisms of Y is fixed, the scalar curvature of an invariant extremal Kähler metric, if it exists, is a holomorphy potential of a well defined vector field called the extremal vector field.In the statement above, the extremal function is encoding the extremal vector field, and a choice of maximal torus of automorphisms of Y is implicitly assumed.We will explain why it reduces to an affine function on the polytope P in Section 3.
We actually prove a much more general sufficient condition, Theorem 2.6, that does not require the fiber to be Fano.Since we obtain already a wealth of new examples with this particular case, and it is a natural generalization of the P 1 -bundle case, we focus on this result for the introduction.
In the case when the fibration is Fano itself, and not only its fiber, then t = 1 and s a = 2 dim(B a )c a so one gets a particularly simple criterion: Corollary 1.2.A Fano semisimple principal toric fibration Y admits an extremal Kähler metric in c 1 (Y ) if its extremal affine function l ext satisfies and the latter obviously needs only be verified at vertices of the moment polytope.
We provide, for the reader's convenience, an elementary Python program implementing the sufficient condition from Theorem 1.1 in the case when there is only one factor in B and the fiber is of dimension one or two.It would be easy to imitate these to allow greater flexibility in the data.It may be used either with all the data given numerically, or some of the data treated as variable.We use this to our advantage to prove the existence of extremal Kähler metrics in a wide range of Kähler classes for some examples of fibrations.
, where B is a Kähler-Einstein Fano threefold, H is the smallest integral divisor of 2πc 1 (B) and 1 ≤ p 1 ≤ p 2 .Then there exists an extremal Kähler metric in the Kähler class c 1 (X) + λc 1 (B) for λ ≥ 7p 2 , where c 1 (X) and c 1 (B) respectively denote the relative first Chern class and the pull-back of the first Chern class, by an abuse of notations.
Here, Y is a semisimple principal fibration over the base B, with fiber the projective space X = P 2 .More generally, projectivizations of direct sums of line bundles can often be considered as semisimple principal fibrations, as explained in Section 3.3.
The article is organized as follows.In Section 2, we prove a general sufficient condition for weighted uniform K-stability of labelled polytopes, and consider the special case of monotone polytopes.Section 3 explains the geometric origin of weighted uniform Kstability of labelled polytopes, with a particular emphasis on semisimple principal toric fibrations.In Section 4, we put together the two aspects to prove Theorem 1.1 and Corollary 1.2 using the monotone case of Section 2, as well as more general statements.We present various examples of applications of the sufficient condition in Section 5, including Proposition 1.3.Finally, we include in an appendix elementary Python programs computing the sufficient condition for fibrations with only one factor in the basis, and a one or two dimensional Fano fiber.
Acknowledgements.The authors are very grateful to Vestislav Apostolov and Eveline Legendre for their valuable comments on the manuscript.The authors also thank the referee for his useful suggestions and corrections.The first author is partially funded by ANR-21-CE40-0011 JCJC project MARGE and ANR-18-CE40-0003-01 JCJC project FIBALGA, as well as PEPS JCJC INSMI CNRS projects 2021 and 2022.The second author was supported by PhD fellowships of the UQAM and of the Université de Toulouse III -Paul Sabatier.
2. Weighted K-stability of labelled polytopes: a sufficient condition 2.1.Weighted K-stability of labelled polytopes.Let V be an affine space of dimension ℓ, equipped with a fixed Lebesgue measure dx.A labelled polytope in V is a pair (P, L) where P is a (compact, convex) polytope in V and L = (L j ) d j=1 is a minimal set of defining affine functions for P , that is, where d is the number of facets (codimension one faces) of P .We denote by F j := {x ∈ P | L j (x) = 0} the facet of P defined by L j .
Definition 2.1.The labelled boundary measure dσ is the measure on ∂P whose restriction to the facet F j is defined by dL j ∧ dσ = −dx.
Note that the labelled boundary measure depends heavily on the choice of labelling (L j ).For example, for any tuple (r j ) of positive real numbers, the tuple (L ′ j ) = (r j L j ) is another labelling of P .The associated labelled boundary measure dσ ′ satisfies dσ ′ = 1 r j dσ on F j .In particular, if the r j are not all equal, there is no obvious relation between dσ and dσ ′ .Similarly, the notion of weighted uniform K-stability that we are about to define depends heavily on the choice of labelling.
Let v be a continuous, positive function on P , and let w be a continuous function on P .Following [17,27,33], we define the (v, w)-Donaldson-Futaki invariant of the labelled polytope (P, L) as the functional F on the space of continuous functions on P such that (1) Following [37,24], we also set where Aff(V ) denotes the space of affine functions on V .
Definition 2.2.A labelled polytope (P, L) is (v, w)-uniformly K-stable if there exists a λ > 0 such that for any continuous convex functions f on P , Remark 2.3.Note that F is linear, and the right-hand side of (2) is always non-negative, hence the following is a necessary condition for (2) to hold: (3) ∀f ∈ Aff(V ), F(f ) = 0.
We will explain in Section 3 the geometric significance of this notion for various choices of v and w, let us for now just highlight that when v and w are constant, the functional F first appeared in [17] as an expression of the (Donaldson-)Futaki invariant for toric test configurations in the study of K-stability of toric manifolds, whence the name.
We denote by CV 0 (P ) the space of continuous convex functions on P , and by CV 1 (P ) the space of all convex functions f on P which are the restrictions to P of a continuously differentiable function defined on an open subset of V containing P .Note that by uniform approximation by smooth functions, it is enough to consider only functions in CV 1 (P ) to check condition (2).
In order to deal more efficiently with the right hand side of (2), following [17], we consider the following normalization of functions.We choose a point x 0 in the interior P 0 of the polytope P .It allows to choose a linear complement CV 1 * (P ) to Aff(V ) in CV 1 (P ), defined by ( 4) Then, any f ∈ CV 1 (P ) can be written uniquely as f = f * + f 0 , where f 0 is affine and f * ∈ CV 1 * (P ), and we will use these notations in the following.By linearity, Lemma 2.4.The labelled polytope (P, L) is (v, w)-uniformly K-stable if and only if there exists λ > 0 such that for all f ∈ CV 1 (P ), where • L 1 denotes the L 1 -norm on P with respect to the Lebesgue measure dx.
Proof.From [37, Proposition 4.1 (3)], there exists a constant C 1 > 0 such that for all continuous convex functions on P , The equivalence between condition (2) and condition (5) follows immediately.
Remark 2.5.Condition (5) is the condition that we will effectively use in the sequel, so one might wonder why we introduced the first definition.The point is that by Lemma 2.4, condition (5) is independent of the choice of x 0 , and condition (2) makes it perfectly clear.In the more familiar unweighted case, the equivalence between various notions of K-stability of polytopes was fully worked out by Nitta and Saito [37].
2.2.The sufficient condition.Let C 0 (P, R) denote the space of continuous functions on P , and let C 1 (P, R) denote the space of functions that are the restriction to P of continuously differentiable functions defined in an open subset of V containing P .
Recall that F j denotes the facet of P defined by L j .For each j, let P j be the cone with basis F j and vertex x 0 as illustrated in Figure 2.2.For a function f ∈ C 1 (P, R), we denote by d x f its differential at x ∈ P .The following is the main technical result of our paper, it imitates quite closely part of the proof by Zhou and Zhu [41] of a coercivity criterion for the modified Mabuchi functional on toric manifolds.Theorem 2.6.Let v ∈ C 1 (P, R) be a positive function on P and let w ∈ C 0 (P, R).Assume that F vanishes on Aff(V ) and that for all j = 1, . . ., d, for all x ∈ P j , Proof.Since L j (x) = 0 for x ∈ F j , we have L j (x 0 ) = d x L j (x 0 − x) for all x ∈ F j .In particular, For each facet F of ∂P j different from F j , and x ∈ F , the interior product ι x−x 0 (dx) vanishes since it vanishes on the affine space spanned by F .If we further use that −dL j ∧ dσ = dx on F j , we obtain Hence by Stokes theorem we obtain Summing the previous identities over j we get Assume condition (6) is satisfied and (P, L) is not (v, w)-uniformly K-stable.We will show contradiction to a stronger condition than condition (5).Namely, assume that condition (6) is satisfied and that there exists a sequence of Recall from [17,Lemma 5.1.3](see [37,Proposition 5.1.2]for a detailed proof and explicit constant C) that there exists a positive constant C > 0 such that for all f ∈ CV 1 (P ), As a consequence, since v > 0 on P , there exists a constant C ′ > 0 such that for all f ∈ CV 1 (P ), In particular, the sequence Since in addition all f * k are smooth and convex we have Then, since condition ( 6) is assumed to hold, all terms of the sum in ( 7) are non-negative.Evaluating (7) at f * k and passing to the limit reveals that The local uniform convergence of {f * k } to the zero function shows that which is in contradiction with condition (8).
From this contradiction it follows that there exists a constant µ > 0 such that for all f ∈ CV 1 (P ), which concludes the proof.
Remark 2.7.We stress that the property of (v, w)-uniform K-stability is independent of the choice of x 0 ∈ P 0 in the previous section, but condition (6) depends on that choice.It is possible and useful in practical uses of the condition to vary this x 0 according to the data of the problem, see 5.3.1.
Remark 2.8.Condition (6) depends continuously on the labelled polytope, the weights v and w, and the choice of x 0 .

Case of monotone polytopes.
Let us recall the terminology of monotone polytopes, used in [29].Definition 2.9.A labelled polytope (P, L) is monotone if there exists an There is thus an obvious choice of x 0 in that case.Our sufficient condition indeed becomes much simpler in that case, since the decomposition of the polytope may essentially be forgotten.
Corollary 2.10.Let (P, L) be a monotone labelled polytope with Assume that F vanishes on Aff(V ) and that for all x ∈ P , The conditions involved form a finite set of conditions to check, contrary to the definition of (v, w)-uniform K-stability.It is furthermore easy to implement in a computer program, via formal or numerical computations depending on the data (P, L, v, w).The same is true for the more general Theorem 2.6, but the decomposition in cones makes it a bit more tedious.

Geometric origin of weighted K-stability of polytopes
3.1.Weighted cscK toric manifolds.The results from Section 2 are motivated by the study of the existence of weighted cscK metrics on toric manifolds, as studied in [26].
Let T be an ℓ-dimensional compact torus.We denote by t its Lie algebra and by Λ ⊂ t the lattice of generators of circle subgroups, so that T = t/2πΛ.Let (X, ω, T) be a compact Kähler toric manifold.Denote by µ the moment map of X with respect to the action of T, and let P = µ(X) ⊂ t * be the moment polytope.The polytope P is a Delzant polytope [16], and in particular, there is a natural choice of labelling L of P such that all the differentials dL j of the defining affine functions L j are primitive elements in the lattice Λ. Remark 3.1.We focus here on smooth manifolds, but let us mention that the cases of orbifolds or pairs would also be natural settings to consider.In these situations, the labelling could be more general, thus justifying our choice to allow arbitrary labellings in the previous section.
In the context of toric manifold, the v-weighted scalar curvature was introduced in [33].To avoid introducing too much notation, we give the definition of [27], which makes sense for general compact Kähler manifold and coincide with the one of [33] in the toric context.Let C ∞ (P, R) denote the space of restrictions to P of smooth functions defined on an open set containing P , and C ∞ (P, R >0 ) the subspace of positive functions.

Definition 3.2 (Weighted cscK metrics).
(1) For v ∈ C ∞ (P, R >0 ), define the v-scalar curvature of ω as the function where Scal(ω) is the usual scalar curvature of the Riemannian metric g ω associated to ω, ∆ ω is the Riemannian Laplacian of g ω , Hess(v) is the Hessian of v viewed as a bilinear form on t * whereas G ω is the bilinear form with smooth coefficients on t, given by the restriction of g ω on fundamental vector fields. (2 In general, no YTD correspondence is proved for the existence of weighted cscK metrics on toric manifolds.However by analogy with the unweighted cscK case, there is a known candidate for the corresponding K-stability condition, which translates on the polytope as Definition 2. In fact, the direction from existence of weighted cscK metrics to K-stability was proved in general by Li, Lian, Sheng [34].
The converse direction is in general much harder, but is known for special choices of weights.
• If v and w are constants, this is the uniform YTD conjecture for cscK metrics on toric manifolds.If v is constant and w is affine, this is the uniform YTD conjecture for extremal metrics on toric manifolds.Both these conjectures were proved recently [24,30,2,31,35,37] thanks to the breakthrough of Chen-Cheng [14], its adaptation by He to the extremal setting [23], and earlier works, notably [17,41].• If only v is constant, the converse of Theorem 3.3 is known for all w ∈ C ∞ (P, R) by [35].• For v-solitons on Fano toric manifolds, which correspond to choosing an arbitrary weight v ∈ C ∞ (P, R >0 ) and w(x) = 2(ℓv(x) + d x v(x)) (see [7, Proposition 1]), it was proved first in [9] that the converse of Theorem 3.3 holds for general weight v, and much earlier in [40] for the weight corresponding to Kähler-Ricci solitons.We note that [32] proved that the general uniform YTD conjecture holds for vsolitons on general Fano manifolds, and refer to Section 4.4 for a discussion of v-solitons on semisimple principal toric fibrations.• Finally, as we shall explain in details in the next sections, the converse of Theorem 3.3 was proven by the second author for weights corresponding to extremal Kähler metrics on semisimple principal toric fibrations [26].
3.2.Construction of semisimple principal toric fibration.In this section we briefly recall the construction of semisimple principal toric fibrations introduced in [4].We take the point of view of [7], which generalized the construction when the fiber is not necessarily toric.
Let T be an ℓ-dimensional compact torus with Lie algebra t.Let where p a ∈ t define one-parameter subgroups of T. Let (X, J X , ω X ) be a toric projective manifold under the action of T. Since T acts on various spaces, to avoid confusion, we under-script the space on which T acts, e.g.T X acts on X.We consider the 2(ℓ + n) dimensional smooth manifold (10) Y := (X × Q)/T, Let H := ann(θ) ⊂ T Q be the horizontal distribution on the principal bundle Q with respect to θ.We consider the smooth section of the endomorphism of where J B acts on Q via the unique horizontal lift of vector fields on B to H.The section J Y is invariant with respect to the T Q×X -action and it is shown in [7, Section 5] that J Y descends to a complex structure (still denoted by J Y ) on Y .Let P ⊂ t * be the Delzant polytope associated to (X, J X , ω X , T) [16].By definition the T-action on X is hamiltonian and we denote by µ : X − P its moment map.We consider the 2-form on X × Q where the k-tuple of real constants (c a ) are such that for all a, p a • µ + c a > 0 and dµ • θ is a 2-form understood as the contraction of the t-valued one form θ and the t * -valued one form dµ. It is shown in [7,Section 5] that ω Y is basic with respect to T X×Q and as such, it is the pullback of a Kähler form (still denoted by ω Y ) on Y .The T X -action on X induces an action on X × Q by the natural T X -action on the first factor.This action commutes with T X×Q and therefore descends to a T Y -action on Y .The Kähler form ω Y on Y defined via the basic 2-form (11) on X × Q is T Y -invariant (see [7,Section 5] for more details).
The Kähler metrics ω a on B a , the connection form θ and the constants p a ∈ t are fixed.The Kähler manifold (Y, J Y , ω Y , T) is a fiber bundle over B with fiber the toric Kähler manifold (X, J X , ω X , T).Following [6] we define: In this setup, the constants c a can vary and they parameterize the compatible Kähler classes.
Let P 0 be the interior of P and X := µ −1 (P 0 ) be the dense open subset of X of regular orbits for the T X -action.The T-action on X extends to an effective holomorphic action of the complexified torus T C := T ⊗ C. Fixing any point x 0 ∈ X, we can identify ( X, J X ) and the orbit Restricting the T X×Q -action to X × Q, we define Remark 3.5.Semisimple principal toric fibrations (Y, J Y , ω Y , T) constructed above correspond to semisimple rigid toric fibration introduced and studied in [3,4,5] when there is no blow-down and the basis B is a global product of cscK Hodge manifolds.
A particular case, which will be of interest for us, is the case of Fano semisimple principal toric fibrations.We recall a characterization of these from [7] Lemma 3.6 ([7, Lemma 5.11]).Assume that each B a is a Fano Kähler-Einstein manifold.Let ω a denote a Kähler-Einstein metric on B a so that I a [ω a ] = 2πc 1 (B a ), where I a denotes the Fano index of B a .We fix a principal bundle with connection (Q, θ) as before (with associated data (p a )).We further assume that (X, ω X ) is a Fano toric manifold with a T-invariant Kähler form ω X ∈ 2πc 1 (X), with the natural choice of moment map µ.If for all a, p a • µ + I a > 0, then the semisimple principal fibration Y associated to the above data is a Fano manifold, and ω Y is in 2πc 1 (Y ) for the k-tuple (c a ) = (I a ).
Note that in the above situation, the scalar curvature of ω a is indeed constant, equal to 2n a I a where n a is the complex dimension of B a .
3.3.Projectivization of sum of line bundles as semisimple principal toric fibration.We now provide an effective way of constructing examples of semisimple principal toric fibrations.
Let (B, ω B ) := k a=1 (B a , ω a ) be a product of compact complex manifolds B a endowed with cscK metrics ω a with [ω a ] primitive element of H 2 (B a , Z).We consider holomorphic line bundles L i − B, i = 1, . . ., ℓ, and we suppose that their first Chern classes satisfy where by definition p ai is the ω a -degree of L i .The natural C * i -action on L i induces an action of i is a compact ℓ-torus.We choose a Hermitian metric h i on L i and consider the norm function r i (u) := (h i (u, u)) 1 2 for any u in L i .On Li , the C * -bundle obtained from L i by removing the zero section, r i is positive and we let t i = log(r i ).We fix a basis ξ = (ξ i ) ℓ i=1 of the Lie algebra t of T and we denote by ξ Li the generator of the S 1 i -action on Li .We then consider the t-valued one-form t := ℓ i=1 t i ξ i and we define a connection one-form θ on Q as the restriction of d c t to Q, seen as the T-bundle of unit element on each (L i , h i ).For all i = 1, . . ., ℓ, it satisfies θ(ξ Li ) = ξ i .We obtain by construction where ω h i is the opposite of the curvature form of the Chern connection of (L i , h i ) and p a = ℓ i=1 p ai ξ i .We consider the ℓ-projective space (P ℓ , ω P ℓ , T) endowed of a toric T-action with respect to a fixed Kähler metric ω P ℓ .We fix the principal T-bundle Q with its connection one form θ, the cscK Kähler manifolds (B a , ω a ) and the toric Kähler manifold (P ℓ , ω P ℓ , T).From these data, we define a semisimple principal toric fibration Y := Q × T P ℓ .By construction, Y is biholomorphic to the total space of the projective bundle P(E), E := O ⊕ ℓ i=1 L i .Suppose ω P ℓ belongs to the first Chern class 2πc 1 (P ℓ ) of P ℓ and denote by P the canonical ℓ-simplex associate to (P ℓ , 2πc 1 (P ℓ ), T ℓ ) via Delzant correspondence [16].By (13), any compatible Kähler metric on Y is of the form 3.4.Extremal metrics on semisimple principal toric fibrations.We begin this section by recalling a general characterisation of extremal metrics on compact Kähler manifold (Y, J Y ) in a fixed Kähler class α Y .By definition a Kähler metric ω is extremal if the symplectic gradient of its scalar curvature is real holomorphic, i.e.
By a well-known result of Calabi [11] an extremal metric needs to be invariant by a maximal torus K in the reduced automorphism group Aut red (Y ) of Y .Let us fix such a torus K.For any K-invariant Kähler metric ω on Y the action of K on (Y, ω) is hamiltonian and we denote by µ ω the corresponding moment map.By a result of Guillemin-Sternberg [22] the image of µ ω is a convex compact polytope P in the dual of the Lie algebra k of K.For any K-invariant metric ω in α Y we normalize µ ω in such way that its image equals P .It then follows from [19] and [27, Lemma 1] that a K-invariant metric ω ∈ α Y is extremal if and only if where l ext is the unique affine extremal function such that the Futaki invariant vanishes ( 17) for any l ∈ Aff(k * ).
We now suppose that (Y, J Y , ω Y , T) is a semisimple principal toric fibration with fiber (X, J X , ω X , T) and that α Y := [ω Y ] is a compatible Kähler class on Y .Let P denotes the Delzant polytope associated to (X, J X , ω X , T).In general, T Y is not maximal in Aut red (Y ).However, by [6, Proposition 1], any compatible Kähler metric (11)  By construction (10), there is an embedding of the space of T-invariant smooth functions where s a are the constant scalar curvatures of ω a and n a := dim(B a ), Scal v (ω X ) is the v-weighted scalar curvature of ω X (see Definition 3.2) with respect to the weight v(x) := k a=1 (p a (x) + c a ) na .It then follows from (18) and ( 20) that ( 17) reduces to an equation on X and is equivalent to for every l ∈ Aff(t * ), where the moment maps µ ω X are normalized such that µ ω X (X) = P and w(x) := l ext (x) − k a=1 sa pa(x)+ca v(x).To summarize we have the following (see [6, Section 3.5] or [7, Lemma 5.14] for more details).
We now state the main existence result of this section, which is one of the main results of [26].Theorem 3.9 ([26, Theorem 3]).Let (Y, J Y , ω Y , T) be a semisimple principal bundle with Kähler toric fiber (X, J X , ω X , T) and denote by P its moment polytope.Then there exists an extremal Kähler metric in [ω Y ] if and only if P is (v, w)-uniformly K-stable, where and l ext is the unique affine function such that (3) holds for (v, w).Equivalently, there exists a (v, w)-cscK metric in [ω X ].
Proof.We only sketch the proof of the direction "(v, w)-uniform K-stability implies existence of a (v, w)-weighted cscK metric" which is key in the present paper, and refer to the original paper [26] for details and the converse direction.We fix the weights (v, w) given by (21).By Proposition 3.7 and ( 20), compatible extremal metrics on Y correspond to (v, w)-cscK metrics on X via (11).Then, the existence of (v, w)-cscK metric in [ω X ] implies the existence of a (compatible) extremal metric in [ω Y ].The main ingredient is the existence result of cscK metric of Chen-Cheng [12,13,14], extended by He [23] to the extremal case: Theorem 3.10 (Chen-Cheng, He [12,13,14,23]).Let α be a Kähler class on a compact Kähler manifold Y and K be a maximal torus in the reduced automorphism group Aut red (Y ).Then there exists an extremal metric in α if and only if the K-relative Mabuchi energy M K is K C -coercive, i.e. there exists C > 0 and D > 0 such that for every ϕ ∈ KK (Y, ω0 ).
In the above statement, ω0 ∈ α is a fixed K-invariant Kähler metric, M K : K K (Y, ω0 ) − R is the K-relative Mabuchi energy defined on the space of K-relative Kähler potential ) is the space of normalized potential (with respect to the vanishing of the Aubin-Mabuchi functional), d 1 is the Darvas distance [16] and K C := K ⊗ C is the complexification of K. Moreover K C acts on KK (Y, ω0 ) via the natural action on K-invariant Kähler metrics in [ω 0 ].Originally [23], the coercivity condition in Theorem 3.10, was expressed in term of the complexification of a maximal compact connected subgroup of Aut red (Y ) (not necessarily commutative).As observed in [26,Section 5], the same arguments as in [14,23] provide this statement.
The proof of Theorem 3.9 is divided in two steps: (1) show that the (v, w)-uniform stability with respect to the weights (21) of the polytope P implies the T C -coercivity of the weighted Mabuchi energy M v,w corresponding to (X, J X , [ω X ], T); (2) adapt the continuity path (22) involved in the proof of Cheng-Cheng and He to obtain the existence of a compatible extremal metric in [ω Y ] (or equivalently a (v, w)-weighted cscK metric in [ω X ]).The weighted Mabuchi energy M v,w mentioned above is the one introduced by Lahdili [27].
Step 1.The first step is an adaptation of [17,41] which established this result in the unweighted case, i.e. when v = 1 and w = l ext .Fix ω 0 ∈ [ω X ] a T-invariant Kähler metric.On a toric Kähler manifold, a T-invariant Kähler metric ω ∈ [ω 0 ] can be defined via two functions: a T-invariant ω 0 -relative Kähler potential ϕ ∈ K T (X, ω 0 ) and a symplectic potential u ∈ S(P,L), which, by definition, S(P,L) is the space of smooth strictly convex functions on P 0 which satisfy the so-called Abreu boundary conditions [1,21].There is a well-known correspondence between a symplectic and a Kähler potential defining the same Kähler metric [3,4,17,21].Via this correspondence we can consider the weighted Mabuchi energy M v,w as a functional on S(P,L).A first step is to show, as in the case v = 1 [17,Proposition 3.3.4],that M v,w extends to the space CV ∞ (P ) of smooth convex functions on P 0 and continuous on P , and that a symplectic potential u ∈ S(P,L) defining a (v, w)-weighted cscK metric realizes the minimum of M v,w over CV ∞ (P ).This is proven in [26,Proposition 7.7].The idea is then, as in [41], to compare M v,w and M v,w 0 , where the weight w 0 is the v-weighted scalar curvature Scal v (u 0 ) of (the Kähler metric defined by) any fixed symplectic potential u 0 .Then u 0 trivially solves Scal v (u) = w 0 , u ∈ S(P,L), In other words, there exists a (v, w 0 )-cscK metric in [ω X ].We deduce that M v,w 0 is bounded from below on CV ∞ (P ) by the discussion above.Consequently, by comparing M v,w and M v,w 0 and using our hypothesis, we can show that [26, Proposition 7.9] (or [41] for v = 1) for any u ∈ S(P,L).There D > 0 and C > 0 are uniform in u and • L 1 is the L 1norm on P with respect to the Lesbegue measure.Denote by d X 1 the Darvas distance on K T (X, ω 0 ).For a (normalized) Kähler potential ϕ ∈ K T (X, ω 0 ) corresponding to (normalized) symplectic potential u ∈ S(P,L) via the correspondence described above, we can show that d X 1 (0, ϕ) ≤ A u + B, for some positive constant A and B. We deduce that for any ϕ (normalized) T-invariant Kähler potential, where C ′ and D ′ are positive constant.We refer to [26,Section 7.6] for the precise normalization of symplectic and Kähler potentials.
Step 2. The proof of the direction "coercivity implies existence" in Theorem 3.10 is based on the resolution of the following continuity path (22) t(Scal(ω ) and Λ ωϕ ( χ) is the symplectic trace of χ with respect to ωϕ .Chen-Cheng and He showed that there exists t 0 ∈ (0, 1) such that We come back to the case of semisimple principal toric fibration.We fix a T-invariant Kähler metric ω 0 ∈ [ω X ] and the induced compatible Kähler metric ω0 ∈ [ω Y ], see (11).It follows from (19) and ( 18) (see [6,Lemma 7] for a proof) that where K is a maximal torus in Aut red (Y ) containing T such that (18) holds.Moreover, for every ϕ ∈ K T (X, ω 0 ), seen as function on Y , the Kähler metric ωϕ := ω0 + dd c ϕ is a compatible Kähler metric in [ω Y ] in the sense of Definition 3.4.Also, ωϕ on Y is the metric induced by ω ϕ on X via (11) (see [6,Lemma 7] or [7, Lemma 5.5]).Following [6], we then refer to the image of ( 23) as the space of compatible Kähler potentials.
Additionally, ( 20), ( 23), [7, (27)] and Proposition 3.7 (see also [7,Lemma 5.10]) show that (24) Let d Y 1 be the Darvas distance on K K (Y, ω0 ) [16].By [7, Corolarry 6.5] and since v > 0, ), where A > 0. By (20), the restriction of ( 22) to the space of compatible Kähler potentials We now want to show that there exists t 0 ∈ (0, 1) such that . Since any T C -orbit is included in a K C -orbit, the T C -coercivity is stronger than the K C -coercivity.Then, by ( 24), ( 25) and Step 1, M K is K C -coercive on the space of compatible potentials K T (X, ω 0 ) suitably normalized.As observed in [26, Lemma 6.3], we can choose χ such that (26) is an equation on X and admits a solution ϕ t 0 ∈ K T (X, ω 0 ) for some t 0 ∈ (0, 1), showing that S is non-empty.The openness follows from an application of the Implicit Function Theorem [26,Proposition 6.4].From the openess, the closedness of Step 1, following the proof of Theorem 3.10, we obtain a sequence of compatible Kähler metric ωϕ j such that ω j := γ * j (ω ϕ j ), γ j ∈ K C , converge to an extremal metric ω 1 .The space of (normalized) compatible Kähler potential KT (X, ω 0 ) is not stable under the action of K C , then either ω j or ω 1 is compatible in general.However, we can show (see [26, Proof of Proposition 6.5]) that ω 1 is of the form of (11), for a possibly different principal connection θ and Kähler metrics ω a .Moreover, we can argue that the Kähler metric ω 1 ∈ [ω X ] defining ω 1 via (11) is a weighted (v, w)-cscK metric for the same weights (21), see [26, Proof of Proposition 6.5].
Note that condition (3) corresponds to the vanishing of the modified Futaki character, and l ext encodes the extremal vector field.In particular the extremal metric above is cscK if and only if l ext is constant.Remark 3.11.It is remarkable that the condition depends on the base only through the constants (s a ) and the existence of a principal T-bundle with connection with corresponding data (p a ).In particular, when we obtain an existence result for extremal Kähler metrics, we usually actually obtain the existence of extremal Kähler metrics over a full deformation family of cscK manifolds.This is exactly this fact which is used in the Proof of Proposition 5.1 to obtain an existence condition of extremal metric on CP 2 -bundles over any Kähler-Einstein Fano threefold depending only on the cohomology class and the degrees of the line bundles.

Geometric applications of the sufficient condition
4.1.The general statement.We use the same notations as in Section 3 for semisimple principal toric fibrations and the same notations as in Section 2 for the decomposition of polytopes.
Corollary 4.1 (of Theorem 2.6).The semisimple principal toric fibration (Y, ω Y ) admits an extremal Kähler metric in [ω Y ] if there exists an x 0 ∈ P 0 and corresponding cone decomposition P = j P j such that for all j and for all x ∈ P j Proof.By Theorem 3.9, the sufficient condition of Theorem 2.6 translates as a sufficient condition of existence of extremal Kähler metrics.To obtain the statement above, it suffices to note that for the weight v involved, we have so that in the condition in Theorem 2.6, we can factor by v(x) which is positive everywhere.

4.2.
Fibrations with Fano fiber.We now turn to the fibrations with Fano fiber, in order to use Corollary 2.10.With the same notations as in Section 3.4, we now assume furthermore that the toric fiber is a Fano manifold, and that the Kähler class [ω X ] is a multiple of the anticanonical class 2πc 1 (X).As a consequence, the moment polytope P is a dilation of a reflexive lattice polytope.This implies that the labelled polytope (P, L) corresponding to the lattice polytope P is monotone, with a preferred point x 0 and Assuming without loss of generality that the (anti-)canonical normalization is used for the moment polytope of the fiber, we may further assume that x 0 = 0, and t = [ω] 2πc 1 (X) .
Proof.Since all L j (x 0 ) are equal to t, the condition from Corollary 4.1 further simplifies to as for Corollary 2.10.Writing 2n a p a (x) = 2n a (p a (x) + c a ) − 2n a c a yields the statement.
While simple enough, and tractable with numerical optimization techniques, the inequality involved is a polynomial inequality in several variables, whose degree can be equal to the dimension of the basis plus one.It is difficult to solve formally, but there is a further reduction that allows to get a simpler condition which can be checked by a finite number of evaluations of polynomial functions.Proof.The inverse of an affine function is convex on the locus where this affine function is positive.Hence under the condition in the statement, the function tsa−2naca pa+ca is concave on P .Condition (27) thus amounts to checking the non-negativity of a concave function on a convex polytope: it is enough to check the non-negativity on vertices.
Remark 4.4.In the case of a simple principal toric fibration, that is, if there is only one factor in the basis, then the condition becomes extremely simple for classes with c a ≥ tsa 2na : it is enough to check a degree two polynomial inequation on every vertex of the moment polytope.This is actually used in most of the examples of Section 5, see e.g.Proposition 5.1 or Proposition 5.3.Remark 4.5.We can write a similar statement for the general case of toric fibrations, by working on the cone decomposition.In that case the conditions to impose are: for all j, for all a, L j (x 0 )s a − 2n a (p a (x 0 ) + c a ) ≤ 0 and condition (4.1) is satisfied at all vertices of P j , that is, some vertices of P and x 0 .4.3.Extremal metrics in the anticanonical class.An important special case when the toric fiber is Fano is given by the semisimple principal toric fibrations which are themselves Fano.By our general sufficient condition, we obtain a very simple condition for the existence of extremal Kähler metrics on Fano toric fibrations.Of course, as in Corollary 4.3, it is enough to check this condition on vertices of the polytope.Furthermore, if l ext is constant, it is equal to 2 dim(Y ) since the class is the anticanonical one.As a consequence, the condition is strictly satisfied: (29) 2 dim(Y ) + 2 − l ext = 2 > 0.
In particular, we recover that a Fano toric fibration with vanishing Futaki invariant admits a Kähler-Einstein metric : 7]).Let (Y, ω Y ) be a Fano semisimple principal toric fibration with vanishing Futaki invariant.Then there exists a Kähler-Einstein metric in 2πc 1 (Y ).
More interestingly, we have the following consequence.Proof.By Remark 2.8, the non-negativity condition in Theorem 2.6 varies continuously with the weight.By equation (29), that condition is strictly satisfied at the anticanonical class, so it is satisfied on a neighborhood of this class.The only added assumption in Theorem 2.6 translates as vanishing of the Futaki invariant.
Remark 4.9.Of course, if the Futaki invariant of the anticanonical class vanishes, this is already known by Lebrun-Simanca [28] and [7].Similarly, if the anticanonical class is strictly K-unstable, nearby classes will be as well.However, in the present setting, working directly with the condition it is not hard to find an explicit neighborhood which works.Furthermore, the statement applies even when we do not know whether there exists an extremal Kähler metric or not in the anticanonical class.In the current state of knowledge, it could also happen that the anticanonical class is K-semistable (for the notion of relative K-stability adapted to extremal Kähler metrics), and the above proposition would still apply in that case.This is a further illustration of a phenomenon observed in [15].
Corollary 4.10.[7, Theorem 3] Let Y be a Fano semisimple principal toric fibration with associate Delzant polytope P .Consider the weighted Donaldson-Futaki invariant F for the weights corresponding to v-solitons defined above.Then, if F vanishes, there exists a v-soliton in 2πc 1 (Y ).

Examples of bases.
In this section, we comment on examples of possibles bases for the semisimple principal toric fibration construction.This allows to determine possible values of s a to plug into the condition.The easiest way to get a cscK basis is to choose a Kähler-Einstein manifold, equipped with a multiple of its first Chern class when it is definite, and with an arbitrary Kähler class for Calabi-Yau manifolds.
For canonically polarized manifolds, there always exists a Kähler-Einstein metric in −2πc 1 (X), and there exists such manifolds in every dimension.In particular, the value s a = − 2na ka are always allowed, for k a ∈ Z >0 .For manifolds with zero first Chern class, there always exist Kähler-Einstein metrics with zero scalar curvature.For the positive curvature case, since the projective space of dimension n is a Kähler-Einstein manifold of index n + 1, all the values s a = 2 na(na+1) ka are allowed, for k a ∈ Z >0 .More generally, for a Kähler-Einstein Fano basis of dimension n a and index I a , then all the values s a = 2 naIa ka are allowed, for k a ∈ Z >0 .Note that the Fano index of an n-dimensional Fano manifold is always an integer between 1 and n + 1.Here are a couple known results on existence of Fano Kähler-Einstein manifolds when n is small or I is large: • if I = n + 1 then X = P n is the n-dimensional projective space, and it is Kähler-Einstein, We proceed analogously for the vertex v 2 and v 3 .We conclude the proof by involving Corollary 4.3.
Remark 5.2.In Proposition 5.1, we obtain a lower bound on c depending only on the degrees p 1 and p 2 of the line bundles L 1 and L 2 .For given values of p 1 and p 2 it is possible to obtain a more optimal result.Indeed, suppose p 1 and p 2 are fixed.Then, the LHS of ( 27) is a rational fraction F depending only on the variable c.We then only need to look for constant α such that F is non-negative for c ≥ α.For example, if B = P 3 , The latter condition is illustrated in Figure 2, and it is obviously less restrictive if one can choose x 0 than the uniform condition corresponding to the obvious choice of x 0 = 0 for the monotone lattice polytope [−1, 1].
We end this paragraph by recalling that (1, w)-uniform stability of the lattice polytope [−1, 1] translates to existence of certain canonical Kähler metrics on P 1 thanks to [35].5.3.2.Extremal metrics on P 1 -bundles.We have focused on applications of our sufficient condition to semisimple principal toric bundles with dimension two toric fiber.This is because in the case of a one-dimensional toric fiber, quite a few strong results have been shown in [5].For example, it is proved in [5,Proposition 11] that if all factors (B a , ω a ) of the basis have non-negative constant scalar curvature, and the fiber is one-dimensional, then there exists an extremal Kähler metric in all compatible Kähler classes.
There cannot be such a result if some factors of the basis have negative constant scalar curvature, as shown by examples in [5].More importantly, some of these examples motivated the initial introduction of the notion of uniform K-stability, as they are likely relatively K-polystable in the sense of [38], but do not admit extremal Kähler metrics.
On the positive side, by [5,Theorem 1], there always exist extremal Kähler metrics on a semisimple principal P 1 -fibration, when all the c a are large enough, an example of existence of extremal Kähler metrics in an adiabatic regime for fibrations.However, it is not so easy to derive explicit Kähler classes with extremal Kähler metrics from this asymptotic proof.A possible approach to get explicit classes with extremal Kähler metrics would be to compute the extremal polynomial (in the terminology of [5]) and check when it is positive.This is less practical than our sufficient condition, which involves only checking the positivity of a polynomial at two points.We provide in the appendix an elementary computer program which checks whether our sufficient condition is satisfied for a simple principal P 1 -fibration, which could easily be adapted to the case of a semisimple principal P 1 -fibration.

5.3.3.
A more explicit example.Consider B a three-dimensional canonically polarized manifold, equipped with its Kähler-Einstein metric in −2πc 1 (X), whose scalar curvature is thus equal to −6.We consider the sufficient condition for existence of extremal Kähler metrics in admissible Kähler classes on the P 1 -bundles P(O B ⊕ K m B ). Up to rescaling and symmetry, this amounts to checking (v, w)-uniform K-stability of the reflexive lattice polytope [−1, 1] ⊂ R with respect to the weights v(x) = (px + c) 3  and w(x) = l ext (x) − −6 px + c (px + c) 3   where p ∈ Q, c ∈ R and c > p > 0. Our sufficient condition allows to obtain the following explicit families of extremal Kähler classes.We only show an example with very rough estimates to illustrate the results, but of course one could get much more classes by using more precise estimates in the proof, and even more classes by using the sufficient condition in Theorem 2.6 in its full generality.Proof.Using Program 1 in the appendix or straightforward but tedious computations, we obtain up to elementary simplifications that the sufficient condition reads as 75c 7 − 300c 6 − 65c 5 p 2 + 160c 4 p 2 − 15c 3 p 4 − 180c 2 p 4 − 27cp 6 + 48p 6 is greater than |−75c 6 p + 5c 4 p 3 + 80c 3 p 3 − 105c 2 p 5 + 15p 7 | Without attempting to give an optimal result, we may as well check that it is greater than 75c 6 p + 5c 4 p 3 + 80c 3 p 3 + 105c 2 p 5 + 15p 7 since c and p are positive.Writing c = αp for some α > 1 and simplifying by p 6 , we get a linear inequation in p

pY
ai x i + c a π * (ω a ) + ω P ℓ , x = (x)In the above formulas, by abuse of notation, ω P d denotes both the Kähler metric on P ℓ and its induced metric in 2πc 1 (O Y (ℓ + 1)).The tuples (c a ) satisfying(15), parametrize the compatible Kähler classes.Furthermore, suppose that B is a product of Kähler-Einstein manifolds (B, ω B ) := k a=1 (B a , ω a ).By Lemma 3.6, if we choose c a equal to the Fano index I a of B a , the corresponding compatible Kähler form ω Y defined in(14) belongs to the first Chern class 2πc 1 (Y ).In particular, if is a Fano manifold with compatible first Chern class.
is invariant by a maximal torus K Y ⊂ Aut red (Y ) containing T Y such that the following exact sequence holds (18) {0} − t − k − k B {0} , where t := Lie(T Y ), k := Lie(K Y ) and k B := Lie(K B ) where K B ⊂ Aut red (B) is a maximal torus such that ω B := k a=1 ω a is K B -invariant (without loss of generality by Lichnerowicz-Matsushima Theorem).

Corollary 4 . 3 .
Assume furthermore that for all a, c a ≥ tsa 2na .Then the semisimple principal toric fibration (Y, [ω Y ]) admits an extremal Kähler metric in [ω Y ] if inequation(27) is satisfied at every vertex of P .

Corollary 4 . 6 .
A Fano semisimple principal toric fibration Y admits an extremal Kähler metric in c 1 (Y ) if its extremal function l ext satisfies: (28) sup l ext ≤ 2(dim(Y ) + 1) Proof.By Lemma 3.6, for all a, s a = 2n a c a and the condition from Corollary 4.2 becomes 2 dim(Y ) + 2 − l ext ≥ 0 on P

Proposition 4 . 8 .
Let (Y, ω Y ) be a Fano semisimple principal fibration.Then on a neighborhood of the anticanonical class, a compatible Kähler class admits a cscK metric if and only if its Futaki invariant vanishes.

Proposition 5 . 1 . 5 − 6537672c 4 + 5624964c 3 − 6 + 382725c 5 +
B ) be a KE Fano threefold with α B := [ω B ] primitive element of H 2 (B, Z) proportional to the first Chern class 2πc 1 (B).Let L i − B be a holomorphic line bundle of degree p i proportional to the anticanonical line bundle −K B , i.e. p i α B = 2πc 1 (L i ).We consider a simple principal toric fibration (i.e. the basis has only one factor) π :Y := P(L 0 ⊕ L 1 ⊕ L 2 ) − B. Since the holomorphic class of Y is invariant by tensoring L 0 ⊕ L 1 ⊕ L 2 witha line bundle, we can suppose without loss of generality that L 0 = O is the trivial line bundle and p i ≥ 0, i = 1, 2. When B is a local Kähler product of nonnegative cscK metric and p 1 = p 2 > 0 or p 2 > p 1 = 0, it is known [5,Proposition 11], that there exists an extremal metric in every compatible Kähler classes.We then suppose p 2 ≥ p 1 > 0.The compatible Kähler classes are parametrized by constants c and are of the form (31)α c := 2πc 1 (O Y (3)) + cπ * (α B ),Asintroduced in Section 5.1, since B is a Fano threefold, the only possible Fano indices I are 1, 2, 3 or 4. In the case where B is the quadric Q 3 or the projective space P 3 (i.e. if I = 3 or I = 4 respectively), Leray-Hirch Theorem shows that H 2 (Y, R) ∼ = R 2 .It follows that, up to scaling, all Kähler classes are compatible, i.e. of the form of (31).It is known that [6, Theorem 4] for c sufficiently large, the class α c is extremal.The following Proposition gives a precise value for c, depending on p 1 and p 2 , from which α c admits an extremal metric.Let Y = P(O ⊕ L 1 ⊕ L 2 ) − B be a simple principal toric fibration over a Kähler-Einstein Fano threefold B, where L 1 and L 2 are holomorphic line bundles of degrees 1 ≤ p 1 ≤ p 2 proportional to the anti-canonical line bundle −K B of B. Then there exists an extremal metric in α c for c ≥ 7p 2 .Proof.Since the arguments are identical for each Fano index I, we give the proof only for I = 4.By Corollary 4.3, for c ≥ 4, it is sufficient to check (27) evaluated in each vertex v 1 := (−1, 2), v 2 := (−1, −1), v 3 := (2, −1) of the polytope P .Using Program 2 in Appendix A, we find that the LHS of (27) evaluated in v 1 is a rational fraction in the variables c, p 1 , p 2 : LHS of (27) = P (c, p 1 , p 2 ) Q(c, p 1 , p 2 ) .We give the explicit expression of the polynomials P and Q in Appendix B. Suppose now c ≥ 7p 2 and p 2 ≥ p 1 ≥ 1.Then we can find two polynomials R(c) :=12250c 10 − 73500c 9 − 295470c 8 + 1296540c 7 − 3657150c 6 + 3776220c 6193584c 2 + 85920232c − 1889568 and S(c) :=6125c 10 + 18375c 9 + 6615c 8 + 19845c 7 + 127575c 17496c 4 + 52488c 3 − 288684c 2 − 866052c such that 0 < R(c) ≤ P (c, p 1 , p 2 ) and 0 < S(c) and S(c) ≥