Thibaut Delcroix

Kähler-Einstein metrics on group compactifications

may come

Introduction

The question of the existence of a Kähler-Einstein metric on a toric Fano manifold, that is, a n-dimensional Fano manifold X acted upon by (C * ) n with an open dense orbit isomorphic to (C * ) n , was fully settled by Wang and Zhu in [START_REF] Wang | Kähler-Ricci solitons on toric manifolds with positive first Chern class[END_REF]. They showed that there exists a Kähler-Einstein metric on X if and only if the Futaki invariant of X vanishes. This invariant was known [START_REF] Mabuchi | Einstein-Kähler forms, Futaki invariants and convex geometry on toric Fano varieties[END_REF] to be linked with the barycenter of the polytope associated to X, so that the condition is equivalent to the fact that this barycenter is the origin.

We consider the existence of Kähler-Einstein metrics on a natural generalization of toric Fano manifolds: the bi-equivariant Fano compactifications of complex connected reductive groups. More precisely, given such a group G, we consider the Fano manifolds X which admit a G × G-action with an open dense orbit isomorphic to G as a G × G-homogeneous space under left and right translations. We will call these, for simplicity, group compactifications. To such a manifold is, as in the toric case, associated a polytope P + , that encodes the information about the boundary X \ G and the anticanonical line bundle.

Let G be a complex connected reductive group, choose T a maximal torus in G. Let M denote the group of characters of T , and let Φ ⊂ M be the root system of (G, T ). Choose a system of positive roots Φ + , denote by 2ρ the sum of positive roots, and by Ξ the relative interior of the cone generated by Φ + . The polytope P + associated to X is in the positive Weyl chamber in M ⊗R. Let bar DH (P + ) denote the barycenter of P + with respect to the measure α∈Φ + α, p 2 dp where dp is the Lebesgue measure on M ⊗R normalized by the lattice M . Our main result is the following characterization of Kähler-Einstein Fano group compactifications in terms of the polytope P + .

Theorem A. Let X be a Fano compactification of G, then X admits a Kähler-Einstein metric if and only if bar DH (P + ) ∈ 2ρ + Ξ.

Let us remark here that P + is related to the moment polytope of X from a symplectic point of view, and that bar DH (P + ) is the barycenter with respect to the Duistermaat-Heckman measure on this moment polytope.

The condition obtained is computable given an explicit polytope, and we provide in this article both a new example of Kähler-Einstein metric and an example of group compactification without any Kähler-Einstein metric. We refer to the author's PhD thesis [START_REF] Delcroix | Métriques de Kähler-Einstein sur les compactifications de groupes[END_REF], or new preprint [Dela], for more examples. The example that does not satisfy the condition is especially interesting because its Futaki invariant vanishes. In light of the proof of the Yau-Tian-Donaldson conjecture in the Fano case [START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities[END_REF][START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π[END_REF][START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof[END_REF] and [START_REF] Tian | K-stability and Kähler-Einstein metrics[END_REF]), and the recent work of Datar and Székelyhidi [DS], it means that there is a non-trivial G × G-equivariant destabilizing test configuration.

In fact, in [Dela] we obtain through essentially algebraic methods a combinatorial criterion for equivariant K-stability of a Fano spherical variety. Combined with [DS] it provides a combinatorial criterion for the existence of Kähler-Einstein metrics on spherical manifolds. Group compactifications are examples of spherical varieties and the criterion from [Dela] specializes to the criterion from Theorem A in this case. Thus the main result of [Dela] provides a broad generalization of Theorem A. On the other hand, the criterion obtained here served as the guiding intuition for [Dela]. Furthermore, the combination of the two articles provide an independent, and low tech, in the sense that no Gromov-Hausdorff convergence result is needed, proof of the Yau-Tian-Donaldson conjecture for Fano group compactifications.

It should be noted that in [Dela] a criterion is also obtained for the existence of Kähler-Ricci solitons on spherical manifolds, hence on group compactifications. The analytic methods from the present article could easily be extended to the case of Kähler-Ricci solitons to provide another proof. For simplicity, we do not provide details for this case.

It was mentioned to the author by Gábor Székelyhidi that the example with no Kähler-Einstein is also interesting in the study of the Kähler-Ricci flow on Fano manifolds. Indeed, it follows from the work of Chen, Sun and Wang in [CSW] that the Kähler-Ricci flow on a Fano manifold X with no Kähler-Einstein metrics should converge to some Kähler-Ricci soliton on the central fiber of a special test configuration for X. It is reasonable to expect, in the situation where X admits an action of a compact group and the initial metric is invariant, that the corresponding test configuration is equivariant. In our example, the possible central fibers are known (see [START_REF] Alexeev | On K-stability of reductive varieties[END_REF][START_REF] Alexeev | Stable reductive varieties. II. Projective case[END_REF]), and are all singular. No example of such a behavior for the Kähler-Ricci flow is known, so it would be interesting to study this in detail.

Let us mention now a consequence of our result in the variational setting. Darvas and Rubinstein proved in [DR] the equivalence between the existence of Kähler-Einstein metrics and various notions of properness of functionals on spaces of Kähler potentials, allowing to take into account automorphisms of the manifold. Our result provides a combinatorial criterion for these notions of properness. More precisely, though we refer to [DR] for details, let ω be a reference Kähler form in 2πc 1 (X), and let H denote the space of Kähler potentials, i.e. functions ψ on X such that ω + i∂∂ψ is still a Kähler form. Assume X is Kähler-Einstein and G = Aut 0 (X) is the neutral component of the automorphism group of X. Let K be a maximal compact subgroup of G, assume that ω is K-invariant and let H K be the space of K-invariant Kähler potentials. Then Darvas and Rubinstein prove that the Mabuchi functional is proper on H modulo the action of G, and that it is proper on H K provided the center of G is finite. In the case of group compactifications, the automorphism group may very well have a center that is not finite, as it is the case for many toric manifolds. It is interesting to note that the methods from [DR] should allow to prove a statement combining the two statements recalled above, valid with no hypotheses on the automorphism group, namely that the Mabuchi functional is proper on H K modulo the action of the normalizer N G (K) of K in G.

Let us now describe our methods. The guiding ideas are to restrict to the open dense orbit G, and consider only K × K-invariant metrics. The last part is possible because of Matsushima's theorem [START_REF] Matsushima | Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne[END_REF] asserting that the isometry group of a Kähler-Einstein metric is a maximal compact subgroup of the connected group of automorphisms of X. We can then combine the two by using the classical KAK decomposition of reductive groups. This decomposition identifies the quotient of G by K × K with the closed positive Weyl chamber. Another point of view is that it translates K × K-invariant geometry on G to W -invariant geometry on M ⊗ R, where W is the Weyl group of Φ.

We used this fact in [Delb] to associate to any positively curved hermitian metric on the anticanonical line bundle -K X a W -invariant convex function, whose asymptotic behavior is controlled by the polytope P + . We use this here to translate the Kähler-Einstein equation into a real Monge-Ampère equation on a real vector space. A key point in this process is to compute the complex Hessian of a K × K-invariant function ϕ on G in terms of the function it determines on the positive Weyl chamber. This computation then provides an expression of the complex Monge-Ampère MA C (ϕ), which is the determinant of the complex Hessian.

Proposition B. Let ϕ be a K × K-invariant smooth function G -→ R, and let u denote the corresponding W -invariant function on M ⊗ R. Then for any x in the interior of the positive Weyl chamber,

MA C (ϕ)(exp x) = MA R (u)(x) α∈Φ + α, ∇u(x) 2 sinh 2 α, x ,
where MA R (u) denotes the determinant of the real Hessian of u.

We use this result to obtain an expression of the potential, with respect to a Haar measure on G, of the restriction to G of a volume form on X determined by a hermitian metric on -K X , in terms of the convex potential of the hermitian metric. This allows to translate the restriction of the Kähler-Einstein equation to G as a real Monge-Ampère equation in a convex function u reading:

MA R (u) α∈Φ + α, ∇u 2 = e -u J,
where J is the function defined by J(x) = α∈Φ + sinh 2 α, x .

Another consequence of our computation of the complex Hessian is an analytic derivation of a formula due to Brion [START_REF] Brion | Groupe de Picard et nombres caractéristiques des variétés sphériques[END_REF] and Kazarnovskii [START_REF] Kazarnovskiȋ | Newton polyhedra and Bezout's formula for matrix functions of finite-dimensional representations[END_REF] for the degree of an ample line bundle on a group compactification, however up to a multiplicative constant, which does not matter for our barycenters considerations.

Let us digress here to mention that the translation of the Kähler-Einstein equation into a real Monge-Ampère equation was already a key step in the toric case, but also in the case of the horospherical manifolds studied by Podestà and Spiro [START_REF] Podestà | Kähler-Ricci solitons on homogeneous toric bundles[END_REF]. These manifolds, along with the group compactifications belong to the large family of spherical manifolds, which was highlighted by Donaldson in his survey [START_REF] Donaldson | Kähler geometry on toric manifolds, and some other manifolds with large symmetry[END_REF] as an interesting family on which to study the existence of Kähler-Einstein metrics. We believe that methods similar to ours can be used for even more spherical manifolds. On another direction, Donaldson suggests not only to study the existence of Kähler-Einstein metrics, but more generally of canonical metrics such as constant scalar curvature. We expect that our computation of the full complex Hessian can be used in relation with this problem. Note that Abreu obtained a formula for the scalar curvature of torus invariant Kähler metrics on toric manifolds [START_REF] Abreu | Kähler geometry of toric varieties and extremal metrics[END_REF] which led to important advances in the study of canonical metrics on toric manifolds.

The main difference between the case of toric or horospherical manifolds and the case of group compactifications here is the term J, which is independent on the function u. Indeed, the toric Monge-Ampère equation is just MA R (u) = e -u and the horospherical case only adds well controlled gradient terms. We thus have to rework in depth the proof of Wang and Zhu to deal with the term J and the walls of the Weyl chamber. This is how we obtained our condition, to replace the information given by the Futaki invariant in the toric case.

The common part with Wang and Zhu is the use of the classical results on the continuity method, and the study of a well chosen proper and strictly convex function ν t defined in terms of the solutions of the continuity method equation. In our case this function involves the term J. The key is then to interpret the absence of a priori estimates along the continuity method as the fact that the point at which the minimum of the function ν t is attained is unbounded as t increases along the continuity method, then to interpret this as a condition on the polytope.

The precise analysis of this behavior further allows to determine the maximal time of existence of solutions along the continuity method. This is called the greatest Ricci lower bound as Székelyhidi [START_REF] Székelyhidi | Greatest lower bounds on the Ricci curvature of Fano manifolds[END_REF] proved that it is also the supremum of all t < 1 such that there exists a Kähler form in c 1 (X) with Ric(ω) ≥ tω where Ric(ω) is the Ricci form of ω.

A B ∂(-Ξ + P + )

P + C Figure 1: Greatest Ricci lower bound
Theorem C. Assume there are no Kähler-Einstein metrics on the group compactification X and let R(X) be the greatest Ricci lower bound of X. Then

R(X) = sup t < 1 ; 1 1 -t 2ρ -t • bar DH (P + ) ∈ -Ξ + P + .
The particular case of toric manifolds was obtained by Li [START_REF] Li | Greatest lower bounds on Ricci curvature for toric Fano manifolds[END_REF]. Indeed, in the case of a Fano toric manifold X, the polytope P + is the usual moment polytope, the root system Φ is trivial, so that 2ρ = 0, Ξ is restricted to the origin {0} and the Duistermaat-Heckman barycenter bar DH (P + ) is just the barycenter of P + with respect to the Lebesgue measure. The condition in Theorem A thus translates as Wang and Zhu's condition, that is, the condition that the barycenter is the origin. To recover Chi Li's result, let us introduce additional notations in the case of groups, as illustrated in Figure 1: let A be bar DH (P + ), let B be 2ρ and let C denote the intersection of the boundary of -Ξ + P + with the half line starting from A in the direction of B if it exists. It follows from Theorem C that the intersection exists if and only if R(X) < 1 and is then the point B + s --→ AB with s = R(X)/(1 -R(X)). We may then check that |BC|/|AC| = s/(1 + s) = R(X). This is precisely the expression given in [START_REF] Li | Greatest lower bounds on Ricci curvature for toric Fano manifolds[END_REF] in the toric case, up to a change in notations.

The structure of the article is as follows. Section 1 is devoted to the computation of the Hessian of a K × K-invariant function on a reductive group G. Proposition B is obtained as a consequence in Corollary 1.3. Section 2 is a short introduction to group compactifications, allowing to introduce the associated polytope and relevant examples. The tool of convex potentials, developed in [Delb], is recalled here and applied to obtain a formula for the degree of an ample linearized line bundle on a group compactification. Section 3 and Section 4 provide the general strategy of the proofs of Theorem A and Theorem B, and preliminary results. Finally, Section 5 contains the proof of the necessity of our condition and of an upper bound on R(X), and Section 6 contains the proof that the condition is sufficient and that the upper bound on R(X) is in fact equal to R(X). Theorem A is a consequence of Proposition 5.3 and Theorem 6.7, and Theorem B is a consequence of Proposition 5.5 and Section 6.4.

The present article along with [Delb] contain the main results of the author's PhD thesis [START_REF] Delcroix | Métriques de Kähler-Einstein sur les compactifications de groupes[END_REF]. The author would like to thank his advisor Philippe Eyssidieux, and also Michel Brion for helpful discussions and the explanations on the automorphism group of the non Kähler-Einstein example.

Notations

Let us introduce here some notations on groups that will be used throughout the article. We use as reference for Lie groups the books [START_REF] Knapp | Lie groups beyond an introduction[END_REF][START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF], and [START_REF] Borel | Linear algebraic groups[END_REF] for algebraic groups.

Let G be a connected complex linear reductive group. Let T be a maximal torus in G. We choose a maximal compact subgroup K in G such that S := K∩T is a maximal compact torus in T . We denote by g, k, t, s the respective Lie algebras of G, K, T , S.

Let Φ denote the root system of (G, T ) and W the corresponding Weyl group. We choose a system of positive roots Φ + . We use the standard notation ρ for the half sum of the positive roots.

Since G is reductive, we have the Cartan decomposition g = k ⊕ ik, and denote by θ the corresponding Cartan involution on g. Define a := t ∩ ik. We identify a with N ⊗ R where N is the group of one parameter subgroups of T . Denote by a + the positive open Weyl chamber defined by Φ + .

Recall the KAK decomposition for reductive groups [Kna02, Theorem 7.39]: every element g ∈ G can be written g = k 1 exp(a)k 2 with a ∈ a + uniquely determined by g, and k 1 , k 2 ∈ K.

We can decompose a in a toric part and a semisimple part:

a = a t ⊕ a ss ,
with a t = z(g) ∩ ik and a ss = a ∩ [g, g]. The Killing form of g, restricted to a ss , gives a scalar product, and we choose a scalar product •, • on a that coincides with the Killing form on a ss and leaves a ss and a t orthogonal. We use this scalar product to identify a and its dual, which is also identified with M ⊗ R, where M is the group of characters of T .

1 Complex Monge-Ampère on G

The aim of this section is to compute, for any

K × K-invariant function ψ on G, the function MA C (ψ) such that i 2π ∂∂ψ n = MA C (ψ)dg,
where dg is a Haar volume form on G and n is the complex dimension of G.

As the notation suggests, this function is the complex Monge-Ampère of ψ in some local coordinates, i.e. the determinant of the complex Hessian of ψ in these coordinates. We express this Hessian and Monge-Ampère in terms of the restriction of ψ to exp(a), using the K × K-invariance and the KAKdecomposition.

Complex Hessian matrix on G

We first choose a special basis of g ≃ T e G. Recall the root decomposition:

g = t ⊕ α∈Φ g α
where, if ad denotes the adjoint representation of g,

g α := {x ∈ g ; ad(h)(x) = α, h x ∀h ∈ t} .
We can choose a set {e α } α∈Φ of elements of g such that e -α = -θ(e α ), [e α , e -α ] = α and e α generates the complex line g α [Hel78, Chapter VI, Lemma 3.1]. We would like to choose a complex basis of g which is also a real basis of k. Define, for α ∈ Φ + ,

k α := x ∈ k ; ad(h) 2 (x) = α, h 2 x ∀h ∈ s .
Then each k α is of real dimension two and we have g α ⊕ g -α = k α ⊕ ik α . We obtain a real basis of k α by considering e α + θ(e α ) = e α -e -α and ie α + θ(ie α ) = ie α + ie -α . We complete this with a real basis of s to obtain a real basis of k = s ⊕ α∈Φ + k α , and thus a complex basis of g. We denote this complex basis of g by (l j ) n j=1 . Since the exponential map is a biholomorphism from a neighborhood of 0 ∈ g to a neighborhood of the neutral element e in G, we get holomorphic coordinates on G near e. Then, using a translation on the right by g ∈ G, this defines holomorphic coordinates on a neighborhood of g. More precisely, the map corresponding to the local coordinates is the map C n → G defined by

(z 1 , . . . , z n ) → exp(z 1 l 1 + • • • z n l n )g.
We will compute the complex Hessian with respect to these coordinates. If ψ is a function on G we denote by Hess C (ψ)(g) the complex Hessian and by MA C (ψ)(g) the determinant of the complex Hessian of ψ at g, called the complex Monge-Ampère, in both cases with respect to the coordinates given above.

If f is a function on a, then we denote by MA R (f )(x) the determinant of its real Hessian at x, and by ∇f the gradient of f with respect to the scalar product •, • on a.

By the KAK decomposition, a K × K-invariant function ψ on G is completely determined by the function f defined on a by f (x) = ψ(exp x).

Let us begin by a remark that shows that even though we use a specific set of coordinates to compute the complex Hessian, the complex Monge-Ampère thus obtained is the potential of i 2π ∂∂ψ with respect to a Haar measure, up to a multiplicative constant.

Remark 1.1. Let dg be a Haar volume form on G (thus a G × G-invariant volume form, because a reductive group is unimodular [Kna02, Section VIII.2]). Then there exists a constant C > 0 such that for any K × K-invariant function ψ on G, we have,

i 2π ∂∂ψ n = CMA C (ψ)dg.
Indeed, in local coordinates z j , we can write

i 2π ∂∂ψ = j,k i 2π ∂ 2 ψ ∂z j ∂z k dz j ∧ dz k ,
so in the coordinates above, we have locally

i 2π ∂∂ψ n = 1 (2π) n MA C (ψ)(i n dz 1 ∧ dz 1 ∧ . . . ∧ dz n ∧ dz n ).
But by the construction of the coordinates, the global form defined locally as i n dz 1 ∧ dz 1 ∧ . . . ∧ dz n ∧ dz n is a Haar volume form. To conclude, remark that all Haar volume forms are positive scalar multiples of one another.

Theorem 1.2. Let ψ be a K × K invariant function on G, and f the associated function on a. Then in the coordinates above and for a ∈ a + , the complex Hessian matrix of ψ is diagonal by blocks, equal to:

Hess C (ψ)(exp(a)) =         1 4 Hess R (f )(a) 0 0 0 M α1 (a) 0 0 0 . . . . . . . . . . . . . . . 0 0 0 M αp (a)        
where the (α i ) i∈{1,...,p} describe the positive roots of Φ and M α is defined by:

M α (a) = 1 2 α, ∇f (a) coth α, a i -i coth α, a .
As a corollary, we obtain Proposition B:

Corollary 1.3. Let ψ be a K ×K invariant function on G, and f the associated function on a. Then in the coordinates above and at a ∈ a + , if r denotes the rank of G and p the number of positive roots, we have Proof. Since MA R (f )(a) = det(Hess R (f )(a)), we just have to compute the determinant of M α . This is

MA C (ψ)(exp(a)) = 1 4 r+p MA R (f )(a) 1 J(a) α∈Φ + α, ∇f (a)
det(M α ) = 1 2 α, ∇f (a) 2 (coth 2 α, a -1) = 1 4 α, ∇f (a) 2 cosh 2 α, a -sinh 2 α, a sinh 2 α, a = 1 4 α, ∇f (a) 2 1 sinh 2 α, a Example 1.4. Consider the case G = PSL 2 (C). Then a + ≃ R *
+ , and there is only one positive root that we can identify with the identity on R. Then

Hess C (ψ)(exp(a)) = 1 2   f ′′ (a)/2 0 0 0 f ′ (a) coth(a) if ′ (a) 0 -if ′ (a) f ′ (a) coth(a)  
and the complex Monge-Ampère reads:

MA C (ψ)(exp(a)) = 1 16 f ′′ (a)(f ′ (a)) 2 1 sinh 2 (a)
.

The rest of the section is devoted to the proof of the theorem. The technique of the proof is based on the work of Bielawski [START_REF] Bielawski | Prescribing Ricci curvature on complexified symmetric spaces[END_REF]. In particular, the idea to use the decomposition in Lemma 1.6 and the Baker-Campbell-Hausdorff formula appears in this article. We begin by introducing these two tools.

The Baker-Campbell-Hausdorff formula

As a formal series in the variables x and y, the logarithm of exp(x) exp(y) is well defined. We denote this by BCH(x, y). The Baker-Campbell-Hausdorff formula is the following.

Proposition 1.5. [Hoc65, Theorem X.3.1] There exists a neighborhood U of 0 in g such that for all x and y in U , BCH(x, y) is convergent and defines an element of g, and we have exp(x) exp(y) = exp(BCH(x, y)).

Furthermore we know explicitly the terms of BCH(x, y). We will only use the following:

BCH(x, y) = x + y + 1 2 [x, y] + O
where O denotes terms of order higher than 2 in x and y.

A decomposition in g

Let a ∈ a + . Let Exp(ad(a)) be the linear application g → g defined by

Exp(ad(a))(x) = ∞ n=0 ad(a) n (x) n! .
Recall that G acts on g through the adjoint action Ad, and that we have the general relation

Exp(ad(a))(x) = Ad(exp a)(x)
for x ∈ g.

Lemma 1.6. Let l ∈ g and a ∈ a + . Then

1. there exists A ∈ k, B ∈ a and C ∈ Ad(exp a) α∈Φ + k α such that l = A + B + C; 2. if l ∈ α∈Φ g α then B = 0; 3. if l ∈ k α , and l ′ denotes ad(a)(il)/ α, a , then l ′ ∈ k α and the decomposi- tion above for the element il ∈ g α ⊕ g -α reads il = -(cosh α, a )l ′ + 1 sinh( α, a ) Ad(exp a)(l ′ ); 4. if l = e α + θ(e α ) then l ′ = ie α -iθ(e α ); 5. if l = ie α -iθ(e α ) then l ′ = -e α -θ(e α ).
In the statement, the result is more and more precise as we know more precisely the element considered. In the proof we will begin by the very precise case of the elements of the basis and work our way up by linearity.

Proof. Let a ∈ a + . We begin by the two last points. By definition of l ′ , we have, if l = e α + θ(e α ),

l ′ = ad(a)(il)/ α, a = ad(a)(ie α )/ α, a + ad(a)(iθ(e α ))/ α, a = ie α -iθ(e α ) and if l = ie α -iθ(e α ), l ′ = ad(a)(il)/ α, a = ad(a)(-e α )/ α, a + ad(a)(θ(e α ))/ α, a = -e α -θ(e α ).
In particular, in both cases, l ′ is in k α . By linearity this is also true of l ′ for any l ∈ k α .

To prove the decomposition in the third point, it suffices to compute that, using the definition of k α ,

Exp(ad(a))(l ′ ) = (cosh α, a )l ′ + (sinh α, a )il = Ad(exp a)(x).
Then by linearity the first point holds true for any l ∈ i α∈Φ + k α , with B = 0. But we have

α∈Φ g α = α∈Φ + k α ⊕ i α∈Φ + k α ,
so we have the decomposition for any l ∈ α∈Φ g α , with B = 0.

Finally for l ∈ t, it suffices to decompose l along t = s ⊕ a. By linearity and the root decomposition, we obtain the proposition for any l ∈ g.

Using the Baker-Campbell-Hausdorff formula

We want to compute the complex Hessian of ψ in the chosen system of coordinates, at a point exp(a) for a in the open Weyl chamber a + . If l 1 and l 2 are two vectors in the chosen basis of k, we thus want to compute:

H l1,l2 (a) := ∂ 2 ∂z 1 ∂z 2 z1,z2=0 ψ(exp(z 1 l 1 + z 2 l 2 ) exp(a)).
There are different cases, according to the subspaces where l 1 and l 2 lie. We will first describe the part of the argument that is used in all cases, which relies on the Baker-Campbell-Hausdorff formula, and then deal with each case separately.

Using the decomposition from Lemma 1.6 on z 1 l 1 + z 2 l 2 we can write

z 1 l 1 + z 2 l 2 = A 1 + B 1 + C 1 with A 1 ∈ k, B 1 ∈ a and C 1 ∈ Ad(exp a)(k)
, and all are of homogeneous degree one in z 1 and z 2 . Let

D 1 = 1 2 ([B 1 , A 1 ] + [C 1 , A 1 ] + [C 1 , B 1 ]),
it is of order two in z 1 and z 2 .

Let us now use again Lemma 1.6 to get

D 1 = A 2 + B 2 + C 2 . with A 2 ∈ k, B 2 ∈ a and C 2 ∈ Ad(exp a)(k)
, and all are of homogeneous degree two in z 1 and z 2 . The Baker-Campbell-Hausdorff formula allows to prove the following lemma, which can be seen as an explicit infinitesimal KAK decomposition. Note that to lighten the notations we do not write the dependence on z 1 , z 2 , but all the terms defined above A j , B j , C j and D 1 are in fact functions of these two complex variables.

Lemma 1.7. We can write

exp(z 1 l 1 + z 2 l 2 ) exp(a) = k 1 exp(B 1 + B 2 + a + O)k 2
where k 1 , k 2 ∈ K, and O denotes terms of order greater than two in z 1 and z 2 .

Proof. We begin by applying Proposition 1.5 to exp(-A 1 ) exp(A 1 + B 1 + C 1 ), and get that this is equal to

exp (B 1 + C 1 + [-A 1 , B 1 + C 1 ]/2 + O 1 ) ,
where O 1 denotes terms of order greater than 2 in z 1 and z 2 .

Then we multiply on the right by exp(-C 1 ) and get, with Proposition 1.5 again,

exp (B 1 + [-A 1 , B 1 + C 1 ]/2 + [B 1 , -C 1 ]/2 + O 2 ) ,
where O 2 denotes terms of order greater than 2 in z 1 and z 2 . By definition of D 1 , we have proved

exp(-A 1 ) exp(z 1 l 1 + z 2 l 2 ) exp(-C 1 ) = exp (B 1 + D 1 + O 2 ) .
Recall that

D 1 = A 2 + B 2 + C 2 ,
and that all of these are of degree two in z 1 and z 2 . We apply another time the Proposition 1.5, to exp(-A 2 ) exp(B 1 + D 1 + O 2 ), but here we only need to use the first term in the development of BCH. We might say that A 2 commutes up to order two with elements of degree greater or equal to one in z 1 , z 2 . We get

exp(-A 2 ) exp(B 1 + D 1 + O 2 ) = exp(B 1 + B 2 + C 2 + O 3 ),
where O 3 denotes terms of order greater than 2 in z 1 and z 2 .

One further use of the Baker-Campbell-Hausdorff formula yields

exp(-A 2 ) exp(B 1 + D 1 + O 2 ) exp(-C 2 ) = exp(B 1 + B 2 + O 4 ),
where O 4 denotes terms of order greater than 2 in z 1 and z 2 . Consider now exp(C 2 ) exp(C 1 ). Since C 1 , C 2 ∈ Ad(exp a)(k), we have

exp(C 2 ) exp(C 1 ) = exp(a)k 2 exp(-a)
for some k 2 ∈ K. On the other hand, we have k 1 := exp(A 1 ) exp(A 2 ) ∈ K.

Summing up we have proved that

exp(z 1 l 1 + z 2 l 2 ) = k 1 exp(B 1 + B 2 + O 4 ) exp(a)k 2 exp(-a).
But then

exp(z 1 l 1 + z 2 l 2 ) exp(a) = k 1 exp(B 1 + B 2 + O 4 ) exp(a)k 2 ,
and one last application of Proposition 1.5 gives the lemma, because B 1 , B 2 and a commute:

exp(z 1 l 1 + z 2 l 2 ) exp(a) = k 1 exp(B 1 + B 2 + a + O)k 2
where O denotes terms of order greater than 2 in z 1 and z 2 .

Lemma 1.8. We have

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ).
Proof. We first use K × K-invariance of ψ and Lemma 1.7 to write

ψ(exp(z 1 l 1 + z 2 l 2 ) exp(a)) = ψ(exp(a + B 1 + B 2 + O)).
Then

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 z1,z2=0 ψ(exp(z 1 l 1 + z 2 l 2 ) exp(a)) = ∂ 2 ∂z 1 ∂z 2 0 ψ(exp(a + B 1 + B 2 + O))
because O is of order greater than two, this becomes

= ∂ 2 ∂z 1 ∂z 2 0 ψ(exp(a + B 1 + B 2 )) since a + B 1 + B 2 ∈ a, this is = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 1 + B 2 )
It remains to determine B 1 + B 2 for all coefficients of the Hessian, and then to compute the coefficient. For that, since we want to reduce to real coordinates, we recall that if

z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 then ∂ 2 ∂z 1 ∂z 2 = 1 4 ∂ 2 ∂x 1 ∂x 2 + ∂ 2 ∂y 1 ∂y 2 + i 4 ∂ 2 ∂x 1 ∂y 2 - ∂ 2 ∂y 1 ∂x 2 . 1.5 Determining H l 1 ,l 2 (a) Lemma 1.9. Assume l 1 , l 2 ∈ s. Then H l1,l2 (a) is the corresponding coefficient of Hess R (f )(a)/4: H l1,l2 (a) = 1 4 ∂ 2 ∂y 1 ∂y 2 0 f (a + y 1 il 1 + y 2 il 2 ).
Proof. In this case we have andA 1 and B 1 commute, so D 1 = 0 and B 2 = 0.

z 1 l 1 + z 2 l 2 = A 1 + B 1 + 0 with A 1 = x 1 l 1 + x 2 + l 2 ∈ s and B 1 = y 1 l 1 + y 2 l 2 ∈ a,
Then by Lemma 1.8,

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + y 1 il 1 + y 2 il 2 ) = 1 4 ∂ 2 ∂y 1 ∂y 2 0 f (a + y 1 il 1 + y 2 il 2 )
Lemma 1.8 now gives

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = ∂ 2 ∂z 1 ∂z 2 0 f a + y 2 il 2 -y 2 1 (coth α, a )[il 1 , l ′ 1 ]/2 = 0 Lemma 1.11. Assume l 1 ∈ k α1 and l 2 ∈ k α2 , with α 1 = α 2 positive roots. Then H l1,l2 (a) = 0.
Proof. Using Lemma 1.6, we write

il 1 = -(coth α 1 , a )l ′ 1 + 1 sinh α 1 , a Ad(exp a)(l ′ 1 ) il 2 = -(coth α 2 , a )l ′ 2 + 1 sinh α 2 , a Ad(exp a)(l ′ 2 )
with l ′ 1 = ad(a)(il 1 )/ α 1 , a and l ′ 2 = ad(a)(il 2 )/ α 2 , a . Then we have

A 1 = x 1 l 1 + x 2 l 2 -y 1 (coth α 1 , a )l ′ 1 -y 2 (coth α 2 , a )l ′ 2 B 1 = 0 C 1 = y 1 1 sinh α 1 , a (Ad(exp a)(l ′ 1 ) + y 2 1 sinh α 2 , a (Ad(exp a)(l ′ 2 ) D 1 = [C 1 , A 1 ]/2 = [y 1 il 1 + y 2 il 2 + y 1 (cosh α 1 , a )l ′ 1 + y 2 (cosh α 2 , a )l ′ 2 , A 1 ]/2 We have y 1 (cosh α 1 , a )l ′
1 + y 2 (cosh α 2 , a )l ′ 2 and A 1 in k, so their bracket remains in k and doesn't appear in B 2 . We compute [y 1 il 1 + y 2 il 2 , A 1 ] which is equal to

x 2 y 1 [il 1 , l 2 ] -y 2 1 (coth α 1 , a )[il 1 , l ′ 1 ] -y 1 y 2 (coth α 1 , a )[il 2 , l ′ 1 ] + x 1 y 2 [il 2 , l 1 ] -y 2 2 (coth α 2 , a )[il 2 , l ′ 2 ] -y 1 y 2 (coth α 2 , a )[il 1 , l ′ 2 ].

Again the properties of the root decomposition tell us that

[il 1 , l 2 ], [il 1 , l ′ 2 ], [il 2 , l 1 ], and [il 2 , l ′ 1 ]
are in α∈Φ g α , so the corresponding terms do not contribute to B 2 . As before, [il 1 , l ′ 1 ] and [il 2 , l ′ 2 ] are in a, so we get

B 2 = 1 2 (-y 2 1 (coth α 1 , a )[il 1 , l ′ 1 ] -y 2 2 (coth α 2 , a )[il 2 , l ′ 2 ]).
Applying Lemma 1.8, we get

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = ∂ 2 ∂z 1 ∂z 2 0 f a -y 2 1 coth( α 1 , a )[il 1 , l ′ 1 ]/2 -y 2 2 coth( α 2 , a )[il 2 , l ′ 2 ]/2 = 0
Suppose now that α 1 = α 2 = α. The subspace k α is two dimensional, and we have chosen a basis formed by the vectors e α + θ(e α ) and ie α -iθ(e α ).

First we deal with the case when l 1 = l 2 .

Lemma 1.12. Suppose l 1 = e α + θ(e α ) and l 2 = ie α -iθ(e α ). Then

H l1,l2 (a) = i 2 α, ∇f (a) = -H l2,l1 (a)
.

Proof. Using Lemma 1.6, we write, just as in the previous proof

il 1 = -(coth α, a )l ′ 1 + 1 sinh α, a Ad(exp a)(l ′ 1 ) il 2 = -(coth α, a )l ′ 2 + 1 sinh α, a Ad(exp a)(l ′ 2 )
with l ′ 1 = l 2 and l ′ 2 = -l 1 . Then we have

A 1 = (x 1 + y 2 coth α, a )l 1 + (x 2 -y 1 coth α, a )l 2 B 1 = 0 C 1 = y 1 1 sinh α, a Ad(exp a)(l ′ 1 ) + y 2 1 sinh α, a Ad(exp a)(l ′ 2 ) D 1 = [C 1 , A 1 ]/2 = [y 1 il 1 + y 2 il 2 + y 1 cosh( α, a )l 2 -y 2 cosh( α, a )l 1 , A 1 ]/2
Once again the bracket of y 1 (cosh α, a )l 2 -y 2 (cosh α, a )l 1 with A 1 yields only terms in k so we compute [y 1 il 1 + y 2 il 2 , A 1 ], which is equal to

y 2 (x 1 + y 2 coth α, a )[il 2 , l 1 ] + y 1 (x 2 -y 1 coth α, a )[il 1 , l 2 ].
Using the explicit choices of l 1 and l 2 we have

-[il 2 , l 1 ] = [il 1 , l 2 ] = [i(e α + θ(e α )), ie α -iθ(e α )] = 2[e α , θ(e α )] = -2[e α , e -α ] = -2α.

Finally we have

B 2 = y 2 x 1 + y 2 2 coth α, a -y 1 x 2 + y 2 1 coth α, a α. Applying Lemma 1.8, we get H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = i 4 ∂ 2 ∂x 1 ∂y 2 - ∂ 2 ∂y 1 ∂x 2 0 f (a + (y 2 x 1 -y 1 x 2 )α) = i 2 (Df ) a (α)
where (Df ) a denotes the differential of f at a, so

H l1,l2 (a) = i 2 α, ∇f (a) .
Lemma 1.13. Suppose now that l 1 = l 2 = e α + θ(e α ), then

H l1,l2 (a) = 1 2 α, ∇f (a) coth α, a .
Proof. Using Lemma 1.6, we write

il 2 = il 1 = -(coth α, a )l ′ 1 + 1 sinh α, a Ad(exp a)(l ′ 1 ) with l ′ 2 = l ′ 1 = ie α -iθ(e α ).

Then we have

A 1 = (x 1 + x 2 )l 1 -(y 1 coth α, a + y 2 coth α, a )l ′ 1 B 1 = 0 C 1 = y 1 1 sinh α, a Ad(exp a)(l ′ 1 ) + y 2 1 sinh α, a Ad(exp a)(l ′ 2 ) D 1 = [C 1 , A 1 ]/2 = [(y 1 + y 2 )il 1 + (y 1 cosh( α, a ) + y 2 cosh( α, a ))l ′ 1 , A 1 ]/2
Once again the bracket of (y 1 cosh α, a + y 2 cosh α, a )l ′ 1 with A 1 yields only terms in k, so we just compute

[(y 1 + y 2 )il 1 , A 1 ] = -(y 1 + y 2 ) 2 (coth α, a )[il 1 , l ′ 1 ].
Using the explicit choices of l 1 we have

[il 1 , l ′ 1 ] = [i(e α + θ(e α )), ie α -iθ(e α )] = -2α.
Finally we have

B 2 = (y 1 + y 2 ) 2 (coth α, a )α.
Applying Lemma 1.8, we get

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = 1 4 ∂ 2 ∂y 1 ∂y 2 0 f a + (y 2 1 + 2y 1 y 2 + y 2 2 )(coth α, a α) = coth α, a 2 (Df ) a (α) = coth α, a 2 α, ∇f (a) .
The last step is to compute the coefficient of the Hessian with l 1 = l 2 = ie α -iθ(e α ), and the result is exactly the same as in the previous case: Lemma 1.14. Assume that l 1 = l 2 = ie α -iθ(e α ), then

H l1,l2 (a) = 1 2 α, ∇f (a) coth α, a .
Proof. Using Lemma 1.6, we write

il 2 = il 1 = -(coth α, a )l ′ 1 + 1 sinh α, a Ad(exp a)(l ′ 1 ) with l ′ 2 = l ′ 1 = -e α -θ(e α ).
The beginning of the computation doesn't change: we have

A 1 = (x 1 + x 2 )l 1 -(y 1 coth α, a + y 2 coth α, a )l ′ 1 B 1 = 0 C 1 = y 1 1 sinh α, a Ad(exp a)(l ′ 1 ) + y 2 1 sinh α, a Ad(exp a)(l ′ 2 ) D 1 = [(y 1 + y 2 )il 1 + (y 1 cosh α, a + y 2 cosh α, a )l ′ 1 , A 1 ]/2
Once again the bracket of (y 1 cosh α, a + y 2 cosh α, a )l ′ 1 with A 1 yields only terms in k, so we just compute

[(y 1 + y 2 )il 1 , A 1 ] = -(y 1 + y 2 ) 2 (coth α, a )[il 1 , l ′ 1 ]
. Now the expression of l 1 has changed, but we have

[il 1 , l ′ 1 ] = [i(ie α -iθ(e α )), -e α -θ(e α )] = 2[e α , θ(e α )]
= -2α.

In other words we have again

B 2 = (y 1 + y 2 ) 2 (coth α, a )α,
and applying Lemma 1.8, we get again

H l1,l2 (a) = ∂ 2 ∂z 1 ∂z 2 0 f (a + B 2 + B 1 ) = coth α, a 2 α, ∇f (a) .

Fano group compactifications

We present here in a concise way group compactifications and the associated polytopes, then recall results from [Delb] about the convex potential of a hermitian metric, giving more precise statements for smooth metrics. We then combine this with the computation of the complex Monge-Ampère (Corollary 1.3) to partially recover the formula of Brion and Kazarnovskii for the degree of an ample line bundle on a group compactification. The reader willing to learn more details about group compactifications can consult, for example, [START_REF] Alexeev | Stable reductive varieties. II. Projective case[END_REF][START_REF] Alexeev | On K-stability of reductive varieties[END_REF] or [START_REF] Brion | Frobenius splitting methods in geometry and representation theory[END_REF]Chapter 6]. For more general references about spherical varieties, see [START_REF] Timashev | Homogeneous spaces and equivariant embeddings[END_REF][START_REF] Perrin | On the geometry of spherical varieties[END_REF].

Definition and examples

Definition 2.1. We say that a projective manifold X is a compactification of G (or a group compactification, when G is not specified) if X admits a holomorphic G × G action with an open and dense orbit isomorphic to G as a G × Ghomogeneous space. We say that (X,

L) is a polarized compactification of G if X is a compactification of G and L is a G × G-linearized ample line bundle on X.
Recall that a G-linearized line bundle over a G-manifold X is a line bundle L on X equipped with an action of G lifting the action on X, and such that the morphisms between the fibers induced by this action are linear.

Example 2.2. If G = T ≃ (C * ) n is a torus, then the compactifications of T are the projective toric manifolds. One goes from the T -action to the T × T action through the morphism

T × T → T, (t 1 , t 2 ) → t 1 t -1 2 .
Example 2.3. Assume that G is an adjoint semisimple group, that is, a reductive group with trivial center. Then De Concini and Procesi [START_REF] Concini | Complete symmetric varieties[END_REF] proved the existence of a special compactification of G, called the wonderful compactification of G. It is the only compactification of G satisfying the following property: its boundary X \ G (where we identify the open and dense orbit in X with G) is a union of simple normal crossing prime divisors D i , i ∈ {1, . . . , r}, such that for any subset I ⊂ {1, . . . , r}, the intersection X ∩ i∈I D i is the closure of a unique G × G-orbit, and all G × G-orbits appear this way. The integer r is equal to the rank of G, which is the dimension of a maximal torus in G.

Polytopes associated to a polarized group compactification

Theorem 2.4. [AB04, Section 2] Let (X, L) be a polarized group compactification of G. Denote by Z the closure of T in X. Then Z is a toric manifold, equipped with a W -action, and L| Z is a W -linearized ample toric line bundle on Z.

We denote by P (X, L), or P for simplicity, the polytope associated to the ample toric line bundle L| Z by the theory of toric varieties [START_REF] Fulton | Introduction to toric varieties[END_REF][START_REF] Oda | Convex bodies and algebraic geometry[END_REF]. The polytope P is a lattice polytope in M ⊗ R, that we identified with a, and it is W -invariant. Define P + (X, L) := P (X, L) ∩ a + . It is a polytope in a, and P (X, L) is the union of the images of P + (X, L) by W . The polytope P + (X, L) encodes the structure of G × G-representation of the space of holomorphic sections of L, generalizing the same property for toric line bundles.

Proposition 2.5. [AB04, Section 2.2] Let (X, L) be a polarized group compactification, then

H 0 (X, L) ≃ {End(V α ) ; α ∈ M ∩ P + (X, L)}
where V α is an irreducible representation of G with highest weight α.

Example 2.6. [BK05, Proposition 6.1.11] The wonderful compactification X of an adjoint semisimple group is Fano. The corresponding polytope P (X, -K X ) is the convex hull of the images by the Weyl group W of the weight 2ρ+ r i=1 α i , where the α i are the simple roots of Φ + , and 2ρ is the sum of all positive roots. We will use the three following examples to illustrate our results.

Example 2.7. The wonderful compactification X 0 of PGL 2 (C) is especially simple: it is P 3 considered as P(Mat 2,2 (C)) equipped with the action of PGL 2 (C)× PGL 2 (C) induced by the multiplication of matrices on the left and on the right.

If we identify a with R by sending the only positive root to 1, then the lattice of characters is (1/2)Z, the polytope P associated to the anticanonical line bundle is [-2, 2] and the polytope P + is [0, 2]. Remark that X 0 is in fact homogeneous under a bigger group. This is in general not the case for wonderful compactifications, as the next example shows.

Example 2.8. Consider the adjoint semisimple group PGL 3 (C). Then its wonderful compactification, that we denote by X 1 , is Fano. Figure 2 represents the polytope P (X 1 , -K X1 ), along with the Weyl chambers decomposition, and the weight lattice M .

Brion computed the automorphism group Aut(X) of all wonderful compactifications X of adjoint semisimple groups [Bri07, Example 2.4.5]. In particular,

Figure 3: Polytope P (X 2 , -K X2 )
the connected component of the identity Aut 0 (X 1 ) ⊂ Aut(X 1 ) is the image of PGL 3 (C) × PGL 3 (C). This shows that the manifold X 1 is not homogeneous under a bigger group, and not toric. The same statement is true for any wonderful compactification of an adjoint semisimple group with no factor of rank one.

Example 2.9. Consider now the simply connected semisimple group Sp 4 (C). It is the subgroup of GL 4 (C) whose elements are the matrices M such that M t ΩM = Ω, where M t denotes the transpose of M and

Ω = 0 1 n 1 n 0 .
It is a semisimple group of type B 2 , which means that its associated root system is the root system B 2 . While it is not an adjoint group, it also admits a wonderful compactification (see for example [START_REF] Gandini | Normality and smoothness of simple linear group compactifications[END_REF]). Let X 2 be the blow up of this wonderful compactification along the closed orbit. Then X 2 is a Fano compactification of Sp 4 (C) [START_REF] Ruzzi | Fano symmetric varieties with low rank[END_REF].

The moment polytope associated to the anticanonical line bundle is obtained easily in the following way: Ruzzi classifies in [START_REF] Ruzzi | Fano symmetric varieties with low rank[END_REF] all Fano compactifications of reductive groups of rank two, by giving for each the usual data identifying a spherical variety. Then we can use the description by Gagliardi and Hofscheier [START_REF] Gagliardi | Gorenstein spherical Fano varieties[END_REF] of the moment polytope of the anticanonical line bundle in these terms. Figure 3 represents the polytope P (X 2 , -K X2 ), along with the Weyl chambers decomposition, and the weight lattice M .

For this manifold also, Aut 0 (X 2 ) is the image of Sp 4 (C) × Sp 4 (C). This is proved using Blanchard's lemma [Bla56, Proposition I.1] applied to the blow up, and the fact that the connected component of the identity in the automorphism group of the wonderful compactification of Sp 4 (C) is also the image of Sp 4 (C) × Sp 4 (C) [START_REF] Pezzini | Automorphisms of wonderful varieties[END_REF].

Convex potentials

Let (X, L) be a polarized compactification of G. We recall in this section how to associate to a K × K-invariant hermitian metric with positive curvature on L a convex function, called its convex potential (see [Delb, Section 2.2]). We let P := P (X, L), P + := P + (X, L), and let v denote the support function of the dilated polytope 2P , defined by v(x) = sup{ p, x ; p ∈ 2P }.

Let s be a G × {e}-equivariant trivialization of L| G , where we identify G with the open G × G-orbit in X. Let h be a smooth, K × K-invariant, positively curved hermitian metric on L. Let ϕ be the local potential of h with respect to the trivialization s, defined for g ∈ G by:

ϕ(g) := -ln |s(g)| 2
h . Then the function ϕ is a smooth strictly plurisubharmonic function on G, K ×Kinvariant by [Delb, Proposition 2.2].

The function u defined on a by u(x) = ϕ(exp x) is then called the convex potential of h and satisfies the following properties.

Proposition 2.10. Let h be a smooth K × K-invariant hermitian metric with positive curvature on L, and let u be its convex potential. Then u is a Winvariant, smooth and strictly convex function such that:

1. u(x) ≥ v(x) + C 1 for all x ∈ a, for some constant C 1 depending on u; 2. given any x 0 ∈ a, u(x) ≤ v(x -x 0 ) + u(x 0 ) for all x ∈ a;
3. the gradient ∇u of u defines a diffeomorphism from a to the interior of 2P ;

4. in particular, there exists a constant d depending only on L such that |∇u(x)| ≤ d for all x ∈ a;

5. the restriction of ∇u to a + is a diffeomorphism to the interior of 2P + .

Proof. The fact that u is a W -invariant smooth and strictly convex function follows from its definition and [START_REF] Azad | Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces[END_REF].

A smooth metric has locally bounded potentials, so [Delb, Theorem 2.4] implies that there exists constants C 1 and C 2 depending on u such that

v(x) + C 1 ≤ u(x) ≤ v(x) + C 2 .
In particular, the first point is proved. Let us prove the second point.

Let x 0 ∈ a. For any 0 = y ∈ a, consider the slope

u(x 0 + ty) -u(x 0 ) t ,
with t > 0. By convexity and the two previous inequalities, we see that this slope increases and converges to v(y) as t tends to infinity. This shows that for any x = x 0 + y ∈ a \ {x 0 }, we have

u(x) ≤ v(x -x 0 ) + u(x 0 ).
This inequality is obviously also satisfied at x 0 , so the second point is proved. Since u is smooth and strictly convex, ∇u is a diffeomorphism. It remains to determine its image. It is clear that this image is open and included in 2P (for example because the domain of the convex conjugate of u is 2P , see [Delb, Proposition 3.12]). Assume now that one point from the interior of 2P is not attained by ∇u. Then by convexity, we can find a 0 = y ∈ a and an ǫ > 0 such that ∀x ∈ a, v(y) -∇u(x), y > ǫ. But considering the ray starting from any x 0 and in the direction y as above leads to a contradiction. This implies that ∇u(a) = Int(2P ). By W -invariance, we also have ∇u(a + ) = Int(2P + ).

Finally, the fourth point is a direct consequence of the third because the polytope 2P is bounded.

Remark 2.11. We fixed a choice of a G × {e}-equivariant trivialization of L on G to define the convex potential. All such trivializations are non zero scalar multiples of one another, so the convex potentials defined with respect to two different such trivializations differ by a constant scalar independent of the metric. In particular, it does not change the asymptotic behavior.

Degree of an ample linearized line bundle

Proposition 2.12. Let (X, L) be a polarized compactification of G, and let

P + = P + (X, L). Then deg(L) = C 2P + α∈Φ + α, p 2 dp
for some constant C depending only on the group G. Furthermore, if u is the convex potential of a smooth positively curved

K × K-invariant metric on L, then deg(L) = C a + α∈Φ + α, ∇u(a) 2 MA R (u)(a)da.
Proof. Let h be a smooth positively curved K × K-invariant hermitian metric on L, with curvature the Kähler form ω ∈ 2πc 1 (L). Then we have

deg(L) = X ω 2π n .
Let s be a G × {e}-equivariant section of L, and ϕ the potential of h with respect to s. We have ω = i∂∂ϕ on G. Using Remark 1.1, Choose a Haar volume form dg on G so that, independently of ϕ,

i 2π ∂∂ϕ n = MA C (ϕ)dg,
where MA C (ϕ) is the Monge-Ampère of ϕ in the coordinates chosen in Section 1.1. Then

deg(L) = X ω 2π n = G ω 2π n = G i 2π ∂∂ϕ n = G MA C (ϕ)dg.
By the KAK integration formula, this is, for a constant C depending only on the choice of Haar measure,

deg(L) = C a + MA C (ϕ)(exp a)J(a)da
where da is the Lebesgue measure on a normalized by N . Let u denote the convex potential of h. From Corollary 1.3 we obtain that this is

deg(L) = C ′ a + α∈Φ + α, ∇u(a) 2 MA R (u)(a)da
for a constant C ′ still depending only on G. We then use the variable change p = ∇u(a), and obtain, by Proposition 2.10,

deg(L) = C ′ 2P + α∈Φ + α, p 2 dp.

Strategy of proof

We describe in this section the global strategy of the proof, starting by recalling classical results on the continuity method. We then determine the expression of the continuity method equation in restriction to G, in terms of convex potentials. We finally introduce the function ν t and gather some estimates on this function or on -ln J that will be used later.

The continuity method

Let X be a Fano manifold. Fix a reference Kähler form ω ref in the class 2πc 1 (X).

The Kähler forms in 2πc 1 (X) can all be written as ω ref + i∂∂ψ with ψ a smooth and ω ref -strictly psh function on X (i.e. such that ω ref + i∂∂ψ > 0). The Kähler-Einstein equation Ric(ω) = ω on X translates, in terms of ω refpsh functions, as the Monge-Ampère equation The following family of equations is the one used in the usual continuity method for the Kähler-Einstein equation:

(ω ref + i∂∂ψ) n = e f ref -ψ ω n ref , (1)
(ω ref + i∂∂ψ t ) n = e f ref -tψt ω n ref . (2)
To show the existence of a Kähler-Einstein metric on X, it is enough to show that the set I of 0 ≤ t ≤ 1 such that this equation admits a solution is exactly

[0, 1].
By the work of Aubin [START_REF] Aubin | Équations du type Monge-Ampère sur les variétés kähleriennes compactes[END_REF] and Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF], 0 ∈ I, and I is open. Furthermore, it is enough to know uniform a priori estimates on the C 0 norm of ψ t , to ensure the closure of I, and thus the existence of a solution at t = 1, i.e. a Kähler-Einstein metric. We recall that by C 0 estimates, we mean, as in most of the literature, a uniform control on sup x∈X |ψ t (x)|. In fact, we can even concentrate only on a uniform upper bound on ψ t (see [Siu88, Proposition 2.1] or [Tia87, pages 235 and 236]). Proposition 3.2. Assume that [t 0 , t 1 [⊂ I, that ψ t denotes the solution at t ∈ [t 0 , t 1 [, and that there exists a constant C such that ψ t ≤ C ∀t ∈ [t 0 , t 1 [. Then t 1 ∈ I.

Reduction to the open orbit

Suppose now that X is a G × G-equivariant smooth and Fano compactification of G. Let P be the polytope associated to the anticanonical bundle -K X .

We choose ω ref and h ref K × K-invariant. Then a solution of equation (2) at t < 1 is K × K-invariant. This follows from the uniqueness result for twisted (or generalized) Kähler-Einstein metrics (see [ZZ14, Corollary 1.4] or [DS]).

By continuity of the solutions ψ t , it is enough to prove a uniform upper bound on the restrictions of ψ t to the open and dense orbit G ⊂ X. Let ϕ t denote the function on a induced by ψ t . It is enough to give an upper bound for ϕ t . We also denote by h t the hermitian metric on -K X whose potential with respect to h ref is ψ t and whose curvature form is

ω ref + i∂∂ψ t .
Any G × {e}-equivariant trivialization s of -K G defines a left G-invariant volume form s -1 ∧ s -1 on G, and so a Haar volume form. All Haar volume forms on G are obtained this way. By Remark 1.1, we can thus choose s and the corresponding Haar volume form dg := s -1 ∧ s -1 such that for all smooth K × K-invariant function ψ on G,

(i∂∂ψ) n = MA C (ψ)dg,
where MA C (ψ) is the complex Monge-Ampère of ψ in the local coordinates given in Section 1.1.

Let u ref be the convex potential of h ref with respect to the trivialization s, defined on a, and denote by u t the convex function u ref + ϕ t which is the convex potential of the metric h t . Finally, we denote by w t the function tu t +(1-t)u ref .

Proposition 3.3. Suppose ψ t is a K × K-invariant solution of equation (2). Then for x ∈ a, MA R (u t )(x) α∈Φ + α, ∇u t (x) 2 = e -wt(x) J(x).
(3)

Proof. Let ψ ref be the potential of the reference metric h ref with respect to the section s. By the choice of s, we have, on G,

(ω ref + i∂∂ψ t )| n G = (i∂∂(ψ ref + ψ t )) n = MA C (ψ ref + ψ t )dg
Now the computation of the complex Monge-Ampère in local coordinates from Section 1.1 gives

(ω ref + ∂∂ψ t ) n (exp x) = MA R (u t )(x) α∈Φ + α, ∇u t (x)
(-tψ t -ψ ref )(exp x) = -tϕ t (x) -u ref (x) = -tu t (x) -(1 -t)u ref (x) = -w t (x).
In conclusion, at a point exp(x) for x ∈ a + , equation (2) reads

MA R (u t )(x) α∈Φ + α, ∇u t (x) 2 1 J(x) dg = e -wt(x) dg.
It is equivalent to the equality of the potentials with respect to dg. Furthermore, by multiplying both sides by J(x), we obtain the equation of the statement, that is well defined on the whole of a, and it is satisfied by W -invariance and smoothness.

Strategy

To find a uniform upper bound for ϕ t we will introduce another function ν t , and study this function, following the strategy used by Wang and Zhu in the toric case. More precisely, let j be the function on the open Weyl chamber a + defined by j(x) = -ln J(x). We consider the function ν t = w t + j defined on a + . We will show that it is a strictly convex function on a + . It is proper in the following sense: As x goes to infinity, or x goes to a wall of a + , ν t (x) goes to infinity. These two properties of ν t imply that it admits a unique minimum. Let m t be the minimum of ν t and x t be the point of a + where this minimum is attained. We will obtain estimates on both the value m t of the minimum and on the distance from the origin |x t | of the point where it is attained. Then we need to relate these estimates with the function that we want to control. Namely we will go from ν t to w t then u t and finally ϕ t .

To summarize, the strategy to prove estimates on ϕ t is in three steps:

• reduce to estimates on |m t | and |x t |,

• find uniform estimates

|m t | ≤ C, • get a uniform control |x t | ≤ C of x t .
We will also have to prove the necessity of the condition in Theorem A and the upper bound on R(X). Before that, we gather some preliminary results.

The function j = -ln J

The half sum of positive roots ρ is in a + , so α, ρ > 0 for all α ∈ Φ + . We will use this as a reference to control the distance to the walls of a + . We also choose a basis {e i } of a, and corresponding coordinates {x i } when necessary.

Lemma 3.4. There exists a constant c such that for any x ∈ a + , we have

j(x) ≥ -4ρ, x + c.
Furthermore, for any b > 0, there exists constants

C = C(b), C ′ = C ′ (b) such that for all x ∈ bρ + a + , j(x) ≤ -4ρ, x + C ≤ C ′ . Proof. Write sinh α, x = e α,x 1 -e -2 α,x 2 ≤ e α,x 2
for x ∈ a + . Then j(x) = -2 α∈Φ + ln sinh α, x ≥ -2 α∈Φ + α, x + c,
where c = 2 ln(2)Card(Φ + ).

If we assume x ∈ bρ + a + , then using the same expression of sinh α, x , we get j(x) ≤ -4ρ, x + C with

C = -2 α∈Φ + ln 1 -e -2b α,ρ 2 . and j(x) ≤ C ′ with C ′ = C -2b α∈Φ + α, ρ .
Let us now turn to the derivatives of j. Recall that the directional derivative of j in the direction ξ is defined by: ∂j ∂ξ (x) := ∇j(x), ξ .

Lemma 3.5. For any ξ ∈ a + , we have, for all x ∈ a + ,

∂j ∂ξ (x) < -4ρ, ξ .
If ξ ∈ a t , then for all x ∈ a + , ∂j ∂ξ (x) = 0.

Proof. The second statement is clear because for any α ∈ Φ, α, x depends only on the projection of x on a ss . For the first statement, we compute

∂j ∂ξ (x) = -2 α∈Φ + α, ξ coth α, x .
Since ξ ∈ a + , we have α, ξ > 0 for all α ∈ Φ + . Since x is also in a + , we have coth α, x > 1. We deduce from this that

∂j ∂ξ (x) < -2 α∈Φ + α, ξ = -4ρ, ξ .
We will also need to control the derivatives of j away from the walls. This is achieved by the following lemma.

Lemma 3.6. For any b > 0, there exists a constant C such that for any

x ∈ bρ + a + , |∇(j)(x)| ≤ C.
Proof. Recall that

∂j ∂x i (x) = -2 α∈Φ +
α, e i coth α, x

For x ∈ bρ + a + , we have 1 < coth α, x < coth(b α, ρ ), so for any i,

∂j ∂x i (x) ≤ 2 α∈Φ + |α| coth(b α, ρ ).
Lemma 3.7. The function j is strictly convex on a + .

Proof. An easy computation shows that

∂ 2 ∂x j ∂x i (-ln sinh α, x ) =
α, e i α, e j sinh 2 α, x .

So the Hessian of j is a sum of semipositive matrices, and it is easy to check that the whole sum is definite, so the Hessian of j is positive definite.

The function ν t

Lemma 3.8. The function ν t is strictly convex and admits a unique minimum.

Proof. From Lemma 3.7, and the fact that w t is strictly convex as a convex combination of strictly convex functions, we obtain that ν t = w t + j is strictly convex. The function w t is bounded below, and j tends to +∞ when x approaches a Weyl wall, so ν t (x) also tends to +∞ when x approaches a Weyl wall. To prove the existence of a minimum x t it remains to explain why ν t goes to infinity at infinity. Proposition 2.10 implies that w t (x) ≥ v(x)+ C 1 for some constant C 1 , where v is the support function of the polytope 2P , so ν t (x) ≥ v(x) + j(x) + C 1 . Then

ν t (x) ≥ v(x) -4ρ, x + c + C 1
by Lemma 3.4. Finally, the fact that X is Fano implies, by [START_REF] Delcroix | Log canonical thresholds on group compactifications[END_REF]Remark 4.10] that 4ρ ∈ Int(2P ), so ν t is indeed proper. Let m t denote the minimum value of ν t and x t be such that ν t (x t ) = m t . Lemma 3.9. There exists a constant b 1 > 0 independent of t such that

x t ∈ b 1 ρ + a +
Proof. By definition of x t , the derivative of ν t at x t vanishes. In particular, we have ∂ν t ∂ρ (x t ) = 0.

Recall that ν t = w t + j, and that the derivatives of w t are bounded by Proposition 2.10. In particular we get a bound

∂w t ∂ρ (x t ) ≤ C.
On the other hand, recall that:

∂j ∂ρ (x t ) = -2 α∈Φ + α, ρ coth α, x t .
So we have

2 α∈Φ + α, ρ coth α, x t ≤ C
but since all the terms of the sum are positive and all the α, ρ are strictly positive, this implies that for all α ∈ Φ + , coth α, x t ≤ C. Observe that the function coth tends to +∞ at 0, so we obtain α, x t ≥ C ′ for all α for a constant C ′ > 0 independent of t.

To conclude, observe that the intersection of the half spaces defined by α, x ≥ C ′ is contained in a translate b 1 ρ + a + for some b 1 > 0 sufficiently small, independent of t.

We will also need to control from below the value of ν t near the walls. This will be achieved by the following technical proposition. For now we cannot control ν t uniformly close to the walls, but we will as soon as we control m t . We will use twice the proposition, first to obtain a lower bound on m t , then to ensure e -νt is uniformly sufficiently small near the walls. Remark also that this proposition can be seen as a precise statement of what we called the properness of ν t near the walls. Proposition 3.10. For any M > 0, there exists a constant b > 0 independent of t such that for any x ∈ a + satisfying α, x < b α, ρ for some root α ∈ Φ + defining a wall of a + , we have

ν t (x) ≥ m t + M.
Recall that the roots defining the walls are also the simple roots of Φ + .

Proof. Let x ∈ a + be such that α, x < b 1 α, ρ for some simple root α ∈ Φ + . Consider the ray {x + sρ, s ≥ 0} starting from x. It meets the boundary

∂(b 1 ρ + a + ) of b 1 ρ + a + at a unique point y = x + s 0 ρ. Furthermore y is in b 1 ρ + α ⊥
for a simple root α. We can then write x = y -s 0 ρ, and s 0 satisfies 0 < b 1 -s 0 < b.

Consider α ∈ Φ + a simple root, and y ∈ (b

1 ρ + α ⊥ ) ∩ ∂(b 1 ρ + a +
). We will show that there exists a constant b > 0 independent of t such that ν t (y -sρ) ≥ m t + M for all s such that 0 < b 1 -s < b, and that this b can be chosen independent of y and α. This is enough to prove the proposition because any x as in the statement is of the form above for some α, y and s as shown at the beginning.

Consider the function g(s) = ν t (y -sρ) on [0, b 1 [. We have g(0) = ν t (y) ≥ m t by definition of m t .

We consider now the derivative of g. Remember that the derivatives of w t are uniformly bounded, by d, in absolute value by Proposition 2.10. Then

g ′ (s) ≥ -d + 2 β∈Φ + β, ρ coth β, y -sρ .
Since all the terms in the sum are positive, we have in particular

g ′ (s) ≥ -d + 2 α, ρ coth α, y -sρ .
From the assumptions, we compute

α, y -sρ = b 1 α, ρ -s α, ρ = (b 1 -s) α, ρ .
Observe that the positive function coth is not integrable near 0 + , so

b1 0 coth((b 1 -s) α, ρ )ds = +∞.
Together with the fact that g is greater than m t at 0, it means that for any M , we can find a b α > 0 such that g(s

) ≥ M + m t for b 1 -s ≤ b α .
Remark that none of what we have done depends on the choice of y. Furthermore, since there are only a finite number of roots α, we can choose a b > 0 such that b < b α for all α, and it concludes the proof.

Reduction to estimates on m t and x t

Lemma 3.11. Suppose we have uniform estimates |m t | < C m and |x t | < C x for t in some interval [t 0 , t 1 [⊂ [0, 1]. Then there is an uniform upper bound for ϕ t on [t 0 , t 1 [. Proof. Recall that it is enough to obtain a uniform upper bound on u t -u ref which is a function defined on a. We have, by Proposition 2.10 with

x 0 = x t , that u t (x) ≤ v(x -x t ) + u t (x t )
where v is the support function of 2P . Using the other inequality for u ref we have

u ref (x) ≥ v(x) + C 1 ≥ v(x -x t ) + C 1 -d|x t |.
Combining these two gives

(u t -u ref )(x) ≤ v(x -x t ) + u t (x t ) -v(x -x t ) -C 1 + d|x t | ≤ u t (x t ) -C 1 + d|x t | ≤ u t (x t ) -C 1 + dC x
so we just have to control u t (x t ).

We have

|m t | = |ν t (x t )| ≤ C m , i.e |tu t (x t ) + (1 -t)u ref (x t ) + j(x t )| ≤ C m .
Now we have:

• t ≥ t 0 > 0, • |j(x t )| ≤ C 2 for some constant C 2 because x t ∈ b 1 ρ + a + ,
• and

u ref (x t ) ≤ sup{u ref (y) ; y ∈ B(0, C x )} =: C 3 . So u t (x t ) ≤ C m + C 2 + C 3 t 0 .
Finally we have proved the uniform upper bound

(u t -u ref )(x) ≤ C 4 := C m + C 2 + C 3 t 0 -C 1 + dC x .

Estimates on |m t |

In this section, we essentially follow the work of Wang and Zhu to obtain estimates on |m t | and a uniform estimates on the (at least linear) growth of ν t .

A key step is proposition 4.3, which is a direct consequence of Proposition 2.12 and will be used in the next sections also. We consider the set

A t := {x ∈ a + ; m t ≤ ν t (x) ≤ m t + 1} ⊂ a + .
We will obtain upper and lower bound for the volume of A t . The upper bound will depend on m t . The key is to obtain an upper bound that is small enough to give information. Note that the set A t is a bounded and convex set. Indeed, since m t is the minimum of ν t , A t is a sublevel set of ν t which is convex, so A t is convex. Furthermore, by the properness of ν t , A t is a bounded set. Lemma 4.1. There is an upper bound on the volume of A t :

Vol(A t ) ≤ Ce mt/2
where the constant C > 0 does not depend on t ≥ t 0 .

Proof. Fritz John proved in [Joh48, Theorem III] that for any convex and bounded subset A of R r , there exists an ellipsoid E such that

1 r E ⊂ A ⊂ E
where (1/r)E is the dilation of E of factor 1/r centered at the center of the ellipsoid E.

We can then find such an ellipsoid E t for A t . Let T be a linear transformation, of determinant one, such that T (E) = B(y, δ) is a ball. We will obtain an upper bound on δ, thus getting an upper bound for the volume of

Vol(A t ) because Vol(A t ) ≤ Vol(E) = Vol(T (E)) = Cδ r .
Let ν ′ t be the function defined by ν ′ t (x) = ν t (T -1 (x)). We want to use a comparison principle on B(y, δ/r). For that we first show that MA R (ν ′ t )(x) ≥ Ce -mt on T (A t ). This is equivalent to showing that MA R (ν t )(x) ≥ Ce -mt on A t . First remark that since the Hessian Hess R ν t of ν t satisfies:

Hess R ν t = tHess R u t + (1 -t)Hess R u ref + Hess R j, we have det(Hess R ν t ) ≥ det(tHess R u t ), i.e. MA R (ν t )(x) ≥ t r MA R (u t )(x).
Using Proposition 3.3 we deduce that

MA R (ν t )(x) ≥ t r J(x)e -wt(x) α∈Φ + 1 α, ∇u t (x) 2 ≥ t r e -νt(x) α∈Φ + 1 α, ∇u t (x) 2 .
We treat the factors separately:

• We have t ≥ t 0 > 0 for t 0 defined in Notation 3.1.

• By definition of A t , we have e -νt(x) ≥ e -mt-1 on A t .

• For any x ∈ a, we have ∇u t (x) ∈ 2P , so for any α ∈ Φ + , α, ∇u t (x) is bounded above independently of t. This implies that

α∈Φ + 1 α, ∇u t (x) 2 ≥ c
for some positive constant c.

In conclusion, we indeed have an inequality MA R (ν t )(x) ≥ Ce -mt on A t , with C a positive constant independent of t ≥ t 0 . Now we use the comparison principle on B(y, δ/r) for real Monge-Ampère equations: let g be the auxiliary function defined by

g(x) = C 1/r e -mt/r |x -y| 2 - δ 2 r 2 + m t + 1.
Then we have

• g(x) = m t + 1 ≥ ν ′ t (x) for x ∈ ∂B(y, δ/r), and • MA R (g)(x) = Ce -mt ≤ MA R (ν ′ t )(x)
on B(y, δ/r). So the comparison principle gives that ν ′ t (x) ≤ g(x) on B(y, δ/r). In particular, we have

m t ≤ ν t (T -1 (y)) ≤ ν ′ t (y) ≤ g(y) ≤ C 1/r e -mt/r - δ 2 r 2 + m t + 1.
We deduce from that the following upper bound for δ:

δ ≤ 1 C 1/r re mt/2r .
Putting everything together, we obtain

Vol(A t ) ≤ Vol(B(y, δ) ≤ C ′ e mt/2 .
We turn now to a lower bound on Vol(A t ).

Lemma 4.2. There exists a constant c > 0 independent of t such that

Vol(A t ) ≥ c.
Proof. There exists a constant b 2 independent of t such that 0 < b 2 < b 1 and

A t ⊂ b 2 ρ + a + .
This is a corollary of Proposition 3.10, taking b 2 corresponding to M = 1.

Then by Lemma 3.6 and Proposition 2.10, on b 2 ρ + a + , |∇ν t | is bounded independently of t, say by C. Then it is clear that the ball B(x t , 1/C) is contained in A t . So Vol(A t ) ≥ c for some c > 0 independent of t.

Proposition 4.3. The following integral is independent of t:

a + e -νt(x) dx = 2P + α∈Φ + α, p 2 dp =: V.
Proof. Applying Proposition 2.12 with the ample line bundle -K X , we have, for some constant C depending only on G, and for any convex potential u of a smooth K × K-invariant positively curved hermitian metric on -K X ,

deg(-K X ) = C a + α∈Φ + α, ∇u(x) 2 MA R (u)(x)dx = C 2P + α∈Φ + α, p 2 dp
We apply this to the convex potential u t , which by Proposition 3.3 satisfies

e -νt(x) = α∈Φ + α, ∇u t (x) 2 MA R (u t )(x)
and obtain the statement, with V = deg(-K X )/C.

We can now prove the main result of this subsection.

Proposition 4.4.

• There exists a constant C independent of t, such that |m t | ≤ C.

• There exist a constant κ > 0 and a constant C, both independent of t, such that for x ∈ a + ,

ν t (x) ≥ κ|x -x t | -C.
Proof. Following here Donaldson [Don08]

ν t (x) ≥ |x -x t |/δ + m t -1
everywhere. Using again the fact that m t is uniformly bounded we get the result with κ = 1/δ.

5 Obstruction, and upper bound on R(X)

We will here explain how our condition appears as a necessary condition, then obtain an upper bound on the greatest Ricci lower bound. Everything relies on the following vanishing statement, which will also be of major importance in the next section.

A vanishing integral

Proposition 5.1. Let u be the convex potential of a smooth positive metric on -K X . Define ν := u + j. Let ξ be any vector in a + . Then

a +
∂ν ∂ξ e -ν dx = 0.

Before we get to the proof, let us remark that the function considered is integrable. More generally, we can remark first that for any potential u 0 , and any vector ξ, the function ∂u0 ∂ξ e -ν is integrable on a + . This is the case because ∇u 0 ∈ 2P , and e -ν ≤ Ce -(v-4ρ)+C (by Proposition 2.10 and Lemma 3.4) is obviously integrable.

Secondly, we have to show that the function ∂j ∂ξ e -ν = ∂j ∂ξ Je -u is integrable. Write

∂j ∂ξ (x)J(x) = -2 α∈Φ + α, ξ coth α, x β∈Φ + sinh 2 β, x = -2 α∈Φ + α, ξ cosh α, x sinh α, x β =α sinh 2 β, x .
Then by a computation similar to Lemma 3.4, we have e 4ρ ∂j ∂ξ J ≤ Ce 4ρ , so again ∂j ∂ξ e -ν is integrable.

Proof. Choose a basis (e i ) i=1..s of the semisimple part a ss which generate a + ∩ a ss as a cone, and a basis (f j ) j=1..r-s of the toric part a t . Consider the following sets, for 0 ≤ ǫ < M :

Q(ǫ, M ) := i x i e i + j y j f j ; ∀i ǫ ≤ x i ≤ M, ∀j -M ≤ y j ≤ M .
Note that ∂Q(ǫ, M ) = S 1 (ǫ, M ) ∪ S 2 (ǫ, M ), where:

S 1 (ǫ, M ) := i x i e i + j y j f j ∈ Q(ǫ, M ) ; ∃i x i = ǫ S 2 (ǫ, M ) := i x i e i + j y j f j ∈ Q(ǫ, M ) ; ∃i x i = M or ∃j |y j | = M .
Remark that ∂ν ∂ξ e -ν = -∂e -ν ∂ξ . Then by the divergence formula applied to e -ν we have for ǫ > 0,

Q(ǫ,M) ∂ν ∂ξ e -ν dx = S1(ǫ,M)∪S2(ǫ,M) e -ν ξ, µ dσ
where µ is the exterior normal and dσ is the surface area.

Write now e -ν = e -u J. This is a continuous function on a, and it vanishes on the Weyl walls. Fixing M , we can thus let ǫ tend to 0, and we have that e -ν tends uniformly to 0 on S 1 (ǫ, M ), so S1(ǫ,M) e -ν ξ, µ dσ tends to 0.

We thus have

Q(0,M) ∂ν ∂ξ e -ν dx = S2(0,M)
e -ν ξ, µ dσ. Now as we have seen before, we have e -ν ≤ Ce -(v-4ρ)+C , so e -ν (x) decreases exponentially as |x| tends to infinity. Since the area of S 2 (0, M ) grows polynomially, this ensures that S2(0,M) e -ν ξ, µ dσ tends to zero as M tends to ∞. This ends the proof.

Obstruction to the existence of a Kähler-Einstein metric

Let us now explain the relation between convex potentials and the barycenter bar DH (P + ). Recall that V denotes the volume of the polytope P + with respect to the Duistermaat-Heckman measure, and is equal, up to a multiplicative constant, to the volume of the anticanonical line bundle.

Proposition 5.2. Assume that u is the convex potential of a smooth, positively curved hermitian metric on -K X , then

a + ∂u ∂ξ α∈Φ + α, ∇u 2 MA R (u)dx = ξ, bar DH (2P + ) V.
Proof. We simply use the change of variable p = ∇u(x), just as in the proof of Proposition 2.12. It gives:

a + ∂u ∂ξ α∈Φ + α, ∇u 2 MA R (u)dx = 2P + p, ξ α∈Φ + α, p 2 dp = ξ, bar DH (2P + ) V.
We now prove an obstruction to the existence of Kähler-Einstein metrics on X. Recall that Ξ denotes the relative interior of the cone generated by the positive roots Φ + . Combining these facts, we have:

ξ, bar DH (2P + ) V -4 ρ, ξ V > a + ∂u ∂ξ e -ν dx + a + ∂j ∂ξ e -ν dx > 0.
Dividing by V we obtain that for any ξ ∈ a + , ξ, bar DH (2P + ) -4ρ > 0.

The large inequality would mean bar DH (2P + ) -4ρ ∈ (a + ) ∨ , and here the strict inequality means that bar DH (2P + )-4ρ is in the relative interior of (a + ) ∨ , which is precisely Ξ.

Example 5.4. This obstruction allows to show that the manifold X 2 does not admit any Kähler-Einstein metric. Recall that this is a compactification of Sp 4 (C) so the corresponding root system is B 2 . Denote by α 1 and α 2 the simple roots, with α 2 the long root. Then the other positive roots are α 1 + α 2 and 2α 1 + α 2 , so 2ρ = 4α 1 + 3α 2 . Choosing a realization of B 2 in the euclidean plane with α 1 = (1, 0) and α 2 = (-1, 1), we compute that, for p

= xα 1 + yα 2 , α∈Φ + α, p 2 = x 2 y 2 (x -y) 2 (-x + 2y) 2 .
We can then compute bar DH (P + ) = 278037566905 66955221696 α 1 + 3043253830 1046175339 α 2 .

In particular the coordinate in α 2 is strictly smaller than 3 so there exists no Kähler-Einstein metric on X 2 . Figure 4 gives a representation of the polytope P + , the barycenter is represented by a cross and the cone delimited by the dashed lines is 2ρ + Ξ.

-νt V + 1 t -1 a + ∂j ∂ξ e -νt V = a + ∂u ref ∂ξ e -νt V .
If v is the support function of 2P , we have for any

x ∈ a + , ∂u ref ∂ξ (x) ≤ v(ξ), so a + ∂u ref ∂ξ e -νt V ≤ v(ξ).
On the other hand, we can use here also Lemma 3.5 to get, for ξ ∈ a + ,

1 t -1 a + ∂j ∂ξ e -νt V > 1 t -1 (-4 ρ, ξ ).
We thus have, using Proposition 5.2,

v(ξ) > t t -1 a + ∂u t ∂ξ e -νt V + 1 t -1 a + ∂j ∂ξ e -νt V > t t -1 ξ, bar DH (2P + ) - 1 t -1 4 ρ, ξ > ξ, 4ρ + t t -1 (bar DH (2P + ) -4ρ)
The fact that this is true for all ξ ∈ a + means, since v is the support function of 2P , that

4ρ + t t -1 (bar DH (2P + ) -4ρ) ∈ -Ξ + 2P + .
6 Absence of estimates on |x t |

Consequence

We will assume now that there are no Kähler-Einstein metrics on X. We will denote by t ∞ := R(X) the greatest Ricci lower bound.

Our assumption implies that |x t | is unbounded as t tends to t ∞ . Indeed if it was not the case, then we would have estimates on |x t | and so by Proposition 3.11 and Lemma 4.4, there would be a solution at time t ∞ and by openness for times greater than t ∞ if t ∞ . This is a contradiction.

We can find a sequence t i such that t i → t ∞ and |x ti | → ∞. Define ξ t ∈ a + as the direction given by the minimum ξ t := x t /|x t |. Up to taking a subsequence, we can also assume that ξ t admits a limit ξ ∞ ∈ a + as t i → t ∞ .

We will consider an integral equality involving ν t and consider the limit as t i → t ∞ . The integral equality follows from Proposition 5.1 applied to w t : a + ∂ν t ∂ξ e -νt dx = 0.

Recall that w t = tu t + (1 -t)u ref by definition, so ∂u ref + j ∂ξ t e -νt dx.

ν t = tu t + (1 -t)u ref + j = t(u t + j) + (1 -t)(u ref + j).
We will compute the limit of each of these terms as t i → t ∞ , and obtain the following result. Proposition 6.1. We have In particular, the limit as

t ∞ bar DH (2P + ) -4ρ, ξ ∞ = (t ∞ -1)(v(ξ ∞ ) -4ρ, ξ ∞ ).

Proof of Proposition

t i → t ∞ is ξ ∞ , bar DH (2P + ) V.
For the other terms we need more work to compute the limits. For simplicity, we will often omit the indices i in t i . We will prove the two following propositions. Proposition 6.2. We have

lim ti→t∞ a + ∂j ∂ξ t e -νt = -4ρ, ξ ∞ V.
Proposition 6.3. We have

lim ti→t∞ a + ∂u ref ∂ξ t e -νt = v(ξ ∞ )V.
Let us first find a domain D(ǫ) of the form B(x t , δ) ∩ (bρ + a + ) where e -νt dx puts all the mass up to 2ǫ > 0. When we write B(x t , δ), we in general mean B(x t , δ) ∩ a + . Lemma 6.4. For any ǫ > 0, there exists a constant δ = δ(ǫ) independent of t such that a + \B(xt,δ) e -νt dx < ǫ and e -κδ+C σ r δ r-1 < ǫ, where σ r is the area of a sphere of radius 1 in R r .

Proof. Recall from Proposition 4.4 that ν t (x) ≥ κ|x -x t | -C, for some κ > 0, C independent of t. Observe that the function e -κ|x-xt|+C is well defined on a, positive and integrable. So there exists a δ > 0 such that a\B(xt,δ)

e -κ|x-xt|+C dx < ǫ.

But then we also have

a + \B(xt,δ) e -νt dx ≤ a + \B(xt,δ) e -κ|x-xt|+C < ǫ.
Of course, since e -κy+C decreases exponentially with respect to y, we can increase δ so as to have the second condition: e -κδ+C σ r δ r-1 < ǫ. Lemma 6.5. For any ǫ > 0, let δ = δ(ǫ) be given by Lemma 6. where dσ is the area measure of ∂D, which is piecewise smooth.

Proof. Here we want to use Proposition 3.10. Now that we know that m t is uniformly bounded, we can choose M and the corresponding b so that:

∀x ∈ a + \ (bρ + a + ), e -νt(x) < max ǫ σ r δ r-1 , ǫ δ r ω r
where ω r is the volume of the ball of radius 1 in R r .

Let us prove that e -νt < ǫ/(σ r δ r-1 ) on ∂D. A point x ∈ ∂D is either on the sphere of radius δ centered at x t , or on ∂(bρ + a + ). In the first case, we have, by Proposition 4.4, e -νt(x) ≤ e -κ|x-xt|+C ≤ e -κδ+C < ǫ σ r δ r-1 by the second consequence of Lemma 6.4. In the second case, x ∈ ∂(bρ + a + ), so by the choice of b above, using the first term in the maximum, we have

e -νt(x) < ǫ σ r δ r-1 . Obviously the volume of ∂D is ≤ σ r δ r-1 , so ∂D e -νt dσ < ∂D ǫ σ r δ r-1 dσ < ǫ.
For the other conclusion, we use the fact that e -νt(x) < ǫ/(δ r ω r ) on B(x t , δ)\ D ⊂ a + \ (bρ + a + ), which implies that B(xt,δ)\D e -νt dx < B(xt,δ)\D ǫ δ r ω r dx < ǫ using the fact that the volume of B(x t , δ) \ D is ≤ δ r ω r .

Lemma 6.6. Let ǫ > 0 and D = D(ǫ) be the domain given by Lemma 6.5. We have We consider now the quantity ∂j ∂ξ t (x) + 4ρ, ξ t for x ∈ D. It is negative by Lemma 3.5 since ξ t ∈ a + . Recall that D(θ) ⊂ b(θ)ρ + a + for some b(θ) > 0, and more precisely that D(θ) = B(x t , δ(θ)) ∩ (b(θ)ρ + a + ). Choose b 0 > 0 such that B(b 0 ρ, δ(θ)) ⊂ b(θ)ρ + a + .

a
We can write v(ξ t )e -νt = v(ξ t )V.

ξ t = x t |x t | = x t -b 0 ρ |x t | + b 0 ρ |x t | = |x t -
Finally, since ξ t converges to ξ ∞ , there exists i 1 such that for i > i 1 ,

|v(ξ t )V -v(ξ ∞ )V | < ǫ 3 .
We have proved that for i > i 0 , i 1 , we have

a + ∂u ref ∂ξ t e -νt -v(ξ ∞ )V < 2 ǫ 3 + 2dθ = ǫ
This concludes the proof of Proposition 6.1, by considering the limit at t ∞ and dividing by V .

Sufficient condition

We can now prove that our condition is sufficient for the existence of a Kähler-Einstein metric and thus conclude the proof of Theorem A. Theorem 6.7. If bar DH (2P + ) ∈ 4ρ + Ξ, then X admits a Kähler-Einstein metric.

Proof. Assume that X admits no Kähler-Einstein metric. Then Proposition 6.1 gives t ∞ bar DH (2P + ) -4ρ, ξ ∞ = (t ∞ -1)(v(ξ ∞ ) -4ρ, ξ ∞ ).

In particular, since v is the support function of 2P and 2ρ ∈ Int(P ), and 0 < t ∞ ≤ 1, we have bar DH (2P + ) -4ρ, ξ ∞ ≤ 0.

Assume that bar DH (2P + ) ∈ 4ρ + Ξ. Then by the definition of Ξ, the only possibility is that ξ ∞ ∈ a t and bar DH (2P + ) -4ρ, ξ ∞ = 0.

To prove that this is impossible, consider the vanishing a + ∂ν t ∂ξ ∞ e -νt dx = 0.

The difference with what we have done before is that we fix ξ ∞ instead of considering ξ t . Since ξ ∞ ∈ a t , we have ∂j ∂ξ∞ = 0 and so we deduce from the vanishing of the integral the following equality, valid for t < t ∞ .

2

  where we denote by J the function defined on a by J(a) := α∈Φ + sinh 2 α, a .

Figure 2 :

 2 Figure 2: Polytope P (X 1 , -K X1 )

Notation 3. 1 .

 1 Let us fix some 0 < t 0 ∈ I, which exists since 0 ∈ I and I is open. Let us summarize the consequence of what we have recalled in this section.

  for x ∈ a + . On the other hand, the definition of the normalized Ricci potential f ref implies that e f ref ω n ref = e -ψ ref dg, which allows to write the right hand side of equation (2) as e f ref -tψt ω n ref = e -tψt-ψ ref dg. For x ∈ a, we have

Proposition 5. 3 .

 3 Assume there exists a Kähler-Einstein metric on X. Then bar DH (2P + ) ∈ 4ρ + Ξ. Proof. By Proposition 3.3, the Kähler-Einstein equation restricted to the open orbit reads:MA R (u)α∈Φ + α, ∇u 2 = e -u J.Suppose that there exists a solution u. Applying Proposition 5.1 to u gives a + ∂ν ∂ξ e -ν dx = 0, so by definition of ν = u + j, dx = 0.Since u is solution to the Kähler-Einstein equation, we havee -ν = e -u J = MA R (u) α∈Φ + α, ρ 2 .In particular, by Proposition 4.3,a + e -ν dx = Vis constant, and by Proposition 5.2, a + ∂u ∂ξ e -ν dx = ξ, bar DH (2P + ) V. By Lemma 3.5, we have for ξ ∈ a + , a + ∂j ∂ξ e -ν dx < -4 ρ, ξ V.

  4. There exists a constant b = b(ǫ) > 0 such that if we denote by D = D(ǫ) the domain B(x t , δ) ∩ (bρ + a + ), we have B(xt,δ)\D e -νt dx < ǫ, and ∂D e -νt dσ < ǫ,

  t

  where f ref is the normalized Ricci potential of ω ref defined as the ω ref -psh function that satisfiesω ref + i∂∂f ref = Ric(ω ref ) and X e f ref ω n ref = X ω n ref . Let h ref be a smooth hermitian metric on -K X with curvature form ω ref .Then it determines a volume form dV on X defined in a local trivialization s of -K X by dV = |s| 2 h ref s -1 ∧ s -1 . Then up to a constant, the Ricci potential f ref is equal to the logarithm of the potential of dV with respect to ω n ref .We choose h ref (by multiplying by a scalar) such that f ref is indeed equal to that.

  rather than Wang and Zhu, we write Vol({ν t ≤ m t + s})ds Now remark that {ν t ≤ m t + s} ⊂ s • A t by convexity of ν t , where s • A t is the s-dilation of A t with center x t . We deduce from that Vol({w t ≤ ν t + s}) ≤ s r Vol(A t ) ≤ Cs r e mt/2 . The left hand side being constant, this inequality gives an upper bound on m t . t is bounded above by a constant independent of t. Thus we have showed estimates on |m t |.Now for linear growth, the estimate on |m t | implies that we know both an upper bound C 1 and a lower bound C 2 independent of t for the volume of A t . Since this set is convex, and contains a ball B(x t , δ 0 ) of fixed radius δ 0 independent of t by the proof of Lemma 4.2, this implies that A t is included in a ball A t ⊂ B(x t , δ) where δ only depends on C 1 and δ 0 .By convexity of ν t , this implies that ν t (x) ≥ |x -x t |/δ + m t outside of the ball, and we can extend this inequality to the whole of a + simply by subtracting 1:

	By Lemma 4.2, Vol(A t ) admits a lower bound c independent of t, so
		V ≥ e -mt c	+∞	e -s ds,
					1
	and we deduce that
		-m t ≤ ln(V ) -ln c	+∞	e -s ds
					1
	so -m
	e -νt(x) dx =			+∞	e -s dsdx
	a +		a +	νt(x)
		=	+∞	e -s	1 {νt(x)≤s} dxds
			-∞		a +
		=	+∞	e -s Vol({ν t ≤ s})ds
			mt
	= e -mt e -s Applying this to the formula above we obtain +∞ 0
		e -νt(x) dx ≤ e -mt Ce mt/2	+∞	e -s s n ds
	a +			0
			≤ C ′ e -mt/2 .
	For the lower bound, remark that
					+∞
	V =	e -νt(x) dx = e -mt	e -s Vol({w t ≤ ν t + s})ds
	a +				0
			≥ e -mt	+∞	e -s Vol({w t ≤ ν t + s})ds
					1
			≥ e -mt Vol(A t )	+∞	e -s ds.
					1

  Proposition 5.5. Assume that X admits no Kähler-Einstein metrics, then the greatest Ricci lower bound of X is lower than or equal to the supremum of all t < 1 such that

							P +
						α 2
									α 1
						Figure 4: Barycenter for X 2
	5.3 Upper bound on R(X)
			4ρ +	t t -1	(bar DH (2P a + ∂ν t ∂ξ e -νt dx = 0.
	This is equivalent to
	t	a +	∂u t ∂ξ	e -νt + (1 -t)	a +	∂u ref ∂ξ	e -νt +	a +	∂j ∂ξ	e -νt = 0.
	Without loss of generality we can assume t < 1 and divide by (t -1)V to
	get
		t		∂u t	e
	t -1 a +	∂ξ

+ ) -4ρ) ∈ -Ξ + 2P + .

Proof. Consider the equation at time t:

MA R (u t ) α∈Φ + α, ∇u t 2 = e -νt .

Apply Proposition 5.1 to w t . This gives for any ξ ∈ a + ,

  6.1 -νt dx = ξ t , bar DH (2P + ) V.

	Let us first consider			a +	∂u t ∂ξ t	e -νt dx.
	Since	e -νt =	α, ∇u t	2 MA R (u t ),
				α∈Φ +
	Proposition 5.2 implies
	a +	∂u t ∂ξ t	e

  The first of these three integrals is zero by Proposition 5.1.For the third term, we have It remains to deal with the second integral. We apply the divergence theorem on D to the function e -νt :Proof of Proposition 6.2. Let ǫ > 0. Set θ = ǫ/(3(2d + 1 + 8|ρ|)) and let D := D(θ). Write -νt + 4ρ,ξ t V + | 4ρ, ξ ∞ -ξ t V |The second term tends to zero so there exists an i 0 such that for all i ≥ i 0 ,| 4ρ, ξ ∞ -ξ t V | ≤ ǫ 3 .

	a +	∂j ∂ξ t	e -νt + 4ρ, ξ ∞ V ≤		a + \D	∂j ∂ξ t	e -νt +	D	∂j ∂ξ t	e -νt + 4ρ, ξ ∞ V
						≤ (2d + 1)θ +	D	∂j ∂ξ t	e -νt + 4ρ, ξ ∞ V
	by Lemma 6.6.
	+ \D a + \D ∂(ν t -w t ) ∂j ∂ξ t e -νt < (2d + 1)ǫ. ∂ξ t e -νt dx a + ∂ν t ∂ξ t e -νt dx -D ∂ν t ∂ξ t e -νt dx -e -νt + 4ρ, ξ ∞ V ≤ Then we have Proof. Let us write: a + \D ∂j ∂ξ t e -νt dx = = D ∂j ∂ξ t D ∂j ∂ξ t e We now deal with the second term:	a + \D	∂ξ t ∂w t	e -νt dx
					a + \D	∂w t ∂ξ t	e -νt dx ≤ d	a + \D	e -νt dx
										< 2dǫ
	by Lemma 6.5.
			D	∂ν t ∂ξ t	e -νt dx = -	D	∂e -νt ∂ξ t	dx =
	Putting everything together, we see that
					a + \D	∂j ∂ξ t	e -νt dx < (2d + 1)ǫ

∂D

e -νt ξ t , n dσ where n is the exterior normal to D and dσ the area measure.

We conclude using Lemma 6.5 that D ∂ν t ∂ξ t e -νt dx ≤ ∂D e -νt dσ < ǫ. D ∂j ∂ξ t e -νt + 4ρ, ξ t V ≤ D ∂j ∂ξ t + 4ρ, ξ t e -νt + a + \D 4ρ, ξ t e -νt ≤ D ∂j ∂ξ t + 4ρ, ξ t e -νt + 4ρ, ξ t a + \D e -νt ≤ D ∂j ∂ξ t + 4ρ, ξ t e -νt + |4ρ| a + \D e -νt and by construction of D we deduce: ≤ D ∂j ∂ξ t + 4ρ, ξ t e -νt + |4ρ| • 2θ

  b 0 ρ| |x t | ξ 1 + |b 0 ρ| |x t | ξ 2where ξ 1 = (x t -b 0 ρ)/|x t -b 0 ρ| and ξ 2 = b 0 ρ/|b 0 ρ|.Let x = x t + y ∈ D, we consider the restriction of j to the line starting from b 0 ρ + y and of direction ξ 1 , which contains x. By convexity, we have+ C = -4ρ, x + C on a + , for some constant C. + C -j(y + b 0 ρ) |x t -b 0 ρ| . + C -j(y + b 0 ρ) + |b 0 ρ| ∂j ∂ξ2 (x) |x t | Now y ∈ B(b 0 ρ, δ(θ))is bounded, j is bounded on b(θ)ρ + a + by Lemma 3.4, and the derivatives of j are bounded on b(θ)ρ + a + by Lemma 3.6, so there is a (negative) constant C ′ such that Applying this to the sequence t i , we find that there exists a i 1 such that for i > i 1 , and unifomrly for x ∈ D, Gathering everything gives, for i > max(i 0 , i 1 ), Proof of Proposition 6.3. Let ǫ > 0. Set θ := ǫ/6d and let δ = δ(θ). First, by Lemma 6.4, we havea + \B(xt,δ) ∂u ref ∂ξ t e -νt < dθ.On B(x t , δ) , we always have ∂u ref ∂ξt ≤ v(ξ t ). Now consider the ray starting from x -x t and going to x. By convexity, we have ∂u ref ∂ξThe last step holds because v is Lipschitz.For i > i 0 for some i 0 , we thus have, for x ∈ B(x t , δ),

	Then for i > i 1 , Applying Lemma 6.4 again gives
	It gives	D	∂ ∂ξ t + 4ρ, ξ t e -νt ≤ = |x t -b 0 ρ| |x t | ∂ ∂ξ 1 B(xt,δ) ∂j ∂ξ t a + v(ξ t )e -νt -	+ D v(ξ t )e -νt < dθ, |b 0 ρ| |x t | ∂ ∂ξ 2 . ∂j ∂ξ t + 4ρ, ξ t e -νt
	with			∂j ∂ξ 1	(x) ≥ a +	j(x) -j(y + b 0 ρ) |x t -b 0 ρ| ≤ ǫ e -νt . 3V D ǫ ≤ 3
	Recall from Lemma 3.4 that j(x) ≥ -2 α∈Φ + α, x Applying this gives a + ∂j ∂ξ t e -νt + 4ρ, ξ ∞ V ≤ (2d + 1)θ + 8|ρ|θ +	ǫ 3	+	ǫ 3	= ǫ.
	∂j ∂ξ 1 -4ρ, x Now going back to ∂j (x) ≥ ∂ξt (x), we have
	∂j ∂ξ t	(x) =	|x t -b 0 ρ| |x t |	∂j ∂ξ 1	(x) +	|b 0 ρ| |x t |	∂j ∂ξ 2	(x)
		≥	|x |x t -b 0 ρ|	+	|b 0 ρ| |x t |	∂j ∂ξ 2	(x)
		≥	-4ρ, x |x t |	+	|b 0 ρ| |x t |	∂j ∂ξ 2	(x)
	so		∂u ref ∂ξ t	(x) ≥ v(ξ t +	x -x t |x t |	) +	C |x t |
	0 ≥ Integrating on the ball gives ∂j ∂ξ t (x) + 4ρ, x t |x t | -4ρ, y 0 ≥ ≥ ≥ v(ξ t ) + ∂j ∂ξ t (x) + 4ρ, ∂u ref ∂ξ t (x) -v(ξ t ) < C ′ |x t | x t |x t |	≥ ǫ V. C ′ 3 |x t |	.
			B(xt,δ)	∂j ∂ξ t ∂u ref ∂ξ t	(x) + 4ρ, ξ t ≤ e -νt -B(xt,δ) v(ξ t )e -νt < 3V . ǫ	ǫ 3	.

t -b 0 ρ| |x t | -4ρ, x + C -j(y + b 0 ρ) t + y + C -j(y + b 0 ρ) t (x) ≥ u ref (x) -u ref (x -x t ) |x t | ≥ v(x) + C |x t |

for some constant C independent of x in B(x t , δ), by Proposition 2.10 and because u ref is bounded on B(0, δ). Then we can write

  Assume first that t ∞ < 1, then we can writet ∞ t ∞ -1 bar DH (2P + ) -4ρ, ξ ∞ = v(ξ ∞ ) -4ρ, ξ ∞ , ∞ (-bar DH (2P + ) + 4ρ), ξ ∞ = v(ξ ∞ ).For t = 0, we have 4ρ ∈ Int(2P ). The function t → t/(1 -t) is strictly increasing and its image is [0, ∞[. Besides, since v is the support function of 2P , the value v(ξ ∞ ) is attained by m, ξ ∞ if and only if m is in the supporting hyperplane of 2P defined by ξ ∞ . We deduce that necessarily t ∞ is the unique value of t for which4ρ + t 1 -t (-bar DH (2P + ) + 4ρ) ∈ ∂(-Ξ + 2P + ), if it exists. If it doesn't exist, then t ∞ = 1.Combining this with the upper bound on R(X), we have proved Theorem C.Example 6.9. This allows to compute exactly the greatest Ricci lower bound for X 2 , which is

	6.4 Lower bound on R(X)
	or	4ρ +	t ∞ 1 -t R(X 2 ) =	1046175339 1236719713	.
			a +	∂u t ∂ξ ∞	e -νt dx = (t -1)	a +	∂u ref ∂ξ ∞	e -νt dx

Lemma 1.10. Assume l 1 ∈ k α and l 2 ∈ s, then H l1,l2 (a) = 0.

Proof. Let us first determine the A 1 , B 1 , C 1 such that z 1 l 1 +z 2 l 2 = A 1 +B 1 +C 1 . Using Lemma 1.6, write

with l ′ 1 = ad(a)(il)/ α, a . Then we have

In fact we must only determine B 2 which is the part of D 1 that lies in a.

We have

and the third point of Lemma 1.6 applies to show that the a component of

For the second part, write

We have here [l ′ 1 , l 1 ], [l ′ 1 , l 2 ] ∈ k, and [il 1 , l 2 ] ∈ g α ⊕ g -α as above, so only [il 1 , l ′ 1 ] matters. By the properties of the root decomposition,

But in fact we don't need to determine it more explicitly because it appears as a term in y 2 1 and these are ignored in the computation of ∂∂.

For the third part,

so there is no contribution to B 2 . We have thus proved that

The left hand side term is zero because we assumed

We thus have, for all t < t ∞ ,

This is a contradiction: let m := min{v(ξ) ; ξ ∈ a, |ξ| = 1} > 0. For any δ > 0 fixed, there exists an

. This is because |x t | goes to ∞ and u ref is asymptotic to v. Choose now δ = δ(m/4) given by Lemma 6.4, then for t close to t ∞ , we obtain

Combined with the obstruction proved earlier, it ends the proof of Theorem A.

Example 6.8. Consider the example X 1 , which we recall is the wonderful compactification of PGL 3 (C). The corresponding root system is A 2 . We denote by α 1 and α 2 the simple roots. The third positive root is then α 1 + α 2 , and

We computed the coordinates x and y of the barycenter bar DH (P + ) and obtained bar DH (P + ) = 24641 9888 (α 1 + α 2 ).

As a consequence, X 1 admits a Kähler-Einstein metric. Figure 5 gives a representation of P + , where the cross is the barycenter, and the convex cone delimited by the dashed lines is 2ρ + Ξ.