Thibaut Delcroix

KÄHLER GEOMETRY OF HOROSYMMETRIC VARIETIES, AND APPLICATION TO MABUCHI'S K-ENERGY FUNCTIONAL

Keywords: 2010 Mathematics Subject Classification. 14M27, 32M12, 32Q15 horosymmetric varieties, Mabuchi K-energy functional, spherical varieties, cscK metrics, log Kahler Einstein metrics. 1

Kähler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functional

Introduction

Toric manifolds are complex manifolds equipped with an action of (C * ) n such that there is a point with dense orbit and trivial stabilizer. The Kähler geometry of toric manifolds plays a fundamental role in Kähler geometry as a major source of examples as well as a testing ground for conjectures. It involves strong interactions with domains as various as convex analysis, real Monge-Ampère equations, combinatorics of polytopes, algebraic geometry, etc. The study of Kähler metrics on toric manifolds relies strongly on works of Guillemin, Abreu, then Donaldson. They have developed, using Legendre transform as a main tool, a very precise setting including:

• a model behavior for smooth Kähler metrics, • a powerful expression of the scalar curvature,

• applications to the study of canonical Kähler metrics via the ubiquitous Mabuchi functional.

This setting allowed Donaldson to prove the Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler (cscK) metrics on toric surfaces. That is, given a toric surface X equipped with an ample line bundle L, he showed that existence of cscK metrics in the Kähler class c 1 (L) is equivalent to torus equivariant K-stability of (X, L). He further translated this condition into a number (in general infinite) of combinatorial conditions on the associated polytope.

Our goal in this article is to generalize this setting to a much larger class of varieties, that we introduce: the class of horosymmetric varieties.

Let G be a complex connected linear reductive group. A normal algebraic Gvariety X is called spherical if any Borel subgroup B of G acts with an open dense orbit on X. Major subclasses of spherical varieties are given by biequivariant group compactifications, and horospherical varieties. A horospherical variety is a Gvariety with an open dense orbit which is a G-homogeneous fibration over a generalized flag manifold with fiber a torus (C * ) r . The author's previous work on spherical varieties [START_REF] Delcroix | Kähler-Einstein metrics on group compactifications[END_REF][START_REF] Delcroix | K-Stability of Fano spherical varieties[END_REF] has highlighted how they provide a richer source of examples than toric varieties, with several examples of behavior which cannot appear for toric varieties. While it was possible to work on the full class of spherical manifolds, from the point of view of algebraic geometry, for the question of the existence of Kähler-Einstein metrics thanks to the proof of the Yau-Tian-Donaldson conjecture for Fano manifolds, it is necessary to develop Guillemin-Abreu-Donaldson theory to treat more general questions. It seems a very challenging problem to do this uniformly for all spherical varieties. On the other hand, the author did develop part of this setting for group compactifications and horospherical varieties.

Group compactifications do not share a nice property that toric, horospherical and spherical varieties possess and frequently used in Kähler geometry: a codimension one invariant irreducible subvariety in a group compactification leaves the class of group compactifications. We introduce the class of horosymmetric varieties as a natural subclass of spherical varieties containing horospherical varieties, group compactifications and more generally equivariant compactifications of symmetric spaces, which possesses the property of being closed under taking a codimension one invariant irreducible subvariety. The definition is modeled on the description of orbits of wonderful compactifications of adjoint (complex) symmetric spaces by De Concini and Procesi: they all have a dense orbit which is a homogeneous fibration over a generalized flag manifold, whose fibers are symmetric spaces. We say that a normal G-variety is horosymmetric if it admits an open dense orbit which is a homogeneous fibration over a generalized flag manifold, whose fibers are symmetric spaces. Such a homogeneous space is sometimes called a parabolic induction from a symmetric space. Here we allow symmetric spaces under reductive groups, thus recovering horospherical varieties by considering (C * ) r as a symmetric space for the group (C * ) r and the involution σ(t) = t -1 . For the sake of giving precise statement in this introduction, let us introduce some notations. A horosymmetric homogeneous space is a homogeneous space G/H such that there exists • a parabolic subgroup P of G, with unipotent radical P u , • a Levi subgroup L of P , • and an involution of complex groups σ : L → L, such that P u ⊂ H and (L σ ) 0 ⊂ L ∩ H as a finite index subgroup, where L σ denotes the subgroup of elements fixed by σ and (L σ ) 0 its neutral connected component.

Spherical varieties in general admit a combinatorial description: there is on one hand a complete combinatorial characterization of spherical homogeneous spaces by Losev [START_REF] Losev | Uniqueness property for spherical homogeneous spaces[END_REF] and on the other hand a combinatorial classification of embeddings of a given spherical homogeneous space by Luna and Vust [START_REF] Luna | Plongements d'espaces homogènes[END_REF][START_REF] Knop | The Luna-Vust theory of spherical embeddings[END_REF]. General results about parabolic induction allow to derive easily the information about a horosymmetric homogeneous spaces from the information about the symmetric space fiber. For symmetric spaces, most of the information is contained in the restricted root system. Choose a torus T s ⊂ L on which the involution acts as the inverse, and maximal for this property. It is contained in a σ-stable maximal torus T in L. Then consider the restriction of roots of G (with respect to T ) to T s . Let Φ s denote the subset of roots whose restriction are not identically zero. The restrictions of the roots in Φ s form a (possibly non reduced) root system called the restricted root system, with corresponding notions of restricted Weyl group W , restricted Weyl chambers, etc. Let Y(T s ) denote the group of one-parameter subgroups of T s , and identify a s = Y(T s ) ⊗ R with the Lie algebra of the non-compact part of the torus T s . The image of a s by the exponential, then by the action on the base point x ∈ X, intersects every orbit of a maximal compact subgroup K of G along restricted Weyl group orbits (see Section 2 for details).

Let L be a G-linearized line bundle on a horosymmetric homogeneous space. It is determined by its isotropy character χ. Fix a maximal compact subgroup K of G. To a K-invariant metric h on L we associate a function u : a s → R, called the toric potential, which together with χ totally encodes the metric. One of our main result is the derivation of an expression of the curvature form ω of h in terms of its toric potential. We achieve this for the general case, but let us only give the statement in the nicer situation where the restriction of L to the symmetric space fiber is trivial. By fixing a choice of basis of a complement of h in g, we may define reference real (1,1)-forms ω ♥,♦ indexed by couples of indices in {1, . . . , r}∪Φ + s ∪Φ Q u (where r = dim(T s ), Φ + s is the intersection of Φ s with some system of positive roots, Φ Q u = -Φ P u is the set of opposite of roots of P u ) that form a point-wise basis, and we express the curvature form in these terms. Given a root α of G, we denote by α ∨ its associated coroot. There is a natural way (see Section 3) to identify both χ and the differential d a u of u at some point a ∈ a s as elements of Y(T ), so that their action on any coroot used in the next statement are well-defined.

Theorem 1.1. Assume that the restriction of L to the symmetric fiber is trivial. Let a ∈ a s be such that β(a) = 0 for all β ∈ Φ s . Then

ω exp(a)H = 1≤j1,j2≤r 1 4 d 2 a u(l j1 , l j2 )ω j1,j2 + α∈Φ Q u -e 2α 2 (d a u -2χ)(α ∨ )ω α,α + β∈Φ + s d a u(β ∨ ) sinh(2β(a)) ω β,β .
The previous theorem concerns only horosymmetric homogeneous spaces. To move on to horosymmetric varieties, we need more input from the general theory of spherical varieties. To a G-linearized line bundle on a horosymmetric variety are associated several polytopes. Of major importance is the (algebraic) moment polytope ∆ + , obtained as the closure of the set of suitably normalized highest weights of the spaces of multi-sections of L, seen as G-representations. It lies in the real vector space X(T ) ⊗ R, where X(T ) denotes the group of characters of T . The main application of the moment polytope to Kähler geometry is that it controls the asymptotic behavior of toric potentials, which in the case of positively curved metrics further allows to give a formula for integration with respect to the Monge-Ampère measure of the curvature form, in conjunction with the previous theorem. Again we do not state our results in their most general form in this introduction but prefer to give the general philosophy in a situation which is simpler than the general one.

Theorem 1.2. Assume that L is an ample G-linearized line bundle on a nonsingular horosymmetric variety X, and that it admits a global Q-semi-invariant holomorphic section, where Q is the parabolic opposite to P with respect to T . Let h be a smooth K-invariant metric on L with positive curvature ω and toric potential u. Then

(1) u is smooth, W -invariant and strictly convex, (2) a → d a u defines a diffeomorphism from Int(-a + s ) onto Int(2χ -2∆ + ). Let ψ denote a K-invariant function on X, integrable with respect to ω n . Let dq denote the Lebesgue measure on the affine span of ∆ + , normalized by the lattice χ + X(T /T ∩ H).

Then there exists a constant C, independent of h and ψ, such that

X ψω n = C ∆ +
ψ(d 2χ-2q u * )P DH (q)dq.

where P DH (q) = α∈Φ Q u ∪Φ + s κ(α, q)/κ(α, ̟) , ̟ is the half sum of positive roots of G and u * is the convex conjugate of u.

The two theorems above form a strong basis to attack Kähler geometry questions on horosymmetric varieties. They are for example all that is needed to study the existence of Fano Kähler-Einstein metrics with the strategy following the lines of Wang and Zhu's work on toric Fano manifolds. They also allow to push further, namely to compute an expression of the scalar curvature, to compute an expression of the (log)-Mabuchi functional, then to obtain a coercivity criterion for this functional, in the line of work of Li-Zhou-Zhu for group compactifications.

The Mabuchi functional is a functional on the space of Kähler metrics in a given class, whose smooth minimizers if they exist should be cscK metrics. There are several extensions of this notion, in particular the log-Mabuchi functional, related to the existence of log-Kähler-Einstein metrics. A natural way to search for minimizers of this functional is to try to prove its properness, or coercivity, with respect to the J-functional. The J-functional is another standard functional in Kähler geometry, which may be considered as a measure of distance from a fixed reference metric in the space of Kähler metrics.

We provide in this paper an application of our setting to this problem of coercivity of the Mabuchi functional, obtaining a very general, but at the same time far from optimal, coercivity criterion for the Mabuchi functional on horosymmetric varieties. We work under several simplifying assumptions to carry out the proof while keeping a reasonable length for the paper, but expect that several of these assumptions can be removed with a little work (see Section 7.1 for a detailed discussion).

Instead of stating these assumptions in this introduction, let us state the result in three examples of situations where they are satisfied. They are as follows. In all cases G is a complex connected linear reductive group and X is a smooth projective G-variety.

(1) The manifold X is a group compactification, that is, G = G 0 × G 0 and there exists a point x ∈ X with stabilizer diag(G 0 ) ⊂ G and dense orbit. We may consider any ample G-linearized line bundle on X.

(2) The manifold X is a homogeneous toric bundle under the action of G, that is there exists a projective homogeneous G/P and a G-equivariant surjective morphism X → G/P with fiber isomorphic to a toric variety, under the action of P which factorizes through a torus (C * ) r . We consider any ample G-linearized line bundles on X.

(3) The manifold X is a toroidal symmetric variety of type AIII(r, m > 2r), that is, m and r are two positive integers with m > 2r, G = SL m , there exists a point x ∈ X with dense orbit, whose orbit is isomorphic to SL m /S(GL r × GL m-r ), and there exists a dominant G-equivariant morphism from X to the wonderful compactification of this symmetric space. We may consider any ample G-linearized line bundle which restricts to a trivial line bundle on the dense orbit.

Let Θ be a G-equivariant boundary, that is, an effective Q-divisor Θ = Y c Y Y where Y runs over all G-stable irreducible codimension one submanifolds of X. We assume furthermore that the support of Θ is simple normal crossing and c Y < 1 for all Y . In particular, the pair (X, Θ) is klt. It follows from the combinatorial description of horosymmetric varieties that to each Y as above is associated an element µ Y of Y(T s ). Let ∆ + be the moment polytope of L, and let λ 0 be a well chosen point in ∆ + (see Section 7). Let ∆+ Y denote the bounded cone with vertex λ 0 and base the face of ∆ + whose outer normal is -µ Y in the affine space χ + X(T /T ∩ H) ⊗ R. Let χ ac denote the restriction of the character α∈Φ Q u α of P to H, and set

Λ Y = -c Y + 1 -α∈Φ Q u ∪Φ + s α(µ Y ) sup{p(µ Y ), p ∈ χ -∆ + } , I H (a) = β∈Φ + s ln sinh(-2β(a)) - α∈Φ Q u 2α(a),
and

SΘ = S -n Y c Y L| n-1 Y /L n .
Let Mab Θ denote the log-Mabuchi functional in this setting, and write Mab Θ (u) for its value on the Hermitian metric with toric potential u. For the question of coercivity, the log-Mabuchi functional matters only up to normalizing additive and multiplicative constants, so we ignore these in the statement.

Theorem 1.3. Let p = p(q) := 2(χq), then we have

Mab Θ (u) = Y Λ Y ∆+ Y (nu * (p) -u * (p) χ(α ∨ ) q(α ∨ ) + d p u * (p))P DH (q)dq + ∆ + u * (p)( χ ac (α ∨ ) q(α ∨ ) -SΘ )P DH (q)dq - ∆ + I H (d p u * )P DH (q)dq - ∆ + ln det(d 2 p u * )P DH (q)dq
where u * denotes the Legendre transform, or convex conjugate, of u.

As an application of this formula, we obtain the following sufficient condition for coercivity. Consider the function F L defined piecewise by

F L (q) = (n + 1)Λ Y -SΘ + (χ ac -Λ Y χ)(α ∨ ) q(α ∨ )
for q in ∆+ Y . Define an element bar of the affine space generated by ∆ + by setting bar = ∆ + qF L (q) P DH (q)dq

∆ + P DH dq .
Despite the notation, it is not in general the barycenter of ∆ + with respect to a positive measure. We will however see how to consider it as a barycenter in the article. The coercivity criterion is stated in terms of F L and bar. Let 2ρ H denote the element of a * s defined by the restriction of α∈Φ Q u ∪Φ + α to a s . Theorem 1.4. Assume that F L > 0 and that the point

(min Y Λ Y ) ∆ + P DH dq ∆ + F L P DH dq (bar -χ) -2ρ H
is in the relative interior of the dual cone of a + s . Then the Mabuchi functional is coercive modulo the action of Z(L) 0 .

Thanks to recent progresses in the field, the above sufficient criterion for properness has strong consequences on the existence of canonical metrics that we illustrate with two corollaries. Since we consider only smooth and invariant potentials, there are no precise statements in the literature which would yield existence of canonical metrics when combined with our coercivity criterion. On the other hand, such statements do hold from carefully following a combination of arguments from [BBE + 16, BDL17, CC18a], for cscK metrics and (weak) log-Kähler-Einstein metrics. We provide in two appendices the outlines of the proofs highlighting the possible obstacles coming from the restriction to smooth K-invariant potentials, and why they are easy to overcome. For cscK metrics this relies heavily on the recent breakthrough of Chen and Cheng [START_REF] Chen | On the constant scalar curvature Kähler metrics[END_REF] and allows to obtain infinite families of new examples of classes with constant scalar curvature metrics.

Corollary 1.5. Under the same combinatorial condition, there exists a constant scalar curvature Kähler metric in c 1 (L).

In the case when L = K -1 X ⊗ O(-Θ) is ample, the pair (X, Θ) is log-Fano and minimizers of the log-Mabuchi functional are log-Kähler-Einstein metrics. We obtain, by combining our coercivity criterion, the reference work [BBE + 16] and an argument from [START_REF] Berman | Convexity of the extended Kenergy and the large time behavior of the weak Calabi flow[END_REF], the following.

Corollary 1.6. Assume L = K -1 X ⊗ O(-Θ), then (X, Θ) admits a log-Kähler-Einstein metric provided bar -α∈Φ Q u ∪Φ + α is in the relative interior of the dual cone of a + s . Note that our proof does not provide conical Kähler-Einstein metrics in a strong sense. We rather obtain weak log-Kähler-Einstein metrics in the sense of [BBE + 16] (and they are weakly conical by [START_REF] Guenancia | Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors[END_REF]). It would be worthwhile to follow a more precise approach to obtain a better regularity for the metrics as it was done for toric manifolds in [START_REF] Ved Datar | Connecting toric manifolds by conical Kähler-Einstein metrics[END_REF] and [START_REF] Wang | Modified Futaki invariant and equivariant Riemann-Roch formula[END_REF].

If one works on, say, a biequivariant compactification of a semisimple group, then it is not hard to check that the condition above is open as L and Θ vary. Starting from an example of Kähler-Einstein Fano manifold obtained in [START_REF] Delcroix | Kähler-Einstein metrics on group compactifications[END_REF], we can extract from this corollary an explicit subset of K -1 X in the ample cone, with non-empty interior, such that each corresponding L writes as L = K -1 (X,Θ) and the pair (X, Θ) admits a log-Kähler-Einstein metric.

While the point of view adopted in this article is definitely in line with the author's earlier work on group compactifications and horospherical varieties, it should be mentioned that there were previous works, and different perspectives on both these classes. Group compactifications have been studied in detail from the algebraic point of view and the first article about the existence of canonical metrics on these was [START_REF] Alexeev | On K-stability of reductive varieties[END_REF] to the author's knowledge, and it built on the extensive study of reductive varieties in [START_REF] Alexeev | Stable reductive varieties. I. Affine varieties[END_REF][START_REF] Alexeev | Stable reductive varieties. II. Projective case[END_REF]. Homogeneous toric bundles have been studied for the Kähler-Einstein metric existence problem by Podestà and Spiro [START_REF] Podestà | Kähler-Ricci solitons on homogeneous toric bundles[END_REF], their point of view on the Kähler geometry of this subclass of horospherical varieties being somewhat different from the author's. Donaldson highlighted in [START_REF] Simon | Kähler geometry on toric manifolds, and some other manifolds with large symmetry[END_REF] the importance and studying these varieties, and there were partly unpublished work of Raza and Nyberg on these subjects in their PhD theses [START_REF] Raza | Scalar curvature and multiplicity-free actions[END_REF][START_REF] Raza | An application of Guillemin-Abreu theory to a non-Abelian group action[END_REF][START_REF] Nyberg | Constant Scalar Curvature of Toric Fibrations[END_REF]. Finally, concerning the application to the Mabuchi functional, we were strongly influenced by Li, Zhou and Zhu's article [START_REF] Li | K-energy on polarized compactifications of Lie groups[END_REF]. The latter in turn used as foundations on one side our work on group compactifications and on the other side a strategy for obtaining coercivity of the Mabuchi functional developed initially by Zhou and Zhu [START_REF] Zhou | Relative K-stability and modified K-energy on toric manifolds[END_REF]. It should be noted that the criterion we obtain for (non-semi-simple) group compactifications is a priori not equivalent to the one given in [START_REF] Li | K-energy on polarized compactifications of Lie groups[END_REF]. We do not claim that ours is better but only that theirs did not generalize naturally to our broader setting.

The paper is organized as follows. Section 2 is devoted to the introduction of horosymmetric homogeneous spaces, and of the combinatorial data associated to them. In Section 3, we introduce the toric potential of a K-invariant metric on a G-linearized line bundle on a horosymmetric homogeneous space, and compute the curvature form of such a metric in terms of this function. Even though the proof is rather technical, involving a lot of Lie bracket computations, it is a central part of the theory to have this precise expression. Theorem 1.1 is Corollary 3.11, a special case of Theorem 3.10. In Section 4, we switch to horosymmetric varieties, we recall their combinatorial classification inherited from the theory of spherical varieties, and we check that a G-invariant irreducible codimension one subvariety remains horosymmetric. Section 5 presents the combinatorial data associated with line bundles on horosymmetric varieties, and in particular the link between several convex polytopes associated to such a line bundle. Section 6 applies the previous sections to Hermitian metrics on polarized horosymmetric varieties, to obtain the behavior of toric potentials and an integration formula. In particular, Theorem 1.2 is proved here (Proposition 6.3 and Proposition 6.9). Finally, we give in Section 7 the application to the Mabuchi functional, starting with a computation of the scalar curvature, then of the Mabuchi functional, to arrive to a coercivity criterion. Theorem 1.3 and Theorem 1.4 are proved in this final section (respectively in Theorem 7.5 and Theorem 7.10). Corollary 1.5 follows from Theorem 1.4 and Appendix B, while Corollary 1.6 follows from Corollary 7.13 and Appendix C.

We tried to illustrate all notions by simple examples (even if they sometimes appear trivial, we believe they are essential to make the link between the theory of spherical varieties and standard examples of complex geometry) and to follow for the whole paper the example of symmetric varieties of type AIII.

Acknowledgements. The several referees for this article deserve special thanks for their valuable comments and corrections. I would also like to warmly thank Chinh Lu who always provided very relevant answers to my questions on several aspects of the variational approach to existence of canonical metrics. The main part of this work was accomplished while I was an FSMP Postdoctoral researcher hosted at the École Normale Supérieure in Paris.

Horosymmetric homogeneous spaces

In this section we introduce horosymmetric homogeneous spaces and their associated combinatorial data, extracting from the literature the results needed for the next sections.

Definition and examples.

We always work over the field C of complex numbers. Given an algebraic group G, we denote by G u its unipotent radical.

A complex algebraic group G is called reductive if G u is trivial. A subgroup L of G is called a Levi subgroup of G if it is a reductive subgroup of G such that G is isomorphic
to the semidirect product of G u and L. There always exists a Levi subgroup, and any two Levi subgroups are conjugate by an element of G u .

From now on and for the whole paper, G will denote a connected, reductive, complex, linear algebraic group. Recall that a parabolic subgroup of G is a closed subgroup P such that the corresponding homogeneous space G/P is a projective manifold, called a generalized flag manifold. Recall also for later use that a Borel subgroup of G is a parabolic subgroup which is minimal with respect to inclusion. Note that any parabolic subgroup of G contains at least one Borel subgroup of G.

Definition 2.1. A closed subgroup H of G is a horosymmetric subgroup if there exists a parabolic subgroup P of G, a Levi subgroup L of P and a complex algebraic group involution σ of L such that • P u ⊂ H ⊂ P and • (L σ ) 0 ⊂ L ∩ H as a finite index subgroup, where L σ denotes the subgroup of elements fixed by σ and (L σ ) 0 its neutral connected component.

Remark 2.2. The condition of being horosymmetric may be read off directly from the Lie algebra of H. As a convention, we denote the Lie algebra of a group by the same letter, in fraktur gothic lower case letter. Then H is horosymmetric if and only if there exists a parabolic subgroup P , a Levi subgroup L of P , and a complex Lie algebra involution σ of l such that h = p u ⊕ l σ From now on, H will denote a horosymmetric subgroup, and P , L, σ will be as in the above definition. We keep the same notation σ for the induced involution of the Lie algebra l. We will also say that G/H is a horosymmetric homogeneous space.

Note that L ∩ H ⊂ N L (L σ ), and we have the following description of N L (L σ ), due to De Concini and Procesi. They assume in their paper that G is semisimple but the proof applies to reductive groups just as well.

Proposition 2.3 ([DP83]

). The normalizer N L (L σ ) is equal to the subgroup of all g such that gσ(g) -1 is in the center of L.

In particular if L = G is semisimple, then N L (L σ )/(L σ ) 0 is finite. Note also that if in addition L is adjoint, N L (L σ ) = L σ and if L is simply connected, then L σ is connected.
Example 2.4. Trivial examples of horosymmetric subgroups are obtained by setting σ = id L . Then H = P is a parabolic subgroup and G/H is a generalized flag manifold. Since we will use them later, let us recall a fundamental example of flag manifold: the Grassmannian Gr r,m of r-dimensional linear subspaces in C m , under the action of SL m . The stabilizer of a point is a proper, maximal (with respect to inclusion) parabolic subgroup of SL m (for 1 ≤ r ≤ m -1).

Example 2.5. Assume that G = (C * ) n , then P = L = G. If we consider the involution defined by σ(g) = g -1 , which is an honest complex algebraic group involution since G is abelian, we obtain {e} ⊂ H ⊂ {±1} n and in any case G/H ≃ (C * ) n . Hence a torus may be considered as a horosymmetric homogeneous space.

Let [L, L] denote the derived subgroup of L and Z(L) the center of L. Then L is a semidirect product of these two subgroups, which means, at the level of Lie algebras, that l = [l, l] ⊕ z(l). Note that any involution of L preserves this decomposition.

Example 2.6. A closed subgroup of G is called horospherical if it contains the unipotent radical of a Borel subgroup of G.

Assume that the involution σ of L restricts to the identity on [l, l]. Then H contains the unipotent radical of any Borel subgroup contained in P . Hence H is horospherical.

Conversely, if H is a horospherical subgroup of G, then taking P := N G (H) which is a parabolic subgroup of G, and letting L be any Levi subgroup of P , we have h = p u ⊕ [l, l] ⊕ r where r = h ∩ z(l) (see [START_REF] Pasquier | Variétés horosphériques de Fano[END_REF]Section 2]). Choose any complement c of r in z(l), and consider the involution of l defined as id on [l, l] ⊕ r and as -id on c. This shows that H is a horosymmetric subgroup of G.

Example 2.7. Consider the linear action of SL 2 on C 2 \ {0}. It is a transitive action and the stabilizer of (1, 0) is the unipotent subgroup B u of the Borel subgroup formed by upper triangular matrices. Under this action, C 2 \ {0} is a horospherical, hence horosymmetric, homogeneous space. Alternatively, one may consider the action of GL 2 instead of the action of SL 2 .

Example 2.8. Assume P = L = G, then σ is an involution of G, and (G σ ) 0 ⊂ H ⊂ N G ((G σ ) 0 ). Such a subgroup is simply called a symmetric subgroups and the associated homogeneous spaces is a (complex reductive) symmetric spaces.

All horosymmetric homogeneous spaces may actually be considered as parabolic inductions from symmetric spaces. Let us recall the definition of parabolic induction.

Definition 2.9. Let G and L be two reductive algebraic groups, then we say that a G-variety X is obtained from an L-variety Y by parabolic induction if there exists a parabolic subgroup P of G, and an surjective group morphism P → L such that X = G * P Y is the G-homogeneous fiber bundle over G/P with fiber Y .

In our situation, G/H admits a natural structure of G-homogeneous fiber bundle over G/P , with fiber P/H. The action of P on P/H factorizes by P/P u and under the natural isomorphism L ≃ P/P u , identifies the fiber with the L-variety L/L∩H, which is a symmetric homogeneous space. Conversely, any parabolic induction from a symmetric space is a horosymmetric homogeneous space. The special case of horospherical homogeneous spaces consists of parabolic inductions from tori.

We will denote by f the G-equivariant map G/H → G/P and by π the quotient map G → G/H.

Let us now give more explicit examples of horosymmetric homogeneous spaces, starting by examples of symmetric spaces.

Example 2.10. Assume g = sl m for some m. Then there are three families of group involutions of g up to conjugation [GW09, Sections 11.3.4 and 11.3.5]. For a nicer presentation we work on the group G = SL m . For an integer p > 0, we define the 2p × 2p block diagonal matrix T p by T 1 = ( 0 1 -1 0 ), and T p = diag(T 1 , . . . , T 1 ). For an integer 0 < r < m/2, we define the m × m matrix J r as follows. Let S r be the r × r matrix with coefficients (δ j+k,r+1 ) j,k , and set

J r =   0 0 S r 0 I m-2r 0 S r 0 0   .
The types of involutions are the following:

(1) (Type AI(m)) Consider the involution of G defined by σ(g) = (g t ) -1 where • t denotes the transposition of matrices. Then G σ = SO m . The symmetric space G/N G (G σ ) may be identified with the space of non-degenerate quadrics in P m-1 , equipped with the action of G induced by its natural action on P m-1 . (2) (Type AII(p)) Assume m = 2p is even. Let σ be the involution defined by σ(g) = T p (g t ) -1 T t p . Then G σ = Sp 2p is the group of elements that preserve the non-degenerate skew-symmetric bilinear form ω(u, v) = u t T p v on C 2p .

(3) (Type AIII(r, m)) Let σ be the involution g → J r gJ r . Then G σ is conjugate to the subgroup S(GL r × GL m-r ).

The space G/G σ may be considered as the set of pairs

(V 1 , V 2 ) of linear subspaces V j ⊂ C m of dimension dim(V 1 ) = r, dim(V 2 ) = m -r, such that V 1 ∩ V 2 = {0}
. This is an (open dense) orbit for the diagonal action of G on the product of Grassmannians Gr r,m × Gr m-r,m .

Example 2.11. Let us illustrate the characterization of the normalizer of a symmetric subgroup in type AIII case. First, since G = SL m is simply connected, G σ is connected. Furthermore, it is easy to check here that

N G (G σ ) is different from G σ if and only if m is even and r = m/2, in which case G σ is of index two in N G (G σ ). For example, if m = 2 and r = 1, N G (G σ ) is generated by G σ and diag(i, -i). In that situation, G/N G (G σ ) is the space of unordered pairs {V 1 , V 2 } of linear subspaces V j ⊂ C m of dimension r for j = 1 and m -r for j = 2, such that V 1 ∩ V 2 = {0}.
Example 2.12. Finally, let us give an explicit example of non trivial parabolic induction from a symmetric space. Consider the subgroup H of SL 3 defined as the set of matrices of the form

  a b 0 b a 0 e f g   .
Then obviously H is contained in the parabolic P composed of matrices with zeroes where the general matrix of H has zeroes, and contains its unipotent radical, which consists of the matrices as above with a = g = 1 and b = 0. The subgroup L = S(GL 2 × C * ) is then a Levi subgroup of P and L ∩ H is the subgroup of elements of L fixed by the involution g → M gM where

M =   0 1 0 1 0 0 0 0 1   .
2.2. Root systems. Example 2.13. The ranks and maximal tori for involutions of SL m are as follows.

Maximally split torus. A torus

T in L is split if σ(t) = t -1 for any t ∈ T . A torus T in L is maximally split if T is a σ-stable
• (Type AI(m)) For σ : g → (g t ) -1 , the rank is m -1 and the torus T of diagonal matrices is a split torus which is also maximal, hence T s = T in this case. • (Type AII(p)) For σ : g → T p (g t ) -1 T t p , with m = 2p, the rank is p -1, and the torus of diagonal matrices provides a maximally split torus. The maximal split subtorus T s is then the subtorus of diagonal matrices of the form diag(a 1 , a 1 , a 2 , a 2 , . . . , a p , a p ) with a 1 , . . . , a p-1 ∈ C * and a p = (a 2 1 • • • a 2 p-1 ) -1 , and T σ is the subtorus of diagonal matrices of the form diag(a 1 , a -1 1 , a 2 , a -1 2 , . . . , a p , a -1 p ) with a 1 , . . . , a p ∈ C * . We record for later use that σ(diag(a 1 , . . . , a n )) = diag(a -1 2 , a -1 1 , a -1 4 , . . . , a -1 m-1 ). • (Type AIII(r, m) Finally, for σ : g → J r gJ r , the rank is r, and the torus T of diagonal matrices is again maximally split. Let υ denote the permutation of {1, . . . , m} defined by

υ(i) = m + 1 -i if 1 ≤ i ≤ r or m + 1 -r ≤ i ≤ m,
and υ(i) = i otherwise. Then σ acts on diagonal matrices as σ(diag(a 1 , . . . , a m )) = diag(a υ(1) , . . . , a υ(m) ).

We then see that the subtorus T σ is the torus of diagonal matrices of the form diag(a 1 , a 2 , . . . , a m-r , a r , a r-1 , . . . , a 1 ) and that T s is the subtorus of diagonal matrices of the form diag(a 1 , . . . , a r , 1, . . . , 1, a -1 r , . . . , a -1 1 ). 2.2.2. Root systems and Lie algebras decompositions. We denote by X(T ) the group of characters of T , that is, algebraic group morphisms from T to C * . We denote by Φ ⊂ X(T ) the root system of (G, T ). Recall the root space decomposition of g:

g = t ⊕ α∈Φ g α , g α = {x ∈ g; Ad(t)(x) = α(t)x ∀t ∈ T }
where Ad denotes the adjoint representation of G on g.

Example 2.14.

In our examples we concentrate on the case when G = SL m , and the root system is of type A m-1 . Let us recall its root system with respect to the maximal torus of diagonal matrices, in order to fix the notations to be used in examples throughout the article. The roots are the group morphisms α j,k : T → C * , for 1 ≤ j = k ≤ m, defined by α j,k (diag(a 1 , . . . , a m )) = a j /a k . The root space g α j,k is then the set of matrices with only one non zero coefficient at the intersection of the j th -line and k th -column.

We denote by Φ L ⊂ Φ the root system of L with respect to T , by Φ P u ⊂ Φ the set of roots of P u , so that

l = t ⊕ α∈ΦL g α , p = l ⊕ α∈Φ P u g α and h = l ∩ h ⊕ α∈Φ P u g α .
Example 2.15. In the case of Example 2.12, G = SL 3 and T is the torus of diagonal matrices. Using notations from Example 2.14, we have Φ = {±α 1,2 , ±α 2,3 , ±α 1,3 },

Φ L = {±α 1,2 }, Φ P u = {-α 1,3 , -α 2,3 }.
2.2.3. Restricted root system. The set of roots in Φ L fixed by σ is a sub root system denoted by Φ σ L . Let Φ s = Φ L \ Φ σ L . Note that Φ s is not a root system in general. Let us now introduce the restricted root system of L/L ∩ H. Given α ∈ Φ L , we set

ᾱ = α -σ(α). It is zero if and only if α ∈ Φ σ L . Proposition 2.16 ([Ric82, Section 4]). The set Φ = { ᾱ; α ∈ Φ s } ⊂ X(T )
is a (possibly non reduced) root system in the linear subspace of X(T ) ⊗ R it generates. The Weyl group W of the root system Φ may be identified with N L (T s )/Z L (T s ) and furthermore any element of W admits a representant in N (L σ ) 0 (T s ).

The root system Φ is called the restricted root system of the symmetric space L/L ∩ H. We will also say that its elements are restricted roots, that W is the restricted Weyl group, etc.

Another interpretation of the restricted root system, which justifies the name, is obtained as follows. For any t ∈ T s , and α ∈ Φ L , we have

σ(α)(t) = α(σ(t)) = α(t -1 ) = (-α)(t).
As a consequence, ᾱ| Ts = 2α| Ts , that is, up to a factor two, ᾱ encodes the restriction of α to T s . More significantly, given γ ∈ X(T s ), let lγ denote the subset of elements x in l such that Ad(t)(x) = γ(t)x for all t ∈ T s . Then by simultaneous diagonalization, we check that l = γ∈X(Ts) lγ . We immediately remark that l0 contains t and all g α for α ∈ Φ σ L , and that lγ contains g α as soon as α ∈ Φ L is such that ᾱ| Ts = 2γ. By the usual root decomposition of l, we check that actually

l0 = t ⊕ α∈Φ σ L g α , lγ = ᾱ|T s =2γ
g α for γ = 0, and l = l0 ⊕ ᾱ∈ Φl ᾱ/2 . A restricted root ᾱ is fully determined by its restriction to T s since T = T s T σ and ᾱ| T σ = 0 since σ(ᾱ) =ᾱ.

Example 2.17. For involutions of SL m , the restricted root systems are as follows.

• In the case of type AI(m), Φ σ L is empty and Φ s = Φ L = Φ. For any α ∈ Φ, we have σ(α) = -α, hence the restricted root system is just the double 2Φ of Φ. • In the case of type AII(p), we check that σ(α j,k ) = α k+(-1) k+1 ,j+(-1) j+1 .

In fact it is easier to identify the restricted root system by analyzing the restriction of roots to T s . We denote an element of

T s by diag(b 1 , b 1 , b 2 , b 2 , . . . , b p , b p ).
We check easily that, for

1 ≤ j = k ≤ p, α 2j,2k-1 | Ts = α 2j-1,2k-1 | Ts = α 2j-1,2k | Ts = α 2j,2k | Ts = b j /b k .
We deduce that Φ σ L = {±α 2j-1,2j ; 1 ≤ j ≤ p} and that the restricted root system is of type A p-1 , with elements ᾱ2j,2k :

diag(b 1 , b 1 , b 2 , b 2 , . . . , b p , b p ) → b 2 j /b 2 k for 1 ≤ j = k ≤ p.
• In the case of type AIII(r, m), finally, we will also identify the root system via restriction to T s . We will denote an element of T s , which is a diagonal matrix, by diag(b 1 , . . . , b r , 1 . . . , 1, b -1 r , . . . , b -1 1 ). In the case when m = 2r, there are no 1 in the middle and the restricted root system will be slightly different. In general, the restriction α j,k | Ts is trivial if and only if r + 1 ≤ j = k ≤ mr, which proves Φ σ L is the subsystem formed by these roots. Since α j,k = -α k,j , it is obviously enough to consider only the case when

j < k. For 1 ≤ j < k ≤ r, we have α j,k | Ts = α m-k+1,m-j+1 | Ts = b j /b k . For 1 ≤ j ≤ r and r + 1 ≤ k ≤ m -r, we have α j,k | Ts = α m-k+1,m-j+1 | Ts = b j . Finally, for 1 ≤ j ≤ r and m + 1 -r ≤ k ≤ m, we have α j,k | Ts = α m-k+1,m-j+1 | Ts = b j b m+1-k .
Remark that in this last case, we may have α m-k+1,m-j+1 = α j,k , namely when j = m + 1k. In this situation we obtain the function b 2 j . Hence, whenever r + 1 ≤ mr, or equivalently r < m/2 since both r and m are integers, the restricted root system is non reduced. It is possible to check that it is of type BC r . In the remaining case, that is when m = 2r, the restricted root system is of type C r .

Cartan involution and fundamental domain.

There always exists a Cartan involution of G such that its restriction to L commutes with σ. We fix such a Cartan involution θ. Denote by K = G θ the corresponding maximal compact subgroup of G. Let a s denote the Lie subalgebra t s ∩ ik of t s .

Consider the group Y(T s ) of one-parameter subgroups of T s , that is, algebraic group morphisms C * → T s . This group naturally embeds in a s : given λ ∈ Y(T s ), it induces a Lie algebra morphism d e λ : C → t s . Here we identified the Lie algebra of C * with C and the exponential map is given by the usual exponential. Then d e λ(1) must be an element of a s and it determines λ completely. This induces an injection of Y(T s ) in a s which actually allows to identify a s with Y(T s ) ⊗ R.

Recall that we may either consider the restricted root system Φ as in X(T ), in which case it lies in the subgroup X(T /T ∩ H), or, via the restriction to T s , we may consider Φ to be in X(T s ). This allows to define a Weyl chambers in a s with respect to the restricted root system. Choose any such Weyl chamber, denote it by a + s and call it the positive restricted Weyl chamber.

Proposition 2.18. The natural map a s → exp(a s )H/H is injective, and the intersection of a K-orbit in G/H with exp(a s )H/H is the image by this map of a W -orbit in a s . As a consequence, the subset exp(a + s )H/H is a fundamental domain for the action of K on G/H. Proof. Remark that K acts transitively on the base G/P of the fibration f : G/H → G/P , since P is parabolic. We are then reduced to finding a fundamental domain for the action of K ∩ P = K ∩ L on the fiber L/L ∩ H.

Flensted-Jensen proves in [FJ80, Section 2] that a fundamental domain is given by the positive Weyl chamber of a root system which is in general different from the restricted root system described above. However, in our situation, the group L and the involution σ are complex, and this allows to show that the two chambers are the same.

More precisely, Flensted-Jensen considers the subspace l ′ of elements fixed by the involution σθ. The positive Weyl chamber he considers is then a positive Weyl chamber for the root system formed by the non zero eigenvalues of the action of ad(a s ) on l ′ . Now remark that the involution σθ stabilizes any of the subspaces lᾱ/2 , which we may decompose as lᾱ/2 = l′ ᾱ/2 ⊕ l′′ ᾱ/2 where l′ ᾱ/2 = lᾱ/2 ∩ l ′ and l′′ ᾱ/2 is the subspace of elements x such that σθ(x) = -x. Furthermore, since σ(it) = iσ(t), multiplication by i induces a bijection between l′ ᾱ/2 and l′′ ᾱ/2 , and in particular l′

ᾱ/2
is not {0} if and only if so is lᾱ/2 . As a consequence, the set of non zero eigenvalues of the action of ad(a s ) on l ′ is precisely Φ.

The reader may find a more detailed account of the results of Flensted-Jensen and of the structure of the action of K on G/H in [vdB05, Section 3].

2.4.

Colored data for horosymmetric homogeneous spaces. As a parabolic induction from a symmetric space, H is a spherical subgroup of G, that is, any Borel subgroup of G acts with an open dense orbit on G/H (see [START_REF] Brion | Variétés sphériques[END_REF][START_REF] Perrin | On the geometry of spherical varieties[END_REF][START_REF] Timashev | Homogeneous spaces and equivariant embeddings[END_REF][START_REF] Knop | The Luna-Vust theory of spherical embeddings[END_REF] for general presentations of spherical homogeneous spaces, and spherical varieties which will appear later).

Given a choice of Borel subgroup B, a spherical homogeneous space G/H is determined by three combinatorial objects (the highly non-trivial theorem that these objects fully determine H up to conjugacy was obtained by Losev [START_REF] Losev | Uniqueness property for spherical homogeneous spaces[END_REF]).

• The first one is its associated lattice M, defined as the subgroup of characters χ ∈ X(B) such that there exists a function

f ∈ C(G/H) with b • f = χ(b)f for all b ∈ B (where b • f (x) = f (b -1 x) by definition). Let us call M the spherical lattice of G/H. Let N = Hom Z (M, Z) denote the dual lattice.
• The second one, the valuation cone V, is defined as the set of elements of N ⊗ Q which are induced by the restriction of G-invariant, Q-valued valuations on C(G/H) to B-semi-invariant functions as in the definition of M. • Finally, the third object needed to characterize the spherical homogeneous space G/H is the color map ρ : D → N , as a map from an abstract finite set D to N , that is, we only need to know the image of ρ and the cardinality of its fibers. The set D is actually the set of codimension one B-orbits in G/H, called colors, and the map ρ is obtained by associating to a color D the element of N induced by the divisorial valuation on C(G/H) defined by D.

In the case of horosymmetric spaces (which are parabolic inductions from symmetric spaces) these data may mostly be interpreted in terms of the restricted root system for a well chosen Borel B. The choice of Borel subgroup is as follows.

First, for the case of the symmetric space L/(L ∩ H), a nice Borel subgroup is provided by: Lemma 2.19. [DP83, Lemma 1.2] There exists a Borel subgroup of L containing T , with corresponding positive roots Φ + L in Φ L , so that for any positive root

α ∈ Φ + L , either σ(α) = α or -σ(α) is in Φ +
L . Now for the horosymmetric space we let Q denote the parabolic subgroup of G opposite to P with respect to L, that is, the only parabolic subgroup of G such that

Q ∩ P = L and L is also a Levi subgroup of Q. First choose any Borel subgroup B ′ of G such that T ⊂ B ′ ⊂ Q. Then B ′ ∩ L is a Borel subgroup of L. Since Borel subgroups of L containing T are conjugate by an element of N L (T ) ⊂ Q, we can choose an element q ∈ N L (T ) such that B = qB ′ q -1 is a still a Borel subgroup satisfying T ⊂ B ⊂ Q and furthermore Φ +
L satisfies the conclusions of the above Lemma.

We fix such a Borel subgroup and denote by Φ + the corresponding positive root system of Φ. We will use the notations

Φ + L = Φ + ∩ Φ L and Φ + s := Φ + L ∩ Φ s . Note also that Φ P u = -Φ + \ Φ L and Φ Q u = -Φ P u . Let S denote the set of simple roots of Φ generating Φ + , and let S L = Φ L ∩ S, S s = Φ s ∩ S.
This induces a natural choice of simple roots in the restricted root system: S = { ᾱ; α ∈ S s }, and corresponding positive roots Φ+ = { ᾱ; α ∈ Φ + s }. Given α ∈ Φ, recall that the coroot α ∨ is defined as the unique element in [g, g]∩t such that for all x ∈ t, α(x) = 2κ(x, α ∨ )/κ(α ∨ , α ∨ ) where κ denotes the Killing form on g. Since α is real on t ∩ ik, the coroot α ∨ is in a = t ∩ ik which we may also identify with Y(T ) ⊗ R.

Example 2.20. In our favorite group SL m , the coroot α ∨ j,k is the diagonal matrix with l th -coefficient equal to δ l,jδ l,k .

We use also the notion of restricted coroots for the restricted root, as defined in [Vus90, Section 2.3]: Definition 2.21. Given α ∈ Φ s the restricted coroot ᾱ∨ is defined as:

• α ∨ /2 if -σ(α) = α (α is then called a real root), • (α ∨ -σ(α ∨ ))/2 = (α ∨ + (-σ(α)) ∨ )/2 if σ(α)(α ∨ ) = 0, • (α -σ(α)) ∨ if σ(α)(α ∨ ) = 1, in which case α -σ(α) ∈ Φ s .
The restricted coroots form a root system dual to the restricted root system, and we thus call simple restricted coroots the basis of this root system corresponding to the choice of positive roots ᾱ∨ for α ∈ Φ + s . One has to be careful here: in general the simple restricted coroots are not the coroots of simple restricted roots.

Example 2.22. Consider the example of type AIII(2, m > 4). Then we already described the restricted root system in Example 2.17. There are two real roots α 1,m and α 2,m-1 . The restricted coroots are diagonal matrices of the form

diag(b 1 , b 2 , 0, . . . , 0, -b 2 , -b 1 )
and we write this more concisely as a point with coordinates (b 1 , b 2 ). The restricted coroot ᾱ∨ 1,m is then (1/2, 0), while ᾱ∨ 2,m-1 = (0, 1/2). The roots α 1,2 and α 1,m-1

Figure 1. Roots and coroots of type AIII(2, m > 4)

• ᾱ1,3 ᾱ2,3 ᾱ1,m 2 ᾱ2,3 = ᾱ2,m-1 ᾱ1,m-1 ᾱ1,2 • ᾱ∨ 1,m ᾱ∨ 2,m-1 ᾱ∨ 1,3 ᾱ∨ 2,3 ᾱ∨ 1,m-1 ᾱ∨ 1,2 satisfy σ(α)(α ∨ ) = 0, hence we have ᾱ∨ 1,2 = (1/2, -1/2) and ᾱ∨ 1,m-1 = (1/2, 1/2).
Finally, the roots α 1,3 and α 2,3 satisfy σ(α)(α ∨ ) = 1, and we have

α 1,3 -σ(α 1,3 ) = α 1,m and α 2,3 -σ(α 2,3 ) = α 2,m-1 , hence ᾱ∨ 1,3 = (1, 0
) and ᾱ∨ 2,3 = (0, 1). We have described that way all (positive) restricted coroots. Figure 1 illustrates the positive restricted roots and coroots in this example.

Recall that f : G/H → G/P denotes the fibration map. The set of colors D(G/P ) of the generalized flag manifold G/P is in bijection with the set Φ Q u ∩ S of simple roots that are also roots of Q u , and any pre-image by f of a color of G/P is a color of G/H. Denote by D α the color of G/P associated with the root

α ∈ Φ Q u ∩ S.
We identify a s with Y(T /T ∩ H) ⊗ R.

Proposition 2.23 ([Tim11, Proposition 20.4] and [START_REF] Vust | Plongements d'espaces symétriques algébriques: Une classification[END_REF]). Assume that G/H is a horosymmetric space. Then

• the spherical lattice M is the lattice X(T /T ∩ H),

• the valuation cone V is the negative restricted Weyl chamber -a + s , • the set of colors may be decomposed as a union of two sets

D = D(L/L ∩ H) ∪ f -1 D(G/P ). The image of the color f -1 (D α ) by ρ is the restriction α ∨ | M of the coroot α ∨ for α ∈ Φ Q u ∩ S. The image ρ(D(L/L ∩ H))
on the other hand is the set of simple restricted coroots.

Remark 2.24. If G = L is semisimple and simply connected, then M is a lattice between the lattice of restricted weights and the lattice of restricted roots determined by the restricted root system [START_REF] Vust | Plongements d'espaces symétriques algébriques: Une classification[END_REF]. More precisely, it is the lattice of restricted weights if and only if H = G σ and it is the lattice of restricted roots if and only if

H = N G (G σ ).
Remark that the proposition does not give here a complete description of ρ in general as it does not give the cardinality of all orbits. There is however a rather general case where the discussion is simply settled. Say that the symmetric space [START_REF] Vust | Plongements d'espaces symétriques algébriques: Une classification[END_REF]). Assume that L/L∩H has no Hermitian factor. Then the color map ρ is injective on D(L/L ∩ H).

L/L ∩ H has no Hermitian factor if [L, L] ∩ Z L (L ∩ H) is finite. Then Vust proved the following full characterization of ρ: Proposition 2.25 ([
Note, and this is a general fact for parabolic inductions, that the images of colors in f -1 D(G/P ) by ρ all lie in the valuation cone V. Indeed, for any two simple roots

• α and β, κ(α, β) ≤ 0. Given α ∈ Φ Q u ∩ S, this implies that κ(α, β) ≤ 0 for any β ∈ Φ +
L and thus κ(α, β) ≤ 0 for β ∈ S s . Example 2.26. We draw here Figure 2 as an example of colored data for the symmetric space of type AIII(2, m > 4). Here the color map is not injective, but is described in details in [Vus90, Section 6.1]. The dotted grid represents the dual of the spherical lattice (which coincides here with the lattice generated by restricted coroots), the cone delimited by the dashed rays represents the valuation cone (the negative restricted Weyl chamber), and the circles are centered on the points in the image of the color map (the simple restricted coroots), the number of circles reflecting the cardinality of the fiber.

Curvature forms

We now begin the study of Kähler geometry on horosymmetric spaces. We first recall how linearized line bundles on homogeneous spaces are encoded by their isotropy characters, then we consider K-invariant Hermitian metrics. We associate two functions to a Hermitian metric: the quasipotential and the toric potential. We express the curvature form of the metric in terms of the isotropy character and toric potential, using the quasipotential as a tool in the proof.

For this section, we use the letter q to denote a metric, as the letter h denotes elements of the group H. Recall that given a Hermitian metric q on a line bundle L, its curvature form ω may be defined locally as follows. Let s be a local trivialization of L and let ϕ denote the function defined by ϕ =ln |s| 2 q . Then the curvature form is the globally defined form which satisfies locally ω = i∂ ∂ϕ.

Linearized line bundles on horosymmetric homogeneous spaces.

Let L be a G-linearized line bundle on G/H. The pulled back line bundle π * L on G is trivial, and we denote by s a G-equivariant trivialization of π * L on G. Denote by χ the character of H defined by h • ξ = χ(h)ξ for any ξ in the fiber L eH . It fully determines the G-linearized line bundle L. The line bundle is trivializable on G/H if and only if χ is the restriction of a character of G.

Example 3.1. The anticanonical line bundle admits a natural linearization, induced by the linearization of the tangent bundle. We may determine the isotropy character χ from the isotropy representation of H on the tangent space at eH. If one identify this tangent space with g/h, then, working at the level of Lie algebras, the isotropy representation is given by h

• (ξ + h) = [h, ξ] + h for h ∈ h, ξ ∈ g.
Taking the determinant of this representation, we obtain for a horosymmetric homogeneous space G/H that the isotropy Lie algebra character for the anticanonical line bundle is the restriction of the character α∈Φ Q u α of p to h.

Example 3.2. On a Hermitian symmetric space, there may be non-trivial line bundles on G/H, as there may exist characters of H which are not restrictions of characters of G. Let us illustrate this with our favorite type AIII example. Consider the matrix

M r =   1 √ 2 I r 0 1 √ 2 S r 0 I m-2r 0 -1 √ 2 S r 0 1 √ 2 I r   so that M r J r M -1 r = I m-r 0 0 -I r , then M r HM -1 r = S(GL m-r × GL r ).
This group obviously has non trivial characters not induced by a character of the (semisimple) group G, for example

( A 0 0 D ) → det(D). Write an element of H as h =   A 11 A 12 A 13 A 21 A 22 A 21 S r S r A 13 S r S r A 12 S r A 11 S r  
then composing with conjugation by M we obtain the non-trivial character

χ : h → det(S r A 11 S r -S r A 13 ).
Example 3.3. Consider the simplest example of type AIII, that is, P 1 ×P 1 \diag(P 1 ) equipped with the diagonal action of SL 2 , and with base point

([1 : 1], [-1 : 1]).
Then we have naturally linearized line bundles given by the restriction of O(k, m) for k, m ∈ N. The character associated to the line bundle O(k, m) is χ m-k with χ as above, which translates here as χ : ( a b b a ) → ab. In particular we recover that it is trivial if and only if k = m.

3.2. Quasipotential and toric potential. Let q be a smooth K-invariant metric on L. Definition 3.4.

• The quasipotential of q is the function φ on G defined by

φ(g) = -2 ln |s(g)| π * q .
• The toric potential of q is the function u : a s → R defined by

u(x) = φ(exp(x)).
Proposition 3.5. The function φ satisfies the following equivariance relation:

φ(kgh) = φ(g) -2 ln |χ(h)|,
for any k ∈ K, g ∈ G and h ∈ H. In particular φ is fully determined by u.

Proof. First, by G-invariance of s, we have

φ(kgh) = -2 ln |kgh • s(e)| π * q = -2 ln |k • g • h • π * s(e)| q by equivariance of π = -2 ln |g • χ(h)π * (s(e))| q by K-invariance of q and by definition of χ = -2 ln |g • s(e)| π * q -2 ln |χ(h)|
Hence the equivariance relation.

Recall from Proposition 2.18 that any K-orbit on G/H intersects the image of a s , so in view of the equivariance formula for φ, we see that φ, hence q, is fully determined by u.

3.3.

Reference (1, 0)-forms. We choose root vectors 0 = e α ∈ g α for α ∈ Φ, such that [θ(e α ), e α ] = α ∨ . In other words, the triples (α ∨ , e α , -θ(e α )) are sl 2 -triples. Using these root vectors, we can give a more explicit decomposition of h:

h = α∈Φ P u Ce α ⊕ t σ ⊕ α∈Φ σ L Ce α ⊕ α∈Φ + s C(e α + σ(e α ))
Choose a basis (l 1 , . . . , l r ) of the real vector space a s . Let us further add the vectors e α for α ∈ Φ Q u and τ β = e βσ(e β ), for β ∈ Φ + s . Then we obtain a family which is the complex basis of a complement of h in g. This also defines local coordinates

g exp   j z j l j + α z α e α + β z β τ β   H near a point gH in G/H, depending on the choice of g. Let γ g ♦ denote the element of Ω (1,0)
gH G/H defined by these coordinates, where ♦ is either some j, some α or some β. Then x → γ exp(x) ♦ provides an exp(a s )-invariant smooth (1, 0)-form on exp(a s )H/H (note that it is well defined since by Proposition 2.18, x → exp(x)H is injective on a s ). From now on we denote by γ ♦ the corresponding (1, 0)-form and by ω ♦, ♥ the (1, 1)-form iγ ♦ ∧ γ♥ .

3.4. Reference volume form and integration. We introduce a reference volume form on G/H. Recall from Example 3.1 that the naturally linearized canonical line bundle K L/L∩H on the symmetric space L/L∩H is L-trivial up to passing to a finite tensor power. For simplicity, we ignore this finite tensor power in the following. The general case follows by considering multisections instead of sections. From now on, we assume that there exists a nowhere vanishing section s 0 : L/L ∩ H → K L/L∩H which is L-equivariant. We can further assume that s 0 coincides with j γ j ∧ β γ β on exp(a s )H/H where j runs from 1 to r and β runs over the set

Φ + s . Recall that f denotes the map G/H → G/P . Let K f = K G/H -f * K G/P
denote the relative canonical bundle. Then the section s 0 above may be considered as a trivialization of K f on the fiber above eP ∈ G/P . Since the map f is G-equivariant, K f admits a natural G-linearization, and we may use the action of the maximal compact group K to build a K-equivariant trivialization s f of K f on G/H. Setting |s f | q f = 1 provides a smooth K-invariant metric q f on K f . Let q P denote the smooth K-invariant metric on K G/P which satisfies |f * ( α γ e α )| qP = 1, where α runs over the set Φ Q u . Pulling it pack provides a smooth K-invariant metric on f * K G/P .

The two metrics together provide a smooth reference metric q H = q f ⊗ f * q P on K G/H = K f ⊗ f * K G/P , which is K-invariant. We denote by dV H the associated smooth volume form on G/H. It is defined point-wise as follows: if ξ is an element of the fiber of K G/H at gH, then (dV

H ) gH = i n 2 |ξ| -2 qH ξ ∧ ξ.
Proposition 3.6. Let a ∈ a s , then

(dV H ) exp(a)H = e 2 α∈Φ Q u α(a) ♦ ω ♦, ♦ exp(a)H
Proof. At a point exp(a)H for a ∈ a s , we can choose

ξ = ♦ γ exp(a) ♦ = exp(-a) * • ♦ γ e ♦ ,
and we get

(dV H ) exp(a)H = |ξ| -2 qH i n 2 ξ ∧ ξ = | exp(-a) * • α γ e α | -2 f * qP i n 2 ξ ∧ ξ by definition of q H and q f , = | exp(-a) * • f * ( α γ e α )| -2 qP i n 2 ξ ∧ ξ = e 2 α α(a) i n 2 ξ ∧ ξ because P acts on the fiber at eP of K G/P via the character -α∈Φ Q u α, = e 2 α α(a) i n (-1) n(n-1)/2 ξ ∧ ξ = e 2 α α(a) i n ♦ γ ♦ ∧ γ♦ = e 2 α α(a) ♦ ω ♦, ♦ by definition.
Remark that dV H depends on the precise choice of basis of the complement of h in g only by a multiplicative constant, as it only changes the element of the fiber of K G/P at eP where q P takes value one, and the element of the fiber of K f at eH where q f takes value one.

Combining fiber integration with respect to the fibration f , and the formula for integration on symmetric spaces from [FJ80, Theorem 2.6], we obtain a formula that reduces integration of a K-invariant function on G/H with respect to dV G/H to integration of its restriction to exp(a + s ) with respect to an explicit measure. Let J H denote the function on a s defined by

J H (x) = α∈Φ + s | sinh(2α(x))|.
Another possible expression of the function J H is:

J H (x) = ᾱ∈ Φ+ | sinh(ᾱ(x)) m ᾱ |
where m ᾱ = dim( lᾱ/2 ) is the number of β ∈ Φ s such that β = ᾱ. The function J H will be given explicitly for some examples following the main result of the current section (see e.g. Example 3.14).

Proposition 3.7. There exists a constant C H > 0 such that for any K-invariant function ψ on G/H which is integrable with respect to dV H , we have

G/H ψdV H = C H a + s ψ(exp(x)H)J H (x)dx
where dx is a fixed Lebesgue measure on a s .

Note that one could choose to integrate on any restricted Weyl chamber (we will later integrate over the negative restricted Weyl chamber). In these situations, the absolute values in the definition of J H are important.

Here again, a more detailed account on the integration formula for symmetric spaces may be found in [vdB05, Section 3].

3.5. Preparation for curvature form. To shorten the formulas, we start using the following notations, for y ∈ g,

ℜ(y) = y -θ(y) 2 ∈ ik and ℑ(y) = y + θ(y) 2 ∈ k.
For y ∈ l, we will also use the notations

H(y) = y + σ(y) 2 ∈ h and P(y) = y -σ(y) 2 .
Remark that τ β = 2P(e β ) and define µ β = 2H(e β ).

Lemma 3.8. Let a ∈ a s be such that β(a) = 0 for all β ∈ Φ s . Consider an element D in g and write

D = 1≤j≤r z j l j + α∈Φ Q u z α e α + β∈Φ + s z β τ β + h
where h ∈ h, and z j for 1 ≤ j ≤ r, z α for α ∈ Φ Q u , and z β for β ∈ Φ + s denote complex numbers. Then we may write

D = A D + B D + C D with A D ∈ Ad(exp(-a))(k), B D ∈ a s and C D ∈ h as follows. A D = 1≤j≤r ℑ(z j l j ) + exp(ad(-a)) β∈Φ + s ℑ(z β τ β ) cosh(β(a)) - ℑ(z β µ β ) sinh(β(a)) + α∈Φ Q u 2e α(a) ℑ(z α e α ) B D = 1≤j≤r ℜ(z j l j ) C D = h + β∈Φ + s tanh(β(a))ℜ(z β µ β ) + coth(β(a))ℑ(z β µ β ) + α∈Φ Q u -e 2α(a) θ(z α e α )
Proof. This is a straightforward rewriting, using the following relations. For α ∈

Φ Q u , exp(ad(-a))(z α e α + θ(z α e α )) = e α(-a) z α e α + e -α(-a) θ(z α e α )
where we remark that z α e α + θ(z α e α ) ∈ k and θ(z α e α ) ∈ h. For the terms in τ β , we use the relations

z β τ β = ℜ(z β τ β ) + ℑ(z β τ β ), exp(ad(-a))(ℑ(z β τ β )) = cosh(β(a))ℑ(z β τ β ) -sinh(β(a))ℜ(z β µ β ), exp(ad(-a))(ℑ(z β µ β )) = cosh(β(a))ℑ(z β µ β ) -sinh(β(a))ℜ(z β τ β ).
Note that the relations hold because a ∈ a s , hence σ(β)(a) = -β(a).

Let a ∈ a s be such that β(a) = 0 for all β ∈ Φ s , and consider now the function

D = D(z) = 1≤j≤r z j l j + α∈Φ + P z α e α + β∈Φ + L \Φ σ L z β τ β ,
where z denotes the tuple obtained by merging the tuples (z j ) j , (z α ) α and (z β ) β . Let A D , B D , C D be the elements provided by Lemma 3.8 applied to D. Let

E = E(z) := ([B D , D] + [C D , B D ] + [C D , D])/2
and introduce also A E , B E , C E the elements provided by Lemma 3.8 applied to E.

Lemma 3.9. For small enough values of z, we have

exp(D) = exp(-a)k exp(a + y + O) exp(h),
where O = O(z) ∈ g is of order strictly higher than two in z, k = k(z) ∈ K, y = y(z) ∈ a s , and h = h(z) ∈ h. Furthermore,

y = B D + B E and exp(h) = exp(C E ) exp(C D ).
Proof. Throughout the proof, O denotes an element of g for z small enough, of order strictly higher than two in z, which may change from line to line. We first write D = A D + B D + C D , with A D ∈ Ad(exp(-a))(k), B D ∈ a s and C D ∈ h given by Lemma 3.8. Remark that they are all of order one in z

Using the Baker-Campbell-Hausdorff formula [Hoc65, Theorem X.3.1] twice, we obtain that

exp(-A D ) exp(D) exp(-C D ) = exp(B D + 1 2 ([C D , B D ]+ [C D , A D ]+ [B D , A D ])+ O).
Writing

A D = D-B D -C D we easily check that 1 2 ([C D , B D ]+[C D , A D ]+[B D , A D ]
) is equal to the E introduced before. We may then decompose again E as

A E + B E + C E where A E ∈ Ad(exp(-a))(k), B E ∈
a s and C E ∈ h given by Lemma 3.8 and all terms are of order two in z. Using again the Baker-Campbell-Hausdorff formula, we get

exp(D) = exp(A D ) exp(A E ) exp(B D + B E + O) exp(C E ) exp(C D ).
In view of the space where A D and A E live, we can write exp(A D ) exp(A E ) = exp(-a)k exp(a) for some k ∈ K which is the one involved in the statement of the lemma. A final application of the Baker-Campbell-Hausdorff formula to exp(a) exp(B D + B E + O) yields the result since a commutes with B D and B E , and any bracket involving at least once O remains negligible.

3.6. Expression of the curvature form. Given a function u : a s → R we may consider its differential d a u ∈ a * s at a given point a ∈ a s as an element of a * by setting d a u(x) = d a u • P(x) and identifying a * s with X(T /T ∩ H) ⊗ R. Let L be a G-linearized line bundle on G/H corresponding to the character χ of H. We also denote by χ the corresponding Lie algebra character h → C. Hoping it will cause no confusion, we will also denote by χ the restriction of χ to a ∩ h and consider it as an element of a * by setting χ(x) = χ • H(x) for x ∈ a.

Let q be a smooth K-invariant metric on L with toric potential u, and let ω denote the curvature form of q.

Theorem 3.10. Let a ∈ a s be such that β(a) = 0 for all β ∈ Φ s . Then

ω exp(a)H = Ω ♦, ♥ω ♦, ♥
where the sum runs over the indices j, α, β, and the coefficients are as follows.

Let

1 ≤ j, j 1 , j 2 ≤ r, α, α 1 , α 2 ∈ Φ Q u , and β, β 1 , β 2 ∈ Φ + s with β 1 = β 2 and α 2 -α 1 ∈ Φ s , then Ω j1, j2 = 1 4 d 2 u(l j1 , l j2 ), Ω j, β = 1 2 β(l j )(1 -tanh 2 (β))χ(θ(µ β )), Ω α, ᾱ = -e 2α 2 (du -2χ)(α ∨ ), Ω α1, ᾱ2 = 2χ([θ(e α2 ), e α1 ]) e -2α1 + e -2α2
,

Ω β1, β2 = tanh(β 2 -β 1 ) 2 1 sinh(2β 1 ) - 1 sinh(2β 2 ) χ([θ(e β2 ), e β1 ]) + tanh(β 1 + β 2 ) 2 1 sinh(2β 2 ) + 1 sinh(2β 1 )
χ([θ(e β2 ), σ(e β1 )])

and

Ω β, β = du(β ∨ ) sinh(2β) - 2 cosh(2β) χ • ℜ([θσ(e β ), e β ])
where all quantities are evaluated at a. Finally, the remaining coefficients except obviously the symmetric of those above are zero.

This very involved description drastically simplifies if the restriction of χ to L ∩ H is trivial on [L, L] ∩ H. It is equivalent to the fact that it coincides with the restriction of a character of L to L ∩ H, or also to the fact that the corresponding line bundle is trivial on the symmetric fiber L/L ∩ H. This particular case in fact covers a wealth of examples, as it is the case for any choice of line bundle whenever the symmetric fiber has no Hermitian factor. In the Hermitian case there are still plenty of line bundle which satisfy this extra assumption. A remarkable example is the anticanonical line bundle.

Corollary 3.11. Assume that the restriction of L to the symmetric fiber L/L ∩ H is trivial. Let a ∈ a s be such that β(a) = 0 for all β ∈ Φ s . Then ω exp(a)H may compactly be written as

1 4 d 2 a u(l j1 , l j2 )ω j1, j2 + -e 2α 2 (d a u -2χ)(α ∨ )ω α, ᾱ + d a u(β ∨ ) sinh(2β(a)) ω β, β
where summands are implicitly taken over {1, . . . , r}, Φ Q u and Φ + s respectively.

Example 3.12. Consider the example of C 2 \ {0}, viewed as a horospherical space under the natural action of SL 2 . Then a s is one-dimensional and we may choose

Φ Q u = {α 2,1 } (Φ +
s is obviously empty). We choose l 1 = α ∨ 2,1 as basis of a s and consider u as a one real variable function, writing d a u = u ′ (y)α 2,1 for a = yl 1 , so that d 2 a u(l 1 , l 1 ) = 4u ′′ (y). Then since H = Stab(1, 0) has no characters, we have at exp(yl 1 )H,

ω = u ′′ (y)ω 1, 1 + e -4y u ′ (y)ω α2,1, ᾱ2,1 .
One major application of this general computation of curvature forms, but not the only one, will be through the Monge-Ampère operator, which reads, with respect to the reference volume form, as follows.

Corollary 3.13. Assume that the restriction of L to the symmetric fiber L/L ∩ H is trivial. Let a ∈ a s be such that β(a) = 0 for all β ∈ Φ s . Then at exp(a)H,

ω n /dV H is equal to n! 2 2r+|Φ Q u | det(((d 2 a u)(l j , l k )) j,k ) J H (a) α∈Φ Q u (2χ -d a u)(α ∨ ) β∈Φ + s |d a u(β ∨ )|
Example 3.14. Consider the example of symmetric space of type AIII(2, m > 4).

We choose as basis l 1 , l 2 the basis dual to (ᾱ 1,2 , ᾱ2,3 ). We write a = a 1 l 1 + a 2 l 2 and d a u = u 1 (a)ᾱ 1 + u 2 (a)ᾱ 2 . We check easily that

P(α ∨ 1,2 ) = ᾱ∨ 1,2 , P(α ∨ 1,m-1 ) = ᾱ∨ 1,m-1 , P(α ∨ 1,m ) = ᾱ∨ 1,3 , P(α ∨ 2,m-1 ) = ᾱ∨ 2,3 , P(α ∨ 1,k ) = ᾱ∨ 1,m and P(α ∨ 2,k ) = ᾱ∨ 2,m-1 for 3 ≤ k ≤ m -2.
Hence we may compute, under the assumption that χ is zero, that at exp(a)H, ω n /dV H is equal to n!/2 times

(u 1,1 u 2,2 -u 2 1,2 )(2u 1 -u 2 ) 2 u 2m-7 1 u 2 2 (u 2 -u 1 ) 2m-7 sinh(a 1 ) 2 sinh(a 1 + a 2 ) 2m-8 sinh(2a 1 + 2a 2 ) sinh(a 1 + 2a 2 ) 2 sinh(a 2 ) 2m-8 sinh(2a 2 )
Let us now illustrate on examples how the other terms in the curvature form may appear.

Example 3.15. Consider again P 1 ×P 1 \diag(P 1 ) equipped with the diagonal action of SL 2 , and with the linearized line bundle O(k, m). Then we can take β = α 1,2 , e β = ( 0 1 0 0 ) and l 1 = β ∨ = ( 1 0 0 -1 ). We may further consider u as a function of a single variable t by writing a = ( t 0 0 -t ) and we get

ω exp(a)H = u ′′ (t) 4 ω 1, 1 + (k -m)(1 -tanh 2 (2t))(ω 1, β + ω β, 1) + u ′ (t) sinh(4t) ω β, β
Example 3.16. Consider the symmetric space of type AIII(1, 3). It admits the non-trivial character χ : (a i,j ) → a 1,1 + a 1,3 . For a metric on the line bundle corresponding to this character, we get for example

d a u(α ∨ 1,3 ) sinh(2α 1,3 (a)) - 1 cosh(2α 1,3 (a))
.

Example 3.17. Consider the symmetric space G/G σ of type AIII(2, 4). It admits the non-trivial character χ : (a i,j ) → a 1,1 + a 2,2 + a 2,3 + a 1,4 , at the Lie algebra level and for (a i,j ) ∈ h. For a metric on the line bundle corresponding to this character, we have for example [θ(e α3,4 ), e α2,4 ] = e α2,3 , [θ(e α3,4 ), σ(e α2,4 )] = -e α4,1 and χ(e α2,3 ) = χ • H(e α2,3 ) = 1/2, χ(-e α4,1 ) = χ • H(-e α4,1 ) = -1/2, hence, writing a = diag(t 1 , t 2 , -t 2 , -t 1 ) we have

Ω α2,4, ᾱ3,4 = -1 2 cosh(2t 1 ) cosh(2t 2 )
.

Example 3.18. Consider again Example 2.12. Using the same notations, we have a non-trivial character χ which associates a + b to any element of H. In this case, since [θ(e α2,3 ), e α1,3 ] = e α1,2 , and χ(e α1,2 ) = χ • H(e α1,2 ) = 1/2, we have

Ω α1,3, ᾱ2,3 = 1 2 cosh(2t)
at the point exp(diag(t, -t, 0))H. We check also that

Ω α2,3, ᾱ2,3 = e 2t (2 -u ′ (t))/4.
The previous examples show that any of the terms written in Theorem 3.10 may be non-zero.

Proof of Theorem 3.10. Step 1

Recall that π denotes the quotient map G → G/H. By definition of the quasipotential φ : G → R of q, i∂∂φ is the curvature form of π * q. Furthermore, this curvature form coincides with π * ω.

Let f ♦ ∈ g be any of the elements l j , e α or τ β for 1

≤ j ≤ r, α ∈ Φ Q u or β ∈ Φ + s . Identifying g with T (1,0) e G, we build a global G-invariant (1, 0) holomorphic vector fields η ♦ by setting (η ♦ ) g = g * f ♦ ∈ T (1,0) g G. Then π * ω g (η ♦ , η♥ ) = i ∂ 2 ∂z ♦ ∂ z♥ 0 φ(g exp(z ♦ f ♦ + z ♥ f ♥ )).
By definition, the set of all direct images π * η ♦ at exp(a)H provides a basis of T

(1,0) exp(a)H G/H which coincides with the dual basis to the basis formed by the

(γ ♦ ) exp(a)H in Ω (1,0) exp(a)H G/H. We thus have Ω ♦, ♥ = -iω exp(a)H (π * η ♦ , π * η♥ ) = -i(π * ω) exp(a) (η ♦ , η♥ ) = ∂ 2 ∂z ♦ ∂ z♥ 0 φ(exp(a) exp(z ♦ f ♦ + z ♥ f ♥ )).
Step 2 We may now write

Set D = z ♦ f ♦ + z ♥ f ♥ .
Ω ♦, ♥ = ∂ 2 ∂z ♦ ∂ z♥ 0 φ(exp(a + B D + B E + O)) -2 ln |χ(exp(C E ) exp(C D ))| = ∂ 2 ∂z ♦ ∂ z♥ 0 φ(exp(a + B D + B E )) -2 ln |χ(exp(C E ) exp(C D ))| = ∂ 2 ∂z ♦ ∂ z♥ 0 (u(a + B D + B E ) -2 Re(χ(C E + C D )).
Note that here the term O denotes terms of order strictly higher than two in (z ♦ , z ♥ ), which become negligible in our computation. Actually, other terms will be negligible and we will now denote by O a sum of terms (which may change from line to line) each with a factor among

z 2 ♦ , z2 ♦ , z ♦ z♦ , z 2 ♥ , z2 ♥ , z ♥ z♥ , z ♦ z ♥ or z♦ z♥ . Step 3
The case by case computation follows. 1) Consider the case D = z 1 l j1 + z 2 l j2 , then we have

B D = (z 1 + z1 )l j1 /2 + (z 2 + z2 )l j2 /2 and B E = C E = C D = 0 hence Ω j1, j2 = 1 4 d 2 a u(l j1 , l j2 ).
2) Consider the case D = z 1 e α + z 2 l j . By Lemma 3.8, we have B D = ℜ(z 2 l j ) and a) θ(e α ) + O.

C D = -e 2α(a) θ(z 1 e α ). We now compute E = ([B D , D] -[B D , C D ] + [C D , D])/2: 2[B D , D] = O -z 1 z2 α(θ(l j ))e α , 2[B D , C D ] = z 2 z1 α(l j )e 2α(a) θ(e α ) + O, [C D , D] = -z 2 z1 α(l j )e 2α(a) θ(e α ) + O, hence E = 1 4 z 1 z2 α(l j )e α - 3 4 z 2 z1 α(l j )e 2α(
Using Lemma 3.8 again we check that B E = O is negligible and a) θ(e α ) + O.

C E = -z 2 z1 α(l j )e 2α(
Since θ(e α ) is in the Lie algebra of the unipotent radical of H, we have χ(θ(e α )) = 0, we may thus end the computation and obtain

Ω j, ᾱ = Ω α, j = 0.
3) Consider the case D = z 1 e α1 + z 2 e α2 . We have B D = 0 and

C D = -e 2α1(a) θ(z 1 e α1 ) -e 2α2(a) θ(z 2 e α2 )
.

Then E = [C D , D]/2 is equal to E = O -e 2α1(a) z1 z 2 [θ(e α1
), e α2 ]/2e 2α2(a) z 1 z2 [θ(e α2 ), e α1 ]/2

We then need to treat several cases separately, depending on

α 1 -α 2 . 3.i) If α 1 = α 2 = α then we have E = - 1 2 e 2α(a) (z 1 z 2 + z 1 z2 )α ∨ + O hence B E = - 1 2 e 2α(a) (z 1 z 2 + z 1 z2 )P(α ∨ ) + O and C E = - 1 2 e 2α(a) (z 1 z 2 + z 1 z2 )H(α ∨ ) + O.
Since χ is trivial on the unipotent radical of H, we have

Re(χ(C D + C E )) = Re(χ(C E )) = χ • ℜ(C E ) = χ(C E ).
We then end the computation to obtain

Ω α, ᾱ = -1 2 e 2α(a) (d a u(α ∨ ) -2χ(α ∨ )) 3.ii) If α 2 -α 1 ∈ Φ σ L then we get B E = O and C E = O+E. Furthermore, we check easily that [θ(e α1 ), e α2 ] ∈ g α2-α1 ⊂ [h, h] (consider [[θ(e α2-α1
), e α2-α1 ], e α2-α1 ]) so χ([θ(e α1 ), e α2 ]) = 0, and the same holds for [θ(e α2 ), e α1 ]. We can then end the computation and obtain

Ω α1, ᾱ2 = 0. 3.iii) If α 2 -α 1 ∈ Φ L \ Φ σ L then
+ tanh((α 1 -α 2 )(a))ℜ(-e 2α2(a) z2 z 1 H([θ(e α2 ), e α1 ])/2)) + coth((α 1 -α 2 )(a))ℑ(-e 2α2(a) z2 z 1 H([θ(e α2 ), e α1 ])/2)).
As a consequence,

ℜ(C E ) =O -e 2α1(a) ℜ(z 1 z 2 H([θ(e α1 ), e α2 ]))/2 -e 2α2(a) ℜ(z 1 z2 H([θ(e α2 ), e α1 ]))/2 + tanh((α 2 -α 1 )(a))ℜ(-e 2α1(a) z1 z 2 H([θ(e α1 ), e α2 ])/2)) + tanh((α 1 -α 2 )(a))ℜ(-e 2α2(a) z2 z 1 H([θ(e α2 ), e α1 ])/2)).
We then check by computation that

Ω α1, ᾱ2 = 1 2 e 2α2(a) (1 + tanh((α 1 -α 2 )(a))) + e 2α1(a) (1 + tanh((α 2 -α 1 )(a))) χ • H([θ(e α2 ), e α1 ]) = 2χ([θ(e α2 , e α1
]) (e -2α1(a) + e -2α2(a) ) .

3.iv) If α

2 -α 1 ∈ Φ \ Φ L , say α 1 -α 2 ∈ Φ P u for example, then B E = O and C E = O + -e 2α2(a) 2 z2 z 1 [θ(e α2 ), e α1 ]
e 2(α2-α1)(a) θ( -e 2α1(a) 2 z1 z 2 [θ(e α1 ), e α2 ]).

Since [θ(e α2 ), e α1 ] is in the Lie algebra of the unipotent radical of H we have the vanishing χ([θ(e α2 ), e α1 ]) = 0 hence

Ω α1, ᾱ2 = 0. 3.v) Finally, if α 2 -α 1 / ∈ Φ,
C D = tanh(β(a))ℜ(z 2 µ β ) + coth(β(a))ℑ(z 2 µ β ).
We compute

[B D , D] = O + z1 z 2 β(l j )µ β /2 [C D , B D ] =O + z 1 z2 4 β(l j )(coth(β(a)) -tanh(β(a)))θ(τ β ) - z1 z 2 4 β(l j )(tanh(β(a)) + coth(β(a))τ β and [C D , D] = O + z 1 z2 2 β(l j )(coth(β(a) -tanh(β(a))θ(τ β ).
From these computations we deduce

E =O + z1 z 2 β(l j ) 4 (µ β - tanh(β(a) + coth(β(a)) 2 τ β ) + z 1 z2 3β(l j ) 8 (coth(β(a)) -tanh(β(a)))θ(τ β )).
We then have B E = O and

ℜ(C E ) = O + β(l j ) 4 ℜ(z 1 z 2 µ β ) + tanh(β(a))ℜ(z 1 z 2 -β(l j ) 8 (tanh(β(a)) + coth(β(a)))µ β ) + tanh(-β(a))ℜ(z 1 z2 3β(l j ) 8 (coth(β(a)) -tanh(β(a)))θ(µ β )) = O + β(l j )(1 -tanh 2 (β(a)))ℜ(z 1 z 2 µ β )/2 since ℜ(z 1 z2 θ(µ β )) = -ℜ(θ(z 1 z2 θ(µ β )) = -ℜ(z 1 z 2 µ β ).
We may thus finish the computation to obtain .

Ω j, β = β(l j )(1 -tanh 2 (β(a)))χ(θ(µ β ))/2.
Then E = [C D , D]/2, which is equal to z 1 z2 2 e 2α(a) [τ β , θ(e α )] + -z 1 z 2 4 (coth(β(a)) -tanh(β(a)))[e α , θ(µ β )] + O.
We then remark that [τ β , θ(e α )] ∈ g -α+β ⊕ g -α+σ(β) and [e α , θ(µ β )] ∈ g α-β ⊕ g α-σ(β) . It is impossible for -α + β as well as for -α + σ(β) to lie in Φ L (write these roots in a basis of simple roots adapted to P to check this assertion). Hence we obtain that C E is a sum of a negligible term and a term in g -α+β ⊕ g -α+σ(β) ⊕ g α-β) ⊕ g α-σ(β) ∩ h. Since this last space is contained in the Lie algebra of the unipotent radical of H, we obtain that χ • ℜ(C E ) is negligible, hence

Ω β, ᾱ = 0.
6) Consider finally the case D = z 1 τ β1 + z 2 τ β2 . Then we have B D = 0 and

C D = tanh(β 1 (a))ℜ(z 1 µ β1 ) + coth(β 1 (a))ℑ(z 1 µ β1 ) + tanh(β 2 (a))ℜ(z 2 µ β2 ) + coth(β 2 (a))ℑ(z 2 µ β2 ).
Then in view of the relation coth(x)tanh(x) = 2/ sinh(2x), we have

E = z2 z 1 [θ(µ β2 ), τ β1 ] 2 sinh(2β 2 (a)) + z1 z 2 [θ(µ β1 ), τ β2 ] 2 sinh(2β 1 (a)) + O
We separate in two cases the end of the computation.

6.i) If β 1 = β 2 then, note that for 1 ≤ j = k ≤ 2, we have [θ(µ β k ), τ βj ] = 2P([θ(e β k ), e βj ]) + 2P([θσ(e β k ), e βj ])
where We may rewrite this as

tanh(β 2 -β 1 )( 1 sinh(2β 1 (a)) - 1 sinh(2β 2 (a)) )ℜ(z 1 z 2 H([θ(e β1 ), e β2 ])) + tanh(β 1 + β 2 )( 1 sinh(2β 1 (a)) + 1 sinh(2β 2 (a)) )ℜ(z 1 z 2 H([θσ(e β1
), e β2 ])).

We compute now

Ω β1, β2 = 1 2 tanh(β 2 (a) -β 1 (a))( 1 sinh(2β 1 (a)) - 1 sinh(2β 2 (a)) )χ([θ(e β2 ), e β1 ]) + 1 2 tanh(β 1 + β 2 )( 1 sinh(2β 1 (a)) + 1 sinh(2β 2 (a))
)χ([θ(e β2 ), σ(e β1 )])

6.ii) If β 1 = β 2 = β then we have E = z 1 z2 + z1 z 2 sinh(2β) (P(β ∨ ) + P([θσ(e β ), e β ]))
and thus

B E = z 1 z2 + z1 z 2 sinh(2β) P(β ∨ )
and

ℜ(C E ) = z 1 z2 + z1 z 2 sinh(2β) tanh(β -σ(β))ℜ • H([θσ(e β ), e β ]) = z 1 z2 + z1 z 2 cosh(2β) H • ℜ([θσ(e β ), e β ]).
Hence

Ω β, β = d a u(β ∨ ) sinh(2β) - 2 cosh(2β) χ • ℜ([θσ(e β ), e β ]).

Horosymmetric varieties

We move on to introduce horosymmetric varieties. We provide several examples and present the classification theory inherited from that of spherical varieties. We then check the property that a G-invariant irreducible codimension one subvariety in a horosymmetric variety is still horosymmetric. Example 4.2. By Example 2.6, any horospherical variety (see [START_REF] Pasquier | Variétés horosphériques de Fano[END_REF]) may be considered as a horosymmetric variety. It includes in particular generalized flag manifolds, toric varieties and homogeneous toric bundles.

Example 4.3. Consider the projective plane P 2 equipped with the action of SL 2 or GL 2 induced by a choice of affine chart C 2 in P 2 . There are three orbits under this action: the fixed point {0}, the open dense orbit C 2 \ {0} and the projective line at infinity P 1 . The GL 2 -variety P 2 is hence a horospherical variety by Example 2.7. We may further consider the blow up of P 2 at the fixed point {0} and lift the action of GL 2 to check that this blow up is also a horospherical variety. More generally, Hirzebruch surfaces have structures of GL 2 -horospherical varieties refining their toric structure.

Assume G = L is semisimple and H = N G (G σ ). Then the wonderful compactifications of G/H constructed by De Concini and Procesi [START_REF] De | Complete symmetric varieties[END_REF] is a horosymmetric variety. It is a particularly nice compactification of G/H characterized by the following properties. Let r denote the rank of G/H and set I = {1, . . . , r}.

Theorem 4.4 ([DP83]

). The wonderful compactification X of G/H is the unique smooth G-equivariant compactification of X such that:

• G-orbit closures X J in X are in bijection with subsets J ⊂ I and • all X J are smooth and intersect transversely, with X J = j∈J X {j} . Furthermore, for each J, there exists a parabolic subgroup P J of G, with σ-stable semisimple Levi factor L ′ J , and an equivariant fibration X J → G/P J with fiber the wonderful compactification of L ′ J /N L ′ J ((L ′ J ) σ ). Example 4.5. Consider the symmetric space G/H of type AIII(2, m > 4). Recall from Example 2.11 that H = N G (H). Using the description of G/H as a dense orbit in the product of Grassmannians X 0 = Gr 2,m × Gr m-2,m as in Example 2.10, we obtain a first example of (horo)symmetric variety with open orbit G/H: this product of Grassmannians X 0 itself. It contains three orbits under the action of G = SL m : the dense orbit of pairs of linear subspaces in direct sum, the closed orbit of flags

(V 1 , V 2 ) with V 1 ⊂ V 2 ,

and the codimension one orbit of pairs

(V 1 , V 2 ) with dim(V 1 ∩ V 2 ) = 1.
One can blow up X 0 along the closed orbit to obtain another G-equivariant compactification X of G/H. This new compactification X is none other than the wonderful compactification of G/H.

In higher ranks, one may still obtain the wonderful compactifications from the product of Grassmannians, but this requires a more involved sequence of blow-ups.

From Theorem 4.4, one sees the first examples of horosymmetric varieties that are neither horospherical nor symmetric. Indeed, the description of orbits in a wonderful compactification show that orbit closures in wonderful compactifications are all horosymmetric, with the only symmetric being the open orbit and the only horospherical being the closed one (actually a generalized flag manifold).

Since it applies only to H = N G (G σ ), the construction of De Concini and Procesi does not exhaust the compactifications of symmetric spaces satisfying the properties of Theorem 4.4, still called wonderful compactifications. The simplest example of wonderful compactification which is not in the examples studied by De Concini and Procesi is the following. Example 4.6. Consider the variety P 1 × P 1 equipped with the diagonal action of SL 2 . There are two orbits under this action: the diagonal embedding of P 1 and its complement. The complement is open dense and isomorphic to the symmetric space SL 2 /T where T is a maximal torus of SL 2 .

The wonderful compactification constructed by De Concini and Procesi and corresponding to this involution on the other hand is P 2 seen as a compactification of SL 2 /N SL 2 (T ) by adding a quadric. Example 4.7. Several papers expanded results valid on the wonderful compactification of a symmetric space to so-called complete symmetric varieties (see e.g. [DP85, Bif90]), that is, smooth G-equivariant compactifications of G/H that dominate the wonderful compactification. Any complete symmetric variety is a horosymmetric variety.

Combinatorial description of horosymmetric varieties.

Colored fans. As spherical varieties, horosymmetric varieties with open orbit

G/H are classified by colored fans for the spherical homogeneous space G/H, which are defined in terms of the combinatorial data M, V and ρ : D → N (Recall these were described in Proposition 2.23). Definition 4.8.

• A colored cone is a pair (C, R), where R ⊂ D, 0 / ∈ ρ(R), and C ⊂ N ⊗ Q is a strictly convex cone generated by ρ(R) and finitely many elements of V such that the intersection of the relative interior of C with V is not empty.

• Given two colored cones (C, R) and (C 0 , R 0 ), we say that

(C 0 , R 0 ) is a face of (C, R) if C 0 is a face of C and R 0 = R ∩ ρ -1 (C 0 ). • A colored
fan is a non-empty finite set F of colored cones such that the face of any colored cone in F is still in F , any v ∈ V is in the relative interior of at most one cone, and the intersection of any two colored cones is a face of both.

An equivariant embedding (X, x) of G/H is the data of a horosymmetric variety X and a base point x ∈ X such that G • x = X and Stab G (x) = H. Theorem 4.9 ([Kno91, Theorem 3.3 and Theorem 4.2]). There is a bijection (X, x) -→ F X between embeddings of G/H up to G-equivariant isomorphism and colored fans. There is a bijection Y → (C Y , R Y ) between the orbits of G in X, and the colored cones in F X . An orbit Y is in the closure of another orbit Z in X if and only if the colored cone

(C Z , R Z ) is a face of (C Y , R Y ). The variety X is complete if and only if the support |F X | = (C,R)∈FX C contains the valuation cone V.
There is a corresponding description of equivariant morphisms between embeddings of G/H in terms of the colored fans. Since the only such morphism we will use is the discoloration in Section 6, let us just refer the reader to [Kno91, Theorem 4.1] for a precise statement.

Example 4.10. Assume G is a semisimple group, and H is a symmetric subgroup of G. Then there is a natural choice of colored fan given by the negative Weyl chamber and its faces. If H = N G (H) then the corresponding variety is the wonderful compactification of G/H. More generally if the valuation cone is strictly convex, then the embedding corresponding to the colored fan given by the valuation cone and its faces is called wonderful if it is in addition smooth. There are criterions of smoothness for spherical varieties [START_REF] Brion | Sur la géométrie des variétés sphériques[END_REF], and some simpler criterions for the case of horospherical [START_REF] Pasquier | Variétés horosphériques de Fano[END_REF] and symmetric [Ruz11, Section 3] varieties. It would certainly be possible and useful to derive such a simpler criterion for the class of horosymmetric varieties. In the case of toroidal horosymmetric varieties, which are introduced in the next section, the criterion is very simple, as it is the case for toroidal spherical varieties in general. 4.2.2. Toroidal horosymmetric varieties. Given an embedding (X, x) of G/H we denote by F X its colored fan and we call the elements of D X = (C,R)∈F R ⊂ D the colors of X. It should be noted that the set of colors does not depend on the base point x, but F X does. We however omit this dependence in the notation. Definition 4.11. An embedding is toroidal if D X is empty, else it is colored.

A toroidal horosymmetric variety is globally a parabolic induction from a symmetric variety. More generally, we record the following elementary statement, easily seen by the classification of horosymmetric varieties by colored fan, and the fact that the colored fan of a parabolic induction is the same as the colored fan of the embedding one starts with. Proposition 4.12. A horosymmetric variety with set of colors D X is globally a parabolic induction from a symmetric variety if and only if

D X ∩ f -1 D(G/P ) = ∅.
Example 4.13. The horospherical variety P 2 under the action of SL 2 is not a global parabolic induction (in particular it is not toroidal), but the blow up of P 2 is.

The following result shows that, in a toroidal horosymmetric variety, there is a well identified toric subvariety which will play an important role in later applications.

Proposition 4.14 ([Kno94, Corollary 8.3 and paragraph after Corollary 6.3]). Let (X, x) be a toroidal embedding of the horosymmetric space G/H, with colored fan F X . Then the closure Z of T • x in X is the T /T ∩ H-toric variety whose fan (as a spherical variety) consists of the images, by elements of the restricted Weyl group W , of the cones C in the colored cone F X .

Remark 4.15. We insist here that we obtain the fan of Z as a spherical variety under the action of T /T ∩ H. It does not exactly coincide in general with the fan of Z as a toric variety with the classical conventions, but to the opposite of this fan. We refer to Pezzini [Pez10, Section 2] for details, but the short explanation is that a character λ of a torus T may be interpreted as a regular B = T -semi-invariant function on T with weight -λ: λ(b -1 t) = (-λ)(b)λ(t). This difference is important to get the correct expression for the asymptotic behavior of metrics in Section 6. This fact was overlooked in some previous works [START_REF] Delcroix | Kähler-Einstein metrics on group compactifications[END_REF][START_REF] Delcroix | Log canonical thresholds on group compactifications[END_REF] of the author, fortunately with no serious consequences.

Example 4.16. Assume X is the wonderful compactification of a symmetric space then the fan of its toric subvariety Z is the fan obtained by considering the collection of all restricted Weyl chambers for Φ and their faces.

Finally, let us mention the criterion of smoothness for toroidal horosymmetric varieties: Proposition 4.17 ([Per14, Corollary 3.3.4]). A toroidal horosymmetric embedding (X, x) is smooth if and only if the toric subvariety Z is smooth, that is, if and only if every cone in the colored fan is generated by a subset of a basis of N .

Facets of a horosymmetric variety.

Definition 4.18. Let X be a horosymmetric variety under the action of G. A facet of X is a G-stable irreducible codimension one subvariety in X.

The goal of this section is to prove the following result.

Proposition 4.19. Let X be a horosymmetric variety under the action of G, then any facet of X is also a horosymmetric variety under the action of G.

We will actually obtain more precise statements describing the corresponding horosymmetric homogeneous spaces, using [START_REF] Brion | Vers une généralisation des espaces symétriques[END_REF]. Let us first introduce some terminology.

Definition 4.20. An elementary embedding (E, x) of G/H is an embedding such that the complement of G/H in E is a single codimension one orbit of G. Equivalently, it is an embedding whose colored fan consists of a single ray C E ⊂ -a + s with no colors.

Elementary embeddings are in bijection with indivisible one parameter subgroups in -a + s ∩ Y(T s ) by selecting the only such one parameter subgroup in the ray associated to the elementary embedding. Given an indivisible µ ∈ -a + s ∩ Y(T s ) we denote by (E µ , x) the corresponding elementary embedding. Furthermore x µ := lim z→0 µ(z) • x exists in E µ and is in the open B-orbit of the codimension one Gorbit [BP87, Section 2.10]. We will use [START_REF] Brion | Vers une généralisation des espaces symétriques[END_REF] to describe the Lie algebra of the isotropy subgroup of x µ .

We have fixed since Section 2 a maximal torus T of G and a Borel subgroup B containing T . Recall that parabolic subgroups containing B are classified by subsets of the set of simple roots S. More precisely, given a subset I ⊂ S, there is a unique parabolic subgroup

Q I of G containing B such that Φ Q u I ∩ S = S \ I. It
σ µ (zµ + m) = zµ + σ(m).
This is a well defined involution of Lie algebras thanks to our choice of complement m.

The following proposition provides a more precise version of Proposition 4.19. Indeed, given a facet Y of X, the union of the open G-orbits in X and Y form an elementary embedding of G/H. Proposition 4.21. Let µ ∈ Y(T s ) indivisible. Then the isotropy subgroup H µ of x µ is horosymmetric as follows:

h µ = p u I(µ) ⊕ l σµ I(µ) .
Proof. An elementary embedding is toroidal, hence, by Proposition 4.12, it is a global parabolic induction E L ֒→ E µ → G/P where (E L , x) is the elementary embedding of L/L ∩ H associated with the same one parameter subgroup µ. Since x µ ∈ E L , we obtain that P u ⊂ H µ ⊂ P and H µ is determined by L ∩ H µ .

We now use the results of [START_REF] Brion | Vers une généralisation des espaces symétriques[END_REF] applied to the case of symmetric spaces to obtain a description of l ∩ h µ . By Proposition 2.4 in [START_REF] Brion | Vers une généralisation des espaces symétriques[END_REF] and the remarks about the case of symmetric spaces in Section 2.2 of the same paper, we have

l ∩ h µ = Cµ ⊕ t σ ⊕ α∈Φ σ L Ce α ⊕ α∈Φ + s ; ᾱ(µ)=0 C(e -α + σ(e -α )) ⊕ α∈Φ + s ; ᾱ(µ) =0 Ce -α .
Since µ ∈ Y(T s ), we have ᾱ(µ) = 2α(µ), so we may write the above expression as

l ∩ h µ = l σµ I(µ) ⊕ α∈Φ P u I(µ) ∩ΦL g α
Putting both results together, we get

h µ = l σµ I(µ) ⊕ α∈Φ P u I(µ)
g α hence the statement.

Example 4.22. Consider the symmetric space of type AIII(2, m > 4). Take µ ∈ -a + s an indivisible one parameter subgroup. Then there are three possibilities for I(µ): we have

I(ᾱ ∨ 1,m ) = S \ {α 1,2 , α m-1,m }, I(ᾱ ∨ 2,m-1 ) = S \ {α 2,3
, α m-2,m-1 }, and I(µ) = S in the other cases. In the first situation, [l I(µ) , l I(µ) ] is isomorphic to the simple Lie algebra sl m-2 and the induced involution is still of type AIII, but now of rank one. In the second situation, [l I(µ) , l I(µ) ] splits as a direct sum of three summands, one fixed by σ and the other two, each isomorphic to sl 2 , exchanged by σ. Finally, in the third scenario, the isotropy group is in fact horospherical.

Remark 4.23. It is in fact very general that if the ray is in the interior of the valuation cone, then the closed orbit in the corresponding elementary embedding is horospherical, with an explicit description of its Lie algebra [BP87, Proposition 3.10].

Remark 4.24. Gagliardi and Hofscheier [START_REF] Gagliardi | Homogeneous spherical data of orbits in spherical embeddings[END_REF] obtained a description of the combinatorial data associated to any orbit in a spherical variety. We could in principle (neglecting the difficulty of describing the color map in general) have used this to show that facets of horosymmetric varieties are horosymmetric. However their result identifies only the conjugacy class of the isotropy subgroup, while we will need the precise knowledge of the isotropy group (actually Lie algebra) of a specific point in the orbit. On the other hand, it is possible to identify precisely the isotropy group of the point we consider, and not only its Lie algebra, by combining our result with that of [START_REF] Gagliardi | Homogeneous spherical data of orbits in spherical embeddings[END_REF]. This could be used to fully recover the description of orbit closures in wonderful compactifications given by De Concini and Procesi. We mention here, as it could be useful for other applications in Kähler geometry, that the work of Gagliardi and Hofscheier [START_REF] Gagliardi | Homogeneous spherical data of orbits in spherical embeddings[END_REF] further allows to identify the colored fan of a facet of a horosymmetric variety.

Linearized line bundles on horosymmetric varieties

In this section, we consider G-linearized line bundles on a G-horosymmetric variety X. We explain how to associate to such a line bundle L a privileged Binvariant Q-divisor, and several convex polytopes. For example, one can associate to L its (algebraic) moment polytope, and in the case X is toroidal, the moment polytope of the restriction of L to the toric subvariety. We determine the relations between these polytopes, and illustrate these notions on some examples, including the anticanonical line bundle.

Note that if G is simply connected, all line bundles on X are G-linearized [KKV89, KKLV89]. Else, if L is not linearized, there exists a tensor power L k which is linearized. Finally, recall that any two linearizations of the same line bundle differ by a character of G.

5.1. Special function associated to a linearized line bundle. Let X be a horosymmetric embedding of G/H. Let L be a G-linearized line bundle on X. The action of G on L induces an action of G on meromorphic sections of L, given explicitly by

(g • s)(x) = g • (s(g -1 • x)).
D s = Y ∈I G (X) v s (µ Y )Y + D∈DX v s (ρ(D))D + D∈D\DX n D D
for some integers n D and a piecewise linear integral function v s on the fan F X .

In general there may not be any privileged B-semi-invariant meromorphic section of L. Given such a section s, with associated weight λ s ∈ X(B), the others are obtained by multiplying by eigenvectors f for the action of B on C(G/H), with eigenvalue an element λ f of M. Assume however that λ s | Ts ∈ X(T s ) is induced by an element of M under the epimorphism X(T s ) → M = X(T /T ∩ H). Then we can choose f so that (λ s λ f )| Ts is trivial.

Definition 5.1. Assume there exists a section s L such that λ s | Ts is trivial. Then we call this section, well defined up to multiplicative scalar, the special section of L, and D L = div(s L ) the special divisor representing L.

In the general case, we may still define D L as a Q-divisor. Indeed, let s be a B-semi-invariant section of L with weight λ. There always exists a tensor power L k of L such that the corresponding multisection s ⊗k has a B-weight kλ whose restriction to T s is induced by an element of M. Thus the previous paragraph defines the special divisor D L k . Definition 5.2. In the situation described above, the special divisor of L is the

Q-divisor D L = D L k /k.
Using the fact that D L k is Cartier, we may write

D L = Y ∈I G (X) v L (µ Y )Y + D∈DX v L (ρ(D))D + D∈D\DX n L,D D.
for some piecewise rational linear function v L on F X .

Definition 5.3. The function v L will be referred to as the special function associated to L.

Remark that it takes non integral values if L admits no special section. Note that all of these notions are relative to the choice of a Borel subgroup B.

Definition 5.4. The unique W -invariant function v t L defined by v t L = v L on -a + s , is the toric special function of L.

Toroidal case.

When X is toroidal, we may identify the restriction of a Glinearized line bundle to the toric subvariety Z in terms of the objects defined in Section 5.1. Proposition 5.5. Let L be a G-linearized line bundle on a toroidal horosymmetric variety X. For a sufficiently divisible integer m, the restriction L m | Z defines a (T /T ∩ H) ⋊ W -linearized toric line bundle on Z, such that the divisor associated with the (T /T ∩ H) ⋊ W -invariant section coincides with

F ∈I T /T ∩H (Z) mv t L (µ F )F.
Proof. The restriction L| Z inherits a linearization of the action of T s as well as of the action of N K (T s )∩H. The line bundle L k for any k divisible enough admits a special section s. By definition of the special section, s is in particular T s ∩ H-invariant, hence T s ∩ H acts trivially on L| Z . Thus the T s -linearization of L k actually comes from a T /T ∩ H-linearization. The section s| Z is obviously T /T ∩ H-invariant.

Let n w be a representant in

N K (T s ) ∩ H of w ∈ W . Then for any t ∈ T s n w • s(n -1 w • t • x) = n w • s(n -1 w tn w • x) since n w ∈ H, = n w n -1 w tn w • s(x) since n ∈ N K (T s ) and s is T s -invariant, = χ(n w )t • s(x)
where χ is the character of H associated to

L k = χ(n w )s(t • x).
Since there is a finite number of n w and they are in K, we may choose k so that χ(n w ) = 1 for all w ∈ W . We now use the local structure of spherical varieties [BP87, Proposition 3.4]. Consider ∆ = D∈D D ⊂ G/H and set U = X \ ∆ and V = Z ∩ U . Then V is the toric subvariety associated to the subfan contained in -a + s , and the toric divisors in V are precisely the Y ∩ V for Y ∈ I G X . By [Bri89, Section 3.2], the restriction of

D L k to V is div(s) ∩ V = Y ∈I G (X) kv L (µ Y )(Y ∩ V ).
By n w -invariance of s, we obtain

div(s) ∩ w • V = Y ∈I G (X) kv L (µ Y )w • (Y ∩ V ).
We deduce that div(s| Z ) =

F ∈I T /T ∩H (Z) kv t L (µ F )F.
Remark that our reasoning with the representatives n w did not endow L k | Z with a W -linearization a priori. However, s| Z is the (up to multiplicative scalar)

T /T ∩ H- invariant section of L k | Z and div(s| Z ) is W -invariant, hence L k | Z admits a natural W -linearization such that w • s| Z = µ(w)
s| Z for all w ∈ W and some character µ : W → S 1 of W . The group W being finite, we may take a multiple m of k such that the character m k µ is trivial, and obtain that the T /T ∩ H-invariant section is also W -invariant.

Polytopes.

To a G-linearized line bundle L on a complete horosymmetric variety X, we may associate several different convex polytopes. The first one is obtained directly from the special divisor of L.

Definition 5.6. The special polytope ∆ L of L is the convex polytope in M ⊗ R defined by the inequalities m+ v L ≥ 0, and m(ρ(D))+ n D,L ≥ 0 for all D ∈ D \ D X .

Definition 5.7. The toric polytope ∆ t L of L is the convex polytope defined by

∆ t L = {m ∈ M ⊗ R; m + v t L ≥ 0}.
Remark that the toric polytope is W -invariant (and independent of the choice of a Borel subgroup B containing T ).

Definition 5.8. The moment polytope ∆ + L is the set defined as the closure in X(T ) ⊗ R of the set of all λ /k such that there exists a non-zero B-semi-invariant global holomorphic section s of

L k with weight λ (that is, b•s = λ(b)s for all b ∈ B).
Note that all of these sets are multiplicative with respect to tensor powers, that is ∆ ♥ L k = k∆ ♥ L for any positive integer k. Even though the definitions above are somewhat different, it becomes clear that they have a strong relationship once we interpret the definitions in terms of sections of the line bundle L. Recall that any irreducible G-representation contains a unique B-stable line, where B acts via a character called the highest weight of the representation. The moment polytope is then by definition the closure of the set of all highest weights of irreducible subrepresentations of G in the space of multisection H 0 (X, L k ), divided by k, for all positive integers k. On the other hand, assuming there exists a special section s of L, then by definition the points in M ∩ ∆ L precisely encode the (weights of the) B-semi-invariant rational functions f on X such that f s is a B-eigenvector in the representation H 0 (X, L). The more precise relationship between the two polytopes is as follows.

Proposition 5.9. Let χ ∈ X(H) be the isotropy character associated to the restriction of L to G/H. Consider χ as before as an element of X(T /T s ) ⊗ R ⊂ X(T ) ⊗ R via its restriction to T ∩ H. Then

∆ + L = χ + ∆ L . Proof.
By multiplicativity, we may as well prove the result for L k . This has two consequences: we may choose k so that L has a special section, and kχ|T s ∩ H = 0.

Let s denote the special section of L k , and denote by λ ∈ X(B) = X(T ) its character. By [Bri89, Proposition 3.3] (see also [START_REF] Brion | Variétés sphériques[END_REF]Section 5.3]), we have

∆ + L k = λ + ∆ L k .
We know that λ| Ts = 0 hence we may consider λ as an element of X(T /T s ) ⊂ X(T ).

From the other consequence of considering L k , we see that we may also consider kχ as an element of

X(T ∩ H/T s ∩ H). The natural epimorphism T ∩ H → T /T s identifies T ∩ H/T s ∩ H with T /T s . Let t ∈ T ∩ H. We have, by definition of s, λ(t)s(eH) = t • s(t -1 H) = t • s(eH) since t ∈ H = (kχ)(t)s(eH)
• M ⊗ R X(T /T s ) ⊗ R • χ ∆ L ∆ + L ∆ t L W Proposition 5.10 ([Bri89, Théorème 3.3]). The G-
• m C1 = m C2 if C 1 = C 2 ∈ F X are two maximal cones, • n D,L = m C (ρ(D)) for all D ∈ D \ D X and maximal cone C ∈ F X .
Definition 5.11. The support function w ∆ : V * -→ R of a convex polytope ∆ in a real vector space V is defined by

w ∆ (x) = sup{m(x); m ∈ ∆}.
One may recover the convex polytope ∆ from its support function by checking ∆ = {m ∈ V ; m ≤ w ∆ }. As a consequence from this definition, we have: Corollary 5.12. If L is globally generated, then v L (y) = w ∆L (-y) for y ∈ |F X |.

Let C + denote the positive Weyl chamber in a * = X(T ) ⊗ R, which may be defined as

C + = {p ∈ a * ; p(α ∨ ) ≥ 0, ∀α ∈ Φ + }.
Similarly, we define the positive restricted Weyl chamber in a * s as C+ = {p ∈ a * s ; p(ᾱ ∨ ) ≥ 0, ∀ᾱ ∈ Φ+ }. The following proposition describes the relationship between the toric polytope and the special polytope (see Figure 3 for an illustration).

Proposition 5.13. The polytope ∆ L is a translate by an element of C+ of a polytope which is the intersection of a W -invariant polytope with C+ . In particular, ∆ L ⊂ C+ and

∆ t L = Conv( W • ∆ L ). Proof.
By definition, the polytope ∆ L has outer normal along codimension one faces which are given by some elements of the valuation cone and the images by ρ of some colors. Since the only images of colors that are not in the valuation cone are simple restricted coroots by Proposition 2.23 we obtain at once that ∆ L is a translate of a polytope which is the intersection of a W -invariant polytope with C+ .

To check that it is a translation by an element of C+ , it is enough to check that ∆ L is included in C+ itself. This is a direct consequence of the relation

∆ + L = χ + ∆ L together with the fact that ∆ + L ⊂ C + by definition. Indeed, given p ∈ ∆ L and α ∈ Φ + s , we have p(ᾱ ∨ ) = (p + χ)(ᾱ ∨ ) -χ(ᾱ ∨ )
which is positive since χ is zero on a s , p+χ ∈ ∆ + L ⊂ C + and ᾱ∨ is a positive multiple of either a positive coroot or the sum of two positive coroots by Definition 2.21.

We have thus proved that ∆ L ⊂ C+ . By definition of the toric polytope, the supporting hyperplanes defining ∆ t L are those supporting ∆ L with normal in C+ , and their images by W . Since ∆ L ⊂ C+ , all images of the special polytope by W are contained in ∆ t L , hence the result. Recall that the linear part of a cone containing the origin is the largest linear subspace included in the cone.

Corollary 5.14. The following conditions are equivalent:

(1) L m admits a global holomorphic Q-semi-invariant section for some m ∈ N * , (2)

∆ + L ∩ X(T /T ∩ [L, L]) ⊗ R = ∅ (3) ∆ L intersects the linear part of C+ , (4) ∆ t L ∩ C+ = ∆ L = -χ + ∆ + L . Proof.
The first condition translates directly into a condition on ∆ + L : it is equivalent to the fact that some L m admits a global holomorphic B-semi-invariant section whose weight is in

X(T /T ∩ [L, L]), that is, ∆ + L ∩ X(T /T ∩ [L, L]) = ∅. One checks easily that the linear part of C+ ⊂ X(T s )⊗ R is X(T s /T s ∩[L, L])⊗ R,
and coincides also with the linear subspace of W -invariant elements of X(T s ) ⊗ R.

Now since ∆ + L = ∆ L + χ and χ ∈ X(T /(([L, L] ∩ T )T s )) ⊗ R ⊂ X(T /[L, L] ∩ T ) ⊗ R, the first condition is equivalent to ∆ L ∩ X(T s /T s ∩ [L, L]) ⊗ R = ∅.
Finally, thanks to Proposition 5.13, we obtain the equivalence with the last condition.

5.4. The anticanonical line bundle. Recall the Weil divisor representing the anticanonical class obtained by Brion on any spherical variety.

Proposition 5.15 ([Bri97, Sections 4.1 and 4.2]). For any spherical G-variety X, there is a B-semi-invariant section of the anticanonical sheaf with Weil divisor

-K X = Y ∈I G (X) Y + D∈D m D D
where the m D are positive integers with an explicit description in terms of the colored data. Furthermore, the B-weight of the section is the sum of α ∈ Φ + such that g -α does not stabilize the open B-orbit in X.

Consider now a horosymmetric variety X. If one considers the subvariety X ⊂ X which consists of all the orbits of codimension strictly less than two in X, it is a smooth variety with a well defined anticanonical line bundle, and this anticanonical line bundle admits a B-semi-invariant section s with weight λ whose divisor is the divisor -K X above. In our horosymmetric situation, the weight λ is equal to α∈Φ Q u ∪Φ + s α. Indeed, if α ∈ Φ Q u then g -α does not even stabilize the open B-orbit in G/P , and if α ∈ Φ L , then g -α stabilizes the open B-orbit in L/L ∩ H if and only if it belongs to Φ σ .

We may reason as if λ|

Ts = α∈Φ Q u ∪Φ + s α • P is induced by an element of X(T /T ∩ H), up to passing to K -k X for some positive integer k if necessary. Let h ∈ C( X) be a B-semi-invariant function with weight -α∈Φ Q u ∪Φ + s α • P. Then hs is the special section of K -1 X . Its B-weight is α∈Φ Q u ∪Φ + s α • H. Note that this is equal to α∈Φ Q u α • H since α∈Φ + s α • P = α∈Φ + s α. The special divisor D ac of K -1
X is thus

D ac = Y ∈I G X   1 - α∈Φ Q u ∪Φ + s α • P(µ Y )   Y + D∈D   m D - α∈Φ Q u ∪Φ + s α • P(ρ(D))   D.
Note that X is a global parabolic induction with respect to the morphism f : X → G/P extending the natural morphism G/H → G/P . We accordingly decompose the anticanonical line bundle as

K -1 X = K -1 f ⊗ f * K -1 G/P
. By definition, the special section is the product of special sections of these naturally linearized line bundles and the divisor D ac on X is the sum of their respective divisors D ac f and D ac P . The special section of

f * K -1 G/P is obviously Q-semi-invariant, with weight precisely equal to α∈Φ Q u α • H. As a consequence, the special section of K -1 f is G-invariant. The special section of K -1 X is thus Q-semi-invariant.
It follows from this discussion that the coefficients of colors coming from L/L ∩ H must vanish.

By normality of X, the special divisors D ac f of K -1 f and D ac P of f * K -1 G/P , defined on X, extend to X as Weil divisors and D ac = D ac f + D ac P holds on X. We further have explicitly

D ac f = Y ∈I G X   1 - α∈Φ Q u ∪Φ + s α • P(µ Y )   Y and
D ac P = α∈Φ Q u ∩S   m Dα - β∈Φ Q u ∪Φ + s β • P(ρ(D α ))   D α .

Examples.

Example 5.16. We consider again the variety X = P 1 × P 1 equipped with the diagonal action of SL 2 . The line bundles on X are the O(k, m) for k, m ∈ Z. They admit natural SL 2 × SL 2 -linearization hence also a natural linearization under the diagonal action. There are two colors D + and D -with same image ᾱ 1,2 ∨ via the color map, the fan of X is the negative Weyl chamber, a single ray generated by -ᾱ∨ 1,2 corresponding to the orbit Y = diag(P 1 ). The line bundle corresponding to D + is, say, O(1, 0) and O(0, 1) corresponds to D -, while the line bundle corresponding to Y is obviously O(1, 1).

The special divisor corresponding to

O(k, m) is k+m 2 Y + k-m 2 (D + -D -)
, its moment polytope is the set of all tα 1,2 for |k-m| 2 ≤ t ≤ k+m 2 . It is the same as the special polytope. The toric polytope is the set of all tα 1,2 for |t| ≤ k+m 2 .

Example 5.17. Consider the wonderful compactification X of Type AIII(2, m > 4), under the action of SL m . Since this group is simple and simply connected, all line bundles admit a unique linearization. It follows from [START_REF] Brion | Groupe de Picard et nombres caractéristiques des variétés sphériques[END_REF] that the Picard group of X is the free abelian group generated by the three colors D + 1 , D - 1 and D 2 whose images under the color map are ρ(D ± 1 ) = ᾱ∨ 2,m-1 and ρ(D 2 ) = ᾱ∨ 1,2 . The two G-invariant prime divisors Y 1 , corresponding to the ray generated by µ 1 = -ᾱ∨ 1,m-1 and Y 2 , corresponding to the ray generated by µ 2 = -ᾱ∨ 1,m write in this basis as

Y 1 = D + 1 + D - 1 -D 2 and Y 2 = 2D 2 -D + 1 -D - 1 . Given a line bundle corresponding to the divisor k + 1 D + 1 + k - 1 D - 1 + k 2 D 2 , the corresponding special divisor is b 1 Y 1 + b 2 Y 2 + b ± (D + 1 + D - 1 ) where b 1 = k + 1 + k - 1 + k 2 , b 2 = k + 1 +k - 1 2 + k 2 and b ± = k + 1 -k - 1 2
. Assume b ± = 0. The polytope ∆ t is the convex hull of the images by W of the point b 2 ᾱ1,2 + b 1 ᾱ2,3 , provided it is in the positive restricted Weyl chamber, that is 2b 2 ≥ b 1 and b 1 ≥ b 2 . Note that Brion's ampleness criterion translates here as the fact that this point is in the interior of the positive restricted Weyl chamber. The polytope ∆ + = ∆ is the intersection of ∆ t with the positive chamber. See Figure 4 in Section 7 for drawings of the moment polytopes. If b ± is non-zero, then we must intersect with another half-plane to get the polytope.

Metrics on line bundles

We will now use the objects introduced in the previous section to study Hermitian metrics on G-linearized line bundles. Given a Hermitian metric q on L, recall that its local potentials are the functions ψ : y →ln |s(y)| 2 where s is a local trivialization of L. We allow for now singular Hermitian metrics, that is the local potentials are only required to be locally integrable. The metric is called locally bounded/continuous/smooth if and only if its local potentials are. It is called non-negatively curved (in the sense of currents) if the local potentials are plurisubharmonic functions and positively curved if its local potentials are strictly plurisubharmonic.

6.1. Asymptotic behavior of toric potentials. Proposition 6.1. Let G/H ⊂ X be a complete horosymmetric variety, and L a G-linearized line bundle on X, with special function v L . Let q be a K-invariant locally bounded metric on L with toric potential u. Then the function

x → u(x) -2v t L (x) is bounded on a s .
The proof will use the process of discoloration, which allows to reduce to the case of a toroidal variety. Definition 6.2. Let (X, x) be an embedding of G/H with colored fan F X . Then the discoloration (X ′ , x ′ ) of (X, x) is the embedding of G/H whose colored fan F X ′ is obtained by taking the collection of all colored cones of the form (C ∩ V, ∅) for (C, R) ∈ F X and their faces.

The discoloration (X ′ , x ′ ) of (X, x) is equipped with a G-equivariant birational proper morphism d ′ : X ′ → X sending x ′ to x. The simplest example of discoloration is given by the blow up of P 2 at 0, seen as a horospherical variety under the action of SL 2 .

Proof of Proposition 6.1. Note that the toric potential is defined only up to an additive scalar, but this does not affect the statement. The choice of toric potential is determined by the choice of a non zero element ξ ∈ L eH . We fix such a choice here.

Since the special function of L m is mv L and the toric potential of q ⊗m is mu, replacing L by a power of L will not affect the result. For example we can already assume that L admits a special section.

Consider the pullback L ′ of L under the discoloration morphism d ′ : X ′ → X, equipped with the metric d ′ * q. The special function v L ′ coincides with the special function v L [Pas17, Proof of Lemma 5.3]. Furthermore, by Proposition 5.5 and up to replacing L by a power of itself, the restriction of L ′ to the toric subvariety Z ′ ⊂ X ′ is a T /T ∩ H-linearized line bundle with divisor

F ∈I T /T ∩H (Z ′ ) v t L (µ F )F.
Consider the Batyrev-Tschinkel metric associated to this line bundle [Mai00, Section 3

.3]. It is a compact torus invariant, continuous metric on L ′ | Z ′ with toric potential u BT : x → -2 ln | exp s (x) • ξ| BT equal to x → 2v t L (x).
Beware that here exp s denotes the exponential map for the Lie group T /T ∩ H, which does not coincide with the exponential map for G. Here however, since the T /T ∩H-linearization of L ′ | Z ′ was obtained via factorization of the T s -linearization, we have exp s (x) • ξ = exp(x) • ξ. We then have

(u -u BT )(x) = -2 ln | exp(x) • ξ| BT | exp(x) • ξ| q .
Since the Batyrev-Tschinkel metric is continuous and the metric q is locally bounded, we obtain that the above function is globally bounded, hence the statement.

6.2. Positive metrics on globally generated line bundles. In this section, X is a horosymmetric variety and L is a globally generated and big line bundle on X.

Proposition 6.3. Let q be a non-negatively curved, K-invariant, locally bounded Hermitian metric on L with toric potential u. Assume in addition that its restriction to L| G/H is smooth and positively curved. Then

(1) u is a smooth, strictly convex, W -invariant function,

(2) there exists a constant C such that w

-2∆ t -C ≤ u ≤ w -2∆ t + u(0), (3) a → d a u defines a diffeomorphism from a s onto Int(-2∆ t ), (4) a → d a u defines a diffeomorphism from Int(a + s ) onto Int(-(2∆ t ∩ C+ )).
Proof. Without loss of generality, we may assume that X is toroidal via the discoloration procedure. Then the first property directly follows from restriction to the toric subvariety. The second property is a translation of Proposition 6.1, with the additional input that by convexity and since w -2∆ t is piecewise linear, we can take u(0) as constant on one side. Then the third property is a consequence of the second, and the fourth follows by W -invariance. Remark 6.4. Note that the open dense orbit is contained in the ample locus of any big line bundle on X, hence there are Hermitian metrics as in the statement of Proposition 6.3. If u is the toric potential of a metric q as in Proposition 6.3, then u * is W -invariant and u * = +∞ on a * s \ -2∆ t . Furthermore, we have u * (p) = p(a)u(a) whenever p = d a u ∈ Int(-2∆ t ). For each facet Y , we choose a complement a Y of Rµ Y in a s as in Section 4.3, corresponding to a torus T s,Y . Note that the torus T s,Y is a maximal split torus for the involution associated to Y as in Section 4.3.

Let h be a Hermitian metric on L and assume that it is smooth on the elementary embedding E Y . Denote by ψ : bH → -2 ln |s(bH)| h the potential of h with respect to the special section s.

There exists a unique λ ∈ X(T

/(T ∩ H)T Y ) ⊗ Q such that λ(µ Y ) = -n Y .
Up to taking a tensor power of L, we may thus find a rational B-semi-invariant function f ∈ C(X) such that ord Y (f ) = -n Y and f (x) = 1. Let s λ = f s denote a B-semiinvariant section obtained by multiplying the section s by f . Then the section s λ does not vanish identically on Y , and its restriction to Y is further a special section for L| Y . The potential ψ λ of h with respect to s λ is defined on the whole For the second limit, fix for clarity a basis l 1 , . . . , l r-1 of a Y , so that together with µ Y it yields an identification of the toric subvariety Z with (C * ) r-1 ×C. Under this identification, the vector field l j corresponds to ∂ /∂ ln |zj |, and µ Y to ∂ /∂ ln |zr| which is not defined on the divisor z r = 0. On the other hand, the vector field

∂ ∂|z r | = 1 |z r | ∂ ∂ ln |z r |
u * Y (d b u Y ) = d b u Y (b) -u Y (b) = lim d tj µY +bj u λ (t j µ Y + b j ) -u λ (t j µ Y + b j )
for any sequences such that lim t j = -∞ and lim b j = b by Proposition 6.5

= lim u * λ (d tj µY +bj u λ ). Note that the convex conjugate u * λ is not a priori continuous up to the boundary, hence it is not enough to conclude. On the other hand, if we choose t ∈ R we have

u * λ (d b u Y ) = lim s→1 u * λ (sd b u Y + (1 -s)d tµY +b u λ ).
We may find t is the toric polytope -2∆ t Y + 2λ of the restriction of L to Y , translated, which is the facet of -2∆ t + 2λ whose outer normal is µ Y . Concerning moment polytopes, we have: Proposition 6.7 ( [Bri]). The moment polytope of L| Y is the codimension one face of ∆ + whose outer normal in the affine space χ + M R is -µ Y .

The special polytope ∆ Y of L| Y is then ∆ + Yχλ and we have χ Y = χ + λ under the usual identifications. 6.4. Volume of a polarized horosymmetric variety. Before applying the results from this section combined with our computation of the Monge-Ampère operator to get an integration formula on horosymmetric varieties, we recall the formula for the degree of a line bundle on a horosymmetric variety. It is a consequence of a general result of Brion, who proved the formula for any spherical variety. Let ̟ = 1 /2 α∈Φ + α denote the half sum of positive roots. Proposition 6.8 (Particular case of [Bri89, Théorème 4.1]). Let X be a projective horosymmetric variety, and L be a G-linearized ample line line bundle on X. Then

L n = n! ∆ + L Φ + \E κ(α, p) κ(α, ̟) dp
where E is the set of roots α ∈ Φ + that are orthogonal to ∆ + L with respect to κ and dp is the Lebesgue measure on the affine span of ∆ + L , normalized by the translated lattice χ + X(T /T ∩ H).

In the case of horosymmetric varieties, the set E is exactly Φ + ∩ Φ σ L . We will use the notation

P DH (p) = Φ Q u ∪Φ + s κ(α, p) κ(α, ̟) = Φ Q u ∪Φ + s κ(α, α) 2κ(α, ̟) p(α ∨ )
6.5. Integration on horosymmetric varieties. Let L be a globally generated and big G-linearized line bundle on a horosymmetric variety X. We assume in this subsection that h is a locally bounded, non-negatively curved metric on L, smooth and positive on the restriction of L to G/H. We denote by ω its curvature current.

Assume furthermore that χ vanishes on [l, l].

Let ψ denote a K-invariant function on X, integrable with respect to ω n , and continuous on G/H. To simplify notations, we denote by ψ(a) the image by ψ of exp(a)H for a ∈ a s . Let ∆ ′ denote the polytope -2∆ t ∩ C-. Proposition 6.9. Let dq denote the Lebesgue measure on the affine span of ∆ + , normalized by the lattice χ + M, let dp denote the Lebesgue measure on M ⊗ R normalized by M. Then there exist a constant C ′ H , independent of h and ψ, such that

X ψω n = C ′ H 2 n ∆ ′ ψ(d p u * )P DH (2χ -p)dp = C ′ H χ+∆ t ∩ C+
ψ(d 2χ-2q u * )P DH (q)dq.

Proof. Since ω n puts no mass on X \ G/H, we may first note that

X ψω n = G/H ψω n
Then by K-invariance and Proposition 3.7, this is equal to

C H -a + s ψ(a)J H (a) ω n dV H (exp(a)H)da.
Now by definition of J H and Corollary 3.13, this is equal to

C H -a + s n! 2 2r+|Φ Q u | ψ(a) α∈Φ Q u (2χ -d a u)(α ∨ ) β∈Φ + s (-d a u)(β ∨ ) det(d 2 u)(a)da
We then use the change of variables 2p = d a u and thanks to Proposition 6.3 we obtain

X ψω n = C H (-∆ t )∩ C- n!2 |Φ + s |-r ψ(d 2p u * ) α∈Φ Q u (χ -p)(α ∨ ) β∈Φ + s (-p)(β ∨ )dp
where dp is for the moment a Lebesgue measure independent of ψ. Actually by considering a constant ψ, hence the volume of L, we see that the Lebesgue measure is further independent of the choice of q.

The assumption that χ vanishes on [l, l] ensures that χ(β ∨ ) = 0 for all β ∈ Φ + s , hence using the change of variables q = χp, we get

X ψω n = n!2 |Φ + s |-r C H χ-(-∆ t )∩ C- ψ(d 2χ-2q u * ) α∈Φ Q u ∪Φ + s q(α ∨ )dq.
Taking the constant C ′ H to be the covolume of the lattice X(T /T ∩ H) under dp times the constant

n!2 2|Φ + s |+|Φ Q u |-r C H α∈Φ Q u ∪Φ + s κ(α, ̟) κ(α, α)
we obtain the result.

Corollary 6.10. Assume that L m admits a global Q-semi-invariant section for some m > 0. Then the constant C ′ H in Proposition 6.9 is equal to n!(2π) n and the integration is over ∆ + in the second equality.

Proof. It follows from applying Proposition 6.9 to the constant function ψ = 1 (2π) n , using Corollary 5.14 to check that the integration is over ∆ + , and comparing with Brion's formula for the degree of a line bundle (Proposition 6.8).

Mabuchi Functional and coercivity criterion

7.1. Setting. We fix in this section a Q-line bundle L on an n-dimensional smooth horosymmetric variety X. We assume that there exists a positive integer m such that L m is an ample line bundle, with a fixed G-linearization, and that it admits a global holomorphic Q-semi-invariant section. Let ∆ + denote the moment polytope of L, and ∆ ′ = -2(∆ + -χ) where χ is the isotropy character of L. As a consequence of Corollary 5.14, we may fix a point λ 0 in the relative interior of ∆ + ∩ X(T /T ∩ [L, L]) ⊗ R. The point 2(χλ 0 ) is then in the interior of ∆ ′ ∩ X(T s /T s ∩ [L, L]) ⊗ R.

We will make the following additional assumptions: (T) the horosymmetric variety X is toroidal, and whenever a facet of ∆ + intersects a Weyl wall, either the facet is fully contained in the wall or its normal belongs to the wall, furthermore, ∆ + intersects only walls defined by roots in Φ L , (R) for any restricted Weyl wall, there are at least two roots in Φ + s that vanish on this Weyl wall. Unlike the assumption that L is trivial on the symmetric fibers, these assumptions are very likely not meaningful. We use these to provide a rather general application of the setting we developed for Kähler geometry on horosymmetric varieties in a paper with reasonable length. We have no claim of giving the most general statement, and expect that at least assumption (R) can be removed without too much difficulties. Once this is achieved, removing assumption (T) should require an analysis similar to that given by Li-Zhou-Zhu in [START_REF] Li | K-energy on polarized compactifications of Lie groups[END_REF] to treat nontoroidal group compactifications. Finally, removing the assumption that L is trivial on the symmetric fibers appears to be a much more challenging problem in view of the expression of the curvature form, as convex conjugacy in this generality seems to be a bit less helpful.

Note that these assumptions are satisfied in a large variety of situations. We expect that the second part of assumption (T), in terms of the moment polytope, is actually implied by the assumption that X is toroidal. This is true at least for symmetric varieties and horospherical varieties. Assumption (R) is satisfied for example when the symmetric fiber is of group type, of type AIII(r, n > 2r), of type AII(p), but unfortunately not when the symmetric fiber is of type AI(m). It is obviously satisfied if the variety X is horospherical.

Recall that we fixed an anticanonical Q-divisor D ac with a decomposition D ac = D ac f + D ac P in Section 5.4 such that O(D ac P ) = f * K -1 G/P on the complement X of codimension ≥ 2 orbits, and

D ac f = Y n Y Y where n Y = 1 - α∈Φ Q u ∪Φ + s α • P(µ Y ).
Let Θ be a G-stable Q-divisor on X with simple normal crossing support. Write

Θ = c Y Y and set n Y,Θ = -c Y + 1 - α∈Φ Q u ∪Φ + s α • P(µ Y ).
We assume all coefficients c Y satisfy c Y < 1. Recall that we denote α∈Φ Q u α • H by χ ac and this is the isotropy character associated to the anticanonical line bundle on G/H.

Fix a smooth K-invariant positive reference metric h ref on L, and denote its curvature form by ω ref . Let rPSH K (X, ω ref ) denote the space of smooth K-invariant strictly ω ref -plurisubharmonic potentials on X. The functions in rPSH K (X, ω ref ) are in one-to-one correspondence with smooth positive Hermitian metrics on L. We denote by h φ the metric corresponding to φ ∈ rPSH K (X, ω ref ) and we write ω φ = ω ref +i∂ ∂φ for the curvature of h φ , which depends on φ only up to an additive constant.

To any φ ∈ rPSH K (X, ω ref ) is associated a toric potential u: the toric potential of h φ . Note that under our assumptions (X is smooth and toroidal) and by Proposition 4.17, X admits a smooth toric submanifold Z, and u is the toric potential of the restriction of h φ to the restricted ample Q-line bundle L| Z , hence the convex potential u * of u satisfies the Guillemin-Abreu regularity conditions in terms of the polytope -∆ t . 7.2. Scalar curvature. The scalar curvature S of a smooth Kähler form ω is defined by the formula S = nRic(ω) ∧ ω n-1 ω n Note that Ric(ω) is the curvature form of the metric on K -1 X corresponding to the volume form ω n , whose toric potential we denote by ũ. Assuming that ω is the curvature form of a metric on L, one can determine this toric potential using Theorem 3.10. Using the liberty to choose the multiplicative constant for the section defining the toric potential (a multiple of the dual of ♦ γ ♦ ), we may assume that,

ũ(a) = -ln det d 2 a u - α∈Φ Q u ∪Φ + s ln((2χ -d a u)(α ∨ )) + β∈Φ + s ln sinh(-2β(a)) - α∈Φ Q u 2α(a),
for a ∈ -a + s .

We will, for the rest of the section, fix a choice of orthonormal basis (l j ) j of a s (with respect to some fixed scalar product whose corresponding Lebesgue measure is normalized by M) and corresponding dual basis (l * j ) of a * s . Write α ∨,j for the coordinates of α ∨ . We use the notations d a u = j u j (a)l * j , (u j,k ) for the inverse matrix of (u j,k ), etc. To simplify notations, summation symbols will sometimes be omitted in which case we sum over repeated indices in a given term. For example ũ =ln det(u l,m ) -

α∈Φ Q u ∪Φ + s ln((2χ l -u l )α ∨,l ) + I H where I H (a) = β∈Φ + s ln sinh(-2β(a)) -α∈Φ Q u 2α(a)
. Set p = d a u and consider both a and p as variables in the dual spaces a s and a * s . Finally, let P ′ DH (p) = P DH (2χp).

Proposition 7.1. The scalar curvature at exp(a)H is equal to

-u * ,i,j i,j (p) + -2u * ,i,j j (p) + (I H ) i (a) P ′ DH,i P ′ DH (p) + u * i,j (p)I H,i,j (a)
u * ,i,j (p) P ′ DH,i,j

P ′ DH (p) + α∈Φ Q u 2χ ac (α ∨ ) (2χ -p)(α ∨ )
Proof. We compute, using Jacobi's formula, ũj = -u l,m u m,l,j -

α∈Φ Q u ∪Φ + s -u l,j α ∨,l (2χ -u l )α ∨,l + I H,j .
Using the variable p = d a u and convex conjugate, we have d p u * = a, d 2 a u = (d 2 p u * ) -1 and thus u * ,i,j j (p) = u k,j (a)u j,k,i (a). We may then give another expression of ũj :

ũj (a) = -u * ,j,i i (p) - α∈Φ Q u ∪Φ + s -u * ,l,j (p)α ∨,l (2χ -p)(α ∨ ) + I H,j (d p u * ). then ũj,k (a) = -u * ,j,i i,s (p)u * ,s,k (p) - α∈Φ Q u ∪Φ + s -u * ,l,j α ∨,l (2χ -p)(α ∨ ) s (p)u * ,s,k (p)
+ I H,j,k (a).

We now compute for a ∈ -a + s ,

nRic(ω φ ) ∧ ω n-1 φ ω n φ = Tr((ũ l,m (a))(u l,m (a))) + α∈Φ Q u ∪Φ + s -ũ l (a)α ∨,l (2χ -p)(α ∨ ) + α∈Φ Q u 2χ ac (α ∨ ) (2χ -p)(α ∨ )
which, by using the previous expressions, is equal to

-u * ,i,j i,j + α∈Φ Q u ∪Φ + s 2 u * ,i,j j α ∨,i (2χ -p)(α ∨ ) - α,β∈Φ Q u ∪Φ + s u * ,i,j α ∨,i β ∨,j (2χ -p)(α ∨ )(2χ -p)(β ∨ ) - α∈Φ Q u ∪Φ + s u * ,i,j α ∨,i α ∨,j ((2χ -p)(α ∨ )) 2 + u * i,j (I H ) i,j (a)
-

α∈Φ Q u ∪Φ + s (I H ) i (a)α ∨,i (2χ -p)(α ∨ ) + α∈Φ Q u 2χ ac (α ∨ ) (2χ -p)(α ∨ )
Recall that P ′ DH (p) = P DH (2χp), and note that

P ′ DH,i (p) = P ′ DH (p) α -α ∨,i (2χ -p)(α ∨ ) and P ′ DH,i,j (p) = P ′ DH (p)   α,β α ∨,i β ∨,j (2χ -p)(α ∨ )(2χ -p)(β ∨ ) + α α ∨,i α ∨,j (2χ -p)(α ∨ ) 2   .
Plugging this into the last expression of the scalar curvature yields the result.

Remark 7.2. The computation of the scalar curvature here is only on the homogeneous space, hence holds under weaker hypothesis than in the setting: we only need to assume that L is a line bundle on G/H whose restriction to the symmetric fiber is trivial, and that h is a smooth and positive metric on L.

Remark 7.3. When G/H is horospherical, I H is linear hence I H,i,j = 0 and I H,i is constant. We can then write the formula in the more concise form -W -1 (W u * ,i,j ) i,j + f where W = P ′ DH and f (p) = I H,i

P ′ DH,i P ′ DH (p) + α∈Φ Q u 2χ ac (α ∨ ) (2χ-p)(α ∨ )
which provides a generalization of Donaldson and Nyberg's formula [START_REF] Simon | Kähler geometry on toric manifolds, and some other manifolds with large symmetry[END_REF] and [Nyb]. 7.3. The functionals. 7.3.1. The J-functional. The J-functional is defined (up to a constant) on the space rPSH K (X, ω ref ) by its variations as follows: if φ t is a smooth path in rPSH K (X, ω ref ) between the origin and φ, then

J(φ) = 1 0 X φt ω n ref -ω n φt (2π) n L n dt.
Proposition 7.4. Let u denote the toric potential of h φ . Then

J(φ) -u(0) - n! L n ∆ + u * (2χ -2q)P DH (q)dq
is bounded independently of φ.

Proof. We have

J(φ) = X φ ω n ref (2π) n L n - 1 0 X φt ω n φt (2π) n L n dt.
The definition of convex conjugate yields u * t (d a u) =ut (a) hence by Proposition 6.9 and Corollary 6.10, we have

1 0 X φt ω n φt dt = - 1 0 C ′ H χ+∆ t ∩ C+ u * t (2χ -2q)P DH (q)dq = -C ′ H χ+∆ t ∩ C+ (u * -u * ref )(2χ -2q)P DH (q)dq = -(2π) n n! ∆ + (u * -u * ref )(2χ -2q)P DH (q)dq
On the other hand, it follows from classical results [GZ05, Proposition 2.7] that

1 (2π) n L n X φω n ref -sup X (φ) is bounded independently of φ ∈ rPSH K (X, ω ref ). We have • sup X (φ) = sup as (u -u ref ), • w -2∆ t -C 1 ≤ u ref ≤ w -2∆ t + u ref (0)
for some constant C 1 by Proposition 6.3, and

• sup as u -w -2∆ t = u(0) by convexity, hence X φω n ref -(2π) n L n u(0)
is bounded independently of φ.

7.3.2. Mabuchi functional. Denote by S the average scalar curvature, defined as

S = X nRic(ω) ∧ ω n-1 X ω n .
The (log)-Mabuchi functional Mab Θ relative to Θ is defined also by integration along smooth path:

Mab Θ (φ) is equal to 1 0 X φt ( SΘ ω φt -nRic(ω φt )) ∧ ω n-1 φt (2πL) n + 2πn Y c Y Y φt ω n-1 φt (2πL) n dt where SΘ = S -n Y c Y L| n-1 Y /L n . Let ∆+
Y denote the bounded cone with vertex λ 0 and base ∆ + Y . Note that in general ∆ + = Y ∆+ Y . It is however the case under assumption (T), that is if we assume that X is toroidal. We set

Λ Y = (n Y -c Y )/v L (µ Y ).
Recall that we treat several quantities as variables: a ∈ -a + s , p ∈ ∆ ′ = -2∆ t ∩ Cand q ∈ ∆ + , with the change of variables formula p = d a u (hence a = d p u * ), and p = 2χ -2q.

Theorem 7.5. Under assumption (T), up to the choice of a normalizing additive constant,

L n n! Mab Θ (φ) is equal to Y Λ Y ∆+ Y nu * (p) -u * (p) χ(α ∨ ) q(α ∨ ) + d p u * (p) P DH (q)dq + ∆ + u * (p) χ ac (α ∨ ) q(α ∨ ) -SΘ P DH (q)dq - ∆ + I H (a)P DH (q)dq - ∆ + ln det(u * i,j )(p)P DH (q)dq Proof. The summands 1 0 X φt SΘ ω n φt dt/(2πL) n and 1 0 2πn Y c Y Y φt ω n-1 φt dt/(2πL) n
may be dealt with as in the computation for J, yielding respectively

SΘ n! L n ∆ + (u * ref -u * )(2χ -2q)P DH (q)dq and Y c Y n! L n ∆ + Y (u * ref -u * )(2χ -2q)P DH (q)dq Y .
The harder part is

1 0 X -n φt Ric(ω φt ) ∧ ω n-1 φt /(2πL) n dt.
Using the integration formula as well as the formula for the scalar curvature, we have

I := X -n φt Ric(ω φt ) ∧ ω n-1 φt = X -n φt Ric(ω φt ) ∧ ω n-1 φt ω n φt ω n φt = C ′ H 2 n ∆ ′ u * t -u * ,i,j t,i,j P ′ DH -2u * ,i,j t,j P ′ DH,i -u * ,i,j t P ′ DH,i,j + u * t,i,j I H,i,j (a)P ′ DH + I H,i (a)P ′ DH,i + α∈Φ Q u 2χ ac (α ∨ ) (2χ -p)(α ∨ ) P ′ DH dp
where the variable, if omitted, is p = d a u.

Apart from the last term, the situation is analogous to Li-Zhou-Zhu [LZZ18] hence we may follow the same steps. However note that translation by χ in the Duistermaat-Heckman polynomial will sometimes introduce extra terms. Denote by ν the unit outer normal to ∂∆ ′ . On a codimension one face corresponding to a facet Y of X, ν is a positive multiple of µ Y .

We will apply several times the divergence theorem as follows, without writing the details every time. For 0 < s < 1, let ∆ ′ s denote the bounded cone with vertex 2(χλ 0 ) and base the dilation by s of the boundary s∂∆ ′ . We may apply the divergence theorem on ∆ ′ s to smooth functions on the interior of -2∆ t , then take the limit as s → 1, applying dominated convergence and using the appropriate convergence result. We let dσ denote the area measure on all boundaries (note that on ∆ Y it is a multiple of dp Y in general).

It follows from Proposition 6.5 applied both to the metric h with toric potential u and to the metric induced on the anticanonical line bundle with toric potential ũ that for p ∈ ∆ ′ Y and any i, lim

s→1 u i,j (d sp u * )ν j = 0 and lim s→1 ũj (d sp u * )(µ Y ) j = n Y . Recall that ũj (a) = -u * ,j,i i (p) - α∈Φ Q u ∪Φ + s -u * ,l,j (p)α ∨,l (2χ -p)(α ∨ ) + I H,j (d p u * ).
Note that the standard way to obtain the above limits is by checking that they hold on the standard Guillemin-Abreu potential, then check that it extends to all smooth potentials. We use Proposition 6.5 to stress that it boils down to a very elementary consequence of the use of log coordinates near a divisor. We deduce from these facts, and the fact that P ′ DH vanishes on restricted Weyl walls, that

lim s→1 ∂∆ ′ s u * t (-u * ,i,j t,j + I H,i (a))ν i P ′ DH dσ = Y ∆ ′ Y 2n Y ν µ Y u * t P ′ DH dσ
where ∆ ′ Y denotes the facet of ∆ ′ whose outer normal is µ Y . A first application of the divergence theorem then yields, by passing to the limit, that the above quantity is equal to

∆ ′ u * t (-u
Y ∆ ′ Y 2n Y ν µ Y u * t P ′ DH dσ - ∆ ′ I H (a)P ′ DH dp - ∆ ′ ln det(u * t,i,j )P ′ DH dp + ∆ ′ u * t α∈Φ Q u 2χ ac (α ∨ ) (2χ -p)(α ∨ ) P ′ DH dp
hence the value of the above expression at t = 1 is the corresponding term of the Mabuchi functional, up to a constant independent of φ.

We finally use the divergence theorem one last time, applied to the vector field u * t p i P ′ DH to obtain, for every Y ,

∆ ′ Y p i ν i u * t P ′ DH dσ = ∆′ Y (u * t P ′ DH,i p i + ru * t P ′ DH + u * t,i p i P ′ DH )dp = ∆′ Y (nu * t -u * t α 2χ i α ∨,i (2χ -p)(α ∨ ) + u * t,i p i )P ′ DH dp Since 2n Y ν µY = nY vL(µY ) p(ν) and dp Y = p(ν) vL(µY ) dσ
, where dp Y is the image of the measure dq Y under translation by -χ then dilation by 2. This allows to transform the remaining integrals on ∆ ′ Y or ∆ + Y to integrals on ∆ + , after the change of variable from p to q. Putting everything together gives the result.

L n n! Mab l Θ (φ) = Y Λ Y ∆+ Y (nu * (p) -u * (p) χ(α ∨ ) q(α ∨ ) + d p u * (p))P DH (q)dq + ∆ + u * (p)( χ ac (α ∨ ) q(α ∨ ) -SΘ )P DH dq + ∆ + 4ρ H (a)P DH (q)dq
where 2ρ H = α∈Φ Q u ∪Φ + s α • P. We will use the notation M l (u * ) to denote the above as a function of u * , where u * is the convex conjugate of the toric potential of h φ . Similarly, we use the notation M nl (u * ) = L n n! Mab Θ (φ) -M l (u * ). Remark 7.7. In the last step of the proof of Theorem 7.5, if we apply the divergence theorem to the vector field 4u * P ′ DH ρ H instead of u * t p i P ′ DH , and recalling that n Y = 1 -2ρ H (µ Y ), we obtain another expression of the linear part of the Mabuchi functional L n n! Mab l Θ (φ):

M l (u * ) = Y 1 -c Y v L (µ Y ) ∆ ′ Y p(ν)u * (p)P ′ DH (p)dσ - ∆ + d p P ′ DH (4ρ H )u * (p)dp + ∆ + u * (p)( χ ac (α ∨ ) q(α ∨ ) -SΘ )P DH dq.
0 = Y ∆+ Y -2q(b) ((n + 1)Λ Y -SΘ )P DH (q) + d q P DH (χ ac -Λ Y χ) dq + ∆ + 2 α∈Φ Q u α(b)P DH (q)dq
for all b ∈ a s ∩ z(l).

The expression in the above statement could naturally be interpreted as a log Calabi-Futaki invariant. 7.4.2. Normalization of potentials. The action of Z(L) 0 allows on the other hand to normalize the psh potentials, as follows. Given φ ∈ rPSH(X, ω ref ), we may add a constant and use the action of an element of Z(L) 0 to obtain another potential φ ∈ rPSH(X, ω ref ), such that if û is the corresponding toric potential, and û * its convex conjugate, we have min -2∆ t û * = û * (2(χλ 0 )) = 0. Definition 7.9. The Mabuchi functional is coercive modulo the action of Z(L) 0 if there exists positive constants ǫ and C such that for any φ ∈ rPSH K (X, ω ref ), there exists g ∈ Z(L) 0 such that

Mab Θ (φ) ≥ ǫJ(g • φ) -C.
Consider the function F L defined piecewise by

F L (q) = (n + 1)Λ Y -SΘ + (χ ac -Λ Y χ)(α ∨ ) q(α ∨ )
for q in the unbounded cone with vertex λ 0χ and generated by ∆+ Y . Note that F λL (q) = 1 λ F L (q). The Mabuchi functional for the line bundle L is coercive if and only if it is coercive for any positive rational multiple of L. As an application of this remark, if F L > 0, we may choose the multiple λL in such a way that

∆ + P DH dq = Y ∆+ Y
F λL (q)P DH (q)dq. We now replace L by its multiple to assume the equality above is satisfied with λ = 1. We then define a barycenter of ∆ + by: bar = Y ∆+ Y qF L (q)P DH (q)dq/ ∆ + P DH dq.

Theorem 7.10. Assume the following:

• F L > 0,
• (min Y Λ Y )(barχ) -2ρ H is in the relative interior of the dual cone of a + s , • assumptions (T) and (R) are satisfied. Then the Mabuchi functional is coercive modulo the action of Z(L) 0 .

Example 7.11. We have determined in previous examples everything that is necessary to check when the criterion apply for the example of the wonderful compactification X of, say, the symmetric space of type AIII(2, 5). We consider the ample Cartier divisors (1 + b)Y 1 + Y 2 for 0 < b < 1 rational, and corresponding uniquely G-linearized Q-line bundles. They run over all ample divisors on X that are trivial on the open orbit, up to rational multiple. We illustrate in Figure 4 the corresponding polytopes and subdivision by ∆+ Y1 and ∆+ Y2 . Then it is easy, with computer assistance, to check when the criterion is satisfied in terms of b, and we obtain bounds b -≃ 0.31 and b + ≃ 0.54 such that when b -< b < b + , the Mabuchi functional (for L = O((1 + b)Y 1 + Y 2 ) and Θ = 0) is coercive modulo the action of Z(L) 0 .

Remark 7.12. The two first assumptions imply readily, from the last section, that the Mabuchi functional is invariant under the action of Z(L) 0 .

If L = K -1 X,Θ then Λ Y = 1 for all Y , SΘ = n, and χ = χ ac (up to changing the linearization of L by a character of G). Furthermore,

χ ac + 2ρ H = α∈Φ Q u ∪Φ + s α • H + α∈Φ Q u ∪Φ + s α • P = α∈Φ Q u ∪Φ + s α.
We then have the corollary: Corollary 7.13. If L = K -1 X,Θ then the Mabuchi functional is coercive modulo the action of Z(L) 0 if assumptions (T) and (R) are satisfied and the translated barycenter bar -α∈Φ Q u ∪Φ + s α is in the relative interior of the dual cone of a + s . Remark 7.14. This generalizes the criterion for existence of Kähler-Einstein metrics obtained in [START_REF] Delcroix | K-Stability of Fano spherical varieties[END_REF] in the sufficient direction. In fact the condition is also necessary in this case, as would follow from a computation of log-Donaldson-Futaki invariants along special equivariant test configurations for example.

Remark 7.15. In the case of group compactifications, assumption (R) is automatically satisfied, and we may use [START_REF] Li | K-energy on polarized compactifications of Lie groups[END_REF] in the later stages of the proof to remove assumption (T).

We prove Theorem 7.10 in the following subsections so unless otherwise stated we make the assumptions as in the statement. 7.5.2. Inequality for Mab l Θ . Assume u * = û * is normalized as in Section 7.4.2. Note the following elementary lemma, following directly from the convexity and normalization of û * . Recall that dσ denotes the area measure induced by the Lebesgue measure with respect to a fixed scalar product on a * s . Lemma 7.16. Assume u * is normalized, then The last summand vanishes by the expression of SΘ obtained in Remark 7.6. The third summand, on the other hand, vanishes by definition of p 0 . The second term is non-negative by the assumptions of Theorem 7.10 and the first term is non-negative by convexity of u * .

Then the fact that M l (u j ) converges to zero implies that v ∞ is an affine function by the first term, and that its linear part is given by an element of Y(T s /[L, L]) ⊗ R by the second term. Since v ∞ is normalized and 2(χλ 0 ) is in the interior of Y(T s /[L, L]) ⊗ R ∩ ∆ ′ , this means that v ∞ = 0. As a consequence, we have Let ǫ > 0 be a fixed positive number, to be determined later. Working for the moment on the integrand, we have ln det(u * i,j ) = ln det(ǫu * i,j )r ln ǫ, then by concavity ofln det applied to the segment between u * ref,i,j and ǫu * i,j we have ln det(u * i,j ) ≥ -ǫu There is no difficulty for the computation on the right hand side. For the left hand side on the other hand, we used several ingredients:

• we used assumption (R) to check that P ′ DH,j vanishes on restricted Weyl walls,

• we used again Proposition 6.5, • we used the fact that u * ,i,j ref u * i ν j converges to zero on the facets of the polytope, which is proved by Donaldson in the proof of [Don02, Lemma 3.3.5].

It is not hard to check that the term u * ,i,j ref,i,j P ′ DH + 2u * ,i,j ref,j P ′ DH,i + u * ,i,j ref P ′ DH,i,j is bounded, so we obtain, for some positive constants Proof of Theorem 7.10. Let φ ∈ rPSH K (X, ω ref ), let φ = g • φ + C be the normalization of φ, obtained via the action of some g ∈ Z(L) 0 and addition of a constant, then since the assumptions imply that the Mabuchi functional is invariant under the action of Z(L) 0 , we have Mab(φ) = Mab( φ). By Proposition 7.4, since the toric potential of a normalized φ satisfies û(0) = 0, we have ∆ ′ û * P ′ DH ≥ L n n! J( φ) -C for some constant C independent of φ. Hence we have

Mab Θ (φ) = n! L n M (û * ) ≥ ǫ n! L n ∆ ′ û * P ′ DH - n! L n C a (ln ǫ + 1) ≥ ǫJ( φ) -C ′
for some constant C ′ independent of φ.

Appendix A. Integration away from the walls

For this appendix we work on a finite dimensional Euclidean vector space (V, , ). Given R ⊂ V , s ≤ t ∈ R ∩ {±∞}, we set Σ(R, s, t) = {x ; ∀α ∈ R, s ≤ α, x ≤ t}.

Let S denote a finite set of unit vectors in V , such that each α ∈ R is the interior pointing unit normal vector to a facet of the cone Σ(S, 0, ∞) (in particular, we implicitly assume that the cone is of the same dimension as V ). Let ∆ denote a convex body contained in the cone Σ(S, 0, ∞). We introduce two assumptions, to be used in later statements. The first one concerns ∆ and S: Assumption A.1. The convex body ∆ satisfies ∆ ∩ σ(S, 0, 0) = ∅ and there exists ǫ > 0 such that for any α ∈ S ∆ ∩ Σ({α}, 0, ǫ) = (∆ ∩ α ⊥ ) × [0, ǫ]α ∩ Σ(S, 0, ∞) (note that the decomposition on the right-hand side is with respect to the orthogonal decomposition V = α ⊥ ⊕ Rα).

The second assumption is an assumption on functions w : ∆ → R:

Assumption A.2. The function w : ∆ → R is non-negative, convex and for any subset R ⊂ S, for any p ∈ Σ(R, 0, 0), the directional derivative of w at p is nonnegative for any direction ξ ∈ Σ ∩ Vect(R).

Let dp denote a Lebesgue measure on V . The goal of this appendix is to provide a proof to the following general version of [LZZ18, Lemma 4.6]. Apart from presentation, the proof is identical to the original, we include it as a courtesy to the reader and since the version we use in the core of the article does not follow from the statement of [LZZ18, Lemma 4.6]. wdp Note that the assumptions are obviously satisfied if S is the set of simple roots of a restricted root system, ∆ is the polytope associated to a polarized horosymmetric variety satisfying assumption (T) as in Section 7 and w is the restriction of the convex conjugate of a normalized toric potential (which is invariant under the restricted Weyl group hence satisfy Assumption A.2). The statement used in the core of the article is rather the following direct consequence, applied to the Duistermaat-Heckman polynomial.

Corollary A.4. Assume (S, ∆) satisfy Assumption A.1, and assume g : ∆ → R is such that for any ǫ > 0, inf ∆∩Σ(S,ǫ,∞) g > 0, then there exists C > 0 such that for any w : ∆ → R satisfying Assumption A.2, ∆ wdp ≤ C ∆ wgdp Proof of Proposition A.3. The proof goes by (strong) induction on the cardinality of S and relies on two ingredients. The first and main ingredient is a decomposition of ∆ into several parts using Assumption A.1, such that the induction hypothesis applies to all but one. The second ingredient is an elementary use of convexity and Assumption A.2 to deal with the remaining part.

Note already that the initialization is trivial: if S is empty, then the statement is also empty. We now describe the decomposition of ∆. Let S denote the set of all subsets R of S satisfying ∆ ∩ σ(R, 0, 0) = ∅. It contains S itself by assumption. Fix an ǫ > 0 small enough so that Assumption A.1 holds with this value of ǫ. Set ∆ S := ∆ ∩ Σ(S, 0, ǫ/2) then in the orthogonal decomposition V = Σ(S, 0, 0) ⊕ Vect(S), Assumption A.1 shows that ∆ S = (∆ ∩ Σ(S, 0, 0)) × (Σ(S, 0, ǫ/2) ∩ Vect(S)).

Figure 2 .

 2 Figure 2. Colored data for type AIII(2, m > 4)

  -trivial contribution in Ω α1,3, ᾱ1,3 which is equal to

  Using Lemma 3.9, we write exp(D) = exp(-a)k exp(a + y + O) exp(h).Then φ(exp(a) exp(D)) = φ(k exp(a + y + O) exp(h)) by the equivariance property of the quasipotential (Proposition 3.5), this is = φ(exp(a + y + O)) -2 ln |χ(exp(h))| Recall from Lemma 3.9 and the notations introduced before this lemma that y = B D + B E and exp(h) = exp(C E ) exp(C D ) where E = 1 2 ([B D , D] + [C D , B D ] + [C D , D]) and B D , C D , B E , C E are provided by Lemma 3.8. Note that ln |χ(exp(C E ) exp(C D ))| = ln |χ(exp(C E ))| + ln |χ(exp(C D ))| = ln |e χ(CE ) | + ln |e χ(CD ) | where we still denote by χ the Lie algebra character h → C induced by χ, = Re(χ(C E ) + χ(C D )) = Re(χ(C E + C D )).

  we have B E = O and C E =Oe 2α1(a) z1 z 2 H([θ(e α1 ), e α2 ])/2 e 2α2(a) z 1 z2 H([θ(e α2 ), e α1 ])/2 + tanh((α 2α 1 )(a))ℜ(-e 2α1(a) z1 z 2 H([θ(e α1 ), e α2 ])/2)) + coth((α 2α 1 )(a))ℑ(-e 2α1(a) z1 z 2 H([θ(e α1 ), e α2 ])/2))

  then we have [θ(e α1 ), e α2 ] = [θ(e α2 ), e α1 ] = 0 hence B E = O and C E = O, and we deduce Ω α1, ᾱ2 = 0. 4) Consider now the case D = z 1 l j + z 2 τ β . Then B D = ℜ(z 1 l j ) and

5)

  Consider the case D = z 1 τ β + z 2 e α . Then we have B D = 0 and C D = -e 2α(a) θ(z 2 e α ) + tanh(β(a))ℜ(z 1 µ β ) + coth(β(a))ℑ(z 1 µ β )

  [e βj , θ(e β k )] ∈ g βj-β k and [e βj , θσ(e β k )] ∈ g βj -σ(β k ) . It shows that B E is negligible, and that ℜ(C E ) is equal to the sum of a negligible term and tanh(β 2 (a)β 1 (a)) sinh(2β 1 (a)) ℜ(z 1 z 2 H([θ(e β1 ), e β2 ])) + tanh(β 2 (a)σ(β 1 )(a)) sinh(2β 1 (a)) ℜ(z 1 z 2 H([θσ(e β1 ), e β2 ])) + tanh(β 1 (a)β 2 (a)) sinh(2β 2 (a)) ℜ(z 2 z 1 H([θ(e β2 ), e β1 ])) + tanh(β 1 (a)σ(β 2 )(a)) sinh(2β 2 (a)) ℜ(z 2 z 1 H([θσ(e β2 ), e β1 ])).

4. 1 .

 1 Definition and examples. Definition 4.1. A horosymmetric variety X is a normal G-variety such that G acts with an open dense orbit which is a horosymmetric homogeneous space.

  further contains a unique Levi subgroup L I containing T , and Φ LI ∩ S = I. We denote by P I the parabolic subgroup opposite to Q I with respect to L I .The subgroup B ∩L is a Borel subgroup of L containing T , and we have the same correspondence between subsets I of S L and parabolic subgroups Q L I of L containing B ∩ L. We have the obvious relationQ L I = Q I ∩ L,and all of these parabolic subgroups are contained inQ SL = Q. The Levi subgroup of Q L I containing T is none other than L I .Given a one parameter subgroup µ ∈ Y(T ), we obtain a subset of S L by settingI(µ) = {α ∈ S L ; α(µ) = 0}. Then the Levi subgroup L I (µ) of Q L I(µ) containing T coincides with the centralizer Z L (µ) of µ.In particular, µ is contained in the radical of the Levi subgroup L I(µ) . Pay attention to the fact that Z L (µ) may be different from Z G (µ) here.Take µ ∈ Y(T s ). Then the restriction of σ to L I(µ) = Z L (µ) is still a well defined involution. Since µ is contained in the radical of L I(µ) , we may choose a σ-stable complement m of Cµ in l I(µ) which contains the derived Lie algebra [l I(µ) , l I(µ) ]. Define a new involution σ µ on L I(µ) by setting, at the level of Lie algebras,

  Such a section is called B-semi-invariant if it is an eigenvector for the action of B, that is, there exists a character λ of B such that b • s = λ(b)s. A meromorphic section s of L defines a Cartier divisor D s = div(s) representing L. If s is B-semi-invariant, then D s is B-invariant. There are two types of irreducible B-stable Weil divisor on X: the closure of colors D ∈ D of G/H, and the facets of X. Denote by I G (X) the set of facets of X. Since D s is by definition Cartier, it writes, by [Bri89, Proposition 3.1],

  by definition of χ, hence the theorem.Recall the characterization of ample and globally generated line bundles proved by Brion. Given a maximal cone C contained in F X , let m C denote the element of M ⊗ Q such that v L (y) = m C (y) for y ∈ C.

Figure 3 .

 3 Figure 3. Relationship between polytopes

The

  convex conjugate u * : a * s → R ∪ {+∞} of u is the convex function defined by u * (p) = sup y∈as (p(y)u(y)).

6. 3 .

 3 Metric induced on a facet. Let L be a G-linearized line bundle on a horosymmetric embedding (X, x) of G/H. Assume that L admits a special section s and write the special divisor as D L = Y n Y Y + D n D D. For every facet Y of X, let µ Y ∈ Y(T s ) denote the indivisible generator of the ray corresponding to Y in the colored fan of X, denote by E Y ⊂ X the corresponding elementary embedding and let x Y = lim z→0 µ Y (z) • x.

  E Y and satisfies, for b ∈ B, ψ λ (bH) = ψ(bH) -2 ln λ(b). The toric potential of h is u(a) = ψ(exp(a) • x) and the toric potential of the restriction of h to L| Y is the function u Y defined by u Y (b) = ψ λ (exp(b) • x Y ) for b ∈ a Y . Let us also define the function u λ on a s by u λ (a) = ψ λ (exp(a) • x) = u(a) -2λ(a). Note that d a u λ = d a u -2λ and that d k u λ = d k u for k ≥ 2. Proposition 6.5. For any integer k ∈ N, for any sequence of real numbers (t j ) such that lim t j = -∞, and for any sequence (b j ) of elements of a Y such that lim b j = b ∈ a Y , we have lim d k tj µY +bj u λ (b 1 , . . . , b k ) = d k b u Y (b 1 , . . . , b k ) and, assuming k ≥ 1 and j < k, lim d k tj µY +bj u λ (µ Y , . . . , µ Y , b 1 , . . . , b j ) = 0. Proof. These statements are essentially direct consequences of the use of log coordinates near a divisor. The first statement follows directly from the smoothness of h, hence of ψ λ , once one remarks that u λ (t j µ Y + b j ) = ψ λ (exp(t j µ Y + b j ) • x) and u Y (b) = ψ λ (exp(b) • x) by definition.

  extends to the whole toric subvariety. Again by smoothness, lim d k exp(tj µY +bj )•x ψ λ ( ∂ /∂|zr|, . . . , ∂ /∂|zr|, b 1 , . . . , b j ) is well defined and finite for any vector fields b k that extend to the divisor z r = 0. Translating in terms of the function u λ yields the result since |z r | r-j converges to zero. Assume now that h is positively curved on the elementary embedding E Y ⊂ X. Then we may consider the convex conjugates u * , u * λ and u * Y . Proposition 6.6. Let b ∈ a Y . Then at p = d b u Y we have u * Y (p) = u * λ (p) = u * (p + 2λ). Proof. The second inequality follows from elementary properties of the convex conjugate. For the other equality, we have

  s and b s such that sd b u Y + (1s)d tµY +b u λ = d tsµY +bs u λ . The fact that lim s→1 d tsµY +bs u λ = d b u Y ensures that lim t s = -∞ and that lim b s = b (it certainly ensures that b s is bounded, then considering converging subsequences yields the limit since u Y is smooth and strictly convex), hence the statement. It is clear from the point of view of the toric subvariety Z that the domain of u * Y

Remark 7. 6 .

 6 As a corollary of the proof, by applying the same steps, we can compute SΘ . We obtainSΘ = Y ∆+ Y (nΛ Y P DH (q) + d q P DH (χ ac -Λ Y χ))dq ∆ +P DH (q)dq. 7.4. Action of Z(L) 0 . Following Donaldson [Don02], let us write the Mabuchi functional as the sum of a linear and a non-linear part Mab Θ = Mab l Θ + Mab nl Θ by setting

7.4. 1 .

 1 Invariance of Mabuchi functional and log-Futaki invariant. Consider the connected center Z(L) 0 of L. It acts on the right on G/H and the action extends to X. The induced action on K-invariant singular Hermitian metrics on L stabilizes the set rPSH K (X, ω ref ). More precisely, for b ∈ a s ∩ z(l), and h a K-invariant, nonnegatively curved singular Hermitian metric on L with toric potential u, the toric potential of the image by exp(b) of h is a → u(a + b). This translates on convex conjugates as replacing u * by u * b = u *b. Note that it is enough to consider only elements of a s ∩ z(l) since Z(L) 0 = T ∩ Z(L) 0 , and T ∩ H as well as T ∩ K act trivially. Since du * b = du *b, α(b) = 0 for α ∈ Φ L , χ(b) = 0 and d 2 u * b = d 2 u * , we have the following proposition. Proposition 7.8. The Mabuchi functional is invariant under the action of Z(L) 0 on the right if and only if

7. 5 .

 5 Coercivity criterion. 7.5.1. Statement of the criterion and examples.

Figure 4 .

 4 Figure 4. Moment polytopes for type AIII(2, 5)

  ∆ ′ u * (p)P ′ DH (p)dp ≤ C ∂ ∂∆ ′ u * P ′ DH (p)dσfor some constant C ∂ independent of u * .Proposition 7.17. Under the assumptions of Theorem 7.10, there exists a positive constant C l such that for any normalized u * ,M l (u * ) ≥ C l ∂∆ ′ u * P ′ DH (p)dσ.Proof. Suppose there exists a sequence of normalized u * j such that∂∆ ′ u * j P ′ DH (p)dσ = 1and M l (u) decreases to 0. By a compactness theorem proved by Donaldson [Don08, Section 5.2], we may assume that u * j converges locally uniformly on-2∆ t to a convex function v ∞ , still satisfying v ∞ (2(χλ 0 )) = min v ∞ = 0.Let p 0 = 2χ -2bar. By convexity of u * we haved p u * (pp 0 ) ≥ u * (p)u * (p 0 ), hence M l (u * ) ≥ Y ∆′ Y F L (q)(u * (p)u * (p 0 )d p0 u * (pp 0 ))P ′ DH (p)dp + Y ∆′ Y d p u * (Λ Y p 0 + 4ρ H )P ′ DH (p)dp + Y ∆′ Y F L (q)d p0 u * (pp 0 )P ′ DH (p)dp + Y ∆′ Y nΛ Y -SΘ + (χ ac -Λ Y χ)(α ∨ ) q(α ∨ ) u * (p 0 )P ′ DH (p)dp

  P ′ DH (p)dp → 0. Let δ = min Y (1c Y )/v L (µ Y ), it is positive by assumption. By the expression of M l given in Remark 7.7, and using again that u * j converges to 0, we obtainlim inf j M l (u * j ) ≥ δ > 0,which provides a contradiction hence proves the proposition.Remark 7.18. Assumption (R) was not used at all here.7.5.3. Proof of coercivity. The strategy to transfer the coercivity result on the linear part to the full functional now follows a general strategy already used by Donaldson in[START_REF] Simon | Scalar curvature and stability of toric varieties[END_REF]. First note that -I H (a) -4ρ H (a) ≥ 0 for a ∈ -a + s , so that M nl (u * ) ≥ -∆ + ln det(u * i,j )(p)P ′ DH dp.

  C b and C c , M nl (u * ) ≥ -C a (ln ǫ + 1) -ǫC b ∂∆ ′ u * P ′ DH (p)dσ -ǫC c ∆ ′ u * dp Assumption (T) finally allows us to use Corollary A.4 to obtain, for some positive constant C d ,M nl (u * ) ≥ -C a (ln ǫ + 1) -ǫC b ∂∆ ′ u * P ′ DH (p)dσ -ǫC c C d ∆ ′ u * P ′ DH dpPutting everything together we haveM (u * ) ≥ M l (u * )ǫ(C c C d ∆ ′ u * (p)P ′ DH (p)dp + C b ∂∆ ′ u * (p)dp) -C a (ln ǫ + 1) ≥ ǫ ∆ ′ u * P ′ DH + M l (u * )ǫ (C c C d + 1) ∆ ′ u * P ′ DH + C b ∂∆ ′ u * (p)dp -C a (ln ǫ + 1) ≥ ǫ ∆ ′ u * P ′ DH + M l (u * )ǫ((C c C d + 1)C ∂ C l + C b C l )M l (u * ) -C a (ln ǫ + 1) ≥ ǫ ∆ ′ u * P ′ DH -C a (ln ǫ + 1), by choosing ǫ = ((C c C d + 1)C ∂ C l + C b C l ) -1 .

  Proposition A.3. Assume (S, ∆) satisfy Assumption A.1, then there exists ǫ > 0 and C > 0 such that for any w : ∆ → R satisfying Assumption A.2, ∆ wdp ≤ C ∆∩Σ(S,ǫ,∞)

  maximal torus in L which contains a split torus T s of maximal dimension among split tori. It turns out that any split torus is contained in a σ-stable maximal torus of L [Vus74] hence maximally split tori exist. From now on, T denotes a maximally split torus in L with respect to σ, and T s denotes its maximal split subtorus. If T σ denotes the subtorus of elements of T fixed by σ, then T σ × T s → T is a surjective morphism, with kernel a finite subgroup. The dimension of T s is called the rank of the symmetric space L/L ∩ H.

  linearized line bundle L is globally generated if and only if • The function v L is convex and • n D,L ≥ m C (ρ(D)) for all D ∈ D \ D X and maximal cone C ∈ F X . It is ample if and only if it is globally generated and furthermore

  From these two applications of the divergence theorem and the expression of the scalar curvature, we deduce by taking the sum that I is C ′

	We may compute
		u * t (-u * ,i,j t,j + I H,i (a))P ′ DH i	= -u * t,i u * ,i,j t,j P ′ DH + u * t,i I H,i (a)P ′ DH
						-u * t u * ,i,j t,i,j P ′ DH + u * t u * j,i I H,i,j (a)P ′ DH
						-u * t u * ,i,j t,j P ′ DH,i + u * t I H,i (a)P ′ DH,i
	and			u * t,i I H,i (a)P ′ DH =	d dt	(I H (a)P ′ DH ).
	Consider now the vector field ( u * t,i P ′ DH -u * t P ′ DH,i )u * ,i,j t	. Applying the divergence
	theorem to this vector field yields
	0 =	∆ ′	( u * t,i,j u * ,i,j t	P ′ DH + u * t,i u * ,i,j t,j P ′ DH -u * t u * ,i,j t,j P ′ DH,i -u * t u * ,i,j t	P ′ DH,i,j )dp
	Note here that	u * t,i,j u * ,i,j t	=	d dt	(ln det(u * t,i,j )).

* ,i,j t,j + I H,i (a))P ′ DH i dp. H /2 n times the derivative with respect to t of

  We thus have, for some positive constant C a , M nl (u * ) ≥ -C a (ln ǫ + 1)ǫ DH We now use a variant of Donaldson's integration by parts [Don02, Lemma 3.3.5]. Namely, we apply the divergence theorem to the vector field u * ,i,j ref,i u * P ′ DHu * ,i,j ref u * i P ′ DH + u * ,i,j ref u * P ′

					∆ +	u * ,i,j ref u * i,j P ′
						DH,i
	to obtain
	Y	∆ ′ Y	u * P ′ DH dσ =	∆ ′	u * u * ,i,j ref,i,j P ′ DH + u * 2u * ,i,j ref,j P ′ DH,i
					+ u * u * ,i,j ref P ′ DH,i,j -u * ,i,j ref u * i,j P ′ DH dp.

* ,i,j ref u * i,ju * ,i,j ref u * ref,i,jln det u * ref,i,jr ln ǫ

We can then define, by induction, the subsets ∆ R for R ∈ S by setting

Each (R, ∆ R ) satisfies Assumption A.1 and the cardinality of R is smaller than that of S when R = S, so by induction hypothesis we may find an ǫ 0 > 0 and C 0 > 0 such that ∆\∆S udp ≤ C 0 (∆\∆S)∩Σ(S,ǫ0,∞) udp.

Finally, we treat the case of ∆ S . Let F denote the set F = {x ∈ Vect(S) ∩ Σ(S, 0, ǫ); ∃α ∈ S, α, x = ǫ} Then we have Appendix B. Properness on invariant potentials and existence of constant scalar curvature metrics Let X be a complex projective manifold, and let be L an ample line bundle on X. Assume (X, L) is equipped with two actions:

• the action of a compact Lie group acting K, and • the action of a connected real Lie group N which normalizes the action of

Recall that we defined in Section 7.3.1 and Section 7.3.2 the functionals J and Mab (in this appendix, Θ is empty) on the space of smooth and invariant Kähler potentials rPSH K (X, ω ref ) for a fixed reference metric ω ref ∈ c 1 (L). Recall that the values of Mab and J depend only on the Kähler metric ω φ = ω ref + i∂ ∂φ defined by φ. In particular it makes sense to define J(n • φ i ) for n ∈ N without fixing a normalization of the potentials.

• it is bounded from below on smooth K-invariant potentials, and

A few remarks are in order before moving on: i) Properness with respect to J, as defined here, is equivalent to properness with respect to Mabuchi's L 1 distance d 1 in restriction to potentials normalized by vanishing of the Aubin-Mabuchi functional thanks to [DR17, Proposition 5.5].

ii) It is standard that boundedness from below of the Mabuchi functional on smooth K-invariant potentials implies N -invariance of the Mabuchi functional by linearity on the families of metrics induced by real one-parameter subgroups of the connected group N (see e.g. [CC18b, Lemma 3.3], the restriction to invariant potentials makes no difference here).

iii) Our definition of coercivity (Definition 7.9) implies properness modulo Z(G) 0 on smooth K-invariant potentials of the Mabuchi functional in the setting of the article.

A close examination of Chen and Cheng's arguments in [START_REF] Chen | On the constant scalar curvature Kähler metrics, general automorphism group[END_REF] allows to obtain the following statement.

Theorem B.2. Assume the Mabuchi functional Mab is proper modulo N on smooth K-invariant Kähler potentials, then there exists a constant scalar curvature metric in c 1 (L).

Proof. Starting from a K-invariant reference metric ω 0 (which exists by averaging) we consider the continuity path of twisted constant scalar curvature equations, for

. By [CC18a, Corollary 4.5] and [BDL17, Theorem 4.7] a solution to the above equation is unique as long as t < 1. Since ω 0 is K-invariant, it implies that any solution for t < 1 is K-invariant. This simple remark allows to apply [CC18a, Theorem 4.1] and [CC18b, Lemma 3.6] in restriction to invariant potentials to obtain solutions for any t < 1.

Let now t i denote a sequence of elements of [0, 1] increasing to 1. Let φi denote the corresponding solutions. By [CC18b, Lemma 3.7], the Mabuchi functional is uniformly bounded along the solutions φi . Properness modulo N on invariant potentials then implies

The conclusion of the theorem then follows directly from [CC18b, Proposition 3.9].

Appendix C. Properness on invariant potentials and existence of log-Kähler-Einstein metrics

Let (X, Θ) denote a log Fano klt pair, and assume that X is smooth, for simplicity in dealing with smooth Kähler metrics. Assume as in the previous appendix that X is equipped with two actions, both stabilizing each component of Θ:

• the action of a compact Lie group acting K,

• and the action of a connected real Lie group N which normalizes the action of K, that is for any n ∈ N , k ∈ K, there exists

In order to follow more closely [BBE + 16], we introduce some notations closer to theirs. We fix a reference metric ω ref in c 1 (X) -Θ. Let E 1 denote the space of finite energy potentials with respect to ω ref as defined in [BBE + 16, Section 1.4]. It maps bijectively to the space T 1 of finite energy currents via the map φ → ω φ = ω ref + i∂ ∂φ and T 1 maps to the space M 1 of finite energy probability measures via the map ω → V -1 ω n where V denotes the volume of (X, L). We further let φ → MA(φ) = V -1 ω n φ denote the composition of the two maps, and let ω → φ ω denote the inverse map from T 1 to E 1 .

Let µ ref denote the adapted measure of the pair (X, Θ) [BBE + 16, Definition 3.1]. Another finite energy probability measure associated to a current ω is obtained as the probability measure µ ω corresponding to the measure defined by e -φω µ ref [BBE + 16, Lemma 3.4].

Definition C.1. [BBE + 16, Definition 3.5] A finite energy current ω is a (weak) log-Kähler-Einstein metric on the pair (X, Θ) if V -1 ω n = µ ω .

We will use the following functionals on E 1

On M 1 , we will use the functionals

The Mabuchi functional M and the Ding functional D are defined on T 1 by

We say analogously as before that the functional M is proper modulo N on smooth K-invariant Kähler metrics if it is bounded from below on smooth K-invariant Kähler metrics and if any sequence

The definition of the (log) Mabuchi functional used in Section 7.3.2 differs from the one above in general, but the difference is bounded independently of the metric [BHJ16, p.24, proof of Theorem 4.2], so our definition of coercivity for the log Mabuchi functional implies properness of M modulo Z(G) 0 on smooth K-invariant Kähler metrics.

Recall that E 1 , T 1 and M 1 are homeomorphic using the previously defined bijections. For a sequence (ω i ) in T 1 , strong convergence translates as weak convergence of ω i to some ω ∞ , together with convergence of J(φ ωi ) to J(φ ω∞ ).

Theorem C.2. Assume M is proper modulo N on smooth K-invariant Kähler metrics in c 1 (X) -Θ, then there exists a log-Kähler-Einstein metric on the pair (X, Θ).

Proof.

Step 1: Properness provides a candidate log-Kähler-Einstein metric. Indeed, the first item of the properness assumption on M implies that it is bounded from below on smooth K-invariant metrics. Let ω i denote a sequence of smooth K-invariant metrics such that M(ω i ) converges to the infimum of M on smooth K-invariant metrics. By the second item of the properness assumption and Ninvariance, we may as well replace the ω i by n i ω i (for a sequence of n i ∈ N ), so that J(φ ωi ) is bounded. It then follows from the comparison between J and E * [BBE + 16, (1.10)] that E * is bounded on the set {V -1 ω n i } i . Since M and E * are bounded, it follows that the entropy H is bounded on the set {V -1 ω n i } i . By [BBE + 16, Theorem 2.17] we can thus replace the sequence (ω i ) by a subsequence strongly converging to some ω ∞ ∈ T 1 . Note that since all ω i are K-invariant, the same is true for the limit ω ∞ .

Step 2: The candidate ω ∞ is a minimizer of M in T K 1 . This claim follows from the fact that the Mabuchi functional on T K 1 is the greatest lower semicontinuous extension of its restriction to smooth K-invariant metrics (with respect to the strong topology). Since E * is continuous, it suffices to consider the entropy H. It is lower semicontinuous and the fact that it is the greatest extension of its restriction to smooth K-invariant metrics is proved in [BDL17, Lemma 3.1]. In [START_REF] Berman | Convexity of the extended Kenergy and the large time behavior of the weak Calabi flow[END_REF], this result is actually proved without the K-invariance property, but the construction preserves K-invariance, thanks to density of K-invariant smooth functions on X in K-invariant L 1 -functions on X, and uniqueness in the Calabi-Yau Theorem.

Step 3: The Ding functional D is also minimized at ω ∞ in T K 1 . Here we use and imitate

where the last steps holds by definition of E * .

Step 4: A minimizer of D in T K 1 is a log-Kähler-Einstein metric. To prove this claim, we follow the proof of ii) ⇒ i) in [BBE + 16, Theorem 4.8]. The only modification is an argument to pass from K-invariant test functions to arbitrary test functions. Let v ∈ C 0 (X) K be a continuous K-invariant function on X. Let φ t denote the ω ref -psh envelope of the function φ ω∞ + tv, then it is a function in E K 1 and the proof of [BBE + 16, Theorem 4.8] shows, by differentiating the Ding functional along t → φ t on both sides of 0, that X vµ ω∞ = V -1 X vω n ∞ Given an arbitrary v ∈ C 0 (X), we may apply Fubini's Theorem to the average of v with respect to the invariant probability measure dk on K to obtain

As a conclusion, ω ∞ is a log-Kähler-Einstein metric on the pair (X, Θ).