Thibaut Delcroix
email: thibaut.delcroix@ujf-grenoble.fr

ALPHA-INVARIANT OF TORIC LINE BUNDLES

We generalize the work of Jian Song to compute the α-invariant of any (nef and big) toric line bundle in terms of the associated polytope. We use the analytic version of the computation of the log canonical threshold of monomial ideals to give the log canonical threshold of any non-negatively curved singular hermitian metric on the line bundle, and deduce the α-invariant from this.

Introduction

The α-invariant of a line bundle L on a complex manifold X is an invariant measuring the singularities of the non-negatively curved singular hermitian metrics on L. It was introduced by Tian in the case of the anticanonical bundle on a Fano manifold. Tian showed in [START_REF] Tian | On Kähler-Einstein metrics on certain Kähler manifolds with C 1 (M ) > 0[END_REF] that if the α-invariant of the anticanonical bundle is strictly greater than n n+1 , then the Fano manifold admits a Kähler-Einstein metric.

The Yau-Tian-Donaldson conjecture asserts in general that X admits an extremal metric in c 1 (L) if and only if the line bundle L is K-stable. It was proved in [START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities[END_REF][START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π[END_REF][START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof[END_REF][START_REF] Tian | K-stability and kähler-einstein metrics[END_REF] that it holds when L is the anticanonical bundle. In particular (as it was shown also in [START_REF] Odaka | Alpha invariant and K-stability of Q-Fano varieties[END_REF]), if the α-invariant of the anticanonical bundle is greater than n n+1 , then the anticanonical bundle is K-stable. Dervan [Der] gave a similar condition of K-stability for a general line bundle, involving again its α-invariant. This is one motivation to compute explicitly the α-invariants of line bundles when possible.

In [START_REF] Chel | Log-canonical thresholds for nonsingular Fano threefolds[END_REF], Chel'tsov and Shramov computed for example the α-invariant of the anticanonical bundle for many Fano manifolds of dimension three. In higher dimensions, Song [START_REF] Song | The α-invariant on toric Fano manifolds[END_REF] proved a formula giving the α-invariant of the anticanonical bundle on a toric Fano manifold in terms of its polytope. The only toric manifolds satisfying Tian's criterion are the symmetric toric manifolds. Batyrev and Selivanova [START_REF] Batyrev | Einstein-Kähler metrics on symmetric toric Fano manifolds[END_REF] proved first that their α-invariant was one, so that they admit a Kähler-Einstein metric. Wang and Zhu [START_REF] Wang | Kähler-Ricci solitons on toric manifolds with positive first Chern class[END_REF] fully settled the question of the existence of Kähler-Einstein metrics on toric Fano manifolds, and an illustration that Tian's criterion is only a sufficient condition can be found in the toric world [START_REF] Nill | Examples of Kähler-Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes[END_REF].

The α-invariant of a line bundle L is strongly related to the log canonical thresholds (lct) of metrics on L. The log canonical threshold was initially an algebraic invariant defined for ideal sheaves, but it was shown to coincide with the complex singularity exponent and Demailly defines the log canonical threshold of any nonnegatively curved singular hermitian metric on a line bundle in [START_REF] Chel | Log-canonical thresholds for nonsingular Fano threefolds[END_REF] for example.

One of the main examples of computation of log canonical threshold is in the case of monomial ideals. Howald carried out the computation of the lct of such an ideal in terms of its Newton polygon [START_REF] Howald | Multiplier ideals of monomial ideals[END_REF]. One can find in Guenancia [START_REF] Guenancia | Toric plurisubharmonic functions and analytic adjoint ideal sheaves[END_REF] an analytic proof of this result, generalized to compute the lct of an ideal generated by a "toric" psh function on a neighborhood of 0 ∈ C n , i.e. a function invariant under rotation in each coordinate.

Since the only smooth affine toric manifolds without torus factor are isomorphic to C n , the computation of Guenancia in fact gives the log canonical threshold of any invariant metric on an affine smooth toric manifold, as we explain in Section 2.

In this note, we give a formula for the α-invariant of any line bundle L on a compact smooth toric manifold in terms of its polytope. We also compute the log canonical threshold of any invariant non-negatively curved singular metric on L.

After this article was accepted, the author was informed that other authors computed similar invariants using other methods (H. Li, Y. Shi, Y. Yao [START_REF] Li | A criterion for the properness of the K-energy in a general Kähler class[END_REF], and F. Ambro [Amb]).

1. Line bundles on smooth toric manifolds 1.1. Toric manifolds. Let us recall some basic facts about toric varieties (see [START_REF] Fulton | Introduction to toric varieties[END_REF], [START_REF] Oda | Convex bodies and algebraic geometry[END_REF], [START_REF] Cox | Toric varieties[END_REF]).

Let T = (C * ) n be an algebraic torus. Denote its group of characters by M , which is isomorphic to Z n through the choice of a basis, and let

M R := M ⊗ R ≃ R n .
The dual N of M consists of the one parameter subgroups of T , and we let also

N R := N ⊗ R ≃ R n .
We denote by T c ≃ (S 1 ) n the compact torus in T .

Considering only cones for the toric setting, we will call σ ⊂ N R a cone if σ is a convex cone generated by a finite set of elements of N . The dual cone σ ∨ is defined as

σ ∨ = {x ∈ M R | x, y ≥ 0 ∀y ∈ σ}.
A fan Σ consists of a finite collection of cones σ ⊂ N R such that every cone is strongly convex (i.e. {0} is a face of σ), the faces of cones in Σ are in Σ and the intersection of two cones in Σ is a union of faces of both. The support of Σ is

|Σ| := σ∈Σ σ ⊂ N R .
Recall that a fan Σ in N R determines a toric variety X Σ , that is, a normal Tvariety with an open and dense orbit isomorphic to T , and every toric variety is obtained this way.

By the orbit-cone correspondence [CLS11, Theorem 3.2.6], a maximal cone σ of Σ corresponds to a fixed point z σ in X Σ . Also, a one-dimensional cone ρ in Σ corresponds to a prime invariant divisor D ρ of X Σ , and these divisors generate the group of Weil divisors of X Σ . Let ρ be such a cone, then we denote by u ρ the primitive vector in N generating this ray. We will denote by Σ(r) the set of r-dimensional cones in Σ.

Many properties of X Σ can be read off from the fan. For example, X Σ is smooth if and only if every cone in the fan Σ is generated by part of a basis of N . We will call a cone smooth if it satisfies this condition. The variety X Σ is complete if and only if |Σ| = N R .

We will assume in general in the following that either |Σ| = N R or that Σ is given by a strongly convex, full dimensional cone σ and its faces, in which case we will denote X σ the corresponding (affine) toric variety.

1.2. Line bundles. Recall that a line bundle L on a G-variety X is called linearized if there is an action of G on L such that for any g ∈ G and x ∈ X, g sends the fiber L x to the fiber L g•x and the map defined this way between L x and L g•x is linear.

To a T -linearized line bundle L on X Σ is associated a set of characters v σ , for σ ∈ Σ(n). We define v σ as the opposite of the character of the action of T on the fiber over the fixed point z σ .

This defines the support function g L of L, which is a function on the support |Σ| of Σ, linear on each cone, which takes integral values at points of N , by x → v σ , x for x ∈ σ.

Another equivalent data is the Weil divisor D L associated to L, which is related to g L by the following:

D L = -ρ g L (u ρ )D ρ .
If L is effective, then to L is associated a polytope P L in M R . This polytope can be defined as

P L = {m ∈ M R |g L (x) ≤ m, x ∀x ∈ |Σ|} .
The properties of the line bundle can be read off from the polytope or the support function. In particular, we can associate to each point of P L ∩ M a global section of L, and the collection of these sections form a basis of the space of algebraic sections of L. Recall also the following, where we assume that |Σ| = N R .

Proposition 1.1. [CLS11, Theorem 6.1.7] The following are equivalent:

• L is nef • L is generated by global sections • {v σ } is the set of vertices of P L • g L is concave.
Proposition 1.2. [CLS11, Lemma 9.3.9] L is big iff P L has nonempty interior.

Proposition 1.3. [CLS11, Lemma 6.1.13] The line bundle L is ample iff g L is concave and

v σ = v σ ′ whenever σ = σ ′ ∈ Σ(n).
Example 1.4. The anticanonical divisor -K XΣ on a toric manifold is given by -K XΣ = ρ D ρ . It is always big on a toric manifold.

1.3. Non-negatively curved singular metrics on line bundles.

1.3.1. Potential on the torus. Let L be a T -linearized line bundle on X Σ . Recall that any linearized line bundle on T ≃ (C * ) n is trivial. Fix an invariant trivialization s of L on T .

Given a hermitian metric h on the line bundle L, we denote by ϕ h the local potential of h on T , which is the function on T defined by:

ϕ h (z) := -ln(||s(z)|| h ).
The local potentials of a smooth hermitian metric are smooth. We will work here with singular metrics, whose local potential are a priori only in L 1 loc . A singular hermitian metric h is said to have non negative curvature (in the sense of currents) if and only if every local potential of h is a psh function.

A T c -invariant function ϕ on T is determined by a function f on N R , identified with the Lie algebra of T c , through the equivariant isomorphism:

T c × N R -→ T ; ((e iθj ) j , (x j ) j ) → (e xj+iθj ) j .
Furthermore, ϕ is psh if and only if f is convex.

So to a non negatively curved, T c -invariant metric h on L is associated a convex function f h , which is the function on N R determined by ϕ h .

Behavior at infinity of the potentials.

Definition 1.5. Let L be a nef line bundle on X Σ . The function f L : x → -g L (-x) is a convex function on N R , and it is the potential of a continuous, T c -invariant, non negatively curved metric on L called the Batyrev-Tschinkel metric (see [START_REF] Maillot | Géométrie d'Arakelov des variétés toriques et fibrés en droites intégrables[END_REF]), which we denote by h L .

Proposition 1.6. The map h → f h defines a bijection between the singular hermitian T c -invariant metrics on L with non-negative curvature, and the convex functions on N R , such that there exists a constant C with

f h ≤ f L + C on N R .
Proof. See also [START_REF] Berman | Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties[END_REF]Proposition 3.3]. Let h be a singular hermitian T c -invariant metrics on L with non-negative curvature. Write h = e -v h L , and let ω L be the curvature current of h L . Then v is a ω L -psh function on X. In particular, v is bounded from above on X. Denote by u the convex function on R n associated to the T c -invariant function v| T . Then we see that

f h (x) -f L (x) = u(x) is bounded above on N R .
Conversely, the standard fact that a psh function, which is bounded from above, extends uniquely over an analytic set, allows one to extend u := f -f L to an ω L -psh function on the whole of X if f satisfies the condition of the proposition.

Log canonical thresholds

2.1. Definition. Let X be a compact complex manifold, and L a line bundle on X. Let h be a singular hermitian metric on L. We recall the definition of the log canonical threshold of h (see the appendix of [START_REF] Chel | Log-canonical thresholds for nonsingular Fano threefolds[END_REF]).

Definition 2.1. Let z ∈ X. The complex singularity exponent c z (h) of h at z is the supremum of the real c > 0 such that e -2cϕ is integrable in a neighborhood of z, where ϕ is a local potential of h near z.

Definition 2.2. The log canonical threshold lct(h) of h is defined as

lct(h) = inf z∈X c z (h).

Newton body of a function.

Definition 2.3. Let σ be a cone. Let f be a function defined on N R . Define the Newton body of f on σ as

N σ (f ) = {m ∈ M R ; f (x) -m, x ≥ O(1), ∀x ∈ σ}. If σ = N R we will write N (f ).
The following properties of the Newton body will be useful.

Proposition 2.4. For any function f , N σ (f ) is convex, and

N σ (f ) = N σ (f ) -σ ∨ . If f is convex, then for any y ∈ N R , N σ (f ) = {m ∈ M R ; f (t) -m, t ≥ O(1), ∀t ∈ y + σ}.
Proof. The first two properties are trivial. Let us briefly prove the last statement.

Let m be in the right-hand set, i.e.

{f (t) -m, t ≥ O(1) ∀t ∈ y + σ}. Let x = t -y ∈ σ for t ∈ y + σ. By convexity, f (x + y) ≤ 1 2 (f (2x) + f (2y)) so we get f (2x) ≥ 2f (x + y) -f (2y) = 2f (t) -f (2y) Subtracting m, 2x gives f (2x) -m, 2x ≥ 2(f (t) -m, t ) + (2 m, y -f (2y)).
The right hand side is the sum of a lower-bounded function of t ∈ y + σ and a constant, so the left hand side is a lower-bounded function of x ∈ σ.

This shows one inclusion and the other is proved by a similar argument.

Given a non negatively curved T c -invariant metric h on L, we define the associated convex subset P h of M R , as the Newton body of f h . Proposition 2.5.

• For the Batyrev-Tschinkel metric h L , we recover the polytope P L .

• For any T c -invariant, non-negatively curved metric h on L, P h ⊂ P L .

• If h is smooth, we also have

P h = P L Proof.
For the first statement, observe that m ∈ P L if and only if for any cone σ ∈ Σ, for all x ∈ σ, g L (x) = v σ , x ≤ m, x . This inequality is equivalent to -v σ , x + m, x ≥ 0 and since the functions involved are linear, it is satisfied for all x ∈ σ if and only if

-v σ , x + m, x is bounded below on σ. Since f L (-x) = -g L (x) = -v σ ,
x for x ∈ σ, we get that m ∈ P L if and only if for every cone σ ∈ Σ, the function f L (-x) -m, -x is bounded below on σ. Finally, this can be translated as: for every cone σ ∈ Σ, the function f L (y) -m, y is bounded below on -σ. To conclude, we note that

N (f L ) = σ N -σ (f L ).
The second statement is an easy consequence of the first and Proposition 1.6 since whenever two functions f and g satisfy f ≤ g + C for a constant C, we have trivially N σ (f ) ⊂ N σ (g).

For the last statement, remark that in this case, f h -f L extends to a continuous function on X Σ , so we have f L -C ≤ f h ≤ f L + C for some constant C. The same property of Newton bodies allows one to conclude.

Integrability condition.

The first result on log canonical thresholds on toric varieties was the computation by Howald [START_REF] Howald | Multiplier ideals of monomial ideals[END_REF] in the case of monomial ideals. Guenancia gave an analytic proof of this result, extending the computation to the case of non algebraic psh functions. The key ingredient in this analytic version is the following integrability condition.

Proposition 2.6. (see [START_REF] Guenancia | Toric plurisubharmonic functions and analytic adjoint ideal sheaves[END_REF]) Let σ be a smooth cone of maximum dimension. Let f be a convex function on N R . Then e -f is integrable on all translates of σ if and only if 0 ∈ Int(N σ (f )). This is essentially the result in Guénancia [START_REF] Guenancia | Toric plurisubharmonic functions and analytic adjoint ideal sheaves[END_REF] because any smooth affine toric manifold with no torus factor is isomorphic to C n . However we describe the change of variables used precisely, to use it later in the compact case.

Proof. Choose a basis of N formed by the generators of the extremal rays of σ, then define S σ to be the isomorphism from N to Z n sending the chosen basis to the canonical basis of Z n .

Let f be a function on N R , and g the function on R n such that f = g • S σ . Then from the definition of Newton body we have N σ (f ) = S * σ (N D (g)), where S * σ is the dual isomorphism from Z n to M and D is the cone generated by the canonical basis of Z n .

Using the change of variables, e -f is integrable on all translates of σ if and only if e -f •S -1 σ is integrable on all translates of D. Apply [Gue12, Proposition 1.9] to the concave function -f • S -1 σ . This proves that we have integrability if and only if 0 ∈ Int(N D (f • S -1 σ )). Using S * σ , which is linear, this indeed translates to 0 ∈ Int(N σ (f )).

Remark that the statement in [Gue12, Proposition 1.9] only mentions integrability on D, but the equivalence with integrability on all translates is easily derived from Proposition 2.4.

lct on an affine smooth toric manifold.

Proposition 2.7. Let σ be a smooth cone of maximum dimension, X σ the corresponding smooth affine toric manifold. Let L be a linearized line bundle on X σ , and h a T c -invariant metric with non-negative curvature. Then

lct(h) = sup{c > 0|cv σ ∈ Int(N -σ (cf h )) -S * σ (1, . . . , 1)}.
Proof. The change of variables for cones S σ in the proof of Proposition 2.6 gives (by [CLS11, Theorem 3.3.4]) an equivariant isomorphism between X σ and C n , which we denote again by S σ . Any linearized line bundle on C n is trivial, so L admits a global equivariant trivialization t on X σ . Remark that, at the fixed point z σ , we have g •t(z σ ) = -v σ (t(z σ )) by definition of v σ . Restricting to T and remembering that s is an invariant trivialization of L on T , we deduce that up to renormalization by a constant, t(z) = v σ (z)s(z) on T .

We can now look at the potential ψ of h with respect to the trivialization t, and remark that, on T , and if ϕ denotes the potential of h with respect to s on T , we have ψ(z) = -v σ , ln |z| + ϕ(z).

Let y ∈ N R . Using again the isomorphism T c × N R ≃ T , we consider T c × (y -σ) as a subset of T , and denote by C y the closure of this set in X σ . Each set C y is a neighborhood of z σ in X σ , and they form a basis of neighborhoods. Observe that the collection of the translates of -σ cover N R and so the corresponding sets cover X σ . More precisely, for any point z in X σ , there is a translate of -σ which corresponds to a neighborhood of z.

We consider first the complex singularity exponent of h at z σ . Suppose c > 0 is such that e -2cψ is integrable in a neighborhood of z σ . Then it is integrable in a neighborhood C y . We have first that, Cy e -2cψ(z) dz ∧ dz = Tc×(y-σ) e -2cψ(z) dz ∧ dz.

Recall that ψ(z) = -v σ , ln |z| + ϕ(z), and that f is the function on N R such that f (x) = ϕ(e x ).

Say we have chosen a basis of N or equivalently of M , and we denote by (x i ) i=1...n the coordinates of x ∈ N R along this basis. This determines local holomorphic coordinates z i = e xi+iθi on T ≃ N R × T c . Using the fact that dzi zi ∧ dzi zi = dx i ∧ dθ i , and T c -invariance, we obtain that, up to a constant, Cy e -2cψ(z) dz ∧ dz = y-σ e -2c(f (x)+ -vσ,x ) e 2 i xi dx.

Since i x i is equal to S * σ (1, . . . , 1), x , we conclude by using Proposition 2.6 that the complex singularity exponent c zσ (h) is the supremum of the c > 0 such that 0 ∈ Int(N -σ (2c(f + -v σ , • ) -2 S * σ (1, . . . , 1), • )). To obtain a simpler condition, remark that for any function g and positive scalar λ, N -σ (λg) = λN -σ (g), and that if g 1 and g 2 are two convex functions then N -σ (g 1 + g 2 ) is the Minkowski sum of N -σ (g 1 ) and N -σ (g 2 ).

So we get c zσ (h) = sup{c > 0|cv σ ∈ Int(N -σ (cf )) -S * σ (1, . . . , 1)}. Furthermore, for any c < c zσ (h), the Proposition 2.6 shows that e -2cψ is integrable on every C y for y ∈ N R . Observe now that for any point z ∈ X σ , there exists a C y containing z. So for any point z ∈ X σ , c z (h) ≥ c zσ (h). This concludes the proof of the proposition.

2.5. lct on a compact smooth toric manifold.

Theorem 2.8. Let X Σ be a smooth compact toric manifold, L a linearized line bundle on X Σ and h a T c -invariant non-negatively curved metric on L. Then

lct(h) = sup{c > 0|cP L ⊂ Int(cP h + P -KX Σ )}.
Proof. The compact manifold X Σ is covered by the affine toric manifolds X σ , for σ ∈ Σ(n). By definition of the log canonical threshold,

lct(h) = min σ∈Σ(n) lct(h| Zσ ).
Another way to say this is that lct(h) is the sup of c > 0 such that c ≤ lct(h| Xσ ) for all σ ∈ Σ(n). Now this condition means, by Proposition 2.7, that for all σ ∈ Σ(n),

cv σ ∈ Int(N -σ (cf h + -S * σ (1, . . . , 1), • )
. By Proposition 2.4, this is equivalent to the condition that for all σ ∈ Σ(n),

cv σ + σ ∨ ⊂ Int(N -σ (cf h + -S *
σ (1, . . . , 1), • ). This is further equivalent to the condition that for all σ ∈ Σ(n),

σ∈Σ(n) (cv σ + σ ∨ ) ⊂ Int(N -σ (cf h + -S * σ (1, . . . , 1), • ).
Recall from Proposition 2.5 that σ∈Σ(n) (v σ + σ ∨ ) = N (f L ) = P L , so that the condition can be written:

N (cf L ) ⊂ σ∈Σ(n) Int(N -σ (cf h + -S * σ (1, . . . , 1), • ) = Int(N (cf h + f -KX Σ )).
Indeed, the support function of the anticanonical bundle is, from Example 1.4,

f -KX Σ (x) = -S * σ (1, . . . , 1), x .

Alpha-invariant

3.1. Log canonical threshold and α-invariant. Let X be a compact Kähler manifold, L a big and nef line bundle on X.

Definition 3.1. Assume that a compact group K acts on X, and that L is Klinearized. The alpha invariant α K (L) of L with respect to the group K is defined as the infimum of the log canonical thresholds of all K-invariant, non negatively curved singular hermitian metrics on L.

The linear systems in a multiple of L give singular metrics on L, that we will call algebraic metrics, in the following way. Let δ 1 , . . . , δ r ∈ H 0 (X, mL) be linearly independent sections, and denote by ∆ the linear system generated by these. Then it defines an algebraic metric h ∆/m on L by setting, in any trivialization,

||ξ|| 2 h ∆/m = |ξ| 2 ( |δ j (z)| 2 ) 1/m , for any ξ ∈ L z . The local potential ϕ ∆/m (z) = 1 2m ln |δ j (z)| 2 is psh. If ∆ is
one dimensional, generated by δ, we denote by h δ/m the corresponding metric.

Recall the following result of Demailly, relating the α-invariant with log canonical thresholds of algebraic metrics: Theorem 3.2. [CS08, Appendix A] Let K be a compact group, let X be a compact complex K-variety and L a big and nef K-linearized line bundle on X. Then

α K (L) = inf m∈N * inf ∆⊂H 0 (X,mL), ∆ K =∆ lct(h ∆/m ).
One can slightly improve this result, and give the following statement, which is only given in the case of a trivial group K by Demailly.

Corollary 3.3. Let K be a compact group, let X be a compact complex K-variety and L a big and nef K-linearized line bundle on X. Then

α K (L) = inf m∈N * inf ∆∈Irr(H 0 (X,mL)) lct(h ∆/m ),
where Irr(H 0 (X, mL)) denotes the set of all irreducible K-subrepresentations of H 0 (X, mL).

Proof. Let ∆ be a K-invariant subspace of H 0 (X, mL), then ∆ = ∆ 1 ⊕ • • • ⊕ ∆ s with ∆ i irreducible subspaces.
For all i, one can choose a basis δ i j of ∆ i . Together they form a basis of ∆ and we can obtain the metric h ∆ this way.

In particular, ϕ ∆/m (z) = 1 2m ln i j |δ i j (z)| 2 . Since the logarithm is increasing we can write

ϕ ∆/m (z) ≥ 1 2m ln |δ 1 j (z)| 2 = ϕ ∆1/m (z)
. This implies, by elementary properties of the complex singularity exponent, [DK01, 1.4] that lct(h ∆/m ) ≥ lct(h ∆1/m ).

We conclude that the log canonical threshold of a metric associated to a Kinvariant linear system is greater than the log canonical threshold of at least one metric associated to an irreducible linear system, so it is enough to consider only these.

3.2. General formula. Let X Σ be a smooth compact toric manifold. Let N (T ) be the normalizer of T in Aut(X Σ ), and denote by W = N (T )/T the Weyl group obtained from T .

The group N (T ) naturally acts on M and since T acts trivially on M , this induces an action of W on M . By duality one also gets an action on N .

From the description of morphisms between toric varieties [CLS11, Theorem 3.3.4], we can see that W is isomorphic to the subgroup of GL(N ) composed of the ρ such that ρ(Σ) = Σ. In particular, W is finite.

Given a subgroup G of W , we denote by T G the preimage in N (T ) of G, and let K G := K ∩ T G . If P is a polytope in M R we let P G be the set of G-invariant points of P .

Finally, if P is a polytope in M R , we denote by P (Q) the set of rational points in P , i.e. points p such that there exists m ∈ N * with mp ∈ M . Theorem 3.4. Let L be a T G -linearized line bundle on X Σ . Then

α KG (L) = inf p∈P G L (Q) sup{c > 0|cP L ⊂ Int(cp + P -KX Σ )}.
Proof. The Corollary 3.3 shows that it is enough to consider algebraic metrics on L associated to K G -irreducible linear system in a multiple of L.

The T c -irreducible subrepresentations of H 0 (X Σ , mL) are the dimension one subspaces corresponding to integral points of the polytope P mL associated to mL. Recall that P mL = mP L . Now a K G -irreducible subrepresentation of H 0 (X Σ , mL) is the union of the images by G of a T c -irreducible representation.

Let p be an integral point in mP L , and denote by ∆ the K G -irreducible linear system generated by the G-orbit of p.

The potential of h ∆/m is

ϕ ∆/m (z) = 1 2m ln   g∈G |(g • p)(z)| 2   .
By arithmetico-geometric inequality,

ϕ ∆/m (z) ≥ 1 2m ln g∈G (g • p) |G| (z) 2 .
The right-hand side of this inequality is the potential of the algebraic metric h g∈G (g•p) m|G| corresponding to the linear system of H 0 (X Σ , m|G|L) generated by the section g∈G (g • p).

Using again the fact that the complex singularity exponent is increasing [DK01, 1.4], we get lct(h ∆/m ) ≥ lct(h g∈G (g•p)

m|G|

).

We have thus shown that it is enough to compute the log canonical thresholds of algebraic metrics associated to one dimensional G-invariant sublinear systems of multiples of L.

We use Theorem 2.8 to conclude. Indeed if p ∈ mP L generates a one dimensional G-invariant sublinear system in H 0 (X Σ , mL), and f p/m denotes the convex function associated to the potential of the corresponding algebraic metric h p/m , we have N (f p/m ) = {p/m}.

Applying Theorem 2.8 gives

lct(h p/m ) = sup{c > 0|cP L ⊂ Int(cp/m + P -KX Σ )}.
Finally, observe that as p and m vary, they describe the set P G L (Q) of G-invariant points of P L with rational coordinates. Remark 3.5. One can also prove, without the use of Corollary 3.3, that we can consider only metrics corresponding to points of P L (not necessarily with rational coordinates), by considering the expression of the log canonical threshold of any metric. p 0 w p Indeed, if f is a convex function on N R , corresponding to a metric h on L, and p is a point in N (f ), then the metric h p associated to the convex function x → p, x is also a non-negatively curved metric on L, and lct(h p ) ≤ lct(h).

3.3.

Case of the anticanonical line bundle. We assume in this section that L = -K XΣ .

This line bundle admits a natural Aut(X)-linearization, and the polytope associated to this linearization contains the origin in its interior, because -K X is big.

For any subgroup G of W , let S G := {p ∈ ∂P L |g • p = p ∀g ∈ G}. If 0 = p ∈ P L , let w p be the point ∂P L ∩ {-tp|t ≥ 0}.

Remark 3.6.

• S G is empty if and only if {0} is the only point fixed by G in P .

• If S W is empty, X Σ is called symmetric. Consider the half-line starting from p and containing the origin. It intersects ∂P at w p . Denote by r its intersection with ∂(p + P ).

Then it is easy to see that the log canonical threshold of h p is equal to the quotient of the distance between p and r by the distance between p and w p . The translation sending 0 to p also sends w p to r, so |r-p| = |w p |. The result follows.

Remark 3.8. If P h = {0} then lct(h) = 1.

Example 3.9. Consider the case P h = {b}, where b is the barycenter of the polytope P L . Then lct(h) is equal to the greatest lower bound for Ricci curvature R(X), introduced by Székelyhidi [Szé11], and computed for toric manifolds by Li [START_REF] Li | Greatest lower bounds on Ricci curvature for toric Fano manifolds[END_REF].

From this formula we recover the previous results of Song and Chel'tsov-Shramov.

Theorem 3.10. [Son05] [CS08, Lemma 6.1] Let X be a smooth Fano toric manifold, and G be a subgroup of W . Then

• if S G is empty, α KG (X) = 1; • else, α KG (X) = 1 1+maxp∈S G |p| |wp | ≤ 1 2 .
Proof. By Theorem 3.4, it is enough to consider only the (rational) G-invariant points of P . The first case follows immediately using Remark 3.8. In the second case, we obtain the formula using Proposition 3.7. Indeed, it is enough to consider points p in S G because if q = 0 is not in ∂P , and p is the intersection of ∂P with the half line starting from the origin and going through q, then lct(h q ) ≥ lct(h p ).

Furthermore, max p∈SG |p| |wp| ≥ 1 because otherwise if p was such a point at which this maximum was attained and it was < 1 then we would have |wp| |p| > 1 with w p ∈ S G , which is a contradiction.

3.4. Example. We compute the α-invariant of any linearized line bundle on the blow up X of P 2 at one point which we denote X in the following.

Identify N with Z 2 . The fan of X has four rays, with generators u 1 = (1, 0), u 2 = (1, 1), u 3 = (0, 1) and u 4 = (-1, -1).

The group W is isomorphic to Z/2Z and acts on M R by exchanging the coordinates (x, y) → (y, x).

We define the polytope P (k, l) to be the polytope whose vertices are (0, k), (0, l), (k, 0) and (l, 0), for k, l ∈ N with l > k. It is easy to see that the polytopes of nef and big divisors are the P (k, l), up to translation by a character. For example, the polytope of the anticanonical bundle is Q := (-1, -1) + P (1, 3).

Proposition 3.11. The α-invariant with respect to K W of the nef and big line bundle corresponding to P (k, l) is equal to inf( 1 l-k , 2 l ). Proof. By Theorem 3.4, it is enough to consider points (with rational coordinates) in the intersection of P (k, l) with the first diagonal. However, one easily remarks that it is enough to consider only the point (l/2, l/2), similarly to the proof of Theorem 3.10.

We want to compute sup{c > 0|cP (k, l) ⊂ Int(c(l/2, l/2) + Q)}.

This is of course equal to sup{c > 0|P (k, l) ⊂ Int((l/2, l/2) + 1 c Q)}.

Observe that l/2 is the least positive constant b such that {(0, l), (l, 0)} ⊂ (l/2, l/2) + bQ.

If k ≥ l/2, then we have also {(0, k), (k, 0)} ⊂ (l/2, l/2) + l/2Q, so P (k, l) ⊂ (l/2, l/2) + l/2Q.

Thus α KW (P (k, l)) = 2/l when k ≥ l/2. For the other case, observe that l -k is the least positive constant b such that (k/2, k/2) ∈ (l/2, l/2) + bQ. If k ≤ l/2, then we have also P (k, l) ⊂ (l/2, l/2) + (l -k)Q.

Thus α KW (P (k, l)) = 1 l-k when k ≥ l/2.

Proposition 3. 7 .

 7 Assume that P h = {p} with 0 = p ∈ P L . Then lct(h) = |w p | |w p | + |p| . Proof. By Theorem 2.8 we have lct(h) = sup{c > 0|cP ⊂ Int(cp + P )}.


Louis Gostiaux

Thierry Dauxois

Laboratory experiments on the generation of internal tidal beams over steep slopes

Keywords: Stratified fluids -Internal waves -Internal tides PACS numbers: 47, 55, Hd Stratified flows, 47, 35, +i Hydrodynamic waves

We designed a simple laboratory experiment to study internal tides generation. We consider a steep continental shelf, for which the internal tide is shown to be emitted from the critical point, which is clearly amphidromic. We also discuss the dependence of the width of the emitted beam on the local curvature of topography and on viscosity. Finally we derive the form of the resulting internal tidal beam by drawing an analogy with an oscillating cylinder in a static fluid.

Over the past two decades it has become apparent that substantial internal tides can be generated by tidal currents over ridges and other rough topography of the ocean floor. This problem is of paramount importance since baroclinic tides, generated by barotropic currents over ocean ridges and seamounts, are an important source of the ocean interior mixing. Indeed, recent observations suggest that mixing in the abyssal oceans is rather weak, except in localized regions near rough topography. It explains the discrepancy between the observed intensity of mixing in the interior of the oceans and is required to satisfy models of ocean circulation [START_REF] Garrett | The dynamic ocean in Perspectives in Fluid Dynamics[END_REF]. This suggests that these topographic effects must be incorporated into realistic climate and circulation models.

This question has been studied theoretically and numerically, while related oceanic observations have been reported. Baines showed [START_REF] Baines | On the internal tide generation models[END_REF] first that points where the topographic slope coincides with the angle of propagation might be an effective generator of internal wave. Such a point is referred to as the critical point (see Fig. 1). Internal beams emanating from the continental slope have been observed recently [START_REF] Azevedo | On the generation and propagation of internal solitary waves in the southern Bay of Biscay[END_REF][START_REF] Gerkema | Internal tides in the Bay of Biscay: conversion rates and seasonal effects[END_REF][START_REF] Jézéquel | Interaction of semidiurnal tide with a continental slope in a continuously stratified ocean[END_REF][START_REF] New | Local generation of internal soliton packets in the central Bay of Biscay[END_REF][START_REF] Pingree | Abyssal penetration and bottom reflection of internal tidal energy in the bay of biscay[END_REF]. On the theoretical side, let us mention several interesting descriptions [START_REF] Petrelis | Tidal conversion at a submarine ridge[END_REF][START_REF] St | The generation of internal tides at abrupt topography[END_REF] of abrupt discontinuity cases such as a topographic step, a knife-edge ridge (zero width ridge as a simple model for tall ridges such as the Hawaiian one), a tent-shaped ridges, or smooth ridges (gaussian, polynomial). Finally, recent numerical reports have also provided some new insight about this mechanism [START_REF] Gerkema | Internal tides in the Bay of Biscay: conversion rates and seasonal effects[END_REF][START_REF] Gerkema | De-cay of semi-diurnal internal tide beams due to subharmonic resonance[END_REF].

All these works converge to conclude that beams of internal tides energy arise due to the interplay between oscillating currents and bathymetric features. Of particular interest is the location where the slope is critical, i.e. where the direction of the wave beam is tangent to the slope. In the remaining of the paper, we experimentally study the location of emission and the form of the internal beam which is emitted. We also explain how this form can be predicted by drawing an analogy between the emission of an internal tide from a static topography, and the emission of internal waves by an oscillating cylinder in a static fluid. The agreement is very good, even quantitatively.

The experiment was performed in a rectangularshaped tank (length 120 cm, height 40 cm and width 10 cm) as sketched in Fig. 1. A curved thin (0.5 cm) PVC plate, introduced before filling, played the role of the continental shelf. The experiment was performed with a linear stratification, pure water at the free surface while highly salted water at the bottom, resulting in a constant Brunt-Vaisälä buoyancy frequency N ≃ 0.81 rad/s. In the absence of rotation, the dispersion relation of internal gravity waves reads ω = N sin θ, where θ is the angle of the beam with respect to the horizontal. The tidal current was generated by a second PVC plate localized far from the continental slope, i.e. in the region corresponding to the abyssal plane or to the open ocean. This plate was fixed at its bottom end, with its top end oscillating horizontally. The oscillation A sin(ωt) was monitored by a position control motor of a printer ap-paratus driven by a sinusoidal electronic signal delivered by a Kepco amplifier. The phase origin at t = 0 corresponds to a vertical position of the paddle and the movement is initiated leftwards towards the topography. Several forcing tidal frequencies and amplitudes were tried out. The tidal frequency ω was satisfying the condition 0 < ω < N , so that internal tides may be freely radiating. Results and figures discussed throughout this paper correspond to ω/(2π) = 0.050 Hz for which θ = 23 • , and A = 0.5 cm.

The density gradient field within the tank was finally obtained with the standard synthetic Schlieren technique [START_REF] Dalziel | Whole density field measurements by synthetic schlieren[END_REF] by acquiring successive side views with a CMOS AVT Marlin F131B Camera. A sheet of paper with points randomly scattered was located 150 cm behind the tank. Successive images (1280 × 300 pixels) of those points obtained with the camera located 280 cm in front of the tank were adequately treated by a homemade software. A correlation image velocimetry algorithm [START_REF] Fincham | Advanced optimization of correlation imaging velocimetry algorithms[END_REF] was applied between a reference image taken prior to the experiment and the different snapshots. As we used 21 × 21 pixels correlation boxes with a 75% overlap, the resolution for the two dimensional density gradient field is 256 × 60.

Figure 2a show a snapshot of the vertical density gradient. To facilitate the visualization, a narrower domain than in the experiments is shown. It corresponds to the dotted rectangle depicted in Fig. 1. To increase the visibility, we filtered at the excitation frequency ω the time series of the density gradient field over one experimental tidal period (see Ref. [START_REF] Gostiaux | Quantitative laboratory observations of internal wave reflection on ascending slopes[END_REF] for more details about this method).

An internal tide is clearly seen to emanate from the upper part of the slope and radiate away from the shelfbreak in both directions. However, it is important to stress the absence of a third beam radiated transversally to the topography. This contradicts Baines analytical theory [START_REF] Baines | On the internal tide generation models[END_REF] in which this third beam reflects on the surface and is present in the general solution for the downward propagating wave. The explanation presumably lie in the presence of a singular point in Baines' case or in Saint Laurent et al numerical simulation [START_REF] St | The generation of internal tides at abrupt topography[END_REF]. In numerical experiments with a smooth slope and a typically shallow continental shelf [START_REF] Gerkema | Internal tides in the Bay of Biscay: conversion rates and seasonal effects[END_REF], no such beam was found and the presence of this third beam was already ambiguous in previous experiments [START_REF] Baines | Internal tide generation at a continental shelf/slope junction: a comparison between theory and a laboratory experiment[END_REF].

The amplitude of the vertical density gradient is given in terms of variations of the squared Brunt-Väisälä frequency. If one considers the original value of N 2 = 0.66 rad 2 /s 2 , the measured amplitude of ∆N 2 = ±0.005 rad 2 /s 2 for the internal wave is a very small perturbation of the original stratification. Using the mass conservation relation

iω(ρ -ρ) = w d ρ dz , ( 1
)
where ρ is the perturbated density, ρ(z) the initial density and w the vertical density, we obtain typical vertical velocities of order ±0.11 mm/s, corresponding to vertical displacements of ±0.15 mm. This is one of the interests of the synthetic Schlieren technique that allows to measure very weak perturbations of the buoyancy field and thus to investigate weakly nonlinear regimes. This vertical amplitude has to be compared to the barotropic elevation of the water induced by the forcing. The paddle oscillates with an A = 0.5 cm amplitude, and the width of the free water volume in the tank is 90 cm for a height of H = 12 cm. The corresponding elevation of the water at the level of the slope is thus ∆H = 0.7 mm. The baroclinic component observed is thus still a perturbation of the barotropic tide.

To the right of the generation point C, energy propagation is downward while, to the left, it is upward. As a consequence of the internal waves propagation law for which group and phase velocities are orthogonal with opposite vertical components, phase propagation is thus upward to the right and downward to the left. One can conclude from this simple observation that the phase has to rotate around C, which is therefore an amphidromic point. Our filtering technique allows to evaluate the phase of the wave [START_REF] Gostiaux | Quantitative laboratory observations of internal wave reflection on ascending slopes[END_REF], which is plotted in Fig. 2b. One can clearly see that the isophase lines converge on a single point previously referred as the generation point C, around which the phase rotates uniformally.

Whereas the location and the inclination of the internal tide are well understood, the selection mechanism of the width of the beam was not yet clearly identified by previous studies. Several length scales can be considered in this problem. The first one corresponds to the thickness of the oscillating boundary layer δ = (ν/ω) 1/2 where ν is the kinematic viscosity. In the present case, δ = 1.8 mm. The second one is the local radius of curvature of the continental shelf R = 3.3 cm. At last, the dimensions of the shelf itself (h, H...) that play a role [START_REF] Baines | On the internal tide generation models[END_REF] in the "flat-bump" geometry for which α < θ happen to be irrelevant in the configuration of a steep topography.

Our understanding is directly inspired from the generation of internal waves by oscillating cylinders. One can therefore try to draw an analogy between the internal tide generation by a curved static topography of a given radius of curvature R and the internal waves generation by an oscillating cylinder of the same radius R in a stratified fluid. has been averaged over one tidal period around 3T . We use the Hurley and Keady analytical solution [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF] for oscillating cylinders to modelize the internal wave beam. We consider the tangential circle at the shelf break as the generating cylinder section and compute the analytical solution isolating the upper right emission point. Vertical dashed lines correspond to the cross section plotted in Fig. 4.

In their seminal theoretical work [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF], Hurley and Keady showed how to get the velocity profile of the beams generated by such a cylinder. First, they proposed to consider a dimensionless parameter λ = ν/(2R 2 ω cot θ) which is proportional to the squared ratio of the oscillatory boundary layer thickness δ that surrounds the cylinder to its radius R. The cylinder emits the four beams of the well known St Andrew's configuration. Each beam is considered independently. Then, the longitudinal velocity component v s for each beam is shown to be given by

v s (s, σ) = V 0 +∞ 0 J 1 (K) exp(-λsK 3 /R + iσK) dK, (2)
where J 1 is the Bessel function of the first kind of order 1 and s (resp. σ) the longitudinal (resp. transversal) coordinate along (resp. across) the beam. V 0 is the projection of the oscillation velocity along the beam. In the domain δ = (ν/ω) 1/2 ≪ R, this function happens to be localized on the characteristics σ = ±R corresponding to the lines of slope tan θ, tangential to the cylinder. In this limit case, 2 × 4 = 8 beams are emitted from the four critical points of the cylinder.

In our case, we only consider the upper characteristic corresponding to the single critical point C of the problem. Hurley and Keady's expression restricted to this single characteristic happens to fit very well our experimental data. In the tidal experiment, λ ≃ 6.3 10 -4 . We estimate the buoyancy perturbation of the stratification by means of Eq. [START_REF] Azevedo | On the generation and propagation of internal solitary waves in the southern Bay of Biscay[END_REF]. By appropriately taking into account the inclination of the beam, one gets a final expression for the vertical density gradient that can be compared to our experimental data.

Fig. 3 compares the vertical density gradient field in the experimental (a) and the theoretical (b) cases. The analytical solution of Hurley and Keady is computed on an imaginary cylinder tangential to the shelf break region. The amplitude of oscillation of the cylinder corresponds to the amplitude of the tidal flow. The agreement is qualitatively very good. The phase opposition between the two parts of the beams is also well described by the model.

To get a more quantitative comparison between experiment and theory, we plot on Fig. 4 five vertical cross sections (indicated in Fig. 3) of the vertical density gradient field at x = -6, -3, 6 and 9 cm relative to the generation point. The quantitative agreement between the experimental and the theoretical profiles is very good, which confirms that the structure of the emitted beam only depends on the viscous boundary layer thickness δ and on the local radius of curvature of the shelf R.

First of all, this experiment proves that the internal tide generation conditions are met when the radiated tidal beam is aligned with the slope of the topography, corresponding to a critical point of emission which happens to be amphidromic. We clearly demonstrate the absence of any transverse emission at that point that is supposed to occur in Baines analytical study [START_REF] Baines | On the internal tide generation models[END_REF]. It is however important to emphasize that this disagreement might come from the absence of any sharp corner in the shelf break, contrary to what has been considered in Refs [START_REF] Baines | On the internal tide generation models[END_REF][START_REF] Baines | Internal tide generation at a continental shelf/slope junction: a comparison between theory and a laboratory experiment[END_REF][START_REF] St | The generation of internal tides at abrupt topography[END_REF]. Following this observation, we derive a simple analytical model derived from the theoretical work of Hurley and Keady [START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF] and show that we can quantitatively estimate the density perturbation profile Since this paper focuses on the internal tide generation, we dot not consider the evolution of the internal tides as they propagate further into the deep ocean. Surface reflections, in particular, are expected to generate internal solitary waves by creating local disturbances at the seasonal thermocline [START_REF] Gerkema | Internal and interfacial tides: beam scattering and local generation of solitary waves[END_REF][START_REF] New | Local generation of internal soliton packets in the central Bay of Biscay[END_REF]. Moreover, the emitted internal wave of tidal period often breaks up into internal waves of shorter period. Indeed, and very interestingly, a recent numerical report has emphasized that parametric subharmonic resonance may come into play in the rotating case [START_REF] Gerkema | De-cay of semi-diurnal internal tide beams due to subharmonic resonance[END_REF]. Experimental work along those lines is in progress.

FIG. 1 :

 1 FIG. 1: Sketch of the experimental set-up. A PVC plate is oscillating between the solid and dashed oblique lines indicated on the right, creating the tide. The continental shelf consists of two planar PVC plates connected by a quarter of cylinder of radius R = 3.3 cm. The upper plate is nearly horizontal, while the downgoing one makes an angle α = 78 • with the horizontal. The aspect ratio h/H is 2/3, for a total water height h + H = 19 cm. The critical point C indicates the location where the internal tide generation is supposed to occur, i.e. where the internal beam inclined at an angle θ = 23 • with the horizontal is tangent to the topography. The dotted rectangle indicates the zone captured by the camera and presented in Fig. 2a.

FIG. 2 :

 2 FIG. 2: (Color online) Panel a presents the two-dimensional vertical density gradient, while panel b shows the phase. Both plots have been averaged over one tidal period of the experiment around t = 3T . Horizontal and vertical distances are in cm. The generation point C appears to be an amphidromic point around which the phase of the wave rotates.

FIG. 3 :

 3 FIG. 3: (Color online) Comparison between experiment (a) and theory (b) for the vertical density gradient field. Panel (a)has been averaged over one tidal period around 3T . We use the Hurley and Keady analytical solution[START_REF] Hurley | The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution[END_REF] for oscillating cylinders to modelize the internal wave beam. We consider the tangential circle at the shelf break as the generating cylinder section and compute the analytical solution isolating the upper right emission point. Vertical dashed lines correspond to the cross section plotted in Fig.4.

FIG. 4 :

 4 FIG.4: Vertical cross sections of the measured vertical density gradient (thin line) and its theoretical prediction using Hurley and Keady's theory (bold line). From top to bottom, the cross sections correspond to the abscisse x = -6, -3, 3, 6 and 9 cm.

  as a function of the viscosity and the local radius of curvature of the topography.

Acknowledgments We thank F. Petrelis and S. Llewellyn Smith for helpful discussions. Comments to the manuscript by Denis Martinand are deeply appreciated. This work has been partially supported by the 2005 PATOM CNRS program and by 2005-ANR project TOPOGI-3D.


Thibaut Delcroix

LOG CANONICAL THRESHOLDS ON GROUP COMPACTIFICATIONS

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical threshold. As a consequence we obtain a formula for the alpha invariant of these line bundles, in terms of the polytope associated to the group compactification.

Introduction

The aim of this article is to begin the study of Kähler metrics on polarized G × G-equivariant compactifications of a connected complex reductive group G. This class of manifolds generalizes the well known class of polarized toric manifolds, and we extend some techniques of toric geometry to this setting. In this article, we study singular hermitian metrics on linearized ample line bundles, which are nonnegatively curved and invariant under the action of K × K, where K is a maximal compact subgroup of G. We associate to such a metric a convex function on some real vector space, that we call the convex potential of the metric, and show how the asymptotic behavior of this function is controlled by a polytope associated to the line bundle. This generalizes the case of toric manifolds and relies on the KAK decomposition of a reductive group.

The correspondence between metrics and their convex potentials is a bijection and provides a description of the set of non-negatively curved, K × K-invariant, singular hermitian metrics. Furthermore, to obtain this description we use a special continuous reference metric that generalizes the Batyrev-Tschinkel metric on toric line bundles, which had already been used as a model for the behavior of continuous metrics in [START_REF] Chambert-Loir | Igusa integrals and volume asymptotics in analytic and adelic geometry[END_REF].

We then proceed to compute the log canonical threshold of such metrics, in terms of the asymptotic behavior of their convex potential. To achieve that goal, we associate a convex body to the metric, that we call the Newton body of the metric, which gives another way to encode the asymptotic behavior of the convex potential, and is well suited to fan decompositions. We should stress at this point that another important ingredient is the existence of a toric subvariety (and corresponding fan) in any group compactification, that contains the information about the compactification. We obtain the following theorem.

Theorem A. Let (X, L) be a polarized G × G-equivariant compactification of G. Assume furthermore that X is Fano. Denote by P the polytope associated to L and by Q the polytope associated to the anticanonical line bundle -K X . Let also H denote the convex hull of the images, by the Weyl group W of G, of the sum of the 1 positive roots of G. Let h be a K × K-invariant hermitian metric with non negative curvature on L, then the log canonical threshold of h is given by: lct(h) = sup{c > 0; 2H + 2cP ⊂ cN (h) + 2Q}, where N (h) is the Newton body of h.

Using this expression of the log canonical threshold, we are able to compute Tian's α invariant for any ample linearized line bundle on a Fano group compactification, with respect to the K × K action.

Theorem B. Let (X, L) be a polarized compactification of G, and P := P (X, L). Assume furthermore that X is Fano and let Q := P (X, -K X ). Then α K×K (L) = sup{c > 0; c(P + (-P W )) ⊂ Q ⊖ H}, where P W denotes the subset of W -invariant points of P and W is the Weyl group of G.

This formula generalizes the formula for the α-invariant of polarized Fano toric manifolds previously obtained in [START_REF] Delcroix | Alpha-invariant of toric line bundles[END_REF], and independently by other authors [START_REF] Ambro | Variation of log canonical thresholds in linear systems[END_REF][START_REF] Li | A criterion for the properness of the K-energy in a general Kähler class[END_REF]. In the case of the anticanonical line bundle on a Fano toric manifold, the formula was initially obtained by Song [START_REF] Song | The α-invariant on toric Fano manifolds[END_REF].

The original motivation of this work was to obtain such an expression, hoping that Tian's criterion for the existence of Kähler-Einstein metrics in terms of this invariant would be satisfied by some group compactifications. Recall that Tian's criterion [START_REF] Tian | On Kähler-Einstein metrics on certain Kähler manifolds with C 1 (M ) > 0[END_REF] is that if the α invariant is strictly greater than n n+1 where n is the dimension of the manifold, then there exists a Kähler-Einstein metric.

For toric manifolds, additional symmetries had to be taken into account for the criterion to be satisfied. For the important examples of wonderful compactifications of semisimple adjoint groups with no rank one factor, Brion computed their automorphism group in [START_REF] Brion | The total coordinate ring of a wonderful variety[END_REF], so that our result allows to compute the alpha invariant with respect to a maximal compact subgroup of the automorphism group, but Tian's criterion is not satisfied. Even though we can in some cases consider additional symmetries, we do not obtain new examples of Kähler-Einstein metrics by this method. Our study of hermitian metrics will be used in [Dela] where we obtain a necessary and sufficient condition for the existence of Kähler-Einstein metrics on a group compactification in terms of the polytope. The present article and [Dela] contain the main results of the author's PhD thesis [Delb].

Group compactifications

1.1. Definition and examples. Let G be a connected complex reductive group. Definition 1.1. Let X be a projective manifold. We say that X is a smooth G×Gequivariant compactification of G (or in short a compactification of G) if X admits a holomorphic G×G-action with an open and dense orbit equivariantly isomorphic to G as a G × G-homogeneous space under the action defined by (g 1 , g 2 ) • g = g 1 gg -1 2 . Let X be a compactification of G. We will always identify G with the open and dense orbit in X. These manifolds belong to the class of spherical manifolds [START_REF] Perrin | On the geometry of spherical varieties[END_REF][START_REF] Timashev | Homogeneous spaces and equivariant embeddings[END_REF]. There is a finite number of G × G-orbits in X and the boundary

X \ G is of codimension one. Example 1.2. If G = T ≃ (C *
) n is a torus, then the compactifications of T are the projective toric manifolds. One goes from the T -action to the T × T action through the morphism T × T → T, (t 1 , t 2 ) → t 1 t -1 2 . Example 1.3. Assume that G is an adjoint semisimple group. Then De Concini and Procesi [START_REF] Concini | Complete symmetric varieties[END_REF] showed the existence of a special compactification of G, called the wonderful compactification of G. It is the only compactification of G satisfying the following property : its boundary X \ G (where we identify the open and dense orbit in X with G) is a union of simple normal crossing prime divisors D i , i ∈ {1, . . . , r}, such that for any subset I ⊂ {1, . . . , r}, the intersection X ∩ i∈I D i is the closure of a unique G × G-orbit, and all G × G-orbits appear this way. The integer r is equal to the rank of G, which is the dimension of a maximal torus in G.

The wonderful compactification of PGL 2 (C) is especially simple : it is P 3 considered as P(Mat 2,2 (C)) equipped with the action of PGL 2 (C) × PGL 2 (C) induced by the multiplication of matrices on the left and on the right. 1.2. Polytopes associated to a polarized group compactification. Recall that a G-linearized line bundle over a G-manifold X is a line bundle L on X equipped with an action of G lifting the action on X, and such that the morphisms between the fibers induced by this action are linear.

Let G be a connected complex reductive group. We call a polarized group compactification a pair (X, L) where X is a compactification of G and L is a G × Glinearized line bundle on X.

Choose T a maximal torus in G, and let S be a maximal compact torus in T . We denote, as usual, by M the lattice of characters of T and by N the lattice of one parameter subgroups of T , naturally dual to each other. Denote by s the Lie algebra of S, by t the Lie algebra of T and by a the Lie algebra is ⊂ t. We identify a with N ⊗ R, and a * with M ⊗ R.

Let Φ ⊂ a * denote the root system of (G, T ). Let W be its Weyl group. Choose a system of positive roots Φ + . It defines a positive Weyl chamber a + in a, resp. a * + in a * . Theorem 1.4. [AB04, Section 2] Let (X, L) be a polarized group compactification of G. Denote by Z the closure of T in X. Then Z is a toric manifold, equipped with a W -action, and L| Z is a W -linearized ample toric line bundle on Z.

We denote by P (X, L), or P for simplicity, the polytope associated to the ample toric line bundle L| Z by the theory of toric varieties [START_REF] Fulton | Introduction to toric varieties[END_REF][START_REF] Oda | Convex bodies and algebraic geometry[END_REF]. The polytope P is a lattice polytope in M ⊗ R, and it is W -invariant. Define P + (X, L) = P (X, L) ∩ a * + . It is a polytope in a * , and P (X, L) is the union of the images of P + (X, L) by W .

The polytope P + (X, L) encodes the structure of G × G-representation of the space of holomorphic sections of L, generalizing the same property for toric line bundles.

Proposition 1.5. [AB04, Section 2.2] Let (X, L) be a polarized group compactification, then

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

Example 1.6. [BK05, Proposition 6.1.11] The wonderful compactification X of an adjoint semisimple group is Fano. The corresponding polytope P (X, -K X ) is the convex hull of the images by the Weyl group W of the weight 2ρ + r i=1 α i , where the α i are the simple roots of Φ + and 2ρ is the sum of the positive roots.

Convex potential

In this section we introduce the convex potential of a K × K-invariant, nonnegatively curved singular hermitian metric on a polarized group compactification. This correspondence gives a bijection between the set of these metrics and the set of the W -invariant convex functions on a which satisfy asymptotic behavior conditions.

2.1. Singular hermitian metrics and potentials. Let X be a compactification of G, and L a linearized ample line bundle on X. Given a hermitian metric h on L and a local trivialization s of L on an open subset U ⊂ X, the local potential of h with respect to s is the function φ defined on U by φ(x) = -ln(|s(x)| 2 h ). We consider here singular hermitian metrics and only require that the potential with respect to any local trivialization is locally integrable. The value +∞ for the potentials is allowed.

We say that a hermitian metric is locally bounded if its potentials with respect to any trivialization on a sufficiently small open subset are bounded. A hermitian metric is smooth (resp. continuous) if and only if its potentials with respect to any local trivialization are. A continuous hermitian metric is locally bounded.

Given a reference metric h 0 , we define the global potential ψ of h with respect to h 0 by

ψ(x) = -ln |y| 2 h |y| 2 h0 ,
for any element y of the fiber L x . This is a function on X that can a priori take the values ±∞. If s is a local trivialization on U , φ (resp. φ 0 ) is the potential of h (resp. h 0 ) with respect to s, then ψ = φφ 0 on U .

2.2. The convex potential. Let (X, L) be a polarized compactification of G.

We identify G with the open dense orbit in X, and first build a trivialization of L on G. Choose 1 e a non-zero element of the fiber L e above the neutral element e of G. Define the section s of L on G by s(g) = (g, e)•1 e . This section is a trivialization over G, equivariant under the action of G × {e}, and any such trivialization is a scalar multiple of s.

Denote by φ the local potential of h on G with respect to s. We are interested in hermitian metrics that are invariant under the action of K × K where K is a maximal compact subgroup of G. We choose K such that K ∩ T = S. We will use the classical KAK decomposition of a complex reductive group.

Proposition 2.1. [Kna02, Theorem 7.39] Any element g ∈ G can be written in the form g = k 1 exp(x)k 2 where k 1 , k 2 ∈ K and x ∈ a + . Furthermore, x is uniquely determined by g. In other words, the set

A + := {exp(x); x ∈ a + } is a fundamental domain for the K × K-action on G.
We first remark that if h is K × K-invariant, then its potential with respect to the section constructed above is still

K × K-invariant. Proposition 2.2. Assume that h is K × K-invariant, then φ is also K × K- invariant. Proof. Let k 1 , k 2 ∈ K and g ∈ G. We can first write s(k 1 gk 2 ) = (k 1 gk 2 , e) • 1 e = (k 1 , k -1 2 )(g, e)(k 2 , k 2 ) • 1 e .
The subgroup diag(G) = {(g, g)|g ∈ G} fixes the neutral element e ∈ G, and thus acts on the fiber L e through a character χ of G, so that (g, g)

• 1 e = χ(g)1 e . We can thus compute

φ(k 1 gk 2 ) = -ln(|s(k 1 gk 2 )| 2 h ) = -ln(|(k 1 , k -1 2 )(g, e)(k 2 , k 2 ) • 1 e | 2 h ) by K × K-invariance of h, this is = -ln(|(g, e)(k 2 , k 2 ) • 1 e | 2 h )
and, by linearity,

= -ln(|χ(k 2 )||(g, e) • 1 e | 2 h ) Since K is compact, |χ(k 2 )| = 1, so we obtain φ(k 1 gk 2 ) = -ln(|(g, e) • 1 e | 2 h ) = φ(g).
Assume that h is in addition non-negatively curved. Then φ is a K ×K-invariant plurisubharmonic function on G. Let u be the function on a defined by

u(x) = φ(exp(x)).
Then Azad and Loeb proved in [START_REF] Azad | Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces[END_REF] that the function u is convex and Winvariant.

In particular, since we assumed that the local potentials of singular hermitian metrics are locally integrable, the K × K-invariance of h ensures that the functions u, respectively φ take finite values on a, resp. G. Indeed, a convex function that takes an infinite value at a point must take an infinite value on a whole octant starting from that point, and then the corresponding K × K-invariant function on G is not locally integrable.

Definition 2.3. We will call u the convex potential of h.

2.3.

Asymptotic behavior of the convex potential.

2.3.1.

A special metric. Let us begin by introducing a continuous, K ×K-invariant, reference hermitian metric on L. We start from the Batyrev-Tschinkel metric defined on toric manifolds, and generalize it to build a reference continuous metric for any polarized group compactification (X, L), with convex potential the support function of the polytope 2P (X, L).

Given a toric manifold Z, equipped with a linearized line bundle D, there is a natural continuous hermitian metric h D , invariant under the action of the compact torus, on D, called the Batyrev-Tschinkel metric (see [Mai00, Section 3.3]). If furthermore the line bundle D is ample, then this metric is non-negatively curved, and its convex potential is the support function v of the polytope 2P (Z, D).

Suppose now that (X, L) is a polarized group compactification, and Z is the toric submanifold. Denote L| Z by D.

Then P (Z, D) = P (X, L) is W -invariant, which implies that the Batyrev-Tschinkel metric h D is W -invariant.
We want to extend h D to a continuous

K × K-invariant metric h L on X. Define h L at ξ ∈ L g by |ξ| hL = |(k 1 , k 2 ) • ξ| hD , for (k 1 , k 2 ) ∈ K × K such that k 1 gk -1 2 ∈ T . We need to check that this is well defined. Since h D is W -invariant we only need to check that, for t ∈ T , if (k 1 , k 2 ) ∈ Stab K×K (t) then |ξ| hD = |(k 1 , k 2 ) • ξ| hD . But Stab K×K (t) acts linearly on the line L t , through a character χ. By compacity, |χ(k 1 , k 2 )| = 1, so |(k 1 , k 2 ) • ξ| hD = |χ(k 1 , k 2 )ξ| hD = |ξ| hD . 2.3.2. Asymptotic behavior.
Theorem 2.4. The singular hermitian K × K-invariant metrics h with non negative current curvature are in bijection with the convex W -invariant functions u : a -→ R satisfying the condition that there exists a constant C 1 ∈ R such that u(x) ≤ v(x) + C 1 on a, where v is the support function of the polytope 2P (X, L). This bijection is obtained by associating to h its convex potential u. Furthermore, h is locally bounded if and only if there exists in addition a constant

C 2 ∈ R such that v(x) + C 2 ≤ ϕ(x) ≤ v(x) + C 1 .
Proof. Let h be a singular hermitian K × K-invariant metric with non negative current curvature on L. Let u be its convex potential. Recall that h L denotes the reference continuous metric constructed above, and let ω L be the curvature current of h L . Denote by ψ the potential of h with respect to h L . It is an ω L -psh function on X. In particular, ψ is bounded from above on X.

Denote by w the function on a associated to the K × K-invariant function ψ| G . Then we see that the function uv is equal to w and thus bounded from above.

If furthermore h is locally bounded then since h L is also locally bounded, the function w is bounded on X. So w = uv is bounded on a.

Conversely, let u be a convex W -invariant function such that u(x) ≤ v(x) + C. We choose any reference metric h 0 on L that is smooth, positively curved and K × K-invariant. Then by the first direction there exist constants C 1 and C 2 such that if u 0 is the potential of h 0 we have

v(x) + C 2 ≤ u 0 (x) ≤ v(x) + C 1 .
Let ω 0 be the curvature form of h 0 .

Consider the function w := uu 0 . It will be enough to show that the function ψ on G corresponding to w extends to an ω 0 -psh function on X.

First remark that ψ = φφ 0 , and by the other direction of the result of Azad and Loeb [START_REF] Azad | Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces[END_REF], φ is psh on G. The assumption on u implies that w, and thus ψ, are bounded from above. Indeed, we have

w = u -u 0 ≤ v + C -u 0 ≤ C -C 2 .
A classical result on psh functions is that a psh function extends over an analytic subset if and only if it is locally bounded from above. Here, applying that with ψ allows to extend ψ to an ω 0 -psh function on X. The corresponding singular hermitian metric h has non-negative curvature, is K × K-invariant, and has convex potential u.

For locally bounded metrics, one just needs to use the refinement that if a psh function is locally bounded then it extends to a bounded psh function.

Newton bodies

In this section we introduce a convex body associated to any non-negatively curved singular K × K-invariant hermitian metric h on an ample linearized line bundle L on a group compactification X. We first define a convex set associated to any function, which is a natural set to consider in the case of convex functions. Applying this construction to the convex potential of a hermitian metric yields a convex body that is contained in 2P (X, L), that will be used to compute the log canonical threshold of h.

Newton set of a function.

Definition 3.1. Let f be a function a → R, and σ a closed convex cone in a. We call Newton set of f the following set in a * .

N σ (f ) := {m ∈ a * ; ∃C, ∀x ∈ σ, f (x) -m(x) ≥ C}.
In the following, we will simply call cone a closed convex cone. For any function f and any cone σ, the Newton set N σ (f ) is clearly convex.

Recall the definition of the dual cone σ ∨ of σ:

σ ∨ = {m ∈ a * ; m(x) ≥ 0 ∀x ∈ σ}.
The Newton set N σ (f ) is by definition stable under addition of an element of the opposite of the dual cone σ ∨ ⊂ a * . We write this also

N σ (f ) = N σ (f ) + (-σ ∨ )
where the plus sign means the Minkowki sum of sets.

Example 3.2. Let f be the affine function

f (x) = m(x) + c where m ∈ a * and c is a constant. Then N σ (f ) = m + (-σ ∨ ).
Let us record the following elementary properties of Newton sets.

Proposition 3.3. Let f and g be two functions on a and c ∈ R. Then

• N σ (cf ) = cN σ (f ) • N σ (f + c) = N σ (f ) • if f ≤ g then N σ (f ) ≤ N σ (g).
• In particular, if for some constants c 1 and c 2 ,

g + c 1 ≤ f ≤ g + c 2 on σ, then N σ (f ) = N σ (g). • Let σ 1 and σ 2 be two convex cones such that σ = σ 1 ∪ σ 2 , then N σ (f ) = N σ1 (f ) ∩ N σ2 (f ).
The last property is very helpful when we want to use a fan decomposition.

Example 3.4. Let v : a → R be a piecewise linear function along a finite fan decomposition σ 0 = ∪ N i=1 σ i where N ∈ N and the σ i are convex cones. For

1 ≤ i ≤ N , denote by v i the element of a * such that v(x) = v i (x) on σ i . Then N σ (v) = N i=1 (v i + (-σ ∨ i ))
.

If furthermore v is convex, then N σ (v) = Conv{v i } + (-σ ∨ ).
3.2. Newton set of convex functions. For this subsection only, we will allow convex functions to take the value +∞. If f is such a function we define its domain by dom(f ) := {x ∈ a; f (x) < ∞}.

We impose however that all functions considered have a non empty domain. In the rest of the section, we always assume dom(f ) = a.

The first remark is that the Newton set of a function f on the whole of a is the domain of its Legendre-Fenchel transform (or convex conjugate) f * defined, for m ∈ a * , by f * (m) := sup{m(x)f (x); x ∈ a}. Let σ be a convex cone, and define the convex function δ σ as the indicator function of σ, i.e.

δ σ (x) = 0 if x ∈ σ and δ σ (x) = ∞ otherwise. Then it is not hard to check that N σ (f ) = N a (f + δ σ ). In other words N σ (f ) is the domain of the convex conjugate of f + δ σ .
We will recall a classical result on convex functions, which allows to express the Newton set of a sum as the Minkowski sum of the Newton sets of the summands. First recall the definition of infimal convolution: Definition 3.5. Let f and g be two convex function. The infimal convolution of f and g is the function f g defined, for x ∈ a, by f g(x) = inf{f (xy) + g(y); y ∈ a}.

Theorem 3.6. [Roc97, Theorem 16.4] Let f and g be two convex functions on a, such that the relative interiors of the domains of f and g have a point in common. Then (f + g) * (m) = f * g * .

Proposition 3.7. Let σ be a convex cone, and f a convex function with dom(f ) = a, then

N σ (f ) = N a (f ) + (-σ ∨ ).
Proof. We have seen that N σ (f ) is the domain of the convex conjugate of f + δ σ , but by Theorem 3.6, this is also the domain of the function f * δ * σ . We can apply this theorem because the intersection of the domains of f and δ σ is σ.

The domain of an infimal convolution is the Minkowski sum of the domains of the two functions involved, so we just need to compute the domain of δ * σ . By definition we check that this is -σ ∨ , and obtain the statement. Proposition 3.8. Let f and g be two convex functions, both with domain a, and σ a convex cone. Then N σ (f + g) = N σ (f ) + N σ (g).

Proof. We have, by the previous proposition,

N σ (f + g) = N a (f + g) + (-σ ∨ ),
and by the same proof,

N a (f + g) = N a (f ) + N a (g), so N σ (f + g) = N a (f ) + N a (g) + (-σ ∨ ) = N σ (f ) + N σ (g).
3.3. Newton body of a metric. Let X be a compactification of G, polarized by L. Let h be a K × K-invariant hermitian metric with non negative curvature on L, and u its convex potential with respect to a fixed left-equivariant trivialization of L on G, which is a function on a. Definition 3.9. We will call Newton body of h the set N (h) := N a (u).

Let P be the polytope corresponding to the polarization L.

Example 3.10. Let h L be the metric constructed in Section 2.3.1. Its convex potential v is the support function of 2P , so N (h L ) = 2P , as in Example 3.4. Remark that the convex potential of h L is piecewise linear with respect to the opposite of the fan of the toric subvariety.

Proposition 3.11. The Newton body of h is stable under the action of the Weyl group W .

Proof. Let u be the convex potential of h, and let m ∈ a * . Suppose that u(x)m(x) ≥ C for some constant C and for all x ∈ a. Let w ∈ W . By W -invariance of u, the inequality is equivalent to

C ≤ u(w • x) -m(x) ≤ u(w • x) -w -1 • m(w • x).
Since w induces a bijection of a, we get that for all w ∈ W , m ∈ N (h) if and only if w • m ∈ N (h), which means that N (h) is W -invariant.

We can finally translate the information about the asymptotic behavior of metrics in terms of their Newton bodies. Proposition 3.12. Let h be a K ×K-invariant hermitian metric with non negative curvature on L. Then N (h) ⊂ 2P . If in addition h is locally bounded, then N (h) = 2P .

Proof. Recall from Section 2.3.1 that the convex potential u of a K × K-invariant hermitian metric h with non negative curvature on L satisfies u ≤ v + C 2 on a for some constant C 2 , and that if h is locally bounded then we have in addition

v + C 1 ≤ u for some constant C 1 .
Now the result easily follows from Proposition 3.3 and Example 3.10.

Log canonical thresholds

In this section, we reduce the computation of the log canonical threshold of a K × K-invariant non-negatively curved metric to an integrability problem involving its convex potential, by using the KAK integration formula. We prove an integrability criterion for exponentials of concave functions, with respect to the measure J(x)dx appearing in the KAK integration formula, and then use it to obtain an expression of the log canonical threshold in terms of the Newton body of the metric. 4.1. Log canonical thresholds on compact manifolds. In this subsection we consider first X a compact complex manifold that is not necessarily a group compactification, and L a line bundle on X.

Definition 4.1. Let x be a point in X, and h a hermitian metric on L. The complex singularity exponent (or local log canonical threshold ) of h at x, which we denote by lct(h, x) is the supremum of all c > 0 such that e -cϕ is integrable with respect to Lebesgue measure in a neighborhood of x, where ϕ is the potential of h with respect to a trivialization s of L in a neighborhood of x.

Remark 4.2. If h is a locally bounded metric then on a sufficiently small neighborhood of any point, the potential ϕ is a bounded function, so it is integrable. It means that for any such metric, lct(h, x) = ∞ at any point x.

Definition 4.3. Let h be a hermitian metric on L, then the log canonical threshold of h is defined as

lct(h) = inf x∈X (lct(h, x)).
Proposition 4.4. Let h be a singular hermitian metric on L, h 0 a locally bounded hermitian metric on L, and ψ the potential of h with respect to h 0 . Let also dV be any smooth volume form on X. Then we have

lct(h) = sup c > 0; X e -cψ dV < ∞ .
Proof. Let x be any point in X, and s a trivialization of L on a neighborhood U of x. Up to shrinking U , we can assume that the local potential ϕ 0 of h 0 with respect to s is bounded.

Let ϕ be the local potential of h with respect to s and ψ the potential of h with respect to h 0 . Then by definition of ψ, we have ψ = ϕϕ 0 on U , and since ϕ 0 is bounded, the integrability of e -cϕ with respect to Lebesgue measure on a neighborhood of x is equivalent to the integrability of e -cψ on the same neighborhood.

Furthermore, in the neighborhood of any point x in X, the integrability with respect to Lebesgue measure is equivalent to integrability with respect to a smooth volume form.

The function ψ is defined everywhere on X, e -cψ is positive, and X is compact, so e -cψ is integrable with respect to dV in the neighborhood of any point in X if and only if X e -cψ dV < ∞.

Take 0 < c < lct(h), then c < lct(h, x) for all x ∈ X, so X e -cψ dV < ∞. This means that

lct(h) ≤ sup c > 0; X e -cψ dV < ∞ .
Conversely, if c > lct(h) then there exists x ∈ X such that c > lct(h, x) but then X e -cψ dV = ∞, so c ≥ sup c > 0; X e -cψ dV < ∞ . Taking the infimum gives the other inequality:

lct(h) ≥ sup c > 0; X e -cψ dV < ∞ .
This proves the proposition. 4.2. Log canonical thresholds on group compactifications. Let X be a Fano compactification of G. Let L be an ample linearized line bundle on X. Using Proposition 4.4, we reduce the computation of log canonical thresholds to integrability conditions on the potentials of metrics, with respect to a smooth volume form. Since volume forms do not put weight on the (codimension one) boundary X \ G, we will restrict to integrability conditions on G. We want to use, in addition, the KAK integration formula, that we recall here : Proposition 4.5. [Kna02, Proposition 5.28] Let dg denote a Haar measure on G, and dx the Lebesgue measure on a, normalized by the lattice of one parameter subgroups N . Then there exists a constant C > 0 such that for any

K ×K-invariant, dg-integrable function on G, G f (g)dg = C a + J(x)f (exp(x))dx, where J(x) = α∈Φ + sinh(α(x)) 2 .
Let us now derive an integrability criterion with respect to J.

Integrability criterion.

4.3.1. Integrability criterion on a cone. We use the following proposition, obtained by Guenancia in [START_REF] Guenancia | Toric plurisubharmonic functions and analytic adjoint ideal sheaves[END_REF]. It is an analytic proof and generalization of the computation by Howald of the log canonical thresholds of monomial ideals. The statement given here is slightly different from the statement in [START_REF] Guenancia | Toric plurisubharmonic functions and analytic adjoint ideal sheaves[END_REF], but is in fact equivalent (see [START_REF] Delcroix | Alpha-invariant of toric line bundles[END_REF] for details).

Proposition 4.6. [Gue12, Proposition 1.9] Let f be a convex function on a. Assume that σ is a smooth polyhedral cone in a = N ⊗ R. Then e -f is integrable on a translate (equivalently on all translates) of σ if and only if 0 is in the interior of the Newton body of f : 0 ∈ Int(N σ (f )).

4.3.2.

Integrability with respect to J. The half sum of the positive roots of Φ is denoted by ρ. We want to prove the following integrability criterion, with respect to the measure J(x)dx.

Proposition 4.7. Assume that a + = i σ i where each σ i is a smooth polyhedral cone of full dimension r. Let l be a function on a, convex on each cone σ i . Then

a + e -l(x) J(x)dx < +∞ if and only if 4ρ ∈ Int(N a + (l)).
Lemma 4.8. Let σ be a smooth full dimensional polyhedral cone in a + , l be a convex function on a, then the following are equivalent:

• σ e -l(x) J(x)dx < ∞; • σ e -l(x)+4ρ(x) dx < ∞; • 4ρ ∈ Int(N σ (l)). Proof. Writing sinh(α(x)) = e α(x) -e -α(x) 2 = 1 2 e α(x) (1 -e -2α(x) ),
we get that

J(x) = 1 2 2Card(Φ + ) e 2 α∈Φ + α(x) α∈Φ + (1 -e -2α(x) ) 2 .
For any x ∈ a + and α ∈ Φ + , α(x) > 0, so 0 ≤ e -2α(x) < 1. This implies

0 < α∈Φ + (1 -e -2α(x) ) 2 ≤ 1, so 0 < J(x) ≤ 1 2 2Card(Φ + ) e 4ρ(x) .
This first inequality allows to say that if σ e -l(x)+4ρ(x) dx < ∞ then σ e -l(x) J(x)dx < ∞.

Let us now prove the converse. Choose γ a point in the interior of σ. Assume that e -l+4ρ is not integrable on σ. Then by the usual integrability criterion (Proposition 4.6) e -l+4ρ is also non integrable on γ + σ.

But now, for x ∈ γ + a + and α ∈ Φ + , we have α(x) ≥ c = min β∈Φ + β(γ) > 0, so 0 ≤ e -2α(x) ≤ e -2c < 1, and this implies

1 -e -2c 2 2Card(Φ + ) e 4ρ(x) ≤ J(x) ≤ 1 2 2Card(Φ + ) e 4ρ(x) .
This gives that σ e -l(x) J(x)dx ≥ γ+σ e -l(x) J(x)dx

≥ γ+σ e -l+4ρ dx ≥ ∞.
We have then shown the equivalence of the two first points in the lemma. By Proposition 4.6 the second point is also equivalent to

0 ∈ Int(N σ (l -4ρ)) = -4ρ + Int(N σ (l)).
Letting 4ρ go to the left, we conclude the proof. Now we can prove the proposition, just by gluing the parts.

Proof. Just remark that since the function e -l(x) J(x) is positive and the cones are full dimensional, a + e -l(x) J(x)dx < +∞ if and only if σi e -l(x) J(x)dx < +∞ for all i.

For each of these integrals we can use the lemma, so the necessary and sufficient condition becomes 4ρ ∈ Int(N σi (l)) for all i, or equivalently 4ρ ∈ Int( i N σi (l)).

To conclude, observe that N a + (l) = i N σi (l) by Proposition 3.3.

4.4. Log canonical thresholds in terms of Newton bodies. Let X be a Fano compactification of G. Let L be a linearized ample line bundle on X, whose associated polytope is P . Denote by Q the polytope associated to the anticanonical bundle -K X . Let also H denote the convex hull of all images of 2ρ by the Weyl group W . We want to prove the following.

Theorem 4.9. Let h be a K × K-invariant hermitian metric with non negative curvature on L, then

lct(h) = sup{c > 0; 2H + 2cP ⊂ cN (h) + 2Q}.
We first introduce some notations. Let us fix s 0 a left G-equivariant trivialization of L on G and s 1 a left G equivariant trivialization of -K X on G.

Let u be the convex potential of h with respect to the section s 0 . Let also u 0 be the support function of P and h 0 be the corresponding metric. It has locally bounded potentials. Denote by ψ the potential of h with respect to h 0 .

Since X is Fano, we can choose h 1 a smooth metric on -K X with positive curvature, and let u 1 be its convex potential with respect to s 1 . This choice determines a smooth volume form on X, which writes, on G, dV = e -u1 dg where dg is the Haar measure s -1 1 ∧ s -1 1 . Remark 4.10. In particular, the integral of this volume form is finite, so applying the KAK integration formula this means that a + e -u1 Jdx < ∞.

By Proposition 4.7, this implies that 4ρ ∈ Int(N (h 1 )) = Int(2Q).

Another way to say that is H ⊂ Int(Q).

Proof. Using Proposition 4.4, then restricting to the dense orbit, we get:

lct(h) = sup c > 0; X e -cψ dV < ∞ = sup c > 0; G e -cψ dV < ∞ .
Since ψ(exp(x)) = u(x)u 0 (x), we can now use the KAK integration formula to write:

lct(h) = sup c > 0; a + e -c(u-u0) e -u1 Jdx < ∞ .
Then Proposition 4.7 gives:

lct(h) = sup {c > 0; 4ρ ∈ Int(N a + (cu -cu 0 + u 1 ))} = sup{c > 0; 4ρ ∈ N a + (cu -cu 0 + u 1 )}.
Let σ i be the cones of full dimension in the fan subdivision of a + corresponding to X (induced by the fan subdivision of a associated to the toric subvariety Z). Then u 0 is linear on each -σ i . We write u i 0 the corresponding element of a * .

We have

lct(h) = sup{c > 0; ∀i, 4ρ ∈ N -σi (cu -cu 0 + u 1 )} = sup{c > 0; ∀i, 4ρ + cu i 0 ∈ N -σi (cu + u 1 )}.
Recall from Example 3.4 that P = N a (u 0 ) ⊂ u i 0 + σ ∨ i , so that

lct(h) = sup{c > 0; ∀i, 4ρ + cP ∈ N -σi (cu + u 1 )} = sup{c > 0; 4ρ + cP ∈ N a + (cu + u 1 )} = sup{c > 0; 2H + 2cP ⊂ N a (cu + u 1 )} by W -invariance.
To conclude it remains to remark that both u and u 1 are convex, so by Proposition 3.8, N a (cu + u 1 ) = cN a (u) + N a (u 1 ) = cN (h) + 2Q.

Alpha invariants

We obtain in this section an expression for the α-invariant of a polarized group compactification in terms of its polytope. We first give the result for general reductive group compactifications, then see how it simplifies when the group is semisimple. We then discuss some some examples and how some additional symmetries can be taken into account for reductive group compactifications. 5.1. General formula. Definition 5.1. Let X be a compact complex manifold, K a compact subgroup of the automorphisms group of X, and L a K-linearized line bundle on X. The alpha invariant of L relative to the group K, denoted by α K (L) is the infimum of the log canonical thesholds of all K-invariant singular hermitian metrics on L with non negative curvature.

Let P and Q be two convex bodies in a * . Recall the definition of the Minkowski difference:

Q ⊖ P = {x|x + P ⊂ Q}.
Another expression of the Minkowski difference is the following, which shows that it is convex if Q is convex:

Q ⊖ P = p∈P (-p + Q).
If P 1 , P 2 and Q are three convex bodies, then P 1 +Q ⊂ P 2 if and only if P 1 ⊂ P 2 ⊖Q.

We can now state our main result.

Theorem 5.2. Let (X, L) be a polarized compactification of G, and P := P (X, L). Assume furthermore that X is Fano and let Q := P (X, -K X ). Then

α K×K (L) = sup{c > 0; c(P + (-P W )) ⊂ Q ⊖ H},
where P W denotes the subset of W -invariant points of P .

Proof. Let h be any K × K-invariant metric on L with non negative curvature. The Newton body of h is convex and W -stable. In particular it contains a W -invariant point p, for example the barycenter of the orbit of any point in N (h). Denote by h p the K × K-invariant metric on L with non negative curvature whose convex potential is the function x → p(x). Then {p} = N (h p ) ⊂ N (h), so by the expression of the log canonical thresholds from Theorem 4.9, lct(h) ≥ lct(h p ).

Since all such h p for p ∈ 2P W define a singular hermitian metric with nonnegative curvature, this remark allows to write the alpha invariant as

α K×K (L) = inf p∈2P W lct(h p ).

Now from the expression of the log canonical threshold we get

lct(h p ) = sup{c > 0; 2H + 2cP ⊂ cN (h p ) + 2Q} = sup{c > 0; -cp + 2cP ⊂ 2Q ⊖ 2H}.
Then the expression of the alpha invariant further simplifies as

α K×K (L) = inf p∈2P W sup{c > 0; -cp + 2cP ⊂ 2Q ⊖ 2H} = sup{c > 0; ∀p ∈ 2P W , -cp + 2cP ⊂ 2Q ⊖ 2H} = sup{c > 0; 2cP + (-2cP W ) ⊂ 2Q ⊖ 2H}.
Dividing by two yields = sup{c > 0; c(P + (-P W )) ⊂ Q ⊖ H} which is the expression in the statement of the Theorem.

Remark 5.3. In the toric case, we recover our previous computation [START_REF] Delcroix | Alpha-invariant of toric line bundles[END_REF]: α (S 1 ) n (L) = sup{c > 0; c(P + (-P )) ⊂ Q}.

Semisimple case.

The alpha invariant of an ample line bundle on a Fano compactification of a semisimple group can be easily expressed in terms of the polytope associated to L as an inradius between two convex bodies.

Definition 5.4. The inradius of Q with respect to P is the number:

inr(P, Q) := sup{c ≥ 0|∃x, x + cP ⊂ Q}. Corollary 5.5. Assume that G is a semisimple group. Then α K×K (L) = inr(P, Q ⊖ H).
Proof. If G is semisimple, we have P W = {0}. In fact, the metric h 0 whose convex potential is the zero function satisfies

α K×K (L) = lct(h 0 ) = sup{c > 0; cP ⊂ Q ⊖ H}.
And this is equal to the inradius inr(P, Q ⊖ H).

Indeed, one inequality is trivial: inr(P, Q ⊖ H) ≥ α K×K (L). Conversely, assume c ≤ inr(P, Q ⊖ H), i.e. there exists an x ∈ a * such that

x + cP ⊂ Q ⊖ H.
Then since P and Q ⊖ H are stable under W -action, we also have

∀w ∈ W, w • x + cP ⊂ Q ⊖ H.
Convexity and the fact that the barycenter of the W -orbit of x is 0 imply that cP ⊂ Q ⊖ H, so c ≤ α K×K (L). We have thus proved the other inequality inr(P, Q ⊖ H) ≤ α K×K (L).

Remark 5.6. In the case of reductive groups, the alpha invariant is not an inradius, but we can bound it from above by an inradius: α K×K (L) ≤ inr(P + (-P ) W , Q ⊖ H).

Additional symmetries.

If the polytopes P and Q admit additional common symmetries, then the value of the alpha invariant can be improved. Indeed, the symmetries of Q translate to a finite subgroup O of the automorphisms group of the variety X, and if P is stable under these symmetries, then it is linearized by O. We can thus consider the alpha invariant with respect to the bigger group generated by K × K and O, that we denote K O .

We then have, adapting the proof of Theorem 5.2, α KO (L) = sup{c > 0; c(P + (-P W,O )) ⊂ Q ⊖ H}.

In particular, if the only fixed point under W, O is the origin, then just as in the semisimple case, we get α KO (L) = inr(P, Q ⊖ H).

Examples.

Let us compute the α invariant of the anticanonical line bundle for some wonderful compactifications of semisimple groups. First remark that in this case we have P = Q and can rewrite the expression of the invariant as :

α K×K (X, -K X ) = sup{c > 0; H ⊂ (1c)Q}, or, by W -invariance, α K×K (X, -K X ) = sup{c > 0; 2ρ ∈ (1c)Q + }.

Remark 5.7. It is interesting to notice that this quantity appeared in the determination of some volume asymptotics by Chambert-Loir and Tschinkel. If σ denotes the quantity the authors compute in the examples of compactifications of semisimple groups [CLT10, Section 5.3], we have σ = 1α K×K (X, -K X ) if the polytope considered is the anticanonical polytope of a Fano compactification. This is because their computation in this special case is equivalent to a computation of the log canonical threshold of a metric on the anticanonical line bundle with constant convex potential.

For wonderful compactifications of semisimple adjoint groups, the polytope of the anticanonical line bundle Q is determined by the root system. Indeed, recall that it is the convex hull of the images by W of the weight 2ρ + r i=1 α i where the α i are the simple roots of Φ + .

In particular, when G = (PSL 2 (C)) n , for any n ≥ 1, the simple roots are the same as the positive roots, so Q = 2H.

Corollary 5.8. Let X be the wonderful compactification of (PSL 2 (C)) n , then α K×K (-K X ) = 1 2 .

Proof. Applying Corollary 5.5 gives α K×K (-K X ) = inr(2H, H) = 1 2 .

More generally for type A n , choosing an appropriate ordering of the simple roots α 1 , . . . , α n , we can write the positive roots as

α i + α i+1 + • • • + α j
for each pair (i, j) with 1 ≤ i ≤ j ≤ n. We see then that the coefficient of α k in the sum of positive roots n l=1 α l is equal to the cardinal of the set {(i, j); 1 ≤ i ≤ k ≤ j ≤ n}. This is k(nk + 1). Adding the sum of simple roots, we see that the k th -coordinate of the vertex defining the polytope of the wonderful compactification of PSL n+1 (C) in the basis of simple roots is 1 + k(nk + 1).

Then from our result, the alpha invariant is easily seen to be the maximum of all c > 0 such that for each k, c(1 + k(nk + 1)) ≤ 1. We deduce the following value for the alpha invariant.

Corollary 5.9. Let X be the wonderful compactification of PSL n+1 (C), then

α K×K (-K X ) = 1 1 + ⌈ n 2 ⌉(⌊ n 2 ⌋ + 1)
.