K\"{a}hler-Einstein metrics on smooth Fano symmetric varieties with Picard number one

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit K\"{a}hler-Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure.


Introduction
A Kähler metric on a complex manifold is said to be Kähler-Einstein if the Riemannian metric defined by its real part has constant Ricci curvature. The existence of Kähler-Einstein metrics on Fano manifolds has become a central topic in complex geometry in recent years. In contrast to Calabi-Yau and general type ( [Aub78,Yau78]), Fano manifolds do not necessarily have a Kähler-Einstein metric in general, and there are obstructions based on the (holomorphic) automorphism group.
The first obstruction was discovered by Matsushima in [Mat57]. He proved that the reductivity of the automorphism group is a necessary condition for the existence of Kähler-Einstein metrics. Later, Futaki [Fut83] proved that the existence of Kähler-Einstein metrics implies that the Futaki invariant, a functional on the Lie algebra of the automorphism group, vanishes. As a generalization of this invariant on test configurations, Tian [Tia94,Tia97] and Donalson [Don02] introduced a certain algebraic stability condition, which is called the K-stability. The famous Yau-Tian-Donaldson conjecture predicts that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the K-stability. Eventually, this conjecture was solved by Chen-Donaldson-Sun [CDS15a,CDS15b,CDS15c] and Tian [Tia15].
Despite of these obstructions, each Fano homogeneous manifold admits a Kähler-Einstein metric [Mat72,Kos55]. Therefore, one can expect the existence of a Kähler-Einstein metric on a Fano manifold if it has large automorphism group. A natural candidate is the almost-homogeneous manifold, that is a manifold with an open dense orbit of a complex Lie group. For the case of toric Fano manifolds, Wang and Zhu [WZ04] proved that the existence of a Kähler-Einstein metric is equivalent to the vanishing of the Futaki invariant. In fact, this was based on the theorem by Mabuchi [Mab87], which says that the Futaki invariant vanishes if and only if the barycenter of the moment polytope is the origin. This gave us a powerful combinatorial criterion for the existence of a Kähler-Einstein metric on a toric Fano manifold, which is much easier to check than the K-stability condition.
An important class of almost-homogeneous varieties is spherical varieties including toric varieties, group compactifications ( [Del17]), and symmetric varieties. A normal variety is called spherical if it admits an action of a reductive group of which a Borel subgroup acts with an open orbit on the variety. As a generalization of Wang and Zhu's work, Delcroix [Del20] extended a combinatorial criterion for K-stability of Fano spherical manifolds, in terms of its moment polytope and spherical data. In particular, this criterion is also applicable to smooth Fano symmetric varieties (see Corollary 5.9 of [Del20]).
By combining the above criterion and Ruzzi's classification [Ruz11] of smooth Fano symmetric varieties with Picard number one, we prove the following.
Theorem 1.1. All smooth Fano symmetric varieties with Picard number one admit Kähler-Einstein metrics.
For this theorem, the condition on the Picard number is crucial because a smooth Fano symmetric variety with higher Picard number may have no Kähler-Einstein metrics. For example, the blow-up of the wonderful compactification of Sp(4, C) along the closed orbit does not admit any Kähler-Einstein metrics (see Example 5.4 of [Del17]). Moreover, we note that Delcroix already provided the existence of Kähler-Einstein metrics on smooth Fano embedding of SL(3, C)/ SO(3, C), and group compactifications of SL(3, C) and G 2 respectively (see Example 5.13 of [Del20]). The above theorem leads us to complete all remaining cases of smooth Fano symmetric varieties with Picard number one also admit Kähler-Einstein metrics.
Acknowledgements. The second author would like to express thanks to Jihun Park and Eunjeong Lee for their interests and helpful comments.
The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1F1A1058962). The second author was supported by the Institute for Basic Science (IBS-R003-D1), and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A2C3010487). The third author was supported by the Institute for Basic Science (IBS-R003-D1).

Criterion for existence of Kähler-Einstein metrics on symmetric varieties
Let G be a connected reductive algebraic group over C.
2.1. Spherical varieties and algebraic moment polytopes. We review general notions and results about spherical varieties. The normal equivariant embeddings of a given spherical homogeneous space are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. We refer [Kno91], [Tim11] and [Gan18] as references for spherical varieties.
Definition 2.1. A normal variety X equipped with an action of G is called spherical if a Borel subgroup B of G acts on X with an open and dense orbit.
Let G/H be an open dense G-orbit of a spherical variety X and T a maximal torus of B. By definition, the spherical weight lattice M of G/H is a subgroup of characters χ ∈ X(B) = X(T ) of (nonzero) B-semiinvariant functions in the function field C(G/H) = C(X), that is, Note that every function f χ in C(G/H) (B) is determined by its weight χ up to constant because C(G/H) B = C, that is, any B-invariant rational function on X is constant. The spherical weight lattice M is a free abelian group of finite rank. We define the rank of G/H as the rank of the lattice M. Let N = Hom(M, Z) denote its dual lattice.
Since the open B-orbit of a spherical variety X is an affine variety, its complement has pure codimension one and is a finite union of B-stable prime divisors.
Definition 2.2. For a spherical variety X, B-stable but not G-stable prime divisors in X are called colors of X. A color of X corresponds to a B-stable prime divisor in the open G-orbit G/H of X. We denote by D = {D 1 , · · · , D k } the set of colors of X (or G/H).
is unique up to constant, we define the color map ρ : D → N by ρ(D), χ = ν D (f χ ) for χ ∈ M, where ν D is the discrete valuation associated to a divisor D, that is, ν D (f ) is the vanishing order of f along D. Unfortunately, the color map is generally not injective. In addition, every valuation on G/H induces a homomorphismρ : {valuations on G/H} → N defined by ρ(ν), χ = ν(f χ ). Luna and Vust [LV83] showed that the restriction ofρ to the set of G-invariant valuations on G/H is injective. From now on, we will regard a G-invariant valuation on G/H as an element of N via the mapρ, and in order to simplify the notationρ(ν E ) will be written asρ(E) for a G-stable divisor E in X.
By the multiplicity-free property of spherical varieties, the algebraic moment polytope ∆(X, L) encodes the structure of representation of G in the spaces of multi-sections of tensor powers of L.
Definition 2.3. Let L be a G-linearized ample line bundle on a spherical G-variety X. The algebraic moment polytope ∆(X, L) of L with respect to B is defined as the closure of k∈N ∆ k /k in M ⊗ R, where ∆ k is a finite set consisting of (dominant) weights λ such that H 0 (X, L ⊗k ) = λ∈∆ k V G (λ). Here, V G (λ) means the irreducible representation of G with highest weight λ.
For a compact connected Lie group K and a compact connected Hamiltonian K-manifold (M, ω, µ), Kirwan [Kir84] proved that the intersection of the image of M through the moment map µ with the positive Weyl chamber with respect to a Borel subgroup B of G is a convex polytope, where G is the complexification of K. The algebraic moment polytope ∆(X, L) for a polarized G-variety X was introduced by Brion in [Bri87] as a purely algebraic version of the Kirwan polytope. This is indeed the convex hull of finitely many points in M ⊗ R (see [Bri87]). Moreover, if X is smooth, then ∆(X, L) can be interpreted as the Kirwan polytope of (X, ω L ) with respect to the action of a maximal compact subgroup K of G, where ω L is a K-invariant Kähler form in c 1 (L).
Example 2.4 (Equivariant compactifications of reductive groups). Any reductive group G is spherical with respect to the action of G×G by left and right multiplication from the Bruhat decomposition. Let us consider the wonderful compactification of a simple algebraic group G of adjoint type constructed by De Concini and Procesi [DCP83]. As a specific example, the wonderful compactification P(Mat 2×2 (C)) ∼ = P 3 of the projective general linear group PGL(2, C) has the action of PGL(2, C) × PGL(2, C) induced by the multiplication of matrices on the left and on the right. It is known that the spherical weight lattice M of the wonderful compactification of a simple algebraic group G of adjoint type coincides with the root lattice of G. Since the anticanonical line bundle where ̟ 1 denotes the fundamental weight of PGL(2, C). Repeating this calculation for tensor powers Hence the algebraic moment polytope ∆(P 3 , where α 1 denotes the simple root of PGL(2, C).

Symmetric spaces and symmetric varieties. For an algebraic group involution
Definition 2.5. A normal G-variety X together with an equivariant open embedding G/H ֒→ X of a symmetric homogeneous space G/H is called a symmetric variety.
Vust proved that a symmetric homogeneous space G/H is spherical (see [Vus74, Theorem 1 in Section 1.3]). By using the Luna-Vust theory on spherical varieties, Ruzzi [Ruz11] classified the smooth projective symmetric varieties with Picard number one from the classification of corresponding colored fans. As a result, there are only 6 nonhomogeneous smooth projective symmetric varieties with Picard number one, and their restricted root systems (see Subsection 2.3 for the definition) are of either type A 2 or type G 2 . Moreover, Ruzzi gave geometric descriptions of them in [Ruz10].
In the case that the restricted root system is of type A 2 (Theorem 3 of [Ruz10]), the symmetric varieties are smooth equivariant completions of symmetric homogeneous spaces SL(3, C)/ SO(3, C), (SL(3, C) × SL(3, C))/ SL(3, C), SL(6, C)/ Sp(6, C), E 6 /F 4 , and are isomorphic to a general hyperplane section of rational homogeneous manifolds which are in the third row of the geometric Freudenthal-Tits magic square.
Remark 2.6. Though all the rational homogeneous manifolds admit Kähler-Einstein metrics, a general hyperplane section of a rational homogeneous manifold is not necessarily the case. For example, a general hyperplane section of the Grassmannian Gr(2, 2n + 1), called an odd symplectic Grassmannian of isotropic planes, does not admit Kähler-Einstein metrics by the Matsushima theorem in [Mat57] because the automorphism group of the odd symplectic Grassmannian is not reductive (see [Mih07]). In the case that the restricted root system is of type G 2 (Theorem 2 of [Ruz10]), the symmetric varieties are the smooth equivariant completions of either G 2 /(SL(2, C) × SL(2, C)) or (G 2 × G 2 )/G 2 . The smooth equivariant completion with Picard number one of the symmetric space G 2 /(SL(2, C) × SL(2, C)), called the Cayley Grassmannian, and the smooth equivariant completion with Picard number one of the symmetric space (G 2 ×G 2 )/G 2 , called the double Cayley Grassmannian, have been studied by Manivel [Man18,Man20].
Their geometric properties including the dimension, the Fano index, the restricted root system are listed in Table 1. For the deformation rigidity properties of smooth projective symmetric varieties with Picard number one, see [KP19].
2.3. Existence of Kähler-Einstein metrics on symmetric varieties. We recall Delcroix's criterion for K-stability of smooth Fano symmetric varieties in [Del20].
For an algebraic group involution θ of G, a torus T in G is split if θ(t) = t −1 for any t ∈ T . A torus T is maximally split if T is θ-stable maximal torus in G which contains a split torus T s of maximal dimension among split tori. Then θ descends to an involution of X(T ) for a maximally split torus T , and the rank of a symmetric homogeneous space G/H is equal to the dimension of a maximal split subtorus T s of T .
Let Φ = Φ(G, T ) be the root system of G with respect to a maximally split torus T . By Lemma 1.2 of [DCP83], we can take a set of positive roots Φ + such that either θ(α) = α or θ(α) is a negative root for all α ∈ Φ + , then we denote is a (possibly non-reduced) root system, which is called the restricted root system. Let C + θ denote the cone generated by positive restricted roots in Φ + θ = {α − θ(α) : α ∈ Φ + \Φ θ }. Proposition 2.7 (Corollary 5.9 of [Del20]). Let X be a smooth Fano embedding of a symmetric homogeneous space G/H. Then X admits a Kähler-Einstein metric if and only if the barycenter of the moment polytope ∆(X, K −1 X ) with respect to the Duistermaat-Heckman measure α∈Φ + \Φ θ κ(α, p) dp is in the relative interior of the translated cone 2ρ θ + C + θ , where κ denotes the Killing form on the Lie algebra g of G.
In fact, this result is a direct consequence of a combinatorial criterion for the existence of a Kähler-Ricci soliton on smooth Fano spherical varieties obtained by Delcroix [Del20, Theorem A]. The proof consists of the existence of a special equivariant test configuration with horospherical central fiber and the explicit computation of the modified Futaki invariant on Fano horospherical varieties.

Moment polytopes of smooth Fano symmetric varieties and their barycenters
We prove in this section our main result Theorem 1.1. The proof combines Proposition 2.7 together with the following result allowing us to compute (algebraic) moment polytopes of Fano symmetric varieties.
Proposition 3.1. Let X be a smooth Fano embedding of a symmetric space G/G θ . Then there exist integers m i such that a Weil divisor −K X = k i=1 m i D i + ℓ j=1 E j represents the anticanonical line bundle K −1 X for colors D i and G-stable divisors E j in X, and the moment polytope ∆(X, in N ⊗ R and its dual polytope Q * X is defined as {m ∈ M ⊗ R : n, m ≥ −1 for every n ∈ Q X }. Proof. Let us recall results about the anticanonical line bundle on a spherical variety from Sections 4.1 and 4.2 of [Bri97]. For a spherical G-variety X, there exists a B-semi-invariant global section s ∈ Γ(X, Furthermore, the B-weight of this section s is the sum of α ∈ Φ such that g −α does not stabilize the open B-orbit in X. Thus, when X is a symmetric variety associated to an involution θ of G, the weight of s is equal to 2ρ θ = α∈Φ + \Φ θ α. For a Gorenstein Fano spherical variety X, Brion obtained the relation between the moment polytope ∆(X, K −1 X ) and a polytope ∆ −KX associated to the anticanonical divisor in Proposition 3.3 of [Bri89]. More precisely, if X is a smooth Fano embedding of G/G θ then the moment polytope ∆(X, K −1 X ) is 2ρ θ + ∆ −KX and a polytope ∆ −KX associated to the anticanonical (Cartier) divisor −K X is the dual polytope Q * X (see Remark 9.1 of [GH15]).

Smooth Fano embedding of SL(3, C)/ SO(3, C) with Picard number one.
Considering the involution θ of SL(n, C) defined by sending g to the inverse of its transpose θ(g) = (g t ) −1 , which is usually called of Type AI, the subgroup fixed by θ is SO(n, C). Since θ(α) = −α for α ∈ Φ = Φ SL 3 , the set Φ θ is empty and the restricted root system Φ θ is the double 2Φ of the root system Φ. The spherical weight lattice M = X(T /T ∩ G θ ) is formed by 2λ for weights λ ∈ X(T ). Thus the dual lattice N is generated by half of the coroots 1 2 α ∨ 1 , 1 2 α ∨ 2 from the relation α ∨ i , ̟ j = δ i,j . In general, Vust [Vus90] proved that when G is semisimple and simply connected, the spherical weight lattice M of the symmetric space G/G θ is the lattice of restricted weights determined by the restricted root system, which implies that N is the lattice generated by restricted coroots forming a root system dual to the restricted root system Φ θ .
Let X 1 be the smooth Fano embedding of SL(3, C)/ SO(3, C) with Picard number one. Using the description in [Ruz10], we know that the two colors D 1 , D 2 and the G-stable divisor E in X 1 have the images Then we have two relations 2D 1 − D 2 − E = 0 and −D 1 + 2D 2 − E = 0, so that D 1 = D 2 = E in the Picard group Pic(X 1 ).
Proof. Choosing a realization of the root system A 2 in the Euclidean plane R 2 with α 1 = (1, 0) and α 2 = − 1 2 , √ 3 2 , for p = (x, y) we obtain its Duistermaat-Heckman measure From Proposition 3.2, we can compute the volume and the barycenter of the moment polytope ∆ 1 with respect to the Duistermaat-Heckman measure. Hence bar DH (∆ 1 ) is in the relative interior of the translated cone 2ρ θ + C + θ (see Figure 1), so X 1 admits a Kähler-Einstein metric by Proposition 2.7.

3.2.
Smooth Fano embedding of (SL(3, C) × SL(3, C))/ SL(3, C) with Picard number one. Any reductive algebraic group L is a symmetric homogeneous space (L × L)/diag(L) under the action of the group G = L × L for the involution θ(g 1 , g 2 ) = (g 2 , g 1 ), g 1 , g 2 ∈ L. If T is a maximal torus of L, then T × T is a maximal torus of G and we get the spherical weight lattice Thus M can be identified with X(T ) by the projection to the first coordinate. Under this identification, the restricted root system Φ θ is identified with the root system Φ L of L with respect to T , and the dual lattice N is generated by the coroots α ∨ 1 , α ∨ 2 , · · · , α ∨ r , where r = dim T . Let X 2 be the smooth Fano embedding of (SL(3, C) × SL(3, C))/ SL(3, C) with Picard number one. Using the description in [Ruz10], we know that the two colors D 1 , D 2 and the G-stable divisor E in X 2 have the images ρ( Proposition 3.4. Let X 2 be the smooth Fano symmetric embedding of (SL(3, C) × SL(3, C))/ SL(3, C) with Picard number one. The moment polytope ∆ 2 = ∆(X 2 , K −1 X2 ) is the convex hull of three points 0, 5̟ 1 , 5̟ 2 in M ⊗ R.

3.3.
Smooth Fano embedding of SL(6, C)/ Sp(6, C) with Picard number one. Recall the involution of Type AII. Let θ be an involution of SL(2m, C) defined by θ(g) = J m (g t ) −1 J t m , where J m is the 2m × 2m block diagonal matrix formed by 0 1 −1 0 . Then G θ = Sp(2m, C) is the group of elements that preserve a nondegenerate skew-symmetric bilinear form ω(v, w) = v t J m w. We can check that the restricted root system Φ θ is the root system of type A 2 with multiplicity four, and the spherical weight lattice M = X(T /T ∩ G θ ) is generated by 2λ for weights λ ∈ X(T s ), where T s denotes a split subtorus of dimension two in a maximal torus T ⊂ SL(6, C). In fact, if we choose the torus of diagonal matrices as T , then the maximal split torus T s consists of diagonal matrices of the form diag (a 1 , a 1 , a 2 , a 2 , a 3 , a 3 ) with a 1 , a 2 , a 3 ∈ C * and a 2 1 a 2 2 a 2 3 = 1. Denoting by α k : T s → C * for k = 1, 2 the characters defined by α k (diag(a 1 , a 1 , a 2 , a 2 , a 3 , a 3 ) we have the restricted root system Φ θ = {±2α 1 , ±2α 2 , ±(2α 1 + 2α 2 )} of type A 2 . Then the dual lattice N is generated by the coroots 1 2 α ∨ 1 , 1 2 α ∨ 2 . Let X 3 be the smooth Fano embedding of SL(6, C)/ Sp(6, C) with Picard number one. Using the description in [Ruz10], we know that the two colors D 1 , D 2 and the G-stable divisor E in X 3 have the images respectively. Proposition 3.6. Let X 3 be the smooth Fano symmetric embedding of SL(6, C)/ Sp(6, C) with Picard number one. The moment polytope ∆ 3 = ∆(X 3 , K −1 X3 ) is the convex hull of three points 0, 18̟ 1 , 18̟ 2 in M ⊗ R. Proof. From the colored data of SL(6, C)/ Sp(6, C) and the G-orbit structure of X 3 , we know the relation −K X3 = 4D 1 + 4D 2 + E of the anticanonical divisor. Using Proposition 3.1, 1 4 ρ(D 1 ), 1 4 ρ(D 2 ) andρ(E) are used as inward-pointing facet normal vectors of the moment polytope ∆(X 3 , K −1 X3 ). Like the previous computations, 1 4 ρ(D 1 ) and 1 4 ρ(D 2 ) determine the positive restricted Weyl chamber. Indeed, 1 4 ρ(D 1 ) = 1 8 α ∨ 1 gives an inequality gives a domain {x · 2̟ 1 + y · 2̟ 2 ∈ M ⊗ R : x + y ≤ 9}, the moment polytope ∆(X 3 , K −1 X3 ) is the intersection of this half-space with the positive restricted Weyl chamber. Thus ∆(X 3 , K −1 X3 ) is the convex hull of three points 0, 18̟ 1 , 18̟ 2 in M ⊗ R.