Gorenstein spherical Fano varieties

We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-dimensional Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}-factorial Gorenstein spherical Fano variety is bounded by 2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2d$$\end{document}. This paper also contains an overview of the description of the natural representative of the anticanonical divisor class of a spherical variety due to Brion.

Gorenstein Fano varieties by convex geometry, namely one has a bijective correspondence between Gorenstein toric Fano varieties and reflexive polytopes, i. e. lattice polytopes whose dual is a lattice polytope as well (see [2,Theorem 4.1.9]). Generalizing the notion of a reflexive polytope, Pasquier established a similar correspondence for horospherical varieties in [28]. In this paper, we extend this correspondence to (arbitrary) spherical varieties.
In [6], Brion determined a natural representative of the anticanonical divisor class for any spherical variety X , on which our combinatorial description of Gorenstein spherical Fano varieties essentially depends. For this reason, we have found it useful to give a self-contained overview of Brion's as well as some related results of Luna, together with some motivation. The new results of this paper, i. e. Theorems 1.9 and 1.10, are stated afterwards.
Let G be a connected reductive complex algebraic group and B ⊆ G a Borel subgroup. A closed subgroup H ⊆ G is called spherical provided that the homogeneous space G/H contains an open B-orbit. Let H ⊆ G be a spherical subgroup. A G-equivariant open embedding G/H → X into a normal irreducible G-variety X is called a spherical embedding, and X is called a spherical variety.
An anticanonical divisor is a Weil divisor −K X of X such that O X (−K X ) =ω X whereω X is the anticanonical sheaf of X , i. e. the reflexive sheaf coinciding with the top exterior power of the tangent sheaf on the smooth locus of X . It is equipped with a natural G-linearization (see Sect. 3).
It is known that the open B-orbit U is isomorphic to (C * ) r × C s (see [29,Theorem 5]) and hence has trivial divisor class group. In particular, the invertible sheafω G/H is trivial on U , and we might ask whether there is a natural choice of a generator s ∈ (U,ω G/H ).
We first recall the well-known case where G = B = T is a torus and H is trivial (see, for instance, [14,Sect. 4.3]). A spherical embedding G/H → X is then simply a toric variety X with embedded torus U = G/H ∼ = T , and it is possible to show that there is a unique generator s ∈ (U,ω G/H ) which is T -invariant. This generator can be explicitly written as where x 1 , . . . , x n is a choice of coordinates for the torus T . In addition to being T -invariant, this section has the important property that it has a zero of order 1 along every T -invariant divisor in X .
In general, however, there need not be any B-invariant section s ∈ (U,ω G/H ) as the following example shows. As we cannot expect a B-invariant section, we have to look for something else. An equivalent characterization for a homogeneous space G/H to be spherical is that the G-module (G/H, L ) is multiplicity-free for every G-linearized invertible sheaf L , i. e. the multiplicity of any simple G-module in the decomposition of the module of global sections is at most 1. We obtain where χ runs over pairwise different dominant weights of B and V χ is the simple G-module of highest weight χ. This decomposition is very simple when H ⊆ G is a parabolic subgroup since then (G/H,ω G/H ) is a simple G-module by the Borel-Weil-Bott theorem (see [10,11]). In particular, exactly one dominant weight occurs (compare this with Example 1.1), and there is a unique choice (up to a constant factor) of a B-semi-invariant section s ∈ (G/H,ω G/H ) which restricts to a generator s ∈ (U,ω G/H ).
Let G/H again be an arbitrary spherical homogeneous space. We denote by P ⊆ G the stabilizer of the open B-orbit U . If H contains a maximal unipotent subgroup of G, the homogeneous space G/H is called horospherical, and the normalizer of H in G is a parabolic subgroup conjugated to the opposite parabolic of P, which we denote by P − . Hence there is a natural morphism π : G/H → G/P − , which is known to be a torus fibration. Therefore π * (ω G/P − ) =ω G/H , the simple G-module (G/P − ,ω G/P − ) is a direct summand of (G/H,ω G/H ), and a unique B-semi-invariant section s ∈ (G/P − ,ω G/P − ) ⊆ (G/H,ω G/H ) exists, whose weight we denote by κ P ∈ X(B). For arbitrary spherical homogeneous spaces G/H , there is no natural morphism π : G/H → G/P − , but the following statement is nevertheless valid. The fact from Theorem 1.2 that the section s has a zero of order 1 along any G-invariant prime divisor actually characterizes this section. This is a straightforward application of basic facts from the embedding theory of spherical homogeneous spaces. Theorem 1.4 Let s ∈ (U,ω G/H ) be a generator (which is automatically B-semiinvariant). Then the following conditions are equivalent: (1) The section s is of B-weight κ P .
(2) For any spherical embedding G/H → X the section s has a zero of order 1 along any G-invariant prime divisor.
We denote by D = {D 1 , . . . , D k } the set of B-invariant prime divisors in G/H , whose elements are called the colors of G/H . In order to obtain an explicit formula for the coefficients m i in Theorem 1.2, Brion divided the colors into several types (see [6, 4.2]). These types are in agreement with the definition of the types of colors due to Luna (see [24, 2.7], [25, 2.3], see also [31,Sect. 30.10]), which we now explain. For additional information on the types of colors, we refer the reader to Sect. 5.
We choose a maximal torus T ⊆ B, denote by R ⊆ X(T ) = X(B) the associated root system, and write S ⊆ R for the set of simple roots corresponding to B. For α ∈ S we denote by P α ⊆ G the corresponding minimal parabolic subgroup containing B, and define As the colors are not G-stable, every color is moved by at least one minimal parabolic subgroup, so that we have D = α∈S D(α). Moreover, the stabilizer P ⊆ G of the open B-orbit is the parabolic subgroup containing B corresponding to the set S p := {α ∈ S : D(α) = ∅}.
We denote by M ⊆ X(B) the weight lattice of B-semi-invariants in the function field C(G/H ) and by N := Hom(M, Z) the dual lattice together with the natural pairing ·, · : N × M → Z. We denote by V the set of G-invariant discrete valuations on C(G/H ), and define the map ι : and unique up to a constant factor. As the map ι is injective, we may consider V as a subset of the vector space N Q := N ⊗ Z Q. It is known that V is a cosimplicial cone (see [4]), called the valuation cone of G/H . In particular, the valuation cone is full-dimensional. By we denote the set of primitive generators in M of the extremal rays of the negative of the dual of the valuation cone V. The elements in are called the spherical roots of G/H . The type of a color D i ∈ D(α) is defined as follows: If α ∈ , we say that D i is of type a. If 2α ∈ , we say that D i is of type 2a. Otherwise, we say that D i is of type b. The type does not depend on the choice α ∈ S such that D i ∈ D(α). Moreover, we have |D(α)| ≤ 2 with |D(α)| = 2 if and only if α ∈ (i. e. the colors in D(α) are of type a).
We can now state the explicit formula for the coefficients m i in the expression for div s due to Brion and Luna.
This concludes the overview, so that we are now able to explain the combinatorial description of Gorenstein spherical Fano varieties. Let m 1 , . . . , m k ∈ Z >0 be the coefficients of the colors in the expression for the anticanonical divisor from Theorem 1.2. We define the map ρ : Definition 1.6 Let G/H → X be a complete spherical embedding, and let X 1 , . . . , X n be the G-invariant prime divisors in X . We define the polytope Remark 1.7 The polytope introduced in Definition 1.6 has been used by Alexeev and Brion to prove boundedness of spherical Fano varieties in [1].
The generalization of the notion of a reflexive polytope to the theory of spherical varieties is the following (it is a generalization of [28,Définition 3.3]). For a polytope Q we denote by Q * its dual polytope, and for a face F Q we denote by F Q * its dual face. Note that cone( v) is a full-dimensional cone in N Q for every vertex v ∈ Q * .

Theorem 1.9 The assignment X → Q X induces a bijection between isomorphism classes of Gorenstein spherical Fano embeddings G/H → X and G/H -reflexive polytopes.
Using Theorem 1.9, one may translate questions about Gorenstein spherical Fano varieties into the realm of convex combinatorics. Applying this approach, we are going to prove the following bound on the Picard number. Theorem 1.10 has been proven by Casagrande in the case of a toric variety X (see [8]) and by Pasquier in the case of a horospherical variety X (see [28]). Our proof is inspired by the two previous works. Observe that Theorem 1.10 does not hold for an arbitrary variety X , e. g. if S is the surface given by blowing-up P 2 in eight general points, the variety S m has Picard number 9m (see [9]).

Notation and generalities
Spherical embeddings admit a combinatorial description due to the Luna-Vust theory (see [21,26]). Similarly to the theory of toric varieties, one obtains a description of spherical embeddings of G/H by colored fans, which are combinatorial objects living in the vector space N Q .

Definition 2.1
A colored cone is a pair (C, F ) where F ⊆ D and C ⊆ N Q is a cone generated by ρ(F) and finitely many elements of V. A colored cone is called supported if relint(C) ∩ V = ∅. A colored cone is called strictly convex if C is strictly convex and 0 / ∈ ρ(F).

Definition 2.2
A face of a colored cone (C, F ) is a colored cone (C , F ) such that C is a face of C and F = F ∩ ρ −1 (C ). It is called a supported face if it is supported as a colored cone.

Definition 2.3
A colored fan is a nonempty finite collection F of strictly convex colored cones such that for every (C, F ) ∈ F every face of (C, F ) is also in F and for every v ∈ N Q there is at most one (C,

Definition 2.4
A supported colored fan is a nonempty finite collection F of strictly convex supported colored cones such that for every (C, F ) ∈ F every supported face of (C, F ) is also in F and for every v ∈ V there is at most one (C, F ) ∈ F with v ∈ relint(C). A supported colored fan F is called complete if supp F ⊇ V.
Remark 2. 5 We have defined the terms "colored cone" and "colored fan" with and without the adjective "supported". In the literature, only the supported versions are usually defined (and without the adjective "supported").

Remark 2.6
There is a natural map If F is complete, every point in V is contained in relint(C) for exactly one (C, F ) ∈ F, which is supported, so F supp is complete as well (as a supported colored fan). We now recall some results on divisors in spherical varieties due to Brion (see [3]). We use [31,Sect. 17] as general reference. Let G/H → X be a complete spherical embedding and F the corresponding supported colored fan. Let E be a B-stable Weil divisor. The Weil divisor E is Q-Cartier (resp. Cartier) if and only if for every maximal supported colored cone (C,  (1) the piecewise linear function ψ E is strictly convex, i. e. for every u ∈ C \ C we have u, v C > u, v C for any two maximal (C, F ), (C , F ) ∈ F, and (2) for every maximal (C, F ) ∈ F and every D ∈ D \ F the multiplicity of D in E is strictly greater than ρ(D), v C .
Finally, we will require the following definition.

Definition 2.10
Let Q ⊆ N Q be a polytope with 0 ∈ int(Q). We associate to it the colored fan F(Q) consisting of the colored cones (cone

The co-and tangent sheaves of a smooth G-variety
In this section, let X be an arbitrary smooth G-variety for an arbitrary algebraic group G. Then G acts on X by an action morphism α : G × X → X . We denote by μ : G × G → G the multiplication morphism of the algebraic group G and by π X : G × X → X (resp. π G×X : G × G × X → G × X ) the natural projection on the second (resp. on the second and third) factor. Let us repeat the definition of a G-linearization of a quasicoherent sheaf (see [31,Definition C.2] or [27, Definition 1.6]). Recall that the cotangent sheaf X of X is, locally on affine open neighbourhoods U , given as the sheaf associated to the module of Kähler differentials O X (U )/C . The pullback of differential forms with respect to the action morphism α yields the inverse of a G-linearization of the cotangent sheaf, namely α −1 : α * X ∼ − → π * X X . As X is smooth, we may dualize α −1 and obtain a G-linearization of the tangent sheaf T X : For an affine open subset U ⊆ X and g ∈ G, let us denote the coordinate rings of U and g · U by A and B respectively. The element g ∈ G acts on a local section Let G = (C, +) or G = (C * , ·) be a one-dimensional connected algebraic group with neutral element e ∈ G. We will recall how one can associate a global vector field u ∈ (X, T X ) to a one-parameter subgroup u : G → G. The coordinate ring of G is either the polynomial ring C[t] or the Laurent polynomial ring C[t ±1 ]. In both cases, we have a natural choice of a basis of the tangent space of G over the point e, namely ∂ ∂t e ∈ T G | e . Let U be an affine open subset of X with coordinate ring A. The restriction of u to U lies in Der C (A, A) and is given by It is straightforward to check that these local sections are well-defined and glue to a global section u ∈ (X, T X ).

Remark 3.2
Assume that X = G and G acts on itself by left translation. It is then straightforward to check that u is an invariant vector field, i. e.
Let G/H → X be a spherical embedding. We denote the G-invariant prime divisors in X by X 1 , . . . , X n . As we are interested in the anticanonical sheaf of X , we may assume that X does not contain G-orbits of codimension two or greater. In particular, X is smooth toroidal and Pic(X ) = Cl(X ).
Since X is a smooth toroidal variety, it has the following local structure (see [31, The kernel of the L-action on Z , which we denote by L 0 , contains (L , L) and Z is a toric embedding of L/L 0 . Every G-orbit intersects Z in a unique L/L 0 -orbit. Any x 0 ∈ U corresponds to some (u 0 , z 0 ) ∈ R u (P) × Z under the isomorphism above. We fix x 0 such that u 0 is the neutral element of R u (P). By replacing H with a conjugate, we may assume that x 0 has stabilizer H ⊆ G.
We denote byω X := dim X T X , i. e. the top exterior power of the tangent sheaf, the anticanonical sheaf of X . It is an invertible sheaf and carries a natural G-linearization induced from the G-linearization of T X .
The variety Z is a toric variety for a quotient torus of T . Let T 0 be the kernel of the T -action on Z and let T 1 be a subtorus of T with T = T 0 T 1 such that T 0 ∩ T 1 is finite. We have a commutative diagram of equivariant morphisms with respect to the action of B 0 := R u (P)T 1 : The arrows are finite coverings. In particular, the tangent space of X at (u 0 , z 0 ) is isomorphic to the direct sum of the tangent spaces of R u (P) and T 1 at the corresponding neutral elements. This is the lie algebra of B 0 which decomposes as where S p denotes the root system generated by S p and g α denotes the subspace of Tsemi-invariant vectors of weight α in g := Lie G. Choose a realization (u α ) α∈R of the root system R (see [30, §8.1]) and choose a basis λ 1 , . . . , λ r of the lattice of one-parameter multiplicative subgroups of T 1 . In Sect. 3 we have seen how to associate a global vector field u α (resp. l 1 , . . . , l r ) to the one parameter subgroup u α (resp. λ 1 , . . . , λ r ). We obtain a global section Since Y has codimension 1 in X and the number of vector fields which have been wedged to obtain s is dim X , the global section s vanishes on Y .
We now show that s vanishes nowhere on the open B-orbit. Since u maps into B 0 , we may define a global vector field u on B 0 . By the local structure theorem, B 0 is a finite covering of B · x 0 . In particular, we have a well-defined pushforward of vector fields, and u| B·x 0 is the pushforward of u . Since u is T 0 ∩ T 1 -invariant (see Remark 3.2), it suffices to show that s ∈ (B 0 ,ω B 0 ), i. e. the global section of the anticanonical sheaf of B 0 which arises by wedging all the global vector fields u , vanishes nowhere.
Since u | e ∈ Lie u(G), it follows that s | e arises by wedging a basis of Lie B 0 . In particular, s does not vanish at e. Since s is invariant, it follows that it vanishes nowhere.
Recall from the introduction that, by the Borel-Weil-Bott theorem, the space (G/P − ,ω G/P − ) is a simple G-module, whose highest weight we denote by κ P . Then the following result implies Theorem 1.2.  Next, we determine the weight of s. Let w ∈ T , f ∈ (U, O U ), and x ∈ U . Then we have .
In particular, the weight of s depends only on the set S p . As G/P − and X have the same stabilizer of the open B-orbit, it follows that s is B-semi-invariant of weight κ P .

Remark 4.3
For I ⊆ S we denote by ρ I the half-sum of the positive roots in the root system generated by I . It is also known that ρ I is the sum of the fundamental dominant weights of the root system generated by I (see [18,Lemma 13.3A]). By Proposition 4.2, we have Remark 4.4 A nonzero B-semi-invariant rational section ofω X is uniquely determined by its weight χ ∈ X(B) up to a constant factor, and such a rational section exists if and only if χ ∈ κ P + M (see also Sect. 9).
r j X j for integers m i and r j . Since s vanishes on every B-invariant divisor, it follows that the m i and r j are positive. To show that r j = 1, we consider the restriction s := s| X 0 . Above, we have seen that the open subset X 0 is isomorphic to the product variety R u (P) × Z . In particular, ω X 0 ∼ = π * 1ω R u (P) ⊗ π * 2ω Z where π i denotes the projection onto the i-th factor. The section s behaves well under this product decomposition. Indeed, set s 1 := α∈R + \ S p u α | R u (P) and s 2 := l 1 | Z ∧ . . . ∧ l r | Z . Then under the isomorphism above the section s corresponds to the section s 1 ⊗ s 2 . Since s 1 does not vanish on R u (P), we obtain n j=1 Now, Z is a toric variety with respect to the quotient torus T 1 /(T 0 ∩ T 1 ), and the pullback under π 2 of its torus-invariant divisors are exactly the G-invariant divisors X j of X . Since σ 2 is invariant under the action of the torus T 1 /(T 0 ∩ T 1 ), it follows, by toric geometry, that r j = 1. Take any spherical embedding G/H → X which contains exactly two orbits, namely G/H and a G-invariant prime divisor X 1 . The colored fan corresponding to X is given by the ray Q ≥0 ν 1 where ν 1 ∈ N denotes the G-invariant valuation induced by X 1 . Since s and s have a zero of order 1 along X 1 , we obtain f / f , ν 1 = 0.
It follows that the valuation cone V of G/H is contained in the subspace {w ∈ N Q : f / f , w = 0}. Since V is a full-dimensional cone, this is only possible if the B-weight of f / f is 0, i. e. f and f coincide up to a scalar multiple.

Types of colors
In [6, 4.2], Brion has defined colors of types II, III, and IV. These types are equivalent to the types b, a, and 2a due to Luna respectively, which means that Theorem 1.5 is equivalent to [6,Theorem 4.2]. We will, however, prove Theorem 1.5 by reducing it to the situation of [24, 3.6], i. e. to the case of a wonderful variety (and afterwards it will be easy to see that the types of colors due to Brion and Luna coincide).
A wonderful variety is a spherical variety which is complete, smooth, simple, and toroidal. We explain (from [25, 6.1]) how to associate to the spherical variety X a wonderful variety Y .
Proof By [24, Proposition 3.6(2)], an anticanonical divisor of Y is given by The result now follows from the fact that D is a basis of Pic(Y ) = Cl(Y ).
Proof of Theorem 1.5 Let s ∈ (G/H,ω G/H ) and s ∈ (G/H ,ω G/H ) be B-semi-invariant sections of weights κ P and κ P respectively. As P = P, we have π * s = s (up to a constant), and hence, for the pullback of Cartier divisors, π * div s = div s. On the other hand, we have the pullback of Cartier divisors π * D i = D i (apply [13, Sect. 2.2, Theorem 2.2], see also [31,Lemma 30.24]), hence m i = m i .

Remark 5.2
Replacing G with a finite cover, we may assume G = G ss × C where G ss is semisimple simply-connected and C is a torus. Then, by [19, Proposition 2.4 and Remark after it], every invertible sheaf on the normal variety X can be G-linearized. There exist (unique) G-linearizations of the invertible sheaves O X (D i ) and O X (X j ) such that their canonical sections are C-invariant. With these linearizations, Brion's description of the anticanonical sheafω of an arbitrary spherical variety X is not only valid inside Pic(X reg ), but even inside the group of isomorphism classes of G-linearized invertible sheaves Pic G (X reg ).

Proposition 5.3 The types of colors II, III, and IV due to Brion coincide with the types of colors b, a, and 2a due to Luna respectively.
Proof A color D i ∈ D is of type II if and only if it is of type b as both situations are characterized by m i ≥ 2. Now let D ∈ D be of type III (resp. of type IV), and let C be a basic curve representing D in the sense of [6, 4.2]. The kernel of the B-action on C is the radical of a (unique) minimal parabolic subgroup P α (see [6, 1.1]), and we have D ∈ D(α) (see [5,Proposition 3.6]). According to [6, Proposition 1.2], the spherical variety G · C contains a color of type a (resp. of type 2a) which is moved by P α . By [15,Theorem 1.1], this is only possible if α ∈ (resp. if 2α ∈ ), i. e. D is of type a (resp. of type 2a).
Finally, it will be helpful to explain one further approach due to Knop to characterize the types of colors, which is easier to apply in certain situations (see the examples in Sect. 6). We fix a point x 0 ∈ U in the open B-orbit. Let α ∈ S such that D(α) = ∅. We have P α /B ∼ = P 1 and the natural action of P α on P 1 yields a morphism φ α : P α → PGL 2 . Let H α be the stabilizer of x 0 inside P α . Then φ α (H α ) is a proper spherical subgroup of PGL 2 (see [  The stabilizer of the open B-orbit in G/H is B itself. Therefore we have S p = ∅ and κ P = 2ρ S . Since ρ S is equal to the sum of the fundamental dominant weights, we obtain To be more precise, let A i j for 1 ≤ i < j ≤ 3 be the 2 × 2 matrix whose rows are given by the first rows of the i-th and j-th factor of G. Then det A i j is an equation for D i j . It is not difficult to see that D(α) = {D 12 , D 13 }, D(β) = {D 12 , D 23 }, and D(γ ) = {D 13 , D 23 }. In particular, it follows that all colors are of type a, and therefore m i j = 1 for 1 ≤ i < j ≤ 3.
By the definition of the types of colors due to Luna (see Sect. 1), it follows that α, β, γ are contained in M, and the valuation cone is given by As in the previous example, it follows that 2α i is contained in M for 1 ≤ i ≤ n − 1, and the valuation cone is given by Let G act on C n × C n by acting naturally on the first factor and with the contragredient action on the second factor. Denoting the coordinates of the first factor by X 1 , . . . , X n and the coordinates of the second factor by Y 1 , . . . Y n , we obtain There are two colors D 1 := V(X n ) and D 2 := V(Y 1 ). It is not difficult to see that D(α n−1 ) = {D 1 }, D(α 1 ) = {D 2 }, and S p = {α 2 , . . . , α n−2 }. We see that both colors are of type b as in the previous example, but we have to suitably conjugate H first since the basepoint does not lie in the open B-orbit. We obtain and similarly m 2 = n − 1.

Q-Gorenstein spherical Fano varieties
We continue to use the notation from the introduction. In particular, G/H is a spherical homogeneous space, D = {D 1 , . . . , D k } is the set of colors, and m i ∈ Z >0 are the coefficients of the colors in the expression for the anticanonical divisor from Theorem 1.2.
Definition 7.1 A polytope Q ⊆ N Q is called Q-G/H -reflexive if the following conditions are satisfied: (1) ρ(D i )/m i ∈ Q for every i = 1, . . . , k.

Proposition 7.2 Let G/H → X be a Q-Gorenstein spherical Fano embedding. Then the polytope Q X ⊆ N Q is Q-G/H -reflexive.
Proof Let X 1 , . . . , X n be the G-invariant prime divisors in X . It follows from the completeness of X that and therefore 0 ∈ int(Q X ).

Proposition 7.3 Let Q ⊆ N Q be a Q-G/H -reflexive polytope. Then F supp (Q) is a complete supported colored fan such that the associated spherical embedding G/H
Proof As Q is full-dimensional, F(Q) is complete (see Definition 2.10), and therefore, by Remark 2.6, F supp (Q) is complete as well. Let X := X F supp (Q) . Every maximal cone We define a piecewise linear function ψ : V → Q by ψ| C∩V := − ·, v C . It is straightforward to check that ψ is the piecewise linear function corresponding to the Q-Cartier divisor −K X . As Q is a polytope, the piecewise linear function ψ is strictly convex. Together with property (1) of Definition 7.1, it follows from Proposition 2.9 that −K X is ample, and hence X = X F supp (Q) is Fano.

Proposition 7.4 The assignments X → Q X and Q → X F supp (Q) define a bijection between isomorphism classes of Q-Gorenstein spherical Fano embeddings of G/H and Q-G/Hreflexive polytopes.
Proof The well-definedness of the two maps follows from Propositions 7.2 and 7.3. It remains to show that the maps are inverse to each other. Let Q ⊆ N Q be a Q-G/H -reflexive polytope and G/H → X the spherical embedding corresponding to the colored fan F supp (Q). Let u ∈ Q be a vertex. By property (3) of Definition 7.1, we have u ∈ Q X , hence Q ⊆ Q X . Now, let u ∈ Q X be a vertex. If u = ρ(D i ) m i for some i ∈ {1, . . . , k}, then u ∈ Q because of property (1) of Definition 7.1. Otherwise, the vertex u is the primitive generator of a ray in F supp (Q), in which case the definition of the face fan also implies u ∈ Q. Hence we have Q X ⊆ Q.
Let G/H → X be a Q-Gorenstein spherical Fano embedding with associated supported colored fan F and ψ : V → Q the strictly convex piecewise linear function associated to the anticanonical divisor −K X . It suffices to show that the maximal cones of F supp (Q X ) and F coincide. Let (C, F ) be a maximal cone in F. Then ψ| C∩V = − ·, v C for some v C ∈ M Q . We obtain that F := {u ∈ Q X : u, v = −1} is a facet of Q X such that cone(F) = C and F = ρ −1 (F). In particular, it follows that all maximal cones of F are in F supp (Q X ).

Gorenstein spherical Fano varieties
Let Q ⊆ N Q be a Q-G/H -reflexive polytope. In this section, we investigate when the associated Q-Gorenstein spherical Fano embedding G/H → X is Gorenstein, i. e. the anticanonical divisor is Cartier. Recall that the vertices of Q * correspond to the colored cones of maximal dimension in F(Q). Definition 8.1 A vertex v ∈ Q * is called supported if the corresponding colored cone in F(Q) is supported. In this case, the corresponding facet v Q is called supported as well. The set of supported vertices of Q * is denoted by V supp (Q * ).
Proof Let v ∈ Q * be a vertex and C := cone( v) ⊆ N Q the corresponding cone in F(Q). We denote by the tangent cone of Q * in v, which is an affine cone with apex v. By [17, Corollary 5.
The last statement follows since any v ∈ T v Q * ∩ (v + cone( )) may be rescaled to be arbitrarily near to v.

Definition 8.4
The Q-G/H -reflexive polytope Q is called G/H -reflexive if every supported vertex of Q * lies in the lattice M.
We will show in Proposition 8.6 that this definition is in agreement with Definition 1.8.

Theorem 8.5 Recall that X is a Q-Gorenstein spherical Fano variety by assumption. The variety X is Gorenstein if and only if the polytope Q is G/H -reflexive. In particular, there is a bijection between isomorphism classes of Gorenstein spherical Fano embeddings of G/H and G/H -reflexive polytopes.
Proof Let ψ : V → Q be the piecewise linear function corresponding to the Q-Cartier divisor (1) ρ(D i )/m i ∈ Q for every i = 1, . . . , k.
(2) 0 ∈ int(Q). Proof Let Q ⊆ N Q be a full-dimensional polytope satisfying the above conditions and u ∈ Q a vertex not contained in {ρ(D i )/m i : i = 1, . . . , k}. Then we have u ∈ V, which implies that there exists a supported full-dimensional cone C in F(Q) having Q ≥0 u as extremal ray.
Let v ∈ Q * be the supported vertex corresponding to C. Then u, v = −1, and, as u ∈ N and v ∈ M, we obtain that u is primitive.

Global sections of the anticanonical sheaf
In this section, we recall from [3, 3.3] the G-module structure of the space of global sections of a B-invariant Cartier divisor on a spherical variety X . For simplicity, we assume that X is complete. We then investigate the special case of the anticanonical sheaf when X is Gorenstein Fano. We use [31,Sect. 17.4] as a general reference. Let G/H → X be an arbitrary spherical embedding with associated supported colored fan F. We denote by D 1 , . . . , D k the colors and by X 1 , . . . , X n the G-invariant prime divisors in X . Consider a B-invariant Cartier divisor For every maximal supported colored cone (C, F ) ∈ F we write v C ∈ M as in the last part of Sect. 2. As in Remark 5.2, we may assume G = G ss × C where G ss is semisimple simply-connected and C is a torus, and then the invertible sheaf O X (δ) can be G-linearized. Let s δ be a rational section of O X (δ) satisfying div s δ = δ. As δ is B-invariant, the section s δ is B-semi-invariant of some weight κ δ ∈ X(B) (not necessarily contained in M). We write F max for the set of maximal cones of F, and we set D X := (C,F)∈F max F . If we define is contained in the set of dominant weights, and we have where V χ denotes the irreducible G-module of highest weight χ.  Example 10.2 Let G := Spin 5 × Spin 5 . Fix some maximal torus contained in some Borel subgroup, and denote by α 1 , α 2 (resp. α 1 , α 2 ) the simple roots of the first (resp. the second) simple factor Spin 5 where α 2 (resp. α 2 ) is the shorter root. According to the third entry in [32, Table B], there exists a spherical homogeneous space G/H with spherical roots γ 1 := α 2 +α 2 and γ 2 := α 1 + α 1 , such that (γ 1 , γ 2 ) is a basis of the lattice M. We denote by (γ * 1 , γ * 2 ) the corresponding dual basis of the lattice N . We have S p = ∅, and there are exactly two colors D 1 , D 2 (both of type b) with ρ(D 1 ) = −γ * 1 + 2γ * 2 and ρ(D 2 ) = 2γ * 1 − 2γ * 2 . As S p = ∅, we have κ P = 2ρ S (see Remark 4.3), so that the coefficients in the expression for the canonical divisor are m 1 = m 2 = 2. Then Q := conv(γ * 1 − γ * 2 , − 1 2 γ * 1 + γ * 2 , −γ * 1 , −γ * 2 ) is a G/Hreflexive polytope, and its dual polytope is Q * = conv(γ 1 − 1 2 γ 2 , γ 1 + γ 2 , γ 2 , −4γ 1 − 3γ 2 ). The polytopes Q and Q * are illustrated in Fig. 3. The valuation cone is shown in grey, and the dashed arrows are the images of the colors under ρ in N . The dotted arrows are translates of the spherical roots γ 1 , γ 2 ∈ M showing that exactly the circled vertex of Q * is supported (see Proposition 8.3).

Polytopes with simplicial facets
The purpose of this section is to prove an auxiliary result on polytopes (Proposition 11.1), which will be used in the proof of Theorem 1.10.
Let V ∼ = Q n be a vector space of dimension n and Q ⊆ V a full-dimensional polytope with 0 ∈ int(Q).
We will use the following result.  Definition 11.5 Let σ ⊆ V be a cone and v ∈ V not contained in σ . Then we call v · σ := cone({v} ∪ σ ) the join of v and σ . Definition 11.6 Let F be a fan in V and v ∈ V . Assume that v · σ is defined for every σ ∈ F and that relint(v · σ ) ∩ relint(v · σ ) = ∅ whenever σ, σ ∈ F are distinct. Then the fan v · F := {v · σ : σ ∈ F} is called the join of v and F. Definition 11.7 Let F be a fan in V and v ∈ V a point such that there exists an (automatically uniquely determined) cone σ ∈ F with v ∈ relint(σ ). Then we call the transition For a full-dimensional polytope Q ⊆ V with 0 ∈ int(Q ) we denote by F(Q ) its face fan in V .

Lemma 11.8 Let F
Q be a non-simplicial face and v ∈ relint(F). Then there exists a polytope Q ⊆ V containing Q such that and the simplicial facets of Q are facets of Q .
Proof Let F s 1 , . . . , F s k be the facets not containing F. Choose t > 1 such that tv, n s j < 1 for j = 1, . . . , k and set v := tv. Then v is beneath the facets not containing F and beyond the facets containing F. We set Q := conv({v } ∪ Q). As simplicial facets of Q do not contain F, it follows from Theorem 11.3(1) that the simplicial facets of Q are facets of Q .
We now verify that F(Q ) = v F(Q). It suffices to check that the sets of maximal cones coincide.
By Theorem 11.3(1), a facet F i Q does not contain F if and only if F i is a facet of Q . Furthermore, the facets of Q not containing F are in correspondence with the maximal cones in F(Q) \ st(Q ≥0 F, F(Q)).
Let F be a facet of Q which is not a facet of Q. By Theorem 11.3(2), we have F = conv({v } ∪ F ) where F Q is a face of codimension 2 such that among the facets of Q containing F there is at least one beneath and at least one beyond v . Such faces F are in correspondence with the maximal cones in st(Q ≥0 F, F(Q)) \ st(Q ≥0 F, F(Q)). The result follows from the equality Q ≥0 F = v · Q ≥0 F .
Proof of Proposition 11.1 We can transform the fan F(Q) into a simplicial one by successively applying stellar subdivision to non-simplicial cones. By Lemma 11.8, we also obtain a corresponding polytope.

Proof of Theorem 1.10: the inequality ρ X ≤ 2d
Let G/H → X be a Gorenstein spherical Fano embedding with associated G/H -reflexive polytope Q. The condition for Q-factoriality from Proposition 2.8 can be straightforwardly translated into the setting of G/H -reflexive polytopes as follows: Now assume that X is Q-factorial, of rank r , and of dimension d. The proof of the inequality ρ X ≤ 2d appearing here is an extended version of the proof of the horospherical case in [28]. Note that, in contrast to the horospherical case, not all facets of the polytope Q are necessarily simplicial (only the facets dual to the supported vertices of Q * are). Let (e * 1 , . . . , e * r ) be the basis of M Q dual to (e 1 , . . . , e r ). Let j = 1, . . . , r . Then e * j , u j = 0 (otherwise u j would be contained in the hyperplane spanned by {e i } i = j ). We define and v j := v + λ j e * j .
Then we have e i , v j = −1 for i = j and u j , v j = −1. It follows that v j is a (not necessarily supported) vertex of Q * and F j = v j . We have λ j > 0 as −1 < e j , v j . Then u, v j = λ j u, e * j and hence u / ∈ F j if and only if u, e * j ≥ 0. If u / ∈ F j for all j = 1, . . . , r , then u, e * j ≥ 0 for all j = 1, . . . , r and therefore u ∈ cone(e 1 , . . . , e r ), which implies u = e j ∈ F j for some j = 1, . . . , r , a contradiction. We obtain u ∈ F j for some j = 1, . . . , r .
Proof As v is a simplex, it has r vertices, which we denote by u 1 , . . . , u n , and the first assertion follows. For the second assertion, by Proposition 11.1, we may replace the polytope Q by a simplicial polytope Q such that Let u ∈ V (Q) ∩ V ∩ N with u, v = 0. By Lemma 12.2, there exists a facet F u Q adjacent to v and containing u. As Q is assumed simplicial, F u has vertices u 1 , . . . , u i−1 , u, u i+1 , . . . , u r for some i = 1, . . . , r . Since there are r facets adjacent to v, the result follows.
For a fixed v the number of vertices u of V with u, v = −1 is equal to the number of vertices of the facet v, which is exactly r . By Corollary 12.3, the number of u ∈ V (Q) ∩ V ∩ N with u, v = 0 is less than or equal to r . Hence the result follows.
Proof As u, γ ≤ 0 for every γ ∈ , u ∈ V, it follows from Proposition 12.5 that Let G/H → X be a Q-factorial Gorenstein spherical Fano embedding of rank r and of dimension d with ρ X = 2d.
Proposition 13. 1 We have |S \ S p | = |R + \ R + S p | where R S p denotes the root system generated by S p .
Proof According to Proposition 12.6, we have d = r + |S \ S p |. On the other hand, we have d = r + dim G/P = r + |R + \ R + S p |.

Proposition 13.2
The root system R is of type A k 1 × R S p for some k ∈ Z ≥0 .
Proof As S ⊆ R + and S p ⊆ R + S p , it follows from Proposition 13.1 that R + \ R + S p = S \ S p , which means that every positive root not contained in R + S p is simple. Let α ∈ S \ S p . Then the irreducible factor of R which contains α must be of type A 1 as it can not contain any non-simple positive root.
As in Remark 5.2, we may assume G = G ss × C where G ss is semisimple simplyconnected and C is a torus. According to Proposition 13.2, we have G ss ∼ = SL k 2 × G where G is the factor corresponding to R S p .