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Global Spherical Tropicalization via Toric Embeddings

The first steps in defining tropicalization for spherical varieties have been taken in the last few years. There are two parts to this theory: tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical embeddings. In this paper, we obtain a new description of spherical tropicalization that is equivalent to the other theories. This works by embedding in a toric variety, tropicalizing there, and then applying a particular piecewise projection map. We use this theory to prove that taking closures commutes with the spherical tropicalization operation.


Introduction
In his thesis [25], Tassos Vogiannou introduces a notion of spherical tropicalization for a spherical homogeneous space G/H. Broadly speaking, this operation records the G-invariant divisors on the field of rational functions while the standard toric tropicalization instead records torus-invariant divisors. Kaveh and Manon in [13] recover this construction using a Gröbner theory for spherical varieties that they develop. In [19], we propose a means for tropicalizing embeddings of spherical homogeneous spaces. There we define two constructions-one that mimics the theory for toric varieties developed by Kajiwara [12] and Payne [21] and one that extends the Gröbner theory definition of Kaveh and Manon-and show that they coincide.
In this paper, we give a third method for obtaining the tropicalization of a spherical embedding. This method is global, by which we mean the tropicalization operation need only be applied once to a single object. It is possible to globally tropicalize a toric variety by tropicalizing the quotient construction of a toric variety via the Cox ring. See [21], Remark 3.5 or [17], §6.2 for a description of this. For details on the quotient construction, refer to [5] and [6] and for a comprehensive overview of Cox rings in general, see [2]. Our aim is to mimic this construction for spherical varieties. This is noteworthy because in the other two constructions of spherical tropicalization from [19], the spherical variety is divided into simple G/H-embeddings, these are tropicalized separately, and then the tropicalizations are glued together.
The construction follows Gagliardi's work in [8], where he describes the Cox ring of a spherical embedding using its combinatorial data (see also [4]). We begin by embedding the spherical variety in a toric variety, which is an example of the embedding of a Mori dream space into a toric variety described by Hu and Keel [11]. Once embedded in the toric variety, we can tropicalize there using the standard toric theory. Then applying a particular piecewise projection map will deliver the spherical tropicalization.
This process does not work for all spherical varieties. When the spherical variety does not have the A 2 -property (Definition 5.1), embedding into a toric variety is not possible. The tropicalization of a spherical variety without the A 2 -property can still be described using this theory, but it cannot be done globally.
The layout of this paper is as follows. In §1, we review some of the theory of spherical varieties and their tropicalization. In §2, we describe how to tropicalize a morphism between two spherical embeddings. We explain a novel means of tropicalizing subvarieties of a spherical homogeneous spaces using toric embeddings in §3. Theorems 3.2 and 3.5 prove that this new method coincides with the original theory of [25]. In §4 we show how to embed toroidal spherical varieties in toric varieties and in §5 we extend this to spherical embeddings with color. Finally, §6 shows how to recover the extended tropicalization of an embedding by using the toric tropicalization. We also show in this section that taking the closure commutes with the tropicalization operation.
Acknowledgments. The author thanks Gary Kennedy for suggesting this direction of research and numerous discussions. Thanks also to Giuliano Gagliardi for providing input on several points.

Background
We review some of the theory of spherical varieties. For additional background, refer to [16], [14], [20], or [22]. If G is a connected reductive group, then the quotient G/H is a (spherical) homogeneous space if it is a normal variety containing a dense orbit of a Borel subgroup B. A spherical embedding X is a normal G-variety with an open equivariant embedding G/H ֒→ X. We will also call X a spherical variety or a G/H-embedding if we wish to highlight the underlying homogeneous space. A spherical variety X is modeled by a combinatorial object called a colored fan. We recall the outline of the theory here; refer to the cited papers for further details.
Denote by X the group of characters B → C * on B. The set C(G/H) (B) of B semi-invariant rational functions on G/H is defined by where the action of g on f is given by gf (x) := f g −1 x for x ∈ G/H. The subset M ⊆ X of characters χ f associated to f ∈ C(G/H) (B) /C * is a lattice. We write N := Hom(M, Z) to denote its dual. Further, M Q := M ⊗ Z Q and N Q := N ⊗ Z Q are the associated Q-vector spaces. We write V(G/H) to denote the set of G-invariant valuations on C(G/H) * that are trivial when restricted to C * . A given v ∈ V(G/H) induces a homomorphism C(G/H) (B) /C * → Q, namely f → v(f ). By identifying f with its associated character χ f , this homomorphism is an element of N Q . Thus, we obtain a map V(G/H) → N Q , which is an inclusion ( [16], Proposition 7.4). We will identify V(G/H) with its image in N Q , which is a cone. We call V(G/H) the valuation cone of G/H and write it as V when G/H is understood.
We also obtain a palette D of colors associated to G/H. These are the prime divisors of G/H that are B-stable but not G-stable. Each D ∈ D induces a G-invariant valuation v D given by vanishing along D. Thus, we obtain a map ρ : D → N given by D → v D ; it need not be injective. We now have enough background to define colored cones and colored fans: consists of a cone σ ⊆ N Q and a subset F ⊆ D such that: (i) σ is generated by ρ(F) and finitely many elements of V; (ii) int(σ) ∩ V = ∅. We call (σ, F) strictly convex if in addition σ is strictly convex and 0 / ∈ ρ(F).
. We write τ σ in this case. Definition 1.3. A colored fan Σ is a collection of colored cones such that: (i) If (σ, F) ∈ Σ, the every face of (σ, F) is in Σ; (ii) For every v ∈ V, there is at most one (σ, F) ∈ Σ with v ∈ int(σ).
We say Σ is strictly convex if in addition every colored cone in Σ is strictly convex.
Note that the relative interiors of two colored cones in a colored fan may intersect nontrivially outside V; we present an instance of this in Example 5.5. There is a bijective correspondence between G/H-embeddings and strictly convex colored fans. The colored cones in a strictly convex colored fan represent the G-orbits of an embedding X. Strictly convex colored cones correspond to simple G/H-embeddings, which have one closed G-orbit. If the colored fan associated to a spherical variety has no colors, we say that the embedding is toroidal.
Throughout this paper, we denote by ν : C{{t}} → Q the valuation that takes a Puiseux series to the lowest power of t appearing with nonzero coefficient. Given a closed subvariety Y ⊆ G/H, we define its spherical tropicalization as follows. For each C{{t}}-point γ ∈ Y (C{{t}}), we obtain the following associated valuation ν γ : C(G/H) * → Q in V: f → ν (γ * (gf )) for f ∈ C(G/H) * and g ∈ G sufficiently general.
We must explain what we mean by sufficiently general g. For g in an open subset of G, the image of γ is in the domain of gf . Furthermore, the minimum of ν (γ * (gf )) over all g ∈ G is met on an open subset. A sufficiently general g lies in both of these open subsets. The tropicalization trop G (Y ) ⊆ V is the collection of ν γ for all γ ∈ Y (C{{t}}). Note that the subscript G records which group action we are tropicalizing with respect to and observe that trop G (G/H) = V(G/H).
We briefly describe the extended tropicalization construction for spherical embeddings from [19]. If X is a G/H-embedding and O ⊆ X is a G-orbit, then O is itself a homogeneous space with respect to the action of G. Thus, O has a valuation cone V(O). As a set, the spherical tropicalization is Note that the subscript G is used to record which group action we are considering. The subset Which elements of C[G/H] that are allowed to be sent to ∞ is determined by which functions in C[G/H] vanish along O. The topology on trop G (X) in essence adds on these extended valuations by considering them as limit points of finitely-valued Q-valuations. If Y ⊆ X is a closed subvariety, we obtain its tropicalization trop G (Y ) by separately tropicalizing its and apply the subspace topology inherited from trop G (X).
We establish some notational conventions. Throughout, X denotes a spherical embedding, Y ⊆ G/H denotes a closed subvariety of a homogeneous space, and Z denotes a toric variety. The combinatorial data associated to spherical varieties is written in non-calligraphic font when referring to toric varieties. That is, a toric variety has the lattices M and N rather than M and N. Finally, we write Q := Q ∪ {∞}.

Tropicalizing Morphisms
In this section, we describe how a morphism between spherical varieties induces a morphism between their tropicalizations. Our main result is that these morphisms commute with tropicalization (Proposition 2.2), which extends Corollary 6.2.17 of [17]. This fact will be used later; we prove it here because it is independent of the other theory in this paper.
Let X and X ′ respectively be a G/H-embedding and a G/H ′ -embedding. If Φ : X → X ′ is a G-morphism induced by a dominant G-equivariant morphism G/H → G/H ′ , then there is an induced continuous map trop(Φ) : trop(X) → trop(X ′ ).
Before describing the map, we collect some facts about morphisms between spherical varieties as stated in §4 of [14]. First, a G-equivariant dominant morphism Φ : G/H → G/H ′ between spherical homogeneous spaces induces an injection Φ * : M(G/H ′ ) ֒→ M(G/H). This in turn induces a surjection Φ * : N(G/H) → N(G/H ′ ), which restricts to Φ * : Since this holds for each G-orbit of X, we can define a map trop(Φ) : i V i → j V ′ j by taking the disjoint union of the pushforwards.
Suppose that we have a morphism Φ : G/H → G ′ /H ′ of homogeneous spaces that is equivariant with respect to some surjective homomorphism of algebraic groups ϕ : G → G ′ . We would like to define the tropicalization of Φ in this setting where G is not necessarily equal to G ′ . The homomorphism ϕ gives an action of G on G ′ /H ′ ; we show that this action makes G ′ /H ′ a spherical homogeneous space with respect to G.
Proposition 2.1. Let X and X ′ be a G/H-embedding and a G ′ /H ′ -embedding, respectively. Suppose Φ : X → X ′ is a dominant morphism equivariant with respect to a surjective homomorphism ϕ : G → G ′ . Then trop(Φ) is continuous.
Proof. By the argument preceding the proposition, we may assume that G = G ′ and ϕ is the identity map.
Proposition 2.2. Let X and X ′ be spherical embeddings with respect to groups G and G ′ , respectively. Let Φ : X → X ′ be a dominant morphism equivariant with respect to a surjective homomorphism ϕ : Proof. As before, we may assume that G = G ′ and ϕ is the identity map.
Let O ⊆ X be an arbitrary G-orbit of X and consider the restriction of trop(Φ) to the valuation cone V(O) associated to O. By definition of trop(Φ), the image of V(O) lies in the valuation cone V(O ′ ) of some orbit O ′ ⊆ X ′ . We will prove the statement in the case that Y ⊆ O is a subvariety of the spherical homogeneous space O. The full statement follows directly by considering the individual intersections of a subvariety with each orbit.
Let γ : Spec C{{t}} → Y be a C{{t}}-point of Y . Then Φ • γ is a C{{t}}-point of Φ(Y ) and all C{{t}}-points of Φ(Y ) arise in this way. This is because Φ : O → O ′ is surjective as it is G-equivariant with respect to the surjective map ϕ. Thus, trop G (Φ(Y )) is defined as follows: where g is chosen to be sufficiently general for a given f . The tropicalization of Y is again for sufficiently general g. Applying trop(Φ) means taking ν γ → ν γ • Φ * , so this gives us trop(Φ)(trop G (Y )): To prove the proposition, we need to show that ν((Φ • γ) * (gf )) = ν(γ * (gΦ * (f ))) for all f ∈ k(Φ(Y )) (B) where g is chosen to be sufficiently general for both f and Φ * (f ). Note that (Φ • γ) * = γ * • Φ * , so it will be sufficient to show that Φ * (gf ) = gΦ * (f ). For x ∈ Y we have the following, using the fact that Φ is G-equivariant: Thus, Φ * (gf ) = gΦ * (f ) and so the proposition holds.

Tropicalizing Homogeneous Spaces via Toric Embeddings
In [8], Gagliardi proves a theorem relating the valuation cone of G/H to tropicalization. Inspired by his result, we find an alternate means for tropicalizing a subvariety of a spherical homogeneous space. The purpose of this section is to set up Gagliardi's theory and show how to recover from it Vogiannou's spherical tropicalization (Theorem 3.2).
Throughout, we will always assume that G is of simply connected type, which means that G = G ss ×C where G ss is semi-simple simply connected and C is a torus. Every connected reductive group has a finite covering by a group of simply connected type with the same embedding, so this is not a restrictive assumption.
We work initially over a spherical homogeneous space G/H with trivial divisor class group; the case of non-trivial divisor class group will be handled later. We will describe how to associate a toric variety Z 0 to G/H. Each color in the palette of G/H is a prime divisor The characters associated to f i and g k span the lattice M of characters of B semi-invariant rational functions k(G/H) (B) . Denote these characters by v * i and w * k , respectively. Define a toric variety with coordinates f ij and g k , matching the basis of M. Denote the lattice of cocharacters of Z 0 This induces a natural action of G on Z 0 commuting with ι. The coordinate ring of Z 0 has coordinates S ij for 1 ≤ i ≤ r and 1 ≤ j ≤ s i and T k for 1 ≤ k ≤ m. The inclusion ι is dual to the map Ψ : Finally, the lattice N dual to M has basis {v 1 , . . . , v r , w 1 , . . . , w m } and we define an inclusion N ֒→ N by v i → v i1 + · · · + v is i and w k → w k . Theorem 3.1 shows how the embedding G/H ֒→ Z 0 can be related to the valuation cone of G/H: We can also write this as trop Ultimately, accounting for subvarieties of G/H requires a new result independent of Theorem 3.1, which we describe now. Note that and define a map ψ : trop T (Z 0 ) → N Q as follows: The following theorem is the core idea of this paper. The further work ultimately simply expands on this result.
The dual lattice M to N is spanned by characters associated to the f i and g k , so we only need to know how ν γ behaves on these functions to completely determine it as an element of N Q . Consider ν γ (f i ) for some i, suppose that g ∈ G is sufficiently general so that ν γ (f i ) = ν(γ * (gf i )), and write gf i = s i j=1 a j f ij where a j ∈ C. Generically, the minimum min j {ν(a j f ij (γ))} is met at only one monomial, so we may write ν γ (f i ) = ν(a j f ij (γ)) for some j.
Note that a j f ij (γ) = 0 as otherwise f i would be zero on all of G/H. Further, observe that by this By Proposition 1.3 of [15], the g k are all G eigenvectors, so it follows that the T-invariant and G-invariant valuations induced by γ act identically on the g k . This implies the first inclusion.
Conversely, suppose that γ is a C{{t}}-point of Y ⊆ Z 0 and consider ψ ( ν γ ), where ν γ is the T-invariant valuation induced by γ. For fixed i, suppose that min j {ν(f ij (γ))} = ν(f ik (γ)) for some k. Note that by definition of Z 0 we cannot have f ij (γ) = 0 for all j, so the minimum is finite and thus the image of Y under ψ is contained in N Q . For generic g ∈ G, gf i has a nonzero coefficient In this setting, m = 0, r = 1, s 1 = 2, and we may write f 11 := y and f 12 := x. The lattice N is spanned by a single ray v 1 . The left-hand side of Figure 3.1 shows the vectors v 1 , v 11 , and v 12 . Then Z 0 ∼ = C 2 \{0} and the inclusion G/H ֒→ Z 0 is an isomorphism given by switching coordinates: This result allows us to recover a description due to Vogiannou of the tropicalization of subvarieties of G/H in this setting ( [25], Example 3.10). Vogiannou considers a curve C in G/H = C 2 \{0} given by a polynomial f (x, y). He shows that trop G (C) is a ray oriented to the left in V ∼ = Q when the constant term of f is nonzero and is all of V when the constant term is zero. The condition that the constant term of f (x, y) is zero is equivalent to saying that the zero vector is contained in the closure of C in C 2 . By Proposition 6.3.5 of [17], this occurs if and only if trop T (C) intersects the interior of the first quadrant. In this case, trop T (C) projects onto the right-hand ray of V under the map ψ, so we conclude that trop G (C) = V.
It follows that G · f 1 has rank four and we may choose as our basis f 11 := f 1 = x 22 , f 12 := x 21 , f 13 := x 12 , and f 14 := x 11 . Finally, Γ G/H, . The tropicalization of this variety is the set of extended valuations µ such that the minimum is met more than once. This forces µ(f 1j ) ≤ 0 for some j. Applying ψ therefore takes us to the ray in N Q ∼ = Q spanned by the valuation We must still consider the case that the spherical homogeneous space has non-trivial divisor class group. Following [8], we will use bold-faced characters G and H when the spherical homogeneous space G/H does not have trivial divisor class group. In general, the spherical data associated to G/H is notated as with G/H but bold-faced. We still assume that G = G ss × C is of semisimple type. For each D i in the palette D := {D 1 , . . . , D r }, we consider the pullback of D i under the projection map G → G/H. By results in [4], there is a unique Then H acts from the right on f i with character χ i ∈ X(H). We will come back to these f i in §5.
Gagliardi defines G := G × (C * ) D and The Borel subgroup is B := B ×(C * ) D . There is a natural isomorphism H ∼ = H given by projection to the first coordinate and a natural morphism π : G/H → G/H with the induced surjective map π * : N Q → N Q . The B-stable prime divisors in the palette D of G/H are precisely the pullbacks under π of the colors in D. The character group of (C * ) D is isomorphic to Z D , and we denote its basis by {η 1 , . . . , η r }. Gagliardi shows that G/H has trivial divisor class group ( [8], Corollary 3.2) and that V = π −1 * (V) ( [8], Proposition 3.3). Theorem 3.2 can now be easily generalized: Proof. By Theorem 3.2, this is equivalent to proving trop G (Y ) = trop(π) trop G π −1 (Y ) , which follows directly from Proposition 2.2. Then G acts on C 2 × C 2 × C * . In this action, G acts by left matrix multiplication on the copies of C 2 and trivially on C * and (C * ) D respectively acts with weights −η 1 and −η 2 on the first and second copy of C 2 and with weight −η 1 − η 2 on C * . If the coordinates of the copies of C 2 are respectively given by S 11 , S 21 and S 12 , S 22 and the coordinate of C * is T , then  1 , a 2 Because a 1 + a 2 ≤ b 1 on ψ (trop G (G/H)), it follows that this valuation is always non-negative on f and so we recover the fact that V = trop G (G/H) is a ray in the positive direction.

Embedding Toroidal Varieties in Toric Varieties
In [8], Gagliardi considers spherical embeddings X of G/H whose associated colored fans only include rays without color. Given such an embedding, he finds an explicit toric variety Z in which G/H embeds such that G/H = X in Z. He restricts himself to such embeddings because to compute the Cox ring of a spherical embedding, only information about the G-stable prime divisors is needed, i.e. rays without color.
In this section we extend his construction to arbitrary toroidal spherical embeddings, whose colored fans contain no colors. We maintain the notation and conventions established in §3. In particular, we start in the case where G/H has trivial divisor class group. Let X be a G/Hembedding given by a fan Σ whose one-dimensional cones are {u 1 , . . . , u n } ⊆ V. For each colored cone (σ, ∅) ∈ Σ, write σ(1) ⊆ {u 1 , . . . , u n } to denote the set of one-dimensional (non-colored) faces of a σ. Define We note that A is defined slightly differently in [8], where the phrase "at least one" is replaced by "exactly one". For each σ and each a ∈ A, we define The dimension of the cone σ a is dim σ a = |a| + dim σ. This follows because for every i at least one v ij is absent from a, which means the rays of σ are linearly independent of the rays in a. Then we define the fan Σ Z to be the set of cones σ a for all (σ, ∅) ∈ Σ and a ∈ A: That Σ Z is well-defined is a corollary of Proposition 5.6, which we state and prove later. Let Z be the toric variety associated to the fan Σ Z . Note that Z 0 ⊆ Z since the fan for Z 0 can be obtained from the definition by taking σ to be the trivial cone and letting a range over A. We claim that the closure of G/H in Z is isomorphic to X. The argument mimics the procedure in [8]; many of the proofs proceed essentially identically. This claim is proved in Proposition 5.7; we delay the proof until §5, where we simultaneously consider non-toroidal varieties.
We now show that the action of G on Z 0 extends to Z. Define N := N ⊕ Z n where Z n is given the basis {e 1 , . . . , e n }. For each σ a ∈ Σ Z , define the cone and let Σ Z := fan ( σ a : σ a ∈ Σ Z ) . In essence, the difference between σ a and σ a is that the rays of σ a in σ(1) are not necessarily orthogonal. In σ a , each u k is replaced with a vector e k that is orthogonal to every vector in a and every other e ℓ .
If Z is the associated toric variety with torus T := T × (C * ) n , then where S is a closed set of codimension at least two. We revisit this theory in more detail in §6.
There is a natural toric morphism p : Z → Z induced by the map N → N defined by v ij → v ij , w k → w k , and e ℓ → u ℓ . By Theorem 4.1 in [23], p is a good quotient. In fact, it is a geometric quotient (cf. [1] Proposition 3.2). The quotient is with respect to a subtorus Γ ∼ = (C * ) n of T × C n , which is parameterized as follows: We may extend the natural action of G on C s 1 +···+sr × (C * ) m to Z by having G act trivially on the additional C n summand. Then the action of G on Z commutes with the action of Γ. As a result, we obtain an action of G on Z. More explicitly, because p is a good quotient, Proposition 5.0.7 of [7] says that for any point x ∈ X, the preimage p −1 (x) contains a unique closed Γ-orbit. If g ∈ G and z ∈ Z, we define g · z := p(g · z), where z is an element of the unique closed Γ-orbit in p −1 (z). Because the actions of G and Γ commute, g takes the closed orbit of p −1 (z) to another Γ-orbit. Since p is constant on orbits, p(g · z) does not depend on the choice of z.
Further, the inclusion N ֒→ N induces a morphism Z 0 → Z that commutes with p : Z → Z to give the inclusion Z 0 ֒→ Z, so the described action of G on Z extends that of G on Z 0 .   If instead we consider Bl 0 (C 2 ) ⊂ C 2 × P 1 with one ray spanned by v 1 , we get the same Z and the map Z → Z = Bl 0 (C 2 ) is given by (x, y, z) → (xz, yz) × [x : y]. The preimage of a point (ac, bc) × [a : b] ∈ Bl 0 (C 2 ) is (at −1 , bt −1 , ct) : t ∈ C * , so it is a geometric quotient by the torus Γ := (t −1 , t −1 , t) : t ∈ C * .
We have not yet discussed the case of a G/H-embedding X where the homogeneous space G/H has nontrivial divisor class group. Given such a homogeneous space, we earlier defined an associated homogeneous space G/H with trivial divisor class group along with a map π : G/H → G/H. Gagliardi shows ( [8], Proposition 3.4) that the pushforward π * : N Q → N Q is an isomorphism when restricted to the subspace (N T ) Q := span {w 1 , . . . , w m }.
If Σ is the colored fan associated to a toroidal embedding X, we define a G/H-embedding X by considering the fan Σ, which is the preimage of Σ under π * | (N T ) Q . Let Z be the toric variety associated to X as constructed earlier in this section.
There is a good geometric quotient of Z by the torus (C * ) D , so we may extend π to π : Z → Z, where Z is a toric variety. Refer again to the theory in [23], as well as [1], Proposition 3.2. By Proposition 5.7, the spherical variety X associated to the fan Σ is the closure of G/H in Z. Then π takes X to the closure of G/H in Z (cf. Definition 5.0.5 and Theorem 5.0.6 of [7]). By the construction of the colored fan, the image of X under π is X and hence X is the closure of G/H in Z. We collect these facts later in Proposition 5.8 where we also consider the possibility of colors.

The Problem of Colors
The methods of §4 are illustrations of Proposition 2.11 of [11], which says that any Mori dream space can be embedded in a projective toric variety. Projective spherical varieties are Mori dream spaces, but not every spherical embedding is a Mori dream space. This issue is addressed in [9], another paper of Gagliardi.  Finally, we make use of the following theorem of W lodarczyk, which Gagliardi cites in [9]: 26]). A normal variety X has the A 2 -property if and only if it admits a closed embedding X ֒→ Z into a toric variety Z.
All of this serves to tell us that the spherical varieties that cannot be embedded in a toric variety are those whose cones are not polyhedral. Intuitively, this is reflected by the fact that a spherical embedding with a non-polyhedral colored fan will be associated by the process described in §4 to a toric variety with a non-polyhedral fan, which is an impossibility. A colored fan can only exhibit non-polyhedral behavior outside of the valuation cone. We illustrate this in the following extended example.  V (y 1 ). The Gmodules G · x 3 and G · y 1 both have rank three and we may define an embedding as follows: The tropicalization trop T (G/H) of G/H in Z 0 is the set of extended valuations µ where the minimum min {µ(x 1 ) + µ(y 1 ), µ(x 2 ) + µ(y 2 ), µ(x 3 ) + µ(y 3 ), 0} is met at least twice. By Theorem 3.1, and µ(y 1 ) = µ(y 2 ) = µ(y 3 ), and so for the above minimum to be met twice, we must have µ( The interest here is that there are two colors that lie outside the valuation cone, so there is potential for valid colored cones that are not polyhedral. Indeed, we may have a simple G/H-embedding corresponding to the colored cone spanned by the color ρ(V (x 3 )) and −2v 1 + v 2 since the interior of this cone intersects the valuation cone. This is illustrated by the red cone in Figure 5.2. Similarly, there exists an embedding whose colored cone is spanned by ρ(V (y 1 )) and v 1 − 2v 2 , the blue cone in Figure 5.2. The union of these cones is a valid colored fan and thus corresponds to a G/Hembedding. Even though this fan is not polyhedral, it is a colored fan because the intersection is outside the valuation cone. Throughout this section we therefore only consider colored fans that are polyhedral. As before, we assume for the moment that G/H has trivial divisor class group. Let X be a G/H-embedding with colored fan Σ and let Z 0 be the toric variety associated to G/H defined in §4. Let (σ, F) ∈ Σ be a colored cone. Then write σ(1) := {u 1 , . . . , u n } for the set of one-dimensional non-colored faces of σ and define Note that A(∅) is the same as A, as defined at the beginning of §4. The set A(F) simply extends A by allowing the entire set {v ij } j to be present when the color D i lies in F. For a given a ∈ A(F), we define a cone in N Q as follows: The fan Σ Z is defined similarly to before: If Σ has no colors, then Σ Z is consistent with the object we described in §4. By a similar argument to the toroidal case, we can see that the action of G on Z 0 also extends to Z in this setting.
Proposition 5.6. If X is a G/H-embedding whose associated colored fan Σ is polyhedral, then the fan Σ Z is well-defined.
Proof. We must check that Σ Z is closed under taking faces and that the intersection of any two cones in Σ Z is a face of each. If σ a is a cone in Σ Z , then its faces are precisely those σ ′ a ′ such that a ′ ⊆ a and σ ′ σ in Σ. Such a cone σ ′ a ′ is in Σ Z , as needed. Suppose σ a , σ ′ a ′ ∈ Σ Z are two cones. Then because Σ is a polyhedral fan, the intersection of σ and σ ′ is a colored cone (σ ′′ , F ′′ ) ∈ Σ that is a colored face of each. The intersection σ a ∩ σ ′ a ′ then equals σ ′′ a∩a ′ , which is in Σ Z .
The following proposition is what this section has been working towards. It is an extension of Lemma 2.13 and Proposition 2.14 in [8], and the proof owes much of its structure to those two results.
Proposition 5.7 (cf. [8], Proposition 2.14). Suppose G/H has trivial divisor class group and X is a G/H-embedding corresponding to a colored fan Σ. Then X is isomorphic to the closure G/H of G/H in the associated toric variety Z.
Proof. Proposition 2.14 of [8] proves the claim when Σ consists solely of non-colored rays. If Σ consists of a single ray σ spanned by the color D i , then on every additional G-orbit of Z outside of Z 0 , f i vanishes and f j and g k do not for every j = i and k. Thus, this embedding corresponds to a ray in the direction of σ. Moreover, this ray must have color since the f ij may vanish with different multiplicities along the orbits added. After gluing together along shared orbits, we conclude that the statement of the proposition holds when Σ consists solely of rays, both colored and not.
We now turn to proving the full claim. It will be sufficient to prove it for simple embeddings since they can then be glued together along shared orbits. Let Σ consist of a single colored cone (σ, F) with non-colored rays u 1 , . . . , u n ∈ σ(1).
Let the maximal cone in Σ Z spanned by the sets σ(1) and D i ∈F {v ij } s i j=1 be denoted τ . The cone τ corresponds to an affine variety U τ . For each ray u ℓ ∈ σ(1), there is an open affine subset U ℓ of U τ with two torus orbits. Similarly, for each D i ∈ F and each 1 ≤ j ≤ s i , there is an open affine subset U ij corresponding to the ray v ij . Let π ℓ ∈ C[U τ ] for 1 ≤ ℓ ≤ n denote prime elements that cut out the closures of the U ℓ in U τ . Similarly choose prime elements π ij ∈ C[U τ ] that cut out the U ij . Then we have the following commutative diagram: In the diagram, R is normalization and L is localization. For each π ℓ and each π ij , there are respectively prime elements π ℓ , π ij ∈ R 2 such that V (π ℓ ) = V ( π ℓ ) and V (π ij ) = V ( π ij ) in Spec R 2 . Now L ∈ {S ij , T k } can be written in the form We may further write where e ℓ , e ij ∈ Z ≥0 and c ∈ R * 2 . The valuations in the colored cone (σ, F) are positive Q-linear combinations of the u ℓ and the valuations induced by the colors D i ∈ F. It follows from the above argument that every such valuation is induced by a torus invariant one in Z. This proves the claim for simple spherical embeddings. Gluing together along shared orbits gives the full result.
It still remains to consider when the homogeneous space G/H has nontrivial divisor class group. Recall that the colors D 1 , . . . , D r of the associated homogeneous space G/H are precisely the pullbacks of the colors D 1 , . . . , D r of G/H. For each D i , Gagliardi defines f i ∈ Γ(G/H, O G/H ) such that V (f i ) = D i . There is an inclusion from the character lattice of (C * ) D to C[G] * that takes a character χ to a monomial ǫ χ ∈ C (C * ) D ⊂ C[G] * . Then we may write f i := f i ǫ −η i , where η i is the character that is trivial on every coordinate not equal to i. The function f i is invariant under the action of H from the right and V (f i ) = D i . Note that as a result (C * ) D has a nontrivial action on f i for all i.
Then let X be a G/H-embedding with polyhedral colored fan Σ. For each color D i appearing in Σ, include in Σ a colored ray corresponding to D i . For each ray without color, take the preimage of this ray under π * | (N T ) Q as we did for toroidal embeddings. Then add in higher-dimensional cones between these rays in N Q if they exist in Σ. As in the toroidal case, there is a good geometric quotient π : Z → Z that extends π : G/H → G/H. We obtain the following result, similarly to the toroidal case: Proposition 5.8. Let G/H ֒→ X be a spherical embedding with associated spherical embedding G/H ֒→ X where G/H has trivial divisor class group. Then G/H ∼ = X, where the closure is taken in Z.

Extended (Global) Tropicalization
This section finally gives a global construction for the tropicalization of a spherical embedding. Because of the issues raised in §5, this construction can only work for a certain class of spherical embeddings, namely those that have the A 2 -property. For many homogeneous spaces, this consideration will not raise any issues. For example, if G/H is horospherical or if it has fewer than two colors lying outside the valuation cone, the global tropicalization will always go through without issue.
As before, we begin in the case that G/H has trivial divisor class group and we maintain the notation introduced previously. Let X be a G/H-embedding with the associated toric varieties Z and Z and morphism p : Z → Z. Our methodology will be to work in the toric world with Z and Z where results are known. Then we will apply an extension of the map ψ. The general theory in the following discussion can be found in more generality in §6.1 of [17]; also refer to [7], §5.1.
There is a short exact sequence where A n−1 Z is the cokernel of the natural inclusion N ֒→ N . Applying the functor Hom(−, C * ), we obtain the following exact sequence: where Q := Hom A n−1 Z , C * . Let E ℓ denote the coordinate in C Z corresponding to the ray e ℓ so that C Z = C[S 11 , . . . , S rsr , T 1 , . . . , T m , E 1 , . . . , E n ].
Then the irrelevant ideal in C Z is for some S of codimension at least two. Finally, we have that This quotient construction of Z tropicalizes in the following sense (see [17], Proposition 6.2.6 and Corollary 6.2.16): Proposition 6.1. Suppose G/H has trivial divisor class group and suppose Y ⊆ G/H is a closed subvariety. Let X be a G/H-embedding with the A 2 -property with associated toric varieties Z and Z and let Y be the closure of Y in Z. Then This result provides a means for globally tropicalizing a toric variety. That is to say, we do not need to consider any separate pieces and then glue together, we may simply tropicalize a single toric variety and take an appropriate quotient. To obtain a universal tropicalization for spherical embeddings, it will be sufficient to describe how to recover trop G Y from trop T Y . When Y is replaced by Y , we described this in Theorem 3.2 using the piecewise projection map ψ: We now define an extension ψ : trop T (Z) → trop G (X) of ψ that takes extended T-invariant valuations to extended G-invariant valuations. Applying this map to the global toric tropicalization above will afford a global spherical tropicalization. Let (σ, F) be a colored cone in the colored fan of the G/H-embedding X and let a ∈ A(F). In the extended tropicalization of the spherical variety, tropicalizing the orbit corresponding to the colored cone (σ, F) corresponds to adding in semigroup homomorphisms in Hom V σ ∨ ∩ M, Q and viewing them as limit points of trop G (G/H) := V ⊆ Hom(M, Q).
Before proceeding, we deal with a slight clash of notation that comes up here. Thought of as a valuation, µ ∈ Hom (σ ∨ a ∩ M, Q) acts on a lattice of torus semi-invariant rational functions, where the group action is multiplicative. As an element of Hom (σ ∨ a ∩ M, Q), however, µ acts on the additive semigroup σ ∨ a ∩ M . This can be addressed this by identifying an element (a 11 , . . . , a rsr , b 1 , . . . , b m ) ∈ σ ∨ a ∩ M with the function f a 11 11 · · · f ars r rsr g b 1 1 · · · g bm m . Similarly, (a 1 , . . . , a r , b 1 , . . . , b m ) ∈ σ ∨ ∩ M is identified with f a 1 1 · · · f sr r g b 1 1 · · · g bm m . The extension ψ will take an extended valuation µ ∈ Hom (σ ∨ a ∩ M, Q) to Hom V (σ ∨ ∩ M, Q). Suppose µ ∈ trop T (O) where O corresponds to the cone σ a . Further let (σ, F) be the associated colored cone corresponding to a G-orbit O. Define a set Ω ⊂ Z r as follows: Then for any µ ∈ Hom σ ∨ a ∩ M, Q , we define ψ(µ) ∈ Hom V σ ∨ ∩ M, Q to be infinite on (σ ∨ \ σ ⊥ ) ∩ M and to act on f a 1 1 · · · f ar r g b 1 1 · · · g bm m ∈ σ ⊥ ∩ M as follows: Remark 6.2. There is tacit assumption here that we only consider ω ∈ Ω such that to ensure that µ is well-defined. When µ = ν γ is induced by a C{{t}}-point γ, then the minimum may be taken over the entirety of Ω. An explanation of this appears in the proof of Theorem 6.5.
Example 6.3. We will extend Example 3.3. Recall that in this setting r = 1, f 11 = y, and f 12 = x. Consider the embedding Bl 0 C 2 of G/H = C 2 \ {0} given by a single non-colored ray in the direction of v 1 , which we call σ. The associated toric variety Z is also Bl 0 C 2 , given by the fan shown in Figure 6.1. In the vector space N Q corresponding to the toric variety, σ is the ray in the fan of Bl 0 C 2 corresponding to the exceptional divisor. There are three possibilities for a: a = ∅, {v 11 }, or {v 12 }. These respectively correspond to three cones σ a : the diagonal ray σ, the two-dimensional cone spanned by σ and v 11 , and the two-dimensional cone spanned by σ and v 12 .
The spherical tropicalization of Bl 0 (C 2 ) is isomorphic to Q, where the tropicalization of E is the point ∞. In other words, σ ⊥ ∩M is the origin in N and corresponds to the map µ ∈ Hom σ ∨ ∩ M, Q sending every non-zero element of σ ∨ ∩ M to ∞, i.e. the extended G-invariant map C[x, y] → Q that is ∞ on all non-constant functions.
Therefore, if µ ∈ σ ∨ a ∩ M for some choice of a, then it must by definition of ψ be sent to ∞ in Q. This can be viewed as collapsing the diagonally-oriented one-dimensional vector space and the two zero-dimensional vector spaces in trop T Bl 0 C 2 to a point. See Figure 6.2 for reference.
In the previous example we saw that ψ takes the extended toric tropicalization to the extended spherical tropicalization. Our goal now is to prove that this holds in general. To accomplish this, we need an additional theorem: Theorem 6.5. If G/H has trivial divisor class group, Y ⊆ G/H is a subvariety, and X is a G/H-embedding with the A 2 -property, then where the closure on the left is taken in Z and the closure on the right is taken in X.
Proof. Let γ : Spec C{{t}} → Y be a C{{t}}-point of Y and suppose ν γ and ν γ are respectively the Tinvariant and G-invariant extended valuations induced by γ. Then γ lies in a G-orbit corresponding to a colored cone (σ, F) and a T-orbit corresponding to a cone σ a . We will show that ψ ( ν γ ) = ν γ . Let f a 1 1 · · · f ar r g b 1 1 · · · g bm m ∈ σ ⊥ ∩ M. Then a sufficiently general element of G takes this B semiinvariant rational function to a function of the form (c 11 f 11 + · · · + c 1s 1 f 1s 1 ) a 1 · · · (c r1 f r1 + · · · + c rsr f rsr ) ar · cg b 1 1 · · · g bm m for c, c ij ∈ C * . Generically, this is a non-zero rational function on the G-orbit corresponding to (σ, F) that is defined at γ, so the valuation ν γ takes it to a finite rational number. By Theorem 6.4, ν γ lies in the closure of trop T (Y ), so there exists a sequence {ν ℓ } ∞ ℓ=1 of T-invariant valuations associated to C{{t}}-points γ ℓ of Y such that lim ℓ→∞ ν ℓ = ν γ in the topology on trop T (Z). Then we have the following, which verifies the claim: Pursuant to Remark 6.2, note that the minimum here is indexed over the entirety of the set Ω since every c ij is nonzero.
Having established the result when the homogeneous space has trivial divisor class group, we turn to the case of non-trivial divisor class group. As before, we write G/H to denote a homogeneous space with non-trivial divisor class group and write G/H for the homogeneous space with trivial divisor class group and dominant map π : G/H → G/H. If X is the G/H-embedding associated to a G/H-embedding X, then the extended map π : X → X is equivariant with respect to the natural surjection G → G. We can therefore extend Theorem 3.5 to spherical embeddings of homogeneous spaces with non-trivial divisor class group; the proof proceeds almost identically. Theorem 6.6. Let G/H ֒→ X be a spherical embedding with associated spherical embedding G/H ֒→ X where G/H has trivial divisor class group. If Y ⊆ G/H is a closed subvariety and X is a G/H-embedding, then where the closures are taken in X.
Proof. By Theorem 6.5, this is equivalent to showing Because π is equivariant with respect to the surjective group homomorphism G → G, the statement follows from Proposition 2.2.
Developing the interplay between spherical tropicalization and toric tropicalization has the potential to give insight into the structure of spherical tropicalizations by translating known results from the toric world. As an example, we can generalize Theorem 6.4 to spherical varieties: Theorem 6.7. If Y ⊆ G/H is a closed subvariety and X is a G/H-embedding, then where the closure on the left is taken in X and the closure on the right is taken in trop G (X).
Proof. We prove the theorem when X is a simple G/H-embedding because X always has the A 2property in this setting. If X is not simple, we may break it up into simple G/H-embeddings, where the result holds, and then glue together along shared orbits. Note that this will work even if X does not have the A 2 -property.
The last equality here follows from the proof of Theorem 6.5. It follows that ν γ is the limit of the sequence ψ ( ν ℓ ) ∞ ℓ=1 . Hence, ν γ ∈ trop G (Y ) and the first inclusion follows. Conversely, suppose that µ ∈ trop G (Y ) and let {γ ℓ } ∞ ℓ=1 be a sequence of C{{t}}-points of Y such that the associated G-invariant valuations µ ℓ := ν γ ℓ satisfy lim ℓ→∞ µ ℓ = µ in the topology on trop G (X). Each γ ℓ induces a T-invariant valuation µ ℓ . By possibly replacing {µ ℓ } ∞ ℓ=1 with a subsequence, there is µ ∈ trop T (Y ) with µ = lim ℓ→∞ µ ℓ . By Theorem 6.4, µ is induced by a C{{t}}-point γ of Y . We claim that µ = ψ ( µ) ∈ trop G Y . Let f a 1 1 · · · f ar r g b 1 1 · · · g bm m ∈ σ ⊥ ∩ M be arbitrary, where σ corresponds to the G-orbit in whose tropicalization µ lies. Then the equality µ f a 1 1 · · · f ar r g b 1 1 · · · g bm m = ψ ( µ) f a 1 1 · · · f ar r g b 1 1 · · · g bm m follows from an argument similar to the previous inclusion. Hence, µ is induced by γ and so µ ∈ trop G Y . Now suppose G/H has nontrivial divisor class group and let G/H be the associated spherical homogeneous space with trivial divisor class group. Then we have the following string of inclusions and equalities: From top to bottom, these containments and equalities respectively follow from Theorem 3.5, Proposition 2.1, the first half of this proof, the continuity of π, and Theorem 6.6. Completing the proof now comes down to showing that the two containments are equalities.