# Spherical tropicalization and Berkovich analytification

Desmond Coles

( The University of Texas at Austin
Austin, TX, United States of America
<EMAIL_ADDRESS>)

Abstract. Let $X$ be a spherical variety. We show that Tevelev and Vogiannou’s
tropicalization map from $X$ to its tropicalization factors through the
Berkovich analytification $X^{\text{an}}$, as in the case for toric varieties.
Furthermore we show that the tropicalization is a strong deformation
retraction of $X^{\text{an}}$. We also give a strong deformation retraction of
Thuillier’s analytification $X^{\beth}$ onto a subspace described using the
colored fan of $X$.

Keywords. Spherical variety. Tropicalization. Berkovich analytification.
Algebraic Geometry.

Statements and Declarations. The author has no competing interests.

Acknowledgements. The author would like to thank their advisor, Sam Payne, for
their support and guidance during this research. The paper also benefited
greatly from conversations with Tom Gannon, Sam Raskin, Yixian Wu, Hernán
Iriarte, Amy Li, and Logan White. The author was grateful to be funded by NSF
DMS–2001502 and NSF DMS–2053261.

## 1 Introduction

Tropical geometry provides a set of tools for assigning combinatorial objects
to algebraic varieties. One then studies the algebraic geometry of the
original variety by looking at the combinatorics. One well known example is
given by embedding a variety into a toric variety and then tropicalizing the
toric variety [Pay09]. In [TV21] Tevelev and Vogiannou introduce
tropicalization for spherical homogenous spaces. Let $k$ be an algebraically
closed field. Recall that if $G$ is a reductive group over $k$, and $X$ is a
normal $G$-variety, then $X$ is spherical if there is a Borel subgroup of $G$
with an open orbit in $X$. Spherical varieties can be considered the
nonabelian analogue of toric varities. Let $\overline{K}=k\\{\\!\\{u\\}\\!\\}$
be Puiseux series, the algebraic closure of Laurent series $K=k(\\!(u)\\!)$.
Equip $\overline{K}$ with the $u$-adic valuation $|\\!|\\!\cdot\\!|\\!|_{u}$.
Given a spherical homogeneous space $G/H$, Tevelev and Vogiannou define a set
theoretic tropicalization map $\operatorname{trop}_{G}\colon
G/H(\overline{K})\rightarrow{\mathcal{V}}$, where ${\mathcal{V}}$ is the set
of $G(k)$-invariant valuations on the function field $k(G/H)$. The set
${\mathcal{V}}$ in fact has the structure of a finitely generated convex cone
[Kno91, Corollary 5.3]. The map $\operatorname{trop}_{G}$ takes $x\in
G/H(\overline{K})$ to the valuation
$|\\!|\\!\cdot\\!|\\!|_{x}\in{\mathcal{V}}$ defined by the formula:

$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{gf(x)}\right|\\!\right|_{u}\quad\text{$g$
in general position}$

where general position means $g\in U_{f}$, $U_{f}\subseteq G(k)$ a nonempty
Zariski open such that $\left|\\!\left|{gf(x)}\right|\\!\right|_{u}$ is
constant for $g\in U_{f}$. For a subvariety $W\subseteq G/H$ one can then
define $\operatorname{Trop}_{G}(W)$ as the closure of
$\operatorname{trop}_{G}(W(\overline{K}))$. This construction agrees with the
usual tropicalization map in the case when $G=T$, $T$ a torus, and $H$
trivial.

In [Nas17] Tevelev and Vogiannou’s construction was extended to all spherical
varieties, generalizing the extension of tropicalization of tori to toric
varities given in [Pay09]. Here the target of the tropicalization map,
$\operatorname{Trop}_{G}(X)$, consists of a gluing of the valuation cones
associated to $G$-orbits in $X$. The tropicalization map is defined as a
disjoint union of the maps given above. In [KM19] Kaveh and Manon develop
tropical geometry for spherical varieties via Gröbner theory and it is shown
that this agrees with the above notion.

Let $X^{\text{an}}$ be the Berkovich analytification of $X$. In the case when
$X$ is a toric variety with dense torus $T$, $\operatorname{Trop}_{T}(X)$ is
canonically a subspace of $X^{\text{an}}$ and the tropicalization map factors
through a continuous retraction
$X^{\text{an}}\rightarrow\operatorname{Trop}_{T}(X)$. It was conjectured in
[TV21] that this is true for tropicalizations of spherical homogeneous spaces.
In [KM19] this is proved for affine spherical varieties. We show this is the
case for all spherical varieties. Let $G^{\beth}$ denote the $k$-analytic
space associated to $G$ as defined in [Thu07, Section 1]. Let
${\mathbf{p}}\colon X^{\text{an}}\rightarrow X^{\text{an}}$ be the map given
by $\ast$-multiplication by the unique point in the Shilov boundary of
$G^{\beth}$.

######  Theorem A.

Let $X$ be a spherical $G$-variety over an algebraically closed and trivially
valued field $k$, and let $\overline{K}$ denote Puiseux series in $k$. Then
there is a canonical homeomorphism
$\iota:\operatorname{Trop}_{G}(X)\rightarrow{\mathbf{p}}(X^{\text{an}})$, and
${\mathbf{p}}\colon X^{\text{an}}\rightarrow\operatorname{Trop}_{G}(X)$ is a
retraction of topological spaces such that the following diagram commutes:

${X(\overline{K})}$${X^{\text{an}}}$
${\operatorname{Trop}_{G}(X).}$$\scriptstyle{\operatorname{trop}_{G}}$$\scriptstyle{{\mathbf{p}}}$

######  Corollary.

For $W$ a subvariety of $X$, there is a strong deformation retraction
$H:[0,1]\times W^{\text{an}}\rightarrow\operatorname{Trop}_{G}(W)$ such that
$H(1,p)={\mathbf{p}}(p)$.

The construction of the map ${\mathbf{p}}$ was inspired by the work of
Thuillier in the toric case, [Thu07, Section 2]. The space $G^{\beth}$ is a
compact analytic subdomain of the $k$-analytic space $G^{\text{an}}$, the
Shilov boundary of this space is a point with important similarities to the
generic point of $G$. The notion of $\ast$-multiplication is an idea
originally due to Berkovich which generalizes left-multiplication by points in
$G^{\text{an}}(k)$.

Our work in Section 3 extends the notion of tropicalization to other
nontrivially valued extensions of $k$ besides $\overline{K}$. We show that for
any field extension $L/k$ with valuation $|\\!|\\!\cdot\\!|\\!|$ there is a
well-defined tropicalization map $\operatorname{trop}_{G,L}\colon
X(L)\rightarrow\operatorname{Trop}_{G}(X)$. For $x\in X(L)$,
$\operatorname{trop}_{G}(x)=|\\!|\\!\cdot\\!|\\!|_{x}$ where
$|\\!|\\!\cdot\\!|\\!|_{x}$ is the valuation
$|\\!|\\!\cdot\\!|\\!|_{x}\in{\mathcal{V}}$ given by the formula:

$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{gf(x)}\right|\\!\right|\quad\text{$g$
in general position}$

where general position means $g\in U_{f}$, $U_{f}\subseteq G(k)$ is a nonempty
Zariski open such that $\left|\\!\left|{gf(x)}\right|\\!\right|$ is constant
for $g\in U_{f}$. It follows from Theorem A that tropicalization is invariant
under the choice of $L$.

######  Corollary.

Let $W$ be a subvariety of $X$, and $L$ be a nontrivially valued and
algebraically closed extension of $k$. Then the closure of
$\operatorname{trop}_{G,L}(W(L))$ in $\operatorname{Trop}_{G}(X)$ is equal to
${\mathbf{p}}(W^{\text{an}})$, where $W^{\text{an}}$ is considered as a
subspace of $X^{\text{an}}$ induced by the inclusion $W\subseteq X$.

Our second main result is on the restriction of the map ${\mathbf{p}}$ to the
subspace $X^{\beth}$. This result generalizes Proposition 2.12 of [Thu07] on
the toric case.

######  Theorem B.

There is a strong deformation retraction from $X^{\beth}$ onto
${\mathbf{p}}(X^{\beth})$. If the colored fan ${\mathfrak{F}}(X)$ satisfies:

$\bigcup_{(\sigma,{\mathcal{F}})\in{\mathfrak{F}}(X)}\sigma\subseteq{\mathcal{V}}.$
($\star$)

Then ${\mathbf{p}}(X^{\beth})$ is homeomorphic to the the canonical
compactification of the fan
$F(X)=\\{\sigma\,|\,(\sigma,{\mathcal{F}})\in{\mathfrak{F}}(X)\\}$.

Recall that just as toric varieties are classified by fans, spherical
varieties are classified by colored fans. For each spherical homogeneous space
$G/H$ there is an associated vector space $N_{{\mathbb{R}}}$. A colored fan,
defined precisely in Section 4, is a finite collection of pairs
$(\sigma,{\mathcal{F}})$ where $\sigma$ is a strictly convex cone in
$N_{{\mathbb{R}}}$ and ${\mathcal{F}}$ is a finite set. The canonical
compactification of a cone $\sigma$ is defined as
$\hom(\sigma^{\vee},[0,\infty])$, where $\sigma^{\vee}$ consists of positive
linear functionals on $\sigma$ and the homomorphisms are semigroup
homomorphisms that preserve scaling by ${\mathbb{R}}_{\geq 0}$. The canonical
compactification of a fan is the polyhedral complex given by gluing the
canonical compactifications of the cones along shared faces. See Remark 6.1
for an explanation of why we impose $\star$ ‣ B and some examples where this
condition does and does not hold. In Remark 6.5 we discuss how to give a
description of ${\mathbf{p}}(X^{\beth})$ when $\star$ ‣ B does not hold.

######  Outline of paper.

Section 2 introduces the necessary notions from Berkovich geometry. Section 3
introduces $\ast$-multiplication and studies the map ${\mathbf{p}}$. Section 4
recalls the Luna-Vust classification of $G/H$-embeddings, and contains
important notation for Sections 5 and 6. Section 5 recalls the work of [TV21]
and [Nas17] and gives a proof of Theorem A. Section 6 recalls the work of
[Thu07, Section 2] and gives a proof of Theorem B.

######  Convention Choice.

Though somewhat nonstandard in the literature of spherical varieties we opt to
use multiplicative notation for valuations to accommodate the conventions of
Berkovich geometry. Our exact conventions are as follows. Let $A$ be a ring.
By a seminorm on $A$ we mean a function $|\\!|\\!\cdot\\!|\\!|_{A}\colon
A\rightarrow{\mathbb{R}}_{\geq 0}$ such that
$\left|\\!\left|{1}\right|\\!\right|_{A}=1$,
$\left|\\!\left|{0}\right|\\!\right|_{A}=0$,
$\left|\\!\left|{fg}\right|\\!\right|_{A}\leq\left|\\!\left|{f}\right|\\!\right|_{A}\left|\\!\left|{g}\right|\\!\right|_{A}$,
and
$\left|\\!\left|{f+g}\right|\\!\right|_{A}\leq\max({\left|\\!\left|{f}\right|\\!\right|_{A},\left|\\!\left|{g}\right|\\!\right|_{A}})$.111These
are non-Archimedean seminorms, which are the only ones of interest because our
base field will always be non-Archimedean. We say that
$|\\!|\\!\cdot\\!|\\!|_{A}$ is a norm if
$\left|\\!\left|{f}\right|\\!\right|_{A}\neq 0$ for all $f\neq 0$. A
(semi)norm will be called a (semi)valuation if it is multiplicative:
$\left|\\!\left|{fg}\right|\\!\right|_{A}=\left|\\!\left|{f}\right|\\!\right|_{A}\left|\\!\left|{g}\right|\\!\right|_{A}$.222More
commonly valuations given as functions $\nu$ into the additive semigroup
${\mathbb{R}}\sqcup\\{\infty\\}$, a multiplicative norm is equivalent to such
a function $\nu$, set $\nu=-\log(|\\!|\\!\cdot\\!|\\!|)$. If $k$ is a field
with valuation $|\\!|\\!\cdot\\!|\\!|_{k}$ and $A$ is a $k$-algebra then a
seminorm on $A$ will always be assumed to extend the valuation on $k$;
$\left|\\!\left|{\lambda
f}\right|\\!\right|_{A}=\left|\\!\left|{\lambda}\right|\\!\right|_{k}\left|\\!\left|{f}\right|\\!\right|_{A}$
for all $\lambda\in k$. By a valued extension $L/k$ we mean a field extension
where $L$ is equipped with a valuation (extending the valuation on $k$ by our
previous convention).

######  Acknowledgments.

The author would like to thank their advisor, Sam Payne, for their support and
guidance during this research. The paper also benefited greatly from
conversations with Tom Gannon, Sam Raskin, Yixian Wu, Hernán Iriarte, Amy Li,
and Logan White.

## 2 Overview of Berkovich geometry

The contents of this Section are an overview of the necessary notions in the
theory of Berkovich spaces, in particular we need to work with
analytifications of varieties and $\beth$-spaces. Berkovich spaces provide a
framework for doing analytic geometry over valued fields. Given a base field
$k$ with valuation, Berkovich spaces are called $k$-analytic spaces. These are
locally ringed spaces equipped with a $k$-analytic atlas. We will not go into
full detail of the theory as we are primarily concerned with the topology of
these spaces, and will only introduce the notions we need. The content in this
Section is laid out in detail in [Ber90], with the exception of
$\beth$-spaces, for this we refer the reader to [Thu07].

Throughout this Section let $k$ be a valued field, and let $X$ be a separated
locally finite type $k$-scheme.

### 2.1 Berkovich analytification

######  Definition 2.1.

The Berkovich analytification of $X$, denoted $X^{\text{an}}$, has points
given by pairs $(x,|\\!|\\!\cdot\\!|\\!|)$ where $x$ is a scheme point of $X$
and $|\\!|\\!\cdot\\!|\\!|$ is a valuation on $k(x)$. There is a map
$\rho\colon X^{\text{an}}\rightarrow X$ given by
$(x,|\\!|\\!\cdot\\!|\\!|)\mapsto x$. The topology on $X^{\text{an}}$ is the
coarsest such that $\rho$ is continuous, and for any open $U\subseteq X$ and
any $f\in{\mathcal{O}}_{X}(U)$, the map $\rho^{-1}(U)\rightarrow{\mathbb{R}}$
given by $(x,|\\!|\\!\cdot\\!|\\!|)\mapsto\left|\\!\left|{f}\right|\\!\right|$
is continuous.

Any valued extension $L/k$ and map of schemes
$x\colon\operatorname{Spec}L\rightarrow X$ defines a point in $X^{\text{an}}$.
The point is given by $(x,|\\!|\\!\cdot\\!|\\!|)$ where $x$ is the point
corresponding to the image of $\operatorname{Spec}L$ and
$|\\!|\\!\cdot\\!|\\!|$ is the valuation on $L$ restricted to $k(x)$. There is
also a sheaf of analytic functions on $X^{\text{an}}$, however we only need a
few details. In particular for $p\in X^{\text{an}}$ the stalk at $p$ is a
local ring. If $\kappa(p)$ is the residue field of the stalk at $p$ then
$\kappa(p)$ has a norm and we let ${\mathcal{H}}(p)$ denote the completion
with respect to this norm. If $p=(x,|\\!|\\!\cdot\\!|\\!|)$ then there is a
canonical isometric isomorphism from the completion of $k(x)$ with respect to
$|\\!|\\!\cdot\\!|\\!|$, to ${\mathcal{H}}(p)$.

When $X=\operatorname{Spec}A$, $A$ a $k$-algebra, we can alternatively
describe $X^{\text{an}}$ as being the semivaluations on $A$. The topology on
$X^{\text{an}}$ in this case will be the coarsest such that evaluation at an
element of $A$ is continuous. One can see that this description of
$X^{\text{an}}$ is equivalent to the original as follows. Let the kernel of a
semivaluation $|\\!|\\!\cdot\\!|\\!|$ on $A$, be
$\ker(|\\!|\\!\cdot\\!|\\!|):=\\{f\in
A\,|\,\left|\\!\left|{f}\right|\\!\right|=0\\}$. Then $|\\!|\\!\cdot\\!|\\!|$
defines a valuation on the fraction field of $A/\ker(|\\!|\\!\cdot\\!|\\!|)$,
which we also denote $|\\!|\\!\cdot\\!|\\!|$. This valuation is characterized
by
$\left|\\!\left|{f+\ker(|\\!|\\!\cdot\\!|\\!|)}\right|\\!\right|:=\left|\\!\left|{f}\right|\\!\right|$.

Berkovich analytification is functorial. For any morphism of $k$-schemes
$f\colon X\rightarrow Y$ there is a morphism of Berkovich spaces over $k$,
$f^{\text{an}}\colon X^{\text{an}}\rightarrow Y^{\text{an}}$. In particular
for a point $(x,|\\!|\\!\cdot\\!|\\!|)\in X^{\text{an}}$ one has that
$f^{\text{an}}((x,|\\!|\\!\cdot\\!|\\!|))=(f(x),\left|\\!\left|{f_{x}(\cdot)}\right|\\!\right|)$,
where $f_{x}\colon k(f(x))\rightarrow k(x)$ is the induced map on function
fields.

### 2.2 The Berkovich spectrum and $\beth$-spaces

We need to introduce one other type of $k$-analytic space. Let ${\mathcal{A}}$
be a Banach $k$-algebra, i.e. ${\mathcal{A}}$ is a $k$-algebra equipped with a
norm $|\\!|\\!\cdot\\!|\\!|_{{\mathcal{A}}}$, and ${\mathcal{A}}$ is complete
with respect to this norm.

######  Definition 2.2.

The Berkovich spectrum, ${\mathcal{M}}({\mathcal{A}})$, is the set of bounded
semivaluations on ${\mathcal{A}}$, i.e. semivaluations $|\\!|\\!\cdot\\!|\\!|$
such that where
$|\\!|\\!\cdot\\!|\\!|\leq|\\!|\\!\cdot\\!|\\!|_{{\mathcal{A}}}$. The topology
is the coarsest such that evaluation at an element of ${\mathcal{A}}$ is
continuous.

######  Example 2.3.

The one-point spaces in the category of $k$-analytic spaces are given by
${\mathcal{M}}(L)$ where $L$ is a complete field over $k$.

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be Banach $k$-algebras with norms
$|\\!|\\!\cdot\\!|\\!|_{{\mathcal{A}}}$ and
$|\\!|\\!\cdot\\!|\\!|_{{\mathcal{B}}}$. One defines the completed tensor
product, ${\mathcal{A}}\hat{\otimes}_{k}{\mathcal{B}}$ to be the completion of
${\mathcal{A}}\otimes_{k}{\mathcal{B}}$, with respect to the norm:

$|\\!|f|\\!|_{{\mathcal{A}},{\mathcal{B}}}:=\inf\left\\{\max_{i}|\\!|g_{i}|\\!|_{{\mathcal{A}}}|\\!|f_{i}|\\!|_{{\mathcal{B}}}\,\bigg{|}\,\text{$\sum\limits_{i}f_{i}\otimes
g_{i}\in{\mathcal{A}}\otimes_{k}{\mathcal{B}}$ such
that}\,f=\sum\limits_{i}f_{i}\otimes g_{i}\right\\}.$

The function $|\\!|\\!\cdot\\!|\\!|_{{\mathcal{A}}{\otimes}_{k}{\mathcal{B}}}$
is clearly a seminorm, that it is a norm and that
${\mathcal{A}}{\otimes}_{k}{\mathcal{B}}$ embeds isometrically into its
completion follows from part 4 of Theorem 1 in [Gru66, Section 3].

######  Proposition 2.4.

[Ber90, Chapter 3] The norm
$|\\!|\\!\cdot\\!|\\!|_{{\mathcal{A}},{\mathcal{B}}}$ makes
${\mathcal{M}}({\mathcal{A}}\hat{\otimes}_{k}{\mathcal{B}})$ the product
${\mathcal{M}}({\mathcal{A}})\times{\mathcal{M}}({\mathcal{B}})$ in the
category of $k$-analytic spaces.

######  Notation.

To avoid cluttered subscripts, when considering $p$ in some analytic space
will choose to write $|\\!|\\!\cdot\\!|\\!|_{p}$ for the norm on
${\mathcal{H}}(p)$ instead of $|\\!|\\!\cdot\\!|\\!|_{{\mathcal{H}}(p)}$.
Similarly for a pair of points $p$ and $q$ we will write
$|\\!|\\!\cdot\\!|\\!|_{p,q}$ for the norm on
${\mathcal{H}}(p)\hat{\otimes}_{k}{\mathcal{H}}(q)$ instead of
$|\\!|\\!\cdot\\!|\\!|_{{\mathcal{H}}(p),{\mathcal{H}}(q)}$.

Consider when $k$ is trivially valued. Let $A$ be a finitely generated
$k$-algebra, $A$ equipped with the trivial valuation is complete. In this case
we can identify ${\mathcal{M}}(A)$ with seminorms $|\\!|\\!\cdot\\!|\\!|$,
that are bounded by $1$. Recall that if $X$ is a variety over $k$, $L/k$ is a
valued field extension, and $R$ is the valuation ring of $L$, then the map of
schemes $\operatorname{Spec}L\rightarrow X$ has center if it factors through
$\operatorname{Spec}L\rightarrow\operatorname{Spec}R$. We say the center is
the image of the closed point of $\operatorname{Spec}R$. Furthermore if
$X=\operatorname{Spec}A$ and $A$ is an integral domain, then a valuation
$|\\!|\\!\cdot\\!|\\!|$ has center in $X$ if and only if
$|\\!|\\!\cdot\\!|\\!|$ is bounded by 1 on $A$. From this it follows that for
any trivially valued finitely generated $k$-algebra $A$, ${\mathcal{M}}(A)$
can be taken to be pairs $(x,|\\!|\\!\cdot\\!|\\!|)$, $x$ a scheme point of
$A$, and $|\\!|\\!\cdot\\!|\\!|$ a valuation on $k(x)$ having center in
$\operatorname{Spec}A$. For a general locally finite type scheme $X$ over a
trivially valued field $k$, we define $X^{\beth}$ to be given by pairs of
points $(x,|\\!|\\!\cdot\\!|\\!|)$ where $|\\!|\\!\cdot\\!|\\!|$ is a
valuation on $k(x)$, and $\operatorname{Spec}k(x)\rightarrow X$ has center in
$X$. So ${\mathcal{M}}(A)=\left(\operatorname{Spec}A\right)^{\beth}$.

######  Proposition 2.5.

[Thu07] The space $X^{\beth}$ is a compact analytic subdomain of
$X^{\text{an}}$, and they are equal when $X$ is proper.

### 2.3 The reduction map

Let
${\mathcal{A}}^{\circ}:=\\{f\in{\mathcal{A}}\,|\,\left|\\!\left|{f}\right|\\!\right|_{{\mathcal{A}}}\leq
1\\}$ and
${\mathcal{A}}^{\circ\circ}:=\\{f\in{\mathcal{A}}\,|\,\left|\\!\left|{f}\right|\\!\right|_{{\mathcal{A}}}<1\\}$.
Then the reduction of ${\mathcal{A}}$ is said to be
$\widetilde{{\mathcal{A}}}:={\mathcal{A}}^{\circ}/{\mathcal{A}}^{\circ\circ}$.
Any map of Banach $k$-algebras ${\mathcal{A}}\rightarrow{\mathcal{B}}$ induces
a map of rings
$\widetilde{{\mathcal{A}}}\rightarrow\widetilde{{\mathcal{B}}}$. In particular
for $p\in{\mathcal{M}}({\mathcal{A}})$ there is a map of rings
$h_{p}\colon\widetilde{{\mathcal{A}}}\rightarrow\widetilde{{\mathcal{H}}(p)}$.
Then there is a map of sets called the reduction map,
$r\colon{\mathcal{M}}({\mathcal{A}})\rightarrow\operatorname{Spec}(\widetilde{{\mathcal{A}}})$
defined by $p\mapsto\ker(h_{p})$.

When $A$ is a finitely generated algebra with the trivial norm we have
$A^{\circ}=A$ and $A^{\circ\circ}=0$, so reduction gives a map of sets
$r\colon{\mathcal{M}}(A)\rightarrow\operatorname{Spec}A$. More generally we
have a map $r\colon X^{\beth}\rightarrow X$. When $X^{\beth}$ is viewed as
pairs $(x,|\\!|\\!\cdot\\!|\\!|)$, reduction maps $(x,|\\!|\\!\cdot\\!|\\!|)$
to its center. This map is anticontinuous [Ber90, Corollary 2.4.2], and
surjective [Ber90, Proposition 2.4.4]. Recall that anticontinuous means that
the preimage of an open set is closed.

### 2.4 The Shilov boundary

A boundary of ${\mathcal{M}}({\mathcal{A}})$ is a closed subset of
${\mathcal{M}}({\mathcal{A}})$, such that for every $f\in{\mathcal{A}}$ the
function
$\text{ev}_{f}\colon{\mathcal{M}}({\mathcal{A}})\rightarrow{\mathbb{R}}$ given
by $|\\!|\\!\cdot\\!|\\!|\mapsto|\\!|f|\\!|$, attains its maximum. A unique
minimal boundary is said to be the Shilov boundary.

######  Proposition 2.6.

[Ber90, Proposition 2.4.4] The reduction map
$r\colon{\mathcal{M}}({\mathcal{A}})\rightarrow\operatorname{Spec}(\widetilde{{\mathcal{A}}})$
gives a bijection between the Shilov boundary of
${\mathcal{M}}({\mathcal{A}})$ and the generic points of the components of
$\operatorname{Spec}(\widetilde{{\mathcal{A}}})$.

## 3 The retraction map

In this Section we assume $k$ is a trivially valued and algebraically closed
field, $G$ is a connected linear algebraic group over $k$, and $X$ is a
$G$-variety. Let $m$ to denote the action map $m\colon G\times X\rightarrow
G$. The objective is to introduce $\ast$-multiplication, use it construct a
retraction of topological spaces ${\mathbf{p}}\colon X^{\text{an}}\rightarrow
X^{\text{an}}$, and give a description of the image of ${\mathbf{p}}$.
Furthermore we show that when $G$ is reductive we can construct a strong
deformation from $X^{\text{an}}$ onto ${\mathbf{p}}(X^{\text{an}})$ using
$\ast$-multiplication.

The notion of $\ast$-multiplication is taken directly from [Ber90, Chapter 5].
Properties specific to $\ast$-multiplication by the unique point in the Shilov
boundary of $G^{\beth}$ are directly inspired by the results on tori from
[Thu07, Section 2].

### 3.1 $\ast$-multiplication

######  Definition 3.1.

We say that a commutative Banach $k$-algebra ${\mathcal{A}}$ is peaked if for
any valued extension $L/k$ the norm on $L\hat{\otimes}_{k}{\mathcal{A}}$ is a
valuation. We say a point $p$ of a $k$-analytic space is peaked if the
completed residue field ${\mathcal{H}}(p)$ is peaked.

######  Proposition 3.2.

All the the points of $G^{\text{an}}$ are peaked.

###### Proof.

This is a particular case of [Poi13, Corollaire 3.14]. ∎

######  Definition 3.3.

Let $g\in G^{\text{an}}$, because $g$ is peaked the norm
$|\\!|\\!\cdot\\!|\\!|_{g,p}$ is a valuation and is thus an element of
${\mathcal{M}}({\mathcal{H}}(g))\times{\mathcal{M}}({\mathcal{H}}(p))$. There
is a map of sets $X^{\text{an}}\rightarrow X^{\text{an}}$ given by $p\mapsto
g\ast p$; where $g\ast p$ is the image of $|\\!|\\!\cdot\\!|\\!|_{g,p}$ under
the composition of maps of $k$-analytic spaces:

${\mathcal{M}}({\mathcal{H}}(g))\times{\mathcal{M}}({\mathcal{H}}(p))\hookrightarrow
G^{\text{an}}\times X^{\text{an}}\xrightarrow{m^{\text{an}}}X^{\text{an}}.$

This is called $\ast$-multiplication by $g$.

######  Example 3.4.

Let $g\in G^{\text{an}}(k)=G(k)$. The map $X^{\text{an}}\rightarrow
X^{\text{an}}$ given by $p\mapsto g\ast p$ is equal to the map induced by left
multiplication by $g$.

Below are some general facts about $\ast$-multiplication.

######  Proposition 3.5.

[Ber90, Proposition 5.2.8] Let $G$ and $H$ be algebraic groups acting on
varieties $X$ and $Y$, respectively. Let $\phi\colon G\rightarrow H$ be a
morphism of algebraic groups, and $\psi\colon X\rightarrow Y$ a morphism of
varieties.

  1. 1.

The map $X^{\text{an}}\rightarrow X^{\text{an}}$ given by
$\ast$-multiplication by a point $g\in G^{\text{an}}$ is continuous.

  2. 2.

If $g$ and $h$ are points of $G^{\text{an}}$ and $H^{\text{an}}$,
respectively, and $p$ a point of $X^{\text{an}}$, then $g\ast(h\ast p)=(g\ast
h)\ast p$

  3. 3.

Given a commutative diagram:

${G^{\text{an}}\times X^{\text{an}}}$${X^{\text{an}}}$${H^{\text{an}}\times
Y^{\text{an}}}$${Y{\text{an}}}$$\scriptstyle{\phi^{\text{an}}\times\psi^{\text{an}}}$$\scriptstyle{\psi^{\text{an}}}$

one has that $\psi^{\text{an}}(g\ast
p)=\phi^{\text{an}}(g)\ast\psi^{\text{an}}(p)$.

Let ${\mathbf{g}}\in G^{\text{an}}$ be the point given by the trivial
valuation. This is the unique point in the Shilov boundary of $G^{\beth}$. Let
${\mathbf{p}}\colon X^{\text{an}}\rightarrow X^{\text{an}}$ be given by
$p\mapsto{\mathbf{g}}\ast p$. By Proposition 3.5 part 1, ${\mathbf{p}}$ is a
continuous map of topological spaces. The remainder of this section will be
devoted to studying this map.

######  Remark 3.6.

We can alternatively describe ${\mathbf{p}}$ in the following way. We have
that ${\mathbf{p}}(p)$ is the image of the unique point in the Shilov boundary
of $G^{\beth}\times{\mathcal{M}}({\mathcal{H}}(p))$ under the map of
$k$-analytic spaces:

$G^{\beth}\times{\mathcal{M}}({\mathcal{H}}(p))\hookrightarrow G^{\beth}\times
X^{\text{an}}\rightarrow X^{\text{an}}.$

So when $G=T$ is a torus, ${\mathbf{p}}$ is the retraction map defined in
[Thu07, Section 2].

######  Example 3.7.

Let $X=T=\operatorname{Spec}k[t_{1}^{\pm 1},\ldots t_{n}^{\pm 1}]$ be the
$n$-dimensional torus, and equip $k[t_{1}^{\pm 1},\ldots t_{n}^{\pm 1}]$ with
the trivial valuation. Let $\lambda\in[0,1]$, and let ${\mathbf{g}}_{\lambda}$
be the point of $T^{\beth}$ corresponding to the semivaluation:

$|\\!|\sum\limits_{I}a_{I}t^{I}|\\!|=\max_{I}\lambda^{|I|}$

where $I$ denotes a multi-index and if $I=(I_{1},\ldots I_{n})$ then
$|I|=I_{1}+\cdots I_{n}$. Then one can see via computation that for any valued
extension $L$, the norm $|\\!|\\!\cdot\\!|\\!|_{{\mathbf{g}}_{\lambda},L}$ on
${\mathcal{H}}({\mathbf{g}}_{\lambda})\hat{\otimes}_{k}L$ is multiplicative.
For $\lambda=0$ this is because ${\mathcal{H}}({\mathbf{g}}_{\lambda})=k$. For
$\lambda\in(0,1]$ ${\mathbf{g}}_{\lambda}$ defines a norm on $k[t_{1}^{\pm
1},\ldots t_{n}^{\pm 1}]$. For $f=\sum_{I}a_{I}t^{I}\in k[t_{1}^{\pm 1},\ldots
t_{n}^{\pm 1}]\otimes_{k}L=L[t_{1}^{\pm 1},\ldots t_{n}^{\pm 1}]$ we compute
that:

$|\\!|f|\\!|_{{\mathbf{g}}_{\lambda},L}=\max_{I}|\\!|a_{I}|\\!|_{L}\lambda^{|I|}$

and this is a valuation.

Let $p\in T^{\text{an}}$ be given by $x=(x_{1},\ldots x_{n})\in T(L)$, where
$L$ is a valued extension of $k$, so $p=(x,|\\!|\\!\cdot\\!|\\!|_{L})$. Let
$m_{x}$ the composition $T\times\operatorname{Spec}k(x)\hookrightarrow T\times
T\xrightarrow{m}T$. Then for
$f=\sum_{I}a_{I}t^{I}\in\operatorname{Spec}k[t_{1}^{\pm 1},\ldots t_{n}^{\pm
1}]$ seminorm ${\mathbf{g}}_{\lambda}\ast p$ is given by:

$|\\!|f|\\!|_{{\mathbf{g}}_{\lambda}\ast
p}=|\\!|m_{x}^{\ast}f|\\!|_{{\mathbf{g}}_{\lambda},p}=|\\!|\sum\limits_{I}a_{I}t^{I}\otimes
x^{I}|\\!|_{{\mathbf{g}}_{\lambda},p}=\max_{I}|\\!|a_{I}x^{I}|\\!|_{L}\lambda^{|I|}.$

######  Corollary 3.8.

The map: ${\mathbf{p}}\colon X^{\text{an}}\rightarrow X^{\text{an}}$ is
idempotent, i.e. ${\mathbf{g}}\ast{\mathbf{g}}={\mathbf{g}}$.

###### Proof.

One has that ${\mathcal{H}}({\mathbf{g}})=k(G)$ and the norm on $k(G)$ defined
by ${\mathbf{g}}$ is the trivial valuation so we have that for
$f\in{\mathcal{H}}({\mathbf{g}})\otimes_{k}{\mathcal{H}}({\mathbf{g}})$:

$|\\!|f|\\!|_{{\mathbf{g}},{\mathbf{g}}}=\inf\\{\max_{i}|\\!|g_{i}|\\!|_{{\mathbf{g}}}|\\!|f_{i}|\\!|_{{\mathbf{g}}}\,|\,f=\sum\limits_{i}g_{i}\otimes
f_{i}\\}$

which is 0 or 1, and 0 if and only if $f=0$. Therefore it is the trivial norm.
so ${\mathbf{g}}\ast{\mathbf{g}}={\mathbf{g}}$. So by Proposition 3.5
${\mathbf{p}}$ is idempotent. ∎

######  Corollary 3.9.

The map ${\mathbf{p}}\colon X^{\text{an}}\rightarrow X^{\text{an}}$ is a
retraction of topological spaces.

######  Proposition 3.10.

Let $p=(x,|\\!|\\!\cdot\\!|\\!|)\in X^{\text{an}}$, then
${\mathbf{p}}(p)=(\eta_{Y},\overline{|\\!|\\!\cdot\\!|\\!|})$. Where
$\eta_{Y}$ is the generic point of $Y$, $Y$ is the Zariski closure of the
image of the composition:

$G\times\operatorname{Spec}(k(x))\hookrightarrow G\times X\xrightarrow{m}X$

and $\overline{|\\!|\\!\cdot\\!|\\!|}$ is a $G(k)$-invariant valuation on
$k(Y)$.

The proof is similar to the proofs of [Thu07, Proposition 2.5 (2)] and [Thu07,
Proposition 2.3 (ii)], which prove the above for $G=T$ a torus.

###### Proof.

Recall that there is a map $\rho:X^{\text{an}}\rightarrow X$ given by
$(x,|\\!|\\!\cdot\\!|\\!|)\mapsto x$. To see that
$\rho({\mathbf{p}}(p))=\eta_{Y}$ notice that there is a commutative diagram:

${{\mathcal{M}}({\mathcal{H}}({\mathbf{g}}))\times{\mathcal{M}}({\mathcal{H}}(p))}$${\operatorname{Spec}k(\eta_{G})\times\operatorname{Spec}k(x)}$${G^{\text{an}}\times
X^{\text{an}}}$${G\times
X}$${X^{\text{an}}}$${X}$$\scriptstyle{\rho}$$\scriptstyle{m^{\text{an}}}$$\scriptstyle{\rho}$$\scriptstyle{m}$$\scriptstyle{\rho}$

where $\eta_{G}$ is the generic point of $G$. In this diagram the image of the
generic point of $\operatorname{Spec}k(\eta_{G})\times\operatorname{Spec}k(x)$
in $X$ is $\eta_{Y}$. But then because
$\rho(|\\!|\\!\cdot\\!|\\!|_{{\mathbf{g}},p})$ is the generic point of
$\operatorname{Spec}k(\eta_{G})\times\operatorname{Spec}k(x)$ we have the
first claim.

To see that $\overline{|\\!|\\!\cdot\\!|\\!|}$ is $G(k)$-invariant notice that
for any $g\in G(k)$, $g\ast{\mathbf{g}}={\mathbf{g}}$. Thus by Proposition 3.5
part 2 we have that $g\ast({\mathbf{g}}\ast p)={\mathbf{g}}\ast p$. But
$p\mapsto g\ast p$ is the map induced by left multiplication by $g$, which for
points of the form $(\eta_{Y},|\\!|\\!\cdot\\!|\\!|)$ is given by precomposing
$|\\!|\\!\cdot\\!|\\!|$ with the left-multiplication by $g$. ∎

Furthermore we have the following.

######  Proposition 3.11.

The image of ${\mathbf{p}}$ is exactly the set of points
$(\eta_{Y},|\\!|\\!\cdot\\!|\\!|)$, where $Y$ is a $G$-invariant subvariety of
$X$ and $|\\!|\\!\cdot\\!|\\!|$ is $G(k)$-invariant.

This proof is the same as the proof of [Thu07, Proposition 2.3 (ii)], which
proves this fact for $G=T$ a torus. We include the argument, except for one
lemma, for completeness.

###### Proof.

By Proposition 3.10 we have one containment. For the other notice that points
of the form $(\eta_{Y},|\\!|\\!\cdot\\!|\\!|)$, where $Y$ is a $G$-invariant
subvariety and $|\\!|\\!\cdot\\!|\\!|$ is $G(k)$-invariant, are exactly the
points $(x,|\\!|\\!\cdot\\!|\\!|)\in X^{\text{an}}$ fixed by the action of
$G(k)$. Let $p=(x,|\\!|\\!\cdot\\!|\\!|)\in X^{\text{an}}$ be a point fixed by
the group action $G(k)$. The morphism:

$G^{\beth}\times{\mathcal{M}}({\mathcal{H}}(p))\hookrightarrow G^{\beth}\times
X^{\text{an}}\xrightarrow{m^{\text{an}}}X^{\text{an}}$

maps the subset $G^{\beth}(k)\times{\mathcal{M}}({\mathcal{H}}(p))$ onto $x$.
The map is continuous and $x$ is closed as Berkovich spaces are locally
hausdorff and thus T1, and so every point in the closure of
$G^{\beth}(k)\times{\mathcal{M}}({\mathcal{H}}(p))$ will have image $p$.
Because ${\mathbf{p}}(p)$ is the image in $X^{\text{an}}$ of the unique Shilov
boundary point of $G^{\beth}\times{\mathcal{M}}({\mathcal{H}}(p))$ under

$G^{\beth}\times{\mathcal{M}}({\mathcal{H}}(p))\hookrightarrow G^{\beth}\times
X^{\text{an}}\xrightarrow{m^{\text{an}}}X^{\text{an}}$

we need to show that the unique point in the Shilov boundary of
$G^{\beth}\times{\mathcal{M}}({\mathcal{H}}(p))$ is in the closure of
$G^{\beth}(k)\times{\mathcal{M}}({\mathcal{H}}(p))$. The field $k$ is
algebraically closed so
$G(k)\times\operatorname{Spec}\widetilde{{\mathcal{H}}(p)}$ is a dense subset
of $G\times\operatorname{Spec}\widetilde{{\mathcal{H}}(p)}$. The following
lemma will complete the proof. ∎

######  Lemma 3.12.

Let $L/k$ be a nonarchimedean field extension, and let $\Sigma$ be a subset of
$G^{\beth}\times_{k}{\mathcal{M}}(L)$ with $r(\Sigma)$ dense in
$G\times_{k}\operatorname{Spec}(\tilde{L})$. Then the Shilov boundary is in
the closure of $\Sigma$.

###### Proof.

The case when $G$ is a torus is proved in [Thu07, lemma 2.4], and the proof
proceeds the same for any connected linear algebraic group $G$. ∎

### 3.2 The action of ${\mathbf{p}}$ on homogeneous spaces

If $G$ is reductive and $X$ is a spherical $G$-variety, then any $G$-invariant
subvariety is in fact the closure of the $G$-orbit of some closed point $y\in
X$ ([Kno91]). Thus we will restrict our attention to when $G$ acts on an
orbit. In this case we will give a formula for ${\mathbf{p}}$ in terms of
valuations that will immediately relate it to the tropicalization map. We
begin by recalling the construction of Luna and Vust which is used in [TV21]
to define tropicaliziation.

Let $\overline{K}=k\\{\\!\\{u\\}\\!\\}$ denote Puiseux series over $k$,
equipped with the $u$-adic valuation $|\\!|\\!\cdot\\!|\\!|_{u}$. In the case
that $G/H$ is spherical for any $\overline{K}$-point $x$, Luna and Vust define
a $G(k)$-invariant valuation $|\\!|\\!\cdot\\!|\\!|_{x}$ ([LV83, Section 4]).
For any $f\in k(G/H)$ there is a nonempty open subset of $U\subseteq G(k)$
such that if $g\in U$ then $gf(x)$ is well defined. Luna and Vust show that
for any $f\in k(G/H)$ there is a dense open $U_{f}\subseteq G(k)$ where
$\left|\\!\left|{gf(x)}\right|\\!\right|_{u}$ is constant as $g$ varies over
$U_{f}$. The valuation $|\\!|\\!\cdot\\!|\\!|_{x}$ is given by
$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{gf(x)}\right|\\!\right|_{u}$
for $g\in U_{f}$.

We will now explain how this generalizes to other valued extensions $L/k$. Let
$x\colon\operatorname{Spec}L\rightarrow G$ and denote the valuation on $L$ by
$|\\!|\\!\cdot\\!|\\!|$. The proof of the following lemma uses the same
technique as the proof of [Kno91, Lemma 1.4].

######  Lemma 3.13.

For each $f\in k(G)$ there is a nonempty Zariski open $U_{f}\subseteq U$ such
that for any $g\in U_{f}$, $\left|\\!\left|{gf(x)}\right|\\!\right|$ is
constant. For $f\in k[G]$ we have
$\left|\\!\left|{gf(x)}\right|\\!\right|=\max_{h\in
G(k)}\left|\\!\left|{hf(x)}\right|\\!\right|$ when $g\in U_{f}$.

###### Proof.

Without loss of generality we can assume $f\in k[G]$. Let $M$ be the
$G$-module generated by $f$, this is a finite dimensional vector space and
thus has basis $g_{1}f,\ldots g_{r}f$. Let
$v=\max_{i}\left|\\!\left|{g_{i}f(x)}\right|\\!\right|$. Let
$V(v)=\\{f^{\prime}\in
k[G]\,|\,\left|\\!\left|{f^{\prime}(x)}\right|\\!\right|<v\\}$. Then set
$U_{f}=\\{g\in G(k)\,|\,gf\notin V(v)\cap M\\}$. ∎

Let $|\\!|\\!\cdot\\!|\\!|_{x}:k(G)\rightarrow(0,\infty)$ be given by:

$f\mapsto\left|\\!\left|{gf(x)}\right|\\!\right|\quad g\in U_{f}.$ (1)

######  Corollary 3.14.

The function $|\\!|\\!\cdot\\!|\\!|_{x}$ defines a $G(k)$-invariant valuation
on $k(G)$.

######  Lemma 3.15.

Let $x:\operatorname{Spec}L\rightarrow G$ be as above and let
$p=(x,|\\!|\\!\cdot\\!|\\!|)$ be the corresponding point in $G^{\text{an}}$.
Then if ${\mathbf{p}}(p)=(\eta_{G},\overline{|\\!|\\!\cdot\\!|\\!|})$, we have
that $\overline{|\\!|\\!\cdot\\!|\\!|}=|\\!|\\!\cdot\\!|\\!|_{x}$.

###### Proof.

We will show that $|\\!|\\!\cdot\\!|\\!|_{x}$ and
$\overline{|\\!|\\!\cdot\\!|\\!|}$ agree for all regular functions $f\in
k[G]$. Let $m_{x}$ be the composition
$G\times\operatorname{Spec}k(x)\hookrightarrow G\times G\xrightarrow{m}G$.
Then if $m_{x}^{*}f=\sum_{i}g_{i}\otimes f_{i}(x)\in k[G]\otimes k(x)$ we have
$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{gf(x)}\right|\\!\right|=\left|\\!\left|{\sum_{i}g_{i}(g)\otimes
f_{i}(x)}\right|\\!\right|\leq\max_{i}(\left|\\!\left|{f_{i}(x))}\right|\\!\right|)$,
where $g\in U_{f}$, $U_{f}$ the open subset of $G(k)$ from Lemma 3.13. But
then
$\overline{\left|\\!\left|{f}\right|\\!\right|}=\left|\\!\left|{m_{x}^{*}f}\right|\\!\right|_{{\mathbf{g}},p}$
and

$\left|\\!\left|{m_{x}^{*}f}\right|\\!\right|_{{\mathbf{g}},p}=\inf\left\\{\max_{i}\left|\\!\left|{g_{i}}\right|\\!\right|_{{\mathbf{g}}}\left|\\!\left|{f_{i}(x)}\right|\\!\right|_{p}\,\,\bigg{|}\,\,\sum\limits_{i}g_{i}\otimes
f_{i}(x)=m_{x}^{*}f\right\\}.$

Because $|\\!|\\!\cdot\\!|\\!|_{{\mathbf{g}}}$ is trivial we have that
$\left|\\!\left|{f}\right|\\!\right|_{x}\leq\overline{\left|\\!\left|{f}\right|\\!\right|}$.

Conversely, to show that
$\left|\\!\left|{f}\right|\\!\right|_{x}\geq\overline{\left|\\!\left|{f}\right|\\!\right|}$
it suffices to show that there is one expression
$m_{x}^{*}f=\sum_{i}g_{i}\otimes f_{i}(x)$ such that
$\left|\\!\left|{f}\right|\\!\right|_{x}=\max_{i}\left|\\!\left|{f_{i}(x)}\right|\\!\right|$.
Let $m_{x}^{*}f=\sum_{i}g_{i}\otimes f_{i}(x)$, by manipulating the sum we can
assume all the $g_{i}$ and $f_{i}(x)$ are nonzero and the functions $g_{i}$
are $k$-linearly independent. We can also assume that
$\max_{i}(\left|\\!\left|{f_{i}(x)}\right|\\!\right|)=\left|\\!\left|{f_{1}(x)}\right|\\!\right|$.
Let $V$ be the $k$-span in $L$ of $f_{i}(x)$. Fix $\alpha>1$. It follows from
[BGR84, Subsection 2.6.2, Proposition 3]333While [BGR84] works with $k$
nontrivially valued, this result holds for trivially valued $k$ as well, the
proof is effectively the same. Such bases are called $\alpha$-Cartesian bases.
there is a $k$-basis of $V$, $v_{1},\ldots v_{r}$ such that $v_{1}$ is a
multiple of $f_{1}(x)$, and for any $\lambda_{1},\ldots\lambda_{r}\in k$

$\left|\\!\left|{\sum\limits_{j}\lambda_{j}v_{j}}\right|\\!\right|\geq\alpha^{-1}\max_{j}(\left|\\!\left|{v_{j}}\right|\\!\right|).$

Then we have for each $i$, $f_{i}(x)=\sum_{j}\lambda^{i}_{j}v_{j}$. So then
$m_{x}^{*}f=\sum_{j}(\sum\limits\lambda^{i}_{j}g_{i})\otimes v_{j}$, by
assumption for each $j$, the function $\sum_{i}\lambda^{i}_{j}g_{i}\in k[G]$
is not identically 0. Take $g\in U_{f}$ and such that
$\sum_{i}\lambda^{i}_{1}g_{i}(g)\neq 0$, which exists because $U_{f}$ is
nonempty and open and so is the set of points such that
$\sum_{i}\lambda^{i}_{1}g_{i}(g)\neq 0$. Now we have that:

$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{\sum\limits_{j}(\sum\limits_{i}\lambda^{i}_{j}g_{i}(g))\otimes
v_{j}}\right|\\!\right|\geq\alpha^{-1}\max_{j}(\left|\\!\left|{v_{j}}\right|\\!\right|)=\alpha^{-1}\left|\\!\left|{f_{1}(x)}\right|\\!\right|.$

But this holds for all $\alpha>1$. So
$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{f_{1}(x)}\right|\\!\right|=\max_{i}(\left|\\!\left|{f_{i}(x))}\right|\\!\right|)$.
So
$\left|\\!\left|{f}\right|\\!\right|_{x}\geq\overline{\left|\\!\left|{f}\right|\\!\right|}$,
and thus the valuations are equal on $k[G]$, and thus are equal on all of
$k(G)$. ∎

Let $Y$ be the $G$-orbit in $X$ of some closed point $y$. The morphism
$G\rightarrow Y$ given by $g\mapsto gy$ defines an inclusion $k(Y)\rightarrow
k(G)$. Given some valued extension $L/k$ and $x\in G(L)$, one can restrict
$|\\!|\\!\cdot\\!|\\!|_{x}$ to $k(Y)$. Let $x^{\prime}$ be the image of $x$
under the map $G(L)\rightarrow Y(L)$. Then $|\\!|\\!\cdot\\!|\\!|_{x}$ is in
fact given by

$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{gf(x^{\prime})}\right|\\!\right|\quad
g\in U_{f}.$ (2)

We denote the restriction of $|\\!|\\!\cdot\\!|\\!|_{x}$ to $k(Y)$ by
$|\\!|\\!\cdot\\!|\\!|_{x^{\prime}}$.

######  Corollary 3.16.

The map ${\mathbf{p}}\colon Y^{\text{an}}\rightarrow Y^{\text{an}}$ is given
by
$(x,|\\!|\\!\cdot\\!|\\!|)\mapsto(\eta_{Y},|\\!|\\!\cdot\\!|\\!|_{x^{\prime}})$.
Furthermore all the $G(k)$-invariant valuations on $k(Y)$ have the form
$|\\!|\\!\cdot\\!|\\!|_{x^{\prime}}$ for some $x^{\prime}\in Y$.

###### Proof.

We have a surjection of sets $G^{\text{an}}\rightarrow Y^{\text{an}}$ and a
commutative diagram of maps of analytic spaces:

${G^{\text{an}}\times G^{\text{an}}}$${G^{\text{an}}}$${G^{\text{an}}\times
Y^{\text{an}}}$${Y^{\text{an}}}$

Thus by Proposition 3.5 part 3 we see that
${\mathbf{p}}((x,|\\!|\\!\cdot\\!|\\!|))=(\eta_{Y},|\\!|\\!\cdot\\!|\\!|_{x^{\prime}})$.

To see that this is all $G(k)$-invariant valuations on $k(Y)$, notice that it
follows from Proposition 3.11 that all $G(k)$-invariant valuations on $k(G)$
are of the form $|\\!|\\!\cdot\\!|\\!|_{x}$ for some $x\in X(L)$ and it
follows from [Kno91, Corollary 1.5] that any $G(k)$-invariant valuation on
$k(Y)$ is the restriction of one on $k(G)$. ∎

### 3.3 A strong deformation retraction when $G$ is reductive

Assume $G$ is a reductive group. We construct a strong deformation retraction
from $X^{\text{an}}$ onto ${\mathbf{p}}(X^{\text{an}})$, using the same method
as the proof of [Ber90, Theorem 6.2.1]. Fix a maximal torus $T$ in $G$, Borel
subgroup $B_{+}$ containing $T$, and let $B_{-}$ be the opposite Borel i.e.
the unique Borel subgroup of $G$ such that $B_{+}\cap B_{-}=T$. Let $U_{+}$ be
the unipotent radical of $B_{+}$, and let $U_{-}$ be the unipotent radical of
$B_{-}$. Then the map:

$U_{-}\times T\times U_{+}\rightarrow G$

given by the multiplication map, is in fact an open immersion [GP11, Exposé
XXII]. Denote the image by $\Omega$. The groups $U_{-}$ and $U_{+}$ are
isomorphic affine $N$-space for some $N$. Let
$U_{+}=\operatorname{Spec}k[u_{1},\ldots u_{N}]$, and
$U_{-}=\operatorname{Spec}k[u_{-1},\ldots u_{-N}]$, and let
$T=\operatorname{Spec}(k[t_{1}^{\pm 1},\ldots t_{n}^{\pm 1})]$. Then the
coordinate ring of $\Omega$, $k[\Omega]$, is given by
$\operatorname{Spec}k[t_{i}^{\pm 1},u_{j}]$. Then set $T_{i}=t_{i}-1$. For
each $\lambda\in[0,1]$ there is a point ${\mathbf{g}}_{\lambda}\in
k[\Omega]^{\beth}$ given by the semivaluation
$|\\!|\\!\cdot\\!|\\!|_{{\mathbf{g}}_{\lambda}}$, which is defined by:

$\left|\\!\left|{\sum_{I,J}\alpha_{I,J}T^{I}u^{J}}\right|\\!\right|_{{\mathbf{g}}_{\lambda}}=\min_{I,J}\left|\\!\left|{\alpha_{I,J}}\right|\\!\right|\lambda^{|I|+|J|}$

where $I$ and $J$ denote multi-indices.Then ${\mathbf{g}}_{1}={\mathbf{g}}$
and ${\mathbf{g}}_{0}=e$, the identity of $G$. Thus we have a map of sets:

$H\colon[0,1]\times X^{\text{an}}\rightarrow X^{\text{an}}$

$(\lambda,p)\mapsto{\mathbf{g}}_{\lambda}\ast p$. Furthermore $H(0,p)=p$ and
$H(1,p)={\mathbf{p}}(X^{\text{an}})$. Notice that because
${\mathbf{g}}_{\lambda}\in X^{\beth}$ for any $\lambda\in[0,1]$, we can
restrict $H$ to a map $[0,1]\times X^{\beth}\rightarrow X^{\beth}$, with
$H(0,p)=p$ and $H(1,p)={\mathbf{p}}(p)$.

######  Proposition 3.17.

The map $H$ is a homotopy, and in fact a strong deformation retraction of
$X^{\text{an}}$ onto ${\mathbf{p}}(X^{\text{an}})$. Similarly $H$ defines a
strong deformation retraction from $X^{\beth}$ onto ${\mathbf{p}}(X^{\beth})$.

###### Proof.

That $H$ is continuous follows directly from [Ber90, Corollary 6.1.2]. It
follows from Proposition 3.5 part 2 that $H$ is strong deformation retract
because we have that
${\mathbf{g}}_{\lambda}\ast{\mathbf{g}}_{1}={\mathbf{g}}_{1}$. Indeed, for
$f\in k[\Omega]$ we will show that
$\left|\\!\left|{m^{\ast}f}\right|\\!\right|_{{\mathbf{g}}_{\lambda},{\mathbf{g}}_{1}}=1$.
Let $m^{\ast}f=\sum a_{I,J}^{L,M}t^{I}u^{J}\otimes t^{L}u^{M}$. Without loss
of generality we can assume that $a_{I,J}^{L,M}=0$ whenever $I$ or $L$ is less
than 0, because otherwise we can factor out a power of $t^{R}\otimes t^{S}$,
and $|\\!|t^{R}\otimes
t^{S}|\\!|_{{\mathbf{g}}_{\lambda},{\mathbf{g}}_{1}}=1$. But then we can
rewrite the sum by substituting $T_{i}$ for $t_{i}-1$, and we have that
$m^{\ast}f=\sum b_{I,J}^{L,M}T^{I}u^{J}\otimes T^{L}u^{M}$. Note that $T^{I}$
and $u^{J}$ both evaluate to 0 on the identity. If $f\neq 0$ then
$e^{\ast}\times\text{id}\circ m^{\ast}f\neq 0$, so there must be some term
$b_{0,0}^{L,M}\neq 0$. But then:

$\left|\\!\left|{m^{\ast}f}\right|\\!\right|_{{\mathbf{g}}_{\lambda},{\mathbf{g}}_{1}}=\left|\\!\left|{\sum
b_{I,J}^{L,M}T^{I}u^{J}\otimes
T^{L}u^{M}}\right|\\!\right|_{{\mathbf{g}}_{\lambda},{\mathbf{g}}_{1}}=\max_{I,J,L,M}(\left|\\!\left|{b_{I,J}^{L,M}}\right|\\!\right|)(\lambda^{|I|+|J|})=1$

as the maximum is achieved when $|I|+|J|=0$.

∎

## 4 The Luna-Vust Classification of Spherical varieties

Here we review the classification of embeddings of spherical homogeneous
spaces via colored fans. The classification was first given by Luna and Vust
in [LV83], though we will follow the exposition of [Kno91]. Much of the
notation for spherical varieties is different than that of toric varieties, we
will attempt to align our notation with that of toric varieties, similar to
the notation in [Nas17].

Let $k$ be an algebraically closed and trivially valued field444In the
literature on spherical varieties one does not generally specify a valuation
on $k$, but it is assumed the valuations on $k(X)$ are trivial when restricted
to $k$. and let $G$ be a reductive group over $k$. Let $H$ be a closed
subgroup such that $G/H$ has an open $B$-orbit, for some Borel subgroup
$B\subseteq G$. A $G/H$-embedding is a normal $G$-variety $X$ with a
$G$-equivariant open embedding $G/H\hookrightarrow X$. Note that if $X$ is a
spherical variety with open $B$-orbit $B_{0}$, then it is a $G_{0}$-embedding
where $G_{0}$ is the open $G$-orbit $GB_{0}$. The Luna-Vust theory is a
generalization of the toric case which puts embeddings of a torus in bijection
with fans. Here $G/H$-embeddings are put in bijection with colored fans. To
define colored fans and state the classification we will need to introduce
some data associated to $G/H$, primarily a lattice $M$, its dual lattice $N$,
a convex cone ${\mathcal{V}}$ in $N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$, and a
finite set of colors ${\mathcal{D}}(G/H)$. We refer the reader to Example 4.8
for an explicit demonstration of the theory.

### 4.1 Data associated to $G/H$

Let ${\mathfrak{X}}(B)$ be the character lattice of $B$ and define:

$k(G/H)^{(B)}=\\{f\in k(G/H)^{\times}\,|\,\text{there exists
$\chi_{f}\in{\mathfrak{X}}(B)$ such that for any $b\in B(k),$ }b\cdot
f=\chi_{f}(b)f\\}.$

We say elements of $k(G/H)^{(B)}$ are $B$ semi-invariant. There is a map
$k(G/H)^{(B)}\rightarrow{\mathfrak{X}}(B)$ given by:

$f\mapsto\chi_{f}$

and the kernel is exactly $k^{\times}$. Let the image of this map be $M$, set
$N=\hom(M,{\mathbb{Z}})$,
$M_{{\mathbb{R}}}=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$, and
$N_{{\mathbb{R}}}=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. We say $M$ is the
weight lattice of $G/H$. Notice that every valuation $|\\!|\\!\cdot\\!|\\!|$
on $k(G/H)$ defines an element $\varrho(|\\!|\\!\cdot\\!|\\!|)\in
N_{{\mathbb{R}}}$ by:

$\varrho(|\\!|\\!\cdot\\!|\\!|)(\chi_{f})=-\log(\left|\\!\left|{f}\right|\\!\right|).$

In particular if ${\mathcal{V}}$ is the set of $G(k)$-invariant valuations on
$k(G/H)$ then the map $\varrho$ restricted to ${\mathcal{V}}$ is an injection
by [Kno91, Corollary 1.8] and the image is a finitely generated convex cone
[Kno91, Corollary 5.3]. In general though the map $\varrho$ is not injective:
if $|\\!|\\!\cdot\\!|\\!|_{1}$ and $|\\!|\\!\cdot\\!|\\!|_{2}$ are distinct
valuations, but not $G(k)$-invariant, then it is possible that
$\varrho(|\\!|\\!\cdot\\!|\\!|_{1})=\varrho(|\\!|\\!\cdot\\!|\\!|_{2})$. We
say ${\mathcal{V}}$ is the valuation cone of $G/H$ and identify it with its
image in $N_{{\mathbb{R}}}$. The cone ${\mathcal{V}}$ spans $N_{{\mathbb{R}}}$
[Kno91, Corollary 5.3] and is cosimplicial, i.e. there exist linearly
independent elements $\chi_{1},\ldots\chi_{r}$ in the dual space
$N_{{\mathbb{R}}}^{\ast}$ such that ${\mathcal{V}}=\\{v\in
N_{{\mathbb{R}}}\,\,|\,\,\chi_{i}(v)\geq 0,\text{ for all $i$}\\}$ [Kno91,
Theorem 5.4].

Now let $X$ be a $G/H$-embedding, and let $Y$ be a $G$-orbit. Then define:

$\displaystyle{\mathcal{D}}(X):=\\{D\,|\,\text{$D$ a $B$-stable prime divisor
in $X$}\\}$ $\displaystyle{\mathcal{D}}_{Y}(X):=\\{D\,|\,\text{$D$ a
$B$-stable prime divisor in $X$ and $D\supseteq Y$}\\}$

We say that the colors of $G/H$ are ${\mathcal{D}}(G/H)$. Notice that every
prime divisor $D$, in particular every color, defines a valuation on $k(G/H)$,
which we denote $|\\!|\\!\cdot\\!|\\!|_{D}$. These valuations then define an
element of $N_{{\mathbb{R}}}$, $\varrho(|\\!|\\!\cdot\\!|\\!|_{D})$, so in
particular each color defines an element of $N_{{\mathbb{R}}}$. For
convenience we may write $\varrho(D)$ when we mean
$\varrho(|\\!|\\!\cdot\\!|\\!|_{D})$.

When we say ‘the data associated to $G/H$’ we mean the vector space
$N_{{\mathbb{R}}}$, the valuation cone ${\mathcal{V}}$ in $N_{{\mathbb{R}}}$,
and the set of colors ${\mathcal{D}}(G/H)$.

### 4.2 Colored fans

We can now define colored fans and colored cones. By a rational point of
$N_{{\mathbb{R}}}$ we mean a point in $N\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$.

######  Definition 4.1.

A colored cone is a pair $(\sigma,{\mathcal{F}})$ where $\sigma$ is a strictly
convex cone in $N_{{\mathbb{R}}}$, and ${\mathcal{F}}$ is a subset of
${\mathcal{D}}(G/H)$, such that the following are satisfied:

  1. CC1

$\sigma$ is generated by $\varrho({\mathcal{F}})$ and finitely many rational
points of ${\mathcal{V}}$

  2. CC2

The relative interior of $\sigma$ intersects ${\mathcal{V}}$

  3. CC3

$0\notin\varrho({\mathcal{F}})$.

We say $(\sigma^{\prime},{\mathcal{F}}^{\prime})$ is a colored face of a
colored cone $(\sigma,{\mathcal{F}})$ if $\sigma^{\prime}$ is a face of the
cone $\sigma$, the relative interior of $\sigma^{\prime}$ intersects
${\mathcal{V}}$, and ${\mathcal{F}}^{\prime}={\mathcal{F}}\cap\
\varrho^{-1}(\sigma)$. A colored fan, ${\mathfrak{F}}$, is a finite set of
colored cones containing all the colored faces of any colored cone in
${\mathfrak{F}}$, and for any $|\\!|\\!\cdot\\!|\\!|\in{\mathcal{V}}$ there is
at most one colored cone $(\sigma,{\mathcal{F}})\in{\mathfrak{F}}$ with
$|\\!|\\!\cdot\\!|\\!|$ in the relative interior of $\sigma$.

We associate a colored cone to each $G$-orbit in $X$, $Y$, as follows. First
define:

$\displaystyle{\mathcal{B}}_{Y}(X)=\\{D\in{\mathcal{D}}_{Y}(X)\,|\,D\text{ is
$G$-stable}\\}$
$\displaystyle{\mathcal{F}}_{Y}(X)=\\{D\in{\mathcal{D}}_{Y}(X)\,|\,D\text{ is
not $G$-stable}\\}$

and let $\sigma_{Y}(X)$ be the cone in $N_{{\mathbb{R}}}$ generated by
$\rho({\mathcal{B}}_{Y}(X))$ and $\rho({\mathcal{F}}_{Y}(X))$. We identify the
elements of ${\mathcal{F}}_{Y}(X)$ with elements of ${\mathcal{D}}(G/H)$ by
intersecting each element $D$ with $G/H$, as $D\cap G/H$ is dense in $D$ when
$D$ isn’t $G$-stable. Then we get a colored cone
$(\sigma_{Y}(X),{\mathcal{F}}_{Y}(X))$. Just as affine toric varieties
correspond to cones, we have a particular type of $G/H$-embedding
corresponding to colored cones. A $G/H$-embedding is simple if it has a unique
closed $G$-orbit.

######  Theorem 4.2.

[Kno91, Theorem 3.1] The map

$X\mapsto(\sigma_{Y}(X),{\mathcal{F}}_{Y}(X))$

gives a bijection between isomorphism classes of simple $G/H$-embeddings and
colored cones.

We can construct all simple subembeddings of $X$ in the following way. Let $Y$
be a $G$-orbit in $X$, then define:

$X_{0}:=X\setminus\left(\bigcup_{D\in{\mathcal{D}}(X)\setminus{\mathcal{D}}_{Y}(X)}D\right).$
(3)

Note that ${\mathcal{D}}(X)\setminus{\mathcal{D}}_{Y}(X)$ consists of all
divisors $D$ such that $|\\!|\\!\cdot\\!|\\!|_{D}$ is not an element of
$\sigma_{Y}(X)$.

######  Theorem 4.3.

[Kno91, Theorem 2.1] The open subvariety $X_{0}$ is a $B$-stable, affine open
subset of $X$. The intersection $Y\cap X_{0}$ is an open $B$-orbit, and
$GX_{0}$ is a simple embedding with closed orbit $Y$.

When $X$ is simple we have that $GX_{0}=X$. The following theorem describes
the cone $\sigma_{Y}(X)$ in this case.

######  Theorem 4.4.

[Kno91, Theorem 2.5] Let $X$ be a simple embedding with closed orbit $Y$, and
let $X_{0}$ be as in 3.

  1. 1.

$k[X_{0}]^{(B)}=\\{f\in
k(G/H)^{(B)}\,|\,\chi_{f}\in(\sigma_{Y}(X))^{\vee}\\}$.

  2. 2.

The center of $|\\!|\\!\cdot\\!|\\!|$ exists if and only if
$|\\!|\\!\cdot\\!|\\!|\in\sigma_{Y}(X)$.

  3. 3.

The center of $|\\!|\\!\cdot\\!|\\!|$ is $Y$ if and only if
$|\\!|\\!\cdot\\!|\\!|$ is in the relative interior of $\sigma_{Y}(X)$.

Furthermore we have the following about the faces of $\sigma_{Y}(X)$ and the
orbits of $X$.

######  Lemma 4.5.

[Kno91, Lemma 3.2] The map

$Z\mapsto(\sigma_{Z}(X),{\mathcal{F}}_{Z}(X))$

gives a bijection between orbits $Z$ whose closure contains $Y$ and the
colored faces of $(\sigma_{Y}(X),{\mathcal{F}}_{Y}(X))$.

Due to the above, and the fact that there are finitely many $G$-orbits in $X$
([Kno91]), we have that the set
${\mathfrak{F}}(X):=\\{(\sigma_{Y}(X),{\mathcal{F}}_{Y}(X))\,|\,\text{$Y$ is a
$G$-orbit}\\}$ is a colored fan. We can now state the main theorem in the
classification.

######  Theorem 4.6.

[Kno91, Theorem 3.3] The map:

$X\mapsto{\mathfrak{F}}(X)$

is a bijection between isomorphism classes of $G/H$-embeddings and colored
fans.

######  Example 4.7.

The classification of toric varieties by fans is a special case of the Luna-
Vust theory. Let $G=T$ be a torus, let $H$ be trivial. Then
${\mathcal{D}}(T)=\emptyset$, $N$ is the cocharacter lattice of $T$, and
$T$-embeddings are classified by fans in $N_{{\mathbb{R}}}$.

######  Example 4.8.

Let $G=\text{SL}_{2}$ and let $H$ be the subgroup of upper triangular matrices
with $1$s on the diagonal. Let $O$ denote the origin of ${\mathbb{A}}^{2}$,
there is an isomorphism $G/H\rightarrow{\mathbb{A}}^{2}_{k}\setminus O$ given
by mapping a matrix to its first column. The complement of the $x$-axis is an
open orbit of the Borel subgroup of upper triangular matrices, $B$. We have
that ${\mathfrak{X}}(B)\cong{\mathbb{Z}}$, where the generator is the
character $\chi$ given by:

$\left[{\begin{array}[]{cc}a&b\\\ 0&a^{-1}\\\ \end{array}}\right]\mapsto a.$

Furthermore $k(G/H)^{(B)}/k^{\times}=\\{y^{n}\,|\,n\in{\mathbb{Z}}\\}$ and
$\chi_{y^{n}}=\chi^{-n}$, so $M={\mathfrak{X}}(B)$. Thus
$N_{{\mathbb{R}}}\cong{\mathbb{R}}$. Let $D$ be the $x$-axis in
${\mathbb{A}}^{2}$. Consider the embedding
${\mathbb{A}}^{2}\hookrightarrow{\mathbb{P}}^{2}$ given by
$(a,c)\mapsto[a:c:1]$. Extend the $\text{SL}_{2}$-action to ${\mathbb{P}}^{2}$
by letting $\text{SL}_{2}$ act on the first two homogeneous coordinates while
preserving the third.

Let $L$ be the line at infinity in ${\mathbb{P}}^{2}$. Then both
$|\\!|\\!\cdot\\!|\\!|_{D}$ and $|\\!|\\!\cdot\\!|\\!|_{L}$ are
$G(k)$-invariant and furthermore $\varrho(|\\!|\\!\cdot\\!|\\!|_{D})=1$ and
$\varrho(|\\!|\\!\cdot\\!|\\!|_{L})=-1$, thus ${\mathcal{V}}=N$. There is one
color, given by the $x$-axis. The table below consists of a complete list of
all $G/H$-embeddings. The hollow circle denotes the color $D$, and $E$ denotes
the exceptional divisor in the blow ups at $O$:
$\text{Bl}_{0}{\mathbb{A}}^{2}$ and $\text{Bl}_{O}{\mathbb{P}}^{2}$.

$X$Closed $G$-orbits${\mathfrak{F}}(X)$${\mathbb{A}}^{2}\setminus
O$$\text{Bl}_{O}{\mathbb{A}}^{2}$${\mathbb{P}}^{2}\setminus
O$$\text{Bl}_{O}{\mathbb{P}}^{2}$${\mathbb{A}}^{2}$${\mathbb{P}}^{2}$$E$$L$$L,\,E$$O$$L,\,O$

### 4.3 Orbits and Orbit Closures

Let $Z$ be the $G$-orbit of a closed point in $X$. Notice that from Theorem
4.3 it follows that $Z$ is spherical. Let $\overline{Z}$ be the closure of $Z$
in $X$, so $\overline{Z}$ is a $Z$-embedding. Let $M(Z)$ be the weight lattice
of $Z$, $N_{{\mathbb{R}}}(Z)=\hom(M(Z),{\mathbb{R}})$, ${\mathcal{V}}(Z)$ be
the valuation cone, and ${\mathcal{D}}(Z)$ be the $B$-stable prime divisors.
We will relate the above to $M$, $N_{{\mathbb{R}}}$, ${\mathcal{V}}$, and
${\mathcal{D}}(G/H)$. Furthermore we will give the relationship between the
colored fan of $X$ and that of $\overline{Z}$. See [Kno91, Section 4] for
further discussion of the following.

Given $z\in Z$ there is a map $\varphi\colon G/H\rightarrow Z$ given by
$gH\mapsto gz$. This induces a map of function fields $k(Z)\hookrightarrow
k(G/H)$. The map $k(Z)\hookrightarrow k(G/H)$ gives an inclusion
$M(Z)\hookrightarrow M$ and after dualizing, a surjection $\varphi_{*}\colon
N_{{\mathbb{R}}}\rightarrow N_{{\mathbb{R}}}(Z)$. Let $\tau=\sigma_{Z}(X)$,
$N(\tau)=N_{{\mathbb{R}}}/\operatorname{span}(\tau)$, and let $M(\tau)$ be the
dual vector space to $N(\tau)$. Consider the following subset of
${\mathcal{D}}$:

${\mathcal{F}}_{\varphi}=\\{D\in{\mathcal{D}}\,|\,\text{$D$ maps dominantly to
$Z$}\\}.$

Let
${\mathcal{F}}_{\phi}^{\text{c}}={\mathcal{D}}\setminus{\mathcal{F}}_{\phi}$.

######  Theorem 4.9.

[Kno91, Theorem 4.4] We have that $N(\tau)=N(Z)$, the map $\varphi_{*}$ is the
projection $N\rightarrow N(\tau)$, $M(\tau)=M(Z)$,
$\varphi_{*}({\mathcal{V}})={\mathcal{V}}(Z)$, and there is a bijection
${\mathcal{F}}_{\varphi}^{\text{c}}\rightarrow{\mathcal{D}}(Z)$ given by
$\varphi$.

To state the relationship between ${\mathfrak{F}}(X)$ and
${\mathfrak{F}}(\overline{Z})$ define the following colored fan:

$\text{Star}(\tau):=\\{(\varphi_{*}(\sigma),\varphi_{*}({\mathcal{F}}\cap{\mathcal{F}}_{\varphi}^{\text{c}}))\,|\,(\sigma,{\mathcal{F}})\in{\mathfrak{F}}(X)\text{
and $(\tau,{\mathcal{F}}_{Z}(X))$ is a colored face of
$(\sigma,{\mathcal{F}})$.}\\}.$

Then as a result of [Kno91, Theorem 4.5] we have that.

######  Theorem 4.10.

Let $X$ be a $G/H$-embedding, and let $Z\subseteq X$ be a $G$-orbit. Let the
Zariski closure of $Z$ in $X$ be $\overline{Z}$. Then
$\text{Star}(\sigma_{Z}(X))={\mathfrak{F}}(\overline{Z})$.

## 5 Spherical Tropicalization

In this Section we review tropicalization for toric varieties, tropicalization
of spherical varieties, and prove Theorem A. We also discuss applications of
Theorem A. Let $k$ be an algebraically closed and trivially valued field, let
$G$ be a reductive group over $k$, let $X$ be a spherical $G$-variety with
open $G$-orbit $G/H$.

### 5.1 Tropicalization of toric varieties

Let $\overline{K}=k\\{\\!\\{u\\}\\!\\}$ be the field of Puiseux series,
equipped with the $u$-adic valuation which we denote
$|\\!|\\!\cdot\\!|\\!|_{u}$. Let $T=\operatorname{Spec}k[t_{1}^{\pm 1},\ldots
t_{n}^{\pm 1}]$ and let be $W$ a subvariety of $T$. There is a map of sets
$\operatorname{trop}_{T}\colon W(\overline{K})\rightarrow{\mathbb{R}}^{n}$
given by:

$(t_{1},\ldots
t_{n})\mapsto(-\log(\left|\\!\left|{t_{1}}\right|\\!\right|_{u}),\ldots-\log(\left|\\!\left|{t_{n}}\right|\\!\right|_{u})).$

We say the closure of $\operatorname{trop}_{T}(W(\overline{K}))$ is the
tropicalization of $W$, denoted $\operatorname{Trop}_{T}(W)$.

This construction extends to subvarieties of toric varieties, as follows. Let
$X$ be a toric variety, with dense torus $T$. Let $N$ be the cocharacter
lattice of $T$, and let $F(X)$ be the fan in
$N_{{\mathbb{R}}}=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ associated to $X$.
Given a cone $\sigma\in F(X)$, consider the vector space
$N(\sigma)=N_{{\mathbb{R}}}/\operatorname{span}(\sigma)$. As a set we define
we the tropicalization of $X$ to be:

$\operatorname{Trop}_{T}(X):=\bigsqcup_{\sigma\in F(X)}N(\sigma).$

To define a topology on $\operatorname{Trop}_{T}(X)$ we first consider the
case when $X$ is affine; so $X$ corresponds to a cone $\sigma\in F(X)$. Let
$M_{{\mathbb{R}}}$ be the dual vector space to $N_{{\mathbb{R}}}$; let
$\sigma^{\vee}\subseteq M_{{\mathbb{R}}}$ denote linear functions on
$N_{{\mathbb{R}}}$ that are nonnegative on $\sigma$. Let
$\overline{{\mathbb{R}}}={\mathbb{R}}\cup\\{\infty\\}$. If
$\hom(\sigma^{\vee},\overline{{\mathbb{R}}})$ denotes semigroup homomorphisms
$\sigma^{\vee}\rightarrow\overline{{\mathbb{R}}}$, then there is a natural
bijection
$\operatorname{Trop}_{T}(X)\rightarrow\hom(\sigma^{\vee},\overline{{\mathbb{R}}})$
defined as follows. The fan $F(X)$ consists of the faces of $\sigma$. Let
$\tau$ be a face of $\sigma$ and let $M(\tau)$ be the dual of $N(\tau)$.
Recall that $M(\tau)=\tau^{\perp}\subseteq M_{{\mathbb{R}}}$. So $\alpha\in
N(\tau)$ defines a semigroup homomorphism
$\tilde{\alpha}\colon\sigma^{\vee}\rightarrow\overline{{\mathbb{R}}}$

$\tilde{\alpha}(v)=\left\\{\begin{array}[]{cc}\alpha(v),&v\in\tau^{\perp}\cap\sigma^{\vee}\\\
\infty,&\text{else}\\\ \end{array}\right.$

This then defines a bijection
$\operatorname{Trop}_{T}(X)\rightarrow\hom(\sigma^{\vee},\overline{{\mathbb{R}}})$
and we give $\operatorname{Trop}_{T}(X)$ the topology
$\hom(\sigma^{\vee},\overline{{\mathbb{R}}})$ inherits from
$\overline{{\mathbb{R}}}^{\sigma^{\vee}}$. For a general toric variety $X$ we
glue the tropicalizations of affine toric subvareities according to the shared
faces of their respective cones in $F(X)$. For further discussion of
tropicalization of toric varieties and their subvarieties see [Pay09].

Recall that $X$ is a disjoint union of tori which are in bijection with the
cones of $F(X)$. Thus there is a tropicalization map
$X(\overline{K})\rightarrow\operatorname{Trop}_{T}(X)$ which is defined by
taking a disjoint union of the tropicalization maps of the tori
$T(\sigma)\rightarrow N(\sigma)$, where $T(\sigma)$ is the torus in $X$
corresponding to $\sigma\in F(X)$. Thus for any subvariety $W$ of a toric
variety $X$ we have a tropicalization map from $W(\overline{K})$ into
$\operatorname{Trop}_{T}(X)$.

### 5.2 Tropicalization of spherical varieties

In [TV21] Tevelev and Vogiannou define a tropicalization map
$\operatorname{trop}_{G}\colon G/H(\overline{K})\rightarrow{\mathcal{V}}$. The
map is given as follows. For each $x\in G/H(\overline{K})$ define
$|\\!|\\!\cdot\\!|\\!|_{x}$ to be the valuation given by:

$\left|\\!\left|{f}\right|\\!\right|_{x}=\left|\\!\left|{gf(x)}\right|\\!\right|_{u}\quad
g\in U_{f}$

as in equation 1. Then define
$\operatorname{trop}_{G}(x)=|\\!|\\!\cdot\\!|\\!|_{x}$, this gives a map
$G/H(\overline{K})\rightarrow{\mathcal{V}}$. For a subvariety $W\subseteq G/H$
the tropicalization is the closure of
$\operatorname{trop}_{G}(W(\overline{K}))$, in particular the tropicalization
of $G/H$ is ${\mathcal{V}}$.

In [Nas17] the above is extended to construct a tropicalization for any
$G/H$-embedding $X$, this is done similarly to the toric case. As a set the
tropicalization, denoted $\operatorname{Trop}_{G}(X)$, is defined to be:

$\operatorname{Trop}_{G}(X):=\bigsqcup_{(\tau,{\mathcal{F}})\in{\mathfrak{F}}(X)}{\mathcal{V}}(\tau)$

where ${\mathcal{V}}(\tau)$ is the valuation cone of the $G$-orbit associated
to $\tau$.

######  Remark 5.1.

If $G$ is a torus and $X$ is a toric variety then the cones
${\mathcal{V}}(\tau)$ are exactly the vector spaces $N(\tau)$ as in the
previous subsection.

We now put a topology on $\operatorname{Trop}_{G}(X)$. Consider first the case
when $X$ is simple with closed orbit $Y$; so $X$ corresponds to the colored
cone $(\sigma_{Y}(X),{\mathcal{F}}_{Y}(X))$. Let $\sigma=\sigma_{Y}(X)$.
Recall that $\hom(\sigma^{\vee},\overline{{\mathbb{R}}})$ is in bijection with
the disjoint union of the vector spaces
$N(\tau)=N_{{\mathbb{R}}}/\operatorname{span}(\tau)$, where $\tau$ ranges over
the faces of $\sigma$. The bijeciton is given by $\alpha\mapsto\tilde{\alpha}$
where

$\tilde{\alpha}(v)=\left\\{\begin{array}[]{cc}\alpha(v),&v\in\tau^{\perp}\cap\sigma^{\vee}\\\
\infty,&\text{else}\\\ \end{array}\right.$

The colored faces of $(\sigma,{\mathcal{F}}_{Y}(X))$ are exactly the faces of
$\sigma$ whose relative interior intersects ${\mathcal{V}}$. Given such a face
$\tau$, $N(\tau)$ contains the valuation cone ${\mathcal{V}}(\tau)$. Thus we
have that
$\operatorname{Trop}_{G}(X)\subseteq\hom(\sigma^{\vee},\overline{{\mathbb{R}}})\subseteq\overline{{\mathbb{R}}}^{\sigma^{\vee}}$
and we give $\operatorname{Trop}_{G}(X)$ the induced topology. For $X$ not
necessarily simple, $\operatorname{Trop}_{G}(X)$ is obtained by tropicalizing
the simple spherical subvarieties and gluing the valuation cones of shared
orbits.

######  Remark 5.2.

Let $X$ be simple with closed orbit $Y$, and $\sigma=\sigma_{Y}(X)$. In
general $\operatorname{Trop}_{G}(X)$ is not equal to
$\hom(\sigma^{\vee},\overline{{\mathbb{R}}})$ as we do not necessarily have
that $\sigma\subseteq{\mathcal{V}}$ or that ${\mathcal{V}}(\tau)=N(\tau)$, for
a given colored face $(\tau,{\mathcal{F}})$.

######  Example 5.3.

Let $X=\text{Bl}_{O}{\mathbb{P}}^{2}$ be the embedding of $\text{SL}_{2}/H$ as
described in example 4.8. Then the tropicalization is given by

The left most point representing the valuation cone of $L$, the right most of
$E$, and the interior representing the valuation cone of $\text{SL}_{2}/H$.
One can find further illustrated examples of this in Sections 4 and 5 of
[Nas17].

We have a tropicalization map $\operatorname{Trop}_{G}\colon
X(\overline{K})\rightarrow\operatorname{Trop}_{G}(X)$ extending the one
defined in [TV21]. This is because all $G$-orbits of $X$ are spherical
homogeneous spaces so we can take the disjoint union of the tropicalization
maps. Explicitly, for $x\in X(\overline{K})$ if $x$ is in the $G$-orbit $Y$
then set $\operatorname{Trop}_{G}(x)$ to be the valuation
$|\\!|\\!\cdot\\!|\\!|_{x}$ on $k(Y)$. Then for a subvariety $W\subseteq X$ we
can again define
$\operatorname{Trop}_{G}(W)\colon=\operatorname{trop}_{G}(W(\overline{K}))$

######  Remark 5.4.

In [Nas17] Nash actually defines a colored tropicalization. This consists of
attaching the finite set of colors associated to a colored face, to the
corresponding valuation cone. However we do not deal with the colors here so
this was not discussed.

We now prove Theorem A. Recall that in Section 3 we showed that we have a
retraction of topological spaces ${\mathbf{p}}\colon X^{\text{an}}\rightarrow
X^{\text{an}}$, given by $\ast$-multiplication by the unique point of the
Shilov boundary of $G^{\beth}$ (Corollary 3.9).

######  Theorem A.

Let $X$ be a spherical $G$-variety over an algebraically closed and trivially
valued field $k$, and let $\overline{K}$ denote Puiseux series in $k$. Then
there is a canonical homeomorphism
$\iota:\operatorname{Trop}_{G}(X)\rightarrow{\mathbf{p}}(X^{\text{an}})$, and
${\mathbf{p}}\colon X^{\text{an}}\rightarrow\operatorname{Trop}_{G}(X)$ is a
retraction of topological spaces such that the following diagram commutes:

${X(\overline{K})}$${X^{\text{an}}}$
${\operatorname{Trop}_{G}(X).}$$\scriptstyle{\operatorname{trop}_{G}}$$\scriptstyle{{\mathbf{p}}}$

###### Proof.

First observe that as sets we have an inclusion
$\iota:\operatorname{Trop}_{G}(X)\hookrightarrow X^{\text{an}}$ given as
follows. For each orbit $Z$ of $X$, and
$|\\!|\\!\cdot\\!|\\!|\in{\mathcal{V}}(Z)$, $|\\!|\\!\cdot\\!|\\!|$ defines an
element of $X^{\text{an}}$ by:

$|\\!|\\!\cdot\\!|\\!|\mapsto(\eta_{Z},|\\!|\\!\cdot\\!|\\!|)$

where $\eta_{Z}$ is the generic point of $Z$. All the $G$-invariant
subvarieties of a spherical variety are closures of $G$-orbits ([Kno91]); thus
by Proposition 3.11 we have that
${\mathbf{p}}(X^{\text{an}})=\iota(\operatorname{Trop}_{G}(X))$ as sets. That
the diagram commutes as maps of sets follows from Corollary 3.16. Finally,
${\mathbf{p}}:X^{\text{an}}\rightarrow{\mathbf{p}}(X^{\text{an}})$ is
continuous by Proposition 3.5. So it remains to show that the topology on
$\operatorname{Trop}_{G}(X)$ agrees with the topology of
${\mathbf{p}}(X^{\text{an}})$, i.e. that $\iota$ is a homeomorphism onto its
image. We will need to temporarily emphasize the difference between
$\operatorname{Trop}_{G}(X)$ and its image in $X^{\text{an}}$. Let
${\mathcal{I}}$ be the set $\iota(\operatorname{Trop}_{G}(X))$, and give
${\mathcal{I}}\subseteq X^{\text{an}}$ the subspace topology.

Consider the case when $X$ is simple with closed orbit $Y$. Let $X_{0}$ be the
$B$-stable affine open defined by equation 3. The open $X_{0}$ meets every
$G$-orbit in $X$ by Theorem 4.3. Thus $X_{0}^{\text{an}}$ contains
${\mathcal{I}}$. The topology on $X_{0}^{\text{an}}$ is generated by the
functions $\text{ev}_{f}\colon
k[X_{0}]^{\text{an}}\rightarrow\overline{{\mathbb{R}}}$, where
$\text{ev}_{f}(|\\!|\\!\cdot\\!|\\!|)=\left|\\!\left|{f}\right|\\!\right|$,
$f\in k[X_{0}]$. The functions $\text{ev}_{f}$ for $f\in k[X_{0}]^{(B)}$
define the topology on $\operatorname{Trop}_{G}(X)$ by part 1 of Theorem 4.4.
So the inclusion
$\iota\colon\operatorname{Trop}_{G}(X)\rightarrow{\mathcal{I}}$ is open. Given
that for a general $G/H$-embedding $\operatorname{Trop}_{G}(X)$ is given by
gluing the tropicalizations of simple subembeddings we have an open bijection
$\iota\colon\operatorname{Trop}_{G}(X)\rightarrow{\mathcal{I}}$ for all $X$.

Consider the case then when $X$ is proper, then $X^{\text{an}}$ is compact. We
know ${\mathcal{I}}$ is a closed subset of $X^{\text{an}}$, and therefore
${\mathcal{I}}$ is compact. Thus
$\iota\colon\operatorname{Trop}_{G}(X)\rightarrow{\mathcal{I}}$ is a
homeomorphism onto its image. Then for a general $X$ there exists a proper
spherical $G$-variety $\overline{X}$ and an open $G$-equivariant embedding
$X\hookrightarrow\overline{X}$ by [Sum74, Theorem 3]. But then we have a
commutative diagram:

${X^{\text{an}}}$${\overline{X}^{\text{an}}}$${\operatorname{Trop}_{G}(X)}$${\operatorname{Trop}_{G}(\overline{X})}$

where the composition
$\operatorname{Trop}_{G}(X)\hookrightarrow\operatorname{Trop}_{G}(\overline{X})\hookrightarrow\overline{X}^{\text{an}}$
is a homeomorphism onto its image by the above. Furthermore the map
$X^{\text{an}}\hookrightarrow\overline{X}^{\text{an}}$ is an open embedding so
the inclusion $\operatorname{Trop}_{G}(X)\hookrightarrow X^{\text{an}}$ is a
homeomorphism onto ${\mathcal{I}}$. ∎

######  Remark 5.5.

In [KM19] it is shown that tropicalization factors through $X^{\text{an}}$
when $X$ is affine. It follows from Corollary 3.16 that ${\mathbf{p}}$ and the
map $X^{\text{an}}\rightarrow\operatorname{Trop}_{G}(X)$ given in Definition
5.18 of [KM19] agree.

######  Corollary 5.6.

For $W$ a subvariety of $X$, $\operatorname{Trop}_{G}(W)$ is a strong
deformation retract of $W^{\text{an}}$.

###### Proof.

We have that $\operatorname{Trop}_{G}(W)={\mathbf{p}}(W^{\text{an}})$ so this
follows from Proposition 3.17. ∎

######  Remark 5.7.

Notice that as a result of Proposition 3.5 one has that
$\operatorname{Trop}_{G}$ defines a functor from the category of spherical
$G$-varieties to the category of topological spaces. One can describe the
morphism as follows. If $X\rightarrow X^{\prime}$ is a morphism of
$G$-varieties, then for a $Z\subseteq X$ a $G$-orbit, $Z$ maps dominantly onto
a $G$-orbit $Z^{\prime}\subseteq X^{\prime}$. Then
$|\\!|\\!\cdot\\!|\\!|\in{\mathcal{V}}(Z)$ is mapped to its restriction to
$k(Z^{\prime})$ under the inclusion $k(Z^{\prime})\hookrightarrow k(Z)$. This
is the same as the construction of tropicalization of morphisms give in
Section 2 of [Nas20].

From Theorem A we can see that tropicalization did not depend on our choice to
use Puiseux series. Let $L/k$ be an algebraically closed, valued extension,
with nontrivial valuation. Define $\operatorname{trop}_{G,L}\colon
X(L)\rightarrow{\mathcal{V}}$ to be the map given
$x\mapsto|\\!|\\!\cdot\\!|\\!|_{x}$, where $x$ is contained in some $G$-orbit
$Y$ and $|\\!|\\!\cdot\\!|\\!|_{x}$ is the valuation on $k(Y)$ defined in 2.

######  Corollary 5.8.

The closure of $\operatorname{trop}_{G,L}(W(L))$ in
$\operatorname{Trop}_{G}(X)$ is equal to
${\mathbf{p}}(W^{\text{an}})=\operatorname{Trop}_{G}(W)$, where
$W^{\text{an}}$ is considered as a subspace of $X^{\text{an}}$ induced by the
inclusion $W\subseteq X$.

###### Proof.

For a subvariety $V\subseteq X$, $W(L)$ is a dense subset of $W^{\text{an}}$,
and we have a commutative diagram

${X(L)}$${X^{\text{an}}}$
${\operatorname{Trop}_{G}(X).}$$\scriptstyle{\operatorname{trop}_{G,L}}$$\scriptstyle{{\mathbf{p}}}$

So the closure of $\operatorname{trop}_{G}(W(L))$ in
$\operatorname{Trop}_{G}(X)$ for any subvariety $W\subseteq X$ must be equal
to ${\mathbf{p}}(W^{\text{an}})$. ∎

In [Nas20] Nash gives a description of tropicalization by embedding $X$ into a
toric variety and then applying the usual tropicalization process for toric
varieties. See [Nas20, Section 6] for details. In particular this process is
used to prove the following result, for which we can give an alternate proof.

######  Theorem 5.9.

[Nas20, Theorem 6.7] If $W\subseteq G/H$ is a subvariety, and $\overline{W}$
is the closure of $W$ in $X$. Then we have that:

$\operatorname{Trop}_{G}(\overline{W})=\overline{\operatorname{Trop}_{G}(W)}$

where $\overline{\operatorname{Trop}_{G}(W)}$ is the closure taken in
$\operatorname{Trop}_{G}(X)$.

###### Proof.

This follows from Theorem A, as ${\mathbf{p}}$ is a retraction. ∎

## 6 Compactification of the colored fan

Recall from subsection 2.2 that when working over a trivially valued field
$X^{\text{an}}$ has a compact analytic subdomain $X^{\beth}$. The space
$X^{\beth}$ is defined as the subspace whose points are pairs
$(x,|\\!|\\!\cdot\\!|\\!|)\in X^{\text{an}}$, where $|\\!|\\!\cdot\\!|\\!|$
has center in $X$. In this Section we study ${\mathbf{p}}(X^{\beth})$ and
prove Theorem B. In the case of toric varieties it is shown in [Thu07, Section
2] that ${\mathbf{p}}(X^{\beth})$ is the canonical compactification of the fan
associated to $X$. We begin by reviewing the work of Thuillier on the toric
case.

Let $X$ be a toric variety over $k$ with dense torus $T$. Let $F(X)$ denote
the fan associated to $X$. Let $\sigma$ be a cone in $F(X)$ and $U_{\sigma}$
the $T$-stable open affine associated to $\sigma$. We will assume all the
cones of $F(X)$ are strictly convex. In [Thu07] Thuillier defines a map of
topological spaces $X^{\beth}\rightarrow X^{\beth}$; which is equal to the map
${\mathbf{p}}$ given by $\ast$-mutliplication by the unique point in the
Shilov boundary of $T^{\beth}$. Let $\sigma^{\vee}$ denote positive linear
functionals on $\sigma$. Thuillier shows that
$\rho^{-1}(T)\cap{\mathbf{p}}(U_{\sigma}^{\beth})$ is naturally homeomorphic
to $\hom(\sigma^{\vee},[0,\infty))$, where $\hom$ denotes semigroup
homomorphisms that preserve scaling by ${\mathbb{R}}_{\geq 0}$ and
$[0,\infty)$ has the additive semigroup structure. The space
$\hom(\sigma^{\vee},[0,\infty))$ is naturally isomorphic to $\sigma$, as
$\sigma$ is strictly convex. Furthermore ${\mathbf{p}}(U^{\beth})$ is compact,
and in fact is canonically homeomorphic to $\hom(\sigma^{\vee},[0,\infty])$,
the canonical compactification of $\sigma$. We will denote the canonical
compactification of a strictly convex cone by $\overline{\sigma}$. In this way
${\mathbf{p}}(X^{\beth})$ is the canonical compactification of the fan $F(X)$.

######  Remark 6.1.

Let $X$ be a spherical $G$-variety with dense $G$-orbit $G/H$. Observe that
${\mathbf{p}}$ has image consisting only of $G(k)$-invariant valuations, and
colored fans may contain points of $N_{{\mathbb{R}}}$ lying outside
${\mathcal{V}}$. Consider Example 6.2. Because of this fact we can give a more
concise description of ${\mathbf{p}}(X^{\beth})$ when the following holds:

$\bigcup_{(\sigma,{\mathcal{F}})\in{\mathfrak{F}}(X)}\sigma\subseteq{\mathcal{V}}$
($\star$)

Horospherical varieties, for example, satisfy this condition. However for a
symmetric homogeneous space $G/H$ there will be colors whose image in
$N_{{\mathbb{R}}}$ lies outside ${\mathcal{V}}$ and thus symmetric varieties
may not satisfy the above condition (see [Vus90] or [Per14, Section 3.4]). In
Remark 6.5 we discuss how to describe ${\mathbf{p}}(X^{\beth})$ when $\star$ ‣
6.1 is not satisfied.

######  Example 6.2.

As in [Nas17, Example 4.4], let $G=\text{GL}_{2}\times\text{GL}_{2}$ act on
$\text{GL}_{2}$ via $(g,h)\cdot x=gxh^{-1}$. Note that $\text{GL}_{2}\cong
G/H$ where $H$ is the diagonal subgroup. Let $B$ be the Borel subgroup given
by elements of the form $(b,b^{\prime})$ where $b$ is an upper triangular
matrix and $b^{\prime}$ is lower triangular; then $B$ has an open orbit in
$\text{GL}_{2}$. Let $X$ be the $\text{GL}_{2}$-embedding given by $2\times 2$
matrices over $k$, and let $X^{\prime}$ be the $\text{GL}_{2}$-embedding given
by $2\times 2$ matrices of rank at least one over $k$. The data associated to
$\text{GL}_{2}$ and the colored cones associated to $X$ and $X^{\prime}$ are
depicted in Figure 1. In particular there is a single color, $D$, given by
matrices where the bottom right entry is 0. Neither variety is horospherical
[Kno91, Corollary 6.2] though both are symmetric varieties and $\star$ ‣ 6.1
does not hold for $X$ but it does for $X^{\prime}$.

${\mathcal{V}}$

Figure 1: To the left is the vector space $N_{{\mathbb{R}}}$, the valuation
cone ${\mathcal{V}}$, and the one color $D$, for the homogeneous space
$\text{GL}_{2}$. The blue circle depicts $\varrho(|\\!|\\!\cdot\\!|\\!|_{D})$,
and the shaded triangle depicts ${\mathcal{V}}$. In the middle is the colored
cone for the $\text{GL}_{2}$-embedding $X$. To the right is the colored cone
for $X^{\prime}$.

######  Lemma 6.3.

Let $X$ be a $G/H$-embedding satisfying $\star$ ‣ 6.1. Let $Z$ be a $G$-orbit
of $X$, and $\overline{Z}$ be the closure of $Z$ in $X$. Then the
$Z$-embedding $\overline{Z}$ satisfies $\star$ ‣ 6.1.

###### Proof.

Without loss of generality we may assume $X$ is simple. Denote the closed
orbit by $Y$. Let $\tau=\sigma_{Z}(X)$, the face of $\sigma_{Y}(X)$
corresponding to $Z$. For $z\in Z$ let $\varphi\colon G/H\rightarrow Z$ be the
map given by $gH\mapsto gz$, and let $\varphi_{*}\colon
N_{{\mathbb{R}}}\rightarrow N_{{\mathbb{R}}}(\tau)$ be the induced map. It
follows from Theorem 4.10 that
$\varphi_{*}(\sigma_{Y}(X))=\sigma_{Y}(\overline{Z})$, and we know from
Theorem 4.9 that $\varphi_{*}({\mathcal{V}})={\mathcal{V}}(Z)$. So
$\sigma_{Y}(\overline{Z})\subseteq{\mathcal{V}}(\tau)$. ∎

Because colored cones correspond to simple embeddings, we will first consider
simple embeddings. Let $X$ be simple with closed $G$-orbit $Y$. Let
$\sigma=\sigma_{Y}(X)$. We begin by defining a map
$\Phi\colon{\mathbf{p}}(X^{\beth})\rightarrow\overline{\sigma}$. The set
$\rho^{-1}(G/H)\cap{\mathbf{p}}(X^{\beth})$ consists of $G(k)$-invariant
valuations on $k(G/H)$ having center in $X$. Given such a valuation
$|\\!|\\!\cdot\\!|\\!|$ we know it uniquely defines a semigroup homomorphism
$\varrho(|\\!|\\!\cdot\\!|\\!|)\colon M\rightarrow{\mathbb{R}}$ by:

$\varrho(|\\!|\\!\cdot\\!|\\!|)(\chi_{f})=-\log(\left|\\!\left|{f}\right|\\!\right|).$

If $p=(\eta_{G/H},|\\!|\\!\cdot\\!|\\!|)\in X^{\beth}$ then denote the
restriction of $\varrho(|\\!|\\!\cdot\\!|\\!|)$ to $\sigma^{\vee}$ by
$\phi_{p}$. By Theorem 4.4 part 2, we have that $\phi_{p}$ takes values in
$[0,\infty)$. Thus we have a map of sets
$\rho^{-1}(G/H)\cap{\mathbf{p}}(X^{\beth})\rightarrow\hom(\sigma^{\vee},[0,\infty))$
given by $p\mapsto\phi_{p}$.

A general point of ${\mathbf{p}}(X^{\beth})$ is a point
$p=(\eta_{Z},|\\!|\\!\cdot\\!|\\!|)$ where $\eta_{Z}$ is the generic point of
a $G$-orbit $Z$ and $|\\!|\\!\cdot\\!|\\!|$ is a $G(k)$-invariant valuation
having center in $\overline{Z}$, where $\overline{Z}$ is the Zariski closure
of $Z$ in $X$. Let $\tau$ be the face of $\sigma$ corresponding to $Z$. Then
$|\\!|\\!\cdot\\!|\\!|$ defines an element of
$\hom(M(\tau),{\mathbb{R}})=\hom(\tau^{\perp}\cap\sigma^{\vee},{\mathbb{R}})$.
This extends uniquely to a morphism of semigroups
$\phi_{p}\colon\sigma^{\vee}\rightarrow\overline{{\mathbb{R}}}$ by setting
$\phi_{p}(u)=\infty$ when $u\notin\tau^{\perp}$. But by Theorem 4.9 and 4.10
we have that $\tau^{\perp}\cap\sigma^{\vee}=\sigma_{Y}(\overline{Z})$, thus by
Theorem 4.4 part 2 we see that $\phi_{p}$ only takes values in $[0,\infty]$.
So we have a map
$\Phi\colon{\mathbf{p}}(X^{\beth})\rightarrow\overline{\sigma}$, given by
$\Phi(p)=\phi_{p}$.

######  Proposition 6.4.

Let $X$ be a simple $G/H$-embedding corresponding to the colored cone
$(\sigma,{\mathcal{F}})$, and satisfying $\star$ ‣ 6.1. Then there is a
commutative diagram:

${{\mathbf{p}}(X^{\beth})}$${\overline{\sigma}}$${{\mathbf{p}}(X^{\beth})\cap\rho^{-1}(G/H)}$${\sigma}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the horizontal arrows are homeomorphisms and are given by $\Phi$.

###### Proof.

Notice that

$\overline{\sigma}=\bigsqcup_{\tau}\hom(\tau^{\perp}\cap\sigma^{\vee},[0,\infty))$

where $\tau$ runs over the faces of $\sigma$. But then because $X$ satisfies
$\star$ ‣ 6.1 we have that:

$\bigsqcup_{\tau}\hom(\tau^{\perp}\cap\sigma^{\vee},[0,\infty))=\bigsqcup_{Z}\hom(\sigma_{Y}(\overline{Z})^{\vee},[0,\infty))$

where $Z$ runs over the orbits of $X$, and $\overline{Z}$ is the Zariski
closure of $Z$ in $X$. Because $X$ satisfies $\star$ ‣ 6.1 all elements of
$\hom(\sigma_{Y}(\overline{Z})^{\vee},[0,\infty))$ are uniquely given by
$G(k)$-invariant valuations on $k(Z)$, and by Theorem 4.4 part 2,
$\hom(\sigma_{Y}(\overline{Z})^{\vee},[0,\infty)$ is exactly those
$G(k)$-invariant valuations with center in $\overline{Z}$. Thus it follows
that $\Phi$ defines a bijection
${\mathbf{p}}(X^{\beth})\rightarrow\overline{\sigma}$ and
${\mathbf{p}}(X^{\beth})\cap\rho^{-1}(G/H)\rightarrow\sigma$.

It remains to show that $\Phi$ is a homeomorphism, and because
$\overline{\sigma}$ is compact it suffices to show $\Phi$ is open. Let $X_{0}$
be the $B$-stable affine open from equation 3, note that $X_{0}$ intersects
all $G$-orbits of $X$. So ${\mathbf{p}}(X^{\beth})\subseteq X_{0}^{\beth}$.
Recall from Theorem 4.4 part 1 that $k[X_{0}]^{(B)}=\\{f\in
k(G/H)^{(B)}\,|\,\chi_{f}\in(\sigma_{Y}(X))^{\vee}\\}$. But then evaluation at
the functions $f\in k[X_{0}]^{(B)}$ generates the topology on
$\overline{\sigma}$, so $\Phi$ is open. ∎

As a corollary we have the following.

######  Theorem B.

There is a strong deformation retraction from $X^{\beth}$ onto
${\mathbf{p}}(X^{\beth})$. If the colored fan ${\mathfrak{F}}(X)$ satisfies:

$\bigcup_{(\sigma,{\mathcal{F}})\in{\mathfrak{F}}(X)}\sigma\subseteq{\mathcal{V}}.$

Then ${\mathbf{p}}(X^{\beth})$ is homeomorphic to the the canonical
compactification of the fan
$F(X)=\\{\sigma\,|\,(\sigma,{\mathcal{F}})\in{\mathfrak{F}}(X)\\}$.

###### Proof.

That $X^{\beth}$ strong deformation retracts onto ${\mathbf{p}}(X^{\beth})$
space follows from Corollary 3.17. By Proposition 6.4 if $X$ satisfies $\star$
‣ 6.1 then ${\mathbf{p}}(X^{\beth})$ is homeomorphic to the canonical
compactification of the fan
$F(X)=\\{\sigma\,|\,(\sigma,{\mathcal{F}})\in{\mathfrak{F}}(X)\\}$. ∎

######  Remark 6.5.

Because the map ${\mathbf{p}}$ maps into the closure of ${\mathcal{V}}$ in
$X^{\beth}$ by construction, one will not have the above for a general
spherical variety. This is because a colored cone need not be contained in
${\mathcal{V}}$. However given a simple embedding $X$, one will have that
${\mathbf{p}}(X^{\beth})=\overline{\sigma\cap{\mathcal{V}}}$, where the
closure is taken in $\hom(\sigma^{\vee},[0,\infty])$. Consider for example the
$\text{GL}_{2}$-embedding ${\mathbf{p}}(X^{\beth})$ as in Figure 1, one can
see ${\mathbf{p}}(X^{\beth})$ depicted in Figure 2.

Figure 2: The canonical compactification of the cone $\sigma$ associated to
$X$, along with ${\mathbf{p}}(X^{\beth})$ depicted as the pink shaded region
to the right.

######  Remark 6.6.

Because of this difference between spherical varieties that satisfy $\star$ ‣
6.1 and those that do not, it may be interesting to describe
$\ast$-multiplication by ${\mathbf{b}}$, the unique point in the Shilov
boundary of $B^{\beth}$. All of the results described in Section 3 regarding
${\mathbf{g}}$ apply to ${\mathbf{b}}$, including the construction of the
homotopy from $X^{\beth}$ to the image of $\ast$-multiplication by
${\mathbf{b}}$.

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