#  Kähler-Ricci flow on $\mathbf{G}$-spherical Fano manifolds

Feng ${\rm Wang}^{\dagger}$ and Xiaohua ${\rm Zhu}^{*}$ SMS, Zhejiang
University, Hangzhou, 310028, China<EMAIL_ADDRESS>SMS, Peking
University, Beijing 100871, China<EMAIL_ADDRESS>
###### Abstract.

We prove that the Gromov-Hausdorff limit of Kähler-Ricci flow on a
$\mathbf{G}$-spherical Fano manifold $X$ is a $\mathbf{G}$-spherical
$\mathbb{Q}$-Fano variety $X_{\infty}$, which admits a (singular) Kähler-Ricci
soliton. Moreover, the $\mathbf{G}$-spherical variety structure of
$X_{\infty}$ can be constructed as a center of torus
$\mathbb{C}^{*}$-degeneration of $X$ induced by an element in the Lie algebra
of Cartan torus of $\mathbf{G}$.

###### Key words and phrases:

$\mathbf{G}$-spherical variety, Kähler-Ricci flow, equivariant
$\mathbb{C}^{*}$-degeneration

###### 2000 Mathematics Subject Classification:

Primary: 53E20, 53C25; Secondary: 32Q20, 53C30, 14L10

${\dagger}$ partially supported by the National Key Research and Development
Program of China No. 2022YFA1005501 and NSFC 12031017.

* partially supported by National Key R&D Program of China 2020YFA0712800 and NSFC 12271009.

###### Contents

  1. 1 Introduction
  2. 2  $\mathbf{G}-$equivariant $1$-PS degenerations
  3. 3 Equivariant Kähler-Ricci flow
    1. 3.1 Algebraic lemmas
    2. 3.2 Proof of Theorem 3.1
  4. 4 Torus $1$-PS degenerations on $\mathbf{G}$-spherical varieties
    1. 4.1 Combinatorial data associated to $\mathbf{G}$-spherical varieties
    2. 4.2 Torus $\mathbb{C}^{*}$-degenerations for $\mathbf{G}$-polarized spherical varieties
  5. 5 H-invariant on $\mathbf{G}$-spherical Fano varieties
    1. 5.1 Special torus $1$-PS degenerations
    2. 5.2 A formula of H-invariant
  6. 6 Proof of Theorem 1.1

##  1\. Introduction

Let $(X,J,\omega_{0})$ be a Fano manifold and $\omega_{t}$ a solution of
normalized Kähler-Ricci (KR) flow,

(1.1) $\displaystyle\frac{\partial\omega_{t}}{\partial t}=-{\rm
Ric}\,(\omega_{t})+\omega_{t},~{}\omega_{0}\in 2\pi c_{1}(M,J).$

It is known that (1.1) has a global solution $\omega_{t}$ for all $t\geq 0$
whenever the initial metric $\omega_{0}$ represents $2\pi c_{1}(M)$ in [9].
The famous Hamilton-Tian (HT) conjecture asserts [35]:

Any sequence of $(X,\omega_{t})$ contains a subsequence converging to a length
space $(X_{\infty},\omega_{\infty})$ in the Gromov-Hausdorff topology and
$(X_{\infty},\omega_{\infty})$ is a smooth KR soliton outside a closed subset
$S$, called the singular set, of codimension at least $4$. Moreover, this
subsequence of $(X,\omega_{t})$ converges locally to the regular part of
$(X_{\infty},\omega_{\infty})$ in the Cheeger-Gromov topology.

This conjecture was first proved by Tian and Zhang in dimension less than $4$
[38]. Moreover, they also proved that $X_{\infty}$ has a structure of
$\mathbb{Q}$-Fano variety. In higher dimensions, the conjecture was proved by
Bamler [4] and Chen-Wang [11], independently. In fact, Bamler proved a
generalized version of the HT conjecture for Ricci flow with bounded scalar
curvature. The authors [41] recently gave an alternative proof to the HT
conjecture by using the Tian’s partial $C^{0}$-estimate established by Zhang
[46] for evolved Kähler metrics of KR flow.

As an application of HT conjecture, Chen-Sun-Wang proved that the limit
$(X_{\infty},\omega_{\infty})$ is independent of choice of subsequence of
$\omega_{t}$ [10]. In fact, they proved that $X_{\infty}$ is a center of
$\mathbb{C}^{*}$-degeneration of another $\mathbb{Q}$-Fano variety
$\bar{X}_{\infty}$, which is a limit of $1$-PS degeneration of $X_{\infty}$
generated by the soliton vector field (VF) $v$ of $X_{\infty}$ (also see [18,
Theorem 3.4] and [42]). Here $v$ can be uniquely determined by the modified
Futaki invariant introduced in [36, 45, 6, 43]. More recently, by showing that
$\bar{X}_{\infty}$ can be regarded as a minimizer of H-invariant for a class
of filtrations associated to $X$ in algebraic geometry, Han-Li further proved
that the limit $X_{\infty}$ is also independent of choice of initial metric in
the class $2\pi c_{1}(X,J)$ [22]. The H-invariant was first introduced by
Tian-Zhang-Zhang-Zhu [39], and latterly generalized by Dervan-Székelyhidi [18]
for $\mathbb{C}^{*}$-degenerations of $X$.

A natural question is how to determine $X_{\infty}$ as well as
$\bar{X}_{\infty}$ effectively. This question have been recently investigated
for Fano group compactifications and horosymmetric Fano manifolds [27, 25,
37]. In particular, by analyzing the singularities for a class of real Monge-
Ampère equations, Tian-Zhu [37] proved that the limit $X_{\infty}$ of KR flow
on a $\mathbf{G}$-horosymmetric Fano manifold is a $\mathbf{G}$-horosymmetric
$\mathbb{Q}$-Fano variety, which can be realized as a center of
$\mathbb{C}^{*}$-degeneration induced by an element in the Lie algebra
$\mathfrak{t}$ of Cartan torus $\mathbf{T}$ of complex reductive Lie group
$\mathbf{G}$ associated to $X$.

The purpose of paper is to generalize the Tian-Zhu’s result above to
$\mathbf{G}$-spherical Fano manifolds. We prove

######  Theorem 1.1.

Let $\mathbf{G}$ be a complex reductive Lie group. Then the Gromov-Hausdorff
limit of KR flow (1.1) on a $\mathbf{G}-$spherical Fano manifold $(X,J)$ is a
$\mathbf{G}$-spherical $\mathbb{Q}$-Fano variety $X_{\infty}$, which admits a
(singular) KR soliton $\omega_{\infty}$. Moreover, the $\mathbf{G}$-spherical
variety structure of $X_{\infty}$ can be constructed as a center of torus
$\mathbb{C}^{*}$-degeneration of $X$ induced by an element $\tilde{\Lambda}$
in $\mathcal{V}_{\mathbb{R}}$. Here $\mathcal{V}_{R}$ is a valuation cone as a
subset of the quotient space of torus Lie algebra $\mathfrak{t}$.

######  Remark 1.2.

The element $\tilde{\Lambda}\in\mathcal{V}_{\mathbb{R}}$ in Theorem 1.1 is
constructed by a small perturbation $\tilde{v}$ of the soliton VF $v$ of
$(X_{\infty},\omega_{\infty})$ (cf. Lemma 6.2). $v$ can be uniquely determined
as a minimizer $\Lambda_{v}$ of convex functional on $\mathcal{V}_{R}$ which
associated to the H-invariant restricted to special torus $1$-PS degenerations
(cf. Proposition 5.7 and Proposition 5.8). In case of horosymmetric manifolds,
it was proved that $\tilde{\Lambda}$ can be chosen as $\Lambda_{v}$ if $v\neq
0$ [37]. The latter implies that both of $\mathbb{Q}$-Fano varieties
$X_{\infty}$ and $\bar{X}_{\infty}$ are same after a projective group
transformation. We guess that the result is also true for
$\mathbf{G}$-spherical manifolds.

By the torus $\mathbb{C}^{*}$-degeneration in Theorem 1.1, $X$ must be
biholomorphic to $X_{\infty}$ if $X$ is modified K-polystable. Thus we prove
the following Yau-Tian-Donaldson conjecture for $\mathbf{G}-$spherical Fano
manifolds via the method of KR flow immediately.

######  Corollary 1.3.

There exits a KR soliton on a $\mathbf{G}-$spherical Fano manifold if and only
it is modified K-polystable in terms of modified Futaki invariant.

The Yau-Tian-Donaldson conjecture for KR solitons has been proved for any Fano
manifolds by Datar-Székelyhidi [12] via the continuity method and by Chen-Sun-
Wang [10] via the KR flow method, respectively. Actually, the necessary part
comes from [6] (also see [13, 23, 7]). We would like to mention that in case
of horosymmetric manifolds the condition of modified K-polystabilty for the
existence part in Corollary 1.3 (in case of KR solitons, but not KE metrics)
can be weakened as the modified K-semistabilty, see Remark 1.2. Thus we guess
that this improvement is still true for $\mathbf{G}-$spherical Fano manifolds
(also see [37, Remark 6.1-1)]).

By Theorem 1.1, we see that the limit $X_{\infty}$ of KR flow (1.1) still
preserves the $\mathbf{G}-$spherical structure in algebraic geometry. On the
other hand, replacing by a class of parabolic complex Monge-Ampère equations
for weak plurisubharmonic functions (cf. [34, 5]), the KR flow can be defined
on any $\mathbb{Q}$-Fano variety with long time existence. We hope that
Theorem 1.1 can be generalized to the KR flow on a spherical $\mathbb{Q}$-Fano
variety. In other words, it is very possible to establish a theory that the
class of $\mathbf{G}-$spherical $\mathbb{Q}$-Fano varieties is a closed set in
the setting of KR flows. 111We are in debt to professor Gang Tian for pointing
this out to us.

The class of $\mathbf{G}$-spherical varieties is a very large one of highly
symmetric varieties, which contains toric varieties, generalized flag
manifolds, group compactifications and horosymmetric Fano manifolds (cf. [1,
14, 15]). In recent years, the existence problem of (singular) KE metrics and
(singular) KR solitons on such $\mathbb{Q}$-Fano varieties has been
extensively studied (cf. [14, 29, 15, 28, 17, 37, 30], etc.). The existence
criterion can be expressed in terms of the barycenter of moment polytope
associated to the Cartan torus $\mathbf{T}$ of $\mathbf{G}$.

At last, we outline the proof of Theorem 1.1. Because of lack of symmetry of
$\mathbf{G}$-spherical manifolds compared with horosymmetric manifolds, we
will use a different approach in [37] to prove Theorem 1.1. The basic idea is
to use the GIT method via the Tian’s partial $C^{0}$-estimate as in [10, 41,
42]. By the uniqueness of $X_{\infty}$ [22, 42], we need to prove the theorem
only for the KR flow with $\mathbf{K}$-invariant initial metrics in $2\pi
c_{1}(X,J)$, where $\mathbf{K}$ is a maximal compact subgroup of $\mathbf{G}$.
We will first establish a $\mathbf{G}_{0}$-equivariant version of Chen-Sun-
Wang’s result in [10] for the KR flow with $\mathbf{K}_{0}$-invariant initial
metrics on any Fano manifold $X$ (cf. Theorem 3.1), where $\mathbf{G}_{0}$ is
a reductive subgroup of automorphisms group ${\rm Aut}(X)$ of $X$ and
$\mathbf{K}_{0}$ is a maximal compact subgroup of $\mathbf{G}_{0}$. The key
point is to show that the $1$-PS degeneration of $X_{\infty}$ induced by the
soliton VF $v$ of $X_{\infty}$ can be approximated by a sequence of
$\mathbf{G}_{0}$-equivariant $\mathbb{C}^{*}$-degenerations (cf. Proposition
2.9, Lemma 3.5). Next, by using the spherical geometry for
$\mathbf{G}$-equivariant $\mathbb{C}^{*}$-degenerations studied by Delcroix
[15], we are able to show that $X_{\infty}$ can be realized as a center of
torus $\mathbb{C}^{*}$-degeneration of $X$ induced by an element in
$\mathcal{V}_{\mathbb{R}}$ (cf. Theorem 5.2 and Lemma 6.2).

The paper is organized as follows. In Section 2, we use the notion of $1$-PS
degeneration on a projective variety introduced in [19, 35] to prove a
$\mathbf{G}_{0}$-equivariant version of approximation result by a sequence of
$\mathbf{G}_{0}$-equivariant $\mathbb{C}^{*}$-degenerations. (cf. Proposition
2.9). Here notions of $1$-PS degeneration and $\mathbb{C}^{*}$-degenerations
are corresponding to ones of $\mathbb{R}$-test configuration and test
configuration introduced in [19] and [20] for the setting of schemes,
respectively. In Section 3, we use the technique of partial $C^{0}$-estimate
to apply the result in Section 2 to the Hilbert points associated to a family
of submanifolds of Kodaira embeddings. Theorem 3.1 is the main result in this
section. In next three sections, Section 4-6, we devote to prove Theorem 1.1
by using the spherical geometry for $\mathbf{G}$-equivariant
$\mathbb{C}^{*}$-degenerations as torus $\mathbb{C}^{*}$-degenerations
described by rational piecewise linear functions on $\mathfrak{t}$ in [15]
(cf. Proposition 4.10, 4.12, 5.1). Theorem 1.1 will be finally proved in
Section 6.

##  2\.  $\mathbf{G}-$equivariant $1$-PS degenerations

Let $(X,L)$ be a polarized normal variety in $\mathbb{C}P^{n}$ such that
$(\mathbb{C}P^{n},\mathcal{O}(1))|_{X}=L$. Let $\mathbf{G}_{0}$ be a reductive
subgroup of automorphisms group ${\rm Aut}(X)$ of $X$ and $\mathbf{K}_{0}$ a
maximal compact subgroup of $\mathbf{G}_{0}$. We recall the definition of
$\mathbb{C}^{*}$-degenerations for $(X,L)$ in [35].

######  Definition 2.1.

A normal variety $\mathcal{X}$ with a $\mathbb{C}^{*}$-action is called a
$\mathbb{C}^{*}$-degeneration of $(X,L)$ if it satisfies:

  1. (1)

there is a flat $\mathbb{C}^{*}$-equivarant map $\pi:\mathcal{X}\to\mathbb{C}$
such that $\pi^{-1}(t)$ is biholomorphic to $X$ for any $t\neq 0$;

  2. (2)

there is a holomorphic line bundle $\mathcal{L}$ on $\mathcal{X}$ such that
$\mathcal{L}|_{\pi^{-1}(t)}$ is isomorphic to $L^{r}$ for some integer $r>0$
and any $t\neq 0$.

We say that $(\mathcal{X},\mathcal{L})$ is $\mathbf{G}_{0}$-equivariant if it
admits an action of $\mathbf{G}_{0}$ which commutes with the
$\mathbb{C}^{*}$-action such that the isomorphism between $\left(X,L\right)$
and $(\pi^{-1}(1),~{}$ $\left.\mathcal{L}\right|_{\pi^{-1}(1)})$ is
$\mathbf{G}_{0}$-equivariant.

For any $\mathbb{C}^{*}$-degeneration $(\mathcal{X},\mathcal{L})$, we can
compactify it by gluing a trivial family to get a family over
$\mathbb{P}^{1}$. We denote the compactification by
$(\bar{\mathcal{X}},\bar{\mathcal{L}})$. Assuming that $L^{r}$ is very ample,
we can embed $M$ into the projective space $\mathbb{P}(H^{0}(X,L^{r})))$ by
sections of $H^{0}(X,L^{r})$. Then $(\bar{\mathcal{X}},\bar{\mathcal{L}})$ is
embedded into $\mathbb{P}(H^{0}(X,L^{r}))\times\mathbb{C}$ and there is a 1-PS
$\tau(s)$ in ${\rm GL}(H^{0}(X,L^{r}),\mathbb{C})$ such that
$\mathcal{X}_{0}=\lim_{s\rightarrow 0}\tau(s)X$ and $\mathcal{L}$ is the
pullback of $\mathcal{O}(1)$(cf. [33]). Here $\mathcal{X}_{0}=\pi^{-1}(0)$
denotes the center of $(\mathcal{X},\mathcal{L})$. It also holds for
$\mathbf{G}_{0}-$equivariant $\mathbb{C}^{*}$-degenerations such that
$\tau(s)$ commute with $\mathbf{G}_{0}$, where $\mathbf{G}_{0}$ can be lifted
as a subgroup of ${\rm GL}(H^{0}(X,L^{r})$. In fact, we have

######  Lemma 2.2.

For any 1-PS $\tau(s)$ $(s\in\mathbb{C}^{*})$ in ${\rm GL}(H^{0}(X,L^{r}))$
commuting with $\mathbf{G}_{0}$, there is a $G_{0}$-equivariant
$\mathbb{C}^{*}$-degeneration $(\mathcal{X},\mathcal{L})$ such that
$\mathcal{X}_{0}=\lim_{s\rightarrow 0}\tau(s)X$. Conversely, any
$\mathbf{G}_{0}$-equivariant $\mathbb{C}^{*}$-degeneration
$(\mathcal{X},\mathcal{L})$ arises this way if $r$ is large enough.

###### Proof.

Given a 1-PS $\tau(s)$ in ${\rm GL}(H^{0}(X,L^{r})$ commuting with
$\mathbf{G}_{0}$, let $\mathcal{X}$ be the closure of $\tau(s)\cdot X$ in
${\rm P}(H^{0}(X,L^{r}))\times\mathbb{C}$. Since $\tau(s)$ commutes with
$\mathbf{G}_{0}$, the action of $\mathbf{G}_{0}$ on ${\rm P}(H^{0}(X,L^{r}))$
preserves each $\tau(s)\cdot X$. Denote by $\mathcal{L}$ be the pull back of
the line bundle $\mathcal{O}(1)$ on ${\rm P}(H^{0}(X,L^{r}))$. Then
$(\mathcal{X},\mathcal{L})$ is a $\mathbb{C}^{*}$-degeneration of $(X,L)$.

Conversely, since $\pi$ is flat, $\pi_{*}\mathcal{L}$ is vector bundle $E$
over $\mathbb{C}$ when $r$ is large enough. This vector bundle is
$\mathbf{G}_{0}\times\mathbb{C}^{*}$ equivariantly isomorphic to
$H^{0}(X,L^{r})\times\mathbb{C}$. The $\mathbb{C}^{*}$ action on the fiber
$E_{0}\cong H^{0}(X,L^{r})$ gives a 1-PS $\tau(s)$ which commutes with
$\mathbf{G}_{0}$. The action of $\tau(s)$ is the original $\mathbb{C}^{*}$
action on $(\mathcal{X},\mathcal{L})$. Thus we have proved the 1-1
correspondence between $\mathbf{G}_{0}$-equivariant
$\mathbb{C}^{*}$-degenerations and 1-PS’s commuting with $\mathbf{G}_{0}$. ∎

In this paper, we will always assume that $X$ is a $\mathbb{Q}$-Fano variety
and $L=K^{-r_{0}}$ for some integer $r_{0}>0$. If the center $\mathcal{X}_{0}$
of $\mathbb{C}^{*}$-degeneration $(\mathcal{X},\mathcal{L})$ is a
$\mathbb{Q}$-Fano variety, we say that $(\mathcal{X},\mathcal{L})$ is special
[35].

Let $\tau(s)$ be the $1$-PS in ${\rm GL}(H^{0}(X,K_{X}^{-rr_{0}}),\mathbb{C})$
associated to $(X,rK^{-r_{0}}_{X})$ in Lemma 2.2 such that

$\mathcal{X}_{0}=\lim_{s\rightarrow 0}\tau(s)X$

is a $\mathbb{Q}$-Fano variety. Then $\tau(s)$ induces a holomorphic VF on
$\mathcal{X}_{0}$. The canonical lifting of $\tau(s)$ is another 1-PS in ${\rm
GL}(H^{0}(M,K_{X}^{-rr_{0}}),\mathbb{C})$, denoted by $\tilde{\tau}(s)$. It is
easy to see that $\tau(s)=\tilde{\tau}(s)s^{c}$ for some integer $c$.
Moreover, we have

######  Lemma 2.3.

$\bar{\mathcal{L}}=K^{-1}_{\mathcal{X}/{\mathbb{P}}^{1}}+c\mathcal{X}_{0}.$

We can write a 1-PS $\tau(s)(s\in\mathbb{C}^{*})$ as

$\displaystyle\tau(s)=\begin{bmatrix}s^{k_{1}}&0&...&0\\\ 0&s^{k_{2}}&...&0\\\
...&...&...&...\\\ 0&0&...&s^{k_{n}}\end{bmatrix}$

where $k_{i}(1\leq i\leq n)$ are integers. Let $t=-\log|s|$. This matrix is
equal to

$\displaystyle\tau(s)=\begin{bmatrix}e^{-tk_{1}}&0&...&0\\\
0&e^{-tk_{2}}&...&0\\\ ...&...&...&...\\\ 0&0&...&e^{-tk_{n}}\end{bmatrix}.$

Now we consider general real $1$-PS $\sigma(t)$ in ${\rm
GL}(H^{0}(X,L^{r}),\mathbb{R})$, where

(2.1) $\displaystyle\sigma(t)=\begin{bmatrix}e^{-\lambda_{1}t}&0&...&0\\\
0&e^{-\lambda_{2}t}&...&0\\\ ...&...&...&...\\\
0&0&...&e^{-\lambda_{n}t}\end{bmatrix}$

with respect to some basis of $H^{0}(X,L^{r})$ and $\lambda_{i}\in\mathbb{R}$
are the eigenvalues of $\sigma(t)$.

######  Definition 2.4.

We call the pair $(X,\sigma(t))$ a 1-PS degeneration of $(X,L)$. For
simplicity, we denote it by $\mathcal{X}^{\sigma}$ and the limit

$\mathcal{X}^{\sigma}_{0}:=\lim_{t\rightarrow\infty}\sigma(t)X$

is called the central fiber of $\mathcal{X}^{\sigma}$. If the algebraic group
$\mathbf{G}_{0}$ acts on $(X,L)$ and $\sigma(t)$ commutes with the induced
action of $\mathbf{G}_{0}$ on $H^{0}(X,L^{r})$, then we say that
$\mathcal{X}^{\sigma}$ is a $\mathbf{G}_{0}$-equivariant 1-PS degeneration.

######  Remark 2.5.

Given any holomorphic VF $v$ on ${\rm P}(H^{0}(X,L^{r}),\mathbb{C})$, $v$
corresponds to an element in ${\rm gl}(H^{0}(X,L^{r}),\mathbb{C})$. Suppose
that $e^{t{\rm Im}(v)}$ generalizes a compact 1-PS of ${\rm
GL}(H^{0}(X,L^{r}),\mathbb{C})$. Then $(X,e^{t{\rm Real}(v)})$ is a 1-PS
degeneration of $(X,L)$. This induced $1$-PS degeneration was first considered
by Ding-Tian in [19] when the limit of $e^{tv}X$ is a normal variety.

######  Remark 2.6.

If a holomorphic VF $v$ is diagonalized as
$v=\sum_{i=1}^{n}a_{i}{z_{i}}\frac{\partial}{\partial z_{i}}$, then $e^{t{\rm
Im}(v)}$ generalizes a compact subgroup if and only if $a_{i}$ are real
numbers. So we denote $e^{t{\rm Real}(v)}$ by $e^{tv}$ without confusion in
the sequel.

The limit in Definition 2.4 is the flat limit which can be defined as follows.
Denote the ideal defining $X$ in ${\rm P}(H^{0}(X,L^{r}),\mathbb{C})$ by $I$.
For $f=\sum_{\alpha}c_{\alpha}x^{\alpha}$, put

$\langle\alpha,\lambda\rangle=\sum\alpha_{i}\lambda_{i},\nu(f)=\min_{c_{\alpha}\neq
0}\langle\alpha,\lambda\rangle$

and $Lt(f)=\sum_{\langle\alpha,\lambda\rangle=\nu(f)}c_{\alpha}x^{\alpha}$.
Then $I_{0}=\\{Lt(f)|f\in I\\}$ is the defining ideal of
$\mathcal{X}^{\sigma}_{0}$. It’s same as
$\lim_{t\rightarrow\infty}\sigma(t)\cdot[X]=[\mathcal{X}^{\sigma}_{0}]$, where
$[\cdot]$ is the Hilbert point of a closed subscheme. A $1$-PS degeneration
can also be defined using filtrations as in [12, 22].

Given $n$ real numbers $\lambda_{1}\leq\lambda_{2}\leq...\leq\lambda_{n}$, we
say it’s of type $(e_{1},e_{2},...,e_{k})$ if
$\lambda_{1}=\lambda_{2}=...=\lambda_{e_{1}}<\lambda_{e_{1}+1}=\lambda_{e_{1}+2}=...=\lambda_{e_{1}+e_{2}}<...$.

######  Lemma 2.7.

Let

$\sigma(t)=\begin{bmatrix}e^{-\lambda_{1}t}&0&...&0\\\
0&e^{-\lambda_{2}t}&...&0\\\ ...&...&...&...\\\
0&0&...&e^{-\lambda_{n}t}\end{bmatrix}$

and

$\tau(t)=\begin{bmatrix}e^{-\lambda^{\prime}_{1}t}&0&...&0\\\
0&e^{-\lambda^{\prime}_{2}t}&...&0\\\ ...&...&...&...\\\
0&0&...&e^{-\lambda^{\prime}_{n}t}\end{bmatrix}$

be two real $1$-PS in $\mathbb{C}P^{n-1}$. If
$\lambda_{1}\leq\lambda_{2}\leq...\leq\lambda_{n}$ and
$\lambda^{\prime}_{1}\leq\lambda^{\prime}_{2}\leq...\leq\lambda^{\prime}_{n}$
are of the same type, then for any
$x=[x_{1},x_{2},...,x_{n}]\in\mathbb{C}P^{n-1}$,

$\lim_{t\rightarrow\infty}\sigma(t)(x)=\lim_{t\rightarrow\infty}\tau(t)(x).$

###### Proof.

Let $i_{0}$ be the smallest index such that $x_{i}\neq 0$ and
$e_{1}+...+e_{s}<i_{0}\leq e_{1}+...+e_{s+1}$. Then

$\lim_{t\rightarrow\infty}\sigma(t)(x)=[0,...,0,x_{i_{0}},x_{i_{0}+1},...,x_{e_{1}+...+e_{s+1}},0,...,0]=\lim_{t\rightarrow\infty}\tau(t)(x).$

∎

Applying the above lemma to the Hilbert point $[X]$ of $X$, we have

######  Lemma 2.8.

Let

$\sigma(t)=\begin{bmatrix}e^{-\lambda_{1}t}&0&...&0\\\
0&e^{-\lambda_{2}t}&...&0\\\ ...&...&...&...\\\
0&0&...&e^{-\lambda_{n}t}\end{bmatrix}$

be a real $1$-PS as in (2.1). Then for any positive number $\epsilon$, there
exists a sequence of $\lambda^{\prime}_{j}\in\mathbb{Q}$ such that
$|\lambda^{\prime}_{j}-\lambda_{j}|\leq\epsilon\,(1\leq j\leq n)$ and

$\lim_{t\rightarrow\infty}\sigma(t)\cdot[X]=\lim_{t\rightarrow\infty}\tau(t)\cdot[X],$

where $\tau(t)$ is the real $1$-PS as in (2.1) with eigenvalues
$\lambda^{\prime}_{i}$.

###### Proof.

At first, we recall the construction of the Hilbert point. For
$X\subseteq\mathbb{C}P^{n-1}$, there is an integer number $m$ depending on the
Hilbert polynomial $p$ of $X$ such that the defining ideal $I$ of $X$ is
generated by homogeneous polynomials in $I$ of degree $m$. Denote the latter
by $I_{m}$ and the whole space of homogeneous polynomials of degree $m$ by
$V$. Then $\dim V=\binom{n+m-1}{n-1}$ and $\dim
I_{m}=\binom{n+m-1}{n-1}-p(m):=k$ by the definition of Hilbert polynomial. So
$I_{m}$ corresponds to a point in the Grassmann manifold $G_{r}(k,V)$. We can
embed the Grassmann into $\mathbb{P}(\Lambda^{k}V)$ using Plücker coordinates
and the corresponding point of $I_{m}$ is the Hilbert point of $X$.

More precisely let $I_{m}=span\\{h_{1},h_{2},...,h_{k}\\}$, then $[X]$ is the
point $[h_{1}\wedge h_{2}\wedge...\wedge h_{k}]\in\mathbb{P}(\Lambda^{k}V)$.
For the standard base
$x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}...x_{n}^{\alpha_{n}}(\sum_{i=1}^{n}\alpha_{i}=m)$
of $V$, the weight of induced action for $\sigma$ is
$\langle\lambda,\alpha\rangle:=\sum_{i=1}^{n}\lambda_{i}\alpha_{i}$. So the
weights on $\Lambda^{k}V$ are given by $\sum_{j\in
I}\langle\lambda,\alpha^{(j)}\rangle$ for $|I|=k$ and $k$ different
$\alpha^{(j)}$. It follows that the type $(f_{1},f_{2},...)$ of the induced
action of $\sigma$ is determined by

(2.2) $\displaystyle\sum_{j\in
I_{1}}\langle\lambda,\alpha^{(j)}\rangle=\sum_{j\in
I_{2}}\langle\lambda,\alpha^{(j)}\rangle=...=\sum_{j\in
I_{f_{1}}}\langle\lambda,\alpha^{(j)}\rangle<\sum_{j\in
I_{f_{1}+1}}\langle\lambda,\alpha^{(j)}\rangle...$

All the equations in (2.2) are of integer coefficients. So we can perturb
$\lambda_{i}$ to rational points $\lambda_{i}^{\prime}$ preserving these
equations and inequalities, i.e., the type of the eigenvalues of the induced
actions of $\sigma(t)$ and $\tau(t)$ are the same. Thus the lemma is proved by
Lemma 2.7. ∎

For the equivariant case, we have

######  Proposition 2.9.

Assume that there is an action of reductive group $\mathbf{G}_{0}$ on $(X,L)$.
Let $\sigma(t)$ be a $\mathbf{G}_{0}$-equivariant real 1-PS. Then there is a
sequence of $\mathbf{G}_{0}$-equivariant $1$-PS degenerations $\tau_{i}(t)$
with rational eigenvalues which converges to the eigenvalues of $\sigma(t)$
such that $\mathcal{X}_{0}^{\tau_{i}}=\mathcal{X}_{0}^{\sigma}$. As a
consequence, there is a $\mathbf{G}_{0}$-equivariant
$\mathbb{C}^{*}$-degeneration $\tau_{i}(t)$ whose central fiber is same as
$\mathcal{X}_{0}^{\sigma}$.

###### Proof.

$\sigma(t)$ commutes with $G_{0}$ if and only if the eigenspace of $\sigma(1)$
are all $\mathbf{G}_{0}$-invariant. In the proof of Lemma 2.8, we can also
require that $\lambda^{\prime}$ and $\lambda$ are of the same type by adding
the corresponding equations and inequalities to (2.2). Then the eigenspaces of
$\lambda^{\prime}$ and $\lambda$ are the same. Consequently the real 1-PS
$\tau(t)$ corresponding to $\lambda^{\prime}$ also commutes with
$\mathbf{G}_{0}$. Since $\lambda^{\prime}$ can be close to $\lambda$ as small
as we want, we get a sequence of $\tau_{i}(t)$ with rational eigenvalues which
converges to the eigenvalues of $\sigma(t)$. For any $\tau_{i}$, there is a
sufficiently divisible integer $N$ such that $\tau_{i}(Nt)$ defines a
$\mathbf{G}_{0}$-equivariant $\mathbb{C}^{*}$-degeneration
$(\mathcal{X},\mathcal{L})$ by Lemma 2.2. Moreover, by Lemma 2.7, the central
fiber of $(\mathcal{X},\mathcal{L})$ is just $\mathcal{X}_{0}^{\sigma}$. The
proposition is proved. ∎

Let $X$ be a $\mathbb{Q}-$Fano variety. For a $1$-PS degeneration
$\mathcal{X}^{\sigma}$ such that $\mathcal{X}^{\sigma}_{0}$ is
$\mathbb{Q}$-Fano, $\sigma(t)$ induces a holomorphic VF $\eta(\sigma)$ on
$\mathcal{X}^{\sigma}_{0}$. Then by the canonical lifting of $\eta(\sigma)$ on
$K_{\mathcal{X}^{\sigma}_{0}}^{-1}$, we get another real 1-PS
$\tilde{\sigma}(t)$. By Lemma 2.3, $\tilde{\sigma}(t)$ is equal to $\sigma(t)$
up to a constant . We say that $\tilde{\sigma}(t)$ is the normalized real 1-PS
corresponding to $\sigma(t)$. We end this section by the following definition.

######  Definition 2.10.

Let $X$ be a $\mathbb{Q}$-Fano variety. A $1$-PS degeneration
$\mathcal{X}^{\sigma}$ of $X$ is called special if $\mathcal{X}^{\sigma}_{0}$
is $\mathbb{Q}$-Fano and $\sigma(t)=\tilde{\sigma}(t)$.

##  3\. Equivariant Kähler-Ricci flow

In this section, we prove an equivariant version of Chen-Sun-Wang’s result
(see an explicit statement in [18, Theorem 3.4]) for the KR flow in [10]. Let
$(X_{\infty},\omega_{\infty})$ be the limit of KR flow (1.1) in the HT
conjecture and $\mathbf{G}_{0}$ the reductive subgroup of automorphisms group
${\rm Aut}(X)$ as in Section 2. We need to prove

######  Theorem 3.1.

$X_{\infty}$ is a center of special
$\mathbf{G_{0}}\times\mathbb{C}^{*}$-equivariant $\mathbb{C}^{*}$-degeneration
of another $\mathbb{Q}$-Fano variety $\bar{X}_{\infty}$ which is a center of
special $\mathbf{G_{0}}$-equivariant $1$-PS degeneration of $X_{\infty}$
induced by the soliton VF $v$ of $(X_{\infty},\omega_{\infty})$.

###  3.1. Algebraic lemmas

As in [10], we use the GIT method to prove Theorem 3.1. Let $E$ be a complex
linear space and $\bar{G}={\rm GL}(E)$. Let $V$ be a representation space of
$\bar{G}$. For a fixed compact subgroup $\mathbf{K}$ of U(N), we define a
subgroup of $\bar{G}$ by

$\bar{G}_{\mathbf{K}}=\\{g\in\bar{G}|~{}g\text{ commutes with }\mathbf{K}\\}.$

Let $A_{i}~{}(i=0,1,...)$ be a sequence of matrices in $\bar{G}_{K}$ and
$\Lambda$ a Hermitian matrix commuting with $\mathbf{K}$. We assume that

(3.1) $\displaystyle\lim_{i\rightarrow\infty}A_{i}A_{i-1}^{-1}=e^{\Lambda}.$

Denote the eigenvalues of $\Lambda$ by
$\mathcal{S}=\\{d_{1}>d_{2}>d_{3}...>d_{k-1}>d_{k}\\}$ and the eigenspaces
corresponding to $d_{j}$ by $U_{j}$. For $v\in V\setminus\\{0\\}$, $[v]$ is
the corresponding point in the projective space $\mathbb{P}(V)$. Denote the
set of limit points of $A_{i_{k}}\cdot[v]$ for any subsequence of
$i_{k}\to\infty$ by ${\rm Lim}_{A_{i}}[v]$. We will give a characterization of
this set.

We fix a U(N)-invariant metric $|\cdot|$ on $V$. Set

$d(v)=\lim_{i\rightarrow\infty}(\log|A_{i+1}v|-\log|A_{i}v|).$

The following lemma is proved in [10].

######  Lemma 3.2.

Assume that (3.1) holds. Then for $v\in V\setminus\\{0\\}$,

$d(v)=\lim_{i\rightarrow\infty}(\log|A_{i+1}v|-\log|A_{i}v|)$

exists and belongs to $\mathcal{S}$. Moreover for any $[w]\in{\rm
Lim}_{A_{i}}[v]$, $w$ belongs to $U_{d(v)}$.

Denote $V_{j}=\\{v\in V|~{}d(v)\leq d_{j}\\}$. It is obvious that
$V_{i},U_{j}$ are $\mathbf{K}$-invariant subspaces. Denote
$\bar{G}_{\mathbf{K},\Lambda}=\\{g\in\bar{G}_{\mathbf{K}}|~{}g\cdot
e^{s\Lambda}=e^{s\Lambda}\cdot g\\}.$

######  Lemma 3.3.

Assume that $V=S^{m}(E)$ for some $m\geq 1$. Then there exists
$C\in\bar{G}_{\mathbf{K}}$, which is independent of $m$ such that $C\cdot
V_{j}=\oplus_{i=j}^{k}U_{i}$ for all $i$. There also exists
$C_{i}\in\bar{G}_{\mathbf{K}}$ with $\lim_{i\rightarrow\infty}C_{i}=Id$ such
that $\tilde{A}_{i}=C_{i}A_{i}C^{-1}\in\bar{G}_{\mathbf{K},\Lambda}$.

###### Proof.

For two $\mathbf{K}-$invariant subspaces with same dimension, we can always
choose the isomorphism between them to be $\mathbf{K}-$invariant. Then the
lemma can be proved as in [10, p3153]. ∎

######  Proposition 3.4.

Assume that $V=S^{m}(E)$ for some $m\geq 1$. Define
$[\bar{v}]=\lim_{i\rightarrow\infty}e^{i\Lambda}C\cdot[v]$. Then ${\rm
Lim}_{A_{i}}[v]\subset\overline{\bar{G}_{\mathbf{K},\Lambda}\cdot[\bar{v}]}$.

###### Proof.

Assuming that $d(v)=d_{j}$, we have $v\in V_{j}$. So $C\cdot
v\in\oplus_{i=j}^{k}U_{i}$, and the $U_{j}$ component $\pi_{j}(C\cdot v)$ of
$C\cdot v$ is not zero. It follows that $[\bar{v}]=[\pi_{j}(C\cdot v)]$. For
$[w]\in{\rm Lim}_{A_{i}}[v]$, by Lemma 3.2, we know that $w\in U_{j}$. Assume
that $[w]=\lim_{i\rightarrow\infty}[A_{\alpha_{i}}v]$, then we have

$\displaystyle\lim_{i\rightarrow\infty}\tilde{A}_{\alpha_{i}}[\bar{v}]$
$\displaystyle=\lim_{i\rightarrow\infty}\tilde{A}_{\alpha_{i}}[\pi_{j}(C\cdot
v)]=\pi_{j}(\lim_{i\rightarrow\infty}\tilde{A}_{\alpha_{i}}[(C\cdot v)])$
$\displaystyle=\pi_{j}(\lim_{i\rightarrow\infty}A_{\alpha_{i}}[v])=\pi_{j}[w]=[w].$

The proposition is proved since
$\tilde{A}_{i}\in\bar{G}_{\mathbf{K},\Lambda}$. ∎

###  3.2. Proof of Theorem 3.1

By the uniqueness of $X_{\infty}$ [22, 42], we may assume that the initial
metric $\omega_{0}$ in (1.1) is a $\mathbf{K}_{0}$-invariant metric. Then all
solutions of $\omega_{t}$ are also $\mathbf{K}_{0}$-invariant. Choose the
multilple $l$ large enough such that the partial $C^{0}$-estimate holds for
$(M,\omega_{t})$ by [46, 41]. Let $h_{t}$ be the $\mathbf{K}_{0}$-invariant
Hermitian metric on $K_{X}^{-1}$ such that the Chern curvature of $h_{t}$ is
$\omega_{t}$. It induces a $K-$invariant metric $H_{t}$ on
$E=H^{0}(X,K_{X}^{-l})$ by

(3.2) $\displaystyle(s^{\alpha},s^{\beta})_{H_{t}}=\int_{M}\langle
s^{\alpha},s^{\beta}\rangle_{h_{t}^{\otimes l}}\omega_{t}^{n}.$

Let $\\{s^{\alpha}(t)\\}$ be an ortho-normal basis of $(E,H_{t})$ and
$\Phi_{t}:M\to\mathbb{C}P^{N}$ be the Kodaira embedding given by
$\\{s^{\alpha}(t)\\}$. By Proposition 2.7 in [41], $\Phi_{t_{i}}(X)$ converges
to a $\mathbb{Q}$-Fano variety $X_{\infty}$ with klt singularities for a
subsequence $t_{i}\rightarrow\infty$. Moreover by Theorem 1.1 in [41],
$X_{\infty}$ admits a weak Kähler-Ricci soliton.

We can make these embeddings equivariant with respect to $\mathbf{K}_{0}$ by
using suitable ortho-normal basis in the following. Choose an ortho-normal
basis $s^{\alpha}(0)$ with respect to $H_{0}$. Then with respect to
$\\{s^{\alpha}(0)\\}$ we can write $H_{t}$ as a matrix. By abuse of notations,
we also denote it by $H_{t}$.

Consider the following ODE for matrix $A_{t}$:

(3.3) $\displaystyle\left\\{\begin{aligned}
\frac{d}{dt}A_{t}&=-\frac{1}{2}A_{t}H_{t}^{\prime}H_{t}^{-1},\\\
A_{0}&=id.\end{aligned}\right.$

It follows that $\frac{d}{dt}(A_{t}H_{t}A_{t}^{*})=0$ and
$A_{t}H_{t}A_{t}^{*}=id$. Define
$s^{\alpha}(t)=\sum_{\beta=0}^{N}A^{\alpha\beta}_{t}s^{\beta}(0)$. Then one
can check that $\\{s^{\alpha}(t)\\}$ is an ortho-normal basis of $E$ with
metric $H_{t}$. Denote by $\Phi_{t}$ the embeddings of $X$ given by
$\\{s^{\alpha}(t)\\}$ as above. Thus we have $\Phi_{t}=A_{t}\cdot\Phi_{0}$.

By (3.2), the inner $(\cdot,\cdot)_{H_{0}}$ on the basis $\\{s^{\alpha}(0)\\}$
induces $\mathbf{K}_{0}$ as a subgroup of U(N+1). Then $\mathbf{G}_{0}$ can be
regarded as a subgroup of GL(N+1, $\mathbb{C}$).

######  Lemma 3.5.

$A_{t}$ commutes with $\mathbf{K}_{0}$ and so it commutes with
$\mathbf{G}_{0}$ .

###### Proof.

Since $H_{t}(g\cdot,g\cdot)=H_{t}$ for $g\in\mathbf{K}_{0}$, we have
$gH_{t}g^{*}=H_{t}$ which is same as $H_{t}g=gH_{t}$ since $g$ is unitary. So
$H_{t}^{\prime}$ also commutes with $\mathbf{K}_{0}$. Now by (3.3) we have

$\frac{d}{dt}[A_{t},g]=-\frac{1}{2}[A_{t},g]H_{t}^{\prime}H^{-1}_{t},$

where $[A,B]=AB-BA$. Hence $[A_{t},g]=0$. By the reductivity of
$\mathbf{G}_{0}$, we also get $[A_{t},g]=0$ for any $g\in\mathbf{G}_{0}$. The
lemma is proved. ∎

###### Completion of proof of Theorem 3.1.

As we know that the ortho-normal basis $\\{s^{\alpha}(0)\\}$ induces the
action of $\mathbf{G}_{0}$ as a subgroup of GL(N+1, $\mathbb{C}$), i.e.,
$\Phi_{0}$ commutes with $\mathbf{G}_{0}$. By the above lemma, $\Phi_{t}$ also
commutes with $\mathbf{G}_{0}$. On the other hand, by Lemma 6.5 in [42],
$A_{t_{i}^{\prime}+s}\cdot A_{t_{i}^{\prime}}^{-1}$ converges to $e^{sv}$
uniformly for any subsequence $t_{i}^{\prime}\rightarrow\infty$, where $v$ is
the soliton vector VF on the limit of Kähler-Ricci flow. Since $v$ is the
limit of $A_{t_{i}^{\prime}+s}\cdot A_{t_{i}^{\prime}}^{-1}$, it commutes with
$\mathbf{K}_{0}$. By Corollary 3.3 in [42], $v$ is unique up to conjugation.
We may assume $v\in gl(N+1,\mathbb{C})$ is diagonal. By [42, Proposition 7.4],
${\rm Im}(v)$ lies in the compact part of $Aut^{v}(X_{\infty})$, where
$X_{\infty}$ is the limit of Kähler Ricci flow. It follows that the
eigenvalues of $v$ are all real numbers. So we can modify $A_{i}$ by unitary
elements commuting with $\mathbf{K}_{0}$ such that the condition (3.1) holds
with $\Lambda=v$. Denote the Hilbert polynomial of $\Phi_{0}(X)$ by $p$ and
the Hilbert scheme of subschemes of $\mathbb{P}(E)$ with Hilbert polynomial
$p$ by $\mathbf{Hilb}^{\mathbb{P}(E),p}$. Then we can apply Proposition 3.4 to
the Hilbert point $[X]=[\Phi_{0}(X)]\in\mathbf{Hilb}^{\mathbb{P}(E),p}$ as in
[42]. Denote $[\bar{X}_{\infty}]=\lim_{t\rightarrow\infty}e^{tv}C\cdot[X]$.
Then by Propsition 3.4 $[X_{\infty}]$ lies in the closure of
$G_{\mathbf{K_{0}},v}\cdot[\bar{X}_{\infty}]$.

Note that both of $C$ and $e^{tv}$ commute with $\mathbf{K}_{0}$. Thus

$\lim_{t\rightarrow\infty}e^{tv}C\cdot X$

is the center of $\mathbf{G}_{0}$-equivariant $1$-PS $\sigma(t)=e^{tv}$
degeneration of $(X,K_{X}^{-1})$. Clearly, the Hilbert point of this limit is
just $[\bar{X}_{\infty}]$. Hence, we may denote the limit by
$\bar{X}_{\infty}$. On the other hand, by [42, Proposition 7.4], ${\rm
Aut}^{v}(X_{\infty})$ is reductive, so $G_{\mathbf{K}_{0},v}\bigcap{\rm
Aut}^{v}(X_{\infty})$ is also a reductive group. So by Luna’s slice lemma,
there is a $1$-PS $\tau(s)$ $(s\in\mathbf{C}^{*})$ in
$G_{\mathbf{K}_{0},v}\bigcap{\rm Aut}^{v}(X_{\infty})$ degenerating
$\bar{X}_{\infty}$ to $X_{\infty}$. Since all the elements in
$G_{\mathbf{K}_{0},v}$ commute with $\mathbf{G}_{0}\times\mathbb{C}^{*}$, by
Lemma 2.2, $\tau(s)$ induces a
$\mathbf{G}_{0}\times\mathbb{C}^{*}$-equivariant $\mathbb{C}^{*}$-degeneration
of $(\bar{X}_{\infty},K_{X_{\infty}}^{-l})$ with the center $X_{\infty}$. The
theorem is proved. ∎

##  4\. Torus $1$-PS degenerations on $\mathbf{G}$-spherical varieties

In this section, we will recall Delcroix’s work [16] on torus
$\mathbb{C}^{*}$-degenerations of spherical varieties and then we generalize
his results for torus $1$-PS degenerations.

###  4.1. Combinatorial data associated to $\mathbf{G}$-spherical varieties

We list the general facts about spherical varieties and give some basic
results of combinatorial data. We refer the reader to the main references [24,
40] in this subsection. Let $\mathbf{G}$ be a complex connected reductive
group and $\mathbf{B}\subseteq\mathbf{G}$ be a Borel subgroup.

######  Definition 4.1.

The homogeneous variety $\mathbf{G}/\mathbf{H}$ is said spherical if there is
an open $\mathbf{B}$-orbit $\mathbf{B}\mathbf{H}$ and the corresponding orbit
map $\pi:\mathbf{G}\rightarrow\mathbf{G}/\mathbf{H}$ is separable. The
$G$-action on $G/H$ is on the left: $\sigma\cdot g\mathbf{H}=(\sigma
g)\mathbf{H}$. We denote by $\mathbf{P}\subseteq\mathbf{G}$ the stabilizer of
the open $\mathbf{B}$-orbit $\mathbf{B}\mathbf{H}$.

An embedding of $\mathbf{G}/\mathbf{H}$ is a normal $\mathbf{G}$-variety $X$
together with a $\mathbf{G}$-equivariant open embedding
$\mathbf{G}/\mathbf{H}\hookrightarrow X$. Let $x_{0}$ be the image of
$\mathbf{H}\in\mathbf{G}/\mathbf{H}$. We say $(X,x_{0})$ or $X$ is a spherical
variety.

The group of characters of any group $\mathbf{S}$ is denoted by
$\mathfrak{X}(\mathbf{S})$:

$\mathfrak{X}(\mathbf{S})=\\{f:S\rightarrow\mathbb{C}^{*}\text{ is a
homomorphism}\\}.$

Note that for a Borel subgroup $\mathbf{B}$ and the maximal subtorus
$\mathbf{T}$ contained in $\mathbf{B}$,
$\mathfrak{X}(\mathbf{T})=\mathfrak{X}(\mathbf{B})$. Let
$\Phi\subset\mathfrak{X}(\mathbf{T})$ be the root system of
$(\mathbf{G},\mathbf{T})$ and denote by $\Phi^{+}$ the positive roots in
$\Phi$ determined by $\mathbf{B}$.

The parabolic subgroup $\mathbf{P}$ of $\mathbf{G}$ containing $\mathbf{B}$
admits a unique Levi subgroup $\mathbf{L}$ containing $\mathbf{T}$. Let
$\Phi_{\mathbf{P}}$ denote the set of roots of $\mathbf{P}$ with respect to
$\mathbf{T}$. We denote by $\Phi_{\mathbf{P}^{u}}$ the set of roots of the
unipotent radical $\mathbf{P}^{u}$ of $\mathbf{P}$. Define

$\kappa_{\mathbf{P}}=\sum_{\alpha\in\Phi_{\mathbf{P}^{u}}}\alpha.$

For a normal variety $X$, denote by $\mathbb{C}(X)$ the rational functions on
$X$. A valuation of a normal variety $X$ is a map

$v:\mathbb{C}(X)^{*}=\mathbb{C}(X)\backslash\\{0\\}\longrightarrow\mathbb{Q}$

with the properties

$\displaystyle
v\left(f_{1}+f_{2}\right)\geq\min\left\\{v\left(f_{1}\right),v\left(f_{2}\right)\right\\}\text{
whenever }f_{1},f_{2},f_{1}+f_{2}\in\mathbb{C}(X)^{*},$ $\displaystyle
v\left(f_{1}f_{2}\right)=v\left(f_{1}\right)+v\left(f_{2}\right)\text{ for all
}f_{1},f_{2}\in\mathbb{C}(X)^{*}\text{ and }$ $\displaystyle
v\left(\mathbb{C}^{*}\right)=0.$

Define

$\displaystyle(\mathfrak{M}:=)\mathfrak{M}(\mathbf{G}/\mathbf{H})=$
$\displaystyle\\{\chi\in\mathfrak{X}(\mathbf{B})|\mathbb{C}(\mathbf{G}/\mathbf{H})^{(\mathbf{B})}_{\chi}\not=\\{0\\}\\},$
$\displaystyle(\mathfrak{N}:=)\mathfrak{N}(\mathbf{G}/\mathbf{H})=$
$\displaystyle{\rm Hom}(\mathfrak{M}(\mathbf{G}/\mathbf{H})),\mathbb{Z}).$

Then $\mathfrak{M}$ is a subgroup of
$\mathfrak{X}(\mathbf{B})=\mathfrak{X}(\mathbf{T})$ and its rank is called the
rank of homogeneous space $\mathbf{G}/\mathbf{H})$.

For any valuations $v$ on $\mathbb{C}(\mathbf{G}/\mathbf{H})$, define
$\rho(v)=\rho_{v}\in\mathfrak{N}\otimes\mathbb{Q}$ by
$\rho_{v}(\chi)=v(f_{\chi})$. Define
$(\mathcal{V}:=)\mathcal{V}(\mathbf{G}/\mathbf{H})=\\{\mathbf{G}\text{
invariant valuations}\\}.$ Then it is known that
$\rho:\mathcal{V}\rightarrow\mathfrak{N}_{\mathbb{Q}}$ is injective. So we can
identify $\mathcal{V}(\mathbf{G}/\mathbf{H})$ with its image and call it the
valuation cone. The valuation cone $\mathcal{V}$ spans
$\mathfrak{N}_{\mathbb{Q}}$. Moreover
$\mathcal{V}\subseteq\mathfrak{N}_{\mathbb{Q}}$ is a cosimplicial cone (cf.
[40, Section 20]). Hence there is a uniquely determined linearly independent
set $\Sigma\subseteq\mathfrak{M}$ of primitive elements such that

$\mathcal{V}=\bigcap_{\gamma\in\Sigma}\left\\{v\in\mathfrak{N}\otimes\mathbb{Q}:\langle
v,\gamma\rangle\leq 0\right\\}$

The elements of $\Sigma$ are called the spherical roots which is a subset of
cone spaned by positive roots of $\mathbf{G}$.

Denote the set of $B$-stable prime divisors in $\mathbf{G}/\mathbf{H}$ by
$\mathcal{D}:=\mathcal{D}(\mathbf{G}/\mathbf{H})$, which are called colors of
$\mathbf{G}/\mathbf{H}$. We also denote the set of $B$-stable prime divisors
in $X$ by $\mathcal{P}(X)$. Let $X$ be an embedding with $\mathbf{G}$-closed
orbit $Y$. Define $\mathcal{D}_{Y}(X)=\\{D\in\mathcal{P}(X),Y\subseteq D\\}$.

The embedding $X$ is called simple if there is only one closed $G$ orbit $Y$.
In this case, we define two sets

$\displaystyle\mathcal{I}_{Y}(X)$
$\displaystyle=\\{v_{D}\in\mathcal{V}|D\in\mathcal{D}_{Y}(X)\text{ is
$G$-stable}\\},$ $\displaystyle\mathcal{J}_{Y}(X)$
$\displaystyle=\\{\emptyset\neq
D\bigcap\mathbf{G}/\mathbf{H}\in\mathcal{D}|D\in\mathcal{D}_{Y}(X)\\}.$

Then $X$ is uniquely determined by $\mathcal{I}_{Y}(X),\mathcal{J}_{Y}(X)$.
Now we want to describe this pair only using information of
$\mathbf{G}/\mathbf{H}$. For any color $D$, $v_{D}$ is a valuation, then we
denote $\rho(v_{D})$ simply by $\rho(D)$. Let
$\mathcal{C}_{Y}(X)\subseteq\mathfrak{N}_{\mathbb{Q}}$ be the cone generated
by $\rho(\mathcal{J}_{Y}(X))$ and $\mathcal{I}_{Y}(X)$.

######  Lemma 4.2.

For a simple embedding $X$, the $\mathbb{Q}_{+}$ span of $\mathcal{I}_{Y}(X)$
are exactly the extremal rays of $\mathcal{C}_{Y}(X)$ which contain no
elements of $\rho(\mathcal{J}_{Y}(X))$.

######  Definition 4.3.

A colored cone is a pair $(\mathcal{C},\mathcal{R})$ with
$\mathcal{C}\subseteq\mathfrak{N}_{\mathbb{Q}}$ and
$\mathcal{R}\subseteq\mathcal{D}$ having the following properties:

  * •

CC1: $\mathcal{C}$ is a cone generated by $\rho(\mathcal{R})$ and finitely
many elements of $\mathcal{V}$.

  * •

CC2: The interior $\mathcal{C}^{\circ}\cap\mathcal{V}\neq\emptyset$

A colored cone is called strictly convex if the following holds:
SCC: $\mathcal{C}$ is strictly convex and $0\notin\rho(\mathcal{R})$.

Denote the pair of $(\mathcal{C}_{Y}(X),\mathcal{J}_{Y}(X))$ by
$\mathcal{C}^{c}(X)$, then we have

######  Theorem 4.4.

The map $X\mapsto\mathcal{C}^{c}(X)$ is a bijection between isomorphism
classes of simple embeddings and strictly convex colored cones.

For the general spherical embedding of $\mathbf{G}/\mathbf{H}$, every
$\mathbf{G}$ orbit gives a colored cone and these colored cones constitute a
colored fan.

######  Definition 4.5.

A colored fan is nonempty finite set $\mathcal{F}$ of colored cones with the
following properties:

  * •

CF1: Every face of $\mathcal{C}^{c}\in\mathcal{F}$ belongs to $\mathcal{F}$.

  * •

CF2: For every $v\in\mathcal{V}$ there is at most one
$(\mathcal{C},\mathcal{R})\in\mathcal{F}$ with $v\in\mathcal{C}^{\circ}$.

A colored fan $\mathcal{F}$ is called strictly convex if
$(0,\varnothing)\in\mathcal{F}$, or equivalently, if all elements of
$\mathcal{F}$ are strictly convex.

######  Theorem 4.6.

The map $X\mapsto\mathcal{F}(X)$ induces a bijection between isomorphism
classes of embeddings and strictly convex colored fans.

Denote the union of all sets $\mathcal{R}$ with
$(\mathcal{C},\mathcal{R})\in\mathcal{F}(X)$ by $\mathcal{D}(X)$, which are
called colors of $X$.

Let $\mathcal{I}_{X}^{G}$ denote the finite set of $G$-stable prime divisors
of $X$. Any divisor $Y\in\mathcal{I}_{X}^{G}$ corresponds to a ray
$(\mathcal{C},\emptyset)\in\mathcal{F}(X)$, and we denote by $u_{Y}$ the
indivisible generator of this ray in $\mathfrak{N}$. For any
$D\in\mathcal{D}$, its closure $\bar{D}$ in $X$ is an irreducible $B$-stable
but not $G$-stable divisor. Any $B$-stable Weil divisor $d$ on $X$ writes

(4.1) $\displaystyle
d=\sum_{Y\in\mathcal{I}_{X}^{G}}n_{Y}Y+\sum_{D\in\mathcal{D}}n_{D}\bar{D}$

for some integers $n_{Y},n_{D}$. In fact, any Weil divisor is linearly
equivalent to a $B$ invariant divisor and Brion proved the following criterion
to characterize Cartier divisors.

######  Proposition 4.7.

[8, Proposition 3.1] A $B$-stable Weil divisor $d$ in $X$ of form (4.1) is
Cartier if and only if there exists an integral piecewise linear function
$l_{d}$ on the support $|\mathcal{F}(X)|$ of the fan $\mathcal{F}(X)$ such
that

(4.2) $\displaystyle
d=\sum_{Y\in\mathcal{I}_{X}^{G}}l_{d}\left(u_{Y}\right)Y+\sum_{D\in\mathcal{D}(X)}l_{d}(\rho(D))\bar{D}+\sum_{D\in\mathcal{D}\backslash\mathcal{D}(X)}n_{D}\bar{D}$

for some integers $n_{D}$.

Let $\mathcal{F}^{\max}(X)$ denote the set of cones $\mathcal{C}\subseteq$
$\mathfrak{N}_{\mathbb{Q}}$ of maximal dimension, such that there exists
$\mathcal{R}\subseteq\mathcal{D}$ with
$(\mathcal{C},\mathcal{R})\in\mathcal{F}(X)$. If $d$ is a Cartier divisor and
$\mathcal{C}\in\mathcal{F}^{\max}(X)$, let $v_{\mathcal{C}}\in\mathfrak{M}$
such that $l_{d}(x)=v_{\mathcal{C}}(x)$ for $x\in\mathcal{C}$. Moreover, $d$
is ample if and only if the following conditions are satisfied:

  * •

the function $l_{d}$ is convex,

  * •

$v_{\mathcal{C}_{1}}\neq v_{\mathcal{C}_{2}}$ if
$\mathcal{C}_{1}\neq\mathcal{C}_{2}\in\mathcal{F}^{\max}(X)$,

  * •

$-n_{D}>v_{\mathcal{C}}(\rho(D))$ for all
$D\in\mathcal{D}\backslash\mathcal{D}(X)$ and
$\mathcal{C}\in\mathcal{F}^{\max}(X)$.

To a Cartier divisor $d$, we associate a polytope
$\Delta_{d}\subseteq\mathfrak{M}_{\mathbb{R}}$ as

$\displaystyle\Delta_{d}:=\left(\cap_{\mathcal{C}\in\mathcal{F}(X)^{\max}}(-v_{\mathcal{C}}+\sigma^{\vee})\right)\bigcap\left(\cap_{D\in\mathcal{D}\setminus\mathcal{D}_{X}}\\{m|m(\rho(D))+n_{D}\geq
0\\}\right).$

If $d$ is ample then $l_{d}$ is the support function of $\Delta_{d}$ for
$x\in\left|\mathcal{F}(X)\right|$.

There is also a moment polytope associated to an ample line bundle. Let $L$ be
a $G$-linearized ample line bundle on a spherical variety $X$. Denote by
$V_{\lambda}$ an irreducible representation of $G$ of highest weight
$\lambda\in\mathfrak{X}(T)$ with respect to $B$. Since $X$ is spherical, for
all $r\in\mathbb{N}$, there exists a finite set
$\Delta_{r}\subset\mathfrak{X}(T)$ such that

$H^{0}\left(X,L^{r}\right)=\bigoplus_{\lambda\in\Delta_{r}}V_{\lambda}$

######  Definition 4.8.

The moment polytope $\Delta_{L}$ of $L$ with respect to $B$ is defined as the
closure of $\bigcup_{r\in\mathbb{N}^{*}}\Delta_{r}/r$ in
$\mathfrak{X}(T)\otimes\mathbb{R}$.

The relationship between the moment polytope and polytope of a $B$-stable
divisor is as follows. Fix a linearization of $L$, and choose a global
$B$-semi-invariant section $s$ of $L$, so that the zero divisor $d$ of $s$ is
an ample $B$-invariant Cartier divisor. Let $\mu_{s}$ be the character of $B$
defined by $s$, that is, such that $b\cdot s\left(b^{-1}\cdot
x\right)=\mu_{s}(b)s(x)$ for all $x\in X$.

######  Proposition 4.9.

([8], Proposition 3.3]) The moment polytope $\Delta_{L}$ of $L$ and the
polytope $\Delta_{d}$ associated to $d$ are related by $\Delta_{L}=$
$\mu_{s}+\Delta_{d}$.

###  4.2. Torus $\mathbb{C}^{*}$-degenerations for $\mathbf{G}$-polarized
spherical varieties

The following classification result is proved in [16] based on works in [20,
2, 3]. We include its proof from [16, Section 4] below for our later use.

######  Proposition 4.10.

[16, Theorem 4.1] Let $(X,L)$ a polarized $\mathbf{G}$-spherical variety. Let
$d$ be any fixed $\mathbf{B}$-stable divisor of $L$. Then the
$\mathbf{G}$-equivariant $\mathbf{C}^{*}$-degenerations of $(X,L)$ are in one-
to-one correspondence with positive rational piecewise linear concave
functions on $\Delta_{d}$, with slopes in $\mathcal{V}$.

###### Proof.

Let $(\mathcal{X},\mathcal{L})$ be a $\mathbf{G}$-equivariant test
configuration for $(X,L)$. Recall the compactification of
$(\mathcal{X},\mathcal{L})$ . That is, we glue the trivial family over
$\mathbb{C}$ to $(\mathcal{X},\mathcal{L})$ along $\mathbb{C}^{*}$ to obtain a
family over $\mathbb{P}^{1}$. We denote the point added to $\mathbb{C}$ by
$\infty$ and still denote by $(\mathcal{X},\mathcal{L})$ the compactified
family.

Up to adding a sufficiently large multiple of the pull back of
$\\{0\\}\in\mathbb{P}^{1}$, we may assume that $\mathcal{L}$ is ample. Then
$(\mathcal{X},\mathcal{L})$ is a polarized spherical variety under the action
of $G\times\mathbb{C}^{*}$. Its open orbit is $G/H\times\mathbb{C}^{*}$, and
the combinatorial data are easily derived from that of $X$.

Suppose that $s$ is the defining section of $d$. Let $\hat{s}$ be the
$\mathbb{C}^{*}$-invariant meromorphic section of $\mathcal{L}$ whose
restriction to $\left(X,L^{r}\right)=$
$\left(\mathcal{X}_{1},\mathcal{L}_{1}\right)$ coincides with $s^{\otimes r}$.
Then the divisor $\hat{d}$ of $\hat{s}$ is
$\left(B\times\mathbb{C}^{*}\right)$-stable, hence an integral linear
combination of the form

(4.3)
$\displaystyle\hat{d}=\sum_{\hat{D}\in\mathcal{P}_{\mathcal{X}}}n_{\hat{D}}\hat{D}$

where $\mathcal{P}_{\mathcal{X}}$ is the set of prime
$\left(B\times\mathbb{C}^{*}\right)$-stable divisors on $\mathcal{X}$. There
are three types of divisors in $\mathcal{P}_{\mathcal{X}}$:

  * (1)

The divisors
$\\{\hat{D}\in\mathcal{P}_{\mathcal{X}}|\hat{\varrho}(\hat{D})\in(\mathfrak{N}_{\mathbb{Q}}\times\\{0\\})\\}$.
Any such $\hat{D}$ satisfies $\hat{D}=\overline{D\times\mathbb{C}^{*}}$ for
some $D\in\mathcal{P}_{X}$. Since $\hat{s}|_{\mathcal{X}_{1}}\cong rL$,
$n_{\hat{D}}=rn_{D}$. In particular, any color of $G/H\times\mathbb{C}^{*}$
must be of this type;

The remaining two types consist of the following
$\left(G\times\mathbb{C}^{*}\right)$-stable divisors of $\mathcal{X}$:

  * (2)

The the fiber $X_{\infty}$ at $\infty\in\mathbb{P}^{1}$. Since
$\hat{s}|_{\mathcal{X}\setminus\mathcal{X}_{0}}\cong
s\times(\mathbb{P}^{1}\setminus\\{0\\})$, $n_{\mathcal{X}_{0}}=0$;

  * (3)

The reduced, irreducible components
$\mathcal{X}_{0,1},...,\mathcal{X}_{0,n_{0}}$ of $\mathcal{X}_{0}$. In this
case $\hat{\rho}(\mathcal{X}_{0,i})=(u_{i},-m_{i})$ is a primitive vector in
$\mathfrak{N}\oplus\mathbb{Z}$ with $u_{i}\in\mathcal{V}$ and $m_{i}>0$. Let
us write also $n_{i}$ for the corresponding coefficient in (4.3).

Let $\hat{\Delta}$ denote the polytope in $\hat{\mathcal{M}}_{\mathbb{R}}$
associated to the divisor (4.3). In view of the previous description of the
divisor, the polytope can be described as

$\hat{\Delta}=\\{(ry,t)\mid y\in\Delta,0\leq t\leq g(x)\\}$

where $g$ is a (positive) rational concave piecewise linear function on
$\Delta$, expressed as

(4.4) $\displaystyle
g(x)=\inf_{i=1,...,n_{0}}\left(\frac{ru_{i}(y)+n_{i}}{m_{i}}\right).$

Note that since each $m_{i}$ is positive, each
$\frac{r}{m_{i}}u_{i}\in\mathcal{V}$.

Conversely, let $g$ be a positive rational piecewise linear concave function
on $\Delta_{d}$ with slopes in the valuation cone of $X$. Then we may assume
that $g>0$ is of form (4.4) for some $r\in\mathbb{N}_{+}$, each
$n_{i}\in\mathbb{Z}$, $u_{i}\in\mathcal{V}\cap\mathfrak{N}$ and
$m_{i}\in\mathfrak{N}_{+}$ so that $(u_{i},-m_{i})$ is a primitive vector in
$\mathfrak{N}\oplus\mathbb{Z}$. Consider the polytope

$\hat{\Delta}=\\{(rx,t)\mid x\in\Delta_{d},0\leq t\leq g(p)\\},$

where $d$ is the $B$-semiinvariant divisor of the line bundle $L$ as in (4.2).

Note that the colors of $\mathbf{G}/\mathbf{H}\times\mathbb{C}^{*}$ can be
identified with colors of $\mathbf{G}/\mathbf{H}$. Indeed, for any color
$\hat{D}\in\hat{\mathcal{D}}(\mathbf{G}/\mathbf{H}\times\mathbb{C}^{*})$,
$\overline{\hat{D}}\cap G/H\in\mathcal{D}(\mathbf{G}/\mathbf{H})$. Conversely,
for every $D\in\mathcal{D}(G/H)$,
$\overline{D\times\mathbb{C}^{*}}\in\hat{\mathcal{D}}(\mathbf{G}/\mathbf{H}\times\mathbb{C}^{*})$.
We are going to define a colored fan consisting of the following maximal
colored cones:

  * (1’)

$(\mathcal{C}\times\\{0\\},\mathcal{R})$, where $(\mathcal{C},\mathcal{R})$ is
a colored cone of $\mathcal{F}(X)$;

  * (2’)

$(\mathcal{C}\times\mathbb{R}_{+},\mathcal{R})$, where
$(\mathcal{C},\mathcal{R})$ is a colored cone of $\mathcal{F}(X)$;

  * (3’)

The inner normal cone of $\hat{\Delta}$ at a vertex $v$ of the graph of $g$.
In this case, we define the colored cone $(\mathcal{C}_{v},\mathcal{R}_{v})$,
where $\mathcal{C}_{v}$ is the inner normal cone at $v$ and

$\mathcal{R}_{v}=\\{\hat{D}=\overline{D\times
C^{*}}|D\in{\mathcal{D}}(\mathbf{G}/\mathbf{H})~{}\text{satisfies}~{}n_{D}-\rho(D)(v)=0\\}.$

Here in the last equality $\rho(D)$ is considered as an vector in
$\mathfrak{N}_{\mathbb{Q}}\times\\{0\\}\subset\hat{\mathfrak{N}}_{\mathbb{Q}}$.

The polytope $\hat{\Delta}$ is the polytope associated to the
$\left(B\times\mathbb{C}^{*}\right)$-stable Cartier divisor

$\hat{d}=\sum_{\hat{D}\in\mathcal{P}_{X}}n_{\hat{D}}\hat{D}$

where

  * (1”)

$n_{\hat{D}}=rn_{D}$ for each divisor $\hat{D}\in\mathcal{P}_{\mathcal{X}}$ so
that $\hat{D}=\overline{D\times\mathbb{C}^{*}}$ for some
$D\in\mathcal{P}_{X}$;

  * (2”)

$n_{X_{\infty}}=0$ for the only divisor $X_{\infty}$;

  * (3”)

$n_{\hat{D}}=n_{i}$ for the $G\times\mathbb{C}^{*}$-stable prime divisor
$\hat{D}_{i}$ corresponding to $(u_{i},-m_{i})$.

In particular, the restriction of $\hat{d}$ to
$\mathcal{X}\setminus\\{\hat{D}_{i}|i-1,...,n_{0}\\}$ is the product of a
divisor $d$ of form (4.1) in $X$ with $\mathbb{C}^{*}\cup\\{\infty\\}$.
Furthermore, $d$ satisfies the ampleness criterion. The associated line bundle
$\mathcal{O}_{\mathcal{X}}(\hat{d})$ is $(G\times\mathbb{C}^{*})$-linearizable
(up to passing to a sufficiently divisible tensor power). Choosing the
linearization such that the natural section $\hat{s}$ of
$\mathcal{O}_{\mathcal{X}}(\hat{d})$ is $\mathbb{C}^{*}$-invariant and has
$B$-weight $r\chi$. It follows
$\mathcal{O}_{\mathcal{X}}(\hat{d})|_{X\times(\mathbb{C}^{*}\cup\\{\infty\\})}\cong(X,rL)\times(\mathbb{C}^{*}\cup\\{\infty\\})$.
Hence $(\mathcal{X},\mathcal{L})$ is a test configuration of $(X,L)$.

∎

According to Proposition 4.10, a $\mathbf{G}$-equivariant
$\mathbb{C}^{*}$-degeneration on a $\mathbf{G}$-spherical variety is
determined by a rational piecewise linear function. As
$\mathbb{C}^{*}$-degenerations on toric manifolds introduced by Donaldson
[20], we will use the notion of torus $\mathbb{C}^{*}$-degenerations for such
degenerations for simplicity.

If $\tau(s)$ is a $1$-PS in ${\rm GL}(H^{0}(X,L^{k}))$ commuting with
$\mathbf{G}$, then the eigenspaces of $\tau(s)$ are $\mathbf{G}$-invariant
subspaces of $H^{0}(X,L^{k})$. Fix a $\mathbf{B}$-semiinvariant divisor of
$L^{k}$ of form (4.1) with associated $\mathbf{B}$-character $\mu_{d}$. The
eigenvalues of $\tau(s)$ is determined by the eigenvalues $s^{(k)}_{\lambda}$
on $V_{\lambda+k\mu_{d}}$ for $\lambda\in
k\Delta_{d}\cap\mathfrak{M}(\mathbf{G}/\mathbf{H})$.

######  Lemma 4.11.

Let $(X,L)$ be a polarized projective variety and $\mathcal{X}$ a torus
$\mathbb{C}^{*}$-degeneration of $(X,L)$. Then the central fiber of
$\mathcal{X}$ is normal if and only if it corresponds to an integral affine
function

(4.5) $\displaystyle f(y)=C+\Lambda(y),~{}y\in\Delta_{d}$

with $\Lambda\in\mathcal{V}\cap\mathfrak{N}$. Moreover, the eigenvalues of the
1-PS subgroup in ${\mathbb{P}}(H^{0}(X,L^{k}))$ corresponding to $\mathcal{X}$
is

(4.6) $\displaystyle
s^{(k)}_{\lambda}=kf(\frac{\lambda}{k}),~{}\forall\lambda\in
k\Delta_{d}\cap\mathfrak{M}~{}.$

###### Proof.

Note that the total space $(\mathcal{X},\mathcal{L})$ is a polarized spherical
embedding of $\mathbf{G}/\mathbf{H}\times\mathbb{C}^{*}$ and the central fibre
$\mathcal{X}_{0}$ is a $G\times\mathbb{C}^{*}$-invariant divisor. By [40,
Theorem 15.20], it suffices to show that $\mathcal{X}_{0}$ is prime.

$\mathcal{X}$ corresponds to a concave, rational piecewise linear function

$f(y)=\min_{a=1,...,N_{f}}\\{C_{a}+\Lambda_{a}(y)\\},~{}y\in\Delta_{d}$

with each $C_{a}\in\mathbb{Q}$ and $\Lambda\in\mathcal{V}\cap\mathfrak{N}$.
Then

$\mathcal{X}_{0}=\sum_{a=1}^{N_{f}}m_{a}\hat{D}_{a},$

where $\hat{D}_{a}$ is the prime
$\mathbf{G}/\mathbf{H}\times\mathbb{C}^{*}$-divisor corresponding to the
$a$-th piece of $f$ and $m_{a}$ is the smallest positive integer so that
$m_{a}\Lambda_{a}\in\mathfrak{N}$. Clearly $\mathcal{X}_{0}$ is prime if
$N_{f}=1$ and $m_{1}=1$. Thus we get (4.5). (4.6) follows from the
$\mathbb{C}^{*}$-action in the construction in Proposition 4.10. ∎

In the case of torus $\mathbb{C}^{*}$-degeneration with reduced center fiber
in Lemma 4.11, the central fiber is a spherical embedding of
$\mathbf{G}/\mathbf{H}_{0}$ for some $\mathbf{H}_{0}$ by Theorem 3.30 in [14].
$\mathbf{H}_{0}$ can be computed as follows: Choose $x_{0}\in X$ such that the
isotropy group of $x_{0}$ is $H$ and take $\hat{x}_{0}=(x_{0},1).$ Put
$\bar{x}_{0}=\lim_{t\rightarrow 0}(\Lambda(t),t)\hat{x}_{0}$. Then
$\mathbf{H}_{0}$ is the isotropy group of $\bar{x}_{0}$. Since the isotropy
group of $(\Lambda(t),t^{m})\hat{x}_{0}$ is $\mathbf{H}_{t}={\rm
Ad}_{(\Lambda(t),t)}\mathbf{H}$ and the limit of $\mathbf{H}_{t}$ is a
subgroup of $\mathbf{H}_{0}$, it must hold by dimension that

$\mathbf{H}_{0}=\lim_{t\rightarrow 0}{\rm Ad}_{(\Lambda(t),t)}\mathbf{H}.$

Now we give a classification of torus $1$-PS degenerations associated to
$\mathbf{G}-$equivariant $1$-PS $\sigma(t)$ in ${\rm GL}(H^{0}(X,L^{k}))$. For
simplicity, we also denote the eigenvalues of $\sigma(t)$ on $V_{\mu}$ by
$s^{(k)}_{\mu}$. Denote by $\mathcal{V}_{\mathbb{R}}$ the closure of
$\mathcal{V}$ in $\mathcal{V}\otimes\mathbb{R}$.

######  Proposition 4.12.

Suppose that the center ${\mathcal{X}}_{0}^{\sigma}$ of torus $1$-PS
degeneration $(X,\sigma(t))$ is normal. Then there exists
$\Lambda\in\mathcal{V}_{\mathbb{R}}$ and a constant $C$ such that
$s^{(k)}_{\mu}=\Lambda(\mu)+C,\forall\mu\in\Delta_{k}$. The converse is also
true.

###### Proof.

By Proposition 2.9, we can find a sequence of $G$-equvariant real 1-PS
$\tau^{i}(t)$ with eigenvalues $s^{(k,i)}_{\mu}\in\mathbb{Q}$ converges to
$s^{(k)}_{\mu}$ such that the central fiber $\mathcal{X}^{\tau^{i}(t)}_{0}$ is
the same as $\mathcal{X}^{\sigma}_{0}$. For a sufficently divisible $N$,
$\tilde{\tau}^{i}(t)=\tau^{i}(Nt)$ is a 1-PS. Applying Lemma 4.11 to
$\tilde{\tau}^{i}(t)$, there is $\tilde{\Lambda}_{i}\in\mathcal{V}$ and
constant $\tilde{C}_{i}$ such that
$Ns^{(k,i)}_{\mu}=\tilde{\Lambda}_{i}(\mu)+\tilde{C}_{i}$. Therefore
$s^{(k,i)}_{\mu}=\Lambda_{i}(\mu)+C_{i}$ for some $\Lambda_{i}\in\mathcal{V}$
and constant $C_{i}$. Since $s^{(k,i)}_{\mu}$ converges to $s^{(k)}_{\mu}$, by
subsequence $\Lambda_{i}$ and $C_{i}$ converge to
$\Lambda\in\mathcal{V}_{\mathbb{R}}$ and $C\in\mathbb{R}$. Then
$s^{(k)}_{\mu}=\Lambda(\mu)+C$ and the Proposition is proved.

We can also prove this as follows. In the decomposition
$H^{0}(X,L^{k})=\oplus_{\mu\in\Delta_{k}}V_{\mu}$, choose sections $x_{\mu}\in
V_{\mu}$ with highest weight $\mu$, i.e.,

$b\cdot x_{\mu}=\mu(b)x_{\mu},\forall b\in B.$

For any equality such as $\sum_{i\in I}\mu_{i}=\sum_{j\in J}\mu_{j}$ with
$|I|=|J|$, there is a constant $c$ such that $\Pi_{i\in
I}x_{\mu_{i}}=c\Pi_{j\in J}x_{\mu_{j}}$ is a defining equation of $X$ in
$\mathbb{P}(H^{0}(X,L^{k}))$. The weight of the two sides of the equation are
$\sum_{i\in I}s^{(k)}_{\mu_{i}}$ and $\sum_{j\in J}s^{(k)}_{\mu_{j}}$. If the
two weights are not the same, there will be a defining equation $\Pi_{i\in
I}x_{\mu_{i}}=0$ or $\Pi_{j\in J}x_{\mu_{j}}=0$ on the limit
${\mathcal{X}}_{0}^{\sigma}$. It contradicts with the normality. So we must
have

$\sum_{i\in I}s^{(k)}_{\mu_{i}}=\sum_{j\in J}s^{(k)}_{\mu_{j}}.$

If follows that there is a affine function $\Lambda(\mu)+C$ such that
$s^{(k)}_{\mu}=\Lambda(\mu)+C$. We are going to show that
$\Lambda\in\mathcal{V}$. Consider the induced action of $\sigma(t)$ on
$H^{0}(X,L^{rk})$ for any $r\geq 1$. The eigenvalues
$s^{(rk)}_{\mu}(\mu\in\Delta_{kr})$ of the induced action is also equal to
$\Lambda_{r}(\mu)+C_{r}$ for some $\Lambda_{r}$ and $C_{r}$. We have the
decompostion

$V_{\mu_{1}}\cdot V_{\mu_{2}}...\cdot
V_{\mu_{r}}=\oplus_{\alpha}V_{\sum_{i=1}^{r}\mu_{i}-\alpha},$

where $\alpha$ is some non-negative linear combination of simple roots of $G$.
The eigenvalue of $\sigma(t)$ on $V_{\sum_{i=1}^{r}\mu_{i}}$ is equal to
$\sum_{i=1}^{r}s^{(k)}_{\mu_{i}}$. So we get
$\sum_{i=1}^{r}(\Lambda({\mu_{i}})+C)=\Lambda_{r}({\sum_{i=1}^{r}\mu_{i}})+C_{r}$.
It follow that $\Lambda_{r}=\Lambda,C_{r}=rC$. On the other component
$V_{\sum_{i=1}^{r}\mu_{i}-\alpha}$, the eigenvalue of $\sigma(t)$ is at least
$\sum_{i=1}^{r}s^{(k)}_{\mu_{i}}$. So $\Lambda(-\alpha)\geq 0$. Since
$\mathcal{V}$ is minus the dual cone of all such $\alpha^{\prime}s$, we know
that $\Lambda\in\mathcal{V}_{\mathbb{R}}$.

Conversely, given any $\Lambda\in\mathcal{V}_{\mathbb{R}}$ and any constant
$C$, the eigenvalues $s^{(k)}_{\mu}=\Lambda(\mu)+C$ define a $G$-equivariant
real 1-PS. Denote it by $\sigma(t)$. We have to show that the central fiber is
normal. The equations determines the type of the eigenvalues of induced action
of $\sigma(t)$ on the Hilbert point have the following form:

$\sum_{i\in I}\Lambda(u_{i})=\sum_{j\in J}\Lambda(u_{j}),|I|=|J|.$

So we can choose a sequence of $\Lambda_{i}\in\mathcal{V}$ which are solutions
of these equations and converges to $\Lambda$. Denote the real 1-PS with
eigenvalues $\Lambda_{i}(\mu)+C$ by $\tau^{i}(t)$. Then the types of the
eigenvalues of induced action of $\tau^{i}(t)$ and $\sigma(t)$ are the same.
So we have $\mathcal{X}_{0}^{\tau^{i}(t)}=\mathcal{X}_{0}^{\sigma(t)}$. By
Lemma 4.11 $\mathcal{X}_{0}^{\tau^{i}(t)}$ is normal. Hence
$\mathcal{X}_{0}^{\sigma(t)}$ is normal and the proposition is proved. ∎

##  5\. H-invariant on $\mathbf{G}$-spherical Fano varieties

###  5.1. Special torus $1$-PS degenerations

For a spherical variety $X$, $K^{-1}_{X}$ is a Weil divisor, which can be
written as

(5.1) $\displaystyle
d=\sum_{Y\in\mathcal{I}_{X}^{G}}Y+\sum_{D\in\mathcal{D}}n_{D}\bar{D}$

where the $n_{D}$ are explicitly obtained in terms of $\kappa_{P}$ and the
types of the roots (see [21] for a precise description of these coefficients).
If this is an ample $\mathbb{Q}$-Cartier divisor, then $X$ is a
$\mathbb{Q}$-Fano spherical variety. In this case, there is a $B$-semi-
invariant section $s$ of $K^{-1}_{X}$ of weight $\kappa_{P}$ such that
$div(s)=d$.

The moment polytope $\Delta_{+}:=\Delta_{K_{X}^{-1}}$ is then
$\kappa_{P}+\Delta_{d}$ by Proposition 4.9, and furthermore the dual polytope
$\Delta_{d}^{*}$ of $\Delta_{d}$ is a $\mathbb{Q}$-$G/H$-reflexive polytope in
the sense of [21], which can be obtained as the convex hull:

$\Delta_{d}^{*}=\operatorname{conv}\left(\left\\{\rho_{D}/n_{D},D\in\mathcal{D}\right\\}\cup\left\\{u_{Y},Y\in\mathcal{I}_{X}^{G}\right\\}\right).$

The $\mathbb{Q}$-Fano variety $X$ can further be recovered from its
$\mathbb{Q}$-$G/H$-reflexive polytope $\Delta_{d}^{*}$ by the following
procedure, detailed in [21]. The colored fan of $X$ is obtained from
$\Delta^{*}_{d}$ as the union of the colored cones $({\rm
Cone}(F),\rho^{-1}(F))$ for all faces $F$ of $\Delta_{d}^{*}$ such that the
intersection of the relative interior of $\operatorname{Cone}(F)$ with the
valuation cone $\mathcal{V}$ is not empty.

By Lemma 4.11, the special torus $\mathbb{C}^{*}$-degeneration of a spherical
Fano variety corresponds to a linear function on
$\Delta_{+}=\Delta_{d}+\kappa_{P}$. More precisely, the constant term can be
determined by the following lemma.

######  Proposition 5.1.

The special torus $\mathbb{C}^{*}$-degeneration on a $\mathbb{Q}$-Fano
spherical variety corresponds to $f(y)=\Lambda y$ for
$\Lambda\in\mathcal{V}\bigcap\mathfrak{N}$ and $y\in\Delta_{d}$.

###### Proof.

For a special torus $\mathbb{C}^{*}$-degeneration, we have
$\mathcal{L}=K^{-1}_{\mathcal{X}/\mathbb{C}}$. Assume that this degeneration
corresponds to the function $f(y)=\Lambda(y)+C$. By the proof of Proposition
4.10 we can write the divisor $div(\hat{s})$ as $\hat{d}+C\mathcal{X}_{0}$.
Since the restriction of the divisor $div(\hat{s})$ on generic fiber is the
canonical divisor, we know that $\hat{d}$ is the divisor of
$K^{-1}_{\mathcal{X}/\mathbb{C}}$. So we have $C=0$. ∎

Like Proposition 4.12, we have the following theorem for special torus $1$-PS
degenerations of spherical Fano manifold.

######  Theorem 5.2.

Assume that $H^{0}(X,K_{X}^{-k})$ is very ample. For any
$\Lambda\in\mathcal{V}_{\mathbb{R}}$, the real 1-PS with eigenvalues

(5.2) $\displaystyle s_{\lambda}^{(k)}=\Lambda(\lambda),~{}\forall\lambda\in
k\Delta_{d}\cap\mathfrak{M}$

defines a special torus $1$-PS degeneration $\mathcal{X}^{\sigma}$ of
$(X,K^{-1}_{X})$. Conversely, any special torus $1$-PS degeneration of
$(X,K^{-1}_{X})$ is given in this way.

Theorem 5.2 is a combination of the following two propositions.

######  Proposition 5.3.

For any $\Lambda\in\mathcal{V}_{\mathbb{R}}$, the real 1-PS $\sigma(t)$ with
eigenvalues given by (5.2) defines a $\mathcal{X}^{\sigma}$ of
$(X,K^{-1}_{X})$.

###### Proof.

Eq. (5.2) defines a special torus $1$-PS degeneration associated to
$\sigma(t)$. By the proof of Proposition 4.12, we can choose a sequence
$\Lambda_{i}\in\mathcal{V}$ so that $\Lambda_{i}\to\Lambda$, and
$\mathcal{X}^{\tau^{i}}_{0}=\mathcal{X}^{\sigma}_{0}$ where $\tau^{i}(t)$ is
the real 1-PS with eigenvalues $\Lambda_{i}(\lambda)$. By Proposition 5.1 each
$\mathcal{X}_{\tau^{i}}$ is special. So $\mathcal{X}^{\sigma}_{0}$ is
$\mathbb{Q}$-Fano. Denote the normalized real 1-PS corresponding to
$\sigma(t)$ by $\tilde{\sigma}(t)$. Since the actions of $\tau^{i}(t)$ on
$\mathcal{X}^{\sigma}_{0}$ also converges to the action of $\sigma(t)$, we
have $\tilde{\sigma}(t)=\lim_{i\rightarrow\infty}\tau^{i}(t)=\sigma(t).$ We
conclude that $\mathcal{X}^{\sigma}$ is special. ∎

Now we prove the inverse direction:

######  Proposition 5.4.

For any special torus 1-ps degeneration of $(X,K^{-1}_{X})$, there is a
$\Lambda\in\mathcal{V}_{\mathbb{R}}$ so that the real 1-PS with eigenvalues
(5.2) corresponds to it.

###### Proof.

Denote the eigenvalues of $\sigma(t)$ on $V_{\lambda}$ by $s^{(k)}_{\lambda}$.
By Proposition 4.12, we can choose a sequence of $\mathbf{G}$-equivariant 1-PS
$\tau^{i}(t)$ that converges to $\sigma(t)$ such that the central fiber
$\mathcal{X}^{\tau^{i}(t)}_{0}$ is the same as $\mathcal{X}^{\sigma}_{0}$.
Denote the normalized real 1-PS corresponding to $\tau^{i}(t)$ by
$\tilde{\tau}^{i}(t)$. So $\mathcal{X}^{\tilde{\tau}^{i}}$ is special. By
Proposition 5.1 there is a $\Lambda_{i}\in\mathcal{V}$ so that the eigenvalues
of $\tilde{\tau}^{i}(t)$ is $\Lambda_{i}(\lambda)$ for $\lambda\in
k\Delta_{d}\bigcap{\mathfrak{M}}$. Considering the action on the central fiber
$\mathcal{X}^{\sigma}_{0}$, we know that

$\lim_{i\rightarrow\infty}\tilde{\tau}^{i}(t)=\sigma(t).$

Thus $\Lambda_{i}$ converges to some $\Lambda\in\mathcal{V}_{\mathbb{R}}$ such
that

$s_{\lambda}^{(k)}=\Lambda(\lambda),~{}\forall\lambda\in
k\Delta_{d}\cap{\mathfrak{M}}.$

∎

###  5.2. A formula of H-invariant

In this subsection, we compute the H-invariant for special torus $1$-PS
degenerations on a $\mathbf{G}-$spherical Fano manifold. A similar formula has
also been obtained by Li-Li by using filtration geometry [26]. Here we give a
simple proof by using the equivariant Riemann-Roch theorem.

As in [39], we introduce the definition of H-invariant for special $1$-PS
degenerations as follows.

######  Definition 5.5.

For a special $1$-PS degeneration $\mathcal{X}^{\sigma}$, $\sigma(t)$ induces
a holomorphic VF $\eta(\sigma)$ on $\mathcal{X}^{\sigma}_{0}$. We define

$V=\int_{\mathcal{X}^{\sigma}_{0}}\omega_{0}^{n},~{}H(\mathcal{X}^{\sigma}):=V\log(\frac{1}{V}\int_{\mathcal{X}^{\sigma}_{0}}e^{\theta}\omega_{0}^{n}),$

where $\omega_{0}$ is the restriction of Fubini-Study metric and $\theta$ is
the potential function corresponding to the canonical lifting of
$\eta(\sigma)$.

In general, the $H$-invariant is defined by

$H(\mathcal{X}^{\sigma})=-\int_{\mathcal{X}^{\sigma}_{0}}\tilde{\theta}e^{h_{0}}\omega_{0}^{n}+V\log\left(\frac{1}{V}\int_{\mathcal{X}^{\sigma}_{0}}e^{\tilde{\theta}}\omega_{0}^{n}\right),$

where $\tilde{\theta}$ is any hamiltonian for vector field $\eta(\sigma)$ and
$h_{0}$ is the Ricci potential of $\omega_{0}$ normalized by
$\int_{\mathcal{X}^{\sigma}_{0}}e^{h_{0}}\omega_{0}^{n}=V$. For the canonical
lifting, it holds that

$\Delta\theta+\theta+\langle\nabla\theta,\nabla h_{0}\rangle=0.$

If follows that $\int_{\mathcal{X}^{\sigma}_{0}}\theta
e^{h_{0}}\omega_{0}^{n}=0$ and the two definitions coincide. By [18] the first
step degeneration from $X$ to $\bar{X}_{\infty}$ is charecterized by the
minimum of the $H$-invariant.

We also need the definition of modified K-stability (cf. [45, 6, 43]). Let $X$
be a Fano manifold and $v$ a holomorphic VF on $X$. Denote by $T$ the torus
generated by $v$.

######  Definition 5.6.

For a special $T$-equivariant $\mathbb{C}^{*}$-degeneration
$(\mathcal{X},\mathcal{L})$, denote by $\theta$ and $\theta_{v}$ the potential
functions corresponding to the cannonical lifing of $\mathbb{C}^{*}$-action
and $v$ on $\mathcal{X}_{0}$ respectively. Put

$Fut_{v}(\mathcal{X},\mathcal{L})=\int_{\mathcal{X}_{0}}\theta
e^{\theta_{v}}\omega_{0}^{n}.$

If $Fut_{v}(\mathcal{X},\mathcal{L})\geq 0$ for any special $T$-equivariant
$\mathbb{C}^{*}$-degeneration $(\mathcal{X},\mathcal{L})$, we say that $(X,v)$
is modified K-semistable. Moreover, if the equality holds only when
$\mathcal{X}\cong X\times\mathbb{C}$ equivariantly, we say that $(X,v)$ is
modified K-polystable.

If $(X,v)$ is modified K-semistable, then $v$ is determined by the modified
Futaki invariant as in [36]. It has proved in [6, 13, 23, 7] that if $X$
admits KR soliton with respect to $v$, then $(X,v)$ is modified K-polystable.

By using the equivariant Riemann-Roch theorem (cf. [43]), we prove

######  Proposition 5.7.

Let $\mathcal{X}$ be a special torus $1$-PS degeneration induced by
$\Lambda\in\mathcal{V}$. Then

$H(\mathcal{X})=V\ln\frac{1}{V}\int_{\Delta_{+}}e^{\Lambda(\kappa_{P}-y)}\pi\,dy,$

where
$\pi(y)=\frac{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,y\rangle}{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,\rho\rangle}$.

###### Proof.

Note that $\Delta_{+}=\Delta_{d}+\kappa_{P}$. By Lemma 4.11 and Proposition
5.1, the weight of action of 1-PS $\tau(t)$ corresponding to $\Lambda$ on
$V_{\lambda}\subseteq H^{0}(X,K_{X}^{-k})$ is $\Lambda(\lambda-k\kappa_{P})$.
Denote the eigenvalues of $\tau(t)$ on $H^{0}(X,K_{X}^{-k})$ by
$\lambda_{i}^{(k)}$. By [43, Section 2], we have

$\sum
e^{t\frac{\lambda_{i}^{(k)}}{k}}=k^{n}\int_{\mathcal{X}_{0}}e^{-t\theta_{0}}\frac{\omega_{0}^{n}}{n!}+O(k^{n-1}),$

where $\theta_{0}$ is the potential function corresponding to the canonical
lifting of $\tau(t)$.

From the decomposition of $H^{0}(X,L^{k})$ and Weyl’s formula of dimension, we
have

$\displaystyle\sum e^{t\frac{\lambda_{i}^{(k)}}{k}}$
$\displaystyle=\sum_{\lambda\in\Delta_{k}}\dim
V_{\lambda}e^{t\frac{\Lambda(\lambda-k\kappa_{P})}{k}}$ (5.3)
$\displaystyle=\sum_{\lambda\in\Delta_{k}}\frac{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,\rho+\lambda\rangle}{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,\rho\rangle}e^{t\frac{\Lambda(\lambda-k\kappa_{P})}{k}}$

Writing $\lambda=ky$ we have

$\frac{1}{k^{n}}\sum
e^{t\frac{\lambda_{i}^{(k)}}{k}}=\sum_{y\in\frac{1}{k}\Delta_{k}}\frac{1}{k^{dim\Delta_{+}}}\frac{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,\frac{\rho}{k}+y\rangle}{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,\rho\rangle}e^{t\Lambda(y-\kappa_{P})}.$

Letting $k\rightarrow\infty$ we get

$\int_{\mathcal{X}_{0}}e^{-t\theta_{0}}\frac{\omega_{0}^{n}}{n!}=\int_{\Delta_{+}}\frac{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,y\rangle}{\prod_{\alpha\in\Phi_{+},\alpha\not\perp\Delta_{+}}\langle\alpha,\rho\rangle}e^{t\Lambda(y-\kappa_{P})}dy.$

The proposition is proved by putting $t=-1$.

∎

######  Proposition 5.8.

Given any real 1-PS $\sigma(t)$ corresponding to
$\Lambda\in\mathcal{V}_{\mathbb{R}}$, we have

(5.4) $\displaystyle
H(\mathcal{X}^{\sigma})=V\ln\frac{1}{V}\int_{\Delta_{+}}e^{\Lambda(\kappa_{P}-y)}\pi\,dy.$

As a consequence, $H$ has a unique minimizer on $\mathcal{V}_{\mathbb{R}}$.

###### Proof.

For the special $1$-PS degeneration $\mathcal{X}^{\sigma}$, we have a sequence
of special $\mathbb{C}^{*}$-degenerations $\mathcal{X}_{i}$ with the same
central fiber $\mathcal{X}_{0}$. Assuming that $\mathcal{X}_{i}$ corresponds
to $\Lambda_{i}\in\mathcal{V}$, we have $\Lambda_{i}\rightarrow\Lambda$.
Denote the holomorphic vector fileds on $\mathcal{X}_{0}$ induced from
$\mathcal{X}$ and $\mathcal{X}_{i}$ by $v$ and $v_{i}$. Then $v_{i}\rightarrow
v$ and $\theta_{v_{i}}\rightarrow\theta_{v}$. So
$H(\mathcal{X}_{i})\rightarrow H(\mathcal{X}^{\sigma})$. By Proposition 5.7,
the proposition is proved. From (5.4), $H$ is a convex function of $\Lambda$.
Hence there is a unique minimizer of $H$. ∎

If $v$ is a holomorphic VF commutting with the $\mathbf{G}-$action, then $v$
corresponds to an element $\Lambda_{0}$ in the linear part of
$\mathcal{V}_{\mathbb{R}}$. Using the same method in Proposition 5.8, we have

######  Proposition 5.9.

Given any real 1-PS $\sigma(t)$ corresponding to
$\Lambda\in\mathcal{V}_{\mathbb{R}}$, we have

(5.5) $\displaystyle
Fut_{v}(\mathcal{X}^{\sigma})=\int_{\Delta_{+}}\Lambda(\kappa_{P}-y)e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy,$

where $\Lambda_{0}$ is the element in $\mathcal{V}_{\mathbb{R}}$ corresponding
to $v$.

######  Corollary 5.10.

If $(X,v)$ is modified K-semistable, then $H$ invariant attains its minimum at
$\Lambda_{0}$, where $\Lambda_{0}$ is the element corresponding to $v$.

###### Proof.

At first $v$ is the holomorphic VF determined by the modified Futaki
invariant, so it commutes with $G$ by [36]. By Proposition 5.8, it only
suffices to prove

(5.6) $\displaystyle\frac{d}{dt}_{t=0^{+}}H(\Lambda_{0}+t\Lambda)\geq
0,\forall\Lambda\in\mathcal{V}_{\mathbb{R}}$

and

(5.7) $\displaystyle\frac{d}{dt}_{t=0}H(\Lambda_{0}+t\Lambda)=0,\text{ if
}\Lambda_{0}+t\Lambda\in\mathcal{V}_{\mathbb{R}}\text{ for small }t.$

By (5.4) and (5.5), we have

$\displaystyle\frac{d}{dt}_{t=0}H(\Lambda_{0}+t\Lambda)$
$\displaystyle=\frac{V}{\int_{\Delta_{+}}e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy}\int_{\Delta_{+}}\Lambda(\kappa_{P}-y)e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy$
$\displaystyle=\frac{V}{\int_{\Delta_{+}}e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy}Fut_{v}(\mathcal{X}^{\sigma}),$

where $\sigma$ is the real 1-PS corresponding to $\Lambda$. So (5.6) is proved
by the assumption. If $\Lambda_{0}+t\Lambda\in\mathcal{V}_{\mathbb{R}}$ for
small $t$, denote by $\tau(t)$ the real 1-PS corresponding to
$\Lambda_{0}-t_{0}\Lambda$ for a fixed small positive numbr $t_{0}$. Then we
have

$\displaystyle\frac{d}{dt}_{t=0}H(\Lambda_{0}+t\Lambda)$
$\displaystyle=\frac{V}{\int_{\Delta_{+}}e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy}\int_{\Delta_{+}}\Lambda(\kappa_{P}-y)e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy$
$\displaystyle=-\frac{V}{t_{0}\int_{\Delta_{+}}e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy}\int_{\Delta_{+}}(\Lambda_{0}-t_{0}\Lambda)(\kappa_{P}-y)e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy$
$\displaystyle=-\frac{V}{t_{0}\int_{\Delta_{+}}e^{\Lambda_{0}(\kappa_{P}-y)}\pi\,dy}Fut_{v}(\mathcal{X}^{\tau})\leq
0,$

since $Fut_{v}(v)=0$. Combined with (5.6), we get (5.7). The corollary is
proved. ∎

##  6\. Proof of Theorem 1.1

In case of $\mathbf{G}$-spherical manifold $X$, we choose
$\mathbf{G}_{0}=\mathbf{G}=\mathbf{K}^{\mathbb{C}}$ in Section 3. Then we can
consider solution $\omega_{t}$ of (1.1) starting with a $\mathbf{K}$-invariant
metric $\omega_{0}$. We will investigate the limit of KR flow on $X$ in more
detail. Using the same notations in section 3, we denote
$E=H^{0}(X,K_{X}^{-l})$, $\bar{G}={\rm GL}(E)$ and

$\bar{G}_{\mathbf{K}}=\\{g|~{}g\text{ commutes with }\mathbf{K}\\}.$

Our key observation comes from the following lemma.

######  Lemma 6.1.

$\bar{G}_{\mathbf{K}}$ is a torus.

###### Proof.

The proof of the lemma essentially uses this property of spherical manifolds.
For a polarized manifold $(X,L)$ with a $\mathbf{G}$-action, by Theorem 25.1
in [40] $X$ is spherical if and only if every irreducible representation in
the decomposition of $H^{0}(X,L^{l})$ has multiplicity one for any $l\geq 1$.
Then we have

$E=H^{0}(X,K_{X}^{-l})=\oplus_{\mu\in\Delta_{l}}V_{\mu}.$

For any $g\in\bar{G}_{\mathbf{K}}$, it must hold that $g(V_{\mu})=V_{\mu}$
since different $\mu^{\prime}s$ correspond to different irreducible
representations of $\mathbf{G}$. By Schur’s lemma, the restriction of $g$ onto
each $V_{\mu}$ is a scalar matrix. So $g$ can be written as
$diag\\{a_{\mu}\\}$. It follows that $\bar{G}_{\mathbf{K}}$ is a torus. ∎

######  Lemma 6.2.

There is a perturbation $\tilde{v}$ of the soliton VF $v$ such that
$X_{\infty}$ is biholomorphic to $\lim_{i\rightarrow\infty}e^{i\tilde{v}}\cdot
X$. Moreover, there is an element $\tilde{\Lambda}\in\mathcal{V}_{\mathbb{R}}$
corresponding to this degeneration.

###### Proof.

As in (3.3), we have defined matrices $A_{t}$. By Lemma 3.5, $A_{t}$ commutes
with $\mathbf{K}$. So from the the lemma above, we can write
$A_{i}=diag\\{a_{i}^{1},...,a_{i}^{N}\\}\in G_{\mathbf{K}}$ and
$v=\\{diag\\{\lambda^{1},...,\lambda^{N}\\}\in Lie(G_{\mathbf{K}})$. Then for
$1\leq k\leq N$,
$\lim_{i\rightarrow\infty}a_{i}^{k}(a_{i-1}^{k})^{-1}=e^{\lambda^{k}}$. It
follows that $a_{i}^{k}=e^{(\lambda^{k}+\epsilon_{i}^{k})i}$ with
$\lim_{i\rightarrow\infty}\epsilon_{i}^{k}=0$. Denote the Hilbert point $[X]$
by $[z_{\alpha}]$ and the vector
$\lambda=(\lambda^{k}),\epsilon_{i}=(\epsilon_{i}^{k})$. As in Lemma 2.8, the
induced action of $A_{i}$ on $[X]$ will be $[e^{-i\langle
w_{\alpha},\lambda+\epsilon_{i}\rangle}z_{\alpha}]$ for some vector
$w_{\alpha}$ with integral components. Let $Q=\\{u\in
Lie(G_{\mathbf{K}})|\langle w_{\alpha},u\rangle=\langle
w_{\beta},u\rangle,\forall\alpha\neq\beta\\}$. For any $\lambda\in
Lie(G_{\mathbf{K}})$,
$e^{\lambda}\cdot[X]=e^{pr_{Q^{\perp}}(\lambda)}\cdot[X]$, where
$pr_{Q^{\perp}}$ means the projection to the complement of $Q$. Now we replace
$\lambda,\epsilon_{i}$ by their projections and keep the notations for
simplicity. Denote $c=min\\{\langle w_{\alpha},\lambda\rangle|z_{\alpha}\neq
0\\}$ and $I=\\{\alpha|\langle w_{\alpha},\lambda\rangle=c,z_{\alpha}\neq
0\\}$. $I$ is a not the whole set. Then we have

(6.1)
$\displaystyle[\bar{X}_{\infty}]=\lim_{i\rightarrow\infty}e^{vi}\cdot[X]$
$\displaystyle=\lim_{i\rightarrow\infty}[e^{-i\langle
w_{\alpha},\lambda\rangle}z_{\alpha}]=[z_{\alpha}(\alpha\in I),0].$

Choose $\alpha_{0}\in I$, we have

$\lim_{i\rightarrow\infty}[e^{-i\langle
w_{\alpha},\lambda+\epsilon_{i}\rangle}z_{\alpha}]=\lim_{i\rightarrow\infty}[e^{-i\langle
w_{\alpha}-w_{\alpha_{0}},\lambda+\epsilon_{i}\rangle}z_{\alpha}].$

Then for $\alpha\notin I$, $\lim_{i\rightarrow\infty}e^{-i\langle
w_{\alpha}-w_{\alpha_{0}},\lambda+\epsilon_{i}\rangle}=0.$ But for $\alpha\in
I$, it holds

$\langle w_{\alpha}-w_{\alpha_{0}},\lambda+\epsilon_{i}\rangle=\langle
w_{\alpha}-w_{\alpha_{0}},\epsilon_{i}\rangle.$

Thus we get

(6.2) $\displaystyle\lim_{i\rightarrow\infty}[e^{-i\langle
w_{\alpha},\lambda+\epsilon_{i}\rangle}z_{\alpha}]=\lim_{i\rightarrow\infty}[e^{-\langle
w_{\alpha},i\epsilon_{i}\rangle}z_{\alpha}(\alpha\in I),0].$

If the norms of $i\epsilon_{i}$ is bounded, we can assume that
$\lim_{j\rightarrow\infty}i_{j}\epsilon_{i_{j}}=\epsilon\in
Lie(G_{\mathbf{K}})$ by subsequence. Then the right hand side of (6.2) is
$[e^{-\langle w_{\alpha},\epsilon\rangle}z_{\alpha}(\alpha\in I),0]$, which is
equal to $e^{\epsilon}\cdot[\bar{X}_{\infty}]$. If the norms of
$i\epsilon_{i}$ goes into infinity, we can assume that
$\lim_{j\rightarrow\infty}\frac{i\epsilon_{i}}{|i\epsilon_{i}|}=\eta\in
Lie(G_{\mathbf{K}})$. Denote $c^{\prime}=min\\{\langle
w_{\alpha},\eta\rangle|\alpha\in I\\}$ and $I^{\prime}=\\{\alpha|\langle
w_{\alpha},\eta\rangle=c^{\prime},\alpha\in I\\}$. Write
$i\epsilon_{i}=|i\epsilon_{i}|(\eta+\eta_{i})$ with $\eta_{i}\rightarrow 0$.
Then by (6.2) we have

$\displaystyle\lim_{i\rightarrow\infty}[e^{-i\langle
w_{\alpha},\lambda+\epsilon_{i}\rangle}z_{\alpha}]$
$\displaystyle=\lim_{i\rightarrow\infty}[e^{-\langle
w_{\alpha},|i\epsilon_{i}|(\eta+\eta_{i})\rangle}z_{\alpha}(\alpha\in I),0]$
$\displaystyle=\lim_{i\rightarrow\infty}[e^{-\langle
w_{\alpha},|i\epsilon_{i}|\eta_{i}\rangle}z_{\alpha}(\alpha\in
I^{\prime}),0].$

The right hand side has the same form as that of (6.2) with a smaller set
$I^{\prime}$. If the sequence $|i\epsilon_{i}|\eta_{i}$ is bounded, we are
done. Otherwise, as above, we can assume that
$\lim_{i\rightarrow\infty}\frac{|i\epsilon_{i}|\eta_{i}}{|i\epsilon_{i}||\eta_{i}|}=\eta_{1}\in
Lie(G_{\mathbf{K}})$ by subsequence. Continuing this process, we will get a
sequence of vectors $\eta,\eta_{1},...,\eta_{l}$ in $Lie(G_{\mathbf{K}})$ for
some $l\leq|I|$. Now for small positive number $s,s_{1},...,s_{l}$, denoting
$\tilde{v}=diag\\{\lambda^{k}+s\eta^{k}+...+s_{l}\eta^{k}_{l}\\}$, we have

$[X_{\infty}]=C\cdot\lim_{i\rightarrow\infty}e^{i\tilde{v}}[X],$

where $C$ is some element in $\bar{G}_{\mathbf{K}}$. Since $e^{\tilde{v}}\in
G_{\mathbf{K}}$, it generates a $\mathbf{G}-$equivariant 1-PS degeneration. By
Theorem 5.2, there is a $\tilde{\Lambda}\in\mathcal{V}_{\mathbb{R}}$
corresponding to this degeneration. The lemma is proved. ∎

###### Completion of proof of Theorem 1.1.

By Theorem 3.1, the soliton VF $v$ of $(X_{\infty},\omega_{\infty})$ induces a
special torus $1$-PS degeneration $e^{tv}$ of $X$ with central fiber
$\bar{X}_{\infty}$. Since $\bar{X}_{\infty}$ can also be a central fiber of
torus $\mathbb{C}^{*}$-degeneration by Proposition 2.9, $\bar{X}_{\infty}$ is
a $\mathbf{G}$-spherical variety by [14, Theorem 3.30]. On the other hand, the
degeneration $(X,e^{tv})$ attains the minimum of H-invariant
$H(\mathcal{X}^{\sigma})$ for special $1$-PS degenerations of $X$ by [18,
Theorem 3.2] (also see [42, Proposition 5.6]), we may restrict the H-invariant
for special torus $1$-PS degenerations. By Theorem 5.2 and Proposition 5.8,
there is a unique $\Lambda_{v}\in\mathcal{V}_{\mathbb{R}}$ as a minimizer of
$H(\mathcal{X}^{\sigma})$ in (5.4) which corresponds to the torus $1$-PS
degeneration $(X,e^{tv})$. Thus by perturbing the soliton VF $v$ to
$\tilde{v}$ in Lemma 6.2, there is another element
$\tilde{\Lambda}\in\mathcal{V}_{\mathbb{R}}$ which induces a special torus
$1$-PS degeneration $e^{i\tilde{v}}$ from $X$ to $X_{\infty}$. Again by
Proposition 2.9 together with Theorem 5.2, we may choose
$\tilde{\Lambda}\in\mathcal{V}_{\mathbb{R}}$ such that the degeneration
$(X,e^{i\tilde{v}})$ induces a special torus $\mathbb{C}^{*}$-degeneration of
$X$ with the central fiber $X_{\infty}$. By [14, Theorem 3.30], it follows
that $X_{\infty}$ is also a $\mathbf{G}$-spherical variety. Therefore, the
proof is completed.

∎

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