# Uniform K-stability of polarized spherical varieties

Thibaut Delcroix  Thibaut Delcroix, Univ Montpellier, CNRS, Montpellier,
France<EMAIL_ADDRESS>http://delcroix.perso.math.cnrs.fr/

(Date: 2020)

###### Abstract.

We express notions of K-stability of polarized spherical varieties in terms of
combinatorial data, vastly generalizing the case of toric varieties. We then
provide a combinatorial sufficient condition of $G$-uniform K-stability by
studying the corresponding convex geometric problem. Thanks to recent work of
Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable
sufficient condition of existence of constant scalar curvature Kähler metrics.
As a side effect, we show that, on several families of spherical varieties,
$G$-uniform K-stability is equivalent to K-polystability with respect to
$G$-equivariant test configurations for polarizations close to the
anticanonical bundle.

##  1\. Introduction

In the seminal article [Don02], Donaldson initiated a study of the existence
of constant scalar curvature Kähler metrics on polarized toric manifolds.
There, he notably introduced the general condition of K-stability, thus
formulating a precise version of the Yau-Tian-Donaldson conjecture whose aim
is to give an algebro-geometric characterization of the existence of constant
scalar curvature Kähler metrics on general polarized varieties. Focusing on
toric varieties, he further translated the condition of K-stability into a
convex geometric problem, and with additional work concluded in [Don09], he
was able to prove the Yau-Tian-Donaldson conjecture for non-singular toric
surfaces. Together with the case of Kähler-Einstein metrics, it is still today
one of the most convincing evidence for the conjecture.

The conjecture is still open in general, and examples (see [ACGTF08]) indicate
that the condition of K-stability should be modified slightly to a condition
of _uniform_ K-stability which has been introduced and refined by several
authors [Szé15, Der16, BHJ17, His20]. However, the work of Donaldson combined
with some more recent advances (notably [CC18b]) shows that the uniform Yau-
Tian-Donaldson conjecture is true for toric manifolds. A different proof of
this fact has been provided very recently by Li [Li20]. Odaka noticed that the
recent work of Li essentially proves the uniform Yau-Tian-Donaldson conjecture
for polarized spherical manifold as well (this is explained in more details in
his appendix to the present article).

Motivated by Odaka’s remark, we translate the uniform K-stability condition
into a convex geometric problem for polarized spherical varieties, in terms of
the combinatorial data encoding these. This task accomplished provides a much
wider playground than the toric case where one can try to show the (non-
uniform) Yau-Tian-Donaldson conjecture for different classes of varieties, or
derive explicit combinatorial conditions of existence of constant scalar
curvature Kähler metrics. We will concentrate here on the second goal and
provide a combinatorial sufficient criterion of uniform K-stability which
applies to a wide range of polarized spherical varieties. We intend to present
progress on the first goal in another article.

To provide the reader with a better flavor of the convex geometric problem
associated to K-stability of spherical varieties, let us first recall the case
of toric varieties. A polarized toric variety $(X,L)$ is a couple formed by a
complex normal $n$-dimensional projective variety $X$ equipped with an
effective action of $(\mathbb{C}^{*})^{n}$, and a
$(\mathbb{C}^{*})^{n}$-linearized ample line bundle. Such a data is fully
encoded by a convex polytope $\Delta$ with integral vertices in
$\mathbb{R}^{n}$, and the correspondence is very explicit: the integral points
of $\Delta$ coincide with the different $(\mathbb{C}^{*})^{n}$-weights of the
action of $(\mathbb{C}^{*})^{n}$ on the space of global holomorphic sections
$H^{0}(X,L)$. Let $\mathop{d\mu}$ denote the standard Lebesgue measure on
$\mathbb{R}^{n}$, and let $\mathop{d\sigma}$ denote the measure on
$\partial\Delta$ which coincides on each facet of $\Delta$ with the Lebesgue
measure on its affine span $V$, normalized to give unit mass to a fundamental
region of the lattice $\mathbb{Z}^{n}\cap V$. Donaldson shows that the
polarized toric variety $(X,L)$ is K-stable if and only if the following
functional is positive for any rational piecewise linear concave function $g$
on $\Delta$

$\operatorname{\mathcal{L}}(g)=2a\int_{\Delta}g\mathop{d\mu}-\int_{\partial\Delta}g\mathop{d\sigma}.$

where $a$ is the real number such that $\operatorname{\mathcal{L}}$ vanishes
on constants.

A polarized spherical variety $(X,L)$ is a couple formed by a normal
projective variety $X$ equipped with an action of a connected complex
reductive group $G$ and a $G$-linearized ample line bundle $L$ on $X$, such
that a Borel subgroup of $G$ acts with an open orbit on $X$. Such a variety is
encoded as well by combinatorial data, including the data of a convex polytope
in a real vector space $\mathbb{R}^{r}$ (of dimension smaller than $X$ in
general), but the definition of these is more involved. Let us just mention
the nature of the problem here and we refer to the body of the paper (mainly
section 3) for precise definitions. The functional
$\operatorname{\mathcal{L}}$ from the toric case is modified as:

(1)
$\operatorname{\mathcal{L}}(g)=\int_{\Delta}2g(aP-Q)\mathop{d\mu}-\int_{\partial\Delta}gP\mathop{d\sigma}$

where $\Delta$ is some convex polytope with rational vertices in
$\mathbb{R}^{r}$, the measure $\mathop{d\mu}$ and $\mathop{d\sigma}$ are
defined as in the toric case, $P$ and $Q$ are polynomials, and the scalar $a$
is still such that $\operatorname{\mathcal{L}}$ vanishes on constants. Let us
consider as well the functional

$\operatorname{\mathcal{J}}(g)=\int_{\Delta}(\max_{\Delta}g-g)P\mathop{d\mu}.$

We add that the polynomial $P$ is positive on the interior of $\Delta$, so
that the functional $\operatorname{\mathcal{J}}$ plays a role of (semi)-norm:
it is non-negative and vanishes only on constant concave functions. Our first
main statement translates conditions of K-stability into conditions on the
functionals above (see section 2 for a recall on these notions). Before
stating these, let us note that there is an additional combinatorial data
associated to a polarized spherical variety: its valuation cone, which may for
now simply be interpreted as the data of some full-dimensional convex cone
$\mathcal{V}$ in $(\mathbb{R}^{n})^{*}$ centered at $0$. We denote by
$\operatorname{Lin}(\mathcal{V})$ the linear part of $\mathcal{V}$, that is,
the largest linear subspace contained in $\mathcal{V}$.

######  Theorem 1.1.

Let $(X,L)$ be a polarized $G$-spherical variety. Then

  1. (1)

it is $G$-equivariantly K-polystable if and only if
$\operatorname{\mathcal{L}}(g)\geq 0$ for any rational piecewise linear
concave function $g:\Delta\to\mathbb{R}$ with slopes in $\mathcal{V}$, and
equality holds only if $g\in\operatorname{Lin}(\mathcal{V})$,

  2. (2)

and it is $G$-uniformly K-stable if and only if there exists an
$\varepsilon>0$ such that

$\operatorname{\mathcal{L}}(g)\geq\varepsilon\inf_{l\in\operatorname{Lin}(\mathcal{V})}\operatorname{\mathcal{J}}(g+l)$

for all rational piecewise linear concave functions $g:\Delta\to\mathbb{R}$
with slopes in $\mathcal{V}$.

We will further give another formulation of the condition of uniform
K-stability, and use this new formulation to obtain a sufficient criterion, by
proving a more general result as follows. Let $\Delta$ be a convex body in
$\mathbb{R}^{n}$, containing the origin in its interior. Let $\mathcal{V}$ be
a full-dimensional convex cone in $(\mathbb{R}^{n})^{*}$, with linear part
$\operatorname{Lin}(\mathcal{V})$, and fix $\mathcal{W}$ a linear complement
to $\operatorname{Lin}(\mathcal{V})$. Let $\mathcal{N}_{0}^{\infty}$ be the
set of all continuous concave functions $f$ on $\Delta$, smooth on the
interior with differentials in $\mathcal{V}$, such that $\max f=0$ and
$d_{0}f\in\mathcal{W}$. Let $P$ be a continuous function on $\Delta$, positive
on the interior. Let $J$ be a non-positive integrable function on $\Delta$
with negative integral, and let $K$ be an integrable function with integral
zero on $\Delta$. Consider the functional $\operatorname{\mathcal{L}}_{s}$
defined on $\mathcal{N}_{0}^{\infty}$ by

(2)
$\operatorname{\mathcal{L}}_{s}(f)=\int_{\Delta}(f(x)K(x)+d_{x}f(x)J(x))\mathop{d\mu}(x)$

We refer to section 6 for the details on how this setting and our second main
theorem that follows relates to the previous uniform K-stability question. Let
us just say as an appetizer that
$\operatorname{\mathcal{L}}(f)=\operatorname{\mathcal{L}}_{s}(f)$ for
functions $f\in\mathcal{N}_{0}^{\infty}$ and well-chosen $K$ and $J$.

######  Theorem 1.2.

Assume that $K+J\leq 0$. Let $b$ be the element of $\Delta$ defined by

$\int_{\Delta}(x-b)(K(x)+J(x))\mathop{d\mu}(x)=0.$

If $b$ is in the relative interior of $-\mathcal{V}^{\vee}$, then there exists
an $\varepsilon>0$ such that

$\operatorname{\mathcal{L}}_{s}(f)\geq\varepsilon\int_{\Delta}(-f)P\mathop{d\mu}$

for all $f\in\mathcal{N}_{0}^{\infty}$.

Despite its simplicity, the above theorem actually provides an explicit and
tractable condition to check for a huge family of polarized spherical
varieties. The main evidence for this follows from two facts. The first one is
that, when specialized to the situation coming from an anti-canonically
polarized spherical variety, all assumptions but the condition on $b$ are
automatically satisfied, and that condition translates to the criterion for
K-polystability with respect to $G$-equivariant special test configurations
obtained in [Del20]. In fact, as we will show in Section 8, the condition on
$b$ translates to $G$-equivariant K-polystability with respect to special test
configurations in all cases.

The second evidence is that we can prove in many cases that if we fix the
variety $X$ but vary the polarization, the condition on $K+J$ is open. These
two facts together show that one obtains an explicit condition to check
existence of constant scalar curvature Kähler metrics on smooth Fano
varieties, for polarizations close to the anti-canonical one (again, for an
explicit range). We will prove this for toroidal horospherical varieties, as
well as for (non-Hermitian) symmetric varieties, but we expect this to hold
much more generally and our proof easily adapts to different situations.

Finally, we note that the condition on $b$ consists of
$\dim(\operatorname{Lin}(\mathcal{V}))$ closed conditions and
$\dim(\mathcal{V})-\dim(\operatorname{Lin}(\mathcal{V}))$ open conditions. In
particular, on a Kähler-Einstein Fano symmetric (non-Hermitian) manifold such
that $\operatorname{Lin}(\mathcal{V})=\\{0\\}$, our theorem shows that there
exists cscK metrics on an explicit neighborhood of the anti-canonical line
bundle.

The paper is organized as follows. Section 2 is devoted to recollection of
notions on K-stability. Section 3 summarizes key combinatorial properties of
spherical varieties. In Section 4, we associate to a spherical test
configuration a piecewise linear concave function, and translate the effect of
twisting a test configuration in terms of tis function. We then express, in
Section 5 the non-Archimedean functionals for spherical test configurations as
functionals on the associated concave functions. We will show how Theorem 1.2
applies to K-stability in Section 6, then we prove this theorem in Section 7.
We provide a full statement of the sufficient condition of $G$-uniform
K-stability thus obtained in Section 8. The remaining three sections are
devoted to illustrating the applicability of the sufficient condition. In
Section 9, we apply our criterion to the blowup of the three dimensional
quadric at a one dimensional subquadric, and obtain from this the existence of
cscK metrics in an explicit neighborhood of the anticanonical line bundle. We
show that the criterion always apply for the anticanonical line bundle on Fano
manifolds in Section 10. In the final Section 11, we provide examples of
statements to the effect that in a wide range of situations, our criterion
shows that $G$-uniform K-stability is equivalent to $G$-equivariant
K-polystability with respect to special test configurations for polarizations
close to the anticanonical line bundle. The appendix by Yuji Odaka shows that
for non-singular spherical varieties, $G$-uniform K-stability is equivalent to
existence of cscK metrics.

Acknowledgements. This research received partial funding from ANR Project
FIBALGA ANR-18-CE40-0003-01.

##  2\. Background on K-stability

Our references for this section are [BHJ17, His20]. We recall the main notions
for the reader’s convenience.

Let $G$ be a complex reductive group. Let $(X,L)$ be a $G$-polarized variety.
A (normal, ample) _$G$ -equivariant test configuration_ for $(X,L)$ consists
of the data of a normal $(G\times\mathbb{C}^{*})$-variety $\mathcal{X}$, a
$(G\times\mathbb{C}^{*})$-linearized ample line bundle $\mathcal{L}$ on
$\mathcal{X}$, and a $\mathbb{C}^{*}$-equivariant flat morphism
$\pi:(\mathcal{X},\mathcal{L})\to\mathbb{C}$ whose fiber
$(X_{1},\mathcal{L}_{1})$ over $1$ is $G$-equivariantly isomorphic to
$(X,L^{r})$ for some $r\in\mathbb{Z}_{>0}$. If the (scheme-theoretic) central
fiber is normal, then the test configuration is called _special_. If the total
space of the test configuration is ($G$-equivariantly) isomorphic to
$X\times\mathbb{C}$, then the test configuration is called a product test
configuration.

The numerical invariants associated to a test configuration
$(\mathcal{X},\mathcal{L})$ may be defined in terms of the central fiber
$(X_{0},\mathcal{L}_{0})$ as follows. First note that it is equipped with a
$\mathbb{C}^{*}$-action induced by the action on the test configuration. For
$k\in\mathbb{N}$, let $d_{k}$ denote the dimension $\dim
H^{0}(X_{0},\mathcal{L}_{0}^{k})$, let
$\lambda_{1,k},\ldots,\lambda_{d_{k},k}$ denote the weights of the
$\mathbb{C}^{*}$-action on $H^{0}(X_{0},\mathcal{L}_{0}^{k})$, let $w_{k}$
denote the sum of the $\lambda_{i,k}$. The quotient $\frac{w_{k}}{kd_{k}}$
admits an expansion in powers of $k$ at infinity, and we will be interested in
the first two terms:

$\frac{w_{k}}{kd_{k}}=F_{0}+F_{1}k^{-1}+o(k^{-1}).$

The _non-Archimedean $J$-functional_ of a test configuration
$(\mathcal{X},\mathcal{L})$ is

$\operatorname{J^{NA}}(\mathcal{X},\mathcal{L})=\sup\\{\lambda_{i,k}/k\mid
k\in\mathbb{Z}_{>0},1\leq i\leq d_{k}\\}-F_{0}$

and the _Donaldson-Futaki invariant_ is

$\operatorname{DF}(\mathcal{X},\mathcal{L})=-F_{1}.$

It is often more convenient to work with the _non-Archimedean Mabuchi
functional_ $\operatorname{M^{NA}}$ instead of the Donaldson-Futaki invariant.
Indeed, the latter does not vary linearly with base changes of the form
$z\mapsto z^{m}$ on the test configurations, while the former does. Their
values coincide when the central fiber is reduced, and the Donaldson-Futaki
invariant is always greater than, or equal to the non-Archimedean Mabuchi
functional. Furthermore, given any test configuration, there exists a base
change such that the resulting test configuration has reduced central fiber,
and base change preserves $G$-equivariance.

The $G$-polarized variety $(X,L)$ is called _$G$ -equivariantly K-semistable_
if $\operatorname{M^{NA}}(\mathcal{X},\mathcal{L})\geq 0$ for all $G$
equivariant test configurations, and _$G$ -equivariantly K-polystable_ if
furthermore $\operatorname{M^{NA}}(\mathcal{X},\mathcal{L})=0$ if and only if
the test configuration is a product test configuration. We also use the self-
explaining terminology _$G$ -equivariantly K-polystable with respect to
special test configurations_, and abbreviate it to _$G$ -stc K-polystable_.

The total space of a given $G$-equivariant test configuration
$(\mathcal{X},\mathcal{L})$ may actually be the underlying total space of
several different $G$-equivariant test configurations. Let $F$ be the group of
$(G\times\mathbb{C}^{*})$-equivariant automorphisms of $\mathcal{X}$. It
contains the factor $\mathbb{C}^{*}$. Let $\mathfrak{Y}(F)$ denote the set of
one-parameter subgroups of $F$. Let $\beta$ denote the character of
$\mathbb{C}^{*}$ of weight $1$, which is identified with a character of $F$.
Any one-parameter subgroup $\eta\in\mathfrak{Y}(F)$ such that $\beta(\eta)=1$
defines a $\mathbb{C}^{*}$-action on $(\mathcal{X},\mathcal{L})$, a projection
to $\mathbb{C}$, and hence a $G$-equivariant test configuration
$(\mathcal{X},\mathcal{L})_{\eta}$, in general different from the initial
$(\mathcal{X},\mathcal{L})$. The test configuration
$(\mathcal{X},\mathcal{L})_{\eta}$ is called the _twist by $\eta$ of the test
configuration $(\mathcal{X},\mathcal{L})$_. Since we can work up to base
change, the same notion makes sense for any element of
$\eta\in\mathfrak{Y}(F)\otimes\mathbb{Q}$ with $\beta(\eta)>0$. One can
actually extend the definition to irrational
$\eta\in\mathfrak{Y}(F)\otimes\mathbb{R}$ though it is not needed for our
paper.

Without a group $G$ involved, _(J-)uniform K-stability_ is defined as the
existence of a positive constant $\varepsilon>0$ such that for all test
configurations,

$\operatorname{M^{NA}}(\mathcal{X},\mathcal{L})\geq\varepsilon\operatorname{J^{NA}}(\mathcal{X},\mathcal{L}).$

The polarized variety $(X,L)$ is called _$G$ -uniformly K-stable_ if there
exists a positive constant $\varepsilon>0$ such that for all $G$-equivariant
test configurations,

$\operatorname{M^{NA}}(\mathcal{X},\mathcal{L})\geq\varepsilon\inf_{\eta\in\mathfrak{Y}(F)\otimes\mathbb{Q},~{}\beta(\eta)=1}\operatorname{J^{NA}}(\mathcal{X},\mathcal{L})_{\eta}.$

In other words and up to base changes, on the right hand side, instead of the
$\operatorname{J^{NA}}$ of the test configuration, we consider the infimum of
$\operatorname{J^{NA}}$ over all test configurations with
$(G\times\mathbb{C}^{*})$-isomorphic polarized total space but different
projections to $\mathbb{C}$.

##  3\. Background on polarized spherical varieties

Our references for this section are [Kno91, Bri89]. We recall the main notions
for the reader’s convenience.

Let $G$ be a complex connected reductive group. We fix a choice of a Borel
subgroup $B\subset G$ and a choice of a maximal torus $T\subset B$. Let $G/H$
be a _spherical $G$-homogeneous space_, that is, such that $B$ acts on $G/H$
with an open orbit. The _spherical lattice_ $M$ of $G/H$ is the subgroup of
the group of characters of $B$ consisting of the weights of $B$-eigenfunctions
in the field $\mathbb{C}(G/H)$. We denote by $N$ the dual lattice:
$N=\operatorname{Hom}(M,\mathbb{Z})$. Since $B$ has an open orbit, the value
of a $B$-invariant valuation on a $B$-eigenfunction depends only on the
eigenvalue, which is an element of $M$. We denote by $\varrho$ the map from
the set of $B$-invariant valuations of $\mathbb{C}(G/H)$ to
$N\otimes\mathbb{R}$. The image of the subset of $G$-invariant valuations
generates a cosimplicial convex cone $\mathcal{V}$ called the _valuation cone_
of $G/H$.

The $G$-equivariant embeddings of $G/H$ are in one-to-one correspondence with
_colored fans_ (see [Kno91] for a detailed exposition of this correspondence).
Let $X$ be a complete $G$-equivariant embedding of $G/H$, with colored fan
$\mathcal{F}_{X}$. Let $\mathcal{P}_{X}$ be the set of $B$-stable prime
divisors in $X$. It is a finite set comprised of closures of codimension one
$B$-orbits in $G/H$ and of closures of codimension one $G$-orbits, the latter
corresponding to colorless rays in $\mathcal{F}_{X}$. We identify such a
divisor $D$ with the induced valuation, and thus get an associated element
$\varrho(D)$ of $N\otimes\mathbb{R}$ via the map $\varrho$.

Let $L$ be an ample $G$-linearized line bundle on $X$. Fix a $B$-semi-
invariant section $s$ of $L$ and denote its $B$-weight by $\chi$. The
$B$-invariant Cartier divisor on $X$ defined by $s$ is of the form

(3) $\sum_{D\in\mathcal{P}_{X}}n_{D}D$

and there exists an integral piecewise linear function $f$ defined on
$N\otimes\mathbb{R}$, such that $(f\circ\varrho)(D)=n_{D}$ whenever $D$
contains a $G$-orbit (in other words, when either $D$ is $G$-invariant or $D$
is a _color_ of $X$). Conversely, a divisor as in (3) with the same property
defines a Cartier divisor.

Since $L$ is ample, the minimal function $f$ which satisfies the above
condition further satisfies:

  1. i)

$f$ is a convex function,

  2. ii)

$(f\circ\varrho)(D)<n_{D}$ for each $D\in\mathcal{P}_{X}$ which does not
contain $G$-orbits,

  3. iii)

the slopes of $f$ on two distinct maximal cones of $\mathcal{F}_{X}$ are
distinct.

Brion defines a polytope $\Delta=\Delta(X,L)$ associated to $s$ as the convex
polytope in $M\otimes\mathbb{R}$ defined by the equations
$\varrho(D)(m)+n_{D}\geq 0$ for all $D\in\mathcal{P}_{X}$. Integral points of
$\Delta$ give the decomposition of $H^{0}(X,L)$ as a $G$-representation. More
precisely, if $V_{\lambda}$ denotes an irreducible representation of $G$ with
highest weight $\lambda$, then $H^{0}(X,L)$ is $G$-isomorphic to

$\bigoplus_{m\in M\cap\Delta}V_{\chi+m}$

where we recall that $\chi$ is the $B$-weight of $s$.

The degree $L^{n}$ is obtained from this polytope by Brion as follows. Let
$R^{+}$ denote the positive root system of $(G,B,T)$ and let $R_{X}^{+}$
denote the set of positive roots that are _not_ orthogonal to $\chi+\Delta$.
Then

(4) $L^{n}=n!\int_{\Delta}P\mathop{d\mu}$

where $\mathop{d\mu}$ is the Lebesgue measure on $M\otimes\mathbb{R}$
normalized by $M$, and $P$ is the Duistermaat-Heckman polynomial defined by:

(5) $P(x)=\prod_{\alpha\in R_{X}^{+}}\frac{\langle
x+\chi,\alpha\rangle}{\langle\varpi,\alpha\rangle}$

where $\varpi$ is the half sum of positive roots of $G$. The above result is
proved by considering the first order asymptotic of the dimensions of the
spaces of pluri-sections $H^{0}(X,L^{k})$ and Weyl’s dimension formula

$\dim(V_{\lambda})=\prod_{\alpha\in
R^{+}}\frac{\langle\alpha,\lambda+\varpi\rangle}{\langle\alpha,\varpi\rangle}.$

Indeed, up to the $n!$ factor, the highest order (in $k$) coefficient of
$\sum_{m\in M\cap k\Delta}\dim(V_{k\chi+m})$ gives the integral in (4) and the
polynomial appearing in this volume is the highest order summand of the
polynomial giving the dimension formula, restricted to the affine space
$\chi+M\otimes\mathbb{R}$.

We will actually need the following refinement, which is a consequence of a
general result of Pukhlikov and Khovanskii [PK92]. The dimension $\dim
H^{0}(X,L^{k})$ admits an expansion in powers of $k$ of the form

(6) $\dim
H^{0}(X,L^{k})=k^{n}\int_{\Delta}P\mathop{d\mu}+k^{n-1}\left(\frac{1}{2}\int_{\partial\Delta}P\mathop{d\sigma}+\int_{\Delta}Q\mathop{d\mu}\right)+o(k^{n-1})$

where

(7) $Q(x)=\sum_{\alpha\in
R_{X}^{+}}\frac{\langle\alpha,\varpi\rangle}{\langle\alpha,x+\chi\rangle}P(x),$

and $\mathop{d\sigma}$ denotes the measure on $\partial\Delta$ which coincides
on each facet with the Lebesgue measure normalized by the intersection of $M$
with the affine space spanned by the face.

##  4\. Test configurations for polarized spherical varieties

###  4.1. Statement

In this section we encode equivariant test configurations for polarized
spherical varieties by certain concave piecewise linear functions. In addition
to Donaldson’s work on toric varieties [Don02], this task has already been
accomplished in different special cases [AB04, Nyb, Del20]. We freely use
notations from section 3.

######  Theorem 4.1.

For a polarized spherical variety $(X,L)$, $G$-equivariant test configurations
are in one-to-one correspondence with positive rational piecewise linear
concave functions on $\Delta(X,L)$, with slopes in the valuation cone of $X$.

The test configuration is furthermore special if the associated function is
integral linear, and it is product if it is integral linear, with slope in
$\operatorname{Lin}(\mathcal{V})$.

Furthermore, rational twists of a given test configuration
$(\mathcal{X},\mathcal{L})$ are in one-to-one correspondence with elements of
$\operatorname{Lin}(\mathcal{V})\cap N\otimes\mathbb{Q}$ and if $g$ is the
function associated to $(\mathcal{X},\mathcal{L})$, then the set of functions
corresponding to the twists is $\\{g+l\mid
l\in\operatorname{Lin}(\mathcal{V})\cap N\otimes\mathbb{Q}\\}$.

The correspondence is explicitly described in the proof below. The key picture
to keep in mind is that the polytope associated (as in [Bri89]) to a trivially
compactified test configuration can be described as the set of point below the
graph of a concave integral piecewise linear function on some multiple of the
polytope associated to the initial polarized variety.

###  4.2. From a test configuration to a concave function…

Let $(X,L)$ be a polarized spherical variety. Let $(\mathcal{X},\mathcal{L})$
be a $G$-equivariant test configuration for $(X,L)$. We still denote by
$(\mathcal{X},\mathcal{L})$ the _trivially compactified_ $G$-equivariant test
configuration for $(X,L)$. 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 keep the notation $(\mathcal{X},\mathcal{L})$ for the family over
$\mathbb{P}^{1}$.

Note that $(\mathcal{X},\mathcal{L})$ is a polarized spherical variety under
the action of $G\times\mathbb{C}^{*}$ (this remark will also be used in
Odaka’s appendix). Its open orbit is $G/H\times\mathbb{C}^{*}$, and the
combinatorial data are easily derived from that of $X$.

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

(8) $\sum_{\hat{D}\in\mathcal{P}_{\mathcal{X}}}n_{\hat{D}}\hat{D}$

where $\mathcal{P}_{\mathcal{X}}$ is the set of prime
$(B\times\mathbb{C}^{*})$-stable divisors on $\mathcal{X}$.

There are three types of such divisors.

  * •

Each divisor $\hat{D}\in\mathcal{P}_{\mathcal{X}}$ with
$\hat{\varrho}(\hat{D})\in(N\otimes\mathbb{R}\times\\{0\\})$ must be of the
form $\hat{D}=\overline{D\times\mathbb{C}^{*}}$ for some
$D\in\mathcal{P}_{X}$. Then by our choice of section, $n_{\hat{D}}=rn_{D}$.
Note that since the Borel subgroup of $\mathbb{C}^{*}$ is $\mathbb{C}^{*}$
itself, all other elements of $\mathcal{P}_{\mathcal{X}}$ must be
$(G\times\mathbb{C}^{*})$-stable.

  * •

There is only one divisor $\hat{D}\in\mathcal{P}_{\mathcal{X}}$ such that
$\hat{\varrho}(\hat{D})\in(\mathcal{V}\times\mathbb{R}_{>0})$, this is the
fiber $X_{\infty}$, because $(\mathcal{X},\mathcal{L})$ is trivial at
$\infty$. The choice of section, on the other hand, implies that for this
divisor, $n_{\hat{D}}=0$.

  * •

Let $\mathcal{A}$ denote the set
$\hat{\varrho}(\mathcal{P}_{\mathcal{X}})\cap(\mathcal{V}\times\mathbb{R}_{<0})$,
and write each element in $\mathcal{A}$ as a couple $(u,s)$ with
$u\in\mathcal{V}$ and $s\in\mathbb{R}_{<0}$. Let us write also $n_{u,s}$ for
the corresponding coefficient in (8).

Let $\hat{\Delta}$ denote the polytope in $\hat{M}\otimes\mathbb{R}$
associated to the divisor (8). In view of the previous description of the
divisor, the polytope can be described as

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

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

$g(x)=\inf_{(u,s)\in\mathcal{A}}\left(\frac{ru(x)+n_{u,s}}{-s}\right).$

Note that since each $s$ is negative, the slopes $(-s)^{-1}ru$ are rational
point of the valuation cone $\mathcal{V}$.

######  Remark 4.2.

It may not seem natural that the positive direction corresponds to $\infty$
for the reader accustomed to a certain point of view on toric varieties. It
stems from the fact that, under the action $w\cdot f(z)=f(w^{-1}z)$, the
function $f:z\mapsto z^{k}$ is a $\mathbb{C}^{*}$-eigenvector with eigenvalue
the one parameter subgroup $\chi:w\mapsto w^{-k}$ rather than $f$ itself.

###  4.3. …and back

We now explain how to reconstruct a test configuration from a concave
function. Let $g$ be a positive rational piecewise linear concave function on
$\Delta(X,L)$, with slopes in the valuation cone of $X$. We can find a
positive integer $r$, a subset $\mathcal{A}$ of
$\mathcal{V}\times\mathbb{Z}_{<0}$, and integers $n_{u,s}$ for all
$(u,s)\in\mathcal{A}$ such that

$g(x)=\inf_{(u,s)\in\mathcal{A}}\left(\frac{ru(x)+n_{u,s}}{-s}\right),$

and $u$ is a primitive element of $N$ for all $(u,s)\in\mathcal{A}$.

Consider the polytope

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

We build a colored fan $\mathcal{F}_{\mathcal{X}}$ for the
$(G\times\mathbb{C}^{*})$-homogeneous space $G/H\times\mathbb{C}^{*}$ from
$\hat{\Delta}$ as follows. Recall that we are given $(X,L)$ and the
corresponding divisor (3), and that colors of $G/H\times\mathbb{C}^{*}$ may be
identified with colors of $G/H$. We first include in
$\mathcal{F}_{\mathcal{X}}$ the colored cones $(\sigma\times\\{0\\},S)$ and
$(\sigma\times\mathbb{R}_{>0},S)$, where $(\sigma,S)$ is a colored cone of
$\mathcal{F}_{X}$. These account for the trivial family over
$\mathbb{C}^{*}\cup\\{\infty\\}$. To complete the fan, we add, for each cone
$\sigma$ in the opposite of the normal fan to $\hat{\Delta}$ which has not
been considered yet and whose intersection with the interior of
$\mathcal{V}\times\mathbb{R}$ is non-empty, a colored cone
$(\sigma,S)\in\mathcal{F}_{\mathcal{X}}$, where $S$ is defined as follows. It
suffices to define it for maximal colored cones. For such a cone $\sigma$, let
$m_{\sigma}$ denote the corresponding vertex of $\hat{\Delta}$. Then $S$ is
the set of all colors $D$ of $G/H\times\mathbb{C}^{*}$ in $\sigma$ such that
$-\varrho(\overline{D})(m_{\sigma})+n_{D}>0$.

We have thus defined a colored fan, hence an embedding $\mathcal{X}$ of
$G/H\times\mathbb{C}^{*}$. As follows from the description of equivariant
morphisms between spherical varieties [Kno91, Theorem 4.1], $\mathcal{X}$
admits a $\mathbb{C}^{*}$-equivariant surjective morphism to $\mathbb{P}^{1}$,
which induces a trivial family with fiber $X$ over the affine chart
$\mathbb{C}^{*}\cup\\{\infty\\}$. We identify this subvariety with
$X\times(\mathbb{C}^{*}\cup\\{\infty\\})$.

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

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

where

  * •

$n_{\hat{D}}=rn_{D}$ for each divisor $\hat{D}\in\mathcal{P}_{\mathcal{X}}$
with $\hat{\varrho}(\hat{D})\in(N\otimes\mathbb{R}\times\\{0\\})$
(equivalently $\hat{D}=\overline{D\times\mathbb{C}^{*}}$ for some
$D\in\mathcal{P}_{X}$).

  * •

$n_{\hat{D}}=0$ for the only divisor $\hat{D}\in\mathcal{P}_{\mathcal{X}}$
such that $\hat{\varrho}(\hat{D})\in(\mathcal{V}\times\mathbb{R}_{>0})$, which
is the fiber $X_{\infty}$.

  * •

$n_{\hat{D}}=n_{u,s}$ for $(u,s)\in\mathcal{A}$ and $\hat{D}$ the $G$-stable
divisor which is the closure of the codimension one $G$-orbit associated to
the colorless ray generated by $(u,s)$.

In particular, the restriction of this divisor to
$X\times(\mathbb{C}^{*}\cup\\{\infty\\})$ is the product of the divisor (3)
with $\mathbb{C}^{*}\cup\\{\infty\\}$. Furthermore, this divisor satisfies the
ampleness assumption. This is not obvious since the colors of $\mathcal{X}$
are different from the colors of $X$ in general, but our choices of colors for
each colored cone was tailored to ensure ampleness. The associated line bundle
$\mathcal{O}(d)$ is $(G\times\mathbb{C}^{*})$-linearizable (maybe up to
passing to a suitable finite tensor power, which does not seriously affect our
statement). Choosing the linearization such that the natural section $s$ of
$\mathcal{O}(d)$ is $\mathbb{C}^{*}$-invariant and has $B$-weight $\chi$
yields the final identification of $\mathcal{O}(d)$ with the pull-back of $L$
by the first projection on $X\times(\mathbb{C}^{*}\cup\\{\infty\\})$. This
concludes the construction of the test configuration
$(\mathcal{X},\mathcal{L})$.

###  4.4. Effect of twisting

We now elucidate the different possible twists of a given test configuration,
as involved in the definition of $G$-uniform K-stability. For a spherical
homogeneous space $G/H\times\mathbb{C}^{*}$, one can easily identify
$\operatorname{Aut}_{G\times\mathbb{C}^{*}}(G/H\times\mathbb{C}^{*})$: it is
the group $N_{G\times\mathbb{C}^{*}}(H\times\\{1\\})/(H\times\\{1\\})$, acting
on the right on $G/H\times\mathbb{C}^{*}$. Furthermore, this group is
diagonalizable and the action of its neutral component $F$ extends to any
embedding. Finally, $\mathfrak{Y}(F)\times\mathbb{R}$ may be identified with
the linear part $\operatorname{Lin}(\mathcal{V})$ of the valuation cone of
$G/H\times\mathbb{C}^{*}$.

The above is not actually necessary since we can identify the possible twists
directly by the theory of spherical embeddings. Indeed, the colored fan of the
twist of a test configuration and of the initial test configurations are the
same, as well as the combinatorial data identifying the line bundle. Note that
what we just wrote is true for the total space of the test configuration
itself, but not the compactification, which depends on the twist. The only
difference is thus that the privileged direction coming from the factor
$\mathbb{C}^{*}$ can be chosen differently, and that will affect the final
expression of $g$.

More precisely, the direction can be chosen arbitrarily among those directions
in $(\operatorname{Lin}(\mathcal{V})\cap N)\times\\{1\\}$, or
$(\operatorname{Lin}(\mathcal{V})\cap(N\otimes\mathbb{Q}))\times\\{1\\}$ to
allow for rational twists. The effect on $g$ is by adding the function $l$ for
some element $l$ of $\operatorname{Lin}(\mathcal{V})\cap N\otimes\mathbb{Q}$.
This concludes the proof of Theorem 4.1. ∎

##  5\. Non-Archimedean functionals for spherical test configurations

###  5.1. Statement

In this section, we compute the non-Archimedean Mabuchi functional and the
non-Archimedean $J$ functional of the test configuration
$(\mathcal{X},\mathcal{L})$ associated to the concave function $g$. The
computation follows the method of [Don02], and was previously used to obtain
sub-cases of our result in [Nyb, AK05].

We use the notations of section 3 and we set the notations

$V:=\int_{\Delta}P\mathop{d\mu}$

and

$a:=\frac{1}{2V}\left(\int_{\partial\Delta}P\mathop{d\sigma}+2\int_{\Delta}Q\mathop{d\mu}\right).$

######  Theorem 5.1.

We have

$\operatorname{M^{NA}}(\mathcal{X},\mathcal{L})=\frac{1}{V}\left(a\int_{\Delta}gP\mathop{d\mu}-\frac{1}{2}\int_{\partial\Delta}gP\mathop{d\sigma}-\int_{\Delta}gQ\mathop{d\mu}\right)$

and

$\operatorname{J^{NA}}(\mathcal{X},\mathcal{L})=\frac{1}{V}\int_{\Delta}(\max_{\Delta}g-g)P\mathop{d\mu}.$

With the notations of the introduction, we thus have
$2V\operatorname{M^{NA}}(\mathcal{X},\mathcal{L})=\operatorname{\mathcal{L}}(g)$
and
$V\operatorname{J^{NA}}(\mathcal{X},\mathcal{L})=\operatorname{\mathcal{J}}(g)$.

###  5.2. Proof

Let $X_{x}$ denote the fiber of the test configuration, for
$x\in\mathbb{P}^{1}=\mathbb{C}\cup\\{\infty\\}$. Donaldson uses the following
exact (for large enough $k$) sequences of $\mathbb{C}^{*}$-representations
obtained by restriction of sections:

$0\to H^{0}(\mathcal{X},\mathcal{L}^{k}\otimes\mathcal{O}(-X_{0}))\to
H^{0}(\mathcal{X},\mathcal{L}^{k})\to H^{0}(X_{0},\mathcal{L}^{k}|_{X_{0}})\to
0$ $0\to H^{0}(\mathcal{X},\mathcal{L}^{k}\otimes\mathcal{O}(-X_{\infty}))\to
H^{0}(\mathcal{X},\mathcal{L}^{k})\to
H^{0}(X_{\infty},\mathcal{L}^{k}|_{X_{\infty}})\to 0$

Donaldson further notes that the $\mathbb{C}^{*}$-action on
$H^{0}(X_{\infty},\mathcal{L}^{k}|_{X_{\infty}})$ is trivial, and that the
family of weights of the $\mathbb{C}^{*}$-representation
$H^{0}(\mathcal{X},\mathcal{L}^{k}\otimes\mathcal{O}(-X_{\infty}))$ is
$(\lambda_{i}+1)_{i\in I}$, if $(\lambda_{i})_{i\in I}$ is the family of
weights of the $\mathbb{C}^{*}$-representation
$H^{0}(\mathcal{X},\mathcal{L}^{k}\otimes\mathcal{O}(-X_{0}))$.

This allows us to express the quantities involved in the definitions of the
Donaldson-Futaki invariant and the non-Archimedean $J$-functional as follows.
The sum $w_{k}$ of weights of $H^{0}(X_{0},\mathcal{L}^{k}|_{X_{0}})$ is given
by

$w_{k}=\dim H^{0}(\mathcal{X},\mathcal{L}^{k})-\dim
H^{0}(X_{0},\mathcal{L}^{k}|_{X_{0}}).$

Furthermore, if $\lambda_{k}$ denotes the maximum of all weights of
$H^{0}(X_{0},\mathcal{L}^{k}|_{X_{0}})$, then

$\sup_{k}\frac{\lambda_{k}}{k}=\max_{\Delta}g$

.

In order to use the expansion (6) applied to $\hat{\Delta}$, we may remark
that for any (say continuous) function $f$ on $\Delta$,

$\int_{\hat{\Delta}}f(x)\mathop{d\mu}(x,t)=\int_{\Delta}f(x)g(x)\mathop{d\mu}(p)$

and

$\int_{\partial\hat{\Delta}}f(x)\mathop{d\sigma}(x,t)=2\int_{\Delta}f(x)\mathop{d\mu}(x)+\int_{\partial\Delta}g(x)f(x)\mathop{d\sigma}(x).$

The latter equality follows from the decomposition of the boundary as the
slice $\hat{\Delta}\cap(M\otimes\mathbb{R}\times\\{0\\})$, the graph
$\\{(x,g(x))\mid p\in\Delta\\}$ of $g$ (giving each one half of the first
summand), and the vertical part $\\{(x,t)\mid x\in\partial\Delta,0\leq t\leq
g(x)\\}$ giving the second summand. For the graph of $g$ to give the right
contribution, given the definition of $\mathop{d\sigma}$, it is actually
_necessary to assume_ that $g$ is defined by _integral_ linear forms. We can
restrict to this case by base change since we are interested in the non-
Archimedean Mabuchi functional rather than the Donaldson-Futaki invariant.

By the expansion (6) applied to both $\Delta$ and $\hat{\Delta}$, we obtain
the following expansions.

$kd_{k}=k^{n+1}\int_{\Delta}P\mathop{d\mu}+k^{n}\left(\frac{1}{2}\int_{\partial\Delta}P\mathop{d\sigma}+\int_{\Delta}Q\mathop{d\mu}\right)+o(k^{n})$
$w_{k}=k^{n+1}\int_{\Delta}gP\mathop{d\mu}+k^{n}\left(\frac{1}{2}\int_{\partial\Delta}gP\mathop{d\sigma}+\int_{\Delta}gQ\mathop{d\mu}\right)+o(k^{n})$

Writing $w_{k}=Ak^{n+1}+Bk^{n}+o(k^{n})$ and
$kd_{k}=Ck^{n+1}+Dk^{n}+o(k^{n})$, we have

$\frac{w_{k}}{kd_{k}}=\frac{A}{C}+\frac{1}{C}\left(B-\frac{AD}{C}\right)\frac{1}{k}+o\left(\frac{1}{k}\right).$

Replacing with the expressions above proves Theorem 5.1. ∎

Putting together the results of section 4 and section 5 proves Theorem 1.1. ∎

##  6\. Restating the problem

In this section we will show how Theorem 1.2 applies to the uniform
K-stability problem. For this we will obtain a new expression of
$\operatorname{\mathcal{L}}$ when applied to smooth functions, and show how to
derive uniform K-stability in these terms. To simplify notations, we assume
(by choosing the global section appropriately) that the origin $0\in
M\otimes\mathbb{R}$ is in the interior of the polytope $\Delta$.

###  6.1. A new expression of $\operatorname{\mathcal{L}}$ on smooth functions

Let $E_{1},\ldots,E_{k}$ denote the facets of $\Delta$, and let $T_{i}$ denote
the pyramid with vertex the origin and base $E_{i}$. This provides in
particular a decomposition $\Delta=\bigcup_{i}T_{i}$. The author learned the
idea of using such a decomposition in [ZZ08, LZZ18].

We will need notations for the set of equations defining $\Delta$ as follows:
for each facet, let $u_{i}$ denote the outward-pointing primitive normal in
$N$, and let $n_{i}$ be the positive number such that

$\Delta=\\{x\in M\otimes\mathbb{R}\mid\forall i,u_{i}(x)\leq n_{i}\\}.$

Let $J$ and $K$ be the functions on $\Delta$ defined (almost-everywhere) by

(9) $J(x)=\frac{-P(x)}{n_{i}}$

and

(10) $K(x)=2aP(x)-2Q(x)-\frac{1}{n_{i}}d_{x}P(x)-\frac{1}{n_{i}}rP(x)$

for $x\in\operatorname{Int}(T_{i})$, where $r$ denotes the dimension of
$M\otimes\mathbb{R}$, also called the _rank_ of $X$. Note that these functions
are not continuous in general, but piecewise polynomial with respect to the
decomposition of the polytope, hence integrable. Furthermore, $J$ is negative
on the interior of $\Delta$.

######  Proposition 6.1.

For any continuous function $f$ on $\Delta$, smooth on the interior, we have

$\operatorname{\mathcal{L}}(f)=\int_{\Delta}(f(x)K(x)+d_{x}f(x)J(x))\mathop{d\mu}(x).$

###### Proof.

We identify $M\otimes\mathbb{R}$ with the Euclidean space $\mathbb{R}^{r}$ by
choosing a basis of $M$. Let $\nu$ denote the unit outward pointing normal
vector to $\partial\Delta$ and let $\mathop{d\sigma_{e}}$ denote the area
measure on $\partial\Delta$. Let also $(x\cdot\nu)$ denote the scalar product
of $x\in M\otimes\mathbb{R}$ with $\nu$ induced by the identification with
$\mathbb{R}^{r}$.

For $f$ smooth, the divergence theorem yields for all $i$,

$\int_{E_{i}}f(x)P(x)(x\cdot\nu)\mathop{d\sigma_{e}}(x)=\int_{T_{i}}\left(f(x)d_{x}P(x)+d_{x}f(x)P(x)+rf(x)P(x)\right)\mathop{d\mu}(x).$

Let $c_{i}$ denote the constant such that $u_{i}(x)=c_{i}(x\cdot\nu|_{E_{i}})$
for $x\in M\otimes\mathbb{R}$. Then
$\mathop{d\sigma_{e}}=c_{i}\mathop{d\sigma}$ on $E_{i}$ and
$(x\cdot\nu)=\frac{n_{i}}{c_{i}}$ for $x\in E_{i}$, hence in the left-hand
side above, we may replace $(x\cdot\nu)\mathop{d\sigma_{e}}$ with
$n_{i}\mathop{d\sigma}$.

Then using the decomposition $\Delta=\bigcup_{i}T_{i}$ to rewrite the boundary
term in $\operatorname{\mathcal{L}}$, we have

$\operatorname{\mathcal{L}}(f)=\sum_{i}\int_{T_{i}}(K(x)f(x)+J(x)d_{x}f(x))\mathop{d\mu}(x)$

by definition of $K$ and $J$. ∎

######  Remark 6.2.

In the case when the restriction of $P$ to the facet $E_{i}$ vanishes, we can
replace the value of $n_{i}$ in the expressions of $J|_{T_{i}}$ and
$K|_{T_{i}}$ by any number or even by $+\infty$, in the sense that one can
take $J=0$ and $K=2aP-2Q$ on $T_{i}$.

###  6.2. Working on smooth functions

As in the introduction, we choose a complement $\mathcal{W}$ of
$\operatorname{Lin}(\mathcal{V})$ in $N\otimes\mathbb{R}$, and we denote by
$\mathcal{N}_{0}^{\infty}$ the space of continuous concave functions on
$\Delta$, smooth in the interior with differentials in $\mathcal{V}$, such
that $\max f=0$ and $d_{0}f\in\mathcal{W}$.

The following proposition shows that Theorem 1.2 in the introduction provides
a criterion of $G$-uniform K-stability.

######  Proposition 6.3.

The polarized $G$-spherical variety $(X,L)$ is $G$-uniformly K-stable if
$\operatorname{\mathcal{L}}$ vanishes on elements of
$\operatorname{Lin}(\mathcal{V})$ and there exists an $\varepsilon>0$ such
that, for all $f\in\mathcal{N}_{0}^{\infty}$,

(11)
$\operatorname{\mathcal{L}}(f)\geq\varepsilon\int_{\Delta}(-f)P\mathop{d\mu}.$

It is K-semistable if $\operatorname{\mathcal{L}}$ is invariant under addition
of an element of $\operatorname{Lin}(\mathcal{V})$ and
$\operatorname{\mathcal{L}}\geq 0$ on $\mathcal{N}_{0}^{\infty}$.

###### Proof.

Assume that $\operatorname{\mathcal{L}}$ is invariant under addition of an
element of $\operatorname{Lin}(\mathcal{V})$ and there exists an
$\varepsilon>0$ such that (11) holds for all $f\in\mathcal{N}_{0}^{\infty}$.

Let $(\mathcal{X},\mathcal{L})$ be a $G$-equivariant test configuration for
$(X,L)$, and let $g$ denote the associated positive concave rational piecewise
linear function, with slopes in $\mathcal{V}$. The first step is to note that
$g$ can be uniformly approximated on $\Delta$ by a sequence $(f_{m})$ of
smooth concave functions with slopes in $\mathcal{V}$.

For each $m$, set

$\hat{f}_{m}:=f_{m}-p_{\operatorname{Lin}(\mathcal{V})}(d_{0}f_{m})-\max_{\Delta}(f_{m}-p_{\operatorname{Lin}(\mathcal{V})}(d_{0}f_{m}))$

where $p_{\operatorname{Lin}(\mathcal{V})}$ is the linear projection on
$\operatorname{Lin}(\mathcal{V})$ relative to $\mathcal{W}$. Then
$\hat{f}_{m}\in\mathcal{N}_{0}^{\infty}$, and since the $f_{m}$ are uniformly
Lipschitz, it subconverges uniformly to a function $g-l-\max_{\Delta}(g-l)$
for some $l\in\operatorname{Lin}(\mathcal{V})$.

By assumption (11) and uniform convergence, we have

$\operatorname{\mathcal{L}}(g-l-\max_{\Delta}(g-l))\geq\varepsilon\int_{\Delta}(\max_{\Delta}(g-l)-g+l)P\mathop{d\mu}.$

By invariance of $\operatorname{\mathcal{L}}$ under addition of a constant, or
an element of $\operatorname{Lin}(\mathcal{V})$, we can replace the right-hand
side by $\operatorname{\mathcal{L}}(g)$. We finally have

$\operatorname{\mathcal{L}}(g)\geq\varepsilon\int_{\Delta}(\max_{\Delta}(g-l)-g+l)P\mathop{d\mu}\geq\varepsilon\inf_{l^{\prime}\in\operatorname{Lin}(\mathcal{V})}\operatorname{\mathcal{J}}(g+l^{\prime}).$

We have proved uniform K-stability by Theorem 1.1.

For K-semistability, it suffices to follow the same arguments with
$\varepsilon=0$. ∎

##  7\. A combinatorial sufficient condition

In this section we will prove Theorem 1.2. The proof is rather elementary and
follows from a well-chosen decomposition of $\operatorname{\mathcal{L}}$ as a
sum of terms, which are each non-negative under the assumptions. We will begin
by a simpler analogue of Theorem 1.2 that uses this decomposition, then
proceed to the proof.

For this section, we place ourselves in the situation presented before the
statement of Theorem 1.2.

###  7.1. A condition for semi-stability

Let for now $b$ be any point in $\Delta$. Write the integrand of the
functional $\operatorname{\mathcal{L}}_{s}$ as

(12) $\displaystyle f(x)K(x)+d_{x}f(x)J(x)=$
$\displaystyle\left(d_{x}f(x-b)-f(x)+f(b)\right)J(x)$ (13)
$\displaystyle\quad+\left(f(x)-f(b)-d_{b}f(x-b)\right)(K+J)(x)$ (14)
$\displaystyle\quad+d_{x}f(b)J(x)$ (15) $\displaystyle\quad+f(b)K(x)$ (16)
$\displaystyle\quad+d_{b}f(x-b)(K+J)(x)$

We thus have a decomposition of $\operatorname{\mathcal{L}}_{s}$ as a sum of
each corresponding integral.

######  Proposition 7.1.

Assume that $K+J\leq 0$. Let $b$ be the element of $\Delta$ defined by

$\int_{\Delta}(x-b)(K(x)+J(x))\mathop{d\mu}(x)=0.$

If $b$ is in $-\mathcal{V}^{\vee}$, then

$\operatorname{\mathcal{L}}_{s}(f)\geq 0$

for any continuous concave function $f$ on $\Delta$, smooth in the interior,
with differentials in $\mathcal{V}$.

###### Proof.

Using $b$ as defined in the statement, we consider the decomposition of
$\operatorname{\mathcal{L}}_{s}$ as above. Then

  * •

the contribution from summands (12) and (13) are non-negative by concavity and
non-positivity of $J$ and $K+J$,

  * •

the contribution (14) is non-negative by the assumption on the barycenter $b$
and the fact that the differentials of $f$ are in $\mathcal{V}$,

  * •

the contribution of the summand (15) is zero since the integral of $K$ is
zero,

  * •

and the contribution (16) is zero by definition of $b$.

∎

###  7.2. A preparatory pre-compactness result

For the full proof of Theorem 1.2, we will use the following pre-compactness
result, which is a generalization of one used by Donaldson [Don02, Corollary
5.2.5].

######  Proposition 7.2.

Let $C$ be a positive real number. Any sequence of non-positive concave
functions $(f_{m})$ on $\Delta$ with $\int_{\Delta}(-f)P\mathop{d\mu}\leq C$
has a sub-sequence which converges to a concave function $f_{\infty}$ on the
interior of $\Delta$, and the convergence is uniform over strict compact
subsets of $\Delta$.

For the proof, we use the standard Euclidean structure on $\mathbb{R}^{n}$.
For any $x\in\Delta$, let $d_{x}$ denote the distance from $x$ to the boundary
$\partial\Delta$. For any positive number $d$, set

$\Delta_{d}:=\\{x\in\Delta\mid d_{x}\geq d\\}.$

Note that by continuity of $P$, for any positive $d$ there exists a positive
constant $\delta_{d}$ such that $P\geq\delta_{d}$ on $\Delta_{d}$. Recall that
if $f$ is a concave function on $\Delta$, a linear function $l$ is a
_superdifferential_ of $f$ at $x$ if for al l $y$, $f(y)\leq f(x)+l(y-x)$.

######  Lemma 7.3.

For any (small enough) positive $d$, there exists a positive constant
$\kappa=\kappa(d)$ such that for any point $x\in\Delta_{d}$, for any non-
positive concave function $f$ on $\Delta$ with finite
$\int_{\Delta}fP\mathop{d\mu}$, and for any superdifferential $l$ of $f$ at
$x$,

$\lVert l\rVert\leq\kappa\int_{\Delta}(-f)P\mathop{d\mu}$

###### Proof.

Consider the ball $B$ of center $x$ and radius $d/2$, which is contained in
$\Delta_{d/2}$ and the half-ball $B^{-}\subset B$ where the affine function
$y\mapsto l(y-x)$ is negative. Then

$\int_{B^{-}}l(y-x)P(y)\mathop{d\mu}(y)\leq-\lVert
l\rVert\delta_{d/2}\left(\frac{d}{2}\right)^{r+1}C$

where $C$ is a positive constant independent of $x$, $d$, $f$. We have
furthermore

$\displaystyle\int_{B^{-}}l(y-x)P(y)\mathop{d\mu}(y)$
$\displaystyle\geq\int_{B^{-}}(l(y-x)+f(x))P(y)\mathop{d\mu}(y)\quad\text{since
}f\text{ is non positive}$
$\displaystyle\geq\int_{B^{-}}f(y)P(y)\mathop{d\mu}(y)\quad\text{by definition
of a superdifferential}$
$\displaystyle\geq\int_{\Delta}f(y)P(y)\mathop{d\mu}(y)\quad\text{by non-
positivity again.}$

This concludes the proof. ∎

###### Proof of Proposition 7.2.

By concavity, we have

$0\geq\max_{\Delta_{d}}f\geq\frac{1}{\int_{\Delta_{d}}P\mathop{d\mu}}\int_{\Delta_{d}}fP\mathop{d\mu}\geq\frac{1}{\int_{\Delta_{d}}P\mathop{d\mu}}\int_{\Delta}fP\mathop{d\mu},$

and we deduce from the lemma that

$\min_{\Delta_{d}}f\geq\max_{\Delta_{d}}f+\mathrm{diam}(\Delta)\kappa(d)\int_{\Delta}fP\mathop{d\mu}.$

As a consequence, a bound on $\int_{\Delta}fP\mathop{d\mu}$ provides a uniform
bound and a Lipschitz bound on any strict compact subset of $\Delta$. The
proposition then follows by applying Arzelà-Ascoli theorem. ∎

###  7.3. End of proof of Theorem 1.2

Let $(f_{m})$ be a sequence of functions in $\mathcal{N}_{0}^{\infty}$. Assume
by contradiction that $\operatorname{\mathcal{L}}(f_{m})\to 0$ while
$\int_{\Delta}f_{m}P=-1$ for all $m$. The second condition implies, by
Proposition 7.2, the existence of a limit $f_{\infty}$ defined on the interior
of the polytope, such that $(f_{m})$ converges to $f_{\infty}$ uniformly on
every compact subset of the interior of $\Delta$.

Furthermore, by using the expression of $\operatorname{\mathcal{L}}(f_{m})$ as
a sum of non-negative terms as in section 7.1, we obtain that each individual
term converges to zero. In particular, that

$\lim_{m\to\infty}f_{m}(x)-f_{m}(b)-d_{b}f_{m}(x-b)=0$

almost-everywhere implies that $f_{\infty}$ is affine. The slope of
$f_{\infty}$ is necessarily in $\mathcal{W}\cap\mathcal{V}$ by the
normalization. Finally, the assumption on the barycenter $b$ and the term (14)
in section 7.1 imply that the slope must be zero. Hence $f_{\infty}=0$, which
contradicts $\int_{\Delta}f_{m}P\mathop{d\mu}\equiv-1$. ∎

##  8\. Full statement

Let us wrap up in this section the statement of the sufficient condition for
uniform K-stability of polarized spherical varieties we proved.

Let us note that $\Delta$ is not the most direct choice of polytope associated
to the polarized spherical variety $(X,L)$. The _moment polytope_ $\Delta_{+}$
is in some sense more natural to consider as it depends on less choices.
Recall that the moment polytope of $(X,L)$ is the convex polytope obtained by
taking the closure of the set of all (normalized) $B$-weights of plurisections
of $L$. It does not lie in the same space as $\Delta$ in general. More
precisely, the relation between the two is a simple translation:
$\Delta_{+}=\chi+\Delta$, and $\Delta_{+}$ lies in the affine space
$\chi+M\otimes\mathbb{R}$. One easily sees from the previous sections that it
is not important in our results for $\chi$ to be in $M$. Hence, the data of
$\Delta_{+}$ alone allows to recover both $\Delta$ and one (several) choice of
$\chi$ to apply our sufficient criterion. On the other hand, the data of $M$
is not readily read from $\Delta_{+}$ alone. The importance of $M$ in the
statement is seen through the integers $n_{i}$.

The full statement for our sufficient condition of $G$-uniform K-stability is
as follows.

######  Theorem 8.1.

Let $(X,L)$ be a polarized $G$-spherical variety with spherical lattice $M$ of
rank $r$, valuation cone $\mathcal{V}$ and moment polytope $\Delta_{+}$. Let
$R_{X}^{+}$ denote the positive roots of $G$ not orthogonal to $\Delta_{+}$.
Choose an element $\chi$ in the interior of $\Delta_{+}$, number the facets of
the translated polytope $\Delta:=-\chi+\Delta_{+}$ by $E_{i}$ for $1\leq i\leq
s$ and let $n_{i}$ be the positive numbers such that

$\Delta=\\{x\in M\otimes\mathbb{R}\mid u_{i}(x)\leq n_{i}\\}$

where $u_{i}\in N=\mathrm{Hom}(M,\mathbb{Z})$ denotes the outward pointing
primitive normal to $E_{i}$. For $m=r$ or $r+1$, let $L_{m}$ be the almost-
everywhere defined function on $\Delta$ such that for $x$ in the interior of
the convex hull of $E_{i}\cup\\{0\\}$,

$L_{m}(x)=\sum_{\alpha\in
R_{X}^{+}}\langle\alpha,\left(mn_{i}^{-1}-2a\right)(x+\chi)+\mathrm{Card}(R_{X}^{+})\left(n_{i}^{-1}x+2\varpi\right)\rangle\prod_{\beta\in
R_{X}^{+}\setminus\\{\alpha\\}}\langle\beta,x+\chi\rangle$

where the constant $a$ is defined by $\int_{\Delta}L_{r}\mathop{d\mu}=0$ for
some Lebesgue measure $\mathop{d\mu}$ on $M\otimes\mathbb{R}$.

Assume that $L_{r+1}$ is non-negative on $\Delta$ and let $b$ denote the
barycenter of $\Delta$ with respect to the measure $L_{r+1}\mathop{d\mu}$.
Assume furthermore that $-b$ is in the relative interior of the valuation cone
$\mathcal{V}$. Then $(X,L)$ is $G$-uniformly K-stable.

If $L_{r+1}$ is non-negative on $\Delta$ and $b\in-\mathcal{V}^{\vee}$, then
$(X,L)$ is $G$-equivariantly K-semistable.

###### Proof.

We work under the assumptions of the theorem, that is, $L_{r+1}$ is non-
negative on $\Delta$ and the barycenter $b$ of $\Delta$ with respect to the
measure $L_{r+1}\mathop{d\mu}$ is in the relative interior of $-\mathcal{V}$.

Consider the functional $\operatorname{\mathcal{L}}$ as defined by (1) in the
introduction. Note that $\operatorname{\mathcal{L}}$ vanishes on constants.
Consider the functional $\operatorname{\mathcal{L}}_{s}$ as defined by (2) in
the introduction, for the functions $K$ and $J$ defined by (10) and (9) in
section 6.

Note that $K=-C_{1}L_{r}$ and $K+J=-C_{2}L_{r+1}$ for some positive constants
$C_{1}$ and $C_{2}$. In particular, the barycenters $b$ involved in Theorem
1.2 and the theorem we are trying to prove are indeed the same.

By Proposition 6.1,
$\operatorname{\mathcal{L}}_{s}=\operatorname{\mathcal{L}}$ for continuous
functions that are smooth on the interior of $\Delta$. In particular, this is
true for any linear function $l\in N\otimes\mathbb{R}$. Furthermore, for $l\in
N\otimes\mathbb{R}$, we have

$\operatorname{\mathcal{L}}(l)=\operatorname{\mathcal{L}}_{s}(l)=\int_{\Delta}l(K+J)\mathop{d\mu}=-C_{3}b$

for some positive constant $C_{3}$.

As a consequence, the condition that $b$ is in the relative interior of
$-\mathcal{V}^{\vee}$ is equivalent to $\operatorname{\mathcal{L}}(l)\geq 0$
for all $l\in\mathcal{V}$, with equality if and only if
$l\in\operatorname{Lin}(\mathcal{V})$. In particular, it implies that
$\operatorname{\mathcal{L}}$ vanishes on $\operatorname{Lin}(\mathcal{V})$.

By Proposition 6.3 and Theorem 1.2, we deduce that $(X,L)$ is $G$-uniformly
K-stable. The same arguments show the K-semistability statement as well. ∎

######  Remark 8.2.

It follows from the relation between $L_{m}$ and $J$ and $K$ in the proof and
Remark 6.2 that the statement of the theorem applies as well if we replace
$(n_{i})^{-1}$ by any non-negative number for the facets $E_{i}$ where $P$
vanishes.

Applying Corollary A.2 from Odaka’s appendix to this paper, we have the
following sufficient condition for existence of cscK metrics.

######  Corollary 8.3.

Assume that $L_{r+1}$ is non-negative, that $b$ is in the relative interior of
$-\mathcal{V}^{\vee}$ and that $X$ is smooth. Then there exists a cscK metric
in $c_{1}(L)$.

On the other hand, from the point of view of K-stability alone, our theorem
can be interpreted in the following way.

######  Corollary 8.4.

Under the assumption that $L_{r+1}$ is non-negative on $\Delta$, $G$-uniform
K-stability is equivalent to $G$-stc K-polystability for $(X,L)$.

In particular, this point of view shows that our barycenter condition is in
fact necessary.

###### Proof.

As noted in the previous proof, the condition that $b$ is in the relative
interior of $-\mathcal{V}^{\vee}$ is equivalent to
$\operatorname{\mathcal{L}}(l)\geq 0$ for all $l\in\mathcal{V}$, with equality
if and only if $l\in\operatorname{Lin}(\mathcal{V})$. By Theorem 4.1, this is
equivalent to $G$-stc K-polystability. ∎

##  9\. Example: blowup of $\mathcal{Q}^{3}$ along $\mathcal{Q}^{1}$

In this section we study the blowup $X$ of the three dimensional quadric along
a $1$-dimensional subquadric. This example was previously considered in [DH18]
and [Del19], where it was presented in more details. The Picard rank of this
variety is two.

The connected reductive group $G$ making $X$ a rank $2$ spherical variety is
$\operatorname{SL}_{2}\times\mathbb{C}^{*}$. We fix a choice of maximal torus
and Borel subgroup. Let $\alpha$ denote the unique positive root, and let $f$
denote the character of weight $1$ of $\mathbb{C}^{*}$. The spherical lattice
$M$ is the lattice generated by $\alpha$ and $\frac{\alpha+f}{2}$. The dual
lattice $N$ is the lattice generated by $\alpha^{\vee}$ and
$\frac{\alpha^{\vee}+f^{\vee}}{2}$, where $\alpha^{\vee}$ is the coroot of
$\alpha$, and $f^{\vee}$ is defined similarly by $f^{\vee}(f)=2$ and
$f^{\vee}(\alpha)=0$. The valuation cone $\mathcal{V}$ is the dual cone to
$\mathbb{R}_{-}\alpha$. Finally, we have $R_{X}^{+}=\\{\alpha\\}$ and
$\varpi=\alpha$.

The moment polytope for an ample line bundle on $X$ is, up to scaling, of the
following form for some $s>3/2$.

$\Delta(s):=\\{x\alpha+yf\in M\otimes\mathbb{R}\mid 0\leq x\leq 3/2,\quad
x-s\leq y\leq s-x\\}$

Its four facets $E_{0},\ldots,E_{3}$ have respective outward pointing
primitive normals $u_{0}=-\alpha^{\vee}$,
$u_{1}=\frac{\alpha^{\vee}+f^{\vee}}{2}$, $u_{2}=\alpha^{\vee}$ and
$u_{3}=\frac{\alpha^{\vee}-f^{\vee}}{2}$.

Figure 1. The polytope $\Delta(s)$ for
$s=2$$\bullet$$0$$\bullet$$\frac{3}{2}\alpha$$\bullet$$sf$

Before trying to apply our theorem, let us compute the important quantity $a$,
and for this it is more convenient to deal with the boundary integral:

$2a=\frac{\int_{\Delta}2Q\mathop{d\mu}+\int_{\partial\Delta}P\mathop{d\sigma}}{\int_{\Delta}P\mathop{d\mu}}.$

We have $P(x\alpha+yf)=2x$, $Q\equiv 1$,
$\mathop{d\mu}=2\mathop{dx}\mathop{dy}$ in the coordinates $x\alpha+yf$,
$\mathop{d\sigma}|_{E_{1}}=\mathop{d\sigma}|_{E_{3}}=2\mathop{dx}$ if we
parametrize by $x$ and
$\mathop{d\sigma}|_{E_{0}}=\mathop{d\sigma}|_{E_{2}}=\mathop{dy}$. We thus
have

$\displaystyle\int_{\Delta}2Q\mathop{d\mu}$
$\displaystyle=\int_{x=0}^{\frac{3}{2}}\int_{y=x-s}^{s-x}4\mathop{dy}\mathop{dx}=12s-9$
$\displaystyle\int_{\Delta}P\mathop{d\mu}$
$\displaystyle=\int_{x=0}^{\frac{3}{2}}\int_{y=x-s}^{s-x}4x\mathop{dy}\mathop{dx}=9(s-1)$
$\displaystyle\int_{E_{0}}P\mathop{d\sigma}$ $\displaystyle=0$
$\displaystyle\int_{E_{1}}P\mathop{d\sigma}=\int_{E_{3}}P\mathop{d\sigma}$
$\displaystyle=\int_{0}^{\frac{3}{2}}4x\mathop{dx}=\frac{9}{2}$
$\displaystyle\int_{E_{2}}P\mathop{d\sigma}$
$\displaystyle=\int_{\frac{3}{2}-s}^{s-\frac{3}{2}}3\mathop{dy}=6s-9$

and

$2a=\frac{2s-1}{s-1}.$

By similar computations, we can check wether the barycenter condition involved
in our theorem, or equivalently, the $G$-stc K-polystability, holds. This
amounts to the two conditions $\operatorname{\mathcal{L}}(f^{\vee})=0$ and
$\operatorname{\mathcal{L}}(-\alpha^{\vee})>0$. The first of these conditions
is automatic by symmetry of the moment polytopes and Duistermaat-Heckman
polynomial. We compute the second, using the expression with a boundary
integral as for $2a$, we obtain

$\operatorname{\mathcal{L}}(-\alpha^{\vee})=\frac{9(8s^{2}-18s+11)}{4(s-1)}$

and it is positive for any $s>3/2$.

We now choose an element of the interior of the polytope (and this is the
tricky part to get the theorem to apply). For reasons related by the general
Fano case to be treated next, we choose $\chi=\frac{s}{2}\alpha$, which can be
considered of course only if $s<3$. Then the translated polytope
$\Delta=-\chi+\Delta_{+}$ is defined by the four equations
$u_{i}(x\alpha+yf)\leq n_{i}$ with

$\displaystyle n_{0}$ $\displaystyle=s$ $\displaystyle n_{1}=n_{3}$
$\displaystyle=\frac{s}{2}$ $\displaystyle n_{2}$ $\displaystyle=3-s$

Figure 2. The decomposition of
$\Delta(s)$$\bullet$$E_{0}$$E_{1}$$E_{2}$$E_{3}$

In order to show that for these polarizations, $G$-uniform K-stability is
equivalent to the barycenter condition, we have to check that for all $i$, for
all $x\alpha+yf\in T_{i}$, we have

$\langle\alpha,(3n_{i}^{-1}-2a)((x+\frac{s}{2})\alpha+yf)+(n_{i}^{-1}x+1)\alpha+n_{i}^{-1}yf\rangle\geq
0.$

Recall that $\langle\alpha,f\rangle=0$, hence it suffices to check

$\displaystyle(4n_{0}^{-1}-2a)x+(3n_{0}^{-1}-2a)\frac{s}{2}+1\geq 0$
$\displaystyle\text{ for }-\frac{s}{2}\leq x\leq 0$
$\displaystyle(4n_{1}^{-1}-2a)x+(3n_{1}^{-1}-2a)\frac{s}{2}+1\geq 0$
$\displaystyle\text{ for }-\frac{s}{2}\leq x\leq\frac{3-s}{2}$
$\displaystyle(4n_{2}^{-1}-2a)x+(3n_{2}^{-1}-2a)\frac{s}{2}+1\geq 0$
$\displaystyle\text{ for }0\leq x\leq\frac{3-s}{2}$
$\displaystyle(4n_{3}^{-1}-2a)x+(3n_{3}^{-1}-2a)\frac{s}{2}+1\geq 0$
$\displaystyle\text{ for }-\frac{s}{2}\leq x\leq\frac{3-s}{2}.$

Since $n_{1}=n_{3}$, the second and fourth conditions are equivalent. Since
$P$ vanishes on $E_{0}$, we can choose any $n_{0}$ we want, for example
$n_{0}=n_{1}$, so that the first condition is implied by the second. We end up
with only two conditions to check.

Replacing $n_{1}$ and $2a$ by their expression in $s$, one of the conditions
is

$\frac{2s(-2s^{2}+9s-8)(2x+s)}{s-1}\geq 0\text{ for }-\frac{s}{2}\leq
x\leq\frac{3-s}{2}.$

The degree two polynomial $-2s^{2}+9s-8$ is non-negative for
$\frac{9-\sqrt{17}}{4}\leq s\leq\frac{9+\sqrt{17}}{4}$, which contains the
range $\frac{3}{2}<s<3$, and the other factors are easily checked to be non-
negative for the values of $x$ and $s$ considered, so the condition is
satisfied.

The other condition, replacing $n_{2}$ and $2a$ by their expression in $s$, is

$\frac{2s^{2}-3s-1}{(s-1)(3-s)}x+\frac{s^{3}-3s^{2}+4s-3}{(3-s)(s-1)}\geq
0\text{ for }0\leq x\leq\frac{3-s}{2}.$

Since $s>\frac{3}{2}>1$, one easily checks that the coefficient of $x$ is
positive. Hence the condition is satisfied if

$0\geq-\frac{s^{3}-3s^{2}+4s-3}{2s^{2}-3s-1}.$

Again, for $s>1$ as here, this is equivalent to

$s^{3}-3s^{2}+4s-3\geq 0$

and one can check that the unique real root $s_{0}$ of this cubic polynomial
is approximately $s_{0}\approx 1.6823$, in the range of $s$ considered.

To sum up, we have shown that $G$-uniform K-stability of the polarized variety
is equivalent to the barycenter condition if $s$ is such that $s_{0}\approx
1.6823\leq s<3$. We have thus proved that $X$ admits a cscK when $s_{0}\leq
s<3$. It is very likely that our choice of $\chi$ was not the optimal one, and
that one can push further the use of our main theorem to get a slightly larger
range of classes with cscK metrics. The natural question regarding this
example is whether it is a Calabi dream space in the sense of Chen and Cheng,
that is, if all classes admit cscK metrics. It is unlikely that our main
theorem is enough for this, but we intend to answer this question in a later
work by studying optimal degenerations for rank two spherical threefolds.

##  10\. Fano case

In this section, we apply our main theorem to the case of a Fano spherical
manifold equipped with its anticanonical polarization.

######  Theorem 10.1.

Assume that $X$ is $\mathbb{Q}$-Fano and that $L$ is (a multiple of) the
anticanonical $\mathbb{Q}$-line bundle of $X$. Then $L_{r+1}$ is positive on
$\Delta$.

This shows, with a very straighforward proof, that for spherical Fano
varieties, $G$-uniform K-stability is equivalent to $G$-stc K-polystability,
and allows to recover the explicit combinatorial condition for this obtained
in [Del20].

###### Proof.

Assume for simplicity that $X$ is Gorenstein and $L=K_{X}^{-1}$. Apart from
notational issues, the general case is the same. Assume furthermore that
$K_{X}^{-1}$ is equipped with its canonical $G$ linearization.

The proposition will follow from the judicious choice
$\chi=2\varpi_{X}=\sum_{\alpha\in R_{X}^{+}}\alpha$ and two steps:

  1. (1)

using Remark 8.2, we can assume that $n_{i}=1$ for all $i$ in the expression
of $L_{m}$,

  2. (2)

we can replace $2\varpi$ with $2\varpi_{X}$ in the expression of $L_{m}$.

Let us begin with the simple case of toric varieties. We obviously have
$\varpi=\varpi_{X}=0$ in the toric case. The torus-invariant section of
$K_{X}^{-1}$ has weight $0$ which corresponds to the unique interior integral
point of the moment polytope $\Delta_{+}$. Finally, the polytope $\Delta_{+}$
is defined by equations $u_{i}(x)\leq 1$ for a set of primitive elements
$u_{i}$ in $N$, hence by choosing $\chi=0$ we have $n_{i}=1$ for all $i$.
Actually, an integral polytope is reflexive if and only if it is defined by
equations $u_{i}(x)\leq 1$ for a set of primitive elements $u_{i}$ in $N$, and
it is well-known that Gorenstein Fano toric varieties correspond to reflexive
polytopes.

For the general case, we will use the description of the anticanonical divisor
of spherical varieties by Brion in [Bri97] as formulated in [GH15a]. Namely,
there exists a section of the anticanonical line bundle, with $B$-weight
$2\varpi_{X}$, whose divisor $\sum_{D\in\mathcal{P}_{X}}n_{D}D$ is such that
$n_{D}=1$ if $D$ is $G$-stable, and the description of the coefficients of
colors (closures of $B$-stable divisors in $G/H$) is explicit, depending on
the type of each color. Since we need the details, let us quickly recall the
possible types of colors.

Let $S$ denote the set of simple roots of $G$. For $\alpha\in S$, let
$P_{\alpha}$ denote the largest parabolic in $G$ containing $B$ such that
$-\alpha$ is not a root of $P_{\alpha}$. Let $\mathcal{D}(\alpha)$ denote the
set of $B$-stable prime divisors of $X$ that are not $P_{\alpha}$-stable. It
turns out that these divisors exhaust the set of divisors in $\mathcal{P}_{X}$
that are not $G$-stable, and that $\mathcal{D}(\alpha)$ is non-empty precisely
if $\alpha\in R_{X}^{+}$. An element $D\in\mathcal{D}(\alpha)$ is

  * •

of type $a$ if $\alpha$ is a primitive element of $M$,

  * •

of type $2a$ if $2\alpha$ is a primitive element of $M$,

  * •

and of type $b$ otherwise.

The coefficient $n_{D}$ is then obtained, depending on the type of
$D\in\mathcal{D}(\alpha)$, as follows:

  * •

$n_{D}=\frac{1}{2}\alpha^{\vee}(2\varpi_{X})=1$ for type $a$ or $2a$,

  * •

$n_{D}=\alpha^{\vee}(2\varpi_{X})$ for type $b$,

where $\alpha^{\vee}$ denotes the coroot of $\alpha$.

Consider now the polytope $\Delta$ associated to the section constructed by
Brion. We want to check that we can take $n_{i}=1$ for each facet. Recall that
the equations defining $\Delta$ from the coefficients of the divisor are the
$\varrho(D)(x)+n_{D}\geq 0$ for $D\in\mathcal{P}_{X}$. In particular, the
equations defining the facets are of the form $\varrho(D)(x)+n_{D}=0$. To put
these in the form $u_{i}(x)=n_{i}$ where $u_{i}$ is a primitive outer normal,
one has to find the positive number $a_{i}$ such that $-\varrho(D)/a_{i}$ is
primitive, in which case one can take $n_{i}=n_{D}/a_{i}$.

Assume now that $D\in\mathcal{P}_{X}$ defines a facet of $\Delta$.

Whenever $D$ is $G$-stable, $\varrho(D)$ is primitive and $n_{D}=1$, so we
have $n_{i}=1$.

Assume now that $D$ is a color of type $a$ and let $\alpha\in S$ be such that
$D\in\mathcal{D}(\alpha)$. Then $\alpha(\varrho(D))=1$ (see [GH15b] for a
convenient summary of the properties of colors by type), which implies that
$\varrho(D)$ is primitive. Since $n_{D}=1$ as well in this case, we indeed
have $n_{i}=1$.

Assume now that $D$ is a color of type $2a$ and let $\alpha\in S$ be such that
$D\in\mathcal{D}(\alpha)$. Then $\varrho(D)=\frac{1}{2}\alpha^{\vee}|_{M}$. If
$x$ is in the facet defined by $\varrho(D)(x)+n_{D}=0$, then we have
$\frac{1}{2}\alpha^{\vee}(x)+\frac{1}{2}\alpha^{\vee}(2\varpi_{X})=0$, which
implies $\langle\alpha,x+2\varpi_{X}\rangle=0$. Since $\chi=2\varpi_{X}$, this
in turn implies that $P$ vanishes on the facet. By remark 8.2, we can then
choose $n_{i}=1$ in the expressions of $L_{r}$ and $L_{r+1}$.

Similarly, if $D$ is a color of type $b$ and $\alpha\in S$ is such that
$D\in\mathcal{D}(\alpha)$ we have $\varrho(D)=\alpha^{\vee}|_{M}$ and
$n_{D}=\alpha^{\vee}(2\varpi_{X})$, hence $P$ must vanish on the facet. Again
by remark 8.2, we can choose $n_{i}=1$ in the expressions of $L_{r}$ and
$L_{r+1}$.

We now turn to the problem of replacing $\varpi$ by $\varpi_{X}$ in the
expression of $L_{m}$. The important property of $R_{X}^{+}$ is that it
consists of all roots of the unipotent radical of some parabolic subgroup
$P_{X}$ of $G$, namely the stabilizer of the open $B$-orbit in $X$. As a
consequence, we may write $\varpi_{X}=\varpi-\varpi_{0}$ where $\varpi_{0}$ is
the sum of positive roots of the Levi subgroup of $P_{X}$. Let $W_{0}$ denote
the Weyl group of this Levi subgroup, which is a subgroup of the Weyl group of
$G$. The action of $W_{0}$ on roots of $G$ induces a permutation of
$R_{X}^{+}$. Consider the linear function

$h:y\mapsto\sum_{\alpha\in R_{X}^{+}}\langle\alpha,w\cdot
y\rangle\prod_{\beta\in
R_{X}^{+}\setminus\\{\alpha\\}}\langle\beta,x+\chi\rangle.$

The discussion above shows that $h$ is invariant under the action of $W_{0}$.
In particular, $h(\varpi_{0})=0$ since there exists $w_{0}\in W_{0}$ such that
$w_{0}(\varpi_{0})=-\varpi_{0}$. We thus have

$\displaystyle L_{m}(x)$
$\displaystyle=h((mn_{i}^{-1}-2a)(x+\chi)+\mathrm{Card}(R_{X}^{+})(n_{i}^{-1}x+2\varpi))$
$\displaystyle=h((mn_{i}^{-1}-2a)(x+\chi)+\mathrm{Card}(R_{X}^{+})(n_{i}^{-1}x+2\varpi_{X})).$

Putting together all ingredients in the case $L=K_{X}^{-1}$
($\chi=2\varpi_{X}$, all $n_{i}=1$ and we can replace $2\varpi$ with
$2\varpi_{X}$), we have

$\displaystyle L_{m}(x)$
$\displaystyle=h((m-2a+\mathrm{Card}(R_{X}^{+}))(x+\chi))$
$\displaystyle=(m-2a+\mathrm{Card}(R_{X}^{+}))\prod_{\alpha\in
R_{X}^{+}}\langle\alpha,x+\chi\rangle$

Since $\prod_{\alpha\in R_{X}^{+}}\langle\alpha,x+\chi\rangle$ is positive on
the interior of $\Delta$, and $\int_{\Delta}L_{r}\mathop{d\mu}=0$, we deduce
that $2a=r+\mathrm{Card}(R_{X}^{+})$. This number actually coincides with the
dimension of the variety, and could be recovered by interpreting directly $2a$
as the average scalar curvature of the Fano variety $X$.

Finally,

$L_{r+1}(x)=\prod_{\alpha\in R_{X}^{+}}\langle\alpha,x+\chi\rangle$

is strictly positive on the interior of $\Delta$, and non-negative on the
whole polytope. ∎

##  11\. Polarizations close to the anticanonical line bundle

In this final section, we illustrate how our main theorem applies to different
situations to give equivalence of $G$-uniform K-stability and $G$-stc
K-polystability for polarizations close to the anticanonical line bundle. We
however believe that the criterion better shows its strength when applied to a
concrete situation as in Section 9.

######  Proposition 11.1.

Let $(X,L)$ be a $G$-spherical variety and assume that there exists a choice
of $\chi\in\Delta_{+}$ and a positive number $\delta$ such that
$L_{r+1}>\delta$ on $\Delta$. Then for polarizations close to $L$, $G$-uniform
K-stability is equivalent to $G$-stc K-polystability, which is equivalent to
the barycenter condition.

###### Proof.

It follows from the simple remark that all the combinatorial data associated
to the polarization vary continuously. From this, one can choose a continuous
family of elements of the varying moment polytopes such that it coincides with
$\chi$ on the given $L$. Then the corresponding function $L_{r+1}$ varies
continuously as well, and the condition $\min L_{r+1}>\delta$ is an open
condition. ∎

######  Corollary 11.2.

Let $(X,L)$ be a Gorenstein Fano toroidal horospherical variety. Then there
exists a neighborhood of the anticanonical line bundle where $G$-uniform
K-stability is equivalent to vanishing of the Futaki invariant.

###### Proof.

For any polarized toroidal horospherical variety, the moment polytope does not
touch the walls of the positive Weyl chamber of $G$ defined by roots not in
$R_{X}^{+}$. Hence the Duistermaat-Heckman polynomial is positive on each
moment polytope. As a consequence from the proof of Theorem 10.1, with the
choice $\chi=2\varpi_{X}$, $L_{r+1}$ is positive on $\Delta$ for the
anticanonical line bundle. As a consequence, we can apply the previous
proposition to obtain that, in a neighborhood of the anticanonical line
bundle, $G$-uniform K-stability is equivalent to $G$-stc K-polystability.
Finally, all special test configurations for horospherical manifolds are
product since the valuation cone is $N\otimes\mathbb{R}$ in this case. Hence
$G$-uniform K-stability with respect to special test configurations is
equivalent to vanishing of the Futaki invariant. ∎

######  Proposition 11.3.

Let $X$ be a Gorenstein Fano $G$-spherical polarized variety, such that the
open orbit $G/H$ is a non-Hermitian symmetric variety. Then on a neighborhood
of the anticanonical line bundle, $G$-uniform K-stability is equivalent to
$G$-stc K-polystability.

###### Proof.

For a non-Hermitian symmetric space $G/H$, the valuation cone is the negative
Weyl chamber defined by a root system in (a subspace of) $M\otimes\mathbb{R}$
(the restricted root system of the symmetric space), and the images of colors
in $N\otimes\mathbb{R}$ are exactly one positive multiple of each simple
coroot of this root system (restricted coroots). As a consequence, the outward
pointing normals of facets of moment polytopes (which are always in
$M\otimes\mathbb{R}$ if the symmetric space is _not_ Hermitian) are either
negative restricted coroots or elements of the positive restricted Weyl
chamber. Furthermore, the name _restricted_ is appropriate in the sense that
restricted roots are exactly (doubles of) restrictions of roots in $R_{X}^{+}$
to $N\otimes\mathbb{R}$. Finally, the restriction of the Duistermaat-Heckman
polynomial to a facet vanishes exactly when the facet is defined by a
restricted coroot.

Consider the anticanonical line bundle on $X$, and instead of
$\chi=2\varpi_{X}$ as in Section 10, consider the element $\chi=2t\varpi_{X}$.
It is still in $\Delta$ for $t$ close to $1$ since in our symmetric situation,
$2\varpi_{X}$ is the half sum of positive restricted roots, and is an element
of $M\otimes\mathbb{R}$.

We then write, still for the anticanonical line bundle,

$\displaystyle L_{r+1}(x)$
$\displaystyle=\left((r+1)n_{i}^{-1}-2a+\mathrm{Card}(R_{X}^{+})n_{i}^{-1}\right)\prod_{\alpha\in
R_{X}^{+}}\langle\alpha,x+\chi\rangle$ $\displaystyle\quad+\sum_{\alpha\in
R_{X}^{+}}\langle\alpha,\mathrm{Card}(R_{X}^{+})\left(2\varpi_{X}-n_{i}^{-1}\chi\right)\rangle\prod_{\beta\in
R_{X}^{+}\setminus\\{\alpha\\}}\langle\beta,x+\chi\rangle.$

We know from Section 10 that for every $i$,

$\left((r+1)n_{i}^{-1}-2a+\mathrm{Card}(R_{X}^{+})n_{i}^{-1}\right)$

is strictly positive if $t$ is close to $1$, since these number vary
continuously with $t$ and are equal to $1$ for $t=1$. For the other term, we
have

$2\varpi_{X}-n_{i}^{-1}\chi=(1-tn_{i}^{-1})2\varpi_{X}.$

The values of $n_{i}$ depend on $t$, but:

  * •

if the Duistermaat-Heckman polynomial vanishes on the facet $E_{i}$ (i.e. it
is defined by a restricted coroot) then we can _choose_ the value of $n_{i}$
to ensure that $1-tn_{i}^{-1}$ is positive for any $t$

  * •

else, the outward pointing normal $u_{i}$ to the facet $E_{i}$ is in the
positive restricted Weyl chamber, and $n_{i}=1-u_{i}((t-1)2\varpi_{X})$ so
that

$1-tn_{i}^{-1}=\frac{(1-t)(1+u_{i}(2\varpi_{X}))}{1-(t-1)u_{i}(2\varpi_{X})}$

is positive when $t<1$.

We can then fix a choice of $t$ and $\chi=2t\varpi_{X}$ so that the
corresponding $L_{r+1}$ is positive on $\Delta$. Applying the same arguments
as for Proposition 11.1 yields the conclusion. ∎

######  Remark 11.4.

Let us stress again that it is very likely that the statements proved above
hold more generally for spherical varieties. It would for example be rather
straightforward to push further the last proposition so that it applies to all
$\mathbb{Q}$-Gorenstein weak Fano spherical varieties whose open orbit is
affine. We leave to further research the exploration of different special
cases or the question of finding an argument applying to general spherical
varieties.

##  Appendix A Uniform Yau-Tian-Donaldson conjecture for polarized spherical
manifolds, by Yuji Odaka

The purpose of this short notes is to clarify the following statements.

######  Theorem A.1.

For any $\mathbb{C}^{*}$-equivariant isotrivial projective family
$\pi\colon\mathcal{X}\to\mathbb{P}^{1}$ whose general fiber is a $G$-spherical
projective variety for a reductive algebraic group $G$, take an arbitrary line
bundle $\mathcal{L}$ which is ample over the general fiber. Note that we do
not assume it is also ample on the central fiber. ([Li20] called
$(\mathcal{X},\mathcal{L})$ a model.)

Then, $\oplus_{m\geq 0}\pi_{*}\mathcal{L}^{\otimes m}$ is a finitely generated
$\mathcal{O}_{\mathbb{P}^{1}}$-algebra.

######  Corollary A.2.

For a polarized smooth projective $G$-spherical varieties $(X,L)$, the
$G$-uniform K-(poly)stability in the sense of [His20, His19, Li20] implies the
existence of a unique cscK metric.

###### proof of Theorem A.1.

By applying the Eakin-Nagata theorem to the normalization of $\mathcal{X}$, we
can and do assume $\mathcal{X}$ is normal. Because of the $G$-action and the
given compatible $\mathbb{C}^{*}$-action in the horizontal direction on
$\mathcal{X}$, it follows that $\mathcal{X}$ has the natural structure as a
$(G\times\mathbb{C}^{*})$-spherical variety. Indeed, the Borel subgroup of
$G\times\mathbb{C}^{*}$ is simply $B(G)\times\mathbb{C}^{*}$ from the
definition, where $B(G)$ denotes the original Borel subgroup of $G$, and it
admits an open dense orbit inside $X\times\mathbb{C}^{*}(\subset\mathcal{X})$
by the $G$-sphericality of $X$.

By taking a $\mathbb{C}^{*}$-equivariant resolution of indeterminacy of
$\mathcal{X}\dashrightarrow X\times\mathbb{P}^{1}$ as in [RT07, Oda13], we can
and do assume $\mathcal{X}$ is the blow up of a flag ideal, i.e., of
dominating type in the terminology of [Li20]. Since replacing $\mathcal{L}$ by
$\mathcal{L}\otimes p_{2}^{*}\mathcal{O}_{\mathbb{P}^{1}}(-c)$ for
$c\in\mathbb{Z}$ does not affect the assertion, we can and do such twist as
follows. Here, $p_{i}$ denotes $i$-th projection from $X\times\mathbb{P}^{1}$.

From the finiteness of the irreducible components of $\mathcal{X}_{0}$, it
easily follows that there exists a large enough $c\in\mathbb{Z}_{>0}$ such
that

$p_{1}^{*}L\otimes
p_{2}^{*}\mathcal{O}_{\mathbb{P}^{1}}(-c^{\prime})\subset\mathcal{L}\subset
p_{1}^{*}L\otimes p_{2}^{*}\mathcal{O}_{\mathbb{P}^{1}}(c^{\prime}).$

Twisting the above by $\mathcal{O}_{\mathbb{P}^{1}}(c^{\prime})$ and letting
$c:=2c^{\prime}$, we can and do assume:

(17) $\displaystyle p_{1}^{*}L\subset\mathcal{L}\subset p_{1}^{*}L\otimes
p_{2}^{*}\mathcal{O}_{\mathbb{P}^{1}}(c).$

In any case, it immediately follows from the above that the filtration
associated to $\mathcal{L}$ is linearly bounded in the sense of [Szé15].

Since we confirmed that $\mathcal{X}$ is a spherical variety, it is also a
Mori Dream space in the sense of [HK00] due to [BK94] (cf., [HK00, p340]).
Therefore, it follows that $\oplus_{m\geq
0}H^{0}(\mathcal{X},\mathcal{L}^{\otimes m})$ is a finitely generated graded
$\mathbb{C}$-algebra. Then, by the natural $\mathbb{C}^{*}$-action on it which
is induced by the $\mathbb{G}_{m}$-action on $(\mathcal{X},\mathcal{L})$, the
complete reductivity of $\mathbb{G}_{m}$ implies that we can take a set of
finite generators as eigenvectors of the form
$\mathcal{S}=\\{(p_{1}^{*}s_{i}^{(m,l)})t^{l}\\}_{l\leq 0,m,i}$. Here, $t$
denotes the homogeneous coordinate of $\mathbb{P}^{1}$ which vanishes at the
origin with order $1$, which we also identify with $p_{2}^{*}t$ on
$\mathcal{X}$, and the indices are of the form $-cm\leq l\leq 0$ and $1\leq
i\leq a_{m,l}$ for a double sequence of positive integers $a_{m,l}$ such that

$0=a_{m,-cm-1}\leq a_{m,-cm}\leq\cdots\leq
a_{m,0}=a_{m,1}=\cdots=h^{0}(X,L^{\otimes m}),$

because of (17). (In the case of test configurations i.e., when $\mathcal{L}$
is relatively ample, these $\\{a_{m,l}\\}$ are determined by
$\\{\lambda_{a,k}\\}$ in the terminology of section 2 of this paper.) The
proof follows from standard arguments so we omit but see and compare with
[Oda13] for instance. Similarly, the following set, in which we allow the
integer index $l$ to be not necessarily negative:

$\displaystyle(\mathcal{S}\subset)\tilde{\mathcal{S}}$
$\displaystyle=\\{(p_{1}^{*}s_{i}^{(m,\min\\{l,0\\})})t^{l}\\}_{m,l\in\mathbb{Z},i}$

generate $\oplus_{m\geq 0}\pi_{*}\mathcal{L}^{\otimes m}$ as a
$\mathcal{O}_{\mathbb{P}^{1}}$-algebra. Note that

$\displaystyle\tilde{\mathcal{S}}$
$\displaystyle=\mathcal{S}\sqcup\\{\\{(p_{1}^{*}s_{i}^{(m,0)})t^{l}\\}_{l>0,m,i}\\}$
$\displaystyle=\mathcal{S}\sqcup(t\mathbb{C}[t]\\{\\{(p_{1}^{*}s_{i}^{(m,0)})\\}_{m,i}\\}).$

Thus, we can particularly take its finite subset

$\overline{\mathcal{S}}:=\mathcal{S}\sqcup\\{\\{(p_{1}^{*}s_{i}^{(m,0)})\\}_{m,i}(\subset\tilde{\mathcal{S}}),$

which still generates $\oplus_{m\geq 0}\pi_{*}\mathcal{L}^{\otimes m}$ as a
graded $\mathcal{O}_{\mathbb{P}^{1}}$-algebra. This completes the proof of
Theorem A.1. ∎

###### proof of Corollary A.2.

The result of [Li20, 1.10] combined with above Theorem A.1 readily imply the
existence part of Corollary A.2. The uniqueness part is due to [BB17] for
general cscK metrics. ∎

Note that in the toric case i.e., when $G$ is an algebraic torus, Corollary
A.2 was known before as a result of [His20] combined with [CC18a, CC18b]. Our
approach above extends [Li20, 1.12] by some part of the theory of Mori dream
space [BK94, HK00].

######  Acknowledgements.

Y.O would like to thank Y.Gongyo, C.Li and S.Okawa for the nice interactions.

Contact: Yuji Odaka, Department of Mathematics, Kyoto University, Kyoto
606-8285. JAPAN
<EMAIL_ADDRESS>
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