# Boundedness of log Fano pairs with certain K-stability

Konstantin Loginov and Chuyu Zhou  École Polytechnique Fédérale de Lausanne
(EPFL), MA C3 615, Station 8, 1015 Lausanne, Switzerland<EMAIL_ADDRESS>Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
<EMAIL_ADDRESS>
###### Abstract.

We prove several boundedness results for log Fano pairs with certain
K-stability. In particular, we prove that K-semistable log Fano pairs of Maeda
type in each dimension form a log bounded family. We also compute K-semistable
domains for some examples.

2010 _Mathematics Subject Classification_ : 14J45.
_Keywords_ : Log Fano pair, Maeda type, boundedness, K-stability, K-semistable
domain.
_Competing interests_ : This work started while the authors were enjoying the
SinG school in Geometry in Trento (Italy) during January 23–27, 2023, where
the hospitality is gratefully acknowledged.

##  1\. Introduction

We work over complex number field $\mathbb{C}$ throughout.

In this paper, we investigate the relation between K-stability and boundedness
of log Fano pairs. A log pair $(X,D)$ where $D=\sum_{i=1}^{k}D_{i}$ is called
a _log Fano manifold_ if it is log smooth and $D_{i}$ are distinct prime
divisors on $X$ such that $-K_{X}-D$ is ample. Log Fano manifolds were
classified in dimension $\leq 3$ by H. Maeda, see [Maeda86, Loginov21]. It is
well known that log Fano manifolds are not bounded in every dimension starting
from $2$. Indeed, consider log Fano manifolds $(\mathbb{F}_{n},s)$ where
$\mathbb{F}_{n}$ is the $n$-th Hirzebruch surface and $s$ is a $(-n)$-section,
it is clear that they are not bounded. However, if the log Fano manifolds have
certain K-stability conditions, they are bounded in dimension $2$.

Indeed, in [Fujita20], the following definition was introduced. A _log Fano
pair of Maeda type_ is a pair $(X,\sum_{i=1}^{k}c_{i}D_{i})$ where
$(X,\sum_{i=1}^{k}D_{i})$ is a log Fano manifold,
$c_{i}\in[0,1)\cap\mathbb{Q}$, and $-K_{X}-\sum_{i=1}^{k}c_{i}D_{i}$ is ample.
Then, in dimension $2$ there is the following

######  Theorem 1.1 ([Fujita20, Theorem 1]).

Let $(X,\sum_{i}c_{i}D_{i})$ be a log Fano pair of Maeda type such that $\dim
X=2$. Then $(X,\sum_{i}c_{i}D_{i})$ is K-semistable if and only if it is
isomorphic to

  * •

$(\mathbb{P}^{2},aC)$ with $C$ a smooth conic and $0\leq a\leq 3/4$, or

  * •

$(\mathbb{P}^{1}\times\mathbb{P}^{1},aC)$ with $C$ the diagonal and $0\leq
a\leq 1/2$.

The same phenomenon appears in dimension three. More precisely, in [Loginov21,
Theorem 1] it is proven that three-dimensional K-semistable log Fano pairs of
Maeda type with at least two components in the boundary are log bounded. Then
it is natural to ask whether K-semistable log Fano pairs of Meada type in
every dimension are bounded. One of the main results of this paper is to
confirm this boundedness.

######  Theorem 1.2.

(= Theorem 3.4 ) K-semistable log Fano pairs of Maeda type in dimension $d$
are log bounded.

We also consider another set of log pairs with certain K-stability. Fix two
positive integers $d$ and $k$, a positive number $v$, and a finite set $I$ of
non-negative rational numbers, we consider the set
$\mathcal{E}:=\mathcal{E}(d,k,v,I)$ of log pairs $(X,\sum_{i=1}^{k}D_{i})$
satisfying the following conditions:

  1. (1)

$X$ is a Fano variety of dimension $d$ and $(-K_{X})^{d}=v$;

  2. (2)

$0\leq D_{i}\sim_{\mathbb{Q}}-K_{X}$ for every $1\leq i\leq k$;

  3. (3)

the coefficients of $D_{i}$ are contained in $I$;

  4. (4)

there exists $(c_{1},...,c_{k})\in\Delta^{k}$ such that
$(X,\sum_{i}c_{i}D_{i})$ is K-semistable, where
$\Delta^{k}:=\\{(c_{1},...,c_{k})\ |\ \textit{$c_{i}\in[0,1)\cap\mathbb{Q}$
and $0\leq\sum_{i}c_{i}<1$}\\}$.

We confirm the boundedness of $\mathcal{E}$.

######  Theorem 1.3.

(= Theorem 4.1 ) The set $\mathcal{E}$ is log bounded.

When there is only one component in the boundary, i.e. $k=1$, the log
boundedness of $\mathcal{E}$ is confirmed in [Zhou23, Section 5].

In the last section, we also compute K-semistable domains (see Definition 5.1)
for various examples. In particular, we describe the domains for two special
classes of pairs as follows.

######  Theorem 1.4.

(Theorem 5.8, Example 5.10) For $n\geq 2$, the K-semistable domain for
$(\mathbb{P}^{n},Q+L)$ is a polytope generated by the following three
K-semistable log pairs

$\mathbb{P}^{n},\ \ \ (\mathbb{P}^{n},\frac{n+1}{2n}Q),\ \ \
(\mathbb{P}^{n},\frac{n}{2(n-1)}Q+\frac{1}{n-1}L).$

Here $Q$ is a smooth quadric hypersurface and $L$ is a hyperplane such that
$(\mathbb{P}^{n},Q+L)$ is log smooth.

######  Theorem 1.5.

(Theorem 5.9, Example 5.11) For $n\geq 3$, the K-semistable domain for
$(\mathbb{P}^{n},Q+Q^{\prime})$ is a polytope generated by the following four
K-semistable log pairs

$\mathbb{P}^{n},\ \ \ (\mathbb{P}^{n},\frac{n+1}{2n}Q),\ \ \
(\mathbb{P}^{n},\frac{n+1}{2n}Q^{\prime}),\ \ \
(\mathbb{P}^{n},\frac{n+1}{2(n-1)}Q+\frac{n+1}{2(n-1)}Q^{\prime}).$

Here $Q,Q^{\prime}$ are smooth quadric hypersurfaces such that
$(\mathbb{P}^{n},Q+Q^{\prime})$ is log smooth.

### Acknowledgement

We are grateful to Julia Schneider for her help on Latex techniques. C. Zhou
is supported by the grant of European Research Council (ERC-804334). K.
Loginov is supported by Russian Science Foundation under grant 21-71-00112.

##  2\. Preliminaries

For the standard definitions of birational geometry, including the concepts of
klt, lc, dlt and $\epsilon$-lc singularities, we refer to [KM98, Kollar13].

We say that $(X,\Delta)$ is a _log pair_ if $X$ is a normal projective variety
and $\Delta$ is an effective $\mathbb{Q}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. The log pair $(X,\Delta)$ is called
_log Fano_ if it admits lc singularities and $-(K_{X}+\Delta)$ is ample; if
$\Delta=0$, we just say $X$ is a _Fano variety_. The log pair $(X,\Delta)$ is
called a _log Calabi-Yau pair_ if $K_{X}+\Delta\sim_{\mathbb{Q}}0$.

###  2.1. K-stability

Let $(X,\Delta)$ be a log pair. Suppose $f\colon Y\to X$ is a proper
birational mophism between normal varieties and $E$ is a prime divisor on $Y$,
we say that $E$ is a prime divisor over $X$ and we define the following
invariant

$A_{(X,\Delta)}(E):=1+{\rm{ord}}_{E}(K_{Y}-f^{*}(K_{X}+\Delta)),$

which is called the _log discrepancy_ of $E$ associated to the log pair
$(X,\Delta)$. If $(X,\Delta)$ is a log Fano pair, we define the following
invariant

$S_{(X,\Delta)}(E):=\frac{1}{{\rm{vol}}(-K_{X}-\Delta)}\int_{0}^{\infty}{\rm{vol}}(-f^{*}(K_{X}+\Delta)-xE){\rm{d}}x.$

Denote $\beta_{(X,\Delta)}(E):=A_{(X,\Delta)}(E)-S_{(X,\Delta)}(E)$. By the
works [Fuj19, Li17], one can define K-stability of a log Fano pair by beta
criterion as follows.

######  Definition 2.1.

Let $(X,\Delta)$ be a klt log Fano pair. We say that $(X,\Delta)$ is _K-
semistable_ if $\beta_{X,\Delta}(E)\geq 0$ for any prime divisor $E$ over $X$.

######  Definition 2.2.

Let $(X,\Delta)$ be a log Calabi-Yau pair, i.e.
$K_{X}+\Delta\sim_{\mathbb{Q}}0$. We say $(X,\Delta)$ is _K-semistable_ if
$(X,\Delta)$ is log canonical (this is equivalent to saying that
$\beta_{(X,\Delta)}(E)\geq 0$ for any prime divisor $E$ over $X$ since
$S_{(X,\Delta)}(E)$=0 in this case, see [BHJ17, Corollary 9.4], [Oda13]).

######  Remark 2.3.

Let $(X,\Delta)$ be a log Fano pair and $0\leq
D\sim_{\mathbb{Q}}-K_{X}-\Delta$. We will use the following well-known
results.

  1. (1)

For a rational number $0\leq c<1$, we have

$S_{(X,\Delta+cD)}(E)=(1-c)S_{(X,\Delta)}(E),$

where $E$ is any prime divisor over $X$.

  2. (2)

Suppose $(X,\Delta)$ is K-semistable and $\dim X=d$, then we have (e.g.
[BJ20])

$\alpha(X,\Delta)\geq\frac{1}{d+1}.$

Here $\alpha(X,\Delta)$ is the alpha-invariant of the log Fano pair
$(X,\Delta)$.

###  2.2. Complements

######  Definition 2.4.

Let $(X,\Delta)$ be a log Fano pair. We say a $\mathbb{Q}$-divisor $D\geq 0$
is a _complement_ of $(X,\Delta)$ if $(X,\Delta+D)$ is log canonical and
$K_{X}+\Delta+D\sim_{\mathbb{Q}}0$; we say $D$ is an _$N$ -complement_ for
some positive integer $N$ if $D$ is a complement and $N(K_{X}+\Delta+D)\sim
0$.

######  Theorem 2.5.

([Birkar19]) Let $(X,\Delta)$ be a klt log Fano pair, then there exists a
positive number $N$ depending only on the dimension of $X$ and the
coefficients of $\Delta$ such that $(X,\Delta)$ admits an $N$-complement.

###  2.3. Boundedness of Fano varieties

We recall the following results on boundedness of Fano varieties.

######  Theorem 2.6.

([Birkar19, Birkar21]) Fix a positive integer $d$ and a positive real number
$\epsilon>0$. Then the following set lies in a bounded family:

$\\{X\ |\ \textit{$X$ is of dimension $d$ and $(X,\Delta)$ is $\epsilon$-lc
log Fano for some $\mathbb{Q}$-divisor $\Delta$ on $X$}\\}.$

######  Theorem 2.7.

([Jiang20]) Fix a positive integer $d$ and a positive real number $v>0$. Then
the following set lies in a bounded family:

$\\{X\ |\ \textit{$X$ is a K-semistable Fano variety of dimension $d$ with
$(-K_{X})^{d}=v$}\\}.$

##  3\. Boundedness I

The goal of this section is to prove Theorem 1.2. We recall the following
definition introduced in [Fujita20]:

######  Definition 3.1.

A _log Fano pair of Maeda type_ is a pair $(X,\sum_{i=1}^{k}c_{i}D_{i})$ where
$(X,\sum_{i=1}^{k}D_{i})$ is a log Fano manifold,
$c_{i}\in[0,1)\cap\mathbb{Q}$, and $-K_{X}-\sum_{i=1}^{k}c_{i}D_{i}$ is ample.
Here by a _log Fano manifold_ me mean a log pair $(X,\sum_{i=1}^{k}D_{i})$
such that it is log smooth and $D_{i}$ are distinct prime divisors on $X$ with
ample $-K_{X}-\sum_{i=1}^{k}D_{i}$.

To give an example, the log pair $(\mathbb{P}^{2},\frac{1}{2}(L_{1}+L_{2}))$,
where $L_{1}$ and $L_{2}$ are two different lines on $\mathbb{P}^{2}$, is of
Maeda type, while the log pair $(\mathbb{P}^{2},\frac{1}{2}(L+Q))$, where $L$
is a general line and $Q$ is a general conic on $\mathbb{P}^{2}$, is not of
Maeda type.

######  Definition 3.2.

Let $X$ be a projective normal variety and $L$ a pseudo-effective
$\mathbb{Q}$-line bundle on $X$. Let $D$ be a prime divisor on $X$, then the
_pseudo-effective threshold_ of $D$ with respect to $L$ is defined as

$\tau(D;L):=\sup\\{t\in\mathbb{R}\ |\ \textit{$L-tD$ is pseudo-effective}\\}.$

######  Lemma 3.3.

Let $(X,D:=\sum_{i=1}^{k}D_{i})$ be a log smooth pair such that $D_{i}$ are
prime divisors on $X$ and $-K_{X}-\sum_{i=1}^{k}D_{i}$ is big and nef, then
there exists a positive number $a_{d}$ depending only on the dimension $d$
such that

$\tau(D;-K_{X}-\sum_{i}D_{i})\geq a_{d}.$

###### Proof.

Suppose there exists a log pair of Maeda type such that

$\tau(D;-K_{X}-\sum_{i}D_{i})<\frac{1}{d+1}.$

Denote by $M:=-K_{X}-D$, then we see that $K_{X}+lM$ is not pseudo-effective
for $1\leq l\leq d+1$. This implies the following vanishing for $1\leq l\leq
d+1$

$H^{0}(X,K_{X}+lM)=0.$

On the other hand, by Kawamata-Viehweg vanishing, we have

$H^{i}(X,K_{X}+lM)=0$

for any $i\geq 1$ and $l\geq 1$. Hence we see

$\chi(X,K_{X}+lM)=0$

for $1\leq l\leq d+1$. By Hirzebruch-Riemann-Roch formula, the Euler
characteristic $\chi(X,K_{X}+lM)$ is a polynomial of degree $d$ with leading
term $\frac{M^{d}}{d!}l^{d}$. This leads to a contradiction as the equation
$\chi(X,K_{X}+lM)=0$ cannot have $d+1$ distinct roots.

We are done by taking $a_{d}=\frac{1}{d+1}.$ ∎

######  Theorem 3.4.

K-semistable log Fano pairs of Maeda type in dimension $d$ form a log bounded
family.

###### Proof.

Let $(X,\sum_{i=1}^{k}c_{i}D_{i})$ be a K-semistable log pair of Maeda type,
we first show that $X$ belongs to a bounded family. We claim that there exists
a positive number $\epsilon_{d}$ depending only on the dimension $d$ such that
$1-c_{i}\geq\epsilon_{d}$ for each $i$. We focus on $c_{1}$.

By Lemma 3.3, there exists a positive number $a_{d}$ depending only on the
dimension $d$ such that

$\tau(D_{1};-K_{X}-\sum_{i}c_{i}D_{i})>\tau(D;-K_{X}-\sum_{i}D_{i})\geq
a_{d}.$

Thus there exists $0\leq\Delta\sim_{\mathbb{Q}}-K_{X}-\sum_{i}c_{i}D_{i}$ such
that ${\rm{ord}}_{D_{1}}(\Delta)\geq a_{d}$. Let
$\alpha:=\alpha(X,\sum_{i}c_{i}D_{i})$ be the alpha invariant of the log Fano
pair $(X,\sum_{i}c_{i}D_{i})$, then the log pair
$(X,\sum_{i}c_{i}D_{i}+\frac{\alpha}{2}\Delta)$ is log canonical. Thus we see

$c_{1}+\frac{\alpha}{2}\cdot a_{d}\leq 1.$

Since $(X,\sum_{i}c_{i}D_{i})$ is K-semistable, then $\alpha\geq\frac{1}{d+1}$
by [BJ20]. Take $\epsilon_{d}:=\frac{a_{d}}{2(d+1)}$, we see that

$1-c_{1}\geq\epsilon_{d}.$

Similarly, this inequality also holds for other $c_{i}$, which implies that
$(X,\sum_{i}c_{i}D_{i})$ is $\epsilon_{d}$-lc. By [Birkar19, Birkar21], we see
that $X$ belongs to a bounded family.

To derive the log boundedness, we take a very ample line bundle $A_{X}$ on $X$
such that $A_{X}^{d}$ is upper bounded by $M(d)$, which only depends on the
dimension $d$. Note that $-K_{X}A_{X}^{d-1}$ is also upper bounded by a
positive number $N(d)$ which depends only on the dimension $d$, then we see

$\sum_{i}D_{i}.A_{X}^{d-1}<-K_{X}A_{X}^{d-1}\leq N(d)$

since $-K_{X}-\sum_{i}D_{i}$ is ample. This means that the degree of
$\sum_{i}D_{i}$ is upper bounded. By a standard argument on Chow scheme we see
that $(X,\sum_{i}D_{i})$ belongs to a log bounded family. ∎

######  Example 3.5.

Note that if a log Fano pair $(X,\sum D_{i})$ is not log smooth, then even if
$(X,\sum a_{i}D_{i})$ is K-semistable for some numbers $a_{i}$, it is not true
that $(X,\sum D_{i})$ is log bounded.

Indeed, for any $n\geq 1$ put $X=\mathbb{P}(1,1,n)$ with coordinates $x,y,z$,
and $D=\\{z=0\\}$. Note that $D\sim nH$ where $H$ is a generator of the class
group of $X$. It is easy to see that the pair $(X,D)$ is a dlt log Fano pair.
Then $(X,(1-\frac{1}{n})D)$ is a K-semistable log Fano pair (e.g. [LZ22, Prop
2.11]) while there is no boundedness.

We conclude by formulating the following question:

######  Question 3.6.

Fix a natural number $d$ and a rational number $\epsilon>0$. Suppose that
$(X,D=\sum D_{i})$ is a dlt log Fano pair of dimension $d$, and
$(X,(1-\epsilon)D)$ is $\epsilon$-lc. Also, assume that $(X,\sum c_{i}D_{i})$
is K-semistable for some rational numbers $c_{i}\in[0,1)$. Is it true that
$(X,D)$ is log bounded?

##  4\. Boundedness II

Fix two positive integers $d$ and $k$, a positive number $v$, and a finite set
$I$ of non-negative rational numbers, we consider the set
$\mathcal{E}:=\mathcal{E}(d,k,v,I)$ of log pairs $(X,\sum_{i=1}^{k}D_{i})$
satisfying the following conditions:

  1. (1)

$X$ is a Fano variety of dimension $d$ and $(-K_{X})^{d}=v$;

  2. (2)

$0\leq D_{i}\sim_{\mathbb{Q}}-K_{X}$ for every $1\leq i\leq k$;

  3. (3)

the coefficients of $D_{i}$ are contained in $I$;

  4. (4)

there exists $(c_{1},...,c_{k})\in\Delta^{k}$ such that
$(X,\sum_{i}c_{i}D_{i})$ is K-semistable, where
$\Delta^{k}:=\\{(c_{1},...,c_{k})\ |\ \textit{$c_{i}\in[0,1)\cap\mathbb{Q}$
and $0\leq\sum_{i}c_{i}<1$}\\}$.

######  Theorem 4.1.

The set $\mathcal{E}$ is log bounded.

###### Proof.

We divide the proof into several steps.

Step 1. In this step, we explain that it is enough to show that the set

$\mathcal{G}:=\\{X\ |\ \textit{$(X,\sum_{i=1}^{k}D_{i})\in\mathcal{E}$}\\}$

is bounded. Suppose $\mathcal{G}$ is bounded, then for each
$(X,\sum_{i}D_{i})\in\mathcal{E}$, there exists a very ample line bundle
$A_{X}$ on $X$ such that $A_{X}^{d}\leq M(d)$ and $-K_{X}A_{X}^{d-1}\leq
N(d)$, where $M(d)$ and $N(d)$ are positive numbers depending only on the
dimension $d$. For each $i$, it is clear that $D_{i}A_{X}^{d-1}\leq N(d)$. As
the coefficients of $D_{i}$ are contained in a fixed set $I$, we see that the
degree of each component of $D_{i}$ is upper bounded. Apply a standard
argument on Chow scheme we see that $D_{i}$ lies in a bounded family, hence
$\mathcal{E}$ is log bounded. The rest of the proof is devoted to the
boundedness of $\mathcal{G}$.

Step 2. In this step, we want to replace
$(X,\sum_{i=1}^{k}D_{i})\in\mathcal{E}$ with
$(X,\sum_{i=1}^{k}D_{i}^{\prime})$ such that
$(X,\sum_{i=1}^{k}D_{i}^{\prime})\in\mathcal{E}$ and $(X,D_{i}^{\prime})$ is a
log canonical Calabi-Yau pair for every $1\leq i\leq k$. By [Birkar19], there
exists a positive integer $m(d)$ depending only on the dimension $d$ such that
$X$ admits $m(d)$-complements. We may choose $m(d)$ properly such that
$m(d)\cdot I\in\mathbb{Z}$. Put $\mathbb{P}:=\frac{1}{m(d)}|-m(d)K_{X}|$, then
we see that $D_{i}\in\mathbb{P}$ for every $1\leq i\leq k$. Since
$(X,\sum_{i=1}^{k}D_{i})\in\mathcal{E}$, there exist some rational numbers
$0\leq c_{i}<1$ such that $(X,\sum_{i}c_{i}D_{i})$ is K-semistable. We are
ready to replace $D_{1}$ with some $D_{1}^{\prime}$. Let $\mathcal{D}\subset
X\times\mathbb{P}$ be the universal divisor associated to the linear system
$\frac{1}{m(d)}|-m(d)K_{X}|$ and consider the universal family

$(X\times\mathbb{P},c_{1}\mathcal{D}+\sum_{j=2}^{k}c_{j}D_{j}\times\mathbb{P})\to\mathbb{P}.$

Since there exists a fiber which is K-semistable, i.e.
$(X,\sum_{i=1}^{k}c_{i}D_{i}$), one could find an open subset
$U\subset\mathbb{P}$ such that
$(X,c_{1}\mathcal{D}_{t}+\sum_{j=2}^{k}c_{j}D_{j})$ is K-semistable for any
$t\in U$ (see [Xu20, BLX22]). Recall that $X$ admits $m(d)$-complements, so we
conclude that $(X,\mathcal{D}_{t})$ is log canonical for general $t\in U$.
Replace $D_{1}$ with such a $\mathcal{D}_{t}$ and expand $I$ properly, we
obtain $D_{1}^{\prime}$ as required. By the same way, we could replace other
$D_{j},2\leq j\leq k$, step by step.

Step 3. By step 2, we assume $(X,\sum_{i=1}^{k}D_{i})\in\mathcal{E}$ satisfies
that $(X,D_{i})$ is log canonical for every $1\leq i\leq k$. For such log pair
$(X,\sum_{i}D_{i})$, we define the following invariant:

$\mu(X,\sum_{i=1}^{k}D_{i}):=\inf\Bigg{\\{}\sum_{i=1}^{k}c_{i}\ |\
\textit{$(X,\sum_{i}c_{i}D_{i})$ is K-semistable}\Bigg{\\}}.$

It is clear that $0\leq\mu(X,\sum_{i}D_{i})<1$. We aim to show that there is a
gap between $\mu(X,\sum_{i}D_{i})$ and $1$. More precisely, there exists a
positive number $0<\epsilon_{0}(d,I)<1$ depending only on $d$ and $I$ such
that

$1-\mu(X,\sum_{i=1}^{k}D_{i})\geq\epsilon_{0}.$

Suppose not, we could find a sequence of pairs $(X_{j},\sum_{i=1}^{k}D_{ji})$
satisfying the following conditions:

  1. (1)

$(X_{j},\sum_{i=1}^{k}D_{ji})\in\mathcal{E}$ for every $j$,

  2. (2)

$(X_{j},D_{ji})$ is log canonical for every $1\leq i\leq k$ and $j$,

  3. (3)

$\mu_{j}:=\mu(X_{j},\sum_{i=1}^{k}D_{ji})$ is an increasing sequence tending
to $1$.

By the definition of $\mu_{j}$, one could find an increasing sequence of
rational numbers $a_{j}<\mu_{j}$ tending to $1$ such that

$(X_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}D_{ji})$

is K-unstable for every $j$. By [BLZ22, LXZ22], for each
$(X_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}D_{ji})$, there exists a prime divisor
$E_{j}$ over $X_{j}$ such that $E_{j}$ computes the delta invariant of
$(X_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}D_{ji})$ and $E_{j}$ induces a special
test configuration

$(\mathcal{X}_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}\mathcal{D}_{ji})\to\mathbb{A}^{1}.$

Subtracting those $a_{j}$ which are not close enough to $1$, we may assume
that the central fiber of the test configuration (after changing the
coefficients), i.e.
$(\mathcal{X}_{j,0},\sum_{i=1}^{k}\frac{1}{k}\mathcal{D}_{ji,0})$, is log
canonical. This means that the test configuration degenerates the log
canonical Calabi-Yau pair $(X_{j},\sum_{i=1}^{k}\frac{1}{k}D_{ji})$ to another
log canonical Calabi-Yau pair
$(\mathcal{X}_{j,0},\sum_{i=1}^{k}\frac{1}{k}\mathcal{D}_{ji,0})$. By [Zhou23,
Lemma 2.8], $E_{j}$ is an lc place of
$(X_{j},\sum_{i=1}^{k}\frac{1}{k}D_{ji})$, which forces that $E_{j}$ is an lc
place of $(X_{j},D_{ji})$ for every $1\leq i\leq k$. Recall that $E_{j}$
computes the delta invariant of the K-unstable log Fano pair
$(X_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}D_{ji})$, we have

$\displaystyle\beta_{(X_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}D_{ji})}(E_{j})\ $
$\displaystyle=\
A_{(X_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}D_{ji})}(E_{j})-S_{(X_{j},\sum_{i=1}^{k}\frac{a_{j}}{k}D_{ji})}(E_{j})$
$\displaystyle=\ (1-a_{j})\\{A_{X_{j}}(E_{j})-S_{X_{j}}(E_{j})\\}$
$\displaystyle<\ 0.$

On the other hand, $(X_{j},\sum_{i=1}^{k}c_{ji}D_{ji})$ is K-semistable for
some rational $0\leq c_{ji}<1$, therefore,

$\displaystyle\beta_{(X_{j},\sum_{i=1}^{k}c_{ji}D_{ji})}(E_{j})\ $
$\displaystyle=\
A_{(X_{j},\sum_{i=1}^{k}c_{ji}D_{ji})}(E_{j})-S_{(X_{j},\sum_{i=1}^{k}c_{ji}D_{ji})}(E_{j})$
$\displaystyle=\
(1-\sum_{i=1}^{k}c_{ji})\\{A_{X_{j}}(E_{j})-S_{X_{j}}(E_{j})\\}$
$\displaystyle\geq\ 0,$

which is a contradiction. This contradiction implies the existence of the gap
we want.

Step 4. Combining step 2 and step 3, we see that for each
$(X,\sum_{i=1}^{k}D_{i})\in\mathcal{E}$, one can find another log pair
$(X,\sum_{i=1}^{k}D_{i}^{\prime})\in\mathcal{E}$ such that
$(X,\sum_{i}c_{i}D_{i}^{\prime})$ is K-semistable for some numbers $0\leq
c_{i}<1$ with

$1-\sum_{i=1}^{k}c_{i}\geq\epsilon_{0}(d,I),$

where $\epsilon_{0}(d,I)$ depends only on $d,I$. Thus we have

$\displaystyle\frac{A_{X}(E)}{(1-\sum_{i}c_{i})S_{X}(E)}\ $
$\displaystyle\geq\
\frac{A_{(X,\sum_{i=1}^{k}c_{i}D_{i})}(E)}{S_{(X,\sum_{i=1}^{k}c_{i}D_{i})}(E)}\geq
1$

for any prime divisor $E$ over $X$. Hence,

$\frac{A_{X}(E)}{S_{X}(E)}\geq 1-\sum_{i=1}^{k}c_{i}\geq\epsilon_{0}(d,I),$

for any prime divisor $E$ over $X$. This says that the delta invariant of $X$
is bounded from below by $\epsilon_{0}(d,I)$. By [Jiang20], the set
$\mathcal{G}$ defined in step 1 lies in a bounded family. The proof is
finished. ∎

##  5\. On K-semistable domains

We have studied the boundedness of two kinds of log Fano pairs with certain
K-stability. It is then natural to ask what are the K-semistable domains for
these log Fano pairs. We present some examples to predict that the domains
should be polytopes for log pairs in $\mathcal{E}$, and we will explore this
problem from theoretical viewpoint in the future work.

######  Definition 5.1.

Let $(X,\sum_{i=1}^{k}D_{i})$ be a log Fano manifold or a log pair in the set
$\mathcal{E}$. We define the _K-semistable domain_ of
$(X,\sum_{i=1}^{k}D_{i})$ as follows:

$\mathrm{Kss}(X,\sum_{i=1}^{k}D_{i}):=\overline{\\{(c_{1},...,c_{k})\ |\
\textit{$c_{i}\in[0,1)\cap\mathbb{Q}$ and $(X,\sum_{i=1}^{k}c_{i}D_{i})$ is
K-semistable}\\}}.$

The overline in the definition means taking the closure.

Before we present the examples, let us first note the following interpolation
property for K-stability, which we will use frequently: if $(X,\Delta_{1})$
and $(X,\Delta_{2})$ are both K-semistable log pairs (log Fano or log Calabi-
Yau), where $\Delta_{i}$ are propotional to $-K_{X}$, then
$(X,t\Delta_{1}+(1-t)\Delta_{2})$ is also K-semistable for any
$t\in[0,1]\cap\mathbb{Q}$.

######  Example 5.2.

Consider a log pair $(X,D_{1}+D_{2}):=(\mathbb{P}^{2},3L_{1}+3L_{2})$, where
$L_{1},L_{2}$ are two distinct lines in $\mathbb{P}^{2}$. Then

$\mathrm{Kss}(X,D_{1}+D_{2})=\\{(0,0)\\}.$

To see this, we suppose $(x,y)\in\mathrm{Kss}(X,D_{1}+D_{2})$, then
$(\mathbb{P}^{2},3xL_{1}+3yL_{2})$ is K-semistable. Applying the beta
criterion we have

$\displaystyle\beta_{(\mathbb{P}^{2},3xL_{1}+3yL_{2})}(L_{1})\ $
$\displaystyle=\
1-3x-\frac{1}{(3-3x-3y)^{2}}\int_{0}^{3-3x-3y}(3-3x-3y-t)^{2}{\rm{dt}}$
$\displaystyle=\ 1-3x-(1-x-y)$ $\displaystyle=\ (x+y)-3x\geq 0.$

Similarly,

$\displaystyle\beta_{(\mathbb{P}^{2},3xL_{1}+3yL_{2})}(L_{2})\ $
$\displaystyle=\
1-3y-\frac{1}{(3-3x-3y)^{2}}\int_{0}^{3-3x-3y}(3-3x-3y-t)^{2}{\rm{dt}}$
$\displaystyle=\ 1-3y-(1-x-y)$ $\displaystyle=\ (x+y)-3y\geq 0.$

The above two equations have the unique solution $(0,0)$, which agrees with
Theorem 1.1.

######  Example 5.3.

Consider a log pair
$(X,D_{1}+D_{2}):=(\mathbb{P}^{2},\frac{3}{2}Q_{1}+\frac{3}{2}Q_{2})$, where
$Q_{1},Q_{2}$ are two general smooth conics in $\mathbb{P}^{2}$. We show that
$\mathrm{Kss}(X,D_{1}+D_{2})$ is given by the following polytope:

$\begin{cases}x\geq 0\\\ y\geq 0\\\ y\leq\frac{1}{2}x+\frac{1}{2}\\\ y\geq
2x-1\\\ x+y\leq 1\\\ \end{cases}$

It is presented by the following picture:

$\frac{1}{3}$$\frac{1}{2}$$\frac{2}{3}$$\frac{1}{3}$$\frac{1}{2}$$\frac{2}{3}$$y$$x$

The extremal points of this polytope correspond to the pairs

$\mathbb{P}^{2},\ \ \ (\mathbb{P}^{2},\frac{3}{4}Q_{2}),\ \ \
(\mathbb{P}^{2},\frac{1}{2}Q_{1}+Q_{2}),\ \ \
(\mathbb{P}^{2},Q_{1}+\frac{1}{2}Q_{2}),\ \ \
(\mathbb{P}^{2},\frac{3}{4}Q_{1}),$

all of which are K-semistable (note that a log Calabi-Yau pair is K-semistable
means that it is log canonical). Thus the polytope is contained in
$\mathrm{Kss}(X,D_{1}+D_{2})$ by the interpolation property of K-stability. To
see that it is exactly the K-semistable domain, it is enough to show that the
points in

$\begin{cases}0<x<\frac{1}{3}\\\ \frac{1}{2}<y<\frac{2}{3}\\\
y>\frac{1}{2}x+\frac{1}{2}\\\ \end{cases}\ \ \text{and}\ \ \ \ \
\begin{cases}\frac{1}{2}<x<\frac{2}{3}\\\ 0<y<\frac{1}{3}\\\ y<2x-1\\\
\end{cases}$

are not K-semistable. These two domains are presented below:

$\frac{1}{3}$$\frac{1}{2}$$\frac{2}{3}$$\frac{1}{3}$$\frac{1}{2}$$\frac{2}{3}$$y$$x$

Suppose $(x,y)\in\mathrm{Kss}(X,D_{1}+D_{2})$, then
$(\mathbb{P}^{2},\frac{3}{2}xQ_{1}+\frac{3}{2}yQ_{2})$ is K-semistable.
Applying the beta criterion, we have

$\displaystyle\beta_{(\mathbb{P}^{2},\frac{3}{2}xQ_{1}+\frac{3}{2}yQ_{2})}(Q_{1})\
$ $\displaystyle=\
1-\frac{3}{2}x-\frac{1}{(3-3x-3y)^{2}}\int_{0}^{\frac{{3-3x-3y}}{2}}(3-3x-3y-2t)^{2}{\rm{dt}}$
$\displaystyle=\ 1-\frac{3}{2}x-\frac{1-x-y}{2}$ $\displaystyle=\
\frac{1}{2}y-x+\frac{1}{2}\geq 0.$

Similarly,

$\displaystyle\beta_{(\mathbb{P}^{2},\frac{3}{2}xQ_{1}+\frac{3}{2}yQ_{2})}(Q_{2})\
$ $\displaystyle=\
1-\frac{3}{2}y-\frac{1}{(3-3x-3y)^{2}}\int_{0}^{\frac{{3-3x-3y}}{2}}(3-3x-3y-2t)^{2}{\rm{dt}}$
$\displaystyle=\ 1-\frac{3}{2}y-\frac{1-x-y}{2}$ $\displaystyle=\
-y+\frac{1}{2}x+\frac{1}{2}\geq 0.$

It is clear that the points of the two domains cannot satisfy the above two
equations, which implies that they are not contained in the K-semistable
domain.

######  Example 5.4.

Consider a log pair $(X,D_{1}+D_{2}):=(\mathbb{P}^{2},\frac{3}{2}Q+3L)$, where
$Q$ is a smooth conic and $L$ is a line such that $(\mathbb{P}^{2},Q+L)$ is
log smooth. We show that $\mathrm{Kss}(X,D_{1}+D_{2})$ is the polytope
generated by the points

$(0,0),\ \ \ (\frac{1}{2},0),\ \ \ (\frac{2}{3},\frac{1}{3}).$

It is clear that the three points correspond to the the three log pairs

$\mathbb{P}^{2},\ \ \ (\mathbb{P}^{2},\frac{3}{4}Q),\ \ \
(\mathbb{P}^{2},Q+L).$

We denote this polytope by $P$. To see $\mathrm{Kss}(X,D_{1}+D_{2})=P$, first
note that the three pairs are all K-semistable, thus
$P\subset\mathrm{Kss}(X,D_{1}+D_{2})$. To see the converse inclusion, we apply
the beta criterion to $Q$, $L$. Suppose $(x,y)\in\mathrm{Kss}(X,D_{1}+D_{2})$,
then $(\mathbb{P}^{2},\frac{3}{2}xQ+3yL)$ is K-semistable. Thus

$\displaystyle\beta_{(\mathbb{P}^{2},\frac{3}{2}xQ+3yL)}(Q)\ $
$\displaystyle=\
1-\frac{3}{2}x-\frac{1}{(3-3x-3y)^{2}}\int_{0}^{\frac{{3-3x-3y}}{2}}(3-3x-3y-2t)^{2}{\rm{dt}}$
$\displaystyle=\ 1-\frac{3}{2}x-\frac{1-x-y}{2}$ $\displaystyle=\
\frac{1}{2}y-x+\frac{1}{2}\geq 0,$

$\displaystyle\beta_{(\mathbb{P}^{2},\frac{3}{2}xQ+3yL)}(L)\ $
$\displaystyle=\
1-3y-\frac{1}{(3-3x-3y)^{2}}\int_{0}^{3-3x-3y}(3-3x-3y-t)^{2}{\rm{dt}}$
$\displaystyle=\ 1-3y-(1-x-y)$ $\displaystyle=\ x-2y\geq 0.$

The polytope defined by

$\begin{cases}0\leq x\leq\frac{2}{3}\\\ 0\leq y\leq\frac{1}{3}\\\
\frac{1}{2}y-x+\frac{1}{2}\geq 0\\\ x-2y\geq 0\\\ \end{cases}$

is exactly $P$, thus $P=\mathrm{Kss}(X,D_{1}+D_{2})$.

We now plan to work out a class of examples in higher dimensions. Before that,
we first list some results we will use.

Let $V$ be a projective Fano manifold of dimension $n$, and $S$ is a smooth
divisor on $V$ such that $S\sim_{\mathbb{Q}}-\lambda K_{V}$ for some positive
rational number $\lambda$. Recall that

$\mathrm{Kss}(V,S)=\overline{\\{a\in[0,1)\cap\mathbb{Q}\ |\ \textit{$(V,aS)$
is K-semistable}\\}}.$

######  Lemma 5.5.

([ZZ22]) Notation as above, suppose V and S are both K-semistable and
$0<\lambda<1$, then $\mathrm{Kss}(V,S)=[0,1-\frac{r}{n}]$, where
$r=\frac{1}{\lambda}-1$. In particular, if $V$ and $S$ are both K-polystable,
then $(V,aS)$ is K-polystable for any $a\in[0,1-\frac{r}{n})$.

As a special case, we have

######  Lemma 5.6.

Let $(\mathbb{P}^{n},S_{d})$ be a log pair where $S_{d}$ is a hypersurface of
degree $1\leq d\leq n$. Suppose $S_{d}$ is K-semistable (this is expected to
be true for smooth hypersurfaces). Then we have
$\mathrm{Kss}(\mathbb{P}^{n},S_{d})=[0,1-\frac{r}{n}]$, where
$r=\frac{n+1-d}{d}$.

The following lemma is also well known, see e.g. [LZ22, Prop 2.11].

######  Lemma 5.7.

Let $(V,\Delta)$ be an n-dimensional log Fano pair, and L an ample line bundle
on V such that $L\sim_{\mathbb{Q}}-\frac{1}{r}(K_{V}+\Delta)$ for some
$0<r\leq n+1$. Suppose Y is the projective cone over V associated to L with
infinite divisor $V_{\infty}$, then $(V,\Delta)$ is K-semistable (resp.
K-polystable) if and only if $(Y,\Delta_{Y}+(1-\frac{r}{n+1})V_{\infty})$ is
K-semistable (resp. K-polystable), where $\Delta_{Y}$ is the divisor on Y
naturally extended by $\Delta$.

Applying Lemma 5.6, a simple computation tells us the following three facts:

  1. (1)

Let $Q\subset\mathbb{P}^{n}$ be a smooth quadric, then $(\mathbb{P}^{n},aQ)$
is K-semistable (resp. K-polystable) for any $a\in[0,\frac{n+1}{2n}]$ (resp.
$a\in[0,\frac{n+1}{2n})$). Moreover, $(\mathbb{P}^{n},\frac{n+1}{2n}Q)$ is
K-semistable but not K-polystable, and it admits a K-polystable degeneration
$(X,\frac{n+1}{2n}D)$. Here $X$ is a hypersurface in $\mathbb{P}(1^{n+1},2)$
(with coordinates $(x_{0},...,x_{n},z)$) defined by
$x_{0}^{2}+x_{1}^{2}+...+x_{n}^{2}=0$, and $D=X\cap\\{z=0\\}$.

  2. (2)

Let $Q_{l}\subset\mathbb{P}^{l+1}$ be a smooth quadric of dimension $l$, then
$(Q_{l},aQ_{l-1})$ is K-semistable for any $a\in[0,\frac{1}{l}]$.

  3. (3)

Let $Q_{l},Q_{l}^{\prime}\subset\mathbb{P}^{l+1}$ be a two smooth quadric
hypersurfaces such that $(\mathbb{P}^{l+1},Q_{l}+Q_{l}^{\prime})$ is log
smooth, then $(Q_{l},a{Q^{\prime}_{l}}|_{Q_{l}})$ is K-semistable for any
$a\in[0,\frac{(l+1)+3}{2(l+1)}]$.

We are ready to prove the following K-semistability.

######  Theorem 5.8.

For $n\geq 2$, the log pair
$(\mathbb{P}^{n},\frac{n}{2(n-1)}Q+\frac{1}{n-1}L)$ is K-semistable, where $Q$
is a smooth quadric hypersurface and $L$ is a hyperplane such that
$(\mathbb{P}^{n},Q+L)$ is log smooth.

###### Proof.

The case $n=2$ is clear. We assume $n\geq 3$. As we have observed,
$(\mathbb{P}^{n},\frac{n+1}{2n}Q)$ is K-semistable and admits a K-polystable
degeneration $(X,\frac{n+1}{2n}D)$, where $X$ is a hypersurface in
$\mathbb{P}(1^{n+1},2)$ (with coordinates $(x_{0},...,x_{n},z)$) defined by
$x_{0}^{2}+x_{1}^{2}+...+x_{n}^{2}=0$, and $D=X\cap\\{z=0\\}$. Denote by
$D^{\prime}$ the corresponding degeneration of $L$ under the test
configuration.

It suffices to show that the log pair
$(X,\frac{n}{2(n-1)}D+\frac{1}{n-1}D^{\prime})$ is a K-semistable log Fano
pair. To see this, first note that
$(X,\frac{n}{2(n-1)}D+\frac{1}{n-1}D^{\prime})$ is the projective cone over
$(Q,\frac{1}{n-1}L\cap Q)$ with respect to the polarization
$\mathcal{O}_{Q}(2):=i^{*}\mathcal{O}_{\mathbb{P}^{n}}(2)$, where
$i:Q\to\mathbb{P}^{n}$ is the natural embedding. We have the following
computation:

$\mathcal{O}_{Q}(2)\sim_{\mathbb{Q}}-\frac{1}{r}(K_{Q}+\frac{1}{n-1}L\cap
Q)\quad\text{and}\quad\frac{n}{2(n-1)}=1-\frac{r}{n}$

for

$r=\frac{n-1-\frac{1}{n-1}}{2}.$

By our notation, we have

$(Q_{n-1},\frac{1}{n-1}Q_{n-2})=(Q,\frac{1}{n-1}L\cap Q),$

which is K-semistable as we have seen before. By Lemma 5.7, we see that
$(X,\frac{n}{2(n-1)}D+\frac{1}{n-1}D^{\prime})$ is K-semistable. ∎

######  Theorem 5.9.

For $n\geq 3$, the log pair
$(\mathbb{P}^{n},\frac{n+1}{2(n-1)}Q+\frac{n+1}{2(n-1)}Q^{\prime})$ is
K-semistable, where $Q,Q^{\prime}$ are smooth quadric hypersurfaces such that
$(\mathbb{P}^{n},Q+Q^{\prime})$ is log smooth.

###### Proof.

The case $n=3$ is clear. We assume $n\geq 4$. The same as before,
$(\mathbb{P}^{n},\frac{n+1}{2n}Q)$ is K-semistable and admits a K-polystable
degeneration $(X,\frac{n+1}{2n}D)$, where $X$ is a hypersurface in
$\mathbb{P}(1^{n+1},2)$ (with coordinates $(x_{0},...,x_{n},z)$) defined by
$x_{0}^{2}+x_{1}^{2}+...+x_{n}^{2}=0$, and $D=X\cap\\{z=0\\}$. Denote by
$D^{\prime}$ the corresponding degeneration of $Q^{\prime}$ under the test
configuration.

It suffices to show that the log pair
$(X,\frac{n+1}{2(n-1)}D+\frac{n+1}{2(n-1)}D^{\prime})$ is a K-semistable log
Fano pair. To see this, first note that
$(X,\frac{n+1}{2(n-1)}D+\frac{n+1}{2(n-1)}D^{\prime})$ is the projective cone
over $(Q,\frac{n+1}{2(n-1)}Q^{\prime}|_{Q})$ with respect to the polarization
$\mathcal{O}_{Q}(2):=i^{*}\mathcal{O}_{\mathbb{P}^{n}}(2)$, where
$i:Q\to\mathbb{P}^{n}$ is the natural embedding. We have the following
computation:

$\mathcal{O}_{Q}(2)\sim_{\mathbb{Q}}-\frac{1}{r}(K_{Q}+\frac{n+1}{2(n-1)}Q^{\prime}|_{Q})\quad\text{and}\quad\frac{n+1}{2(n-1)}=1-\frac{r}{n}$

for

$r=\frac{n-1-\frac{n+1}{n-1}}{2}.$

Since $\frac{n+1}{2(n-1)}\leq\frac{n+3}{2n}$, we know that
$(Q,\frac{n+1}{2(n-1)}Q^{\prime}|_{Q})$ is K-semistable. By Lemma 5.7, we see
that $(X,\frac{n+1}{2(n-1)}D+\frac{n+1}{2(n-1)}D^{\prime})$ is K-semistable. ∎

Armed by Theorem 5.8, 5.9 , we consider the following two classes of examples.

######  Example 5.10.

Consider a pair $(\mathbb{P}^{n},Q+L)$ for $n\geq 2$, where $Q$ is a smooth
quadric hypersurface and $L$ is a hyperplane such that $(\mathbb{P}^{n},Q+L)$
is log smooth. We want to compute $\mathrm{Kss}(\mathbb{P}^{n},Q+L)$.

Suppose $(x,y)\in\mathrm{Kss}(\mathbb{P}^{n},Q+L)$, then
$(\mathbb{P}^{n},xQ+yL)$ is K-semistable. Applying beta criterion, we have

$\displaystyle\beta_{(\mathbb{P}^{n},xQ+yL)}(Q)\ $ $\displaystyle=\
1-x-\frac{1}{(n+1-2x-y)^{n}}\int_{0}^{\frac{{n+1-2x-y}}{2}}(n+1-2x-y-2t)^{n}{\rm{dt}}$
$\displaystyle=\ 1-x-\frac{n+1-2x-y}{2(n+1)}$ $\displaystyle=\
\frac{1}{2}-\frac{nx}{n+1}+\frac{y}{2(n+1)}\geq 0.$

Similarly,

$\displaystyle\beta_{(\mathbb{P}^{n},xQ+yL)}(L)\ $ $\displaystyle=\
1-y-\frac{1}{(n+1-2x-y)^{n}}\int_{0}^{n+1-2x-y}(n+1-2x-y-t)^{n}{\rm{dt}}$
$\displaystyle=\ 1-y-\frac{n+1-2x-y}{(n+1)}$ $\displaystyle=\
\frac{2x}{n+1}-\frac{ny}{n+1}\geq 0.$

It is clear that the polytope given by

$\begin{cases}0\leq x\leq 1\\\ 0\leq y\leq 1\\\
\frac{1}{2}-\frac{nx}{n+1}+\frac{y}{2(n+1)}\geq 0\\\
\frac{2x}{n+1}-\frac{ny}{n+1}\geq 0\\\ \end{cases}$

is generated by the extremal points

$(0,0),\ \ \ (\frac{n+1}{2n},0),\ \ \ (\frac{n}{2(n-1)},\frac{1}{n-1}).$

These three points correspond to log pairs

$\mathbb{P}^{n},\ \ \ (\mathbb{P}^{n},\frac{n+1}{2n}Q),\ \ \
(\mathbb{P}^{n},\frac{n}{2(n-1)}Q+\frac{1}{n-1}L),$

which are all K-semistable by Theorem 5.8. Thus the polytope is exactly
$\mathrm{Kss}(\mathbb{P}^{n},Q+L)$. We also mention that when $n=3$, the
example corresponds to [Loginov21, Theorem 1 (1)].

######  Example 5.11.

Consider a log pair $(\mathbb{P}^{n},Q+Q^{\prime})$ for $n\geq 3$, where
$Q,Q^{\prime}$ are smooth quadric hypersurfaces such that
$(\mathbb{P}^{n},Q+Q^{\prime})$ is log smooth. We want to compute
$\mathrm{Kss}(\mathbb{P}^{n},Q+Q^{\prime})$.

Suppose $(x,y)\in\mathrm{Kss}(\mathbb{P}^{n},Q+Q^{\prime})$, then
$(\mathbb{P}^{n},xQ+yQ^{\prime})$ is K-semistable. Applying the beta
criterion, we have

$\displaystyle\beta_{(\mathbb{P}^{n},xQ+yQ^{\prime})}(Q)\ $ $\displaystyle=\
1-x-\frac{1}{(n+1-2x-2y)^{n}}\int_{0}^{\frac{{n+1-2x-2y}}{2}}(n+1-2x-2y-2t)^{n}{\rm{dt}}$
$\displaystyle=\ 1-x-\frac{n+1-2x-2y}{2(n+1)}$ $\displaystyle=\
\frac{1}{2}-\frac{nx}{n+1}+\frac{y}{n+1}\geq 0.$

Similarly,

$\displaystyle\beta_{(\mathbb{P}^{n},xQ+yQ^{\prime})}(Q^{\prime})\ $
$\displaystyle=\
1-y-\frac{1}{(n+1-2x-2y)^{n}}\int_{0}^{\frac{{n+1-2x-2y}}{2}}(n+1-2x-2y-2t)^{n}{\rm{dt}}$
$\displaystyle=\ 1-y-\frac{n+1-2x-2y}{2(n+1)}$ $\displaystyle=\
\frac{1}{2}-\frac{ny}{n+1}+\frac{x}{n+1}\geq 0.$

It is clear that the polytope given by

$\begin{cases}0\leq x\leq 1\\\ 0\leq y\leq 1\\\
\frac{1}{2}-\frac{nx}{n+1}+\frac{y}{n+1}\geq 0\\\
\frac{1}{2}-\frac{ny}{n+1}+\frac{x}{n+1}\geq 0\\\ \end{cases}$

is generated by the extremal points

$(0,0),\ \ \ (\frac{n+1}{2n},0),\ \ \ (0,\frac{n+1}{2n}),\ \ \
(\frac{n+1}{2(n-1)},\frac{n+1}{2(n-1)}).$

These four points correspond to log pairs

$\mathbb{P}^{n},\ \ \ (\mathbb{P}^{n},\frac{n+1}{2n}Q),\ \ \
(\mathbb{P}^{n},\frac{n+1}{2n}Q^{\prime}),\ \ \
(\mathbb{P}^{n},\frac{n+1}{2(n-1)}Q+\frac{n+1}{2(n-1)}Q^{\prime}),$

which are all K-semistable by Theorem 5.9. Thus the polytope is exactly
$\mathrm{Kss}(\mathbb{P}^{n},Q+Q^{\prime})$.

######  Remark 5.12.

For all examples we treat above, the K-semistable domains are polytopes. In
[Loginov21], the K-semistable domains for some log Fano pairs of Maeda type in
dimension three are computed, but it is hard to say whether these sets are
convex or polytopes.

## References