2.1 GIT and Chow stabilities

In this section, we briefly recall some basic knowledge of GIT. For details, please read [Reference Gelfand, Kapranov and ZelevinskyGKZ94].

Let $(X,L)$ be an $n$ -dimensional polarized variety of degree $d$ , i.e. $X$ is a complex irreducible variety with degree $d$ , and $L$ is an ample line bundle. Then there exists $k_0\in \mathbb {N}$ such that, for every integer $k\geqslant k_0$ , there exists an embedding map $\iota _k\colon X\rightarrow \mathbb {P}(H^0(X,kL))\cong \mathbb {P}^{N_k}$ defined by

\begin{equation*}\iota _k(x):= [s_0(x),\ldots, s_{n_K(x)}],\end{equation*}

where $\{s_0,\cdots, s_{N_k}\}$ is a basis of $H^0(X,kL)$ . Consider $\underline {\iota _k(X)}$ be the corresponding image on $H^0(X,kL)$ , i.e. $p\in \underline {\iota _k(X)}$ if and only if $[p]\in \iota _k(X)$ . Then $\underline {\iota _k(X)}$ is an $(n+1)$ -dimensional cone in $H^0(X,kL)\cong {\mathbb {C}}^{N_k+1}$ . Then, for a generic linear $N_k-n+1$ subspace $W\subset H^0(X,kL)$ , the set of intersections between $W$ and $\underline {\iota _k(X)}$ is in dimension 1, but for a generic linear $N_k-n$ subspace $L\subset H^0(X,kL)$ , the set of intersections between $L$ and $\underline {\iota _k(X)}$ is $\{0\}$ .

Therefore, we can define a degree $d$ divisor $Z_X$ on the Grassmannian $Gr(N_k-n, N_k+1)$ by

\begin{equation*}Z_k:=\{L^{N_k-n}\subset H^0(X,kL) \mid L^{N_k-n}\cap (\underline {\iota _k(X)}-\{0\})\neq \phi \}.\end{equation*}

This induces a point $R_X\in \mathbb {P}(H^0(Gr(N_k-n, N_k+1)), \mathcal {O}(d))$ , which is called the Chow point.

Also, under the embedding $Gr(N_k-n, N_k+1)\rightarrow \wedge ^{n+1}(H^0(X,kL))$ , where $V=H^0(X,kL)$ , we have

\begin{equation*}R_k\in \textrm{Sym}^d(\wedge ^{n+1}H^0(X,kL)):=V_k,\end{equation*}

and then the $SL(N_k+1,{\mathbb {C}})$ action on $H^0(X,kL)$ induces an action on $V_k$ .

2.2 Toric varieties

We now recall some background knowledge of toric varieties.

We will focus on polarized toric varieties $(X,L)$ . There exists $k_0$ such that for all $k\geqslant k_0$ , the map $\iota _k\colon X \rightarrow \mathbb {P}(H^0(X,kL))$ we defined above is embedding. Moreover, we can choose the basis $\{s_0,\ldots, s_{N_k}\}$ such that for any $\lambda :=(e^{\lambda _1},\ldots, e^{\lambda _n})\in ({\mathbb {C}}^*)^n$ ,

\begin{equation*}\lambda \cdot s_i=e^{\alpha _i^1\lambda _1+\cdots +\alpha _i^n\lambda _n}s_i.\end{equation*}

Then for $\lambda \in ({\mathbb {C}}^*)^n$ ,

\begin{equation*}\lambda \cdot \iota _k(x)=\iota _k(\lambda \cdot x),\end{equation*}

and, in particular, the $(S^1)^n\lt ({\mathbb {C}}^*)^n$ is a subgroup of the Hamiltonian group of $\iota _k^*\omega _{FS}$ . Thus we have the moment polytopes $\mu _k\colon X \rightarrow \textrm{Lie}((S^1)^n)\cong \mathbb {R}^n$ , and the images are defined as the moment map polytope $P_k$ . Notice that we have

\begin{equation*}P_{kl}=lP_k, \end{equation*}

if $\iota _k$ defines an embedding. Moreover,

\begin{equation*}P_k= \textrm{conv}\{(\alpha _i^1,\ldots, \alpha _i^n)\in \mathbb {R}^n\mid i=0,\ldots, N_k\}.\end{equation*}

The reason is we have a moment map $\mu _{\mathbb {P}^{N_k}}\colon \mathbb {P}^{N_k}\rightarrow \textrm{Lie}((S^1)^{N_k})^*$ , and the image is the standard simplex. Then the moment map

\begin{equation*}\mu = \iota _k^*\circ \mu _{\mathbb {P}^{N_k}}\mid_{\iota _k(X)},\end{equation*}

where

\begin{equation*}\iota _k^*\colon \textrm{Lie}((S^1)^{N_k})^*\rightarrow \textrm{Lie}((S^1)^{n})^*, \end{equation*}

is the induced homomorphism from $\iota _k$ ,

\begin{equation*}(\iota _k^*\eta ^*)(\xi )=\eta ^*((\iota _k)_*(\xi )).\end{equation*}

In terms of a matrix,

\begin{align*}(\iota _k)_*=\left(\begin{matrix} \alpha _0^1 & & \cdots & & \alpha _0^n \\[3pt]\vdots & & \ddots & & \vdots \\[3pt] \alpha _{N_k}^1 & & \cdots & & \alpha _{N_k}^n \end{matrix}\right),\end{align*}

and as a result, $\iota _k^*=((\iota _k)_*)^T$ is the transpose and therefore

\begin{equation*}Im(\mu )=\textrm{conv}\{(\alpha _i^1,\ldots, \alpha _i^n)\in \mathbb {R}^n\mid i=0,\ldots, N_k\}.\end{equation*}

2.3 Chow stability of toric variety and criteria

Recall that by Fataki and Ono ([Reference FutakiFut04, Reference OnoOno13, Reference Ono, Sano and YotsutaniOSY12], and also see [Reference Lee, Li, Sturm and WangLLSW19]), a toric variety $X_P$ is asymptotically Chow semistable if there exists $C$ such that for any $k\geqslant C$ , and for any convex $G$ invariant function $f\colon kP \rightarrow \mathbb {R}$ , we have

(1) \begin{align} \frac {1}{\textrm{Vol}(kP)}\int _{kP} f\, dV\leqslant \frac {1}{\chi (kP \cap \mathbb {Z}^n)} \sum _{k p\in kP \cap \mathbb {Z}^n} f(p), \end{align}

and $X_P$ is polystable if the equality holds only when $v$ is affine. (In [Reference OnoOno13] and [Reference Lee, Li, Sturm and WangLLSW19], the inequality is on the opposite side as the inputs are concave functions.) Here $G\lt SL(n, \mathbb {Z})$ is the biggest group fixing $P$ , which is a discrete group.

Notice that if there exists a toric equivariant ${\mathbb {C}}^*$ action on $X_P$ , then it corresponds to an affine function on $P$ (see [Reference DonaldsonDon02]). So we can write the following definition.

Definition 2.5. Let $P$ be an integral convex polytope. The Futaki–Ono invariant of an affine function $v(x)=a_1x_1+\cdots +a_nx_n+a_0$ is given by

\begin{equation*}FO_P(a,k):=\frac {1}{\chi (kP \cap \mathbb {Z}^n)} \sum _{k p\in kP \cap \mathbb {Z}^n} a(p)-\frac {1}{\textrm{Vol}(P)}\int _P a(x)\, dV.\end{equation*}

We can rephrase Corollary 4.7 in [Reference OnoOno13] as the following lemma.

Lemma 2.6 (Corollary 4.7 in [Reference OnoOno13]; also see [Reference FutakiFut04]). Suppose $P$ is asymptotically Chow semistable. Then there exists $C$ such that for any $k\geqslant C$ , and for any affine function $a$ on $kP$ , we have

\begin{equation*}FO_P(a,k)=0.\end{equation*}

Recall the following definition.

In particular, any $G$ invariant affine function on symmetric polytopes must be constant; hence it must vanish. We also define the following.

Definition 2.8. A polytope $P$ is weakly symmetric if for any $k$ , and for any affine function $a$ on $kP$ ,

\begin{equation*}FO_P(a,k)=0.\end{equation*}

Lemma 2.10. A weakly symmetric integral polytopes $P$ is (asymptotically ) Chow semistable if for any $k\in \mathbb {N}$ ( $k \geqslant C$ for some fix $C$ ), and for any convex function $f\colon kP\rightarrow \mathbb {R}$ which $\min _{x\in kP}f(x)=f(0)=0$ , we have

\begin{equation*}\frac {1}{\textrm{Vol}(kP)}\int _{kP} f\, dV\leqslant \frac {1}{\chi (kP \cap \mathbb {Z}^n)} \sum _{k v\in kP \cap \mathbb {Z}^n} f(v).\end{equation*}

Proof. For any convex function $f\colon kP\rightarrow \mathbb {R}$ , there exists an affine function $a_k$ such that

\begin{equation*}\min _{x\in kP}(f(x)-a(x))=f(0)=0.\end{equation*}

Therefore, we have

\begin{equation*}\frac {\sum _{k p\in kP \cap \mathbb {Z}^n} f(p)}{\chi (kP \cap \mathbb {Z}^n)} -\frac {\int _P f(x)\, dV}{\textrm{Vol}(P)}=\frac {\sum _{k p\in kP \cap \mathbb {Z}^n} (f-a)(p)}{\chi (kP \cap \mathbb {Z}^n)} -\frac {\int _P (f-a)(x)\, dV}{\textrm{Vol}(P)}.\end{equation*}

The result follows.

3.1 Product class

The first class of polytopes is in the form $P_1\times \cdots \times P_r$ , where $P_1,\ldots, P_r$ , and is Chow stable.

Proof. Consider $f_{P_2}(tx_1+(1-t)x_2)$ , where $0\leqslant t\leqslant 1$ . We have

\begin{align*} f_{P_2}(tx_1+(1-t)x_2)=&\int _{P_2}f(tx_1+(1-t)x_2,y)dV_y\\ \leqslant &\int _{P_2}tf(tx_1,y)dV_y+\int _{P_2}f((1-t)x_2,y)dV_y\\ =& tf_{P_2}(x_1)+(1-t)f_{P_2}(x_2). \end{align*}

Proof. Suppose for any $k\geqslant C_1$ and $k \geqslant C_2$ and for any convex function $f_1, f_2$ on $P_1$ and $P_2$ , we have

\begin{equation*}\frac {1}{\textrm{Vol}(kP_1)}\int _{kP_1}f_1(x)dV\leqslant \frac {1}{\chi (kP_1)}\sum _{p\in P_1}f_2(p);\end{equation*}

\begin{equation*}\frac {1}{\textrm{Vol}(kP_2)}\int _{kP_2}f_2(x)dV\leqslant \frac {1}{\chi (kP_2)}\sum _{p\in P_2}f_2(p).\end{equation*}

Then for any $k\geqslant \max \{C_1,C_2\}$ , and for any convex function $f$ , we have

\begin{eqnarray*} \frac {1}{\textrm{Vol}(kP_1\times kP_2)}\int _{kP_1\times kP_2}f(x,y)dV_xdV_y &=&\frac {1}{\textrm{Vol}(kP_1)}\int _{kP_1} \frac {1}{\textrm{Vol}(kV_2)}\int _{kP_2}f(x,y)dV_ydV_x\\ &=&\frac {1}{\textrm{Vol}(kP_1)}\int _{kP_1} \frac {1}{\textrm{Vol}(kV_2)} f_{kP_2}(x)dV_x \qquad (\text {Lemma 3.1})\\ &\leqslant &\frac {1}{\textrm{Vol}(kP_2)} \frac {1}{\chi (kP_1)}\sum _{p_1\in kP_1\cap \mathbb {Z}^{n_1}}f_{P_2}(p_1)\\ &=&\frac {1}{\chi (kP_1)}\sum _{p_1\in kP_1\cap \mathbb {Z}^{n_1}}\left (\frac {1}{\textrm{Vol}(kP_2)}\int _{kP_2}f(p_1,y)dV_y\right )\\ &\leqslant & \frac {1}{\chi (kP_1)}\sum _{p_1\in kP_1\cap \mathbb {Z}^{n_1}}\frac {1}{\chi (kP_2)}\sum _{p_2\in kP_2\cap \mathbb {Z}^{n_2}}f(p_1,p_2)\\ &=& \frac {1}{\chi (k(P_1\times P_2))}\sum _{p\in k(P_1\times P_2)\cap \mathbb {Z}^{n_1}\times \mathbb {Z}^{n_2}}f(p). \end{eqnarray*}

In particular, if $C_1=C_2=1$ , then this inequality holds for any convex function and any $k$ .

For the opposite, without loss of generality, assume $P_1$ is unstable. Then there exists a sequence of convex functions $f_k$ on $kP_1$ such that for any $k\gt \gt 0$ ,

\begin{equation*}\frac {1}{\textrm{Vol}(kP_1)}\int _{kP_1}f_k(x)dV \geqslant \frac {1}{\chi (kP_1)}\sum _{p\in kP_1\cap \mathbb {Z}^{n_1}}f(p).\end{equation*}

Define $f_k\colon kP_1\times kP_2\rightarrow \mathbb {R}$ such that

\begin{equation*}f_k(x,y)=f_k(x).\end{equation*}

Then

\begin{align*} &\frac {1}{\textrm{Vol}(kP_1\times kP_2)}\int _{kP_1\times kP_2}f_k(x,y)dV=\frac {1}{\textrm{Vol} (kP_1)} \int _{kP_1}f_k(x)dv \geqslant \frac {1}{\chi (kP_1)}\sum _{p\in kP_1\cap \mathbb {Z}^{n_1}}f_k(p)\\ \qquad =& \frac {1}{\chi (kP_1\times kP_2)}\sum _{p\in kP_1\cap \mathbb {Z}^{n_1}} \chi (kP_2)f_k(p)=\frac {1}{\chi (k(P_1\times P_2))}\sum _{p\in k(P_1\times P_2)\cap \mathbb {Z}^{n_1}\times \mathbb {Z}^{n_2}} f_k(p). \end{align*}

As a quick check, we have a computational proof of the following well-known fact.

3.2 Symmetric double cone type

We now consider a class of examples where the members are reflexive and symmetric, but not asymptotically Chow semistable. Also, it is arguably one of the simplest and yet non-trivial classes to study.

Definition 3.4. Let $P$ be an $n$ -dimensional integral polytope. Then we define the double cone

\begin{equation*}D(P):=\textrm{conv}\{0,\ldots, 0,1), (0,\ldots, 0,-1), (p,0)\mid p\in P\}.\end{equation*}

Notice that

\begin{equation*}kD(P)=\{(p, q)\in \mathbb {R}^n\times \mathbb {R}\mid p\in (k-q)P, -k\leqslant q\leqslant k\}.\end{equation*}

Proof. If $G$ acts on $P$ , then $G\times \mathbb {Z}/ 2\mathbb {Z}$ acts on $D(P)$ by

\begin{equation*}(g, \pm 1)\cdot (p,q)=(g\cdot p, \pm q). \end{equation*}

Hence if $P$ is symmetric, then $D(P)$ is symmetric.

To give a counterexample, first we have the following well-known fact.

Lemma 3.6 (See [Ehr77] or [Reference Beck, De Loera, Develin, Pfeifle and StanleyBDLD⁺05]). Let $P$ be a convex integral polytope with dim $\geqslant 2$ . Then the number of points

\begin{equation*}\chi (kP):=\mid kP\cap \mathbb {Z}^n\mid=\textrm{Vol}(P)k^n+\frac {1}{2}\textrm{Vol}(\partial P)k^{n-1}+p(k),\end{equation*}

where $p(k)$ is a polynomial in $k$ of degree $n-2$ which depends on $P$ only. For $n=1$ ,

\begin{equation*}\chi (kP)=\textrm{Vol}(P)k+1;\end{equation*}

and for $n=2$ , we have the Pick theorem (see [Reference PickPic99]),

\begin{equation*}\chi (kP):=\mid kP\cap \mathbb {Z}^n\mid =\textrm{Vol}(P)k^2+\frac {1}{2}\textrm{Vol}(\partial P)k+1.\end{equation*}

In particular, for $k\gt \gt 0$ ,

\begin{equation*}\chi (kP)-\textrm{Vol}(kP)=\frac {\textrm{Vol}(\partial P)}{2}k^{n-1}+p(k)\gt 0.\end{equation*}

Proof. For $kD(P)$ , denote the point in $kD(P)$ to be $(p,q)$ , where $p\in \mathbb {R}^n$ , $q\in \mathbb {R}$ . Consider the function

\begin{equation*}f(p,q)=\left \{ \begin{matrix} 0 & \text { if } & \mid q\mid \leqslant k-1,\\ t & \text { if } & \mid q\mid = (1-t)(k-1)+tk =k-1+t, 0\leqslant t\leqslant 1. \end{matrix}\right . \end{equation*}

Then

\begin{equation*}\sum _{p\in kD(P)}f(p)=2.\end{equation*}

Let $\textrm{Vol}(P)= (n+2)(n+1)(1+\delta )$ for some $\delta \geqslant 0$ . Then

\begin{align*} &\int _{kD(P)} f(x)\, dV=2\int _{0}^1 t(1-t)^n\textrm{Vol}(P) dt=2\textrm{Vol}(P)\int _{0}^1 t^n(1-t)dt\\ =& 2\textrm{Vol}(P)\left (\frac {1}{n+1}-\frac {1}{n+2}\right ) = 2 \frac {\textrm{Vol}(P)}{(n+1)(n+2)}=2+2\delta, \end{align*}

for some fix $\delta \gt 0$ . Therefore,

\begin{equation*}\frac {1}{\textrm{Vol}(kD(P))}\int _{kD(P)} f(x)\, dV=\frac {2+2\delta }{\textrm{Vol}(D(P))k^{n+1}}, \end{equation*}

and

\begin{equation*} \frac {1}{\chi (kD(P))}\sum _{p\in kD(P)}f(p)=\frac {2}{\chi (kD(P))}.\end{equation*}

As a result,

\begin{eqnarray*} \frac {1}{\chi (kD(P))}\sum _{p\in kD(P)}f(p)- \frac {1}{\textrm{Vol}(kD(P))}\int _{kD(P)} f(x)\, dV &=& \frac {2}{\chi (kD(P))}- \frac {2+2\delta }{\textrm{Vol}(kD(P))}\\ &\lt & \frac {2}{\textrm{Vol}(kD(P))}- \frac {2+2\delta }{\textrm{Vol}(kD(P))}\\ &=& \frac {-2\delta }{\textrm{Vol}(k D(P))}\\ &\leqslant & 0. \end{eqnarray*}

In the following example, we construct a toric variety which is defined by a reflexive and symmetric polytope, but it is not asymptotically Chow semistable.

Example 3.9. Consider $P=[-1,1]^6= ((\mathbb {P}^1)^6, O(2,2,2,2,2,2))$ , so then

\begin{equation*}\textrm{Vol}(P)=2^6=64\gt 56=8\times 7=(6+2)(6+1). \end{equation*}

Indeed, as $2^x-(x+2)(x+1)$ is increasing when $x\geqslant 6$ , so for all $n\geqslant 6$ ,

\begin{equation*}2^n-(n-2)(n-1)\geqslant 64-56=8\gt 0,\end{equation*}

which implies that $D([-1,1]^n)$ are not asymptotically Chow semistable for all $n\geqslant 6$ .

Remark 3.10. In the previous example, we provided a sequence of functions $f_k\colon kD([-1,1]^n) \rightarrow \mathbb {R}$ such that the inequality (1) does not hold for all $k\gt \gt 0$ , and hence we show that $D([-1,1]^n)$ are not asymptotically Chow semistable for all $n\geqslant 6$ . We can generalize this construction to any $d$ -dimensional toric variety with the polytope $\Delta$ . To be precise, for any $p\in \Delta$ , we define a sequence of piecewise linear functions $f_{p,k}\colon k\Delta \rightarrow \mathbb {R}$ such that:

1. (i) $f_k(kp)=1$ and $f_k(q)=0$ for any $q\in k\Delta \cap \mathbb {Z}^n$ ; and

2. (ii) for any piecewise linear function $g_k$ satisfies the condition (i),

   \begin{equation*}\int _{k\Delta }f_{p,k}(x)dV \geqslant \int _{k\Delta }g_k(x)dV.\end{equation*}

This is equivalent to defining hyperplanes $H_k:=\left \{L\left (\frac {x}{k}\right )=0\right \}$ with $L(p)\gt 0$ such that:

1. (i) $R_{p,k}^o:=\{L\left (\frac {x}{k}\right )\gt 0\}\cap k\Delta =\{kp\}$ ; and

2. (ii) for any ${R_{p,k}^{\prime}}^o:=\{\hat {L}\left (\frac {x}{k}\right )\gt 0\}\cap k\Delta =\{kp\}$ ,

   \begin{equation*}\textrm{Vol}(R_k^o)\geqslant \textrm{Vol}({R_k^{\prime}}^o).\end{equation*}

We denote

\begin{equation*}Q_p:=\{L\left (x\right )=0\}\cap \Delta, \end{equation*}

which is the base of the cone $\overline {R_{p,1}^o}$ . By the same argument as in the proof of Proposition 3.7, if there exists $p$ such that $\textrm{Vol}(Q_p)\geqslant d(d+1)$ , then $f_k$ do not satisfy the inequality (1) for all $k\gt \gt 0$ . The whole construction is called the cut a vertex technique as $k\Delta =R_{p,k}^o \cap \{f_k=0\}$ , where we separate $k\Delta$ into a cone $\overline {R_{p,k}}$ near the vertex $kp$ and the remaining, and we can show that $\Delta$ is not asymptotically Chow semistable by studying the properties of $\overline {R_{p,k}}$ or even only $\overline {R_{p,1}}$ .

Additionally, for $(\mathbb {P}^{n}, O(n+1))$ and $((\mathbb {P}^1)^{n}, O(2,\ldots, 2))$ , under the above construction, $\overline {R_k^o}$ must be an $n$ -dimensional simplex. Hence $Q_p$ are $(n-1)$ -dimensional simplices for all $p$ , and the volume of $Q_p$ is

\begin{equation*}\textrm{Vol}(Q_p)\frac {1}{(n)!}\lt (n+2)(n+1),\end{equation*}

which is expected as we know that they are asymptotically Chow polystable.

In § 4, we will define a more restrictive type of polytopes, which are asymptotically Chow polystable.

8.1 $D(X_8)$ and $D(X_9)$

Recall that $D(X_8)$ and $D(X_9)$ are given by

\begin{gather*}D(X_8):=\mathrm {conv}\{(1,1,0), (-1,1,0), (1,-1,0), (-1,-1,0), (0,0,1), (0,0,-1)\},\\D(X_9):=\mathrm {conv}\{(-1,-1,0), (2,-1,0), (-1,2,0), (0,0,1), (0,0,-1)\}.\end{gather*}

In Example 4.8, we showed that $D(X_8)$ and $D(X_9)$ are not special. Therefore, to show that $D(X_8)$ and $D(X_9)$ are asymptotically Chow polystable, we have to triangulate the whole polytopes and compute the inequality directly.

Notice that the only way to triangulate $D(X_8)$ and $D(X_9)$ into simplices is the following. We triangulate $X_8$ and $X_9$ as follows.

Then we connect any small triangle to $(0,0,1)$ and $(0,0,-1)$ to get 3-simplex. Therefore, we can triangulate $D(X_8)$ into 16 simplices and $D(X_9)$ into 18 simplices. Then, by triangulation of each simplex, we have a triangulation of $kD(X_8)$ and $kD(X_9)$ .

As a consequence of Lemma 7.1, we have the following lemma.

Lemma 8.3. For $kD(X_i)$ , under the above triangulation,

\begin{equation*}n(p) \left \{ \begin{array}{cl} =i &\text { if } p=(0,0, \pm k),\\ \leqslant 24 & \text { if } p\in kD(X_i)^o,\\ \leqslant 12 & \text { if } p\in \partial kD(X_i). \end{array}\right . \end{equation*}

Moreover, the triangulation on $kT_2$ combined with the induced triangulation on $D(X_i)$ onto $\partial kD(X_i)$ gives

\begin{equation*}m(p) \left \{ \begin{array}{cl} =i &\text { if } p=(0,0, \pm k),\\ \leqslant 6 & \text { otherwise.} \end{array}\right .\end{equation*}

As a remark, for each $D(X_i)$ , $n_{kP}(0,\ldots, 0)=2i$ , also for the triangulation of $\partial kD_i$ , $n(p)=4$ for $p \in \partial kP$ intersect with the red line.

Proof. For $p \in \partial kP$ such that $p\neq (0,\ldots, \pm 1)$ , $n(p;k)\leqslant \frac {(n+1)!}{2}$ and $m(p;k)\leqslant n!$ , the inequality becomes

\begin{equation*}n(n!) \leqslant (n+1)!, \end{equation*}

which is true. Also, at $p=(0,\ldots, 0, \pm 1)$ , we have

\begin{equation*}n(p;k)=m(p;k)=i,\end{equation*}

and then we need

\begin{equation*}(n+1)!\gt \frac {(n+2)}{2}i.\end{equation*}

That is,

\begin{equation*}1\gt \frac {(n+2)i}{2(n+1)!}.\end{equation*}

If $n=3$ , then it becomes

\begin{equation*}1\gt \frac {5i}{48},\end{equation*}

and hence this inequality holds for $i\leqslant 9$ . Therefore, by Lemma 8.3 and Theorem 8.1, for $i=3,4,6,8,9$ , $D(X_i)$ are asymptotically Chow polystable.

9.1 One- and two-dimensional symmetric reflexive polytopes

Example 9.2. Suppose $P$ is a two-dimensional symmetric reflexive polytope, so then it is special, and hence it is asymptotic Chow stable. Indeed, by Theorem 1.2 and Corollary 3.3 in [Reference Lee, Li, Sturm and WangLLSW19], combined with the fact that $\mathbb {P}^2$ and $\mathbb {P}^1\times \mathbb {P}^1$ (by Proposition 3.2 ) are Chow stable, so indeed, all two-dimensional special polytopes are Chow stable.

As a remark, they are given by

\begin{equation*}X_3:= \mathbb {P}^2/ (\mathbb {Z}^/3\mathbb {Z}), X_4:= \mathbb {P}^1\times \mathbb {P}^1/ (\mathbb {Z}/2\mathbb {Z}), X_6:=\mathbb {P}^2\; \text{blow up 3 points}, X_8:=\mathbb {P}^1\times \mathbb {P}^1, X_9:=\mathbb {P}^2,\end{equation*}

and all the line bundles to define the polytopes are $-K_{X_i}$ . The polytopes are given in Figure 4.

Figure 4. $X_i$ for $i=3,4,6,8,9$ .

Notice that in [Reference Lee, Li, Sturm and WangLLSW19] there are some examples of the non-reflexive polytopes which we will not discuss in detail in this note.

9.2 Three-dimensional polytopes

To study the higher-dimensional polytopes, we first recall that, given a reflexive polytope $P$ , we can define the dual polytope $\hat {P}$ as follows. Let

\begin{equation*}\hat {P}:=\{y\in \mathbb {R}^n\ \langle x, y\rangle \geqslant -1, \text { for all }x\in P\}. \end{equation*}

If $P$ is symmetric and reflexive, so is $\hat {P}$ . For example, $D(P)=\widehat {\hat {P}\times \mathbb {P}^1}$ . However, the duality may not share the stability.

By Lemma 7.1, for faces that are 2-simplices or rectangles, then $m(p)=6$ for $p\in P^o$ , $m(p)=3$ if $p\in (\partial P)^o$ , and $m(p)=1$ if $p$ is the vertex. For any $p\in (\partial P)^o$ , there are at most two faces connected to $p$ , so in order to check if a polytope is special, we only need that there are $i\leqslant 6$ simplices connecting each vertex under the triangulation on the boundary.

Denote $\Delta _0$ as the triangulation of a 2-simplex. As in Figure 5, if the faces are given by $X_i$ , for $i=3,4,6$ , then $n(0)=i$ , $n(p)=6$ for $p\in P^o-\{0\}$ , $n(p)=3$ if $p\in (\partial P)^o$ , and $n(p)=2$ if $p$ is the vertex using the rotation of $\Delta _0$ as the triangulation.

Figure 5. $\triangle _0\subset X_3$ , $\triangle _0\subset X_4$ and $\triangle _0\subset X_6$ .

Therefore, if the polytopes whose faces are a combination of the above, then the only possible problem is the vertex, and we can study those polytopes case by case.

Figure 6. $X_i \times [-1,1]$ .

Figure 7. $D(X_i)$ .

Figure 8. Other special polytopes, where the red line indicates part of the triangulation.

9.3 Higher-dimensional polytopes

One knows that in high dimensions that not every symmetric reflexive polytope is asymptotic Chow stable, for example $ D([-1,1]^n)$ for $n\geqslant 6$ . On the other hand, we can provide two classes of polytopes that are special.

Example 9.5. Consider $A_n:=\{[z_0,\ldots, z_{n+1}]\in \mathbb {P}^{n+1}\mid z_1\ldots z_{n+1}=z_{0}^{n+1}\}$ . The corresponding polytopes, also denoted as $A_n$ , are given by

\begin{equation*}A_n=\textrm{conv}\{(1,\ldots, 0),(0,1,\ldots, 0), \ldots, (0,\ldots, 0,1), (-1,-1,\ldots, -1)\}.\end{equation*}

As all the faces of $A_n$ are simplices, with all the codimension 2 or above boundary intersecting with less than $n$ , it is asymptotic Chow polystable.

Notice that $A_2= X_3$ in our notation on symmetric reflexive polygons. Also, as a polytope, each $A_n$ is the dual polytope of the polytope corresponding to $(\mathbb {P}^n, O(n+1))$ .

To show $A_n$ are special for all $n\geqslant 3$ , notice that the boundry of $A_n$ is given by $n$ piece of $(n-1)$ -simplices. So we only need to know how many simplicies will be attached to a point in the codimension k skeleton.

Let $a_i=e_i$ and $a_{n-1}= (-1,\ldots, -1)$ , so then we can represent any codimensional k piece by the following: $a_1$ represents the point $e_1$ , $\{a_1,a_2\}$ represents the segment containing $a_1,a_2$ , etc. Also, as the symmetric group $S_{n+1}$ acts on $A_n$ , we only need to consider how many faces contain the $n-r$ skeleton $\{a_1\ldots a_{n-r}\}$ . But the faces containing $a_1\ldots a_{n-r}$ are represented by the set $\{a_{n-r+1}\ldots a_{n+1}\}$ removing one element. Therefore, there are $r+1$ faces connecting the skeleton containing $a_1\ldots a_{n-r}$ . Therefore, Lemma 7.1 implies that, for any point $p\in \partial kP$ which is in the interior of the $n-r$ skeleton,

\begin{equation*}n(p)=\frac {(n)!}{(r)!}(r)=\frac {n!}{(r-1)!}\leqslant n!.\end{equation*}

Therefore, $A_n$ has a regular triangulation. Also, all $A_n$ are symmetric and reflexive, and hence they are special, which implies all $A_n$ for $n\geqslant 2$ are asymptotically Chow polystable.

Example 9.6. Consider $(\mathbb {P}^n, O(n+1))$ . The boundary of $k(\mathbb {P}^n, O(n+1))$ is defined by

\begin{equation*}\bigcap _{i=1}^n\{x_i=-k\}\cap \{x_1+\cdots + x_n=k\}.\end{equation*}

Up to an $S_{n+1}$ action, a point $p$ is in the interior of a codimensional $r$ skeleton if

\begin{equation*}p=(-k,\ldots, -k,v),\end{equation*}

where $v\in \mathbb {R}^{n-r}$ such that

\begin{gather*}-r+v_1+\cdots +v_{n-r}\lt k,\\v_i\gt -k, \end{gather*}

for all $i=1,\ldots, n-k$ . Hence $n(p)=r$ . So we have the same calculation of $A_n$ , which implies it is special, and therefore it is asymptotically Chow polystable.

Proof. Notice that $k D_n=\mathbb {Z}_2^n \cdot kT_n$ , where $\mathbb {Z}_2^n=\{1,-1\}^n$ with the group action to be multiplication, and the action is multiplication to the corresponding coordinate. $D_n$ is symmetric and reflexive. To show that $D_n$ has a regular boundary, $p$ is in the interior of the codimension $r+1$ skeleton if $p=(x_1,\ldots, x_n) \in \partial k D_n$ with

\begin{equation*}x_{i_1}=\cdots =x_{i_r}=0, \end{equation*}

for $r=0,\ldots, n-1$ . We denote these points as $p_r$ . Hence, similar to Example 9.6, as a consequence of Lemma 7.1, we have

\begin{equation*}n(p_r)=\frac {(n)!}{(r+1)!}(2^r)=\left (\frac {2}{r+1}\right ) \cdots \left (\frac {2}{2}\right ) n! \leqslant n!,\end{equation*}

and hence $D_n$ is special.