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1 Introduction↩︎

The famous Yau-Tian-Donaldson conjecture asserts that the existence of Kähler-Einstein metrics on a Fano manifold is equivalent to the K-stability. The conjecture has been proved by Tian [1] and Chen-Donaldson-Sun [2] alternatively. In recent years, the conjecture has been widely extended to other canonical metrics (such as solitons, cscK metrics on polarized vareities), log pairs, and a uniform version for singular \(\mathbb{Q}\)-Fano varieties, see [3]–[5], etc.

The notion of K-stability was first introduced by Tian [6] in terms of special degenerations and then reformulated by Donaldson [7] in an algebro-geometric way via test-configurations. In general, to test K-stability one has to study infinitely many possible degenerations. A natural question is how to reduce it to a finitely dimensional process. The reduction would be possible when considering varieties with large group actions and the respective equivariant K-stability.

Let \({\rm k}\) be an algebraically closed field of characteristic 0 and \(G\) a connected, reductive, linear algebraic group over \({\rm k}\). Let \(X\) be a normal variety with a regular \(G\)-action. Fix any Borel subgroup \(B\) of \(G\). The complexity of the \(G\)-action on \(X\) is the codimension of a \(B\)-orbit in general position (cf. [8]). The varieties of complexity 0 are called spherical varieties, which include the well-known toric varieties. The geometry of spherical varieties has been comprehensively studied by various of authors, we refer to [9]–[12], and references therein for more knowledge. The spherical variety has a nice combinatorial description which brings great convenience to the study of its geometric structure. Roughly speaking, given a spherical variety \(X\), one may begin with the lattice \(\Gamma\) of \(B\)-semiinvariant rational functions³, which is contained in the lattice \(\mathfrak X(B)\) of \(B\)-weights, and its (\(\mathbb{Q}\)-)dual \(\Gamma_\mathbb{Q}^*\) in which the set of \(G\)-valuations (called its valuation cone, which is a solid cosimplicial cone) is embedded. The variety \(X\) itself is fully characterised by a fan of convex cones (more precisely, coloured cones, see [9], [11], [12] for details), while line bundles are described by piecewise linear functions on this fan (cf. [13], [14]). In particular, when \(X\) is polarized by some ample line bundle \(L\), there is a nice moment polytope in \(\Gamma_\mathbb{R}:=\Gamma\otimes_\mathbb{Q}\mathbb{R}\) that encodes the structure of \(G\)-modules on the spaces of sections \({\rm H}^0(X,L^k)\) for \(k\in\mathbb{N}\). In terms of these data, in recent years several combinatorial criterions of K-stability of spherical varieties were established, which give a practical way to test K-stability and provide rich examples. For instance, the K-stability criterion for toric varieties was studied by Donaldson [7], for spherical varieties established by Delcroix [15], [16], etc. It turns out that in these cases K-stability is characterised by certain barycenter of the moment polytope.

It is then natural to consider \(G\)-varieties of complexity 1. In fact, many examples in the literatures belong to this class, see for instance [17], [18]. The classification theory of \(G\)-varieties of complexity 1 was established by Timashëv [19]. In particular [19] gives a combinatorial description of these varieties analogous to the spherical cases. Unlike the spherical case, a \(B\)-semiinvariant rational function can not be completely determined only by its weight. In this case we need to consider the hyperspace instead of \(\Gamma_\mathbb{Q}^*\) in which the \(G\)-valuations are embedded. Roughly speaking, the hyperspace is a family of half-spaces \(\{\mathscr Q_{x,+}|x\in C\}\) parameterized by points of a smooth projective curve \(C\), each isomorphic to \(\Gamma_\mathbb{Q}^*\times \mathbb{Q}_+\), that glued together along common boundary \(\mathscr Q=\Gamma_\mathbb{Q}^*\times\{0\}\). The valuation cone is obtained by gluing a collection of solid convex cones in \(\mathscr Q_{x,+}\) along common face in \(\mathscr Q\). Varieties with given field of rational functions are classified by fans of coloured cones and hypercones (a hypercone is again a family of cones that glued along common boundary). The combinatorial theory of line bundles on \(G\)-varieties of complexity 1 was then studied in [14]. Line bundles are described by collection of linear functionals defined on \(\mathscr Q_{x,+}\) with common restriction on \(\mathscr Q\). See Section 2.2 for details. For \(T\)-varieties of complexity 1 with \(T\) an algebraic torus, Ilten-Süß [20] developed the classification theory of polarized \(T\)-varieties of complexity 1. They introduced the “divisorial polytope" which turns out to be a very useful tool in the study of polarized \(T\)-varieties of complexity 1. Based on this theory, they obtained a combinatorial criterion of equivariantly K-(poly)stability of Fano \(T\)-varieties of complexity 1 in [21]. Recently Rogers [22] studied K-(poly)stability of smooth Fano \({\rm SL_2}\)-varieties of complexity 1.

Let \(G\) be an arbitrary connected, reductive, linear algebraic group of simply connected type, and \(X\) a projective \(G\)-variety of complexity 1. Motivated by the works cited above, in this paper we give a criterion of \(G\)-equivariantly uniform K-stability in terms of its combinatorial data. To this end, we will first classify \(G\)-equivariant normal test configuration of \(X\) with integral central fibre. This class in particular includes the special test configurations, which are enough for testing K-stability of a \(\mathbb{Q}\)-Fano variety (cf. [4], [23]). We call \(X\) a \(\mathbb{Q}\)-Fano variety if \(X\) is \(\mathbb{Q}\)-Gorenstein and \(K_X^{-1}\) is ample. When \(X\) is \(\mathbb{Q}\)-Fano, we then consider a family of polytopes \(\{\Delta_x^O(K_X^{-1})\subset\mathfrak X_\mathbb{R}(B)\times\mathbb{R}|x\in C\}\) that encodes the information of K-stability of \(X\), where \(C\cong\mathbb{P}^1\) is birational to the rational quotient for the \(B\)-action. These polytopes were first introduced in [21] to characterize K-stability of \(T\)-varieties of complexity 1. Based on a formula of \(B\)-stable anti-canonical divisors of \(X\) derived in Section 3, it turns out that the Futaki invariant can be expressed in a very simple way on this family of polytopes. Finally we prove a combinatorial criterion of K-stability if \(X\) in addition has at most klt singularities. Let \(\mathscr V\) be the valuation cone of \(X\) and \(\mathscr V_x\) its intersection with the hyperspace over \(x\in C\), \(\mathscr A_x\subset\mathscr V_x\) the space parameterizing one-parameter subgroups in the \(G\)-equivariant automorphism group \({\rm Aut}_G(X)\) of \(X\) (see Section 4.2.1 below). Denote by \(\kappa_P\) the sum of unipotent roots of the associated parabolic subgroup \(P\) of \(X\) (see Section 2.3.2 below). Then each \(\Delta_{x}^O(K_X^{-1})\) is in fact a solid convex polytope in \((\Gamma_\mathbb{R}+\kappa_P)\times\mathbb{R}\). Also set \[{\mathbf{b}}(\Delta_{x}^O(K_X^{-1})):=\frac{1}{V}\int_{\Delta_{x}^O(K_X^{-1})}(\lambda,t)\pi(\lambda)d\lambda\wedge dt\] the barycenter of a polytope \(\Delta_{x}^O(K_X^{-1})(\subset(\Gamma_\mathbb{R}+\kappa_P)\times\mathbb{R})\), where \((\lambda,t)\) are coordinates on \(\Delta_{x}^O(K_X^{-1})\), \(\pi(\lambda)\) is a non-negative polynomial function depends on \(X\), \(d\lambda\) the standard Lebesgue measure on \(\Gamma_\mathbb{R}+\kappa_P\) which is normalized by the lattice \(\Gamma\), and \[V=\int_{\Delta_{x}^O(K_X^{-1})}\pi(\lambda)d\lambda\wedge dt\] is the volume of \(\Delta_{x}^O(K_X^{-1})\) which is in fact independent of \(x\) (see Section 5.2 below for details). We have the following

Precise definitions and useful properties of the terms involved here can be found in the preliminary section and indicated above places. By the famous Yau-Tian-Donaldson conjecture [1], [4], when \({\rm k}=\mathbb{C}\) the above K-stability criterion then provides an existence criterion of Kähler-Einstein metrics.

As mentioned above, we need to derive a formula of \(B\)-stable anti-canonical divisors which helps to simplify the Futaki invariant. The following is a counterpart of a result of Brion [24] for spherical varieties.

As will be seen below, there are two classes of \(G\)-varieties of complexity one, the one-parameter and quasihomogeneous ones. For this reason Theorem 3 will be proved separately in Sections 3.1 and 3.2 (under slightly weaker assumptions, respectively). In fact the proof in the quasihomogeneous case relies on that in the one-parameter case. We would also like to briefly explain here the relation between Theorem 3 with results derived by different authors in the literatures. The \(T\)-varieties, or more general, horospherical varieties of complexity 1 are of one-parameter type. There such a formula has been proved by Petersen-Süss [25] and Langlois-Terpereau [26], respectively. Later Langlois [27] proved a general formula, which in particular include the one-parameter case here, in a different framework. However, up to the authors’ knowledge, there is not a unified formula like ?? for the quasihomogeneous cases in the literatures.

Concerning applications of the above results, we give examples in Section 6. In Section 6.1 we illustrate our frame on the well-known Mukai-Umemura threefold, which is a \({\rm SL}_2\)-variety of complexity 1. Then in Section 6.2 we apply the above results to the following example \[X = \{(p_1, p_2, p_3, l_1, l_2, l_3)|p_j\in \mathbb{P}^2,~l_i \in \mathbb{P}^{2^*},~p_j \in l_i~\text{whenever}~i\not=j\},\] which can be realized as a \({\rm SL}_3\)-variety of complexity 1 (see Section 6.2 for details on group actions). We will prove

The paper is organized as following: In Section 2 we give preliminaries. In Section 3 we derive a formula of anti-canonical divisors on \(X\). This formula will be crucial when simplifying the expression of Futaki invariants later. In Section 4 we classify \(G\)-equivariant normal test configuration of \(X\) with integral central fibre. We also study the central fibre and give combinatorial parameterizations of \({\rm Aut}_G(X)\) via studying the associated filtration in Section 4.2. In Section 5 we give a formula of Futaki invariant in terms of the combinatorial data, based on the formula of total weights. In particular we simplify the formula on \(\mathbb{Q}\)-Fano varieties. In Section 5.4 we prove Theorem 1. In Section 6 we give two examples where Theorem 1 is applied. In the Appendix we collect useful Lemmas, and give an alternative computation of the Futaki invariant for \(\mathbb{Q}\)-Fano varieties via a formula of intersection numbers. Also some discussions on \(\Delta_{x}^O(K_X^{-1})\)’s are included by the end of the Appendix.

Notations and conventions↩︎

In this paper we use the following conventions:

• \({\rm k}\)-an algebraically closed field of characteristic 0;

• \(G\)-a connected, reductive, linear algebraic group over \({\rm k}\). \(G\) is assumed to be of simply connected type from Section 2.3;

• \(\Phi_G\)-root system of \(G\);

• \(\Pi_G\)-the set of positive roots in \(\Phi_G\) of some group \(G\) with respect to certain Borel subgroup;

• \(\rho:=\frac{1}{2}\sum_{\alpha\in\Pi_G}\alpha\)-half the sum of positive roots of \(G\);

• \(P=L_P\cdot P_u\)-Levi decomposition of a parabolic subgroup \(P\subset G\), where \(L_P\) is a Levi subgroup of \(P\) and \(P_u\) the unipotent radical of \(P\);

• \(\Pi_{L_P}\subset\Pi_G\)-positive roots of \(L_P\) in \(\Pi_G\);

• \(\Pi_{P_u}\subset\Pi_G\)-the unipotent roots of a parabolic subgroup \(P\);

• \(\mathfrak X(\cdot)\)-characters of a group;

• \(V^H\)-the subset of all \(H\)-invariants in a linear \(H\)-representation \(V\);

• \(V^{(H)}_\chi\)-the subset of all \(H\)-semiinvariants with weight \(\chi\) in a linear \(H\)-representation \(V\);

• \(\langle S\rangle\)-linear span of a set \(S\) in a linear space;

• \(\langle v,w^*\rangle\)-linear pairing of \(v\in V\) and \(w^*\in V^*\), where \(V\) is a linear space;

• \({\rm Conv}(S)\)-convex hull of a set \(S\);

• \(S^\vee\)-dual cone of a set \(S\) in \(V^*\), where \(V\) is a linear space containing \(S\);

• \(S^\perp\)-elements in \(V^*\) that is orthogonal to \(S\), where \(V\) is a linear space containing \(S\);

• \({\rm Stab}_G(S)\)-the group of normalizer in \(G\) of a set \(S\). That is, the set of elements in \(g\) that keep the set \(S\) invariant. If \(S=\{D\}\) for a single divisor \(D\), we write \({\rm Stab}_G(D)\) in short of \({\rm Stab}_G(\{D\})\);

• \(K\)-a function field which admits a \(G\)-action. We assume in this paper that \(K^B\cong {\rm k}(\mathbb{P}^1)\) unless otherwise stated;

• \(X\)-a \(G\)-model of \(K\), see Section 2.2;

• \(\kappa_P\)-sum of unipotent roots of the associated parabolic subgroup of \(X\) (see Section 2.3 for this parabolic subgroup). It is also the \(B\)-weight of a canonically chosen rational section of \(K_X^{-1}\), see Theorems 29 and ?? below;

• \(L^n\)-the \(n\)-th tensor power of a line bundle \(L\);

• \(L^{\cdot k}\)-the \(k\)-th self-intersection of a line bundle \(L\);

• If \(\mathfrak M\) is a lattice, then we denote \(\mathfrak M_\mathbb{Q}=\mathfrak M\otimes_\mathbb{Z}\mathbb{Q}\) and \(\mathfrak M_\mathbb{R}=\mathfrak M\otimes_\mathbb{Q}\mathbb{R}\);

• \(d\lambda\)-the standard Lebesgue measure on \(\Gamma_\mathbb{R}\) or its translation \(\Gamma_\mathbb{R}+\lambda_0\) for some \(\lambda_0\in\mathfrak X(B)\), which is normalized by \(\Gamma\).

2 Preliminaries↩︎

2.1 Test configurations and K-stability↩︎

Let \(X\) be a projective variety. Then the pair \((X,L)\) with \(L\) an ample line bundle on \(X\) is called a polarized variety (polarized by the line bundle \(L\)). The notation of K-stability of a polarized variety are usually stated in terms of test configurations.

There is also a geometric way to construct test configuration of \((X,L)\) (cf. [7], [28] or [3]). Suppose that there is a Kodaira embedding of \(X\) by \(|L ^{r_0}|\) for some \(r_0\in\mathbb{N}_+\), \[i:X\to\mathbb{P}({\rm H}^0(X,L ^{r_0}))=:\mathbb{P}^{N-1}.\] Choose a vector \(\Lambda\in\mathfrak{psl}_N\) so that \(\Lambda\) generates a rank \(1\) torus of \({\rm PSL}_N\). Then it defines a test configuration \((\mathcal{X},\mathcal{L})\) via \[\mathcal{X}_{t}:=\exp(z\Lambda)\cdot i(X),~t=e^z\in{\rm k}^\times,\] and \[\mathcal{L}|_{\mathcal{X}_t}:=\mathcal{O}_{\mathbb{P}^{N-1}}(1)|_{\mathcal{X}_t}.\] Also define \(\mathcal{X}_0:=\lim_{t\to0}\mathcal{X}_t\) in the sense of the flat limit of \(\mathcal{X}_t\) as \(t\to0\), and \(\mathcal{L}_0:=\mathcal{L}|_{\mathcal{X}_0}\). Moreover, the total space of \((\mathcal{X},\mathcal{L})\) can be interpreted as the Zariski closure in \({\mathbb{P}}^{N-1}\times{\rm k}\) of the image of the closed embedding \(X\times{\rm k}^\times\to\mathbb{P}^{N-1}\times{\rm k}\) mapping \((x,t)\) to \((\exp(t\Lambda)\cdot i(x),t)\). Indeed, any test configuration can be realized in this way (cf. [3]). Product test configurations are precisely those with \(\Lambda\in\mathfrak{aut}(X)\).

For our later use, in the following we compactify \((\mathcal{X},\mathcal{L})\) to a family over \((\mathcal{X},\mathcal{L})\overset{{\rm pr}}{\to}\mathbb{P}^1\) by adding a trivial fibre \((X,L^{r_0})\) at \(\infty\in\mathbb{P}^1\). Alternatively, we glue \((\mathcal{X},\mathcal{L})\) with \((X,L ^{r_0})\times{\rm k}\cong(X,L ^{r_0})\times({\rm k}^\times\cup\{\infty\})\) along the common part \((X,L ^{r_0})\times{\rm k}^\times\). Also, if \(\mathcal{L}\) is relative ample with respect to \({\rm pr}\), then by replacing \(\mathcal{L}\) by \(\mathcal{L}+c{\rm pr}^*\mathscr O_{\mathbb{P}^1}(1)\) with sufficiently large \(c\gg1\), we can always assume that \(\mathcal{L}\) is ample (cf. [29]). From now on, by \((\mathcal{X},\mathcal{L})\) we always refer to the compactified family with ample \(\mathcal{L}\). Finally, in the following, we always assume in addition that any test configurations under consideration is normal. That is, its total space \(\mathcal{X}\) is a normal variety. Note that by Hironaka’s lemma (cf. [30]), a special test configuration is always normal.

Without loss of generality we assume that \(r_0=1\) and denote \(d_k:=\dim{\rm H}^0(X,L^k)\). Otherwise, we replace \(L\) by \(L ^{r_0}\). Choose a basis \(\{e_p\}_{p=1}^{d_1}\) of \({\rm H}^0(\mathcal{X}_0,\mathcal{L}_0)\) so that each \(e_p\) is an eigenvector of the \(\exp(t\Lambda)\)-action. Denote by \(\{e^{\Lambda_p^k}\}_{p=1}^{d_k}\) the eigenvalues of the canonical lifting of the \(\exp(t\Lambda)\)-action on \({\rm H}^0(\mathcal{X}_0,\mathcal{L}^k_0)\). Here we use the fact that \(d_k=\dim{\rm H}^0(\mathcal{X}_0,\mathcal{L}_0^k)\). Define the total weight \[\begin{align} w_k(\mathcal{X},\mathcal{L}):=&\sum_{p=1}^{d_k}\Lambda_p^k. \end{align}\] It is showed in [7] that there are constants \(A,B,C,D\) so that \[w_k(\mathcal{X},\mathcal{L})=Ak^{n+1}+Bk^{n}+O(k^{n-1}),~k\to+\infty,\] and \[\dim{\rm H}^0(X,L^k)=Ck^n+Dk^{n-1}+O(k^{n-2}),~k\to+\infty.\] The Futaki invariant of \((\mathcal{X},\mathcal{L})\) is then defined as \[\begin{align} \label{Fut-def} {\rm Fut}(\mathcal{X},\mathcal{L}):=\frac{AD-BC}{C^2}. \end{align}\tag{1}\] The K-stability is defined as

Let \((X,L)\) be a polarized variety. Denote by \({\rm Aut}(X)\) the automorphism group of \(X\). Let \({\mathbf{G}}\subset{\rm Aut}(X)\) be any of its connected, reductive subgroup and \({\mathbf{T}}\) the identity component of its centre. Hisamoto [31] introduced the twist of equivariant test configurations and \({\mathbf{G}}\)-uniform K-stability. We briefly recall the construction below and refer to [31] for details.

Let \((\mathcal{X},\mathcal{L})\) be a \({\mathbf{G}}\)-equivariant test configuration of \((X,L)\). Denote by \(\Lambda\) the vector field induced by the \({\rm k}^\times\)-action, it defines a grading (still denoted by \(\Lambda\)) on \[\begin{align} \label{Kodaira-ring} R(X,L):=\bigoplus_{k=0}^{+\infty}R_k(X,L),~R_k(X,L):={\rm H}^0(X,L^k), \end{align}\tag{2}\] the Kodaira ring of \((X,L)\). Let \(\sigma\in{\rm Lie}({\mathbf{T}})\). The twist \((\mathcal{X}_\sigma,\mathcal{L}_\sigma)\) of \((\mathcal{X},\mathcal{L})\) is defined as following: The uncompactified total space \((\mathcal{X}_\sigma\setminus\mathcal{X}_{\sigma\,\infty},\mathcal{L}_\sigma|_{\mathcal{X}\setminus\mathcal{X}_{\sigma\,\infty}})\) is isomorphic to that of \((\mathcal{X},\mathcal{L})\), but grading induced by \((\mathcal{X}_\sigma,\mathcal{L}_\sigma)\) is \(\Lambda+\sigma\). That is, if the grading \(\Lambda\) on \(R_k\) has weight \(\Lambda_1,...,\Lambda_{n_k}\), then the same basis diagonalizes \(\sigma\) so that the weights for the grading \(\Lambda+\sigma\) are \(\Lambda_1+\sigma_1,...,\Lambda_{n_k}+\sigma_{n_k}\). When \(\sigma\) is integral, \((\mathcal{X}_\sigma,\mathcal{L}_\sigma)\) is a test configuration so that the \({\rm k}^\times\)-action induces a vector field \(\Lambda+\sigma\). However, the compactified total space of \((\mathcal{X}_\sigma,\mathcal{L}_\sigma)\) is different from that of \((\mathcal{X},\mathcal{L})\). When \(\sigma\) is irrational, \((\mathcal{X}_\sigma,\mathcal{L}_\sigma)\) in general may not be a test configuration in the above sense, and is called an \(\mathbb{R}\)-test configuration (cf. [4], [31]).

The \({\mathbf{G}}\)-uniform K-stability is phrased in terms of non-Archimedean functionals. For any test configuration \((\mathcal{X},\mathcal{L})\), denote by \({\rm DH}(\mathcal{X},\mathcal{L})\) its induced Duistermaat-Heckman measure, and \(\Lambda_{\max}(\mathcal{X}, \mathcal{L})\) the upper bound of the support of \({\rm DH}(\mathcal{X},\mathcal{L})\) (cf. [3]). Define the non-Archimedean J-functional \[\begin{align} \label{J-NA-def} {\rm J}^{\rm NA}(\mathcal{X}, \mathcal{L}):=&\Lambda_{\max}(\mathcal{X}, \mathcal{L})-\frac{1}{(n+1)V}\mathcal{L}^{\cdot(n+1)}, \end{align}\tag{3}\] where \(\mathcal{L}^{\cdot(n+1)}\) denotes the top intersection number of \(\mathcal{L}\). Also, the non-Archimedean Mabuchi functional \[\begin{align} {\rm M}^{\rm NA}(\mathcal{X},\mathcal{L}):=\frac{1}{V}K^{\rm log}_{\mathcal{X}/\mathbb{P}^1}\cdot\mathcal{L}^{\cdot n}+\frac{\bar S}{V(n+1)}\mathcal{L}^{\cdot(n+1)}, \end{align}\] where \(V=L^{\cdot n}\) is the volume of \((X,L)\), \[\begin{align} \bar S=\frac{n}{V}(K_X^{-1}\cdot L^{\cdot n}) \end{align}\] the mean value of the scalar curvature of \((X,L)\), and \[\begin{align} K^{\log}_{\mathcal{X}/\mathbb{P}^1}=K_{\mathcal{X}}+\mathcal{X}_0^{\rm red}-{\rm pr}^*(K_{\mathbb{P}^1}+[0]) \end{align}\] a Weil divisor on \(\mathcal{X}\). Here by \(\mathcal{X}_0^{\rm red}\) we denote the reduced structure of the central fibre \({\rm pr}^{-1}(0)\). We have

When there is no confusion, we may say “uniform K-stability" in short of”\({\mathbf{G}}\)-uniform K-stability" when the group \({\mathbf{G}}\) is fixed.

Let us give a few remarks. In general it holds \({\rm Fut}(\mathcal{X},\mathcal{L})\geq{\rm M}^{\rm NA}(\mathcal{X},\mathcal{L})\). However they coincide on test configurations with reduced central fibre. It is also proved that a test configuration can always have reduced central fibre after a suitable base change \(t\to t^d\) on \({\rm k}^\times\) (and then taking normalization of the total space). Both \({\rm M}^{\rm NA}(\cdot)\) and \({\rm J}^{\rm NA}(\cdot)\) vary linearly under base change. That is, if \((\mathcal{X}^{(d)},\mathcal{L}^{(d)})\) is the base change of \((\mathcal{X},\mathcal{L})\), then \[\begin{align} \label{base-change-of-NA} {\rm M}^{\rm NA}(\mathcal{X}^{(d)},\mathcal{L}^{(d)})=d{\rm M}^{\rm NA}(\mathcal{X},\mathcal{L})~\text{and}~{\rm J}^{\rm NA}(\mathcal{X}^{(d)},\mathcal{L}^{(d)})=d{\rm J}^{\rm NA}(\mathcal{X},\mathcal{L}). \end{align}\tag{4}\] For a detailed study of the non-Archimedean functionals and K-stability, we refer to the readers [3], [4]. On the other hand, we recall that for a \(\mathbb{Q}\)-Fano variety with klt singularities, Li-Xu [23] (Li [4], resp.) proved that to check K-stability (equivariantly uniform K-stability, resp.) of \((X,K_X^{-1})\), it suffices to consider only special test configurations, which always has reduced central fibre.

The K-stability is closely related to the existence of Kähler-Einstein metrics when \({\rm k}=\mathbb{C}\). The Yau-Tian-Donaldson conjecture for Fano manifolds with discrete auto morphism group was proved by Tian [1] (see also [2]). For the equivariant version, Datar-Székelyhidi [32] proved that equivariant K-stability is equivalent to the existence of Kähler-Einstein metrics for Fano manifolds. Later Li [4] proved the uniform Yau-Tian-Donaldson conjecture (where the conjecture is stated for a stronger uniform K-stability condition) for \(\mathbb{Q}\)-Fano varieties with klt-singularities. We refer to the readers [1], [3], [4], [32] and references therein for further knowledge.

2.2 \(G\)-varieties of complexity 1↩︎

2.2.1 General Luna-Vust theory↩︎

In this section we recall the general theory of normal varieties with a reductive group action, which is now referred as Luna-Vust theory. This theory was originally developed by Luna-Vust [9] and has been extended and modified by Timashëv [19]. In the following we use [12], [19] as our main references and collect useful to us information.

Let \(G\) be a connected, reductive, linear algebraic group over an algebraic closed field \(\rm k\) of characteristic 0. Suppose that \(G\) acts on a function field \(K\). A \(G\)-model of \(K\) is a normal variety \(X\) with regular \(G\)-action so that \({\rm k}(X)=K\). A \(G\)-model \(X\) is uniquely determined by the set of local rings \(\mathscr O_{X,Y}\subset K\) of all \(G\)-stable subvarieties \(Y\subset X\) (cf. [19]).

Fix a Borel subgroup \(B\) in \(G\). Denote by \(\Gamma\subset\mathfrak X(B)\) the lattice of weights of all (non-zero) \(B\)-semiinvariant functions \(f\in K^{(B)}\subset K\). Denote by \(\mathscr V\) the set of \(G\)-invariant discrete \(\mathbb{Q}\)-valued geometric valuations (called \(G\)-valuation) of \(K\). Knop [33] proved that such a valuation is uniquely determined by its restriction on \(K^{(B)}\). The valuation ring of a valuation \(v\) on \(K\) will be denoted by \(\mathscr O_v\) below.

We would like to recall a characteristic of \(\mathscr V\) for our later use, which we learned from [12]. Let us begin with some definitions there. Note that we have an exact sequence \[\begin{align} \label{ex-seq-func} 1\to(K^B)^\times\to K^{(B)}\to\Gamma\to0. \end{align}\tag{5}\] Let \(\nu\) be any geometric valuation of \(K^B\). Factoring the sequence 5 by \(\mathscr O_\nu^\times(\subset\mathscr O_\nu,~\text{the valuation ring of}~\nu)\) we get an exact sequence of lattices \[1\to\mathbb{Z}_\nu\to\Gamma_\nu\to\Gamma\to0,\] where \(\mathbb{Z}_\nu\cong\mathbb{Z}\) or \(0\) is the value group of \(\nu\). Taking \(\mathbb{Q}\)-dual we get \[\begin{align} &0& &\longleftarrow& &\mathbb{Q}_\nu& &\longleftarrow& &\mathscr Q_\nu& &\longleftarrow& &\mathscr Q_{0}& &\longleftarrow& &0&\\ &\,& &\,& &\cup& &\,& &\cup& &\,& &||& &\,& &\,&\\ &0& &\longleftarrow& &\mathbb{Q}_{\nu,+}& &\longleftarrow& &\mathscr Q_{\nu,+}& &\longleftarrow& &\mathscr Q_{0,+}& &\longleftarrow& &0.& \end{align}\] Here \(\mathbb{Q}_\nu=\mathbb{Q}\) and \(\mathscr Q_{\nu,+}\) is the preimage of the positive ray \(\mathbb{Q}_{\nu,+}\) for \(\nu\not=0\), and \(\mathbb{Q}_0=\mathbb{Q}_{0,+}=0\), \(\mathscr Q_{0,+}=\mathscr Q_0\cong{\rm Hom}(\Gamma,\mathbb{Q})\) (which will be simply denoted by \(\mathscr Q\) below). The hyperspace of \(K\) is defined as \[\mathscr E:=\cup_\nu\mathscr Q_{\nu,+},\] where \(\nu\) runs over all a geometric valuation of \(K^B\) up to proportionality. More precisely, \(\mathscr E=\mathscr Q\) if \(c(X)=0\), while when \(c(X)>0\), each \(\mathscr Q_{\nu,+}\) is a half-space, and \(\mathscr E\) is obtained by gluing together the \(\mathscr Q_{\nu,+}\)’s along their common boundary hyperplane \(\mathscr Q\). This boundary \(\mathscr Q\) is called the centre of \(\mathscr E\). The valuation cone \(\mathscr V\) can be embedded into \(\mathscr E\) as follows: Suppose that \(v\in\mathscr V\) is a geometric valuation dominating \(\nu\). Then via restriction \(v|_{K^{(B)}}\), \(v\) is mapped to an element in \(\mathscr Q_{\nu,+}\) which is linear on \(\Gamma_\nu\) and non-negative on \(\mathbb{Z}_\nu\). It is further proved by [33] that this map is injective.

Fix any \(G\)-model \(X\) of \(K\) and denote by \(\mathscr D\) the set of all prime divisors that is not \(G\)-stable. This set in fact depends only on \(K\) but not on the choice of \(X\). \(B\)-stable elements \(\mathscr D^B\) are called the colours. For any \(D\in\mathscr D\), denote by \(v_D={\rm ord}_D\) the corresponding normalized valuation of \(K\). Similarly, there is a restriction map \(\varrho:\mathscr D^B\to\mathscr E\) that maps a colour \(D\) to \(v_D|_{K^{(B)}}\), which can be identified with an element of \(\mathscr E\). In the following we often denote this element by \(v_D\) when there is no confusions. The map \(\varrho\) is in general not injective. For more knowledge on hyperspaces, we refer to [12].

We may assume that \(K={\rm Quot}R\) for some rational \(G\)-algebra \(R\). For example we can take \(R={\rm k}[X]\) if \(X\) is a quasi-affine model of \(K\) or the Kodaira ring of a complete model. Let \(f_1,...,f_s\in R^{(B)}\) and \(f\not=f_1...f_s\) be any \(B\)-eigenvector in \(\langle G\cdot f_1\rangle...\langle G\cdot f_s\rangle\). Here by \(\langle G\cdot f\rangle\) we mean the linear span of the orbit \(G\cdot f\) in \(R\). Then \(f/f_1...f_s\) is called a tail vector of \(R\) and its weight is called a tail. The tails are negative linear combination of simple roots of \(G\) (with respect to \(B\)). For an affine \(G\)-variety, as proved by [34] (see also [12]), the tails span a finitely generated semigroup. It is showed that (cf. [12])

Moreover, it is known that each \(\mathscr V_\nu=\mathscr V\cap\mathscr Q_{\nu,+}\) is a finitely generated solid convex cone in \(\mathscr Q_{\nu,+}\) (cf. [12]).

Let \(X\) be a \(G\)-model of \(K\). \(B\)-stable affine open subsets of \(X\) are called \(B\)-charts. Recall the sets \(\mathscr D\) and \(\mathscr D^B\) defined above. For any \(D\in\mathscr D\), denote by \(\mathscr O_D(=\mathscr O_{v_D})\) its valuation ring. Then given any \(B\)-chart \(\mathring X\), there are subsets \(\mathscr W\subset\mathscr V\) and \(\mathscr R\subset\mathscr D^B\) so that (cf. [19]) \[\begin{align} {\rm k}[\mathring X]=(\bigcap_{w\in\mathscr W}\mathscr O_w)\cap(\bigcap_{D\in\mathscr R\sqcup(\mathscr D\setminus\mathscr D^B)}\mathscr O_D). \end{align}\] It is of great interest to find a criterion on when the ring \[\begin{align} \label{A40W44R41} A(\mathscr W,\mathscr R)=(\bigcap_{w\in\mathscr W}\mathscr O_w)\cap(\bigcap_{D\in\mathscr R\sqcup(\mathscr D\setminus\mathscr D^B)}\mathscr O_D) \end{align}\tag{6}\] of a given pair \((\mathscr W,\mathscr R)\) defines a \(B\)-chart. It turns out

However, the pair \((\mathscr W,\mathscr R)\) may still contains some redundant data that different choices of \((\mathscr W,\mathscr R)\) may lead to same \(A(\mathscr W,\mathscr R)\) (and consequently the resulting \(B\)-chart). This can be overcame if we further that every valuation \(w\in\mathscr W\) is normalized and essential. By normalized we mean the group of values of a valuation \(w\) is precisely \(\mathbb{Z}\), and by essential we mean removing \(w\) from the right-hand side of 6 will change \(A(\mathscr W,\mathscr R)\). We recall that in 6 any valuation in \(\mathscr R\sqcup(\mathscr D\setminus\mathscr D^B)\) is essential (cf. [9], [33] and [12]). All valuations in \(\mathscr W\) are essential if and only if

• For any \(w\in\mathscr W\), there exists \(f\in K^{(B)}\) so that for any \(w'\in\mathscr W\setminus\{w\}\), \(D\in\mathscr R\), it holds \[w'(f),~v_D(f)\geq0,~w(f)<0.\]

Theorem 9 then can be refined to

In the following, unless otherwise stated, when referring to a data \((\mathscr W,\mathscr R)\) we always assume it satisfies the conditions (C), (F) and (W).

The \(G\)-model \(X\) can be recovered by patching the \(G\)-spans of finitely many \(B\)-charts that intersect \(G\)-invariant subvarieties in it (cf. [19]). For any \(G\)-invariant subvariety \(Y\subset X\), denote by \(\mathscr V_Y\) (\(\mathscr D^B_Y\), resp.) the set of normalized valuations corresponding to \(G\)-stable prime divisors of \(X\) containing \(Y\) (colours containing \(Y\), resp.). Then it is showed by [9] (see also [19]) that the local ring \[\mathscr O_{X,Y}=A(\mathscr V_Y,\mathscr D^B_Y).\] The support \(\mathscr S_Y\) of \(Y\) is \[\mathscr S_Y:=\{v\in\mathscr V|\mathscr O_v~\text{dominates}~\mathscr O_{X,Y}\}.\] Given a \(B\)-chart \(\mathring X\) associated to the data \((\mathscr W,\mathscr R)\), we can recover \((\mathscr V_Y,\mathscr D^B_Y)\) for any \(G\)-invariant subvariety \(Y\) intersecting \(\mathring X\) according to the conditions (V), (V’), (D’) introduced in [19]. Finally, we have

2.2.2 The case of complexity 1↩︎

When \(c(K)={\rm tr.deg}_{\rm k}(K^B)=1\), \(K^B={\rm k}(C)\) for a unique smooth projective curve \(C\) and there is a rational quotient \({\rm pr}_B:X\dasharrow C\) by the \(B\)-action. In the following we mainly consider the case when \({\rm k}(X)^B\cong{\rm k}(\mathbb{P}^1)\). This holds, for example when \(X\) is unirational (by Lüroth’s Theorem).

Let us recall the reduction in [19] of the general settings above to the complexity 1 case. As in [19], by fixing any monomorphism \(e:\Gamma\to K^{(B)}\) that maps a weight \(\lambda\) to a non-zero \(B\)-semiinvariant function \(e_\lambda\in K^{(B)}_\lambda\), the sequence 5 splits. That is, any \(f\in K^{(B)}_\lambda\) can be decomposed as \(f=f_0e_\lambda\) with \(f_0\in K^B\cong{\rm k}(C)\). On the other hand, any valuation \(\nu\) on \({\rm k}(C)\) can be written as \(\nu=h{\rm ord}_x\), where \(h\in\mathbb{Q}_+\) and \(x\in C\) a point. The hyperspace \(\mathscr E\) introduced in the previous section then can be written more precisely as \[\mathscr E=\bigsqcup_{x\in C}\mathscr Q_{x,+}/\sim,\] where \(\mathscr Q_{x,+}=\{x\}\times\mathscr Q\times\mathbb{Q}_+\), and \((x,\ell,h)\sim(x',\ell',h')\) if and only if \((x,\ell,h)=(x',\ell',h')\) or \(h=h'=0\), \(\ell=\ell'\). A valuation \(v\in\mathscr V\) (when embedded in \(\mathscr E\) via \(v|_{K^{(B)}}\)) is decomposed as \[v=h_vq_{x_v}+\ell_v,\] so that \(v|_{K^B}=h_vq_{x_v}\) with \(h_v\geq0\), \(q_{x_v}=(x_v,O,1)\) the valuation at a point \(x_v\in C\), and \(\ell_v=v|_{e(\Gamma)}\). We call \(h_v\) the jump of \(v\). Elements in \(\mathscr Q\) (i.e. with zero jump) are said to be central. For any central valuations \(v\in\mathscr Q\), by the relation \(v(f)=v(e_\lambda)\), we get \(v(e_\lambda)=\ell_v(\lambda)\), and consequently \(\mathscr Q\cong\Gamma_\mathbb{Q}^*\). Denote by \(\mathscr B(X)\) the set of \(B\)-stable divisors in \(X\). For any \(D\in\mathscr B(X)\), by restricting the valuation corresponding to \(D\) on \(K^{(B)}\), \(D\) is mapped to \(v_D\in\mathscr E\) by \[v_D=h_Dq_{x_D}+\ell_D.\] It is proved in [19] that for any \(x\in C\), the set of divisors in \(\mathscr B(X)\) that maps to \(\mathscr Q_{x,+}\) is finite, and for almost every (that is, by ruling out finite exceptions) \(x\in C\) there is exactly one \(D\in\mathscr B(X)\) maps to \(\mathscr Q_{x,+}\setminus\mathscr Q\), and this divisor satisfies \(v_D=q_x\).

The above setting leads to a combinatorial description of functions in \(K^{(B)}\) and \(B\)-charts on a \(G\)-model of \(K\). For any \(f\in K^B\), it induces a collection of linear functionals \[\varphi=\{\varphi_x:\mathscr Q_{x,+}\to\mathbb{Q}|x\in C\}\] so that \(\varphi_x|_\mathscr Q\equiv0\) and \(\varphi_x(q_x)=q_x(f)\) for any \(x\in C\). In particular \(\sum_{x\in C}\varphi_x(q_x)\cdot x\) is a principle divisor on \(C\). A function \(f=f_0e_\lambda\in K^{(B)}_\lambda\) with \(f_0\in K^B\) defines the functionals \[\varphi=\{\varphi_{0,x}+\lambda|x\in C\},\] where \(\varphi_0\) is the functional of \(f_0\).

Given a \(B\)-chart \(\mathring X\) with data \((\mathscr W,\mathscr R)\), a function \(f\in K^{(B)}\) lies in \({\rm k}[\mathring X]\) if and only if \((\mathscr W\sqcup\mathscr R)(f)\geq0\), or equivalently, the functional of \(f\), \(\varphi\in(\mathscr W\sqcup\mathscr R)^\vee\). This condition selects regular functions in \(K^{(B)}\). There are two types of \(B\)-charts:

• Type : \({\rm k}[\mathring X]^{(B)}\not={\rm k}\). In this case there is some \(x\in C\) so that no element in \(\mathscr W\sqcup\mathscr R\) falls into \(\mathscr Q_{x,+}\setminus\mathscr Q\);

• Type : \({\rm k}[\mathring X]^{(B)}={\rm k}\). In this case any \(\mathscr Q_{x,+}\) contains a non-central element in \(\mathscr W\sqcup\mathscr R\).

Accordingly, a \(B\)-chart of type (or type resp.) is characterised by the image of the rational quotient \({\rm pr}_B\) is a proper subset of \(C\) (or equals to \(C\), resp.).

For \(\mathring X=\mathring X(\mathscr W,\mathscr R)\), it is proved in [19] the set \[\mathscr C(\mathscr W,\mathscr R)=\{v\in\mathscr E|v(\varphi)\geq0,~\forall \varphi\in(\mathscr W\sqcup\mathscr R)^\vee\}\] together with \(\mathscr R\) forms a coloured hypercone in \(\mathscr E\) defined as

The admissibility conditions of a coloured hypercone are obtained in [19], which are equivalent to the Conditions (C), (F) and (W) (cf. [19]). It is proved in [19] that \(B\)-charts of type (or type , resp.) are determined by coloured hypercones of type (or type , resp.).

In [19], the \(G\)-invariant subvarieties (referred as \(G\)-germs in [19]) are classified by \(B\)-charts intersecting them. A \(G\)-germ adimitting a \(B\)-chart of type is called a \(G\)-germ of type . \(G\)-germs admitting only \(B\)-charts of type are called \(G\)-germs of type . Also, a face of a coloured hypercone \((\mathscr C, \mathscr R)\) is either

• a coloured cone \((\mathscr C',\mathscr R')\subset\mathscr Q_{x,+}\), where \(\mathscr C'\) is a face of \(\mathscr C_x\) intersecting \(\mathscr K\) at a true face and \(\mathscr R'\) are colours in \(\mathscr R\) that maps to \(\mathscr C'\);

or

• a hyperface \((\mathscr C',\mathscr R')\) of type . In this case there is a linear functional \(\varphi\in(\mathscr W\sqcup\mathscr R)^\vee\) so that \(\mathscr C'=\mathscr C\cap\ker(\varphi)\) that intersects \(\mathscr K\) at a pseudoface, and \(\mathscr R'\) are colours in \(\mathscr R\) that maps to \(\mathscr C'\).

For a coloured cone \((\mathscr C,\mathscr R)\subset\mathscr Q_{x,+}\), we define its relative interior to be \({\rm RelInt}(\mathscr C)\) in the usual sense; for a coloured hypercone \((\mathscr C,\mathscr R)\) of type , we define \[{\rm RelInt}(\mathscr C):=(\cup_{x\in C}{\rm RelInt}(\mathscr C_x))\bigcup{\rm RelInt(\mathscr C\cap\mathscr Q)}.\] It is proved in [19] that

A collection \(\mathfrak F_X\) of coloured cones and hypercones of type that defines a \(G\)-model \(X\) above is called the coloured fan of \(X\). A \(G\)-model \(X\) is said to be of type if its coloured fan consists of only coloured cones.⁴

Finally we recall the fact that there are two kinds of \(G\)-varieties of complexity 1:

• The one-parameter case: \(K^G=K^B\). In this case, a \(G\)-orbit in general position has codimension 1 and contains a dense \(B\)-orbit;

• The quasihomogeneous case: \(K^G={\rm k}\). In this case, there is a homogeneous space \(G/H\) so that any \(G\)-model of \(K\) contains a dense \(G\)-orbit isomorphic to \(G/H\). A quasihomogeneous variety is always unirational (cf. [19]).

2.2.3 The central automorphism group↩︎

Let \(X\) be a \(G\)-variety. The central \(G\)-equivariant automorphism group is defined as \[\mathfrak A(X):=\{\sigma\in{\rm Aut}_G(X)|\sigma\cdot f= f,~\forall f\in{\rm k}(X)^{B}\}.\] Then central valuation cone \(\mathscr V\cap\mathscr Q\) is closely related to the Lie algebra of \(\mathfrak A(X)\) (cf. [33] and [12]). It is showed in [33] that there is a torus \({\mathbf{T}}\subset \mathfrak A(X)\) whose Lie algebra is the linear part \(\mathscr A(:=\mathscr V\cap(-\mathscr V)\cap\mathscr Q)\) of \(\mathscr V\) (identified with a subalgebra of \({\rm Lie}(T)\), where \(T=B\cap B^-\) is a maximal torus of \(G\)). Consider the group \({\mathbf{G}}\subset{\rm Aut}(X)\) generate by \({\mathbf{T}}\) and the \(G\)-action on \(X\). Then \({\mathbf{G}}\) is a reductive group with \({\mathbf{T}}\) contained in its centre.

2.3 Cartier divisors on \(G\)-varieties↩︎

Let \(X\) be a \(G\)-model of \(K\) and \(L\) a line bundle on it. Up to replacing \(G\) by a finite covering, we may assume that \(G\) is of simply connected type. Then by [35], \(L\) is \(G\)-linearized, and by [36] there always exists a \(B\)-stable divisor \(\mathfrak d\) of \(L\). In this section, we recall some basic results on Cartier divisors on \(X\) established by [14], in particular the combinatorial data associated to ample divisors. Finally we give a formula of \(B\)-stable anti-canonical divisors.

Let \(X\) be a \(G\)-model of \(K\) and \(\mathscr B(X)\) the set of all \(B\)-stable prime divisors of \(X\). Let \[\begin{align} \label{B-stable-divisor} \mathfrak d=\sum_{D\in\mathscr B(X)}m_DD \end{align}\tag{7}\] be a \(B\)-stable Weil divisor on \(X\). The following criterions of Cartier property and ampleness were proved in [14], which is a generalization of [13] for spherical varieties.

Given a \(B\)-stable Cartier divisor, we have the following ampleness criterion:

2.3.1 The ample divisors↩︎

Suppose that the divisor \(\mathfrak d\) in 7 is ample, and \(L=L_\mathfrak d\) its line bundle. Also suppose that \(\mathfrak d\) is the divisor of some section \({\rm H}^0(X,L)^{(B)}_{\lambda_0}\) for some \(\lambda_0\in\mathfrak X(B)\). In this section we recall results on the space of global sections \({\rm H}^0(X,L^k)\) from [14]. For our later use, we only focus on the results when \(K^B\) is rational, in which \(C=\mathbb{P}^1\). For results in general cases we refer to the readers [14].

Recall the functions \[\begin{align} A_x(\mathfrak d,\lambda):=\min_{\{v_D=h_Dq_{x_D}+\ell_D\in\mathscr B(X)|x_D=x,h_D>0\}}\frac{\ell_D(\lambda)+m_D}{h_D},~\lambda\in\Gamma_\mathbb{R},~x\in C, \end{align}\] defined in [14]. Note that the minimum taken here is well-defined since there are only finitely many divisors in \(\mathscr B(X)\) satisfy \(x_D=x\) for every \(x\in C\). For almost every \(x\in C\) (that is, there are at most finitely many exceptions), \(A_x(\mathfrak d,\lambda)\equiv0\). Thus \[\begin{align} \label{A40d44lambda41} A(\mathfrak d,\lambda):=\sum_{x\in C}A_x(\mathfrak d,\lambda) \end{align}\tag{8}\] is well-defined. [14] (see also [12]) introduced the following polytope \[\begin{align} \Delta_\mathscr Z(\mathfrak d):=\{\lambda\in\Gamma_\mathbb{R}|v_D(\lambda)+m_D\geq0,~\forall{D\in\mathscr B(X),h_D=0},~\text{and}~A(\mathfrak d, \lambda)\geq0\}, \end{align}\] and proved

From Proposition 16 we directly get the following expression of \(\dim{\rm H}^0(X,L^k)\), which will be frequently used later. Set \[\begin{align} \label{pi-def} \pi(\lambda)=\prod_{\alpha^\vee\in\Pi_{G}^\vee\setminus(\Gamma+\mathbb{Z}\lambda_0)^\perp}\frac{\langle\lambda,\alpha^\vee\rangle}{\langle\rho,\alpha^\vee\rangle},~\lambda\in\mathfrak X_\mathbb{R}(B). \end{align}\tag{9}\] We have the following asymptotic expression of \(\dim{\rm H}^0(X,L^k)\) which is a refinement of [14], and also a counterpart of the intersection formulas established in [15], [16] for spherical varieties.

2.3.2 Associated parabolic subgroup and properties of the function \(\pi\)↩︎

The set \(\Pi_{G}^\vee\setminus(\Gamma+\mathbb{Z}\lambda_0)^\perp\) in 9 is in fact independent with the choice of the ample line bundle \(L\) (cf. [13], [14]). Indeed it consists of unipotent coroots of the associated parabolic subgroup of \(X\). This seems to be well-known to the experts, while for the readers’ convenience we give a short explanation below. Recall that for any \(G\)-variety \(X\), its associated parabolic subgroup is defined by (see [11] for definition) \[\begin{align} \label{ass-para} P=\{g\in G|gBx=Bx~\text{for general}~x\in X\}, \end{align}\tag{11}\] or equivalently, the maximal common stabilizer group of all colours. Let \(L\) be any ample line bundle on \(X\). Then \(P\) stabilizes any line in \({\rm H^0}(X,L^k)\) generated by a \(B\)-semiinvariant section for all \(k\in\mathbb{N}_+\), since the divisor of such a section is also \(P\)-stable. Moreover, since \(L\) is ample, \(P\) in particular equals to the stabilizer of one \(B\)-semiinvariant section in some \({\rm H^0}(X,L^{k_0})\) ([33]). Thus for every \(k\in\mathbb{N}_+\), the roots of the Levi group \(L_P\) of \(P\) are precisely roots of \(G\) that are orthogonal to the set \[\bigcup_{k\in\mathbb{N}_+}\{\lambda\in\mathfrak X(B)|\dim{\rm H}^0(X,L^k)^{(B)}_\lambda\not=0\}.\] By Proposition 16, they are precisely the roots of \(G\) that are orthogonal to \(\Delta_\mathscr Z(L)\), which is a solid polytope in \(\Gamma_\mathbb{R}+\lambda_0\). Hence

For our later use, we introduce a further property of the gradient \(\nabla\pi\) of \(\pi\), following the argument of [16] in the spherical case. Let \(L\) be any ample line bundle on \(X\) and fix a section \(s_L\in{\rm H}^0(X,L)^{(B)}_{\lambda_0}\) for some \(\lambda_0\in\mathfrak X(B)\). Consider the Weyl group \(W(L_P)\) of \(L_P\) with respect to \(B\). As discussed above, \(P\) stabilizes any point \(\lambda\in \Gamma\) and the weight \(\lambda_0\). Thus \(\lambda\) and \(\lambda_0\) are \(L_P\)-character and hence are \(W(L_P)\)-invariant. On the other hand, \(W(L_P)\) acts on \(\Pi_G\) and permutes elements in \(\Pi_G\setminus\Pi_{L_P}=\Pi_{P_u}\). By Lemma 18, \(W(L_P)\) precisely permutes factors \(\alpha^\vee\)’s in 9 . It follows \[(w^*\pi)(\lambda)=\pi(\lambda),~\forall w\in W(L_P),~\forall\lambda\in\mathfrak X_\mathbb{R}(B).\] Denote \[\kappa_P:=\sum_{\alpha\in\Pi_{P_u}}\alpha=2\rho-\sum_{\alpha\in\Pi_{L_P}}\alpha.\] Note that there is a longest element \(w_o\in W(L_P)\) that maps every \(\alpha\in\Pi_{L_P}\) to \(-\alpha\). In particular, \[w_o(\kappa_P-2\rho)=2\rho-\kappa_P.\] Combining with the fact \[w_0(\lambda+\lambda_0)=\lambda+\lambda_0,~\forall\lambda\in\Gamma_\mathbb{R},\] we get

Here in the second line we used Proposition 16.

2.3.3 Relation with the divisorial polytope construction↩︎

Finally we introduce some combinatorial properties of ample line bundles. When \(G\) is a torus, Ilten-Süß  classified polarized \(T\)-varieties of complexity 1 via the “divisorial polytopes" introduced in [20]. This is a family of concave functions on \(\Delta_\mathscr Z(L)\) satisfying certain conditions. In particular, in the case of \(T\)-varieties of complexity 1 there is no colour and the fan of a polarized variety can be recovered from its divisorial polytopes. The following Proposition is a partial generalization of this direction in the case of a general reductive \(G\).

For each \(x\in C\), consider the following polyhedron \[\begin{align} \Delta_x(\mathfrak d):=\{(t,\lambda)\in\mathbb{R}\times\Delta_\mathscr Z(\mathfrak d)|t\geq -A_x(\mathfrak d,\lambda)\}. \end{align}\] Then we have

2.4 The affine cone over a projectively normal embedding↩︎

Let \(X\) be a projective \(G\)-variety of complexity 1, and \(L\) a \(G\)-linearized ample line bundle on it. As the coloured fan of \(X\) is a bit complicated, in this section we will study \(X\) via its affine cone under certain embedding. This approach was used in [21] in the study of \(T\)-varieties.

Suppose that \(L=L_\mathfrak d\) is a \(G\)-linearized ample line bundle on \(X\) which has a divisor \(\mathfrak d\) given in 7 . Up to replacing \(L\) by its tensor power, we may assume that \(X\) can be embedded into a projective space by \(|L|\) and the embedding is projectively normal. In this case, the affine cone \(\hat{X}\) over \(X\) is an affine normal variety \[\hat{X}={\rm Spec}R(X,L),\] where \(R(X,L)\) is the Kodaira ring of \((X,L)\) given by 2 . As \(X\) is complete, it holds \({\rm H}^0(X,L^0)={\rm k}\). Hence the vertex of the cone \(\hat{X}\) is precisely the origin point \(O\). Let \({\rm pr}:\hat{X}\setminus\{O\}\to X\) be the projection. Then for any subvariety \(Y\subset X\), \(\hat{Y}:={\rm pr}^{-1}(Y)\) is the affine cone over \(Y\) and it also contains \(O\) as its vertex.

2.4.1 Combinatorial data of the affine cone↩︎

Obviously that the affine cone \(\hat{X}\) is a \((G\times {\rm k}^\times)\)-variety. In fact, the group \(\hat{G}:=G\times {\rm k}^\times\) acts on \(\hat{X}\) as following: \(G\) acts on each piece \(R_k(X,L)\) naturally and \(t\in k^\times\) acts on \(R_k(X,L)\) by \[(t\cdot s)(\hat{x})(=:s(t^{-1}\cdot\hat{x}))=t^{-k}(s(\hat{x})),~\forall s\in R_k(X,L),~\hat{x}\in \hat{X}~\text{and}~k\in\mathbb{N}.\] In particular, \(s\) has \({\rm k}^\times\)-weight \(-k\).⁵ In the following we denote objects of \(\hat{X}\) by adding a hat.

It is direct to see that \({\rm k}(\hat{X})^{B\times{\rm k}^\times}={\rm k}(X)^B\), the lattice \(\hat{\Gamma}\cong\Gamma\oplus\mathbb{Z}\) and the hyperspace \(\hat{\mathscr E}\cong\mathscr E\times\mathbb{Q}\). In fact, by choosing \(B\)-semiinvariant \(s_*\in R_1(X,L)_{\lambda_*}^{(B)}\) so that \(\lambda_*\in(\Delta_\mathscr Z(\mathfrak d)+\lambda_0)\), the map \[\begin{align} {\rm k}(X)^{(B)}\times\mathbb{Z}&\rightarrow {\rm k}(\hat{X})^{(\hat{B})}\label{k40hatX41-k40X41}\\ (f_\lambda,p)&\rightarrow f_\lambda s_*^{p}\in {\rm k}(\hat{X})^{\hat{B}}_{(\lambda+p\lambda_*,p)},\notag \end{align}\tag{14}\] gives a desired isomorphism.

Then we determine the combinatorial data of \(\hat{B}\)-stable prime divisors \(\hat{\mathscr B}(\hat{X})\) in \(\hat{X}\). The \(\hat{B}\)-stable divisors in \(\hat{X}\) consist of all \(\hat{D}\)’s, where each \(\hat{D}\) is the affine cone over \(D\) with \(D\in\mathscr B(X)\). For convenience, without loss of generality we can assume that the divisor 7 is effective and choose \(s_*=s_0\). Then for every \(D\in\mathscr B(X)\), it holds \[v_{\hat{D}}(f_\lambda)=v_D(f_\lambda),~\forall f_\lambda\in {\rm k}(X)^{(B)}_\lambda~\text{and}~v_{\hat{D}}(s_0)=-m_D.\] Thus we get \[\begin{align} v_{\hat{D}}=(v_{D},-m_D)\in\mathscr E\times\mathbb{Q}(=:\hat{\mathscr E}). \end{align}\]

Finally we derive the combinatorial data of \(\hat{X}\). The affine cone \(\hat{X}\) itself is a \(\hat{B}\)-chart determined by \(\{\hat{D}|D\in\mathscr B(X)\}\). In fact, \(\hat{X}\) is the \(\hat{B}\)-chart intersecting \(O\in\hat{X}\). Since \(X\) is complete, its coloured fan \(\mathfrak F_X\) covers \(\mathscr V\). In particular, for each \(x\in C\), there is at least one \(D\in\mathscr B(X)\) so that \(v_D\in\mathscr Q_{x,+}\setminus\mathscr Q\). Clearly such a \(D\) satisfies \(v_{\hat{D}}\in(\mathscr Q_{x,+}\times\mathbb{Q})\setminus(\mathscr Q\times\mathbb{Q})\). Hence the coloured hypercone \((\hat{\mathscr C},\hat{\mathscr R})\) is a coloured hypercone of type . We then determine its pseudofaces. For each \(D\in\mathscr B(X)\) with \(v_D\in\mathscr Q_{x_D,+}\), we have \(x_{\hat{D}}=x_D\) and \(h_{\hat{D}}=h_D\). Set \[\hat{\mathscr P_x}:={\rm Conv}(\{\frac{v_{\hat{D}}}{h_D}-q_{x_D}|D\in\mathscr B(X)~\text{so that}~x_D=x~\text{and}~h_D\not=0\}),\] and \[\begin{align} \label{affine-cone-P-eq} \hat{\mathscr P}=\sum_{x\in C}\hat{\mathscr P_x}, \end{align}\tag{15}\] where the right-hand side is the Minkowski sum. Then the pseudofaces of \(\hat{\mathscr C}\) are exactly those intersect \(\hat{\mathscr P}\). We get

2.4.2 Admissibility↩︎

It is also direct to check that the combinatorial data ?? ?? is admissible. More precisely, in the following we show that they fulfill the conditions in [19].

For each \(x\in C\), consider the cone⁶ \[\tilde{\mathscr C}_x=\{(\epsilon,m)\in\Gamma_\mathbb{R}\times\mathbb{R}\times\mathbb{R}|-\frac{\epsilon}{m}\in\Delta_x(\mathfrak d)\}.\] In fact, it is the cone over \(\Delta_x(\mathfrak d)\times\{-1\}\subset(\Gamma_\mathbb{R}\times\mathbb{R}\times\mathbb{R})\). We will see that under the ampleness assumption, \(\hat{\mathscr C}_x\) is the dual cone of \(\tilde{\mathscr C}_x\), as showed in [21] for \(T\)-varieties.

It is clear that a point \((\epsilon,m)\in(\Gamma_\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{\leq0})\) lies in the dual cone \((\hat{\mathscr C}_x)^\vee\) of \(\hat{\mathscr C}_x\) if and only if \[\begin{align} \label{hatC-dual-eq-1} v_D(\epsilon)-m_Dm\geq0,~\forall D\in\mathscr B(X)~\text{so that}~x_D=x, \end{align}\tag{16}\] and \[\begin{align} \label{hatC-dual-eq-2} p(\epsilon)-m_pm\geq0,~\forall p=\sum_{y\in C}p_y\in\hat{\mathscr P}, \end{align}\tag{17}\] where each \[\begin{align} \label{hatC-py} p_y=\sum_{D_y}c_{D_y}(\frac{v_{\hat{D}_y}}{h_{D_y}}-q_y)\in\hat{\mathscr P}_y \end{align}\tag{18}\] is a convex combination of some \(D_y\in\mathscr B(X)\) so that \(h_{D_y}\not=0\) and \(x_{D_y}=y\), and \[m_p=\sum_{y\in C}\sum_{D_y}c_{D_y}\frac{m_{D_y}}{h_{D_y}}.\]

Let us show that some inequalities in the system 16 17 can be removed while keeping the solution set unchanged. In fact, suppose that there are two points \((v_i,m_i)\), \(i=1,2\), which lie on edges of two different cones \(\mathscr C_{i,x}\in\mathfrak F_{X,x},~i=1,2\), respectively, and \(m_i=m_D\) if \(v_i=v_D\) or \(m_i=m_p\) if \(v_i\) lies in a pseudoface. Then \[\Delta_x(\mathfrak d)\subset\{\epsilon\in\mathscr Q_{x,+}|v_i(\epsilon)+m_i\geq0,~i=1,2\}.\] Consequently for their convex combination \(v=tv_1+(1-t)v_2\) with \(t\in[0,1]\), it holds \(m_v=tm_1+(1-t)m_2\) and \[\Delta_x(\mathfrak d)\subset\{\epsilon\in\mathscr Q_{x,+}|v(\epsilon)+m_v\geq0\}.\] The same holds true for convex combination of arbitrarily many points \(\{v_i\}\) where each \(v_i\) lies on an edge of some \(\mathscr C_{i,x}\in\mathfrak F_{X,x}\). Hence in 17 it suffices to consider only inequalities given by \(p\)’s for which all \(D_y\)’s in 18 belong to a same hypercone of type in \(\mathfrak F_X\) (i.e. removing other inequalities in 17 does not change the solution set). In this way the system 16 17 then reduces to ?? and we get that \((\hat{\mathscr C}_x)^\vee=\tilde{\mathscr C}_x\). Thus \(\hat{\mathscr C}_x=(\tilde{\mathscr C}_x)^\vee\) is a convex cone lies in \(\mathscr Q_{x,+}\times\mathbb{Q}_{\leq0}\). Clearly \(O\not\in\hat{\mathscr P}\) and the conditions in [19] are fulfilled.

2.4.3 Relationship with the coloured fan of \(X\)↩︎

Let \(\hat{X}\) be the affine cone as before. We see that except the origin \(O\), any \(\hat{G}\)-invariant subvariety of \(\hat{X}\) is the affine cone over a \(G\)-invariant subvariety in \(X\). Thus by [19], \(G\)-invariant subvarieties in \(X\) are in one-one correspondence with faces (coloured cone intersecting only true faces or hypercones of type that intersecting pseudofaces) of \((\hat{\mathscr C},\hat{\mathscr R})\), whose relative interior intersects the valuation cone of \(\hat{X}\).

Let \(\hat{Y}\subset\hat{X}\) be a \(\hat{G}\)-invariant subvariety and \((\hat{\mathscr C}',\hat{\mathscr R}')\) the face of \((\hat{\mathscr C},\hat{\mathscr R})\) that is associated to \(\hat{Y}\). Then it defines a \(\hat{B}\)-chart \(\widehat{X'}\subset\hat{X}\setminus\{O\}\) intersecting \(\hat{Y}\) (when \((\hat{\mathscr C}',\hat{\mathscr R}')\) is a coloured cone, we choose \(\widehat{X'}\) any \(\hat{B}\)-chart that corresponds to a coloured hypercone of type containing \((\hat{\mathscr C}',\hat{\mathscr R}')\)). Then \(\widehat{X'}\) is a normal affine open set and \(\mathring X':={\rm Proj}(\widehat{X'})\) is a \(B\)-chart of \(X\) intersecting \(Y={\rm Proj}(\hat{Y})\). It remains to determine the coloured hypercone \(({\mathscr C}',{\mathscr R}')\) defining \(\mathring X'\). From 14 we see that the cone \({\mathscr C}'\) associated to \(\mathring X'\) is the image of \(\hat{\mathscr C}'\) under the canonical projection from \(\hat{\mathscr E}\) to \(\mathscr E\). We then select the colours. Suppose that \(D\in\mathscr B(X)\setminus\mathscr B(X)^G\) is a colour with \(x_D=x\). By Lemma 20, \(\mathscr C'_x\) is the inner normal cone of \(\Delta_x(\mathfrak d)\) at some face \(F\). From the ampleness condition, \(D\in\mathscr R'\) if and only if the hyperplane \(\{\epsilon\in\Gamma_\mathbb{R}\times\mathbb{R}|v_D(\epsilon)+m_D=0\}\) intersects \(F\).

As showed in the previous subsection, \((\hat{\mathscr C})_x^\vee\subset\{(\epsilon,m)\in(\Gamma_\mathbb{R}\times\mathbb{R}\times\mathbb{R})|m\leq0\}\) for each \(x\in C\). Hence, \((0,-1)\in\hat{\mathscr C}\subset\hat{\mathscr E}\cong\mathscr E\times\mathbb{Q}\). We conclude that the image of proper faces of \((\hat{\mathscr C},\hat{\mathscr R})\) under the canonical projection to \(\mathscr E\) forms a coloured fan in \(\mathscr E\), and it is indeed the coloured fan of \(X\).

3 The anti-canonical divisors↩︎

Let \(X\) be a normal variety. The anti-canonical sheaf is defined by \(\check{\omega}_X=i_*\check{\omega}_{X_{\rm reg}}\), where \(X_{\rm reg}\) is the regular locus, \(\check{\omega}_{X_{\rm reg}}\) its anti-canonical sheaf, and \(i\) denotes the inclusion. Then \(\check{\omega}_X\) is isomorphic to \(\mathscr O_X(\mathfrak d)\) for some Weil divisor \(\mathfrak d\) on \(X\), which we call anti-canonical divisor. In this section, we derive a formula of \(B\)-stable anti-canonical divisor on a \(G\)-variety of complexity 1. The one-parameter and quasihomogeneous cases will be treated separately.

3.1 The one-parameter case↩︎

In Section 3.1 we do not need to assume that \({\rm k}(X)^B\) is rational. For our later use, we first study the structure of a generic \(G\)-orbit in \(X\), and derive some coloured data of \(X\) from that of this orbit.

3.1.1 Generic \(G\)-orbits↩︎

In the one-parameter case, consider the rational quotient for the \(B\)-action \[{\rm pr}_B: X\dashrightarrow C.\] Then the fibre \(X_z:={\rm pr}_B^{-1}(z)\) over a generic \(z\in C\) contains a \(G\)-spherical homogeneous space \(O_z\). In fact, by [34] there exists a \(G\)-stable dense open subset \(X_O\subset X\) and a spherical subgroup \(H\subset G\) such that in an open set \(\mathring C\subset C\), any \(G\)-orbit \(X_z\cap X_O=O_z\) is \(G\)-equivariantly isomorphic to a fixed spherical homogeneous space \(O:=G/H\). It is further showed in [27] that

The first part of Theorem 25 was originally proved in [38]. Here we use its reformulation [27]. The statement on the \(A\)-action on \(\tilde{X}\) was proved in [27]. The last point is due to [27].

We want to compute central birational invariants of \(X\). By Theorem 25, it suffices to consider the open dense set \(X_O\) and with out loss of generality we can assume that ?? holds.

First we consider the lattice \(\Gamma\) of weights of \(B\)-semiinvariant functions on \(X\). It suffices to find \({\rm k}(\tilde{X})^{(B)\times A}\). Denote by \(\mathfrak M(O)\) the lattice of \(B\)-semiinvariant functions of \(O\). Suppose that \(e_\lambda\in{\rm k}(O)^{(B)}_\lambda\) for \(\lambda\in\mathfrak M(O)\). Then \(e_\lambda\) pulls back to a function (still denoted by \(e_\lambda\)) in \({\rm k}(\tilde{X})^{(B)}_\lambda\). Since \(A\) acts on \(O\) as a subgroup in \({\rm Aut}_G(O)\), there is an \(A\)-character \(\chi_\lambda\in\mathfrak X(A)\) such that \[a\cdot e_\lambda=\chi_\lambda(a)e_\lambda,~\forall a\in A.\] On the other hand, since the field extension \({\rm k}(\tilde{C}):{\rm k}(C)\) is Galois, \({\rm k}(\tilde{C})\) is a regular \(A\)-module (cf. [39]). There always exists an \(\tilde{f}_0\in{\rm k}(\tilde{C})^{(A)}_{-\chi_\lambda}\), and consequently \(\tilde{f}_0e_\lambda\in{\rm k}(\tilde{X})^{(B)\times A}_\lambda\). Also, since \({\rm k}(\tilde{X}):{\rm k}(X)\) is a Galois extension, \({\rm k}(X)={\rm k}(\tilde{X})^A\). We get \(\tilde{f}_0e_\lambda\in{\rm k}(X)^{(B)}_\lambda\), whence \(\mathfrak M(O)\subset \Gamma\). The converse inclusion is obvious (cf. [19]). Hence we can identify \(\Gamma\) with \(\mathfrak M(O)\). Also, the central part \(\mathscr V\cap\mathscr Q\) of the valuation cone \(\mathscr V\) of \(X\) can be identified with the valuation cone \(\mathscr V(O)\) of \(O\), as showed in [19].

It remains to deal with the colours. By [19], all colours in one-parameter case are central. Let \(D\in\mathscr D^B\) be a colour of \(X\). We assign to \(D\) any colour \(\hat{D}\) in \(O\) so that \(\hat{D}\times \tilde{C}\) (which is a colour in \(\tilde{X}\)) that maps to \(D\) via \(\Psi\). Suppose that \(\hat{D}\) and \(\hat{D}'\) are any two such divisors in \(O\). Then they differ from each other by an \(A\)-action. By Lemma 55 in the Appendix, both \(\hat{D}\) and \(\hat{D}'\) map to a same point \(v_{\hat{D}}=v_{\hat{D}'}\) in the hyperspace of \(G/H\), which can be identified with \(\Gamma^*_\mathbb{Q}\). The assignment \[\mathscr D^B\ni D\to v_{\hat{D}}\in\Gamma^*_\mathbb{Q}\] is well-defined. Note that for any \(\tilde{f}_0e_\lambda\in{\rm k}(\tilde{X})^{(B)\times A}\) as above, \[{\rm ord}_D(\tilde{f}_0e_\lambda)={\rm ord}_{\hat{D}}(e_\lambda).\] We can further identify \(v_D\) with \(v_{\hat{D}}\). Thus we have

Recall that any colour in a spherical variety is assigned to one of certain types (type-a, a’, b) according to the minimal parabolic subgroup of \(G\) that moves it. We refer to the readers [40] and [12] for a convenient survey on properties of colours of different types. By Lemma 55 in the Appendix, different colours in \(O\) that are assigned to a same colour \(D\in\mathscr D^B\) of \(X\) have the same type. Thus we have

For our later use, we make the following remarks:

3.1.2 The anti-canonical divisor↩︎

In Theorem 29 below we give a formula of \(B\)-stable anti-canonical divisors on a one-parameter \(G\)-variety \(X\). The first part of Theorem 29 (i.e. equation ?? ) for \(T\)-varieties of complexity 1 was proved by Petersen-Süß [25], and Langlois-Terpereau [26] for horospherical complexity 1 varieties. Langlois [27] gives a formula of canonical divisors for general normal \(G\)-varieties with spherical orbits, which in fact leads to ?? in our case. We remark that [27] uses a different approach and the formula there is given in terms of certain Galois covering of \(X\).

Recall Remark 28. For a one-parameter \(G\)-variety \(X\), its associated parabolic subgroup \(P\) equals to that of \(O\). In the following we fix a Levi decomposition \(P=L_PP_u\), where \(P_u\) is the unipotent radical and \(L_P\) is the Levi subgroup that contains the maximal torus \(B\cap B^-\). We have

3.2 The quasihomogeneous case↩︎

3.2.1 Reduction to the one-parameter case↩︎

In this section we deal with the quasihomogeneous case. Let \(X\) be a quasihomogeneous \(G\)-variety of complexity 1, which contains the homogeneous space \(G/H\) as an open \(G\)-orbit. To find an anti-canonical divisor of \(X\) we first reduce the problem to the one-parameter case, which has already been solved above. Put \[P':=\{g\in G|gD=D~\text{for any B-stable divisor D in general position}\}.\] Then \(P'\) is a parabolic subgroup satisfying \[\begin{align} \label{stab-parabolics} B\subset P\subset P'\subsetneq G, \end{align}\tag{20}\] where the last inequality is strict since \(X\) is quasihomogeneous. We will show that

Before the proof of Lemma 31, let us recall some facts on colours with non-zero jump from [19]. Note that via pull-back, there is a one-to-one correspondence between \(B\)-stable divisors in \(G/H\) and \(B\)-semiinvariant sections (up to multiplication by an invertible function) of homogeneous line bundles on \(G\). Here a homogeneous line bundle is \(L_\chi:=G\times^H{\rm k}_\chi\), where \({\rm k}_\chi\) is the 1-dimensional \(H\)-representation with \(\chi\in\mathfrak X(H)\). Recall that we have assumed \(G\) is of simply connected type, then by [35] the \(B\)-semiinvariant section further reduces to a function in \({\rm k}[G]^{(B\times H)}\), which is a defining equation of the pull-back of this \(B\)-stable divisor.

With this correspondence, as showed in [19], the rational quotient for the \(B\)-action \({\rm pr}_B\) is defined by a 1-dimensional system of colours as follows: There is a \(G\)-linearized line bundle \(L\) on \(G/H\) and a 2-dimensional subspace \(M\subset{\rm H}^0(G/H,L)^{(B)}_{\mu_0}\cong{\rm k}[G]^{(B\times H)}_{(\mu_0,\chi_0)}\) with \(\mu_0\in\mathfrak X(B)\) and \(\chi_0\in\mathfrak X(H)\) so that \({\rm pr}_B\) is given by \[{\rm pr}_B: G/H\dashrightarrow C:=\mathbb{P}^1(M^*)(\cong\mathbb{P}^1).\] Clearly \({\rm pr}_B\) is a morphism on \(X^o:=X\setminus{\rm Bs}(M)\), i.e. outside the base locus \({\rm Bs}(M)\) of \(M\). Moreover, it is further shown in [19] that \(M\) (considered as a subset of \({\rm k}[G]\)) consists of elements of the following types:

• Generic members in \(M\) are indecomposable, which correspond to colours at a general position. These colours are called the regular colours. Any regular colour \(D\) satisfies \(D={\rm pr}_B^*([x])\) (on \(X^o\)) for some \(x\in C\), and is mapped to \(v_D=q_{x}+\ell_D\in\mathscr E\). Moreover, \(\ell_D=O\) for all but finitely many regular colours. Such a regular colour will be denoted by \(X_{x}\). Conversely, for all but finitely many \(x\) in \(C\), the closure of \({\rm pr}_B^*([x])\) (considered as a divisor in \(X^o\)) in \(G/H\) is the regular colour \(X_x\).

• There are finitely many lines in \(M\) such that elements on these lines are decomposable. Components of their corresponding \(B\)-stable divisors are the remaining colours with non-zero jump. These colours are called the subregular colours, and there are finitely many subregular colours. A subregular colour \(D\) occurs in \({\rm pr}_B^*([x])\) (on \(X^o\)) for some \(x\in C\) with multiplicity \(h_D\), and is mapped to \(v_D=h_Dq_{x}+\ell_D\).

The remaining colours in \(G/H\) are precisely the central ones, i.e. those \(D\) with \(v_D\in\mathscr Q\).

Consider the variety \(X\) that contains \(G/H\). For any colour \(D\) of \(G/H\), its closure in \(X\) is a colour of \(X\), which will still be denoted by \(D\) for short. Any colour in \(X\) arises in this way. Obviously, a colour \(D\) of \(G/H\) is central if and only if its closure in \(X\) is. Thus for convenience, below we will call a colour \(D\) in a quasihomogeneous \(G\)-variety \(X\) regular (subregular, central, resp.) if \(D\cap(G/H)\) is regular (subregular, central, resp.).

Denote by \(P'=L_{P'}P'_u\) its Levi decomposition so that \(L_{P'}\) is a standard Levi subgroup. Take any \(x\in C\) so that the only member in \(\mathscr B(X)\) with non-zero jump that mapped into \(\mathscr Q_{x,+}\) is the regular colour \(X_x\), and consider \[X'=X\setminus \{\text{G-orbits contained in}~X_x\}.\] Then by [41], \(X_x\) is an effective \(B\)-stable Cartier divisor on \(X'\). By [41], \[X'\setminus X_x\cong P'_u\times^{L_{P'}}Z'\cong P'_u\times Z',\] where \(Z'\) is an \(L_{P'}\)-variety of complexity 1. Note that any colour \(D\) in \(X'\) of non-zero jump except \(X_x\) (in fact every divisor in \(\mathscr B(X)\setminus\{X_x\}\) with non-zero jump) descends to a \(L_{P'}\)-stable divisor in \(Z'\) since \(D\) is \(P'\)-stable. We conclude \(Z'\) is a one-parameter \(L_{P'}\)-variety. Also any central colour \(D\) in \(X\) descends to a central \(B\cap L_{P'}\)-stable prime divisor \(D'\) in \(Z'\). Set \[\begin{align} \label{bar-m-D-quasi} \bar m_D=\left\{\begin{aligned}&1,~\text{if}~D'~\text{is}~L_{P'}\text{-stable},\\ &\bar m_{D'},~\text{if}~D'~\text{is a colour of}~Z', \end{aligned}\right. \end{align}\tag{21}\] where \(\bar m_{D'}\) is given by ?? for \(D'\subset Z'\). We say that a central colour \(D\subset X\) is of type-a (a’, b, resp.) if \(D'\) is a central colour of \(Z'\) of type-a (a’, b, resp.).

3.2.2 The anti-canonical divisor↩︎

4 Equivariant test configurations↩︎

4.1 The structure of an equivariant test configuration↩︎

Let \((X,L)\) be a polarized, projective \(G\)-variety of complexity 1 with \({\rm k}(X)^B\cong{\rm k}(\mathbb{P}^1)\), and \(\mathfrak d\) an effective divisor of \(L\). In this section we classify \(G\)-equivariant normal test configurations of \((X,L)\) with irreducible central fibres. The case of \(T\)-varieties of complexity 1 has been solved by [21].

Let \((\mathcal{X},\mathcal{L})\overset{{\rm pr}}{\to}\mathbb{P}^1\) be a \(G\)-equivariant normal test configuration of \((X,L)\) with index \(r_0\). Then \((\mathcal{X},\mathcal{L})\) is a polarized projective \(G\times{\rm k}^\times\)-variety of complexity 1.⁹ It is clear that the hyperspace \(\bar{\mathscr E}\) of \(\mathcal{X}\) is \(\bar{\mathscr E}\cong\mathscr E\times\mathbb{Q}\), and the valuation cone \(\bar{\mathscr V}\cong\mathscr V\times\mathbb{Q}\). We have

4.2 Filtration induced by the test configuration↩︎

Suppose that \((\mathcal{X},\mathcal{L})\) is the \(G\)-equivariant normal test configuration of \((X,L)\) with associated data \((v_0,m)\) as in Proposition 34, and 25 gives a divisor of \(\mathcal{L}\) (with \(m_\infty=0\)). Then by [45], [46] \((\mathcal{X},\mathcal{L})\) induces a \(G\)-equivariant (\(\mathbb{Z}\)-)filtration \(\mathscr F_{(\mathcal{X},\mathcal{L})}\) on the Kodaira ring \(R(X,L)\) by setting \[\begin{align} \label{Fil-XL} \mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k:=\{\sigma\in R_k|t^{-\tau}\sigma\in R(\mathcal{X},\mathcal{L})\},~\forall k,\tau\in\mathbb{N}, \end{align}\tag{26}\] and \[\begin{align} \mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k:=\mathscr F_{(\mathcal{X},\mathcal{L})}^{\lceil\tau\rceil}R_k,~\forall k\in\mathbb{N}~\text{and}~\tau\in\mathbb{R}. \end{align}\] Then \(\mathscr F_{(\mathcal{X},\mathcal{L})}\) is left-continuous and decreasing. It is further multiplicative, pointwise left-bounded at \(\tau=0\) and linearly right-bounded by the finite generation of \(R(X,L)\) (cf. [46]). Since each \(\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k\) is a finite dimensional \(G\)-invariant linear space, to determine \(\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k\) it suffices to determine all \(B\)-semiinvariants in it.

Suppose that \(R(X,L)\) is generated by the first piece \(R_1\) over \(R_0\cong{\rm k}\). Fix the section \(s_0\in (R_1)^{(B)}_{\lambda_0}\) so that the divisor of \(s_0\) is \(\mathfrak d\) given by 7 . Then any \(s_\lambda\in{\rm H}^0(X,L^k)^{(B)}_\lambda\) can be written as \[\begin{align} \label{sigma-lambda-decomp} s_\lambda=f_0e_{\lambda-k\lambda_0}s_0^k \end{align}\tag{27}\] for some \(f_0\in{\rm k}(X)^B\). By Lemma 36 we have

It is known that the Kodaira ring of the central fibre \(\mathcal{X}_0\) with respect to \(\mathcal{L}_0:=\mathcal{L}|_{\mathcal{X}_0}\) is the graded algebra of the filtration \(\mathscr F_{(\mathcal{X},\mathcal{L})}\), \[\begin{align} \label{Gr40F41} {\rm Gr}(\mathscr F_{(\mathcal{X},\mathcal{L})})=\bigoplus_{k=0}^{+\infty}\bigoplus_{\tau=0}^{+\infty}\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k/\mathscr F_{(\mathcal{X},\mathcal{L})}^{>\tau}R_k, \end{align}\tag{28}\] and the induced \(\rm k^\times\)-action acts on the \((\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k/\mathscr F_{(\mathcal{X},\mathcal{L})}^{>\tau}R_k)\)-piece with weight \(\tau\). In the following we also say that a non-zero element \(s\in\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k\setminus\mathscr F_{(\mathcal{X},\mathcal{L})}^{>\tau}R_k\) has weight \(\tau\) (induced by \((\mathcal{X},\mathcal{L})\)). Note that each \((\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k/\mathscr F_{(\mathcal{X},\mathcal{L})}^{>\tau}R_k)\)-piece in 28 is also a \({\rm k}^\times\)-module of weight \(\tau\). Thus \[(\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k/\mathscr F_{(\mathcal{X},\mathcal{L})}^{>\tau}R_k)^{(B)}_\lambda\subset{\rm Gr}(\mathscr F_{(\mathcal{X},\mathcal{L})})\] is a \(B\times{\rm k}^\times\)-module of weight \((\lambda,\tau)\) in the degree \(k\)-piece of 28 .

For the study of K-stability, its suffices to consider only special test configurations (cf. [23]). In this case, \(v_0\in\mathscr E\) is an integral element and \(m=-1\). Using Proposition 37 we get

4.2.1 Product test configurations↩︎

Combining with Proposition 38, we get

4.2.2 Some combinatorial data↩︎

We hope to compute the polytope \(\Delta_\mathscr Z(\mathcal{L})\) and related combinatorial data of \((\mathcal{X},\mathcal{L})\) as done for \((X,L)\) before. We mainly interest in the case when \(m=-1\), that is, \(\mathcal{X}\) has integral central fibre.

Keep the notations above. By direct computation we have \[\begin{align} \label{A-of-D} A_{x}(\mathfrak D,\lambda,t)=\left\{\begin{aligned}&\min\{A_{x_0}(\mathfrak d,\lambda),\frac{-t+m_0+\ell_0(\lambda)}{h_0}\},~\text{when}~x=x_0,\\ &A_{x}(\mathfrak d,\lambda),~\text{when}~x\not=x_0,\end{aligned}\right.~\text{for}~\lambda\in\Gamma_\mathbb{R}, \end{align}\tag{33}\] and \[\begin{align} \label{A-sum-of-D} A(\mathfrak D,\lambda,t)=A_{x_0}(\mathfrak D,\lambda,t)+\sum_{x\not=x_0}A_x(\mathfrak d,\lambda),~\lambda\in\Delta_\mathscr Z(\mathfrak d). \end{align}\tag{34}\] Here when \(h_0=0\), we formally take \(\frac{c}{h_0}=\pm\infty\) or \(0\) according to the sign of \(c\). Under this convention \(A_{x_0}(\mathfrak D,\lambda,t)\) reduces to \[A_{x_0}(\mathfrak D,\lambda,t)=A_{x_0}(\mathfrak d,\lambda)\chi_{\{-t+m_0+\ell_0(\lambda)\geq0\}},~\lambda\in\Delta_\mathscr Z(\mathfrak d),\] where by \(\chi_S\) we denote the characteristic function of a set \(S\), and \[A(\mathfrak D,\lambda,t)=A(\mathfrak d,\lambda)\chi_{\{-t+m_0+\ell_0(\lambda)\geq0\}},~\lambda\in\Delta_\mathscr Z(\mathfrak d).\] Thus, we get \[\begin{align} \label{tilde-DZ40d41} {\Delta}_\mathscr Z(\mathcal{L})=&\{(\lambda,t)\in\Delta_\mathscr Z(L)\times\mathbb{R}|0\leq t\leq \tau^0(\lambda-\lambda_0)\}\notag\\=&(\Delta_\mathscr Z(L)\times\mathbb{R})\cap\{A(\mathfrak D,\lambda-\lambda_0,\tau)\geq0\}, \end{align}\tag{35}\] and \({\Delta}_\mathscr Z(\mathfrak D)={\Delta}_\mathscr Z(\mathcal{L})-(\lambda_0,0)\), where \[\begin{align} \label{tau-0} \tau^0(\lambda):=m_0+\ell_0(\lambda)+h_0(A(\mathfrak d,\lambda)-A_{x_0}(\mathfrak d,\lambda)),~\lambda\in\Gamma_\mathbb{R}. \end{align}\tag{36}\]

Also, for each \(x\in C\), \[\begin{align} {\Delta}_x(\mathfrak D)=\{(\lambda,h, t)\in\Gamma_\mathbb{R}\times\mathbb{R}_+\times\mathbb{R}|(\lambda,t)\in {\Delta}_\mathscr Z(\mathfrak D),~h\geq-A(\mathfrak D,\lambda,t)\}, \end{align}\] and the coloured fan \(\mathfrak F_\mathcal{X}\) of \(\mathcal{X}\) at each \(x\in C\) consists of inner normal cones of \({\Delta}_x(\mathfrak D)\) whose relative interior intersects \(\bar{\mathscr V}\).

4.3 The classification↩︎

In this section we consider the inverse direction of Proposition 34. Given any integral \(v_0\) in \(\mathscr V_{x_0}\), we will construct a \(G\)-equivariant normal test configuration of \((X,L)\) whose central fibre is integral.

Suppose that \(v_0=\ell_0+h_0q_{x_0}\in\mathscr V_{x_0}\) (when \(h_0=0\), we can choose \(x_0\) to be any point in \(C\)). We can choose a sufficiently large integer \(m_0\) so that the function \(\tau^0(\cdot)\) defined by 36 is positive on \(\Delta_\mathscr Z(\mathfrak d)=\Delta_\mathscr Z(L)-\lambda_0\). As in 33 , set \[\begin{align} \label{A-of-v0} \tilde{A}_{x}(\lambda,t):=\left\{\begin{aligned}&\min\{A_{x_0}(\mathfrak d,\lambda),\frac{-t+m_0+\ell_0(\lambda)}{h_0}\},~\text{when}~x=x_0,\\ &A_{x}(\mathfrak d,\lambda),~\text{when}~x\not=x_0,\end{aligned}\right.~\text{for}~\lambda\in\Gamma_\mathbb{R}, \end{align}\tag{37}\] and \[\tilde{A}(\lambda,t):=\tilde{A}_{x_0}(\lambda,t)+\sum_{x\not=x_0}A_x(\mathfrak d,\lambda),~\lambda\in\Delta_\mathscr Z(\mathfrak d).\] Then \(\tilde{A}(\lambda,t)\geq0\) for any \((\lambda,t)\) lies in \[\begin{align} \tilde{\Delta}_\mathscr Z:=\{(\lambda,t)\in\Delta_\mathscr Z(\mathfrak d)\times\mathbb{R}|0\leq t\leq \tau^0(\lambda)\}. \end{align}\] Also, for each \(x\in C\), set \[\begin{align} \tilde{\Delta}_x=\{(\lambda+hq_x, t)\in\Gamma_\mathbb{R}\times\mathbb{R}_+\times\mathbb{R}|(\lambda,t)\in\tilde{\Delta}_\mathscr Z,~h\geq-\tilde{A}_x(\lambda,t)\}, \end{align}\] We call \(\tilde{\Delta}_x\cap\{h=-\tilde{A}_x(\lambda,t)\}\) the bottom of \(\tilde{\Delta}_x\).

Now we construct a coloured fan \(\bar{\mathfrak F}\). Since the total space \(\mathcal{X}\) is assumed to be complete, it suffices to give its maximal coloured cones and coloured hypercones of type . We say a coloured hypercone \(\mathscr C\) is maximal if at least one \(\mathscr C_x=\mathscr C\cap\mathscr Q_{x,+}\) is of maximal dimension. Consider the inner normal cone of every vertex of \(\tilde{\Delta}_x\) for all \(x\in C\). If the relative interior of such a cone \(\mathscr C\) intersects \(\bar{\mathscr V}\) and all its generators lies in \[\tilde{\mathscr B}:=\{(v_D,0)|D\in\mathscr B(X)\}\cup\{(v_0,-1)\},\] then \(\mathscr C\) will be a coloured cone in \(\bar{\mathfrak F}\). It remains to select colours. Suppose that \(\mathscr C\) is the inner normal cone of \(\tilde{\Delta}_x\) at a vertex \(p_0\). Then we choose \[\mathscr R=\{\bar D|D\in\mathscr B(X)\setminus\mathscr B(X)^G,~m_D=-p_0(v_D)\},\] and put the coloured cone \((\mathscr C,\mathscr R)\) in \(\bar{\mathfrak F}\).

It remains to construct hypercones of type . Consider a remaining normal cone as above which has generators out of \(\tilde{\mathscr B}\). If \(\mathscr C\) is the inner normal cone of \(\tilde{\Delta}_x\) at a vertex \(p_*\), then \(p_*\) projects to some \(p_*'\), where \(p_*'=(\lambda_*,t_*)\) is a boundary point of \(\tilde{\Delta}_\mathscr Z\) such \(\tilde{A}(p_*')=0\). For any \(x\in C\), set \[\mathscr S(x,\lambda_*):=\{D\in\mathscr B(X)|v_D\in\mathscr Q_{x,+},~\tilde{A}_x(\lambda_*,t)=\frac{m_D+\ell_D(\lambda_*)}{h_D}\}.\] Define a coloured hypercone \((\mathscr C,\mathscr R)\) by setting \[\begin{align} \mathscr C_x=&\text{the inner normal cone of \tilde{\Delta}_x at the point on its bottom that projects to}~p_*'\in\tilde{\Delta}_\mathscr Z,\\ \mathscr W_x=&\{\bar D|D\in\mathscr B(X)^G,h_D=0,m_D=-\lambda_*(v_D)\}\cup\left\{\begin{aligned}&\mathscr B(X)^G\cap\mathscr S(x,\lambda_*),~\text{if}~x\not=x_0,\\&(\mathscr B(X)^G\cap\mathscr S(x_0,\lambda_*))\cup\{\mathcal{X}_0\},~\text{if}~x=x_0,\end{aligned}\right.\\ \mathscr R_x=&\{\bar D|D\in\mathscr B(X)\setminus\mathscr B(X)^G,h_D=0,m_D=-\lambda_*(v_D)\}\cup\left((\mathscr B(X)\setminus\mathscr B(X)^G)\cap\mathscr S(x,\lambda_*)\right). \end{align}\] Then \((\mathscr C,\mathscr R)=\{(\mathscr C_x,\mathscr R_x)|x\in C\}\) is the coloured hypercone defined by the data \((\mathscr W,\mathscr R):=(\cup_{x\in C}\mathscr W_x,\cup_{x\in C}\mathscr R_x)\). Such a cone will also be taken into account in \(\bar{\mathfrak F}\) if its relative interior intersects \(\bar{\mathscr V}\).

Together with the set \[\{(\mathscr C\times\mathbb{Q}_+,\mathscr R\times\{0\})|(\mathscr C,\mathscr R)\in\mathfrak F_X\},\] and all their faces, we get a coloured fan \(\bar{\mathfrak F}\) that covers \(\bar{\mathscr V}\). Let \(\mathcal{X}\) be the \(G\times{\rm k}^\times\)-variety defined by \(\bar{\mathfrak F}\). From our contraction, we can directly check that the divisor \(\mathfrak D\) given by 25 is ample on \(\mathcal{X}\) by Theorem 15. On the other hand, by removing the divisor \(\mathcal{X}_0\) that corresponds to \((v_0,-1)\), there is a \({\rm k}^\times\)-equivariant morphism \({\rm pr}:\mathcal{X}\setminus\{\mathcal{X}_0\}(\cong X\times{\rm k})\to{\rm k}(\cong\mathbb{P}^1\setminus\{0\})\). We then show \({\rm pr}\) extends to a \({\rm k}^\times\)-equivariant projection \({\rm pr}:\mathcal{X}\to\mathbb{P}^1\). It suffices to look at \({\rm pr}\) near \(\mathcal{X}_0\). Recall that the hyperspace \(\tilde{\mathscr E}\) of the \(G\times{\rm k}^\times\)-variety \(\mathcal{X}\) is isomorphic to \(\mathscr E\times\mathbb{Q}\), where \(\mathscr E\) is the hyperspace of \(X\) and \(\mathbb{Q}\) stands for the \({\rm k}^\times\)-factor. Under the projection \({\rm pr}:\mathscr E\times\mathbb{Q}\to\mathbb{Q}\) to the second factor, the coloured cones and hypercones of type in \(\bar{\mathfrak F}\) are mapped into either \(\mathbb{Q}_{\geq0}\) or \(\mathbb{Q}_{\leq0}\). In particular, if \(\mathcal{Y}\subset\mathcal{X}_0\) is a \(G\times{\rm k}^\times\)-orbit, then its coloured cone/hypercone of type in \(\bar{\mathfrak F}\) maps surjectively to \(\mathbb{Q}_{\leq0}\). Also, the colours in \(\mathcal{X}\) are precisely \(\{\overline{D\times{\rm k}^\times}|D\in\mathscr D^B\}\), which all map dominantly to \({\rm k}^\times\). Following the argument of [11] we get \(\mathscr O_{\mathcal{X},\mathcal{Y}}\) dominates \(\mathscr O_{\mathbb{P}^1,0}\). Hence \({\rm pr}\) extends to a regular map on the whole \(\mathcal{X}\) (cf. [12]), and \(\mathcal{X}\) is indeed a test configuration associated to \((v_0,-1)\) in the sense of Proposition 34.

Combining with Proposition 34, we get

In the following, we often say a test configuration \((\mathcal{X},\mathcal{L})\) is associated to \(v_0\) if it corresponds to some \((v_0,m_0)\in\mathscr T\).

4.4 Twist of a test configuration↩︎

Given a test configuration \((\mathcal{X},\mathcal{L})\) as above. It is proved by [31] that \({\rm J}^{\rm NA}(\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\) is a rational, convex piecewise linear and proper function of \(\ell'\in{\rm Lie}({\mathbf{T}})\). In particular, it is continuous. With the help of the continuity, for our latter use it suffices to study the twist \((\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\) of \((\mathcal{X},\mathcal{L})\) when \(\ell'\) is rational.

Let \((\mathcal{X},\mathcal{L})\) be a \(G\)-equivariant normal test configuration associated to \(v_0\) as above, and \(\ell'\) a rational element in the linear part \(\mathscr A(\cong{\rm Lie}({\mathbf{T}}))\) of \(\mathscr V\) so that \(q\ell'\) in primitive for some \(q\in\mathbb{N}_+\). Recall that the uncompactified total space \((\mathcal{X}_{\ell'}\setminus\mathcal{X}_{\ell'\,\infty},\mathcal{L}_{\ell'}|_{\mathcal{X}\setminus\mathcal{X}_{\ell'\,\infty}})\) is isomorphic to \((\mathcal{X}\setminus\mathcal{X}_{\infty},\mathcal{L}|_{\mathcal{X}\setminus\mathcal{X}_{\infty}})\), but the grading on \(R(X,L)\) is shifted by \(\ell'\). More precisely, if the \(s\) in the \(G\)-span of \({\rm H}^0(X,L^k)^{(B)}_\lambda\subset R(X,L)\) and has grading \(\tau(s)\) induced by \((\mathcal{X},\mathcal{L})\), then the grading of \(s\) induced by \((\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\) is \[\begin{align} \label{grading-twist} \tau'(s)=\tau(s)+\ell'(\lambda). \end{align}\tag{38}\]

We first deal with integral \(\ell'\). In this case \(q=1\), \((\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\) is a test configuration and we shall determine the compactified total space of \((\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\). By the above discussion, the coloured cones and hypercones of type in \(\mathfrak F_{\mathcal{X}}\) that lies in \(\mathscr E\times\mathbb{Q}_{\leq0}\), which precisely give the coloured fan of \((\mathcal{X}_{\ell'}\setminus\mathcal{X}_{\ell'\,\infty},\mathcal{L}_{\ell'}|_{\mathcal{X}\setminus\mathcal{X}_{\ell'\,\infty}})\cong(\mathcal{X}\setminus\mathcal{X}_{\infty},\mathcal{L}|_{\mathcal{X}\setminus\mathcal{X}_{\infty}})\), keep the same in \(\mathfrak F_{\mathcal{X}_{\ell'}}\). On the other hand, the \({\rm k}^\times\)-action of \((\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\) corresponds to \((\ell',1)\in{\rm Lie}(\mathbf{T}\times{\rm k}^\times)\cong\mathscr A\times\mathbb{Q}\). Thus the fibre \(\mathcal{X}_{\ell'\,\infty}\) at \(\infty\) in \(\mathbb{P}^1\) corresponds to the ray \(\mathbb{Q}_{\geq0}(\ell',q)\in\bar{\mathscr E}\), and the other coloured cones and hypercones of type in \(\mathfrak F_{\mathcal{X}_\ell}\) are \[\begin{align} \label{fan-trivial-part} \{{\rm Cone}((\mathscr C,\mathscr R),(\ell',1))|~(\mathscr C,\mathscr R)\in\mathfrak F_{\mathcal{X}}~\text{is a coloured cone or hypercone of type \uppercase{\romannumeral2} in}~\mathfrak F_{\mathcal{X}}\}. \end{align}\tag{39}\] Taking an \({\rm SL}_{{\rm rk}(\Gamma)+1}\)-transformation \[(v,u)\to(v-u\ell',u),~\forall (v,u)\in\mathscr E\times \mathbb{Q},\] which in particular transforms \((\ell',1)\) to \((0,1)\), we see that \((\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\) is isomorphic to the test configuration associated to \(v_0+\ell'\).

Then we turn to the general rational case. Denote by \((\mathcal{X}^{(q)},\mathcal{L}^{(q)})\) the base change (and then a normalization) \(t\to t^q\), \(t\in{\rm k}^\times\) of \((\mathcal{X},\mathcal{L})\). In this case, the base change \((\mathcal{X}_{\ell'}^{(q)},\mathcal{L}_{\ell'}^{(q)})\) of the twist \((\mathcal{X}_{\ell'},\mathcal{L}_{\ell'})\) is a test configuration of \((X,L)\), which is defined as the twist of \((\mathcal{X}^{(q)},\mathcal{L}^{(q)})\) by the integral element \(q\ell'\). Let \(\mathfrak F_{\mathcal{X}}\) be the coloured fan of \(\mathcal{L}\). The coloured fan of \(\mathcal{X}^{(q)}\) consists of the rescalling coloured cones and hypercones of type in \(\mathfrak F_{\mathcal{X}}\). The rescalling maps each ray \(\mathbb{Q}_{\geq0}(v,\pm1)\) to \(\mathbb{Q}_{\geq0}(qv,\pm1)\). In particular, \(\mathbb{Q}_{\geq0}(v_0,-1)\) is mapped to \(\mathbb{Q}_{\geq0}(qv_0,-1)\) with primitive generator \((qv_0,-1)\), and \(\mathbb{Q}_{\geq0}(\ell',1)\) is mapped to \(\mathbb{Q}_{\geq0}(q\ell',1)\) with primitive generator \((q\ell',1)\). From the previous case, we see that \((\mathcal{X}_{\ell'}^{(q)},\mathcal{L}_{\ell'}^{(q)})\) is associated to \(q(v_0+\ell')\).

There is an alternative approach to see the relation. Taking a base change \(t\to t^p\), we get the corresponding grading of \(s\) induced by \((\mathcal{X}^{(q)}_{\ell'},\mathcal{L}^{(q)}_{\ell'})\) is \(q\tau(s)+q\ell'(\lambda)\). We then concludes that \((\mathcal{X}^{(q)}_{\ell'},\mathcal{L}^{(q)}_{\ell'})\) coincides with the twist of \((\mathcal{X}^{(q)},\mathcal{L}^{(q)})\) by the integral element \(q\ell'\). By the previous case we see that

5 The Futaki invariant and K-stability↩︎

In this section we compute the Futaki invariant of \(G\)-equivariant normal test configurations with \(m=-1\), and derive a K-stability criterion of \(X\).

5.1 Preparations↩︎

Let \((X,L)\) be a polarized \(G\)-variety of complexity 1 with \({\rm k}(X)^B\cong{\rm k}(\mathbb{P}^1)\), and \((\mathcal{X},\mathcal{L})\) a test configuration of \((X,L)\) associated to some integral \(v_0=h_0q_{x_0}+\ell_0\in\mathscr V_{x_0}(\subset\mathscr V)\). Recall that we have already calculated \(\dim{\rm H}^0(X,L^k)\) in Lemma 17. In this section we calculate the total weight \(w_k(\mathcal{X},\mathcal{L})\) of \((\mathcal{X},\mathcal{L})\) for general cases. Both expressions will be simplified in the next section when \(X\) is \(\mathbb{Q}\)-Fano and \(L\) equals to a multiple of \(K_X^{-1}\).

We shall first introduce some notations here. By Proposition 37, when \(m=-1\), we directly conclude that \(\mathscr F_{(\mathcal{X},\mathcal{L})}^\tau R_k=0\) whenever \[\begin{align} \tau\geq[km_0+\ell_0(\lambda-k\lambda_0)+h_0k(A(\mathfrak d,\frac{\lambda}{k}-\lambda_0)-A_{x_0}(\mathfrak d,\frac{\lambda}{k}-\lambda_0))]+1=[k\tau^0(\frac{\lambda}{k}-\lambda_0)]+1, \end{align}\] where the function \(\tau^0(\cdot)\) is defined by 36 . Also, recall the functions \(A_x(\mathfrak D,\cdot,\cdot)\) defined by 33 34 and the polytope \(\Delta_\mathscr Z(\mathcal{L})\) defined by 35 . For our later use, we need to figure out domains of linearity of all \(\{A_x(\mathfrak D,\cdot,\cdot)\}_{x\in C}\). When \(h_0=0\), they are precisely those \(\{\Omega_a\}\) defined above (see ?? ). It remains to consider the cases when \(h_0\not=0\). Note that when \(h_0\not=0\) and \(m_0\gg1\), \[\begin{align} \label{tilde-A} A(\mathfrak D,\lambda,\tau)=\left\{\begin{aligned}&A(\mathfrak d,\lambda),~&&\text{when}~0\leq\tau\leq\tilde{\tau}^0(\lambda),\\ &\sum_{x\not=x_0}A_x(\mathfrak d,\lambda)+\left(\frac{-\tau+m_0+\ell_0(\lambda)}{h_0}\right),~&&\text{when}~\tilde{\tau}^0(\lambda)\leq\tau\leq\tau^0(\lambda),\end{aligned} \right. \end{align}\tag{40}\] where \[\tilde{\tau}^0(\lambda):=m_0+\ell_0(\lambda)-h_0A_{x_0}(\mathfrak d,\lambda),~\lambda\in\Delta_{\mathscr Z}(\mathfrak d).\] We divide \({\Delta}_\mathscr Z(\mathcal{L})\) into two parts \[\begin{align} \label{polytope-D-Zo40mathcal-L41} {\Delta}_\mathscr Z^o(\mathcal{L}):={\Delta}_\mathscr Z(\mathcal{L})\cap\{\tilde{\tau}(\lambda-\lambda_0)\leq\tau\leq\tau(\lambda-\lambda_0)\}, \end{align}\tag{41}\] and \({\Delta}_\mathscr Z'(\mathcal{L}):={\Delta}_\mathscr Z(\mathcal{L})\setminus{\Delta}_\mathscr Z^o(\mathcal{L})\). Then by concavity of \(A_{x_0}(\mathfrak d,\cdot)\), \({\Delta}_\mathscr Z^o(\mathcal{L})\) is a convex polytope. The common domains of linearity of all \(\{A_x(\mathfrak D,\cdot,\cdot)\}_{x\in C}\) precisely consist of \[\begin{align} \label{linear-domains} \tilde{\Omega}_a^o:=(\Omega_a\times\mathbb{R})\cap{\Delta}_\mathscr Z^o(\mathcal{L})~\text{and}~\tilde{\Omega}_a':=(\Omega_a\times\mathbb{R})\cap{\Delta}_\mathscr Z'(\mathcal{L})~\text{for}~a=1,...,N. \end{align}\tag{42}\] Sometimes we write those domains in total as \(\{\tilde{\Omega}_{\tilde{a}}\}_{\tilde{a}}^{\tilde{N}}\) and denote \[\tilde{D}_{\tilde{a}}(x_0)=(h_0,\ell_0),~\tilde{D}_{\tilde{a}}(x_0)=m_0~\text{if}~\tilde{\Omega}_{\tilde{a}}=\tilde{\Omega}_a^o~\text{for some}~a,\] when there is no confusions.

With the conventions introduced above, we have

5.2 The Futaki invariant of \(\mathbb{Q}\)-Fano \(G\)-varieties↩︎

In this section we will express the Futaki invariant in terms of purely combinatorial data. For simplicity we consider the case when \(X\) is Gorenstein and choose the divisor \(\mathfrak d\) in [anti-can-div-thm] with weight \(\lambda_0=\kappa_P\) for \(L=K_X^{-1}\). In general cases, we may choose \(L=K_X^{-m}\) and \(m\mathfrak d\) with weight \(\lambda_0=m\kappa_P\) for a sufficiently divisible \(m\in\mathbb{N}_+\).

In the following we mainly focus on the case \(h_0\not=0\) since the case \(h_0=0\) is much simpler. In this case we first do some reductions on \(\partial\Delta_\mathscr Z(\mathcal{L})\). The boundary \(\partial\Delta_\mathscr Z(\mathcal{L})\) contains three parts (see Fig-1):

• The base \(\Delta_\mathscr Z(K_X^{-1})\times\{0\}\), on which \(A(\mathfrak D,\lambda-\kappa_P,\tau)=A(\mathfrak d,\lambda-\kappa_P)\) and \(d\tilde{\sigma}=d\lambda\);

• The walls \(\tilde{F}:=(F\times\mathbb{R})\cap\Delta_\mathscr Z(\mathcal{L})\), where \(F\) is a facet of \(\Delta_\mathscr Z(K_X^{-1})\). Note that if \(F\subset\{A(\mathfrak d,\lambda-\kappa_P)=0\}\), then \(\tilde{\tau}^0=\tau^0\) on \(\tilde{F}\), and \(\tilde{F}\) intersects \(\Delta_\mathscr Z^o(\mathcal{L})\) on a face of codimension at least 2;

• The graph of \(t=\tau^0(\lambda-\kappa_P)\) over \(\Delta_\mathscr Z(K_X^{-1})\), on which \(A(\mathfrak D,\lambda-\kappa_P,\tau)=0\).

We need a few more discussions on the \(\tilde{F}\)’s with \(F\subset\{A(\mathfrak d,\lambda-\kappa_P)\not=0\}\). Such an \(F\) is defined by \[\begin{align} \label{eq-F} v_D(\lambda)+m_D-v_D(\kappa_P)=0. \end{align}\tag{44}\] We have the following cases:

• \(D\) is a central \(G\)-stable divisor, or a colour of type-a, or a central colour in the quasihomogeneous case that descends to a \(B\cap L'\)-stable divisor in Section 3.2. In all these three cases \(m_D=1\) and \(v_D\) is primitive. The unit outer normal vector of \(F\) is \(\nu=-v_D/|v_D|\), and the induced lattice measure on \(F\) is \[\begin{align} d\sigma=\frac{\langle\lambda,\nu\rangle}{1-v_D(\kappa_P)}d\sigma_0=\langle\lambda-\kappa_P,\nu\rangle d\sigma_0, \end{align}\] where \(d\sigma_0\) is the standard induced Lebesgue measure. Consequently, the induced lattice measure on \(\tilde{F}\) is \[\begin{align} d\tilde{\sigma}=\langle\lambda-\kappa_P,\nu\rangle d\sigma_0\wedge dt; \end{align}\]

• \(D\) is a colour of type-a’ or b. By Remarks 28 and 33, 44 reduces to \(v_D(\lambda)=0\) in both cases. As for a colour of type-a’ or b, \(v_D\) is proportional to \(\alpha^\vee|_{\mathscr Q}\), we get \(\pi(\cdot)|_F\equiv0\) in both cases.

From the above discussions we get

Also we introduce a family of convex polytopes that describes the K-stability of \(X\). The polytope \(\Delta_\mathscr Z(K_X^{-1})\), although independent with the choice of divisor \(\mathfrak d\), itself hardly represents full information of \({\rm H}^0(X,K_X^{-1})^{(B)}_\lambda\) for a fixed \(\lambda\). In [21], Ilten-Süß introduced a family of polytopes for Fano \(T\)-varieties of complexity 1 that fully encodes the information of K-stability. In the following we define its counterpart for a \(\mathbb{Q}\)-Fano \(G\)-variety of complexity 1.

Now we prove the main result in this section, which holds for both cases \(h_0\not=0\) and \(h_0=0\):

Using Theorem 48, Remark 35 and 4 , one directly gets

5.3 The non-Archimedean J-functional↩︎

Let \(v_0=\ell_0+h_0q_{x_0}\in\mathscr Q_{x_0,+}\) for some \(x_0\in C\), and \(\mathfrak d=\mathfrak d_\mathfrak a\) be an anti-canonical divisor given by ?? or ?? . We have

5.4 K-stability criterion↩︎

Recall Definition 6. Theorem 1 is then a direct consequence of Theorem 48.

There is a variant of Theorem 1 for horospherical \(G\)-varieties of complexity 1, which is much more simplified. The following is a generalization of the K-polystability criterion [21] of Fano \(T\)-varieties of complexity 1 to horospherical \(G\)-varieties of complexity 1.

6 Examples↩︎

In this section we apply Theorem 1 to concrete examples.

6.1 The Mukai-umemura threefold↩︎

The Mukai-Umemura threefold was constructed in [17] as a smooth Fano \({\rm SL}_2\)-variety. The existence of Kähler-Einstein metrics on this threefold (and equivalently, uniform K-stability) has been proved in [48] by computing the \(\alpha\)-invariant. In the following we view the Mukai-Umemura threefold as an equivariant completion of \({\rm SL}_2/H\), where \(H\) is the binary icosahedral group, and present concretely how the theorems established in the previous sections apply on this well-known example. We also remark that the equivariant K-polystability of smooth, Fano quasihomogeneous \({\rm SL}_2\)-varieties of complexity 1 has been studied in [22] via a different approach.

Let us recall the construction in [17]: Denote by \(s^{k}\) the irreducible \((G:=){\rm SL}_2\)-representation which consists of homogeneous symmetric polynomials of two variables of degree \(k\in\mathbb{N}\). We identify \(\mathbb{P}^1\) with the Riemann sphere \(S^2\) so that \(G\) acts on it, and \({\rm SU}_2\subset{\rm SL}_2\) acts on \(S^2\) via the epimorphism \({\rm SU}_2\to{\rm SO}_3\). Also we identify a polynomial in \(s^{12}\) with its roots located on \(S^2\). There is an element \(f_v\in s^{12}\) so that its roots are the 12 vertices of an icosahedron inscribed in \(S^2\), the stabilizer subgroup of \(f_v\) in \(G\) is exactly \(H\). Then the closure \(X\) of the \(G\)-orbit of \([f_v]\in\mathbb{P}(s^{12})\) is the Mukai-Umemura threefold, which is a smooth Fano threefold (cf. [17], [48]).

As shown in [17], there are three \(G\)-orbits in \(X\). First, we have

• The unique open orbit \(\mathcal{O}:=G\cdot[f_v]\cong G/H\).

The complement of \(\mathcal{O}\) is a \(G\)-divisor \(D_\infty\) which is the closure of

• The orbit \(\mathcal{O}_1:=G\cdot[f_1]\), where \(f_1\in s^{12}\) is a polynomial which has a root of multiplicity 11 and a simple root.

The divisor \(D_\infty\) contains \(\mathcal{O}_1\) as its unique open \(G\)-orbit and the complement \(\Delta_\infty\) of \(\mathcal{O}_1\) in \(D_\infty\) is

• The unique closed \(G\)-orbit in \(X\), which consists of \(G\cdot[f_2]\), where \(f_2\in s^{12}\) is a polynomial which has a root of multiplicity

The coloured data of \(G/H\) are given in [19]. Let us briefly recall the construction in [19] here. Recall the action of \(G\) on \(S^2\) and an icosahedron \(\mathbf{I}\) inscribed in this \(S^2\). Then \(H\) is the pre-image of the symmetry group of \(\mathbf{I}\) in \(G\). The vertices, edge midpoints, and face centers of \(\mathbf{I}\) give those points on \(S^2\) so that \(H\) acts with non-trivial subgroup of stabilizers. In fact, they form three \(H\)-orbits, respectively. The corresponding points are denoted by \(x_v\), \(x_e\) and \(x_f\), respectively. Also they define three polynomials \(f_v\in s^{12}\), \(f_e\in s^{30}\) and \(f_f\in s^{20}\).

Denote by \(B\) the subgroup of upper-triangular matrixes in \(G\) and fix it as the positive Borel subgroup. Also let \(\omega\) be the corresponding fundamental weight of \(G\). It is showed in [19] that the subregular semiinvariants are \(f_v\), \(f_e\) and \(f_f\) with corresponding \(B\times H\)-biweights \((12\omega; 1)\), \((30\omega; 1)\) and \((20\omega; 1)\). The regular semiinvariants fill the two-dimensional subspace \({\rm k}[G]^{(B\times H)}_{(60\omega;1)}\) spanned by \(f_v^5\), \(f_e^2\) and \(f_f^3\), where \[f_v^5+f_e^2+f_f^3=0.\] Functions in \({\rm k}[G]^{(B\times H)}_{(60\omega;1)}\) give all the colours in \(X\), and there is no colour of central type (cf. [19]). Also \(\Gamma=\mathbb{Z}\cdot2\omega\) with generator \[e_{2\omega}:=\frac{f_vf_f}{f_e}.\] In particular, \({\rm rk}(X)=1\).

The valuation cone \(\mathscr V\) is given as follows: an element \(hq_x+a\frac{1}{2}\omega^*\in\mathscr Q_{x,+}\)¹¹ is contained in \(\mathscr V_x\) if and only if \[\begin{align} (a,h)~\text{satisfies}~\left\{ \begin{aligned} &-a\geq h\geq0,~\text{if}~x\not=x_v,x_f,\\ &h,-a\geq0,~\text{if}~x=x_v,x_f. \end{aligned} \right. \end{align}\]

Colours are computed in [19], and the coloured data of \(X\) by [22]. We list the data of \(B\)-stable divisors in Table-1.1.

Table-1.1: \(B\)-stable divisors in \(X\)

Clearly the associated parabolic subgroup of \(X\) is \(B\) and \(\kappa_P=2\omega\).

The coloured fan of \(X\) consists of one maximal hypercone \(\mathscr C\) of type . More precisely, it is a hypercone of type A\(_1\) (in the sense of [9]) with data \[\mathscr W=\{q_{x_v}\},~\mathscr R=\{X_x|x\not=x_v\}.\] This hypercone is shown below in Fig-2, where its intersection with \(\mathscr Q_{x_+}\) is marked as the dark areas, and the valuation cones as hatched. Colors are represented by circles and \(D_\infty\) by a dark dot.

Now we compute anti-canonical divisors. By Theorem ?? , for each anti-canonical divisor \(\mathfrak a=\sum_{x\in \mathbb{P}^1}a_x\cdot x\) on \(\mathbb{P}^1\), there is a anti-canonical divisor \[\begin{align} \label{anti-can-div-mukai} \mathfrak d_\mathfrak a=\sum_{x\not= x_v,x_e,x_f}a_xX_{x}+(-4+5a_{x_v})X_{x_v}+(-1+2a_{x_e})X_{x_e}+(-2+3a_{x_f})X_{x_f}+a_{x_v}D_\infty \end{align}\tag{67}\] of \(X\), which is the divisor of a rational section of \(K_X^{-1}\) of weight \(2\omega\).

However, there is no effective divisor in 67 . In fact, it is showed in [48] that \[{\rm H}^0(X,K_X^{-1})\cong s^0\oplus s^{12},\] which implies \[{\rm H}^0(X,K_X^{-1})^{(B)}_{2\omega}=0.\] Also, [48] gives an effective divisor \[\mathfrak d_0=D_\infty\] on \(X\), which is the divisor of a \(G\)-invariant section in the component \(s^0\subset K_X^{-1}\). The divisor \(\mathfrak d_0\) can also be expressed as \[\mathfrak d_0=\mathfrak d_\mathfrak a-{\rm div}(e_{2\omega})\] with \[\mathfrak a=[x_v]+[x_f].\] Also by choosing \[\mathfrak a'=3[x_e]-[x_f],\] we get another effective anti-canonical divisor \[\mathfrak d_{\mathfrak a'}+5{\rm div}(e_{2\omega})=X_{x_v},\] which is the divisor of a section in \({\rm H}^0(X,K_X^{-1})^{(B)}_{12\omega}\subset s^{12}\).

For convenience, below we work with the divisor \(\mathfrak d_0\). Since it is \(G\)-invariant, we have \(\Delta_{\mathscr Z}(\mathfrak d_0)=\Delta_{\mathscr Z}(K_X^{-1})\) and \(A_x(\mathfrak d_0,\lambda)=A_x(K_X^{-1},\lambda-\kappa_P)\) for all \(x\). Set \(\lambda=k\omega\). We have \[\Delta_{\mathscr Z}(K_X^{-1})=\{k\omega|k\in\mathbb{Q},~0\leq k\leq 12\},\] and the data are listed in Table-1.2. The ploytopes \(\Delta^O_x(K_X^{-1})\) are drawn in Fig-3. Also, \[A(\mathfrak d_0,\lambda)=\min\{\frac{1}{60}k,1-\frac{1}{12}k\}.\]

Table-1.2: Data of \(K_X^{-1}\)

It is direct to check that ?? holds and \[(\kappa_P-\mathbf{b}(\Delta_{x}^O(K_X^{-1})))^\perp\cap \mathscr V_x=\{O\}\] for all \(x\). This implies that \(X\) is not \({\rm SL}_2\times{\rm k}^\times\)-spherical. Otherwise the \(\{\rm Id\}\times{\rm k}^\times\)-action would induce a non-trivial, \(G\)-equivariant product test configuration with zero Futaki invariant that is associated to some non-central \(v_0\in\mathscr V\). A contradiction. Thus by Theorem 1, \(X\) is \({\mathbf{G}}\)-uniform K-stable.

6.2 A completion of space of ordered triangles in \(\mathbb{P}^2\)↩︎

In this section we apply our results to study the equivariant uniform K-stability of a \(\mathbb{Q}\)-Fano quasihomogeneous \({\rm SL}_3\)-variety given by [12]. In the following we use the notations and conventions in [12]. Let \(G={\rm SL}_3\) and \(T\subset G\) the torus of diagonal matrixes. Then \(G/T\) is quasihomogeneous, which is referred as the space of ordered triangles in \(\mathbb{P}^2\). Choose \(B\) the group of upper triangular matrixes. Combinatorial data, in particular, the colours \(\mathscr D^B\), the lattice \(\Gamma\) and the embedding \(e\) of \(\Gamma\) into \(K^{(B)}\), and the valuation cone \(\mathscr V\) of \(G/H\) are established in [19] (see also [12]). We recall some of them for our later use.

As usual, for \(g=(g_{ij})\in G\), we choose \(\epsilon_i(g)=g_{ii}\), \(\omega_1 =\epsilon_1\), \(\omega_2 =\epsilon_1+\epsilon_2\) are the fundamental weights, and \(\alpha_1 =\epsilon_1-\epsilon_2\), \(\alpha_2 =\epsilon_2-\epsilon_3\) are the simple roots. Denote by \(\omega^\vee_i\) the fundamental coweights. Then for the weight lattice \(\Gamma\) we have \[\Gamma=\mathbb{Z}\alpha_1\oplus\mathbb{Z}\alpha_2.\]

The data of colours are listed as in Table-2.1 (according to the notation of [12]).

Table-2.1: Colours in \(G/H\)

Clearly, the associated parabolic subgroup \[P(G/H)={\rm Stab}_G(D_2)\cap{\rm Stab}_G(\tilde{D}_2)=B.\] Thus \(\kappa_P=2(\alpha_1+\alpha_2)\).

The valuation cone of \(G/H\) is given as following: an element \(a_1\omega_1^\vee+a_2\omega_2^\vee+hq_x\) is contained in \(\mathscr V_x\) if and only if \[\begin{align} (a_1,a_2,h)~\text{satisfies}~\left\{ \begin{aligned} &a_1,a_2\leq0\leq h,~\text{if}~x=x_1, x_2,x_3,\\ &a_1,a_2\leq-2h\leq0,~\text{if}~x=\infty,\\ &a_1,a_2\leq-h\leq0,~\text{otherwise}. \end{aligned} \right. \end{align}\]

We have the following completion of \(G/H\), \[X = \{(p_1, p_2, p_3, l_1, l_2, l_3)|p_j\in \mathbb{P}^2,~l_i \in \mathbb{P}^{2^*},~p_j \in l_i~\text{whenever}~i\not=j\}.\] Here \(G\) acts on \(\mathbb{P}^2\) via the tautological representation on \({\rm k}^3\) and on the dual space \(\mathbb{P}^{2^*}\) via the corresponding dual representation. Let us recall the list of \(G\)-orbits in \(X\) given in [12],

• The open \(G\)-orbit of non-degenerate triangles.

• The divisors \(W_i\), \(i=1,2,3\) which consists of degenerate triangles with \(p_j=p_k\) and \(l_j=l_k\), where \(\{i,j,k\}=\{1,2,3\}\).

• The divisor \(\tilde{W}\) which consists of degenerate triangles with \(p_1\), \(p_2\), \(p_3\) collinear and \(l_1=l_2=l_3\).

• The divisor \(W\) which consists of degenerate triangles with \(p_1=p_2=p_3\) and \(l_1\), \(l_2\), \(l_3\) pass through this point.

• Codimension 2 orbits \(\tilde{Y}_i\) which consists of degenerate triangles with \(p_j=p_k\) and \(l_1=l_2=l_3\), where \(\{i,j,k\}=\{1,2,3\}\).

• Codimension 2 orbits \(Y_i\) which consists of degenerate triangles with \(l_j=l_k\) and \(p_1=p_2=p_3\), where \(\{i,j,k\}=\{1,2,3\}\). It has codimension 3.

and

• A minimal \(G\)-germ (the closed orbit) \(Y\) which consists of degenerate triangles with \(p_1=p_2=p_3\) and \(l_1=l_2=l_3\).

Thus \(X = G\mathring X\), where \(\mathring X\) is the minimal \(B\)-chart of \(Y\) determined by the colored hypercone \(\mathscr C=\mathscr C(\mathscr W,\mathscr R)\) with \[\begin{align} \mathscr W=\{W,\tilde{W}, W_1,W_2,W_3\},~ \mathscr R=\{D_x|x\not=x_1,x_2,x_3\}, \end{align}\] where the \(G\)-valuations in \(\mathscr W\) are given in Table-2.2.

Table-2.2: \(G\)-valuations in \(\mathscr W\)

There is a \(G\)-equivariant small resolution of \(X\) (cf. [49] and [14]). By “small" we mean the exceptional locus has codimension at least 2 (it is showed in [49] that in case of \(X\) the exceptional locus has dimension 4). A small resolution is in particular crepant. This implies that \(X\) has klt singularities (cf. [44]).

By Theorem 32, there is a \(B\)-stable anti-canonical \(\mathbb{Q}\)-divisor \[\mathfrak d=W+\tilde{W}+\frac{2}{3}\sum_{i=1}^3(D_i+\tilde{D}_i+W_i),\] where the corresponding \(\mathfrak a=\frac{2}{3}([x_1]+[x_2]+[x_3])\). Choose the function \(f=f_0e_{-\alpha_1-\alpha_2}\in K^{(B)}_{-\alpha_1-\alpha_2}\), where \(f_0\in K^B\) with \({\rm div}(f_0)=\frac{2}{3}([x_1]+[x_2]+[x_3])-2[\infty]\) on \(\mathbb{P}^1\), it is direct to check by Theorem 15 that \(3\mathfrak d\) is an ample divisor on \(X\), which is the divisor of a section in \({\rm H}^0(X,K_X^{-3})^{(B)}_{3\kappa_P}\). It is showed that \(\Gamma=\mathbb{Z}\alpha_1\oplus\mathbb{Z}\alpha_2\). Choose a coordinate \(\lambda=\lambda_1\alpha_1+\lambda_2\alpha_2\) for \(\lambda\in\Gamma_\mathbb{Q}\). We have \[\Delta_\mathscr Z(\mathfrak d)=\{\lambda|2+2\lambda_1-\lambda_2\geq0,~2+2\lambda_2-\lambda_1\geq0,~2-\lambda_1-\lambda_2\geq0,~1-\lambda_1\geq0,~1-\lambda_2\geq0\}.\]

The functions \(\{A_x(\mathfrak d,\lambda):\Delta_\mathscr Z(\mathfrak d)|x\in C\}\), and barycenter of \(\Delta_x^O(K_X^{-1})\) are given in Table-2.3. The concave piecewise linear function \(A(\mathfrak d,\lambda)\) is marked in Fig-4.

Table-2.3: Data of \(K_X^{-1}\)

It is direct to check that ?? holds and \[(\kappa_P-\mathbf{b}(\Delta_{x}^O(K_X^{-1})))^\perp\cap \mathscr V_x=\{O\},~\forall x\in \mathbb{P}^1,\] which in particular implies that \(X\) can not be \(G\times{\rm k}^\times\)-spherical again. By Theorem 1, we have

7 Appendix↩︎

7.1 Lemmas on colours in spherical homogeneous spaces↩︎

The following lemma is a combination of several known results (cf. [12] and [50]), we include it here for readers’ convenience.

7.1.1 A combinatorial property of \(B\)-stable ample divisors↩︎

Let \((X,L)\) be a polarized \(G\)-variety of complexity 1, and \[\mathfrak d=\sum_{D\in\mathscr B(X)}m_DD\] a divisor of \(L\), which is the divisor of some \(B\)-semiinvariant rational section \(s\) of \(L\) with \(B\)-weight \(\lambda_0\). We have

7.2 Lemmas on one-parameter \(G\)-varieties of type↩︎

The following Lemma is a special case of a general result [12]. We include an elementary proof below in our case only for readers’ convenience.

7.3 Lemmas on counting integral points↩︎

From the above Lemma we get the following estimates

The following Lemma can be derived from a general result of Pukhlikov-Khovanskij [55]. We include it here for reads’ convenience.

7.4 An alternative proof to Theorem 48↩︎

In the following we give an alternative proof of via an intersection formula of the Futaki invariant (cf. [3], [23]). Given a normal test configuration \((\mathcal{X},\mathcal{L})\) of \((X,L)\), the Futaki invariant can also be interpreted as intersection numbers \[\begin{align} \label{Fut-def-int} {\rm Fut}(\mathcal{X},\mathcal{L})=\frac{1}{V}K_{\mathcal{X}/\mathbb{P}^1}\cdot\mathcal{L}^{\cdot n}+\frac{\bar S}{V(n+1)}\mathcal{L}^{\cdot(n+1)}, \end{align}\tag{75}\] where \(V=L^n\) is the volume of \((X,L)\), \(\bar S\) the mean value of the scalar curvature of \((X,L)\), and \[\begin{align} K_{\mathcal{X}/\mathbb{P}^1}=K_{\mathcal{X}}-{\rm pr}^*K_{\mathbb{P}^1} \end{align}\] a Weil divisor on \(\mathcal{X}\).

Assume that \((X,L)\) is a polarized spherical variety with \(L\) has a divisor \(\mathfrak d_0\) given by 7 , and \[\mathfrak d_K=\sum_{D\in\mathscr B(X)}\bar m_DD\] a \(B\)-stable anti-canonical divisor of \(X\) corresponds to weight \(\kappa_P\). Let \((\mathcal{X},\mathcal{L})\) be a normal test configuration associated to \(v_0\in\mathcal{Q}_{x_0,+}\) and \(m=-1\). Without loss of generality we may also assume that \(\mathcal{L}\) has a divisor \(\mathfrak D\) given by 25 with \(r_0=1\) and \(m_\infty=0\) (otherwise one needs to divide 75 by \(r_0^{n+1}\)). Then by Theorems 29 (in the one-parameter case) and 32 (in the quasihomogeneous case), it is direct to see \[\begin{align} \mathfrak d_{\mathcal{X}/\mathbb{P}^1}=-\sum_{D\in\mathscr B(X)}\bar m_D\overline{D}-h_0(a_{x_0}-1)\mathcal{X}_0. \end{align}\] In particular, if \(X\) is \(\mathbb{Q}\)-Fano and \(K=K_X^{-1}\), we can take \(\mathfrak d_K=\mathfrak d_0\) (in particular, \(\bar m_D=m_D\) for all \(D\in\mathscr B(X)\)) and \[\mathfrak d_{\mathcal{X}/\mathbb{P}^1}=-(\mathfrak D+(h_0(a_{x_0}-1)-m_0){\rm pr}^*([0]))\] is also a \(\mathbb{Q}\)-Cartier divisor that corresponds to the weight \(-\kappa_P\).

The line bundle \[\begin{align} \mathcal{L}_\epsilon:=\mathcal{L}+\epsilon K_{\mathcal{X}/\mathbb{P}^1} \end{align}\] is a \(\mathbb{Q}\)-line bundle on \(\mathcal{X}\), and is ample when \(0<\epsilon\ll1\). The divisor \[\begin{align} \mathfrak D_\epsilon:=\mathfrak D+\epsilon \mathfrak d_{\mathcal{X}/\mathbb{P}^1}=\sum_{D\in\mathscr B(X)}(m_D-\epsilon m_D)\overline{D}+(m_0-\epsilon(a_{x_0}-1))\mathcal{X}_0 \end{align}\] is a divisor of \(\mathcal{L}_\epsilon\) corresponds to weight \((1-\epsilon)\kappa_P\), and the associated function is \[\begin{align} \tilde{A}_{x}^\epsilon(\lambda,t):=\left\{\begin{aligned}&(1-\epsilon)\min\{A_{x_0}(\mathfrak d,\frac{\lambda{1-\epsilon}}{)},\frac{-t+m_0-\epsilon h_0(a_{x_0}-1)+\ell_0(\lambda)}{h_0(1-\epsilon)}\},~\text{when}~x=x_0,\\ &(1-\epsilon)A_{x}(\mathfrak d,\frac{\lambda}{1-\epsilon}),~\text{when}~x\not=x_0,\end{aligned}\right. \end{align}\] for \(\lambda\in\Gamma_\mathbb{R}\) and \(\tilde{A}^\epsilon(\lambda,t)=\sum_{x\in C}\tilde{A}_{x}^\epsilon(\lambda,t)\). The polytope \[\begin{align} \label{Delta-eps} \tilde{\Delta}_\mathscr Z^\epsilon(\mathfrak D_\epsilon)=&\{(\lambda,t)|\tilde{A}^\epsilon(\lambda,t)\geq0\}\notag\\ =&\{(\lambda,t)|\lambda\in(1-\epsilon)\Delta_\mathscr Z(\mathfrak d_0),~0\leq t\leq\tau^0_\epsilon(\lambda)\}, \end{align}\tag{76}\] where \[\begin{align} \tau^0_\epsilon(\lambda)=m_0-\epsilon h_0(a_{x_0}-1)+\ell_0(\lambda)+h_0(1-\epsilon)(A(\frac{\lambda{1-\epsilon}}{)}-A_{x_0}(\frac{\lambda{1-\epsilon}}{)}). \end{align}\] Set \[\begin{align} \tilde{\tau}^0_\epsilon(\lambda)=m_0-\epsilon h_0(a_{x_0}-1)+\ell_0(\lambda)-h_0(1-\epsilon)A_{x_0}(\frac{\lambda{1-\epsilon}}{)}. \end{align}\] Then for \(\lambda\in\Gamma_\mathbb{R}\), \[\begin{align} \label{A-x-eps-piece} \tilde{A}^\epsilon(\lambda,t):=\left\{\begin{aligned} &(1-\epsilon)A(\mathfrak d_0,\frac{\lambda}{1-\epsilon}),~\text{when}~0\leq t\leq\tilde{\tau}_0^\epsilon(\lambda),\\ &(1-\epsilon)\sum_{x\not=x_0}A_{x}(\mathfrak d_0,\frac{\lambda}{1-\epsilon})+\frac{m_0-\epsilon h_0(a_{x_0}-1)-t+\ell_0(\lambda)}{h_0},~\text{when}~\tilde{\tau}^0_\epsilon(\lambda)\leq t\leq\tau^0_\epsilon(\lambda).\end{aligned}\right. \end{align}\tag{77}\]

Note that when \(\epsilon\in\mathbb{Q}\), \[\mathcal{L}_\epsilon^{\cdot(n+1)}=\mathcal{L}^{\cdot(n+1)}+(n+1)\epsilon K_{\mathcal{X}/\mathbb{P}^1}\cdot \mathcal{L}^{\cdot n}+O(\epsilon^2),~\epsilon\to0^+.\] Apply the intersection formula [14] to \(\mathcal{L}_\epsilon\), we get \[\begin{align} \label{mathcal-L-n431} \mathcal{L}^{\cdot(n+1)}=&(n+1)!\int_{\tilde{\Delta}_\mathscr Z(\mathfrak D)}\tilde{A}(\lambda,\tau)\pi(\lambda+\kappa_P) d\lambda\wedge d\tau\notag\\ =&(n+1)!\int_{\Delta_\mathscr Z(\mathfrak d_0)}\int_{0}^{\tau^0(\lambda)}\tilde{A}(\lambda,\tau)\pi(\lambda+\kappa_P) d\lambda\wedge d\tau, \end{align}\tag{78}\] and \[\begin{align} \label{KX47P-term} K_{\mathcal{X}/\mathbb{P}^1}\cdot \mathcal{L}^{\cdot n}&=\frac{1}{n+1}\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}\mathcal{L}_\epsilon^{\cdot(n+1)}\notag\\ &=n!\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}\int_{\tilde{\Delta}_\mathscr Z^\epsilon(\mathfrak D_\epsilon)}\tilde{A}^\epsilon(\lambda,\tau)\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge d\tau. \end{align}\tag{79}\] By 76 and 77 , \[\begin{align} &\int_{\tilde{\Delta}_\mathscr Z^\epsilon(\mathfrak D_\epsilon)}\tilde{A}^\epsilon(\lambda,\tau)\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge d\tau\notag\\ =&\int_{(1-\epsilon)\Delta_\mathscr Z(\mathfrak d_0)}\int_{0}^{\tilde{\tau}^0_\epsilon(\lambda)}(1-\epsilon)A(\mathfrak d_0,\frac{\lambda}{1-\epsilon})\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge dt\notag\\ &+\int_{(1-\epsilon)\Delta_\mathscr Z(\mathfrak d_0)}\int_{\tilde{\tau}^0_\epsilon(\lambda)}^{\tau^0_\epsilon(\lambda)}(1-\epsilon)\sum_{x\not=x_0}A_x(\mathfrak d_0,\frac{\lambda}{1-\epsilon})\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge dt\\ &+\int_{(1-\epsilon)\Delta_\mathscr Z(\mathfrak d_0)}\int_{\tilde{\tau}^0_\epsilon(\lambda)}^{\tau^0_\epsilon(\lambda)}\frac{m_0-\epsilon h_0(a_{x_0}-1)-t+\ell_0(\lambda)}{h_0}\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge dt. \end{align}\] Using change of variables \(\lambda\to(1-\epsilon)\lambda\), \(t\to(1-\epsilon)t\), and the relation 10 , we can rewrite the first two terms as \[\begin{align} &\int_{(1-\epsilon)\Delta_\mathscr Z(\mathfrak d_0)}\int_{0}^{\tilde{\tau}^0_\epsilon(\lambda)}(1-\epsilon)A(\mathfrak d_0,\frac{\lambda}{1-\epsilon})\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge dt\notag\\ &+\int_{(1-\epsilon)\Delta_\mathscr Z(\mathfrak d_0)}\int_{\tilde{\tau}^0_\epsilon(\lambda)}^{\tau^0_\epsilon(\lambda)}(1-\epsilon)\sum_{x\not=x_0}A_x(\mathfrak d_0,\frac{\lambda}{1-\epsilon})\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge dt\notag\\ =&(1-\epsilon)^{n+1}\int_{\Delta_\mathscr Z(\mathfrak d_0)}\int_{0}^{\frac{1}{1-\epsilon}\tilde{\tau}^0_\epsilon((1-\epsilon)\lambda)}A(\mathfrak d_0,\lambda)\pi(\lambda+\kappa_P) d\lambda\wedge dt\notag\\ &+(1-\epsilon)^{n+1}\int_{\Delta_\mathscr Z(\mathfrak d_0)}\int_{\frac{1}{1-\epsilon}\tilde{\tau}^0_\epsilon((1-\epsilon)\lambda)}^{\frac{1}{1-\epsilon}\tau^0_\epsilon((1-\epsilon)\lambda)}\sum_{x\not=x_0}A_x(\mathfrak d_0,\lambda)\pi(\lambda+\kappa_P) d\lambda\wedge dt, \end{align}\] and the last term can be rewritten as \[\begin{align} &\int_{(1-\epsilon)\Delta_\mathscr Z(\mathfrak d_0)}\int_{\tilde{\tau}^0_\epsilon((1-\epsilon)\lambda)}^{\tau^0_\epsilon((1-\epsilon)\lambda)}\frac{m_0-\epsilon h_0(a_{x_0}-1)-t+\ell_0(\lambda)}{h_0}\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge dt\\ =&(1-\epsilon)^{n+1}\int_{\Delta_\mathscr Z(\mathfrak d_0)}\int_{\frac{1}{1-\epsilon}\tilde{\tau}^0_\epsilon((1-\epsilon)\lambda)}^{\frac{1}{1-\epsilon}\tau^0_\epsilon((1-\epsilon)\lambda)}\frac{\frac{1}{1-\epsilon}({m_0-\epsilon h_0(a_{x_0}-1)})-t+\ell_0(\lambda)}{h_0}\pi(\lambda+\kappa_P) d\lambda\wedge dt \end{align}\] Taking variation and using Lemma 45, we get \[\begin{align} &\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}\int_{\tilde{\Delta}_\mathscr Z^\epsilon(\mathfrak D_\epsilon)}\tilde{A}^\epsilon(\lambda,\tau)\pi(\lambda+(1-\epsilon)\kappa_P) d\lambda\wedge d\tau\notag\\ =&-(n+1)\int_{\tilde{\Delta}_\mathscr Z(\mathfrak D)}\tilde{A}(\mathfrak D,\lambda,\tau)\pi(\lambda+\kappa_P) d\lambda\wedge d\tau+\int_{\Delta_\mathscr Z(\mathfrak d_0)}\int_{\tilde{\tau}^0(\lambda)}^{\tau^0(\lambda)}(\frac{m_0}{h_0}-a_{x_0}+1)\pi(\lambda+\kappa_P)d\lambda\wedge dt. \end{align}\] Combining with 75 , 78 79 , we get \[\begin{align} {\rm Fut}(\mathcal{X},\mathcal{L})=&\frac{1}{V}\int_{\Delta_\mathscr Z(\mathfrak d_0)}\int_{\tilde{\tau}^0(\lambda)}^{\tau^0(\lambda)}\frac{m_0-h_0(a_{x_0}-1)}{h_0}\pi(\lambda+\kappa_P)d\lambda\wedge dt\\ &-\frac{1}{V}\int_{\Delta_\mathscr Z(\mathfrak d_0)}\int_{0}^{\tau^0(\lambda)}\tilde{A}(\mathfrak D,\lambda,t)\pi(\lambda+\kappa_P)d\lambda\wedge dt\notag\\ =&\frac{1}{V}\int_{\Delta_{x_0}^O(K_X^{-1})}\langle(\kappa_P-\lambda,-t),v_0\rangle\pi(\lambda)d\lambda\wedge dt, \end{align}\] where in the last line we used 59 and the fact that \[\begin{align} V=\frac{(K_X^{-1})^{\cdot n}}{n!}=\int_{\Delta_\mathscr X(\mathfrak d_0)}A(\mathfrak d_0,\lambda)\pi(\lambda+\kappa_P)d\lambda=\int_{\Delta_{x_0}^O(K_X^{-1})}\pi(\lambda)d\lambda\wedge dt. \end{align}\]

7.5 One the polytope \(\Delta_{x_0}^O(K_X^{-1})\)↩︎

The polytope \(\Delta_{x_0}^O(K_X^{-1})\) has another geometric meaning

We remark that Proposition 63 for \(T\)-varieties of complexity 1 has been proved in [21]. Also the above method applies to \(G\)-equivariant special test configurations of a general polarized \((X,L)\) (not necessarily \(L=K_X^{-1}\)).

Acknowledgement↩︎

We sincerely thank Prof. D.A. Timashëv for kindly introduce us his book [12] and many helpful discussions. We also thank the referees for careful reading and valuable comments, which improve this paper a lot.

References↩︎

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1. \(^{*1}\)Partially supported by NSFC Grant 12101043 and the Beijing Institute of Technology Research Fund Program for Young Scholars.↩︎

2. \(^{*2}\)Partially supported by NSFC Grant 12001032.↩︎

3. More precisely, the lattice of \(B\)-weights of semiinvariant functions. Note that on a spherical variety, up to multiplication by a non-zero constant, a \(B\)-semiinvariant function is uniquely determined by its \(B\)-weight.↩︎

4. In Section 6.1 below there is explicit data of a concrete example, the Mukai-Umemura threefold. We also refer to the readers Fig-2 there for an example of coloured hypercone of type .↩︎

5. Note that this is opposite to the usual convention in the case of torus actions.↩︎

6. Again, note the opposite sign convention used here compared with the case of torus actions.↩︎

7. Thus \(a_x=0\) for almost every \(x\in C\).↩︎

8. To be precise, we add a short explanation to 24 . In the last summation term it may happen \(F_{D}=F_{D'}\) for different \(D, D'\in\mathscr B(X)\) with \(h_D=h_{D'}=0\). If one of them is of type-a’ or b, then \(\pi(\lambda+\lambda_0)=0\) on this facet. Otherwise \(n_D\bar m_D=n_{D'}\bar m_{D'}=1\). Thus each term in the last sum does not depend on the choice in \(\{D,D'\}\).↩︎

9. We do not write \(\hat{G}\) instead of \(G\times{\rm k}^\times\) here in order not to be confounded with the previous section.↩︎

10. More precisely, the image of \(G\) in \({\rm Aut}(X)\) of the homomorphism \(G\to{\rm Aut}(X)\).↩︎

11. We put \(\frac{1}{2}\) here since the dual lattice \(\Gamma^*=\frac{1}{2}\mathbb{Z}\omega^*\)↩︎