Singular Kähler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups

1.   Introduction

Let $G$ be an $n$-dimensional connected, linear complex reductive Lie group which is the complexification of a compact Lie group $K$. Let $T$ be a maximal torus of $K$, which has dimension $r$ and Lie algebra $\mathfrak t$. Then $T^\mathbb C$ is a maximal complex torus of $G$. Denote by $\Phi_+$ a chosen positive roots system associated to $T^{\mathbb C}$. Put

It can be regarded as a character in $\mathfrak a^*$, where $\mathfrak a^*$ is the dual space of the non-compact part $\mathfrak a = \sqrt{-1}\mathfrak t$ of $\mathfrak t^\mathbb C$. Let $\pi$ be a function on ${\mathfrak a^*}$ defined by

where $\langle\cdot, \cdot \rangle$^* denotes the Cartan-Killing inner product on ${\mathfrak a^*}$.

^* Without of confusion, we also write it as $\alpha(y)$ for simplicity.

Let $M$ be a $\mathbb Q$-Fano $G$-compactification (cf. ^[4]). Denote by $Z$ the closure of $T^{\mathbb C}$ in $M$. Then $(Z, -K_M|_Z)$ is a polarized toric variety. Hence there is an associated moment polytope $P$ of $(Z, -K_M|_Z)$ induced by $(M, -K_M)$ ^[3,4]. Let $P_+$ be the positive part of $P$ defined by $P_+ = P\cap\overline{\mathfrak a^*_+}$, where

is the positive Weyl chamber in $\mathfrak a^*$ defined by $\Phi_+$. Denote by $2P_+$ a dilation of $P_+$ at rate $2$. We define the barycenter of $2P_+$ with respect to the weighted measure $\pi(y)dy$ by

In ^[20], Delcroix proved the following existence theorem for Kähler-Einstein metrics on smooth Fano $G$-compactifications.

Theorem 1.1. Let $M$ be a smooth Fano $G$-compactification. Then $M$ admits a Kähler-Einstein metric if and only if

where $\Xi$ is the relative interior of the cone generated by $\Phi_+$.

There is an alternative proof of Theorem 1.1 given by Li, Zhou and Zhu via the variation method ^[36]. They also showed that (1.2) is actually equivalent to the K-stability condition in terms of ^[44] and ^[24] by constructing $\mathbb C^*$-action through piecewisely rationally linear function which is invariant under the Weyl group action. In particular, it implies that $M$ is K-unstable if ${\rm bar}(2P_+) \not\in \overline { 4\rho+\Xi}$. A more general construction of $\mathbb C^*$-action was also discussed in ^[21].

In the present paper, we extend the above theorem to $\mathbb Q$-Fano compactifications of $G$ which may be singular. It is well known that any $\mathbb Q$-Fano $G$-compactification has klt-singularities [4,Section 5]. For a $\mathbb Q$-Fano variety $M$ with klt-singularities, there is naturally a class of admissible Kähler metrics induced by the Fubini-Study metric (cf. ^[23]). In ^[10], Berman, Boucksom, Eyssidieux, Guedj and Zeriahi introduce a class of Kähler potentials associated to admissible Kähler metrics and refer it as the $\mathcal E^1(M, -K_M)$ space. Then they define the singular Kähler-Einstein metric on $M$ with the Kähler potential in $\mathcal E^1(M, -K_M)$ via the complex Monge-Ampère equation, which is the usual Kähler-Einstein metric on the smooth part of $M$. It is a natural problem to establish an extension of the Yau-Tian-Donaldson conjecture we have solved for smooth Fano manifolds ^[44,46], that is, an equivalence relation between the existence of such singular Kähler-Einstein metrics and the K-stability on a $\mathbb Q$-Fano variety $M$ with klt-singularities. Actually, there are many recent works on this fundamental problem in terms of uniform K-stability. We refer the reader to ^{[10,11,32,33,34]}, etc..

We assume that the associated polytope $P$ of $(Z, K^{-1}_M|_Z)$ is fine in sense of ^[25], namely, each vertex of $P$ is the intersection of precisely $r$ facets. Then we prove

Theorem 1.2. Let $M$ be a $\mathbb Q$-Fano $G$-compactification such that the associated polytope $P$ is $\textrm{fine}$.Then $M$ admits a singular Kähler-Einstein metric if and only if (1.2) holds.

By a result of Abreu ^[1], the assumption that the polytope $P$ being fine is equivalent to that the metric induced by the Guillemin function can be extended to a Kähler orbifold metric on $Z$.^† It follows that the Guillemin function of $2P$ in Theorem 1.2 induces a $K\times K$-invariant singular metric $\omega_{2P}$ in $\mathcal E^1(M, -K_M)$ (cf. Lemma 3.4). Moreover, we can prove that the Ricci potential of $\omega_{2P}$ on $M$ is uniformly bounded above. We note that $P$ is always fine when rank$(G) = 2$. Thus for a $\mathbb Q$-Fano compactification of $G$ with rank$(G) = 2$, $M$ admits a singular Kähler-Einstein metric if and only if (1.2) holds. As an application of Theorem 1.2, we show that there is only one example of non-smooth Gorenstein Fano ${\rm SO}_4(\mathbb C)$-compactifications which admits a singular Kähler-Einstein metric (cf. Section 7.1).

^† It can not be guaranteed that the $G$–compactification is smooth even if $Z$ is smooth, see an examples for $G = SO_{2n+1}(\mathbb C)$ in [47,Section 11].

On the other hand, it has been shown in ^[41] that there are only three smooth Fano compactifications of ${\rm SO}_4(\mathbb C)$, i.e., Case-1.1.2, Case-1.2.1 and Case-2 in Section 7.1.

The first two manifolds do not admit any Kähler-Einstein metric ^[20,36]. By Theorem 1.2, we further prove

Theorem 1.3. There is no $\mathbb Q$-Fano compactification of ${\rm SO}_4(\mathbb C)$ which admits a singular Kähler-Einstein metric with the same volume as Case-1.1.2 or Case-1.2.1 in Section 7.1.

Theorem 1.3 gives a partial answer to a question proposed in ^[38] about limit of Kähler-Ricci flow on either Case-1.1.2 or Case-1.2.1. It has been proved there that the flow has type II singularities on each of Case-1.1.2 and Case-1.2.1. By the Hamilton-Tian conjecture ^[7,16,44,48], the limit should be a singular Kähler-Ricci soliton on a $\mathbb Q$-Fano variety with the same volume as that of initial metric. However, by Theorem 1.3, the limit can not be a $\mathbb Q$-Fano compactification of ${\rm SO}_4(\mathbb C)$ with a singular Kähler-Einstein metric. This implies that the limiting soliton of flow on either Case-1.1.2 or Case-1.2.1 will have less symmetric than the initial one, which is totally different from the situation of smooth convergence of $K\times K$-invariant metrics on a smooth compactification of Lie group ^[38].

As in ^[9,36,49], we use the variational method to prove Theorem 1.2. More precisely, we will prove that a modified version of the Ding functional $\mathcal D(\cdot)$ is proper under the condition (1.2). This functional is defined for a class of convex functions $\mathcal E^1_{K\times K}(2P)$ associated to $K\times K$-invariant metrics on the orbit of $G$ (cf. Section 4, 6). The key point is that the Ricci potential $h_0$ of the Guillemin metric $\omega_{2P}$ is bounded from above when $P$ is fine (cf. Proposition 5.1). This enables us to control the nonlinear part $\mathcal F(\cdot)$ of $\mathcal D(\cdot)$ by modifying $\mathcal D(\cdot)$ as done in ^[24,37] (cf. Section 6.1). We shall note that it is in general impossible to get a lower bound of $h_0$ if the compactification is a singular variety (cf. Remark 5.2). However, by a recent deep result of Li in ^[32], the additional $fine$ condition in Theorem 1.2 can be actually dropped. For the completeness, we will also improve the theorem at the end of paper, Appendix 2.

The minimizer of $\mathcal D(\cdot)$ corresponds to a singular Kähler-Einstein metric. We will prove the semi-continuity of $\mathcal D(\cdot)$ and derive the Kähler-Einstein equation for the minimizer (cf. Proposition 6.6). Our proof is similar with what Berman and Berndtsson studied on toric varieties in ^[9].

The proof of the necessity part of Theorem 1.2 is the same as one in Theorem 1.1. In fact, a $\mathbb Q$-Fano compactification of $G$ is not K-polystable if (1.2) is not satisfied [36,Proposition 3.4] (also see ^[21]). This will be a contradiction to the K-polystability of $\mathbb Q$-Fano variety with a singular Kähler-Einstein metric (cf. ^[8,33]). We omit this part.

The organization of paper is as follows. In Section 2, we recall some notations in ^[10] for singular Kähler-Einstein metrics on $\mathbb Q$-Fano varieties. In Section 3, we introduce a subspace $\mathcal E^1_{K\times K}(M, -K_M)$ of $\mathcal E^1(M, -K_M)$ and prove that the Guillemin function lies in this space (cf. Lemma 3.4). In Section 4, we prove that $\mathcal E^1_{K\times K}(M, -K_M)$ is equivalent to a dual space $\mathcal E^1_{K\times K}(2P)$ of Legendre functions (cf. Theorem 4.2). In Section 5, we compute the Ricci potential $h_0$ of $\omega_{2P}$ and show that it is bounded from above (cf. Proposition 5.1). The sufficient part of Theorem 1.2 will be proved in Section 6. In Section 7, we construct many $\mathbb Q$-Fano compactifications of ${SO}_4(\mathbb C)$ and in particular, we will prove Theorem 1.3.

2.   Preliminary on $\mathbb Q$-Fano varieties

For a $\mathbb Q$-Fano variety $M$, by Kodaira's embedding, there is an integer $\ell > 0$ such that we can embed $M$ into a projective space $\mathbb {CP}^N$ by a basis of $H^0(M, K_M^{-\ell})$, for simplicity, we assume $M\subset \mathbb {CP}^N$ and $K_M^{-\ell} = \mathcal O_{\mathbb {CP}^N}(1)$. Then we have a metric

where $\omega_{FS}$ is the Fubini-Study metric of $\mathbb {CP}^N$. Moreover, there is a Ricci potential $h_0$ of $\omega_0$ such that

In the case that $M$ has only klt-singularities, $e^{h_0}$ is $L^p$-integrable for some $p > 1$ (cf. ^[10,23]). We call $\omega$ an admissible Kähler metric on $M$ if there are an embedding $M\subset \mathbb {CP}^N$ as above and a smooth function $\phi$ on $\mathbb {CP}^N$ such that

In particular, $\phi$ is a function on $M$ with $\phi\in L^\infty(M)\cap C^\infty(M_{\text{reg}})$, called an admissible Kähler potential associated to $\omega_0$.^‡

^‡ For simplicity, we will denote a Kähler metric by its Kähler form thereafter.

For a general (possibly unbounded) Kähler potential $\phi$, we define its complex Monge-Ampère measure $\omega_{\phi}^n$ by

where $\phi_j\, = \max\{\phi, -j\}$. According to ^[10], we say that $\phi$ (or $\omega_{\phi}^n$) has full Monge-Ampère (MA) mass if

The MA-measure $\omega_{\phi}^n$ with a full MA-mass has no mass on the pluripolar set of $\phi$ in $M$. Thus we need to consider the measure on $M_{\rm reg}$. Moreover, $e^{-\phi}$ is $L^p$-integrable for any $p > 0$ associated to $\omega_0^n$.

Definition 2.1. We call $\omega_{\phi}$ a (singular) Kähler-Einstein metric on $M$ with full MA-mass if $\phi$ satisfies the following complex Monge-Ampère equation,

It has been shown in ^[10] that $\phi$ is $C^\infty$ on $M_{\rm reg}$ if it is a solution of (2.1). Thus $\omega_{\phi}$ satisfies the Kähler-Einstein equation on $M_{\rm reg}$,

2.1.   The space $\mathcal E^1(M, -K_M)$ and the Ding functional

On a smooth Fano manifold, there is a well-known Euler-Lagrange functional for Kähler potentials associated to (2.1), often referred as the Ding functional or F-functional, defined by (cf. ^[22,43]),

In case of $\mathbb Q$-Fano manifold with klt-singularities, Berman-Boucksom-Eyssidieux-Guedj-Zeriahi ^[10] extended $F(\cdot)$ to the space $\mathcal E^1(M, -K_M)$ defined by

They showed that $\mathcal E^1(M, -K_M)$ is compact in certain weak topology. By a result of Darvas ^[18], $\mathcal E^1(M, -K_M)$ is in fact compact in the topology of $L^1$-distance. It provides a variational approach to study (2.1).

Definition 2.2. ^[10,44] The functional $F(\cdot)$ is called proper if there is a continuous function $p(t)$ on $\mathbb R$ with the property $\lim_{t\to+\infty} p(t)\, = \, +\infty$, such that

In ^[10], Berman-Boucksom-Eyssidieux-Guedj-Zeriahi proved the existence of solutions for (2.1) under the properness assumption (2.3) of $F(\cdot)$. However, this assumption does not hold in the case of existence of non-zero holomorphic vector fields such as in our case of $\mathbb Q$-Fano $G$-compactifications. So we need to consider a modified version of properness of the Ding functional instead to overcome this new difficulty as done on toric varieties ^[9,37].

3.   Associated polytopes and $K\times K$-invariant metrics

Let $M$ be a $\mathbb Q$-Fano compactification of $G$ with $Z$ being the closure of a maximal complex torus $T^{\mathbb C}$-orbit. We first characterize the polytope $P$ of $Z$ associated to $(M, -K_M)$. Denote by $\mathfrak M$ the lattice of $G$-weights and $W$ the Weyl group of $(G, T^{\mathbb C})$. Then $P$ is an $r$-dimensional $W$-invariant, convex, rational polytope in $\mathfrak a^* = \mathfrak M_\mathbb R$. Let $\{F_A\}_{A = 1, ..., d_0}$ be the facets of $P$ and $\{F_A\}_{A = 1, ..., d_+}$ be those whose relative interior intersects $\mathfrak a_+^*$. Suppose that

for some primitive vector $u_A\in\mathfrak N$ and the facet

By the $W$-invariance, for each $A\in\{1, ..., d_0\}$, there is some $w_A\in W$ such that $w_A(F_A)\in\{F_B\}_{B = 1, ..., d_+}$. Denote by $\rho_A\, = \, w_A^{-1}(\rho)$, where $\rho\in\mathfrak a_+^*$ is given by (1.1). Then $\rho_A(u_A)$ is independent of the choice of $w_A\in W$ and hence it is well-defined.

The following Lemma can be derived from a general result [26,Theorem 1.9] on polytopes of $\mathbb Q$-Fano spherical varieties. For readers' convenience, we sketch a direct proof in cases of group compactifications below by using [15,Section 4].

Lemma 3.1. Let $M$ be a $\mathbb Q$-Fano compactification of $G$ with $P$ being the associated moment polytope. Then for each $A = 1, ..., d_0$, it holds

Conversely, a $W$-invariant convex polytope $P$ given by $(3.1)$ is the associated polytope of some $\mathbb Q$-Fano $G$-compactification if $(3.2)$ holds.

Sketch of proof. Suppose that $-mK_M$ is a Cartier divisor for some $m\in\mathbb N_{+}$. Up to a dilation of the polytope, it suffices to consider the case when $m = 1$. Denote by $B^+$ the (positive) Borel subgroup of $G$ corresponding to $(T^\mathbb C, \Phi_+)$ and $B^-$ be the opposite one. Then by [15,Section 4], there exists a $B^+\times B^-$-semiinvariant section of $-K_M$ whose divisor is

where $\{X_{A'}\}$ is the set of $G\times G$-invariant prime divisors and $Y_{\alpha_i}$ is the prime $B^+\times B^-$-invariant divisor corresponds to the weight $\alpha_i$ in $\Phi_{+, s}$, the set of simple roots in $\Phi_+$. Note that the corresponding $B^+\times B^-$-weight of this divisor is $2\rho$ (cf. [21,Section 3.2.4]). Thus by adding the divisor of a $B^+\times B^-$-semi-invariant rational function $f_o$ with weight $-2\rho$, we have

is a $G\times G$-invariant divisor. On the other hand, by [5,Theorem 2.4 (3)], the prime $G\times G$-invariant divisors of $M$ are in bijections with $W$-orbits of prime toric divisors of $Z$. Restricting the above divisor to $Z$, we get (3.2).

Conversely, suppose that there is a $W$-invariant polytope $P$ given by (3.1) and (3.2). Then $P$ is rational and there is an $m\in\mathbb N_+$ so that $mP$ is integral. The support function of $mP$ is $W$-invariant and strictly convex. Hence it corresponds to an ample Cartier divisor $m\mathfrak d'$, where (cf. [14,Section 3.3])

Obviously $\mathfrak d'-{\rm div}(f_o)$ equals to the divisor $\mathfrak d$ defined by (3.3), which is a divisor of $-K_M$ by [15,Section 4]. We conclude the Lemma.

3.1.   $K\times K$-invariant metrics

On a $\mathbb Q$-Fano compactification of $G$, we may regard the $G\times G$-action as a subgroup of ${PGL}_{N+1}(\mathbb C)$ which acts holomorphically on the hyperplane bundle $L\, = \, {\mathcal O}_{\mathbb {CP}^N}(1)$. Then any admissible $K\times K$-invariant Kähler metric $\omega_\phi\in\frac{ 2\pi }{\ell} c_1(L)$ can be regarded as a restriction of $K\times K$-invariant Kähler metric of $\mathbb {CP}^N$. Thus the moment polytope $P$ associated to $(Z, L|_Z)$ is a $W$-invariant rational polytope in $\mathfrak a^*$. By the $K\times K$-invariance, the restriction of $\omega_\phi$ on $T^{\mathbb C}$ is an open toric Kähler metric. Hence, it induces a strictly convex, $W$-invariant function $\psi_\phi$ on ${\mathfrak a}$ ^[6] (also see Lemma 3.3 below) such that

By the $KAK$-decomposition ([31,Theorem 7.39]), for any $g\in G$, there are $k_1, \, k_2\in K$ and $x\in\mathfrak a$ such that $g = k_1\exp(x)k_2$. Here $x$ is uniquely determined up to a $W$-action. This means that $x$ is unique in $\bar{\mathfrak a}_+$. Thus there is a bijection between $K\times K$-invariant functions $\Psi$ on $G$ and $W$-invariant functions $\psi$ on $\mathfrak a$ which is given by

Without of confusion, we will not distinguish $\psi$ and $\Psi$, and call $\Psi$ (or $\psi$) convex on $G$ if $\psi$ is convex on ${\mathfrak a}$.

The following $KAK$-integration formula can be found in [31,Proposition 5.28].

Proposition 3.2. Let $dV_G$ be a Haar measure on $G$ and $dx$ the Lebesgue measureon $\mathfrak{a}$.Then there exists a constant $C_H > 0$ such that for any$K\times K$-invariant, $dV_G$-integrable function $\psi$ on $G$,

where

Without loss of generality, we may normalize $C_H = 1$ for simplicity.

Next we recall a local holomorphic coordinate system on $G$ used in ^[20]. By the standard Cartan decomposition, we can decompose $\mathfrak g$ as

where $\mathfrak t$ is the Lie algebra of $T$ and

is the root space of complex dimension $1$ with respect to $\alpha$. By ^[28], one can choose $X_{\alpha}\in V_{\alpha}$ such that $X_{-\alpha} = -\iota(X_{\alpha})$ and $[X_{\alpha}, X_{-\alpha}] = \alpha^{\vee},$ where $\iota$ is the Cartan involution and $\alpha^{\vee}$ is the dual of $\alpha$ by the Killing form. Let $E_{\alpha} = X_{\alpha}-X_{-\alpha}$ and $E_{-\alpha} = J(X_{\alpha}+X_{-\alpha})$. Denoted by $\mathfrak k_{\alpha}, \, \mathfrak k_{-\alpha}$ the real line spanned by $E_\alpha, \, E_{-\alpha}$, respectively. Then we get the Cartan decomposition of Lie algebra $\mathfrak k$ of $K$ as follows,

Choose a real basis $\{E^0_1, ..., E^0_r\}$ of $\mathfrak t$, where $r$ is the dimension of $T$. Then $\{E^0_1, ..., E^0_r\}$ together with $\{E_{\alpha}, E_{-\alpha}\}_{\alpha\in\Phi_+}$ forms a real basis of $\mathfrak k$, which is indexed by $\{E_1, ..., E_n\}$. We can also regard $\{E_1, ..., E_n\}$ as a complex basis of $\mathfrak g$. For any $g\in G$, we define local coordinates $\{z_{(g)}^i\}_{i = 1, ..., n}$ on a neighborhood of $g$ by

It is easy to see that $\theta^i|_g\, = \, dz_{(g)}^i|_g$, where the dual $\theta^i$ of $E_i$ is a right-invariant holomorphic $1$-form. Thus

is also a right-invariant $(n, n)$-form, which defines a Haar measure.

For a $K\times K$-invariant function $\psi$, Delcroix computed the Hessian of $\psi$ in the above local coordinates as follows [20,Theorem 1.2].

Lemma 3.2. Let $\psi$ be a $K\times K$ invariant function on $G$.Then for any $x\in \mathfrak{a}_+$, the complex Hessian matrix of $\psi$ in the above coordinates is diagonal by blocks, and equals to

where $\Phi_+ = \{\alpha_{(1)}, ..., \alpha_{(\frac{n-r}{2})}\}$ is the set of positive roots and

By (3.7) in Lemma 3.3, we see that $\psi_\phi$ induced by an admissible $K\times K$-invariant Kähler form $\omega_\phi$ is convex on $\mathfrak{a}$. The complex Monge-Ampère measure is given by

By (3.6), for any $x\in\mathfrak a_+$ we have

in (3.8). In particular, by Proposition 3.2,

Clearly, (3.9) also holds for any Kähler potential in $\mathcal E^1(M, -K_M)$, which is smooth and $K\times K$-invariant on $G$. For the completeness, we introduce a subspace of $\mathcal E^1(M, -K_M)$ by

Here $\psi_0$ is a convex function on $\mathfrak{a}$ associated to a background admissible $K\times K$-invariant metric $\omega_0$ as in (3.5). $\mathcal E^1_{K\times K}(M, -K_M)$ is locally precompact in terms of convex functions on $\mathfrak{a}$. In Sections 4 and 6, we will prove its completeness by using the Legendre dual.

3.2.   Fine polytope $P$

In this subsection, we show that the Legendre dual of Guillemin function $u_{2P}$ on $2P$ lies in $\mathcal E^1_{K\times K}(M, -K_M)$ when $P$ is fine.

Recall (3.1). For convenience, we set

Then

Thus, $u_{2P}$ is given by (cf. ^[1])

Clearly, it is $W$-invariant, so its Legendre function $\psi_{2P}$ is also $W$-invariant, where

Hence, by [1,Theorem 2] and ^[6] (also see Lemma 3.3),^$\S$ we can extend

^$\S$ The corresponding moment map is given by ${\frac12}\nabla \psi_{2P}$, whose image is $P$.

to a $K\times K$-invariant metric on $G$.

Lemma 3.4. Let $\psi_0$ be the background $K\times K$-invariant Kähler potential in (3.11). Assume that $P$ is fine. Then the Kähler potential $(\psi_{2P}-\psi_0)$ of $\omega_{2P}$ lies in $\phi\in L^\infty(M)\cap C^\infty(M_{\rm reg})$. In particular, $(\psi_{2P}-\psi_0) \in\mathcal E^1_{K\times K}(M, -K_M).$

Proof. Fix an $m_0\in\mathbb Z_+$ such that $-m_0 K_X$ is very ample. We consider the projective embedding

given by $|-m_0K_M|$, where $N = h^0(M, -m_0K_M)-1$. By [39,Section 2.3], the pull back of the Fubini-Study metric on $\mathbb{CP}^N$ gives a $K\times K$-invariant, Hermitian metric $h$ on $L = {\mathcal O}_{\mathbb CP^N}(-1)|_M$. Moreover, we have

where $\bar n(\lambda)\in\mathbb Z_+$. Thus we have a Kähler potential on $T^{\mathbb C}$ by

Since $P$ is fine, one can show directly that

where $v_{2P}(\cdot)$ is the support function on $\mathfrak a$ defined by

Recall that the Legendre function $u_\psi$ of $\psi$ is defined as in (3.12) by

It is known that $\psi\in\mathcal V(2P)$ if and only if $u_\psi$ is uniformly bounded on $2P$ since the Legendre function of $v_{2P}$ is zero (cf. ^[42]). Thus the Legendre function $u_{h}$ of $h|_{T^\mathbb C}(x)$ is uniformly bounded on $2P$. It follows that

Hence, we get

Consequently,

By (3.10), $(\psi_{2P}-\psi_{0})$ has full MA-mass, so we have completed the proof.

4.   The space $\mathcal E^1_{K\times K}(2P)$

In this section, we describe the space $\mathcal E^1_{K\times K}(M, -K_M)$ in (3.11) via Legendre functions as in ^[17] for $\mathbb Q$-Fano toric varieties. Recall the background $K\times K$-invariant Kähler potential $\psi_0$ in Lemma 3.4. Then we can normalize $\psi_0$ up to an action of the centre $Z(G)$ of $G$ as follows (cf. ^[38]),

where $O$ is the origin of ${\mathfrak a}$. Similarly, for any $\phi\in \mathcal E^1_{K\times K}(M, -K_M)$, $\psi_\phi = \psi_0+\phi$ can be also normalized as in (4.1).

The following lemma is elementary.

Lemma 4.1. For any $K\times K$-invariant potential $\phi$ normalized as in $(4.1)$, it holds

where $\partial (\psi_\phi) (\cdot)$ is the normal mapping of $\psi_\phi$.

Proof. We choose a sequence of decreasing and uniformly bounded $K\times K$-invariant potential $\phi_i$ normalized as in $(4.1)$ such that

and

Then

It follows that

This implies that $\partial\psi_\phi\subseteq 2P$. By the convexity, we also get $\psi_\phi\leq v_{2P}$.

It is easy to see that the Legendre function $u_\phi$ of $\psi_\phi$ with $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$ satisfies

We set a class of $W$-invariant convex functions on $2P$ by

Our main goal in this section is to prove

Theorem 4.2. A Kähler potential $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$ with normalized $\psi_\phi$ satisfying $(4.1)$ if and only if the Legendre function $u_\phi$ of $\psi_\phi$ lies in $\mathcal E^1_{K\times K}(2P)$. In particular, $u_\phi$ is locally bounded in ${\rm Int}(2P)$ if $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$.

As in ^[17], we need to establish a comparison principle for the complex Monge-Ampère measure in $\mathcal E^1_{K\times K}(M, -K_M)$. For our purpose, we will introduce a weighted Monge-Ampère measure on $\mathfrak a$ in the following.

4.1.   Weighted Monge-Ampère measure

Definition 4.3. Let $\Omega\subseteq \mathfrak a$ be a $W$-invariant domain and $\psi$ any $W$-invariant convex function on $\Omega$. Define a weighted Monge-Ampère measure on $\Omega$ by

where $\partial \psi(\cdot)$ is the normal mapping of $\psi$.

Remark 4.4. Let $\{\psi_k\}$ be a sequence of convex functions which converges locally uniformly to $\psi$ on $\Omega$, then $\text{MA}_{\mathbb R; \pi}(\psi_k)$ converges to $\text{MA}_{\mathbb R; \pi}(\psi)$ (cf. [2,Section 15]).

We have the following $KAK$-integration for the measure $\omega_\phi^n$ with $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$.

Lemma 4.5. Let $\omega_\phi\, = \, \sqrt{-1}\partial\bar\partial \psi_\phi$ with $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$. Then for any $K\times K$-invariant continuous uniformly bounded function $f$ on $G$, it holds

Proof. First we assume that $f$ is a $K\times K$-invariant continuous function with compact support on ${\mathfrak a}$. We take a sequence of smooth $W$-invariant convex functions $\psi_k\searrow\psi$ and let $\omega_k = \sqrt{-1}\partial\bar\partial\psi_k$. Then for any $W$-invariant $\Omega'\Subset {\mathfrak a}$, it holds

By the standard $KAK$-integration formula in Proposition 3.2, it follows that

Since

we get (4.3) by Remark 4.4.

Next we choose a sequence of exhausting $W$-invariant convex domains $\Omega_k$ in ${\mathfrak a}$ and a sequence of $W$-invariant convex functions with the support on $\Omega_{k+1}$ such that $f_k = f|_{\Omega_k}$. Since $\omega^n$ has full MA-mass, we get

4.2.   Comparison principles

In this subsection, we establish an ordinary comparison principle for the weighted Monge-Ampère measure $\text{MA}_{\mathbb R; \pi}(\psi)$. As showed in ^[17], this will then lead a global comparison principle (see Proposition 4.7 below) which can be used to estimate the MA mass of Kähler potential. We will only prove Proposition 4.6 and omit other proofs, since the others follow directly from Proposition 4.6 by corresponding arguments in ^[17].

Proposition 4.6. Let $\Omega\subseteq\mathfrak a$ be a $W$-invariant domain and $\varphi, \psi$ be two convex functions on $\Omega$ such that

Then

Proof. It is sufficient to prove (4.5) when $\varphi$ and $\psi$ are smooth, since we can approximate general $\varphi$ and $\psi$ by smooth $W$-invariant convex functions by Lemma 4.5. Let

Then

and

Using the fact that

and integration by parts, we have

Also

Note that

Plugging (4.7)–(4.9) into (4.6) and using the boundary condition (4.4), we have

Hence we get (4.5).

By the above Proposition, we get the following analogue of [17,Lemma 2.3], which gives a global comparison principle for the weighted Monge-Ampère measures.

Proposition 4.7. Let $\varphi, \psi$ be two $W$-invariant convex functions on $\mathfrak a$ so that

and

Then

As an application of Proposition 4.7, we get by the argument of [17,Lemma 2.7],

Lemma 4.6. Let $\psi$ be a $W$-invariant convex function on $\mathfrak a$ and $u$ its Legendre function. Suppose that for some constant $C$,

where $v_{2P}$ is the support function of $2P$. Then

if $u < +\infty$ everywhere in the interior of $2P$.

The inverse of Lemma 4.8 is also true as an analogue of [17,Theorem 3.6]. In fact, we have

Proposition 4.9. Let $\phi$ be a $K\times K$-invariant potential. Then $\psi_\phi$ satisfies (4.11) if and only if$u_\phi$ is finite everywhere in ${\rm Int}(2P)$.

Proposition 4.9 will be used in the proof of Theorem 4.2 in next subsection.

4.3.   Proof of Theorem 4.2

It is easy to see that (4.1) is equivalent to (4.2). Thus, to prove Theorem 4.2, we only need to show that

The following lemma can be found in [9,Lemma 2.7] (proved in [9,Appendix]).

Lemma 4.10. Let $\psi$ be a convex function on ${\mathfrak a}$ and $u_\psi$ its Legendre dual on $P$.

(1) $u_\psi$ is differentiable at $p$ if and only if the sup defining $u_\psi$ is attained at a unique point $x_p\in {\mathfrak a}$ and $x_p = \nabla u_\psi(p)$;

(2) Suppose that $(\psi-\psi_0)\in \mathcal E^1_{K\times K}(M, -K_M)$. Let $p\in P$ at which $u_\psi$ is differentiable. Then for any continuous uniformly bounded function $v$on ${\mathfrak a}$, it holds

where $u_{\psi+tv}$ is the Legendre function of $\psi+tv$ as in (3.15) which is well-defined since $v$ is continuous and uniformly bounded on ${\mathfrak a}$.

Remark 4.11. By Lemma 4.5 and Part (1) in Lemma 4.7, we can prove the following: Let $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$, then for any $K\times K$-invariant continuous uniformly bounded function $f$ on $G$, it holds

Proof of Theorem 4.2. We will follow the arguments in [17,Proposition 3.9] to prove the theorem.

Necessary part. First we show that $\phi$ has full MA-mass by Proposition 4.9. In fact, by a result in [36,Lemma 4.5], we see that for any $W$-invariant convex polytope $2P'\subseteq 2P$, there is a constant $C = C(P')$ such that for any $W$-invariant convex $u_\phi\geq0,$

This implies that $u_\phi$ is finite everywhere in Int$(2P)$ by the convexity of $u_\phi$. Thus we get what we want from Proposition 4.9.

Next we prove that $\phi$ is $L^1$-integrate associated to the MA-measure $\omega_{\phi}^n$. Let $\psi_1 = \psi_0+\phi$ ($\phi$ may be different to a constant) be satisfying (4.1). We define a distance between $\psi_0$ and $\psi_1$ for $p\geq1$,

where $\phi_t\in\mathcal E^1(M, -K_M)$ $(t\in[0, 1])$ runs over all curves joining $0$ and $\phi$ with $\omega_{\phi_t}\ge 0$. Choose a special path ${\phi_t}$ such that the corresponding Legendre functions of $\psi_t = \psi_0+\phi_t$ are given by

where $u_1$ and $u_0$ are the Legendre functions of $\psi_1$ and $\psi_0$, respectively. Note that by Lemma 4.10,

Then by Lemma 4.5 (or Remark 4.11), we get

On the other hand, by a result of Darvas-Rubinstein ^[19], there are uniform constant $C_0$ and $C_1$ such that for any Kähler potential $\phi$ with full MA-measure it holds,

Thus we obtain

Hence, $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$.

Sufficient part. Assume that $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$. We first deal with the case of $\phi\in L^\infty(M)\cap C^\infty(G)$. Then

and

is a bijection. Thus

Moreover,

Hence,

Next for an arbitrary $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$, we choose a sequence of smooth $K\times K$-invariant functions $\{\phi_j\}$ decreasing to $\phi$

such that $\phi_j\in C^\infty(G)$ and

Then as in (4.17), we have

where $u_j$ is the Legendre function of $\psi_j = \psi_0 +\phi_j.$ Note that

and $u_{j}\nearrow u_\phi$. Thus by taking the above limit as $j\to+\infty$, we also get (4.17).

5.   Computation of the Ricci potential

In this section, we assume that the associated polytope $P$ is fine. Then by Lemma 3.4, $(\psi_{2P}-\psi_0)\in\mathcal E^1_{K\times K}(M, -K_M)$ is a smooth $K\times K$-invariant Kähler potential on $G$. It follows that

gives a Ricci potential $h_0$ of $\omega_{2P}$, which is smooth and $K\times K$-invariant on $G$.

The following proposition gives an upper bound of $h_0$.

Proposition 5.1. The Ricci potential $h_0$ of $\omega_{2P}$ is uniformly bounded from above on $G$. In particular, $e^{h_0}$ is uniformly bounded on $G$.

Proof. As in [36,Sections 3.2 and 4.3], the proof is based on a direct computation of asymptotic behavior of $h_0$ near every point of $\partial(2P_+)$. Recall that

Since the Ricci potential of $h_0$ is also $K\times K$-invariant, by (5.1) and (3.9),

Here in the second line we take the Legendre transformation

Note that

and

Thus we have

By (5.3), $h_0$ is locally bounded in the interior of $2P_+$. Thus we need to prove that $h_0$ is bounded above near each $y_0\in\partial(2P_+)$. There will be three cases as follows.

Case-1. $y_0\in\partial(2P_+)$ and is away from any Weyl wall (see Figure 1).

Note that

Then we get as $y\to y_0$,

By (5.3), it follows that

However, by Lemma 3.1, we have

Hence $h_0$ is bounded near $y_0$.

Case-2. $y_0$ lies on some Weyl walls but away from any facet of $2P$ (see Figure 2).

In this case it is direct to see that $h_0$ is bounded near $y_0$ since

are all bounded.

Case-3. $y_0$ lies on the intersection of $\partial (2P)$ with some Weyl walls. In this case, by (3.1), we rewrite (5.3) as

Here we used a fact that

Set

for each $\alpha\in\Phi_+$. Then

Note that each $I_\alpha(y)$ involves only one root $\alpha$. Thus, without loss of generality, we may assume that $y_0$ lies on only one Weyl wall.

Assume that $y_0\in \partial (2P)\cap W_{\alpha_0}$ for some simple Weyl wall $W_{\alpha_0}$, $\alpha_0\in\Phi_+$ and it is away from other Weyl walls. Now we estimate each $I_\alpha(y)$ in (5.5). When $\beta\not = \alpha_0$, it is easy to see that

Then, by (5.4), we have

Note that $y_0\in\{\beta(y) > 0\}$. Thus any facet $F_A$ passing through $y_0$ lies in $\{\beta(y) > 0\}$ or is orthogonal to $W_\beta$. Since $2P$ is convex and $s_\beta$-invariant, where $s_\beta$ is the reflection with respect to $W_\beta$, these facets must satisfy

Hence, for any $\beta\not = \alpha_0,$ we get

It remains to estimate the second term in $I_{\alpha_0}(y)$,

We first consider a simple case that $y_0$ lies on the intersection of $W_{\alpha_0}$ with at most two facets of $2P$. Then there will be two subcases: $y_0\in W_{\alpha_0} \cap F_1$ or $y_0\in W_{\alpha_0}\cap F_1\cap F_2$, where $F_1, F_2$ are two facets of $P$.

Case-3.1. $y_0\in W_{\alpha_0} \cap F_1$ is away from other facets of $2P$. Then $F_1$ is orthogonal to $W_{\alpha_0}$ (see Figure 3).

It follows that $l_A(y_0)\not = 0$ for any $A\not = 1$. Thus

Let $\{F_1, ..., F_{d_1}\}$ be all facets of $P$ such that $\alpha_0(u_{A})\geq0, A = 1, ..., d_1$. Let $s_{\alpha_0}$ be the reflection with respect to $W_{\alpha_0}$. Then by $s_{\alpha_0}$-invariance of $P$, for each $A'\not\in\{1, ..., d_1\}$ there is some $A\in\{1, ..., d_1\}$ such that

It follows that

Thus, by (5.8) and the fact that $\alpha_0(u_1) = 0$, we obtain

Hence

Together with (5.6), we see that $h_0$ is bounded near $y_0$.

Case-3.2. $y_0\in W_{\alpha_0} \cap F_1\cap F_2$ and is away from other facets of $2P$ (see Figure 4).

By the $W$-invariance of $2P$, it must hold $F_1 = s_{\alpha_0}(F_2)$. We may assume that $F_2\subseteq\overline{\mathfrak a_+}$ and then

As $y\to y_0$ we have

It follows that

Then the second term in $I_{\alpha_0}(y)$ becomes

We will settle it down according to the different rate of $\frac{\alpha_0(y)}{l_2(y)}$ below.

Case-3.2.1. $\alpha_0(y) = o(l_2(y))$. Then

Note that $s_{\alpha_0}(u_1) = u_2\in\overline{\mathfrak a_+}$, we have

Using the above relation, (5.9) and (5.10), we get

Here we used $l_1 = l_2(1+o(1))$ in the last equality.

Note that by our assumption $\alpha_0(u_2) > 0$. Then

since $\alpha_0(u_2)\in\mathbb Z$. Hence, as $l_1(y), l_2(y) \to0^+$, by (5.5), (5.6) and (5.11), we see that $h_0$ is bounded from above in this case.

Case-3.2.2. $c\leq\frac{\alpha_0(y)}{l_2(y)}\leq C$ for some constants $C, c > 0$. Then

and the right hand side of (5.5) becomes

Again $h_0$ is also bounded from above.

Case-3.2.3. $\frac{\alpha_0(y)}{l_2(y)}\to+\infty$. Then

and the right hand side of (5.5) becomes

Hence $h_0$ is bounded from above as in Case-3.2.1.

Next we consider the case that there are facets $F_1..., F_s\; (s \ge 3)$ such that

and it is away from any other facet of $2P$. We only need to control the term (5.7) as above. If $F_1, ..., F_s$ are all orthogonal to $W_{\alpha_0}$ as in Case-3.1, we see that $h_0(y)$ is uniformly bounded. Otherwise, for any $y$ nearby $y_0$ there is a facet $F = F_{i'}$ for some $i'\in \{1, ..., s\}$ such that

As $y\to y_0$, up to passing to a subsequence, we can fix this $i'$. Clearly, $y_0\in W_{\alpha_0}\cap F_1\cap F_2$ as in Case-3.2, where $F_2 = F\subseteq\overline{\mathfrak a_+}$ and $F_1 = s_{\alpha_0}(F)$ for the reflection $s_{\alpha_0}$. Hence by following the argument in Case-3.2, we can also prove that $h_0(y)$ is uniformly bounded from above. Therefore, the proposition is true in Case-3. The proof of our proposition is completed.

Remark 5.2. We note that $h_0$ is always uniformly bounded in Case-1, Case-2 and Case-3.1. Furthermore, if rank$(G) = 2$, there are at most two facets $F_1, F_2$ intersecting at a same point $y_0$ of $W_{\alpha_0}$ as in Cases-3.2.1–3.2.3, thus, by the asymptotic expressions of $h_0$ in $(5.11)$, $(5.12)$ and $(5.13)$, respectively, we see that $h_0$ is uniformly bounded if and only if the following relation holds,

In other words, in Cases-3.2.1–3.2.3,

if $(5.14)$ does not hold.

Remark 5.3. $(5.14)$ always holds when $M$ is a smooth $G$-compactification. This is because the Guillemin metric can be extended to a global one on $M$ and so $h_0$ is uniformly bounded (cf. [5,Proposition 3.2]). We also note that in this case $(5.14)$ can come from Lemma 6.1 in ^[39] directly.

6.   Reduced Ding functional and existence criterion

By Theorem 4.2, we see that for any $u\in{\mathcal E}^1_{K\times K}(2P)$, its Legendre function

corresponds to a $K\times K$-invariant weak Kähler potential $\phi_u = \psi_u-\psi_0$ which belongs to $\mathcal E^1_{K\times K}(M, -K_M)$. Here we can choose $\psi_0$ to be the Legendre function $\psi_{2P}$ of Guillemin function $u_{2P}$ as in (3.12). As we know, $e^{-\phi_u}\in L^p(\omega_0)$ for any $p\ge 0$. Thus $\int_{\mathfrak a_+}e^{-\psi_u}\mathbf J(x)dx$ is well-defined.

We introduce the following functional on $\mathcal E^1_{K\times K}(2P)$ by

where

and

It is easy to see that on a smooth Fano compactification of $G$,

and $\mathcal D(u_\phi)$ is just the Ding functional $F(\phi)$. We note that a similar functional on such Fano manifolds has been studied for Mabuchi solitons in [37,Section 4]). Hence, for convenience, we call $\mathcal D(\cdot)$ the reduced Ding functional on a $\mathbb Q$-Fano compactifications of $G$.

In this section, we will use the variation method to prove Theorem 1.2 by verifying the properness of $\mathcal D(\cdot)$. We assume that the associated polytope $P$ is fine so that the Ricci potential $h_0$ is uniformly bounded above by Proposition 5.1.

6.1.   A criterion for the properness of $\mathcal D(\cdot)$

In this subsection, we establish a properness criterion for $\mathcal D(u_\phi)$, namely,

Proposition 6.1. Let $M$ be a $\mathbb Q$-Fano compactification of $G$. Suppose that the associated polytope $P$ is fine and satisfies $(1.2)$. Then there are constants $\delta$ and $C_\delta$ such that

The proof follows the line of argument in ^[37]. We note that $u_\phi$ satisfies the normalized condition $u\geq u(O) = 0$. Then we have the following estimate for the linear term $\mathcal L(\cdot)$ as in [36,Proposition 4.2].

Lemma 6.2. Under the assumption (1.2), there exists a constant $\lambda > 0$ such that

For the non-linear term $\mathcal F(\cdot)$, we can also get an analogy of [37,Lemma 4.8] as follows.

Lemma 6.3. For any $\phi\in\mathcal E^1_{K\times K}(M, -K_M)$, let

Then

Consequently, for any $c > 0$,

Let $\phi_0, \phi_1\in\mathcal E^1_{K\times K}(M, -K_M)$ and $u_0, u_1$ be two Legendre functions of $\psi_0+\phi_0$ and $\psi_0+\phi_1$, respectively. Let $u_t$ $(t\in [0, 1])$ be a linear path connecting $u_0$ to $u_1$ as in (4.14). Then by Theorem 4.2, the corresponding Legendre functions $\psi_t$ of $u_t$ give a path in $\mathcal E^1_{K\times K}(M, -K_M)$. The following lemma shows that $\mathcal F(\psi_t)$ is convex in $t$.

Lemma 6.4. Let

Then $\hat{\mathcal F}(t)$ is convex in $t$ and so is $\mathcal F(\psi_t)$.

Proof. By definition, we have

On the other hand,

Combining these two inequalities, we get

Hence, by applying the Prekopa-Leindler inequality to three functions $e^{-\psi_t}\mathbf J, e^{-\psi_1}\mathbf J$ and $e^{-\psi_0}\mathbf J$ (cf. [27,Theorem 7.1]), we prove

This means that $\hat{\mathcal F}(t)$ is convex.

Proof of Proposition 6.1. By Proposition 5.1,

is bounded, where $y(x) = \nabla\psi_0(x)$. Then the functional

is well-defined on $\mathcal E^1_{K\times K}(2P)$, where

It is easy to see that $u_0$ is a critical point of $\mathcal D_A(\cdot)$. On the other hand, by Lemma 6.4, $\mathcal F(\cdot)$ is convex along any path in $\mathcal E^1_{K\times K}(M, -K_M)$ determined by their Legendre functions as in (4.14). Note that $\mathcal L_A^0(\cdot)$ is convex in $\mathcal E^1_{K\times K}(2P)$. Hence

Now together with Lemma 6.2 and Lemma 6.3, we can apply arguments in the proof of [37,Proposition 4.9] to proving that there is a constant $C > 0$ such that for any $u\in\mathcal E^1_{K\times K}(2P)$,

Therefore, we get (6.1).

6.2.   Semi-continuity

Write ${\mathcal E}^1_{K\times K}(2P)$ as

where

By [36,Lemma 6.1] and Fatou's lemma, it is easy to see that any sequence $\{u_n\}\subseteq {\mathcal E}^1_{K\times K}(2P;\kappa)$ has a subsequence which converges locally uniformly to some $u_\infty\in {\mathcal E}^1_{K\times K}(2P;\kappa)$. Thus each ${\mathcal E}^1_{K\times K}(2P;\kappa)$, and so ${\mathcal E}^1_{K\times K}(2P)$ is complete. Moreover, we have

Proposition 6.5. The reduced Ding functional $\mathcal D(\cdot)$ is lower semi-continuous on the space ${\mathcal E}^1_{K\times K}(2P)$. Namely, for any sequence $\{u_n\}\subseteq{\mathcal E}^1_{K\times K}(2P)$, which converges locally uniformly to some $u_\infty$, we have $u_\infty\in {\mathcal E}^1_{K\times K}(2P)$ and it holds

Proof. By Fatou's lemma, we have

Then $u_\infty\in {\mathcal E}^1_{K\times K}(2P)$ and

It remains to estimate $\mathcal F(u_\infty)$. Note that $u_\infty$ is finite everywhere in Int$(2P)$ by the locally uniform convergence and its Legendre function $\psi_\infty\le v_{2P}$. Thus, for any $\epsilon_0\in(0, 1)$ there is a constant $M_{\epsilon_0} > 0$ such that (cf. [17,Lemma 2.3]),

On the other hand, the Legendre function $\psi_n$ of $u_n$ also converges locally uniformly to $\psi_\infty$. Then

almost everywhere. Since

we have

as long as $n\gg1$. Note that

By choosing an $\epsilon_0$ such that $4\rho\in(1-\epsilon_0)\text{Int}(2P)$, we get

Hence, combining this with (6.8) and (6.9) and using Fatou's lemma, we derive

Therefore, we have proved (6.6) by (6.7).

6.3.   Proof of Theorem 1.2

Now we prove the sufficient part of Theorem 1.2. Suppose that (1.2) holds. Then by Propositions 6.1 and 6.5, there is a minimizing sequence $\{u_n\}$ of $\mathcal D(\cdot)$ on ${\mathcal E}^1_{K\times K}(2P)$, which converges locally uniformly to some $u_\star\in{\mathcal E}^1_{K\times K}(2P)$ such that

Let $\psi_\star$ be the Legendre function of $u_\star$. Then by Theorem 4.2, we have

We need to show that $\phi_\star$ satisfies the Kähler-Einstein equation (2.1).

Proposition 6.6. $\phi_\star$ satisfies the Kähler-Einstein equation $(2.1)$.

Proof. Let $\{u_t\}_{t\in[0, 1]}\subseteq{\mathcal E}^1_{K\times K}(2P)$ be a family of convex functions with $u_0 = u_\star$ and $\psi_t$ the corresponding Legendre functions of $u_t$. Then by Part (2) in Lemma 4.10,

Note that

Thus by (4.11) in Lemma 4.8, we get

For any continuous, compactly supported $W$-invariant function $\eta\in C_0({\mathfrak a})$, we consider a family of functions $u_\star+t\eta$. In general, it may not be convex for $t\not = 0$ since $u_\star$ is just weakly convex. In the following, we use a trick to modify the function $\mathcal D(u_t)$ as in [9,Section 2.6]. Define a family of $W$-invariant functions by

Then it is easy to see that the Legendre function $\hat u_t$ of $\hat\psi_t$ satisfies

By Theorem 4.2, we see that $(\hat\psi_t-\psi_0)\in\mathcal E^1_{K\times K}(M, -K_M)$. Without loss of generality, we may assume that $\hat\psi_t$ satisfies (4.1).

Let

Then

and

Claim 6.7. ${\mathcal L}(\hat u_t)+\hat u_t(4\rho)$ is differentiable for $t$. Moreover,

To prove this claim, we let a convex function $g(t) = \hat u_t(p)$ for each fixed $p\in2P$. Then it has left and right derivatives $g'_-(t; p), g'_+(t; p)$, respectively. Moreover, they are monotone and $g'_-(t; p)\leq g'_+(t; p)$. Thus, $g'_-, g'_+\in L^\infty_{\text{loc}}$. It follows that

and by the Lebesgue monotone convergence theorem,

Recall that $g'_-(t; p) = g'_+(t; p)$ holds almost everywhere. Thus we see that

is differentiable.

Note that

where $u_{\psi_\star+t\eta}$ is the Legendre function of ${\psi_\star+t\eta}$. It follows from Part (2) in Lemma 4.10 that

Hence by Lemma 4.5 (or Remark 4.11), we get

where $\phi_\star = \psi_*-\psi_0$. The claim is proved.

Similar with Claim 6.7, we have

Thus, by (6.12)–(6.15), we derive

As a consequence,

Therefore, by Lemma 4.5 and $KAK$-integration formula, we prove that $\phi_\star$ satisfies (2.1) on $G$.

Next we show that $\omega_{\phi_\star}$ can be extended to a singular Kähler-Einstein metric on $M$. Choose an $\epsilon_0$ such that $4\rho\in\text{Int}(2(1-\epsilon_0)P)$. Since $u_\star$ is locally uniformly bounded on $2P$, there is a constant $C_\star > 0$ such that

Thus

is bounded on $\mathfrak a_+$. Also $\pi(\partial\psi_\star)$ is bounded. Therefore, by (4.11), for any $\epsilon > 0$, we can find a neighborhood $U_\epsilon$ of $M\setminus G$ such that

This implies that $\phi_\star$ can be extended to be a global solution of (2.1) on $M$. The proposition is proved.

7.   $\mathbb Q$-Fano compactifications of ${SO}_4(\mathbb C)$

In this section, we will construct $\mathbb Q$-Fano compactifications of ${SO}_4(\mathbb C)$ as examples and in particular, we will prove Theorem 1.3. Note that in this case rank$(G) = 2$. Thus we can use Theorem 1.2 to verify whether there exists a Kähler-Einstein metric on a $\mathbb Q$-Fano ${SO}_4(\mathbb C)$-compactification by computing the barycenter of their associated polytopes $P_+$. For convenience, we will work with $P_+$ instead of $2P_+$ throughout this section. It is easy to see that the condition (1.2) is equivalent to

Let

Then we can choose a maximal torus of ${\rm SO}_4(\mathbb C)$ in $GL_4(\mathbb C)$ as follows,

Recall $\mathfrak M$ the lattice of ${\rm SO}_4(\mathbb C)$-weights. Denote the basis of $\mathfrak N = {\rm Hom}_\mathbb Z(\mathfrak M, \mathbb Z)$ by $E_1, E_2$ which generates the actions of $R(z^1)$ and $R(z^2)$. Thus we have the two positive roots in $\mathfrak M$,

Also we get

and

7.1.   Gorenstein Fano ${{\rm{SO}}}_4(\mathbb C)$-compactifications

In this subsection, we use Lemma 3.1 to exhaust all polytopes associated to the Gorenstein Fano compactifications. Here by Gorenstein, we mean that $K_{M_{reg}}^{-1}$ can be extended to a holomorphic line bundle on $M$. In this case, the whole polytope $P$ is a lattice polytope. Also, since $2\rho = (2, 0)$, each outer edge of $P_+$ must lies on some line

An edge of $P_+$ is called an outer one if it does not lie in any Weyl wall, cf. ^[36].

for some coprime pair $(p, q)$. Assume that $l_{p, q}\geq 0$ on $P$. By the convexity and $W$-invariance of $P$, $(p, q)$ must satisfy

Let us start at the outer edge $F_1$ of $P_+$ which intersects the Weyl wall

There are two cases: Case-1. $F_1$ is orthogonal to $W_1$; Case-2. $F_1$ is not orthogonal to $W_1$.

Case-1. $F_1$ is orthogonal to $W_1$. Then $F_1$ lies on

Consider the vertex $A_1 = (x_1, 3-x_1)$ of $P_+$ on this edge and suppose that the other edge $F_2$ at this point lies on

Thus

and by the convexity of $P$,

We will have two subcases according to the possible choice $A_1 = (2, 1)$ or $(3, 0)$.

Case-1.1. $A_1 = (2, 1)$. Then by (7.4),

Thus $q_2 = 1$ and $p_2\geq2$.

On the other hand, $l_{p_2, q_2}$ must pass another lattice point $A_2 = (x_2, y_2)$ as the other endpoint of $F_2$. It is direct to see that there are only two possible choices $p_2 = 2, 4$ and three choices of $A_2 = (5, -5)$, $(3, -1)$ and $(3, -3)$.

Case-1.1.1. $A_2 = (5, -5)$ which lies on the other Weyl wall $W_2 = \{x+y = 0\}$. There can not be any other outer edges of $P_+$, and $P_+$ is given by the first case in Figure 5 (we denote it by $P_+^{(1)}$). By Theorem 1.2 (or equivalently (7.1)), this compactification admits no Kähler-Einstein metric.

Case-1.1.2. $A_2 = (3, -1)$. Then we exhaust the third edge $F_3$ which lies on

so that

Hence the only possible choice is $p_3 = 1, q_3 = 0$ and the other endpoint of $F_3$ is $A_3 = (3, -3)$. Then $P_+$ is given by the second case in Figure 5. Again, this compactification admits no Kähler-Einstein metric.

Case-1.1.3. $A_2 = (3, -3)$ which lies on the other Weyl wall $W_2 = \{x+y = 0\}$. There can not be any other outer edges of $P_+$, and $P_+$ is given by the third one in Figure 5. By Theorem 1.2, this compactification admits no Kähler-Einstein metric.

Case-1.2. $A_1 = (3, 0)$. By the same exhausting progress as in Case-1.1. There are two possible polytopes $P_+$, Case-1.2.1 and Case-1.2.2 (see the first two cases of Figure 6).

Case-1.2.1. This compactification admits no Kähler-Einstein metric.

Case-1.2.2. This compactification admits a Kähler-Einstein metric.

Case-2. $F_1$ is not orthogonal to $W_1$. Then its intersection $A_1 = (x_1, x_1)$ with $W_1$ is a vertex of $P$. We see that $F_1$ lies on $l_{p_1, q_1}$ and

So the only choice is

and $A_1 = (3, 3)$. The only new polytope $P_+$ is given by the last one of Figure 6, which admits a Kähler-Einstein metric.

It is known that Case-1.1.2, Case-1.2.1 and Case-2 are the only smooth ${\rm SO}_4(\mathbb C)$-compactifications as shown in ^[41]. We summarize results of this subsection in Table 1.

7.2.   $\mathbb Q$-Fano ${ {\rm{SO}}}_4(\mathbb C)$-compactifications

In general, for a fixed integer $m > 0$, it will be hard to give a classification of all $\mathbb Q$-Fano compactifications such that $-mK_X$ is Cartier. This is because when $m$ is sufficiently divisible, there will be too many repeated polytopes according to Lemma 3.1. In the following, we give a way to exhaust all $\mathbb Q$-Fano polytopes according to the intersection point of $\partial P_+$ with $x$-axis.

We will adopt the notations from the previous subsection. We consider the intersection of $P_+$ with the positive part of the $x$-axis, namely $(x_0, 0)$. Then

for some $p_0\in\mathbb N_+$, and there is an edge which lies on some $\{l_{p_0, q_0} = 0\}$. Without loss of generality, we may also assume that $\{l_{p_0, q_0} = 0\}\cap\{y > 0\}\not = \emptyset$. Thus by symmetry, it suffices to consider the case

Indeed, by the prime condition, $q_0\not = 0, \pm p_0$ if $p_0\not = 1$. Hence, we may assume

We associate this number $p_0$ to determine each $\mathbb Q$-Fano polytope $P$ (and hence $\mathbb Q$-Fano compactifications of ${\rm SO}_4(\mathbb C)$). By the convexity, other edges determined by $l_{p, q}$ must satisfy (see Figure 7 below)

since we assume that

Thus, once $p_0$ is fixed, there are only finitely possible $\mathbb Q$-Fano compactifications of ${\rm SO}_4(\mathbb C)$ associated to it. In Table 2, we list all possible $\mathbb Q$-Fano compactifications with $p_0\leq2$, and test the existence of Kähler-Einstein metrics on these compactifications. In Appendix 1 we figure out the associated polytopes $P$ of nine non-smooth examples in Table 2.

7.3.   Proof of Theorem 1.3

Proof. We introduce some notations for convenience: For any domain $\Omega\subset\overline{\mathfrak a_+^*}$, define

and

By Theorem 1.2 and (7.2), we have $\bar c(P_+) > 2$ whenever the $\mathbb Q$-Fano compactification of ${\rm SO}_4(\mathbb C)$ admits a Kähler-Einstein metric.

Recall the numbers $p_0, q_0$ introduced in Section 7.2. Consider the line segment $I_t$ cut by $P_+$ on $\{y = x-2t\}$ for $t\geq0$ (see Figure 8 below). Set

where $ds$ is the standard Lebesgue measure on $I_t$. Then $\bar C(t), l(t)$ are the mean value of $\bar c(\cdot)$ and length of $I_t$ against the weight $\pi$, respectively. Also, since $P_+$ is bounded, there is some $0\leq t_0 < +\infty$ so that $I_t$ is non-empty only on $[0, t_0]$. Thus we get

On the other hand, for each $0\leq t\leq t_0$, the line segment $I_{t} = \{(x_0, -x_0)+s(1, 1)|\; 0\leq s\leq S_t\}$, where $(x_0, -x_0)\in P_+$ and $(x_0, -x_0)+S_t(1, 1)\in\partial P$, satisfies

Here in the first equality we use the relation (7.5), and the second follows from the fact that the endpoint of $I_0$ can not exceed the intersection point

Thus by (7.7), we get

By the fact $\bar c(P_+) > 2$, we derive from the above upper bound of $\bar c(P_+)$,

Then

It turns that for $p_0\geq9$,

However,

where $\text{Vol}(P^{(2)}_+)$ and $\text{Vol}(P^{(4)}_+)$ are volumes of polytopes in Case-1.1.2 and Case-1.2.1, respectively. Hence, there is no desired Kähler-Einstein polytope with its volume equals to ${\rm Vol}(P^{(2)}_+)$ or ${\rm Vol}(P^{(4)}_+)$ when $p_0\geq9$.

Since $q_0\in\mathbb N$, we can improve (7.9) to

Here $[x] = \max_{n\in\mathbb Z}\{n\leq x\}$. By the above estimation, when $p_0 = 4, 6, 7, 8$, we have

As a consequence, they are not Kähler-Einstein polytopes. Hence, it remains to deal with the cases when $p_0 = 3, 5$. In these two cases, we shall rule out polytopes that may not satisfy (7.10).

When $p_0 = 5$, there are three possible choices of $q_0$, i.e., $q_0 = 1, 2, 3$ by (7.8). It is easy to see that (7.10) still holds for the first two cases by the second relation in (7.9). Thus we only need to consider all possible polytopes when $q_0 = 3$. In this case, $\{l_{5, 3} = 0\}$ is an edge of $P_+$.

Case-7.3.1. $P_+$ has only one outer face which lies on $\{l_{5, 3} = 0\}$. Then

Case-7.3.2. $P_+$ has two outer edges. Assume that the second one lies on $\{l_{p_1, q_1} = 0\}$. Then

By a direct computation, we see that (7.10) holds except the following two subcases:

Case-7.3.2.1. $p_1 = 4, q_1 = 3$,

Case-7.3.2.2. $p_1 = 2, q_1 = 1$,

Case-7.3.3. $P_+$ has three outer edges. Then $P_+$ is obtained by cutting one of polytopes in Case-7.3.2 with adding new edge $\{l_{p_2, q_2} = 0\}$. In fact we only need to consider $P_+$ obtained by cutting Case-7.3.2.1 and Case-7.3.2.2 above, since it obviously satisfies (7.10) in the other cases. By our construction, we can assume that $|q_2|\leq p_2\leq p_1$. The only possible $P$ which does not satisfy (7.10) is the case that $p_1 = 4, q_1 = 3$ and $p_2 = 2, q_2 = 1$. However,

Case-7.3.4. $P_+$ has four outer edges. We only need to consider $P_+$ which is obtained by cutting Case-7.3.3 with adding new edge $\{l_{p_3, q_3} = 0\}$ with $|q_3|\leq p_3\leq 2$. One can show that all of these possible $P_+$ satisfy (7.10). Thus we do not need to consider more polytopes with more than four outer edges in case of $p_0 = 5$. Hence we conclude that for all polytopes $P$ with $p_0 = 5$,

Theorem 1.3 is true when $p_0 = 5$.

The case $p_0 = 3$ can be ruled out in the same way. We only list the exceptional polytopes such that the volumes of $P_+$ do not satisfy (7.10):

Case-7.3.1'. $P_+$ has only one outer face $\{l_{3, 2} = 0\}$. Then

Case-7.3.2'. $P_+$ has two outer face $\{l_{3, 2} = 0\}$ and $\{l_{2, 1} = 0\}$. Then

In summary, when $p_0\ge 3$, the volume of $P_+$ is not equal to either ${\rm Vol}(P^{(2)}_+)$ or ${\rm Vol}(P^{(3)}_+)$. Finally by exhausting all possible compactifications for $p_0 = 1, 2$ (see Table-2), we finish the proof of Theorem 1.3.

Remark 7.1. If $P_+$ is further symmetric under the reflection with respect to the $x$-axis, it is easy to see its barycenter is $(\bar x(P_+), 0)$ and

Thus a Kähler-Einstein polytope of this type must satisfy

Acknowledgments

Yan Li was partially supported by NSFC Grant 12101043 and Beijing Institute of Technology Research Fund Program for Young Scholars. Xiaohua Zhu was partially supported by BJSF Grants Z180004. Both Gang Tian and Xiaohua Zhu were partially supported by National Key R & D Program of China 2020YFA0712800.

Conflict of interest

The authors declare no conflict of interest.

A.   Appendix 1: Non-smooth $\mathbb Q$-Fano ${{\rm{SO}}}_4(\mathbb C)$-compactifications with $p_0\leq2$

In this appendix we list all polytopes $P_+$ of non-smooth $\mathbb Q$-Fano ${\rm SO}_4(\mathbb C)$-compactifications with $p_0\leq2$, namely, (3)–(8) and (10)–(12) labeled as in Table 2 (see Figure 9 below).

B.   Appendix 2: An improvement of Theorem 1.2

In this appendix, we show that the fine condition in Theorem 1.2 can be dropped by a recent result of Li in [32,Theorem 1.2]. Namely, we can generalize Theorem 1.1 to a $\mathbb Q$-Fano $G$-compactification $M$.

Let $\mathbb G$ be the image of $G\times G$ embedded in ${\rm Aut}_0(M)$, the identity component of ${\rm Aut}(M)$. Then the image $\mathbb T'$ of $Z(G)\times\{e\}$ is a subtorus of $\mathbb T = Z(\mathbb G)$. Let $\mathbf{M}^{\rm NA}(\cdot)$ and $\mathbf{J}^{\rm NA}_\mathbb T(\cdot)$ be the non-Archimedean Mabuchi K-energy and $J$-functional defined in ^[13,32], respectively. By [32,Theorem 1.2], it suffices to check that $M$ is $\mathbb G$-uniformly K-stable under the assumption (1.2). More precisely, by [36,Proposition 4.2] (an analogous version of Lemma 6.2), we prove

Theorem B.1. Suppose that $(1.2)$ holds. Then there is a constant $c_0 > 0$ such that for any $\mathbb G$-equivariant normal test configuration $(\mathcal X, \mathcal L)$ of $(M, -K_M)$, it holds

Consequently, Proposition 6.1 holds and so $M$ admits a (singular) Kähler-Einstein metric.

Proof. We need to compute the two functionals $\mathbf{M}^{\rm NA}(\cdot)$ and $\mathbf{J}^{\rm NA}_\mathbb T(\cdot)$. Note that these two functionals can be computed via the $\mathbb C^*$-weights on $(\mathcal X, \mathcal L)$ (cf. ^[13,40]). Moreover, the first one is also same with the CM-weight in ^[40,44,45] when the central fiber is reduced. In our case for a general $\mathbb G$-equivariant test configuration, it can be normalized with a reduced central fiber via a base change as follows.

Recall that the $\mathbb G$-equivariant normal test configurations are in one-one correspondence with $W$-invariant, convex, piecewise linear functions with rational coefficients on $P$ (cf. [5,Section 2.4]). In particular, when $M$ is a toric manifold with torus action $\mathbb T$, $\mathbb T$-equivariant normal test configurations are same with toric degenerations. Thus there is such a function $f$ associated to $\pi:(\mathcal X, \mathcal L)\to\mathbb {C}$. By a base change $z\to z^d$ on $\mathbb C$ for sufficiently divisible $d\in\mathbb N_+$, the normalization

has a reduced central fibre (cf. [13,Proposition 7.16]). In fact, from the proof of [35,Theorem 4.1], $(\mathcal X^{(d)}, \mathcal L^{(d)})$ is still a $\mathbb G$-equivariant normal test configuration associated to $df$.

By [13,Proposition 2.8], we have

Note that $\mathbf{M}^{\rm NA}(\cdot)$ is linear under the base change. It follows

On the other hand, by (3.11) and (3.13) in ^[36],

This formula was proved by Donaldson for $f$ with integral coefficients on a toric manifold [24,Proposition 4.2.1], but the arguments in his proof do not generalize to general cases.^|| Thus

^|| In fact, by using (B.3) and ^{[13,40,44,45]}, Proposition 4.2.1 in ^[24] is equivalent to that the central fiber of $(\mathcal X, \mathcal L)$ is reduced. Clearly, this condition on the central fiber does not hold in general cases.

Also, by an analogous argument for any toric degeneration on a toric manifold in ^[29], we can get

and for a twisted test configuration it holds

where $\xi$ lies in the Lie algebra ${\rm Lie}(\mathbb T')$ of $\mathbb T'$ ^[29,32].

By [36,Proposition 4.2], we see

Since $u$ is convex and $W$-invariant,

It implies that $u(ty)$ is non-decreasing for $t\geq0$. Thus by (4.2),

Combining with (B.6), we get

For the function $f$ in (B.3), by the $W$-invariance, there is always an $\xi\in{\rm Lie}(\mathbb T')$ such that

satisfies (4.2). Thus applying (B.7) to $f_{\xi}$ together with (B.3) and (B.5), we obtain

Hence (B.1) holds.