Singular K\"ahler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups

In this paper, we prove an existence result for K\"ahler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups. As an application, we classify $\mathbb Q$-Fano compactifications of $SO_4(\mathbb C)$ which admit a K\"ahler-Einstein metric with the same volume as that of a smooth Fano compactification of $SO_4(\mathbb C)$.


Introduction
Let G be an n-dimensional connected, complex reductive Lie group which is the complexification of a compact Lie group K. Let T C be a maximal Cartan torus of G whose dimension is r. Denote by Φ + a positive roots system associated to T C . Put It can be regarded as a character in a * , where a * is the dual space of real part a of Lie algebra of T C . Let π be a function on a * defined by π(y) = α∈Φ+ α, y 2 , y ∈ a * , where ·, · 1 denotes the Cartan-Killing inner product on a * .
Let M be a Q-Fano compactification of G. Since M contains a closure Z of T C -orbit, there is an associated moment polytope P of Z induced by (M, −K M ) [3,4]. Let P + be the positive part of P defined by P + = {y ∈ P | α, y > 0, ∀ α ∈ Φ + }.
Another proof of Theorem 1.1 was given by Li, Zhou and Zhu [31]. They also showed that (1.2) is actually equivalent to the K-stability condition in terms of [37] and [21] by constructing 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 bar(2P + ) ∈ 4ρ + Ξ. A more general construction of C * -action was also discussed in [18].
In the present paper, we extend the above theorem to Q-Fano compactifications of G which may be singular. It is well known that any Q-Fano compactification of G has klt-singularities [5]. For a Q-Fano variety M with klt-singularities, there is naturally a class of admissible Kähler metrics induced by the Fubini-Study metric (cf. [20]). In [10], Berman, Boucksom, Eyssidieux, Guedj and Zeriahi introduce a class of Kähler potentials associated to admissible Kähler metrics and refer it as the E 1 (M, −K M ) space. Then they define the singular Kähler-Einstein metric on M with the Kähler potential in 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 an natural problem to establish an extension of the Yau-Tian-Donaldson conjecture we have solved for smooth Fano manifolds [37,38], that is, an equivalence relation between the existence of such singular Kähler-Einstein metrics and the K-stability on a Q-Fano variety M with klt-singularities. There are many recent works on this fundamental problem. We refer the readers to [9,13,29,30,28], etc..
In this paper, we will assume that the moment polytope P of Z is fine in sense of [22], namely, each vertex of P is the intersection of precisely r facets. We will prove Theorem 1.2. Let M be a Q-Fano compactification of G such that the moment polytope P of Z is fine. Then M admits a singular Kähler-Einstein metric if and only if (1.2) holds.
By a result of Abreu [1], the polytope P of Z being fine is equivalent to that the metric induced by the Guillemin function can be extended to a Kähler orbifold metric on Z. 2 It follows from the fineness assumption of P in Theorem 1.2 that the Guillemin function of 2P induces a K × K-invariant singular metric ω 2P in E 1 (M, −K M ) (cf. Lemma 3.4). Moreover, we can prove that the Ricci potential of ω 2P on M is uniformly bounded above. We note that P is always fine when rank(G) = 2 [23,Chapter 3]. Thus for a 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 SO 4 (C)-compactifications which admits a singular Kähler-Einstein metric (cf. Section 7.1).
On the other hand, it has been shown in [17] and [33] that there are only three smooth Fano compactifications of SO 4 (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. By Theorem 1.2, we further prove Theorem 1.3. There is no Q-Fano compactification of SO 4 (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 [33] 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 whenever the initial metric is K × K-invariant. By the Hamilton-Tian conjecture [37,7,14], the limit should be a Q-Fano variety with a singular Kähler-Ricci soliton of the same volume as that of initial metric. However, by Theorem 1.3, the limit can not be a Q-Fano compactification of SO 4 (C) with a singular Kähler-Einstein metric. This implies that the limiting soliton will has less homogeneity than the initial one, which is totally different from the situation of smooth convergence of K ×K-metrics on a smooth compactification of Lie group [33].
As in [10], we use the variation method to prove Theorem 1.2, more precisely, we will prove that a modified version of the Ding functional D(·) is proper under the condition (1.2). This functional is defined for a class of convex functions E 1 K×K (2P ) associated to K × 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 ω 2P is bounded from above when P is fine (cf. Proposition 5.1). This enables us to control the nonlinear part F(·) of D(·) by modifying D(·) as done in [21,32] (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). On the other hand, we expect that the "fine" condition in Theorem 1.2 can be dropped.
The minimizer of D(·) corresponds to a singular Kähler-Einstein metric. We will prove the semi-continuity of D(·) and derive the Kähler-Einstein equation for the minimizer (cf. Proposition 6.6). Our proof is similar to what Berman and Berndtsson studied on toric varieties in [9].
The proof of the necessity part of Theorem 1.2 is same as one in Theorem 1.1. In fact, a Q-Fano compactification of G is K-unstable if bar(2P + ) ∈ 4ρ + Ξ [31]. This will be a contradiction to the semi-stability of Q-Fano variety with a singular Kähler-Einstein metric (cf. [29]). We will 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 Q-Fano varieties. In Section 3, we introduce a subspace E 1 K×K (M, −K M ) of 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 Theorem 4.2). In Section 5, we compute the Ricci potential h 0 of ω 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 Q-Fano compactifications of SO 4 (C) and in particular, we will prove Theorem 1.3.

Preliminary on Q-Fano varieties
For a Q-Fano variety M , by Kodaira's embedding theorem, there is an integer > 0 such that we can embed M into a projective space CP N by a basis of H 0 (M, K − M ), for simplicity, we assume M ⊂ CP N . Then we have a metric where ω F S is the Fubini-Study metric of CP N . Moreover, there is a Ricci potential h 0 of ω 0 such that In the case that M has only klt-singularities, e h0 is L p -integrate for some p > 1 (cf. [20,10]). We call ω an admissible Kähler metric on M if there are an embedding M ⊂ CP N as above and a smooth function ϕ on CP N satisfying: In particular, ϕ is a function on M with ϕ ∈ L ∞ (M ) ∩ C ∞ (M reg ), called an admissible Kähler potential associated to ω 0 . 3 For a general (possibly unbounded) Kähler potential ϕ, we define its complex Monge-Ampère measure ω n ϕ by ω n ϕ = lim j→∞ ω n ϕj , where ϕ j = max{ϕ, −j}. According to [10], we say that ϕ (or ω n ϕ ) has full Monge- The MA-measure ω n ϕ with a full MA-mass has no mass on the pluripolar set of ϕ in M . Thus we need to consider the measure on M reg . Moreover, e −ϕ is L p -integrable for any p > 0 associated to ω n 0 . Definition 2.1. We call ω ϕ a (singular) Kähler-Einstein metric on M with full MA-mass if ϕ satisfies the following complex Monge-Ampére equation, It has been shown in [10] that ϕ is C ∞ on M reg if it is a solution of (2.1). Thus ω ϕ satisfies the Kähler-Einstein equation on M reg , Ric(ω ϕ ) = ω ϕ .
2.1. The space E 1 (M, −K M ) and the Ding functional. On a smooth Fano manifold, there is a well-known Euler-Langrange functional for Kähler potentials associated to (2.1), often referred as the Ding functional or F-functional, defined by (cf. [19,36]) In case of Q-Fano manifold with klt-singularities, Berman, Boucksom, Eyssidieux, Guedj and Zeriahi [10] extended F (·) to the space E 1 (M, −K M ) defined by In [10], Berman, Boucksom, Eyssidieux, Guedj and Zeriahi proved the existence of solutions for (2.1) under the properness assumption (2.3) of F (·). However, this assumption does not hold in the case of existence of non-zero holomorphic vector fields such as in our case of Q-Fano G-compactifications. So we need to consider the reduced Ding functional instead to overcome this new difficulty as done on toric varieties [9,32].

Moment polytope and K × K-invariant metrics
Let M be a Q-Fano compactification of G with Z being the closure of a maximal complex torus T C -orbit. We first characterize the moment polytope P of Z associated to (M, K −1 M ). Let {F A } A=1,...,d0 be the facets of P and {F A } A=1,...,d+ be those whose interior intersects a * + . Suppose that for some prime vector u A ∈ N and the facet Let W be the Weyl group of (G, T C ). By the W -invariance, for each , where ρ ∈ a * + is given by (1.1). Then ρ A (u A ) is independent of the choice of w A ∈ W and hence it is well-defined.
The following is due to [12].
Lemma 3.1. Let M be a Q-Fano compactification of G with P being the associated moment polytope. Then for each A = 1, ..., d 0 , it holds Proof. Suppose that −mK M is a Cartier divisor for some m ∈ N. Then by [12,Section 3], where D A is the toric divisor of Z associated to u A . Thus the associated polytope of (Z, −mK M | Z ) is given by On a Q-Fano compactification of G, we may regard the G × G action as a subgroup of P GL N +1 (C) which acts holomorphically on the hyperplane bundle L = O CP N (−1). Then any admissible K × K-invariant Kähler metric ω φ ∈ 2π c 1 (L) can be regarded as a restriction of K × K-invariant Kähler metric of CP N . Thus the moment polytope P associated to (Z, L| Z ) is a Winvariant rational polytope in a * . By the K × K-invariance, the restriction of ω φ on T C is an open toric Kähler metric. Hence, it induces a strictly convex, W -invariant function ψ φ on a [6] (also see Lemma 3.3 below) such that By the KAK-decomposition ([27, Theorem 7.39]), for any g ∈ G, there are k 1 , k 2 ∈ K and x ∈ 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 a + . Thus there is a bijection between K ×K-invariant functions Ψ on G and W -invariant functions ψ on a which is given by Ψ(exp(·)) = ψ(·) : a → R.
Clearly, when a W -invariant ψ is given, Ψ is well-defined. Without of confusion, we will not distinguish ψ and Ψ, and call Ψ convex on G if ψ is convex on a.
The following KAK-integral formula can be found in [27,Proposition 5.28].
Proposition 3.2. Let dV G be a Haar measure on G and dx the Lebesgue measure on a. Then there exists a constant C H > 0 such that for any K × K-invariant, dV G -integrable function ψ 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 [17]. By the standard Cartan decomposition, we can decompose g as where t is the Lie algebra of T and is the root space of complex dimension 1 with respect to α. By [24], one can choose respectively. Then we get the Cartan decomposition of Lie algebra k of K as follows, For a K × K-invariant function ψ, Delcroix computed the Hassian of ψ in the above local coordinates as follows [17, Theorem 1.2]. Lemma 3.3. Let ψ be a K × K invariant function on G. Then for any x ∈ a + , the complex Hessian matrix of ψ in the above coordinates is diagonal by blocks, and equals to By (3.4) in Lemma 3.3, we see that ψ φ induced by an admissible K ×K-invariant Kähler form ω φ is convex on a. The complex Monge-Ampére measure is given by In particular, by Proposition 3.2, Clearly, (3.5) also holds for any Kähler potential in E 1 (M, −K M ), which is smooth and K × K-invariant on G. For the completeness, we introduce a subspace of is locally precompact in terms of convex functions on G. We will also prove its completeness by using the Legendre dual in subsequent Sections 4, 6.
3.2. Fine polytope P . In this subsection, we show that the Legendre dual of Hence, by [1, Theorem 2] and [6] (also see Lemma 3.3), 4 we can extend Lemma 3.4. Let ψ 0 be a Kähler potential of admissible K × K-invariant metric ω 0 as in (3.3). Assume that P is fine. Then the Kähler potential [35]). Moreover, we have wheren(λ) ∈ Z + . Thus we have a Kähler potential on T C by The corresponding moment map is given by 1 2 ∇ψ 2P , whose image is P .
Since P is fine, one can show directly that Recall that the Legendre function u ψ of ψ is defined as in (3.8) by It is known that ψ(x) ∈ V(2P ) if and only if u ψ is uniformly bounded on 2P since the Legendre function of v 2P is zero (cf. [34]). Thus the Legendre function u h of h| T C (x) is uniformly bounded on 2P . It follows that By (3.6), (ψ 2P − ψ 0 ) has full MA-mass, so we have completed the proof.
The following lemma is elementary.
Proof. We choose a sequence of decreasing and uniformly bounded K ×K-invariant potential φ i normalized as in (4.1) such that
It is easy to see that the Legendre function We set a class of W -invariant convex functions on 2P by The main goal in this section is to prove . As in [15], we need to establish a comparison principle for the complex Monge-Ampère measure in E 1 K×K (M, −K M ). For our purpose, we will introduce a weighted Monge-Ampère measure in the following.  where ∂ψ(·) is the normal mapping of ψ.
Remark 4.4. Let ψ k be a sequence of convex functions which converge locally uniformly to ψ on Ω, then MA R;π (ψ k ) converge to MA R;π (ψ) (cf. [2,Section 15]). This follows from the fact: Proof. First we assume that f is a K × K-invariant continuous function with compact support on a. We take a sequence of smooth W -invariant convex functions By the standard KAK-integration formula, it follows that it follows from Remark 4.4 that (4.3) is true. Next we choose a sequence of exhausting W -invariant convex domains Ω k in a and a sequence of W -invariant convex functions with the support on Ω k+1 such that f k = f | Ω k . Since ω n has full MA-mass, we get

Comparison principles.
We establish the following comparison principle for the weighted Monge-Ampére measure MA R;π (ψ).
Let Ω ⊆ a be a W -invariant domain and ϕ, ψ be two convex functions on Ω such that Proof. It is sufficient to prove (4.5) when ϕ and ψ are smooth, since we can approximate general ϕ and ψ by smooth W -invariant convex functions by Lemma 4.5. Let Using the fact that and integration by parts, we have . (4.9) 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 [ where v 2P is the support function of 2P . Then a+ MA R;π (ψ)dx = 2P π dy, (4.11) if u < +∞ everywhere in the interior of 2P .
The inverse of Lemma 4.8 is also true as an analogue of [15,Theorem 3.6]. In fact, we have  (1) u ψ is differentiable at p if and only if u ψ is attained at a unique point x p ∈ a and x p = ∇u ψ (p); Let p ∈ P at which u ψ is differentiable. Then for any continuous uniformly bounded function v on a, it holds where u ψ+tv is the Legendre function of ψ + tv as in (3.11) which is welldefined since v is continuous and uniformly bounded on a. Proof of Theorem 4.2. Necessary part. First we show that φ has full MA-mass by Proposition 4.9. In fact, by a result in [31,Lemma 4.5], we see that for any W -invariant convex polytope 2P ⊆ 2P , there is a constant C = C(P ) such that for any W -invariant convex u φ ≥ 0, This implies that u φ is finite everywhere in Int(2P ) by the convexity of u φ . Thus we get what we want from Proposition 4.9.
Next we prove that φ is L 1 -integrate associated to the MA-measure ω n φ . Let ψ 1 = ψ 0 + φ (φ may be different to a constant). We define a distance between ψ 0 and ψ 1 for p ≥ 1, ) runs over all curves joining 0 and φ with ω φt ≥ 0. Choose a special path φ t such that the corresponding Legendre functions of ψ t = ψ 0 + φ t are given by where u 1 and u 0 are the Legendre functions of ψ 1 and ψ 0 , respectively. Note that by Lemma 4.10,ψ t = −u t = u 0 − u 1 , almost everywhere. Then by Lemma 4.5 (or Remark 4.11), we get On the other hand, by a result of Darvas [16], there are uniform constant C 0 and C 1 such that for any Kähler potential φ with full MA-measure it holds, Thus we obtain − M φω n φ ≤ C. and ∇ψ φ : a → 2P is a bijection. Thus Moreover, Hence, Then as in (4.17), we have where u j is the Legendre functions of ψ j = ψ 0 + φ j . Note that M |φ j |ω n j → M |φ|ω n φ and u j u φ . Thus by taking the above limit as j → +∞, we get (4.17) for φ. In particular, 2P+ u φ π dy < +∞.

Computation of Ricci potential
In this section, we assume that the moment polytope P of Z is fine. Then by gives a Ricci potential h 0 of ω 2P , which is smooth and K × K-invariant on G.
The following proposition gives an upper bound on h 0 .
Proposition 5.1. The Ricci potential h 0 of ω 2P is uniformly bounded from above on G. In particular, e h0 is uniformly bounded on G.
Since the Ricci potential of h 0 is also K × K-invariant, by (5.1) and (3.5), Note that 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 from above near each y 0 ∈ ∂(2P + ). There will be three cases as follows.
Case-1. y 0 ∈ ∂(2P + ) and is away from any Weyl wall. Then we get as y → 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 . In this case it is direct to see that h 0 is bounded near y 0 since log det(u 2P ,ij ), y i u 2P ,i , J(∇u 2P ) π(y) are all bounded. Case-3. y 0 lie on the intersection of ∂(2P ) with some Weyl walls. In this case, by (3.1), we rewrite (5.3) as Here we used a fact that for each α ∈ Φ + . Then Note that each I α (y) involves only one root α. Thus, without loss of generality, we may assume that y 0 lies on only one Weyl wall.
Then, by (5.4), we have Note that y 0 ∈ {β(y) > 0}. Thus any facet F A passing through y 0 lies in {β(y) > 0} or is orthogonal to W β . Since 2P is convex and s β -invariant, where s β is the reflection with respect to W β , these facets must satisfy Hence, for any β = α 0 , we get =O (1), as y → y 0 . (5.6) It remains to estimate the second term in I α0 (y), We first consider a simple case that y 0 lies on the intersection of W α0 with at most two facets of 2P . Then there will be two subcases: Case-3.1. y 0 ∈ W α0 ∩ F 1 is away from other facet of 2P . Then F 1 is orthogonal to W α0 . It follows that l A (y 0 ) = 0 for any A = 1. Thus α 0 , y = o(l A (y)), y → y 0 , A = 1. Let {F 1 , ..., F d1 } be all facets of P such that α 0 (u A ) ≥ 0, A = 1, ..., d 1 . Let s α0 be the reflection with respect to W α0 . Then by s α0 -invariance of P , for each A ∈ {1, ..., d 1 } there is some A ∈ {1, ..., d 1 } such that It follows that Thus, by (5.8) and the fact that α 0 (u 1 ) = 0, we obtain = log α 0 , y + O(1).
Next we consider the case that there are facets F 1 ..., F s (s ≥ 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 α0 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 ∈ {1, ..., s} such that l i (y) = min{l i (y)| i = 1, ..., s such that α 0 (u i ) = 0}.
As y → y 0 , up to passing to a subsequence, we can fix this i . Clearly, y 0 ∈ W α0 ∩ F 1 ∩ F 2 as in Case-3.2, where F 2 = F ⊆ a + and F 1 = s α0 (F ) for the reflection s α0 . 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. In the other words, in Cases-3.2.1-3.2.3, if (5.14) does not hold.

Reduced Ding functional and existence criterion
By Lemma 4.8 and Theorem 4.2, we see that for any u ∈ E 1 K×K (2P ), its Legendre function corresponds to a K ×K-invariant weak Kähler potential φ u = ψ u −ψ 0 which belongs to E 1 K×K (M, −K M ). Here we can choose ψ 0 to be the Legendre function ψ 2P of Guillemin function u 2P as in (3.8). As we know, e −φu ∈ L p (ω 0 ) for any p ≥ 0. Thus a+ e −ψu J(x)dx is well-defined.
We introduce the following functional on E 1 K×K (2P ) by and It is easy to see that on a smooth Fano compactification of G, and D(u φ ) is just the Ding functional F (φ). We note that a similar functional on such Fano manifolds has been studied for Mabuchi solitons in [32,Section 4]).
Hence, for convenience, we call D(·) the reduced Ding functional on a Q-Fano compactifications of G.
In this section, we will use the variation method to prove Theorem 1.2 by verifying the properness of D(·). We assume that the moment polytope P is fine so that the Ricci potential h 0 is uniformly bounded above by Proposition 5.1.

A criterion for the properness of D(·).
In this subsection, we establish a properness criterion for D(u φ ), namely, Proposition 6.1. Let M be a Q-Fano compactification of G. Suppose that the moment polytope P is fine and it satisfies (1.2). Then there are constants δ and C δ such that The proof goes almost the same as in [32]. We sketch the arguments here for completeness. First we note that u φ satisfies the normalized condition u ≥ u(O) = 0. Then we have the following estimate for the linear term L(·) as in [32,Proposition 4.5].
For the non-linear term F(·), we can also get an analogy of [32, Lemma 4.8] as follows. Then

SINGULAR KÄHLER-EINSTEIN METRICS ON Q-FANO COMPACTIFICATIONS OF LIE GROUPS 23
Consequently, for any c > 0, K×K (−K M ) and u 0 , u 1 be two Legendre functions of ψ 0 + φ 0 and ψ 0 + φ 1 , respectively. Let u t (t ∈ [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 ψ t of u t give a path in E 1 K×K (−K M , −K M ). The following lemma shows that F(ψ t ) is convex in t.

ThenF(t) is convex in t and so is F(ψ 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 −ψt J, e −ψ1 J and e −ψ0 J (cf. [39]), we prove − log

This means thatF(t) is convex.
Proof of Proposition 6.1. By Proposition 5.1, is bounded, where y(x) = ∇ψ 0 (x). Then the functional It is easy to see that that u 0 is a critical point of D A (·). On the other hand, by Lemma 6.4, F(·) is convex along any path in E 1 K×K (M, −K M ) determined by their Legendre functions as in (4.14). Note that L 0 . Now together with Lemma 6.2 and Lemma 6.3, we can apply arguments in the proof of [32,Proposition 4.9] to proving that there is a constant C > 0 such that for any u ∈ E 1 K×K (2P ), Therefore, we get (6.1).
It remains to estimate F(u ∞ ). Note that u ∞ is finite everywhere in Int(2P ) by the locally uniformly convergence and its Legendre function ψ ∞ ≤ v 2P . Thus, for any On the other hand, the Legendre function ψ n of u n also converges locally uniformly to ψ ∞ . Then ∂ψ n → ∂ψ ∞ almost everywhere. Since as long as n 1. Note that By choosing an 0 such that 4ρ ∈ (1 − 0 )Int(2P ), we get Hence, combining this with (6.8) and (6.9) and using Fatou's lemma, we derive − log Therefore, we have proved (6.6) by (6.7).
For any continuous, compactly supported W -invariant function η ∈ C 0 (a), we consider a family of functions u + tη. In general, it may not be convex for t = 0 since u is just weakly convex. In the following, we use a trick to modify the function D(u t ) as in [9,Section 2.6]. Define a family of W -invariant functions bŷ Then it is easy to see that the Legendre functionû t ofψ t satisfies |û t − u 0 | ≤ C, ∀|t| 1.
Next we show that ω φ can be extended as a singular Kähler-Einstein metric on M . Choose an 0 such that 4ρ ∈ Int(2(1 − )P ). Since u ∈ E 1 K×K (2P ), by Lemma 4.8, there is a constant C > 0 such that is bounded on a + . Also π(∂ψ ) is bounded. Therefore, by (4.11), for any > 0, we can find a neighborhood U of M \ G such that This implies that φ can be extended to be a global solution of (2.1) on M . The proposition is proved.
7. Q-Fano compactification of SO 4 (C) In this section, we will construct Q-Fano compactifications of SO 4 (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 Q-Fano SO 4 (C)-compactification by computing the barycenter of their moment polytopes P + . For convenience, we will work with P + instead of 2P + throughout this section. Then it is easy to see that the existence criterion (1.2) is equivalent to bar(P + ) ∈ 2ρ + Ξ.
Consider the canonical embedding of SO 4 (C) into GL 4 (C) and choose the maximal torus Choose the basis of N as E 1 , E 2 , which generates the R(z 1 ) and R(z 2 )-action. Then we have two positive roots in M, Also we have a * + = {(x, y)| − x ≤ y ≤ x}, , 2ρ = (2, 0) and 7.1. Gorenstein Fano SO 4 (C)-compactifications. In this subsection, we use Lemma 3.1 to exhaust all polytopes associated to Gorenstein Fano compactifications. Here by Gorenstein, we mean that K −1 Mreg can be extended as a holomorphic vector line bundle on M . In this case, the whole polytope P is a lattice polytope. Also, since 2ρ = (2, 0), each outer edge 5 of P + must lies on some line for some coprime pair (p, q). Assume that l p,q ≥ 0 on P . By convexity and Winvariance of P , (p, q) must satisfy p ≥ |q| ≥ 0.
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 {(x, y)| l p2,q2 (x, y) = 0}.
It is known that Case-1.1.2, Case-1.2.1 and Case-2 are the only smooth SO 4 (C)compactifications as shown in [33]. We summarize results of this subsection in Table-1. 7.2. Q-Fano SO 4 (C)-compactifications. In general, for a fixed integer m > 0, it may be hard to give a classification of all Q-Fano compactifications such that −mK X is Cartier. This is because when m is sufficiently divisible, there will be too many repeated polytopes directly using Lemma 3.1. To avoid this problem, we give a way to exhaust all Q-Fano polytopes according to the intersection point of ∂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 x 0 = 2 + 1 p 0 for some p 0 ∈ N + , and there is an edge which lies on some {l p0,q0 = 0}. Without loss of generality, we may also assume that {l p0,q0 = 0} ∩ {y > 0} = ∅. Thus by symmetry, it suffices to consider the case Indeed, by the prime condition, q 0 = 0, ±p 0 if p 0 = 1. Hence, we may assume We associate this number p 0 to each Q-Fano polytope P (and hence Q-Fano compactifications of SO 4 (C)). By the convexity, other edges determined by l p,q must satisfy (see the figure below) p ≤ p 0 , since we assume that P + ⊆ ({l p0,q0 ≥ 0} ∩ a + ). (7.6) Thus, once p 0 is fixed, there are only finitely possible Q-Fano compactifications of SO 4 (C) associated to it. In the following table, we list all possible Q-Fano compactifications with p 0 ≤ 2. We also test the existence of Kähler-Einstein metrics on these compactifications. In the appendix we list the nine non-smooth examples above labeled as in Table- Thus by (7.7) and (7.8), we derive q 0 < 1 2 p 0 + 3 4 . (7.9) By (7.9), we have Vol(P + ) ≤ Vol({l p0,q0 ≥ 0, x ≥ y ≥ −x})  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 (7.11).
When p 0 = 5, there are three possible choices of q 0 , i.e. q 0 = 1, 2, 3 by (7.9). It is easy to see that (7.11) still holds for the first two cases by the second relation in (7.10). 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 Vol(P + ) = 1771561 23040 .
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 p3,q3 = 0} with |q 3 | ≤ p 3 ≤ 2. One can show that all of these possible P + satisfy (7.11). 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, Vol(P + ) = Vol(P In summary, when p 0 ≥ 3, the volume of P + is not equal to either Vol(P  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 (x(P + ), 0) and Thus a Kähler-Einstein polytope of this type must satisfy p 0 ≤ 3.