In this subsection we establish the fact used above, that the Mukai-Umemura manifold has a Kahler-Einstein metric¹¹ 1 This material appeared in the preprint A note on the $\alpha$-invariant of the Mukai-Umemura 3-fold arxiv DG 07114357., which is . For this we appeal to the theory of the $\alpha$-invariant, developed by Tian [30]. We begin by recalling the definition. Let $Z$ be a Fano manifold on which a compact group $G$ acts by holomorphic automorphisms and fix a $G$-invariant Kahler metric $\omega_{0}$ in the cohomology class $-c_{1}(K_{Z})$. Let ${\cal P}$ be the set of $G$-invariant Kahler potentials $\psi$ on $X$ such that $\omega_{\psi}=\omega_{0}+i\partial\overline{\partial}\psi>0$ and $\max_{Z}\psi=0$. Thus ${\cal P}$ can be identified with the set of all $G$-invariant Kahler metrics in the given Kahler class. Let $A\subset{\bf R}$ be the set defined by the condition that $\beta\in A$ if there exists a $C_{\beta}\in{\bf R}$ such that

$$\int_{Z}e^{-\beta\psi}d\mu_{0}\leq C_{\beta},$$

for all $\psi\in{\cal P}$. Here $d\mu_{0}$ is the volume form defined by the fixed metric $\omega_{0}$. Then Tian sets

$$\alpha_{G}(Z)=\sup\{\beta:\beta\in A\},$$

and shows that this does not depend on the choice of $\omega_{0}$. He shows that $\alpha_{G}(Z)$ is always strictly positive and that if $\alpha_{G}(Z)>\frac{n}{n+1}$ then $Z$ has a Kahler-Einstein metric. What we really show in this subsection is that if we take the Mukai-Umemura manifold $X$ with the action of $SO(3)$ then,

So, since $5/6>3/4$, Tian’s theory proves the existence of a Kahler-Einstein metric. We should say straightaway that this is not really a new result. Alessio Corti has explained to the author that, given the facts above, it can be obtained from the more general theories of [13]. But our argument is extremely simple and fits well into the general framework of this article.

We will only write down the proof that $\alpha\geq 5/6$, which is what is relevant to Corollary 1. The proof that $\alpha=5/6$ is an easy extension of this.

This is a step in Tian’s proof that $\alpha>0$ and we repeat his argument. If $\psi\in{\cal P}$ we have

$$\Delta_{0}\psi=2\Lambda(i\partial\overline{\partial}\psi)\geq-2n.$$

Let $K$ be the Green’s function for $\Delta_{0}$, so that for all functions $f$ on $Z$

$$f(x)=-\int_{Z}K(x,y)(\Delta_{0}f)(y)d\mu_{0}(y)+\frac{1}{V}\int f(y)d\mu_{0}(y),$$

where $V$ is the volume of the manifold. With our sign conventions, $K$ is bounded below and, since we can change $K$ by the addition of a constant without affecting the identity, we may suppose that $K\geq 0$. While $K$ is singular along the diagonal it is integrable in each variable. Let $x$ be the point where $\psi$ vanishes. Then applying the Green’s identity to $\psi$ we have

$$\int_{Z}\psi(y)d\mu_{0}(y)=V\int_{Z}K(x,y)\Delta_{0}\psi d\mu_{0}(y)\geq-2nV\int_{Z}K(x,y)d\mu_{0}(y).$$

So we can take

$$M=2nV\max_{x}\int_{Z}K(x,y)d\mu_{0}(y).$$

For the rest of this section we work with the Mukai-Umemura manifold, which we denote by $X$. Let $\sigma$ be the $SO(3)$-invariant section of the anticanonical bundle $K^{-1}$ cutting out the divisor $D$. There is a Hermitian metric on this line bundle such that the curvature of the associated unitary connection is $-i\omega_{0}$. Set

$$f_{0}=-\log\left(|\sigma|^{2}\right).$$

This is a smooth function on $X\setminus D$ and $i\partial\overline{\partial}f_{0}=\omega_{0}$.

This is also standard. The integral in question is

$$\int_{Z}|\sigma|^{-2\beta}d\mu_{0}.$$

By what we know about the singlarities of $D$, we can reduce to considering the integrals

$$\int_{B}|z^{2}-w^{3}|^{-2\beta},$$

where $B$ is the unit ball in ${\bf C}^{2}$ and $z,w$ are complex co-ordinates. Let $T$ be the linear map $T(z,w)=(z/8,w/4)$ and for $r\geq 1$ set

$$\Omega_{r}=T^{r}(B)\setminus T^{r-1}(B).$$

Set

$$I_{r}=\int_{\Omega_{r}}|z^{2}-w^{3}|^{-2\beta}.$$

The substitution $(z^{\prime},w^{\prime})=T(z,w)$ shows that

$$I_{r+1}=2^{(12\beta-10)}I_{r}.$$

Thus $\sum_{r}I_{r}$ is finite if $\beta<5/6$ and the union of the $\Omega_{r}$ cover $B^{4}\setminus\{0\}$.

Now we give the main proof. Let $x_{0}\in X$ be the point with stabiliser $\Gamma$. We identify $SO(3)$-invariant functions on $X\setminus D$ with $\Gamma$-invariant functions on $M=PSL(2,{\bf C})/SO(3)$ as in (4.2). The function $f_{0}$ on $x\setminus D$ corresponds to a convex function $\phi_{0}$ on $M=PSL(2,{\bf C})/SO(3)$ which is an “admissible potential” in the language of (4.2). For any other admissible potential $\phi$ the difference $\phi-\phi_{0}$ corresponds to $\psi$, restricted to $X\setminus D$. The normalisation that $\max\psi=0$ becomes the condition that ${\rm sup}\ \phi-\phi_{0}=0$, and in particular $\phi\leq\phi_{0}$.

Let $P_{0}\in M$ be the identity coset. It is the unique point fixed by the action of $\Gamma$. Any admissible potential function $\phi$ on $M$ is proper and bounded below so achieves a minimum in $M$. By the convexity and $\Gamma$-invariance this minimum must occur at $P_{0}$. Set $\phi(P_{0})=-b$. Then the inequality $\phi_{0}\geq-b$ translates back into the statement that $\psi\geq f_{0}-b$. So

$$\int_{Z}e^{-\beta\psi}d\mu_{0}\leq e^{b\beta}\int_{Z}f_{0}^{-\beta}d\mu_{0}.$$

By Lemma 1, it suffices to obtain an upper bound on $b$. Let $B$ be the geodesic ball in $M$ centred on $P_{0}$, of radius $1$ say, and let $\overline{a}$ be the maximum value of $\phi_{0}$ on $B$, so for any $\phi$ we have $\phi\leq\overline{a}$ on $B$. Convexity along geodesics emanating from $P_{0}$ implies that

$$\phi(Q)\leq-b+(\overline{a}+b)\ {\rm dist}(Q,P_{0}),$$

for any point $Q$ in $B$. In particular, on the ball $\frac{1}{2}B$ of radius $1/2$ about $P_{0}$ we have $\phi\leq(\overline{a}-b)/2$.

Take the inverse image in $PSL(2,{\bf C})$ of the ball $\frac{1}{2}B$ and map this to $X$ by $g\rightarrow g(x_{0})$. The image obviously contains a neighbourhood $N$ of $x_{0}$ and on $N$ we have $\psi\leq\frac{\overline{a}-b}{2}+f_{0}$. Then Lemma 1 implies that $b$ cannot be very large. In fact, if the minimum of $f_{0}$ on $N$ is $\underline{a}$, we have $\psi\leq(\frac{\overline{a}}{2}-\underline{a})-\frac{b}{2}$ on $N$, so

$$-M\leq\int_{N}\psi\ d\mu_{0}\leq((\frac{\overline{a}}{2}-\underline{a})-\frac{b}{2}){\rm Vol}(N),$$

hence

$$b\leq(\overline{a}-2\underline{a})+\frac{2M}{{\rm Vol}(N)}$$

where $M$ is as in Lemma 1. This completes the proof of Theorem 3.

Notice that the same argument can be applied in the toric case, when the polytope $P$ has a group $\Gamma$ of symmetries, as discussed in (4.2). We should suppose that $\Gamma$ has a unique fixed point in $P$: then the proof proceeds exactly as before. The analogue of Lemma 1 holds with $\beta<1$ since the local models for the zeros of $s$ are $f_{p}(z_{1},\dots,z_{n})=0$ where $f_{p}(z_{1},\dots,z_{n})=z_{1}\dots z_{p}$ and $|f_{p}|^{-2\beta}$ is locally integrable for $\beta<1$. the conclusion is that the $\alpha$-invariant in this case is $1$. which is a theorem of Batyrev and Selinova [4]. Song gave another proof in [26], and showed conversely that for polytopes which do not have such a symmetry group the $\alpha$-invariant never exceeds $n/n+1$. The fact that such toric manifolds nevertheless have Kahler-Einstein metrics illustrates the point that Tian’s $\alpha$-invariant criterion is sufficient but not necessary.