We start with a $7$-dimensional complex vector space $V$ and write $Gr_{3}(V)$ for the Grassmann manifold of 3-dimensional subspaces of $V$. So $Gr_{3}(V)$ has dimension $3.(7-3)=12$. A form $\Omega\in\Lambda^{2}(V^{*})$ defines a subset $Z_{\Omega}\subset Gr_{3}(V)$ consisting of the $3$-planes $P$ such that $\Omega|_{P}$ vanishes. In other language we consider the tautological rank 3 vector bundle $U\rightarrow Gr_{3}(V)$; the form $\Omega$ defines a section $s_{\Omega}$ of $\Lambda^{2}U^{*}$ with zero set $Z_{\Omega}$. For generic $\Omega$ this zero set is a smooth subvariety of codimension $3$. Now let $\Omega_{1},\Omega_{2},\Omega_{3}$ be three such forms and consider

$$X=Z_{\Omega_{1}}\cap Z_{\Omega_{2}}\cap Z_{\Omega_{3}}\subset Gr_{3}(V).$$

Of course this only depends on the $3$-plane $\Pi$ in $\Lambda^{2}V^{*}$ spanned by the $\Omega_{i}$, so we may sometimes write $X_{\Pi}$. Obviously there is a Zariski-open subset ${\cal U}$ in the Grassmannian $Gr_{3}(\Lambda^{2}V^{*})$ of $3$-planes $\Pi$ such that $X_{\Pi}$ is a smooth subvariety of dimension $12-3.3=3$. This set ${\cal U}$ is non-empty, as we will see later. The group $SL(V)$ acts on the whole construction and obviously different subspaces $\Pi$ which lie in the same $SL(V)$ orbit define isomorphic manifolds $X_{\Pi}$, so we get a set of equivalence classes of manifolds constructed in this way, parametrised by the quotient ${\cal U}/SL(V)$. (Mukai shows further that this parametrisation is effective: i.e. $X_{\Pi_{1}}$ is isomorphic to $X_{\Pi_{2}}$ if and only if $\Pi_{1},\Pi_{2}$ lie in the same $SL(V)$ orbit. Moreover, he shows that all “prime Fano $3$-folds of genus 12” arise in this way.)

We compute the canonical bundle $K_{X}$ of the variety $X=X_{\Pi}$ for some $\Pi\in{\cal U}$. We have

$$\Lambda^{2}U^{*}=U\otimes H$$

where $H$ is the ample line bundle $\Lambda^{3}U^{*}$. So, writing $\det$ for the the top exterior power of a vector bundle, we have

$$\det\Lambda^{2}U^{*}=H^{\otimes 2}.$$

The tangent bundle of the Grassmannian at a $3$-plane $P\subset V$ can be identified with $P^{*}\otimes V/P$. So

$$\det TGr_{3}=H^{\otimes 7}.$$

Now since the tangent bundle of $X$ is the kernel of a surjective map from $TGr_{3}(V)$ to $\Lambda^{2}U^{*}\oplus\Lambda^{2}U^{*}\oplus\Lambda^{2}U^{*}$ we have

$$K_{X}^{-1}=\det TX=H^{\otimes(7-3.2)}=H.$$

Thus $X$ is a Fano manifold. The sections of $H$ over $Gr_{3}$ give the Plucker embedding

$$Gr_{3}(V)\rightarrow{\bf P}(\Lambda^{3}V)={\bf P}^{34}$$

For any $3$-form $A\in\Lambda^{3}V^{*}$ we get a hyperplane section $Y_{A}\subset Gr_{3}(V)$ which just consists of the $3$-planes $P$ such $A|_{P}=0$. By definition this occurs if $P$ is in $X$ and $A$ is in the image of the wedge product map $\Pi\otimes V^{*}\rightarrow\Lambda^{3}V^{*}$. We expect this map to have an image of dimension $7.3=21$ in which case the image of the composite

$$X\rightarrow Gr_{3}(V)\rightarrow{\bf P}^{34}$$

lies in a linear subspace ${\bf P}^{34-21}={\bf P}^{13}$. Certainly this map is defined by sections of $K_{X}^{-1}$, we will see later that $H^{0}(X,K_{X}^{-1})$ has dimension $14$ and that this embedding is that given by the anticanonical system.

To make this more concrete we show now that $X$ is a rational variety; that is, we construct an explicit parametrisation of a dense open set in $X$. Suppose we have a pair of $3$-dimensional subspaces $P_{0},Q_{0}\subset V$ with $P_{0}\cap Q_{0}=0$. We ask what $3$-planes $P$ in the $6$-dimensional subspace $P_{0}\oplus Q_{0}$ lie in $X$. In matrix notation, we can write the restriction of a form $\Omega$ to $P_{0}\oplus Q_{0}$ as

$$\left(\begin{array}[]{cc}\sigma&A\\ -A^{T}&\tau\end{array}\right)$$

Now consider the $3$-dimensional subspaces $P$ which arise as the graphs of linear maps $M:P_{0}\rightarrow Q_{0}$. The condition becomes

$$\sigma+M^{T}\tau M+(AM-(AM)^{T})=0.$$

(33)

So our three forms $\Omega_{i}$ give us three triples $A_{i},\sigma_{i},\tau_{i}$ and we have three equations of the form (33) to solve to find a point of $X$. We have $9$ unknowns: the entries of the matrix $M$. The left hand side of (33) takes values in the $3$-dimensional space of skew symmetric $3\times 3$ matrices so we obtain a total of $3.3=9$ equations in these $9$ unknowns and we expect a finite number of solutions. These equations are quadratic and one can solve them explicitly, to see that there are generically two solutions. However it is easier to suppose that we are in the case when $P_{0}$ itself lies in $X$. This means that all the $\tau_{i}$ are zero, so the equations (33) become linear. Generically this system of $9$ linear equations in $9$ unknowns is nondegerate and there is a unique solution. Now suppose we have found one point $P_{0}$ in $X$ and consider the space of $6$-planes in $V$ which contain $P_{0}$. This is a copy of projective $3$-space ${\bf P}^{3}$. Given a point in ${\bf P}^{3}$, that is to say a 6 dimensional subspace $E$ of $V$, we choose a complementary subspace to write is as $E=P_{0}\oplus Q_{0}$. Then we can proceed as above and, by solving linear equations, find the points of $X\cap Gr_{3}(E)$. Generically there is just one, $P_{E}$ say, different from the original $P_{0}$. Conversely for any $P^{\prime}\in X$ the sum $P\oplus P^{\prime}$ lies in a $6$-dimensional subspace. Of course there will be various exceptional cases, but the upshot is that we get a birational map from ${\bf P}^{3}$ to $X$ which takes a subspace $E$ containing $P_{0}$ to $P_{E}$.

We now consider a special manifold in this family. Take the vector space $V$ to be the sixth symmetric power $s^{6}$ of the fundamental representation of $SL(2,{\bf C})$. Then $\Lambda^{2}V^{*}=\Lambda^{2}s^{6}$ decomposes into distinct irreducible representations

$$\Lambda^{2}s^{6}=s^{10}\oplus s^{6}\oplus s^{2}.$$

The $s^{2}$ summand is a $3$-plane $\Pi_{0}$ invariant under $SL(2,{\bf C})$, so there is a natural $SL(2,{\bf C})$ action on the corresponding variety, the Mukai-Umemura manifold, $X_{0}=X_{\Pi_{0}}$. We will see below that $X_{0}$ admits a Kahler-Einstein metric. The representation $s^{6}$ has a standard invariant symmetric form $(\ ,\ )$ and the inclusion $s^{2}\rightarrow\Lambda^{2}s^{6}$ is just the map from the Lie algebra of $SL(2,{\bf C})$ given by the action on $s^{6}$. This comes down to saying that a $3$-plane $P$ is in $X_{0}$ if and only if

$$(\delta p,q)=0$$

(34)

for all $p,q\in P$ and $\delta\in\mathfrak{s}\mathfrak{l}_{2}$. Notice that the action of $SL(2,{\bf C})$ on all the spaces involved actually factors through $PSL(2,{\bf C})$.

Identify the projectivisation of the fundamental representation $s^{1}={\bf C}^{2}$ with the standard round sphere and fix an icosahedron, which can be regarded as a set of 12 vertices in in this sphere. Thus we get a symmetry group $\Gamma\subset SO(3)\subset PSL(2,{\bf C})$ of order $60$. There is a simple way to see that the $7$-dimensional representation $s^{6}$ of $PSL(2,{\bf C})$ becomes reducible when restricted to $\Gamma$. There are $6$ pairs of antipodal vertices and for each such pair $p,\overline{p}$ we have a $1$-dimensional subspace consisting of polynomials which vanish to order $3$ at $p,\overline{p}$. The sum of these $6$ subspaces is obviously invariant under $\Gamma$ and is a proper subspace of $s^{6}$ since it has codimension at least $1$. A little calculation shows that this invariant subspace is of dimension $3$ and satisfies the criterion (34). So this subspace gives a point $P_{0}$ in $X_{0}$ fixed by $\Gamma$. On the other hand the stabiliser of $P_{0}$ is obviously not the whole of $SO(3)$ and, since there is no finite subgroup of $SO(3)$ strictly larger than $\Gamma$, the stabiliser must be exactly $\Gamma$.

Now go back to the wedge product $P_{0}\wedge V^{*}\rightarrow\Lambda^{3}V^{*}$. In terms of representations this is an $SL(2,{\bf C})$-map

$$s^{2}\otimes s^{6}\rightarrow\Lambda^{3}s^{6}.$$

It is an exercise in representation theory to show that

$$\Lambda^{3}s^{6}=s^{12}\oplus s^{8}\oplus s^{6}\oplus s^{4}\oplus s^{2}\oplus s^{0}\oplus s^{0}.$$

So comparing with

$$s^{2}\otimes s^{6}=s^{8}\oplus s^{6}\oplus s^{4}\oplus s^{2}\oplus s^{0}$$

we see that the embedding $X_{0}\subset Gr_{3}(V)\subset{\bf P}(\Lambda^{3}V)$ gives rise to an $SL(2,{\bf C})$-equivariant embedding

$$X_{0}\rightarrow{\bf P}(s^{0}\oplus s^{12}).$$

(35)

In other words, by our identification of the anticanonical bundle $K^{-1}$ we have

$$H^{0}(X_{0},K^{-1})=s^{0}\oplus s^{12},$$

as a representation of $SL(2,{\bf C})$.In particular, there is an $SL(2,{\bf C})$-invariant section $\sigma$ of $K^{-1}$. Explicitly, if we identify $\Lambda^{3}s^{6}$ with $\Lambda^{4}s^{6}$ then $\sigma$ corresponds to the $4$-form on $V=s^{6}$ defined as follows. We choose any orthonormal basis $\Omega_{1},\Omega_{2},\Omega_{3}$ of $P_{0}$ and write down the $4$-form

$$*\sigma=\Omega_{1}^{2}+\Omega_{2}^{2}+\Omega_{3}^{2}.$$

In this way, we get another description of the manifold $X_{0}$. Our point $P_{0}\in X_{0}$ cannot lie in the zero set of $\sigma$ (since its orbit is $3$-dimensional). So, in the embedding (35), we have

$$P_{0}=[1,v_{0}]\in{\bf P}({\bf C}\oplus s^{12}).$$

Thus $v_{0}$ is an element of $s^{12}$ whose stabiliser in $PSL(2,{\bf C})$ is exactly $\Gamma$.Now there is an obvious element of the projective space ${\bf P}(s^{12})$ with stabiliser $\Gamma$, just the configuration of vertices of the icosahedron, regarded as an element of the symmetric product. Since $\Gamma$ is a perfect group it must act trivially on the corresponding line in $s^{12}$, so we get a vector in $s^{12}$ with stabiliser $\Gamma$. It is easy to see that, up to a multiple, this in the only element of $s^{12}$ with stabiliser $\Gamma$, and thus we have identified $v_{0}$. Then we can simply define $X_{0}$ to be the closure in ${\bf P}({\bf C}\oplus s^{12})$ of the $PSL(2,{\bf C})$-orbit of $v_{0}$ in $s^{12}$. (Here we are regarding the vector space $s^{12}$ as being a subset of the projective space ${\bf P}({\bf C}\oplus s^{12})$ in the familiar way.)

In this description, the intersection of $X_{0}$ with the hyperplane at infinity

$$D={\bf P}(s^{12})\subset{\bf P}({\bf C}\oplus s^{12}),$$

is, by definition, the zero set of the invariant section $\sigma$ of $K^{-1}$. Consider a $1$-parameter subgroup $\lambda_{t}$ in $PSL(2,{\bf C})$. Thus we have a pair of distinct point $z_{+},z_{-}$ such that when $t$ is large positive the map $\lambda_{t}$ contracts most of the sphere to a small neighbourhood of $z_{+}$, and when $t$ is large negative to a small neighbourhood of $z_{-}$. If $y_{1},\dots y_{12}$ is any configuration of distinct points it is not hard to see that the limit as $t\rightarrow\infty$ of

$$\lambda_{t}(\underline{y})=\left(\lambda_{t}(y_{1}),\lambda_{t}(y_{2}\dots\lambda_{t}(y_{12})\right)$$

in the symmetric product ${\bf P}(s^{12})$ is either $12z_{+}=(z_{+},z_{+},\dots,z_{+})$ (in the generic case) or $11z_{+}+z_{-}=(z_{+},\dots,z_{+},z_{-})$ (in the case when one of the $y_{i}$ is $z_{-}$). Using this, Mukai and Umemura show that the divisor at infinity $D$ consists precisely of the union of points of the form $12z_{+}$ or $11z_{+}+z_{-}$ in ${\bf P}(s^{12})$. It is easy to identify this geometrically. The points of the form $12z_{+}$ make up the rational normal curve in ${\bf P}(s^{12})$. Our divisor $D$ is the surface swept out by the lines in ${\bf P}(s^{12})$ tangent to the rational normal curve. As a set we can identify $D$ with ${\bf P}^{1}\times{\bf P}^{1}$: we just map $(z_{+},z_{-})\in{\bf P}^{1}\times{\bf P}^{1}$ to $11z_{+}+z_{-}\in D$. But the surface $D$ is singular and a more precise statement is that the map above is a holomorphic map $\nu:{\bf P}^{1}\times{\bf P}^{1}\rightarrow D$ which is the normalisation of $D$. The singular set of $D$ is the image of the diagonal in ${\bf P}^{1}\times{\bf P}^{1}$, and it is easy to check that the singularity has the form of a cusp transverse to the diagonal. That is to say, we can choose local co-ordinates $z_{1}z_{2},z_{3}$ in $X_{0}$ around a singular point of $D$ such that $D$ is defined by the equation $z_{1}^{2}=z_{2}^{3}$.

We now have a rather explicit description of $X_{0}$, as the compactification of $PSL(2,{\bf C})/\Gamma$ formed by adjoining the divisor $D$. We can use this to compute the action of $PSL(2,{\bf C})$ on all of the spaces of sections $H^{0}(X_{0},K^{-p})$. For the pull back $\nu^{*}(K^{-1})$ is isomorphic to the line bundle ${\cal O}(11,1)$ over ${\bf P}^{1}\times{\bf P}^{1}$. We can regard the structure sheaf of $D$ as a subsheaf of that of ${\bf P}^{1}\times{\bf P}^{1}$. From the local model of the singularity along the diagonal one sees that the quotient can be identified with sections of ${\cal O}(2)$ along the diagonal. This means that $H^{0}(D,K^{-p}|_{D})$ is the kernel of a map $H^{0}({\bf P}^{1}\times{\bf P}^{1};{\cal O}(11p,p))\rightarrow H^{0}({\bf P}^{1};{\cal O}(12p-2))$. As representations of $PSL(2,{\bf C})$ this is a map

$$s^{11p}\otimes s^{p}\rightarrow s^{12p-2}.$$

Now

$$s^{11p}\otimes s^{p}=s^{12p}+s^{12p-2}\dots\oplus s^{10p}$$

and the map above is just the projection to the second factor. So

$$H^{0}(D;K^{-p}|_{D})=s^{12p}\oplus s^{12p-4}\oplus s^{12p-6}\dots\oplus s^{10p+2}\oplus s^{10p}.$$

Then the exact cohomology sequence of

$$0\rightarrow K^{-(p-1)}\rightarrow K^{-p}\rightarrow K^{-p}|_{D}\rightarrow 0$$

together with Kodaira vanishing on $X_{0}$ gives

$$H^{0}(X_{0},K^{-p})=H^{0}(X_{0},K^{-(p-1)})\oplus s^{12p}\oplus s^{12p-4}\dots s^{10p},$$

and inductively we get a description of each $H^{0}(X_{0},K^{-p})$. Thus

$$H^{0}(X_{0},K^{-1})=s^{0}\oplus s^{12},$$

$$H^{0}(X_{0},K^{-2})=s^{0}\oplus s^{12}\oplus s^{24}\oplus s^{20}.$$

For $p\geq 6$ we get multiplicities: $H^{0}(X_{0},K^{-6})$ contains two copies of $s^{60}$. This illustrates the difference with the multiplicity-free case discussed above. (Although since the multiplicities are small until $p$ becomes quite large, once is tempted to think of $X_{0}$ as being “close” to multiplicity-free. )

We will now get another explicit picture of $X_{0}$, taking the point of view of symplectic geometry. Recall that all Kahler metrics in the cohomology class $c_{1}(X_{0})$ define equivalent symplectic structures, so we have a well-defined symplectic manifold $(X_{0},\omega)$ with an $SO(3)$-action. Thus we have an equivariant moment map

$$\mu:X_{0}\rightarrow{\bf R}^{3}={\rm Lie}(SO(3))^{*}.$$

whose image is clearly a ball in ${\bf R}^{3}$. We can understand the structure of this moment map by restricting to a subgroup $S^{1}\subset SO(3)$, say that corresponding to the $x_{1}$-axis in ${\bf R}^{3}$. Then the Hamiltonian $H$ for this circle action on $X_{0}$ is the composite of $\mu$ with projection to the $x_{1}$-axis. The critical points of $H$ are the fixed points of the circle action and we can find these explicitly. We can suppose that our circle subgroups corresponds to the standard action of

$$\left(\begin{array}[]{cc}\lambda^{1/2}&0\\ 0&\lambda^{-1/2}\end{array}\right),$$

acting on ${\bf C}^{7}$ with weights $\lambda^{3},\dots\lambda^{-3}$. We write $e_{i}$ for the basis vector belonging to the weight $\lambda^{i}$. This induces an action on the Grassmannian $Gr_{3}(V)$ whose fixed points are just invariant $3$-dimensional subspaces of ${\bf C}^{7}$ and these are just the spans $P_{ijk}=\langle e_{i},e_{j},e_{k}\rangle$ for distinct $i,j,k$. By checking the 35 different cases, or otherwise, one finds that the only $P_{ijk}$ which satisfy the criterion (34) to lie in $X_{0}$ are $P_{123},P_{023},P_{0-2-3},P_{-1-2-3}$. There is an action of the Weyl group $\{\pm 1\}$ on the whole situation which commutes, up to sign, with the circle action, takes $H$ to $-H$ and takes $P_{ijk}$ to $P_{-i-j-k}$. So there are four fixed points of the circle action but to analyse the local structure around them it suffices to consider the two cases $P_{123},P_{023}$. Notice that, by considering $H$ as a Morse function we immediately see that $X_{0}$ has the same additive homology as ${\bf C}{\bf P}^{3}$. Notice also that the value of $H$ at a critical point is just given by the weight of the action on the fibre of $K^{-1}$ over this point, which is just $i+j+k$ at $P_{ijk}$.

We next compute the weights of the circle action on the tangent spaces at the fixed points. This is similar to the calculation of the canonical bundle. At a fixed point the tangent space $TX_{0}$, viewed as a representation of $S^{1}$, can be written as the formal difference

$$TGr_{3}(V)-\left(\Lambda^{2}U^{*}\otimes{\rm Lie}(SO(3))\right).$$

Computing the weights of these two terms and subtracting we find that the weights of the action on the tangent space at $P_{123}$ are $(1,2,3)$ and on the tangent space at $P_{023}$ are $(1,-1,5)$. In either case the orbit of the fixed point is a copy of $SO(3)/S^{1}=S^{2}$ in $X_{0}$ and the weight $1$ in the action on $TX_{0}$ just corresponds to the tangent space of this orbit. The weights normal to the orbit are $(2,3)$ in the case of $P_{123}$ and $(-1,5)$ in the case of $P_{023}$.

With these calculations we can get a good picture of the map $\mu$. Write $\Sigma,\Sigma^{\prime}$ for the orbits of $P_{123}$ and $P_{023}$ respectively. Then $\mu$ restricts to an $SO(3)$-equivariant equivalence between $\Sigma$ and the sphere of radius $1+2+3=6$ in ${\bf R}^{3}$ and between $\Sigma^{\prime}$ and the sphere of radius $0+2+3=5$. The image of $\mu$ is the ball of radius $6$ and the critical values of $\mu$ are precisely these two spheres. So $\mu$ is a fibration away from these spheres. For $\underline{x}\in{\bf R}^{3}$, write $F_{\underline{x}}$ for the preimage $\mu^{-1}(\underline{x})$. If $|\underline{x}|\neq 5,6$ the fibre $F_{\underline{x}}$ is a $3$-manifold. If also $|\underline{x}|>0$ then this $3$-manifold has a natural circle action defined by the circle subgroup of $SO(3)$ fixing $\underline{x}$. When $\underline{x}=0$ the fibre has an $SO(3)$ action. As $\underline{x}$ varies in ${\bf R}^{3}$ the fibre only “changes”—in the obvious sense—when $|\underline{x}|$ crosses the special values $5,6$. Thus we understand the full topological picture if we understand the changes in the fibre as $\underline{x}$ moves along the positive $x_{1}$-axis, say. Let $V\subset X_{0}$ be the pre-image by $\mu$ of the positive $x_{1}$-axis. This is a smooth $4$-manifold, with a circle action, and the fibres $F_{\underline{x}}$, for $\underline{x}$ on the axis, are the level sets of the Hamiltonian $H$, restricted to $V$. Then we have the usual Morse-theory description of these changes, from the Hessian of $H$ on $V$, which is determined by the weights of the circle action. As $\underline{x}$ moves across the point $(6,0,0)$ the situation is modelled by the level sets

$$2|z_{1}|^{2}+3|z_{3}|^{2}=\epsilon,$$

for $(z_{1},z_{2})\in{\bf C}^{2}$, with the circle action of weight $(2,3)$. Thus the fibre changes from the empty set to a $3$-sphere with an action given by these weights. As $\underline{x}$ moves across the point $(5,0,0)$ the situation is modelled, locally, by the level sets

$$-|z_{1}|^{2}+5|z_{2}|^{2}=\epsilon,$$

with the circle action of weight $(-1,5)$. The effect on the fibres is to perform a “Dehn surgery” on an $S^{1}$-orbit. Thus the fibres $F_{\underline{x}}$ for $|\underline{x}|<5$ are obtained by performing this surgery on a knot $\Gamma\subset S^{3}$. Now $\Gamma$ is a free orbit of the $(2,3)$ action so it is the $(2,3)$ “torus knot” which is just a trefoil. To nail down the Dehn surgery completely we need to specify a framing of the knot but this is determined by the fact that the linking number of a nearby orbit with $\Gamma$ is the weight $5$, from which one concludes that the framing is $+1$. This is a well-known description of the Poincaré homology sphere (the result of $+1$-surgery on a trefoil), and ties in with our previous discussion since the fibre $F_{0}$ is the $SO(3)$-orbit $SO(3)/\Gamma$. (Another way of expressing this is that the fibres $F_{\underline{x}}$ are Seifert-fibred $3$-manifolds: for $5<|\underline{x}|<6$ we have two multiple fibres with multiplicity $(2,3)$ and the surgery across $|\underline{x}|=5$ introduces another multiple fibre with multiplicity $5$, so for $|\underline{x}|<5$ we get the Seifert manifold with multiplicities $(2,3,5)$, which is another well-known description of the Poincaré manifold.)

It is interesting to match this picture up with the algebro-geometric description. This illustrates the general theory of Kirwan [19]. The $2$-sphere $\Sigma$ at which $|\mu|$ attains its maximal value $6$ is a holomorphic sphere in $X_{0}$: it is just the rational normal curve in our divisor $D\subset{\bf P}(s^{12})$. The other sphere $\Sigma^{\prime}$ is not holomorphic. It is a critical manifold for the function $|\mu|^{2}$ on $X_{0}$ and the divisor $D$ appears as the associated “ascending set”: the closure of the set of points which flow to $\Sigma^{\prime}$ under the decreasing gradient flow of $|\mu|^{2}$. In our description of $D$ as $S^{2}\times S^{2}$ the holomorphic curve $\Sigma$ is the diagonal and $\Sigma^{\prime}$ is the “anti-diagonal”consisting of pairs of antipodal points. One can also see the cusp singularity in $D$, transverse to $\Sigma$, from the weights $(2,3)$ of the circle action on the normal bundle.

Notice that if we write ${\bf C}{\bf P}^{3}={\bf P}(s^{3})$, for the $4$-dimensional representation $s^{3}$ of $SU(2)$, the moment map $\mu:{\bf C}{\bf P}^{3}\rightarrow{\bf R}^{3}$ for the action gives a description of ${\bf C}{\bf P}^{3}$ very similar to that above. In this case $\mu^{-1}(0)$ is $SO(3)/H$ where $H\subset SO(3)$ is the group of symmetries of an equilateral triangle, and we see this $3$-manifold described as the Seifert fibration with multiple fibres $(2,2,3)$.

Here we study the deformations of Mukai’s construction about the special solution $X_{0}$. Recall that a manifold in this family is specified by a $3$-plane in $\Lambda^{2}{\bf C}^{7}$. We start with the $3$-plane $s^{2}\subset\Lambda^{2}s^{6}=s^{10}\oplus s^{6}\oplus s^{2}$. The tangent space of the Grassmannian at this point is given by the linear maps from $s^{2}$ to the complementary subspace $s^{10}\oplus s^{6}$, that is (using the fact that all these representations are isomorphic to their duals)

$$TGr_{3}(\Lambda^{2}{\bf C}^{7})=(s^{10}\oplus s^{6})\otimes s^{2}=s^{12}\oplus 2s^{8}\oplus s^{6}\oplus s^{4}.$$

The action of the group $SL({\bf C}^{7})=SL(s^{6})$ gives a linear map

$$\mathfrak{s}\mathfrak{l}(7)\rightarrow TGr_{3}(\Lambda^{2}{\bf C}^{7}),$$

which we know has kernel the Lie algebra $\mathfrak{s}\mathfrak{l}(2)$ of the stabiliser. Now the Lie algebra of $GL({\bf C}^{7})=GL(s^{6})$ is

$$s^{6}\otimes s^{6}=s^{12}\oplus s^{10}\oplus s^{8}\oplus s^{6}\oplus s^{4}\oplus s^{2}\oplus s^{0}$$

so the Lie algebra of $SL(s^{6})$ is $s^{12}\oplus\dots s^{2}$. It is clear then that the quotient of the tangent space by the tangent space to the orbit is just $s^{8}$, as a representation of $PSL(2,{\bf C})$. By general theory there is an equivariant slice: a $PSL(2,{\bf C})$ equivariant embedding $j$ from a neighbourhood of $0$ in $s^{8}$ into $Gr_{3}(\Lambda^{2})$, mapping $0$ to our fixed subspace $s^{2}$, such that two points $j(p),j(q)$ in the same $SL(7)$ orbit if and only $p,q$ are in the same $PSL(2)$ orbit. In fact, although we do not really need this, what we are describing is the versal deformation of $X_{0}$, so $H^{1}(TX_{0})=s^{8}$, as a representation of $PSL(2,{\bf C})$.

One can gain a lot of insight from this simple calculation. The structure of the orbits of $PSL(2,{\bf C})$ on $s^{8}$ (or any $s^{p}$) is a standard example in Geometric Invariant Theory. There are five cases

1. 1.

   The trivial orbit $\{0\}$.

2. 2.

   The orbits of polynomials having no zero of multiplicity $\geq 4$. These are closed in $s^{8}$.

3. 3.

   The orbit of polynomials having two distinct zeros, each of multiplicity four. This orbit is closed and each point in it has stabiliser ${\bf C}^{*}\subset PSL(2,{\bf C})$.

4. 4.

   The orbits of polynomials having a zero of multiplicity four and other zeros each of multiplicity less than four. These orbits are not closed but there closure contains the orbit of type (3).

5. 5.

   The orbits of polynomials having a zero of multiplicity $\geq 5$. these are not closed and contain $0$ in their closure.

This is the source of the famous example of Tian of Fano manifolds without Kahler-Einstein (or Ricci soliton) metrics [31]. Tian shows that the manifolds corresponding to any $PSL(2,{\bf C})$ orbit of type (5) cannot have such metrics. Tian’s general results also show the same for the manifolds corresponding to orbits of type (4). Tian’s results are of course deep and difficult but we note now that a weaker statement is rather obviously true. For this we need to recall some background.

In general, the linearisation of the Kahler-Einstein equations on a complex manifold $Z$ at a solution $\omega_{0}$ is given by the self-adjoint operator $\Delta+1$ and, much as we have seen in Section 3, the kernel of this can be identified with the Lie algebra of the isometry group $G$ of $\omega_{0}$. Suppose we have a $G$-equivariant deformation of $Z_{0}$: i.e. a complex manifold ${\cal Z}$ with a $G$-action, an action of $G$ on a ball $B\subset{\bf C}^{m}$ and a $G$-equivariant submersion $\pi:{\cal Z}\rightarrow B$. In this situation we automatically get “local actions” of the complexified group $G^{c}$ on ${\cal Z}$ and $B$, compatible with $\pi$. The standard “Kuranishi method”, which depends only on the formal properties of the situation, yields the following structure (after possibly restricting to a smaller ball $B$).

• •

  A $G$-invariant family of Kahler metrics $\omega_{t}$ on the fibres $Z_{t}=\pi^{-1}(t)$ such that $\omega_{t}$ is isometric to $\omega_{t^{\prime}}$ if and only if $t$ and $t^{\prime}$ are in the same $G$-orbit.

• •

  A smooth map $\nu:B\rightarrow\mathfrak{g}^{*}$, equivariant for the action of $G$ on $B$ and the co-adjoint action on $\mathfrak{g}^{*}$, such that $\omega_{t}$ is Kahler-Einstein if and only if $\nu(t)=0$.

Now in this general situation we can see that, if the $G$-action on $B$ is non-trivial the map $\nu$ cannot be identically zero. For if $t,t^{\prime}$ are in the same orbit of the local $G^{c}$ action on $B$ then $Z_{t}$ and $Z_{t^{\prime}}$ are isomorphic complex manifolds. But if $\nu(t)$ and $\nu(t^{\prime})$ both vanish then $\omega_{t}$ and $\omega_{t^{\prime}}$ are Kahler-Einstein and, by the uniqueness of the Kahler-Einstein solution, they must be isometric and this only happens if $t,t^{\prime}$ are in the same $G$-orbit. Thus what we see from this elementary argument is that as we deform $Z_{0}$ in the smooth family $Z_{t}$ we cannot deform the metric $\omega_{0}$ in a smooth family of Kahler-Einstein metrics, for all small $t$. Tian’s much stronger result is that if the Futaki invariant of $Z_{0}$ vanishes (say), and if $0$ lies in the closure of the the $G^{c}$-orbit of a point $t\in B$ then $Z_{t}$ does not admit any Kahler-Einstein metric at all. This is an example of the “jumping of structures” phenomenon discussed in Section 1: there are arbitrarily small deformations of $Z_{0}$ which are equivalent to a different structure $Z_{t}$.

Returning to our special case of the Mukai-Umemura manifold, we can see conversely that there are some deformations of $X_{0}$ which do admit Kahler-Einstein metrics. The general theory of these “obstruction maps” $\nu$ is being developed by T. Brönnle, in his Ph.D thesis, but in this special case we can make some simple deductions from symmetry arguments. Let $p$ be a point in $s^{8}$ which is fixed by a subgroup $J\subset SO(3)$. Then $J$ acts on ${\bf R}^{3}=\mathfrak{s}\mathfrak{u}(2)$ and if $\nu$ is any equivariant map from $s^{8}$ to ${\bf R}^{3}$ then $J$ must fix $\nu(p)$. So if the origin is the only point in ${\bf R}^{3}$ fixed by $J$ then we must have $\nu(p)=0$. Consider, for example,

$$p=C(z^{4}-\alpha w^{4})(w^{4}-\alpha z^{4}),$$

with any $\alpha,C\in{\bf C}$. This is fixed by a dihedral group $J$ of order $8$ which has the desired property, so we see that the deformations corresponding such elements of $s^{8}$ admit Kahler-Einstein metrics, for small $C$. For $\alpha,C\neq 0$ the element $p$ has a discrete stabiliser in $SO(3)$ and it follows that the corresponding metrics have discrete isometry groups. But then the deformation theory implies that all small deformations of these manifolds admit Kahler-Einstein metrics. So we conclude that there is a non-empty open set in ${\cal U}$ where the manifolds admit Kahler-Einstein metrics.

Taking $\alpha=0$ above we get a special family of deformations, admitting Kahler-Einstein metrics, where we can take $J=O(2)\subset SO(3)$. It follows that the corresponding manifolds have a ${\bf C}^{*}$-action. We can see this family of manifolds explicitly as follows. Fix the action on ${\bf C}^{7}$ with weights $\lambda^{3},\dots,\lambda^{-3}$ as usual. Then we want to look at $3$-dimensional subspaces $\Pi$ of $\Lambda^{2}{\bf C}^{7}$ preserved by the action and we just consider those on which the action has weights $1,0-1$. Now the weight $1$-subspace of $\lambda^{2}$ has a basis $e_{3}\wedge e_{-2},e_{2}\wedge e_{-1},e_{1}\wedge e_{0}$ and our space $\Pi$ must contain a vector

$$u=u_{3,-2}e_{3}\wedge e_{-2}+u_{2,-1}e_{2}\wedge e_{-1}+u_{1,0}e_{1}\wedge e_{0},$$

for scalars $u_{3,-2}$ etc. Similarly $\Pi$ must contain a vector

$$v=v_{1,-1}e_{1}\wedge e_{-1}+v_{2,-2}e_{2}\wedge e_{-2}+v_{3,-3}e_{3}\wedge e_{-3}$$

and a vector

$$w=w_{-3,2}e_{-3}\wedge e_{2}+w_{-2,1}e_{-2}\wedge e_{1}+w_{-1,0}e_{-1}\wedge e_{0}.$$

The vector space $\Pi$ is determined by these three vectors $u,v,w$. The coefficients are not unique. We could change $u,v,w$ to $\mu_{1}u,\mu_{2}v,\mu_{3}w$. Also we could change our basis vectors $e_{i}$ to $\lambda_{i}e_{i}$ to give an equivalent $3$-plane. This would change the coefficients, for example $u_{3,-2}$ would change to $\lambda_{3}\lambda_{-2}u_{3,-2}$. However the expression

$$\tau=\frac{u_{3,-2}w_{-3,2}v_{1,-1}}{u_{2,-1}w_{-2,1}v_{3,-3}}$$

is invariant under all these changes and gives a “modulus” for this family. The Mukai-Umemura manifold has $\tau=1$. When $\tau$ is close to $1$ we have seen that the corresponding manifold admits a Kahler-Einstein metric. It seems likely that this true for all $\tau$ but, as far the author is aware, this is not known. It seems an interesting test case for future developments in the existence theory.

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 $B^{4}\setminus\{0\}$.