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.