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 $(2,2,3)$.