Take the 3-dimensional complex projective space $\mathbb{P}^3$. Consider the action of the group $SU(2)\times SU(2)$. I have read in physics related articles that these group gives a singular foliation of $\mathbb{P}^3$ in three types of orbits: one is the Segré submanifold $\mathbb{P}^1\times\mathbb{P}^1$, another is the real projective space $\mathbb{RP}^3\cong SO(3)$ and then a family of 5-dimensional surfaces that are non-trivial $SO(3)$ fiber bundles over $S^2$. I am trying to recover this foliation by working dircetly in $\mathbb{C}^4$. The idea (which could be incorrect) is to use the homogenous polynomial $P(z)=z_1z_4-z_2z_3$ in $\mathbb{C}^4$. For $P(z)=0$ this is the equation of a well known 6-dimensional singular cone (it is fact a Conifold), it is easy to see that the base (the angular part) of the cone is topologically $S^2\times S^3$ and is a U(1) fiber bundle over $S^2\times S^2$, i.e. the segré orbit of $\mathbb{P}^3$ is recovered in the base of this particular cone. The Kähler metric of this cone is of the form

$ds^2=dr^2+r^2d\Sigma^2$

where $r$ is the radial coordinate and $d\Sigma^2$ is the metric of the base of the cone. My question is the following: can I recover the remaining leaves of $\mathbb{P}^3$ in a similar way? I am almost sure that they can be recovered by "deforming" the equation of the cone by $P(z)=\frac{1}{2}\epsilon$ in $\mathbb{C}^4$ (there is a nice paper of Candelas *et al.* "comments on conifolds" were this is explained very well). The surfaces obtained for fixed $\epsilon\in\mathbb{R}^*$ are everywhere smooth cones and I believe that the remaining orbits of $SU(2)\times SU(2)$ appear in the bases of these cones. Nevertheless, I am having some trouble to recover the 5-dimensional orbits. Is all my approach wrong??! This is kind of new for me. Any known literature or article that can help me with this?

This post imported from StackExchange MathOverflow at 2015-03-02 12:58 (UTC), posted by SE-user Darius Alexander