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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


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?

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