Explicit bundle projection

Question

Consider the principal bundle $S^7$ with base space $\mathbb{C}P^3$ (3-dimensional complex projective space) and fiber $S^1\cong U(1)$. Can someone write to me the bundle projection $\pi:S^7\rightarrow\mathbb{C}P^3$ explicitly?

Answer 1

Regard $S^7$ as the set of points in $\mathbb C^4$ of length $1$. If $p:\mathbb C^4\to\mathbb CP^3$ is the usual quotient map, then the restriction $p|_{S^7}:S^7\to\mathbb CP^3$ is the projection you are after.