It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see some similar identification of the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$, is it a Lens space $S^{2n+1}/\mathbb{Z}_{2n+1}$?
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Write $X:=\mathbb{CP}^{n}$, the space of lines in $\mathbb C^{n+1}$. For each $k\ge1$ the total space of circle bundle $\mathcal O(-k)\to X$ is $\{\ell\in X,v\in(\mathbb C^{n+1})^{\otimes k}:v\in\ell^{\otimes k},\|v\|=1\}$. Now there is a homeomorphism $$ \begin{align*} S^{2n+1}/\mathbb Z_k=\{\ell\in X,v\in\ell:\|v\|=1\}/(z\sim \zeta_kz)&\simeq\{\ell\in X,w\in\ell^{\otimes k}:\|w\|=1\} \\ (\ell,z)&\mapsto (\ell,z\otimes\cdots\otimes z). \end{align*}$$
