Properness implies $f(v)\ne0$ for $v\ne0$: otherwise the whole noncompact complex line $\mathbb Cv$ would lie in $f^{-1}(0)$. Homogeneity therefore defines
$$
\bar f:\mathbb{CP}^{n-1}\longrightarrow\mathbb{CP}^{n-1},
\qquad [v]\longmapsto[f(v)].
$$
On the fiber over $[v]$, define
$$
([v],zv)\longmapsto([v],zf(v)).
$$
The identity $f(\lambda v)=\lambda f(v)$ makes this independent of the representative $v$, and it is a nonzero complex-linear map on every fiber. It is therefore the bundle isomorphism
$$
\gamma_{1,n}\cong\bar f^*\gamma_{1,n}.
$$

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-114/3/c/solution