= Solution
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.
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