Problem. Fix an $ n \in \mathbb{N} $. Is it true that there exists a $ k \in \mathbb{N} $ such that every smooth complex line bundle over a smooth $ n $-dimensional real manifold is the pullback of the tautological line bundle $ \gamma_{k}^{1} $ over $ \mathbb{CP}^{k} = \mathbf{Gr}_{1} \! \left( \mathbb{C}^{k + 1} \right) $? (Note: The pullback is assumed to be smooth as well.)
I do know that:
By the Whitney Embedding Theorem, we can find a fixed $ k \in \mathbb{N} $ such that every smooth complex line bundle over a smooth $ n $-dimensional real manifold is the (smooth) pullback of the universal rank-$ 1 $ sub-bundle $ E^{1} $ of the tautological $ k $-bundle $ \gamma_{1}^{k} $ over $ \mathbf{Gr}_{k} \! \left( \mathbb{C}^{k + 1} \right) $.
My guess: The problem would have an affirmative answer if two things were true:
- There exists a diffeomorphism $ \phi: \mathbf{Gr}_{1} \! \left( \mathbb{C}^{k + 1} \right) \to \mathbf{Gr}_{k} \! \left( \mathbb{C}^{k + 1} \right) $.
- We can arrange for $ \phi $ to satisfy $ \gamma_{k}^{1} = {\phi^{*}}(E^{1}) $.
The first point seems plausible and might be a well-known fact. However, I am doubtful about the second point.
Thanks to anyone willing to take a stab at this!