Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$.

Does $R_\mathbb{R}$ send an algebraic vector bundle $p:V\to X$ to a real topological vector bundle $R_\mathbb{R}(p)$ and does $R_\mathbb{C}$ send $p$ to a complex topological vector bundle $R_\mathbb{C}(p)$?

If this is actually the case I wonder if with $X=\mathbb{P^1}$ the line bundles $\mathcal{O}(m)$ over $X$ are mapped to the trivial bundle if $m$ is even and to the Moebius bundle if $m$ is odd.

Thank you!