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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!

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Which topology are you putting on the topological realisations? Also note that even a trivial vector bundle, i.e. a product, may run into trouble because the functor from schemes to spaces that I'm thinking of doesn't commute with products. –  David Roberts Apr 15 '11 at 7:42
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It's presumably the Hausdorff topology, i.e. the one induced by the Euclidean topology on $\mathbb{R}^n$ or $\mathbb{C}^n$ for subschemes. This would preserve products, and I think the answer to the first question is yes. The second seems to yes as well: the $1$-cocycle for $O(m)$ in the standard covering is $x^m$... –  Donu Arapura Apr 15 '11 at 13:06
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