Vector bundles of schemes and their topological realizations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:37:03Z http://mathoverflow.net/feeds/question/61516 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61516/vector-bundles-of-schemes-and-their-topological-realizations Vector bundles of schemes and their topological realizations mustafa-kava 2011-04-13T07:40:03Z 2011-04-15T07:13:45Z <p>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 $R_\mathbb{C}$.</p> <blockquote> <p>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 $R_\mathbb{C}(p)$?</p> </blockquote> <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.</p> <p>Thank you!</p>