Let $X$ be a scheme defined over $\mathbb{C}$ with an involution $\sigma$. How to get
a $X\_{\mathbb{R}}$ scheme defined over $\mathbb{R}$ such that $X\_{\mathbb{R}}\times\_\mathbb{R} \mathbb{C} = X$ ? How are the real algebraic bundle on $X\_{\mathbb{R}}$ and complex bundle on
$X$ related?



Real algebraic variety and real algebraic bundle
The paper by
Atiyah
on [K-theory and reality](http://qjmath.oxfordjournals.org/cgi/reprint/17/1/367)
Quart. J. Math. Oxford Ser. (2) 17 1966 367--386. MR0206940 discusses the topological analogue of this question, showing how to relate complex vector bundles over a space with vector bundles over the fixed points of an involution. The idea is to define a sort of intermediate K-group KR of vector bundles with involution, and compare it to the two other K-groups. 


