TITLE: Any real algebraic variety is diffeomorphic to a real algebraic variety defined over $\mathbb{Q}$
QUESTION [17 upvotes]: Given a smooth proper real algebraic variety can you find a smooth proper real algebraic variety defined over $\mathbb{Q}$ that is diffeomorphic to it?

REPLY [9 votes]: Edoardo Ballico and Alberto Tognoli proved in their paper "Algebraic models defined over $\mathbb{Q}$ of differential manifolds" (Geom. dedicata 42, 155-161, 1992) that every compact differential manifold is diffeomorphic to the real points of a regular affine variety defined over $\mathbb{Q}$. 
For non-smooth algebraic varieties there are obstructions to descend from $\mathbb{R}$ to $\mathbb{Q}$, there is a recent paper by Adam Parusinski and Guillaume Rond "Algebraic varieties are homeomorphic to varieties defined over number fields" arXiv 1810.00808 on this subject.
Edit : let me recall some basic facts about real algebraic sets (I refer to "Real algebraic geometry" Bochnak, Coste and Roy).

A complete nonsingular affine real algebraic variety is projective (see BCR 3.4 p.74-75)
An algebraic subset of $\mathbb{R}^n$ is complete if and only if it is closed and bounded (3.4.9 and 3.4.10).