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? For projective varieties you can approximate the defining equations (though even in this case I don't think I can rigorously write down the details).

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?

Given a smooth proper complex variety can you find a smooth proper complex variety defined over $\mathbb{Q}$ that is diffeomorphic to it? For projective varieties you can approximate the defining equations (though even in this case I don't think I can rigorously write down the details).

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? For projective varieties you can approximate the defining equations (though even in this case I don't think I can rigorously write down the details).