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I was reading Barry Mazur's biography and come across this part:

Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first conversations with me, he raised the question (asked of him by Washnitzer) of whether a smooth proper algebraic variety defined over a real quadratic field could yield topologically different differentiable manifolds realized by the two possible imbeddings of the number field into the reals. What a perfect question, at least for me! Not that I answered it. But it was surely one of the very few algebro-geometric questions that I then had the background to appreciate. ... the question provided quite an incentive for a topologist to look at algebraic geometry. I began to learn the elements of algebraic geometry working with Mike Artin.

Is the problem still open? I am an algebraic topology student so it feels very surprising someone will come up with a question like this. But I am at a loss how to experimentally find some toy examples one can work by hand.

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Dear user, There are examples of varieties defined over number fields which give non-homeomorphic manifolds for different embeddings of the number field into $\mathbb C$. I think that Serre gave one of the first such examples, and the theory of Shimura varieties provides others. If I can recall the details, I'll post an answer. I'm not sure if I can give an example defined over a real quad. field, though; I'll have to think about it. Regards, – Matt E Dec 30 '12 at 12:48
Dear Emerton: I see. I should ask my professor (Adrian Vasiu) next semester. – Bombyx mori Dec 30 '12 at 12:55
Dear user, Yes, Adrian will know many examples, probably off the top of his head! Regards, – Matt E Dec 30 '12 at 12:56
Dear Emerton: You know him! Yes, he is very quick and is usually correct. – Bombyx mori Dec 30 '12 at 12:57
Dear Emerton: I have met Professor Vasiu, and he said he cannot recall such an example right away. He said this is just taking quotients of schimura varieties in a subtle way. My level is too low to appreciate his explanation, so the question remains open (to me). – Bombyx mori Feb 7 '13 at 15:49

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