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This question is migrated from math.stackexchange. I ask because it is still unclear to me and I did not receive an answer.

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.

(Edit: as comments pointed out this problem is partly resolved by Serre in the case of imaginary quadratic field, and there is a post by David Speyer on this. But the original real quadratic problem is still unclear)

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    $\begingroup$ I believe that this question was answered in the affirmative by Jean-Pierre Serre, but no longer remember where I read that, and may be misremembering. As for the corresponding question for cohomology groups, see pg. 6 of arxiv.org/pdf/math/0210327v1.pdf $\endgroup$
    – Jonah Sinick
    Commented Apr 29, 2013 at 4:28
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    $\begingroup$ If all you want is an exposition of Serre's theorem, then David Speyer wrote up a nice account on the Secret Blogging Seminar : sbseminar.wordpress.com/2009/07/28/… $\endgroup$
    – Andy Putman
    Commented Apr 29, 2013 at 4:40
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    $\begingroup$ I still do not have time to read the paper, but now everything is making sense. The question is around 1958-1959, and Serre's paper is around 1964. $\endgroup$
    – Bombyx mori
    Commented Apr 29, 2013 at 5:23
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    $\begingroup$ In Serre's example, the base field is the Hilbert class field of an imaginary quadratic field. Is there an example with a real quadratic field? $\endgroup$
    – François Brunault
    Commented Apr 29, 2013 at 9:38
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    $\begingroup$ Will an email to Professor Mazur be offensive? $\endgroup$
    – Kerry
    Commented Apr 30, 2013 at 21:41

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