Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by complex conjugation be trivial?

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    $\begingroup$ How does conjugation act on $V$ or $G$? $\endgroup$ – Piotr Achinger Jun 25 at 20:30
  • $\begingroup$ @PiotrAchinger I forgot to add a condition $\endgroup$ – vrz Jun 25 at 20:43
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    $\begingroup$ Don't forget about base points. Usually it can be ignored, but in this case, you probably need to pick one in $V(\mathbb{R})$. In particular, the set of real points should be nonempty. $\endgroup$ – Donu Arapura Jun 26 at 12:34
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    $\begingroup$ You might be able to concoct an example from a finite group representation. Suppose one has a complex representation which is conjugation equivariant, so that conjugation acts nontrivially. The action on a Flag variety should give a complex projective variety with the appropriate properties I think. $\endgroup$ – Ian Agol Jun 27 at 23:54
  • $\begingroup$ Where are going to get variety over $\mathbb C$ with fundamental group $G$? Have you heard of Godeaux-Serre varieties? I think that if you just follow the construction over $\mathbb R$, you get what you seek. I googled Godeaux-Serre to see if there was a useful reference, and I didn't find one. But I did find this paper saying that you can have whatever action you want, over any field. $\endgroup$ – Ben Wieland Jun 29 at 0:06

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