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?
$\begingroup$
$\endgroup$
7
-
1$\begingroup$ How does conjugation act on $V$ or $G$? $\endgroup$– Piotr AchingerCommented Jun 25, 2020 at 20:30
-
$\begingroup$ @PiotrAchinger I forgot to add a condition $\endgroup$– user145520Commented Jun 25, 2020 at 20:43
-
1$\begingroup$ 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 ArapuraCommented Jun 26, 2020 at 12:34
-
1$\begingroup$ 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 AgolCommented Jun 27, 2020 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 WielandCommented Jun 29, 2020 at 0:06
|
Show 2 more comments