Timeline for The direct product of the geometric fundamental group and the absolute Galois group
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 25, 2020 at 23:03 | comment | added | Will Sawin | Both $SL_2$ and many Enriques surfaces are examples, because they have a geometrically nontrivial double cover defined over $\mathbb Q$. | |
| Dec 26, 2019 at 22:01 | comment | added | Will Sawin | There are non-smooth examples, like the cubic $y^2 =x^3 -x^2$. | |
| Dec 26, 2019 at 21:26 | answer | added | Felipe Voloch | timeline score: 4 | |
| Dec 26, 2019 at 18:06 | comment | added | Jef | @TheoJohnson-Freyd An Enriques surface has geometric fundamental group of order $2$. | |
| Dec 26, 2019 at 17:53 | comment | added | Kevin Casto | @vrz Yes, that was part of it. Mostly I just couldn't think of any examples where it acts trivially! And analytically, conjugation certainly acts nontrivially on the loop space, so you'd have to cook things up so that it takes loops to homotopic loops. | |
| Dec 26, 2019 at 16:26 | comment | added | LSpice | @TheoJohnson-Freyd, the fundamental group of $\operatorname{SL}_2$ is $\mathbb Z/2\mathbb Z$ (but it's not proper). | |
| Dec 26, 2019 at 10:57 | comment | added | user145520 | @KevinCasto is your objection that conjugation has to interchange $H^{1, 0}$ and $H^{0, 1}$? Could the fundamental group be perfect to avoid that? | |
| Dec 26, 2019 at 4:37 | comment | added | Kevin Casto | @TheoJohnson-Freyd Fair enough -- I guess I really had in mind the setting of the question, with an infinite fundamental group | |
| Dec 26, 2019 at 4:27 | comment | added | Theo Johnson-Freyd | @KevinCasto Maybe there is a variety with $\pi_1 = \mathbb{Z}/2$? | |
| Dec 26, 2019 at 4:04 | comment | added | Kevin Casto | Does complex conjugation ever act trivially on a nontrivial fundamental group of a variety? | |
| Dec 26, 2019 at 1:45 | history | asked | user145520 | CC BY-SA 4.0 |