Timeline for Fundamental group of the moduli space of parabolic bundles with fixed determinant
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2023 at 15:20 | comment | added | yors | Thank you so much. I got it. | |
| May 1, 2023 at 10:56 | comment | added | Jason Starr | That follows from the Hurewicz Isomorphism. | |
| May 1, 2023 at 2:21 | comment | added | yors | Thank you for your response. So since the moduli space of parabolic bundles is rational, it is simply connected. Is the isomorphism from $\pi_2$ to $H_2$ true for the parabolic case (in characteristic zero)? | |
| Apr 30, 2023 at 16:42 | comment | added | Jason Starr | The moduli space is still rational (not just unirational), hence simply connected (in fact, all separably rationally connected, smooth projective varieties over an algebraically closed field are simply connected -- char 0 by Campana, char p by Koll'ar and Debarre). In characteristic 0, by Hurewicz, the map from $\pi_2$ to $H_2$ is an isomorphism, and these are free Abelian groups of rank $1$ (if no parabolic structure). For $\pi_3$, already the Jacobian of the curve intercedes. I believe this is worked out in the textbook of Griffiths-Morgan. | |
| Apr 30, 2023 at 16:20 | history | asked | yors | CC BY-SA 4.0 |