Timeline for Finiteness property of fundamental groups
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 14, 2011 at 12:55 | answer | added | Sebastian Petersen | timeline score: 1 | |
| Jan 31, 2011 at 23:41 | comment | added | Minhyong Kim | Donu's argument works by the invariance of the etale $pi_1$ under base-change of algebraically closed fields in characteristic zero. I believe the required specialization argument can be found in Serre's lectures on Galois theory. The key point is that $\pi_1(X\times Y)=\pi_1(X)\times \pi_1(Y)$, which fails in positive characteristic. An amusing (or vexing) fact is that an 'algebraic proof', in any reasonable sense, of finite-generation is unknown even for curves. | |
| Jan 31, 2011 at 21:37 | comment | added | Donu Arapura | Martin, in general, you can choose alg. closed fields $K\supset K_0\subset \mathbb{C}$ so that $X$ is defined over $K_0$. You would need to verify that the (tame) fundamental group is unchanged in the process. This may require digging through SGA and/or supplying a proof. | |
| Jan 31, 2011 at 20:25 | comment | added | Martin Brandenburg | @Donu: This only covers the case $K = \mathbb{C}$, right? Or can we generalize this using model theory? | |
| Jan 31, 2011 at 18:09 | comment | added | Donu Arapura | Mariano, I don't know of a purely algebraic argument, but it would certainly be interesting. As an aside, I might point out that the "wild" part of $\pi_1(\matbb{A}^1)$ in positive characteristic need not be finitely generated. | |
| Jan 31, 2011 at 16:35 | comment | added | Mariano Suárez-Álvarez | @Donu: can it be proved 'purely algebraicly'? | |
| Jan 31, 2011 at 16:21 | comment | added | Donu Arapura | Yes. Use the finite generation for the topological fund. grp (by using finite triangulability of complex varieties) + the comparison theorem SGA1 exp XII cor 5.2. | |
| Jan 31, 2011 at 16:11 | history | asked | Sebastian Petersen | CC BY-SA 2.5 |