Timeline for Picard Groups of Moduli Problems
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 22, 2010 at 12:11 | vote | accept | Charles Siegel | ||
| May 27, 2010 at 4:28 | comment | added | Andy Putman | @Charles : That's right. The picture for A_g was known long before the work on M_g and was an important inspiration for that work. | |
| May 27, 2010 at 4:15 | comment | added | Charles Siegel | @Andy, thanks for the detailed answer for the case of curves (and especially the references, going through Hain now). The case for abelian varieties is fairly similar, with the Symplectic group and $\mathbb{H}_g$ replacing Teichmuller space and and the mapping class group? | |
| May 20, 2010 at 15:07 | comment | added | Andy Putman | Actually, here's an example of how to use central extensions to prove things about line bundles. Recently Funar proved that the universal central extension of Mod_g is residually finite (a thm of Deligne shows that the universal central extension of Sp_2g(Z) is not residually finite). This has a nice implication for the Picard group of moduli space : by passing to high enough finite covers of moduli space, you can make the Hodge bundle divisible by as much as you like. Deligne's thm implies by passing to finite covers of A_g, the best you can do is make the hodge bundle divisible by 2. | |
| May 20, 2010 at 14:56 | comment | added | Andy Putman | That an interesting question. Central extensions by Z of Mod_g correspond to classes in H^2(Mod_g;Z). The calculations of H^2(Mod_g;Z) that I know of first use topology to give upper bounds on how large H^2(Mod_g;Z) is, and then prove that this bound is realized. One can either do this using line bundles on moduli space or by using the so-called "Meyer cocycle". Thus line bundles are more used to study central extensions than the other way around. | |
| May 20, 2010 at 14:22 | comment | added | S. Carnahan♦ | Is it profitable to think of these line bundles in terms of central extensions of $\operatorname{Mod}_g$? | |
| May 20, 2010 at 13:03 | history | answered | Andy Putman | CC BY-SA 2.5 |