Timeline for Deligne-Mumford space defined in complex geometry category
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Oct 13, 2010 at 19:10 | answer | added | Johannes Ebert | timeline score: 6 | |
| Jul 7, 2010 at 14:32 | comment | added | Andy Putman | @BCnrd : I'm not a historian, so don't take the following too seriously. Ahlfors and Bers didn't really consider M_g to be a space w/ a universal property. AFAIK, they had 2 motivations. One, they wanted to make rigorous Riemann's parameter count for M_g. Two, they were interested in understanding Fuchsian and Kleinian groups. The latter might explain why they studied Teichmuller space more than M_g, as Teichmuller space really is just a space of representations. I actually think they found it pretty surprising when they discovered that Teichmuller space had a complex structure... | |
| Jul 6, 2010 at 5:12 | comment | added | BCnrd | Dear HYYY: I don't know symplectic geometry, so I suppose your comment is indicating a context for which the approach of Bers and Alfhors is more appropriate. But are you also asking again the question about the compactness? I briefly sketched in my 2nd comment about why the analytic moduli space is compact and Hausdorff (in the orbifold sense). There is no adequate literature reference (as far as I know); sorry. Hopefully you can get something useful out of my sketch. I had to figure out all of this stuff for myself, and assumed everyone else has to do the same. | |
| Jul 6, 2010 at 2:37 | comment | added | HYYY | Hi,BCnrd, in symplectic GW theory, we have a Gromov compactness on the space of stable maps, if we allow the target symplectic manifold be a point, then can we say we can the answer,that is, the space of smooth or nodal Riemann surface with genus $g$ and $n$ marked points and satisfying stability condition is a compact orbifold? but it seems that Gromov compactness is for Gromov Topology, it doesn't fit into the concept of moduli space! | |
| Jul 5, 2010 at 5:05 | comment | added | BCnrd | Andy, perhaps you can answer some questions I've wondered for a long time: from the pre-Grothedieck viewpoint of Bers and Alfhors, was the moduli space (or Teichmuller space) just "a construction" (in which case I wonder their reason to study it beyond "seems a good thing to do") or was it characterized by an abstract property that endowed it with intrinsic significance relative to general "families" (expressed in some concrete terms)? And from their point of view was it possible to study the space without knowing how it was constructed (as is often done with Grothendieck's approach)? | |
| Jul 5, 2010 at 4:09 | comment | added | Andy Putman | Bers's stuff is quite different from the algebro-geometric approach. It is more in line with the classical approach to the moduli space of Riemann surfaces that Bers and Ahlfors developed in the early '50's, following ideas of Teichmuller (using tools like hyperbolic geometry and quasiconformal maps). While this approach isn't the best approach for studying algebro-geometric problems, it has been very influential among other "users" of the moduli space of curves (eg in Kleinian groups, complex dynamics, hyperbolic geometry, ...). | |
| Jul 5, 2010 at 1:24 | comment | added | BCnrd | One addendum to my previous comment: I've never read anything by Bers (apart from the nifty mnemonic to remember the quotient rule for derivatives in his calculus book), so I have no idea how his stuff interacts with the method I outline above. It is not necessary to know anything about Bers' work to develop the theory (which isn't to say that there's anything wrong with looking at his stuff for inspriration; I simply never did so). | |
| Jul 5, 2010 at 1:19 | comment | added | BCnrd | All "basic", but no ref. with proofs (e.g., not the nice Encyclopedia of Math book on complex spaces). Hand-waving not enough? :) To rigorously prove analytification of alg. moduli space is analytic mod. space, key ingredients (beyond GAGA) are: analytic Proj and analytic relative ampleness. I wrote "Rel. ampleness in rigid-analytic geometry" (includes analytic Proj) so works in the complex-analytic case; sec. 4 handles Hilbert & Quot functors, and methods adapt to $\overline{M}_{g,n}$ using rel. ample line bundles. Chow's Lemma for DM-stacks + alg. properness yields compact & Hausdorff. QED | |
| Jul 5, 2010 at 0:05 | comment | added | HYYY | Hi,Bcnrd,thanks! So $\bar{M_{g,n}}$ can be defined in complex geometry,it is the space of smooth or nodal Riemann surface with genus $g$ and $n$ marked points and satisfying stability condition, is there any reference showing that it is compact and hausdorff? I didn't see any referece, but it should be basic material! | |
| Jul 4, 2010 at 22:33 | comment | added | BCnrd | Do you want to know if the "analytification" of the DM-stacks of stable marked curves (equipped with their "universal" structure) enjoy an analogous universal property in the complex-analytic category? This is not a tautology, so you probably won't see it in the work of Bers (whom I assume doesn't work with non-smooth bases, for instance), but it can be deduced by analytic analogues of certain algebraic constructions (ultimately deriving from universality of analytified Hilbert schemes, which also requires a proof). Please clarify your question. | |
| Jul 4, 2010 at 21:06 | vote | accept | HYYY | ||
| Jul 4, 2010 at 20:43 | answer | added | Andy Putman | timeline score: 5 | |
| Jul 4, 2010 at 20:20 | history | asked | HYYY | CC BY-SA 2.5 |