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Is Deligne-Mumford space could also be defined in the complex geometry context? I check wiki, it says we can similarly define Riemann surface with nodes and stability condition, I am wondering if there is any reference providing more details about this aspect. Thanks!

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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. –  BCnrd Jul 4 '10 at 22:33
    
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! –  HYYY Jul 5 '10 at 0:05
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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 –  BCnrd Jul 5 '10 at 1:19
    
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). –  BCnrd Jul 5 '10 at 1:24
    
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, ...). –  Andy Putman Jul 5 '10 at 4:09
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2 Answers

up vote 5 down vote accepted

I'm not sure where to point you for full details of this, but quite a few details are in some old research announcements of Bers. See his papers

MR0361051 (50 #13497) Bers, Lipman Spaces of degenerating Riemann surfaces. Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 43--55. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974.

and

MR0361165 (50 #13611) Bers, Lipman On spaces of Riemann surfaces with nodes. Bull. Amer. Math. Soc. 80 (1974), 1219--1222.

and

MR0374496 (51 #10696) Bers, Lipman Deformations and moduli of Riemann surfaces with nodes and signatures. Collection of articles dedicated to Werner Fenchel on his 70th birthday. Math. Scand. 36 (1975), 12--16.


EDIT : Sorry to resurrect this ancient thread, but I heard a lovely talk from Sarah Koch a few weeks ago in which she described a recent paper that she wrote with John Hubbard in which they give a complex-analytic construction of the DM compactification of the moduli space of curves and prove that (as a complex analytic space) it is isomorphic to the analyticification of the usual one. In particular, this gives all the missing details in Bers's papers above (along with much more). See their paper "An analytic construction of the Deligne-Mumford compactification of the moduli space of curves" available from Sarah's webpage here.

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Another construction of the Deligne-Mumford compactification in the complex-analytic category was carried some years ago by Robbin and Salamon; "A construction of the Deligne-Mumford orbifold", JEMS 8 (2006), 611-699 or http://www.math.ethz.ch/~salamon/PREPRINTS/dmETH.pdf It contains full details, but their approach is completely different from Bers's one. Robbin and Salamon use techniques from geometric analysis. If you are not afraid of such analysis, take a look.

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