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! 



Deligne-Mumford space defined in complex geometry category
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. 




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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](http://www.math.harvard.edu/~kochs/papers.html).


