Is DeligneMumford 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!

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. 4355. 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), 12191222. 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), 1216. 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 complexanalytic 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 DeligneMumford compactification of the moduli space of curves" available from Sarah's webpage here. 


Another construction of the DeligneMumford compactification in the complexanalytic category was carried some years ago by Robbin and Salamon; "A construction of the DeligneMumford orbifold", JEMS 8 (2006), 611699 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. 

