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An analytic construction of the Deligne-Mumford compactification of the moduli space of curves. (English) Zbl 1318.32019

Denote by \(\mathcal{M}_{g,n}\) the moduli space of curves of genus \(g\) and \(n\) marked points such that \(2 - 2g - n < 0\). Deligne and Mumford compactified \(\mathcal{M}_{g,n}\), obtaining a projective algebraic variety denoted by \(\overline{\mathcal{M}}_{g,n}\). Alternatively, one can interpret \(\mathcal{M}_{g,n}\) as the quotient of the Teichmüller space \(\mathcal{T}_{g,n}\) of an \(n\)-punctured genus \(g\) surface, by its mapping class group \(MCG_{g,n}\). The Teichmüller space sits inside the augmented Teichmüller space \(\widehat{\mathcal{T}}_{g,n}\) introduced by Abikoff. This is \(\mathcal{T}_{g,n}\) with a stratified boundary attached to it. The group \(MCG_{g,n}\) acts on \(\widehat{\mathcal{T}}_{g,n}\) and the quotient \(\widehat{\mathcal{M}}_{g,n}\) provides also a compactification of the moduli space, homeomorphic to \(\overline{\mathcal{M}}_{g,n}\). Nevertheless, \(\widehat{\mathcal{M}}_{g,n}\) does not carry, a priori, a natural analytic structure since the augmented Teichmüller space is not a manifold. The paper under review provides a canonical analytic structure on \(\widehat{\mathcal{M}}_{g,n}\) and proves that \(\widehat{\mathcal{M}}_{g,n}\) and \(\overline{\mathcal{M}}_{g,n}\) are canonically isomorphic.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14D22 Fine and coarse moduli spaces
30F60 Teichmüller theory for Riemann surfaces
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