## 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

### Keywords:

moduli space of curves; compactification
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