An analytic construction of the Deligne-Mumford compactification of the moduli space of curves

Abstract

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves $\mathcal{M}_{g,n}$. Their compactification $\overline{\mathcal{M}}_{g,n}$ is a projective algebraic variety, and as such it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmüller space by the action of the mapping class group gives a compactification of $\mathcal{M}_{g,n}$. We put an analytic structure on this quotient and prove that with respect to this structure, the compactification is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$.