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

TL;DR

This paper provides an analytic construction of the Deligne-Mumford compactification of the moduli space of curves, establishing a canonical isomorphism with the quotient of augmented Teichmueller space.

Contribution

It introduces an analytic structure on the quotient of augmented Teichmueller space and proves its equivalence to the Deligne-Mumford compactification as an analytic space.

Findings

Established a canonical isomorphism between two compactifications.

Provided an analytic structure on the quotient space.

Unified algebraic and analytic perspectives of the moduli space.

Abstract

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