The moduli space of multi-scale differentials

TL;DR

This paper introduces a new compactification of the moduli space of abelian differentials on Riemann surfaces, called the moduli space of multi-scale differentials, which has a normal crossing boundary and extends the GL2(R) action continuously.

Contribution

It constructs the moduli space of multi-scale differentials as a complex orbifold with a detailed local description and extends the GL2(R) action to the boundary.

Findings

The compactification is a normal crossing boundary orbifold.

The moduli space is a proper Deligne-Mumford stack.

The GL2(R) action extends continuously to the boundary.

Abstract

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossing boundary. Locally, our compactification can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined in [BCGGM18] as the closure of the stratum of abelian differentials in the closure of the Hodge bundle. We also define families of projectivized multi-scale differentials, which gives a proper Deligne-Mumford stack, and our compactification is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized moduli space of multi-scale differentials such that the $\mathrm{GL}_2(\mathbb R)$-action in the interior of the moduli space extends continuously to the boundary.