The moduli space of multi-scale differentials

We construct a compactification P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) 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, P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined by Bainbridge, Chen, Gendron, Grushevsky, and Möller [Duke Math. J. 167 (2018), pp. 2347–2416] 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 smooth Deligne-Mumford stack, and P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized space Ξ M ¯ g , n ( μ ) \Xi \overline {\mathcal {M}}_{g,n}(\mu ) such that the G L 2 + (