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The moduli space of multi-scale differentials
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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.
Forward citations
Cited by 2 Pith papers
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The geometry of totally geodesic subvarieties of moduli spaces of Riemann surfaces
Totally geodesic subvarieties of moduli space have semisimple Deligne-Mumford boundary and are hierarchically hyperbolic.
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The boundary of a totally geodesic subvariety of moduli space
The boundary of a totally geodesic subvariety of moduli space is itself totally geodesic in each Deligne-Mumford boundary stratum and decomposes into prime pieces with locally isometric projections.
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