The moduli space of multi-scale differentials

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.

Major revision: Many details in the proofs added or fixed. All main results unchanged

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Cited by in corpus (14)

• Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
• Infinitesimal structure of the pluricanonical double ramification locus
• A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves
• Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
• Pixton's formula and Abel-Jacobi theory on the Picard stack
• Equations of linear subvarieties of strata of differentials
• The boundary of linear subvarieties
• The Hurwitz space Picard rank conjecture for $d>g-1$
• The WYSIWYG compactification
• Reconstructing orbit closures from their boundaries
• Measure bound for translation surfaces with short saddle connections
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Realizability of tropical pluri-canonical divisors
• Reduction and lifting problem for differential forms on Berkovich curves