We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

We construct a compactification PΞMg,n(µ) 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, PΞMg,n(µ) 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 PΞMg,n(µ) is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized space ΞMg,n(µ) such that the SL2(R)-action in the interior of the moduli space extends continuously to the boundary.A multi-scale differential on a pointed stable curve is the data of an enhanced level structure on the dual graph, prescribing the orders of poles and zeroes at the nodes, together with a collection of meromorphic differentials on the irreducible components satisfying certain conditions. Additionally, the multi-scale differential encodes the data of a prong-matching at the nodes, matching the incoming and outgoing horizontal trajectories in the flat structure. The construction of PΞMg,n(µ) furthermore requires defining families of multi-scale differentials, where the underlying curve can degenerate, and understanding the notion of equivalence of multi-scale differentials under various rescalings.Our construction of the compactification proceeds via first constructing an augmented Teichmüller space of flat surfaces, and then taking its suitable quotient. Along the way, we give a complete proof of the fact that the conformal and quasiconformal topologies on the (usual) augmented Teichmüller space agree.

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of the odd minimal stratum in genus three.

Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.

A k-differential on a Riemann surface is a section of the kth power of the canonical line bundle. Loci of k-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of k-differentials. In this paper, we give a complete description for the compactification of the strata of k-differentials in terms of pointed stable k-differentials, for all k. The upshot is a global k-residue condition that can also be reformulated in terms of admissible covers of stable curves. Moreover, we study properties of k-differentials regarding their deformations, residues, and flat geometric structure.