The moduli space of multi-scale differentials

Abstract

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.

Compactifying the rank two Hitchin system via spectral data on semistable curves

“…We recall the construction of a smooth compactification of strata of k-differentials from [CMZ18] and [BCGGM3], specialized to the case of quadratic differentials and later moreover to the principal stratum where all zeros are simple. The fiber of this compactification over a smooth curve is a modification on the space of quadratic differentials, which we analyze in the subsequent Section 5.…”

“…The complete definition for families is lengthy as it requires to encode the passage from the smooth situation (where no prong-matching is present) to boundary points (where prong-matchings are part of the datum). The notion of rescaling ensemble in [BCGGM3] serves this purpose. We only give the consequences here.…”

“…The construction of the modified Hitchin base is motivated by the compactification of strata of quadratic differentials in [BCGGM3] and works in families as the Riemann surface X varies. In particular, it allows to study the (modified) Higgs moduli space on any stable degeneration of X.…”

“…A 'compactification' of strata of differentials with prescribed type of zeros is the space of multi-scale differentials that was introduced in [BCGGM3] for abelian differentials and extended to k-differentials in [CMZ19]. (Strictly speaking, only the C * -quotient by the action of rescaling the differentials is a compact space.)…”

“…To begin with we specify certain Higgs pairs that appear. As in [BCGGM3] and as in the definition of SD Xst these Higgs pairs will be defined level by level on some pointed stable curve (X, z) that stabilizes after forgetting the marked points to its top level curve, which is X st . We define a trace-free multi-scale GL(2, C)-Higgs pair to be a tuple consisting of the following objects.…”