Compactification of strata of abelian differentials

TL;DR

This paper characterizes the boundary of strata of abelian differentials in the moduli space, providing explicit descriptions and proofs for smoothing boundary differentials, with a focus on global residue conditions.

Contribution

It introduces a new global residue condition based on dual graph ordering to describe the closure of abelian differential strata.

Findings

Explicit boundary description of abelian differential strata

Complex analytic and geometric proofs for smoothing boundary differentials

Examples illustrating the boundary and smoothing processes

Abstract

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.