Compactification of strata of Abelian differentials

Abstract

We describe the closure of the strata of abelian differentials with pre- scribed 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.