Duke Mathematical Journal

Compactification of strata of Abelian differentials

Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin Möller

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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.

Article information

Source
Duke Math. J., Volume 167, Number 12 (2018), 2347-2416.

Dates
Received: 23 March 2017
Revised: 12 March 2018
First available in Project Euclid: 10 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1533866574

Digital Object Identifier
doi:10.1215/00127094-2018-0012

Mathematical Reviews number (MathSciNet)
MR3848392

Zentralblatt MATH identifier
06966873

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Keywords
Abelian differentials flat surfaces moduli space of stable curves Teichmüller dynamics plumbing

Citation

Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin. Compactification of strata of Abelian differentials. Duke Math. J. 167 (2018), no. 12, 2347--2416. doi:10.1215/00127094-2018-0012. https://projecteuclid.org/euclid.dmj/1533866574


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