The moduli space of multi-scale differentials
Abstract
We construct a compactification P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) 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, P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined by Bainbridge, Chen, Gendron, Grushevsky, and Möller [Duke Math. J. 167 (2018), pp. 2347–2416] 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 smooth Deligne-Mumford stack, and P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized space Ξ M ¯ g , n ( μ ) \Xi \overline {\mathcal {M}}_{g,n}(\mu ) such that the G L 2 + (
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BibTeX
@article{bainbridge2026moduli,
title = {{The moduli space of multi-scale differentials}},
author = {Matt Bainbridge and Dawei Chen and Quentin Gendron and Samuel Grushevsky and Martin Möller},
journal = {Communications of the American Mathematical Society},
year = {2026},
month = oct,
doi = {10.1090/cams/78},
url = {https://doi.org/10.1090/cams/78},
note = {CC BY},
}
RIS
TY - JOUR
TI - The moduli space of multi-scale differentials
AU - Matt Bainbridge
AU - Dawei Chen
AU - Quentin Gendron
AU - Samuel Grushevsky
AU - Martin Möller
PY - 2026
DA - 2026/10/20
JO - Communications of the American Mathematical Society
DO - 10.1090/cams/78
UR - https://doi.org/10.1090/cams/78
AB - We construct a compactification P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) 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, P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined by Bainbridge, Chen, Gendron, Grushevsky, and Möller [Duke Math. J. 167 (2018), pp. 2347–2416] 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 smooth Deligne-Mumford stack, and P Ξ M ¯ g , n ( μ ) \mathbb {P}\Xi \overline {\mathcal {M}}_{g,n}(\mu ) is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized space Ξ M ¯ g , n ( μ ) \Xi \overline {\mathcal {M}}_{g,n}(\mu ) such that the G L 2 + (
SN - 2692-3688
ER -