The moduli space of multi-scale differentials
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| Acceso en línea: | |
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| _version_ | 1866916497535795200 |
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| author | Bainbridge, Matt Chen, Dawei Gendron, Quentin Grushevsky, Samuel Möller, Martin |
| author_facet | Bainbridge, Matt Chen, Dawei Gendron, Quentin Grushevsky, Samuel Möller, Martin |
| contents | We construct a compactification 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, our compactification 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 our compactification is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized moduli space of multi-scale differentials such that the $\mathrm{GL}_2(\mathbb R)$-action in the interior of the moduli space extends continuously to the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_13492 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The moduli space of multi-scale differentials Bainbridge, Matt Chen, Dawei Gendron, Quentin Grushevsky, Samuel Möller, Martin Algebraic Geometry Dynamical Systems Geometric Topology We construct a compactification 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, our compactification 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 our compactification is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized moduli space of multi-scale differentials such that the $\mathrm{GL}_2(\mathbb R)$-action in the interior of the moduli space extends continuously to the boundary. |
| title | The moduli space of multi-scale differentials |
| topic | Algebraic Geometry Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/1910.13492 |