Symmetric products of Galois-Maximal varieties
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929491027165184 |
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| author | Orts, Javier |
| author_facet | Orts, Javier |
| contents | The main result of this paper is the proof that all the symmetric products of a (finite) Galois-Maximal space are also Galois-Maximal spaces. This applies to the special case of real algebraic varieties, solving the problem first stated by Biswas and D'Mello in \cite{biswas&d'mello:symmetric_products_M-curves} about symmetric products of Maximal curves, and then generalised by Baird in \cite{baird:symmetric_products_GM-curves} to Galois-Maximal curves. We also give characterisations of these spaces and state a new definition that generalises to a larger class of spaces. Then, we extend the characterisation in terms of the Borel cohomology given in \cite{us} to the new family. Finally, we introduce the notion of cohomological stability and cohomological splitting, provide a systematic treatment and relate them with the properties of being a Maximal or Galois-Maximal space. These cohomological properties play an important role in the proof of our main theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09934 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric products of Galois-Maximal varieties Orts, Javier Algebraic Geometry Algebraic Topology 14F45, 14P25, 55N91 The main result of this paper is the proof that all the symmetric products of a (finite) Galois-Maximal space are also Galois-Maximal spaces. This applies to the special case of real algebraic varieties, solving the problem first stated by Biswas and D'Mello in \cite{biswas&d'mello:symmetric_products_M-curves} about symmetric products of Maximal curves, and then generalised by Baird in \cite{baird:symmetric_products_GM-curves} to Galois-Maximal curves. We also give characterisations of these spaces and state a new definition that generalises to a larger class of spaces. Then, we extend the characterisation in terms of the Borel cohomology given in \cite{us} to the new family. Finally, we introduce the notion of cohomological stability and cohomological splitting, provide a systematic treatment and relate them with the properties of being a Maximal or Galois-Maximal space. These cohomological properties play an important role in the proof of our main theorem. |
| title | Symmetric products of Galois-Maximal varieties |
| topic | Algebraic Geometry Algebraic Topology 14F45, 14P25, 55N91 |
| url | https://arxiv.org/abs/2403.09934 |