The $\mathbb{A}^1$-Euler characteristic of symmetric powers
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914246839762944 |
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| author | Bröring, Louisa F. Pajwani, Jesse Viergever, Anna M. |
| author_facet | Bröring, Louisa F. Pajwani, Jesse Viergever, Anna M. |
| contents | The $\mathbb{A}^1$-Euler characteristic is a refinement in algebraic geometry of the classical topological Euler characteristic, which can be constructed using motivic homotopy theory. This invariant is a quadratic form rather than an integer, which carries a lot of information, but is difficult to compute in practice. In this survey, we discuss a conjectural way for computing the $\mathbb{A}^1$-Euler characteristic of the symmetric powers of a variety in terms of the $\mathbb{A}^1$-Euler characteristic of the variety itself formulated using the theory of power structures. We discuss evidence towards the conjecture so far, techniques to approach it, and applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $\mathbb{A}^1$-Euler characteristic of symmetric powers Bröring, Louisa F. Pajwani, Jesse Viergever, Anna M. Algebraic Geometry 14G27, 14N10, 14F42 The $\mathbb{A}^1$-Euler characteristic is a refinement in algebraic geometry of the classical topological Euler characteristic, which can be constructed using motivic homotopy theory. This invariant is a quadratic form rather than an integer, which carries a lot of information, but is difficult to compute in practice. In this survey, we discuss a conjectural way for computing the $\mathbb{A}^1$-Euler characteristic of the symmetric powers of a variety in terms of the $\mathbb{A}^1$-Euler characteristic of the variety itself formulated using the theory of power structures. We discuss evidence towards the conjecture so far, techniques to approach it, and applications. |
| title | The $\mathbb{A}^1$-Euler characteristic of symmetric powers |
| topic | Algebraic Geometry 14G27, 14N10, 14F42 |
| url | https://arxiv.org/abs/2510.06922 |