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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2302.00038 |
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| _version_ | 1866910898377981952 |
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| author | Bu, Chenjing |
| author_facet | Bu, Chenjing |
| contents | In this series of papers, we propose a theory of enumerative invariants counting self-dual objects in self-dual categories. Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group $\mathrm{GL} (n)$, and our theory is an extension of this to structure groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$. Examples of our invariants include invariants counting principal orthogonal or symplectic bundles, and invariants counting self-dual quiver representations.
In the present paper, we take the motivic approach, and define our invariants as elements in a ring of motives. We also extract numerical invariants by taking Euler characteristics of these elements. We prove wall-crossing formulae relating our invariants for different stability conditions. We also provide an explicit algorithm computing invariants for quiver representations, and we present some numerical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00038 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Enumerative invariants in self-dual categories. I. Motivic invariants Bu, Chenjing Algebraic Geometry In this series of papers, we propose a theory of enumerative invariants counting self-dual objects in self-dual categories. Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group $\mathrm{GL} (n)$, and our theory is an extension of this to structure groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$. Examples of our invariants include invariants counting principal orthogonal or symplectic bundles, and invariants counting self-dual quiver representations. In the present paper, we take the motivic approach, and define our invariants as elements in a ring of motives. We also extract numerical invariants by taking Euler characteristics of these elements. We prove wall-crossing formulae relating our invariants for different stability conditions. We also provide an explicit algorithm computing invariants for quiver representations, and we present some numerical results. |
| title | Enumerative invariants in self-dual categories. I. Motivic invariants |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2302.00038 |