Singularities in Calogero--Moser Varieties
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909616966729728 |
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| author | Bellamy, Gwyn Maksimau, Ruslan Schedler, Travis |
| author_facet | Bellamy, Gwyn Maksimau, Ruslan Schedler, Travis |
| contents | In this article we describe completely the singularities appearing in Calogero--Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero--Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnafé for these groups. We use the fact that the Calogero--Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularities in Calogero--Moser Varieties Bellamy, Gwyn Maksimau, Ruslan Schedler, Travis Algebraic Geometry Representation Theory Symplectic Geometry In this article we describe completely the singularities appearing in Calogero--Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero--Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnafé for these groups. We use the fact that the Calogero--Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties. |
| title | Singularities in Calogero--Moser Varieties |
| topic | Algebraic Geometry Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2505.13779 |