Nonprojective crepant resolutions of quiver varieties
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909609031106560 |
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| author | Kaplan, Daniel Schedler, Travis |
| author_facet | Kaplan, Daniel Schedler, Travis |
| contents | In this paper, we construct a large class of examples of proper, nonprojective crepant resolutions of singularities for Nakajima quiver varieties. These include four and six dimensional examples and examples with $Q$ containing only three vertices. There are two main techniques: by taking a locally projective resolution of a projective partial resolution as in our previous work arXiv:2311.07539, and more generally by taking quotients of open subsets of representation space which are not stable loci, related to Arzhantsev--Derental--Hausen--Laface's construction in the setting of Cox rings. By the latter method we exhibit a proper crepant resolution that does not factor through a projective partial resolution. Most of our quiver settings involve one-dimensional vector spaces, hence the resolutions are toric hyperkähler, which were studied from a different point of view in Arbo and Proudfoot arXiv:1511.09138.
This builds on the classification of projective crepant resolutions of a large class of quiver varieties in arXiv:2212.09623 and the classification of proper crepant resolutions for the hyperpolygon quiver varieties in arXiv:2406.04117. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonprojective crepant resolutions of quiver varieties Kaplan, Daniel Schedler, Travis Algebraic Geometry Representation Theory Symplectic Geometry 14E15, 14J42 In this paper, we construct a large class of examples of proper, nonprojective crepant resolutions of singularities for Nakajima quiver varieties. These include four and six dimensional examples and examples with $Q$ containing only three vertices. There are two main techniques: by taking a locally projective resolution of a projective partial resolution as in our previous work arXiv:2311.07539, and more generally by taking quotients of open subsets of representation space which are not stable loci, related to Arzhantsev--Derental--Hausen--Laface's construction in the setting of Cox rings. By the latter method we exhibit a proper crepant resolution that does not factor through a projective partial resolution. Most of our quiver settings involve one-dimensional vector spaces, hence the resolutions are toric hyperkähler, which were studied from a different point of view in Arbo and Proudfoot arXiv:1511.09138. This builds on the classification of projective crepant resolutions of a large class of quiver varieties in arXiv:2212.09623 and the classification of proper crepant resolutions for the hyperpolygon quiver varieties in arXiv:2406.04117. |
| title | Nonprojective crepant resolutions of quiver varieties |
| topic | Algebraic Geometry Representation Theory Symplectic Geometry 14E15, 14J42 |
| url | https://arxiv.org/abs/2505.08624 |