Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917261145538560 |
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| author | Bellamy, Gwyn Schmitt, Johannes Thiel, Ulrich |
| author_facet | Bellamy, Gwyn Schmitt, Johannes Thiel, Ulrich |
| contents | Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_00880 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution Bellamy, Gwyn Schmitt, Johannes Thiel, Ulrich Algebraic Geometry Representation Theory Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution. |
| title | Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2010.00880 |