Noncommutative resolution of $SU_C(2)$
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arXiv
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| Format: | Preprint |
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2024
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| author | Sink, Elias Tevelev, Jenia |
| author_facet | Sink, Elias Tevelev, Jenia |
| contents | We study the derived category of the moduli space $SU_C(2)$ of rank $2$ vector bundles on a smooth projective curve $C$ of genus $g\ge 2$ with trivial determinant. This generalizes the recent work by Tevelev and Torres on the case with fixed odd determinant. Since $SU_C(2)$ is singular, we work with its resolution of singularities, specifically with the noncommutative resolution constructed by Pădurariu and Špenko--Van den Bergh (in the more general setting of symmetric stacks). We show that this noncommutative resolution admits a semiorthogonal decomposition into derived categories of symmetric powers $Sym^{2k}C$ for $2k\le g-1$. In the case of even genus, each block appears four times. This is also true in the case of odd genus, except that the top symmetric power $Sym^{g-1}C$ appears twice. In the case of even genus, the noncommutative resolution is strongly crepant in the sense of Kuznetsov and categorifies the intersection cohomology of $SU_C(2)$. Since all of its components are "geometric," our semiorthogonal decomposition provides evidence for the expectation, which dates back to the work of Newstead and Tyurin, that $SU_C(2)$ is a rational variety. Finally, we study mutations of semiorthogonal decompositions on the Hecke correspondence, answering a question of Pădurariu and Toda. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_08891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noncommutative resolution of $SU_C(2)$ Sink, Elias Tevelev, Jenia Algebraic Geometry We study the derived category of the moduli space $SU_C(2)$ of rank $2$ vector bundles on a smooth projective curve $C$ of genus $g\ge 2$ with trivial determinant. This generalizes the recent work by Tevelev and Torres on the case with fixed odd determinant. Since $SU_C(2)$ is singular, we work with its resolution of singularities, specifically with the noncommutative resolution constructed by Pădurariu and Špenko--Van den Bergh (in the more general setting of symmetric stacks). We show that this noncommutative resolution admits a semiorthogonal decomposition into derived categories of symmetric powers $Sym^{2k}C$ for $2k\le g-1$. In the case of even genus, each block appears four times. This is also true in the case of odd genus, except that the top symmetric power $Sym^{g-1}C$ appears twice. In the case of even genus, the noncommutative resolution is strongly crepant in the sense of Kuznetsov and categorifies the intersection cohomology of $SU_C(2)$. Since all of its components are "geometric," our semiorthogonal decomposition provides evidence for the expectation, which dates back to the work of Newstead and Tyurin, that $SU_C(2)$ is a rational variety. Finally, we study mutations of semiorthogonal decompositions on the Hecke correspondence, answering a question of Pădurariu and Toda. |
| title | Noncommutative resolution of $SU_C(2)$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2405.08891 |