Noncommutative crepant resolutions of $cA_n$ singularities via Fukaya categories
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913993960980480 |
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| author | Evans, Jonathan David Lekili, Yanki |
| author_facet | Evans, Jonathan David Lekili, Yanki |
| contents | We compute the wrapped Fukaya category $\mathcal{W}(T^*S^1, D)$ of a cylinder relative to a divisor $D= \{p_1,\ldots, p_n\}$ of $n$ points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over $k[t_0,\ldots, t_n]$) of the singularity $uv=t_0t_1\ldots t_n$. Upon making the base-change $t_i= f_i(x,y)$, we obtain the derived category of any crepant resolution of the $cA_{n}$ singularity given by the equation $uv= f_0\ldots f_n$. These categories inherit braid group actions via the action on $\mathcal{W}(T^*S^1,D)$ of the mapping class group of $T^*S^1$ fixing $D$. We also give a geometric model of the derived contraction algebra of a $cA_n$ singularity in terms of the relative Fukaya category of the disc. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06592 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Noncommutative crepant resolutions of $cA_n$ singularities via Fukaya categories Evans, Jonathan David Lekili, Yanki Symplectic Geometry Algebraic Geometry Representation Theory 53D37, 14J33, 16S38, 14A22 We compute the wrapped Fukaya category $\mathcal{W}(T^*S^1, D)$ of a cylinder relative to a divisor $D= \{p_1,\ldots, p_n\}$ of $n$ points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over $k[t_0,\ldots, t_n]$) of the singularity $uv=t_0t_1\ldots t_n$. Upon making the base-change $t_i= f_i(x,y)$, we obtain the derived category of any crepant resolution of the $cA_{n}$ singularity given by the equation $uv= f_0\ldots f_n$. These categories inherit braid group actions via the action on $\mathcal{W}(T^*S^1,D)$ of the mapping class group of $T^*S^1$ fixing $D$. We also give a geometric model of the derived contraction algebra of a $cA_n$ singularity in terms of the relative Fukaya category of the disc. |
| title | Noncommutative crepant resolutions of $cA_n$ singularities via Fukaya categories |
| topic | Symplectic Geometry Algebraic Geometry Representation Theory 53D37, 14J33, 16S38, 14A22 |
| url | https://arxiv.org/abs/2307.06592 |