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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2011.06114 |
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| _version_ | 1866910593512898560 |
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| author | Huang, Jesse Zhou, Peng |
| author_facet | Huang, Jesse Zhou, Peng |
| contents | Consider $(\mathbb{C}^*)^k$ acting on $\mathbb{C}^N$ satisfying certain 'quasi-symmetric' condition which produces a class of toric Calabi-Yau GIT quotient stacks. Using subcategories of $Coh([\mathbb{C}^N / (\mathbb{C}^*)^k])$ generated by line bundles whose weights are inside certain zonotope called the 'magic window', Halpern-Leistner and Sam give a combinatorial construction of equivalences between derived categories of coherent sheaves for various GIT quotients. We apply the coherent-constructible correspondence for toric varieties to the magic windows and obtain a non-characteristic deformation of Lagrangian skeletons in $\mathbb{R}^{N-k}$ parameterized by $\mathbb{R}^k$, exhibiting derived equivalences between A-models of the various phases. Moreover, by translating the magic window zonotope in $\mathbb{R}^k$, we obtain a universal skeleton over $\mathbb{R}^k \times \mathbb{R}^k \setminus \mathcal{D}$ for some fattening of hyperplane arrangements $\mathcal{D}$, and we show that the the universal skeleton induces a local system of categories over $\mathbb{R}^k \times \mathbb{R}^k \setminus \mathcal{D}$. We also connect our results to the perverse schober structure identified by Špenko and Van den Bergh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_06114 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Variation of GIT and Variation of Lagrangian Skeletons II: Quasi-Symmetric Case Huang, Jesse Zhou, Peng Symplectic Geometry Consider $(\mathbb{C}^*)^k$ acting on $\mathbb{C}^N$ satisfying certain 'quasi-symmetric' condition which produces a class of toric Calabi-Yau GIT quotient stacks. Using subcategories of $Coh([\mathbb{C}^N / (\mathbb{C}^*)^k])$ generated by line bundles whose weights are inside certain zonotope called the 'magic window', Halpern-Leistner and Sam give a combinatorial construction of equivalences between derived categories of coherent sheaves for various GIT quotients. We apply the coherent-constructible correspondence for toric varieties to the magic windows and obtain a non-characteristic deformation of Lagrangian skeletons in $\mathbb{R}^{N-k}$ parameterized by $\mathbb{R}^k$, exhibiting derived equivalences between A-models of the various phases. Moreover, by translating the magic window zonotope in $\mathbb{R}^k$, we obtain a universal skeleton over $\mathbb{R}^k \times \mathbb{R}^k \setminus \mathcal{D}$ for some fattening of hyperplane arrangements $\mathcal{D}$, and we show that the the universal skeleton induces a local system of categories over $\mathbb{R}^k \times \mathbb{R}^k \setminus \mathcal{D}$. We also connect our results to the perverse schober structure identified by Špenko and Van den Bergh. |
| title | Variation of GIT and Variation of Lagrangian Skeletons II: Quasi-Symmetric Case |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2011.06114 |