Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909446957957120 |
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| author | Gammage, Benjamin Jeffs, Maxim |
| author_facet | Gammage, Benjamin Jeffs, Maxim |
| contents | We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. These two mirrors are related via a non-geometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03227 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces Gammage, Benjamin Jeffs, Maxim Symplectic Geometry Algebraic Geometry 53D37 (Primary) 14J33, 53D40 (Secondary) We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. These two mirrors are related via a non-geometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors. |
| title | Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces |
| topic | Symplectic Geometry Algebraic Geometry 53D37 (Primary) 14J33, 53D40 (Secondary) |
| url | https://arxiv.org/abs/2210.03227 |