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Homological mirror
symmetry for hypertoric varieties, I: Conic equivariant
sheaves
Michael McBreen and Ben Webster |
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Geometry & Topology 28 (2024) 1005–1063
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Abstract |
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We consider homological mirror symmetry in the context of hypertoric varieties, showing that an appropriate category of –branes (that is, coherent sheaves) on an additive hypertoric variety matches a category of –branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data. For technical reasons, the –branes we consider are modules over a deformation quantization (that is, DQ–modules). We consider objects in this category equipped with an analogue of a Hodge structure, which corresponds to a –action on the dual side of the mirror symmetry. This result is based on hands-on calculations in both categories. We analyze coherent sheaves by constructing a tilting generator, using the characteristic approach of Kaledin; the result is a sum of line bundles, which can be described using a simple combinatorial rule. The endomorphism algebra of this tilting generator has a simple quadratic presentation in the grading induced by –equivariance. In fact, we can confirm it is Koszul, and compute its Koszul dual . We then show that this same algebra appears as an Ext–algebra of simple –branes in a Dolbeault hypertoric manifold. The –equivariant grading on coherent sheaves matches a Hodge grading in this category. |
Keywords
mirror symmetry, hypertoric, characteristic $p$,
deformation quantization
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Mathematical Subject Classification
Primary: 14J33, 47A67
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References |
Publication
Received: 9 October 2020
Revised: 22 August 2022
Accepted: 24 September 2022
Published: 10 May 2024
Proposed: Mark Gross
Seconded: Richard P Thomas, Dan Abramovich
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Authors |
| Michael McBreen | |
| Department of Mathematics Chinese University of Hong Kong Hong Kong |
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| Ben Webster | |
| Department of Pure Mathematics University of Waterloo Waterloo, ON Canada |
Perimeter Institute for Theoretical
Physics Waterloo, ON Canada |
| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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