Homological Mirror Symmetry for Hypertoric Varieties I
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arXiv
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| Format: | Preprint |
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2018
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| author | McBreen, Michael Webster, Ben |
| author_facet | McBreen, Michael Webster, Ben |
| contents | We consider homological mirror symmetry in the context of hypertoric varieties, showing that appropriate categories of B-branes (that is, coherent sheaves) on an additive hypertoric variety match a category of A-branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data. For technical reasons, the category of A-branes we consider is the 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 $\mathbb{G}_m$-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 $p$ approach of Kaledin; the result is a sum of line bundles, which can be described using a simple combinatorial rule. The endomorphism algebra $H$ of this tilting generator has a simple quadratic presentation in the grading induced by $\mathbb{G}_m$-equivariance. In fact, we can confirm it is Koszul, and compute its Koszul dual $H^!$.
We then show that this same algebra appears as an Ext-algebra of simple A-branes in a Dolbeault hypertoric manifold. The $\mathbb{G}_m$-equivariant grading on coherent sheaves matches a Hodge grading in this category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_10646 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Homological Mirror Symmetry for Hypertoric Varieties I McBreen, Michael Webster, Ben Algebraic Geometry Representation Theory Symplectic Geometry 14J33, 47A67 We consider homological mirror symmetry in the context of hypertoric varieties, showing that appropriate categories of B-branes (that is, coherent sheaves) on an additive hypertoric variety match a category of A-branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data. For technical reasons, the category of A-branes we consider is the 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 $\mathbb{G}_m$-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 $p$ approach of Kaledin; the result is a sum of line bundles, which can be described using a simple combinatorial rule. The endomorphism algebra $H$ of this tilting generator has a simple quadratic presentation in the grading induced by $\mathbb{G}_m$-equivariance. In fact, we can confirm it is Koszul, and compute its Koszul dual $H^!$. We then show that this same algebra appears as an Ext-algebra of simple A-branes in a Dolbeault hypertoric manifold. The $\mathbb{G}_m$-equivariant grading on coherent sheaves matches a Hodge grading in this category. |
| title | Homological Mirror Symmetry for Hypertoric Varieties I |
| topic | Algebraic Geometry Representation Theory Symplectic Geometry 14J33, 47A67 |
| url | https://arxiv.org/abs/1804.10646 |