Homological mirror symmetry for hypertoric varieties I: Conic equivariant sheaves

We consider homological mirror symmetry in the context of hypertoric varieties, showing that an appropriate category of B-branes (that is, coherent sheaves) on an additive hypertoric variety matches a category of A-branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data. For technical reasons, the A-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 Gm-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 Gm-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 Gm-equivariant grading on coherent sheaves matches a Hodge grading in this category.