Mirror Symmetry for Perverse Schobers from Birational Geometry

Abstract

Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent–constructible correspondence and a recent result of Ganatra et al. (Microlocal morse theory of wrapped fukaya categories. arXiv:1809.08807) to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of Bondal et al. (Selecta Math 24(1):85–143, 2018) on our mirror schobers.