Toric mirror symmetry and GIT windows

Abstract

This thesis, partially based on the author's joint work [HZ20], studies mirror symmetry for toric varieties from the perspective of geometric invariant theory. We translate GIT constructions of toric varieties into skeletal terms and study them in the context of wrapped Fukaya categories and wrapped constructible sheaves. We explain how the mirror image of the Halpern-Leistner-Sam "magic window" for quasi-symmetric torus actions computes the category associated to the equivariant semistable skeleton upon stop removal, where the generating cotangent fibers are described by a generic shift of a lattice zonotope in the real character space of the torus. After identifying the "FI parameter space" of the B-model GLSM with the GKZ complex parameter space of the skeletal LG A-model, homological mirror symmetry in the quasi-symmetric case becomes two geometric incarnations of the Špenko-Van den Bergh schober over C^k stratified by a complexified periodic hyperplane arrangement whose complement models both parameter spaces. We obtain a universal deformation of skeletons interpolating variation of parameters.