Variation of GIT and Variation of Lagrangian Skeletons II: Quasi-Symmetric Case

Abstract

Consider $({\mathbb{C}}^{\ast }{)}^k$ acting on ${\mathbb{C}}^N$ satisfying certain 'quasi-symmetric' condition which produces a class of toric Calabi-Yau GIT quotient stacks. Using subcategories of $Coh([{\mathbb{C}}^N/({\mathbb{C}}^{\ast }{)}^k])$ generated by line bundles whose weights are inside certain zonotope called the 'magic window', Halpern-Leistner and Sam give a combinatorial construction of equivalences between derived categories of coherent sheaves for various GIT quotients. We apply the coherent-constructible correspondence for toric varieties to the magic windows and obtain a non-characteristic deformation of Lagrangian skeletons in ${\mathbb{R}}^{N-k}$ parameterized by ${\mathbb{R}}^k$, exhibiting derived equivalences between A-models of the various phases. Moreover, by translating the magic window zonotope in ${\mathbb{R}}^k$, we obtain a universal skeleton over ${\mathbb{R}}^k\times {\mathbb{R}}^k\setminus \mathcal{D}$ for some fattening of hyperplane arrangements $\mathcal{D}$, and we show that the the universal skeleton induces a local system of categories over ${\mathbb{R}}^k\times {\mathbb{R}}^k\setminus \mathcal{D}$. We also connect our results to the perverse schober structure identified by Špenko and Van den Bergh.

47 pages