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

TL;DR

This paper explores the relationship between GIT quotients, derived categories, and Lagrangian skeletons in the quasi-symmetric case, revealing new equivalences and structures in the context of toric Calabi-Yau stacks.

Contribution

It introduces a non-characteristic deformation of Lagrangian skeletons and constructs a universal skeleton over parameter spaces, linking derived categories and perverse schobers.

Findings

Derived equivalences between GIT phases via Lagrangian skeletons

Construction of a universal skeleton over parameter spaces

Connection to perverse schober structures

Abstract

Consider $(\mathbb{C}^*)^k$ acting on $\mathbb{C}^N$ satisfying certain 'quasi-symmetric' condition which produces a class of toric Calabi-Yau GIT quotient stacks. Using subcategories of $Coh([\mathbb{C}^N / (\mathbb{C}^*)^k])$ 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 \v{S}penko and Van den Bergh.