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

Jesse Huang, Peng Zhou

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 Špenko and Van den Bergh.

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

Abstract

Consider acting on satisfying certain 'quasi-symmetric' condition which produces a class of toric Calabi-Yau GIT quotient stacks. Using subcategories of 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 parameterized by , exhibiting derived equivalences between A-models of the various phases. Moreover, by translating the magic window zonotope in , we obtain a universal skeleton over for some fattening of hyperplane arrangements , and we show that the the universal skeleton induces a local system of categories over . We also connect our results to the perverse schober structure identified by Špenko and Van den Bergh.

Paper Structure

This paper contains 38 sections, 64 theorems, 252 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Toric GIT
  3. Zonotopes and Windows
  4. Constructible sheaf of categories
  5. Window Skeleton and Perverse Schober
  6. Universal Skeleton and Local System of Categories
  7. Related works
  8. Mirror symmetry for Toric Calabi-Yau
  9. Windows and Quasi-Symmetric GIT
  10. Outline
  11. Acknowledgements
  12. Notation and Conventions
  13. Geometry of the Window Skeleton: Stable Region
  14. GKZ fan and GIT skeleton
  15. Window skeleton over stable regions
  16. ...and 23 more sections

Key Result

Theorem 1.4

For any $\delta \in \mathbb{R}^k$ and any GKZ chamber $C$, if we define and the skeleton for GIT quotient in chamber $C$, then we have In particular, if $l \in C_{\nabla_\delta} \cap \mathbb{Z}^k$, then $\overline \Lambda_{\delta, l} \subset T^* T^n$ is the (non-equivariant) FLTZ skeleton mirror to the GIT quotient $\mathbb{C}^N //_l (\mathbb{C}^*)^k$.

Figures (13)

  • Figure 1: Stratification of the shift parameter space $\mathbb{R}^k_\delta$.
  • Figure 2: The zonotope $\nabla_\delta$ (yellow), the windows points $W_\delta$ (red) and the singular supports (blue hairy lines and blue arcs) of the sheaf of categories on $\mathbb{R}^k_l \simeq \mathbb{R}^2$.
  • Figure 3: This is a local picture of a skeleton $\Lambda_{W_\delta}$ in $\mathbb{R}^2$ such that $\mathcal{C}_\delta$ has jumping loci in $\mathbb{R}^1_l$. The fiber skeleton over the green dashed line has one stop, and the one over the red line has two stops, a zero-dimensional Legendrian vanishing sphere $S^0$. The yellow region is the support of a sheaf that vanishes when restricted to the right of the green line, hence $SS_{Hom}(\mathcal{C})$ has a covector pointing to the left.
  • Figure 4: Window skeleton of $\Lambda_{W_\delta} \subset T^* \mathbb{R}^N$ for $N=2$. From left to right, we have $W_\delta=\{0\}, \{0,1\}, \{1\}$ respectively. In the middle figure, the yellow region is an example of a 'vanishing cycle sheaf' whose tip has $\mu$ image at $l=1$. Similarly, the green region is the support of a 'vanishing cycle sheaf' whose tip has $\mu$ image at $l=0$. They are the images of the structure sheaf of the unstable loci in the action of $\mathbb{C}^*$ on $\mathbb{C}^2$ with weight $(1,-1)$. See Example \ref{['ex:N2k1']} for details.
  • Figure 5: An increasing sequence of open sets $U_t$ (the interior regions of the green curves). See Figure \ref{['fig:d2']}, where we only keep the window $W_\delta$ (red dots) and the singular support $SS(\mathcal{C}_\delta)$ (blue hairy line). This is a non-characteristic expansion of open sets (since when the green line become tangent to the blue lines, the hairs are in opposite direction), hence the categories $\mathcal{C}_\delta(U_t)$ are invariant.
  • ...and 8 more figures

Theorems & Definitions (144)

  • Definition 1.1: Quasi-Symmetric Condition
  • Remark 1.2
  • Definition 1.3: Window skeletons
  • Theorem 1.4: Theorem \ref{['t:LaWLaC']}
  • Theorem 1.5: Theorem \ref{['t:l-indep']}
  • Theorem 1.6
  • Remark 1.7
  • Corollary 1.8
  • Proposition 1.9
  • proof
  • ...and 134 more