Variation of GIT and Variation of Lagrangian Skeletons I: Flip and Flop
Peng Zhou
TL;DR
The paper develops a framework to translate toric flips and flops, described via variation of GIT, into variation of Lagrangian skeleta using the coherent-constructible correspondence. By introducing window skeletons Λ_W and analyzing their behavior across a VGIT-parameterized real line, it shows that the B-model window subcategory is equivalent to the A-model window category when the window is large enough, producing a controlled semi-orthogonal decomposition across chambers. Central to the approach are wrapped microlocal sheaves, the CCC for toric stacks, and a detailed microlocal analysis of singular supports and non-characteristic deformations. The results provide a concrete, sheaf-theoretic realization of how toric birational changes induce variations in Lagrangian skeleta and associated categories, with potential to extend to higher-dimensional VGIT and broader birational phenomena.
Abstract
Coherent-Constructible Correspondence for toric variety assigns to each $n$-dimensional toric variety $X_Σ$ a Lagrangian skeleton $Λ_Σ\subset T^*T^n$, such that the derived category of coherent sheaves $Coh(X_Σ)$ is equivalent to the (wrapped) constructible sheaves $Sh^w(T^n, Λ_Σ)$. In this paper, we extend this correspondence, so that flip and flop between toric varieties corresponds to variation of Lagrangian skeletons. The main idea is to translate window subcategory in variation of GIT to a window skeleton.