Variation of GIT and Variation of Lagrangian Skeletons I: Flip and Flop

Abstract: Coherent-Constructible Correspondence for toric variety assigns to each $n$-dimensional toric variety $X_\Sigma$ a Lagrangian skeleton $\Lambda_\Sigma \subset T^*T^n$, such that the derived category of coherent sheaves $Coh(X_\Sigma)$ is equivalent to the (wrapped) constructible sheaves $Sh^w(T^n, \Lambda_\Sigma)$. 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.