Twisted Polytope Sheaves and Coherent-Constructible Correspondence for Toric Varieties

TL;DR

This paper provides a new proof of the coherent-constructible correspondence for smooth projective toric varieties, utilizing non-characteristic deformation and twisted polytope sheaves to establish the equivalence of derived categories.

Contribution

It introduces a novel proof method for the correspondence, emphasizing the use of non-characteristic deformation and twisted polytope sheaves in the smooth projective case.

Findings

Established a new proof of the quasi-equivalence for smooth projective toric varieties.

Demonstrated the use of twisted polytope sheaves to co-represent stalk functors.

Provided insights into the structure of derived categories for toric varieties.

Abstract

Given a smooth projective toric variety $X_\Sigma$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_\Sigma)$ into the dg derived category of constructible sheaves on a torus $Sh(T^n, \Lambda_\Sigma)$. Recently, Kuwagaki \cite{Ku2} proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors.