The Coherent-Constructible Correspondence and Fourier-Mukai Transforms

TL;DR

This paper explores the relationship between derived categories of toric orbifolds and constructible sheaves, providing new insights and reproofs of Kawamata's results within the framework of the coherent-constructible correspondence and Fourier-Mukai transforms.

Contribution

It offers a novel perspective on Kawamata's derived equivalences by connecting them with constructible sheaves and the coherent-constructible correspondence.

Findings

Reproves Kawamata's derived equivalences using constructible sheaves.

Establishes a natural identification between derived categories and sheaf categories.

Enhances understanding of the geometric and categorical structures of toric orbifolds.

Abstract

In arXiv:math/0311139, as evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In arXiv:0911.4711, we showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on R^n. In this paper we investigate and reprove some of Kawamata's results from this perspective.