Perverse sheaves on smooth toric varieties and stacks

TL;DR

This paper provides an explicit algebraic description of perverse sheaves on smooth toric varieties and stacks, facilitating their study through module categories and comparing derived categories.

Contribution

It introduces a novel algebraic framework for perverse sheaves on toric varieties and stacks, including equivariant cases, and relates their derived categories to constructible sheaves.

Findings

Perverse sheaves on smooth toric varieties are equivalent to modules over a finite-dimensional algebra.

The paper extends the description to equivariant perverse sheaves on toric orbifolds.

It establishes a comparison between the derived categories of perverse sheaves and constructible sheaves.

Abstract

It is usually not straightforward to work with the category of perverse sheaves on a variety using only its definition as a heart of a $t$-structure. In this paper, the category of perverse sheaves on a smooth toric variety with its orbit stratification is described explicitly as a category of finite-dimensional modules over an algebra. An analogous result is also established for various categories of equivariant perverse sheaves, which in particular gives a description of perverse sheaves on toric orbifolds, and we also compare the derived category of the category of perverse sheaves to the derived category of constructible sheaves.