Tilting objects in singularity categories of toric Gorenstein varieties

TL;DR

This paper demonstrates that certain toric Gorenstein varieties with isolated singularities have singularity categories that admit tilting objects, establishing an equivalence with perfect categories of finite-dimensional algebras.

Contribution

It extends the understanding of singularity categories for specific toric Gorenstein varieties by proving the existence of tilting objects, linking geometric singularities to algebraic categories.

Findings

Singularity categories admit tilting objects.

Equivalence to perfect categories of finite-dimensional algebras.

Applicable to varieties as quotients of unimodular representations.

Abstract

We study certain toric Gorenstein varieties with isolated singularities which are the quotient spaces of generic unimodular representations by the one-dimensional torus, or by the product of the one-dimensional torus with a finite abelian group. Based on the works of \v{S}penko and Van den Bergh [Invent. Math. 210 (2017), no. 1, 3-67] and Mori and Ueyama [Adv. Math. 297 (2016), 54-92], we show that the singularity categories of these varieties admit tilting objects, and hence are triangle equivalent to the perfect categories of some finite dimensional algebras.