Note on the Bondal-Orlov Functors for Toric DM Stacks

TL;DR

This paper provides explicit formulas for equivariant Bondal-Orlov functors on localized K-theory for crepant birational transformations of toric Deligne-Mumford stacks, linking them to derived category equivalences and spherical twists.

Contribution

It offers new explicit formulas for Bondal-Orlov functors in the context of toric DM stacks and explores their relation to derived category autoequivalences.

Findings

Explicit formulas for equivariant Bondal-Orlov functors

Demonstration of these functors as derived category equivalences

Connection to Seidel-Thomas spherical twists

Abstract

We calculate explicit formulas for the general equivariant Bondal-Orlov functors on the localized K-theory groups for a crepant birational transformation of toric DM stacks. We recall some facts that the Bondal-Orlov functors give equivalences on the bounded derived categories. Applying twice of these functors we get the Seidel-Thomas spherical twists for the derived category.