Semi-Homogeneous Sheaves and Twisted Derived Categories

TL;DR

This paper explores twisted derived equivalences involving torsors under abelian varieties and their moduli spaces, establishing new connections and criteria that extend classical results in derived category theory.

Contribution

It introduces a framework for twisted derived equivalences between torsors and moduli spaces, including a converse result and extensions of derived equivalence criteria.

Findings

Established twisted derived equivalences between torsors and moduli spaces.

Proved the natural converse to the main equivalence result.

Extended the derived equivalence criterion for abelian varieties.

Abstract

We produce twisted derived equivalences between torsors under abelian varieties and their moduli spaces of simple semi-homogeneous sheaves. We also establish the natural converse to this result and show that a large class of twisted derived equivalences, including all derived equivalences, between torsors arise in this way. As corollaries, we obtain partial extensions of the usual derived equivalence criterion for abelian varieties established by Orlov and Polishchuk.