Derived Categories of Twisted Sheaves on Elliptic Threefolds

TL;DR

This paper establishes an equivalence between the derived category of sheaves on an elliptic threefold without a section and a derived category of twisted sheaves on its Jacobian's resolution, advancing understanding of their categorical relationships.

Contribution

It introduces a new equivalence between derived categories of sheaves and twisted sheaves on elliptic threefolds and their Jacobians, expanding the framework for studying such geometric objects.

Findings

Constructed an explicit equivalence of derived categories.

Extended the understanding of twisted sheaves on elliptic threefolds.

Provided tools for future research in algebraic geometry and string theory.

Abstract

We construct an equivalence between the derived category of sheaves on an elliptic threefold without a section and a derived category of twisted sheaves (modules over an Azumaya algebra) on any small resolution of its relative Jacobian.