Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds

TL;DR

This paper establishes Bridgeland stability conditions on a specific class of abelian threefolds by proving a key inequality and demonstrating the preservation of stability under Fourier-Mukai transforms, also characterizing certain reflexive sheaves.

Contribution

It proves a conjectural inequality for tilt stable objects and shows that Fourier-Mukai transforms preserve the stability condition's heart on abelian threefolds.

Findings

Validated the generalized Bogomolov-Gieseker inequality for tilt stable objects.

Demonstrated that Fourier-Mukai transforms preserve the stability condition.

Characterized reflexive sheaves with zero Chern classes as flat line bundles.

Abstract

We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized Bogomolov-Gieseker inequality for certain tilt stable objects. We do this by proving that a suitable Fourier-Mukai transform preserves the heart of a particular conjectural stability condition. We also show that the only reflexive sheaves with zero first and second Chern classes are the flat line bundles.