Stability Conditions Under the Fourier-Mukai Transforms on Abelian Threefolds

TL;DR

This paper establishes explicit symmetries of Bridgeland stability conditions on abelian threefolds via Fourier-Mukai transforms, extending previous work to prove the Bogomolov-Gieseker inequality and constructing stability conditions in this setting.

Contribution

It demonstrates that certain Fourier-Mukai transforms induce equivalences of stability condition hearts and proves the Bogomolov-Gieseker inequality for all abelian threefolds, generalizing prior results.

Findings

Fourier-Mukai transforms induce stability condition equivalences.

Proof of Bogomolov-Gieseker inequality for any abelian threefold.

Explicit cohomological Fourier-Mukai transform in anti-diagonal form.

Abstract

We realize explicit symmetries of Bridgeland stability conditions on any abelian threefold given by Fourier-Mukai transforms. In particular, we extend the previous joint work with Maciocia to study the slope and tilt stabilities of sheaves and complexes under the Fourier-Mukai transforms, and then to show that certain Fourier-Mukai transforms give equivalences of the stability condition hearts of bounded t-structures which are double tilts of coherent sheaves. Consequently, we show that the conjectural construction proposed by Bayer, Macri and Toda gives rise to Bridgeland stability conditions on any abelian threefold by proving that tilt stable objects satisfy the Bogomolov-Gieseker type inequality. Our proof of the Bogomolov-Gieseker type inequality conjecture for any abelian threefold is a generalization of the previous joint work with Maciocia for a principally polarized abelian threefold with Picard rank one case, and also this gives an alternative proof of the same result in full generality due to Bayer, Macri and Stellari. Moreover, we realize the induced cohomological Fourier-Mukai transform explicitly in anti-diagonal form, and consequently, we describe a polarization on the derived equivalent abelian variety by using Fourier-Mukai theory.