Fourier-Mukai Transforms and Stability Conditions on Abelian Varieties

TL;DR

This paper explores how Fourier-Mukai transforms relate to stability conditions on abelian varieties, establishing conjectures and results for abelian surfaces and 3-folds, with implications for tilt stability and Bogomolov-Gieseker inequalities.

Contribution

It formulates and proves a conjecture linking Fourier-Mukai transforms to Bridgeland stability conditions on abelian varieties, extending previous work to 3-folds.

Findings

Conjecture on Fourier-Mukai transforms and stability conditions formulated.

Proved for abelian surfaces and 3-folds.

Strong Bogomolov-Gieseker inequalities confirmed for tilt stable objects.

Abstract

This article is based on a talk given at the Kinosaki Symposium on Algebraic Geometry in 2015, about a work in progress. We describe a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. We explicitly formulate a conjecture which says certain Fourier-Mukai transforms between derived categories give equivalences of some hearts of Bridgeland stability conditions. We establish it for abelian surfaces, which is already known due to D. Huybrechts, and for abelian 3-folds. This generalizes the author's previous joint work with A. Maciocia on principally polarized abelian 3-folds with Picard rank one. Consequently, we see that the strong Bogomolov-Gieseker type inequalities hold for tilt stable objects on abelian 3-folds.