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This article is based on a talk given at the Kinosaki Symposium on AlgebraicGeometry in 2015, about a work in progress. We describe a polarization on aderived equivalent abelian variety by using Fourier-Mukai theory. We explicitlyformulate a conjecture which says certain Fourier-Mukai transforms betweenderived categories give equivalences of some hearts of Bridgeland stabilityconditions. We establish it for abelian surfaces, which is already known due toD. Huybrechts, and for abelian 3-folds. This generalizes the author's previousjoint work with A. Maciocia on principally polarized abelian 3-folds withPicard rank one. Consequently, we see that the strong Bogomolov-Gieseker typeinequalities hold for tilt stable objects on abelian 3-folds.

Fourier-Mukai Transforms and Stability Conditions on Abelian Varieties