  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.
