Abstract. We use the ideas of Bayer, Bertram, Macrí and Toda to construct a Bridgeland stability condition on a principally polarized abelian threefold (X, L) with NS(X) = Z[ℓ] by establishing their Bogomolov-Gieseker type inequality for certain tilt stable objects associated to the pair (A √

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian threefold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov-Gieseker type inequality. This is done by showing certain Fourier-Mukai transforms give equivalences of abelian categories which are double tilts of coherent sheaves.

Abstract. We show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type, if they satisfy the Bogomolov-Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson-Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi-Yau 3-folds satisfying the BG inequality conjecture, for example onétale quotients of abelian 3-folds.