  We describe a connected component of the space of stability conditions on
abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant
resolution of) a finite quotient of an abelian threefold. Our proof includes
the following essential steps:
  1. We simultaneously strengthen a conjecture by the first two authors and
Toda, and prove that it follows from a more natural and seemingly weaker
statement. This conjecture is a Bogomolov-Gieseker type inequality involving
the third Chern character of "tilt-stable" two-term complexes on smooth
projective threefolds; we extend it from complexes of tilt-slope zero to
arbitrary tilt-slope.
  2. We show that this stronger conjecture implies the so-called support
property of Bridgeland stability conditions, and the existence of an explicit
open subset of the space of stability conditions.
  3. We prove our conjecture for abelian threefolds, thereby giving reproving
and generalizing a result by Maciocia and Piyaratne.
  Important in our approach is a more systematic understanding on the behaviour
of quadratic inequalities for semistable objects under wall-crossing, closely
related to the support property.
