  We construct new t-structures on the derived category of coherent sheaves on
smooth projective threefolds. We conjecture that they give Bridgeland stability
conditions near the large volume limit. We show that this conjecture is
equivalent to a Bogomolov-Gieseker type inequality for the third Chern
character of certain stable complexes. We also conjecture a stronger
inequality, and prove it in the case of projective space, and for various
examples.
  Finally, we prove a version of the classical Bogomolov-Gieseker inequality,
not involving the third Chern character, for stable complexes.
