  We modify the conjectural Bogomolov-Gieseker type inequality introduced by
Bayer, Macri and Toda to construct a family of geometric Bridgeland stability
conditions on smooth projective 3-folds. We give an equivalent conjecture which
needs to check these inequalities for a small class of tilt stable objects. We
extend some of the techniques in Li's work for Fano 3-folds of Picard rank one
to establish our modified Bogomolov-Gieseker type inequality conjecture for
general Fano 3-folds.

Algebraic Geometry