Bogomolov-type inequalities in higher dimension

In this seminar (based on joint work in progress with A. Bayer, Y. Toda, and A. Bertram), we will present a conjectural approach to the construction of Bridgeland stability conditions on the derived category of a higher dimensional variety. The main ingredient is a generalization to complexes of the classical Bogomolov inequality for sheaves.
We will also discuss an application of this inequality to the Fujita Conjecture for threefolds.