Stronger Bogomolov-Gieseker type Inequality and stability condition

The classical Bogomolov inequality gives a bound for the second Chern character of slope stable sheaves on smooth projective varieties. The inequality is known to be sharp for some varieties (e.g. Abelian varieties), as well as non-sharp for some others (e.g. the projective plane). Besides Fano and K3 surfaces, it is always difficult to get stronger Bogomolov type inequalities for other surfaces and higher dimensional varieties. I will talk about the method to set up such inequalities via the Bridgeland stability condition. The stronger Bogomolov type inequality has several implications. One upshot will be the existence of stability condition on smooth quintic threefolds. They are the first examples of Calabi-Yau threefolds with trivial fundamental group known to have stability conditions.