DG-category of coherent sheaves over a complex manifold

In the first part of this talk, I will introduce the construction of certain differential graded (dg) category of coherent sheaves over a given complex manifold by Jonathan Block, which serves as an enhanced version of the derived category of the manifold. In the second part, I will explain how this dg-category and its variations could help us understand the Ext and Tor functor of the derived category in some special yet interesting cases.