Definition of Bridgeland stability

In this section we want to give the definition of a Bridgeland stability condition. In the previous talks we have given an introduction to bounded derived categories and, more in general, to triangulated categories. With this achieved, we are ready to give basic definitions of Bridgeland stability conditions following Section 5.1 and Section 5.2 of Macrì-Schmidt's notes.

For this purpose, first introduce the key notion of heart of a bounded t-structure and show that it is an abelian subcategory. Then, introduce the notion of a slicing and explain its relation with the heart of a bounded t-structure. Here we are ready to give the definition of a Bridgeland stability condition} on the derived category of coherent sheaves of a smooth projective variety. Finally we show the close relation between the abstract definition of a Bridgeland stability condition on the derived category of coherent sheaves of a smooth projective variety and the definition of a stability condition for an arbitrary abelian category.