Stability Conditions On Derived Categories and Kuznetsov Components

The construction of stability conditions on the bounded derived category of higher dimensional varieties is a difficult task starting from dimension three. Nevertheless, once a stability condition is given, many interesting applications are known, in particular concerning the study of the geometry of the associated moduli spaces of semistable objects. In this course, we will introduce the notion of derived category and semiorthogonal decompositions. Then we will define stability conditions and the known construction technique via tilt stability in lower dimension. Then we will focus on the case of Kuznetsov components in semiorthogonal decompositions, explaining a criterion to induce stability conditions, and applications to the geometry of certain Fano threefolds and fourfolds.