K3 ploskve z vidika izpeljanih kategorij

We construct the derived category of an abelian category, equip it with a triangulated structure and define derived functors. In particular we address the bounded derived category of coherent sheaves on a smooth projective variety and introduce derived functors of geometric origin. Utilizing them we define and study Fourier–Mukai transforms first at the level of derived categories, then at the level of K-groups and lastly at the level of rational cohomology. Focusing on complex K3 surfaces we describe some of their invariants, most notably their intersection form and associated Hodge structure. Lastly we present Orlov's proof of the derived Torelli theorem, which characterizes derived equivalent K3 surfaces in terms of their Mukai lattices and the associated moduli spaces of stable sheaves.