To what extent is a smooth projective variety determined by its bounded derived category of coherent sheaves? The question, for minimal complex surfaces, reduces to surfaces of K3 or abelian type and elliptic surfaces. If X and Y are two derived equivalent K3 surfaces, do they share the same geometry? For example, if X has infinitely many rational curves, does the same hold for Y? At the moment a direct geometric approach seems not available. As a first step in this direction, Huybrechts recently proved that X and Y can be compared motivically.
In the talk we will recall the theory of Fourier-Mukai transforms and discuss which properties are preserved under derived equivalence, and which ones are not. In particular we will present some classical results concerning non-isomorphic derived equivalent K3s. If time permits we will introduce the theory of Chow motives to prove Huybrechts result and show some applications of this.