Derived categories of singular varieties

We give a short introduction to derived categories of projective varieties with singularities. We discuss necessary conditions for when such derived categories admit a semiorthogonal decomposition into two components; a "small" component containing the information of the singularity and a "big" component encoding the smooth information. Our main result, however, considers sufficient conditions for such decompositions. Particularly, we study the derived category of varieties with 1/n(1,...,1) singularities. We give various examples satisfying our condition, but we also give criteria which do not allow such decompositions. This is joint work in progress with M. Kalck and Y. Kawamata.