Semiorthogonal decompositions on singular varieties

Inspired by recent work of Kawamata, we introduce and study certain semiorthogonal decompositions on singular projective varieties, which generalize full exceptional sequences for smooth varieties. We will first present constructions of such decompositions. Using singularity categories, negative K-theory, NCCRs and cluster theory, we will then discuss obstructions to the existence of such decompositions. This is joint work with N.Pavic and E.Shinder.