Abstract : Given a finite-dimensional representation M of a Dynkin quiver Q (which is the orientation of a simply-laced Dynkin diagram) we attach to it the variety of its subrepresentations. This variety is strati ed according to the possible dimension vectors of the subresentations of M. Every piece is called a quiver Grassmannian. Those varieties were introduced by Schofield and Crawley Boevey for the study of general representations of quivers. As pointed out by Ringel, they also appeared previously in works of Auslander.
They reappered in the literature in 2006, when Caldero and Chapoton proved that they can be used to categorify the cluster algebras associated with Q. A special case is when M is generic. In this case all the quiver Grassmannians are smooth and irreducible of "minimal dimension". On the other hand, in collaboration with Markus Reineke and Evgeny Feigin, we showed that interesting varieties appear as quiver Grassmannians associated with non-rigid modules. In this talk I will survey on recent progresses on the subject. In particular I will provide another proof of the key result of Caldero and Chapoton.