Ricerc@Sapienza

The paper includes a new proof of the fact that quiver Grassmannians associated
with rigid representations of Dynkin quivers do not have cohomology in odd degrees.
Moreover, it is shown that they do not have torsion in homology. A new proof of the
Caldero-Chapoton formula is provided. As a consequence a new proof of the positivity of
cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The
methods used here are based on joint works with Markus Reineke and Evgeny Feigin.