  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.
