Résume | In a recent paper in collaboration with E. Feigin and M. Reineke we investigate the connection beetween flag varieties and their degenerations, with quiver Grassmannians associated with representations of a Dynkin quiver. In previous works, E. Feigin introduced a natural degeneration of (partial) flag varieties. He showed that these varieties are (typically singular) irreducible, normal, local complete intersection which are flat degenerations of the usual flag varieties. Moreover they admit a group action with finitely many orbits and a cellular decomposition. The number of cells equals the (median) Genocchi numbers. In the paper we observed that these varieties are naturally isomorphic to quiver Grassmannians of the form $Gr_{\dim A}(A+A*)$, where $A$ is the path algebra of an equioriented quiver of type A. We hence consider quiver Grassmannians of the form $Gr_{\dim P} (P+I)$, where $P$ and $I$ are respectively a projective and an injective representation of a Dynkin quiver. We find the same properties as for type A. Moreover we compute the Poincaré polynomials of these varieties, finding a natural $q$-version of the median Genocchi numbers. |