Quiver Grassmannians associated to nilpotent cyclic representations defined by a single matrix

We study the geometry of the closed Białynicki-Birula cells of the quiver Grassmannians associated to a nilpotent representation of a cyclic quiver defined by a single matrix. The main result of this paper is that for the special case where we choose subrepresentations of dimension 1=(1,…,1), the closed Białynicki-Birula cells are smooth. We also discuss their cohomology ring. Namely, we describe the Knutson–Tao basis of equivariant cohomology that is dual to fundamental classes in equivariant homology.