Realization of Cohomology Classes in Grassmannians

The study of irreducible subvarieties has recently seen a surge of interest due to connections with convex geometry. In this paper, we study cohomology classes of Grassmannians that can be realized by irreducible subvarieties. We give a complete classification of such classes in dimensions 2 and 3 and codimensions 2 and 3. We also classify all cohomology classes that are realizable by irreducible subvarieties for the Grassmannians $G(2,n)$ for n ≤ 6 and $G(3,6)$. We classify the cohomology classes in all dimensions that are, up to positive multiple, realizable by an irreducible subvariety for the Grassmannians $G(2,n)$.