Cell decompositions and algebraicity of cohomology for quiver Grassmannians

We show that the cohomology ring of a quiver Grassmannian associated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated with an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular decomposition for quiver Grassmannians associated with representations with rigid regular part. Finally, we study the geometry behind the cluster multiplication formula of Caldero and Keller, providing a new proof of a slightly more general result. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

The article demonstrates properties of quiver Grassmannians associated with rigid quiver representations, including properties of the cohomology ring, Chow ring generators, and polynomial point count. By restricting to finite or affine type quivers, it is shown that quiver Grassmannians associated with indecomposable representations have a cellular decomposition. Additionally, the geometry behind the cluster multiplication formula of Caldero and Keller is studied, providing a new proof of a slightly more general result.