Cell decompositions and algebraicity of cohomology for quiver Grassmannians

TL;DR

This paper proves that quiver Grassmannians have well-behaved cohomology and Chow rings, with explicit generators and cellular decompositions in certain cases, and offers new insights into their geometric structure and cluster algebra relations.

Contribution

It establishes the algebraic and geometric properties of quiver Grassmannians, including cohomology, Chow rings, and cellular decompositions, extending to non-rigid and regular representations.

Findings

Cohomology ring has no odd cohomology and cycle map is an isomorphism.

Chow ring admits explicit generators over any field.

Quiver Grassmannians have cellular decompositions in finite or affine types.

Abstract

We show that the cohomology ring of a quiver Grassmannian asssociated 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 to 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.