Geometry of Quiver Grassmannians of Dynkin type with applications to cluster algebras

TL;DR

This paper provides new proofs for properties of quiver Grassmannians of Dynkin type, demonstrating their cohomological and homological characteristics, and offers a novel proof of the Caldero--Chapoton formula, impacting cluster algebra positivity.

Contribution

It introduces new proofs for the cohomology and torsion properties of quiver Grassmannians and the Caldero--Chapoton formula, enhancing understanding of cluster algebra structures.

Findings

Quiver Grassmannians of Dynkin type have no odd-degree cohomology.

They do not have torsion in homology.

A new proof of the positivity of cluster monomials is established.

Abstract

The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero--Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin.