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 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.

Cell decompositions and algebraicity of cohomology for quiver Grassmannians / Cerulli Irelli, Giovanni; Esposito, Francesco; Franzen, Hans; Reineke, Markus. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 379(2021). [10.1016/j.aim.2020.107544]

Cell decompositions and algebraicity of cohomology for quiver Grassmannians

Cerulli Irelli, Giovanni
;
2021

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 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11573/1472718
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