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2022 Cohomology of cluster varieties I: Locally acyclic case
Thomas Lam, David E. Speyer
Algebra Number Theory 16(1): 179-230 (2022). DOI: 10.2140/ant.2022.16.179

Abstract

We initiate a systematic study of the cohomology of cluster varieties. We introduce the Louise property for cluster algebras that holds for all acyclic cluster algebras, and for most cluster algebras arising from marked surfaces. For cluster varieties satisfying the Louise property and of full rank, we show that the cohomology satisfies the curious Lefschetz property of Hausel and Rodriguez-Villegas, and that the mixed Hodge structure is split over . We give a complete description of the highest weight part of the mixed Hodge structure of these cluster varieties, and develop the notion of a standard differential form on a cluster variety. We show that the point counts of these cluster varieties over finite fields can be expressed in terms of Dirichlet characters. Under an additional integrality hypothesis, the point counts are shown to be polynomials in the order of the finite field.

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Thomas Lam. David E. Speyer. "Cohomology of cluster varieties I: Locally acyclic case." Algebra Number Theory 16 (1) 179 - 230, 2022. https://doi.org/10.2140/ant.2022.16.179

Information

Received: 4 January 2021; Accepted: 21 April 2021; Published: 2022
First available in Project Euclid: 9 May 2022

Digital Object Identifier: 10.2140/ant.2022.16.179

Subjects:
Primary: 13F60
Secondary: 14F40

Keywords: cluster algebras , cluster varieties , Cohomology , mixed Hodge structure

Rights: Copyright © 2022 Mathematical Sciences Publishers

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Vol.16 • No. 1 • 2022
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