#### Vol. 16, No. 1, 2022

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Cohomology of cluster varieties, I: Locally acyclic case

### Thomas Lam and David E. Speyer

Vol. 16 (2022), No. 1, 179–230
##### 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 $ℚ$. 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.

##### Keywords
cluster algebras, cluster varieties, cohomology, mixed Hodge structure
Primary: 13F60
Secondary: 14F40
##### Milestones
Accepted: 21 April 2021
Published: 22 February 2022
##### Authors
 Thomas Lam Department of Mathematics University of Michigan Ann Arbor, MI United States David E. Speyer Department of Mathematics University of Michigan Ann Arbor, MI United States