2016/04/30 by Thomas Lam, David E Speyer, David E. Speyer
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Cluster (spacecraft) #Cluster algebra #Cohomology #Combinatorics #Computer science #Field (mathematics) #Mathematics #Property (philosophy) #Pure mathematics #Rank (graph theory) #Variety (cybernetics) #math.AG #math.CO
paper · pdf · doi:10.2140/ant.2022.16.179
published as Alg. Number Th. 16 (2022) 179-230 · v2: 46 pages
arxiv created 2021/11/24 · openalex publication_date 2022/02/22 · arxiv updated 2022/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 the rationals. 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.