Cohomology of cluster varieties. I. Locally acyclic case

TL;DR

This paper systematically studies the cohomology of cluster varieties, introducing the Louise property and revealing their cohomological and point-counting structures, especially for acyclic and surface-related cases.

Contribution

It introduces the Louise property for cluster algebras, proves the curious Lefschetz property for certain cluster varieties, and describes their mixed Hodge structures and point counts.

Findings

Cohomology satisfies the curious Lefschetz property.

Mixed Hodge structure is split over the rationals.

Point counts over finite fields relate to Dirichlet characters.

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 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.