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 $\mathbb{Q}$. 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.

我们启动了集群品种上同调的系统研究。我们引入了簇代数的路易丝性质，它适用于所有无环簇代数，以及大多数由标记表面产生的簇代数。对于满足 Louise 属性和满秩的簇簇，我们证明了上同调满足 Hausel 和 Rodriguez-Villegas 的好奇 Lefschetz 属性，并且混合 Hodge 结构被分裂$\mathbb{Q}$. 我们对这些集群品种的混合霍奇结构的最高权重部分进行了完整的描述，并发展了集群品种的标准微分形式的概念。我们表明，这些簇簇在有限域上的点数可以用狄利克雷特征表示。在一个额外的完整性假设下，点计数显示为有限域阶的多项式。