Cohomology of cluster varieties. II. Acyclic case

TL;DR

This paper explores the cohomology of acyclic cluster varieties, revealing that their mixed Hodge structures and point counts are governed by quiver combinatorics, with significant vanishing results for certain cases.

Contribution

It establishes a link between the combinatorics of quivers and the cohomological properties of acyclic cluster varieties, advancing understanding of their structure.

Findings

Mixed Hodge numbers are determined by quiver independent sets.

Point counts are essentially governed by quiver combinatorics.

Strong vanishing conditions for mixed Hodge numbers in full rank cases.

Abstract

In previous work, we initiated the study of the cohomology of locally acyclic cluster varieties. In the present work, we show that the mixed Hodge structure and point counts of acyclic cluster varieties are essentially determined by the combinatorics of the independent sets of the quiver. We use this to show that the mixed Hodge numbers of acyclic cluster varieties of really full rank satisfy a strong vanishing condition.