{"ID":23510315,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-18T02:21:44.056544415Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19744","arxiv_id":"2609.19744","title":"Chow Vanishing and Motives of Cluster Varieties","abstract":"We prove that the integral Chow groups $CH^i$ and mixed Hodge degree $H^{2i, (i, i)}$ cohomology groups of really full rank (RFR) sink-recurrent cluster varieties vanish for $i \u003e 0$. In particular this applies to braid varieties and open Richardson varieties in any Lie type. Our main tool is the construction of a stratification of any RFR sink-recurrent cluster variety $\\mathcal{A}(Σ)$ into (affine spaces times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than $Σ$. We employ the theory of Voevodsky motives, and towards this end we prove that the cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. We then show that RFR sink-recurrent cluster varieties have mixed Tate and, in fact, split motives. Finally, we use our results to deduce vanishing theorems about the Khovanov-Rozansky homology groups of closures of positive braids and generation properties of the cohomology of closed Richardson, projected Richardson, and brick varieties.","short_abstract":"We prove that the integral Chow groups $CH^i$ and mixed Hodge degree $H^{2i, (i, i)}$ cohomology groups of really full rank (RFR) sink-recurrent cluster varieties vanish for $i \u003e 0$. In particular this applies to braid varieties and open Richardson varieties in any Lie type. Our main tool is the construction of a strat...","url_abs":"https://arxiv.org/abs/2609.19744","url_pdf":"https://arxiv.org/pdf/2609.19744v1","authors":"[\"Josephine Hlavinka\"]","published":"2026-09-17T06:09:36Z","proceeding":"math.AG","tasks":"[\"math.AG\",\"math.CO\",\"math.GT\",\"math.RT\"]","methods":"[]","has_code":false}