Chow Vanishing and Motives of Cluster Varieties
Abstract
We prove that the integral Chow groups CHi and mixed Hodge degree H2i, (i, i) cohomology groups of a really full rank (RFR) sink-recurrent cluster variety vanish for i > 0. In particular this applies to braid varieties and open projected Richardson varieties in any Lie type. Our main technical tool is the construction of a stratification of any RFR sink-recurrent cluster variety 𝒜(Σ) over any field k into (affine space times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than Σ, with which we show that RFR sink-recurrent cluster varieties are linear in the sense of Totaro. We also show that cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. Finally, we use our results to deduce vanishing theorems for the Khovanov-Rozansky homology groups of closures of positive braids and generation properties for the Chow groups of flag Richardson, projected Richardson, and brick varieties: namely, that these groups are generated integrally by the fundamental classes of their closed Richardson, positroid, and braid strata, respectively.
Related
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BibTeX
@misc{hlavinka2026chow,
title = {{Chow Vanishing and Motives of Cluster Varieties}},
author = {Josephine Hlavinka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19744},
eprint = {2609.19744},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19744},
}
RIS
TY - UNPB
TI - Chow Vanishing and Motives of Cluster Varieties
AU - Josephine Hlavinka
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.19744
UR - https://arxiv.org/abs/2609.19744
AB - We prove that the integral Chow groups CH_i and mixed Hodge degree H^2i, (i, i) cohomology groups of a really full rank (RFR) sink-recurrent cluster variety vanish for i > 0. In particular this applies to braid varieties and open projected Richardson varieties in any Lie type. Our main technical tool is the construction of a stratification of any RFR sink-recurrent cluster variety 𝒜(Σ) over any field k into (affine space times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than Σ, with which we show that RFR sink-recurrent cluster varieties are linear in the sense of Totaro. We also show that cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. Finally, we use our results to deduce vanishing theorems for the Khovanov-Rozansky homology groups of closures of positive braids and generation properties for the Chow groups of flag Richardson, projected Richardson, and brick varieties: namely, that these groups are generated integrally by the fundamental classes of their closed Richardson, positroid, and braid strata, respectively.
ER -