Year: 2026
Type: preprint
Abstract: 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 $\mathcal{A}(Σ)$ 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.
Source: arXiv (Cornell University)
Author: Hlavinka, Josephine
Institution: University of California, Berkeley
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213593670