arXiv subjects

Josephine Hlavinka

Publications and source records attributed to Josephine Hlavinka.

2 recordsLinked to original sources

arXiv2026

Chow Vanishing and Motives of Cluster Varieties

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 $i > 0$. 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.

math.AG↗

arXiv2026

Fence Complexes and Toric Degenerations of Positroid Varieties

We associate to each positroid variety in the Grassmannian $\mathrm{Gr}(k,n)$ a polyhedral complex, which we call a fence complex. Fence complexes consist of unions of faces of the Gelfand-Tsetlin polytope $P_{k,n}$ associated to a fundamental weight $ω_k$. We show that these fence complexes are homeomorphic to closed balls. Furthermore, they endow the Gelfand-Tsetlin polytope with the structure of a regular CW complex, giving a polyhedral complex presentation of the regular CW complex structure on $\mathrm{Gr}(k,n)_{\geq 0}$. We also show that the Ehrhart polynomial of a fence complex equals the Hilbert polynomial of the associated positroid variety. We prove that under the Sturmfels-Gonciulea-Lakshmibai degeneration of $\mathrm{Gr}(k,n)$ to the toric variety of the Gelfand-Tsetlin polytope, positroid varieties degenerate to the reduced union of toric varieties corresponding to their fence complexes. As an application, we classify when positroid varieties contained inside hook Schubert varieties are arithmetically Gorenstein. We also derive a recursive character formula for cyclic Demazure modules, which we show is equivalent to a formula of Almousa, Gao and Huang.

math.AG↗