Chow 消没与簇变体的 motive
Chow Vanishing and Motives of Cluster Varieties
- University of California, Berkeley（加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明真正满秩汇点递归簇变体的 Chow 群和混合 Hodge 上同调在正次数消没，通过构造分层和运用 Voevodsky motive 理论，并应用于辫子簇、Richardson 簇等，推导出 Khovanov-Rozansky 同调消没定理。
AI中文摘要：
我们证明了真正满秩（RFR）汇点递归簇变体的积分 Chow 群 $CH^i$ 和混合 Hodge 度 $H^{2i, (i, i)}$ 上同调群在 $i > 0$ 时消没。特别地，这适用于任何 Lie 型中的辫子簇和开 Richardson 簇。我们的主要工具是构造任何 RFR 汇点递归簇变体 $\mathcal{A}(\Sigma)$ 的一个分层，该分层由（仿射空间乘以）具有比 $\Sigma$ 更少可变质点的种子的 RFR 汇点递归簇变体组成。我们运用 Voevodsky motive 理论，为此我们证明了对于数域上的任何混合 Tate 簇变体，环类映射是同构到有理 Borel-Moore 同调的最低权部分。然后我们证明 RFR 汇点递归簇变体具有混合 Tate 性质，事实上具有分裂 motive。最后，我们利用我们的结果推导出关于正辫子闭包的 Khovanov-Rozansky 同调群的消没定理，以及闭 Richardson 簇、投影 Richardson 簇和砖簇的上同调的生成性质。
英文摘要：
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.
